Generalized self-consistent method: Modeling and computation of effective elastic properties of composites with composite or hollow inclusions

A new approach to the generalized self-consistent method [1,2] has been developed for problems of the statistical mechanics of composites with composite or hollow inclusions. The approach can reduce the problem of predicting the effective elastic properties of composites to a simpler averaged problem of a single, composite, or hollow inclusion with inhomogeneous elastic surrounding in a homogeneous effective elastic medium. The problem of predicting the effective elastic properties of composites with unidirectional hollow fibers or hollow spherical inclusions are studied by using the new approach.Submitted to the 10th International Conference on Mechanics of Composite Materials, April 20–23, 1998, Riga, Latvia.Perm' State Technical University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 2, pp. 173–183, March–April, 1998.     

A generalized self-consistent method for composites with random elastic properties of inclusions

The generalized self-consistent method is extended to the problems of statistical mechanics of composites with random elastic properties of inclusions. This approach makes it possible to reduce the problem of predicting the effective elastic properties of composites with random structures to a sequence of simpler homogenized boundary-value problems for solitary inclusions with inhomogeneous elastic transition layers in a homogeneous effective elastic medium and with the corresponding boundary conditions. The elastic properties of a solitary inclusion for the gth homogenized problem are found from the solutions of the gth and (g+1)th homogenized problems. The elastic properties and sizes of the transition layers account for the random distribution, random sizes, and random elastic properties of inclusions in the composite. A test problem of predicting the effective elastic properties of a transversely isotropic layer composite with random elastic properties of some layers is solved by using the method proposed. The solution obtained coincides with the known exact solution [1].Perm State Technical University, Perm, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 6, pp. 785–796, November–December, 1999.     

Shear of orthogonally reinforced spherofibrous composites

The problem of determining the shear characteristics and interphase stress concentration of fibrous composites with spherical inclusions is examined on the basis of a three-phase model. Stress fields caused by diffusion interaction of phases are neglected. The elastic moduli of the composite are investigated and compared with those obtained from a two-phase model. The general formula for determination of the shear modulus of triorthogonally reinforced compsites is derived using previously investigated relationships for averaged stress fields. The matrix of these compsites contained spherical cavities. The dependence of integral characteristics of three-phase composites on their bulk phase concentration was investigated. The stresses between phases were studied as a function of composite structure.A. A. Blagonravov Machine-Science Institute, Russian Academy of Sciences, Moscow, Russia. Translated from Mekhanika Kompozitnykh Materialov, No. 1, 104–111, January–February, 1997.     

Generalized self-consistent method for predicting the effective elastic properties of composites with random hybrid structures

The feasibility of using a generalized self-consistent method for predicting the effective elastic properties of composites with random hybrid structures has been examined. Using this method, the problem is reduced to solution of simpler special averaged problems for composites with single inclusions and corresponding transition layers in the medium examined. The dimensions of the transition layers are defined by correlation radii of the composite random structure of the composite, while the heterogeneous elastic properties of the transition layers take account of the probabilities for variation of the size and configuration of the inclusions using averaged special indicator functions. Results are given for a numerical calculation of the averaged indicator functions and analysis of the effect of the micropores in the matrix-fiber interface region on the effective elastic properties of unidirectional fiberglass—epoxy using the generalized self-consistent method and compared with experimental data and reported solutions.Perm State Technical University. Translated from Mekhanika Kompozitmykh Materialov, Vol. 33, No. 3, pp. 289–299, May–June, 1997.     

Thermoelastic expansion of spherical fiber composites

A refined solution is constructed for thermoelastic expansion of spherical fiber composites with a three-dimensional structure on the basis of existing hypotheses about the longitudinal state of fibers in a matrix strengthened with spherical inclusions. Relationships defining the dependence in explicit form of thermoelastic coefficients on structural parameters are obtained in analytical form. Thermal expansion coefficients for composites with cubic symmetry are discussed in detail.Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 2, pp. 251–257, March–April, 1997.     

Solution of the stochastic boundary-value problem of elasticity theory for composites with disordered structures in the correlative approximation of the method of quasi-periodic components

The method of quasi-periodic components, a new method of statistical mechanics of composites, is presented. In correlative approximation, this method makes it possible to reduce the problem of solving the stochastic boundary-value problem of elasticity theory for composites with disordered structures to a simpler problem for an individual cell with one inclusion in a homogeneous elastic medium. The generalized volumetric forces on the cell boundary take into account the random distribution of inclusions in the composite fragment studied. The problem for one inclusion cell can be solved, for example, by the boundary element method. The numerical solution in the correlative approximation of the method of quasi-periodic components for inhomogeneous interphase stress fields in the matrix of an epoxy composite containing randomly distributed unidirectional fibers is given. A comparison with the known analytical solutions obtained by other authors confirms the high accuracy of the correlative approximation.Perm' State Technical University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 4, pp. 465–478, July–August, 1999.     

Virial expansions in problems of effective characteristics. 2. Antiplanar deformation of a fiber composite. analysis of self-consistent methods

Conclusion The virial expansion for the effective shear modulus (condition of antiplanar deformation) was constructed for an isotropic material with parallel cylindrical inclusions and the terms which are quadratic with respect to the concentration of inclusions were precisely calculated. A comparison of the results obtained with the results found with self-consistent methods showed that the differential (step) method gives a precise solution in the case of inclusions which strongly differ in size and a relatively small error (under 25% in the quadratic term) in the case of identical inclusions. The algebraic method and Lorentz method give more significant errors (up to 100 and 50%) in the quadratic term, respectively.See [1] for Communication 1.Translated from Mekhanika Kompozitnykh Materialov, Vol. 30, No. 2, pp. 329–342, March–April, 1994.     

Non-linear deformation properties of materials with stochastically distributed anisotropic inclusions

In the present contribution, the problem of non-linear deformation of materials with stochastically distributed anisotropic inclusions is considered on the basis of the methods of mechanics of stochastically non-homogeneous media. The homogenization model of materials of stochastic structure with physically non-linear components is developed for the case of a matrix which is strengthened by unidirectional ellipsoidal inclusions. It is assumed that the matrix is isotropic, deforms non-linearly; inclusions are linear-elastic and have transversally-isotropic symmetry of physical and mechanical properties. Stochastic differential equations of physically non-linear elasticity theory form the underlying equations. Transformation of these equations into integral equations by using the Green's function and application of the method of conditional moments allow us to reduce the problem to a system of non-linear algebraic equations. This system of non-linear algebraic equations is solved by the Newton-Raphson method. On the analytical as well as the numerical basis, the algorithm for determination of the non-linear effective characteristics of such a material is introduced. The non-linear behavior of such a material is caused by the non-linear matrix deformations. On the basis of the numerical solution, the dependences of homogenized Poisson's coefficients on macro-strains and the non-linear stress-strain diagrams for a material with randomly distributed unidirectional ellipsoidal pores are predicted and discussed for different volume fractions of pores. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Singular approximation of the method of periodic components in statistical mechanics of composite materials

A quasi-periodic model is developed for random structures of composites, when the locations of inclusions are given in terms of random deviations from nodes of an ideal periodic lattice. Solution of the stochastic boundary problem of the theory of elasticity is examined for a quasi-periodic component by the method of periodic components, which is reduced to determination of the field of deviations from the known solution for a corresponding periodic composite. The solution is presented for the tensor of effective elastic properties of a quasi-periodic composite in singular approximation of the method of periodic components in terms of familiar solutions for tensors of the effective elastic properties of composites with periodic and chaotic structures and the parameters of the quasi-periodic structure: the coefficient of periodicity and the tensor of the anisotropy of inclusion disorder. A numerical calculation is performed for the effective transversally isotropic elastic properties of unidirectional fibrous composites with different degrees of fiber disorder.Perm' State Technical University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 460–473, July–August, 1997.     

On the approximation of solutions of stochastic differential inclusions

A sequence of approximating equations is constructed for stochastic differential inclusions, and the properties of the measures corresponding to solutions of the approximating equations are studied for the class of stochastic differential inclusions.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 43–48, 1986.     

Vibrations of a semicircular membrane containing thin rigid inclusions

This article examines problems concerning steady-state vibrations of a semicircular membrane containing thin rigid inclusions of different configurations. The generalized method of integral transforms is used to formulate the problem in the form of a system of singular integral equations in each specific case. With the use of the asymptote of the sought functions as a basis, these equations are solved approximately by the method of orthogonal polynomials. A study is made of the validity of using the reduction method to approximately solve the infinite linear algebraic matrix system which is obtained. The results of calculations are analyzed.Translated from Dinamicheskie Sistemy, No. 5, pp. 49–55, 1986.     

Thermal stresses in a granular composite of regular structure

The problem is considered of thermal stresses in a composite material which is an elastic isotropic matrix with uniform spherical inclusions placed regularly within it. A solution is presented in the form of series for a system of double-periodic solutions of the equilibrium equation built up in a special way. An infinite set of linear algebraic equations of the normal type is obtained from the boundary conditions. Numerical studies are carried out for the stress distribution at the matrix-grain contact surface, and their nature in relation to the volume content of dispersed phase and geometric structure parameters of the composite is determined.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 90–93, 1988.     

Nonstationary monotone iterative methods for nonlinear partial differential equations

A simple technique is given in this paper for the construction and analysis of monotone iterative methods for a class of nonlinear partial differential equations. With the help of the special nonlinear property we can construct nonstationary parameters which can speed up the iterative process in solving the nonlinear system. Picard, Gauss–Seidel, and Jacobi monotone iterative methods are presented and analyzed for the adaptive solutions. The adaptive meshes are generated by the 1-irregular mesh refinement scheme which together with the M-matrix of the finite element stiffness matrix lead to existence–uniqueness–comparison theorems with simple upper and lower solutions as initial iterates. Some numerical examples, including a test problem with known analytical solution, are presented to demonstrate the accuracy and efficiency of the adaptive and monotone properties. Numerical results of simulations on a MOSFET with the gate length down to 34 nm are also given.     

Analysis of elastomeric composites based on fiber-reinforced systems. 1. Development of design methods for composite materials

A method for calculating the elastic properties of fiber-reinforced composites is discussed. The method is based on the structural macroscopic theory for reinforced media [1, 2], which can be used for analysis of stiff and soft composites. As a measure of the elastic properties of composites, the parameters of macroscopic deformations of the base system of Cartesian coordinates are used, with the axes oriented in a certain direction relative to the general reinforcement and loading field. The corresponding macrostresses in the loaded composites are found by a solution of the microboundary problem for a composite macroelement with sides parallel to reinforcement planes of the system. The microboundary-value problem is multiply connected and is formulated based on the information about the homogeneous field of macroscopic displacements specified by the parameters of macroscopic deformation. The problem is solved using the local system of coordinates whose axes are directed along some of the reinforcement trajectories.State Metallurgical Academy of Ukraine, Dniepropetrovsk, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 6, pp. 733–745, November–December, 1998.     

Generalized Bezoutian and the inversion problem for block matrices,I. General scheme

The unified approach to the matrix inversion problem initiated in this work is based on the concept of the generalized Bezoutian for several matrix polynomials introduced earlier by the authors. The inverse X^–1 of a given block matrix X is shown to generate a set of matrix polynomials satisfying certain conditions and such that X^–1 coincides with the Bezoutian associated with that set. Thus the inversion of X is reduced to determining the underlying set of polynomials. This approach provides a fruitful tool for obtaining new results as well as an adequate interpretation of the known ones.     

The riemann boundary problem with a complex orthogonal matrix

The Riemann boundary problem is studied under the assumption that the coefficient of the problem is a complex orthogonal matrix. In this case a property of the partial indices of the problem is established together with certain properties of the canonical matrices, which are then used to construct the canonical matrix of a complex orthogonal matrix of second order.Translated from Matematicheskie Zametki, Vol. 23, No. 3, pp. 405–416, March, 1978.The authors thank É. I. Zverovich, G. S. Litvinchuk, and I. M. Spitkovskii for helpful advice and remarks.     

Nonlinear Deformational Properties of Dispersely Strengthened Materials

A method and an algorithm for determining the effective deformational properties of dispersely strengthened materials with a physically nonlinear matrix and quasi-spheroidal linearly elastic inclusions are elaborated based on the stochastic differential equations of the physically nonlinear theory of elasticity. Their transformation to integral equations and the application of the method of conditional moments reduce the problem to a system of nonlinear algebraic equations, whose solution is constructed by the iteration method. The deformation diagrams as functions of the volume content of inclusions are investigated.     

Scalar and matrix Riemann–Hilbert approach to the strong asymptotics of Padé approximants and complex orthogonal polynomials with varying weight

We describe methods for the derivation of strong asymptotics for the denominator polynomials and the remainder of Padé approximants for a Markov function with a complex and varying weight. Two approaches, both based on a Riemann–Hilbert problem, are presented. The first method uses a scalar Riemann–Hilbert boundary value problem on a two-sheeted Riemann surface, the second approach uses a matrix Riemann–Hilbert problem. The result for a varying weight is not with the most general conditions possible, but the loss of generality is compensated by an easier and transparent proof.     

Modal analysis of piezoelectric bodies with voids. II. Finite element simulation

The paper is concerned with the eigenvalue problems for piezoelectric bodies with voids in contact with massive rigid plane punches and coved by the system of open-circuited and short-circuited electrodes. The linear theory of piezoelectric materials with voids for porosity change properties according to Cowin–Nunziato model is used. The generalized statements for eigenvalue problem are obtained in the extended and reduced forms. A variational principle is constructed which has the properties of minimality, similar to the well-known variational principle for problems with pure elastic media. The discreteness of the spectrum and completeness of the eigenfunctions are proved. The orthogonality relations for eigenvectors are obtained in different forms. As a consequence of variational principles, the properties of an increase or a decrease in the natural frequencies, when the mechanical, electric and “porous” boundary conditions and the moduli of piezoelectric solid with voids change, are established.     

Solution of the nonlinear seepage problem with weakly permeable inclusions

An algorithm for numerical solution of the problem is proposed. Calculations performed by the corresponding programs are reported. The effect of inclusions on seepage characteristics is analyzed.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 92–99, 1988.