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Remarks on the use and misuse of the semi-inverse method in the nonlinear theory of elasticity. (English) Zbl 1255.74009
Summary: We show, by considering some specific examples, that in the nonlinear theory of elasticity the application of the semi-inverse method has to be carried out very carefully. Usually, the strategy followed in applying the semi-inverse method while dealing with complex models is to generalize forms of solutions already known within the framework of a simpler theory. This is a smart strategy from a mathematical point of view, but from the physical point of view, it may be imprudent. For example, in the theory of nonlinear elasticity, we are at times motivated by the results obtained in the incompressible case to arrive at an understanding of what happens in the compressible case, and by doing so, many important exact solutions for special classes of compressible elastic materials have been obtained successfully. Sometimes, the admissibility of a given deformation field is considered to delineate special classes of constitutive laws. We wish to point out that the classes of constitutive equations thus identified from the standpoint that it may admit a type of deformation may lead to models that exhibit physically unacceptable mechanical behavior. To illustrate the dangers inherent to merely turning the mathematical crank to determine classes of constitutive equations where a certain class of deformations are possible, we consider the torsion of a cylindrical shaft and the propagation of transverse waves in a compressible nonlinear elastic material and show that care has to be exercised in appealing to the semi-inverse method.

MSC:
74B20 Nonlinear elasticity
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