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.     

Stochastic Models of Composite Mechanics in the Problems of Designing Structural Elements in Aerospace Engineering

The present-day trends and development prospects of stochastic averaging methods in composite mechanics are analyzed. The following methods are considered: the traditional one (the medium of comparison is homogeneous), the method of periodic components (the medium of comparison has a regular structure), the numerical method of local approximation with regard for the short-range order in the arrangement and interaction of inhomogeneities, and a synthesis of the periodic-component and local-approximation methods. A detailed procedure is presented for calculating the functionals of second, third, and higher orders of the stochastic problem. The results obtained are used for evaluating, in an explicit form, corrections to the effective elastic moduli found for quasi-isotropic and unidirectional fiber composites by the traditional averaging method and the method of periodic components. Analytical formulas for the second-order moment functions of structural stresses and strains are derived. It is shown that the fields of structural stresses and strains are locally ergodic. A new multilevel approach is proposed for designing composites, which takes into account the effect of structurally technological factors. Thus, unidirectional fiber composites are calculated by a two-level model, layered structures - by a three-level model, and carbon-carbon structures - by a five-level model. A stage-by-stage solution procedure is suggested for the boundary-value problem of micromechanics of composites. For a wide class of composites, the effective elastic moduli are calculated and the strength surfaces are constructed.     

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.     

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.     

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 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.     

Spectral problem for an anisotropic elastic waveguide

We consider the homogeneous boundary-value problem of propagation of elastic waves in a hollow anisotropic cylinder with free walls. The problem is solved by a method based on series expansion of traveling-wave amplitudes in powers of the radial coordinate. A dispersion relationship is obtained. The different types of wave motion are analyzed as a function of the degree of anisotropy of the material.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 57, pp. 76–80, 1985.     

Microstress fields and the mechanical properties of disordered fiber composites

Conclusion The effective elastic moduli and Poisson's ratios and the mean characteristics of the stress fields in the components of unidirectional fiber composites with a stochastic structure are nearly the same as the corresponding values calculated for a regular model of the composite. Relatively small increase (up to 6%) is seen in the transverse shear moduli with the transition from a regular structure to a stochastic structure. In the latter, there is a substantial increase in the stress concentration factor. Here, the difference between the stochastic structure and the regular structure increases with an increase in fiber stiffness and is particularly great (with a difference of two to three orders of magnitude) in the case of shear loading. The probability of the occurrence of microscopic fracture in the binder of the investigated materials is higher in transverse tension, but the difference from the results obtained for the regular models is more significant in the case of shear loading. Microscopic fracture nuclei will be formed in the matrix of the composite with the stochastic structure at considerably lower macroscopic stresses than are required for the regular structure.Translated from Mekhanika Kompozitnykh Materialov, No. 5, pp. 860–865, September–October, 1990.     

Use of Green's functions of the Laplace equation in the solution of problems of thermoelasticity of anisotropic bodies

The solution of a boundary-value problem concerning the thermoleasticity of homogeneous anisotropic bodies under several general conditions has been represented by a uniformly converging quadrature series. An example is given.S. M. Kirov Ural Polytechnic Institute, Sverdlovsk. Translated from Mekhanika Polimerov, No. 5, pp. 823–829, Semptember–October, 1973.     

Boundary-value problems for an elliptic equation with complex coefficients and a certain moment problem

Elliptic systems of two second-order equations, which can be written as a single equation with complex coefficients and a homogeneous operator, are studied. The necessary and sufficient conditions for the connection of traces of a solution are obtained for an arbitrary bounded domain with a smooth boundary. These conditions are formulated in the form of a certain moment problem on the boundary of a domain; they are applied to the study of boundary-value problems. In particular, it is shown that the Dirichlet problem and the Neumann problem are solvable only together. In the case where the domain is a disk, the indicated moment problem is solved together with the Dirichlet problem and the Neumann problem. The third boundary-value problem in a disk is also investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 11, pp. 1476–1483, November, 1993.     

On the steady propagation of a semi-infinite crack

We consider the rectilinear propagation of a semi-infinite crack with constant velocity in a crystal structure. We obtain the solutions of homogeneous boundary-value problems for the corresponding difference-differential operators in spaces of one and two dimensions. We give a justification of the computational aspect of the problem. Bibliography: 8 titles.Translated fromProblemy Matematicheskogo Analiza, No. 12, 1992, pp. 127–153.     

Analysis of elastomeric composites based on fiber-reinforced systems. 2. Unidirectional composites

The elastic properties of unidirectionally reinforced composite materials under large deformations are studied. The applied model for deformation of materials is based on the structural macroscopic theory of stiff and soft composites, including micro- and macromechanical levels of analysis of composite media. The properties of unidirectional elastomeric composites are studied in tension and shear in the plane of reinforcement. The microscopic fields in the structural components of composites having poorly compressible and compressible matrices are also analyzed. Changes in the parameters of macroscopic deformation of the composites are examined as functions of the loading parameters and initial conditions of the structure. The evolution of the structural changes in deformed composite materials is described.State Metallurgical Academy of Ukraine, Dnepropetrovsk, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 1, pp. 29–50, January–February, 1999.     

Averaging of electrical properties of anisotropic fiber-metal composites

The electrical properties of fiber-metal composites with anisotropic components were averaged within the framework of a regular structure model. The corresponding current problem was reduced to a system of two integral equations with elliptical kernels. The effective electrical conductivity tensor of such materials was defined as several functionals based on integral equation solutions for the boundary problem. The calculation results are given.Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 2, pp. 263–268, March–April, 1997.     

Numerical solutions of the first boundary-value problem for the telegraph equation on open contours

The first plane initial—boundary-value problem for the telegraph equation is reduced by a Chebyshev—Laguerre temporal integral transform to a sequence of stationary boundary-value problems for elliptic equations. Their solutions are sought in integral form. This leads to a recursive sequence of integral equations of the first kind that are solved by the collocation method with isolation of singularities. The sought function is determined by the inverse transform.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 57–62, 1990.     

Separation of variables as a means of solving the mixed boundary-value problem of viscoelasticity

The quasistatic mixed boundary-value problem is considered for a viscoelastic body with constant Poisson's ratio. A method of reducing viscoelastic problems to the solution of a series of uniform elastic problems is described. The convergence of the method is proved. The method is illustrated with reference to the solution of a specific viscoelastic problem. Tabulated influence functions of the Abel type are used in the solutions.Moscow M. V. Lomonosov State University. Translated from Mekhanika Polimerov, No. 4, pp. 588–596, July–August, 1969.     

Locally one-dimensional difference scheme for the convective diffusion equation

We consider the mixed boundary-value problem for the nonstationary convective diffusion equation in a rectangular region. The summation approximation method is applied to construct a locally homogeneous difference scheme with O(t ^1/2 + h^3/2) rate of convergence in the L₂ grid metric.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 74, pp. 64–67, 1992;     

Numerical solution of multipoint problems in statics of shells

We purpose an approach to solving multipoint boundary-value problems for a system of ordinary differential equations in the theory of shells. The technique is based on reduction of the original problem to several two-point boundary-value problems, which are solved by a stable numerical method. Examples of calculation of variable-thickness cylindrical shells are given.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 64, 58–65, 1988.     

Predicting the transverse strength of unidirectional composites under combination loading

This article presents a mathematical model for predicting the transverse strength of unidirectional fiber composites subjected to combination transverse loading under different conditions. The behavior of the matrix is described by nonlinear physical equations consistent with the strain theory of plasticity for the active loading section. The fibers are assumed to be isotropic and elastic. The boundary-value problem of micromechanics that is formulated includes strength criteria for the matrix and fibers that mark the beginning of their possible failure. The modeling of the fracture process is taken farther through the use of a scheme that reduces the stiffness of the matrix and fibers in the failed regions in relation to the sign of the first invariant of the stress tensor. The method of local approximation is used together with the finite-element method to calculate the stress and strain fields in unidirectional composites with cylindrical fibers in a tetragonal layup. The model is used to study the behavior of an epoxy-based organic-fiber-reinforced plastic subjected to transverse loading in different simple paths — including simultaneous compressive and tensile loads, as well as transverse shear.Paper to be presented at the Ninth International Conference on the Mechanics of Composite Materials (Riga, October 1995).Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 4, pp. 473–481, July–August, 1995.     

What Is a Quantum Stochastic Differential Equation from the Point of View of Functional Analysis?

We prove that a quantum stochastic differential equation is the interaction representation of the Cauchy problem for the Schrödinger equation with Hamiltonian given by a certain operator restricted by a boundary condition. If the deficiency index of the boundary-value problem is trivial, then the corresponding quantum stochastic differential equation has a unique unitary solution. Therefore, by the deficiency index of a quantum stochastic differential equation we mean the deficiency index of the related symmetric boundary-value problem.In this paper, conditions sufficient for the essential self-adjointness of the symmetric boundary-value problem are obtained. These conditions are closely related to nonexplosion conditions for the pair of master Markov equations that we canonically assign to the quantum stochastic differential equation.     

Electroviscoelastic oscillations of compound rod-shaped piezoelectric transducers

An analytical method for calculating rod-shaped piezoelectric transducers, consisting of N components, is proposed. Each component is polarized in the axial direction, and neighboring components are oppositely polarized. Electrodes are deposited on the interface of the components and at the ends, and a fixed electric potential is applied to them. The approach is based on reducing the starting boundary-value problem with the help of transfer matrices to a system of algebraic equations. The solution of the system in the case of identical components is found with the help of Chebyshev polynomials of the second kind. The characteristic distributions of the displacements and potential along transducers consisting of five and ten components at their first resonance frequencies are presented.Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 66–71, 1990.