On the growth of longitudinal waves in anisotropic elastic materials

1970 ◽  
Vol 36 (5) ◽  
pp. 381-389 ◽  
Author(s):  
Peter J. Chen
Keyword(s):  
Elastic Materials ◽  
Longitudinal Waves ◽  
Anisotropic Elastic

Anisotropic elastic materials that have one or two sheets of spherical slowness surface

2006 ◽  
Vol 462 (2074) ◽  
pp. 3133-3149 ◽  
Author(s):  
T.C.T Ting
Keyword(s):  
Wave Surface ◽  
Elastic Materials ◽  
Slowness Surface ◽  
Transverse Waves ◽  
Longitudinal Waves ◽  
Velocity Surface ◽  
Unusual Phenomenon ◽  
Anisotropic Elastic ◽  
The Waves ◽  
Special Case

It is shown that certain anisotropic elastic materials can have one or two sheets of spherical slowness surface. The waves associated with a spherical slowness sheet can be longitudinal, transverse or neither. However, a longitudinal wave can propagate in any direction if and only if the slowness sheet is a sphere . The same cannot be said of transverse waves. A transverse wave can propagate in any direction without having a spherical slowness sheet. If a spherical slowness sheet exists, the waves need not be transverse. The existence of a spherical slowness sheet means that the associated velocity surface and the wave surface also have a sphere. Thus, one sheet of the wave front due to a point source is a sphere, a rather unusual phenomenon for anisotropic elastic materials. Particularly interesting anisotropic elastic materials are the ones in which one longitudinal and two transverse waves can propagate in any direction. They have one spherical slowness sheet for the longitudinal waves. In the special case, they have a second spherical slowness sheet which is disjoint from the spherical slowness sheet . The third slowness sheet is a spheroid.

Small Scale Contact Conditions for the Linear-Elastic Interface Crack

1988 ◽  
Vol 55 (4) ◽  
pp. 814-817 ◽  
Author(s):  
Peter M. Anderson
Keyword(s):  
Interface Crack ◽  
Growth Parameter ◽  
Small Scale ◽  
Elastic Materials ◽  
Small Scale Yielding ◽  
Contact Conditions ◽  
Linear Elastic ◽  
Elastic Interface ◽  
Scale Yielding ◽  
Anisotropic Elastic

Conditions are discussed for which the contact zone at the tip of a two-dimensional interface crack between anisotropic elastic materials is small. For such “small scale contact” conditions combined with small scale yielding conditions, a stress concentration vector uniquely characterizes the near tip field, and may be used as a crack growth parameter. Representative calculations for an interface crack on a representative Cu grain boundary show small contact conditions to prevail, except possibly under large shearing loads.

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.

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). . .

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.

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