Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 

 

On extraordinary semisimple matrix $ {\bf N}(v)$ for anisotropic elastic materials


Author: T. C. T. Ting
Journal: Quart. Appl. Math. 55 (1997), 723-738
MSC: Primary 73B40; Secondary 73C02, 73D20
DOI: https://doi.org/10.1090/qam/1486545
MathSciNet review: MR1486545
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The $ 6 \times 6$ real matrix $ N\left( v \right)$ for anisotropic elastic materials under a two-dimensional steady-state motion with speed $ v$ is extraordinary semisimple when $ N\left( v \right)$ has three identical complex eigenvalues $ p$ and three independent associated eigenvectors. We show that such an $ N\left( v \right)$ exists when $ v \ne 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 \ne 0$ are presented. However, $ N\left( v \right)$ cannot be extraordinary semisimple for surface waves. When $ v = 0$, $ N\left( 0 \right)$ 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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 73B40, 73C02, 73D20

Retrieve articles in all journals with MSC: 73B40, 73C02, 73D20


Additional Information

DOI: https://doi.org/10.1090/qam/1486545
Article copyright: © Copyright 1997 American Mathematical Society


Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2017 Brown University
Comments: qam-query@ams.org
AMS Website