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On Saint-Venant’s principle for elasto-plastic bodies. (English) Zbl 1197.74038

Summary: This paper concerns Zanaboni’s version of Saint-Venant’s principle, which states that an elongated body in equilibrium subject to a self-equilibrated load on a small part of its smooth but otherwise arbitrary surface, possesses a stored energy that in regions of the body remote from the load surface decreases with increasing distance from the load surface. We here prove this formulation of Saint-Venant’s principle for elastic-plastic bodies. The present proof, which for linear elasticity considerably simplifies that developed by Zanaboni, depends crucially upon the principle of minimum strain energy to obtain a fundamental inequality that leads to the required result. Differential inequalities are not involved. The conclusion is not restricted to cylinders but is valid for plastic bodies of general geometries. Although no conditions are imposed on the plastic theories discussed here, counter-examples indicate that in certain circumstances the fundamental inequality, and hence the result, may be valid only for restricted data that includes the body’s minimum length.

MSC:

74G50 Saint-Venant’s principle
74C15 Large-strain, rate-independent theories of plasticity (including nonlinear plasticity)
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