The influence of inclusive deformation mapping on the kinematic equations of nonlinear elasticity

A model of continuum is often used in statics and dynamics modeling with material and spatial description. In this paper, we describe body deformation using an innovative approach employing embedding. The mapping ϕ maps a domain from the space Rⁿ to the image of space R^m, where the dimensions of the spaces m and n are necessarily different, usually n < m. Due to the different dimensions m and n, the determinant of the mapping is not defined, nor is the mapping reversible. This complicates the calculation of the fundamental kinematic quantities, the determinant J of the displacement gradient F, the transposed inverse of F, which occur in the Nanson's equation for deformed surface. In this paper, we derive terms for these quantities. The usage of the Moore-Penrose generalized inverse is essential here. This approach is a generalization of the established approach and thus opens up new possibilities in describing deformation. Displacements often appear as one of the basic unknowns, especially when using the finite element method. In the teaching of nonlinear mechanics, we found that the displacements are not adequately defined. In the paper, we therefore attempt to rectify this shortcoming.