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A series solution is presented for a spherical inclusion embedded in an infinite matrix under a remotely applied uniform intensity. Particularly, the interface between the inclusion and the matrix is considered to be inhomegeneously bonded. We examine the axisymmetric case in which the interface parameter varies with the cone angle θ. Two kinds of imperfect interfaces are considered: an imperfect interface which models a thin interphase of low conductivity and an imperfect interface which models a thin interphase of high conductivity. We show that, by expanding the solutions of terms of Legendre polynomials, the field solution is governed by a linear set of algebraic equations with an infinite number of unknowns. The key step of the formulation relies on algebraic identities between coefficients of products of Legendre series. Some numerical illustrations are presented to show the correctness of the presented procedures. Further, solutions of the boundary-value problem are employed to estimate the effective conductivity tensor of a composite consisting of dispersions of spherical inclusions with equal size. The effective conductivity solely depends on one particular constant among an infinite number of unknowns.
The thermal stress for a penny-shaped crack contained in an infinite isotropic elastic solid initially subjected to an axisymmetrical tension of any amount at infinity is investigated using the techniques of Hankel transforms and multiplying factors. The effect that the lateral normal stress has on the thermal stresses is studied on the basis of the theory of small deformations superposed on finite deformation. Symmetrical thermal loadings are applied over the crack surfaces. For the case of constant temperature over the crack surfaces, expressions for the crack shape and thermal stresses in the crack plane are obtained in closed forms. The stress intensity factor is also obtained and shown to be dependent on the lateral stress.
The general approximate solutions for the two-dimensional thermoelastic problems with a nearly circular hole are provided in this study. Based on Stroh formalism and the method of conformal mapping, the boundary perturbation analysis is applied to solve the problems of a hole with arbitrary shape. The radius of the hole considered here is represented as a sum of a reference constant and a perturbation magnitude that is expanded into a Fourier series. In order to illustrate the applicability and efficiency of the present approach, special examples associated with polygonal hole problems are solved explicitly and discussed in detail. Since the general solutions have not been found in the literature, comparison is made with some special cases for which the analytical solutions exist, which shows that our proposed method is effective and general.
An approximate anisotropic yield function is presented for anisotropic sheet metals containing spherical voids. Hill's quadratic anisotropic yield function is used to describe the anisotropy of the matrix. The proposed yield function is validated using a three-dimensional finite element analysis of a unit cell model under different straining paths. The results of the finite element computations are shown in good agreement with those based on the yield function with three fitting parameters. For demonstration of applicability, the anisotropic Gurson yield function is adopted in a combined necking and shear localization analysis to model the failure of AA6111 aluminum sheets under biaxial stretching conditions.
In this paper, a discrete system model and its equation of motion for beams with arbitrary supports at two ends are established. These supports include elastic, rigid and free supports in translation and rotation directions. Based on theory of oscillatory matrices, a series of qualitative properties of frequencies and modes of this system are derived. The basic properties include: non-zero frequencies are distinct; the ith displacement mode has i - 1 nodes; nodes of ith mode and (i + 1)th mode interlace.
Some additional important qualitative properties owned by rotation modes and strain modes are given as well.
Without knowing the dynamic constitutive relation of materials under high strain rates, no wave propagation can be correctly analyzed. A Series of experimental and theoretical investigation at high strain rates revealed that the nonlinear viscoelastic behavior of polymers and the related composites are well described by the Zhu-Wang-Tang (ZWT) nonlinear viscoelastic constitutive equation. The impulsive reponse of ZWT materials consists of a rate independent nonlinear elastic response and a high frequency linear viscoelastic response. The dispersion and attenuation of nonlinear viscoelastic waves mainly depend on the effective nonlinearity and the high frequency relaxation time θ2. An “effective influence distance” or “effective influence time” is defined to characterize the wave propagation range where θ2 dominates the impact relaxation process.
With the aid of a six-dimensional special eigenvector q, T.C.T.Ting finds five new invariants of anisotropic elasticity constants. The purpose of this paper is to consider some character of the eigenvector q. It is pointed that the six-dimensional special eigenvector q is unique, if it is independent of the coordinate transformation, and the general form of a three-rank orthogonal matrix is given if it has a three-dimensional special eigenvector like q. In addition, the concept of the special eigenvector q is extended and 20 invariants of anisotropic elasticity constants are obtained under rotation about x3-axis.
Shooting angle of an inverse ray for imaging 2D multi-layered structures from reflected travel-times is derived in a closed form. By considering the normal incidence of two neighboring rays reflected at interfaces when sources are at the same locations as receivers, the traveling distance and direction of two inverse-rays are determined successively from the lowermost layer to the uppermost layer. This approach is similar to and also confirmed with the Huygens' principle that the equal travel-time along a wave front (perpendicular to the rays) is conserved. The closed-form solution of the inverse rays is further applied to image a complex structure of a ramp-flat fault with eleven layers. The results demonstrate that the inverse-ray imaging from travel-time picks of all layers is superior to that picked by a layer-stripping approach.
When an elastic medium containing an elliptic inclusion with a sliding interface is subjected to a remote pure shear, it was found that the inclusion behaves like a cavity. Since a circle is a special case of an ellipse, the solution should be applicable to a circular inclusion as well. However, it breaks down when the ellipse degenerates into a circle. This implies that the solution is questionable. In this paper the problem is examined by considering a rigid elliptic inclusion in an elastic medium with sliding interface between them. By taking account of a large rotation of the inclusion instead of a small rotation, we obtain a uniformly valid solution applicable to a circular inclusion as well as to an elliptic inclusion. The solution reveals a remarkable snapping behavior of the inclusion under a critical load. A simple condition for its occurrence is derived.
The effect of an imperfect interface on the stress singularity of anisotropic bimaterial wedges subjected to traction free boundary conditions are investigated. The interfacial tractions are assumed to be continuous, directly proportional to the displacement jumps and inversely proportional to the radial coordinate. The characteristic equation for the order of singularity is obtained and numerical results are given for the angle-ply bimaterial composite wedge.
The pairing of a chemical potential and its associated concentration rate in the thermodynamic identity is well known. The existence of an experimentally determinable molar volume as a function of molar concentrations is also widely used in chemical engineering. What is perhaps less known and infrequently used is the fact that a spatially nonuniform molar volume leads to a field of geometrically incompatible eigenstrain, or eigentransformation in finite deformation. This incompatibility forces the material environment to deform, and the result is a strain energy trapped inside the material body. The change of this energy with respect to the eigentransformation is a generalized configurational stress, which, in the limit as the eigentransformation tends to the identity transformation, tends to the classical energy momentum tensor of Eshelby , or the so-called configurational stress. It is shown that the generalized configurational stress is an integral part of the chemical potentials that are responsible for atomic diffusion. This cycle of cause and effect is demonstrated in an axially symmetric setting where the material configuration is taken to be cylindrically orthotropic.
Symmetry can be used in many occasions to bare the simple meaning of a complex mathematical expression hiding under the disguise of tensors and index notation. We used it to elucidate the “missing” term in surface chemical potential in 1996  and are now applying it to identify the chemical potential in the bulk. Both Professor Thomas C. T. Ting and I are civil engineering graduates of Tai-Da, the world-renowned National Taiwan University, but did not know each other until we joined UIC. It has been a wonderful friendship of many stimulating anisotropic discussions and numerous delicious potluck dinners. Happy birthday, Tom!
The transient motion in an anisotropic elastic half-space due to a moving surface line load is considered. The load is applied suddenly on the surface and moves off in a fixed direction with nonuniform speed. Integral expressions for the displacements are derived using the reciprocal theorem. The waves generated by the moving load are discussed. Special attention is paid to the singularities in surface displacements generated as the load moves through the Rayleigh wave speed. Explicit expression is obtained for the particle velocity due to a constant load moving with constant speed.
In this paper, we utilized a Stroh based formulation for solving problems of surface waves in layered piezoelectric media, and then, applied it to analyze surface acoustic wave (SAW) devices. The determination of the optimal cut of a piezoelectric crystal and the choice of the best propagation of SAW devices were given. The dispersion induced by a thin metal layer on SAW propagation in a SAW device was analyzed and discussed. Finally, we applied the formulation to calculate the effective permittivity and phase velocity dispersion of a LiNbO3/Diamond layered SAW device. Both of the null frequency bandwidth and the insertion loss of the dispersive SAW device were obtained.
This brief note discusses some issues related to the calculation of energy release rate for elliptical cracks in anisotropic solids. By using the Stroh formalism, analytical expressions of the energy release rate are obtained for elliptical cracks in an unbounded anisotropic solid. Because of material anisotropy and geometric asymmetry of the crack, the local energy release rate varies along the crack front. The average energy release rate can be obtained by integrating the local energy release rate over the entire crack front. On the other hand, the total work done by the crack-surface traction on the entire crack opening displacement can be easily evaluated once the crack opening displacement is known. It is shown that the average energy release rate is equal to the rate of change per unit crack area increment of the work done by the external load on the crack opening displacement.
Based on Stroh's formalism, analytic solutions are derived for a half-infinite coupled crack in piezoelectric bimaterials without oscillation by using analytical function technique. Four intensity factors related to the crack tip fields are obtained. It is found that these intensity factors are independent of the bimaterials constants when no oscillation occurs. Some numerical calculations about the stress tensor and the electric field ahead of the crack tip are conducted finally.
In this paper, a modified version of method of steepest descent combining with Durbin's method is proposed to study the transient motion in either an elastic or a viscoelastic half-space. The causal condition is satisfied based on the Durbin's method while the wavenumber integral for any range of frequency is evaluated by applying the modified method of steepest descent. The validity and accuracy of the proposed method is tested by studying the transient response generated by a buried dilatational line source in an elastic half-space, for which the exact solution (Garvin's solution) can be obtained. Then the same formalism is extended to Kelvin-Voigt half-space, and the transient surface motions in elastic or viscoelastic half-spaces media are studied and discussed in details.
The Green's function for the damped wave equation on a finite interval subject to two Robin boundary conditions is studied. The problem for a semi-infinite interval with one Robin boundary is considered first by the Laplace transform method to establish the reflection principle at a Robin boundary. It is seen that the reflected wave generated by an exiting wave at a Robin boundary is a convolution involving the exiting wave and some kernel function. This generalizes the well known classical results for reflections at Dirichlet and Neumann boundaries. The reflection principle at a Robin boundary is then used for the finite interval case by considering multiple reflections and this leads to an infinite sequence of waves. In order to justify the results obtained here we also study the finite interval case by the Laplace transform method. The Laplace transform of the Green's function is expanded, for large values of the transform parameter, into an infinite series of negative exponentials and then inverted term by term. Agreement is reached between these two approaches.
A modified image method is presented for obtaining the solutions of the fundamental singularities in the neighborhood of a plane interface between two semi-infinite, immiscible, and incompressible viscous fluids. The fundamental singularities considered are the stokeslet, rotlet, stresslet, stokes-doublet, source, and source-doublet. The Galerkin vector function representation introduced reduces the complexity of the expressions for the solutions. Moreover, the physical meaning of each solution is clearly identified by these new expressions.