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Notes on Softening and Local Instability

  • Gilbert Strang
  • M. Abdel-Naby
Conference paper
Part of the Lecture Notes in Engineering book series (LNENG, volume 3)

Summary

We discuss some of the consequences of a loss of monotonicity in the constitutive law. In penetration problems this occurs for adiabatic heating, and shear bands appear; they have been analyzed by Clifton, Needleman, Hutchinson, Bodner, and many others, and we mention a forthcoming paper of Merzer. In a different context we consider the model equation ut = (σ(ux))x, and describe Höllig’s construction of a

family of solutions. Where σ’ < 0 this is a backward heat equation, but the numerical calculations remain stable (convergence is not so clear); ux jumps into neighboring regions where σ’ > 0. We reproduce in a simple form the Brezis-Ekeland variational principle for the heat equation, and we follow Nohel in speculating on its convexification for a law that has a “softening range” σ’ < 0.

Keywords

Shear Band Heat Equation Adiabatic Shear Adiabatic Shear Band Local Instability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Knowles, J. K., and Sternberg, E.: On the failure of ellipticity of the equations for finite elastostatic plane strain. Arch. Rat. Mech. Anal. 63 (1977) 321–336.CrossRefMATHMathSciNetGoogle Scholar
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    Merzer, A. M.: Modelling of adiabatic shear band development from small imperfections. J. Mech. Phys. Solids, to appear.Google Scholar
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    Gurtin, M. E., and Temam, R.: On the anti-plane shear problem in finite elasticity. J. Elasticity 11 (l98l) 197–206.CrossRefMATHMathSciNetGoogle Scholar
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    HÖllig, K.: Existence of infinitely many solutions for a forward backward heat equation. Trans. Amer. Math. Soc., to appear.Google Scholar
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    Brézis, H., and Ekeland, I.: Un principe variationnel associe à certaines équations paraboliques. Comptes Rendus Acad. Sc. Paris 282 (1976) 971–974.MATHGoogle Scholar
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    Ekeland, I., and Temam, R.: Convex Analysis and Variational Problems.- Amsterdam: North-Holland 1976.MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1983

Authors and Affiliations

  • Gilbert Strang
    • 1
  • M. Abdel-Naby
    • 2
  1. 1.Department of MathematicsM.I.T.CambridgeUSA
  2. 2.Department of MathematicsAin Shams UniversityCairoEgypt

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