Legendre-Hadamard inequalities for Cosserat elastic shells with a single deformable director

We consider the general nonlinear theory of Cosserat elastic shells with a single deformable director and apply the methods of variational calculus to obtain the relevant Legendre-Hadamard inequalities. These are pointwise necessary conditions for stable equilibrium states, which are obtained from the property that the second variation of the total potential energy is non-negative. Thus, we establish a stability criterion which must be satisfied by any stable solution of the equilibrium equations for shells.