Notes on Softening and Local Instability
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.
KeywordsShear Band Heat Equation Adiabatic Shear Adiabatic Shear Band Local Instability
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© Springer-Verlag Berlin Heidelberg 1983