LOCALIZATION2

This localisation object evaluates the overall elastic stiffness \(\boldsymbol{L}\).

Syntax

The syntax depends on the localization rule.

**localization name \(~\,~\,\) …

voigt

Using the Voigt scheme, the elastic stiffness is \(\boldsymbol{L} = 2 \mu \boldsymbol{I}\)

**localization  voigt \(~\,~\,\) mu <double>

matrix

The elastic stiffness is the Fourth-order compliance tensor \(\boldsymbol{L} = \boldsymbol{S}\)

**localization  matrix \(~\,~\,\) young <double> \(~\,~\,\) poisson <double>

kroner

The overall elastic stiffness is \(\displaystyle{\boldsymbol{L} = \alpha (\boldsymbol{L_r} (\boldsymbol{I} - \boldsymbol{S}))}\) where \(S\) is the effective Eshelby tensor and \(\boldsymbol{L_r}\) is the elasticity matrix of the sub-volume \(r\).

**localization  kroner \(~\,~\,\) alpha <double> \(~\,~\,\) poisson <double> \(~\,~\,\) ratio <double>

tangente

The elastic stiffness is \(\boldsymbol{L} = A \boldsymbol{I}\):

(648)\[{\rm where} \qquad A = \frac{2 \mu \mu_p (7 -5 \nu_p)}{\mu_p(7 -5 \nu_p)+2 \mu (4 - 5 \nu_p)}\]

(649)\[{\rm and} \quad f = \frac{\sqrt{\Delta \ten{E}^p: \Delta \ten{E}^p}}{\sqrt{\Delta \ten{\sigma}: \Delta \ten{\sigma}}}, \qquad \mu_p = \frac{\mu}{1 + 2 \mu f} \qquad {\rm and} \qquad \nu_p = \frac{\nu+2\mu (1+\nu)f/3}{1+4\mu(1+\nu)f/3}\]

**localization  tangente \(~\,~\,\) mu <double> \(~\,~\,\) poisson <double>