A Saint-Venant principle for a theory of nonlinear plane elasticity

A formulation of Saint-Venant’s principle within the context of a restricted theory of nonlinear plane elasticity is described. The theory assumes small displacement gradients while the stress-strain relations are nonlinear. We consider plane deformations of such an elastic material occupying a rectangular region. The lateral sides are traction-free while the far end is subjected to a uniformly distributed tensile traction $\tau \ge 0$. The near end is subjected to a prescribed normal and shear traction. If this end loading were also that of uniform tension, then one possible corresponding stress state throughout the rectangle is that of uniform tension. When the near end loading is not uniform, the resulting stress field is expected to approach a uniform tensile state with increasing distance from the near end. This result is established here using differential inequality techniques for quadratic functionals. It is shown that, under certain constitutive assumptions, an energy-like quadratic functional, defined on the difference between the deformation field and the uniform tensile state, decays exponentially with distance from the near end. The estimated decay rate (which is a lower bound on the actual rate of decay) is characterized in terms of the load level $\tau$ , the domain geometry, and material properties. The results predict a progressively slower decay of end effects with increasing load level $\tau$. The mathematical issues of concern involve spatial decay of solutions of a fourth-order nonlinear elliptic partial differential equation.