This paper presents a combined approach to the solution of boundary value problems. The approach implies the joint application of a macro-phenomenological model and multilevel physical constitutive models for describing the behavior of a polycrystalline material. Within the framework of the proposed approach, the boundary value problem is solved for a certain product (structure) using a macro-phenomenological model of the material and the response in the chosen area is clarified using multilevel modeling techniques. The capabilities of the approach were demonstrated in a series of numerical experiments covering metal forming processes. In order to test and verify the proposed technique, in the first numerical experiment, the process of an upsetting of the sample was considered using only a multilevel physical model of the material and using the formulated combined approach. The response in the chosen domain was compared to the results obtained by different modeling approaches. The comparison showed good agreement in terms of the variables of the stress–strain state at the macro level, as well as in terms of the variables characterizing the internal structure of the material. At the same time, using the combined approach, the required computation time was reduced by more than 12 times. In the second and third numerical experiments, the process of sheet rolling and sample pressing was successfully simulated using a combined approach. The formulated approach is computationally efficient and applicable for solving the relevant boundary value problems of design and the optimization of technological processes.

The products made by the forming of polycrystalline metals and alloys, which are in high demand in modern industries, have pronounced inhomogeneous distribution of grain orientations. The presence of specific orientation modes in such materials, i.e., crystallographic texture, is responsible for anisotropy of their physical and mechanical properties, e.g., elasticity. A type of anisotropy is usually unknown a priori, and possible ways of its determination is of considerable interest both from theoretical and practical viewpoints. In this work, emphasis is placed on the identification of elasticity classes of polycrystalline materials. By the newly introduced concept of “elasticity class” the union of congruent tensor subspaces of a special form is understood. In particular, it makes it possible to consider the so-called symmetry classification, which is widely spread in solid mechanics. The problem of identification of linear elasticity class for anisotropic material with elastic moduli given in an arbitrary orthonormal basis is formulated. To solve this problem, a general procedure based on constructing the hierarchy of approximations of elasticity tensor in different classes is formulated. This approach is then applied to analyze changes in the elastic symmetry of a representative volume element of polycrystalline copper during numerical experiments on severe plastic deformation. The microstructure evolution is described using a two-level crystal elasto-visco-plasticity model. The well-defined structures, which are indicative of the existence of essentially inhomogeneous distribution of crystallite orientations, were obtained in each experiment. However, the texture obtained in the quasi-axial upsetting experiment demonstrates the absence of significant macroscopic elastic anisotropy. Using the identification framework, it has been shown that the elasticity tensor corresponding to the resultant microstructure proves to be almost isotropic.

As proven in numerous experimental and theoretical studies, physical and mechanical properties of materials are determined by their internal structure. In the particular case of polycrystalline metals and alloys, an important role is given to the orientation distributions of crystalline lattices, or, in other words, crystallographic textures. Physically reasonable models of texture formation are highly demanded in modern Material Science and Engineering since they can provide an efficient tool for designing polycrystalline products with improved operational characteristics. Models of interest can be obtained on the basis of statistical formulations of multilevel approaches and crystal elasto–visco–plasticity theories (in particular, Taylor–Bishop–Hill models and their various modifications are appropriate here). In such a framework, a representative volume element of a polycrystal is numerically implemented as a finite aggregate of crystallites (grains or subgrains) with a homogenized response at the macro-scale. Quantitative texture analysis of this aggregate requires estimating statistically stable features of the orientation distribution. The present paper introduces a clustering-based approach for executing this task with regard to preferred orientations. The proposed procedure operates with a weighted sample of orientations representing the aggregate and divides it into clusters, i.e., disjoint subsets of close elements. The closeness criterion is supposed to be defined with the help of a special pseudometric distance, which takes rotational symmetry of the crystalline lattice into account. A specific illustrative example is provided for better understanding the developed procedure. The texture in the clustered aggregate can be described reductively in terms of effective characteristics of distinguished clusters. Several possible reduced-form representations are considered and investigated from the viewpoint of aggregating elastic properties in application to some numerically simulated textures.

A quasi-static geometrically non-linear initial-boundary value problem is considered in a rate form for investigating deformation of a solid. Within the framework of the finite element method, an approach is proposed to computationally implement unilateral contact conditions for this problem with friction defined by the Coulomb–Siebel‘s law. Such formulations are often encountered in simulating technological processing operations based on severe inelastic deformations. The approach consider nodes and facets as contacting discrete fragments, where the nodes correspond to a body representing a deformable billet, and the facets refer to a rigid tool performing a processing program. Modeling contact in this way reduces to imposing special kinematic and quasi-static constraints in the nodes at each time slice. Procedurally, these constraints can be realized by the well-known standard modifications of a resolving system of linear algebraic equations which establish relationships between nodal velocities and force rates. In this paper, we provide a detailed description of main techniques required for this task and formulate a variant of an algorithm executing them in an appropriate sequence.

An approach based on Monte Carlo simulations for sampling crystallite orientations from known texture data is developed. A randomized algorithm is proposed to reconstruct the orientation distribution function by a given set of pole densities. The algorithm fills up the orientation space with lots of uniformly distributed points and tries to evaluate the reconstructable function specifically at these points. This leads to a sparse quadratic programming problem, which is a discrete statistical analogue of the fundamental equation of Texture Analysis. The resulted distribution is represented as a weighted sample of orientations, with the weights equal to the reconstructed distribution function values. By means of randomized weight-based selections, such a sample can be reduced to an unweighted one with appropriately distributed orientations. A theoretical substantiation of the methods is presented in the paper in the weak probabilistic sense. Several examples of practical applications are provided as well.

This article describes a modified multiscale constitutive model for analyzing the stress-strain behavior of polycrystalline materials, which explicitly accounts for the influence of grain boundary misorientation on the mechanical response. The model was implemented as a user subroutine and added to the SIMULIA Abaqus FEA software suite. A slip system compatibility factor was introduced into the hardening law via a relation for critical shear stresses, enabling the capture of interactions between dislocations and grain boundaries. The model was used in a numerical study of a generated polycrystalline structure of 308LSi austenitic stainless steel. The results of the study reveal the localization of stresses and plastic strains near high-angle grain boundaries, demonstrating the significant influence of crystallo-geometric factors on material hardening.