Davide Bigoni

Uniqueness and localization—I. Associative and non-associative elastoplasticity

Abstract Localization of deformation into a planar band in the incremental response of elasto-plastic material is studied in the case of small strains and rotations. The critical hardening modulus for localization is given in an explicit form (uncoupled from the band normal) for an arbitrary rate independent non-associative plasticity. Loss of uniqueness of the response is investigated in terms of positiveness of the second order work density. Criteria for loss of second order work positiveness and localization are compared for plane stress and plane strain. In these cases, for the associative flow rule, the threshold for the second order work positiveness coincides with the threshold for shear band formation. This coincidence may not, however, occur if localization into splitting mode is attained.

Analytical Derivation of Cosserat Moduli via Homogenization of Heterogeneous Elastic Materials

Why do experiments detect Cosserat-elastic effects for porous, but not for stiff-particlereinforced, materials? Does homogenization of a heterogeneous Cauchy-elastic material lead to micropolar (Cosserat) effects, and if so, is this true for every type of heterogeneity? Can homogenization determine micropolar elastic constants? If so, is the homogeneous (effective) Cosserat material determined in this way a more accurate representation of composite material response than the usual effective Cauchy material? Direct answers to these questions are provided in this paper for both two- (2D) and three-dimensional (3D) deformations, wherein we derive closed-form formulae for Cosserat moduli via homogenization of a dilute suspension of elastic spherical inclusions in 3D (and circular cylindrical inclusions in 2D) embedded in an isotropic elastic matrix. It is shown that the characteristic length for a homogeneous Cosserat material that best mimics the heterogeneous Cauchy material can be derived (resulting in surprisingly simple formulae) when the inclusions are less stiff than the matrix, but when these are equal to or stiffer than the matrix, Cosserat effects are shown to be excluded. These analytical results explain published experimental findings, correct, resolve and extend prior contradictory theoretical (mainly numerical and limited to two-dimensional deformations) investigations, and provide both a general methodology and specific results for determination of simple higher-order homogeneous effective materials that more accurately represent heterogeneous material response under general loading conditions. In particular, it is shown that no standard (Cauchy) homogenized material can accurately represent the response of a heterogeneous material subjected to a uniform plus linearly varying applied traction, while a homogenized Cosserat material can do so (when inclusions are less stiff than the matrix).

Asymptotic models of dilute composites with imperfectly bonded inclusions

The asymptotic scheme for the analysis of dilute elastic composites, which includes circular inclusions with imperfect bonding at the interface is presented. Interface is characterized by a discontinuous displacement field across it, linearly related to the tractions. The problem of a linear-elastic, circular inclusion with generic loading condition at infinity is solved, and used to analyze effective elastic moduli of composite materials. Effects due to the interaction of a small circular defect and a crack are investigated. It is shown that interfacial stiffness has a strong effect on the crack path and therefore may be an important design parameter for composites.

Band-gap shift and defect-induced annihilation in prestressed elastic structures

Design of filters for electromagnetic, acoustic, and elastic waves involves structures possessing photonic/phononic band gaps for certain ranges of frequencies. Controlling the filtering properties implies the control over the position and the width of the band gaps in question. With reference to piecewise homogeneous elasticbeams on elastic foundation, these are shown to be strongly affected by prestress (usually neglected in these analyses) that (i) “shifts” band gaps toward higher (lower) frequencies for tensile (compressive) prestress and (ii) may “annihilate” certain band gaps in structures with defects. The mechanism in which frequency is controlled by prestress is revealed by employing a Green’s-function-based analysis of localized vibration of a concentrated mass, located at a generic position along the beam axis. For a mass perturbing the system, our analysis addresses the important issue of the so-called effective negative mass effect for frequencies within the stop bands of the unperturbed structure. We propose a constructive algorithm of controlling the stop bands and hence filtering properties and resonance modes for a class of elastic periodic structures via prestress incorporated into the model through the coefficients in the corresponding governing equations.

Statics and dynamics of structural interfaces in elasticity

The concept of structural interface possessing finite width and joining continuous media is presented. Mechanical effects differentiating this model from conventional zero-thickness interfaces are explored for static and dynamic problems. In the static case, the thickness of the interface introduces an additional characteristic length, providing a parameter which can serve different design needs, for instance, may be employed to obtain neutral coated inclusions. In the dynamic case, a number of effects arise, related to the inertia of the interface. In particular, examples demonstrate that the design of inertial properties of the interface may be useful to produce sharp filters of elastic waves.

Effect of interfacial compliance on bifurcation of a layer bonded to a substrate

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.

Bifurcation and Instability of Non-Associative Elastoplastic Solids

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.

Effects of temperature and thermo-mechanical couplings on material instabilities and strain localization of inelastic materials

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.

An elastoplastic framework for granular materials becoming cohesive through mechanical densification. Part I – small strain formulation

Mechanical densification of granular bodies is a process in which a loose material becomes increasingly cohesive as the applied pressure increases. A constitutive description of this process faces the formidable problem that granular and dense materials have completely different mechanical behaviours (nonlinear elastic properties, yield limit, plastic flow and hardening laws), which must both be, in a sense, included in the formulation. A treatment of this problem is provided here, so that a new phenomenological, elastoplastic constitutive model is formulated, calibrated by experimental data, implemented and tested, that is capable of describing the transition between granular and fully dense states of a given material. The formulation involves a novel use of elastoplastic coupling to describe the dependence of cohesion and elastic properties on the plastic strain. The treatment falls within small strain theory, which is thought to be appropriate in several situations; however, a generalization of the model to large strain is provided in Part II of this paper.

Green's function for incremental nonlinear elasticity: shear bands and boundary integral formulation

Abstract 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 Greens 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.