Load capacity of bodies

For the stress analysis in a plastic body Ω, we prove that there exists a maximal positive number C, the load capacity ratio, such that the body will not collapse under any external traction field t bounded by CY₀, where Y₀ is the yield stress. The load capacity ratio depends only on the geometry of the body and is given byfrac(1, C) = under(sup, w ∈ LD (Ω)_D) frac(∫_{∂ Ω} | w | d A, ∫_Ω | ε{lunate} (w) | d V) = ∥ γ_D ∥ .Here, LD (Ω)_D is the space of incompressible vector fields w for which the corresponding linear strains ε{lunate} (w) are assumed to be integrable and γ_D is the trace mapping assigning the boundary value γ_D (w) to any w ∈ LD (Ω)_D.