A dynamic theory of material instability

Béda, Péter (2002) A dynamic theory of material instability. JOURNAL OF COMPUTATIONAL AND APPLIED MECHANICS, 3 (1). pp. 5-13. ISSN 1586-2070

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Abstract

In continuum mechanics dynamical effects are usually connected to waves. Such studies lead to the wave dynamical theory of constitutive equations. In stability analysis, the terms of the theory of dynamical systems are valuable tools. Most of the material instability investigations published deal with small deformations and static or quasi-static loading conditions. To study dynamical effects and finite deformations as well we need first of all appropriate constitutive equations. In order to take second gradient dependent materials widely used for numeric investigations of post-localization into account we shall assume that a jump exists in the derivative of the acceleration field and this singular surface propagates with finite velocities (generalized wave). From that assumption conditions are obtained for example for the second order derivatives of the variables of the constitutive equations. In material instability problems we prescribe that the loss of stability should be a generic one in terms of the theory of dynamical systems. There are two main points. Firstly, in the generic case the multiplicity of the critical eigenvalue should be one or at least finite, moreover the two basic types of stability loss should not be coexistent. Secondly, we would require a finite dimensional critical eigenspace. These lead to further conditions for the constitutive equations.

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