Breakage mechanics—Part II: Modelling granular materials

The compression of granular materials has been traditionally modelled with the limitations of classical elasto-plasticity. The energy was implicitly assumed to dissipate from the frictional interaction of particles. However, the fact that brittle granular materials crush suggests that energy must also be dissipated from the fracturing of the grains, as in fracture mechanics. The concept of breakage as a thermomechanical internal variable was introduced in Part I [Einav, I., 2006. Breakage mechanics—Part I: theory. J. Mech. Phys. Solids 00,000-000] to describe the fracturing mechanisms. The theory allows to treat ideal theoretical materials that undergo dissipation purely from breakage with no other mechanism allowed for the energy consumption. However, as accounted for in elasto-plasticity, dissipation must also occur from the frictional rearrangement of grains. The combination of the two dissipative mechanisms of breakage and plasticity must therefore be investigated, as we do in this paper. Those two mechanisms are generally coupled, in the sense that one inevitably appears when the other develops. Plastic dissipation emerges as a by-product of breakage dissipation because after grains crush, local rearrangement must occur. This scenario may be termed an ‘active breakage mechanism’, and typifies compression deformations. In shear the plastic dissipation is dominant but breakage appears inevitably from grains abrasion. This scenario may be called a ‘passive breakage mechanism’. Based on the coupling assumption, models are developed for granular materials. In particular, we show that in compression isotropic hardening of sands may appear without involving plastic strains, i.e., independent of frictional dissipation. This interpretation of hardening is different from the one used in classical critical state soil mechanics. However, frictional dissipation leads to plastic straining that are necessary for the models to be predictive in unloading.     

Plasticity size effects in tension and compression of single crystals

The effect of size and loading conditions on the tension and compression stress-strain response of micron-sized planar crystals is investigated using discrete dislocation plasticity. The crystals are taken to have a single active slip system and both small-strain and finite-strain analyses are carried out. When rotation of the tensile axis is constrained, the build-up of geometrically necessary dislocations results in a weak size dependence but a strong Bauschinger effect. On the other hand, when rotation of the tensile axis is unconstrained, there is a strong size dependence, with the flow strength increasing with decreasing specimen size, and a negligible Bauschinger effect. Below a certain specimen size, the flow strength of the crystals is set by the nucleation strength of the initially present Frank-Read sources. The main features of the size dependence are the same for the small-strain and finite-strain analyses. However, the predicted hardening rates differ and the finite-strain analyses give rise to some tension-compression asymmetry.     

The Role of a Vanishing Interfacial Layer in Perfect Elasto-Plasticity

A two-phase elasto-plastic material is investigated. It is shown that, if the interface is modeled as the limit of a vanishing layer of a third material, then the resulting two-phase material will exhibit a smaller interfacial dissipation than that of a pure two-phase model.     

Distortional hardening plasticity model for paperboard

A distortional hardening elasto-plastic model at finite strains suitable for modeling of orthotropic materials is presented. As a prototype material, paperboard is considered. An in-plane model is established. The model developed is motivated from non-proportional loading tests on paperboard where the paperboard is pre-strained in one direction and then loaded in the perpendicular direction. A softening effect is revealed in the pre-strained samples. The observed experimental findings cannot be accurately predicted by current models for paperboard. To be able to model the softening effects, a yield surface based on multiple hardening variables is introduced. It is shown that the model parameters can be obtained from simple uniaxial experiments. The model is implemented in a finite element framework which is used to illustrate the behavior of the model at some specific loading situations and is compared with strain fields obtained from Digital Image Correlation experiments.     

Mesoscale plastic flow generation and development for polycrystals

The traditional yield criteria of plasticity such as Mises, Tresca, etc. make use of averaged macroparameters while mesomechanics consideration is based on the physical notion of plastic deformation mechanisms. They may involve the development of plastic shears on the surfaces and interfaces of internal structure elements involving stress concentration and relaxation. A criterion of plastic flow is proposed; it is based on the stress–strain state in a cell of computational grid as well as in the neighboring cells. An algorithm of plastic shear generation is developed for the progressive propagation of the plastic shears over the crystal. Test calculations of the crystal behavior under tension are made and the results are presented.     

On Generalised and Implicit Normality Hypotheses

The restrictive framework of generalised standard materials can be suitably extended to model non-associative constitutive equations, by exploiting the concept of implicit standard materials, based on the use of a bipotential of dissipation. As presented in this study, it allows one to incorporate, in an easy and elegant way, non-linear kinematic and isotropic hardening in the constitutive equations and to recover useful flow rule normality for these non-associative behaviours. Sommario. Il ristretto ambito dei materiali standard generalizzati è adeguatamente esteso nel lavoro utilizzando il concetto di materiale standard implicito e di bipotenziale di dissipazione. Ciò consente di incorporare nelle equazioni costitutive, in modo semplice ed elegante, leggi non lineari di incrudimento cinematico ed isotropo e di riottenere formalmente, per questi comportamenti di tipo non associato, la condizione di normalità nella legge di flusso.     

Assessment of composite failure and ultimate strength without experiment on composite

This paper attempts to estimate the ultimate strength of a laminated composite only based on its con- stituent properties measured independently. Three important issues involved have been systematically addressed, i.e., stress calculation for the constituent fiber and matrix materials, failure detection for the lamina and laminate upon the internal stresses in their constituents, and input data determination of the constituents from monolithic measurements. There are three important factors to influence the accuracy of the strength prediction. One is the stress concentration factor （SCF） in the matrix. Another is matrix plasticity. The third is thermal residual stresses in the constituents. It is these three factors, however, that have not been sufficiently well realized in the composite community. One can easily find out the elastic and strength parameters of a great many laminae and laminates in the current literature. Unfortunately, necessary information to determine the SCF, the matrix plasticity, and the thermal residual stresses of the composites is rare or incomplete. A useful design methodology is demonstrated in the paper.     

Bounds on the hydrostatic plastic strength of voided polycrystals and implications for linear-comparison homogenization techniques

A linear-comparison homogenization technique and its relaxed version are used to compute bounds of the Hashin–Shtrikman and the self-consistent types for the hydrostatic strength of ideally plastic voided polycrystals. Closed-form analytical results are derived for isotropic aggregates of various cubic symmetries (fcc, bcc, ionic). The impact of the variational relaxation on the bounds is found to be significantly larger than that previously observed in fully dense polycrystals. So much so that, quite surprisingly, relaxed self-consistent bounds are found to be weaker than non-relaxed Hashin–Shtrikman bounds in some of the material systems considered.     

Limit Analysis of Structures with Stochastic Strengths by a Static Approach

A procedure for the evaluation of the conditional probability of collapse of elastic–plastic structures with stochastic strengths is presented. The procedure represents an extension to the stochastic framework of the static approach, coupled with the method of redundant unknowns, well established in the classical deterministic limit analysis. In this paper, a kinematic approach for probabilistic limit analysis, provided in the literature, is also recalled in order to show the duality with the proposed static approach.A beam of elastic perfectly plastic material with correlated stochastic strengths is studied and the influence of the degree of correlation between cross-sections is evidenced. When a low correlation is encountered an accurate discretisation is necessary. Furthermore, investigation of the influence of different collapse events on the conditional probability of collapse is conducted for a frame structure.