On the “Symmetry” of Tangent Operators in Nonlinear Mechanics

In the nonlinear theory of thin elastic shells with the displacement vector as the only independent variable we are concerned with the functional of total potential energy defined on a normed linear space, where the second Gâteaux differential of the functional is symmetric leading to a symmetric tangent stiffness matrix in finite element approximation procedures. On the other hand, in shell theories with displacement vector and rotation tensor as independent kinematical variables the corresponding functional is defined on a manifold without vector space structure. The question whether or not the second derivative and the associated tangent stiffness matrix are symmetric is controversely discussed in the recent literature. Using notions and methods of differential geometry we consider in this paper definition and symmetry property of the second differential of a functional defined on a finite dimensional manifold and Lie group (rotation group), respectively, and we discuss related problems and implications in detail. As it is shown the results obtained for linear spaces cannot be generalized to manifolds and Lie groups in a straightforward way.