Given μ, κ, c >0, we consider the functional $$F(u) = \int_{\Omega \backslash S_u } {\left( {\mu |E^D u|^2 + \frac{\kappa }{2}(div u)^2 } \right)} dx + c\int_{S_u } {|u^ + - u^ - |} d\mathcal{H}^{n - 1} ,$$ defined on all R n -valued functions u on the open subset Ω 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 discontin uities of the displacement. The lower semicontinuous envelope $$\bar F$$ is studied, with respect to the L 1 (Ω; R n )-topology, on the space P (Ω) of the functions of bounded deformation with distributional divergence in L 2 (Ω) ( F is extended with value +∞ on the whole P (Ω)). The following integral representation is proved: $$\bar F(u) = \int_\Omega {\left( {\varphi (\varepsilon ^D u) + \frac{\kappa }{2}(div u)^2 } \right)} dx + \int_\Omega {\varphi ^\infty } \left( {\frac{{E_s^D u}}{{|E_s^D u|}}} \right)|E_s^D u|, u \in P(\Omega ),$$ where ϕ is a convex function with linear growth at infinity. Now Eu is a measure, ɛ D u represents the density of the absolutely continuous part of the absolutely continuous part of E D u, while E s D u denotes the singular part and ϕ ∞ the recession function of ϕ . Finally, we show that $$\bar F$$ 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.
Applied Mathematics and Optimization – Springer Journals
Published: Jan 1, 1997
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