Generalized self-adjustment method for statistical mechanics of composite materials

A new method is developed for the statistical mechanics of composite materials — the generalized selfadjustment method — which makes it possible to reduce the problem of predicting effective elastic properties of composites with random structures to the solution of two simpler “averaged” problems of an inclusion with transitional layers in a medium with the desired effective elastic properties. The inhomogeneous elastic properties and dimensions of the transitional layers take into account both the “approximate” order of mutual positioning, and also the variation in the dimensions and elastics properties of inclusions through appropriate special averaged indicator functions of the random structure of the composite. A numerical calculation of averaged indicator functions and effective elastic characteristics is performed by the generalized self-adjustment method for a unidirectional fiberglass on the basis of various models of actual random structures in the plane of isotropy.     

Quasicontinuous approximation in classical statistical mechanics

Within the framework of classical statistical mechanics, we consider infinite continuous systems of point particles with strong superstable interaction. A family of approximate correlation functions is defined to take into account solely the configurations of particles in the space \mathbb R^d {{\mathbb R}^d} that contain at most one particle in each cube of a given partition of the space \mathbb R^d {{\mathbb R}^d} into disjoint hypercubes of volume a ^d : It is shown that the approximations of correlation functions thus defined are pointwise convergent to the exact correlation functions of the system if the parameter of approximation a approaches zero for any positive values of the inverse temperature β and fugacity z: This result is obtained both for two-body interaction potentials and for many-body interaction potentials.     

Fundamentals of the statistical mechanics of composite systems

Conclusions The proposed variant of the statistical theory of composite media makes it possible to derive relations between the effective parameters of the medium and the dispersion characteristics of the structure, and also to account for the effect of specimen shape and variable structural heterogeneity on these parameters. In the limiting case of an infinitely large specimen, all relationships comply with the results of the traditional theory of composite systems.Presented at the Sixth All-Union Conference on the Mechanics of Polymer and Composite Materials (Riga, November, 1986).Translated from Mekhanika Kompozitnykh Materialov, No. 1, pp. 21–30, January–February, 1988.     

Boundary-value problem in the correlative approximation of the method of quasi-periodic components for a unidirectional fiber composite

The boundary-value problem in the correlative approximation of the method of quasi-periodic components and a numerical algorithm based on the boundary element method for determining the nonuniform stress fields in the matrix of a unidirectional fiber composite with a disordered structure are considered. The numerical results and analysis of the probability density function, for example, for normal stresses at some points of the interface of absolutely rigid fibers of the composite are presented. Perm State Technical University, Russia. Translated from Mekhanika Kompozytnykh Materialov, Vol. 35, No. 5, pp. 629–642, September–October, 1999.     

Multiscale computational method for thermoelastic problems of composite materials with orthogonal periodic configurations

This study develops a novel multiscale computational method for thermoelastic problems of composite materials with orthogonal periodic configurations. Firstly, the multiscale asymptotic analysis for these multiscale problems is given successfully, and the formal second-order two-scale approximate solutions for these multiscale problems are constructed based on the above-mentioned analysis. Then, the error estimates for the second-order two-scale (SOTS) solutions are obtained. Furthermore, the corresponding SOTS numerical algorithm based on finite element method (FEM) is brought forward in details. Finally, some numerical examples are presented to verify the feasibility and effectiveness of our multiscale computational method. Moreover, our multiscale computational method can accurately capture the local thermoelastic responses in composite block structure, plate, cylindrical and doubly-curved shallow shells.     

Second-order two-scale method for bending behavior analysis of composite plate with 3-D periodic configuration and its approximation

This paper considers the bending behaviors of composite plate with 3-D periodic configuration.A second-order two-scale(SOTS)computational method is designed by means of construction way.First,by 3-D elastic composite plate model,the cell functions which are defined on the reference cell are constructed.Then the effective homogenization parameters of composites are calculated,and the homogenized plate problem on original domain is defined.Based on the Reissner-Mindlin deformation pattern,the homogenization solution is obtained.And then the SOTS’s approximate solution is obtained by the cell functions and the homogenization solution.Second,the approximation of the SOTS’s solution in energy norm is analyzed and the residual of SOTS’s solution for 3-D original in the pointwise sense is investigated.Finally,the procedure of SOTS’s method is given.A set of numerical results are demonstrated for predicting the effective parameters and the displacement and strains of composite plate.It shows that SOTS’s method can capture the 3-D local behaviors caused by3-D micro-structures well.