matmod

Description

This model is an implementation of the MATMOD equations due to Miller [M6]. The model accounts for isotropic/kinematic hardening and presents a particular form of temperature dependence in the material coefficients.

(287)\[\dot{\ten{\varepsilon}}_{in} = B\theta^\prime\left\{\sinh\left[ A f_1 \left(\frac{J(\ten \sigma-\ten R)}{D}\right)^{3/2} \right]\right\}^n\frac{3}{2}\frac{\bf s-\ten R}{J(\ten \sigma-\ten R)}\]

(288)\[\begin{split}\begin{aligned} &\theta^\prime= \exp\left\{ \left[ -\frac{Q}{0.6kT_m}\right] \left[ \ln\left(\frac{0.6T_m}{T}\right)+1\right] \right\}\hskip1cm \hbox{for $T\leq 0.6T_m$} \\ &\theta^\prime= \exp-Q/kT) \hskip1cm \hbox{for $T\geq 0.6T_m$} \\ \end{aligned}\end{split}\]

(289)\[\dot{\ten R} = H_1 (\dot{\ten{\varepsilon}}_{in} - B\theta^\prime[\sinh(A_1\vert\ten R\vert)]^n) \frac{\ten R}{\vert\ten R\vert}\]

(290)\[\dot{D} = H_2 \vert \dot{\ten{\varepsilon}}_{in}\vert\left[ C_2 + \vert\ten R\vert-(A_2/A_1)D^3\right] - H_2 C_2 B\theta^\prime\left[ \sinh(A_2 D^3)\right]^n\]

Syntax

***behavior matmod [ **thermal_strain <THERMAL_STRAIN> ] [ **elasticity <ELASTICITY> ] \(~\,\) **model_coef

Stored Variables

prefix	size	description	default
eto	T-2	total strain	yes
sig	T-2	Cauchy stress	yes
ein	T-2	inelastic strain tensor	yes
D	S	isotropic drag stress	yes
R	T-2	kinematic rest stress	yes

Example

The following is a simple example of the MATMOD material corresponding to the material in Miller’s original paper:

%
%  As given by Mill76 (Tests matmod#)
%
***behavior matmod
 **elasticity
    young   temperature
     1.93e5  23.0
     1.55e5  538.0
    poisson 0.3
***model_coef
    A1   0.108     % MPa^-1
    A2   2.27e-7   % MPa^-1
    B    1.e15     % sec^-1
    C2   0.69      % MPa
    H1   1930      % MPa
    H2   100       %
    n    5.8       %
    Q    91000.0   % cal/mol
    Tm   1800.0    % Degree K
***return