Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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



On wave propagation problems in which $ c_f=c_s=c_2$ occurs

Author: T. C. T. Ting
Journal: Quart. Appl. Math. 31 (1973), 275-286
DOI: https://doi.org/10.1090/qam/99700
MathSciNet review: QAM99700
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Abstract | References | Additional Information

Abstract: The combined longitudinal and torsional plastic waves in a thin-walled tube of rate-independent isotropic work-hardening material are used to illustrate the problems involved when the situation $ {c_f} = {c_8} = {c_2}$ occurs. Two examples are presented. In the first example, the stress paths in the $ \sigma \sim \tau $ plane for the fast and slow simple waves are examined in the region near the singular point $ \left( {\sigma *,0} \right)$ where $ {c_f} = {c_8} = {c_2}$. For $ \eta \ge \frac{1}{2}$, where $ \eta $ is a nondimensional material constant defined in the paper, there is no stress path passing through the singular point $ (\sigma *,0)$ other than the $ \sigma $-axis itself. For $ 0 < \eta < \frac{1}{2}$, there is a family of stress paths emanated from $ (\sigma *,0)$ which span an angle of $ {\tan ^{ - 1}}{\left( {1 - 2\eta } \right)^{1/2}}$ with the $ \sigma $-axis. In any case, the stress paths for the fast and slow simple waves are not orthogonal to each other at the singular point. In the second example, a study is made of the propagation of the plastic wave front into the tube which is initially prestressed at the stress state $ \left( {\sigma *,0} \right)$. It is shown that the solution in the region next to a region of constant state is not necessarily a simple wave solution. In fact, an unloading can occur at the plastic wave front which changes its speed from $ {c_2}$ to $ {c_0}$ at the onset of the unloading.

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

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Additional Information

DOI: https://doi.org/10.1090/qam/99700
Article copyright: © Copyright 1973 American Mathematical Society

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