/MAT/PLAS_ZERIL

Block Format Keyword This law defines an isotropic elasto-plastic material using the Zerilli-Armstrong plasticity model.

Format

(1)	(3)	(5)	(6)	(7)	(9)
/MAT/PLAS_ZERIL/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$ $ρ_{i}$
`E`	`v`
`C`₀	`C`₅	`n`		$ε_{p}^{m a x}$ $ε_{p}^{m a x}$	$σ_{max 0}$ $σ_{\max 0}$
`C`₁	${˙ ε}_{0}$ ${\dot{ε}}_{0}$	`ICC`	`F`_smooth	`F`_cut
`C`₃	`C`₄	$ρ C_{p}$ $ρ C_{p}$		`T`_r

Definitions

Field	Contents	SI Unit Example
`mat_ID`	Material identifier (Integer, maximum 10 digits)
`unit_ID`	Unit Identifier (Integer, maximum 10 digits)
`mat_title`	Material title (Character, maximum 100 characters)
$ρ_{i}$ $ρ_{i}$	Initial density (Real)	$[\frac{kg}{m^{3}}]$ $[\frac{kg}{m^{3}}]$
`E`	Young's modulus (Real)	$[Pa]$ $[Pa]$
`v`	Poisson's ratio (Real)
`C`₀	Plasticity yield stress (Real)	$[Pa]$ $[Pa]$
`C`₅	Plasticity hardening parameter (Real)	$[Pa]$ $[Pa]$
`n`	Plasticity hardening exponent. 5 Default = 1.0 (Real)
$ε_{p}^{m a x}$ $ε_{p}^{m a x}$	Failure plastic strain. Default = 10³⁰ (Real)
$σ_{max 0}$ $σ_{\max 0}$	Plasticity maximum stress. Default = 10³⁰ (Real)	$[Pa]$ $[Pa]$
`C`₁	Strain rate formulation coefficient. (Real)	$[Pa]$ $[Pa]$
${˙ ε}_{0}$ ${\dot{ε}}_{0}$	Reference strain rate (must be 1 s^-1 converted into user's units). (Real)	$[\frac{1}{s}]$ $[\frac{1}{s}]$
`ICC`	Strain rate computation flag. 7 = 0 (Default) Set to 1 = 1 Strain rate effect on $σ_{max}$ $σ_{max}$ . = 2 No strain rate effect on $σ_{max}$ $σ_{max}$ . (Integer)
`F`_smooth	Smooth strain rate option flag. = 0 (Default) No strain rate smoothing. = 1 Strain rate smoothing active. (Integer)
`F`_cut	Cutoff frequency for strain rate filtering. 8 Default = 10³⁰ (Real)	$[Hz]$ $[Hz]$
`C`₃	Temperature effect coefficient. (Real)	$[\frac{1}{K}]$ $[\frac{1}{K}]$
`C`₄	Temperature effect coefficient. = 0 No strain rate effect. (Real)	$[\frac{1}{K}]$ $[\frac{1}{K}]$
$ρ C_{p}$ $ρ C_{p}$	Specific heat per unit of volume. = 0 Temperature is constant: `T` = `T`_r (Real)	$[\frac{J}{m^{3} \cdot K}]$ $[\frac{J}{m^{3} \cdot K}]$
`T`_r	Reference temperature. Default = 298 K (Real)	$[K]$ $[K]$

Comments

The Zerilli-Armstrong law is applicable only to shells and solids.
The equation that describes stress during plastic deformation is: (1)
$σ = C_{0} + (C_{1} exp ((- C_{3} T + C_{4} T ln (\frac{˙ ε}{{˙ ε}_{0}})))) + C_{5} ε_{p}^{n}$ $σ = C_{0} + (C_{1} \exp ((- C_{3} T + C_{4} T \ln (\frac{\dot{ε}}{{\dot{ε}}_{0}})))) + C_{5} ε_{p}^{n}$

Where,

$ε_{p}$ $ε_{p}$

Plastic strain

$˙ ε$ $\dot{ε}$

Strain rate

$T$ $T$

Temperature
Yield stress should be strictly positive.
When ${¯ ε}_{p}$ ${\bar{ε}}_{p}$ reaches $ε_{p}^{m a x}$ $ε_{p}^{m a x}$ in one integration point, then based on the element type:
- Shell elements:
  The corresponding shell element is deleted.
- Solid elements:
  The deviatoric stress of the corresponding integral point is permanently set to 0; however, the solid element is not deleted.
n must be less than 1.
If ${˙ ε}_{0}$ ${\dot{ε}}_{0}$ is 0, there is no strain rate effect.

ICC is a flag of the strain rate effect on material maximum stress

σ_{max}

$σ_{\max}$ :

Strain rate filtering input (F_cut) is only available for shell and solid elements.
The strain rate filtering is used to smooth strain rates.
Temperature is computed assuming adiabatic conditions:(2)
$T = T_{r} + \frac{E_{i n t}}{ρ C_{ρ} (V o l u m e)}$ $Τ = Τ_{r} + \frac{E_{i n t}}{ρ C_{ρ} (V o l u m e)}$

Where, E_int is the internal energy computed by Radioss.
When the temperature is not initialized using /HEAT/MAT or /INITEMP, the reference temperature (T_r) is also the initial temperature.

$σ = σ_{y} (1 + c ln (\frac{˙ ε}{{˙ ε}_{o}}))$ $σ = σ_{y} (1 + c ln (\frac{\dot{ε}}{{\dot{ε}}_{o}}))$	$σ = σ_{y} (1 + c ln (\frac{˙ ε}{{˙ ε}_{o}}))$ $σ = σ_{y} (1 + c ln (\frac{\dot{ε}}{{\dot{ε}}_{o}}))$
$σ_{max} = σ_{max 0} (1 + c ln (\frac{˙ ε}{{˙ ε}_{o}}))$ $σ_{\max} = σ_{\max 0} (1 + c \ln (\frac{\dot{ε}}{{\dot{ε}}_{o}}))$	$σ_{max} = σ_{max 0}$ $σ_{\max} = σ_{\max 0}$