# Brittle Damage: Johnson-Cook Plasticity Model (LAW27)

Johnson-Cook plasticity model is presented in Johnson-Cook Plasticity Model (LAW2). For shell applications, a simple damage model can be associated to this law to take into account the brittle failure. The crack propagation occurs in the plan of shell in the case of mono-layer property and through the thickness if a multi-layer property is defined (図 1).
The elastic-plastic behavior of the material is defined by Johnson-Cook model. However, the stress-strain curve for the material incorporates a last part related to damage phase as shown in 図 2. The damage parameters are:
${\epsilon }_{t1}$
Tensile rupture strain in direction 1
${\epsilon }_{m1}$
Maximum strain in direction 1
dmax1
Maximum damage in direction 1
${\epsilon }_{f1}$
Maximum strain for element deletion in direction 1
The element is removed if one layer of element reaches the failure tensile strain, ${\epsilon }_{f1}$ . The nominal and effective stresses developed in an element are related by:(1)
${\sigma }_{n}={\sigma }_{eff}\left(1-d\right)$
Where,
Damage factor
The strains and the stresses in each direction are given by:(2)
${\epsilon }_{1}=\frac{{\sigma }_{1}}{\left(1-{d}_{1}\right)E}-\frac{\nu {\sigma }_{2}}{E}$
(3)
${\epsilon }_{2}=\frac{{\sigma }_{2}}{E}-\frac{\nu {\sigma }_{1}}{E}$
(4)
${\gamma }_{12}=\frac{{\sigma }_{12}}{\left(1-{d}_{1}\right)G}$
(5)
${\sigma }_{1}=\frac{E\left(1-{d}_{1}\right)}{\left[1-\left(1-{d}_{1}\right){\nu }^{2}\right]}\left({\epsilon }_{1}+\nu {\epsilon }_{2}\right)$
(6)
${\sigma }_{2}=\frac{E}{\left[1-\left(1-{d}_{1}\right){\nu }^{2}\right]}\left({\epsilon }_{2}+\left(1-{d}_{1}\right)\nu {\epsilon }_{1}\right)$

The conditions for these equations are:

$0

$\epsilon ={\epsilon }_{t}$ ;

$\epsilon ={\epsilon }_{m}$ ;

A linear damage model is used to compute the damage factor in function of material strain.(7)
$d=\frac{\epsilon -{\epsilon }_{t}}{{\epsilon }_{m}-{\epsilon }_{t}}$
The stress-strain curve is then modified to take into account the damage by 式 1. Therefore:(8)
$\sigma =E\frac{{\epsilon }_{m}-\epsilon }{{\epsilon }_{m}-{\epsilon }_{t}}\left(\epsilon -{\epsilon }_{t}^{p}\right)$
The softening condition is given by:(9)
${\epsilon }_{m}-{\epsilon }_{t}\le {\epsilon }_{t}-{\epsilon }_{t}^{p}$
The mathematical approach described here can be applied to the modeling of rivets. Predit law in Radioss allows achievement of this end by a simple model where for the elastic-plastic behavior a Johnson-Cook model or a tabulated law (LAW36) may be used.