On the characterizations of matrix fields as linearized strain tensor fields

Abstract

Saint Venant's and Donati's theorems constitute two classical characterizations of smooth matrix fields as linearized strain tensor fields. Donati's characterization has been extended to matrix fields with components in L² by T.W. Ting in 1974 and by J.J. Moreau in 1979, and Saint Venant's characterization has been extended likewise by the second author and P. Ciarlet, Jr. in 2005. The first objective of this paper is to further extend both characterizations, notably to matrix fields whose components are only in H⁻¹, by means of different, and to a large extent simpler and more natural, proofs. The second objective is to show that some of our extensions of Donati's theorem allow to reformulate in a novel fashion the pure traction and pure displacement problems of linearized three-dimensional elasticity as quadratic minimization problems with the strains as the primary unknowns. The third objective is to demonstrate that, when properly interpreted, such characterizations are "matrix analogs" of well-known characterizations of vector fields. In particular, it is shown that Saint Venant's theorem is in fact nothing but the matrix analog of Poincare's lemma.