Existence of Solutions in Finite Elasticity

Abstract

Very few exact solutions are known to static and dynamic problems of finite elasticity, particularly in the case when the material is compressible. General theorems on existence of solutions provide reassurance that the theory is mathematically sound; for example it is important to understand whether or not solutions of the basic equations have singularities consistent with assumptions used in deriving the equations. But there are several other, equally important, reasons for studying questions of existence of solutions. One such is the establishment of convergence properties for numerical methods in elasticity (in this connection it should be noted that finite-difference schemes for certain partial differential equations may converge to solutions of different equations). Experience with other partial differential equations has also taught us that existence theorems are an essential prerequisite for the study of various qualitative properties of solutions (for example, bifurcation, stability and asymptotic behaviour). In a broader context, we today face problems in elasticity similar to unsolved questions in other branches of mechanics and physics, and the unifying nature of the theory of partial differential equations can thus lead us to hope, as has been the case in the past, that advances in elasticity will lead to corresponding progress in other fields. Here, however, we concentrate on a more specific reason for proving, or attempting to prove, existence theorems in elasticity, namely that it leads to information concerning the relationship between constitutive hypotheses (i.e. assumptions on the stored-energy function, or stress-strain law of the material) and smoothness properties of solutions.