A finite strain theory of elastoplasticity and its application to wave propagation
A constitutive theory of finite strain plasticity is developed by using the methods of convex analysis. The theory abstracts and extends the classical assumptions of a convex region of admissible stresses, and the normality law. The overall effects of plastic behaviour are contained in the theory through the presence of one or more internal variables. The thermodynamic restrictions of the second law together with the use of results of convex analysis lead in a natural way to the evolution equation or flow law. Non-smooth yield surfaces are included in the theory; nevertheless, the form of this theory makes a study of propagation of singular surfaces awkward. With a view to carrying out such a study, an alternative means of treating non-smooth convex yield surfaces is developed. This alternative theory is essentially a synthesis of the theory of Sewell, and that presented earlier in the thesis. The theory of singular surfaces is reviewed in the context of finite strain elastoplasticity, and necessary conditions for the propagation of acceleration waves are derived. A comparison of elastic and plastic wave speeds is made, and inequalities similar to those of Mandel for the small-strain case are derived. The propagation conditions for principal waves in both longitudinal and transverse directions, and the corresponding wave speeds, are found and compared for solids obeying a neo-Hookean elastic law, and with either the von Mises or Tresca yield criteria.
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