Kinematics of Continua

Abstract

Let us consider a continuum $\mathcal{B}$. Due to Axiom2, at time t=0 there is a one-to-one correspondence between every material point $\mathcal{M}\in \mathcal{B}$and its radius-vector $\mathop{\mathbf{x}}^{\circ } =\overrightarrow{ O\mathcal{M}}$in a Cartesian coordinate system $O\bar{{\mathbf{e}}}_{i}$. Denote Cartesian coordinates of the radius-vector by $\mathop{x}^ {\circ }{}^{i}$($\mathop{\mathbf{x}}^{\circ } =\mathop{ x}^ {\circ }{}^{i}\bar{{\mathbf{e}}}_{i}$) and introduce curvilinear coordinates X ⁱof the same material point $\mathcal{M}$in the form of some differentiable one-to-one functions

$$\mathop{x}^ {\circ }{}^{i} =\mathop{ x}^ {\circ }{}^{i}({X}^{k}).$$

(2.1)