Load Capacity of Bodies

For the stress analysis in a plastic body $\mathrm{\Omega }$, 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 $$\frac{1}{C}=\underset{w\in LD(\mathrm{\Omega }{)}_D}{\sup }\frac{{\int }_{\mathrm{\partial }\mathrm{\Omega }}\mathrm{\mid }w\mathrm{\mid }dA}{{\int }_{\mathrm{\Omega }}\mathrm{\mid }ϵ(w)\mathrm{\mid }dV}=\parallel {\gamma }_D\parallel \mathrm{.}$$ Here, $LD(\mathrm{\Omega }{)}_D$ is the space of isochoric vector fields $w$ for which the corresponding stretchings $ϵ(w)$ are assumed to be integrable and ${\gamma }_D$ is the trace mapping assigning the boundary value ${\gamma }_D(w)$ to any $w\in LD(\mathrm{\Omega }{)}_D$.