Kinematics. Changes of Frame

Abstract

Since the motion X̰_κ̰is continuous, F̰ is non-singular, so the polar decomposition theorem of CAUCHY enables us to write it in the two forms

$$\mathop F\limits_ \sim = \mathop R\limits_ \sim \mathop U\limits_ \sim = \mathop V\limits_ \sim \mathop R\limits_ \sim ,$$

((2.1))

where R̰ is an orthogonal tensor, while Ṵ and V̰ are positive-definite symmetric tensors. R̰, Ṵ, and V̰ are unique. CAUCHY’S decomposition tells us that the deformation corresponding locally to F̰ may be obtained by effecting pure stretches of amounts, say, v_i, along three suitable mutually orthogonal directions ḛ_i, followed by a rigid rotation of those directions or by performing the same rotation first and then effecting the same stretches along the resulting directions. The quantities v_i are the principal stretches; corresponding unit proper vectors of Ṵ and V̰ point along the principal axes of strain in the reference configuration and the present configuration X̰, respectively. Indeed, if

$${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{U}}{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{e}}}_{i}} = {{v}_{i}}{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{e}}}_{i}},$$

(2.2)

then by (2.1)

$$\mathop V\limits_ \sim \left( {\mathop R\limits_ \sim {{\mathop e\limits_ \sim }_i}} \right) = \left( {\mathop R\limits_ \sim \mathop U\limits_ \sim {{\mathop R\limits_ \sim }^T}} \right)\left( {\mathop R\limits_ \sim {{\mathop e\limits_ \sim }_i}} \right) = {v_i}\left( {\mathop R\limits_ \sim {{\mathop e\limits_ \sim }_i}} \right).$$

((2.3))