Material properties
Attention: This command is in the beta stage and the format may change over time.
"Optional title"
mid, $\rho_0$, $G$
$\sigma_c$, $\sigma_x$, $p_x$, $\sigma_{cap}$, $c_{1,f}$, $\sigma_{cap,f}$, $\varepsilon_{p,f}$, $s$
$K_1$, $K_2$, $K_3$, $yield$, $\beta$, $\varepsilon_{v,max}$, $c$, $\dot{\varepsilon}_0$
$\psi$, $d$, $d_{dec}$, nsplit
Parameter definition
mid Unique material identification number
$\rho_0$ Density
$G$ Shear modulus
$\sigma_c$ Uniaxial compressive strength
$\sigma_x$ Compressive strength at pressure x
$p_x$ Pressure x
$\sigma_{cap}$ Upper bound for yield surface
$c_{1,f}$ Flow stress pressure hardening parameter for failed material
$\sigma_{cap,f}$ Upper bound for flow stress of failed material
$\varepsilon_{p,f}$ Effective plastic strain at failure
options: constant, fcn
$s$ Tensile damage softening exponent
default: no damage softening
$K_1$ Linear bulk stiffness term
$K_2$ Quadratic bulk stiffness term
$K_3$ Cubic bulk stiffness term
$yield$ Type of yield surface (= 0.0 $\rightarrow$ von Mises, = 1.0 $\rightarrow$ Rankine)
default: 0.0 (von Mises)
$\beta$ Parameter controlling the plastic flow direction ($0.0 \leq \beta \leq 1.0$).
default: 0.0 (no increase of pressure with plastic flow)
$\varepsilon_{v,max}$ Cap on volumetric strain of bulking
default: not used
$c$ Strain rate parameter
$\dot{\varepsilon}_0$ Reference strain rate
$\psi$ Parameter controlling the rate dependency on the hydrostatic tensile strength
default: 0.0 (no rate effects on hydrostatic tensile strength)
$d$ Damping coefficient
$d_{dec}$ Damping decay coefficient
nsplit Node splitting activation flag
0 $\rightarrow$ node splitting inactive
1 $\rightarrow$ node splitting active

This is a ceramic model (by Marcus Menchawi and Thomas Öst) with a pressure dependent shear resistance. The pressure dependency of undamaged material is defined from two points on the yield strength vs. pressure curve and a yield strength cap.


The pressure-volume relationship is cubic in compression:

$p = K_1 \mu + K_2 \mu^2 + K_3 \mu^3 \;\;\; \mu > 0$

and linear in tension:

$p = K_1 \mu \;\;\; \mu \lt 0$


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

Yield strength

The yield strength vs. pressure curve is defined as a combination of two functions. $C^1$ continuity prevails in the transition between the two functions and the transition is defined at a pressure $p_t$, defined as:

$\displaystyle{p_t = max\left(\frac{\sigma_c}{3}, p_x\right)}$

The yield strength, $\sigma_y$ vs. pressure, $p$, is defined as:

$\sigma_y(p) = \left\{ \begin{array}{cc} \displaystyle{c_1 \cdot p + c_2} &: p \leq p_t\\ \displaystyle{\sigma_{cap} \cdot \left(1.0 - exp\left(-c_3 \cdot p + c_4\right)\right)} &: p \gt p_t \end{array} \right. $

Parameters $c_1$ to $c_4$ are calculated and set in the solver based on given input.

Yield strength vs. pressure (Lode angle $\theta=60^\circ$)
Yield strength vs. pressure (Lode angle $\theta=60^\circ$)

By default, von Mises yield surface is used (yield = 0.0). A Rankine surface is defined by setting yield = 1.0. Values between 0.0 and 1.0 are accepted to achieve a combination of the two surfaces.

Yield surfaces
Yield surfaces

Plastic flow is optional and included by setting $\varepsilon_{p,f} \gt 0$ or by providing a CURVE defining failure strain as function of pressure. With zero as input, the material becomes linear-elastic up to failure and the material strength vs. pressure curve coincides with the yield strength vs. pressure curve. Dilatation is optional during plastic flow and limited by $\varepsilon_{v,max}$. Dilatation is included by defining $\beta$ and $\varepsilon_{v,max} \gt 0.0$. Associated plastic flow is achieved with $\beta = 1.0$.

Direction of plastic flow
Direction of plastic flow

The shear resistance (flow stress) of failed material is defined from parameters $c_{1,f}$ and $\sigma_{cap,f}$.

Flow stress for fully damaged material
Flow stress for fully damaged material

Rate effects

Strain rate effects are included by increasing the quasi-static yield strength by a factor $rfac$, calculated as:

$\displaystyle{rfac = \left(1.0 + \frac{\dot{\varepsilon}^{eff}}{\varepsilon_0}\right)^c}$

With $\psi = 0.0$ (default), the hydrostatic tensile strength is not affected by the strain rate effects. $\psi = 1.0$ means that the hydrostatic tensile strength also is scaled by the factor $rfac$.


Damage, $D$, is a function of the effective plastic strain, $\varepsilon^{eff}_p$, and develops gradually from 0 to 1:

$\displaystyle{D = min \left(1.0, \frac{\varepsilon^{eff}_p}{\varepsilon_{p,f}}\right)}$

$\varepsilon_{p,f}$ is either a constant or a user defined function of pressure.

If the parameter $s$ is defined, then the hydrostatic tensile strength is reduced with a damage softneing factor $sfac$:

$\displaystyle{sfac = 1 - D^s}$