Load Capacity of Bodies

For the stress analysis in a plastic body $\Omega$, 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 $$\frac{1}{C}=\sup_{w\in LD(\Omega)_D} \frac{\int_{\partial\Omega}|w|dA} {\int_{\Omega}|\epsilon(w)|dV}=\left\|\gamma_D\right\|.$$ Here, $LD(\Omega)_D$ is the space of isochoric vector fields $w$ for which the corresponding stretchings $\epsilon(w)$ are assumed to be integrable and $\gamma_D$ is the trace mapping assigning the boundary value $\gamma_D(w)$ to any $w\in LD(\Omega)_D$.