Discussion on the conformal mapping of a half-plane onto a unit disk in anisotropic elasticity and related applications

2020 ◽  
Vol 26 (1) ◽  
pp. 110-117
Author(s):  
Ming Dai ◽  
Jian Hua
Keyword(s):  
Thin Films ◽  
Conformal Mapping ◽  
Unit Disk ◽  
Half Plane ◽  
Complex Variable ◽  
Plane Deformation ◽  
Anisotropic Elasticity ◽  
Two Dimensional ◽  
Plane Shear ◽  
Anisotropic Elastic

The conformal mapping, which transforms a half-plane into a unit disk, has been used widely in studies involving an isotropic elastic half-plane under anti-plane shear or plane deformation. However, very little attention has been paid to the possibility of utilizing this mapping in the study of an anisotropic elastic half-plane under the same deformation. In this paper, we discuss a general case of an arbitrarily located anisotropic elastic half-plane that corresponds to several affine counterparts (resulting from corresponding complex variable formalism). We show that this mapping is indeed applicable to each of the affine half-planes if and only if the key parameters in the mapping satisfy simple geometrical conditions. In addition, we introduce the application of this mapping with the corresponding geometrical conditions to the related study of anisotropic thin films under two-dimensional deformation.

The Stroh Formalism

1996 ◽  
Author(s):  
T. T. C. Ting
Keyword(s):  
General Solution ◽  
Elastic Body ◽  
Three Dimensional ◽  
Anisotropic Elasticity ◽  
Stroh Formalism ◽  
Two Dimensional ◽  
Stroh's Formalism ◽  
Elasticity Problems ◽  
Anisotropic Elastic ◽  
Stroh’S Formalism

In this chapter we study Stroh's sextic formalism for two-dimensional deformations of an anisotropic elastic body. The Stroh formalism can be traced to the work of Eshelby, Read, and Shockley (1953). We therefore present the latter first. Not all results presented in this chapter are due to Stroh (1958, 1962). Nevertheless we name the sextic formalism after Stroh because he laid the foundations for researchers who followed him. The derivation of Stroh's formalism is rather simple and straightforward. The general solution resembles that obtained by the Lekhnitskii formalism. However, the resemblance between the two formalisms stops there. As we will see in the rest of the book, the Stroh formalism is indeed mathematically elegant and technically powerful in solving two-dimensional anisotropic elasticity problems. The possibility of extending the formalism to three-dimensional deformations is explored in Chapter 15.

scholarly journals Conservation laws in anisotropic elasticity I. Basic framework

1993 ◽  
Vol 443 (1917) ◽  
pp. 139-151 ◽  
Keyword(s):  
Conservation Laws ◽  
General Procedure ◽  
Anisotropic Elasticity ◽  
Two Dimensional ◽  
Real Form ◽  
Elastic Materials ◽  
Complete Class ◽  
First Order ◽  
Cauchy's Theorem ◽  
Anisotropic Elastic

A complete class of first order conservation laws for two dimensional deformations in general anisotropic elastic materials is derived. The derivations are based on Stroh’s formalism for anisotropic elasticity. The general procedure proposed by P. J. Olver for the construction of conservation integrals is followed. It is shown that the conservation laws are intimately connected with Cauchy’s theorem for complex analytic functions. Real-form conservation laws that are valid for degenerate or non-degenerate materials are given.

Contact Problems of Two Dissimilar Anisotropic Elastic Bodies

1998 ◽  
Vol 65 (3) ◽  
pp. 580-587 ◽  
Author(s):  
Chyanbin Hwu ◽  
C. W. Fan
Keyword(s):  
Boundary Conditions ◽  
Contact Region ◽  
Complex Function ◽  
Contact Problems ◽  
Anisotropic Elasticity ◽  
Analytical Continuation ◽  
Two Dimensional ◽  
Different Boundary Conditions ◽  
Elastic Bodies ◽  
Anisotropic Elastic

In this paper, a two-dimensional contact problem of two dissimilar anisotropic elastic bodies is studied. The shapes of the boundaries of these two elastic bodies have been assumed to be approximately straight, but the contact region is not necessary to be small and the contact surface can be nonsmooth. Base upon these assumptions, three different boundary conditions are considered and solved. They are: the contact in the presence of friction, the contact in the absence of friction, and the contact in complete adhesion. By applying the Stroh’s formalism for anisotropic elasticity and the method of analytical continuation for complex function manipulation, general solutions satisfying these different boundary conditions are obtained in analytical forms. When one of the elastic bodies is rigid and the boundary shape of the other elastic body is considered to be fiat, the reduced solutions can be proved to be identical to those presented in the literature for the problems of rigid punches indenting into (or sliding along) the anisotropic elastic halfplane. For the purpose of illustration, examples are also given when the shapes of the boundaries of the elastic bodies are approximated by the parabolic curves.

Integral characteristics of elastic inclusions and cavities in the two-dimensional theory of elasticity

1992 ◽  
Vol 3 (1) ◽  
pp. 21-30 ◽  
Author(s):  
A. B. Movchan
Keyword(s):  
Conformal Mapping ◽  
Unit Disk ◽  
Theory Of Elasticity ◽  
Two Dimensional ◽  
Dimensional Theory ◽  
The Matrix ◽  
Integral Characteristics ◽  
Elastic Inclusions

Integral characteristics, such as elastic polarization matrices of elastic inclusions and cavities, are described. The matrix of elastic polarization of a finite cavity is constructed in the case of the two-dimensional Lamé operator under the assumption that the geometry of the domain occupied by the cavity is defined by a conformal mapping from the unit disk. Examples and applications of these integral characteristics in the theory of cracks are considered.

Three-Dimensional Deformations

1996 ◽  
Author(s):  
T. T. C. Ting
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Anisotropic Elasticity ◽  
Two Dimensional ◽  
Elastic Solids ◽  
Elastic Materials ◽  
Dimensional Solution ◽  
Line Integral ◽  
Isotropic Elasticity ◽  
Anisotropic Elastic

There appears to be very little study, if any, on the extension of Stroh's formalism to three-dimensional deformations of anisotropic elastic materials. In most three-dimensional problems the analyses employ approaches that are remotely related to Stroh's two-dimensional formalism. This is not unexpected, since this has been the situation between two-dimensional and three-dimensional isotropic elasticity. However it needs not be the case for three-dimensional anisotropic elasticity. Much can be gained if a connection to the Stroh formalism can be established. Barnett and Lothe (1975a) appeared to be the only ones who made a connection between a three-dimensional solution and Stroh's two-dimensional formalism. Earlier, several investigators obtained the Green's function for the infinite anisotropic medium in term of a line integral on an oblique plane in the three-dimensional space. That line integral, as we will see here, is one of Barnett-Lothe tensors on an oblique plane. We propose in this chapter extensions and applications of Stroh's two-dimensional formalism to certain three-dimensional deformations of anisotropic elastic solids.

Stroh Finite Element for Two-Dimensional Linear Anisotropic Elastic Solids

2000 ◽  
Vol 68 (3) ◽  
pp. 468-475
Author(s):  
Chyanbin Hwu ◽  
J. Y. Wu ◽  
C. W. Fan ◽  
M. C. Hsieh
Keyword(s):  
Finite Element ◽  
General Solution ◽  
Anisotropic Elasticity ◽  
Element Formulation ◽  
Two Dimensional ◽  
Elastic Solids ◽  
Field Problem ◽  
Concentration Problem ◽  
Uniform Stress Field ◽  
Anisotropic Elastic

A general solution satisfying the strain-displacement relation, the stress-strain laws and the equilibrium conditions has been obtained in Stroh formalism for the generalized two-dimensional anisotropic elasticity. The general solution contains three arbitrary complex functions which are the basis of the whole field stresses and deformations. By selecting these arbitrary functions to be linear or quadratic, and following the direct finite element formulation, a new finite element satisfying both the compatibility and equilibrium within each element is developed in this paper. A computer windows program is then coded by using the FORTRAN and Visual Basic languages. Two numerical examples are shown to illustrate the performance of this newly developed finite element. One is the uniform stress field problem, the other is the stress concentration problem.

Thermoelastic Problems for the Anisotropic Elastic Half-Plane

2004 ◽  
Vol 126 (3) ◽  
pp. 459-465 ◽  
Author(s):  
Yuan Lin ◽  
Timothy C. Ovaert
Keyword(s):  
Half Plane ◽  
Hilbert Problem ◽  
Thermoelastic Problem ◽  
Two Dimensional ◽  
Transfer Condition ◽  
Isotropic Media ◽  
Anisotropic Elastic ◽  
Steady State Heat ◽  
Steady State Heat Transfer ◽  
Thermoelastic Problems

By applying the extended version of Stroh’s formalism, the two-dimensional thermoelastic problem for a semi-infinite anisotropic elastic half-plane is formulated. The steady-state heat transfer condition is assumed and the technique of analytical continuation is employed; the formulation leads to the Hilbert problem, which can be solved in closed form. The general solutions due to different kinds of thermal and mechanical boundary conditions are obtained. The results show that unlike the two-dimensional thermoelastic problem for an isotropic media, where a simply-connected elastic body in a state of plane strain or plane stress remains stress free if the temperature distribution is harmonic and the boundaries are free of traction, the stress within the semi-infinite anisotropic media will generally not equal zero even if the boundary is free of traction.

Stroh-Like Complex Variable Formalism for the Bending Theory of Anisotropic Plates

2003 ◽  
Vol 70 (5) ◽  
pp. 696-707 ◽  
Author(s):  
C. Hwu
Keyword(s):  
Complex Variable ◽  
Anisotropic Elasticity ◽  
Stroh Formalism ◽  
Plate Bending ◽  
Two Dimensional ◽  
New Formalism ◽  
Anisotropic Plates ◽  
Present Formalism ◽  
Bending Problems ◽  
Almost All

Based upon the knowledge of the Stroh formalism and the Lekhnitskii formalism for two-dimensional anisotropic elasticity as well as the complex variable formalism developed by Lekhnitskii for plate bending problems, in this paper a Stroh-like formalism for the bending theory of anisotropic plates is established. The key feature that makes the Stroh formalism more attractive than the Lekhnitskii formalism is that the former possesses the eigenrelation that relates the eigenmodes of stress functions and displacements to the material properties. To retain this special feature, the associated eigenrelation and orthogonality relation have also been obtained for the present formalism. By intentional rearrangement, this new formalism and its associated relations look almost the same as those for the two-dimensional problems. Therefore, almost all the techniques developed for the two-dimensional problems can now be applied to the plate bending problems. Thus, many unsolved plate bending problems can now be solved if their corresponding two-dimensional problems have been solved successfully. To illustrate this benefit, two simple examples are shown in this paper. They are anisotropic plates containing elliptic holes or inclusions subjected to out-of-plane bending moments. The results are simple, exact and general. Note that the anisotropic plates treated in this paper consider only the homogeneous anisotropic plates. If a composite laminate is considered, it should be a symmetric laminate to avoid the coupling between stretching and bending behaviors.

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