Stress rate and incremental principle of virtual work in finite deformations

Solution of finite deformation problems is sought in the space of all deformation tensor fields. Representation of a deformation process here as a trajectory makes us possible to further classify symmetric second-order tensor fields either as points, vectors, or covectors, and, as a consequence, assign them the corresponding time derivatives. However, as the space of all deformation tensor fields has proved non-euclidean, the time derivative of vector, and covector fields along the trajectory should be defined by the covariant derivative. This approach enables us coherently to formulate an incremental principle of virtual work, and propose the corresponding procedure in solving finite deformation problems