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In this paper, a variational principle of Lagrange and the Ritz method with generalized reduced and selective integration for mixed piecewise polynomial functions are used to obtain a stiffness matrix and a system of linear algebraic equations for micropolar theory of elasticity. This approach is implemented for anisotropic, isotropic, and centrally symmetric material in case of nonisothermal process. The cube problem is considered. The performance for finite element with mixed piecewise polynomial functions is exposed. [ABSTRACT FROM AUTHOR]

Liquid crystal elastomers (LCEs) are a class of materials which exhibit an anisotropic behavior in their nematic state due to the main orientation of their rod-like molecules called mesogens. The reorientation of mesogens leads to the well-known actuation properties of LCEs, i.e. exceptionally large deformations as a consequence of particular external stimuli, such as temperature increase. Another key feature of nematic LCEs is the capability to undergo deformation by constant stresses while being stretched in a direction perpendicular to the orientation of mesogens. During this plateau stage, the mesogens rotate towards the stretching direction. Such characteristic is defined as semisoft elastic response of nematic LCEs. We aim at modeling the semisoft behavior in a dynamic finite element method based on a variational-based mixed finite element formulation. The reorientation process of the rigid mesogens relative to the continuum rotation is introduced by micropolar drilling degrees of freedom. Responsible for the above-mentioned characteristics is an appropriate free energy function. Starting from an isothermal free energy function based on the small strain theory, we aim to widen it into the framework of large strains by identifying tensor invariants. In this work, we analyze the isothermal influence of the tensor invariants on the mechanical response of the finite element formulation and show that its space-time discretization preserves mechanical balance laws in the discrete setting. [ABSTRACT FROM AUTHOR]

Strain localization simulation of overconsolidated soils with high overconsolidation ratio (OCR) has been a long‐standing challenge. Some critical state soil models, including the modified Cam‐clay (MCC) model, have been widely applied, but they may not predict the shear dilatancy of overconsolidated soils well in some cases. Hence, the unified hardening (UH) model, which may be viewed as a generalized version of the MCC model, is implemented. It has been recognized, nonetheless, that without resorting to the regularization mechanism, the standard finite element method (FEM) or the second‐order cone programming optimized finite element method (FEM‐SOCP) often experiences instability or interruption of calculating the hardening‐softening responses of overconsolidated soils. To resolve the aforementioned difficulty, the UH model is developed and implemented in the framework of FEM‐SOCP based on the micropolar continuum (mpcFEM‐SOCP) to predict strain localizations of overconsolidated soils. Furthermore, to obviate non‐convexity of mpcFEM‐SOCP induced by material softening, an effective composite update scheme of hardening variable pc, which refers to the implicit variable (IV) scheme for the hardening stage and then refers to the explicit variable (EV) scheme for the softening stage, is proposed. Based on one biaxial compression problem and one rigid strip footing problem, numerical analyses disclose that by applying mpcFEM‐SOCP in conjunction with the composite update scheme of pc, the UH model of micropolar continuum can effectively predict the strain localization behavior of overconsolidated soil during its failure stage, and the stable hardening‐softening responses of overconsolidated soils can be readily attained. [ABSTRACT FROM AUTHOR]

In this work, preconditioning strategies are developed in the context of generalized continuum formulations used to regularize multifield models for simulating localized failure of quasi‐brittle materials. Specifically, a micropolar continuum extended by a nonlocal damage formulation is considered for regularizing both, shear dominated failure and tensile cracking. For such models, additional microrotation and nonlocal damage fields, and their interactions, increase the complexity and size of the arising linear systems. This increases the demand for specialized preconditioning strategies when iterative solvers are adopted. Herein, a block preconditioning strategy, employing algebraic multigrid methods (AMG) for approximating the application of sub‐block inverses, is developed and tested in three steps. First, a block preconditioner is introduced for linear systems resulting from micropolar models. For this case, a simple sparse Schur complement approximation, which is practical to compute, is proposed and analyzed. It is tested for three different discretizations. Second, the developed preconditioner is extended to reflect the additional nonlocal damage field. This extended three‐field preconditioner is tested on the simulation of a compression test on a sandstone sample. All numerical tests show an improved performance of the block preconditioning approach in comparison to a black‐box monolithic AMG approach. Finally, a problem‐adapted preconditioner setup strategy is proposed, which involves a reuse of the multigrid hierarchy during nonlinear iterations, and additionally accounts for the different stages occurring in the simulation of localized failure. The problem‐adapted preconditioning strategy has the potential to further reduce the total computation time. [ABSTRACT FROM AUTHOR]

The spotlight of this research is on the application of the fixed‐pole interpolation, sometimes used in the analysis of three‐dimensional (3D) geometrically non‐linear beams, but for which no attempts have been made to apply it to linear analysis so far. Particular attention is given to the correlation between the linearised forms of the fixed‐pole and helicoidal interpolation with the linked interpolation. Knowledge of this interdependence is crucial for identifying paths for possible enhancement and extension from the Timoshenko beam of arbitrary order to the new hexahedral finite element of arbitrary order for linear analysis of 3D micropolar continuum. After ensuring the convergence of the newly developed micropolar element through a set of patch tests, three numerical examples of a 3D micropolar continuum in static equilibrium and free vibration of 3D micropolar plates with different geometric properties and boundary conditions have been analysed. Based on these results, the newly proposed finite elements have been critically assessed against the conventional elements. [ABSTRACT FROM AUTHOR]

Elasto-plastic deformations of micropolar continuum are discussed by a non-Riemannian geometry. The non-locality of micropolar continuum is described in a second-order vector bundle of displacements and microrotations. With a decomposition of total elasto-plastic field, geometric quantities are divided into the elastic and plastic components independently. Especially, when an intrinsic parallelism of displacements and microrotations holds, integrability conditions of the elasto-plastic field are represented by a torsion tensor or the curvature of nonlinear connection. Then, Burgers and Frank vectors and an energy release rate around crack tips are related to the torsion tensor or the curvature of nonlinear connection. Moreover, the non-locality of microrotation is discussed based on a kink band as a disclination. It is found a generalized expression of Burgers vector which can describe the kink interface including the disclination. [ABSTRACT FROM AUTHOR]

In this paper, a variational principle of Lagrange, the Ritz method, and piecewise polynomial shape functions of brick family "linear" element are used to obtain reduced and selective integration techniques in a form of the tensor-block stiffness matrices to prevent the locking effect for nearly incompressible, isotropic, and centrally symmetric material of the micropolar theory of elasticity. [ABSTRACT FROM AUTHOR]

To address the problems of strain localization, the exact Mohr-Coulomb (MC) model is used based on second-order cone programming (mpcFEM-SOCP) in the framework of micropolar continuum finite element method. Using the uniaxial compression test, we focused on the earth pressure problem of rigid wall segment involving non-associated plasticity. The numerical results reveal that when mpcFEM-SOCP is applied, the problems of mesh dependency can be effectively addressed. For geotechnical strain localization analysis involving non-associated MC plasticity, mpcFEM-SOCP in conjunction with the pseudo-time discrete scheme can improve the numerical stability and avoid the unreasonable softening issue in the pressure-displacement curves, which may be encountered in the conventional FEM. It also shows that the pressure-displacement responses calculated by mpcFEM-SOCP with the pseudo-time discrete scheme are higher than those calculated by mpcFEM-SOCP with the Davis scheme. The inclination angle of shear band predicted by mpcFEM-SOCP with the pseudo-time discrete scheme agrees well with the theoretical solution of non-associated MC plasticity.

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In this work, we aim to model the reorientation process of mesogens in nematic liquid crystal elastomers within the context of dynamics. We consider a continuum model with separate mappings for the deformation of the monolithic material and the orientation of the nematic director, where the latter describes the inclination of the mesogens. We achieve the inextensibility of the nematic director through the introduction of drilling degrees of freedom. We combine this approach with the application of the principle of virtual power and a mixed finite element formulation, in order to formulate distinct momentum and angular momentum balance laws for the two separate mappings. Furthermore, we include in our continuum model a volume load and a surface load associated only with the orientation mapping. We show in the presented three numerical examples that our formulation enables the fulfillment of all momentum and angular momentum balance laws. [ABSTRACT FROM AUTHOR]

In this paper, a variational principle of Lagrange, the Ritz method, and piecewise polynomial serendipity shape functions are used to obtain a stiffness matrix and a system of linear algebraic equations in the micropolar theory of elasticity for anisotropic, isotropic, and centrally symmetric material in case of a nonisothermal process. [ABSTRACT FROM AUTHOR]

Study of strain localization in geomaterials generally consists of two aspects, the pre-localization regime and the post-localization regime. The former refers to the onset and the orientation of the shear band, which corresponds to the bifurcation theory. The latter deals with the width and the evolution of the shear band. The purpose of this paper is to make a full-range analysis of the shear band in structural level. By comparing the bifurcation conditions in classical continuum an analogical bifurcation condition in micropolar theory is defined. The factors which may affect the shear band orientation in the numerical modeling process are emphasized, and a basic principle of the geometric modeling is concluded. Furthermore, the existing shear band width measurement methods are summarized, and some other more effective methods considering the singularity of the discontinuous surface is suggested. At the last, the ratio between the node distance of the element and the internal length less than 5 is suggested as reasonable choice. [ABSTRACT FROM AUTHOR]

In this paper, the variational principle of Lagrange, the Ritz method and piecewise polynomial serendipity shape functions are used to obtain the stiffness matrix and a system of linear algebraic equations in the micropolar theory of elasticity for orthotropic and centrally symmetric material. [ABSTRACT FROM AUTHOR]

February 10, 2022 the famous scientist in mechanics of solids and applied mathematics, teacher, organizer of science and higher education in Russia YuriN.Radayev is celebrating his60thanniversary. YuriN.Radayev is known as a prominent scientist in the field of mechanics and applied mathematics. The principal directions of his academic activity are the Mathematical Theory of Plasticity, Fracture Mechanics, the Theory of Cracks and Microdamages, Coupled Hyperbolic Thermoelasticity and Thermomechanics, Micropolar Elasticity, Mechanics of Granular Solids, Mechanics of Growing Solids. In this biographical background we discuss the scientific and educational work of Prof. Yuri N.Radayev, give an information on his achievements and a list of his main publications.

The article is dedicated to the 60th anniversary of professor Yu. N. Radaev, the well-known scientist in the field of mechanics of deformable solids and applied mathematics, the member of the editorial board of the journal “Izvestiya of Saratov University. Mathematics. Mechanics. Informatics”.

The paper focuses on the dependence of the micro-inertia tensor on continuum deformations. Modeling of the micropolar medium is based on the spatial description that opens the possibility to model situations and materials with a continuum point that on the microscale consists no longer of the same elementary units during a physical process. The tensor of inertia of the polar particle is obtained by averaging the inertia tensors of microparticles within a representative volume. Because of the medium deformation, the representative volume contains different micro-particles as the medium moves, and the inertia tensor of the volume will change due to the incoming or outgoing flux of inertia. A possibility of dependence of the tensor of inertia on the strain measures is demonstrated, and suitable forms of the dependence are suggested. [ABSTRACT FROM AUTHOR]

We develop a linear theory of the Cosserat continuum of a special type. This continuum possesses only rotational degrees of freedom. The constitutive equation for the moment stress tensor is the same as for the elastic continuum. The main feature of our model is that the differential equation relating the wryness tensor to the angular velocity vector contains a source term. Thanks to a special choice of the constitutive equation for the source term, we obtain a model of continuum that has some properties of a viscoelastic continuum. Considering such a continuum, we associate the main variables characterizing its stress–strain state with quantities characterizing electrodynamic and thermodynamic processes. Our new model describes all physical processes that were described by using our previous models, but at the same time, it gives us some important results that cannot be obtained in the framework of our previous models. In contrast to the previous model, which is based on Maxwell's model of a viscoelastic material, the new model allows us to arrive at Maxwell's equations for conductors without modifying constitutive equations. In addition, the new model describes the conversion of electrical energy into thermal energy due to Joule heat. Furthermore, the new model allows us to obtain the entropy balance equation. [ABSTRACT FROM AUTHOR]

Within the framework of second-order cone programming optimized micropolar continuum finite-element method (CosFEM-SOCP), the geotechnical strain localization can be adequately modeled. In most existing literatures, however, the constant internal characteristic length lc has been adopted and less attention has been paid to the evolution of lc. To more accurately predict the strain localization, response, and stability of a geotechnical system, one relationship for evolving lv that relies on the equivalent plastic strain is implemented and investigated. Based on one homogeneous slope example and one rigid strip footing example, it was found that geotechnical stability may not be significantly affected by evolving lv, indicating that constant lc can be simply applied to geotechnical stability analysis. For the rigid strip footing problem, nevertheless, the effects of evolving lv on the pressure–displacement response curves should not be ignored, and the influence range of the shear band predicted by CosFEM-SOCP with evolving lv is generally smaller than that predicted by CosFEM-SOCP with constant lc. Consequently, in order to more accurately predict the pressure–displacement response curves and the failure zone, the evolving lv will be adequately assessed and modeled. [ABSTRACT FROM AUTHOR]

A model is proposed for the description of a highly inhomogeneous distribution of hydrogen within a saturated metal specimen (the so-called skin effect due to hydrogen saturation). The model is based on the micropolar continuum approach and results in a nonuniform stress–strain state of a cylindrical metal specimen due to distributed couples or microrotations. The dependence of the diffusion coefficient on the strain energy is considered in order to model stress-induced diffusion. Accumulation of hydrogen within a thin boundary layer results in a highly nonuniform distribution of hydrogen across the specimen. The mutual influence of the stress–strain state and hydrogen accumulation is taken into account. The estimated thickness of the surface layer containing hydrogen is comparable to the thickness observed in experiments. The predicted average concentration coincides with experimental data. [ABSTRACT FROM AUTHOR]

The conventional flow rule and plasticity constitutive models implicitly assume that the principal stress and plastic strain rate are coaxial, which often result in various setbacks in modeling, for example, the bifurcation and the evolution of shear band cannot be accurately and adequately captured. In this work, we investigate the non-coaxial effects of an elastoplastic constitutive model on strain localization in the framework of micropolar plasticity theory. The vertex-like yield surface is introduced in the flow rule, and a non-coaxial Drucker–Prager model is implemented in a user-defined finite element subroutine (UEL) in ABAQUS. The simple shear test and the plane strain compression test have been conducted to study the non-coaxial stress–strain response and the strain localization behavior. Effects of boundary condition on bifurcation, shear band orientation and thickness are investigated. Results indicate that the proposed model shows a good performance in the simulation of the non-coaxial behavior of geomaterials. On the premise of overcoming mesh dependency, the shear band orientation and width predicted by the non-coaxial model possess higher values than those predicted by the coaxial model, and a better agreement with the experimental results are achieved by the non-coaxial model. Moreover, by analyzing stress–strain responses, we have found that material strength may be systematically overestimated by the coaxial model. [ABSTRACT FROM AUTHOR]

We consider a linear theory of a viscoelastic Cosserat continuum of a special type. In doing so, we associate the main variables characterizing the stress–strain state of the continuum with quantities characterizing the electrodynamic and thermal processes. Taking into account the suggested analogues, we interpret equations describing the continuum as equations of thermodynamics and electrodynamics. We identify parameters of our model by comparing the obtained equations with Maxwell's equations and the hyperbolic heat conduction equation. As a result, we arrive at two three-dimensional telegrapher's equations: one for temperature and the other for the electric field vector. These equations are novel. They describe electromagnetic and thermal processes and also how they affect each other more accurately compared to the classical theory. In particular, these telegrapher's equations account for not only the skin effect described in many literature sources on electrodynamics, but also the so-called static skin effects observed in a number of experiments. In contrast to classical electrodynamics, which contains two mutually orthogonal vectors: the electric field vector and the magnetic induction vector, the proposed theory contains three mutually orthogonal vectors: the electric field vector, the magnetic induction vector and the temperature gradient. It agrees with experimental facts discovered by Ettingshausen and Nernst (the Ettingshausen effect and the Nernst–Ettingshausen effect). If thermal component is ignored, the proposed theory reduces to the system of equations, which is a generalization of Maxwell's equations. This system of equations is novel. It is a three-dimensional analogue of Kirchhoff's laws for electric circuits, while Maxwell's equations are not. [ABSTRACT FROM AUTHOR]

Motivated by the abundance of fractals in the natural world, this paper further develops continuum-type models for product-like fractals. The theory is based on a version of the non-integer dimensional space approach, in which global balance laws written for fractal media are expressed in terms of conventional (integer-order) integrals. Key relations of calculus for finite strain kinematics of fractal media are obtained, especially clarifying the fractal Jacobian and the fractal Reynolds transport theorem. The local forms are then written in terms of partial differential equations with derivatives of integer order. Hence, fractal versions of local continuity, linear and angular momenta and energy balance are derived. The angular momentum balance implies that the approximating continuum is micropolar rather than classical. Accordingly, the Cauchy postulate, lemma and theorem for Cauchy force-stress and couple-stress are re-formulated. The corresponding partial differential equations for finite as well as infinitesimal elasticity are given explicitly in both the displacement and stress formulations. The invariance of the stress field in planar fractal elastic media is shown to hold just like the one in planar micropolar elasticity. [ABSTRACT FROM AUTHOR]

In this paper, we study analytically the three-dimensional (3D) stress–strain relationships at the elastic stage with regard to the anisotropic granular materials composed of regularly arranged bonded ellipsoidal particles by the micro-structural mechanics approach, where the macroscopic elastic constants are expressed in closed form with respect to the microscopic parameters. The bonded ellipsoidal particle assembly is first equivalent to a 3D lattice network composed of lattice beams with different contact properties. Based on the principle of energy balance, the macroscopic elastic stress–strain relationships for the equivalent micropolar continuum are obtained by analysing the unit cells of the lattice beam system. Using the proposed closed-form expressions of the anisotropic elastic constants, the elastic moduli, Poisson's ratios, Cauchy and Cosserat shear moduli and bending moduli can be expressed as functions of the microscopic parameters pertaining to the particle shape, size and contact properties. The elastic moduli and Poisson's ratios from closed-form expressions for regular particle arrangement and modified expressions for irregular particle arrangement are noted to be in agreement with the results obtained using distinct element method (DEM). The results suggest that the regularly arranged ellipsoid particle assembly shows a typical 3D orthotropic feature, and the elastic constants change significantly as the anisotropy of particle increases. [ABSTRACT FROM AUTHOR]

In this paper, new representations of three-dimensional asymmetric stress tensor and the corresponding form of the differential equilibrium equations are given. Asymmetric theories of solid mechanics continues to attract attention in connection with the necessity of mathematical modelling of the mechanical behaviour of the advanced materials. The study is restricted to such asymmetric second rank tensors, for which it is still possible to keep the notion of real eigenvalues, but not to accept the mutual orthogonality of the directors of the principal trihedron. The exact algebraic formulation of these asymmetry conditions is discussed. The study extends the dyadic tensor representations of the symmetric stress tensor based on the notion of asymptotic directions. The obtained results are a clear evidence in favor of algebraic hyperbolicity both the symmetric and asymmetric second rank tensors in three-dimensional space.

This paper is concerned with a mechanical explanation of a highly inhomogeneous distribution of hydrogen within metal specimens, based on the micropolar continuum approach. The primary focus is on the modeling of the nonuniform stress–strain state of a cylindrical metal specimen that rapidly fades away from the border and changes the inner structure of the material near the lateral surface. The boundary condition used in the considered boundary value problem reflects the influence of the structural defects located on the boundary. Thus, this model considers inner stresses and strains due to the structural inhomogeneity. Large values of the strain energy within the area comparable to the size of the structural inhomogeneity lead to a significant increase in the diffusion coefficient in the vicinity of the border. As a result, fast accumulation of hydrogen within a thin boundary layer produces a highly nonuniform distribution of hydrogen across the specimen. The comparison between the concentrations of hydrogen measured experimentally and estimated analytically was made. [ABSTRACT FROM AUTHOR]

By the micro-structural mechanics approach, this study establishes the quantitative stress–strain relationships at the elastic stage with respect to the microscopic features of the particles for anisotropic granular materials that are composed of regularly arranged elliptical particles with the same size. Firstly, the elliptical particle assembly is equated with a lattice network described by beam elements attached to the center of particles. Then, the elastic stress–strain relationships, which exactly show the features of orthotropic micropolar continuum, are established through analyzing the triangular and hexagonal cells based on the principle of energy balance. Finally, the analytical expressions of eight independent parameters in stress–strain relationships are obtained as the functions of particle shape, size, and microscopic contact stiffnesses. Further, the relationship between microscopic parameters and macroscopic elastic constants for anisotropic granular materials is proposed. The analytical expressions are verified by comparison between theoretical and discrete element method (DEM) results for elliptical particle assembly, and the influences of microscopic parameters on the macroscopic elastic constants are investigated in detail according to the proposed analytical expressions. Our work provides some useful insights for the microscopic explanation and the influences of microscopic parameters on the macroscopic mechanical behavior of granular materials. [ABSTRACT FROM AUTHOR]

This study continues the line of earlier research in mechanical models of electrodynamic processes suggested in previous works. The basic steps we take to construct these models are: to formulate equations of a special type Cosserat continuum, and then to suggest analogies between quantities characterizing the stress–strain state of the continuum and quantities characterizing electrodynamic processes. In addition to the previously introduced mechanical analogies of the electric field vector and the magnetic induction vector, in this paper we provide mechanical analogies of the magnetic field vector, the electric induction vector, the electric current density and the electric charge density. In the framework of the suggested model, we obtain a set of differential equations that coincide with the first Maxwell's equation (the one with the displacement current term), the Gauss law for electric field and the charge conservation law. As a debatable question, we discuss the possibility of modifying the Maxwell–Faraday equation and the Gauss law for magnetic field. This study continues the line of earlier research in mechanical models of electrodynamic processes suggested in previous works. The basic steps we take to construct these models are: to formulate equations of a special type Cosserat continuum, and then to suggest analogies between quantities characterizing the stress–strain state of the continuum and quantities characterizing electrodynamic processes.... [ABSTRACT FROM AUTHOR]

Linear model of micropolar elastic continuum (known also as the Cosserat continuum) is considered. Kinematics and strain measures are discussed. The symmetric small strains tensor, relative microrotation vector and spatial gradient of the total microrotation vector (the wryness tensor) are then employed for a covariant formulation of the micropolar theory. By means of the principle of virtual displacements much simplified by the lack of internal forces and couples contributions to the virtual work and the Lagrange multipliers method the micropolar theory of elasticity is developed. Hemitropic micropolar continuum model is investigated in further details. The paper is to be considered as a universal covariant script of equations of the linear micropolar theory of elasticity derived from the virtual displacements principle.

An isolated surface that moves relative to the micropolar media and across which the first derivatives of variables are discontinuous is considered. The reduced Cosserat continuum is an elastic medium where all translations and rotations are independent. Moreover, the force stress tensor is asymmetric and the couple stress tensor is equal to zero. Continuity conditions were established and it is shown that the first derivative of the rotation vector cannot have discontinuities. It is demonstrated that the solution in this case is unique. [ABSTRACT FROM AUTHOR]

The main purpose of this contribution is to develop a three-dimensional (3D) nonlinear beam model based on the micropolar continuum theory. To do so, a kinematic model based on the deformation of three directors and accounting for the micro-rotation tensor of the micropolar theory is introduced. One of the main characteristics of the present beam model is that 3D constitutive equations without any modification can be directly used in the formulation. Furthermore, it is known that a body couple field is induced in hard-magnetic soft materials (HMSMs) when subjected to external magnetic fluxes. Therefore, the stress tensor in HMSMs is asymmetric, in general. Since the asymmetry of stress is one of the main features of the micropolar theory, the present formulation can be used for analyzing the deformation of beams made of HMSMs. Accordingly, the virtual external work of the present model is formulated so that it accounts for the contribution from uniform or constant-gradient external magnetic fluxes on the beam. Moreover, a Total Lagrangian (TL) nonlinear finite element (FE) formulation to provide numerical solutions of the related problems is developed. Several numerical examples are solved to investigate the capability of the developed formulation. It is shown that the present formulation can model the size-dependent behavior of beam-like structures if the material length-scale parameter of the micropolar constitutive model is comparable to the thickness of the beam. Moreover, the proposed model can successfully predict the finite deformation of 3D beams made of HMSMs subjected to magnetic loading. [Display omitted] • A 3D beam model based upon the micropolar theory is developed. • A nonlinear FE formulation for the present beam model is elaborated. • Both size effect and magnetic loading can be modeled by the present formulation. • Numerical results reveal the good performance of the present formulation. [ABSTRACT FROM AUTHOR]

In this paper the effect of grain fragmentation of a cohesionless granular material on the change of microstructure quantities is investigated using a micropolar continuum model. To this end the change of the grain size distribution is related in a simplified manner to a reduction of the mean grain diameter, which enters the constitutive equation as the internal length. The additional densification of the grain skeleton is modelled by a reduction of the incremental stiffness, which is related to the so-called solid hardness defined in the sense of a continuum description. The reduction of the mean grain diameter and the solid hardness caused by grain fragmentation is described by corresponding evolution equations, which depend on an increasing mean pressure and an increasing deviatoric stress. The focus of the numerical investigations is on the evolution of microstructure quantities like the void ratio, mean grain diameter and couple stresses under monotonic compression. It is shown that for an initially inhomogeneous distribution of the void ratio polar quantities like microrotations and couple stresses can develop even outside of shear bands. With increasing compression the polar quantities can increase, while the range of fluctuation of the void ratio decreases. [ABSTRACT FROM AUTHOR]