
Enrico RADI
Professore Ordinario Dipartimento di Scienze e Metodi dell'Ingegneria

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2022
 2D Green’s functions for an elastic layer on a rigid support loaded by an internal point force
[Articolo su rivista]
Falope, Federico; Lanzoni, Luca; Radi, Enrico
abstract
The problem of a homogeneous isotropic elastic layer resting on a rigid base and loaded by an internal point force is analytically investigated under plane strain conditions. The displacement field is sought as the superposition of a fundamental solution (doublet state) and a homogeneous solution which allows satisfying the boundary conditions at the upper and lower boundaries of the layer. The displacement field is represented through convergent Fourier integral transforms.
Once the closed form solution in the transformed domain is found, the displacement and stress fields are assessed by numerical inversion of transforms. Results concerning both the displacements and stresses for different positions of the load application point are reported and
compared to FEM solution, finding very good agreement. The solutions obtained for horizontal and vertical point forces define the Green’s functions for the layer, which can be used to describe the mechanical interaction between the layer and an embedded body. As a striking application,
the interaction between an elastic layer and an embedded laterally loaded wall or diaphragm is finally addressed, finding the contact pressure and, in turn, the internal forces in the diaphragm.
2022
 Electrostatic pull‑in instability for tweezer architectures
[Articolo su rivista]
Bianchi, G.; Sorrentino, A.; Radi, E.; Castagnetti, D.
abstract
The work investigates the static pullin instability of electrostatically actuated tweezers with tubular electrodes. At a critical voltage, named pullin voltage, the attraction force between the two electrodes causes the unexpected pullin of the tubular cantilevers, which defines the limit of the elastic region of the system, especially in the case of carbonnano tubes (CNTs) applications. The work aims to evaluate the lower and upper bounds of pullin parameters of a tweezer device with the use of an accurate analytical model which allows to calculate the critical voltage and deflection values of the system. In order to assess the accuracy of the analytical model, we built a prototype and measured the critical pullin voltage for different geometrical configurations of the device. The experimental results confirm the analytical predictions, with a maximum relative difference between the experimental and analytical values of the pullin voltage lower than 13%.
2022
 Frictionless contact problem between a loaded beam and a couple stress elastic halfplane
[Abstract in Atti di Convegno]
Radi, E; Nobili, A
abstract
2022
 Green functions for an elastic layer on a rigid base and related problems
[Abstract in Atti di Convegno]
Falope, FEDERICO OYEDEJI; Lanzoni, Luca; Radi, Enrico
abstract
Abstract
2022
 Indentation of a free beam resting on an elastic substrate with an internal lengthscale
[Articolo su rivista]
Radi, E.; Nobili, A.; Guler, M. A.
abstract
The plane strain problem of a slender and weightless beamplate loaded by a transversal point force in unilateral contact with a couple stress elastic foundation is investigated. The study aims to explore the consequences of the material internal lengthscale on the contact mechanics. In particular, compatibility between the beam and the foundation surface demands that both displacement and rotation match along the contact line. To this aim, couple tractions are exchanged besides the traditional contact pressure until separation between the beam and the foundation occurs. The problem is formulated making use of the Green's functions for a point force and a point couple acting atop of a couple stress elastic halfplane. A pair of coupled integral equations is thus derived, that governs the distribution of contact pressure and couple tractions, with one of them being immediately solved to provide an explicit relation between the two unknowns. In this sense, we retrieve the concept of a mechanically equivalent action, as it is the case of the Kirchhoff shear for plates. The remaining integral equation sets a cubic eigenvalue problem, whose linear term accounts for the microstructure. Its numerical solution is sought by expanding the equivalent contact pressure in series of Chebyshev polynomials vanishing at the contact region ends points, namely the liftoff points, and then applying a collocation strategy. The contact length, the distributions of contact pressure and couple tractions under the beam and the shearing force and bending moment along the beam are then obtained as a function of the material characteristic length. Results clearly indicate that accounting for the material internal lengthscale is mainly realized through exchange of the couple tractions, in the lack of which results much resemble those of the classical solution. Specifically, greater contact lengths and a stronger focusing effect about the loading point are encountered, which become very significant when the contact length approaches the internal lengthscale.
2022
 Performance of concrete reinforced with synthetic fibres obtained from recycling endoflife sport pitches
[Articolo su rivista]
Signorini, Cesare; Marinelli, Simona; Volpini, Valentina; Nobili, Andrea; Radi, Enrico; Rimini, Bianca
abstract
Microplastics pollution has risen at an alarming pace over the last decades and it is now recognised as a leading environmental emergency. Indeed, only a very small fraction of annual plastic production is successfully reused, while the vast majority is either disposed of (mainly through incineration or landfilling) or dispersed into the environment. In this paper, polyolefins synthetic fibres, obtained from processing disposed artificial turf pitches aimed at paving sport facilities, are studied. Focus is set on assessing their potential for the Fibre Reinforced Concrete (FRC) technology. Mechanical performance is discussed at two fibre volume fractions, namely 3% and 5% vol., alongside environmental impact. The former is assessed in bending and reveals a significant enhancement of the postcrack energy dissipation capability, whose extent is compatible with what is usually obtained by the adoption of virgin fibres. This is especially significant in consideration of the light processing operated on the waste material. Indeed, life cycle assessment is adopted to evaluate the environmental impact of fibre reuse against fibre manufacturing from either virgin materials or plastic waste. It clearly appears that fibre reuse brings a double environmental benefit: on the one side, it decreases the need for new plastics and, on the other, it reduces plastic waste, whose traditional disposal technique, through incineration, entails a considerable footprint.
2022
 Pullin Instability Analysis of a Nanocantilever Based on the TwoPhase Nonlocal Theory of Elasticity
[Articolo su rivista]
Mikhasev, G.; Radi, E.; Misnik, V.
abstract
This paper deals with the pullin instability of cantilever nanoswitches subjected to electrostatic and intermolecular forces in the framework of the twophase nonlocal theory of elasticity. The problem is governed by a nonlinear integrodifferential equation accounting for the external forces and nonlocal effects. Assuming the Helmholtz kernel in the constitutive equation, we reduce the original integrodifferential equation to a sixthorder differential one and derive a pair of additional boundary conditions. Aiming to obtain a closedform solution of the boundaryvalue problem and to estimate the critical intermolecular forces and pullin voltage, we approximate the resultant lateral force by a linear or quadratic function of the axial coordinate. The pullin behavior of a freestanding nanocantilever as well as its instability under application of a critical voltage versus the local model fraction are examined within two models of the load distribution. It is shown that the critical voltages calculated in the framework of the twophase nonlocal theory of elasticity are in very good agreement with the available data of atomistic simulation.
2022
 Resistivity contribution tensor for nonconductive sphere doublets
[Articolo su rivista]
Lanzoni, L; Radi, E.
abstract
The distribution of the temperature and heat flux fields around a couple of unequal nonconductive tangent spherical inhomogeneities (or pores) embedded in an infinite medium under a steadystate and remotely applied heat flux is addressed in the present work. Owing to the 3D geometrical layout of the inhomogeneity, use is made of the tangent sphere coordinate system. A corrective temperature field expressed in terms of convergent integrals is superposed to the fundamental one to fulfill the BCs at the surfaces of the spheres. When the heat flux is aligned to the symmetry axis (axisymmetric problem), the solution can be found straightforwardly by introducing a stream function, which allows for transforming the Neumann BCs into a Dirichlet boundary value problem. Conversely, for the transversal heat flux (nonaxisymmetric problem), the problem is formulated in terms of temperature, thus leading to a system of two ODEs which is handled numerically through a Euler shooting method, after preliminary asymptotic expansions.
Once the temperature fields are known, the components of the resistivity contribution tensor are assessed varying the aspect ratio of the two spheres. It is found that the extrema of the thermal resistivity are achieved for spheres of equal size. The study allows assessing the effective thermal conductivity of a wide range of smart composites involving insulating inhomogeneities resembling sphere doublets.
2022
 Resistivity contribution tensor for two nonconductive overlapping spheres having different radii
[Articolo su rivista]
Lanzoni, Luca; Radi, Enrico; Sevostianov, Igor
abstract
We consider here the problem of a threedimensional (3D) body subjected to an arbitrarily oriented and remotely applied stationary heat flux. The body includes a nonconductive inhomogeneity (or pore) having the shape of two intersecting spheres with different radii. Using toroidal coordinates, the steadystate temperature field and the heat flux have been expressed in terms of Mehler–Fock transforms. Then, by imposing Neumann BCs at the surface of the spheres, a system of two Fredholm integral equations is obtained and solved based on Gauss–Laguerre quadrature rule. It is shown
that the components of the resistivity contribution tensor exhibit a nonmonotonic trend with the distance between sphere centers. In particular, if the inhomogeneity has a symmetric dumbbellshape, then the extrema of the resistivity contribution tensor components occur when the two overlapping spheres have the same size. Differently, when the inhomogeneity has a lenticular shape, then these extrema are attained for a nonsymmetric configuration, namely, for different radii of the intersecting spheres.
2022
 Stroh formulation of straingradient elasticity.
[Abstract in Atti di Convegno]
Nobili, A; Radi, E
abstract
2021
 A loaded beam in full frictionless contact with a couple stress elastic halfplane: Effects of nonstandard contact conditions
[Articolo su rivista]
Radi, E.
abstract
The plane problem of a loaded EulerBernoulli beam of finite length in frictionless bilateral contact with a microstructured halfplane modelled by the couple stress theory of elasticity is considered here. The study is aimed to investigate the size effects induced on the beam internal forces and moments by the contact pressure and couple stress tractions transmitted across the contact region. Use is made of the Green’s functions for point force and point couple applied at the surface of the couple stress elastic halfplane. The problem is formulated by imposing compatibility of strain between the beam and the halfplane along the contact region and three alternative types of microstructural contact conditions, namely vanishing of couple stress tractions, vanishing of microrotations and compatibility between rotations of the beam cross sections and microrotations of the halfplane surface. The first two types of boundary conditions are usually assumed in the technical literature on micropolar materials, without any sound motivation, although the third boundary condition seems the most correct one. The problem is thus reduced to one or two (singular) integral equations for the unknown distributions of contact pressure and couple stress tractions, which are expanded in series of Chebyshev orthogonal polynomials displaying squareroot singularity at the beam ends. By using a collocation method, the integral equations are reduced to a linear algebraic system of equations for the unknown coefficients of the series. The contact pressure and couple stress along the contact region and the shear force and bending moment along the beam are then calculated under various loading conditions applied to the beam, varying the flexural stiffness of the beam and the characteristic length of the elastic halfplane. The size effects due to the characteristic length.
2021
 Analytical estimates of the pullin voltage in MEMS and NEMS
[Relazione in Atti di Convegno]
Bianchi, G.; Radi, E.
abstract
Micro or NanoElectroMechanical Systems, MEMSNEMS, are currently employed in a wide variety of applications, ranging from mechanical or electronic engineering to chemistry or biology. The growing interest in this technology is due to notable need for accurate ultrasmall instruments and equipment characterized by very diminutive size, low power consumption, high precision, reliability and compatibility with the integrated circuits [1]. The micro or nanocantilever beam electrode, suspended above a conductive substrate and actuated by a voltage difference, is the fundamental component of many MEMS and NEMS devices. Moreover, due to their smart mechanical and electronic properties and the recent progress in their fabrication, carbon nanotubes are significantly exploited in industrial applications, such as sensors, nanoactuators, memory devices and nanotweezers, becoming essential components in NEMS [2]. Recent research remarks the role of micropumps in drug delivery systems able to regulate very small and accurate volumes in various industrial, chemical and biomedical applications. Electrostatic micropumps typically are composed of two parallel, thin, circular micro or nanoplates. The membrane of the electrostatic micropump can be actuated and displaced towards the fixed electrode by applying a voltage across the electrodes. When the actuation voltage is removed, the displaced membrane releases and returns to its original position. In general, under the action of the electrostatic force and intermolecular surface forces, particularly significant at the micro or nanoscale, the movable electrode deflects toward to the substrate, thus reducing the separation distance between the electrodes. Correspondingly, the magnitude of the attractive forces increases until at a critical voltage, named the pullin voltage, the flexible electrode collapses onto the substrate. In this work, an analytical method is proposed for estimating the pullin voltage and the correspondent deflection accurately, thus providing a useful tool for the effective design of innovative MEMS and NEMS devices [3].
2021
 Analytical modeling of the shape memory effect in SMA beams with rectangular cross section under reversed pure bending
[Articolo su rivista]
Radi, E.
abstract
An analytical model is developed for a prismatic SMA beam with rectangular cross section subjected to alternating bending at temperature below the austenitic transformations. The loading path consists in a loadingunloading cycle under bending and reversed bending. Two opposite martensitic variants take place, whose volume fractions evolve linearly with the axial stress. Different Young’s moduli are taken for the austenitic and martensitic phases. As the bending moment is increased, the martensitic transformation starts from the top and bottom and then it extends inwards. If the maximum applied bending moment is large enough, then the complete Martensitic transformation takes place at the upper and lower parts of the cross section. During unloading and reversed bending, reorientation of the Martensite variants into the opposite ones takes place starting from the boundary between the fully martensitic region and the intermediate transforming region. Special attention is devoted to calculate analytically the axial stress and Martensite variant distributions within the cross section at each stage of the process. A closed form momentcurvature relation is provided for loading and elastic unloading and in integral form for the rest of the process. The approach is then validated by comparison with analytical results available in the literature.
2021
 Bounds to the pullin voltage of a mems/nems beam with surface elasticity
[Articolo su rivista]
Radi, E.; Bianchi, G.; Nobili, A.
abstract
The problem of pullin instability of a cantilever micro or nanoswitch under electrostatic forces has attracted considerable attention in the literature, given its importance in designing micro and nanoelectromechanical systems (MEMS and NEMS). The nonlinear nature of the problem supports the typical approach that relies on numerical or semianalytical tools to approximate the solution.
By contrast, we determine fully analytical upper and lower bounds to the pullin instability phenomenon for a cantilever beam under the action of electrostatic, van der Waals or Casimir forces. In particular, the novel contribution of this works consists in accounting for size effects analytically, in the spirit of surface elasticity, which adds considerable complication to the problem, allowing for a nonconvex beam deflection. Surface energy effects are generally ignored in classical elasticity. However they become relevant for nanoscale structures due to their high surface/volume ratio. Closed form lower and upper bounds are given for the pullin characteristics, that allow to discuss the role of several tuneable parameters. Indeed, the evolution of the cantilever tip deflection is presented as a function of the applied voltage up to the occurrence of pullin and the contribution of van der Waals and Casimir intermolecular interactions is discussed. It is found that intermolecular forces strongly decrease the pullin voltage, while surface elasticity works in the opposite direction and stabilizes the system. The accuracy of the bounding solutions is generally very good, given that upper and lower analytical bounds are very close to each other, although it decreases as the effect of surface elasticity becomes more pronounced. Finally, approximated closedform relations are developed to yield simple and accurate design formulae: in particular, they provide estimates for the minimum theoretical gap and for the maximum operable length for a freestanding cantilever in the presence of the effects of surface elasticity and intermolecular interactions. Results may be especially useful for designing and optimizing NEMS switches.
2021
 Effect of pore coalescence on the effective conductivity of an isotropic material
[Relazione in Atti di Convegno]
Radi, E.; Lanzoni, L.; Sevostianov, I.
abstract
The purpose of this work is to evaluate effect of two coalesced pores or insulating inhomogeneities on the overall conductive properties of an isotropic material. Analytical modeling of the effective properties of materials with microstructures formed by inhomogeneities of nonellipsoidal shape has not been well developed. The inhomogeneities are typically assumed to be ellipsoids of identical aspect ratios. This unrealistic assumption is largely responsible for insufficient linkage between methods of micromechanics and material science applications. The resistivity contribution tensor gives the extra temperature gradient produced by introduction of the inhomogeneity into a material subjected to otherwise uniform heat flux. The main goal of this work is to obtain an analytical solution for the components of the resistivity contribution tensor of two overlapping pores, in the 2D and 3D frameworks [1, 2].
2021
 Exact solutions for isothermal cyclic torsional loading of a circular SMA bar exploiting the shape memory effect
[Articolo su rivista]
Radi, Enrico
abstract
Two simple and fully analytical models are presented for a SMA bar or wire of circular cross section subjected to fullyreversed cyclic torsional loading, by taking into consideration the reorientation of the martensitic variants occurring during unloading and reverse loading. The process is assumed to take place at constant temperature between the start temperatures of the martensitic and austenitic transformations. In the first part of this work, the same shear modulus is taken both for Martensite and Austenite. In the second part, different elastic shear moduli are considered for the two phases. The volume fractions of both positive and negative twisted Martensite are assumed to evolve linearly with the shear stress. The bar is initially in a state of Austenite. As the applied torque is increased the martensitic transformation starts from the outer surface and then it extends inwards. If the maximum applied torque is large enough, then the complete Martensitic transformation takes place in the outer region of the cross section. During unloading and subsequent reverse loading the martensitic reorientation process may occur starting from the boundary between the fully martensitic outer region and the intermediate transforming region. Particular attention is focused on modeling the unloading and reverse loading processes. At each stage of the process, the radial distributions of shear stress and Martensite variant are calculated analytically. A closed form relation between the applied torque and the angle of twist is presented for the entire process in the case of equal shear moduli, and only for the loading and elastic unloading stages in the case of different shear moduli. The approach is then validated against analytical, numerical and experimental results available in the literature for the direct loadingunloading process. Application to the seismic response of dissipative systems based on SMA helical springs is also envisaged.
2021
 Nonstandard contact conditions between a beam and a couple stress elastic halfplane
[Abstract in Atti di Convegno]
Radi, E
abstract
In the present work, the problem of a deformable EulerBernoulli beam of length 2a in bilateral frictionless contact with a couple stress elastic halfplane, whose constitutive parameters are the shear modulus , the Poisson coefficient and the material characteristic length l, is investigated by assuming that both contact pressure and couple stress tractions are transmitted across the contact zone. The present study is aimed to investigate the size effects induced on the beam internal forces and moments by the contact pressure and couple stress tractions transmitted across the contact region. It may be considered an extension of the works on beams in contact with an elastic halfplane performed by Shield and Kim (1992), Lanzoni and Radi (2016), and on rigid indenters in contact with an elastic couplestress halfplane developed by Guorgiotis and Zisis (2016) and Zisis et al. (2018).
The couple stress theory of elasticity requires boundary conditions on the microrotation and couple stress tractions in addition to the usual boundary conditions of the classic nonpolar continuum on displacements and stress tractions. A challenging problem is thus how to extend the classic contact conditions to include the effects of the microrotation and couple stress tractions. In the proposed approach, the classical strain compatibility condition between the slope of the beam and that of the halfplane surface is imposed along the contact region. Moreover, three alternative kinds of microstructural contact conditions are considered and discussed, namely, vanishing of couple stress tractions, vanishing of microrotations and compatibility between microrotatons of the halfplane surface and slope of the beam. The first two types of boundary conditions are usually assumed in the technical literature on micropolar materials, although the third boundary condition seems the most correct one. Use is made of the Green’s functions for point force and point couple applied at the surface of the couple stress elastic halfplane. The problem is thus reduced to one or two (singular) integral equations for the unknown distributions of contact pressure and couple stress tractions, which are expanded in series of Chebyshev orthogonal polynomials of the first kind displaying the classical squareroot singularity at the beam ends. By using a collocation method, the integral equations are reduced to a linear algebraic system of equations for the unknown coefficients of the Chebyshev series expansion adopted for the contact pressure and couple stress tractions. The contact pressure and couple stress along the contact region and the shear force and bending moment along the beam are then calculated under various loading conditions applied to the beam, varying the flexural stiffness EI of the beam and the characteristic length l of the elastic halfplane. The three alternative conditions lead to significantly different results in term of bending moment along the beam. The size effects due to the characteristic length of the halfplane and the implications of the generalized contact conditions are illustrated and discussed.
The classical elastic solution is recovered as the characteristic length becomes vanishing small. Generally, the magnitude of the couple stress tractions is found to increase with the characteristic length. Although its contribution is usually smaller than that of the contact pressure and mainly restricted to the edges of the beam, it may provide a significant influence on the shear force and bending moment along the beam. Therefore, the obtained results show that the couple stress tractions exhibit a large influence on the beam internal forces and moments and display size dependent behavior when the beam length is comparable to the intrinsic characteristic length scale of the ground.
Moreover, we show that accounting for the micropolar behavior of the ground, but neglecting the moment traction
2021
 Thermal conductivity of solids with coalescing spherical pores
[Abstract in Atti di Convegno]
Lanzoni, Luca; Radi, Enrico; Sevostianov, Igor
abstract
The overall thermal properties of media containing insulating cylindrical inhomogeneities has been addressed recently in [1] making reference to a 2D layout. There, the cross section of the fibers is formed by two intersecting circles to simulate a variety of nonconductive fibers (e.g. electrospinned polystyrene fibers). On the other hand, intersecting circles is a relevant layout to assess the physical properties of a variety of porous materials (e.g. Gasar metals) during the processes of pore coalescence and growth. In this work we extend the analysis addressed in [1] to a 3D framework by considering the effect of insulating inhomogeneities having the shape of intersecting spheres. The analysis aims at assessing the secondorder resistivity contribution tensor, which provides the corrective temperature gradient induced by the volume V * of the inhomogeneity over the reference volume V of the background material subjected the a remotely applied heat flux q. Owing to the geometric setting, reference is made to toroidal coordinates and Mehler–Fock transforms are used to represents the perturbation temperature field due to the inhomogeneity [2]. As remarked in [3] for coalescing spheres having the same diameter, the components of tensor R display a nonmonotonic trend varying the distance between their centers. In particular, unlike what is observed for spheroids, the stationary values of the components of tensor R occur when spherical pores are slightly intersecting.
2020
 A new Rayleighlike wave in guided propagation of antiplane waves in couple stress materials
[Articolo su rivista]
Nobili, A.; Radi, E.; Signorini, C.
abstract
Motivated by the unexpected appearance of shear horizontal Rayleigh surface waves, we investigate the mechanics of antiplane wave reflection and propagation in couple stress (CS) elastic materials.
Surface waves arise by mode conversion at a free surface, whereby bulk travelling waves trigger inhomogeneous modes. Indeed, Rayleigh waves are perturbations of the travelling mode and stem from its reflection at grazing incidence. As well known, they correspond to the real zeros of the Rayleigh function.
Interestingly, we show that the same generating mechanism sustains a new inhomogeneous wave, corresponding to a purely imaginary zero of the Rayleigh function. This wave emerges from "reflection"
of a bulk standing mode: This produces a new type of Rayleighlike wave that travels away from, as opposed to along, the free surface, with a speed lower than that of bulk shear waves. Besides, a third zero of the Rayleigh function may exist, which represents waves attenuating/exploding both along and away from the surface. Since none of these zeros correspond to leaky waves, a new classification of the Rayleigh zeros is proposed. Furthermore, we extend to CS elasticity Mindlin’s boundary conditions, by which partial waves are identified, whose interference lends RayleighLamb guided waves. Finally, asymptotic analysis in the thinplate limit provides equivalent 1D models.
2020
 Analytical estimates of the pullin voltage for carbon nanotubes considering tipcharge concentration and intermolecular forces
[Articolo su rivista]
Bianchi, G.; Radi, E.
abstract
Twoside accurate analytical estimates of the pullin parameters of a carbon nanotube switch clamped at one end under electrostatic actuation are provided by considering the proper expressions of the electrostatic force and van der Waals interactions for a carbon nanotube, as well as the contribution of the charge concentration at the free end. According to the EulerBernoulli beam theory, the problem is governed by a fourthorder nonlinear boundary value problem. Twoside estimates on the centreline deflection are derived. Then, very accurate lower and upper bounds to the pullin voltage and deflection are obtained as function of the geometrical and material parameters. The analytical predictions are found to agree remarkably well with the numerical results provided by the shooting method, thus validating the proposed approach. Finally, a simple closedform relation is proposed for the minimum feasible gap and maximum realizable length for a freestanding CNT cantilever.
2020
 Buckling of a Timoshenko beam bonded to an elastic halfplane: Effects of sharp and smooth beam edges
[Articolo su rivista]
Falope, FEDERICO OYEDEJI; Lanzoni, Luca; Radi, Enrico
abstract
The problem of a compressed Timoshenko beam of finite length in frictionless and bilateral contact with an elastic halfplane is investigated here. The problem formulation leads to an integrodifferential equation which can be transformed into an algebraic system by expanding the rotation of the beam cross sections in series of Chebyshev polynomials. An eigenvalue problem is then obtained, whose solution provides the buckling loads of the beam and, in turn, the corresponding buckling mode shapes. Beams with sharp or smooth edges are considered in detail, founding relevant differences. In particular, it is proofed that beams with smooth edges cannot exhibit a rigidbody buckling mode. A characteristic value of the stiffness ratio dimensionless parameter has been found for sharp edges, under which without loss of reliability, an analytic buckling load formula is provided. Finally, in agreement with the Galin solution for the rigid flat punch on a halfplane, a simple relation between the halfplane elastic modulus and the Winkler soil constant is found. Thus, a straightforward formula predicting the buckling loads of high stiff beams resting on elastic compliant substrates is proposed.
2020
 Diffraction and Reflection of Antiplane Shear Waves in a Cracked Couple Stress Elastic Material
[Capitolo/Saggio]
Nobili, Andrea; Radi, Enrico; Mishuris, Gennady
abstract
We investigate the effect of a semiinﬁnite rectilinear crack on diffraction and reﬂection of antiplane shear waves in an elastic solid with microstructure. Waves are induced by moving shear traction vectors applied at the faces of the crack. The material behavior is described by the indeterminate theory of couple stress elasticity considering micro inertia. This elastic constitutive model accounts for the material microstructure and it is a special case of the micropolar theory; it was developed by Koiter [3] for the quasistatic regime and later extended by Eringen [1] to include dynamic effects. The fullﬁeld solution is obtained through integral transforms and the WienerHopf technique [5] and it may be used as a bulding block to solve general wave propagation problems in a cracked halfspace in antiplane deformation. The solution differs signiﬁcantly from the classical result given in [2] for isotropic elastic materials. Indeed, unlike classical elasticity, antiplane shear Rayleigh waves are supported in couple stress materials [4]. A complicated wave pattern appears which consists of entrained waves extending away from the crack, reﬂected Rayleigh waves, localized waves irradiating from and body waves scattered by the cracktip. Wave diffraction and interference brings an important contribution to the stress intensity factors originally presented in [6] in the static framework. Resonance is triggered when the applied loading is fed into the cracktip at Rayleigh speed and this result is conﬁrmed by the evaluation of the energy release rate.
2020
 Effect of spherical pores coalescence on the overall conductivity of a material
[Articolo su rivista]
Lanzoni, L.; Radi, E.; Sevostianov, I.
abstract
The problem about steadystate temperature distribution in a homogeneous isotropic medium containing a pore or an insulating inhomogeneity formed by two coalesced spheres of the same radius, under arbitrarily oriented uniform heat flux, is solved analytically. The limiting case of two touching spheres is analyzed separately. The solution is obtained in the form of converged integrals that can be calculated using GaussLaguerre quadrature rule. The temperature on the inhomogeneity’s surface is used to determine components of the resistivity contribution tensor for the insulating inhomogeneity of the mentioned shape. An interesting observation is that the extreme values of these components are achieved when the spheres are already slightly coalesced
2020
 Experimental characterization of pullin parameters for an electrostatically actuated cantilever
[Articolo su rivista]
Sorrentino, A.; Bianchi, G.; Castagnetti, D.; Radi, E.
abstract
MEMSNEMS applications extensively use micronano cantilever structures as actuation system, thanks to their intrinsically simple end efficient configuration. Under the action of an electrostatic actuation voltage the can tilever deflects, until it reaches the maximum value of the electrostatic actuation voltage, namely the pullin voltage. This limits its operating point and is a critical issue for the switching of the actuator.
The present work aims to experimentally measure the variation of the pullin voltage and the tip deflection for different geometri cal parameters of an electrostatically actuated cantilever. First, by relying on a nonlinear differential model from the literature, we designed and built a macroscale cantilever switch, which can be simply adapted to different configurations. Second, we experimentally investigated the effect of the free length of the suspended electrode, and of the gap from the ground, on the pullin response. The experimental results always showed a close agree ment with the analytical predictions, with a maximum relative error lower that 10% for the pullin voltage, and a relative difference lower than 18% for the pullin deflection.
2019
 Analytical bounds for the pullin voltage of carbon nanotubes
[Abstract in Atti di Convegno]
Bianchi, G.; Radi, E.
abstract
Carbon nanotubes (CNTs) display a number of attractive electronic and mechanical properties that are currently exploited in a wide variety of industrial applications, such as sensors, nanoactuators, memory devices, switches, high frequncy nanoresonators and nanotweezers. Due to their tiny size they indeed display ultralow mass and very high resonance frequency as well as the capability to carry huge electrical currents and to sustain high current densities. These properties, in conjunction with the significant progress recently made in the fabrication of carbon nanostructures, allow
CNTs to become essential components in the fabrication of enhanced nanoelectromechanical systems (NEMS)
An accurate determination of the stable actuating range and the pullin instability threshold is a crucial issues for designing reliable CNT based NEMS. Despite the amount of numerical or approximated investigations, analytical models and closed form expressions for pullin instability analysis of CNT still appears to be limited. An analytical methodology for assessing accurate lower and upper bounds to the pullin parameters of an electrostatically actuated micro or
nanocantilever has been provided in two previous works [1, 2], taking into consideration the contributions of flexible support and compressive axial load.
In the present work, attention is paid to investigate the pullin phenomenon in CNT with circular crosssection, by considering the proper expressions of the electrostatic and van der Waals forces per unit length acting on a CNT, as well as the significant reduction of the pullin voltage induced by the charge concentration at the free end [3]. Twoside accurate analytical estimates of the pullin parameters of a carbon nanotube switch clamped at one end under electrostatic
actuation are provided by considering the effects of van der Waals interactions and charge concentration at the free end. The problem is governed by a fourthorder nonlinear boundary value problem, according to the EuleroBernoulli beam theory. Twoside estimates on the deflection are first derived, then very accurate lower and upper bounds to the pullin voltage and deflection are obtained as functions of the geometrical and material parameters. The analytical predictions are then found to agree remarkably well with the numerical results provided by the shooting method.
2019
 Diffraction and reflection of antiplane shear waves in a cracked couple stress elastic material
[Abstract in Atti di Convegno]
Nobili, A.; Radi, E.; Mishuris, G.
abstract
We investigate the effect of a semiinfinite rectilinear crack on diffraction and reflection of antiplane shear waves in an elastic solid with microstructure. Waves are induced by moving shear traction vectors applied at the faces of the crack. The material behavior is described by the indeterminate theory of couple stress elasticity considering micro inertia. This elastic constitutive model accounts for the material microstructure and it is a special case of the micropolar theory; it was developed by Koiter [3] for the quasistatic regime and later extended by Eringen [1] to include dynamic effects.
The fullfield solution is obtained through integral transforms and the WienerHopf technique [5] and it may be used as a bulding block to solve general wave propagation problems in a cracked halfspace in antiplane deformation. The solution differs significantly from the classical result given in [2] for isotropic elastic materials. Indeed, unlike classical elasticity, antiplane shear Rayleigh waves are supported in couple stress materials [4]. A complicated wave pattern appears which consists of entrained waves extending away from the crack, reflected Rayleigh waves, localized waves irradiating from and body waves scattered by the cracktip. Wave diffraction and interference brings an important contribution to the stress intensity factors originally presented in [6] in the static framework. Resonance is triggered when the applied loading is fed into the cracktip at Rayleigh speed and this result is confirmed by the evaluation of the energy release rate.
2019
 Diffraction of antiplane shear waves and stress concentration in a cracked couple stress elastic material with micro inertia
[Articolo su rivista]
Nobili, A.; Radi, E.; Wellender, A.
abstract
Scattering of waves impinging along a rectilinear semiillimitate crack is investigated in an elastic solid with microstructure under antiplane deformation. The material behavior is described by the indeterminate theory of couple stress elasticity. The propagation of Rayleigh waves is investigated first for, in contrast to classical elasticity, it may occur also in the absence of rotatory inertia. The fullfield solution is obtained through the WienerHopf technique and integral transform inversion. This solution provides dynamic stress intensity factors which appear as generalization of the corresponding results obtained in statics. Resonance, wave transmission and energy release rate are also discussed. Finally, it is shown that the maximum stress criterion can be hardly generalized in a timeharmonic framework.
2019
 Effect Of Pair Coalescence Of Circular Pores On The Overall Elastic Properties
[Articolo su rivista]
Lanzoni, L.; Radi, E.; Sevostianov, I.
abstract
The paper focuses on the effect of the pair coalescence of circular pores on the overall elastic properties. An analytic solution for the stress and displacement fields in an infinite elastic medium, containing cylindrical pore with the crosssection formed by two circles, and subjected to remotely applied uniform stresses is obtained. The displacement field on the surface of the pore is then determined as a function of the geometrical parameters. This result is used to calculate compliance contribution tensor for the pore and to evaluate effective elastic properties of a material containing multiple pores of such a shape.
2019
 Effect of a rigid toroidal inhomogeneity on the elastic properties of a composite
[Articolo su rivista]
Krasnitskii, Stanislav; Trofimov, Anton; Radi, Enrico; Sevestianov, Igor
abstract
An analytical solution is obtained for the problem of an infinite elastic medium containing a rigid toroidal inhomogeneity under remotely applied uniform strain. The traction on the torus surface is determined as a function of torus parameters and strain components applied at infinity. The results are utilized to calculate components of the stiffness contribution tensor of the rigid toroidal inhomogeneity that is required for calculation of the overall elastic properties of a material containing multiple toroidal inhomogeneities. The analytical results are verified by comparison with finite element model calculations.
2019
 Effective elastic properties of media containing coalescing holes
[Abstract in Atti di Convegno]
Lanzoni, Luca; Radi, Enrico; Sevostianov, Igor
abstract
A recent study about the temperature and heat flux distributions around two nonconductive (separate or intersecting) circular holes in a plane system recently appeared in Literature [1]. These results have been used to construct the secondrank resistivity contribution tensor which allows assessing the effective thermal properties of a composite including circular inhomogeneities.
Here, that study is extended to assess the overall elastic properties of an isotropic elastic matrix with two separate circular cavities or a cavity obtained by the union of two circles of generally different diameters (Figure 1). The problem is formulated in terms of stress functions expressed in Fourier series or Fourier transforms. Reference is made to bipolar cylindrical coordinates [2]. Once the displacement field u has been calculated, the extra strain due to the inhomogeneity is assessed according to (1), being n the normal vector and V the volume reference. Finally, the extra strain is used to assess the fourthrank compliance contribution tensor varying the size of the circular arcs.
2019
 Experimental characterization of pullin parameters for an electrostatically actuated cantilever
[Relazione in Atti di Convegno]
Sorrentino, Andrea; Bianchi, Giovanni; Castagnetti, Davide; Radi, Enrico
abstract
Microelectromechanical systems (MEMS) are a promising research frontier thanks to their multiple physical fields properties. In the field of microcantilever actuators, Radi et al., 2017, proposed an accurate analytical approach for estimating the pullin characteristics of microcantilever actuators subject to electrostatic actuation. The present work assesses this previous analytical model via experimental tests with the use of a simple millimeterscale device. The aim of the work is to measure the critical pullin voltage and the deflection of an actuated cantilever beam for different configurations in order to validate the variation of the pullin voltage with the geometrical parameters of the device provided by theoretical investigations. Preliminary tests show that the experimental pullin voltage and deflection are in good agreement with the results provided by the analytical model. Specifically, the relative difference between experimental and analytical values of pullin voltage is in the range between 0.7% and 10%.
2019
 Pullout modelling of viscoelastic synthetic fibres for cementitious composites
[Articolo su rivista]
Sorzia, Andrea; Lanzoni, Luca; Radi, E
abstract
The problem of the pullout of a viscoelastic synthetic fibre embedded in a cementitious matrix and subjected to an external timedependent axial load is considered in the present work. A 1D phenomenological model able to simulate the contribution of viscoelastic relaxation as well as the hardening behavior due to abrasion phenomena during slippage is developed. The cement matrix compliance is neglected with respect to the fibre elongation. The interfacial shear stress between the fibre and the surrounding matrix is assumed to depend on the slippage distance through a second degree polynomial law, thus involving three constitutive parameters. Two distinct phases are recognized: An earlier debonding stage followed by the effective fibre pullout process. Two different creep functions have been assumed for modelling the viscous response of polymeric fibres: A function based on the fractionexponential Rabotnov operator and a classical exponential model. Identification of the governing constitutive parameters allows obtaining the relation between the external strain and the axial displacement, which has been compared with experimental results provided by pullout tests both on plain and treated fibres, finding a good agreement. It is shown that the proposed approach can predict the whole pullout process of discrete synthetic macrofibres.
2019
 Timeharmonic analysis of antiplane crack in couple stress elastic materials
[Abstract in Atti di Convegno]
Radi, E; Nobili, A; Mishuris, G
abstract
The time harmonic response of a rectilinear and semiinfinite crack in a couple stress (CS) elastic solid under Mode III loading conditions is investigated in the present work. The fullfield solution of the dynamic crack problem obtained in [1] through Fourier integral transforms and the Wiener–Hopf technique is generalized here by considering more general loading conditions, consisting in arbitrary reduced stress and couple stress tractions applied at the crack faces.
The solution for quasistatic Mode III crack in indeterminate CS elastic materials was given in [2]. Later, the problem of steadystate Mode III crack propagation was investigated in [3]. In the present work, a travelling wave loading, applied in the form of generalized reduced tractions at the crack faces, is considered as the forcing term. As a result, a complex wave pattern appears, which differs significantly from the Mode III classical elastic solution. The results of the present analysis may be used as a building block to address, by means of superposition, the problem of arbitrary antiplane wave propagation in a cracked CS solid. Resonance is triggered when the applied loading is fed into the cracktip at Rayleigh speed. Elastodynamic stress intensity factors are given, which generalize the corresponding results presented in [2] for the qusistatic framework. They incorporate the effect of the applied loading frequency and thereby account for the interplay of the diffracted waves.
A remarkable wave pattern appears which consists of entrained waves extending away from the crack, reflected Rayleigh waves moving along the crack surfaces, localized waves irradiating from the cracktip and body waves scattered around the cracktip. Interestingly, the localized wave solution may be greatly advantageous for defect detection through acoustic emission.
2018
 A piezoelectric based energy harvester with dynamic magnification: modelling, design and experimental assessment
[Articolo su rivista]
Castagnetti, Davide; Radi, Enrico
abstract
This work presents a simple and innovative piezoelectric energy harvester, inspired by fractal geometry and intrinsically including dynamic magnification. Energy harvesting from ambient vibrations exploiting piezoelectric materials is an efficient solution for the development of selfsustainable electronic nodes. After an initial design step, the present work investigates the eigenfrequencies of the proposed harvester, both through a simple free vibration analysis model and through a computational modal analysis. The experimental validation performed on a prototype, confirms the accurate frequency response predicted by these models with five eigenfrequencies below 100 Hz. Despite the harvester has piezoelectric transducers only on a symmetric half of the top surface of the lamina, the rate of energy conversion is significant for all the investigated eigenfrequencies. Moreover, by adding a small ballast mass on the structure, it is possible to excite specific eigenfrequencies and thus improving the energy conversion.
2018
 Adhesively bonded disk under compressive diametrical load
[Articolo su rivista]
Radi, E.; Dragoni, E.; Spaggiari, A.
abstract
A closedform fullfield solution is presented for stresses and displacement in a circular disk containing a diametrical adhesive thin layer induced by two opposite compressive loads acting along an arbitrary di ametrical direction. For the sake of simplicity, the adhesive layer is treated as a tangential displacement discontinuity between the two disk halves. The problem is split into symmetric and skewsymmetric loading conditions. No contribution is expected from the adhesive layer for the symmetric problem. For the skewsymmetric loading condition, a general integral solution in bipolar coordinates has been as sumed for the Airy stress function in the form of a Fourier sine transform. The imposition of the boundary conditions then allows us to reduce the problem to a Fredholm integral equation of the first kind defined on the halfline or equivalently to a singular integrodifferential equation defined on a bounded interval. A preliminary asymptotic analysis of the stress and displacement fields at the edges of the adhesive thin layer shows that the stress field is finite therein, but the rotation displays a logarithmic singularity. A numerical solution of the singular integrodifferential equation is then provided by assuming a power se ries expansion for the shear stress, whose coefficients are determined by using a collocation method. An approximate closedform solution is also derived by exploiting a perturbation method that assumes the ratio between the shear modulus of the disk material and the shear stiffness of the adhesive thin layer as small parameter. The shear stress distribution along the thin layer turns out to be more and more uni form as the adhesive shear stiffness decreases. In order to validate the analytical results, FE investigations and also experimental results obtained by using Digital Image Correlation (DIC) techniques are presented for varying loading orientation and material parameters.
2018
 Analytical bounds for the electromechanical buckling of a compressed nanocantilever
[Articolo su rivista]
Radi, E.; Bianchi, G.; di Ruvo, L.
abstract
An analytical approach is presented for the accurate definition of lower and upper bounds for the pullin voltage and tip displacement of a micro or nano¬cantilever beam subject to compressive axial load, electrostatic actuation and intermolecular surface forces. The problem is formulated as a nonlinear twopoint boundary value problem that has been transformed into an equivalent nonlinear integral equation. Initially, new analytical estimates are found for the beam deflection, which are then employed for assessing accurate bounds from both sides for the pullin parameters, taking into account for the effects of the compressive axial load. The analytical predictions are found to closely agree with the numerical results provided by the shooting method. The effects of surface elasticity and residual stresses, which are of significant importance when the physical dimensions of structures descend to nanosize, can also be included in the proposed approach.
2018
 Brazilian test for the characterization of adhesively bonded joints
[Abstract in Atti di Convegno]
Radi, E.; Dragoni, E.; Spaggiari, A.
abstract
In the present work, we propose the use of the Brazilian test on a adhesively bonded disk for the characterization of adhesion properties of the adhesive. The main advantage of this test is that any combination of shear and normal loading can be achieved by appropriate choice of the bonding inclination angle with respect to the loading direction. A closedform fullfield solution is presented for stresses and displacement in a circular disk containing a diametrical adhesive thin layer induced by two opposite compressive loads acting along an arbitrary diametrical direction. For the sake of simplicity, the adhesive layer is treated as a tangential displacement discontinuity between the two disk halves. The problem is split into symmetric and skew symmetric loading conditions. No contribution is expected from the layer for the symmetric problem. For the skewsymmetric loading condition, a general integral solution in bipolar coordinates has been assumed for the Airy stress function in the form of a Fourier sine transform [1, 2]. The imposition of the boundary conditions then allows us to reduce the problem to a Fredholm integral equation of the first kind defined on the halfline or equivalently to a singular integrodifferential equation defined on a bounded interval. A preliminary asymptotic analysis of the stress and displacement fields at the edges of the adhesive thin layer shows that the stress field is regular therein, but the rotation displays a logarithmic singularity [3]. A numerical solution of the singular integrodifferential equation is then provided by assuming a power series expansion for the shear stress distribution, whose coefficients are found by means of a collocation method. An approximate closedform solution is also derived by exploiting a perturbation method that assumes the ratio between the shear modulus of the disk material and the shear stiffness of the adhesive thin layer as small parameter [4]. The shear stress distribution along the thin layer turns out to be more and more uniform as the adhesive shear stiffness decreases. In order to validate the analytical results, FE investigations and also experimental results obtained by using Digital Image Correlation (DIC) techniques are presented for varying loading orientation and material parameters. The present investigation thus provides some fundamental understandings of the effects of adhesive compliance on the distribution of the shear stress along the adhesive bonding. The analytical solution presented here may be considered particularly valuable, since it allows for the validation of numerical methods as well as for a preliminary design of adhesively bonded connections employed in many structural engineering applications.
2018
 Effect of cylindrical fibers, with crosssections formed by two circular arcs, on the overall conductivity of a composite
[Articolo su rivista]
Lanzoni, L.; Radi, E.; Sevostianov, I.
abstract
An analytic solution for the steadystate temperature distribution in an infinite conductive medium, containing nonconductive fiber with the crosssection of irregular shape formed by two circles, and subjected to remotely applied uniform heat flux is obtained. The temperature flux on the surface of the inhomogeneity is then determined as a function of the geometrical parameters. This result is used to calculate resistivity contribution tensor for the fiber and to evaluate effective conductive properties of a material containing multiple inhomogeneities of this shape.
2018
 Effective properties of composites containing toroidal inhomogeneities
[Relazione in Atti di Convegno]
Radi, Enrico; Sevostianov, Igor; Lanzoni, Luca
abstract
The present work focuses on the problem of a rigid inhomogeneity of toroidal shape embedded in an elastic matrix. Inhomogeneities of this kind occur in both natural and manmade materials. Barium titanate nanotori are used as nonvolatile memory devices, transducers, optical modulators, sensors and possible energy storage in supercapacitors. Toroidal particles represent preferred morphology of Li2O2 deposition on porous carbon electrode in lithiumoxygen batteries. Polymeric “microdonuts” are used in bioengineering; toroidal shape of nanoparticles is preferred for microwave absorption properties of BaTiO3. Toroidal particles of SiO2 may form in a Cu matrix due to internal oxidation of a CuSi solidsolution polycrystal. Analytical modeling of materials with such microstructure has not been well developed. In the homogenization schemes, the inhomogeneities are usually assumed to be of ellipsoidal shape. This unrealistic assumption is responsible for insufficient linkage between micromechanics and materials science applications. While for 2D nonelliptical inhomogeneities many analytical and numerical results have been obtained, only a limited number of approximate estimates are available for nonellipsoidal 3D shapes. Asymptotic methods have been used in [1] to evaluate the contribution of a thin rigid toroidal inhomogeneity into overall stiffness. Eshelby tensor for a toroidal inclusion has been also derived by Onata. However, Eshelby tensor for nonellipsoidal inhomogeneities is irrelevant to the problem of effective properties of a heterogeneous material. The effective conductivity of a material containing toroidal insulating inhomogeneities has been addressed in [2].
We first consider a homogeneous elastic material, with isotropic stiffness tensor C0, containing a rigid inhomogeneity of volume V(1). The contribution of the inhomogeneity to the overall stress per representative volume V (the extra stress Δσ, as compared to the homogeneous matrix) is given by the fourthrank stiffness contribution tensor N, defined by the following relation
where ε ∞ is the remotely applied strain, n is the outward unit normal to the inhomogeneity surface S. To calculate the components of N, a displacement boundary value problem has been solved for 3D elastic space containing a rigid toroidal inhomogeneity.
2018
 Effects of nanosilica treatment on the flexural post cracking behaviour of polypropylene macrosynthetic fibre reinforced concrete
[Articolo su rivista]
DI MAIDA, Pietro; Sciancalepore, Corrado; Radi, Enrico; Bondioli, Federica
abstract
The effects of a surface nanosilica treatment, carried out with the sol gel method, on the postcracking behaviour of polypropylene macrosynthetic fibre reinforced concrete are experimentally investigated here for the first time. The present study extends previous experimental and analytical investigations on the corresponding improvement of the bonding properties of a single synthetic macro fibre, performed by means of pullout test. Scanning electron microscopy is adopted here to explore the changes in the morphological characteristics of polypropylene macro synthetic fibres, before and after mixing in the concrete matrix. A comparative analysis, carried out with threepoint bending tests on notched beam specimens, is used to evaluate the effects of the nanosilica treatment on the concrete post cracking behaviour. Increase in concrete toughness and residual postcracking strength is recorded due to improved adhesion between fibres and the concrete matrix and to the consequent increase in the frictional shear stress generated during the fibre pullout, especially for large crack opening. As shown by the SEM images, the nanotreatment favours the bonding of the concrete hydration products to the surface of the treated fibres, thus ensuring strengthening of the interface transition zone. In addition, the links between the nanosilica coating and the concrete hydration products improve the frictional shear stress and thus the overall energy absorption, as denoted by the increase of the residual strength during the postcracking phase.
2018
 Effects of toroidal inhomogeneities on the effective properties of a composite
[Relazione in Atti di Convegno]
Radi, E.; Lanzoni, L.; Sevostianov, I.
abstract
The present work focuses on the problem of rigid inhomogeneity of toroidal shape embedded in an elastic matrix. Inhomogeneities of this kind occur both in natural and manmade materials. Analytical modeling of materials with such microstructure has not been well developed. In the homogenization schemes, the inhomogeneities are usually assumed to be of ellipsoidal shape. This unrealistic assumption is largely responsible for insuffcient linkage between methods of micromechanics and materials science applications. While for 2D nonelliptical inhomogeneities many analytical and numerical results have been obtained, only a limited number of numerical results and approximate estimates are available for nonellipsoidal 3D shapes. Most of them are related to pores and cracks. The problem of the effective conductivity (thermal or electric) of a material containing toroidal insulating inhomogeneities has been addressed in a pèrevious work, where an analytic solution is presented for the steadystate temperature distribution in an infinite conductive medium containing an insulated toroidal inclusion, under uniform heat flux in an arbitrary direction. The temperature flux on the torus surface is then determined as a function of torus parameters. This result is then used to determine resistivity contribution tensor for the toroidal inhomogeneity and for calculation of effective conductive
properties of a material containing multiple inhomogeneities of this shape.
A general analytical solution is developed here for the problem of an infinite elastic medium containing a rigid toroidal inhomogeneity, under remotely applied uniform strain. The traction vector on the torus surface is determined as a function of torus parameters and remote strain components. The results are utilized to calculate the components of the fourthrank stiffness contribution tensor of the rigid toroidal inhomogeneity that are required for calculation of the overall elastic properties of a material containing multiple toroidal inhomogeneities. The analytical results are verified by comparison with FEM calculations.
2018
 Evaluation of the probability density of inhomogeneous fiber orientations by computed tomography and its application to the calculation of the effective properties of a fiberreinforced composite
[Articolo su rivista]
Mishurova, Tatiana; Rachmatulin, Natalia; Fontana, Patrick; Oesch, Tyler; Bruno, Giovanni; Radi, Enrico; Sevostianov, Igor
abstract
This paper focuses on the experimental evaluation of one of the key microstructural parameters of a shortfiber reinforced composite –the orientation distribution of fibers. It is shown that computed tomography (CT) produces results suitable for reconstruction of the orientation distribution function. This function is used for calculation of the effective elastic properties of polymerfiber reinforced concrete. Explicit formulas are derived for overall elastic moduli accounting for orientation distribution in the frameworks of the non interaction approximation, the Mori–Tanaka–Benveniste scheme, and the Maxwell scheme. The approach illustrated can be applied to any kind of composite material.
2018
 Evolution of multiple Martensite variants in a SMA thickwalled cylinder loaded by internal pressure
[Articolo su rivista]
Radi, Enrico
abstract
The stress and deformation fields in a SMA ring or a thickwalled cylinder loaded by internal pressure at constant temperature (over the start temperature of the martensitic transformation) are determined in closed form under plane stress loading conditions. The phenomenological SMA constitutive model incorporates the volume fractions of multivariants Martensite, which are assumed to evolve linearly with the Tresca effective stress, according to the associative flow rule and the corner flow rule. Initially, the cylinder is everywhere in a state of Austenite. The application of an internal pressure then triggers the martensitic transformation starting from the inner radius of the cylinder wall and extending towards the outer radius. If the wall thickness is large enough, the tangential stress may vanish at the inner radius and correspondingly the stress state may reach a corner of the Tresca transformation condition, thus originating two different Martensite variants according to the corner transformation rule. The admissible phase partitions within the wall thickness originating during the loading process have been systematically investigated according to the ratio between the outer and inner radii. The results obtained here suggest that the loading process should be interrupted soon after the complete martensitic transformation is achieved at the inner radius of the cylinder to avoid permanent plastic deformations.
2018
 Microstructural analysis and mechanical properties of concrete reinforced with polymer short ﬁbers
[Articolo su rivista]
Troﬁmov, Anton; Mishurova, Tatiana; Lanzoni, Luca; Radi, Enrico; Bruno, Giovanni; Sevostianov, Igor
abstract
The paper focuses on the development of a methodology for quantitative characterization of a concrete containing polymer ﬁbers and pores. Computed tomography (CT) characterization technique is used to provide input data for Finite Element Method (FEM) simulations and analytical modeling based on micromechanical homogenization via the compliance contribution tensor formalism. Effective elastic properties of reinforced concrete are obtained experimentally using compression testing, analytically in the framework of NonInteraction approximation and numerically performing direct FEM simulations on specimen with reconstructed microstructure. It is shown that CT produces results suitable for implementation in numerical and analytical models. The results of analytical and numerical modeling are in a good agreement with experimental measurements providing maximum discrepancy of ∼ 2.5%.
2018
 Overall elastic properties of a plate containing inhomogeneities of irregular shape
[Relazione in Atti di Convegno]
Lanzoni, L.; Radi, E.; Sevostianov, I.
abstract
The present work deals with the stiffness properties of an infinite 2D isotropic elastic system containing inhomogeneities having a circular contour. Starting from this general layout, the cases of a matrix with lenticular, perfectly circular, semicircular, “Cshaped” and thin straight inclusions can be obtained as limit cases. Owing to the geometry of the system, reference is made to bipolar cylindrical coordinates ( ), which are linked to the Cartesian ones (x1, x2) through the conformal map [2]. The effective elastic properties of the system is analytically investigated by introducing a fourthorder compliance contribution tensor H, which represents the effect induced by the inhomogeneity on the compliance of the system according to [1], being S the compliance tensor for the homogeneous elastic matrix and e the stress field. It is remarked that the last term in eq (1) denotes the correction acting on the strain field owing to the presence of the inclusions. The system without inhomogeneities and subjected to a remote stress field is considered first. The corresponding fundamental stress field (0) within the matrix does not accomplish the BCs at the contour of the inhomogeneities. Thus, following the Jeffery approach, an auxiliary stress field deduced by a biharmonic stress function in bipolar coordinates is introduced and tensor H is then evaluated by performing proper contour integrals involving the total stress distribution along the contours of the inclusions. The study allows evaluating the effective elastic properties of a wide class of inhomogeneous materials, with particular reference to composites reinforced with natural or synthetic fibres having optimized cross sections.
References
[1] Sevostianov, I., and Kachanov, M., “Explicit crossproperty correlations for anisotropic
twophase composite materials” Journal of the Mechanics and Physics of Solids, 50, 253282 (2002).
[2] Korn, G.A. and Korn, T.M., Mathematical handbook for scientists and engineers. Definitions, Theorems and Formulas for Reference and Review, Dover, New York (1968).
2018
 Overall thermal conductivity of fibre reinforced materials
[Abstract in Atti di Convegno]
Lanzoni, Luca; Radi, Enrico; Sevostianov, Igor
abstract
The overall thermal conductivity of composites involving cylindrical fibres of irregular shape is investigated in the present work. Isotropic and homogeneous thermal conductivity is assumed for both the matrix and fibre. The system consists of an infinite plate with an embedded fibre subjected to a remotely applied steady state heat flux q acting along a given
direction. Once the alteration of the heat flux and temperature field T due to the presence of the inclusion is assessed, the homogeneized thermal properties of the composite material can be computed following the procedure reported in [1].
As an example, the dimensionless temperature distribution Tk/Rq and heat flow q/q in an infinite with a nonconductive
circula fiber is sketched in Figure 1, being k the thermal conductivity of the matrix and R denotes the radius of the fiber.
The study extends the results reported in [2] performed for nonconductive inclusions accounting for the real thermal conductivity of the fibres. The analysis allows assessing the effective thermal properties of a fibre reinforced material based on fibres with cross section formed by circular arcs, as polystyrene, polyacrylonitrile and sisal fibres.
2018
 Shafthub press fit subjected to couples and radial forces: analytical evaluation of the shafthub detachment loading
[Articolo su rivista]
Bertocchi, Enrico; Lanzoni, Luca; Mantovani, Sara; Radi, Enrico; Strozzi, Antonio
abstract
A shafthub press fit subjected to two nonaxisymmetric loading conditions is examined and the situation
of incipient detachment between the shaft and the hub is determined. The first condition consists of a
central radial load P applied to the hub, balanced by two lateral forces P=2 applied to the shaft at a
distance d from the hub lateral walls. In the second condition, a central couple C is applied to the hub,
and it is balanced by two lateral opposite loads withstood by the shaft at a distance d from the hub
lateral walls. The shafthub contact is modelled in terms of two elastic Timoshenko beams connected by
distributed elastic springs (Winkler foundation), whose constant is analytically evaluated. Based upon
this enhanced beamlike modelling, the loading inducing an undesired shafthub incipient detachment
is theoretically determined in terms of the shafthub geometry, of the initial shafthub interference, and
of the elastic constants. Finite element forecasts are presented to quantify the error of this beamlike
approximate analytical approach.
2018
 Shear deformable beams in contact with an elastic halfplane
[Abstract in Atti di Convegno]
Falope, F. O.; Lanzoni, L.; Radi, E.
abstract
The present work deals with the contact problem of a Timoshenko beam bonded to an elastic semiinfinite substrate under different loading conditions. The analysis allows investigating the effects induced by shear compliance of the beam, the stress intensity factors ad the beam edges as well as the singular nature of the interfacial stresses.
2018
 Stress analysis around a tunnel in a gravitating poroelastic half plane
[Relazione in Atti di Convegno]
Lanzoni, Luca; Radi, Enrico; Nobili, Andrea
abstract
The present work deals with the mechanical behaviour of a circular tunnel embedded in a semiinfinite
poroelastic half plane under gravitational body forces. Owing to the geometric layout,
reference is made to bipolar cylindrical coordinates (, ) and symmetry occurs with respect to
coordinate. The fluid flux is assumed stationary, thus making the fluid pressure p a harmonic field, being Cn, Dn unkonwn arbitrary constants.
The problem is governed by the following Navier equation in terms of the displacement field the solid phase, where denotes the elastic shear modulus of the solid phase, g is the ground acceleration, e1 denotes the unit vector of the vertical axis, n is the porosity and the (real and positive) parameters and sw characterize completely the mechanical response of the poroelastic soil according to
the Bowen formalism [1]. A particular solution of eqn (2) is found in closed form by introducing a Helmholtz potential
for the displacement. However such a solution does not accomplish the BCs at the ring of the
tunnel and at the free surface of the half plane. Then, by following the Jeffery procedure [2], an
auxiliary Airy stress function is introduced that, added to the fundamental solution, allows
accomplish the BCs.
Two limit situations are considered at the ring of the tunnel. In particular, a given radial
pressure acting at the ring of the tunnel (Dirichlet BCs) and a given fluid flux across the contour of
the tunnel (Neumann BCs) have been considered [3]. The latter situation allows investigating the
effect of a tunnel having an impermeable surface embedded in a gravitating poroelastic soil.
Results in terms of fluid pressure and stress near the rim of the hole as well as at the free
surface of the half plane are analysed in detail varying both the geometric and constitutive
parameters of the system.
2018
 Stress and pressure fields around two wellbores in a poroelastic medium
[Articolo su rivista]
Lanzoni, Luca; Radi, Enrico; Nobili, Andrea
abstract
The problem of two circular wellbores of different size in a poroelastic medium is considered in the present work. The constitutive behaviour of the poroelastic medium is assumed to comply with the classical Biot model for isotropic porous materials infiltrated by compressible fluid. The wellbores are assumed infinitely long and the fluid flow is taken stationary, thus making it possible to perform a plane strain analysis. Owing to the geometrical layout of the system, bipolar cylindrical coordinates have been adopted. Three different sets of BCs on the pressure field and on the fluid flux have been considered, founding the corresponding forms of the pressure field. Based on Helmholtz representation, a displacement potential has been introduced, and the corresponding stress field in the poroelastic medium has been assessed. However, such a solution does not satisfy the BCs at the edges of the wells. Then, an auxiliary stress function, which allows accomplishing the BCs, is introduced, leading to the complete solution of the problem. The cases of two coaxial wellbores (eccentric annulus), a single hole bored in a poroelastic half plane and two intersecting holes have been considered also. The proposed approach allows evaluating the pore pressure and the stress and strain fields in the system varying the amplitude of the wells and the physical parameters of the porous material. In particular, the evaluation of the peak values of the stress components around the circular boreholes plays a key role in a variety of engineering contexts, with particular reference to the stability analysis of wellbores and tunnels and failure of vascular vessels in biological tissues.
2017
 A WienerHopf System of Equations in the SteadyState Propagation of a Rectilinear Crack in an Infinite Elastic Plate
[Relazione in Atti di Convegno]
Nobili, Andrea; Radi, Enrico; Lanzoni, Luca
abstract
A WienerHopf system of functional equations is shown to govern the steadystate propagation of a semiinfinite rectilinear crack in an infinite elastic Kirchhoff–Love plate. Solution is presented in terms of Fourier transforms via kernel factorization
2017
 Effective thermal properties of fibre reinforced materials
[Relazione in Atti di Convegno]
Lanzoni, L.; Radi, E.; Tarantino, A. M.
abstract
The thermal behaviour of an elastic matrix reinforced with synthetic micro or macro fibres subjected to a constant heat flow is investigated in the present work. Steadystate condition for the heat flux is considered and isotropic thermal conductivity for both the matrix and fibres is assumed. Owing to the geometry of the system, reference is made to bipolar cylindrical coordinates.
Various boundary conditions can be considered on the contours of the fibres. In particular, for a matrix reinforced with two fibres taken as insulated inclusions, a vanishing heat flow across the contour of the fibres must be imposed.
After the temperature field has benn determined analytically, a homogeneization procedure is performed in order to find the equivalent thermal properties of the fibre reinforced composite material.
2017
 Flexural edge waves generated by steadystate propagation of a loaded rectilinear crack in an elastically supported thin plate
[Articolo su rivista]
Nobili, Andrea; Radi, Enrico; Lanzoni, Luca
abstract
The problem of a rectilinear crack propagating at constant speed in an elastically supported thin plate and acted upon by an equally moving load is considered. The fullfield solution is obtained and the spotlight is set on flexural edge wave generation.
Below the critical speed for the appearance of travelling waves, a threshold speed is met which marks the transformation of decaying edge waves into edge waves propagating along the crack and dying away from it. Yet, besides these, and for any propagation speed, a pair of localized edge waves, which rapidly decay behind the crack tip, is also shown to exist. These waves are characterized by a novel dispersion relation and fade off from the crack line in an oscillatory manner, whence they play an important role in the far field behaviour. Dynamic stress intensity factors are obtained and, for speed close to the critical speed, they show a resonant behaviour which expresses the most efficient way to channel external work into the crack. Indeed, this
behaviour is justified through energy considerations regarding the work of the applied load and the energy release rate. Results might be useful in a wide array of applications, ranging from fracturing and machining to acoustic emission and defect detection.
2017
 On the edgewave of a thin elastic plate supported by an elastic halfspace
[Relazione in Atti di Convegno]
Nobili, A.; Kaplunov, J.; Radi, E.; Tarantino, A. M.
abstract
In this contribution, we consider edgewave propagating in a thin elastic semiinfinite
plate which is bilaterally supported by a homogenenous isotropic elastic halfspace. The
problem is formulated in terms of a eigenproblem constituted by a system of five linear PDEs
in the plate transverse displacement and in the scalar and vector elastic potentials subject
to mixed boundary conditions accounting for platefundation displacement continuity under
the plate and zero normal stress outside. Zero tangential stress is envisaged throughout. The
problem could be reduced to an inhomogenenous WienerHopf functional equation in terms of
the halfspace surface displacement and of the platetofundation contact pressure only. The
kernel function is analyzed and the Rayleigh wave speed is obtained together with a novel
dispersion equation. Finally, kernel factorization is performed.
2017
 Shafthub press fit subjected to bending couples: Analytical evaluation of the shafthub detachment couple
[Articolo su rivista]
Radi, Enrico; Lanzoni, Luca; Strozzi, Antonio; Bertocchi, Enrico
abstract
A mathematical modelling of a shafthub pressfit subjected to bending couples applied to the shaft extremities is developed, and the value of the bending couple inducing an un desired shafthub incipient detachment is analytically determined. The shafthub contact is modelled in terms of two elastic Timoshenko beams connected by a distributed elastic spring, whose stiffness is analytically evaluated. Two models of the distributed spring are considered. The first model expresses the combined deformability of both the shaft and the hub cross sections. The second model accounts for the stiffening effect exerted by the shaft portion protruding from the hub on the adjacent shaft part that is in contact with the hub, and, consequently, it assumes only a rigid body motion of the shaft cross section, thus neglecting its deformability. Based upon this beamlike model, the bending couple producing the incipient detach ment between the shaft and the hub is theoretically determined in term of the shafthub geometry, of the initial shafthub interference, and of the elastic constants. Comparisons with selected Finite Element (FE) forecasts indicate that the first modelling produces an incipient detachment couple that appreciably overrates the FE forecasts, whereas the sec ond modelling lowers the error down to technically acceptable predictions.
2017
 Upper and lower bounds for the pullin parameters of a micro or nanocantilever on a flexible support
[Articolo su rivista]
Radi, Enrico; Bianchi, Giovanni; DI RUVO, Lorenzo
abstract
An analytical method is proposed to accurately estimate the pullin parameters of a micro or nanocantilever
beam elastically constrained by a rotational spring at one end. The system is actuated by electrostatic force and
subject to Casimir or van der Waals forces according to the beam size. The deflection of the beam is described by
a fourthorder nonlinear boundary value problem, or equivalently in terms of a nonlinear integral equation. New
a priori analytical estimates on the solution from both sides are first derived and then lower and upper bounds
for the pullin parameters are obtained, with no need of solving the nonlinear boundary value problem. The
lower and upper bounds turn out to be very close each other and in excellent agreement with the numerical
results provided by the shooting method. The approach also provides accurate predictions for the pullin
parameters of a freestanding nanoactuator.
2016
 A loaded Timoshenko beam bonded to an elastic half plane
[Articolo su rivista]
Lanzoni, Luca; Radi, Enrico
abstract
The contact problem of a Timoshenko beam of finite length loaded by concentrated forces and couples and perfectly bonded to a homogeneous elastic and isotropic half plane is considered in the present work. The study is aimed to investigate the effects induced by shear deformation of the beam on the contact stresses arising at the interface between the beam and the underlying half plane. The asymptotic analysis of the stress field at the beam edges and in the neighborhood of the loaded section of the beam allows us to characterize the singular nature of the peeling and shear stresses. The problem is formulated by imposing the strain compatibility condition between the beam and the half plane, thus leading to a system of two singular integral equations with Cauchy kernel. The unknown interfacial stresses are expanded in series of Jacobi orthogonal polynomials displaying complex singularity. This approach allows us to handle the oscillatory singularity and to reduce the integral equations to a linear algebraic system of equations for the unknown coefficients of the interfacial stresses, which is solved through a method of collocation. The interfacial peeling and shear stresses and, in turn, the displacement field along the contact region have been calculated under various loading conditions acting on the beam. The internal forces and moments along the beam have been evaluated varying the shear and flexural stiffness of the beam. The complex stress intensity factors and the strength of the stress singularities have been assessed in detail.
2016
 Conservation integrals for two circular holes kept at different temperatures in a thermoelastic solid
[Articolo su rivista]
Radi, Enrico; Morini, L.; Sevostianov, I.
abstract
An explicit analytic solution for thermal stresses in an infinite thermoelastic medium with two circular cylindrical holes of different sizes kept at different constant temperatures, under steadystate heat flux is presented. The solution is obtained by using the most general representation of a biharmonic function in bipolar coordinates. The stress field is decomposed into the sum of a particular stress field induced by the steadystate temperature distribution and an auxiliary isothermal stress field required to satisfy the boundary conditions on the holes. The variations of the stress concentration factor on the surface of the holes are determined for varying geometry of the holes. The concept of the conservation integrals Jk, M and L is extended to steady state thermoelasticity and the integrals are proved to be pathindependent. These integrals are calculated on closed contours encircling one or both holes. The geometries of a hole in a halfspace and an eccentric annular cylinder are considered as particular cases.
2016
 Effective properties of materials containing multiple toroidal inhomogeneities
[Relazione in Atti di Convegno]
Radi, Enrico; Sevostianov, Igor
abstract
We calulate effective conductive properties of composite materials containing toroidal inclusion
2016
 Effective viscoelastic properties of shortfiber reinforced composites
[Articolo su rivista]
Igor, Sevostianov; Valery, Levin; Radi, Enrico
abstract
The paper focuses on the calculation of the effective viscoelastic properties of a short fiber reinforced composite. The orientation distribution of the fibers is described by a scatter parameter, varying from perfectly aligned fibers to randomly oriented ones. Both matrix and fibers are assumed to be isotropic.The viscoelastic behavior is described using fractionexponential operators of Scott Blair–Rabotnov. Results are obtained in closed form.
2016
 Failure mechanism of FRC slabs on nonlocal ground
[Articolo su rivista]
Lanzoni, Luca; Nobili, Andrea; Radi, Enrico; Sorzia, Andrea
abstract
The present work deals with the mechanical behavior of a large FRC slab bilaterally supported by a nonlocal soil. The slab is modelled as a ductile Kirchhoff plate laying on a twoparameter elastic and transversally loaded by a uniform
pressure applied on a circular area, thus making the problem axisymmetric. This layout covers a wide array of practical applications of fiber reinforced concrete in structural and civil engineering related to the assessment of the load carrying capacity of industrial floors, roads, airfield pavements and building foundations. The problem is governed by a fourth
order linear ODE with variable coefficients, whose solution has been obtained in power series by using the Frobenius method. The analysis allows us to evaluate the influence of the size of the loaded area and the relative stiffness of the slab/subgrade system on the collapse mechanism and the corresponding load carrying capacity, as well as on the distributions of displacement, rotation, bending moments, shear force and contact pressure at the onset of collapse
2016
 Finite Thin Cover on an Orthotropic Elastic Half Plane
[Articolo su rivista]
Falope, FEDERICO OYEDEJI; Radi, Enrico
abstract
The present work deals with the mechanical behaviour of thin films bonded to a homogeneous elastic orthotropic half plane under plain strain condition and infinitesimal strain. Both the film and semiinfinite substrate display linear elastic orthotropic behaviour. By assuming perfect adhesion between film and half plane together with membrane behaviour of the film, the compatibility condition between the coating and substrate leads to a singular integral equation with Cauchy kernel. Such an equation is straightforwardly solved by expanding the unknown interfacial stress in series of Chebyshev polynomials displaying squareroot singularity at the film edges. This approach allows handling the singular behaviour of the shear stress and, in turn, reducing the problem to a linear algebraic system of infinite terms. Results are found for two loading cases, with particular reference to concentrated axial forces acting at the edges of the film. The corresponding mode II stress intensity factor has been assessed, thus providing the stress concentrations at both ends of the covering. Possible applications of the results here obtained range from MEMS, NEMS, and solar Silicon cell for energy harvesting to welded joint and building foundation.
2016
 On the effect of the backup plate stiffness on the brittle failure of a ceramic armor
[Articolo su rivista]
Nobili, Andrea; Radi, Enrico; Lanzoni, Luca
abstract
The present investigation enquires the role of the backup plate mechanical properties in the brittle failure of a ceramic tile. It provides a fullfield solution for the elastostatic problem of an infinite Kirchhoff plate containing a semiinfinite rectilinear crack (the tile) resting on a twoparameter elastic foundation (the backup plate) and subjected to general transverse loading condition.
The backup plate is modeled as a weakly nonlocal (Pasternaktype) foundation, which reduces to the familiar local (Winkler) model once the Pasternak modulus is set to zero. The same governing equations are obtained for a curved plate (shell) subjected to inplane equibiaxial loading.
Fourier transforms and the WienerHopf technique are employed. The solution is obtained for the case when the Pasternak modulus is greater than the Winkler modulus.
Superposition and a two step procedure are employed: first, an infinite uncracked plate subjected to general loading is considered, then the bending moment and shearing force distribution acting along the crack line is adopted as the (continuous) loading condition to be fed in the solution for the cracked plate.
Results are obtained as a function of the ratio of the Pasternak over the Winkler foundation stiffness times the tile flexural rigidity. It is established that the elastic foundation significantly affects the mechanical behavior of elastic plate. In particular, the Winkler model substantially underestimates the stress condition. Stress intensity factors are determined and they are employed as a guideline for increasing the composite toughness. The analytical solution presented in this paper may serve as a benchmark for a more refined numerical analysis.
2016
 Steady state propagation of a rectilinear crack in a thin elastic plate supported by a Winkler elastic foundation
[Relazione in Atti di Convegno]
Nobili, Andrea; Radi, Enrico; Lanzoni, Luca
abstract
We consider steady state propagation of a rectilinear crack in a infinite thin elastic Kirchhoff plate bilaterally supported by an elastic Winkler foundation. The crack flanks are subjected to a continuous (between the flanks) harmonic load. The problem's governing equation features the biharmonic operator together with a curvature (along the crack) term. Through application of the Fourier transforms to the even/odd part of the problem, a pair of inhomogenenous uncoupled WeinerHopf equations is met. Solution is obtained through numeric factorization
of the kernel function. The fullfield solution is given, together with conditions on the energy radiation. The special case of stationary crack is also retrieved.
2016
 Stress and pore pressure fields around two boreholes in a poroelastic medium
[Relazione in Atti di Convegno]
Lanzoni, Luca; Nobili, Andrea; Radi, Enrico
abstract
We calculate the stress and pore pressure field around two circular holes in a poroelastic fluidsaturated media
2016
 Thin film bonded to elastic orthotropic substrate under thermal loading
[Articolo su rivista]
Falope, FEDERICO OYEDEJI; Lanzoni, Luca; Radi, Enrico; Tarantino, Angelo Marcello
abstract
The problem of thin elastic films bonded on an elastic orthotropic substrate under thermal load is investigated in this work. Differently from past studies on the same topic, the effects induced by anisotropic behavior of the elastic substrate will be taken into account. Particular attention will also be paid to the determination of the displacement and stress fields induced by thermal loading. In particular, it is assumed that the thin films are deposed on the substrate at high temperature, and then the mismatch occurring during the cooling process, due to the difference between the thermal expansion coefficients of the two materials, is responsible for the permanent deformation assumed by the system. This phenomenon can be exploited for realizing a crystalline undulator. To this aim, the permanent deformation must be optimized in order to encourage the channeling phenomenon. By imposing equilibrium conditions and perfect adhesion between the film and the substrate, a singular integral equation is derived. A closedform solution is achieved by expanding the interfacial shear stress fields in Chebyshev series. The unknown coefficients in the series expansion are then determined by transforming the integral equation into an infinite algebraic system.
2016
 Toroidal insulating inhomogeneity in an infinite space and related problems
[Articolo su rivista]
RADI, Enrico; Sevostianov, I.
abstract
An analytic solution for the steadystate temperature distribution in an infinite conductive medium containing an insulated toroidal inhomogeneity and subjected to remotely applied uniform heat flux is obtained. The temperature flux on the torus surface is then determined as a function of torus parameters. This result is used to calculate the resistivity contribution tensor for the toroidal inhomogeneity required to evaluate the effective conductive properties of a material containing multiple inhomogeneities of this shape.
2016
 Uniqueness condition for the (local) solution of the electrostatic microcantilever beam
[Relazione in Atti di Convegno]
DI RUVO, Lorenzo; Radi, Enrico
abstract
A typical micro electromechanical system (MEMS) actuator is formed by a micro cantilever beam electrode suspended above a conductive substrate and subject to a voltage difference.
Due to electrostatic forces the microbeam deflects toward to the substrate and pullsin onto
the substrate plane at a critical voltage, named pullin voltage, thus causing a short circuit.
Under the pullin voltage the microbeam leaves the principal equilibrium path, which becomes
unstable due to the increase of the electrostatic force with the beam deflection, and a kind of
snapthrough instability occurs. The deflection of an elastic cantilever beam is described by a
fourthorder nonlinear ODE. Using fixed point Theorem, we prove the local uniqueness of the solution with respect to L2 norm.
2015
 Analytical modelling of the pullout behavior of synthetic fibres treated with nanosilica
[Relazione in Atti di Convegno]
Radi, Enrico; Lanzoni, Luca; Sorzia, Andrea
abstract
An accurate onedimensional analytical model for simulating the pullout process of synthetic fibres from a cement matrix is proposed in the present study in order to shed light on the ductile behavior exhibited by Fibre Reinforced Concrete (FRC) members.The proposed model is able to predict the non linear relation between the applied tensile load and the fibre displacement and is particularly suitable for synthetic fibres that may exhibit large axial elongation and sliphardening interface behavior. Indeed, the balance conditions between the axial load and the shear stress arising on the fibre surface in frictional contact with the matrix are imposed on the deformed configuration. The frictional bond strength is assumed to increase with slippage distance as a consequence of the increasing abrasion of the fibre surface occurring for polymeric fibres that have been subjected to surface treatments. The model is also suitable for metallic fibres if constant friction or slipsoftening interface behavior is assumed instead. The results provided by the proposed model are compared with the results obtained from pullout tests performed on polymeric fibres embedded in a cement matrix, both for treated and untreated fibres. After conveniently setting the constitutive parameters, the model proves to be able to predict the experimental curves accurately.
2015
 Axisymmetric loading of an elasticplastic plate on a general twoparameter foundation
[Articolo su rivista]
Lanzoni, Luca; Nobili, Andrea; Radi, Enrico; Sorzia, Andrea
abstract
The load carrying capacity and collapse scenarios for an infinite elasticplastic plate resting on a twoparameter
elastic foundation uniformly loaded on a small circular footprint are investigated in a general
framework of stiffness and yield parameters. The present work extends the study already presented for
a specific value of the Pasternak modulus and it allows the investigation of the influence of the stiffness
property of the underlying soil and the amplitude of the loaded region on the load carrying capacity
of the plate and the corresponding collapse mechanism. Moreover, the present analysis allows for the evaluation of the transverse deflection, slope, radial and circumferential bending moments, shearing force within the plate and the reactive pressure of the elastic subgrade at the onset of the plastic collapse together with their dependence on the foundation moduli. The effect of the ratio between negative and positive yield moments is also investigated. The amplitude and assembly of plastic regions at the onset of the plastic collapse are discussed in some detail.
2015
 Effective properties of linear viscoelastic microcracked materials: Application of Maxwell homogenization scheme
[Articolo su rivista]
Sevostianov, I.; Levin, V.; Radi, Enrico
abstract
The present paper focuses on the effect of microcracks on the overall properties of viscoelastic materials. For this goal we extend Maxwell homogenization scheme to the case of viscoelasticity and derive explicit formulas for components of the anisotropic creep operator in dependence on scatter parameter characterizing orientation distribution of cracks.
Microcracks can have any orientation distribution with randomly oriented and strictly parallel being the limiting cases.
Viscoelastic behavior is described using fractionexponential operators. The results are illustrated on example of microckracked polymethylmethacrylate (PMMA).
2015
 Elasticplastic plates on a nonlocal subgrade
[Abstract in Atti di Convegno]
Sorzia, Andrea; Lanzoni, Luca; Nobili, Andrea; Radi, Enrico
abstract
A thin elasticplastic Kirchhoff plate resting on a nonlocal foundation subjected to an axisymmetric load distribution is investigated in the present work. The plate is assumed to obey to the Johansen yield criterion with associative flow rule, whereas the subgrade is modeled like an elastic nonlocal twoparameter foundation. The analysis of an elasticplastic plate resting on a Winkler soil has been developed previously. Later, a generalization has been proposed by introducing a two parameter foundation, but only for a specific ratio between the soil parameters. The present study extends the previous analysis by considering an arbitrary Pasternak soil.
A method based on a contour integral is adopted here to solve in closed form the fourthorder linear ODE governing the elasticplastic region of the plate. The boundary conditions are set by imposing proper continuity conditions for displacement, rotation, bending moment and shear force across the boundaries between the annular regions of the plate. Two different mechanisms at the onset of collapse are found, depending mainly on the amplitude of the loaded region as well as on the stiffness of the plate. The study allows to investigate the effect induced by the Pasternak soil parameters on the ultimate bearing capacity of the plate and the corresponding collapse mechanism. In particular, it is found that a circular elasticplastic region with positive yield lines in both radial and circumferential directions occurs under the loaded area of the plate, whereas such a region degenerates into a point size plastic hinge for a Winkler foundation. Moreover, the amplitude of the loaded region is found to significantly affect the response of the system, particularly in terms of the reactive pressure distribution arising in the underlying foundation.
The study covers a wide array of practical applications in structural and civil engineering, ranging from the assessment of the ultimate bearing capacity of fiber reinforced industrial floors to the behavior of rigid road pavements near its limit state and the ultimate conditions of a building foundation.
2015
 FRC plates on nonlocal subgrade
[Relazione in Atti di Convegno]
Lanzoni, Luca; Nobili, Andrea; Radi, Enrico; Sorzia, Andrea
abstract
The present work deals with the mechanical behavior of a large FRC slab bilaterally supported by a nonlocal elastic soil. The slab is modelled as a ductile Kirchhoff plate laying on a twoparameter elastic foundation and transversally loaded by a uniform pressure applied on a circular area, thus making the problem axisymmetric. This layout covers a wide array of practical applications in the framework of structural and civil engineering, ranging from the assessment of the ultimate bearing capacity of industrial floors, road pavements and building raft foundation. The analysis allows to evaluate the effects induced by the nonlocal character of the foundation on the ultimate carrying capacity and collapse mechanism of FRC slabs.
2015
 On the problem of a Timoshenko beam bonded to an elastic halfplane
[Abstract in Atti di Convegno]
Lanzoni, Luca; Radi, Enrico; Sorzia, Andrea
abstract
The contact problem of beams, rods, ribs and plates bonded to a halfplane has been widely investigated by many Authors. In particular, the problem of prismatic beams resting on a finite or semiinfinite elastic substrate deserves great interest because its practical applications in many engineering application. As an example, Shield and Kim (1992) investigated the problem of an EuleroBernoulli beam resting on an elastic halfplane under symmetric loading conditions, founding the interfacial stresses as well as the SIFs at the edges of the beam. The Authors also studied the effect induced by an elasticperfectly plastic cohesive interface. Nonetheless, a complete analytical study of the contact problem of a Timoshenko beam bonded to a halfplane cannot be found in literature.
The present study concerns the contact problem of a Timoshenko beam resting on an elastic halfplane under general edge loading. The problem is solved by imposing the strain compatibility condition between the beam and the halfplane leading to a. system of 2 integral equations, which is transformed to an algebraic system equations by using series expansions in Jacobi polynomials for the displacement field and Chebyshev polynomials for shear and peeling stresses along the interface.
2015
 Partially coated ceramic layer under thermal stress
[Relazione in Atti di Convegno]
Falope, FEDERICO OYEDEJI; Lanzoni, Luca; Radi, Enrico
abstract
Thin films and coatings technology has known a large development in the last decades due to the large number of devices involving thin films employed in hightech industries, mainly in microelectronics, electrochemistry, semiconductors and optical electronics. Indeed, realization of MEMS and NEMS used into biomedical components, chemical reactors, integrated circuit, solar cells, flat panels displays, sensors, insulator and protection systems, transducers, highprecision measuring instruments, etc. are examples of important applications having significant commercial implication. Recently, many theoretical and experimental studies have been focused on the feasibility of a crystalline undulator (CU), that is a special kind of MEMS realized by covering a ceramic substrate. This microdevice can be used to produce a coherent beam of Xray at high energy levels by exploiting the channelling phenomenon [1]. The substrate generally consists of a Silicon or Germanium crystalline plate covered by a thin film deposed on both surfaces by a proper chemical process (e.g. LPCVD) at high temperature. Through a suitable photolithographic process, the film is properly patterned in order to impart a periodic deformation to the crystalline substrate, suitable to produce coherent interaction with a beam particles. The system adopts a periodic curvature as a result of the misfit strain due to the different thermal expansivities of the layer and the film
The present work provides an extension of the paper [2] by taking into account the anisotropy of the substrate and coatings. The substrate is modelled as a 2D orthotropic elastic layer under plane strain conditions, whereas the film is assumed to behave like a membrane, thus neglecting its flexural stiffness. The problem is formulated by imposing perfect adhesion between the film and the substrate, thus leading to a singular integral equation. The problem can be reduced to a linear algebraic system by using a series expansion of Chebyshev polynomials for the interfacial shear stress and Fourier series expansion for the displacement field. The effects of anisotropy of the materials are then examined and discussed.
2015
 Pathindependent integrals around two circular holes in a thermoelastic medium
[Abstract in Atti di Convegno]
Radi, Enrico; Morini, Lorenzo
abstract
An analytic solution obtained by using bipolar coordinates is presented for thermal stresses in an infinite thermoelastic medium with two unequal circular holes kept at different temperatures. The Jkintegral vector and M and L integrals are derived for steady thermoelasticity and calculated on a closed contour encircling one or both holes.
2015
 Pullout behavior of polypropylene macrosynthetic fibers treated with nanosilica
[Articolo su rivista]
DI MAIDA, Pietro; Radi, Enrico; Sciancalepore, Corrado; Bondioli, Federica
abstract
A study of the effects of nanosilica treatment on the bonding properties of macro synthetic polypropylene fibers embedded in a cement matrix is provided in the present paper as a step to improve interfacial properties of the fiber reinforced cementitious composites (FRCC). Polypropylene fibers were treated by sol–gel technique, allowing to obtain a nanosilica coating. Scanning electron microscopy was used to
observe the morphological features of PP fibers surfaces before and after the pullout test. The effects of the treatment were investigated by comparative pullout tests on treated and untreated fibers. An increase in maximum load and energy necessary for the complete extraction of the fiber was observed, as a consequence of the improvement of the interface properties due to the nanosilica hydration activity.
These two parameters control the crackresistance and ductility properties of FRCC and are deeply affected by bonding and friction phenomena. The hydration products act as chemical and physical anchors, thus producing a densification of the interface transition zone (ITZ). The abrasion phenomena occurring on the fiber surface during the pullout test are responsible of hardening behavior, consisting in the
increase in the frictional shear stress with the fiber slip and thus in the energy required for fiber extraction.
2015
 Thermal stress fields between two unequal circular holes in a ceramic medium
[Abstract in Atti di Convegno]
Radi, Enrico; Morini, Lorenzo
abstract
Thermal stresses play a significant role in a number of engineering problems ranging from the design of heat engines, nuclear plants and aircrafts to the enhancement of electronic devices and MEMS performance. In particular, the determination of stress concentrations due to thermal loadings is a main issue for the accurate design of many electronic devices, where a large number of conductive electric wires are embedded in a ceramic or Silicon matrix at a small distance from each other. In this case, the heat production due to the Joule effect may create high enough thermal stresses to cause cracking and rupture of the insulating ligament between the wires, thus reducing the performance of the device. Since cracks often initiate and propagate from the locations of stress concentration, such as holes and inclusions, then, an accurate evaluation of the stress concentration factor (SCF) in proximity of these defects is a prerequisite to assure the structural integrity of a number of ceramic components and to guarantee the proper functionality of many electronic devices.
An analytic solution is presented here for thermal stresses in an infinite thermoelastic medium with two unequal circular cylindrical holes held at different temperatures, under steadystate heat flux. The most general representation for a biharmonic function in bipolar coordinates has been used. The stress field is decomposed in the sum of a particular stress field induced by the steadystate temperature distribution, which does not satisfy the conditions of vanishing tractions on the surfaces of the holes and vanishing remote stress field, and an auxiliary stress field required to satisfy these boundary conditions, which has been obtained for isothermal elasticity. The corresponding variations of the stress concentration factor, are determined in terms of the holes geometry and temperatures. Moreover, the Jkintegral vector and the Mintegral are first generalized for steady state thermoelasticity and then calculated on a closed contour encircling one or both holes. Results are then presented for varying geometry of the holes.
2014
 A cracked infinite Kirchhoff plate supported by a twoparameter elastic foundation
[Articolo su rivista]
Nobili, Andrea; Radi, Enrico; Lanzoni, Luca
abstract
This paper presents a fullfield solution for the linear elastostatic problem of a homogeneous infinite Kirchhoff plate with a semiinfinite line crack resting on a twoparameter elastic foundation. The same model describes the problem of a cracked plate equibiaxially loaded in its midplane by a constant normal force and, as a limiting case, the problem
of a cracked spherical shell. The fullfield solution is obtained in closed form through the WienerHopf method in terms of Fourier integrals. The stressintensity factor (SIF) for the case of symmetric and skewsymmetric loading conditions are obtained and the role of the soil parameters discussed. In particular, it is shown that a purely local model (Winkler) is unable to provide a safeproof design limit.
2014
 Analytical solution for ductile and FRC plates on elastic ground loaded on a small circular area
[Articolo su rivista]
Radi, Enrico; DI MAIDA, Pietro
abstract
The problem of an infinite FRC slab resting on a Winklertype elastic foundation and subject to a transversal load distributed over a small circular area is investigated in the present work. The mechanical behavior is described by the Kirchhoff theory of elasticperfectly plastic plates obeying Johansen’s yield condition and associative flow rule. The governing equations within both the inner elasticplastic circular region near to the loaded area and the outer elastic region are found in terms of the transversal displacement and solved in closed form. After the formation of radial positive yield lines, namely cracks at the bottom side of the slab, the onset of a circumferential crack at the top of the slab defines its loadcarrying capacity. Two possible configurations are envisaged, depending on whether the circumferential crack occurs within the inner elasticplastic region, where radial cracks take place on the bottom side, or the outer uncracked elastic region. The ratio between the subgrade modulus and flexural rigidity of the plate allows introducing a characteristic length. The influence of both material and geometrical parameters on the ultimate load is then investigated. Based on the analytical results, a simplified method for the calculation of the loadcarrying capacity of FRC slabs on grade is also proposed and compared with previously developed models.
2014
 Full field solution for a rectilinear crack in an infinite Kirchhoff plate supported by a Pasternak elastic foundation.
[Abstract in Atti di Convegno]
Nobili, Andrea; Radi, Enrico; Lanzoni, Luca
abstract
The failure of cracked ceramic components is governed by the stresses in the neighbouring of the crack tip, which is described by the stress intensity factor (SIF).
Despite the availability of several handbooks for SIFs, very few fullfield solutions are available for cracked plates resting on an elastic foundation. This lack of results is problematic, since this situation often occur in practice (e.g. roadways, pavements, floorings, etc.).
Furthermore, when some results are available, they never involve the foundation's mechanical properties alone.
For instance, the problem of a finite crack in an infinite Kirchhoff plate supported by a Winkler foundation is considered in [1] and it is reduced to a singular integral equation. However, since two length scales exist in the problem (the crack length and the foundation relative Winkler modulus), the SIF may be related to some dimensionless ratio of them and not directly to the foundation's mechanical property. In actual facts, this outcome stems from the Winkler approximation to the foundation and not from the physical feature of the problem. The fullfield solution for a semiinfinite rectilinear crack in an infinite Kirchhoff plate resting on a Winkler foundation is found in [2]. Since this is a selfsimilar problem, no characteristic length scale exists. Application of the above results to road and airport pavements is given in [3]. As a result, the influence of the pavement foundation on the SIF cannot be properly assessed.
Several papers address crack problems in plate theory and a literature review of the stress field at the crack tip in thin plates and shells is given in [4] along with comparison with the available experimental results.
The present work deals with the elastostatic problem of a semiinfinite rectilinear crack in an infinite Kirchhoff plate resting on a twoparameter elastic foundation under very general loading conditions. The foundation, also termed Pasternaktype, is weakly nonlocal, as it accommodates for coupling among the independent springs of a purely local model (i.e. the Winkler model).
The same model governs the problem of a Kirchhoff plate equibiaxially loaded in its midplane. The Pasternak foundation accounts for two length scales such that the whole problem is governed by a parameter η expressing the soil to plate relative stiffness. The discussion was addressed in a previous paper [5] but therein limited to the range 0 < η < 1, where the limiting case as η → 0 recovers the nonlocal Winkler model. In the present work the analysis has been extended to the full range of values for the paremeter η. The problem is formulated in terms of a pair of dual integral equations solved through the Wiener–Hopf technique. Numerical results are given for the full field bending and shear stress field within the plate, the corresponding SIFs are obtained and some conclusions drawn.
2014
 Ultimate carrying capacity of elasticplastic plates on Pasternak foundation.
[Articolo su rivista]
Lanzoni, Luca; Radi, Enrico; Nobili, Andrea
abstract
In the present work, the problem of an infinite elasticperfectly plastic plate under axisymmetrical loading conditions resting on a Pasternak elastic foundation is considered. The plate is assumed thin, thus making possible to neglect the shear deformation according to the classical Kirchhoff theory. Yielding is governed by the Johansen’s yield criterion with associative flow rule. An uniformly distributed load is applied on a circular area on the top of the plate. As the load is increased, a circular elasticplastic region spreads out starting from the center of the loaded area, whereas the outer unbounded region behaves elastically. Depending on the size of the loaded area, under the collapse load the inner elasticplastic region can be separated into two or three different regions, corresponding to different yield loci. A closed form solution of the governing equations for each region is found for a special value of the ratio between Pasternak soil moduli. The performed analysis allows to estimate the elasticplastic behavior of the plate up to the onset of collapse, thus providing the collapse load of the plate, the corresponding plastic mechanism and the size of the elasticplastic regions. The influence of the soil moduli, plate bending stiffness, and size of the loaded area on the ultimate bearing capacity of the plate is then investigated in detail.
2013
 Fracture in quasicrystals: vistas
[Abstract in Atti di Convegno]
Paolo Maria, Mariano; Radi, Enrico
abstract
Quasicrystals are aluminiumbased alloys – intermetallic solids, indeed – characterized by longrange order and absence of periodicity in the distribution of atomic maximum probability locations. And periodicity is typical of standard crystals.
In contrast, quasiperiodicity is the characteristic feature of quasicrystal lattices: we have, in fact, prevailing atomic clusters – typically with icosahedral symmetry in 3D space, which is incompatible with lattice translations – and additional atomic arrangements determining the quasiperiodic structure. When we evaluate Fourier representation of mass density over such a quasiperiodic lattice of atoms in 3D ambient space, the related wave vectors result six dimensional to render compatible the expansion with the quasiperiodic geometry. Three of these six degrees of freedom can be attributed to finer spatial scale flips of atoms which annihilate and reconstruct quasiperiodicity in going from metastable states to stable ones and vice versa, depending on the interaction with the surrounding environment. These microstructural events influence even drastically the macroscopic mechanical behaviour of quasicrystals and nonstandard actions (in the sense that they do not admit a representation in terms of standard Cauchy’s stress alone) are power conjugated with them.
The microtomacro coupling is essentially stochastic [1]. In presence of fractures growing dynamically, such microstructural actions influence the force driving the crack tip.
Here, we present an overview of questions connected with the dynamic propagation of fractures in quasicrystals.
1. In finite strain setting, we show (by following the general theoretical structure presented in [2] for the mechanics of simple and complex materials undergoing macroscopic mutations such as fractures) how the laws of dynamics crack propagation in quasicrystals derive from a requirement of invariance of what is called relative power. The procedure puts in evidence the existence of a bulk conservative selfaction, in contrast with common formulations of the mechanics of quasicrystals where the conservative part of the selfaction is neglected.
2. In the limit of vanishing conservative selfaction, and within the limitations of small strain regime, by using Stroh’s formalism we provide solutions to boundary value problems involving dynamic steadystate crack propagation in quasicrystals [3].
3. Finally, in small strain regime we present solutions to boundary value problems involving subsonic crack propagation between dissimilar quasicrystals [4] and allow comparison between the cases of presence and absence of conservative microstructural selfaction.
2013
 Integral identities for a semiinfinite interfacial crack in anisotropic elastic bimaterials
[Articolo su rivista]
Morini, Lorenzo; Piccolroaz, A.; Mishuris, G.; Radi, Enrico
abstract
The focus of the article is on the analysis of a semiinfinite crack at the interface between two dissimilar
anisotropic elastic materials, loaded by a general asymmetrical system of forces acting on the crack faces.
Recently derived symmetric and skewsymmetric weight function matrices are introduced for both plane
strain and antiplane shear cracks, and used together with the fundamental reciprocal identity (Betti formula)
in order to formulate the elastic fracture problem in terms of singular integral equations relating
the applied loading and the resulting crack opening. The proposed compact formulation can be used to
solve many problems in linear elastic fracture mechanics (for example various classic crack problems
in homogeneous and heterogeneous anisotropic media, as piezoceramics or composite materials). This
formulation is also fundamental in many multifield theories, where the elastic problem is coupled with
other concurrent physical phenomena.
2013
 Interface fracture phenomena in quasicrystals.
[Abstract in Atti di Convegno]
Radi, Enrico; Mariano, P. M.
abstract
The twodimensional setting selected here allows the use of Stroh formalism, leading to a complex variable representation of generalized stress and displacement fields in terms of four complex potentials. We adapt it to the mechanics of QCs by following the path used to describe the steadystate crack propagation of straight cracks in bodies constituted by a single type of QCs. When applied directly to the mechanics of QCs, the procedure based on the standard Stroh formalism involves a degenerate eigenvalue problem, so appropriate modifications have to be considered. The bimaterial nature of the body under scrutiny here is accounted for by generalizing the approach used for classical elastic bimaterial.
The results shows that macroscopic and substructural stresses display oscillatory behaviour or classical square root singularity at the crack tip, depending on the constitutive parameters of both QCs. Finally, the energy release rate is evaluated for subsonic subRayleigh crack propagation.
2013
 On fracture criteria for dynamic crack propagation in elastic materials with couple stresses
[Articolo su rivista]
Morini, Lorenzo; Piccolroaz, A.; Mishuris, G.; Radi, Enrico
abstract
The focus of the article is on fracture criteria for dynamic crack propagation in elastic materials with microstructures. Steadystate propagation of a Mode III semiinfinite crack subject to loading applied on the crack surfaces is considered. The micropolar behavior of the material is described by the theory of couplestress elasticity developed by Koiter. This constitutive model includes the characteristic lengths in bending and torsion, and thus it is able to account for the underlying microstructures of the material. Both translational and microrotational inertial terms are included in the balance equations, and the behavior of the solution near to the crack tip is investigated by means of an asymptotic analysis. The asymptotic fields are used to evaluate the dynamic Jintegral for a couplestress material, and the energy release rate is derived by the corresponding conservation law. The propagation stability is studied according to the energybased Griffith criterion and the obtained results are compared to those derived by the application of the maximum total shear stress criterion.
2013
 Stroh formalism in analysis of skewsymmetric and symmetric weight functions for interfacial cracks
[Articolo su rivista]
Morini, Lorenzo; Radi, Enrico; A. B., Movchan; N. V., Movchan
abstract
The focus of the paper is on the analysis of skewsymmetric weight functions for interfacial cracks in twodimensional anisotropic solids. It is shown that the Stroh formalism proves to be an efficient tool for this challenging task.
Conventionally, the weight functions, both symmetric and skewsymmetric, can be identified as nontrivial singular solutions of a homogeneous boundaryvalue problem for a solid with a crack. For a semiinfinite crack, the problem can be reduced to solving a matrix Wiener–Hopf functional equation. Instead, the Stroh matrix representation of displacements and tractions, combined with a Riemann–Hilbert formulation, is used to obtain an algebraic eigenvalue problem, which is solved in a closed form.
The proposed general method is applied to the case of a quasistatic semiinfinite crack propagating between two dissimilar orthotropic media: explicit expressions for the weight functions are evaluated and then used in the computation of the complex stress intensity factor corresponding to an asymmetric load acting on the crack faces.
2013
 The bending stress in a cracked Kirchhoff plate resting on a Pasternak foundation
[Relazione in Atti di Convegno]
Lanzoni, Luca; Nobili, Andrea; Radi, Enrico
abstract
The problem of a rectilinear semiillimitate crack in a Kirchhoff elastic plate laying on a weakly nonlocal elastic twoparameter (Pasternak) foundation and subject to transversal loadings applied on the plate surfaces is studied in the present work. By extending the approach developed
by Ang et al. [1] for a plate on a Winkler elastic foundation, the analytical fullfield solution is obtained by solving a system of dual integral equations. In particular, the kernel function is factorized by using a procedure based on the Cauchy theorem, similar to the WienerHopf method
used in the solution of antiplane fracture problems [2, 3].
Closedform solutions are thus provided for displacement, bending moment and shear force per unit length along the crack line. The ratioes between the subgrade parameters and flexural rigidity of the plate allow introducing two
characteristic lengths, whose influence on the bending stress intensity factor and fracture toughness of the plate is then investigated.
The governing equation is closely related to the bending problem of initially curved thin sheets under transversal loading and subjected to a constant normal stress, so that results apply to such situation as well. Accordingly, this is one of the few closedform solutions available for initially
curved cracked thin sheets.
2013
 The bending stress in a cracked ceramic plate resting on a two parameter elastic grade
[Abstract in Atti di Convegno]
Lanzoni, Luca; Nobili, Andrea; Radi, Enrico
abstract
The problem of a rectilinear semiillimitate crack in a ceramic plate laying on a weakly nonlocal elastic twoparameter (Pasternak) foundation and subject to transversal loadings applied on the plate surfaces is studied in the present work.
The analytical fullfield solution is obtained by solving a system of dual integral equations. In particular, the kernel function is factorized by using a procedure based on the Cauchy theorem, similar to the WienerHopf method used in the solution of antiplane fracture problems.
Closedform solutions are thus provided for displacement, bending moment and shear force per unit length along the crack line.
The ratioes between the subgrade parameters and flexural rigidity of the plate allow introducing two characteristic lengths, whose influence on the bending stress intensity factor and fracture toughness of the plate is then investigated.
2012
 Interfacial cracks in bimaterial solids: Stroh formalism and skewsymmetric weight functions
[Abstract in Atti di Convegno]
Morini, Lorenzo; Radi, Enrico; A. B., Movchan; N. V., Movchan
abstract
A new general approach for deriving the weight functions for 2D interfacial cracks in anisotropic bimaterials has been developed.For perfect interface conditions, the new method avoid the use of WienerHopf technique and the challenging factorization problem connected. Both symmetric and skewsymmetric weight functions can be derived by means of the new approach. Weight functions can be used for deriving singular integral formulation of interfacial cracks in anisotropic media. The proposed method can be applied for studying interfacial cracks problems in many materials:monoclinic, orthotropic, cubic, piezoelectrics, poroelastics, quasicrystals.
2012
 Mode III crack propagation in couple stress elastic materials.
[Abstract in Atti di Convegno]
Piccolroaz, A.; Mishuris, G.; Radi, Enrico
abstract
The mechanical behaviour of most materials with microstructure, like composites, cellular materials, foams, masonry, bone tissues, glassy and semicrystalline polymers, is strongly influenced by the microstructural characteristic lengths, especially in the presence of large (or strain) gradients (Lakes, 1986). These findings stimulated the development of generalized theories of continuum mechanics such as micropolar elasticity and indeterminate couple stress elasticity (Koiter, 1964). Due to the effects of the characteristic lengths, these generalized theories predict dispersive wave propagation in microstructured media. Moreover, they predict different asymptotic behaviour for some components of the deformation and stress fields near singular points, compared to the classical elasticity theory. Due to the complexity of the equations of motion provided by the couple stress elastic theory with rotational inertia, only few crack propagation problems have been considered in the literature, most of them have been solved numerically and very few fullfield solution have been worked out in closed form. However, a simple asymptotic characterization of the crack tip fields is not sufficient to analyse the fracture behaviour of materials with microstructure. This is due to the fact that the asymptotic solution is valid in a region close to the crack tip which is smaller than the characteristic lengths of the material, and thus it is of scarce physical relevance. A full field analysis is then required in order to grasp the qualitative and quantitative behaviour of the solution in a larger region and to be able to judge on the stress level supported by the material.The full field solution obtained by Radi (2008) for a Mode III stationary crack by using Fourier transforms and Wiener–Hopf technique, shows that ahead of the crack tip within a zone smaller than the characteristic length in torsion, the total shear stress and reduced tractions occur with the opposite sign with respect to the classical LEFM solution. However, this zone is found to have limited physical relevance and to become vanishing small for a characteristic length in torsion of zero. In this limit, the solution recovers the classical KIII field with square root stress singularity. Outside the zone ahead of the crack tip where the total shear stress is negative, the full field solution exhibits a bounded maximum for the total shear stress, whose magnitude was adopted as a measure of the critical stress level for crack advancing. The introduction of a stress based fracture criterion thus defines a critical stress intensity factor increasing with the characteristic length in torsion.In this paper the previous analysis is extended to the case of dynamic Mode III crack propagation, subject to antiplane loading applied on the crack surfaces, in order to study the effects of inertia and cracktip speed on the stress and deformation fields, as well as the variation of the fracture toughness due to the presence of microstructure. In particular, we assume that the crack tip speed is smaller than the shear wave velocity. The constitutive behaviour is described by the indeterminate theory of couple stress elasticity developed by Koiter. Differently from classical elastic theories, this reasonably simple model is able to account for the underlying microstructure as well as for the strong size effects arising at small scales. The analytical fullfield solution is addressed making use of Fourier transform and Wiener–Hopf technique. Closedform solutions are provided for vanishing or small contribution from rotational inertia. The analysis confirms and extends earlier results on stationary cracks (Radi, 2008) by including the effects of crack velocity and rotational inertia. By adopting the criterion of maximum total shear stress previously introduced, we also discuss the effects of microstructural parameters on the stability of crack prop
2012
 Mode III interfacial crack in the presence of couple stress elastic materials
[Articolo su rivista]
A., Piccolroaz; G., Mishuris; Radi, Enrico
abstract
In this paper we are concerned with the problem of a crack lying at the interface between dissimilar materials with microstructure undergoing antiplane deformations. The micropolar behaviour of the materials is described by the theory of couple stress elasticity developed by Koiter (1964). This constitutive model includes the characteristic lengths in bending and torsion and thus it is able to account for the underlying microstructure of the two materials. We perform an asymptotic analysis to investigate the behaviour of the solution near the crack tip. It turns out that the stress singularity at the crack tip is strongly influenced by the microstructural parameters and it may or may not showoscillatory behaviour depending on the ratio between the characteristic lengths
2012
 Steadystate propagation of a Mode III crack in couple stress elastic materials
[Articolo su rivista]
G., Mishuris; A., Piccolroaz; Radi, Enrico
abstract
This paper is concerned with the problem of a semiinfinite crack steadily propagating in an elastic solidwith microstructures subject to antiplane loading applied on the crack surfaces. The loading is movingwith the same constant velocity as that of the crack tip. We assume subsonic regime, that is the crackvelocity is smaller than the shear wave velocity. The material behaviour is described by the indeterminatetheory of couple stress elasticity developed by Koiter. This constitutive model includes the characteristiclengths in bending and torsion and thus it is able to account for the underlying microstructure of thematerial as well as for the strong size effects arising at small scales and observed when the representativescale of the deformation field becomes comparable with the length scale of the microstructure, such asthe grain size in a polycrystalline or granular aggregate.The present analysis confirms and extends earlier results on the static case by including the effectsof crack velocity and rotational inertia. By adopting the criterion of maximum total shear stress, wediscuss the effects of microstructural parameters on the stability of crack propagation.
2011
 Antiplane crack in couple stress elastic materials under dynamic loadings
[Abstract in Atti di Convegno]
Radi, Enrico; G., Mishuris; A., Piccolroaz
abstract
The problem of a rectilinear crack in an elastic solid with microstructures subject to harmonic loadings applied on the crack surfaces is investigated in the present work. The material behavior is described by the indeterminate theory of couple stress elasticity developed by Koiter. This constitutive model includes the characteristic lengths in bending and torsion and thus it is able to account for the underlying microstructure of the material as well as for the strong size effects arising at small scales and observed when the representative scale of the deformation field becomes comparable to the length scale of the microstructure, such as the grain size in a polycrystalline or granular aggregate. It is sufficiently accurate to simulate the behavior of materials at the micron scale as well as the size effects occurring at distances to the crack tip comparable to characteristic lengths, but it is also simple enough to allow the achievement of closedform solutions.The quasistatic fullfield solution, obtained by using Fourier transforms and WienerHopf technique and the asymptotic analysis of the interface crack problem, showed that ahead of the crack tip within a zone smaller than the characteristic length in torsion, the total shear stress and reduced tractions occur with the opposite sign with respect to the classical LEFM solution. However, this zone was found to have limited physical relevance and to become vanishing small for a characteristic length in torsion of zero. In this limit case, the solution recovers the classical square root stress singularity. Outside the zone where the total shear stress is negative, the full field solution exhibits a bounded maximum for the total shear stress ahead of the crack tip, whose magnitude was adopted as a measure of the critical stress level for crack advancing. The corresponding fracture criterion defines a critical stress intensity factor, which increases with the characteristic length in torsion.In the present work the analysis is extended to the case of time harmonical loading of the crack faces in order to study the effects of inertia and dynamic loading on the stress and deformation fields, as well as the variation of the fracture toughness due to the presence of microstructures.
2011
 Dynamic steadystate crack propagation in quasicrystals
[Articolo su rivista]
Radi, Enrico; P. M., Mariano
abstract
A closedform solution is provided for the stress, strain and velocity fields due to a planar crack steadily propagating in an elastic quasicrystal with fivefold symmetry at speed lower than the bulk wavespeeds. Both cases of a semiinfinite rectilinear crack and a Griffith crack which propagates mantaining a constant length, according to the Yoffe model, are considered. Crack face loading and remote loading conditions are taken into consideration. The dynamic theory of quasicrystal with inertia forces, but neglecting dissipative phonon activity, is assumed to govern the motion of the medium. The phonon and phason stress fields turn out to be squareroot singular at crack tip. The energy release rate is positive for subsonic and subRayleigh crack propagation.
2011
 M and Lintegrals enclosing two circular holes in an infinite plate
[Abstract in Atti di Convegno]
Radi, Enrico
abstract
An analytic solution is presented for stresses induced in an infinite plate with two unequal circular holes by remote uniform loadings and arbitrary internal pressures in the holes. The solution has been obtained by using the general expression for a biharmonic function in bipolar coordinates provided by Jeffery (1921). The Airy stress function is decomposed in the sum of a fundamental stress function for an infinite plate remotely loaded, which gives non vanishing tractions on the circular boundaries, and an auxiliary stress function required to satisfy the boundary conditions on the pressures at the edges of the holes, which produces vanishing stresses at infinity. By using the Jeffery solution, the problem of a circular disk containing a sliding eccentric circular inclusion has been recently solved by Radi and Strozzi (2009).Once the stress and displacement fields are obtained in closed form, the path independent Jk (k = 1, 2), M and Lintegrals introduced by Knowles and Sternberg (1972) and Budiansky and Rice (1973) are analytically calculated on a closed contour encircling both holes by considering traction free hole surfaces. These integrals play an important role in the description of multiple defects damaged brittle materials. Physically, the Jk, M and Lintegrals can be interpreted as the energy release rate for uniform movements, expansion, and rotation of the defects, respectively. Results are here presented for varying loading orientation angle ζ and holes geometry. The J1 and J2integrals calculated for a closed contour enclosing both holes are found to vanish, whatever be the remote loading orientation and holes geometry. These results confirm the conservation laws proposed by Chen and Hasabe (1998) and then proved by Chen (2001) for multiple discontinuities, such as cracks, voids and inclusions, subject to remote uniform loading conditions, when the integration contour encloses all the discontinuities. Differently from the J1 and J2integrals, the M and Lintegrals do not vanish when the integration contour encloses both holes. In particular, the Mintegral attains a maximum for a certain loading orientation angle ζ0, and, correspondingly, the Lintegral becomes vanishing small. For ζ < ζ0 the Lintegral turns out to be negative, whereas for ζ > ζ0 it assumes positive values. Chen (2001) and Hu and Chen (2009, 2011) observed that an implicit relation exists between the Mintegral and the reduction in the effective elastic modulus. These authors showed that the loading direction along which the Mintegral becomes maximum coincides with the direction corresponding to the minimum of the effective elastic modulus, due to the presence of the holes. Conversely, the loading direction along which the Mintegral becomes minimum is just the direction of the maximum effective elastic modulus. This occurrence allowed these authors to conjecture the possiblity of formulating the effective elastic properties and describing the damage level induced by interacting holes in terms of the Mintegral, although the proper mathematical formulation has not been adequately investigated yet.The purpose of the present contribution is to provide some basic understanding for the role played by conservation laws in multiple defects analysis.
2011
 Mode III crack in couple stress elastic materials under timeharmonic loadings
[Relazione in Atti di Convegno]
Radi, Enrico; G., Mishuris; A., Piccolroaz
abstract
The problem of a rectilinear crack in an elastic solid with microstructures subject to harmonic loadings applied on the crack surfaces is investigated. The material behavior is described by the indeterminate theory of couple stress elasticity developed by Koiter.This constitutive model includes the characteristic lengths in bending and torsion and thus it is able to account for the underlying microstructure of the material as well as for the strong size effects arising at small scales and observed when the representative scale of the deformation field becomes comparable to the length scale of the microstructure. It is sufficiently accurate to simulate the behavior ofmaterials at the micron scale as well as the size effects occurring at distances to the crack tip comparable to characteristic lengths, but it is also simple enough to allow the achievement of closedform solutions.The stress and displacement fields near the tip of a Mode III crack are strongly influenced by the microstructural characteristic lengths. The quasistatic fullfield solution, obtained by using Fourier transforms and WienerHopf technique and the asymptotic analysis of the interface crack problem, showed that ahead of the crack tip within a zone smaller than the characteristic length in torsion, the total shear stress and reduced tractions occur with the opposite sign with respect to the classical LEFM solution. However, this zone was found to have limited physical relevance and to become vanishing small for a characteristic length in torsion of zero. Within this limit case, the solution recovers the classical KIII field with square root stress singularity. Outside the zone where the total shear stress is negative, the full field solution exhibits a bounded maximum for the total shear stress ahead of the crack tip, whose magnitude was adopted as a measure of the critical stress level for crack advancing. The corresponding fracture criterion defines a critical stress intensity factor, which increases with the characteristic length in torsion.In the proposed research the previous analysis will be extended to the dynamic case in order to study the effects of inertia on the stress and deformation fields, as well as the variation of the fracture toughness due to the presence of microstructures.
2011
 Pathindependent integrals around two circular holes in an infinite plate under biaxial loading conditions
[Articolo su rivista]
Radi, Enrico
abstract
An analytic solution is presented for stresses induced in an infinite plate with two unequal circular holes by remote uniform loadings and arbitrary internal pressures in the holes. The solution has been obtained by using the general expression for a biharmonic function in bipolar coordinates. The Airy stress function is decomposed in the sum of a fundamental stress function for an infinite plate remotely loaded, which gives non vanishing tractions on the circular boundaries, and an auxiliary stress function required to satisfy the boundary conditions on the pressures at the edges of the holes, which produces vanishing stresses at infinity. Correspondingly, the variations of the stress concentration factor are determined in terms of the holes geometry and loading conditions. The path independent Jk (k = 1, 2), M and Lintegrals are analytically calculated on a closed contour encircling the two holes, under remote loading, in order to evaluate the energy release rates accompanying unit translation, self similar expansion and rotation of the holes, respectively. Results are then presented for varying loading orientation angle, biaxial loading ratio and holes geometry.
2011
 Steadystate moving dislocations in quasicrystals
[Relazione in Atti di Convegno]
Radi, Enrico; P. M., Mariano
abstract
We summarize in these pages results on gliding and climbing dislocation steadystate motion in twodimensional quasicrystals that we have presented elsewhere. The analysis is developed in infinitesimal deformation setting. Linear elastic constitutive structures are involved. We furnish closedform solutions to the problems that we analyze. Emphasis is done on the coupling between macroscopic and microstructural events.
2011
 Steadystate propagation of dislocations in quasicrystals
[Articolo su rivista]
Radi, Enrico; P. M., Mariano
abstract
We analyze the steady propagation at constant speed, lower than the shear wavespeed, of a straight dislocation in an unbounded elastic quasicrystal with fivefold symmetry. We discuss only the ideal elastic behavior, neglecting the dissipation associated with the atomic rearrangements. Under these conditions, we provide the expressions of standard and microscopic (phason) fields in closed form. Both standard and phason stresses appear to be singular near to the dislocation core. We also find the explicit expression of the energy per unit length around a moving dislocation.
2011
 Surface waves propagation in 2D icosahedral quasicrystals
[Abstract in Atti di Convegno]
Morini, Lorenzo; Radi, Enrico
abstract
Surface waves propagation in icosahedral quasicrystals is investigated by means of planar steady state problem setting in generalized linear elasticity. By using the Stroh formalism, the dispersion relation for surface waves propagating in a halfplane of quasicristalline material is derived. Analytical solutions for the dispersion relation are evaluated for small phasonic diffusive coefficient as well as for vanishing small phason diffusion. The influence of phasonic diffusion on the surface wave propagation is analyzed and discussed in detail.
2010
 Effects of microstructure on antiplane crack growth in couplestress elastic materials
[Relazione in Atti di Convegno]
Radi, Enrico
abstract
The problem of a propagating rectilinear crack in an elastic solid with microstructures subject to remote classical KIII field is investigated in the present work. The material behavior is described by the indeterminate theory of couple stress elasticity developed by Koiter. This constitutive model includes the characteristic lengths in bending and torsion and thus it is able to account for the underlying microstructure of the material as well as for the strong size effects arising at small scales and observed when the representative scale of the deformation field becomes comparable to the length scale of the microstructure, such as the grain size in a polycrystalline or granular aggregate.
The stress and displacement fields near the tip of a Mode III propagting crack are thus expected to be strongly influenced by the microstructural characteristic lengths. The stationary fullfield solution, already obtained [1] by using Fourier transforms and WienerHopf technique, showed that ahead of the crack tip within a zone smaller than the characteristic length in torsion, the total shear stress and reduced tractions occur with the opposite sign with respect to the classical LEFM solution, due to the relative rotation of the microstructural particles currently at the crack tip. However, this zone was found to have limited physical relevance and to become vanishing small for a characteristic length in torsion of zero. In this limit case, the solution recovers the classical KIII field with square root stress singularity. Outside the zone where the total shear stress is negative, the full field solution exhibits a bounded maximum for the total shear stress ahead of the crack tip, whose magnitude was adopted as a measure of the critical stress level for crack advancing. The corresponding fracture criterion defines a critical stress intensity factor, which increases with the characteristic length in torsion.
In the proposed research the previous analysis will be extended in order to consider the effects of crack speed and inertia terms on the stress and deformation fields, as well as on the stability of the crack propagation in the presence of microstructures.
2010
 Mechanics of interfacial cracks between dissimilar quasicrystals
[Relazione in Atti di Convegno]
J., Planas; Radi, Enrico; M. M., Stickle; P. M., Mariano
abstract
We analize the steady propagation of a straight interfacial crack between two dissimilar quasicrystals in the infinitesimal deformation regime. A closed form solution to the balance equations is furnished. Inertia is attributed only to the macroscopic motion.The twodimensional setting selected here allows the use of Stroh formalism, thus generalizing the method used to describe the propagation of straight cracks in bodies constituted by a single type of quasicrystals. Boundary conditions are prescribed at infinityand along the interface, including the crack.
2010
 Preliminary plane mechanical modeling of hexagonal contact
[Relazione in Atti di Convegno]
Strozzi, Antonio; Radi, Enrico; Baldini, Andrea; Giacopini, Matteo; Campioni, Eleonora
abstract
A hexagonal joint is mechanically analysed. A cross section of the contact between male and female components is modelled as a plane strain problem, and the contact and detachment zones are investigated with two approaches, a) an analytical study
formulated in terms of an integral equation; b) a FE analysis. Preliminary results refer to the situation of null initial clearance and coefficient of friction. For each side of the hexagonal contact, the contact zone constitutes a small portion of the length of the
hexagonal side, since separation occurs along a sizeable side length.
2010
 Stationary straight cracks in quasicrystals
[Articolo su rivista]
Radi, Enrico; Mariano, P. M.
abstract
Stationary straight cracks in quasicrystals in linear elastic setting are under scrutiny. The analysis is developed by using Stroh formalism which is modified to account for a totally degenerate eigenvalue problem: in fact, the fundamental matrix of the governing equations of motion admits a repeated eigenvalue corresponding to a single eigenvector.
Cases of a semiinfinite rectilinear crack loaded along its margins and a crack of finite length under remote loading conditions are considered.
Standard and phason stresses display squareroot singularities at crack tip. The latter stresses represent peculiar microstructural inner actions occurring
in quasicrystals and are determined by rearrangements
assuring quasiperiodicity of the atomic tiling modes described by a vector field, called phason field, collecting the local degrees of freedom exploited by the atoms within the material elements.
Energy release rate increases with the coupling parameter between displacement and phason fields.
2009
 Jeffery solution for an elastic disk containing a sliding eccentric circular inclusion assembled by interference fit
[Articolo su rivista]
Radi, Enrico; Strozzi, Antonio
abstract
An analytic solution is presented for stresses induced in an elastic and isotropic disk by an eccentric pressfitted circular inclusion. The disk is also subject to uniform normal stress applied at its outer border. The inclusion is assumed to be of the same material as the annular disk and both elements are in a plane stress or plane strain state. A frictionless contact condition is assumed between the two members. The solution is obtained by using the general expression for a biharmonic stress function in bipolar coordinates. The results show that the maximum of the von Mises effective stress due to the inclusion interference occurs in the ligament for large eccentricity, but it deviates from the symmetry axis for small eccentricity. Moreover, along the border of the circular inclusion the hoop stress locally coincides with the contact pressure, in agreement with a similar classical result valid for a half plane.
2009
 Propagation of cracks and dislocations in 2D quasicrystals
[Relazione in Atti di Convegno]
Radi, Enrico; Mariano, P. M.
abstract
A closedform solution is provided for the stress, strain and velocity fields due to a planar crack steadily propagating in an elastic quasicrystal with fivefold symmetry at speed lower than the bulk wavespeeds. The cases of a semiinfinite rectilinear crack and a Griffith crack which propagates maintaining a constant length, according to the Yoffe model, are considered. Crack face loading and remote loading conditions are taken into consideration. The dynamic theory of quasicrystal with inertia forces, but neglecting dissipative phonon activity, is assumed to govern the motion of the medium. The phonon and phason stress fields turn out to be squareroot singular at crack tip. The energy release rate is positive for subsonic and subRayleigh crack propagation.
2009
 Thermally induced deformations in a partially coated elastic layer
[Articolo su rivista]
Lanzoni, L.; Radi, Enrico
abstract
The problem of a thin film coated on an elastic layer and subject to a thermal variation is analytically investigated in the present work. The analysis is developed in order to assess the mechanical behaviour of a crystalline undulator designed for obtaining high emission radiations through channelling phenomenon.It consists in a plane silicon wafer alternately patterned with thin films in silicon nitride on both surfaces. The system adopts a periodic curvature as a result of the misfit strain due to the different thermal expansivities of the layer and the film. The problem is governed by an integral equation which can be reduced to a linear algebraic system by approximating the unknown interfacial shear stress via series expansion of Chebyshev polynomials.
2008
 A steadily propagating crack in planar quasicristals with fivefold symmetry
[Relazione in Atti di Convegno]
Radi, Enrico; P. M., Mariano
abstract
A closedform solution is provided for the stress, strain and velocity fields due to a planar crack steadily propagating in an elastic quasicrystal with fivefold symmetry at speed lower than the bulk wavespeeds. The case of a semiinfinite rectilinear crack loaded on its surfaces is considered. The dynamic theory of quasicrystal with inertia forces, but neglecting dissipative phonon activity, is assumed to govern the motion of the medium. Both phonon and phason stress fields display squareroot singular at crack tip. The energy release rate is positive for subsonic and subRayleigh crack propagation. The limit case of a stationary crack is then recovered as the crack tip speed becomes vanishing small.
2008
 A stedily propagating crack in planar quasicrystal with fivefold symmetry
[Relazione in Atti di Convegno]
Radi, E.; Mariano, P. M.
abstract
A closedform solution is provided for the stress, strain and velocity fields due to a planar crack steadily propagating in an elastic quasicrystal with fivefold symmetry at speed lower than the bulk wavespeeds. The case of a semiinfinite rectilinear crack loaded on its surfaces is considered. The dynamic theory of quasicrystal with inertia forces, but neglecting dissipative phonon activity, is assumed to govern the motion of the medium. Both phonon and phason stress fields display squareroot singular at crack tip. The energy release rate is positive for subsonic and subRayleigh crack propagation. The limit case of a stationary crack is then recovered as the crack tip speed becomes vanishing small.
2008
 Mode I intersonic crack propagation in poroelastic media
[Articolo su rivista]
Radi, Enrico; B., Loret
abstract
Intersonic crack propagation takes place in an elastic fluidsaturated porous solid under Mode I loading conditions.The crack tip speeds of interest c are constant and bounded below by the slower between the slow longitudinal wavespeed and the shear wavespeed, and above by the fast longitudinal wavespeed. Biot’s theory of poroelasticity with inertia forces governs the motion of the mixture. The poroelastic moduli depend on the porosity, and the complete range of porosities [0, 1] is investigated. Three characteristic regions in the plane (n, c) are delineated, depending on the relative order of the body wavespeeds.The crack surface is considered to be permeable. Cracks with and without a process zone are envisaged.In each region of the plane (n, c), the analytical solution to a Riemann–Hilbert problem provides the stress, pore pressureand velocity fields near the tip of the crack. The effective length scale introduced by the process zone is found to depend strongly on both the actual length of the process zone and the singularity exponent.Intersonic crack propagation may occur with squareroot singularity of the stress and velocity fields for crack tip speeds slower than the shear wavespeed and faster than the slow longitudinal wavespeed. In this region of the (n, c)plane, the energy release rate is finite and positive, for cracks with and without a process zone. As an uncommon feature, linked to the fact that the effective singularity exponent is 1/2 for both crack types, the stress and velocity fields are continuous across the Mach ray.
2008
 On the effects of characteristic lengths in bending and torsion on Mode III crack in couple stress elasticity
[Articolo su rivista]
Radi, Enrico
abstract
The problem of a stationary semiinfinite crack in an elastic solid with microstructures subject to remote classical KIII field is investigated in the present work. The material behavior is described by the indeterminate theory of couple stress elasticity developed by Koiter. This constitutive model includes the characteristic lengths in bending and torsion and thus it is able to account for the underlying microstructure of the material as well as for the strong size effects arising at small scales. The stress and displacement fields turn out to be strongly influenced by the ratio between the characteristic lengths.Moreover, the symmetric stress field turns out to be finite at the crack tip, whereas the skewsymmetric stress field displays a strong singularity. Ahead of the crack tip within a zone smaller than the characteristic length in torsion, the total shear stress and reduced tractions occur with the opposite sign with respect to the classical LEFM solution, due to the relative rotation of the microstructural particles currently at the crack tip. The asymptotic fields dominate within this zone, which however has limited physical relevance and becomes vanishing small for a characteristic length in torsion of zero. In this limiting case the fullfield solution recovers the classical KIII field with squareroot stress singularity. Outside the zone where the total shear stress is negative, the fullfield solution exhibits a bounded maximum for the total shear stress ahead of the crack tip, whose magnitude can be adopted as a measure of the critical stress level for crack advancing. The corresponding fracture criterion defines a critical stress intensity factor, which increases with the characteristic length in torsion.Moreover, the occurrence of a sharp crack profile denotes that the crack becomes stiffer with respect to the classical elasticresponse, thus revealing that the presence of microstructures may shield the crack tip from fracture.
2007
 Effects of characteristic material lengths on mode III crack propagation in couple stress elasticplastic materials
[Articolo su rivista]
Radi, Enrico
abstract
The asymptotic fields near the tip of a crack steadily propagating in a ductile material under Mode III loading conditions are investigated by adopting an incremental version of the indeterminate theory of couple stress plasticity displaying linear and isotropic strain hardening. The adopted constitutive model is able to account for the microstructure of the material by incorporating two distinct material characteristic lengths. It can also capture the strong size effects arising at small scales, which results from the underlying microstructures. According to the asymptotic crack tip fields for a stationary crack provided by the indeterminate theory of couple stress elasticity, the effects of microstructure mainly consist in a switch in the sign of tractions and displacement and in a substantial increase in the singularity of tractions ahead of the cracktip, with respect to the classical solution of LEFM and EPFM. The increase in the stress singularity also occurs for small values of the strain hardening coefficient and is essentially due to the skewsymmetric stress field, since the symmetric stress field turns out to be nonsingular. Moreover, the obtained results show that the ratio g introduced by Koiter has a limited effect on the strength of the stress singularity. However, it displays a strong influence on the angular distribution of the asymptotic crack tip fields.
2007
 Fullfield solution for Mode III crack in couple stress elastic materials with two characteristic lengths
[Relazione in Atti di Convegno]
Radi, Enrico
abstract
The problem of a stationary semi infinite crack in an elastic solid with microstructures subject to remote classical KIII field is investigated in the present work. The material behaviour is described by the indeterminate theory of couple stress elasticity. The adopted constitutive model incorporates the characteristic lengths in bending and torsion of the material and thus it is able to account for the underlying microstructure as well as for the strong size effects arising at small scales. The stress and displacement fields turn out to be strongly influenced by the ratio between the characteristic lengths. In particular, due to the relative rotation of the microstructural particles currently at the crack tip the total shear stress and reduced tractions ahead of the crack tip display the opposite sign with respect to the classical LEFM solution within a zone smaller than the characteristic length in torsion. However, this zone has limited physical relevance and becomes vanishing small for a characteristic length in torsion of zero. Outside this zone, the full field solution exhibits a bounded maximum for the shear stress ahead of the crack tip, whose magnitude can be adopted as a measure of the critical stress level for crack advancing. The corresponding fracture criterion defines a critical stress intensity factor which increases with the characteristic length in torsion. Moreover, the occurrence of a sharp crack profile indicates that the crack becomes stiffer with respect to the classical elastic response, thus revealing that the presence of microstructures may shield the crack tip from fracture.
2007
 Fullfield solution for an antiplane shear crack in elastic materials with microstructures
[Relazione in Atti di Convegno]
Radi, Enrico
abstract
In questo lavoro si studia il problema di una frattura semillimitata in condizioni di Modo III in un mezzo elastico caratterizzato dalla presenza di microstruttura. Il comportamento del materiale viene rappresentato attraverso il modello costitutivo elastico micropolare sviluppato da Koiter. Il modello considerato include le lunghezze caratteristiche a flessione e a torsione, proprie della microstruttura del materiale, e risulta pertanto in grado di simulare i rilevanti effetti di scala che si riscontrano in prossimità dell’apice di una frattura nei materiali con microstruttura, a distanze comparabili alle lunghezze caratteristiche. I campi di tensione e spostamento risultano notevolmente influenzati dal rapporto tra le lunghezze caratteristiche. In particolare, le tensioni tangenziali davanti all’apice a distanza inferiore alla lunghezza caratteristica a torsione, risultano di segno opposto rispetto alla soluzione elastica classica, a causa della rotazione relativa tra le particelle materiali situate in corrispondenza dell’apice. La soluzione completa mostra che la zona in cui si verifica tale inversione ha un’estensione molto ridotta e tende ad annullarsi con la lunghezza caratteristica a torsione del materiale. Al di fuori di tale zona, la tensione tangenziale assume un valore massimo, positivo e limitato. Tale valore può adottarsi come misura del livello critico di tensione necessario per far propagare la frattura. Il manifestarsi di un profilo di frattura a cuspide rivela inoltre che la tenacità a frattura dei materiali con microstruttura è in generale più elevata di quella dei materiali elastici classici, indicando quindi che la presenza di microstruttura può inibire il processo di frattura.
2007
 Fullfield solution for mode III crack in couple stress elastic materials with two characteristic lengths
[Relazione in Atti di Convegno]
Radi, Enrico
abstract
In questo lavoro si studia il problema di una frattura semiillimitata in condizioni di Modo III in un mezzo infinito dotato di microstruttura.Il materiale viene rappresentato attraverso il modello elastico micropolare di Koiter, in grado di rappresentare il comportamento meccanico di materiali granulari, materiali cellulari, compositi fibrorinforzati, laminati e muratura.
Tale modello ingloba le lunghezze caratteristiche a flessione e torsione del materiale, e risulta pertanto in grado di simulare il comportamento dei materiali a scala microscopica ed, in particolare, in prossimità dell’apice di una frattura, a distanze comparabili alle lunghezze caratteristiche.
La soluzione completa si ottiene utilizzando il metodo di Wiener Hopf e mostra una transizione continua dalla soluzione elastica classica, valida a sufficiente distanza dall’apice, alla soluzione asintotica con tensione tangenziale negativa, valida in prossimità dell’apice a distanze inferiori alla lunghezza caratteristica a torsione. Tale circostanza si verifica a causa della rotazione relativa tra le particelle in corrispondenza dell’apice, che produce spostamenti di segno opposto davanti all’apice (effetto forbici).
La zona in cui avviene l’inversione di segno è molto ridotta. Al di fuori di tale zona, la tensione tangenziale attinge un valore massimo positivo. Per tali materiali si può quindi formulare un criterio per propagazione della frattura basato sul valore massimo della tensione tangenziale.
Anche il profilo di frattura a cuspide rivela che i materiali considerati hanno una rigidezza più elevata dei materiali elastici classici, indicando che la presenza di microstruttura può inibire il processo di frattura.
L’approccio adottato fornisce inoltre un possibile collegamento tra i punti di vista atomistico e macroscopico, atto a favorire la comprensione dei meccanismi di frattura nei materiali con microstruttura, sino alla scala microscopica.
2007
 Mode II intersonic crack propagation in poroelastic media
[Relazione in Atti di Convegno]
B., Loret; Radi, Enrico
abstract
A crack is steadily running in an elastic isotropic fluidsaturated porous solid at an intersonic constant speed c. The crack tip speeds of interest are bounded belowby the slower between the slow longitudinal wavespeed and the shear wavespeed, and above by the fast longitudinal wavespeed. Biot’s theory ofporoelasticity with inertia forces governs the motion of the mixture. The poroelastic moduli depend on the porosity, and the complete range of porosities n ∈ [0, 1] is investigated. Solids are obtained as the limit case n = 0, and the continuity of the energy release rate as the porosity vanishes is addressed. Three characteristic regions in the plane (n, c) are delineated, dependingon the relative order of the body wavespeeds. Mode II loading conditions are considered, with a permeable crack surface. Cracks with and without process zones are envisaged. In each region, the analytical solution to a Riemann–Hilbert problem provides the stress, pore pressure and velocity fields near the tip of the crack. For subsonic propagation, the asymptotic crack tip fields are known to be continuous in the body [Loret and Radi (2001) J Mech Phys Solids 49(5):995–1020]. In contrast, for intersonic crack propagation without a process zone, the asymptotic stress and pore pressure might display a discontinuity across two or four symmetric rays emanating from themoving crack tip. Under Mode II loading condition, the singularity exponent for energetically admissible tip speeds turns out to be weaker than 1/2, except at a special point and along special curves of the (n, c)plane. The introduction of a finite length process zone is required so that 1. the energy release rate at the crack tip is strictly positive and finite; 2. the relative sliding of the crack surfaces has the same direction as the applied loading. The presence of the process zone is shown to wipe out possible first order discontinuities.
2007
 On the problem of a coated elastic layer subjected to residual thermal stress
[Relazione in Atti di Convegno]
Lanzoni, L.; Radi, Enrico
abstract
In the present work the problem of thin surface coatings of an elastic layer subject to a residual thermal stress is analytically investigated. The system adopts a curvature as a result of the misfit strain due to the different thermal expansivities of the layer and coatings. The problem is governed by a singular integral equation which can be reduced to a linear algebraic system by approximating the unknown interfacial shear stress via series expansion of Chebyshev polynomials. The present analysis is developed in order to assess the mechanical behaviour of a crystalline undulator designed for obtaining high emission radiations through channelling phenomenon.
2007
 On the seismic response of a flexible wall retaining a viscous poroelastic soil
[Articolo su rivista]
Lanzoni, L.; Radi, Enrico; Tralli, A.
abstract
A simple and reliable method is presented for the seismic analysis of a flexible wall retaining a layer of fluidsaturated viscous and poroelastic soil. A viscous version of the linear poroelastic Biot model is adopted for the description of the soil dissipative behavior. The effects of the wall flexibility and the mechanical properties of the soil on the amplitude and distribution of the pressures and the associated forces acting on the wall under harmonic loadings are firstly analyzed. The pseudostatic response is then recovered as a particular case for a vanishing small frequency of excitation. Finally, the response of the soil–wall system to generic seismic excitation is obtained using the Discrete Fourier Transform (FFT) method through the superposition of the contribution of each harmonic component of the ground acceleration spectrum. The analysis of the dynamic response obtained for different geometries of the wall and mechanical soil behavior allowed the relative importance of the various parameters involved in the seismic response of the system to be assessed.
2007
 Ricoprimento sottile periodico di un mezzo elastico soggetto a stress termico residuo
[Abstract in Atti di Convegno]
Lanzoni, Luca; Radi, Enrico
abstract
Nel presente lavoro viene studiato il problema di contatto e adesione tra uno strato di silicio parzialmente ricoperto da un film sottile di nitruro di silicio, soggetto ad uno stress termico residuo. Questo tipo di microstruttura trova rilevanti applicazioni nel processo di channeling di fasci di particelle ad alta energia. In particolare, si considera un modello periodico con film disposti ad intervalli regolari sia sulla superficie di un semispazio elastico che di uno strato di spessore finito. Utilizzando il metodo delle trasformate integrali, il problema si può formulare attraverso un sistema di equazioni integrali duali. Tale sistema può quindi ricondursi ad un'unica equazione integrale di Fredholm, che può risolversi attraverso tecniche di approssimazione basate sull'impiego dei polinomi di Chebyshev.
2006
 Ductile crack propagation at the micron scale
[Relazione in Atti di Convegno]
Radi, Enrico
abstract
The asymptotic fields near the tip of a crack steadily propagating in a ductile material under Mode III loading conditions are investigated by adopting an incremental version of the indeterminate theory of couple stress plasticity displaying linear and isotropic strain hardening. The adopted constitutive model is able to account for the microstructure of the material by incorporating two distinct material characteristic lengths. It can also capture the strong size effects arising at small scales, which results from the underlying microstructures. The effects of microstructure on Mode III crack tip fields mainly consist in a switch in the sign of tractions and displacement and in a substantial increase in the singularity of tractions ahead of the cracktip, with respect to the classical solution of LEFM and EPFM. The increase in the stress singularity is essentially due to the skewsymmetric stress field, since the symmetric stress field turns out to be nonsingular. Moreover, the obtained results show that the ratio between the characteristic lengths has a limited effect on the strength of the stress singularity. However, it displays a strong influence on the angular distribution of the asymptotic crack tip fields.
2006
 Effects of characteristic material lengths on ductile crack propagation.
[Relazione in Atti di Convegno]
Radi, Enrico
abstract
The asymptotic fields near the tip of a crack steadily propagating in a ductile material under Mode III loading conditions areinvestigated by adopting an incremental version of the indeterminate theory of couple stress plasticity displaying linear strainhardening. The adopted constitutive model is able to account for the microstructure of the material by incorporating two distinctmaterial characteristic lengths. It can also capture the strong size effects arising at small scales, which results from theunderlying microstructures. The effects of microstructure on Mode III crack tip fields mainly consist in a substantial increase inthe singularities of the skewsymmetric stress and couple stress fields, which occurs also for small values of the strainhardening coefficient, whereas the symmetric stress field turns out to be nonsingular according to the asymptotic crack tipfields for a stationary crack provided by the indeterminate theory of couple stress elasticity. The performed asymptotic analysisthus predicts a significant increase of the tractions level ahead of the cracktip, due to the contribution of the rotation gradient.
2006
 Mode III crack tip fields in couple stress elastic materials with two characteristic lengths
[Relazione in Atti di Convegno]
Radi, Enrico
abstract
In questo lavoro viene effettuata un’analisi asintotica dei campi di tensione e deformazione in prossimità dell’apice di una frattura in un materiale con microstruttura, in condizioni di modo III. Per rappresentare il comportamento del materiale è stato utilizzato il modello costitutivo elastico polare sviluppato da Koiter, che include due distinte lunghezze caratteristiche proprie della microstruttura del materiale. Pertanto, il modello considerato risulta in grado di simulare i rilevanti effetti di scala che si riscontrano in prossimità dell’apice di una frattura nei materiali con microstruttura, a distanze comparabili a quelle delle lunghezze caratteristiche. In particolare si osserva che la presenza di microstruttura comporta un notevole incremento della singolarità delle componenti antisimmetriche di tensione e, quindi, delle trazioni davanti all’apice, mentre le componenti simmetriche di tensione risultano limitate. Tali osservazioni si presentano in accordo con i risultati ottenuti considerando una sola lunghezza caratteristica del materiale. Nel presente lavoro viene inoltre evidenziata l’influenza esercitata dal rapporto tra le due lunghezze caratteristiche sugli andamenti dei campi asintotici in funzione della coordinata angolare e sul rilascio di energia in prossimità dell’apice.
2006
 Modelli analitici per lo studio della risposta dinamica di paratie flessibili soggette ad azioni sismiche
[Articolo su rivista]
Lanzoni, L.; Fioravante, V.; Radi, Enrico; Tralli, A.
abstract
Nel lavoro vengono presentati due modelli analitici sviluppati per descrivere la complessa interazione dinamica fra il terreno ed una paratia incastrata alla base. Per superare i limiti dei modelli pseudoelastici suggeriti dagli Eurocodici, i modelli sviluppati consentono di valutare il contenuto in frequenza del sisma di progetto, la rigidezza della paratia e le proprietà elastiche e reologiche del terreno. Si rende inoltre possibile simulare il comportamento dinamico dell'insieme terrenoopera di sostegno in presenza di una forzante comunque variabile nel tempo, applicando la trasformata discreta di Fourier. La risposta dinamica relativa all'interazione con terreni poco permeabili viene simulata con un modello costitutivo viscoelastico, mentre per terreni aventi granulometria maggiore si adotta un modello poroviscoelastico. La risposta dei modelli ad alcuni accelerogrammi reali viene quindi comparata con i risultati riportati nella letteratura tecnica.
2005
 Crack propagation in poroelastic fluidsaturated solids at intersonic velocities
[Relazione in Atti di Convegno]
Radi, E.; Loret, B.
abstract
A closedform asymptotic solution is provided for the stress, pore pressure and displacement fields near the tip of a crack, steadily running in an elastic fluidsaturated porous solid at crack tip speed ranging between the faster longitudinal wavespeed and the lower between the longitudinal Biot second wavespeed and the shear wavespeed. Mode I and Mode II loading conditions with permeable crack surfaces have been considered. The Biot theory of poroelasticity with inertia forces is assumed to govern the motion of the medium. At variance with the subsonic case where the asymptotic crack tip fields are continuous in the body, for intersonic crack propagation, the stress and pore pressure asymptotic fields display a strong discontinuity (shock wave) across two symmetric rays emanating from the moving crack tip. The obtained solution also reveals that favorable velocity regimes with crack face opening and positive normal tractions ahead of the crack tip exist. The associated singularity of the stress and pore pressure fields turns out to be weaker than the squareroot singularity which characterizes the subsonic case. The introduction of a finite length cohesive zone allows to obtain an energy release rate at the crack tip that does not vanish, unlike for a point size process zone
2005
 On the dynamic response of flexible walls retaining a dissipative, dried or fluidsaturated porous media
[Relazione in Atti di Convegno]
Lanzoni, L; Radi, Enrico; Tralli, A.
abstract
In the present work the problem of thin surface coatings of an elastic layer subject to a residual thermal stress is analytically investigated. The system adopts a curvature as a result of the misfit strain due to the different thermal expansivities of the layer and coatings. The problem is governed by a singular integral equation which can be reduced to a linear algebraic system by approximating the unknown interfacial shear stress via series expansion of Chebyshev polynomials. The present analysis is developed in order to assess the mechanical behaviour of a crystalline undulator designed for obtaining high emission radiations through channelling phenomenon
2005
 Sulla risposta dinamica di paratie flessibili incastrate soggette a sollecitazioni sismiche in depositi alluvionali
[Articolo su rivista]
Lanzoni, L.; Fioravante, V.; Radi, Enrico; Tralli, A.
abstract
Nel lavoro sono illustrati alcuni risultati di una ricerca volta a valutare la complessa risposta dinamica di una paratia flessibile incastrata, mediante due modelli lineari, uno viscoelastico, applicabile a terreni poco permeabili, ed uno poroviscoelastico,per terreni a granulometria maggiore.A differenza dei modelli pseudostatici suggeriti dagli Eurocodici, i modelli sviluppati forniscono una anlaisi dinamica completa, in quanto consentono di valutare il contenuto energetico del sisma di progetto, e la reale rigidezza rigidezza della paratia. Al fine di valutare l'efficacia dei modelli proposti e per quantificare le differenze delle sollecitazioni ottenute rispetto al metodo esemplificato di MononobeOkabe, sono riportati i risultati di due simulazioni finalizzate ai progetti di due opere di sostegno in due siti alluvionali interessati dal progetto dell'Idrovia Ferrarese. Le analisi sono state condotte utilizzando gli accellerogrammi registrati durante due eventi sismici recenti (Novellara (RE), 1996 e Assisi, 1997).Si rende inoltre possibile simulare il comportamento dinamico dell'insieme terrenoopera di sostegno in presenza di una forzante comunque variabile nel tempo, applicando la trasformata discreta di Fourier.
2004
 Mode III crack growth in linear hardening materials with strain gradient effects
[Articolo su rivista]
Radi, Enrico; M., Gei
abstract
The flowtheory version of couple stress strain gradient plasticity is adopted for investigating the asymptotic fields near a steadily propagating cracktip, under Mode III loading conditions. By incorporating a material characteristic length, typically of the order of few microns for ductile metals, the adopted constitutive model accounts for the microstructure of the material and can capture the strong size effects arising at small scales. The effects of microstructure result in a substantial increase in the singularities of the skewsymmetric stress and couple stress fields, which occurs also for a small hardening coefficient. The symmetric stress field turns out to be nonsingular according to the asymptotic solution for the stationary crack problem in linear elastic couple stress materials. The performed asymptotic analysis can provide useful predictions about the increase of the traction level ahead of the cracktip due to the sole contribution of the rotation gradient, which has been found relevant and nonnegligible at the micron scale.
2004
 Sulla risposta dinamica di paratie soggette a sollecitazioni sismiche
[Relazione in Atti di Convegno]
Radi, Enrico; Fioravante, V; Tralli, A.
abstract
Nel presente lavoro vengono presentati i primi risultati di una ricerca volta a valutare la risposta dinamica di una paratia flessibile infissa a contenimento di uno strato di terreno. Nei depositi alluvionali della pianura padana si incontrano sia terreni argillosi (saturi e non) che possono essere considerati praticamente impermeabili, sia terreni granulari (sabbie, ghiaie) caratterizzati da un’elevata permeabilità. Nel primo caso è possibile valutare la risposta dinamica dello strato facendo ricorso ad un modello viscoelastico lineare, nel secondo caso un modello poroelastico lineare appare sicuramente più appropriato. Vengono, nel primo caso, analizzati gli effetti della flessibilità della parete sull’ampiezza e sulla distribuzione delle pressioni dinamiche agenti sulla paratia in seguito ad azioni sismiche orizzontali, e sulle corrispondenti sollecitazioni. A tal fine viene formulato un modello di analisi semplice ed affidabile che consente di valutare l’importanza relativa dei diversi fattori che caratterizzano il comportamento del terreno e che influiscono sulla risposta dinamica della paratia.
2003
 On intersonic crack propagation in poroelastic fluidsaturated solids
[Relazione in Atti di Convegno]
Radi, Enrico; Loret, B.
abstract
A closedform solution is provided for the stress, pore pressure and displacement fields near the tip of a dynamic crack steadily propagating in an elastic fluidsaturated porous solid at crack tip speed ranging between the faster longitudinal wavespeed and the lower between the longitudinal Biot second wavespeed and the shear wavespeed. The Biot theory of poroelasticity with inertia forces is assumed to govern the motion of the medium. At variance with the subsonic case, for intersonic crack propagation the stress and pore pressure fields display a strong discontinuity (shock wave) across symmetric rays emanating from the moving crack tip. However, for favorable velocity regimes the stress and pore pressure singularity turns out to be weaker than squareroot, leading to a vanishing small energy release rate at the crack tip. The introduction of a finite length cohesive zone remedies the vanishing of fracture energy, which is due to the point size model of the process zone.
2003
 Response to: Comments on E. Radi, D. Bigoni and D. Capuani, Effects of prestress on cracktip fields in elastic, incompressible solids [International Journal of Solids and Structures 39 (2002) 39713996]
[Articolo su rivista]
Radi, Enrico; D., Bigoni; D., Capuani
abstract
We sincerely thank A.N. Guz for having brought to our attention his large contribution to the topic analyzed by us, of which we were obviously unaware [with the exception mentioned in our footnote (2)].As far as the initial observations made by A.N. Guz are concerned, we specify that our formulation pertains to small deformations superimposed upon arbitrary large, homogeneous initial deformation. The similarity of the localization patterns found by us to real experiments [see for instance Vasko, G.M., Leo,P.H. and Shield T.W., Prediction and observation of crack tip microstructure in shape memory CuAlNi single crystals, J. Mech. Phys. Solids 50 (2002) 1843–1867] points out the physical sense of our results.
2003
 Straingradient effects on steadystate crack growth in linear hardening materials
[Articolo su rivista]
Radi, Enrico
abstract
Steadystate rectilinear crack propagation is analysed in couple stress elasticplastic solids, displaying linear isotropic hardening. The flowtheory version of couple stress plasticity is adopted for the constitutive description of the material. A higher order asymptotic analysis of cracktip fields is performed under mode I and mode II loading conditions, both for plane strain and plane stress problems. In particular, the stress and couple stress fields are assumed to display distinct strengths of their singularity, so that, although the most singular term of the velocity field turns out to be irrotational, the leading order terms of couple stress and rotation gradient fields do not vanish, but couple with higher order terms of the strain and velocity fields. It follows that, under mode I crack propagation, the rotation gradients produce a substantial increase of the stress singularity and, thus, of the traction level ahead of the cracktip, with no need to invoke stretch gradients, whereas an increase in the shear traction ahead of the cracktip is observed under mode II loading conditions. These results may contribute to explaining the occurrence of cleavage fracture in ductile metals from the point of view of atomistic fracture mechanics.
2002
 Asymptotic tip fields for a steadily growing crack in pressuresensitive materials
[Capitolo/Saggio]
Bigoni, D; Radi, Enrico
abstract
Mode I steadystate, quasistatic crack propagation is analysed in elastoplastic pressuresensitive solids obeying the DruckerPrager yield condition. The asymptotic cracktip fields are numerically obtained with reference to the incremental small strain theory, in the case of linearisotropic hardening, under plane stress and plane strain conditions.
The determination of the stress and strain aneartip fields is fundamental for the understanding of fracture in ceramics, amorphous rocks at low temperature and concrete.
2002
 Crack growth in elasticplastic materials with straingradient effects
[Relazione in Atti di Convegno]
Radi, Enrico; Gei, M.
abstract
Straingradient effects on the asymptotic neartip stress and velocity fields of a crack propagating steadily and quasistatically in an elasticplastic material displaying linear hardening are highlighted. The flow theory version of couple stress plasticity is adopted for the constitutive description of the material. Under mode I crack propagation a substantial increase of the stress singularity is observed and, thus, of the traction level ahead of the cracktip, whereas an increase in the shear traction ahead of the cracktip is noticed under mode II loading conditions.
2002
 Effects of microstructure on ductile crack growth
[Relazione in Atti di Convegno]
Radi, Enrico; M., Gei
abstract
The flowtheory version of couple stress plasticity is employed to investigate the asymptotic fields near a steadily propagating cracktip, under Mode III loading conditions. By incorporating a material characteristic length, typically of the order of few microns for ductile metals, the adopted constitutive model accounts for the microstructure of the material and can capture the strong size effects arising at small scales. Due to the effects of microstructure, the singularities of the stress and couple stress fields increase substantially, also for small hardening. The skewsymmetric stress field is found to be more singular then couple stress fields, whereas the symmetric stress components are regular at the cracktip. The performed asymptotic analysis can provide useful predictions about the increase of the traction level ahead of the cracktip due to the strain gradient effects, which have been found relevant and non negligible at the micron scale.
2002
 Effects of prestress on cracktip fields in elastic, incompressible solids
[Articolo su rivista]
Radi, Enrico; D., Bigoni; D., Capuani
abstract
A closedform asymptotic solution is provided for velocity fields and the nominal stress rates near the tip of a stationary crack in a homogeneously prestressed configuration of a nonlinear elastic, incompressible material. In particular, a biaxial prestress is assumed with stress axes parallel and orthogonal to the crack faces. Two boundary conditions are considered on the crack faces, namely a constant pressure or a constant dead loading, both preserving all homogeneous ground state. Starting from this configuration, small Superimposed Mode I or Mode II deformations are solved, in the framework of Biot's incremental theory of elasticity. In this way a definition of an incremental stress intensity factor is introduced, slightly different for pressure or dead loading conditions on crack faces. Specific examples are finally developed for various hyperelastic materials, including the J2deformation theory of plasticity. The presence of prestress is shown to strongly influence the angular variation of the asymptotic cracktip fields, even if the nominal stress rate displays a square root singularity as in the infinitesimal theory. Relationships between the solution with shear band formation at the crack tip and instability of the crack surfaces are given in evidence.
2002
 Instabilities and near tip crack fields in elastic, incompressible solids
[Relazione in Atti di Convegno]
D., Bigoni; Radi, Enrico; D., Capuani
abstract
A closedform asymptotic solution is provided for velocity fields and the nominal stress rates near the tip of a stationary crack in a homogeneously prestressed configuration of a nonlinear elastic, incompressible material. In particular, a biaxial prestress is assumed with stress axes parallel and orthogonal to the crack faces. Two boundary conditions are considered on the crack faces, namely a constant pressure or a constant dead loading, both preserving an homogeneous ground state. Starting from this configuration, small superimposed Mode I or Mode II deformations are solved, in the framework of Biot’s incremental theory of elasticity. In this way a definition of an incremental stress intensity factor is introduced, slightly different for pressure or dead loading conditions on crack faces. Specific examples are finally developed for various hyperelastic materials, including the J2deformation theory of plasticity. The presence of prestress is shown to strongly influence the angular variation of the asymptotic cracktip fields, even if the nominal stress rate displays a square root singularity as in the infinitesimal theory. Relationships between the solution with shear band formation at the crack tip and instability of the crack surfaces are given in evidence.
2002
 Mode III crack growth in elasticplastic strain gradient solids
[Relazione in Atti di Convegno]
Radi, Enrico; Gei, M.
abstract
The flowtheory version of couple stress plasticity developed by Fleck and Hutchinson is employed to investigate the asymptotic fields near a cracktip steadily propagating under mode III loading condition.
2002
 NearTip fields of mode III steadystate crack propagation in elasticplastic strain gradient solids
[Relazione in Atti di Convegno]
Gei, Massimiliano; Radi, Enrico
abstract
The flowtheory version of couple stress strain gradient plasticity is adopted for investigating the asymptotic fields near a steadily propagating cracktip, under Mode III loading conditions. By incorporating a material characteristic length, typically of the order of few microns for ductile metals, the adopted constitutive model accounts for the micro¬structure of the material and can capture the strong size effects arising at small scales. The effects of microstructure result in a substantial increase in the singularities of the skewsymmetric stress and couple stress fields, also for small hardening. The symmetric stress field turns out to be nonsingular according to the asymptotic solution for the stationary crack problem in linear elastic couple stress materials. The performed asymptotic analysis can provide useful predictions about the increase of the traction level ahead of the cracktip due to the sole contribution of the rotation gradient, which has been found relevant and non negligible at the micron scale.
2002
 On toughening in Zirconia containing ceramics
[Capitolo/Saggio]
Bigoni, D.; Radi, Enrico
abstract
Transformation toughening in zirconiacontaining ceramics is related to dilatational, inelastic volumetric strain. A model for steadystate, Mode I crack propagation in a pressuresensitive, dilatational elasticplastic material is presented, based on the DruckerPrager yield criterion. In the framework of asymptotic analysis, results demonstrate a toughening effect related to pressuresensitivity and volumetric inelastic strain. Asymptotic field representations may yield a deep understanding of nearcrack tip stressdeformation phenomena.
2002
 Steady crackgrowth in elasticplastic fluidsaturated porous media
[Articolo su rivista]
Radi, Enrico; D., Bigoni; B., Loret
abstract
An asymptotic solution is obtained for stress and pore pressure fields near the tip of a crack steadily propagating in an elasticplastic fluidsaturated porous material displaying linear isotropic hardening. Quasistatic crack growth is considered under plane strain and Mode I loading conditions. In particular, the effective stress is assumed to obey the DruckerPrager yield condition with associative or nonassociative flowrule and linear isotropic hardening is assumed. Both permeable and impermeable crack faces are considered. As for the problem of crack propagation in poroelastic media, the behavior is asymptotically drained at the cracktip. Plastic dilatancy is observed to have a strong effect on the distribution and intensity of pore water pressure and to increase its flux towards the cracktip.
2001
 Neartip fields for quasistatic crack growth along the interface between a porousductile material and a rigid substrate
[Articolo su rivista]
Radi, Enrico; M. C., Porcu
abstract
A numerical asymptotic solution is provided for stress and velocity fields near the tip of an interface crack steadily propagating between a porous elasticplastic material and a rigid substrate, under plane strain conditions. The constitutive description of the ductile material is defined by the Gurson model with constant and uniform porosity, both for isotropic hardening and for perfectly plastic behavior as a limit case. Solutions are obtained by numerically integrating the field equations within elastic and plastic asymptotic sectors and by imposing full stress and velocity continuity. If the hardening coefficient is lower than a critical value two distinct kinds of solution can be found in variableseparable form, corresponding to predominantly tensile or shear mixed mode. The elasticperfectly plastic solution is constructed by means of an appropriate assembly of generalized centered fan and nonsingular plastic sectors and an elastic unloading sector. The results show that the porosity mainly influences the stress fields of the tensile mode rather than the shear mode, due to the higher hydrostatic stress level. In particular, for high porosities the maximum of the hoop stress deviates from the interface line ahead of the cracktip, causing possible kinking of the crack trajectory. The performed analysis of the debonding process of this kind of interface is essential for the determination of the overall strength, toughness and reliability of many advanced composite materials and structural components.
2001
 The effects of inertia on crack growth in poroelastic fluidsaturated media
[Articolo su rivista]
B., Loret; Radi, Enrico
abstract
A closedform asymptotic solution is provided for the stress, pore pressure and displacement fields near the tip of a Mode I crack, dynamically running in an elastic fluidsaturated porous solid. The Biot theory of poroelasticity with inertia forces is assumed to govern the motion of the medium. At variance with the quasistatic case where the cracktip is effectively drained, for rapid transient crack propagation, the pore fluid has no time to diffuse away from the cracktip.Both a qualitative analysis and the obtained asymptotic solution reveal that the pore pressure near the cracktip displays the same square root singularity as stress in the solid skeleton. Previous analyses have neglected the inertia of the fluid and obtained a regular pore pressure.
2000
 Interface Crack Propagation between a PorousDuctile Material and a Rigid Substrate
[Relazione in Atti di Convegno]
Porcu, M. C.; Radi, E.
abstract
Along the interfaces between ductile and brittle materials a slow, stable crack growth is often observed before the crack propagates into one of the two materials. In the present work, a numerical asymptotic solution is provided for the stress and velocity fields near the tip of an interface crack, steadily propagating between a porous elasticplastic material and a rigid substrate, under plane strain conditions. The Gurson model with constant and uniform porosity distribution and isotropic hardening is assumed for the constitutive description of the ductile material. This model may accurately describe the behavior of incompletely sintered porous metals and particulatereinforced metal matrix composites. In analogy with the problem of interface crack growth in fully dense elasticplastic materials, two distinct kinds of solution can be found in variableseparable form, corresponding to predominantly tensile or shear mixed mode. These solutions exist only if the
hardening coefficient is lower than a critical value. For higher values the solution may display a complex stress singularity, as for the problem of an interface crack between linear elastic materials. In any case, if the ductile material is elastically incompressible then the Dugdale parameter vanishes and variableseparable cracktip fields can be found for every
set of the material parameters.Due to the higher hydrostatic stress state, the porosity influences only the stress fields of
the tensile mode significantly. In particular, for high porosities the maximum of the hoop stress deviates from the interface line ahead of the cracktip towards the porous ductile material, causing possible kinking of the fracture, so that the toughness of the interface crack may increase significantly. Therefore, the performed analysis of debonding process of this kind of interface results to be essential for the determination of the overall strength, toughness and reliability of many advanced composite materials.
2000
 Localization of deformation in plane elasticplastic solid with anisotropic elasticity
[Articolo su rivista]
D., Bigoni; B., Loret; Radi, Enrico
abstract
Localization of deformation is analyzed in elasticplastic solids endowed with elastic anisotropy and nonassociative flow rules. A particular form of elastic anisotropy is considered, for which the localization analysis can be performed with reference to an elasticplastic solid endowed with isotropic elasticity but whose normals to the yield function and plastic potential are not coaxial. On the other hand, so far, available analytical solutions for the onset of strain localization in elasticplastic solids assume isotropic elasticity and coaxial plastic properties. Here, a new analytical solution is presented when the plastic normals are not coaxial but the analysis is restricted to plane strain and plane stress loadings. As an illustration, for a material with transverse elastic isotropy and with pressuredependent yield surface and plastic potential, this solution provides explicit expressions at the onset of strain localization for the plastic modulus, for the orientation of the shearband and for the slip mode. The numerical results highlight the importance of the coupled influence of elastic anisotropy and nonassociativity on the onset of strain localization.
1999
 On uniqueness for frictional contact rate problems
[Articolo su rivista]
Radi, Enrico; D., Bigoni; A., Tralli
abstract
A linear elastic solid having part of the boundary in unilateral frictional contact with a stiffer constraint is considered. Bifurcations of the quasistatic velocity problem are analyzed making use of methods developed for elastoplasticity. An exclusion principle for bifurcation is proposed which is similar, in essence, to the well!known exclusion principle given by Hill (1958). Sufficient conditions for uniqueness are given for a broad class of contact constitutive equations. The uniqueness criteria are based on the introduction of "linear comparison interfaces" defined both where the contact rate constitutive equation are piece!wise incrementally linear and where these are thoroughly nonlinear. Structural examples are proposed which give evidence to the applicability of the exclusion criteria.
1999
 Strain localization for a class of anisotropic elasticplastic solids
[Relazione in Atti di Convegno]
Bigoni, D.; Loret, B.; Radi, E.
abstract
Localization of deformation is analyzed in elasticanisotropic plastic solids with nonassociative flow law. A particular form of elastic anisotropy is considered, for which the localization analysis can be performed with reference to isotropicelastic plastic solids having noncoaxial yield function and plastic potential. In this framework, a new analytical solution is presented restricted to plane strain or plane stress assumption.
1998
 A note on uniqueness in frictional contact rate problem
[Abstract in Atti di Convegno]
Radi, Enrico
abstract
1996
 Asymptotic solution for Mode III crack growth in J2elastoplasticity with mixed isotropickinematic strain hardening
[Articolo su rivista]
D., Bigoni; Radi, Enrico
abstract
Mode III fracture propagation is analyzed in a J2flow theory elastoplastic material characterized by a mixed isotropic/kinematic law of hardening. The asymptotic stress, back stress and velocity fields are determined under smallstrain, steadystate, fracture propagation conditions. The increase in the hardening anisotropy is shown to be connected with a decrease in the thickness of the elastic sector in the crack wake and with an increase of the strength of the singularity. A second order analytical solution for the crack fields is finally proposed for the limiting case of pure kinematic hardening. It is shown that the singular terms of this solution correspond to fully plastic fields (without any elastic unloading sector), which formally are identical to the leading order terms of a crack steadily propagating in an elastic medium with shear modulus equal to the plastic tangent modulus in shear
1996
 Effects of anisotropic hardening on crack propagation in porousductile materials
[Articolo su rivista]
Radi, E.; Bigoni, D.
abstract
Slow, stable, rectilinear crack propagation is investigated for porous, elastoplastic solids displaying combined isotropic and kinematic hardening. The Gurson model with constant porosity, and therefore the associated flow rule, is used as the constitutive description. An asymptotic analysis of cracktip fields is performed under Mode I, steadystate, smallstrain, plane stress and plane strain conditions. A continuous distribution of asymptotic neartip fields is found. However, the possibility of the appearance of stress jumps in the asymptotic fields is analysed in detail. The results show many interesting features, which are related to the presence of both porosity and Bauschinger effect.
1995
 Shear banding and crack initiation in Zirconia ceramics
[Abstract in Atti di Convegno]
Bigoni, D.; Esposito, L.; Laudiero, F; Radi, E.; Tucci, A.
abstract
Experimental results obtained by Chen and Reyes Morel show that the behavior of Zirconia ceramics is highly pressuresensitive and, in particular, fits coherently with the DruckerPrager plasticity model. This is the macroscopic counterpart of shear banding into grains, interaction of shear bands with grain boundaries and subsequent microcracking.
These results have strong implications in the design of Zirconia structural elements. In fact, design philosophy based on yield strength is much different from that based on flaw strength. Moreover, the DruckerPrager yield criterion predicts a material behavior substantially different from the von Mises based plasticity. In particular, the DruckerPrager criterion allows for the introduction of a nonassociated flow rule, so giving the possibility of material instabilities (cavitation, surface exfoliations and macro shear banding) in the hardening range. Moreover, material pressuresensitivity and nonassociativity have strong effects on crack growth, which may be significant to develop sound design criteria.
1994
 Crack propagation in porous hardening metals
[Articolo su rivista]
Radi, Enrico; D., Bigoni
abstract
Steadystate and quasistatic rectilinear crack propagation is analyzed in porous elastoplastic solids obeying the Gurson yield condition and flowlaw. Both plane strain and plane stress conditions are considered under Mode I and Mode II loading conditions. The asymptotic cracktip fields are obtained with reference to the incremental small strain theory in the case of linear isotropic hardening behavior of the matrix material. The porosity of the material is assumed constant, therefore the elastoplastic constitutive operator results in being selfadjoint.Elastic unloading and plastic reloading on crack flanks are taken into account.
1993
 Asymptotic fields of mode I steadystate crack propagation in nonassociative elastoplastic solid
[Articolo su rivista]
Radi, E.; Bigoni, D.
abstract
The quasistatic, steadystate propagation of a crack running in an elastoplastic solid with volumetricnonassociative flowlaw is analyzed. The adopted constitutive model corresponds to the small strain version of that proposed by Rudnicki andRice. The asymptotic cracktip fields are numerically obtained for the case of the incremental theory with linear isotropichardening, under mode I planestress conditions. A relevant conclusion of the study is that the singularity of the neartipfields appears to be mainly governed by the flowrule, rather than by the yield surface gradient.
1993
 Mode I crack propagation in elastoplastic pressuresensitive materials
[Articolo su rivista]
Bigoni, D.; Radi, E.
abstract
Mode I steadystate, quasistatic crack propagation is analysed for elastoplastic pressuresensitive solids. Reference is made to the incremental small strain elastoplasticity obeying the DruckerPrager yield condition with associative flow law. The asymptotic cracktip fields are numerically obtained for the case of linearisotropic hardening, under plane stress and plane strain conditions.
1991
 Elastodynamic local fields for a crack running in an orthotropic medium
[Articolo su rivista]
A., Piva; Radi, Enrico
abstract
The dynamic stress and displacement fields in the neighborhood of the tip of a crack propagating in an orthotropic medium are obtained. The approach deals with the methods of linear algebra to transform the equations of motion into a firstorder elliptic system whose solution is sought under the assumption that the local displacement field may be represented under a scheme of separated variables. The analytical approach has enabled the distinction between two kinds of orthotropic materials for which explicit espressions of the neartip stress fields are obtained. Some results are presented graphically also in order to compare them with the numerical solution given in a quoted reference.
1990
 Analisi del comportamento dinamico di strutture a mensola mediante il modello di trave di Timoshenko
[Articolo su rivista]
Radi, Enrico; E., Viola
abstract
Nella presente memoria viene sviluppato un modello di trave a mensola (trave di Timoshenko) in grado di descrivere il comportamento dinamico di pareti di taglio o strutture analoghe.Gli effetti della cedevolezza del vincolo di fondazione sulle prime cinque frequenze naturali di vibrazione vengono esaminati anche nel campo delle frequenze superiori ad un valore critico ben definito, ove risulta rappresentato il comportamento dinamico della struttura per i modi di vibrare superiori al primo.Vengono inoltre riportati dei grafici di immediata utilizzazione per la determinazione delle prime cinque frequenze naturali di vibrazione, in funzione dei parametri geometrici e di rigidezza delle molle con cui si è modellato il terreno di fondazione.
1990
 Modellazione e verifica sperimentale del comportamento dinamico di un edificio in muratura e C.A. in scala 1:3.
[Articolo su rivista]
Radi, Enrico; A., Di Tommaso; E., Viola
abstract
Il presente lavoro propone uno strumento semplice ed affidabile per lo studio degli edifici a struttura mista soggetti ad azioni sismiche.La memoria trae spunto dalle prove dinamiche eseguite attraverso la tavola vibrante dell'IZIIS di Skopje su di un edificio realizzato in scala 1:3 in muratura e C.A. I sistemi misti sono strutture complesse, con elementi resistenti eterogenei, a comportamento meccanico certamente non lineare e di incerta valutazione, soprattutto nei confronti di azioni cicliche, quali quelle sismiche.A fronte della complessità del problema è sorta l'esigenza di formulare modelli semplici, che tuttavia colgano gli aspetti essenziali della risposta dinamica di tali strutture. Per valutare il danneggiamento dei materiali vengono posti a confronto modelli elastoplastici tradizionali, modelli analoghi con degrado e modelli isteretici. I telai e le pareti murarie sono stati schematizzati come oscillatori nonlineari equivalenti. Tale approccio ha consentito consente di ottenere l'integrazione nel tempo delle equazioni del moto dell'intera struttura, con un impegno computazionale minimo, superando le difficoltà normalmente incontrate in analisi dinamica nel caso di comportamento non lineare dei materiali.Il procedimento proposto permette una valutazione più attendibile del degrado della struttura sotto le azioni sismiche, della variazione del suo periodo proprio di oscillazione, della sua duttilità e, conseguentemente, della sua resistenza al sisma.
1989
 Crack propagation in an orthotropic medium under general loading
[Articolo su rivista]
E., Viola; A., Piva; Radi, Enrico
abstract
The elastodynamic response of a finite crack steadily propagating in an orthotropic medium, acted upon at infinity by uniform biaxial and shear loads, is studied. In particular, the effects of varying crack velocity as well as the ratio of shear load to tensile load are described. The action of material orthotropy on various quantities describing the crack propagation characteristics is pointed out
1989
 Influenza della deformabilità tagliante e dei cedimenti di fondazione sulle frequenze di vibrazione della struttura a mensola
[Articolo su rivista]
E., Viola; Radi, Enrico
abstract
In questo articolo è mostrata l'influenza della deformabilità tagliante e dell'inerzia rotatoria sulle frequenze naturali dei primi cinque modi normali di vibrazione di una trave a mensola con cui può essere modellato il comportamento dinamico di un edificio o struttura analoga.L'effetto di interazione suolostruttura viene studiato considerando il vincolo della mensola cedevole elasticamente alla rotazione ed alla traslazione.Come casi particolari della trattazione sono riportati i risultati della struttura a mensola deformabile solo a flessione e quelli della trave deformabile a solo taglio, per le varie condizioni di vincolo.