Formulation of Variational Theorems in Geometrically Nonliear Elasticity

Abstract

The writer considers finite‐deformation elasticity theory with particular emphasis on the form of variational principles. For the sake of clarity the analysis is first undertaken in two dimensions, with subsequent reformulation in three dimensions. Beginning with a discussion of the known systems of Kirchhoff‐Trefftz and Piola stress components, and of Green and displacement gradient strain components, a case is made for the introduction of generalized, and distinguished generalized Piola components, and for the introduction of generalized displacement gradient components, involving the concept of rotational in addition to translational displacement components. With these various concepts consideration is given to the question of invertability or non‐invertability of constitutive equations and to the significance of this question in connection with the formulation of variational principles. In the discussion of various possible forms of variational statements particular attention is devoted to statements involving independent stress, translational and rotational displacement component variations, and to a statement involving translational and rotational displacement component variations in conjunction with certain “reactive” stress measure component variations.