Kinematics of Continua

Part of the Solid Mechanics and Its Applications book series (SMIA, volume 174)


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 i of the same material point \(\mathcal{M}\)in the form of some differentiable one-to-one functions
$$\mathop{x}^ {\circ }{}^{i} =\mathop{ x}^ {\circ }{}^{i}({X}^{k}).$$


Material Point Deformation Gradient Tensor Field Total Derivative Deformation Tensor 
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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Bauman Moscow State Technical UniversityMoscowRussia

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