The intrinsic theory of linearly elastic plates

International audienceIn an intrinsic approach to a problem in elasticity, the only unknown is a tensor field representing an appropriate 'measure of strain', instead of the displacement vector field in the classical approach. The objective of this paper is to study the displacement traction-problem in the special case where the elastic body is a linearly elastic plate of constant thickness, clamped over a portion of its lateral face. In this respect, we first explicitly compute the intrinsic three-dimensional boundary condition of place in terms of the Cartesian components of the linearized strain tensor field, thus avoiding the recourse to covari-ant components in curvilinear coordinates and providing an interesting example of actual computation of an intrinsic boundary condition of place in three-dimensional elasticity. Second, we perform a rigorous as-ymptotic analysis of the three-dimensional equations as the thickness of the plate, considered as a parameter, approaches zero. As a result, we identify the intrinsic two-dimensional equations of a linearly elastic plate modeled by the Kirchhoff-Love theory, with the linearized change of metric and change of curvature tensor fields of the middle surface of the plate as the new unknowns, instead of the displacement field of the middle surface in the classical approach. Keywords: Displacement-traction problem, intrinsic elasticity, intrinsic boundary condition of place 1. The classical and intrinsic three-dimensional equations of a linearly elastic body In what follows, Latin indices and exponents range in the set {1, 2, 3} save when they are used for indexing sequences; while Greek indices and exponents range in the set {1, 2} save in the notations ∂ ν and ∂ τ , and the summation convention with respect to repeated indices is systematically used. For brevity, "three-dimensional" and "two-dimensional" will be usually abbreviated as "3d" and "2d", respectively. All functions, vector fields, etc., considered here are real. As usual, δ j i = δ ij := 1 if i = j and δ j i = δ ij := 0 if i = j. Spaces of vector fields are denoted by boldface letters while spaces of symmetric 3 × 3 or 2 × 2 matrix fields are denoted by special Roman capital letters