The use of strain energy to quantify the effect of residual stress on mechanical behavior

Residual stresses are common in engineering and biological materials, and their effect on the mechanical behavior of a body can be significant. In this article, the authors use the change in strain energy density for deformations out of the residually stressed configuration to quantify the contribution of a residual stress to the mechanical behavior of an elastic body and to examine the relation between the form of the residual stress field and the mechanical response. The change in strain energy density at a point is a measure of the extent to which the material at that point is loaded or unloaded by a deformation. The authors show that the strain energy density can be used in conjunction with the stress and strain fields to develop a clear physical interpretation of the global mechanical behavior of a residually stressed body. These ideas are illustrated in two simple examples in which the presence of the residual stress field has the effect of making the body more compliant than an equivalent stress-free body under some deformations and less compliant under other deformations.