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)


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.


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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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