Elastodynamic analysis of inhomogeneous anisotropic bodies

The integral equation method is presented for elastodynamic problems of inhomogeneous anisotropic bodies. Since fundamental solutions are not available for general inhomogeneous anisotropic media, we employ the fundamental solution for homogeneous elastostatics. The terms induced by material inhomogeneity and inertia force are regarded as body forces in elastostatics, and evaluated in the form of volume integrals. The scattering problems of elastic waves by inhomogeneous anisotropic inclusions are investigated for some test cases. Numerical results show the significant effects of inhomogeneity and anisotropy of materials on wave propagations.     

Discontinuous deformation gradients in the finite twisting of an incompressible elastic tube

Summary It is known that the type of the system of partial differential equations governing finite elastostatics can change from elliptic to non-elliptic at sufficiently large deformations for certain materials. This introduces the possibility that the elastostatic field may exhibit certain discontinuities. Some aspects of the general theory associated with these issues were examined in a recent series of studies by Knowles and Sternberg. In this paper we illustrate the occurrence of elastostatic fields with discontinuous deformation gradients in a physical problem. The body is assumed to be composed of a material which belongs to a particular class of isotropic, incompressible, elastic materials which allow for a loss of ellipticity. It is shown that no solution which is smooth in the classical sense exists to this problem for certain ranges of the applied loading. Next, we admit solutions involving elastostatic shocks into the discussion and find that the problem may then be solved completely. When this is done, however, there results a lack of uniqueness of solutions to the boundary-value problem. In order to resolve this non-uniqueness, the dissipativity and stability of the solutions are investigated.The results communicated in this paper were obtained in the course of an investigation supported by Contract N00014-75-C-0196 with the Office of Naval Research in Washington D.C.     

A New Method for an Inhomogeneity with Stepwise Graded Interphase under Thermomechanical Loadings

The solution for a circular inhomogeneity embedded in an infinite elastic matrix with a multilayered interphase plays a fundamental role in many practical and theoretical problems. Therefore, improved analysis methods for this problem are of great interest. In this paper, a new procedure is presented to obtain the exact stress fields within the inhomogeneity and the matrix under thermomechanical loadings, without the need of solving the full multiphase composite problem. With this short-cut method, the problem is reduced to a single linear algebraic equation and two coupled linear algebraic equations which determine the only three real coefficients of the stress field within the inhomogeneity. In particular, the average stresses within the inhomogeneity can be calculated directly from the three real coefficients. Further, the other three unknown real coefficients associated with the stress field in the matrix can be determined subsequently. Hence, the influence of the stepwise graded interphase on the stress fields is manifested by its effect on the six real coefficients. All these results hold for stepwise graded interphase composed of any number of interphase layers. Several examples serve to illustrate the method and its advantages over other existing approaches. The explicit solutions are used to study the design of harmonic elastic inclusions, and the effect of a compliant interphase layer on thermal-mismatch induced residual stresses. This revised version was published online in August 2006 with corrections to the Cover Date.     

Elliptical inhomogeneity with polynomial eigenstrains embedded in orthotropic materials

A closed-form solution for elastic field of an elliptical inhomogeneity with polynomial eigenstrains in orthotropic media having complex roots is presented. The distribution of eigenstrains is assumed to be in the form of quadratic functions in Cartesian coordinates of the points of the inhomogeneity. Elastic energy of inhomogeneity–matrix system is expressed in terms of 18 real unknown coefficients that are analytically evaluated by means of the principle of minimum potential energy and the corresponding elastic field in the inhomogeneity is obtained. Results indicate that quadratic terms in the eigenstrains induce zeroth-order elastic strain components, which reflect the coupling effect of the zeroth- and second-order terms in the polynomial expressions on the elastic field. In contrast, the first-order terms in the eigenstrains only produce corresponding elastic fields in the form of the first-order terms. Numerical examples are given to demonstrate the normal and shear stresses at the interface between the inhomogeneity and the matrix. Furthermore, the solution reduces to known results for the special cases.     

Application of transition matrix to scattering of elastic waves in half-space by a surface scatterer

A transition matrix formulism for the scattering problems of elastic waves using a surface scatterer of three-dimensional half-space is derived by applying Betti's third identity and orthogonality conditions. The basis functions and their corresponding regular parts are derived from Lamb's singular solutions. The new orthogonal functions are constructed based on the linear transform. The intrinsic properties of the scattering and transition matrices derived in this paper are investigated. Numerical results for verification are presented and discussed.     

On the anti-plane shear of an elliptic nano inhomogeneity

The elastic field of an elliptic nano inhomogeneity embedded in an infinite matrix under anti-plane shear is studied with the complex variable method. The interface stress effects of the nano inhomogeneity are accounted for with the Gurtin–Murdoch model. The conformal mapping method is then applied to solve the formulated boundary value problem. The obtained numerical results are compared with the existing closed form solutions for a circular nano inhomogeneity and a traditional elliptic inhomogeneity under anti-plane. It shows that the proposed semi-analytic method is effective and accurate. The stress fields inside the inhomogeneity and matrix are then systematically studied for different interfacial and geometrical parameters. It is found that the stress field inside the elliptic nano inhomogeneity is no longer uniform due to the interface effects. The shear stress distributions inside the inhomogeneity and matrix are size dependent when the size of the inhomogeneity is on the order of nanometers. The numerical results also show that the interface effects are highly influenced by the local curvature of the interface. The elastic field around an elliptic nano hole is also investigated in this paper. It is found that the traction free boundary condition breaks down at the elliptic nano hole surface. As the aspect ratio of the elliptic hole increases, it can be seen as a Mode-III blunt crack. Even for long blunt cracks, the surface effects can still be significant around the blunt crack tip. Finally, the equivalence between the uniform eigenstrain inside the inhomogeneity and the remote loading is discussed.     

Antiplane deformation of a partially bonded elliptical inclusion

A new method that introduces two holomorphic potential functions (the two-phase potentials) is applied to analyze the antiplane deformation of an elliptical inhomogeneity partially-bonded to an infinite matrix. Elastic fields are obtained when either the matrix is subject to a uniform longitudinal shear or the inhomogeneity undergoes a uniform shear transformation. The stress field possesses the square-root singularity of a Mode III interface crack, which, in the special case of a rigid line inhomogeneity, changes in order, as the crack tip approaches the inhomogeneity end. In the latter situation the crack-tip elastic fields are linear in two real stress intensity factors related to a strong and a weak singularity of the stress field.     

CLOSED-FORM SOLUTIONS FOR ELASTOPLASTIC PURE BENDING OF A CURVED BEAM WITH MATERIAL INHOMOGENEITY

The elastoplastic pure bending problem of a curved beam with material inhomo- geneity is investigated based on Tresca＇s yield criterion and its associated flow rule. Suppose that the material is elastically isotropic, ideally elastic-plastic and its elastic modulus and yield limit vary radially according to exponential functions. Closed-form solutions to the stresses and radial displacement in both purely elastic stress state and partially plastic stress state are presented. Numerical examples reveal the distinct characteristics of elastoplastic bending of a curved beam composed of inhomogeneous materials. Due to the inhomogeneity of materials, the bearing capac- ity of the curved beam can be improved greatly and the initial yield mode can also be dominated. Closed-form solutions presented here can serve as benchmark results for evaluating numerical solutions.     

Two circular inclusions with inhomogeneously imperfect interfaces in plane elasticity

This paper is concerned with the problem of two circular inclusions with circumferentially inhomogeneously imperfect interfaces embedded in an infinite matrix in plane elastostatics. Infinite series form solutions to this problem are derived by applying complex variable techniques. The numerical results demonstrate that the interface imperfection, interface inhomogeneity, and interaction among neighboring inclusions (fibers) will exert a significant influence on the stresses along the interfaces and average stresses within the inclusions.     

Elastic field of an isotropic matrix with a nanoscale elliptical inhomogeneity

In traditional continuum mechanics, the effect of surface energy is ignored as it is small compared to the bulk energy. For nanoscale materials and structures, however, the surface effects become significant due to the high surface/volume ratio. In this paper, two-dimensional elastic field of a nanoscale elliptical inhomogeneity embedded in an infinite matrix under arbitrary remote loading and a uniform eigenstrain in the inhomogeneity is investigated. The Gurtin–Murdoch surface/interface elasticity model is applied to take into account the surface/interface stress effects. By using the complex variable technique of Muskhelishvili, the analytic potential functions are obtained in the form of an infinite series. Selected numerical results are presented to study the size-dependency of the elastic field and the effects of surface elastic moduli and residual surface stress. It is found that the elastic field of an elliptic inhomogeneity under uniform eigenstrain is no longer uniform when the interfacial stress effects are taken into account.     

A screw dislocation near a circular nano-inhomogeneity in gradient elasticity

A screw dislocation outside an infinite cylindrical nano-inhomogeneity of circular cross section is considered within the isotropic theory of gradient elasticity. Fields of total displacements, elastic and plastic distortions, elastic strains and stresses are derived and analyzed in detail. In contrast with the case of classical elasticity, the gradient solutions are shown to possess no singularities at the dislocation line. Moreover, all stress components are continuous and smooth at the interface unlike the classical solution. As a result, the image force exerted on the dislocation due to the differences in elastic and gradient constants of the matrix and inhomogeneity, remains finite when the dislocation approaches the interface. The gradient solution demonstrates a non-classical size-effect in such a way that the stress level inside the inhomogeneity decreases with its size. The gradient and classical solutions coincide when the distances from the dislocation line and the interface exceed several atomic spacings.     

Discontinuous deformation gradients in plane finite elastostatics of incompressible materials

Summary This investigation is concerned with the possibility of the change of type of the differential equations governing finite plane elastostatics for incompressible elastic materials, and the related issue of the existence of equilibrium fields with discontinuous deformation gradients. Explicit necessary and sufficient conditions on the deformation invariants and the material for the ellipticity of the plane displacement equations of equilibrium are established. The issue of the existence, locally, of elastostatic shocks—elastostatic fields with continuous displacements and discontinuous deformation gradients—is then investigated. It is shown that an elastostatic shock exists only if the governing field equations suffer a loss of ellipticity at some deformation. Conversely, if the governing field equations have lost ellipicity at a given deformation at some point, an elastostatic shock can exist, locally, at that point. The results obtained are valid for an arbitrary homogeneous, isotropic, incompressible, elastic material.The results communicated in this paper were obtained in the course of an investigation supported by Contract N00014-75-C-0196 with the Office of Naval Research in Washington D.C.     

The interaction between a screw dislocation and a rigid wedge inhomogeneity with an elastic circular inhomogeneity at the tip

The interaction between a screw dislocation and an elastic semi-cylindrical inhomogeneity abutting on a rigid half-plane is investigated. Utilizing the image dislocations method, the closed form solutions of the stress fields in the matrix and the inhomogeneity region are derived. The image force acting on the dislocation is also calculated. The results were used to study the interaction between a screw dislocation and a rigid wedge inhomogeneity with an elastic circular inhomogeneity at the tip by means of conformal mapping. The results show that an unstable equilibrium point of the dislocation near the semi-cylindrical inhomogeneity is found when the inhomogeneity is softer than the matrix. Moreover, the force on the dislocation is strongly affected by the position of the dislocation and the shear modulus of the semi-circular inhomogeneity. Positive screw dislocations can reduce the SIF of the rigid wedge inhomogeneity (shielding effect) only when it located in the lower half-plane. The shielding effect increases with the increase of the shear modulu of both the matrix and the inhomogeneity and increases with the increase of the wedge angle. The shielding effect (or anti-shielding effect) reaches the maximum when the dislocation tends to the wedge inhomogeneity interface.     

An edge dislocation and a heat source at the center of a multicoated circular inhomogeneity

Closed-form solutions are derived to the problem of an edge dislocation or a steady line heat source at the center of a multicoated circular inhomogeneity by using the complex variable method and the transfer matrix method. The problem is reduced to a single linear algebraic equation which determines the single unknown real coefficient appearing in the complex stress functions defined in the surrounding matrix. The other unknown real coefficient in the complex stress functions in the inhomogeneity can then be conveniently determined.     

A circular inclusion with circumferentially inhomogeneous imperfect interface in harmonic materials

In the following analysis, we present a rigorous solution for the problem of a circular elastic inclusion surrounded by an infinite elastic matrix in finite plane elastostatics. The inclusion and matrix are separated by a circumferentially inhomogeneous imperfect interface characterized by the linear spring-type imperfect interface model where the interface is such that the same degree of imperfection is realized in both the normal and tangential directions. Through the use of analytic continuation, a set of first-order coupled ordinary differential equations with variable coefficients are developed for two analytic potential functions. The unknown coefficients of the potential functions are determined from their analyticity requirements and some additional problem-specific constraints. An example is then presented for a specific class of interface where the inclusion mean stress is contrasted between the homogeneous interface and inhomogeneous interface models. It is shown that, for circumstances where a homogeneously imperfect interface may not be warranted, the inhomogeneous model has a pronounced effect on the mean stress within the inclusion.     

Elastic Energies in Circular Inhomogeneities: Imperfect Versus Perfect Interfaces

The change in the total potential energy in a stressed elastic plane system, consisting of an unbounded matrix containing a cylindrical inhomogeneity of circular cross-section, is studied, when an imperfect bonding is formed across the interface. The imperfect bonding is simulated by linearly elastic springs distributed over the interface. Two loading cases are examined: an equilibrium system of fixed uniform tractions acting in the remote boundary of the matrix, and a phase transformation in the inhomogeneity prescribed by stress free uniform eigenstrains distributed in the inhomogeneity region. For both loadings, the fully elastic fields in explicit forms are derived involving the spring compliances and three new two-phase parameters depending on the elastic properties of the two materials. The elastic energies stored in the whole system and in its constituents are determined in simple and compact forms. It is shown that, in both loading cases, the total potential energy of the system is reduced. It is found that, in nanoscale, the ratio of the elastic energy stored in interface to the elastic energy stored in inhomogeneity increases rapidly for small values of the circular radius and tends to zero for large values. Also, this ratio increases as the matrix becomes softer compared to the inhomogeneity.     

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.     

A three-dimensional inverse problem for inhomogeneities in elastic solids

The Newtonian potential is used to solve an inverse problem in which we seek the shape of an inhomogeneity in an infinite elastic matrix under uniform applied stresses at infinity such that certain stress components are uniform on the boundary of the inhomogeneity. It is shown that ellipsoids furnish the solution of this inverse problem. Exact and general expressions for the stress and displacement are given explicitly for points in the elastic matrix outside the inhomogeneity. The solution of the corresponding plane deformation problem is found as a limiting case. Several applications are presented, and results from the literature are confirmed as special cases.     

Non-uniform eigenstrain induced stress field in an elliptic inhomogeneity embedded in orthotropic media with complex roots

This paper deals with an elastic orthotropic inhomogeneity problem due to non-uniform eigenstrains. The specific form of the distribution of eigenstrains is assumed to be a linear function in Cartesian coordinates of the points of the inhomogeneity. Based on the polynomial conservation theorem, the induced stress field inside the inhomogeneity which is also linear, is determined by the evaluation of 10 unknown real coefficients. These coefficients are derived analytically based on the principle of minimum potential energy of the elastic inhomogeneity/matrix system together with the complex function method and conformal transformation. The resulting stress field in the inhomogeneity is verified using the continuity conditions for the normal and shear stresses on the boundary. In addition, the present analytic solution can be reduced to known results for the case of uniform eigenstrain.     

增强相/夹杂内螺位错偶极子与含共焦钝裂纹椭圆夹杂的干涉效应

运用弹性力学的复势方法,研究了纵向剪切下增强相/夹杂内螺型位错偶极子与含共焦钝裂纹椭圆夹杂的干涉效应,得到了该问题复势函数的封闭形式解答,由此推导出了夹杂区域的应力场、作用在螺型位错偶极子中心的像力和像力偶矩以及裂纹尖端应力强度因子级数形式解。并分析了位错偶极子倾角 、钝裂纹尺寸和材料常数对位错像力、像力偶矩以及应力强度因子的影响。数值计算结果表明：位错像力、像力偶矩以及应力强度因子均随位错偶极子倾角做周期变化;夹杂内部的椭圆钝裂纹明显增强了硬基体对位错的排斥,减弱了软基体对位错的吸引,且对于硬夹杂,位错出现了一个不稳定平衡位置,该平衡位置随钝裂纹曲率的增大不断向界面靠近;变化 值将出现改变位错偶极子对应力强度因子作用方向的临界值。