Micromechanical definition of an entropy for quasi-static deformation of granular materials

A micromechanical theory is formulated for quasi-static deformation of granular materials, which is based on information theory. A reasoning is presented that leads to the definition of an information entropy that is appropriate for quasi-static deformation of granular materials. This definition is based on the hypothesis that relative displacements at contacts with similar orientations are independent realisations of a random variable. This hypothesis is made plausible based on the results of Discrete Element simulations. The developed theory is then used to predict the elastic behaviour of granular materials in terms of micromechanical quantities. The case considered is that of two-dimensional assemblies consisting of non-rotating particles with an elastic contact constitutive relation. Applications of this case are the initial elastic (small-strain) deformation of granular materials. Theoretical results for the elastic moduli, relative displacements, energy distribution and probability density functions are compared with results obtained from the Discrete Element simulations for isotropic assemblies with various average numbers of contacts per particle and various ratios of tangential to normal contact stiffness. This comparison shows that the developed information theory is valid for loose systems, while a theory based on the uniform-strain assumption is appropriate for dense systems.     

Coupled continuum and discrete analysis of random heterogeneous materials: Elasticity and fracture

Recent work has suggested that the heterogeneous distribution of mechanical properties in natural and synthetic materials induces a toughening mechanism that leads to a more robust structural response in the presence of cracks, defects or other types of flaws. Motivated by this, we model an elastic solid with a Young′s modulus distribution described by a Gaussian process. We study the pristine system using both a continuum and a discrete model to establish a link between the microscale and the macroscale in the presence of disorder. Furthermore, we analyze a flawed discrete particle system and investigate the influence of heterogeneity on the fracture mechanical properties of the solid. We vary the variability and correlation length of the Gaussian process, thereby gaining fundamental insights into the effect of heterogeneity and the essential length scales of heterogeneity critical to enhanced fracture properties. As previously shown for composites with complex hierarchical architectures, we find that materials with disordered elastic fields toughen by a ‘distribution-of-weakness’ mechanism inducing crack arrest and stress delocalization. In our systems, the toughness modulus can increase by up to 30% due to an increase in variability in the elastic field. Our work presents a foundation for stochastic modeling in a particle-based micromechanical environment that can find broad applications within natural and synthetic materials.     

Non-local energetics of random heterogeneous lattices

In this paper, we study the mechanics of statistically non-uniform two-phase elastic discrete structures. In particular, following the methodology proposed in Luciano and Willis (2005) [Journal of the Mechanics and Physics of Solids 53, 1505-1522], energetic bounds and estimates of the Hashin-Shtrikman-Willis type are developed for discrete systems with a heterogeneity distribution quantified by second-order spatial statistics. As illustrated by three numerical case studies, the resulting expressions for the ensemble average of the potential energy are fully explicit, computationally feasible and free of adjustable parameters. Moreover, the comparison with reference Monte-Carlo simulations confirms a notable improvement in accuracy with respect to approaches based solely on the first-order statistics.     

Well-posedness of a model of strain gradient plasticity for plastically irrotational materials

The initial boundary value problem corresponding to a model of strain gradient plasticity due to [Gurtin, M., Anand, L., 2005. A theory of strain gradient plasticity for isotropic, plastically irrotational materials. Part I: Small deformations. J. Mech. Phys. Solids 53, 1624–1649] is formulated as a variational inequality, and analysed. The formulation is a primal one, in that the unknown variables are the displacement, plastic strain, and the hardening parameter. The focus of the analysis is on those properties of the problem that would ensure existence of a unique solution. It is shown that this is the case when hardening takes place. A similar property does not hold for the case of softening. The model is therefore extended by adding to it terms involving the divergence of plastic strain. For this extended model the desired property of coercivity holds, albeit only on the boundary of the set of admissible functions.     

Two exact micromechanics-based nonlocal constitutive equations for random linear elastic composite materials

A Hashin-Shtrikman-Willis variational principle is employed to derive two exact micromechanics-based nonlocal constitutive equations relating ensemble averages of stress and strain for two-phase, and also many types of multi-phase, random linear elastic composite materials. By exact is meant that the constitutive equations employ the complete spatially-varying ensemble-average strain field, not gradient approximations to it as were employed in the previous, related work of Drugan and Willis (J. Mech. Phys. Solids 44 (1996) 497) and Drugan (J. Mech. Phys. Solids 48 (2000) 1359) (and in other, more phenomenological works). Thus, the nonlocal constitutive equations obtained here are valid for arbitrary ensemble-average strain fields, not restricted to slowly-varying ones as is the case for gradient-approximate nonlocal constitutive equations. One approach presented shows how to solve the integral equations arising from the variational principle directly and exactly, for a special, physically reasonable choice of the homogeneous comparison material. The resulting nonlocal constitutive equation is applicable to composites of arbitrary anisotropy, and arbitrary phase contrast and volume fraction. One exact nonlocal constitutive equation derived using this approach is valid for two-phase composites having any statistically uniform distribution of phases, accounting for up through two-point statistics and arbitrary phase shape. It is also shown that the same approach can be used to derive exact nonlocal constitutive equations for a large class of composites comprised of more than two phases, still permitting arbitrary elastic anisotropy. The second approach presented employs three-dimensional Fourier transforms, resulting in a nonlocal constitutive equation valid for arbitrary choices of the comparison modulus for isotropic composites. This approach is based on use of the general representation of an isotropic fourth-rank tensor function of a vector variable, and its inverse. The exact nonlocal constitutive equations derived from these two approaches are applied to some example cases, directly rationalizing some recently-obtained numerical simulation results and assessing the accuracy of previous results based on gradient-approximate nonlocal constitutive equations.     

Micromechanical study of elastic moduli of loose granular materials

In micromechanics of the elastic behaviour of granular materials, the macro-scale continuum elastic moduli are expressed in terms of micro-scale parameters, such as coordination number (the average number of contacts per particle) and interparticle contact stiffnesses in normal and tangential directions. It is well-known that mean-field theory gives inaccurate micromechanical predictions of the elastic moduli, especially for loose systems with low coordination number. Improved predictions of the moduli are obtained here for loose two-dimensional, isotropic assemblies. This is achieved by determining approximate displacement and rotation fields from the force and moment equilibrium conditions for small sub-assemblies of various sizes. It is assumed that the outer particles of these sub-assemblies move according to the mean field. From the particle displacement and rotation fields thus obtained, approximate elastic moduli are determined. The resulting predictions are compared with the true moduli, as determined from the discrete element method simulations for low coordination numbers and for various values of the tangential stiffness (at fixed value of the normal stiffness). Using this approach, accurate predictions of the moduli are obtained, especially when larger sub-assemblies are considered. As a step towards an analytical formulation of the present approach, it is investigated whether it is possible to replace the local contact stiffness matrices by a suitable average stiffness matrix. It is found that this generally leads to a deterioration of the accuracy of the predictions. Many micromechanical studies predict that the macroscopic bulk modulus is hardly influenced by the value of the tangential stiffness. It is shown here from the discrete element method simulations of hydrostatic compression that for loose systems, the bulk modulus strongly depends on the stiffness ratio for small stiffness ratios.     

Elastic properties of model random three-dimensional open-cell solids

Most cellular solids are random materials, while practically all theoretical structure-property relations are for periodic models. To generate theoretical results for random models the finite element method (FEM) was used to study the elastic properties of open-cell solids. We have computed the density (ρ) and microstructure dependence of the Young's modulus (E) and Poisson's ratio (ν) for four different isotropic random models. The models were based on Voronoi tessellations, level-cut Gaussian random fields, and nearest neighbour node-bond rules. These models were chosen to broadly represent the structure of foamed solids and other (non-foamed) cellular materials. At low densities, the Young's modulus can be described by the relation E∝ρⁿ. The exponent n and constant of proportionality depend on microstructure. We find 1.3<n<3, indicating a more complex dependence than indicated by periodic cell theories, which predict n=1 or 2. The observed variance in the exponent was found to be consistent with experimental data. At low densities we found that ν≈0.25 for three of the four models studied. In contrast, the Voronoi tessellation, which is a common model of foams, became approximately incompressible (ν≈0.5). This behaviour is not commonly observed experimentally. Our studies showed the result was robust to polydispersity and that a relatively large number (15%) of the bonds must be broken to significantly reduce the low-density Poission's ratio to ν≈0.33.     

A Binary Model of textile composites: III high failure strain and work of fracture in 3D weaves

Prior experiments have revealed exceptionally high values of the work of fracture (0.4-) in carbon/epoxy 3D interlock woven composites. Detailed destructive examination of specimens suggested that much of the work of fracture arose when the specimens were strained well beyond the failure of individual tows yet still carried loads . A mechanism of lockup amongst broken tows sliding across the final tensile fracture surface was suggested as the means by which high loads could still be transferred after tow failure. In this paper, the roles of weave architecture and the distribution of flaws in the mechanics of tow lockup are investigated by Monte Carlo simulations using the so-called Binary Model. The Binary Model was introduced previously as a finite element formulation specialised to the problem of simulating relatively large, three-dimensional segments of textile composites, without any assumption of periodicity or other symmetry, while preserving the architecture and topology of the tow arrangement. The simulations succeed in reproducing all qualitative aspects of measured stress-strain curves. They reveal that lockup can indeed account for high loads being sustained beyond tow failure, provided flaws in tows have certain spatial distributions. The importance of the interlock architecture in enhancing friction by holding asperities on sliding fibre tows into firm contact is highlighted.     

Global and local large-deformation response of sub-micron,soft- and hard-particle filled polycarbonate

Since polymers play an increasingly important role in both structural and tribological applications, understanding their intrinsic mechanical response is key. Therefore in the last few decades much effort has been devoted into the development of constitutive models that capture the polymers' intrinsic mechanical response quantitatively. An example is the Eindhoven Glassy Polymer model. In practice most polymers are filled, e.g. with hard particles or fibers, with colorants, or with soft particles that serve as impact modifiers. To characterize the influence of type and amount of filler particles on the intrinsic mechanical response, we designed model systems of polycarbonate with different volume fractions of small, order 100 nm sized, either hard or soft particles, and tested them in lubricated uniaxial compression experiments. To reveal the local effects on interparticle level, three-dimensional representative volume elements (RVEs) were constructed. The matrix material is modeled with the EGP model and the fillers with their individual mechanical properties. It is first shown that (only) 32 particles are sufficient to capture the statistical variations in these systems. Comparing the simulated response of the RVEs with the experiments demonstrates that in the small strain regime the stress is under-predicted since the polymer matrix is modeled by using only one single relaxation time. The yield- and the large strain response is captured well for the soft-particle filled systems while, for the hard-particles at increased filler loadings, the predictions are less accurate. This is likely caused by polymer–filler interactions that result in accelerated physical aging of the polymer matrix close to the surfaces. Modifying the S_a-parameter, that captures the thermodynamic state of the polymer matrix, allows us to correctly predict the macroscopic response after yield. The simulations reveal that all rate-dependencies of the different filled systems originate from that of the polymer matrix. Finally, an onset is presented to predict local and global failure based on critical events on the microlevel, that are likely to cause the over-prediction in the large-strain response of the hard-particle filled systems.     

FE analysis of stress and strain fields in finite random composite bodies

In this paper the mechanical behaviour of finite random heterogeneous bodies is considered. The analysis of non-local interactions between heterogeneities in microscopically heterogeneous materials is necessary when the spatial variation of the load or the dimensions of the body, relative to the scale of the microstructure, cannot be ignored. Microstructures can be periodic but generically they are random. In the first case, an exact calculation can be performed but in the second case recourse has to be made either to simulation or to some scheme of approximation. One such scheme is based on a stochastic variational principle. The novelty of the present work is that a stochastic variational principle is projected directly onto a finite-element basis so that all subsequent analysis is performed within a finite-element framework. The proposed formulation provides expressions for the local stress and strain fields in any realization of the medium, from which expressions for statistically-averaged quantities can be derived. Then an approximation of Hashin-Shtrikman type is developed, which generates a FE-based numerical procedure able to take account of interactions between random inclusions and boundary layer effects in finite composite structures. Finally, two examples are presented, namely a cylinder with square cross-section subjected to mixed boundary conditions of different types on different faces and a rectangular body containing a centre crack. The results show that in the vicinity of the boundary or close to the crack tip, the strain and the stress in the matrix and in the inclusions differ considerably from those obtained by the formal application of conventional homogenization.     

Incorporation of yield surface distortion in finite deformation constitutive modeling of rigid-plastic hardening materials based on the Hencky logarithmic strain

In this paper a constitutive model for rigid-plastic hardening materials based on the Hencky logarithmic strain tensor and its corotational rates is introduced. The distortional hardening is incorporated in the model using a distortional yield function. The flow rule of this model relates the corotational rate of the logarithmic strain to the difference of the Cauchy stress and the back stress tensors employing deformation-induced anisotropy tensor. Based on the Armstrong–Fredrick evolution equation the kinematic hardening constitutive equation of the proposed model expresses the corotational rate of the back stress tensor in terms of the same corotational rate of the logarithmic strain. Using logarithmic, Green–Naghdi and Jaumann corotational rates in the proposed constitutive model, the Cauchy and back stress tensors as well as subsequent yield surfaces are determined for rigid-plastic kinematic, isotropic and distortional hardening materials in the simple shear deformation. The ability of the model to properly represent the sign and magnitude of the normal stress in the simple shear deformation as well as the flattening of yield surface at the loading point and its orientation towards the loading direction are investigated. It is shown that among the different cases of using corotational rates and plastic deformation parameters in the constitutive equations, the results of the model based on the logarithmic rate and accumulated logarithmic strain are in good agreement with anticipated response of the simple shear deformation.     

Modelling of laminated microstructures in stress-induced martensitic transformations

This paper is concerned with micromechanical modelling of stress-induced martensitic transformations in crystalline solids, with the focus on distinct elastic anisotropy of the phases and the associated redistribution of internal stresses. Micro-macro transition in stresses and strains is analysed for a laminated microstructure of austenite and martensite phases. Propagation of a phase transformation front is governed by a time-independent thermodynamic criterion. Plasticity-like macroscopic constitutive rate equations are derived in which the transformed volume fraction is incrementally related to the overall strain or stress. As an application, numerical simulations are performed for cubic β₁ (austenite) to orthorhombic γ₁′ (martensite) phase transformation in a single crystal of Cu-Al-Ni shape memory alloy. The pseudoelasticity effect in tension and compression is investigated along with the corresponding evolution of internal stresses and microstructure.     

Constitutive behaviour of anisotropic materials under shock loading

Thermodynamically and mathematically consistent constitutive equations suitable for shock wave propagation in an anisotropic material are presented in this paper. Two fundamental tensors α_ij and β_ij which represent anisotropic material properties are defined and can be considered as generalisations of the Kronecker delta symbol, which plays the main role in the theory of isotropic materials. Using two fundamental tensors α_ij and β_ij, the concept of total generalised “pressure” and pressure corresponding to the thermodynamic (equation of state) response are redefined. The equation of state represents mathematical and physical generalisation of the classical Mie–Grüneisen equation of state for isotropic material and reduces to the Mie–Grüneisen equation of state in the limit of isotropy. Based on the generalised decomposition of the stress tensor, the modified equation of state for anisotropic materials, and the modified Hill criteria, combined with the associated flow rule, a system of constitutive equations suitable for shock wave propagation is formulated. The behaviour of aluminium alloy 7010-T6 under shock loading conditions is considered. A comparison of numerical simulations with existing experimental data shows good agreement of the general pulse shape, Hugoniot Elastic Limits (HELs), and Hugoniot stress levels, and suggests that the constitutive equations are performing satisfactorily. The results are presented and discussed, and future studies are outlined.     

Inhomogeneity effects on the crack driving force in elastic and elastic-plastic materials

This article evaluates the effect of material inhomogeneities on the crack-tip driving force in general inhomogeneous bodies and reports results for bimaterial composites. The theoretical model, based on Eshelby material forces, makes no assumptions about the distribution of the inhomogeneities or the constitutive properties of the materials. Inhomogeneities are modeled by making the stored energy have an explicit dependence on the reference coordinates. Then the material inhomogeneity effect on the crack-tip driving force is quantified by the term C_inh, which is the integral of the gradient of the stored energy in the direction of crack growth. The model is demonstrated by two model problems: (i) bimaterial elastic composite using asymptotic solutions and (ii) graded elastic and elastic-plastic compact tension specimen using numerical methods for stress analysis.     

A unified treatment of strain gradient plasticity

A theoretical framework is presented that has potential to cover a large range of strain gradient plasticity effects in isotropic materials. Both incremental plasticity and viscoplasticity models are presented. Many of the alternative models that have been presented in the literature are included as special cases. Based on the expression for plastic dissipation, it is in accordance with Gurtin (J. Mech. Phys. Solids 48 (2000) 989; Int. J. Plast. 19 (2003) 47) argued that the plastic flow direction is governed by a microstress q_ij and not the deviatoric Cauchy stress σ_ij′ that has been assumed by many others. The structure of the governing equations is of second order in the displacements and the plastic strains which makes it comparatively easy to implement in a finite element programme. In addition, a framework for the formulation of consistent boundary conditions is presented. It is shown that there is a close connection between surface energy of an interface and boundary conditions in terms of plastic strains and moment stresses. This should make it possible to study boundary layer effects at the interface between grains or phases. Consistent boundary conditions for an expanding elastic-plastic boundary are as well formulated. As examples, biaxial tension of a thin film on a thick substrate, torsion of a thin wire and a spherical void under remote hydrostatic tension are investigated.     

Bounds and estimates for the effect of strain gradients upon the effective plastic properties of an isotropic two-phase composite

Predictions are made for the size effect on strength of a random, isotropic two-phase composite. Each phase is treated as an isotropic, elastic-plastic solid, with a response described by a modified deformation theory version of the Fleck-Hutchinson strain gradient plasticity formulation (Fleck and Hutchinson, J. Mech. Phys. Solids 49 (2001) 2245). The essential feature of the new theory is that the plastic strain tensor is treated as a primary unknown on the same footing as the displacement. Minimum principles for the energy and for the complementary energy are stated for a composite, and these lead directly to elementary bounds analogous to those of Reuss and Voigt. For the case of a linear hardening solid, Hashin-Shtrikman bounds and self-consistent estimates are derived. A non-linear variational principle is constructed by generalising that of Ponte Castañeda (J. Mech. Phys. Solids 40 (1992) 1757). The minimum principle is used to derive an upper bound, a lower estimate and a self-consistent estimate for the overall plastic response of a statistically homogeneous and isotropic strain gradient composite. Sample numerical calculations are performed to explore the dependence of the macroscopic uniaxial response upon the size scale of the microstructure, and upon the relative volume fraction of the two phases.     

Micromechanical analysis of kinematic hardening in natural clay

This paper presents a micromechanical analysis of the macroscopic behaviour of natural clay. A microstructural stress–strain model for clayey material has been developed which considers clay as a collection of clusters. The deformation of a representative volume of the material is generated by mobilizing and compressing all the clusters along their contact planes. Numerical simulations of multistage drained triaxial stress paths on Otaniemi clay have been performed and compared the numerical results to the experimental ones in order to validate the modelling approach. Then, the numerical results obtained at the microscopic level were analysed in order to explain the induced anisotropy observed in the clay behaviour at the macroscopic level. The evolution of the state variables at each contact plane during loading can explain the changes in shape and position in the stress space of the yield surface at the macroscopic level, as well as the rotation of the axes of anisotropy of the material.     

Mechanical properties of nanostructure of biological materials

Natural biological materials such as bone, teeth and nacre are nanocomposites of protein and mineral with superior strength. It is quite a marvel that nature produces hard and tough materials out of protein as soft as human skin and mineral as brittle as classroom chalk. What are the secrets of nature? Can we learn from this to produce bio-inspired materials in the laboratory? These questions have motivated us to investigate the mechanics of protein-mineral nanocomposite structure. Large aspect ratios and a staggered alignment of mineral platelets are found to be the key factors contributing to the large stiffness of biomaterials. A tension-shear chain (TSC) model of biological nanostructure reveals that the strength of biomaterials hinges upon optimizing the tensile strength of the mineral crystals. As the size of the mineral crystals is reduced to nanoscale, they become insensitive to flaws with strength approaching the theoretical strength of atomic bonds. The optimized tensile strength of mineral crystals thus allows a large amount of fracture energy to be dissipated in protein via shear deformation and consequently enhances the fracture toughness of biocomposites. We derive viscoelastic properties of the protein-mineral nanostructure and show that the toughness of biocomposite can be further enhanced by the viscoelastic properties of protein.