DC Drucker-Prager¶

Module: solid

Category: materialprop

Type string: "DC Drucker-Prager"

Parameters¶

Name	Description	Default	Units
`b`	b	0	[]

Description¶

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

\[ \Xi\left(\mathbf{F}\right)=\sigma_{e}-b\sigma_{m} \]

where \(\sigma_{e}\) is the von Mises stress, \(\sigma_{m}=\frac{1}{3}tr\boldsymbol{\sigma}_{0}\) is the hydrostatic stress, \(b\) is a parameter that depends on the yield (or failure) stress \(\sigma_{t}\) in tension, and \(\sigma_{c}\) in compression \((\sigma_{c}\ge0)\),

\[ b=3\frac{\sigma_{t}-\sigma_{c}}{\sigma_{c}+\sigma_{t}} \]

This parameter \(b\) is negative when \(\sigma_{c}>\sigma_{t}\). In the special case when \(b=0\) the Drucker-Prager criterion reduces to the von Mises criterion. When used as a plastic yield criterion (see Section [sec:Reactive-Plasticity]), the yield stress \(\sigma_{y}\) for this type of material is given by

\[ \sigma_{y}=2\frac{\sigma_{c}\sigma_{t}}{\sigma_{c}+\sigma_{t}} \]

which reduces to \(\sigma_{y}=\sigma_{t}\) for uniaxial tension, \(\sigma_{y}=-\sigma_{c}\) in uniaxial compression, and more generally, \(\sigma_{y}=\sigma_{c}=\sigma_{t}\) when \(\sigma_{t}=\sigma_{c}\). The parameter \(a\) and \(b\) are related to the parameters A and B appearing on Wikipedia's description of the Drucker-Prager yield criterion via

\[ \begin{aligned}a & =\sqrt{3}A & b & =3\sqrt{3}B\end{aligned} \]

Given these relationships, users may consult that web page for finding the relation between these parameters and cohesion c and angle of internal friction \(\phi\) (as well as the Mohr-Coulomb yield surface).

Drucker, Daniel Charles and Prager, William, "Soil mechanics and plastic analysis or limit design", Quarterly of applied mathematics 10, 2 (1952), pp. 157--165. ↩