Load Capacity of Bodies
Abstract
Description
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 $w\in LD(Ω)_D$.
The earlier version was replaced because: 1. there are problems with Section 5 (Formal variational approach probably meaningless), 2. the notion of "load capacity ratio" and relation of the previous analysis to limit analysis in plasticity theory where added. Thanks to anonymous reviewers for pointing out the relation to limit analysis
The earlier version was replaced because: 1. there are problems with Section 5 (Formal variational approach probably meaningless), 2. the notion of "load capacity ratio" and relation of the previous analysis to limit analysis in plasticity theory where added. Thanks to anonymous reviewers for pointing out the relation to limit analysis