Elastoplasticity¶

This chapter describes FrontISTR's elastoplastic framework. Selection and input details are in the Material Data chapter.

Elastoplastic Constitutive Equation¶

FrontISTR adopts an associated-flow elastoplastic formulation. The constitutive equation is written between the Jaumann rate of the Kirchhoff stress and the rate of deformation tensor, and the analysis uses the Updated Lagrange formulation.

The yield condition is given as

\[ F(\sigma, \sigma_{y_0}) \]

initially, and

\[ F(\sigma, \sigma_y(\overline{e}^p)) \]

subsequently, where

\(F\): yield function
\(\sigma_{y_0}\): initial yield stress
\(\sigma_y\): subsequent yield stress
\(\sigma\): stress tensor
\(e\): infinitesimal strain tensor
\(e^p\): plastic strain tensor
\(\overline{e}^p\): equivalent plastic strain.

The yield-stress vs equivalent plastic strain relation is taken to coincide with the uniaxial stress vs plastic strain relation:

\[ \sigma = H(e^p), \qquad \frac{d\sigma}{de^p} = H', \]

where \(H'\) is the strain-hardening modulus, and equivalently

\[ \overline{\sigma} = H(\overline{e}^p), \qquad \dot{\overline{\sigma}} = H' \dot{\overline{e}^p}. \]

For brevity the subsequent yield function is taken as a function of \(\overline{e}^p\) only (in general it also depends on temperature and plastic work). During plastic loading \(F = 0\) must persist, hence

\[ \dot{F} = \frac{\partial F}{\partial \sigma} : \dot{\sigma} + \frac{\partial F}{\partial e^p} : \dot{e}^p = 0. \]

Assuming the existence of a plastic potential \(\Theta\), the plastic strain rate is

\[ \dot{e}^p = \dot{\lambda} \frac{\partial \Theta}{\partial \sigma}. \]

Setting \(\Theta = F\) gives the associated flow rule

\[ \dot{e}^p = \dot{\lambda} \frac{\partial F}{\partial \sigma}, \]

and substitution into the consistency condition yields

\[ \dot{\lambda} = \frac{a^T : d_D}{A + a^T : D : a} \, \dot{e}, \]

where \(D\) is the elasticity matrix and

\[ a^T = \frac{\partial F}{\partial \sigma}, \qquad d_D = D a^T, \qquad A = -\frac{a}{\dot{\lambda}} \frac{\partial F}{\partial e^p} : \dot{e}^p. \]

The elastoplastic stress-strain relation is then

\[ \dot{\sigma} = \left\{ D - \frac{d_D \otimes d_D^T}{A + d_D^T a} \right\} : \dot{e}. \]

Once the yield function \(F\) is specified, the constitutive equation follows from this expression.

Yield Functions¶

The yield functions supported by FrontISTR are listed below.

Von Mises¶

\[ F = \sqrt{3 J_2} - \sigma_y = 0, \]

where \(J_2\) is the second invariant of the deviatoric stress.

Mohr-Coulomb¶

\[ F = \sigma_1 - \sigma_3 + (\sigma_1 + \sigma_3) \sin\phi - 2 c \cos\phi = 0, \]

where \(\sigma_1, \sigma_3\) are the maximum and minimum principal stresses, \(c\) is the cohesion, and \(\phi\) is the internal friction angle.

Drucker-Prager¶

\[ F = \sqrt{J_2} - \alpha \, \sigma : I - \sigma_y = 0, \]

with the constants \(\alpha\) and \(\sigma_y\) given in terms of cohesion and friction angle by

\[ \alpha = \frac{2 \sin\phi}{3 + \sin\phi}, \qquad \sigma_y = \frac{6 c \cos\phi}{3 + \sin\phi}. \]