cast_iron_brittle

Description

This model is a simplified version of the more general cast_iron model, as it implements brittle damage mechanism only. The implementation of the model has been revisited to improve local and global convergences. The same manner as cast_iron model, 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, viscoplastic mechanisms and hardening models.

This model uses a scalar damage variable which is brittle mode damage (\(d_b\)) based on maximum principal stress values, like that described by [].

The scalar damage value is 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.

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_brittle [ **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, M1, m1

nonlinear Armstrong-Frederick kinematic hardening with the possibility to activate static recovery mechanism. Q1 is the saturation stress level and D1 is the saturation rate. The coefficients M1 and m1 are the standard Norton-like static recovery coefficients. In the absence of these latter two, there will be no static recovery calculation.

Any number of kinematic hardening terms may be added, but both coefficients Qn and Dn must be entered always, and the numbering n is sequential from 1 to N.

dmax

damage limit preventing total failure.

e_0, e, E

Brittle mode damage equation takes the following form:

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

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:

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

closure:

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

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 \(\delta_d \times (d+\delta_{d0})\)

d_b_ini

Initial damage convenient for identification purposes

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_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_brittle
 **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 limit
     dmax   0.25
***return