Protoelastic bodies with large deformation

Abstract

The ability to obtain exact solutions of the equations of equilibrium of isotropically elastic bodies with arbitrary strain energy functions can be largely credited with the interest and drive that nonlinear continuum theories have exhibited since the middle fifties. Since this ability stems largely from the fact that the stress tensor admits a potential that is a function of the strain tensor, it is reasonable to look for extensions of such potential-like representations. Now, a potential representation for the stress tensor in terms of the strain tensor may be looked upon as the Euler-Lagrange derivative* if we view the strain energy function as the Lagrangian function and the displacement gradients as the field variables. With this in mind, it is a simple extension to consider strain energy functions that contain functionals of the strain as arguments in addition to the usual arguments of a strain energy function. In this way we arrive at a theory for a class of materials that we term protoelastic. The essential aspects of such materials is that the stress tensor at a given point of the body will depend on the displacement gradients at all points within the body; the stress becomes a nonlocal quantity. The essential aspects of the calculus of variations for Lagrangian functions that have functionals as arguments is given in Section 2. After a development of the variational statements that are equivalent to the law of balance of linear momentum for a deformable body in Section 3, a protoelastic body is defined in Section 4 in terms of a protopotential function and the associated variational forms for such bodies are examined. The implications of the laws of balance of moment of momentum and of energy for protoelastic bodies are given in Section 5, where it is shown that the protopotential function can depend on the displacement gradients only through a dependence on the material Cauchy-Green deformation tensor. Section 6 briefly examines the types of structure that are included under the general heading of protoelastic bodies; elastic bodies, bodies with rate effects, bodies with dependence on gradients of the deformation tensor, and bodies that exhibit intrinsically nonlocal properties. Suitable extensions of RIVLIN'S inverse method of obtaining exact solutions are shown to be possible for protoelastic bodies. Exact solutions for the biaxial shear of a rectangular protoelastic body and for the torsion of a protoelastic cylinder are given; both problems giving rise to nonlocal effects.