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Example 19: Nonlocal plasticity in softening bar.

A bar of length $1$ is clamped at the left edge and is subjected at the right edge to a step velocity of $0.7$. The Young's modulus is $1$ and the density is $1$. The velocity induces a stress step of $0.7$ in the bar which travels with a speed of $1$ to the left. At the left edge it re-bounces, and wants to build a stress of $1.4$. The yield stress in plasticity model is $1$. We soften the yield stress with the plasticity parameter $\kappa$. This maximum stress softens from $1$ at $\kappa ~=~ 0$ down to $0$ at $\kappa ~=~ 0.7$. To prevent unlimited localization of the plasticity zone, viscoplasticity is used, so that stresses a bit above the yield stress are allowed.

With 64 linear elements we find after time $1.5$ the following result for the stress:

\begin{figure}
\centerline {\epsfig{file=ps/ex19sxx.ps,width=6cm}}\end{figure}

Notice that the stress is a bit higher then 1, which is due to the viscoplastic law. The result for $\kappa$ is:

\begin{figure}
\centerline {\epsfig{file=ps/ex19kap.ps,width=6cm}}\end{figure}

The plasticity occurs over a length larger then the element size. Without viscoplasticity, the plasticity would have localised to 1 element, because of the softening.


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Next: Example 20: Three dimensional Up: Examples Previous: Example 18: Thermally induced   Contents
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1999-04-23