/MAT/LAW84

Block Format Keyword Swift-Voce elastoplastic law with Johnson-Cook strain rate hardening and temperature softening. This law allows modeling a quadratic non-associated flow rule.

Format

(1)	(3)	(5)	(7)	(9)
/MAT/LAW84/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$
`E`	$ν$
`P`₁₂	`P`₂₂	`P`₃₃	`Q`	`B`
`G`₁₂	`G`₂₂	`G`₃₃	`K`₀	$α$
`A`	$ε_{0}$	`n`	`C`	${\dot{ε}}_{0}$
$η$	`C`_p	`T`_ini	`T`_ref	`T`_melt
`m`	${\dot{ε}}_{α}$

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)
$ρ_{i}$	Initial density (Real)	$[\frac{kg}{m^{3}}]$
`E`	Young's modulus (Real)	$[Pa]$
$ν$	Poisson's ratio (Real)
`P`₁₂	Yield parameter Default = -0.5 (Real)
`P`₂₂	Yield parameter Default = 1.0 (Real)
`P`₃₃	Yield parameter Default = 3.0 (Real)
`G`₁₂	Flow rule parameter Default = `P`₁₂ (Real)
`G`₂₂	Flow rule parameter Default = `P`₂₂ (Real)
`G`₃₃	Flow rule parameter Default = `P`₃₃ (Real)
`Q`	Voce hardening coefficient (Real)	$[Pa]$
`B`	Voce plastic strain coefficient Default = 0.0 (Real)
`K`₀	Voce parameter (Real)
$α$	Yield weighting coefficient. =1 Swift hardening law. =0 Voce hardening law. Default = 0.0 (Real)
`A`	Swift hardening coefficient. (Real)	$[Pa]$
`n`	Swift hardening exponent. Default = 1.0 (Real)
$ε_{0}$	Swift hardening parameter. Default = 0.00 (Real)
`C`	Strain rate coefficient. = 0 No strain rate effect. Default = 0.00 (Real)
${\dot{ε}}_{0}$	Reference strain rate. Default = 10³⁰, no strain rate effect (Real)	$[\frac{1}{s}]$
$η$	Taylor-Quinney coefficient quantifies the fraction of plastic work converted to heat. (Real)
`C`_p	Specific heat. (Real)	$[\frac{J}{kg \cdot K}]$
`T`_ini	Initial temperature used in initialization when time=0. (Real)	$[K]$
`T`_ref	Reference temperature. (Real)	$[K]$
`T`_melt	Melting temperature. (Real)	$[K]$
`m`	Temperature exponent. (Real)
${\dot{ε}}_{α}$	Strain rate optimization parameter for temperature dependency. (Real)	$[\frac{1}{s}]$

Example (Metal)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW84/1/1
Swift-voce (metal)
#              Rho_i
                8E-9
#                  E                  Nu
              206000                  .3
#                P12                 P22                 P33                   Q                   B
                 -.5                   1                   3                 524                  25
#                G12                 G22                 G33                  K0               ALPHA
                 -.5                   1                   3                 100                  .5
#                  A                EPS0                   n                   C              EPSDOT
                1000              .00128                  .2                .014               .0011
#                ETA                  CP                Tini                Tref               Tmelt
                  .9         42000000000                 293                 293                1700
#                  m             EPSDOTA
                .921               1.379
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

Yield stress is computed using an analytic expression with a combination of both Swift and Voce models, the strain rate dependency and temperature dependency following Johnson-Cook law.(1)
$σ_{y} = {α [A {({\bar{ε}}_{p} + ε_{0})}^{n}] + (1 - α) [K_{0} + Q (1 - \exp (- B {\bar{ε}}_{p}))]} (1 + C \ln \frac{{\dot{\bar{ε}}}_{p}}{{\dot{ε}}_{0}}) [1 - {(\frac{T - T_{r e f}}{T_{m e l t} - T_{r e f}})}^{m}]$
The effective stress is computed as:(2)
$\begin{array}{l} \bar{σ} = f (σ) \\ \begin{matrix} \end{matrix} = \sqrt{σ^{T} P σ} \\ \begin{matrix} \end{matrix} = σ_{11}^{2} + P_{22} σ_{22}^{2} + (1 + P_{12} + P_{22}) σ_{33}^{2} \\ \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} + 2 P_{12} σ_{11} σ_{22} - 2 (1 + P_{12}) σ_{11} σ_{33} - 2 (P_{22} + P_{12}) σ_{22} σ_{33} \\ \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} + (P_{33} + 3) σ_{12}^{2} + 3 σ_{23}^{2} \end{array}$
The plastic non-associated flow rule is computed as:(3)
$Δ ε_{p} = Δ {\bar{ε}}_{p} \frac{\partial g (σ)}{\partial σ}$

Where,(4)
$\begin{array}{l} g (σ) = \sqrt{σ^{T} G σ} \\ \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} = σ_{11}^{2} + G_{22} σ_{22}^{2} + (1 + G_{12} + G_{22}) σ_{33}^{2} \\ \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} + 2 G_{12} σ_{11} σ_{22} - 2 (1 + G_{12}) σ_{11} σ_{33} - 2 (G_{22} + G_{12}) σ_{22} σ_{33} \\ \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} + (G_{33} + 3) σ_{12}^{2} + 3 σ_{23}^{2} \end{array}$
Temperature is updated using:(5)
$Δ T = ω ({\dot{\bar{ε}}}_{p}) \frac{η}{ρ C_{p}} \bar{σ} d {\bar{ε}}_{p}$

Where, $ω ({\dot{\bar{ε}}}_{p}) = {\begin{matrix} 0 & if & {\dot{\bar{ε}}}_{p} < {\dot{ε}}_{0} \\ 1 & if & {\dot{\bar{ε}}}_{p} > {\dot{ε}}_{α} \\ \frac{{({\dot{\bar{ε}}}_{p} - {\dot{ε}}_{0})}^{2} (3 {\dot{ε}}_{α} - 2 {\dot{\bar{ε}}}_{p} - {\dot{ε}}_{0})}{{({\dot{ε}}_{α} - {\dot{ε}}_{0})}^{3}} & if & {\dot{ε}}_{0} \leq {\dot{\bar{ε}}}_{p} \leq {\dot{ε}}_{α} \end{matrix}$