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Explicit solutions for the formation of discontinuity bands are obtained, for a class of non-associative flow rules. Specialization to particular yield functions for pressure sensitive, dilatant or compactive materials is given.

Etude de la plasticite associee, avec une surface de plasticite rendant compte du durcissement et de l'adoucissement mecanique sous l'hypothese d'un petit gradient de deplacement. Determination d'un module critique de durcissement pour la localisation et pour l'inclinaison de la bande

A rod covering a fixed span is loaded at the middle with a transverse force, such that with increasing load a progressive deflection occurs. After a certain initial deflection, a phenomenon is observed where two points of the rod come in contact with each other. This is defined as the ‘dripping point’ and is when ‘self-encapsulation’ of the elastic rod occurs. Dripping seems at a first glance to be impossible and definitely cannot occur in the presence of ‘ordinary’ constraints (such as simple supports or clamps) at the ends of the span. However, the elastica governs oscillating pendulums, buckling rods and pendant drops, so that a possibility for self-encapsulation might be imagined. This phenomenon is indeed demonstrated (both theoretically and experimentally) to occur when at least one of the constraints at the ends of the rod is a sliding sleeve. This mechanical device generates a configurational force, causing the dripping of the rod, in a fully elastic set-up.

Perturbations in terms of small elastic deformations superimposed upon a given homogeneous strain are analysed within a boundary element framework. This is based on a recently-developed Green’s function and boundary integral equations for non-linear incremental elastic deformations. Plane strain deformations are considered of an incompressible hyperelastic solid within the elliptic range. The proposed approach is shown to yield bifurcation loads and modes via a perturbative approach. Numerical treatment of the problem is detailed and applications to multilayers are shown. Relations between shear band formation and global instabilities are given evidence.

Superimposed dynamic, time-harmonic incremental deformations are considered in an elastic, orthotropic and incompressible, infinite body, subject to plane, homogeneous—but otherwise arbitrary—deformation. The dynamic, infinite body Green's function is found and, in addition, new boundary integral equations are obtained for incremental in-plane hydrostatic stress and displacements. These findings open the way to integral methods in incremental, dynamic elasticity. Moreover, the Green's function is employed as a dynamic perturbation to analyze interaction between wave propagation and shear band formation. Depending on anisotropy and pre-stress level, peculiar wave patterns emerge with focussing and shadowing effects of signals, which may remain undetected by the usual criteria based on analysis of weak discontinuity surfaces.

A general framework for rate-independent, small-strain, thermoinelastic material behaviour is presented, which includes thermo-plasticity and -damage as particular cases. In this context, strain localization and the development of material instabilities are investigated to highlight the roles of thermal effects and thermomechanical couplings. During a loading process, it is shown that two conditions play the essential roles and correspond to the singularity of the isothermal and the adiabatic acoustic tensors. Under quasi-static conditions, strain localization (in a classical sense) may occur when either of these two conditions is met. It involves a jump in temperature rate in the latter case, whereas temperature rate remains continuous in the former, but a discontinuity in the spatial derivatives of the heat flux must occur. This is consistent with the condition of stationarity of acceleration waves, which are shown to be homothermal and propagate with a velocity related to the eigenvalues of the isothermal acoustic tensor. A linear perturbation analysis further clarifies the above findings. In particular, for a quasi-static path of an infinite medium, failure of positive definiteness of either of the acoustic tensors corresponds to bifurcations in wave-like modes of arbitrary wave-length and infinite rate of growth. Under dynamic conditions, unbounded growth of perturbations is associated only to the short wavelength regime and corresponds to divergence growth or flutter phenomena relative to the isothermal acoustic tensor.

Failure modes of silicon nitride cylinders have been investigated under uniaxial compression at 1200 °C in air. Samples with different aspect ratios (h/d=5/2, 4/2, 2/2, and 1/2 mm/mm) have been tested. In all cases, the stress/strain curves evidence an initial linear portion followed by a peak and a slight softening, denoting a plastic behaviour. Surface exfoliation is the dominant failure mode, although traces of localized patterns of deformations––which initiated and propagated macrocracks––can be found in some samples. A bifurcation analysis has been carried out in order to describe the onset of the specific failure mode. The first failure mode predicted by this approach is an antisymmetric mode, while symmetric modes almost immediately follow. However, antisymmetric modes may be partially hampered by friction at the specimen/cushion contact, while symmetric modes could be triggered by residual stress. Therefore, an interpretation of the observed failure mode is that the exfoliation mechanism may result as an evolution of a first antisymmetric mode into a symmetric one and that localized deformations follow to produce final macrocracks growth.

A framework for thermoelastic analysis of wave propagation in multilaminated structures is given. The elastic material is subject to an arbitrary, homogeneous deformation and to a condition of uniform temperature. Small-amplitude vibrations are analyzed starting from this state, in a fully coupled thermomechanical formulation.

A closed-form asymptotic solution is provided for velocity fields and the nominal stress rates near the tip of a stationary crack in a homogeneously pre-stressed configuration of a nonlinear elastic, incompressible material. In particular, a biaxial pre-stress 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 J2-deformation theory of plasticity. The presence of pre-stress is shown to strongly influence the angular variation of the asymptotic crack-tip 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.

A nonlinear interface constitutive law is formulated for modeling the mechanical behavior of the periodontal ligament. This gives an accurate interpolation of the few available experimental results and provides a reasonably simple model for mechanical applications. The model is analyzed from the viewpoints of both mathematical consistency and effectiveness in numerical calculations. In order to demonstrate the latter, suitable two- and three-dimensional nonlinear interface finite elements have been implemented.

An elastic, incompressible, infinite body is considered subject to plane and homogeneous deformation. At a certain value of the loading, when the material is still in the elliptic range, an incremental concentrated line load is considered acting at an arbitrary location in the body and extending orthogonally to the plane of deformation. This plane strain problem is solved, so that a Green's function for incremental, nonlinear elastic deformation is obtained. This is used in two different ways: to quantify the decay rate of self-equilibrated loads in a homogeneously stretched elastic solid; and to give a boundary element formulation for incremental deformations superimposed upon a given homogeneous strain. The former result provides a perturbative approach to shear bands, which are shown to develop in the elliptic range, induced by self-equilibrated perturbations. The latter result lays the foundations for a rigorous approach to boundary element techniques in finite strain elasticity.

The infinite-body three-dimensional Green's function set (for incremental displacement and mean stress) is derived for the incremental deformation of a uniformly strained incompressible, nonlinear elastic body. Particular cases of the developed formulation are the Mooney–Rivlin elasticity and the J 2 -deformation theory of plasticity. These Green's functions are used to develop a boundary integral equation framework, by introducing an ad hoc potential, which paves the way for a boundary element formulation of three-dimensional problems of incremental elasticity. Results are used to investigate the behaviour of a material deformed near the limit of ellipticity and to reveal patterns of shear failure. In fact, within the investigated three-dimensional framework, localized deformations emanating from a perturbation are shown to be organized in conical geometries rather than in planar bands, so that failure is predicted to develop through curved and thin surfaces of intense shearing, as can for instance be observed in the cup–cone rupture of ductile metal bars.

The concept of 'deformable arm scale' (completely different from a traditional rigid arm balance) is theoretically introduced and experimentally validated. The idea is not intuitive, but is the result of nonlinear equilibrium kinematics of rods inducing configurational forces, so that deflection of the arms becomes necessary for the equilibrium, which would be impossible for a rigid system. In particular, the rigid arms of usual scales are replaced by a flexible elastic lamina, free of sliding in a frictionless and inclined sliding sleeve, which can reach a unique equilibrium configuration when two vertical dead loads are applied. Prototypes realized to demonstrate the feasibility of the system show a high accuracy in the measure of load within a certain range of use. It is finally shown that the presented results are strongly related to snaking of confined beams, with implications on locomotion of serpents, plumbing, and smart oil drilling., Comment: 11 pages, 5 figures

Bifurcations in velocities from a state of homogeneous axisymmetric deformation are investigated for a coated elastic cylinder subject to axial tension or compression. The cylinder and the finite-thickness coating have circular cross sections. At the coating/core contact, a linear interface is introduced to simulate imperfect bonding. The particular case in which the thickness of the coating becomes infinite is also addressed. This may model the behaviour of a fiber embedded in an infinite matrix. Generic modes of bifurcations are investigated in the elliptic range, comprised axi- and anti-symmetric modes. Incompressible, hyperelastic materials, including Ogden, Mooney–Rivlin, and J2-deformation theory of plasticity, are considered in the applications.

Global and local uniqueness and stability criteria for elastoplastic solids with non-associative flow rules are presented. Hill’s general theory is developed in the form generalized by Raniecki to non-associativity. Local stability criteria are presented and systematically discussed in a critical way. These are: positive definiteness and non-singularity of the constitutive operator, and positive definiteness (strong ellipticity) and non-singularity (ellipticity) of the acoustic tensor. The former criteria are particularly relevant for homogeneous deformation of solids subject to all-round controlled nominal surface tractions. Dually, the latter criteria are particularly relevant for homogeneous deformation of solids subject to displacements prescribed on the entire boundary. Flutter instability as related to complex conjugate eigenvalues of the acoustic tensor is also briefly discussed.

Localization of deformation is analyzed in elastic–plastic solids endowed with elastic anisotropy and non-associative flow rules. A particular form of elastic anisotropy is considered, for which the localization analysis can be performed with reference to an elastic–plastic 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 elastic–plastic 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 pressure-dependent 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 shear-band and for the slip mode. The numerical results highlight the importance of the coupled influence of elastic anisotropy and non-associativity on the onset of strain localization.

The effect of prestress on dislocation (and inclusion) fields in nonlinear elastic solids is analysed by extending previous solutions by Eshelby and Willis. Using a plane-strain constitutive model (for incompressible incremental nonlinear elasticity) to describe the behaviour of ductile metals ( J 2 -deformation theory of plasticity), we show that when the level of prestress is high enough that shear band formation is approached, strongly localized strain patterns emerge, when a dislocation dipole is emitted by a source. These may explain cascade activation of dislocation clustering along slip band directions.

Mode III fracture propagation is analyzed in a J2-flow theory elastoplastic material characterized by a mixed isotropic/kinematic law of hardening. The asymptotic stress, back stress and velocity fields are determined under small-strain, steady-state, 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.

Steady-state and quasi-static rectilinear crack propagation is analyzed in porouss elastoplastic solids obeying the Gurson yield condition and flow-law. Both plane strain and plane stress conditions are considered under Mode I and Mode II loading conditions. The asymptotic crack-tip 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 self-adjoint. Elastic unloading and plastic reloading on crack flanks are taken into account.

The quasi-static, steady-state propagation of a crack running in an elastoplastic solid with volumetric-non-associative flow law is analyzed. The adopted constitutive model corresponds to the small strain version of that proposed by Rudnicki and Rice. The asymptotic crack-tip fields are numerically obtained for the case of the incremental theory with linear isotropic hardening, under mode I plane-stress conditions. A relevant conclusion of the study is that the singularity of the near-tip fields appears to be mainly governed by the flow-rule, rather than by the yield surface gradient.

The constitutive modelling of granular, porous and quasi-brittle materials is based on yield (or damage) functions, which may exhibit features (for instance, lack of convexity, or branches where the values go to infinity, or `false elastic domains') preventing the use of efficient return-mapping integration schemes. This problem is solved by proposing a general construction strategy to define an implicitly defined convex yield function starting from any convex yield surface. Based on this implicit definition of the yield function, a return-mapping integration scheme is implemented and tested for elastic---plastic (or -damaging) rate equations. The scheme is general and, although it introduces a numerical cost when compared to situations where the scheme is not needed, is demonstrated to perform correctly and accurately.

Starting from a Cauchy elastic composite with a dilute suspension of randomly distributed inclusions and characterized at first-order by a certain discrepancy tensor (see part I of the present article), it is shown that the equivalent second-gradient Mindlin elastic solid: (i.) is positive definite only when the discrepancy tensor is negative defined; (ii.) the non-local material symmetries are the same of the discrepancy tensor, and (iii.) the nonlocal effective behaviour is affected by the shape of the RVE, which does not influence the first-order homogenized response. Furthermore, explicit derivations of non-local parameters from heterogeneous Cauchy elastic composites are obtained in the particular cases of: (a) circular cylindrical and spherical isotropic inclusions embedded in an isotropic matrix, (b) n-polygonal cylindrical voids in an isotropic matrix, and (c) circular cylindrical voids in an orthortropic matrix., 19 pages, 10 figures, 3 tables

A unified approach is presented for the analysis of the finite static expansion of a spherical or cylindrical cavity in an indefinite elastic-perfectly plastic medium. Mohr-Coulomb yield criterion is adopted with an arbitrary degree of non-associativity of the volumetric dilatational component of the plastic flow. This idealization makes the present analysis particularly appropriate for rock-like materials. The general solution obtained requires a numerical integration over the plastic zone. Some numerical examples, referring to both the spherical and the cylindrical cavity emphasize the determining role of the elastic deformations in the plastic region.

The effect of interfacial compliance on the bifurcation of a layer bonded to a substrate is analyzed. The bifurcation problem is formulated for hyperelastic, layered solids in plane strain. Attention is then confined to the problem of a layer of finite thickness on a half-space. The layer and substrate are subject to plane strain compression, with the compression axis parallel to the bond line. The materials in the layer and in the half-space are taken to be incrementally linear, incompressible solids, with most results presented for Mooney-Rivlin and J2-deformation theory constitutive relations. The limiting case of an undeforming half-space is also considered. The interface between the layer and the substrate is characterized by an incrementally linear traction rate vs velocity jump relation, so that a characteristic length is introduced. A variety of bifurcation modes are possible depending on the layer thickness, on the constitutive parameters of the layer and the substrate, and on the interface compliance. These include shear band modes for the layer and the substrate, and diffuse instability modes involving deformation in the layer and the substrate. For a sufficiently compliant interface, the mode with the lowest critical stress is a long (relative to the layer thickness) wavelength plate-like bending mode for the layer.

A linear elastic solid having part of the boundary in unilateral frictional contact witha stiffer constraint is considered. Bifurcations of the quasistatic velocity problem are analyzed,making use of methods developed for elastoplasticity. An exclusion principle for bifurcation isproposed 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 contactconstitutive equations. The uniqueness criteria are based on the introduction of linear comparisoninterfaces defined both where the contact rate constitutive equation are piece-wise incrementallylinear and where these are thoroughly nonlinear. Structural examples are proposed which giveevidence to the applicability of the exclusion criteria.

A model for serpentine locomotion is derived from a novel perspective based on concepts from configurational mechanics. The motion is realized through the release of the elastic energy of a deformable rod, sliding inside a frictionless channel, which represents a snake moving against lateral restraints. A new formulation is presented, correcting previous results and including situations never analysed so far, as in the cases when the serpent's body lies only partially inside the restraining channel or when the body has a muscle relaxation localized in a small zone. Micromechanical considerations show that propulsion is the result of reactions tangential to the frictionless constraint and acting on the snake's body, a counter-intuitive feature in mechanics. It is also experimentally demonstrated that the propulsive force driving serpentine motion can be directly measured on a designed apparatus in which flexible bars sweep a frictionless channel. Experiments fully confirm the theoretical modelling, so that the presented results open the way to exploration of effects, such as variability in the bending stiffness or channel geometry or friction, on the propulsive force of snake models made up of elastic rods.

Small axial and flexural oscillations are analyzed for a periodic and infinite structure, constrained by sliding sleeves and composed of elastic beams. A nested Bloch-Floquet technique is introduced to treat the non-linear coupling between longitudinal and transverse displacements induced by the configurational forces generated at the sliding sleeve ends. The action of configurational forces is shown to play an important role from two perspectives. First, the band gap structure for purely longitudinal vibration is broken so that axial propagation may occur at frequencies that are forbidden in the absence of a transverse oscillation and, second, a flexural oscillation may induce axial resonance, a situation in which the longitudinal vibrations tend to become unbounded. The presented results disclose the possibility of exploiting configurational forces in the design of mechanical devices towards longitudinal actuation from flexural vibrations of small amplitude at given frequency., Comment: 20 pages, 5 figures

A ductile matrix material containing an arbitrary distribution of parallel and stiff lamellar (‘rigid-line’) inclusions is considered, subject to a prestress state provided by a simple shear aligned parallel to the inclusion lines. Because the lamellae have negligible thickness, the simple shear prestress state remains uniform and its amount can be high enough to drive the matrix material on the verge of ellipticity loss. Close to this critical stage, a uniform remote Mode I perturbation realizes shear band formation, growth, interaction, thickening or thinning. This two-dimensional problem is solved through the derivation of specific boundary integral equations, holding for a nonlinear elastic matrix material uniformly prestressed; the related numerical treatment is specifically tailored to capture the stress singularity present at the inclusion tips. Results show how complex localized deformation patterns form, so explaining features related to the failure mechanisms of ductile materials reinforced with stiff and thin inclusions. In particular, the influence of the inclusion distribution on the shear bands pattern is disclosed. Conditions for the magnification (the attenuation) of the localized deformations are revealed, fostering the progress (the setback) of the failure process.

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In the last years, 2D nanomaterials, and in particular graphene, have received considerable attention, due to the wide number of demonstrated or potential applications in electronic nanodevices, energy generation and storage, biotechnologies, composite materials. The study of their mechanical behaviour plays an important role for their manufacturing and integration in devices, for tuning their performances as well as for controlling the mechanics of their composites. For this reason, considerable research efforts have been done for modelling their mechanical behaviour through a variety of approaches, ranging from the solid state physics methods to the use of equivalent continua. The most of the existing literature has focused only the in-plane behaviour of these materials, while there is a lack of studies of their transverse behaviour, relevant for buckling, wrinkling and rippling phenomena or simply to establish the bending stiffness of the continua. With this in mind, in this paper, the Gillis sticks-and-springs model for the in-plane behaviour of graphene is extended to account also for transverse deformation. To this aim, the contribution of dihedral angles is included. Numerical analyses regarding the out-of plane buckling of graphene sheets are addressed.