Volume integral equation method for multiple circular and elliptical inclusion problems in antiplane elastostatics

A Volume Integral Equation Method (VIEM) is introduced for the solution of elastostatic problems in an unbounded isotropic elastic solid containing interacting multiple isotropic and anisotropic circular/elliptical inclusions subject to remote antiplane shear. This method is applied to two-dimensional problems involving long parallel cylindrical inclusions. A detailed analysis of the stress field at the interface between the matrix and the central inclusion is carried out for square and hexagonal packing of isotropic and anisotropic inclusions. The effects of the number of isotropic and anisotropic inclusions and various fiber volume fractions on the stress field at the interface between the matrix and the central circular/elliptical inclusion are also investigated in detail. The accuracy of the method is validated by solving single isotropic and orthotropic circular/elliptical inclusion problems and multiple isotropic circular and elliptical inclusion problems for which solutions are available in the literature.     

Generalised plane strain problem for a circular inclusion in anisotropic elasticity

This paper is concerned with a generalised plane deformation problem in the linear theory of anisotropic elasticity. As is well known, the generalised plane deformation is the deformation of a body of infinite length bounded by a cylindrical surface, when all the stress and strain components exist but they are functions of two co-ordinates x₁, and x₂ only. It may be shown that if u₃ = 0, it is impossible to satisfy all the three equations of equilibrium of anisotropic elastic body. One has to choose u₃ as a non-zero function of x₁, x₂ for satisfying equations of equilibrium. In isotropic elasticity, u₃ = 0, makes the third equation of equilibrium identically equal to zero.The problem in this paper concerns an elastic circular cylindrical inclusion embedded in a matrix of different anisotropic material. The matrix and the inclusion are perfectly bonded at the interface. Each of the two materials possesses anisotropy of a general form with all the 21 elastic constants. The matrix is subjected to a uniform stress at infinity. The equations of elasticity theory demand that the rotation component ω₃ must also be prescribed at infinity. The complex variable technique is used and exact analytical expressions are derived for the elastic field in both the regions.     

Uniform antiplane shear stress inside an anisotropic elastic inclusion of arbitrary shape with perfect or imperfect interface bonding

We consider an anisotropic elastic inclusion of arbitrary shape embedded inside an infinite dissimilar anisotropic elastic medium (matrix) subjected to a uniform antiplane shear loading at infinity. In contrast to the corresponding results from linear isotropic elasticity, we show that for certain anisotropic materials, despite the limitation of perfect bonding between the inclusion and its surrounding matrix, it is possible to design an arbitrarily shaped (not necessarily elliptic) inclusion so that the interior stress distribution is uniform provided the shear stress in the matrix (of dissimilar anisotropic material) is also uniform. Further, in the case when the bonding between the inclusion and the matrix is assumed to be imperfect, we show that for the stress distribution inside the inclusion to be uniform, the inclusion must be elliptical.     

Characterization of the anisotropic materials capable of exhibiting an isotropic Young or shear or area modulus

By means of the decomposition of an anisotropic elastic tensor into symmetric traceless tensors, the general intrinsic expressions involving no redundant elastic coefficients are obtained for the orientation distribution functions of the Young, shear and area moduli of an anisotropic material. Necessary and sufficient conditions are established for each of these moduli to be isotropic. It is found that an anisotropic material exhibiting an isotropic Young or shear or area modulus can be only either transversely isotropic or orthotropic.     

Temperature field near the singular line of the interface of anisotropic media

A representation of the temperature fields and the components of the vector of heat-flux density in both the base material (matrix) and the conjugate medium (inclusion) has been found for anisotropic media in the case of the interface with a singular line on condition of ideal thermal contact. It has been shown that the anisotropy of thermal properties makes it possible to do away with the singularity of the components of the heatflux-density vector. Particular cases of isotropy of the media and of heat-insulated and isothermal inclusions have been investigated. The results obtained are applicable for studying the nonstationary heat conduction of an anisotropic body with an uneven anisotropic inclusion. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 82, No. 1, pp. 157–162, January–February, 2009.     

A hybrid complex-variable solution for piezoelectric/isotropic elastic interfacial cracks

In many practical applications, piezoelectric ceramics are bonded to non-piezoelectric and insulating isotropic elastic materials such as polymer. Since the conventional form of Stroh’s formulation, on which almost all of existing works on interfacial cracks in piezoelectric media have been based, breaks down or becomes complicated for isotropic elastic materials, many solutions available in the literature cannot be directly applied to interfacial cracks between a piezoelectric material and an isotropic elastic material. The present paper is devoted to a hybrid complex-variable method which combines the Stroh’s method of piezoelectric materials with the well-known Muskhelishvili’s method of isotropic elastic materials. This method is illustrated in detail for an insulating interfacial crack between a piezoelectric half-plane and an isotropic elastic half-plane, although interface cracks between piezoelectric and isotropic elastic conductor can be analyzed in a similar way. The solution obtained generally exhibits oscillatory singularity, in agreement with a previous known result based on the Stroh’s formulation. A simple explicit condition is obtained for the bimaterial constants under which the oscillatory singularity disappears. It is expected that the hybrid complex-variable method could more conveniently handle other possible complications (such as a hole or an inclusion) inside the isotropic elastic material, because it offers explicit solutions of a single complex variable rather than several different complex-variables associated with the Stroh’s formulation.     

Eshelby inclusion of arbitrary shape in an anisotropic plane or half-plane

Summary.  Analytical solution for Eshelby's problem of an anisotropic non-elliptical inclusion remains a challenging problem. In this paper, a simple method is presented to obtain an analytical solution for Eshelby's problem of an inclusion of arbitrary shape within an anisotropic plane or half-plane of the same elastic constants. The method is based on an observation that the interface conditions for arbitrary inclusion-shape can be written in a decoupled form in which three unknown Stroh's functions are decoupled from each other. The solution is given in terms of three auxiliary functions constructed by three conformal mappings which map the exteriors of three image curves of the inclusion boundary, defined by three Stroh's variables, onto the exterior of the unit circle. With aid of these auxiliary functions, techniques of analytical continuation can be applied to the inclusion of any shape. The solution is given in the physical plane rather than in the image plane, and is exact provided that the expansion of every mapping function includes only a finite number of terms. On the other hand, if at least one of the mapping functions includes infinite terms, a truncated polynomial mapping should be used, and thus the method gives an approximate solution. A remarkable feature of the present method is that it gives elementary expressions for the internal stress field within an inclusion in an anisotropic entire plane. Elliptical and polygonal inclusions are used to illustrate the construction of the auxiliary functions and the details of the method. Received February 18, 2002; revised July 22, 2002 Published online: February 10, 2003 Acknowledgement The financial support of the Natural Science and Engineering Research Council of Canada is gratefully acknowledged.     

Prediction of branching (or relaxation) angle in anisotropic or isotropic elastic bimaterials with rigid substrate

This paper focuses on isotropic and anisotropic materials welded to a rigid substrate with a debond crack on its interface. The contact zone model is used in all the cases obtaining results for the near-tip displacement and stress fields. Using these results, it is possible to predict the angle at which the crack may branch or plastic relaxation may occur.     

Arbitrarily oriented microcracks near the tip of an interface macrocrack in bonded dissimilar anisotropic materials

It is well known that microcracking in brittle materials results in a reduction of the stress intensity factor (SIF) and energy release rate (ERR). The reduced SIF or ERR represents crack tip shielding which is of significant interest to micromechanics and material science researchers. However, the effect of microcracking on the SIF and ERR is a complicated subject even for isotropic homogeneous materials, and becomes much more formidable in case of interface cracks in bonded dissimilar solids. To unravel the micromechanics of interface crack tip shielding in bonded dissimilar anisotropic solids, an interface crack interacting with arbitrarily oriented subinterface microcracks in bonded dissimilar anisotropic materials is studied. After deducing the fundamental solutions for a subinterface crack under concentrated normal and tangential tractions, the present interaction problem is reduced to a system of integral equations which is then solved numerically. A J‐integral analysis is then performed with special attention focused on the J₂‐integral in a local coordinate system attached to the microcracks. Theoretical and numerical results reassert the conservation law of the J‐integral derived for isotropic materials 1 , 2 also to be valid for bonded dissimilar anisotropic materials. It is further concluded that there is a wastage when the remote J‐integral transmits across the microcracking zone from infinity to the interface macrocrack tip. In order to highlight the influence of microstructure on the interfacial crack tip stress field, the crack tip SIF and ERR in several typical cases are presented. It is interesting to note that the Mode I SIF at the interface crack tip is quite different from the ERR in bonded dissimilar anisotropic materials.     

Path-independent integral analyses for microcrack damage in dissimilar anisotropic materials

Summary This paper deals with microcrack damage below an interface in a dissimilar anisotropic material. The interaction problem among arbitrarily oriented and located subinterface microcracks is studied in detail by solving the associated singular integral equations derived from the dislocation technique and superimposing technique. The path-independent integral analyses are performed by adopting theJ _k -integral vector and theM-integral, which are evaluated, respectively, along specially introduced contours. It is found that the total contributions of the subinterface microcracks to the first component of theJ _k -vector vanish, provided that the chosen contour encloses all the microcracks. This leads to a conservation law of theJ ₁-integral in the microcrack damage problem, from which a consistency check or a necessary condition for this kind of problem is presented. Similar conclusions are also given for the second component of the vector under the more strict assumption that the closed contour chosen to calculate theJ ₂-integral is infinite large and encloses not only all the microcracks, but also the whole interface. Therefore, theM-integral analysis could be performed, which is proved to be divided into two parts. One of them, called the net part, is the simple summation of the distinct contribution induced from the stress intensity factors at all microcrack tips, while the other part, called the additional part, is induced from the global coordinates of each microcrack center and theJ _k -vectorevaluated along a specially introduced contour surrounding only one single microcrack completely. Numerical results are given for a particular dissimilar anistropic material whose upper half part shows anisotropy (the fiber direction is parallel to the interface) and the lower half part shows isotropic (the fiber direction is perpendicular to the global coordinate plane). It is concluded that the numerical results do actually meet the consistency check. Moreover, it is found that the values of theM-integral for the present dissimilar material is always smaller than those for the corresponding isotropic material, no matter how the subinterface matrix microcracks are located and oriented. It is concluded that theM-integral plays an important role in the present microcrack damage problem and that any aray of matrix microcracks below the interface shows less unstable nature than the same array in the corresponding isotropic material.     

Numerical modeling of development of fracture in anisotropic composite materials at low-velocity loading

The problem of normal interaction of the steel isotropic compact cylindrical projectile with the orthotropic plate on ballistic limit in range of velocities of impact from 50 to 400 m/s is considered. Target material is as organoplastic with initial orientation of mechanical properties, and the material, which properties received by turn on 90° relative to the axis OY of an initial material. Fracture of targets is investigated; the comparative analysis of efficiency of their protective properties depending on orientation of elastic and strength properties of an anisotropic material is carried out. The task is solved numerically, using the method of finite elements in three-dimensional statement. The behavior of a material of the projectile is described by elastic–plastic model; the behavior of anisotropic material of the target is described by elastic–brittle model with various ultimate strength limits on pressure and tension.     

Fracture problems of rubbers: J-integral estimation based upon η factors and an investigation on the strain energy density distribution as a local criterion

The plane elasticity problem studied is of a circular inclusion having a circular arc-crack along the interface and a crack of arbitrary shape in an infinite matrix of different material subjected to uniform stresses at infinity. The solution of the problem is given using Muskhelishvili's complex variable method with sectionally holomorphic functions. First, the solution to the (auxiliary) problem of a dislocation (or force) applied at a point in the matrix with the circular inclusion partially bonded is derived fully in its general form by solving the appropriate Rieman-Hilbert problem. It is subsequently used as the Green's function for the initial problem by introducing an unknown density function associated with a distribution of dislocations along the crack in the matrix. The initial problem is then reduced to a singular integral equation (SIE) over the crack in the matrix only. The SIE is solved numerically by appropriate quadratures and the stress intensity factors reported for the arc-cut and a straight crack in the matrix for a range of values of the geometrical parameters.     

The problem of two plastic and heterogeneous inclusions in an anisotropic medium

We demonstrate an integral equation for the total local strain ε^T in an anisotropic heterogeneous medium with incompatible strain ε^p and which is at the same time submitted to an exterior field. The integral equation is solved in the case of an heterogeneous and plastic pair of inclusions, for which we calculate the average fields in each inclusion as well as the different parts of the elastic energy stocked in the medium.The solution is applied to the case of two isotropic and spherical inclusions in an isotropic matrix loaded in shear. The results are compared with those deduced from a more approximate method based on Horn's approximation of the integral equation. In appendix we give a numerical method for calculating the interaction tensors between anisotropic inclusions in an anisotropic medium as well as the analytic solution in the case of two spherical inclusions located in an isotropic medium.     

Method of perturbations of the shape of the boundary in fracture mechanics. Part 2: Inclusions

A brief history of solutions of the elastic problem for a plate with inclusions is given. The method of perturbation of the shape of the boundary for this problem solution is developed for two cases. In the case of an isotropic plate containing isotropic inclusions with turning points on their contours, expansion in terms of a small parameter ∈ and function ζ(z) is used. For the case of an anisotropic plate with an anisotropic or isotropic inclusion, expansion of complex stress functions in terms of the parameter ∈ and Faber polynomialsP _n for some ellipse is applied. The algorithm and its numerical realization are described in detail for the case of noncanonical elastic isotropic inclusions with small, but finite curvature radii at the tips. The limits of applicability of this method concerning both the defects geometry and elastic characteristics of the composition have been established. The convergence of series for the generalized SIF has been studied numerically. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 33, No. 6, pp. 44–54, November–December, 1997.     

Screw dislocations interacting with a partially debonded interface in cylindrically anisotropic composites

Interaction between screw dislocations and a partially debonded interface in cylindrically anisotropic composites subjected to uniform stress at infinity is investigated in this paper. Using Muskhelishvili’s complex variable method, the closed forms of complex potentials are obtained for a screw dislocation and a screw dislocation dipole located inside either matrix or inhomogeneity. Explicit expressions of stress intensity factors at the crack tips, image forces and image torques acting on dislocation or the center of dipole are provided. The results show that the crack and dislocation geometry combination plays an important role in the interaction between screw dislocations and interface crack. Furthermore, it is found that the anisotropy of solids may change the shielding and anti-shielding effects arising from screw dislocations and the equilibrium position of screw dislocations. The presented solutions are valid for anisotropic, orthotropic or isotropic composites and can be reduced to some novel or previously known results.     

Multiple circular nano-inhomogeneities and/or nano-pores in one of two joined isotropic elastic half-planes

The paper considers the problem of multiple interacting circular nano-inhomogeneities or/and nano-pores located in one of two joined, dissimilar isotropic elastic half-planes. The analysis is based on the solutions of the elastostatic problems for (i) the bulk material of two bonded, dissimilar elastic half-planes and (ii) the bulk material of a circular disc. These solutions are coupled with the Gurtin and Murdoch model of material surfaces [Gurtin ME, Murdoch AI. A continuum theory of elastic material surfaces. Arch Ration Mech Anal 1975;57:291–323; Gurtin ME, Murdoch AI. Surface stress in solids. Int J Solids Struct 1978;14:431–40.]. Each elastostatic problem is solved with the use of complex Somigliana traction identity [Mogilevskaya SG, Linkov AM. Complex fundamental solutions and complex variables boundary element method in elasticity. Comput Mech 1998;22:88–92]. The complex boundary displacements and tractions at each circular boundary are approximated by a truncated complex Fourier series, and the unknown Fourier coefficients are found from a system of linear algebraic equations obtained by using a Taylor series expansion. The resulting semi-analytical method allows one to calculate the elastic fields everywhere in the half-planes and inside the nano-inhomogeneities. Numerical examples demonstrate that (i) the method is effective in solving the problems with multiple nano-inhomogeneities, and (ii) the elastic response of a composite system is profoundly influenced by the sizes of the nano-features.     

Interaction between an interface crack and a parallel subinterface crack in dissimilar anisotropic composite materials

In this paper, the pseudo-traction method is combined with the edge-dislocation method (i.e. PTDM) to solve the interaction problem between an interface crack and a parallel subinterface crack in dissimilar anisotropic materials. After deriving the fundamental solutions for an interface crack loaded by normal or tangential tractions on both crack surfaces and the fundamental solutions for an edge dislocation beneath the interface in the lower anisotropic material, the interaction problem is reduced to a system of a singular integral equations by adopting the well-known superposition technique. The equations are then solved numerically with the aid of the Chebyshev numerical integration and the Chebyshev polynomial expansion technique. Several typical examples are calculated and numerical results are shown in figures and tables from which a series of valuable conclusions is obtained. Since the present results should be verified and since no previous results exist to compare them with a consistency check in introduced which starts from the conservation law of the J-integral in anisotropic cases. It is shown that the check provides a powerful tool to examine the results, although it really presents a necessary condition rather than a sufficient way to the crack-tip parameters of the interface crack and the subinterface crack in the dissimilar anisotropic materials.     

Interface waves along an anisotropic imperfect interface between anisotropic solids

First and second order asymptotic boundary conditions are introduced to model a thin anisotropic layer between two generally anisotropic solids. Such boundary conditions can be used to describe wave interaction with a solid-solid imperfect anisotropic interface. The wave solutions for the second order boundary conditions satisfy energy balance and give zero scattering from a homogeneous substrate/layer/substrate system. They couple the in-plane and out-of-plane stresses and displacements on the interface even for isotropic substrates. Interface imperfections are modeled by an interfacial multiphase orthotropic layer with effective elastic properties. This model determines the transfer matrix which includes interfacial stiffness and inertial and coupling terms. The present results are a generalization of previous work valid for either an isotropic viscoelastic layer or an orthotropic layer with a plane of symmetry coinciding with the wave incident plane. The problem of localization of interface waves is considered. It is shown that the conditions for the existence of such interface waves are less restrictive than those for Stoneley waves. The results are illustrated by calculation of the interface wave velocity as a function of normalized layer thickness and angle of propagation. The applicability of the asymptotic boundary conditions is analyzed by comparison with an exact solution for an interfacial anisotropic layer. It is shown that the asymptotic boundary conditions are applicable not only for small thickness-to-wavelength ratios, but for much broader frequency ranges than one might expect. The existence of symmetric and SH-type interface waves is also discussed.     

Boundary collocation method for multiple defect interactions in an anisotropic finite region

The modified mapping collocation method is extended for the solution of plane problems of anisotropic elasticity in the presence of multiple defects in the form of holes, cracks, and inclusions under general loading conditions. The approach is applied to examine the stress and strain fields in an anisotropic finite region including an elliptical and a circular hole, an elliptical flexible inclusion, and a line crack. It can be readily incorporated into micro-mechanics models, capturing the relative importance of the matrix, the fiber/matrix interface, and reinforcement geometry and arrangement while estimating the effective elastic properties of composite materials. The accuracy and robustness of this method is established through comparison with results obtained from finite element analysis.     

Guided waves in laminated isotropic circular cylinder

This paper investigates, analytically and numerically, the dispersion characteristics of a laminated isotropic circular cylinder. The propagator matrix, which relates the stresses and displacements of one interface of a layer to those of another interface, is formulated based upon the three-dimensional theory of elasticity. The dispersion relation of the cylinder is implicitly established from this propagator matrix. The numerical evaluation is carried out by the Muller's method with an initial guess from a Rayleigh-Ritz type approximate method. Examples of an elastic rod and a two layered isotropic cylinder are presented and discussed to illustrate the accuracy and effectiveness of the method.