A RELAXATION APPROACH TO HENCKYS PLASTICITY

Given mu, kappa, c > 0, We consider the functional F(u) = integral(Ome ga\Su) (mu\E(D)u\(2) + kappa/2(div u)(2)) dx + c integral(Su) \u(+) - u(-)\ dH(n-1), defined on all R(n)-valued functions u on the open subs et Omega of R(n) which are smooth outside a free discontinuity set S-u , on which the traces u(+), u(-) on both sides have equal normal compo nent (i.e., u has a tangential jump along S-u). E(D)u = Eu - 1/3(div u ) I, with Eu denoting the linearized strain tensor. The functional F i s obtained from the usual strain energy of linearized elasticity by ad dition of a term (the second integral) which penalizes the jump discon tinuities of the displacement. The lower semicontinuous envelope (F) o ver bar is studied, with respect to the L(1) (Omega; R(n))-topology, o n the space P (Omega) of the functions of bounded deformation with dis tributional divergence in L(2)(Omega) (F is extended with value +infin ity on the whole P (Omega)). The following integral representation is proved: (F) over bar(u) = integral(Omega)(phi(epsilon(D)u) + kappa/2(d iv u)(2)) dx + integral(Omega)phi(infinity) (E(s)(D)u/\E(s)(D)u\)\E(s) (D)u\, u is an element of P(Omega), where phi is a convex function wit h linear growth at infinity. Now Eu is a measure, epsilon(D)u represen ts 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 functio nal which intervenes in the minimum problem for the displacement in th e theory of Hencky's plasticity with Tresca's yield conditions.