Alternatively, and as the only means of defining rate-dependent yield stress
for the Johnson-Cook and the crushable foam plasticity models, the strain rate
behavior can be assumed to be separable, so that the stress-strain dependence
is similar at all strain rate levels:

$$\overline{\sigma}={\sigma}^{0}\left({\overline{\epsilon}}^{pl},\theta ,{f}_{i}\right)R\left({\dot{\overline{\epsilon}}}^{pl},\theta ,{f}_{i}\right),$$

where ${\sigma}^{0}\left({\overline{\epsilon}}^{pl},\theta ,{f}_{i}\right)$
(or $B\left({\overline{\epsilon}}^{pl},\theta ,{f}_{i}\right)$
in the foam model) is the static stress-strain behavior and
$R\left({\dot{\overline{\epsilon}}}^{pl},\theta ,{f}_{i}\right)$
is the ratio of the yield stress at nonzero strain rate to the static yield
stress (so that $R\left(0,\theta ,{f}_{i}\right)=1.0$).

Three methods are offered to define R in
Abaqus:
specifying an overstress power law, defining R directly as
a tabular function, or specifying an analytical Johnson-Cook form to define
R.

#### Overstress power law

The Cowper-Symonds overstress power law has the form

$${\dot{\overline{\epsilon}}}^{pl}=D{(R-1)}^{n}\mathit{}\text{for}\mathit{}\overline{\sigma}\ge {\sigma}^{0}\mathit{}(\text{or}\mathit{}\overline{B}\ge B\mathit{}\text{in the crushable foam model}),$$

where $D\left(\theta ,{f}_{i}\right)$
and $n\left(\theta ,{f}_{i}\right)$
are material parameters that can be functions of temperature and, possibly, of
other predefined field variables.

Abaqus/CAE Usage

Property module: material editor: : Hardening: Power Law (available for valid plasticity models)

#### Tabular function

Alternatively, R can be entered directly as a tabular
function of the equivalent plastic strain rate (or the axial plastic strain
rate in a uniaxial compression test for the crushable foam model),
${\dot{\overline{\epsilon}}}^{pl}$;
temperature, $\theta $;
and field variables, ${f}_{i}$.

Abaqus/CAE Usage

Property module: material editor: : Hardening: Yield Ratio (available for valid plasticity models)

#### Johnson-Cook rate dependence

Johnson-Cook rate dependence has the form

$${\dot{\overline{\epsilon}}}^{pl}={\dot{\epsilon}}_{0}\mathrm{exp}\left[\frac{1}{C}\left(R-1\right)\right]\text{for}\overline{\sigma}\ge {\sigma}^{0},$$

where ${\dot{\epsilon}}_{0}$
and C are material constants that do not depend on
temperature and are assumed not to depend on predefined field variables.
Johnson-Cook rate dependence can be used in conjunction with the Johnson-Cook
plasticity model, the isotropic hardening metal plasticity models, and the
extended Drucker-Prager plasticity model (it cannot be used in conjunction with
the crushable foam plasticity model).

This is the only form of rate dependence available for the Johnson-Cook
plasticity model. For more details, see
Johnson-Cook plasticity.

Abaqus/CAE Usage

Property module: material editor: : Hardening: Johnson-Cook (available for valid plasticity models)