aging

Description

This behavior is a viscoplastic model intended for modeling the thermally activated aging behavior of aluminum for applications such as cast Al cylinder heads in TMF loading. The model includes classical combined nonlinear isotropic-kinematic hardening (only 2 kinematic terms are now allowed). The model is programmed for both Runge-Kutta and theta-method implicit integration, and for both general FEA and simulation modes.

The model has a standard additive strain decomposition with elastic, viscoplastic and thermal strain parts.

(253)\[\dot{\ten{\varepsilon}}_{el} = \dot{\ten{\varepsilon}}_{to} -\dot{\ten{ \varepsilon}}_{th}-\dot{\ten{\varepsilon}}_{vp}\]

The stress is computed from linear elasticity depending on the elastic matrix component selected:

(254)\[\ten \sigma = \tenf D_{el}:\ten \varepsilon_{el}\]

There is an integrated aging parameter \(\bf \eta\) which will cause aging effects on the hardening. This parameter evolves from 0 (unaged) to 1 (fully aged) with a saturating nonlinear form.

(255)\[\dot{\bf \eta} = \left<\frac{\bf \eta_\infty - \bf \eta}{\tau}\right>\]

and the coefficient \(\tau\) is the time constant for saturation of the aging. We expect that the two coefficients \(\bf \eta_\infty\) and \(\tau\) are functions of temperature to be able to handle overheating effects.

The hardening parameters are computed with aging effects as follows:

(256)\[\begin{split}\begin{aligned} &\ten X_1 = \dfrac{2}{3}C_1 \ten\alpha_1 \\ &\ten X_2 = (1-{\bf \eta})\dfrac{2}{3}C_2 \ten\alpha_2 \\ &R = R_0 + Q(1-e^{-b \lambda}) + (1-{\bf \eta})R_0^* \end{aligned}\end{split}\]

The remainder of the model is classical viscoplasticity:

(257)\[\begin{split} \begin{aligned} &f = J(\ten s - \ten X_1 - \ten X_2) - R\qquad \ten n = \pder{f}{\ten \sigma} \\ &\dot{\lambda} = \left<\frac{f}{K}\right>^n \\ &\dot{\ten{\varepsilon}_{vp}} = \dot{\lambda}\ten n \\ &\dot{\ten{\alpha}}_i = \dot{\lambda} \left[\ten n - \dfrac{3}{2}\dfrac{D_i}{C_i}\ten X_i\right] \\ \end{aligned}\end{split}\]

Note the aging effects are in the recall term for \(\ten X_2\).

Syntax

The basic input syntax here is:

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

The following coefficients are available:

K, n

viscoplastic Norton law coefficients for \(\dot{\lambda} = \left<f/K\right>^n\)

C1, D1

nonlinear Armstrong-Frederick kinematic hardening. C1 is the kinematic modulus and D1 is the saturation rate. The saturation back stress is occurs at \(C1/D1\).

C2, D2

nonlinear kinematic coefficients which have the aging effects applied to them.

R0, Q, b

nonlinear isotropic hardening (or softening with \(Q<0\)) having the same meaning as the nonlinear isotropic model.

R0_star

isotropic aging effect on the yield radius.

Tau, a_inf

the two aging variable coefficients.

alpha

optional thermal strain expansion coefficient.

Stored Variables

The following variables are stored with this model:

prefix	size	description	default
eto	T-2	total strain	yes
sig	T-2	Cauchy stress	yes
eel	T-2	elastic strain tensor	yes
evi	T-2	viscoplastic strain tensor	yes
eme	T-2	mechanical strain tensor	yes
evcum	S	cumulated viscoplastic strain magnitude	yes
age	S	aging variable & yes
alpha(i)	T-2	kinematic back strain, i=1,2	yes

Note

Early releases of this model misspell the model name as ageing

Example

There is an example file in the test database directory ZebFront_test/INP named aging*.inp

These tests are normally deprecated compared to the equivalent but more general capability in the tests Simulator_test/INP/aging_tmf*.