Stress Optimization for Supported Bodies

For a surface traction t, acting on a region Γt of the boundary of a given body Ω supported on Γ0 ⊂ ∂Ω, we consider the infimum σ opt t = infσ {ess supx |σ(x)|} over all stress fields σ in equilibrium with t, i.e., the smallest essential bound on all conceivable stress fields. Using the space of LD-vector fields on Ω, we show that σ t is attainable by some stress field σ̂ and that σ opt t = sup w∈C∞(Ω,R3) ∣∣ ∫ Γt t · w dA ∣∣ ∫ Ω |ε| dV , where ε is the linear strain associated with the vector field w. Varying the traction, it is shown that K = sup t σ opt t ‖t‖∞ is given by K = ‖γ0‖, where γ0 is the trace mapping. The boundary value problems associated with the suprema in the expressions for σ t and ‖γ0‖ are derived.