Existence of solid mechanics problems that cannot be solved through Lax-Milgram approaches

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?

• I have no idea really, but it strikes me that it wouldn't work for the Navier-Stokes equations (fluid dynamics) because somebody would be very rich if they could prove a solution exists for those equations. The NS equations and the equations for solid mechanics are the same (I think) for a material that is isotropic and does not support shear waves. Another example I can think of would be any solid mechanics problem with fracture. FEM does not work in those cases, and something special has to be done like Cohesive FEM. – tpg2114 Jun 16 '18 at 23:27
• Though the equations for solid and fluid mechanics are derived from the same place (the Cauchy momentum equation), their difference arises with the fact that the stress tensor is a function of the strain tensor in solids whereas it’s a function of the strain rate tensor in fluids. This means you can work with displacements rather than velocities, leading to different equations that have different solvability. You’re completely right on the second point though, although in that case I think it’s more “bad geometry/BC’s” than it is the equation itself. Happy to be proved wrong though! – aghostinthefigures Jun 17 '18 at 0:08