The use of a virtual configuration in formulating constitutive equations for residually stressed elastic materials.

*(English)*Zbl 0857.73007The authors present a method by which one can utilise data obtained from standard destructive experiments to derive constitutive equations that describe the mechanical behaviour of residually stressed elastic materials. In contrast to previous methods, the one given here leads to a constitutive equation that is an explicit function of the residual stress, and includes only material parameters that are required to describe the stress free material. The authors first obtain the constitutive equation for a special class of those residually stressed bodies which can be cut up into a finite number of parts, each of which is entirely free of residual stress. Then they introduce the idea of pointwise stress free configurations to obtain the constitutive equation for a more general class of residually stressed bodies. The derivation for this latter class is based on the assumption that each infinitesimally small neighbourhood of the body has an associated stress free configuration called its ‘virtual configuration’. It also requires that the constitutive equation for the stress free or natural material is known and invertible.

The authors go on to illustrate the use of their theory by considering two particular bodies where the undeformed configuration is a thick walled incompressible spherical shell which supports a residual stress. One is an example of the special class and the other an example of the more general class, and in both cases the natural material is a Mooney-Rivlin material. They conclude with a discussion of the relationship between appropriate virtual configurations and destructive experiments. In particular, the use of the theory in improving the design of such experiments and the use of the experiments in obtaining data required to formulate the appropriate constitutive equation are considered.

The authors go on to illustrate the use of their theory by considering two particular bodies where the undeformed configuration is a thick walled incompressible spherical shell which supports a residual stress. One is an example of the special class and the other an example of the more general class, and in both cases the natural material is a Mooney-Rivlin material. They conclude with a discussion of the relationship between appropriate virtual configurations and destructive experiments. In particular, the use of the theory in improving the design of such experiments and the use of the experiments in obtaining data required to formulate the appropriate constitutive equation are considered.

Reviewer: P.J.Barratt (Glasgow)

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\textit{B. E. Johnson} and \textit{A. Hoger}, J. Elasticity 41, No. 3, 177--215 (1995; Zbl 0857.73007)

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