#### MAT_JH_CERAMIC

###### Material properties
*MAT_JH_CERAMIC
"Optional title"
mid, $\rho$, $G$
$A$, $B$, $C$, $m$, $n$, $\dot\varepsilon_0$, $T$
$HEL$, $p_{\mathrm{HEL}}$, $\beta$, $D_1$, $D_2$, $K_1$, $K_2$, $K_3$
erode
##### Parameter definition
VariableDescription
mid Unique material identification number
$\rho$ Density
$G$ Shear modulus
$A$ Yield surface parameter
$B$ Failure surface parameter
$C$ Strain rate parameter
$m$ Failure surface parameter
$n$ Yield surface parameter
$\dot\varepsilon_0$ Reference strain rate
$T$ Strength in hydrostatic tension
$HEL$ Axial stress for uni-axial strain at Hugoniot elastic limit
$p_{\mathrm{HEL}}$ Pressure at Hugoniot elastic limit (only needed if $HEL$ is not defined)
$\beta$ Fraction of elastic energy loss due to damage that is converted to hydrostatic energy (pressure)
$D_1$ Failure strain parameter
$D_2$ Failure strain parameter
$K_1$ Linear bulk stiffness term
$K_2$ Quadratic bulk stiffness term
$K_3$ Cubic bulk stiffness term
erode Element erosion flag
options:
0 $\rightarrow$ failed element is not eroded
1 $\rightarrow$ failed element is eroded
##### Description

The Johnson-Holmquist ceramic model (JH-2) is used to model brittle materials having a higher strength in compression than in tension. The yield criterion and flow rule follow J2 plasticity. A bulking pressure term is added to account for dilatation during deviatoric plastic flow. The figure below shows the von Mises plastic flow stress as function of pressure and damage.

Damage evolves according to:

$\dot D = \frac{\dot\varepsilon_{eff}^p}{\varepsilon_{fail}}$

where:

$\varepsilon_{fail} = D_1 \left( \frac{T + p}{p_{HEL}} \right)^{D_2}$

The elastic pressure-volume relationship is cubic in compression:

$p_{elastic} = K_1 \mu + K_2 \mu^2 + K_3 \mu^3 \;\;\; \mu > 0$

and linear in tension:

$p_{elastic} = K_1 \mu \;\;\; \mu \lt 0$

where:

$\mu = \rho/\rho_0 - 1$

It is assumed that a fraction $\beta$ of the elastic deviatoric strain energy that is being released as damage grows is transformed into pressure (the rest dissipates into heat). This is the so called bulking pressure. Assuming an incremental release of elastic deviatoric strain energy $\Delta U$, the incremental bulking pressure is defined as:

$\Delta p_{bulk} = \sqrt{ p^2 + 2 \beta K_1 \Delta U} - p$

Here $p$ is the total pressure:

$p = p_{elastic} + p_{bulk}$