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Elastoplasticity

This chapter describes the framework of the elastoplastic constitutive law adopted by FrontISTR. For details on selection and input specifications, see 03_material in the Functions section.

Phenomenon (Plastic Deformation)

TBD (to be written in the next phase).

Constitutive-Law Framework

FrontISTR adopts an elastoplastic constitutive equation based on an associated flow rule. The constitutive equation expresses the relationship between the Jaumann rate of Kirchhoff stress and the rate-of-deformation tensor, and the deformation analysis uses the updated Lagrange formulation.

Assume that the yield condition of an elastoplastic material is given as follows.

Initial yield condition:

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

Subsequent yield condition:

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

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–equivalent-plastic-strain relation is assumed to coincide with the stress–plastic-strain relation under a uniaxial state.

Stress–plastic-strain relation under a uniaxial state

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

where \(H'\) is the strain-hardening modulus.

Equivalent-stress–equivalent-plastic-strain relation

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

In general, the subsequent yield function is a function of temperature and plastic-strain work, but for simplicity it is assumed here to be a function only of the equivalent plastic strain \(\overline{e}^p\). During the progress of plastic deformation, \(F = 0\) must continue to hold, and therefore the following equation must be satisfied.

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

Here, \(\dot{F}\) denotes the time derivative of \(F\). Hereafter, the time derivative of a quantity \(A\) is denoted by \(\dot{A}\).

Assume the existence of a plastic potential \(\Theta\) and express the plastic strain rate by the following equation.

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

Here, \(\dot{\lambda}\) is a coefficient.

Further, assuming that the plastic potential \(\Theta\) is equal to the yield function \(F\), the following associated flow rule is used.

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

Substituting this equation into the consistency condition gives the following equation.

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

Here, \(D\) 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 can be written as follows.

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

Once the yield function \(F\) of the elastoplastic material is known, its constitutive equation is obtained from this expression.

Yield Functions

The elastoplastic yield functions adopted by FrontISTR are listed below.

Von Mises Yield Function

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

Here, \(J_2\) is the second invariant of the deviatoric stress tensor.

Mohr-Coulomb Yield Function

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

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

Drucker-Prager Yield Function

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

Here, the material constants \(\alpha\) and \(\sigma_y\) are calculated from the material cohesion and friction angle as follows.

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