cast_iron

Description

This behavior is a combined damage-viscoplasticity model to simulate the nonlinear and fatigue behavior of cast iron materials. It allows significant flexibility in terms of the combination of damage and plastic mechanisms, hardening models, and driving force for damage.

This model uses a scalar damage variable which is the summation of a brittle mode based on maximum principal stress values, and a fatigue damage model like that described by [M5]. The total effective damage at any time is a sum of brittle (\(d_b\)) and fatigue (\(d_f\)) damage contributions:

\[d = d_b + d_f\]

The scalar damage value is however applied to an anisotropic closure treatment utilizing 2 separate closure/opening criteria. The use of a scalar variable is an approximation assuming close to cyclically proportional strain paths. The closure point is based on a strain criterion which shifts into compression as damage accumulates, and the opening criterion is a sole function of the principle stress being positive. Both terms are smoothed using an interpolation function with adjustable width. Note that the opening and closing combined with the general difficulties of damage based models can lead to convergence difficulties. The damage effects here are not really meant for predicting true failure, but rather are needed to predict the LCF hysteresis loops. Other methods of cast iron modeling are available using the cast_iron yield criterion and non-symmetric kinematic hardening model.

Syntax

The basic input syntax accepts a number of sub-objects: thermal_strain, elasticity, criterion, flow and isotropic hardening and can be represented as shown below:

***behavior cast_iron [ **thermal_strain <THERMAL_STRAIN> ] [ **elasticity <ELASTICITY> ] [ **flow <FLOW> ] [ **criterion <CRITERION> ] [ **isotropic <ISOTROPIC_HARDENING> ] \(~\,\) **model_coef \(~\,~\,~\,\) …

With the specific model’s coefficients (**model_coef) available:

Q1, D1

nonlinear Armstrong-Frederick kinematic hardening. Q1 is the saturation stress level and D1 is the saturation rate. Any number of these terms may be added, but both coefficients must be entered always, and the numbering is sequential from 1 to N.

dmax, dmax_b, dmax_f

damage limits preventing total failure (dmax_b and dmax_f allow to control brittle and fatigue damage limits independently).

e_0, e, E

Brittle mode damage equation takes the following form:

\[d_b=\left( \frac{\sqrt{<s_1>^2 + <s_2>^2 + <s_3>^2}-e_0}{E} \right)^{e}\]

delta_e, delta_s

Controls for the width of closure in strain dimensions (\(\delta_e\) of 0.004 recommended as a start) and damage “opening” in stress dimensions (\(\delta_s\) of 25 recommended). The interpolation functions used for the closure and opening which allows for smooth transitioning between damage active and damage closure are:

opening:

\[h_i(\sigma) = \frac12 \left[ 1 + tanh\left(\frac{-4\sigma_i}{\delta_s}\right) \right]\]

closure:

\[h_i(\epsilon) = \frac12 \left[ 1 + tanh\left(\frac{-4(\epsilon_i+\delta_d\times(d+\delta_{d0})}{\delta_e}\right) \right]\]

delta_d0

Initial shift in the strain damage closing effect

delta_d

Shift parameter for progressively compressive closure as damage increases. The closure point is \(\delta_d \times (d+\delta_{d0})\)

a, b, A, r

Simple cumulated plastic strain fatigue mode damage:

\[\chi(\ten{\sigma})=\alpha J_0+\beta J_1+(1-\alpha-\beta)J_2\]

\[\dot d_f = \left( \frac{\chi}{A} \right)^{r}\dot{p}\]

where \(p\) is the cumulated plastic strain

Stored Variables

prefix	size	description	default
eto	T-2	total strain	yes
sig	T-2	Cauchy stress	yes
eel	T-2	elastic strain	yes
eps_me	T-2	mechanical strain	yes
eps_th	T-2	thermal strain	yes
evi	T-2	viscoplastic strain	yes
evcum	S	inelastic strain equivalent	yes
d_f	S	fatigue damage	yes
d_b	S	brittle damage	yes
Dsum	S	total effective damage	yes
alpha(i)	T-2	kinematic hardening variable	yes

Example

The following is a short example material file for gray iron.

***behavior cast_iron
 **elasticity isotropic
     young   130000.
     poisson 0.26
 **flow norton
     K  400.
     n  4.
 **criterion mises
 **isotropic constant
     R0  150.
 **model_coef
   % kinematic hardening
     Q1 300.0
     D1 80.
   % brittle damage
     e   1.0  % exponent
     e_0 2.0  % criterion
     E  1000. % approx critical stress
   % progressive closure
     delta_d   0.006 % shift in strain closure according to d
     delta_e   0.008 % width of strain transition
   % damage limits
     dmax   0.25
     dmax_b 0.25
***return