On anholonomic deformation, geometry, and differentiation

Abstract

In geometrically nonlinear theories of inelasticity of solids, the deformation gradient is typically split multiplicatively into two (or more terms), none of which need be integrable to a motion or displacement field. Such terms, when not integrable, are termed anholonomic, and can be associated with intermediate configuration(s) of a deformed material element. In this work, aspects of tensor calculus associated with anholonomic deformation are analyzed in general curvilinear coordinates. Various linear connection coefficients for intermediate configurations are posited or derived; of particular interest are those mapped coefficients corresponding to the choice of identical basis vectors in multiple configurations. It is shown that torsion and curvature associated with such mapped coefficients do not necessarily vanish, even though torsion and curvature tensors of the original connections vanish by definition in reference or current configurations. Intermediate connection coefficients defined in this way exhibit vanishing covariant derivatives of corresponding metric tensors, but are time dependent even when reference (current) configuration connections are fixed in time at a given material (spatial) location. Formulae are derived for total covariant derivatives of two- and three-point tensors with one or more components referred to the intermediate configuration. It is shown that in intermediate coordinates, neither the divergence of the curl of a vector field nor the curl of the gradient of a scalar field need always vanish. The balance of linear momentum for a hyperelastic–plastic material is examined in the context of curvilinear intermediate coordinates.