Two-dimensional Intrinsic Elasticity

One of the major objectives of elasticity theory is to compute the stress tensor field arising in an elasticbody subjected to specific boundary conditions and to given applied forces. This objective is achievedin the classical approach to elasticity theory by first computing the displacement vector field inducedby these forces inside the body, then by computing the strain tensor field by direct differentiation, andfinally by determining the stress tensor field by means of the constitutive equation of the elastic materialconstituting the body. Alternatively, one may use a mixed approach, in which both the displacementvector field and the strain tensor field are simultaneously computed.By contrast, an intrinsic approach achieves this objective in one single step, by solving a boundaryvalue problem whose unknown is the strain tensor field, or equivalently (by means of the constitutiveequation), the stress tensor field itself. The equations found in this boundary value problem thereforeprovide a direct way of computing the stress tensor field inside an elastic body, often the unknown ofprimary interest in the numerical simulation of elastic structures.The mathematical foundations of the intrinsic approach to linear three-dimensional elasticity havebeen recently established and are now on firm grounds. By contrast, the mathematical foundations of theintrinsic approach to two-dimensional linear elasticity, i.e., to two-dimensional linear plate theory and totwo-dimensional shell theory, have been laid down only in the restricted (and to a large extent physicallyunrealistic) case of the pure traction problem, i.e., when no Dirichlet boundary conditions are imposedon the displacement field.It thus remains to identify the boundary value problems of the intrinsic approach to two-dimensionallinear elasticity for the more general (and physically realistic) displacement-traction problems, i.e., whenin the classical approach a Dirichlet boundary condition is imposed on the displacement field on a portionof the boundary.The solution of this completely open problem constitutes the main theme of this Proposal, which aimsat finding and justifying the boundary value problems of intrinsic two-dimensional linear elasticity in theirfull generality.