Material properties
"Optional title"
mid, $\rho$, $G$
$\sigma_c$, $\sigma_x$, $P_x$, $\sigma_{cap}$, $\alpha$, $\varepsilon_{p,fail}$, $yield$
$K$, $\beta$, $\varepsilon_{v,max}$, $c$, $\dot{\varepsilon}_0$, $\psi$, $d$, $d_{dec}$
Parameter definition
mid Unique material identification number
$\rho$ Density
$G$ Shear modulus
$\sigma_c$ Uniaxial compressive strength
$\sigma_x$ Compressive strength at pressure x
$P_x$ Pressure x
$\sigma_{cap}$ Cap on yield surface
default: no cap
$\alpha$ Failed-to-intact material strength ratio
$\varepsilon_{p,fail}$ Effective plastic strain at failure
$yield$ Type of yield surface (= 0.0 $\rightarrow$ von Mises, = 1.0 $\rightarrow$ Rankine)
default: 0.0 (von Mises)
$K$ Bulk modulus
$\beta$ 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

This ceramic model is assumed to have a pressure dependent shear resistance. The pressure dependency is defined by two points on the yield strength vs. pressure curve and an optional cap on the 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
Yield strength vs. pressure

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 $\varepsilon_{p,fail} \gt 0.0$. 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

Failed material maintains a fraction of the intact shear resistance, $\alpha$, in the positive pressure regime.

Yield strength for intact and failed material
Yield strength for intact and failed material

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 0.0 to 1.0:

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