Deformation of composites with arbitrarily oriented orthotropic fibers under matrix microdamages

In the present work, a model of nonlinear deformation of stochastic composites under microdamaging is developed for the case of a composite with orthotropic inclusions, when microdefects are accumulated in the matrix. The composite is treated as an isotropic matrix strengthened by triaxial arbitrarily oriented ellipsoidal inclusions with orthotropic symmetry of the elastic properties. It is assumed that the process of loading leads to accumulation of damage in the matrix. Fractured microvolumes are modeled by a system of randomly distributed quasispherical pores. The porosity balance equation and relations for determining the effective elastic modules in the case of orthotropic components are taken as basic relations. The fracture criterion is specified as the limiting value of the intensity of average shear stresses acting in the intact part of the material. On the basis of the analytic and numerical approach, we propose an algorithm for the determination of nonlinear deformation properties of the investigated material. The nonlinearity of composite deformations is caused by the finiteness of deformations. By using the numerical solution, the nonlinear stress–strain diagrams are predicted and discussed for an orthotropic composite material for various cases of orientation of inclusions in the matrix.     

Homogenization of Unidirectional and Arbitrarily Oriented Fiber-Reinforced Materials by the Method of Conditional Moments

In this paper the method of conditional moments is developed for the case of a two–component matrix composite with randomly distributed unidirectional and arbitrarily oriented ellipsoidal inclusions. The algorithm for determination of the effective elastic properties of composites from the given elastic constants of the components and geometrical parameters and orientation of inclusions is discussed. It is assumed that the components of the composite show orthotropic symmetry of thermoelastic properties. As a numerical example arbolite (straw particle inclusions in a cement matrix) is considered. The dependencies of Young's moduli, Poisson's ratios and shear moduli from the concentration of inclusions and for certain orientations of the inclusions are predicted and discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Thermally Stressed State of Bodies with Two Ellipsoidal Inclusions

Solutions are obtained for the interaction of two ellipsoidal inclusions in an elastic isotropic matrix with polynomial external athermal and temperature fields. Perfect mechanical and temperature contact is assumed at the phase interface. A solution to the problem is constructed. When the perturbations in the temperature field and stresses in the matrix owing to one inclusion are re-expanded in a Taylor series about the center of the second inclusion, and vice versa, and a finite number of expansion terms is retained, one obtains a finite system of linear algebraic equations in the unknown constants. The effect of a force free boundary of the half space on the stressed state of a material with a triaxial ellipsoidal inhomogeneity (inclusion) is investigated for uniform heating. Here it was assumed that the elastic properties of the inclusions and matrix are the same, but the coefficients of thermal expansion of the phases differ. Studies are made of the way the stress perturbations in the matrix increase and the of the deviation from a uniform stressed state inside an inclusion as it approaches the force free boundary.     

Deformation of orthotropic composites with unidirectional ellipsoidal inclusions under matrix microdamages

In the present paper, a model of deformation of stochastic composites under microdamaging is developed for the case of orthotropic composite, when the microdamages are accumulated in the matrix. The composite is treated as an isotropic matrix strengthened by three-axial ellipsoidal inclusions with orthotropic symmetry of elastic properties. It is assumed that the loading process leads to accumulation of damages in the matrix. Fractured microvolumes are modeled by a system of randomly distributed quasispherical pores. The porosity balance equation and relations for determining the effective elastic moduli for the case of a composite with orthotropic components are taken as the basic relations. The fracture criterion is assumed to be given as the limit value of the intensity of average shear stresses occurring in the undamaged part of the material. Based on the analytical and numerical approach, an algorithm for the determination of nonlinear deformation properties of such a material is constructed. The nonlinearity of composite deformations is caused by the accumulation of microdamages in the matrix. Using the numerical solution, nonlinear stress-strain diagrams for the orthotropic composite in the case of biaxial extension are obtained. Published in Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 121–130, January–March, 2008.     

Deformation and Damaging of Composites with Transversally-isotropic Components under Compressive Loading

In the present paper, the problem of deformation and damage of composites with a porous isotropic matrix and transversally-isotropic unidirectional fibers under compressive loading is considered when microdamages are accumulated in the fiber. Fractured micro-volumes are modelled by a system of randomly distributed quasi spherical pores. The Shleicher-Nadai fracture criterion is used as a condition for the origin of micro-pores (micro-damage) based on the assumption of a rigid material. The limit value of the strength of the material is assumed as a stochastic function of coordinates. By using a numerical procedure, the solution of the above problem is found. The nonlinear stress-strain diagrams for a transversally-isotropic composite are obtained for the case of uniaxial compression-tension along the fibers. The nonlinearity of the deformations of the composite is caused by accumulation of micro-damages in the matrix. The influence of the physical-mechanical properties of materials, of the volume concentration, of the porosity of components, of the geometrical parameters of the structure, and of the character of the scatter of the strength in the material on the micro-damage of the material and, as a consequence, the influence on the macro-stress-macro-strain diagram is analyzed. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Non-Linear Ordinary Differential Equations of Boundary Layer Type

A numerical method previously applied to linear two-point boundary value problems of boundary layer type is extended to some non-linear problems. Discretization of the differential equation leads to a set of non-linear algebraic equations, which is solved by a modified Newton's method; both the mesh spacing and the boundary layer parameter are iteratively adjusted during the solution process. Several examples are discussed; one of these concerns the problem of shock wave formation in a supersonic nozzle.     

Effective bulk moduli of materials containing stochastically distributed nano-inhomogeneities with surface stresses

In the present contribution, a mathematical model for the investigation of the effective properties of a material with randomly distributed nano-particles is proposed. The surface effect is introduced via Gurtin-Murdoch equations describing properties of the matrix/nano-particle interface. They are added to the system of stochastic differential equations formulated within the framework of linear elasticity. The homogenization problem is reduced to finding a statistically averaged solution of the system of stochastic differential equations. These equations are based on the fundamental equations of linear elasticity, which are coupled with surface/interface elasticity accounting for the presence of surface tension. Using Green's function this system is transformed to a system of statistically non-linear integral equations. It is solved by the method of conditional moments. Closed-form expressions are derived for the effective moduli of a composite consisting of a matrix with randomly distributed spherical inhomogeneities. The radius of the nano-particles is included in the expression for the bulk moduli. As numerical examples, nano-porous aluminum and nano-porous gold are investigated assuming that only the influence of the interface effects on the effective bulk modulus is of interest. The dependence of the normalized bulk moduli of nano-porous aluminum on the pore volume fraction (for certain radii of nano-pores) are compared to and discussed in the context of other theoretical predictions. The effective Young's modulus of nano-porous gold as a function of pore radius (for fixed void volume fraction) and the normalized Young's modulus vs. the pore volume fraction for different pore radii are analyzed. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Implementation of sparse matrix algorithms in an advection-diffusion-chemistry module

A two-dimensional advection-diffusion-chemistry module of a large-scale environmental model is taken. The module is described mathematically by a system of partial differential equations. Sequential splitting is used in the numerical treatment. The non-linear chemistry is most time-consuming part and it is handled by six implicit algorithms for solving ordinary differential equations. This leads to the solution of very long sequences of systems of linear algebraic equations. It is crucial to solve these systems efficiently. This is achieved by applying four different algorithms. The numerical results indicate that the algorithms based on a preconditioned sparse matrix technique and on a specially designed algorithm for the particular problem under consideration perform best.     

New exact multi-coated ellipsoidal inclusion model for anisotropic thermal conductivity of composite materials

The present study deals with a new micromechanical modeling of the thermal conductivity of multi-coated inclusion-reinforced composites. The proposed approach has been developed in the general frame of anisotropic thermal behavior per phase and arbitrary ellipsoidal inclusions. Based on the Green's function technique, a new formulation of the problem of multi-coated inclusion is proposed. This formulation consists in constructing a system of integral equations, each associated to the thermal conductivity of each coating and the reference medium. Thanks to the concept of interior- and exterior-point Eshelby's conduction tensors, the exact solution of the problem of multicoated inclusion is obtained. Analytical expressions of the intensity in each phase and the effective thermal conductivity of the composite, through homogenizations schemes such as Generalized self-consistent and Mori-Tanaka models are provided. Results of the present model are successfully compared with those issued from both analytical models and finite elements methods for composites with doubly coated inclusions. Moreover, the developed micromechanical model has been applied to a three phase composite materials in order to analyze combined effects of the aspect ratio and the volume fraction of the ellipsoidal inclusions, the anisotropy of the thermal conductivity of interphase, the thermal conductivity contrast between local phases on the predicted effective thermal conductivity.     

Quasi-Newton Methods for Discretized Non-linear Boundary Problems

The discretization of non-linear boundary problems generallyleads to a finite system of non-linear algebraic equations,and it is to be expected that this latter has special structurearising both from the boundary problem and the method of discretizationused. The numerical solution of the algebraic system representsa serious numerical problem, and it is the point of this paperto indicate that, in certain important cases, special purposequasi-Newton methods can be constructed. We illustrate by consideringa single nonlinear differential equation discretized by collocationand present experimental results which indicate that an improvementin performance can be expected from the special methods.     

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.     

Microstructure inducted stress concentrations in CVI-CFC materials

Carbon/carbon materials produced by chemical vapour infiltration consist of carbon fibers embedded in an anisotropic matrix of pyrocarbon. In this study we propose: 1) an approach for hierarchical homogenization of material parameters; 2) an approach for prediction of the stress concentrations in this material. The model estimates material properties on two scales: nano and micro scale. The microstructural morphology of CFC-material on the nano scale can be represented as distribution of mono crystals of pyrographite. For modeling the response at this scale we utilize the Eshelbi's theory for continuously distributed inclusions. The orientations distribution functions of inclusions (mono crystals) are used for calculation of the homogenized elasticity tensor of pyrographite. The numerical calculations of the stress fields in the samples characterized by different types of pyrocarbon coatings provides us the information about the (failure) regions with maximal (critical) values of stress. The obtained results demonstrate a good coincidence with experimentally identified failure regions. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical computation of the Tau approximation for the Volterra-Hammerstein integral equations

In this work, we propose an extension of the algebraic formulation of the Tau method for the numerical solution of the nonlinear Volterra-Hammerstein integral equations. This extension is based on the operational Tau method with arbitrary polynomial basis functions for constructing the algebraic equivalent representation of the problem. This representation is an special semi lower triangular system whose solution gives the components of the vector solution. We will show that the operational Tau matrix representation for the integration of the nonlinear function can be represented by an upper triangular Toeplitz matrix. Finally, numerical results are included to demonstrate the validity and applicability of the method and some comparisons are made with existing results. Our numerical experiments show that the accuracy of the Tau approximate solution is independent of the selection of the basis functions.     

一种考虑孔洞形状变化的损伤细观模型及其在孔洞闭合过程中的应用

本文在一种特殊的坐标系下,建立了非线性的基体材料,有限大的椭球体中含椭球形孔洞的损伤细观模型,考虑了孔洞形状的影响.得出的粘性约束方程(或称屈服面方程)除应力∑_ij,孔隙度f,幂硬化指数m外,还与孔洞的形状有关.通过曲线拟合的方法,对Gurson方程进行了修正,使之适合于非线性的基体材料、变形状孔洞的情形.最后将此模型用于分析非线性材料内部孔洞的闭合过程.     

Optimization of a patch

We study optimal patterns of a patch made of an elastic anisotropic homogeneous material for covering a hole in a two-dimensional body possessing different physical characteristics. In addition to the optimization problem for inclusions in two-dimensional and three-dimensional elastic and piezoelectric bodies, we also consider similar problems for an arbitrary formally selfadjoint elliptic system of differential equations in multidimensional domains. A condition for the stationarity of the energy functional is obtained; for a free parameter the matrix of orthogonal transformations of the Euclidean space is taken; the result is based on an algebraic fact about small increments of orthogonal and unitary matrices. Bibliography: 23 titles. Illustrations: 1 figure.     

Investigation of the Stress-Strain State of Viscoelastic Piecewise-Homogeneous Bodies by the Method of Boundary Integral Equations

The problem on the stress state of a viscoelastic half-plane containing a finite number of inclusions of arbitrary shape and subjected to the action of distributed tangential and normal loads on its boundary is considered. Integral representations for the displacement vector and stress tensor are obtained for the case of an ideal mechanical contact on the conjugation contour of the regions. Discrete analogues of the boundary-temporal integral equations are constructed with account for the singularities of the stress field near the corner points. A numerical calculation is performed and the mechanical effects for an epoxy matrix with metal inclusions are analyzed.     

小垂度粘弹性索非线性响应及振动主动控制

研究具有初始应力的小垂度粘弹性索的非线性动态响应及振动主动控制。在假定索材料的本构关系为一般微分本构类型的基础上,建立小垂度粘弹性索的运动微分方程;应用Galerkin方法将其转化为可用Runge-Kutta数值积分方法求解的一系列三阶非线性常微分方程。在仅考虑面内的横向振动及忽略非线性的情况下得到了连续状态空间中的状态方程,将状态方程离散为差分方程形式,并用矩阵指数来逐步近似状态转移矩阵;基于二次性能指标的最小化得到了最优的控制力与状态向量。最后通过数值仿真研究说明了粘性参数对索动态响应的影响。     

An exponential matrix method for solving systems of linear differential equations

This paper presents an exponential matrix method for the solutions of systems of high‐order linear differential equations with variable coefficients. The problem is considered with the mixed conditions. On the basis of the method, the matrix forms of exponential functions and their derivatives are constructed, and then by substituting the collocation points into the matrix forms, the fundamental matrix equation is formed. This matrix equation corresponds to a system of linear algebraic equations. By solving this system, the unknown coefficients are determined and thus the approximate solutions are obtained. Also, an error estimation based on the residual functions is presented for the method. The approximate solutions are improved by using this error estimation. To demonstrate the efficiency of the method, some numerical examples are given and the comparisons are made with the results of other methods. Copyright © 2012 John Wiley & Sons, Ltd.     

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.