A relaxation approach to Hencky's plasticity

Given mu, kappa, c > 0, We consider the functional F(u) = integral(Omega\Su) (mu\E(D)u\(2) + kappa/2(div u)(2)) dx + c integral(Su) \u(+) - u(-)\ dH(n-1), defined on all R(n)-valued functions u on the open subset Omega of R(n) which are smooth outside a free discontinuity set S-u, on which the traces u(+), u(-) on both sides have equal normal component (i.e., u has a tangential jump along S-u). E(D)u = Eu - 1/3(div u) I, with Eu denoting the linearized strain tensor. The functional F is obtained from the usual strain energy of linearized elasticity by addition of a term (the second integral) which penalizes the jump discontinuities of the displacement. The lower semicontinuous envelope (F) over bar is studied, with respect to the L(1) (Omega; R(n))-topology, on the space P (Omega) of the functions of bounded deformation with distributional divergence in L(2)(Omega) (F is extended with value +infinity on the whole P (Omega)). The following integral representation is proved: (F) over bar(u) = integral(Omega)(phi(epsilon(D)u) + kappa/2(div u)(2)) dx + integral(Omega)phi(infinity) (E(s)(D)u/\E(s)(D)u\)\E(s)(D)u\, u is an element of P(Omega), where phi is a convex function with linear growth at infinity. Now Eu is a measure, epsilon(D)u represents the density of the absolutely continuous part of E(D)u, while E(s)(D)u denotes the singular part and phi(infinity) the recession function of phi. Finally, we show that (F) over bar coincides with the functional which intervenes in the minimum problem for the displacement in the theory of Hencky's plasticity with Tresca's yield conditions.