bodner_partom

Description

This behavior is an implementation of the classical model due to Bodner and Partom. The model is viscoplastic and incorporates scalar and tensorial hardening variables. There is no initial yield radius thereby allowing inelastic deformation at very low stress levels over long periods of time. The state and evolution equations are the following:

(276)\[Z = Z_i + Z_0 + \ten \beta:\ten u \hskip1cm \ten u = \dfrac{\ten\sigma}{\sqrt{\ten \sigma:\ten \sigma}}\]

(277)\[D_{p_2} = D_0^2 \exp\left[ -\left(\frac{Z^2}{3J_2}\right)^n \right]\]

(278)\[\dot{\ten \varepsilon}_{vi} = \sqrt{\lambda_2}\ten S \hskip1cm \ten S = \tenf U:\ten \sigma \hskip1cm \lambda_2 = D_{p_2}/J_2 \hskip1cm J_2 = \dfrac{1}{2}\ten S:\ten S\]

(279)\[W_p = \ten \sigma:\dot{\ten \varepsilon}_{vi} = \sqrt{\lambda_2}\ten S:\ten \sigma\]

(280)\[\begin{split}\begin{aligned} \dot{\ten{\varepsilon}}_{el} &= \dot{\boldsymbol{\varepsilon}}_{to} - \dot{\ten \varepsilon}_{vi} \\ \dot{Z}_i &= m_1(Z_1 - Z_i - Z_0 ) W_p \\ \dot{\ten \beta}&= m_2(Z_3 \ten u - \ten \beta )W_p \\ \end{aligned}\end{split}\]

Syntax

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

Coefficient names are n, Z0, Z2, D0, Z1, Z3, m1, m2, A1, A2, r1, r2

Stored Variables

prefix	size	description	default
eto	T-2	total strain	yes
sig	T-2	Cauchy stress	yes
evi	T-2	inelastic strain tensor	yes
Zi	S	isotropic drag stress	yes
beta	T-2	kinematic variable	yes
p	S	inelastic strain equivalent	yes
Ztot	S	Sum of \(Z\) parts	yes

Example

The following is a simple example of the Bodner-partom material using room temperature coefs for HASTELLOY-X as given by Rowley and Thornton [M5].

 ***behavior bodner_partom
  **elasticity isotropic
    young  196.6e3
    poisson 0.33
  **model_coef
    n   1.0
    Z0  1860.
    Z2  1860.
    D0  10000.0
    Z1  2390.
    Z3  603.
    m1  0.139
    m2  3.49
    A1  1.0e-9
    A2  1.0e-9
    r1  1.
    r2  1.
***return