Modeling of effective elastic-plastic properties of layered composites with a periodic structure in the framework of the anisotropic flow theory

. The article is devoted to the development of a method for constructing theoretical strain diagrams. The method is based on the use of a model of effective constitutive relations for approximating the deformation diagrams of layered composites obtained using the asymptotic averaging method. To find the elastic constants of the model of a transversally isotropic composite, the method of minimizing the deviation of the approximation deformation diagrams from the diagrams obtained by the asymptotic homogenization (AH) method is used for a series of standard problems of deformation at small deformations. Minimization problems were solved using the Hooke-Jeeves method. The results of numerical simulation by the proposed method for layered composites are presented, which showed good approximation accuracy, which is achieved due to the proposed method for separating the coupled problems of micro-and macroscopic deformation.


Introduction
Currently, there are many works devoted to modeling the effective mechanical characteristics of composite materials.For practical purposes, the problem of determining the effective elastic characteristics of composites based on information about the microstructure and properties of the constituent phases is of great importance.There are quite a few methods for this problem, but most of them are inapplicable for composites with small deformations [1,2].
To calculate the effective characteristics of composites, the most promising method is the homogenization method (AH), proposed by N.S.Bakhvalov, G.P. Panasenko, E. Sanchez-Palencia.The method of asymptotic averaging is well developed at present and has been successfully implemented numerically for various problems in mechanics, but mainly for linear problems [3−12].
To solve the problems of macroscopic deformation of structural elements made of composites with small deformations, it is necessary to apply constitutive relations for composites, taking into account anisotropy.The direct use of the asymptotic averaging method for constructing constitutive relations is possible and was done in [13], the method is based on solving special problems of microscopic deformation on periodicity cells (PC).Calculations of macroscopic deformation, for example, by the finite element method, when at each node of the grid it is necessary to solve the problem on a periodicity cell, lead to very large amounts of calculations.
The purpose of this work is to develop a method that will allow us to separate the problems of macro-and microscopic deformation of linear elastic composites without using the asymptotic averaging method.

Model of transversely isotropic elastic -plastic media with small deformations
Since the solution of local problems on the PC requires large amounts of calculations, we will move from the exact solution of these problems in each specific case of loading the PC with a system of averaged stresses to the construction of analytically effective constitutive relations for the composite as a whole.The constants included in these constitutive relations will be found by approximating the deformation diagrams of composites for particular deformation problems, while the deformation diagrams themselves are calculated with a limited number of options for solving problems on the PC of the composite [14,15].
The proposed method is in fact a numerical experiment, in which, instead of finding deformation diagrams experimentally, a numerical solution of problems on the PC is used.In the future, both in a numerical and in a real physical experiment, the most successful analytical version of the constitutive relations is selected, which best describe the maximum number of deformation diagrams in a standard verification set of experiments.
Let us consider an elastic-plastic layered composite material, which we will assume to be a transversally isotropic medium.We accept the basic model about the additivity of elastic and plastic deformations .
e P (1) The elastic strain tensor will be proposed with a tensor density Here E -metric tensor, 2 J in this case have the following form: where 2 ( ) I principal invariants of the tensor [19], O is given in [19].
Substituting these expressions into the formula of the constitutive relations presented in the tensor basis and grouping them by tensor powers, we obtain the representation of the constitutive relations of a transversally isotropic medium in the tensor basis: ˆ, where   G -elastic constants.
For the plastic strain tensor, we accept the associated plastic flow model [19], according to which .Equations (10) determine the position of the plasticity surface.
Then, substituting the derivative tensors ( 5) into ( 9), we obtain the following expression for the plastic strain rate tensor : p : For a transversely isotropic medium, we will assume that there are only 2 plastic potentials 2 k where joint invariants, where 0 0 , H n -constants.This model generalizes the well-known Huber-Mises model for isotropic media.
Calculating the derivatives of f and substituting them into (12), we obtain where From ( 16) we obtain expressions for the loading parameters 1 1 and 2 2 .
Functions f are chosen in a quadratic form similar to the Mises model: The functions 1s are called the yield strengths in longitudinal tension and compression, respectively, and 4 s are the shear yield strengths in the plane of transversal isotropy.The functions 2s are called the yield strength in transverse tension and compression, and 3sthe yield strength in interlayer shear.These functions are usually determined experimentally.For anisotropic media, the difference between the tensile and compressive yield strengths is usually quite significant, so the s and s functions can differ significantly [19].Explicit formulas for invariants (14) have the form

Method for identification of the model parameters for transversally isotropic composites
Using the method of asymptotic averaging for a linearly elastic layered composite material, it is possible to construct constitutive relations based on those for individual layers.However, these relations do not have an explicit analytical expression; they are calculated in the form of a numerical algorithm for solving a local problem in PC.This method is very accurate from a mathematical point of view, but leads to high costs for calculating.Let us consider another method for constructing deformation diagrams, when the constitutive relations are given in the form of explicit analytical relations (9), and the constants 3

F
obtained by direct numerical solution of the problem in PC for some standard problems of macroscopic deformation, in which a homogeneous stress-strain state with ij and ij independent of coordinates is realized.A numerical method for solving a local problem in PL for a linearly elastic layered composite was implemented in [13].
As standard problems of macroscopic deformation, we consider the class of problems of uniaxial tension-compression of a plate in the form of a parallelepiped, the faces of which are parallel to the coordinate axes Oe .The formulation of these problems can be represented as a system with different boundary conditions

Consider the case of tension-compression of the plate along the axis 3
Oe .Under such loading Let us express 33 e in terms of 33 using the system (25) Let us now find the dependence of the plastic strain tensor 33 p on the stress tensor 33 according to the Huber-Mises model (18), in which the cases of plastic tension and plastic compression should be considered separately., obtained as a result of solving local problems by the method of asymptotic homogenization, arising during transverse uniaxial tension-compression of a layered composite material.

Composite interlayer shear
Let us consider the case of interlaminar shift of the plate between the axes ( ) , obtained as a result of solving local problems with the help of AH method, arising from the interlayer shear of the composite.obtained as a result of solving local problems with the help of AH method, arising from shear in the plane of the composite layer.

Formulation of the problem of searching for model parameters
Let us now combine the obtained solutions of four problems: (29), ( 33) and (37).From the solution of the corresponding local problems with the help of AH method, we have several curves.For each specific set of constants , can be calculated by fitting experimental tensile and compressive strain curves.To do this, the problem of minimizing the functional of the standard deviation of the experimental and theoretical curves at N points is solved: To solve minimization problems (38), the Hooke-Jeeves method was used.

Results of numerical simulation of strain diagram of laminated composite
With the help of the developed algorithm, averaged strain diagrams ij ij mn F of a layered composite were calculated and built under uniaxial loading in different directions for tension and compression.Also, using the problem of minimizing the functional (38), the constants 3 1.The calculation was carried out according to formula (29), and the unknown constants were preliminarily determined from the strain curve.
On Fig. 2 shows diagrams of deformation during interlaminar shear σ ε , the values for their constants 13 , G 0 3 , n 0 3 , H 3s are given in Table 2; and in Fig. 3at 3.   The maximum relative error among the obtained diagrams is 14% , which indicates a quite satisfactory quality of the model of an effective transversally isotropic medium.

Conclusions
A model of an effective transversally isotropic linear elastic medium with small deformations, which belongs to the class of universal models, is proposed.The model is applied to deformation diagrams of layered composite materials with small deformations and a periodic structure using a universal representation of constitutive relations for composite layers.
A method for finding the effective constants of the composite model by solving the problem of minimizing the functional of the standard deviation of the experimental deformation diagrams obtained by numerically solving problems on the PC and theoretical deformation diagrams obtained by approximation is proposed.
Numerical modeling of deformation diagrams of layered composite materials with small deformations was performed using the AH method, which showed fairly good results.
the expression for the tensor 43 consider a model in which the elastic potential does not depend on the implementation of the invariants(3then relation (6) takes the form: parameters, h -indicator function that determines active plastic loading 1 h and unloading 0 h

C
-components of the elastic modulus tensor, inverse to the elastic compliance tensor 4 П .

.
Then we look for the solution of this problem in the following form: Dependence of the elastic strain tensor 13 e on the stress tensor 13 : search is carried out by comparison with the experimental deformation diagram ( ) ( ) 13 13 a shift in the plane of the layer σ ε , and their unknown constants 12 ,

Table 1 .
Values of the constants of composite models in uniaxial tension and compression in the transverse direction.

Table 2 .
Values of constants of the composite model at interlayer shear.