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
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.