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.

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.

The method of caustics for the determination of normal loads acting on surfaces of elastic bodies

1980 ◽  
Vol 15 (1) ◽  
pp. 37-41 ◽  
Author(s):  
P S Theocaris ◽  
N I Ioakimidis
Keyword(s):  
Three Dimensional ◽  
Contact Problems ◽  
Two Dimensional ◽  
Concentrated Force ◽  
Elastic Bodies ◽  
Plate And Shell ◽  
Elasticity Problems ◽  
Three Dimensional Elasticity ◽  
Quantities Of Interest

The optical method of caustics constitutes an efficient experimental technique for the determination of quantities of interest in elasticity problems. Up to now, this method has been applied only to two-dimensional elasticity problems (including plate and shell problems). In this paper, the method of caustics is extended to the case of three-dimensional elasticity problems. The particular problems of a concentrated force and a uniformly distributed loading acting normally on a half-space (on a circular region) are treated in detail. Experimentally obtained caustics for the first of these problems were seen to be in satisfactory agreement with the corresponding theoretical forms. The treatment of various, more complicated, three-dimensional elasticity problems, including contact problems, by the method of caustics is also possible.

A Modified Conjugate Gradient Method for Frictionless Contact Problems

1983 ◽  
Vol 105 (2) ◽  
pp. 242-246 ◽  
Author(s):  
W. R. Marks ◽  
N. J. Salamon
Keyword(s):  
Finite Element ◽  
Conjugate Gradient ◽  
Solution Method ◽  
Contact Region ◽  
Contact Problems ◽  
Computer Code ◽  
Frictionless Contact ◽  
Elastic Bodies ◽  
Gradient Technique ◽  
Modified Conjugate Gradient Method

The solution of elastic bodies in contact through the application of a conjugate gradient technique integrated with a finite element computer code is discussed. This approach is general, easily applied, and reasonably efficient. Furthermore the solution method is compatible with existing finite element computer programs. The necessary algorithm for use of the technique is described in detail. Numerical examples of two-dimensional frictionless contact problems are presented. It is found that the extent of the contact region and the displacements and stresses throughout the contacting bodies can be economically computed with precision.

scholarly journals CONTINUOUS ABELIAN SANDPILE MODEL IN TWO DIMENSIONAL LATTICE

2011 ◽  
Vol 25 (32) ◽  
pp. 4709-4720 ◽  
Author(s):  
N. AZIMI-TAFRESHI ◽  
E. LOTFI ◽  
S. MOGHIMI-ARAGHI
Keyword(s):  
Boundary Conditions ◽  
Small Mass ◽  
Two Dimensional ◽  
Dimensional Lattice ◽  
New Model ◽  
Different Boundary Conditions ◽  
Sandpile Model ◽  
Abelian Sandpile Model ◽  
General Properties ◽  
Abelian Sandpile

We investigate a new version of sandpile model which is very similar to Abelian Sandpile Model (ASM), but the height variables are continuous ones. With the toppling rule we define in our model, we show that the model can be mapped to ASM, so the general properties of the two models are identical. Yet the new model allows us to investigate some problems such as the effect of very small mass on the height probabilities, different boundary conditions, etc.

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.

Antiplane Deformations

1996 ◽  
Author(s):  
T. T. C. Ting
Keyword(s):  
Plane Strain ◽  
Plane Stress ◽  
Antiplane Deformation ◽  
Two Dimensional ◽  
Elastic Materials ◽  
Rectangular Coordinate System ◽  
Elastic Bodies ◽  
Basic Solutions ◽  
Anisotropic Elastic ◽  
Anisotropic Elastic Material

As a starter for anisotropic elastostatics we study special two-dimensional deformations of anisotropic elastic bodies, namely, antiplane deformations. Not all anisotropic elastic materials are capable of an antiplane deformation. When they are, the inplane displacement and the antiplane displacement are uncoupled. The deformations due to inplane displacement are plane strain deformations. Associated with plane strain deformations are plane stress deformations. After defining these special deformations in Sections 3.1 and 3.2 we present some basic solutions of antiplane deformations. They provide useful references for more general deformations we will study in Chapters 8, 10, and 11. The derivation and motivation in solving more general deformations in those chapters become more transparent if the reader reads this chapter first. The solutions obtained in those chapters reduce to the solutions presented here when the materials are restricted to special materials and the deformations are limited to antiplane deformations. In a fixed rectangular coordinate system xi (i=1, 2, 3), let ui, σij, and εij be the displacement, stress, and strain, respectively. The strain-displacement relations and the equations of equilibrium are . . .εij = 1/2 (ui,j + uj,i),. . . . . . (3.1 -1) . . . . . .σij,j =0,. . . . . . (3.1 - 2). . . in which repeated indices imply summation and a comma stands for differentiation. The stress-strain laws for an anisotropic elastic material can be written as σij = Cijks εks or εij = Sijksσks, . . .(3.1 - 3). . . where Cijks and Sijks are, respectively, the elastic stiffnesses and compliances.

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.

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