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.

Linear Anisotropic Elastic Materials

1996 ◽  
Author(s):  
T. T. C. Ting
Keyword(s):  
Elastic Constants ◽  
Elastic Material ◽  
Positive Definite ◽  
Material Symmetry ◽  
Two Dimensional ◽  
Elastic Materials ◽  
Rectangular Coordinate System ◽  
Full Symmetry ◽  
Anisotropic Elastic ◽  
Anisotropic Elastic Material

The relations between stresses and strains in an anisotropic elastic material are presented in this chapter. A linear anisotropic elastic material can have as many as 21 elastic constants. This number is reduced when the material possesses a certain material symmetry. The number of elastic constants is also reduced, in most cases, when a two-dimensional deformation is considered. An important condition on elastic constants is that the strain energy must be positive. This condition implies that the 6×6 matrices of elastic constants presented herein must be positive definite. Referring to a fixed rectangular coordinate system x1, x2, x3, let σij and εks be the stress and strain, respectively, in an anisotropic elastic material. The stress-strain law can be written as . . . σij = Cijksεks . . . . . .(2.1-1). . . in which Cijks are the elastic stiffnesses which are components of a fourth rank tensor. They satisfy the full symmetry conditions . . . Cijks = Cjiks, Cijks = Cijsk, Cijks = Cksij. . . . . . .(2.1-2). . .

Steady State Motion and Surface Waves

1996 ◽  
Author(s):  
T. T. C. Ting
Keyword(s):  
Steady State ◽  
Surface Wave ◽  
Surface Waves ◽  
Existence And Uniqueness ◽  
Wave Speed ◽  
Wave Theory ◽  
Two Dimensional ◽  
Elastic Materials ◽  
Steady State Motion ◽  
Anisotropic Elastic

The Stroh formalism for two-dimensional elastostatics can be extended to elastodynamics when the problem is a steady state motion. Most of the identities in Chapters 6 and 7 remain applicable. The Barnett-Lothe tensors S, H, L now depend on the speed υ of the steady state motion. However S(υ), H(υ), L(υ) are no longer tensors because they do not obey the laws of tensor transformation when υ≠0. Depending on the problems the speed υ may not be prescribed arbitrarily. This is particularly the case for surface waves in a half-space where υ is the surface wave speed. The problem of the existence and uniqueness of a surface wave speed in anisotropic materials is the crux of surface wave theory. It is a subject that has been extensively studied since the pioneer work of Stroh (1962). Excellent expositions on surface waves for anisotropic elastic materials have been given by Farnell (1970), Chadwick and Smith (1977), Barnett and Lothe (1985), and more recently, by Chadwick (1989d).

The Lekhnitskii Formalism

1996 ◽  
Author(s):  
T. T. C. Ting
Keyword(s):  
Symmetry Plane ◽  
Stroh Formalism ◽  
Two Dimensional ◽  
Elastic Materials ◽  
Engineering Community ◽  
Anisotropic Elastic

A two-dimensional deformation means that the displacements ui, (i= 1,2,3) or the stresses σij depend on x1 and x2 only. Among several formalisms for two-dimensional deformations of anisotropic elastic materials the Lekhnitskii (1950, 1957) formalism is the oldest, and has been extensively employed by the engineering community. The Lekhnitskii formalism essentially generalizes the Muskhelishvili (1953) approach for solving two-dimensional deformations of isotropic elastic materials. The formalism begins with the stresses and assumes that they depend on x1 and x2 only. The Stroh formalism, to be introduced in the next chapter, starts with the displacements and assumes that they depend on x1 and x2 only. Therefore the Lekhnitskii formalism is in terms of the reduced elastic compliances while the Stroh formalism is in terms of the elastic stiffnesses. It should be noted that Green and Zerna (1960) also proposed a formalism for two-dimensional deformations of anisotropic elastic materials. Their approach however is limited to materials that possess a symmetry plane at x3=0. The derivations presented below do not follow exactly those of Lekhnitskii.

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.

Dispersion equations for steady waves in an anisotropic elastic plate or a layered plate

2007 ◽  
Vol 464 (2091) ◽  
pp. 613-629 ◽  
Author(s):  
T.C.T Ting
Keyword(s):  
Sandwich Plate ◽  
Sliding Contact ◽  
Interface Conditions ◽  
Elastic Materials ◽  
General Anisotropy ◽  
Dispersion Equations ◽  
Slippery Surface ◽  
Anisotropic Elastic ◽  
Anisotropic Elastic Material ◽  
Special Case

Steady waves propagating in a plate that consists of one or more layers of general anisotropic elastic material are studied. The surface of the plate can be a traction-free (F), rigid (R) or slippery surface (S). The interface between any two layers in the plate can be perfectly bonded (b) or in sliding contact (s). The thickness of the layers need not be the same. The purpose of this paper is to present dispersion equations for all possible combinations of the boundary and interface conditions. If the thickness h of one of the layers is very small, the dispersion equation allows us to expand the solution in an infinite series in the power of h from which an approximate solution can be obtained by keeping the terms up to O ( h n ) for any n . The special case of a sandwich plate that consists of a centre layer and two identical outside layers is studied. In the literature, the dispersion equations for a sandwich plate were studied for special elastic materials. The results presented here are for elastic materials of general anisotropy.

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.

Note on the shearing motion of fluid past a projection

1944 ◽  
Vol 40 (3) ◽  
pp. 214-222 ◽  
Author(s):  
W. R. Dean
Keyword(s):  
Rational Function ◽  
Plane Strain ◽  
Viscous Liquid ◽  
Plane Stress ◽  
Two Dimensional ◽  
Close Approximation ◽  
Shearing Motion

1. A method has lately been developed by N. Muschelišvili (1) for the solution of problems of the slow two-dimensional motion of viscous liquid and of the corresponding problems of plane stress and plane strain, in cases in which the area in the x, y-plane that is concerned can be represented conformally on the interior of the circle |ζ| = 1 in the ζ-plane by a relation of the form z = x + iy = r(ζ), where r(ζ) is a rational function of ζ. In most problems in which the method has been used the function r(ζ) has been a simple one, but it is of importance to consider a rational function of as general a form as possible since, given any relation z = f(ζ), it will usually be possible to find a rational function that approximates to f(ζ) throughout the circle |ζ| = 1 and for a close approximation a complicated function r(ζ) will in general be required.

Three-dimensional thermo-mechanical analysis of continuous casting and comparison with two-dimensional models

2018 ◽  
Vol 53 (6) ◽  
pp. 421-434
Author(s):  
Reza Vaghefi ◽  
MR Hematiyan ◽  
Ali Nayebi
Keyword(s):  
Continuous Casting ◽  
Phase Change ◽  
Galerkin Method ◽  
Plane Strain ◽  
Plane Stress ◽  
Three Dimensional ◽  
Casting Process ◽  
Mechanical Analysis ◽  
Two Dimensional ◽  
Generalized Plane Strain

In this study, a three-dimensional thermo-elasto-plastic model is developed for simulating a continuous casting process. The obtained results are compared with those from different two-dimensional analyses, which are based on plane stress, plane strain, and generalized plane strain assumptions. All analyses are carried out using the meshless local Petrov–Galerkin method. The effective heat capacity method is employed to simulate the phase change process. The von Mises yield criterion and elastic–perfectly-plastic model are used to simulate the stress state during the casting process; while, material parameters are assumed to be temperature-dependent. Based on the three-dimensional and two-dimensional models, numerical results are provided to determine the stress, displacement, and temperature fields induced in the cast material. It is observed that the present meshless local Petrov–Galerkin method is accurate in three-dimensional thermo-mechanical analysis of highly nonlinear phase change problems. Reasonable agreements are observed between the results obtained from the three-dimensional analysis with those retrieved by the generalized plane strain assumption. However, it is observed that the results obtained under plane stress/strain conditions have some significant differences with the results obtained from three-dimensional modeling of continuous casting.

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.

Sign in / Sign up

Export Citation Format

Share Document