New Formulation of the Boundary Value Problems of Elasticity

Classically, the displacement vector field of the reference configuration is the unknown(or one of the unknowns) appearing in the mathematical models of linearized and nonlinear three-dimensional elasticity.Yet another approach, called the “intrinsic” approach, is slowly coming of age, where the strain tensor field, or equivalently the stress tensor field (by means of the constitutive equation), is the only unknown. Since the knowledge of the stress tensor field is of primary importance in computational mechanics, an intrinsic approach is thus particularly attractive from a practical viewpoint.It is only recently, however, that the mathematical analysis and numerical analysis of the intrinsic approach have been undertaken. The objective of this proposal is to contribute to its mathematical analysis, in both the linear and nonlinear cases.In the linear case, its mathematical analysis has been so far limited to pure traction problems, i.e., where no boundary condition on the unknown displacement is imposed on the boundary of the reference configuration. The first objective of this proposal consists in finding explicit expressions of boundary conditions that only involve the strains for pure displacement, or displacement-traction problems, i.e., where a more realistic boundary condition of place is imposed on the whole boundary of the reference configuration, or on a subset thereof (this means that the unknown displacement should vanish there). It is also planned to establish the existence of a solution of the corresponding minimization problems, which are thus expressed only in terms of strains(or equivalently, only in terms of stresses).In the nonlinear case, it was already shown that the pure displacement problem can be recast as a minimization problem where the only unknown is the Cauchy-Green strain tensor. The second objective of this proposal is to find in this case an explicit formula for the corresponding boundary condition, expressed only in terms of the components of the unknown Cauchy-Green strain tensor.In addition to its mathematical interest per se, the proposed research would pave the way in the linear case for developing and analyzing new finite element approximations of the intrinsic approach applied to the more realistic displacement-traction problems (preliminary computations indicate that such numerical methods seem to be quite effective).In the nonlinear case, the proposed research would constitute a welcome complement to the landmark existence theory of John Ball, which could thus be completely recast in terms of the Cauchy-Green strain tensor field as the sole unknown, at least in some special cases.