A nonlinear Cauchy strain tensor and its applications

The theory of nonlinear elasticity has a long history, with contributions from great mathematicians including Euler, Lagrange, Cauchy, and many others and with critical applications in engineering and technology. The complex equations required for stress analyses in elastic solids have been the subject of extensive studies over many years, and effective and efficient systematic procedures have been developed to solve these equations. More recently, the rapid growth of computing technology and numerical methods has also guaranteed accurate and reliable solutions. In the formulation of nonlinear static and dynamic problems involving elastic solids, the most widely used form of the nonlinear strain tensor, namely, the Green strain tensor, is defined in terms of the deformation gradient. Another nonlinear strain tensor, the Euler–Almansi strain tensor, is also defined in a similar manner. It has now been found that on the basis of the linear Cauchy strain tensor and considering changes of the coordinate framework under finite deformations, a new nonlinear strain tensor can be defined consistently and differs from previously known nonlinear strain tensors. This new strain tensor, the nonlinear Cauchy strain tensor, offers a novel formulation of nonlinear elasticity in addition to the existing nonlinear strain tensors.