On kinematic conditions of compatibility (Q1197271)

Let \(C\) and \(\overline C\) denote two symmetric positive definite constant matrices. It follows from linear algebra that there are matrices \(F\) and \(\overline F\) such that: (*) \(F^ T F=C\), \(\text{det} F>0\); \(\overline{F}^ T\overline{F}=\overline C\), \(\text{det} \overline F>0\). If there are two vectors \(A\) and \(N\) such that (**) \(\overline F =F(I+A\otimes N)\), then \(F\) and \(\overline F\) can be interpreted as constant deformation gradients relative to a homogeneous reference configuration, and (**) represents a ``kinematic condition of compatibility'', permitting us to construct a continuous deformation with the deformation gradient jumping across a plane with normal \(N\). The author claims that such types of deformation occur in a theory of phase transitions in crystals, and he provides a number of conditions under which the matrices \(C\) and \(\overline C\) related by some rotation \(R\) through: \(\overline C=R^ T CR\), are compatible with equation (**).