Very often, solid mechanicians employ finite-element analyses to solve problems in linear solid mechanics. This approach is guaranteed to work because the Lax-Milgram theorem, along with some corollaries, ensure that the finite-element formulation of certain linear PDEs generates a unique solution that approximates the solution to the original PDE to arbitrary accuracy in a well-defined way.
The equations of linear solid mechanics fall into the category of PDEs for which these theorems are true—and the finite element approach therefore verifiably works—but the equations of nonlinear solid mechanics don't. With this in mind:
Are there any basic/elementary examples of nonlinear solid mechanics problems for which a Lax-Milgram based solution approach demonstrably fails?