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.     

Stressed state of an anisotropic body with elliptical holes, elastic inclusions, and cracks

A method is proposed for studying the two-dimensional stressed state of a multiply connected anisotropic body with cavities and elastic and rigid inclusions, as well as planar cracks and rigid laminar inclusions. Generalized complex potentials, conformal mapping, and the method of least squares are used. The problem is reduced to solving a system of linear algebraic equations. Formulas are given for finding the stress intensity factors in the case of cracks and laminar inclusions. For an anisotropic plate with a single elliptical hole or a crack and an elastic (rigid) inclusion, some numerical results are presented from a study of the effect of the rigidity of the inclusion and the closeness of the contours to one another on the distribution of stresses and the stress intensity factor. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 175–187, 1999.     

Influence of inclusion on the stress intensity factors under thermomechanical loads in a PM superalloy

The interaction between a round inclusion and a crack under thermomechanical loading is analyzed based on a modified body force method. The traction-free condition on the crack line is mended by adding the resultant force induced by thermal stress to the force equilibrium equations, so that the coupling of mechanical and thermal loads could be taken into account. The series of integral equations can be discretized to a set of linear equations. Stress intensity factors (SIFs) are obtained through solving the linear equations. The calculated results in this paper are compared to those in open references to validate the method and code. The method is applied to a case of FGH95 PM superalloy containing Al₂O₃ inclusions under mechanical and thermal loads. The results show that the thermal load has little effect on SIF, while the mechanical load is the dominant factor.     

The equivalent inclusion technique and its convergence

This paper addresses the convergence characteristics of an iterative solution scheme of the Neumann‐type useful for obtaining homogenized mechanical material properties within an RVE. The analysis is based on the idea of “equivalent inclusions” and, within the context of stress/strain analysis, allows modeling of elastically highly heterogeneous bodies with the aid of discrete Fourier transforms. Within the iterative scheme the proof of convergence depends critically upon the choice of an appropriate, auxiliary stiffness matrix, which also determines the speed of convergence. Mathematically speaking it is based on Banach's fixpoint theorem and only results in necessary convergence conditions. However, for all cases of elastic heterogeneity that are of practical importance convergence can be demonstrated.     

Method of Riemann surfaces for an inverse antiplane problem in an n-connected domain

ABSTRACT

An inverse problem of the theory of harmonic functions for an n-connected domain is analyzed. The problem is equivalent to a problem of antiplane elasticity on determination of the profiles of n uniformly stressed inclusions. The inclusions are in ideal contact with the surrounding matrix, the stress field inside the inclusions is uniform, and at infinity the body is subjected to antiplane uniform shear. The exterior of the inclusions, an n-connected domain, is treated as the image by a conformal map of an n-connected slit domain with the slits lying in the same line. The inverse problem is solved by quadratures by reducing it to two Riemann-Hilbert problems on a Riemann surface of genus n?1. Samples of two and three symmetric and non-symmetric uniformly stressed inclusions are reported.     

Redistribution of finite elastic strains after the formation of inclusions. Approximate analytical solution

A class of two-dimensional static problems of the stress-strain state of non-linearly elastic bodies, in which domains with different elastic properties (inclusions) arise after preloading, is considered. Problems are formulated and solved using the theory of the repeated superposition of finite strains. The mechanical properties of the initial material and the material of the inclusions are described by Murnaghan-type or Mooney-type constitutive relations. Two ways of specifying the constitutive relations for the material of an inclusion are considered: when there are inherent strains in this material and when there are not. Approximate analytical methods are used for the solution.     

Boundary element methods in problems of thermoelasticity for piecewise-homogeneous bodies

Integral representations of the components of the displacement vector and the stress tensor and the corresponding system of boundary integral equations are derived for a piecewise-homogeneous body. A numerical scheme is developed that allows for the specific behavior of the stress fields in the neighborhood of the corner points of the inclusions. As an example, we consider the thermally stressed state of an elastic half-plane with inclusions of various shapes.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 75, pp. 87–98, 1991.     

Determination of the viscoelastic state of an anisotropic plate with rigid inclusions

A method for the determination of the viscoelastic state of multiconnected anisotropic plates with absolutely rigid inclusions is proposed. As an example, we consider the solution of the problem of viscoelasticity for a plate with an elliptic inclusion which, in a special case, turns into a linear one. For the case of tension of the plate, results of numerical investigations of the stress state depending on geometric parameters of the inclusion and on the time of application of load are described.     

Quadruple Trigonometrical Series Equations and Their Application to an Inclusion Problem in the Theory of Elasticity

Closed form solution of quadruple series equations involving cosine kernels has been obtained by reducing the series equations into triple Abel's type integral equations which in turn are reduced to a single integral equation. Making use of finite Hilbert transforms the solution of the single integral equation is obtained in closed form. This solution is used to solve an electrostatic problem. The results of this paper have also been used in a two-dimensional elastostatic problem under anti-plane shear and the effect of rigid line inclusions with thickness on the Griffith cracks has been examined. The expressions for shear stress and stress intensity factor at the tip of the crack are obtained. Finally, some numerical results for the stress intensity factor and shear stress distribution are obtained.     

Collinear periodic cracks and/or rigid line inclusions ofantiplane sliding mode in one-dimensional hexagonal quasicrystal

For a one-dimensional (1D) hexagonal quasicrystal (QC), there is the periodic (x₁,x₂)-plane of atomic structures with the quasiperiodic direction x₃-axis along which there exists a phason displacement. The macroscopically collinear periodic cracks and/or rigid line inclusions are placed on the periodic (x₁,x₂)-plane for finding out the influence of phason displacement on the related physical quantities. These two models are reduced to the Riemann–Hilbert problem of periodic analytic functions to obtain the closed-form solutions for the antiplane sliding mode. It is found that the phonon and phason stress intensity factors of cracks as well as the phonon and phason stress field intensity factors of rigid line inclusions are not related to the coupling of phonon and phason fields. These mean that there is not the influence of phason displacement on both the phonon stress intensity factor (usual stress intensity factor) of cracks and the phonon stress field intensity factor of rigid line inclusions. However, the energy release rates of periodic cracks and/or rigid line inclusions are obtained and affected not only by the periodicity of cracks and/or rigid line inclusions but also by the phason displacement.     

Calculation of deformations in nanocomposites using the block multipole method with the analytical-numerical account of the scale effects

Local scale effects for linear continuous media are investigated as applied to the composites reinforced by nanoparticles. A mathematical model of the interphase layer is proposed that describes the specific nature of deformations in the neighborhood of the interface between different phases in an inhomogeneous material. The characteristic length of the interphase layer is determined formally in terms of the parameters of the mathematical model. The local stress state in the neighborhood of the phase boundaries in the interphase layer is examined. This stress can cause a significant change of the integral macromechanical characteristics of the material as a whole if the interphase boundaries are long. Such a situation is observed in composite materials reinforced by microparticles and nanoparticles even when the volume concentration of the inclusions is small. A numerical simulation of the stress state is performed on the basis of the block analytical-numerical multipole method with regard for the local effects related to the special nature of the deformation of the interphase layer in the vicinity of the interface.     

Antiplane shear of a strip containing a staggered array of rigid line inclusions

Motivated by the increased use of fibre-reinforced materials, we illustrate how the effective elastic modulus of an Isotropic and homogeneous material can be increased by the insertion of rigid inclusions. Specifically, we consider the two-dimensional antiplane shear problem for a strip of material. The strip is reinforced by introducing two sets of ribbon-like, rigid inclusions perpendicular to the faces of the strip. The strip is then subjected to a prescribed uniform displacement difference between its faces, see Figure 1. It should be noted that the problem posed is equivalent to that of the uniform antiplane shear problem for an infinite two-dimensional material containing a staggered array of rigid inclusions (see [1] for a review of antiplane problems in the literature). The problem is reduced in standard fashion [2–6] to a mixed boundary value problem in a rectangular domain, whose closed form solution given in terms of integrals of Weierstrassian Elliptic functions, is obtained via triple sine series techniques. The effective shear modulus of the reinforced strip can now be calculated and compared with the shear modulus of a strip without inclusions. Also obtained are the stress singularity factors at the end tips of the inclusions. Numerical results are presented for several different reinforcement geometries.     

Theory of spherofibrous composites. 1. Input relationships: Hypotheses and models

A theory is proposed for fibrous composites with a matrix reinforced with spherical hollow and solid inclusions based on an internal stress field and structural models. The problem solutions are obtained for the fiber-averaged matrix level. The matrix properties are determined assuming a regular distribution of the matrix inclusions. The problem of accounting for the scatter of inclusion properties on the effective composite parameters is examined.Presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, Latvia (October, 1995).A. A. Blagonravov Mechanical Engineering Institute. Russian Academy of Sciences. Moscow, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 32, No. 3, pp. 291–305. May–June, 1996.     

Multiscale modeling of multiphase "in situ" composite

The elastic-plastic behaviour of rapidly solidified Al based (FeSi)-enriched alloys containing intermetallic compounds is considered. A new multilevel mechanical model for the “in situ” composite is proposed considering the aluminium matrix as a micropolar elastic plastic Cosserat material and the hardening phases as pure elastic ones. A two steps homogenization procedure is applied to obtain the overall properties of the multiphase “in situ“ composite, taking into account the existence of different sizes of intermetallic inclusions. A variational approach is applied to evaluate the equivalent stress on macro level at the transition from micro to macro scale. The model is developed using information provided by microstructural investigations and EDX analysis. The multistage bulk material manufacturing process from rapid solidified powders or ribbons is simulated using the Finite Element Method. The model is implemented as user subroutines into the FE code MARC. Numerical simulations are provided, corresponding to different values of metal forming parameters. The influence of the different inclusions sizes on the hardening behavior is discussed. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effective Elastic Characteristics of a Piecewise-Homogeneous Isotropic Body with a System of Rectilinear Thin Inclusions under Conditions of Longitudinal Shear

We study the problem of determination of effective elastic characteristics of a piecewise-homogeneous isotropic body containing a twice-periodic system of tunnel thin rectilinear inclusions. The body is subjected to antiplane strain, and the conditions of ideal mechanical contact between the matrix and inclusions are satisfied. We reduce the solution of the problem to a system of integro-differential equations which enable us to find the expressions for the effective elastic characteristics of the composite. We also present the results of numerical analysis of these characteristics for a rectangular grid of periods and for various geometric and mechanical parameters of the problem.     

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)     

Elastic stress concentration near boundary defects in the case of a longitudinal shear deformation

We study the concentration of stresses due to two boundary defects located in an elastic half-space with stress-free boundary loaded at infinity with a constant shear load. The problem is reduced to solving singular integral equations for the cases in which the half-space contains defects of different types: two cracks, two inclusions, and a crack and an inclusion, whose solutions are sought by the method of mechanical quadratures. The interaction of the defects as they approach each other and the influence of their relative sizes are studied numerically. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

On the uniform stress state inside an inclusion of arbitrary shape in a three-phase composite

The stress field inside a two-dimensional arbitrary-shape elastic inclusion bonded through an interphase layer to an infinite elastic matrix subjected to uniform stresses at infinity is analytically studied using the complex variable method in elasticity. Both in-plane and anti-plane shear loading cases are considered. It is shown that the stress field within the inclusion can be uniform and hydrostatic under remote constant in-plane stresses and can be uniform under remote constant anti-plane shear stresses. Both of these uniform stress states can be achieved when the shape of the inclusion, the elastic properties of each phase, and the thickness of the interphase layer are properly designed. Possible non-elliptical shapes of inclusions with uniform hydrostatic stresses induced by in-plane loading are identified and divided into three groups. For each group, two conditions that ensure a uniform hydrostatic stress state are obtained. One condition relates the thickness of the interphase layer to elastic properties of the composite phases, while the other links the remote stresses to geometrical and material parameters of the three-phase composite. Similar conditions are analytically obtained for enabling a uniform stress state inside an arbitrary-shape inclusion in a three-phase composite loaded by remote uniform anti-plane shear stresses.     

A multiscale analysis of the mechanical behavior of a thermo-oxidized c/epoxy lamina

The mechanical behavior of carbon-fiber-reinforced polymer matrix composites having undergone a thermo-oxidation process is studied. The purpose is to perform a multiscale analysis of the consequences of oxidation on the intrinsic mechanical properties of the external composite ply and on the internal mechanical states experienced by the structure under mechanical loads. The effective mechanical properties of oxidized composite plies are determined according to the Eshelby–Kr?ner self-consistent homogenization procedure, depending on evolution of the oxidation process. The results obtained are compared with estimates found by the finite-element method. The macroscopic mechanical states are calculated for a unidirectional composite and laminates. The macroscopic stresses in each ply of the structure are determined by the classical lamination theory and the finite-element method, whereas the local stresses in the carbon fiber and epoxy matrix are calculated by using an analytical stress concentration relation.