The Weak Compatibility Equations of Nonlinear Elasticity and the Insufficiency of the Hadamard Jump Condition for Non-Simply Connected Bodies

We derive the compatibility equations of L² displacement gradients on non-simply-connected bodies. These compatibility equations are useful for non-smooth strains such as those associated with deformations of multi-phase materials. As an application of these compatibility equations, we study some configurations of different phases around a hole and show that, in general, the classical Hadamard jump condition is not a sufficient compatibility condition.