and Computational Methods

doi: 10.18698/2309-3684-2017-1-3254

The article considers the modeling results of incompressible layered composites with finite strains deformation according to the individual layers characteristics. The article proposes an asymptotic averaging method version for layered nonlinearly elastic incompressible composites with finite deformations and periodic structure. We are using a universal representation of the defining relations for incompressible composite layers, proposed by Yu.I. Dimitrienko, which allows us to simulate simultaneously for a complex of various nonlinear elastic media models characterizedby the choice of a pair of energy tensors. We proved that if all composite layers are incompressible, the composite as a whole is also an incompressible, but anisotropic, medium. The article considers the problem of laminated plate uniaxial stretching from incompressible layers with finite deformations. Using the developed method, we calculated the effective deformation diagrams connecting the averaged Piola — Kirchhoff stress tensors components and the strain gradient, as well as the stress distribution in the composite layers.

The developed method for calculating effective deformation diagrams and stresses in composite layers can be used in the design of elastomeric composites with specified properties.

Dimitrienko Y., Gubareva E., Kolzhanova D., Karimov S. Incompressible layered composites with finite deformations on the basis of the asymptotic averaging method. Маthematical Modeling and Coтputational Methods, 2017, №1 (13), pp. 32-54

doi: 10.18698/2309-3684-2024-2-1734

The paper is devoted to modeling the deformation of composite materials with finite deformations. The so-called universal models of constitutive relations for composite components are considered, defining several classes of nonlinear relationship between the Piola-Kirchhoff stress tensor and the strain gradient within different energy pairs of stress-strain tensors. The method of asymptotic averaging is applied and local problems are formulated to solve the problem of determining the averaged properties of composites with finite deformations. A variational formulation of the original deformation problem, the so-called local problems on a periodicity cell and the averaged problem for a composite is considered, which makes it possible to use FEM for the numerical solution of these classes of problems. A software module has been developed as part of the SMCM software package, which implements the proposed numerical algorithm. An example of the numerical solution of problems on a periodicity cell for a 3D orthogonally reinforced composite is given, taking into account large deformations of the matrix and fibers, and composite deformation diagrams are calculated for various variants of universal models of constitutive relations.

Димитриенко Ю.И., Каримов С.Б., Димитриенко А.Ю. Моделирование конечных деформаций композиционных материалов на основе универсальных моделей Аn и метода асимптотического осреднения. Математическое моделирование и численные методы, 2024, № 2, с. 17–34.

doi: 10.18698/2309-3684-2021-1-330

The problem of calculating the stress–strain state of a composite laminated panel during cylindrical bending under conditions of finite deformations is considered. To solve the problem, the method of asymptotic averaging of periodic nonlinear elastic structures with finite deformations was applied, which was developed in detail earlier in the previous works of the authors. A feature of this problem is the use of universal models of constitutive relations for isotropic components of the composite, as well as for the composite as a whole, which is a transversely isotropic nonlinear elastic medium. Universal models make it possible to obtain solution of problems within the framework of a single solution algorithm simultaneously for several classes of models of nonlinear elastic media corresponding to different conjugate pairs of stress tensors–deformation. An analytical solution is obtained for the problem of cylindrical bending of a composite panel. A numerical analysis of the solution is carried out using the example of a composite, the periodicity cell of which consists of two layers: polyurethane and rubber. It is shown that for thin panels the stresses, both averaged and true, practically do not depend on the class of the model of the constitutive relations. At the same time, for thicker panels, the stresses differ significantly for different classes of models of composite layers.

Димитриенко Ю.И., Губарева Е.А., Каримов С.Б., Кольжанова Д.Ю. Моделирование напряжений в композитной нелинейно упругой панели при цилиндрическом изгибе. Математическое моделирование и численные методы, 2021, № 1, с. 3–30.

doi: 10.18698/2309-3684-2018-4-7292

The paper considers a model of effective constitutive relations for a transversal-isotropic incompressible composite with finite strains. The model belongs to the so-called class of universal models that connect several pairs of energy stress and strain tensors simultaneously. A method is proposed for separating coupled problems of micro- and macroscopic deforming composites with finite strains that arise when the method of asymptotic homogenization (AH) of periodic structures is used. The method is based on the application of the effective constitutive relation model as an approximation dependence of the results of numerical simulation of the composite deformation curves obtained using the exact AH method. To find the elastic constants of the transversely isotropic composite model the method of minimizing the deviation of the approximation of deformation diagrams from the AH diagrams is used for a series of problems of standard deforming at finite strains. To solve minimization problems, the Nelder—Mead method is used. The results of numerical simulation by the proposed method for nonlinear elastic layered composites are presented, which showed good approximation accuracy, achieved due to application of the proposed method for the separation of coupled problems of micro- and macroscopic deforming.

Димитриенко Ю.И., Губарева Е.А., Каримов С.Б., Кольжанова Д.Ю. Модели-рование эффективных характеристик трансверсально-изотропных несжимаемых композитов с конечными деформациями. Математическое моделирование и чис-ленные методы, 2018, № 4, с. 72–92.