A Cesàro–Volterra formula with little regularity

Abstract

If a symmetric matrix field e of order three satisfies the Saint-Venant compatibility conditions in a simply-connected domain Ω in R 3 , there then exists a displacement field u of Ω with e as its associated linearized strain tensor, i.e., e = 1 2 ( ∇ u T + ∇ u ) in Ω. A classical result, due to Cesàro and Volterra, asserts that, if the field e is sufficiently smooth, the displacement u ( x ) at any point x ∈ Ω can be explicitly computed as a function of the matrix fields e and CURL e , by means of a path integral inside Ω with endpoint x. We assume here that the components of the field e are only in L 2 ( Ω ) (as in the classical variational formulation of three-dimensional linearized elasticity), in which case the classical path integral formula of Cesàro and Volterra becomes meaningless. We then establish the existence of a “Cesàro–Volterra formula with little regularity”, which again provides an explicit solution u to the equation e = 1 2 ( ∇ u T + ∇ u ) in this case. We also show how the classical Cesàro–Volterra formula can be recovered from the formula with little regularity when the field e is smooth. Interestingly, our analysis also provides as a by-product a variational problem that satisfies all the assumptions of the Lax–Milgram lemma, and whose solution is precisely the unknown displacement field u . It is also shown how such results may be used in the mathematical analysis of “intrinsic” linearized elasticity, where the linearized strain tensor e (instead of the displacement vector u as is customary) is regarded as the primary unknown.