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MAT_MMC

Material properties

Attention: This command is in the beta stage and the format may change over time.

*MAT_MMC "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

Variable	Description
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

Description

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.

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.

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 $\beta = 1.0$.

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

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:

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