Time-dependent Green's functions for an anisotropic bimaterial with viscous interface

By virtue of the Stroh formalism, we derive the exact closed-form solutions for the time-dependent two-dimensional Green's functions due to a line force and line dislocation in an anisotropic bimaterial with a viscous interface. We first reduce the boundary value problem to two coupled homogeneous first-order partial differential equations, which can be solved using a decoupling technique. The full-field expressions of the time-dependent displacements and stresses due to the line force and line dislocation interacting with the viscous interface are obtained.     

Three-dimensional solutions for general anisotropy

The Stroh formalism is extended to provide a new class of three-dimensional solutions for the generally anisotropic elastic material that have polynomial dependence on x₃, but which have quite general form in x₁,x₂. The solutions are obtained by a sequence of partial integrations with respect to x₃, starting from Stroh's two-dimensional solution. At each stage, certain special functions have to be introduced in order to satisfy the equilibrium equation. The method provides a general analytical technique for the solution of the problem of the prismatic bar with tractions or displacements prescribed on its lateral surfaces. It also provides a particularly efficient solution for three-dimensional boundary-value problems for the half-space. The method is illustrated by the example of a half-space loaded by a linearly varying line force.     

Some half-space problems of cubic piezoelectric materials

Half-space problems of a cubic piezoelectric material subjected to anti-plane deformation and in-plane electric field are studied. A general solution in terms of the integration of the boundary data prescribed over the surface of the semi-infinite domain is derived. Based on the general solution, the problem of a concentrated line force acting on the surface is treated and ensuing electromechanical response is determined. The solution to the problem of a screw dislocation in the half-space is also obtained, and the result is exploited to study a sub-surface crack problem by simulating the crack as a continuous distribution of dislocations.     

Generalization of the Stroh Formalism to 3-Dimensional Anisotropic Elasticity

A method of analysis for three-dimensional anisotropic elasticity is developed, based on Stroh's two-dimensional formalism. By use of the present method a three-dimensional solution can be constructed by the corresponding two-dimensional solution. The method is applied to solve several problems concerning a point force or a dislocation loop. Existing results for these problems are recovered in a unified manner and additional new results are obtained. This revised version was published online in August 2006 with corrections to the Cover Date.     

Elastic Green’s Function in Anisotropic Bimaterials Considering Interfacial Elasticity

The two-dimensional elastic Green’s function is calculated for a general anisotropic elastic bimaterial containing a line dislocation and a concentrated force while accounting for the interfacial structure by means of a generalized interfacial elasticity paradigm. The introduction of the interface elasticity model gives rise to boundary conditions that are effectively equivalent to those of a weakly bounded interface. The equations of elastic equilibrium are solved by complex variable techniques and the method of analytical continuation. The solution is decomposed into the sum of the Green’s function corresponding to the perfectly bonded interface and a perturbation term corresponding to the complex coupling nature between the interface structure and a line dislocation/concentrated force. Such construct can be implemented into the boundary integral equations and the boundary element method for analysis of nano-layered structures and epitaxial systems where the interface structure plays an important role.     

Transient motion of an anisotropic elastic bimaterial due to a line source

The transient motion of an anisotropic elastic bimaterial due to a line force or a line dislocation is studied. The bimaterial is assumed to be at rest and stress-free for t < 0. The line source is applied at t = 0 and maintained for t > 0. A formulation which is an extension to Stroh’s formalism for anisotropic elastostatics is employed. The general solution is expressed in terms of the eigenvalues and eigenvectors of a related eigenvalue problem. The method is used to obtain the analytic solutions without the need of performing integral transforms. Numerical examples of the GaAs bimaterial due to a line force or dislocation are presented for illustration.     

Green’s functions of two-dimensional piezoelectric quasicrystal in half-space and bimaterials

In this paper, we obtain Green’s functions of two-dimensional (2D) piezoelectric quasicrystal (PQC) in half-space and bimaterials. Based on the elastic theory of QCs, the Stroh formalism is used to derive the general solutions of displacements and stresses. Then, we obtain the analytical solutions of half-space and bimaterial Green’s functions. Besides, the interfacial Green’s function for bimaterials is also obtained in the analytical form. Before numerical studies, a comparative study is carried out to validate the present solutions. Typical numerical examples are performed to investigate the effects of multi-physics loadings such as the line force, the line dislocation, the line charge, and the phason line force. As a result, the coupling effect among the phonon field, the phason field, and the electric field is prominent, and the butterfly-shaped contours are characteristic in 2D PQCs. In addition, the changes of material parameters cause variations in physical quantities to a certain degree.     

Interaction of an anti-plane singularity with interfacial anti-cracks in cylindrically anisotropic composites

Anti-plane problem for a singularity interacting with interfacial anti-cracks (rigid lines) under uniform shear stress at infinity in cylindrically anisotropic composites is investigated by utilizing a complex potential technique in this paper. After obtaining the general solution for this problem, the closed solution for the interface containing one anti-crack is presented analytically. In addition, the complex potentials for a screw dislocation dipole inside matrix are obtained by the superimposing method. Expressions of stress singularities around the anti-crack tips, image forces and torques acting on the dislocation or the center of dipole are given explicitly. The results indicate that the anisotropy properties of materials may weaken the stress singularity near the anti-crack tip for the singularity being a concentrated force but enhance the one for the singularity being a screw dislocation and change the equilibrium position of screw dislocation. The presented solutions are valid for anisotropic, orthotropic or isotropic composites and can be reduced to some new or previously known results.     

Independence of the Presence of Cracks or Inclusions of the Net Interaction Force Acting on a Dislocation by a Skewed Line Singularity

Orlov and Indenbom [1] have shown that the net (integrated) interaction force F between two skew dislocations with Burgers vectors separated by a distance h in an infinite anisotropic elastic medium is independent of h. Nix [2] computed numerically the net interaction force F between two skew dislocations that are parallel to the traction-free surface X2=0 of an isotropic elastic half-space. His numerical results showed that F was independent of h; a partial result of what Barnett [3] called Nix"s theorem. The separation-independence portion of Nix"s theorem has been proved to hold for a general anisotropic elastic half-space with a traction-free, rigid, or slippery surface, and for bimaterials [3-5]. In this paper, we show that the net interaction force is independent of the presence of inclusions. We will consider the case in which the line dislocation b is a more general line singularity which can include a coincident line force with strength f per unit length of the line singularity. An inclusion is an infinitely long dissimilar anisotropic elastic cylinder of an arbitrary cross-section whose axis is parallel to the line singularity (f, b). The (skew) line dislocation does not intersect the inclusion. The special cases of an inclusion are a void, crack, or rigid inclusion. There can be more than one inclusion of different cross sections and different materials. The line singularity (f, b) can be outside the inclusions or inside one of the inclusions. The inclusions and the matrix need not have a perfect bonding. One can have a debonding with or without friction. This revised version was published online in August 2006 with corrections to the Cover Date.     

Two-dimensional Green's functions in anisotropic multiferroic bimaterials with a viscous interface

We derive, by virtue of the unified Stroh formalism, the extremely concise and elegant solutions for two-dimensional and (quasi-static) time-dependent Green's functions in anisotropic magnetoelectroelastic multiferroic bimaterials with a viscous interface subjected to an extended line force and an extended line dislocation located in the upper half-plane. It is found for the first time that, in the multiferroic bimaterial Green's functions, there are 25 static image singularities and 50 moving image singularities in the form of the extended line force and extended line dislocation in the upper or lower half-plane. It is further observed that, as time evolves, the moving image singularities, which originate from the locations of the static image singularities, will move further away from the viscous interface with explicit time-dependent locations. Moreover, explicit expression of the time-dependent image force on the extended line dislocation due to its interaction with the viscous interface is derived, which is also valid for mathematically degenerate materials. Several special cases are discussed in detail for the image force expression to illustrate the influence of the viscous interface on the mobility of the extended line dislocation, and various interesting features are observed. These Green's functions can not only be directly applied to the study of dislocation mobility in the novel multiferroic bimaterials, they can also be utilized as kernel functions in a boundary integral formulation to investigate more complicated boundary value problems where multiferroic materials/composites are involved.     

A moving screw dislocation near interfacial rigid lines in two dissimilar anisotropic media

This paper attempts to investigate the problem for the interaction between a uniformly moving screw dislocation and interface rigid lines in two dissimilar.anisotropic. materials. Integrating Riemann-Schwarz＇s symmetry principle with the analysis singularity of complex functions, we present the general elastic solutions of this problem and the closed form solutions for interfaces containing one and two rigid lines. The expressions of stress intensity factors, at the rigid line tips and image force acting on moving dislocation are derived explicitly. The results show that dislocation velocity has an antishielding effect on the rigid line tip and a larger dislocation velocity leads to the equilibrium position of dislocation closing with the rigid line. The presented solutions contain previously known results as the special cases.     

Bonded contact of a flexible elliptical disk with a transversely isotropic half-space

The article concerns the problem of bonded contact of a thin, flexible elliptical disk with a transversely isotropic half-space. Three different cases of loading have been considered: (a) the disk is loaded by a transverse force, whose line of action passes through the center of the disk and lies in the plane of the disk; (b) the disk is subjected to a rotation by a torque, whose axis is perpendicular to the surface of the half-space; (c) the half-space with the bonded disk is under uniform stress field at infinity. The problem corresponding to all three cases is reduced, in a unified manner, to a set of coupled two-dimensional integral equations. Closed-form solutions for these equations have been obtained by using Galins theorem.     

Thermal stress intensity factors for a crack in an anisotropic half plane

On the basis of the two-dimensional theory of anisotropic thermoelasticity, a solution is given for the thermal stress intensity factors due to the obstruction of a uniform heat flux by an insulated line crack in a generally anisotropic half plane. The crack is replaced by continuous distributions of sources of temperature discontinuity and dislocations. First, the particular thermoelastic dislocation solutions for the half plane are obtained; then the corresponding isothermal solutions are superposed to satisfy the traction-free conditions on the crack surfaces. The dislocation solutions are applied to calculate the thermal stress intensity factors, which are validated by the exact solutions. The effects of the uniform heat flux, the ply angle and the crack length are investigated.     

Rapid sliding indentation with friction on a transversely isotropic thermoelastic half-space

A rigid insulated die slides at a constant sub-critical speed on a transversely isotropic half-space in the presence of friction. In a two-dimensional analysis of the dynamic steady-state, the coupled equations of thermoelasticity are invoked. All elements of the Coulomb friction model are strictly enforced, thus giving rise to auxiliary conditions, including two unilateral constraints.Robust asymptotic forms of an exact solution to a related problem with unmixed boundary conditions lead to analytical solutions for the sliding indentation problem. The solution expressions, abetted by calculations for zinc, show the role of frictional heating on the half-space surface. The effects of friction and sliding speed on contact zone size and location and average contact zone temperature are also studied.The analysis is aided by factoring procedures that simplify the complicated forms that arise in anisotropic elasticity. A scheme that renders expressions for roots of certain irrational functions analytic to within a single quadrature also plays a role.     

Problem on an inhomogeneity in a linearly elastic space or half-space

For a weakly contrasting anisotropic inhomogeneity in a linearly elastic homogeneous space or half-space, using the perturbation method, we obtain an approximate solution and estimate its accuracy. In the case of inhomogeneity of arbitrary contrast, we reduce the problem to a system of integral equations. In the general case, it is easy to compose the procedure for solving this problem approximately. In the special case of a homogeneous anisotropic ellipsoidal inhomogeneity in space, the strain state inside the inhomogeneity turns out to be homogeneous, and we thus obtain the exact solution of the problem.     

On the theory of reflection from a wire grid parallel to an interface between homogeneous media

Summary The reflection from a wire grid parallel to a plane interface is considered. The respective media are homogeneous and either or both can be dissipative. The grid is composed of thin equi-spaced wires of finite conductivity. The plane wave solution for arbitrary incidence is then generalized for cylindricalwave excitation. The energy absorbed from a magnetic line source by a grid situated on the surface of a dissipative half-space is treated in some detail. This latter problem is a two-dimensional analogy of a vertical antenna with a radial wire ground system.     

On linking n-dimensional anisotropic and isotropic Green’s functions for infinite space,half-space,bimaterial, and multilayer for conduction

We establish exact mathematical links between the n-dimensional anisotropic and isotropic Green’s functions for diffusion phenomena for an infinite space, a half-space, a bimaterial and a multilayered space. The purpose of this work is not to attempt to present a solution procedure, but to focus on the general conditions and situations in which the anisotropic physical problems can be directly linked with the Green’s functions of a similar configuration with isotropic constituents. We show that, for Green’s functions of an infinite and a half-space and for all two-dimensional configurations, the exact correspondences between the anisotropic and isotropic ones can always be established without any regard to the constituent conductivities or any other information. And thus knowing the isotropic Green’s functions will readily provide explicit expressions for anisotropic Green’s functions upon back transformation. For three- and higher-dimensional bimaterials and layered spaces, the correspondence can also be found but the constituent conductivities need to satisfy further algebraic constraints. When these constraints are fully satisfied, then the anisotropic Green’s functions can also be obtained from those of the isotropic ones, or at least in principle.     

GREEN FUNCTIONS FOR A DECAGONAL QUASICRYSTALLINE MATERIAL WITH A PARABOLIC BOUNDARY

This investigation presents the Green functions for a decagonal quasicrystalline material with a parabolic boundary subject to a line force and a line dislocation by means of the complex variable method. The surface Green functions are treated as a special case, and the explicit expressions of displacements and hoop stress at the parabolic boundary are also given.Finally, the stresses and displacements induced by a phonon line force acting at the origin of the lower half-space are presented.     

Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution

We study a mathematical model describing dislocation dynamics in crystals. We consider a single dislocation line moving in its slip plane. The normal velocity is given by the Peach-Koehler force created by the dislocation line itself. The mathematical model is an eikonal equation with a velocity which is a non-local quantity depending on the whole shape of the dislocation line. We study the special case where the dislocation line is assumed to be a graph or a closed loop. In the framework of discontinuous viscosity solutions for Hamilton–Jacobi equations, we prove the existence and uniqueness of a solution for small time. We also give physical explanations and a formal derivation of the mathematical model. Finally, we present numerical results based on a level-sets formulation of the problem. These results illustrate in particular the fact that there is no general inclusion principle for this model.     

压电螺型位错和共线界面刚性线夹杂的干涉效应

研究了压电材料中压电螺型位错和共线界面导电刚性线夹杂的电弹干涉效应．运用复变函数解析延拓技术与奇性主部分析方法,获得了该问题的一般解答．作为算例,求出了界面含一条刚性线夹杂时两种压电介质区域广义应力函数的封闭形式解．导出了作用在位错上的像力和刚性线夹杂表面剪应力和电位移的解析表达式．讨论了界面刚性线长度,两种材料的剪切模量比和压电系数比对位错力和刚性线表面剪应力的影响规律．为进一步研究该类问题提供了一个基本解。