Foundations of the Theory of Surface Waves in Anisotropic Elastic Materials

Abstract.The 6x6 real matrix N(i>) for anisotropic elastic materials under a twodimensional steady-state motion with speed v is extraordinary semisimple when N(t>) has three identical complex eigenvalues p and three independent associated eigenvectors. We show that such an N(i,) exists when v ^ 0. The eigenvalues are purely imaginary. The material can sustain a steady-state motion such as a moving line dislocation. Explicit expressions of the Barnett-Lothe tensors for v ^ 0 are presented. However, N(i>) cannot be extraordinary semisimple for surface waves. When v -0, N(0) can be extraordinary semisimple if the strain energy of the material is allowed to be positive semidefinite. Explicit expressions of the Barnett-Lothe tensors and Green's functions for the infinite space and half-space are presented. An unusual phenomenon for the material with positive semidefinite strain energy considered here is that it can support an edge dislocation with zero stresses everywhere.In the special case when p = i is a triple eigenvalue, this material is an un-pressurable material in the sense that it can change its (two-dimensional) volume with zero pressure. It is a counterpart of an incompressible material (whose strain energy is also positive semidefinite) that can support pressure with zero volume change.

“…The vectors bk are determined from (4). They automatically satisfy the orthogonality relations [1,2] ak ■ bs + as ■ bk = 0 (k ^ s).…”

Section: Extraordinary Semisimple N(v) Equation (1) Is Equivalent Tomentioning

confidence: 99%

Section: Extraordinary Semisimple N(v) Equation (1) Is Equivalent Tomentioning

confidence: 99%

See 1 more Smart Citation

On extraordinary semisimple matrix 𝑁(𝑣) for anisotropic elastic materials

Ting¹

1997

“…(5.11) of [22] and it is called the fundamental elasticity tensor. It will be shown that the Jordan structure of this matrix essentially determines the nature of the conservation laws.…”

Section: The Stroh Formalismmentioning

confidence: 99%

Conservation laws and canonical forms in the Stroh formalism of anisotropic elasticity

Hatfield¹

1996

Abstract.We find and classify all first-order conservation laws in the Stroh formalism. All possible non-semisimple degeneracies are considered. The laws are found to depend on three arbitrary analytic functions.In some instances, there is an "extra" law which is quadratic in Vu. Separable and inseparable canonical forms for the stored energy function are given for each type of degeneracy and they are used to compute the conservation laws. The existence of a real Stroh eigenvector is found to be a necessary and sufficient condition for separability. The laws themselves are stated in terms of the Stroh eigenvectors.

“…Therefore, there are only two independent eigenvectors a for the triple eigenvalue p = i. The system is nonsemisimple [7,8]. The general solution for a is aT = (kl, iakl + (a + 1 )y/k2, k2), (3.20) where kx and k2 are arbitrary constants.…”

mentioning

confidence: 99%

On anisotropic elastic materials that possess three identical Stroh eigenvalues as do isotropic materials

Ting¹

1994

Abstract. For anisotropic elastic materials for which the displacements w( depend on xx and x2 only, a general solution for m( depends on one variable z = x, + px2 where p is an eigenvalue of the fundamental elasticity tensor of Stroh. There are six p's which consist of three pairs of complex conjugates. For isotropic materials, p = ±i are the eigenvalues of multiplicity three. We point out trivial cases in which a completely anisotropic material has the eigenvalues p = ±i and has the solutions to two-dimensional elasticity problems that are identical to the solutions for isotropic materials. Excluding these trivial cases, we show that p = ±i can be the eigenvalues of multiplicity three for monoclinic materials with the symmetry plane at x, = 0, at x2 = 0, or at any plane that contains the x3-axis. If the symmetry plane is at x3 = 0, then p = ±i occur only when the material is transversely isotropic with the axis of symmetry at the x3-axis. We also consider the general case in which the eigenvalues are arbitrary and are of multiplicity three. The eigenrelation associated with the triple eigenvalues is nonsemisimple for all cases studied here. There are only two independent eigenvectors associated with the triple eigenvalues.