Load Capacity of Bodies

Abstract

For the stress analysis in a plastic body $Ω$, we prove that there exists a maximal positive number $C$, the \emph{load capacity ratio,} such that the body will not collapse under any external traction field $t$ bounded by $Y_{0}C$, where $Y_0$ is the elastic limit. The load capacity ratio depends only on the geometry of the body and is given by $$ \frac{1}{C}=\sup_{w\in LD(Ω)_D} \frac{\int_{\partialΩ}|w|dA} {\int_Ω|ε(w)|dV}=\left\|γ_D\right\|. $$ Here, $LD(Ω)_D$ is the space of isochoric vector fields $w$ for which the corresponding stretchings $ε(w)$ are assumed to be integrable and $γ_D$ is the trace mapping assigning the boundary value $γ_D(w)$ to any $w\in LD(Ω)_D$.