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On material and geometrical instabilities in finite elasticity and elastoplasticity

Reese S.

Archives of Mechanics

2000

Vol. 52, nr 6

969-999

Material instability phenomena arise in homogeneous stress states if nonlinear stress-strain relations are considered. The stability behaviour is investigated by looking at the Gateaux derivative of the first Piola-Kirchhoff stress tensor in the direction of the deformation gradient. This requires to solve a nine-dimensional matrix eigenvalue problem. In the present contribution, it is shown that material instabilities can be clearly differentiated from instabilities of geometrical character. The latter aspect is especially important for the design of new materials, since unstable solution paths under common loading conditions are not desirable. Geometrical instabilities, however, can usually be avoided by choosing appropriate boundary conditions. The derivation in this work leads to a simple stability criterion which allows to describe the stability behaviour of many materials in a very general context.