Stability analysis of thermo-visco-plastic materials undergoing high-rate shear deformations
Authors:
Reza Malek-Madani and Raouf Ali Raouf
Journal:
Quart. Appl. Math. 57 (1999), 213-227
MSC:
Primary 74C10; Secondary 35Q72, 74F05, 74H55
DOI:
https://doi.org/10.1090/qam/1686186
MathSciNet review:
MR1686186
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Abstract: In this paper we present an energy estimate for two systems of partial differential equations that govern thermo-mechanical behavior of materials undergoing simple and anti-plane shear deformations. A linearized stability analysis is then carried out for the case of simple shear deformation with an exponentially softening stress-strain law. A series of numerical experiments on the fully nonlinear one-dimensional system of partial differential equations constitutes the final section of the paper.
- John H. Maddocks and Reza Malek-Madani, Steady-state shear-bands in thermoplasticity. I. Vanishing yield stress, Internat. J. Solids Structures 29 (1992), no. 16, 2039–2061. MR 1165415, DOI https://doi.org/10.1016/0020-7683%2892%2990193-W
M. Reiner, Rheology, Elasticity and Plasticity, Handbuch der Physik 6, Editor S. Flugge, 434–549 (1958)
- A. E. Tzavaras, Effect of thermal softening in shearing of strain-rate dependent materials, Arch. Rational Mech. Anal. 99 (1987), no. 4, 349–374. MR 898715, DOI https://doi.org/10.1007/BF00282051
- Donald A. French and Sonia M. F. Garcia, Finite element approximation of an evolution problem modeling shear band formation, Comput. Methods Appl. Mech. Engrg. 118 (1994), no. 1-2, 153–161. MR 1292106, DOI https://doi.org/10.1016/0045-7825%2894%2900021-2
- Song Jiang, Asymptotic stability of shearing flows of incompressible thermoviscous fluids, Nonlinear Anal. 24 (1995), no. 8, 1207–1220. MR 1325643, DOI https://doi.org/10.1016/0362-546X%2894%2900192-K
J. H. Maddocks and R. Malek-Madani, Steady-state shear bands in thermo plasticity - I. Vanishing yield stress, Internat. J. Solids Struct. 29, 2039–2061 (1992)
M. Reiner, Rheology, Elasticity and Plasticity, Handbuch der Physik 6, Editor S. Flugge, 434–549 (1958)
A. Tzavaras, Effect of thermal softening in shearing of strain-rate dependent materials, Archive for Rational Mechanics, vol. 99, 1987, pp. 349–374
D. A. French and S. M. F. Garcia, Finite element approximation of an evolution problem modeling shear band formation, Comput. Methods Appl. Mech. Engrg., vol. 118, 1994, pp. 153–163
S. Jiang, Asymptotic stability of shearing flows of incompressible thermoviscous fluids, Nonlinear Analysis, vol. 24, no. 8, 1995, pp. 1207–1220
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Article copyright:
© Copyright 1999
American Mathematical Society