Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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

   
 

 

Further study on one-dimensional shock waves in nonlinear elastic media


Author: T. C. T. Ting
Journal: Quart. Appl. Math. 37 (1980), 421-429
MSC: Primary 73D05; Secondary 35L67
DOI: https://doi.org/10.1090/qam/564733
MathSciNet review: 564733
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: One may define the growth of a shock wave by the growth in the amplitude of discontinuity in the velocity (denoted by $ \left[ v \right]$) across the shock wave as the shock wave propagates. One may also define the growth of a shock wave by the growth in the amplitude of discontinuity in the stress $ \left[ \sigma \right]$, strain $ \left[ \epsilon \right]$, or entropy $ \left[ \eta \right]$, It is shown that one definition predicts the growth of the shock wave while others may predict its decay. In this paper we derive the transport equations for one-dimensional shock waves in nonlinear elastic media in which the shock wave can be defined as the amplitude of either $ {\left[ \epsilon \right]^2}$, $ \left( b \right)$, $ \left[ v \right]$ or $ \left[ \eta \right]$. Moreover, the dependent quantity can be any one of, or a linear combination of, the seven quantities behind the shock wave. It is shown that when the region ahead of the shock wave is under a homogeneous deformation, the amplitudes of $ \left[ v \right]$, $ \left[ \sigma \right]$ and $ \left[ \epsilon \right]$ grow or decay simultaneously if $ \left( a \right)$ $ {\left[ \epsilon \right]^2}$ is a strictly increasing function of $ \left[ \eta \right]$, or $ \left( b \right)$ the purely mechanical theory of shock waves is employed in which the effect of the entropy is ignored. Regardless of whether the effect of the entropy is ignored or not, there is no assurance that the amplitudes of $ \left[ v \right]$, $ \left[ \sigma \right]$ and $ \left[ \epsilon \right]$ grow or decay simultaneously if the region ahead of the shock wave is not under a homogeneous deformation.


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

  • [1] P. J. Chen and M. E. Gurtin, The growth of one-dimensional shock waves in elastic non-conductors, Int. J. Solids Structures 7, 5-10 (1971)
  • [2] Peter J. Chen, One dimensional shock waves in elastic non-conductors, Arch. Rational Mech. Anal. 43 (1971), 350–362. MR 0347194, https://doi.org/10.1007/BF00252001
  • [3] T. W. Wright, An intrinsic description of unsteady shock waves, Q. J. Appl. Math. Mech. 29, 311-324 (1976)
  • [4] C. Truesdell and R. Toupin, The classical field theories, Handbuch der Physik, Bd. III/1, Springer, Berlin, 1960, pp. 226–793; appendix, pp. 794–858. With an appendix on tensor fields by J. L. Ericksen. MR 0118005

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 73D05, 35L67

Retrieve articles in all journals with MSC: 73D05, 35L67


Additional Information

DOI: https://doi.org/10.1090/qam/564733
Article copyright: © Copyright 1980 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