Inhomogeneity-matrix Interface Research Articles

Elastic spherical inhomogeneity in an infinite elastic solid: an exact analysis by an engineering treatment of the problem based on the corresponding cavity solution

In the present work, solutions are recapitulated according to the theory of elasticity for the deformations of an adhesive spherical inhomogeneity in an infinite matrix under remote uniform axial and axial-symmetrical radial tension. Stress fields in the inhomogeneity and at the interface in the matrix are provided, too. It is shown that the sphere is deformed to a spheroid under any of the loading cases considered. Due to the axial-symmetric setup of the problem, the deformation is fully described by the two displacement values at line segments on the principal axes of the spheroid. The displacements depend on the applied remote load and on two traction fields at the inhomogeneity-matrix interface. For any combination of inhomogeneity and matrix stiffness, the condition of compatibility of deformations yields a system of two linear equations with the two magnitudes of the tractions as unknowns. Thus, the problem is reduced to a formulation for solving a twofold statically indetermined structure. The system is solved and the exact solution of the general spherical inhomogeneity problem with differing stiffness in terms of Young’s moduli and Poisson’s ratios of inclusion and matrix is presented.

Mechanical Performance of a Thermoelectric Composite in the Vicinity of an Elliptic Inhomogeneity

Summary Thermal stress induced by an uneven temperature field and mismatched thermal expansion is known to be a dominating factor in the debonding mechanism that threatens reliability and ultimately leads to failure in thermoelectric (TE) composites. Accordingly, we analyse the stress distributions in a TE composite induced by the presence of an elliptic inhomogeneity embedded in the surrounding matrix material. Using complex variable methods, we obtain closed-form representations of the thermal–electric and thermal–elastic fields and find that the temperature field around the inhomogeneity is reduced dramatically by the application of a remote electric current density without affecting the temperature difference across the inhomogeneity–matrix interface. This ensures the conversion efficiency of the TE composite while improving its reliability. Numerical results illustrate that a suitable choice of electric current density can prevent interfacial debonding via the suppression of the maximum positive normal stress on the interface.

A screw dislocation near a damaged arbitrary inhomogeneity–matrix interface

In the literature, the analytical solutions concerned with the interaction between screw dislocation and surfaces/interfaces have been mainly limited to simple geometries and perfect interfaces. The focus of the current work is to provide an approach based on a rigorous semi-analytical theory suitable for treatment of such surfaces/interfaces that concurrently have complex geometry and imperfect bonding. The proposed approach captures the singularity of the elastic fields exactly. A vast variety of the pertinent interaction problems such as dislocation near a multi-inhomogeneity with arbitrary geometry bonded imperfectly to a matrix, dislocation near the free boundaries of a finite elastic medium of arbitrary geometry, and so on is considered. In the present approach the out-of-plane component of the displacement in each domain is decomposed as the displacement corresponding to a screw dislocation in a homogeneous elastic body of infinite extent and the disturbance displacement due to the interaction. Subsequently, the disturbance displacement in each medium is expressed in terms of eigenfunction expansion. Damaged interfaces are modeled by a spring layer of vanishing thickness, and the amount of damage is controlled via the stiffness of the spring. For the illustration of the robustness of the proposed methodology a variety of examples including the interaction of a screw dislocation with a circular as well as a star-shaped inhomogeneities, two interacting inhomogeneities, imperfectly bonded to an unbounded medium are given. Also, examples for highlighting the effect of free surfaces in the case of finite domains are provided. It is revealed that in the cases where matrix is stiffer than the inhomogeneity and the dislocation is inside the inhomogeneity, or the other way around, then the amount of interface damage can change the sign of the image force.

Analytic solution for a circular nano-inhomogeneity in a finite matrix

Abstract The Gurtin-Murdoch model has found wide applications in analyzing the mechanical behaviors of nanocomposites with surface/interface effect. In the existing literature, the matrix is usually assumed to be infinite and the surface/interface effect is considered only at the inhomogeneity-matrix interface. This assumption is indeed valid as the matrix is usually at macroscale rather than nanoscale. However, if the size of the matrix decreases to the nanoscale too, the surface/interface effect will have to be considered at the outer boundary of the matrix. In this paper, the plane deformation of a circular nano-inhomogeneity embedded inside a finite circular matrix (which implies the matrix is also at nanoscale) is investigated. The stress boundary conditions are given at the inhomogeneity-matrix interface and the outer boundary of the matrix by the G-M model. The analytic solution for the stress field is finally obtained through the complex variable method. The results show that the stress field inside the inhomogeneity is still uniform (size-dependent) when the surface/interface effect is considered. In addition, the stress field inside the bulk (including the inhomogeneity and the matrix) can be influenced not only by the size and elastic constant of the inhomogeneity, but also by those of the matrix.

Analytic solution for a circular nano-inhomogeneity with interface stretching and bending resistance in plane strain deformations

In the mechanical analysis of composites containing nano-inhomogeneities, it is customary to consider only the stretching resistance of the inhomogeneity-matrix interface but neglect the bending resistance of the interface. In this paper, we consider a circular nano-inhomogeneity in an infinite elastic plane subjected to an arbitrary uniform remote in-plane loading with both stretching and bending resistance incorporated on the interface. Analytic solutions are obtained for the stress field both inside and outside the inhomogeneity by using an integral-type boundary condition representing the jump in traction across the interface. We show that the presence of interface bending resistance has no influence on the average of the mean stress in the inhomogeneity, and for certain interface stretching and bending rigidities the stress field inside the inhomogeneity can remain uniform regardless of the specific uniform remote loading. Numerical examples are presented to examine the influence of the interface bending resistance on the interfacial tractions imposed on the inhomogeneity and matrix for a uniform remote uniaxial loading. It is found that the introduction of interface bending resistance perturbs the (interfacial) tractions imposed on the inhomogeneity only slightly whether the inhomogeneity is softer or harder than the matrix, while it may influence the (interfacial) tractions imposed on the matrix significantly when the inhomogeneity is much softer than the matrix. Moreover, it is shown that the peak of the interface bending resistance-induced jump in traction across the interface initially increases and then decreases as the inhomogeneity becomes harder (from an initial state in which the inhomogeneity is softer than the matrix).

An edge dislocation near a nanosized circular inhomogeneity with interface slip and diffusion

Abstract We study the transient elastic field induced by an edge dislocation near a nanosized circular elastic inhomogeneity in which the effects of interface slip and diffusion are incorporated into the model of deformation. Separate Gurtin-Murdoch surface elasticities are specified on the surface of the inhomogeneity and on the adjoining surface of the surrounding matrix. In addition, rate-dependent interface slip and diffusion are assumed to occur concurrently on the inhomogeneity-matrix interface. The ensuing interaction problem is solved using a simple yet effective method based on analytic continuation and a convenient decomposition of the proposed solution. In particular, our method allows us to circumvent the second-order tangential derivative taken with respect to the interfacial normal stress, typically a source of additional complication and often an obstacle to the solution of such problems. The original problem is reduced to two coupled linear algebraic equations and a number of mutually independent sets of state-space equations, the general solutions of which can be obtained by solving the associated generalized eigenvalue problem. The image force acting on the edge dislocation is derived using the Peach-Koehler formula. Corresponding stress and displacement fields as well as the image force are found to be dependent on four size-dependent dimensionless parameters (arising from the surface elasticities) and on two size-dependent parameters (having the dimension of time) arising from the incorporation of interface slip and diffusion and they evolve with an infinite number of size-dependent relaxation times.

Shape optimizations of inhomogeneities of two dimensional (2D) and three dimensional (3D) steady state heat conduction problems by the boundary element method

The shape optimizations of inhomogeneities for 2D and 3D steady state heat conductions in an infinite medium are respectively studied by the boundary element method (BEM). Interest in the shape optimization by the BEM is mainly due to its high computation accuracy and simplicity in meshing. The boundary integral equations and the heat energy formulations in this paper only contain the temperature on each inhomogeneity–matrix interface. The heat energy increment in the inhomogeneous medium is taken as the objective function. The method of moving asymptotes (MMA) is adopted to carry out numerical implementation of the shape optimizations of 2D and 3D inhomogeneities.

Size-dependent interaction of an edge dislocation with an elliptical nano-inhomogeneity incorporating interface effects

The elastic behavior of an edge dislocation, which is positioned outside of a nanoscale elliptical inhomogeneity, is studied within the interface elasticity approach incorporating the elastic moduli and surface tension of the interface. The complex potential function method is used. The dislocation stress field and the image force acting on the dislocation are found and analyzed in detail. The difference between the solutions obtained within the classical-elasticity and interface-elasticity approaches is discussed. It is shown that for the stress field, this difference can be significant in those points of the inhomogeneity-matrix interface, where the radius of curvature is smaller and which are closer to the dislocation. For the image force, this difference can be considerable or dispensable in dependence on the dislocation position, its Burgers vector orientation, and relations between the elastic moduli of the matrix, inhomogeneity and their interface. Under some special conditions, the dislocation can occupy a stable equilibrium position in atomically close vicinity of the interface. The size effect is demonstrated that the normalized image force strongly depends on the inhomogeneity size when it is in the range of several tens of nanometers, in contrast with the classical solution where this force is always constant. The general issue is that the interface elasticity effects become more evident when the characteristic sizes of the problem (inhomogeneity size, interface curvature radius and dislocation-interface spacing) reduce to the nanoscale.

Analysis of cracked transversely isotropic and inhomogeneous solids by a special BIE formulation

In this paper, a special boundary integral equation (BIE) formation is proposed to analyze the fracture problem in transversely isotropic and inhomogeneous solids. In this formulation, the single-domain boundary element method (BEM) is utilized to discretize the cracked matrix and the displacement BEM to the surface of the embedded inhomogeneity. The two regions are then connected through the continuity conditions along their joint interface. The conventional and three special nine-node quadrilateral elements are utilized to discretize the inhomogeneity–matrix interface and the crack surface. From the crack-opening displacements on the crack surface, the mixed-mode stress intensity factors (SIFs) are calculated, using the well-known asymptotic expression in terms of the Barnett–Lothe tensor. In the numerical analysis, the distance between the inhomogeneity and the crack as well as the orientation of the isotropic plane of the transversely isotropic media is varied to show their influences on the mixed-mode SIFs along the crack fronts.

Uniform Stresses Inside an Elliptical Inhomogeneity With an Imperfect Interface in Plane Elasticity

We consider an elliptical inhomogeneity embedded in an infinite isotropic elastic matrix subjected to in-plane deformations under the assumption of remote uniform loading. The inhomogeneity-matrix interface is assumed to be imperfect, which is simulated by the spring-layer model with vanishing thickness. Its behavior is based on the assumption that tractions are continuous but displacements are discontinuous across the interface. We further assume that the same degree of imperfection on the interface is realized in both the normal and tangential directions. We find a form of interface function, which leads to uniform stress field within the elliptical inhomogeneity. The explicit expressions for the uniform stress field within the elliptical inhomogeneity are derived. The obtained results are verified by comparison with existing solutions. The condition under which the internal stress field is not only uniform but also hydrostatic is also presented.

Closed-form solutions for a mode III radial matrix crack penetrating a circular inhomogeneity

In this research we address in detail a mode III radial matrix crack penetrating a circular inhomogeneity. One tip of the radial crack lies in the matrix, while the other tip of the radial crack lies in the circular inhomogeneity. In addition the two tips of the crack are mutually image points (or inverse points) with respect to the circular inhomogeneity-matrix interface. First we conformally map the crack onto a unit circle C a in the new ζ -plane. Meanwhile the inhomogeneity-matrix interface is mapped onto C b , a part of another circle in the ζ -plane. In addition C a and C b intersect at a vertex angle π /2. By using the method of image in the ζ -plane, closed-form solutions in terms of elementary functions are derived for three loading cases: (1) remote uniform antiplane shearing; (2) a screw dislocation located in the unbounded matrix; and (3) a radial Zener–Stroh crack.

Analysis of a partially debonded elliptic inhomogeneity in piezoelectric materials

A generalized solution was obtained for the partially debonded elliptic inhomogeneity problem in piezoelectric materials under antiplane shear and inplane electric loading using the complex variable method. It was assumed that the interfacial debonding induced an electrically impermeable crack at the interface. The principle of conformal transformation and analytical continuation were employed to reduce the formulation into two Riemann-Hilbert problems. This enabled the determination of the complex potentials in the inhomogeneity and the matrix by means of series of expressions. The resulting solution was then used to obtain the electroelastic fields and the energy release rate involving the debonding at the inhomogeneity-matrix interface. The validity and versatility of the current general solution have been demonstrated through some specific examples such as the problems of perfectly bonded elliptic inhomogeneity, totally debonded elliptic inhomogeneity, partially debonded rigid and conducting elliptic inhomogeneity, and partially debonded circular inhomogeneity.

Neutrality of the elliptic inhomogeneity in the case of non-uniform loading

In this paper, we consider the problem of a single elliptic elastic inhomogeneity embedded within an infinite elastic matrix in antiplane shear. In particular, we examine the (stress) neutrality of this inhomogeneity when a non-uniform stress field is prescribed in the surrounding matrix. Since it is known that neutral elastic inhomogeneities do not exist when the inhomogeneity is assumed to be perfectly bonded to the matrix, the method presented here is based on the assumption of imperfect interface and the appropriate choice of the (single) interface parameter (characterizing the imperfect interface) to achieve the desired neutrality. Specifically, neutrality is established for specific (polynomial) classes of prescribed states of stress in the surrounding matrix. The results in this paper affirm the feasibility of designing a neutral elastic inhomogeneity by controlling the (imperfect) interface parameter describing the inhomogeneity–matrix interface.

On the uniformity of stresses inside an inhomogeneity of arbitrary shape

We consider an inhomogeneity of 'arbitrary' shape embedded within an infinite isotropic elastic medium (matrix) subjected to antiplane shear deformations under the assumption of uniform remote loading. The inhomogeneity-matrix interface is assumed to be imperfect, characterized by a single interface function. Under these assumptions, we present a novel method leading to the solution of the problem concerned with identifying the shape of the inhomogeneity and the form of the corresponding interface function which leads to a uniform interior stress field. The analysis is based on complex variable methods. Specific solutions are derived in closed form and verified by comparison with existing solutions. As a consequence of our analysis, we present an interesting result on the uniqueness (within a certain class of smooth curves) of the circular inhomogeneity as the only inhomogeneity which, under the given conditions, leads to a uniform interior stress field when the interface function is constant (homogeneously imperfect interface).

On the Design of Neutral Elastic Inhomogeneities in the Case of Non-Uniform Loading

In this paper, we consider the problem of a single elastic inhomogeneity embedded within an infinite elastic matrix in anti-plane shear. In particular, we examine the design of this inhomogeneity to achieve (stress) neutrality when a non-uniform stress field is prescribed in the surrounding matrix. Since it is known that neutral elastic inhomogeneities do not exist when the inhomogeneity is assumed to be perfectly bonded to the matrix, the design method presented here is based on the assumption of an imperfect interface and the appropriate choice of the (single) interface parameter (characterizing the imperfect interface) to achieve the desired neutrality. Specifically, in the case of a homogeneously imperfect interface, it is shown that the circular inhomogeneity is neutral if and only if the prescribed non-uniform stress field in the surrounding matrix belongs to a certain class of polynomial functions. In the case of an inhomogeneously imperfect interface, neutrality is established for circular and elliptic inhomogeneities for specific classes of prescribed states of stress in the surrounding matrix. The results in this paper affirm the feasibility of designing a neutral elastic inhomogeneity by controlling the (imperfect) interface parameter describing the inhomogeneity-matrix interface.

On the interaction between a dislocation and a circular inhomogeneity with imperfect interface in antiplane shear

The solution of appropriate elasticity problems involving the interaction between inclusions and dislocations plays a fundamental role in many practical and theoretical applications, namely, it increases the understanding of material defects thereby providing valuable insight into the mechanical behavior of composite materials. Although the problem of a three-phase circular inclusion interacting with a dislocation in antiplane shear has been presented [Xiao and Chen, Mech. Mater. 32 (2000) 485], the analysis is limited to the classical perfect bonding condition. The current paper considers the solution for a homogeneous circular inclusion interacting with a dislocation under thermal loadings in antiplane shear. The bonding along the inhomogeneity–matrix interface is considered to be imperfect with the assumption that the interface imperfections are constant. It is found that when the inhomogeneity is soft, regardless of the level of interface imperfection, the inhomogeneity will always attract the dislocation. As a result, no equilibrium positions are available. Alternatively, when the inhomogeneity is hard, an unstable equilibrium position is found which depends on the imperfect interface condition and the shear moduli ratio μ 2/ μ 1.

Interfacial debonding of a circular inhomogeneity in piezoelectric materials

A generalized and mathematically rigorous model is developed to treat the partially-debonded circular inhomogeneity problem in piezoelectric materials under antiplane shear and in plane electric field using the complex variable method. The principle of analytical continuation and the complex series expansion method were employed to reduce the formulations into Riemann-Hilbert problems. This enabled the explicit determination of the complex potentials inside the inhomogeneity and the matrix. The resulting closed form expressions were then used to obtain the energy release rate for several interesting cases involving partial-debonding at the inhomogeneity-matrix interface.

Chemical Inhomogeneities and Square B-H Loops

It is shown that in ferrites exhibiting spontaneously square B-H loops, reverse-domain nucleation may occur either at a grain boundary or at a chemical inhomogeneity having a volume roughly 0.1 of a grain volume, a magnetization about 2% different from that of the parent matrix, and no static crystallographic discontinuity at the inhomogeneity-matrix interface. It is pointed out that such chemical inhomogeneities may occur in spinels having Jahn-Teller ions in concentrations that are too small to produce a cooperative, static distortion of the entire crystal from cubic symmetry. It is postulated that the formation of such inhomogeneities is responsible for the sharp knee in spontaneously square B-H loops found in many ferrites containing Mn3+ and Cu2+ ions, in particular the magnesium-manganese composition used in memory-core ferrites.