DC Drucker shear stress¶

Module: solid

Category: materialprop

Type string: "DC Drucker shear stress"

Parameters¶

Name	Description	Default	Units
`c`	c	0	[]

Description¶

The material type for the Drucker shear stress criterion is DC Drucker shear stress. It is based on the yield criterion for plasticity introduced in ¹. For this criterion,

\[ \Xi\left(\mathbf{F}\right)=k=\left(J_{2}^{3}-cJ_{3}^{2}\right)^{1/6} \]

where \(J_{2}=\frac{1}{2}\text{dev}\boldsymbol{\sigma}_{0}:\text{dev}\boldsymbol{\sigma}_{0}\), \(J_{3}=\det\left(\text{dev}\boldsymbol{\sigma}_{0}\right)\), \(k\) is the yield limit in simple shear and \(c\) is a user-specified non-dimensional material constant which must lie in the range \(-\frac{27}{8}\le c\le\frac{9}{4}\). To better understand the meaning of \(k\), consider uniaxial loading of a bar which yields at the normal stress value of \(\sigma_{y}\). In this case,

\[ k=\frac{\sigma_{y}}{\sqrt{3}}\left(1-\frac{4}{27}c\right)^{1/6}\quad\frac{\sigma_{y}}{\sqrt{3}}\left(\frac{2}{3}\right)^{1/6}\le k\le\frac{\sigma_{y}}{\sqrt{3}}\left(\frac{3}{2}\right)^{1/6} \]

In the special case when \(c=0\) the Drucker criterion reduces to the von Mises criterion, with \(k=\sigma_{y}/\sqrt{3}\).

Example:

<criterion type="DC Drucker shear stress">
  <c>2.25</c>
</criterion>

Drucker, Daniel Charles, "Relation of experiments to mathematical theories of plasticity", Journal of Applied Mechanics (1949). ↩