/MAT/LAW101

Block Format Keyword This law is a time and temperature dependent material model for thermoplastic polymers using a thermodynamic approach with physically-based multiscale internal state variables. This law is only available for solid elements.

Format

(1)	(3)	(5)	(7)
/MAT/LAW101/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$
$E_{r e f}$	`E`₁	$ν$	$V E_{1}$
$V E_{2}$	${\dot{ε}}_{r e f}$	${\dot{γ}}_{0}^{p}$	$α_{p}$
$Δ H$	$V$	$m$	`C`₃
`C`₄	$α_{k 1}$	$α_{k 2}$	$h_{0}$
${\bar{z}}_{1 i}$	`C`₅	`C`₆	`C`₇
`C`₈	`C`₉	`C`₁₀	$h_{1}$
${\bar{z}}_{2 i}$	`C`₁₁	`C`₁₂	`C`₁₃
`C`₁₄	`C`₁	`C`₂	$λ_{L}$
$ρ (θ_{r e f})$	$c_{v} (θ_{r e f})$	$θ_{r e f}$	$α_{t h}$
$θ_{g l a s s}$	$ω$	$θ_{f l a g}$	$θ_{i}$

Definition

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)	$[\frac{kg}{m^{3}}]$
$E_{r e f}$	Young's modulus at the reference temperature. 6 (Real)	$[Pa]$
`E`₁	Material parameter. 6 (Real)	$[\frac{Pa}{K}]$
$ν$	Poisson's ratio. (Real)
`VE`₁, `VE`₂	Material parameter for temperature dependent Young's modulus. 6 (Real)
${\dot{ε}}_{r e f}$	Material parameter for temperature dependent Young's modulus. 6 (Real)	$[\frac{1}{s}]$
${\dot{γ}}_{0}^{p}$	Viscous flow. 8 (Real)	$[\frac{1}{s}]$
$α_{p}$	Pressure sensitivity parameter. 8 (Real)
$Δ H$	Activation energy. 8 No matter what units are used in the model, the activation energy is always entered as $[\frac{k J}{m o l}]$ . (Real)	$[\frac{J}{m o l}]$
`V`	Activation volume. 8 (Real)	$[m^{3}]$
`m`	Viscous flow exponent. 8 (Real)
`C`₃	Material parameter. 8 (Real)	$[\frac{Pa}{K}]$
`C`₄	Material parameter. 8 (Real)	$[Pa]$
$α_{k 1}$ , $α_{k 2}$	Material parameter. 7 (Real)
$h_{0}$	Hardening modulus. 7 (Real)
${\bar{z}}_{1 i}$	Initial value for state variable ${\bar{z}}_{1}$ . 7 (Real)
${\bar{z}}_{2 i}$	Intial value for state variable ${\bar{z}}_{2}$ . 7 (Real)
`C`₅	Material parameter. 7 (Real)	$[\frac{1}{K}]$
`C`₆	Material parameter. 7 (Real)
`C`₇	Material parameter. 7 (Real)	$[\frac{1}{K}]$
`C`₈	Material parameter. 7 (Real)
`C`₉	Material parameter. 7 (Real)	$[\frac{1}{K}]$
`C`₁₀	Material parameter. 7 (Real)
$h_{1}$	Hardening modulus. 7 (Real)
`C`₁₁	Material parameter. 7 (Real)	$[\frac{1}{K}]$
`C`₁₂	Material parameter. 7 (Real)
`C`₁₃	Material parameter. 7 (Real)	$[\frac{1}{K}]$
`C`₁₄	Material parameter. 7 (Real)
`C`₁	Material parameter. 7 (Real)	$[\frac{Pa}{K}]$
`C`₂	Material parameter. 7 (Real)	$[Pa]$
$λ_{L}$	Network locking stretch. 7 (Real)
$ρ (θ_{r e f})$	Density at the reference temperature. 9 (Real)	$[\frac{kg}{m^{3}}]$
$c_{v} (θ_{r e f})$	Heat capacity at the reference temperature. 9 (Real)	$[\frac{J}{K}]$
$θ_{r e f}$	Reference temperature. (Real)	$[K]$
$α_{t h}$	Thermal expansion. (Real)	$[\frac{1}{K}]$
$θ_{g l a s s}$	Glass transition temperature. 9 (Real)	$[K]$
$ω$	Material conversion factor for temperature calculation when the adiabatic condition is set. 9 (Real)
$θ_{f l a g}$	Temperature activation flag. = 0.0 Isothermal (temp = $θ_{i}$ ). = 1.0 Thermo-mechanical problems. = 2.0 Adiabatic (initial temperature = $θ_{i}$ ). (Real)
$θ_{i}$	Initial temperature. (Real)	$[K]$

Example

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                  kg                  mm                  ms
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW101/1/1
Talc filled Polypropylene
#              RHO_I
            9.05E-3
#               EREF                  E1                  Nu                 VE1
              2700.0                                  0.2600              0.0E-1
#                VE2            EDOT_REF        GAMA_DOT_REF              ALPHAP
              1.0E-2               1.0E3           3.464E+18            8.313E-2
#            delta_H                   V                   m                  C3
            109000.0           1.325E-27               5.000            
#                 C4             ALPHAK1             ALPHAK2                  H0
                  12              1.0E-2                 0.0                  40
#            ZETA1_i                  C5                  C6                  C7
               0.000             -9.0E-3              0.6600                   0  
#                 C8                  C9                 C10                  h1
               0.200              -0.300               6.700               0.000
#            ZETA2_i                 C11                 C12                 C13
               0.000                 0.0              12.000             -7.0E-3
#                C14                  C1                  C2            LAMBDA_L
               0.600             -0.1000               8.000               4.800
#        RHO_theta_0          CV_theta_0              THETA0            ALPHA_TH
            9.05E-10               2.0E9             298.000             7.70E-5
#        THETA_GLASS         TEMP_FACTOR          THETA_FLAG              THETAi
             373.000               0.000               0.000             298.000
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#enddata

Comments

This material is only compatible with solid elements, with Lagrange type total strain. The strain formulation flag defaults to I_smstr =10 in /PROP/SOLID.
The computational cost of this material law is high. However, this model gives very good results for polypropylene materials.
This model is proposed by Bouvard ¹ based on a thermodynamic framework in which physically based Internal State Variables (ISVs) are selected to accurately represent the underlying physics of the polymer chain deformation. These ISVs describe the current energetic state of the polymer network and are included in the Helmholtz free energy. The model is based on three ISVs:

${\bar{z}}_{1}$

Internal strain-like scalar induced by entanglement points. These points represent a sort of obstacle to the chain motion and, during the chain entanglement slippage at a certain amount of stress leads to a strain softening (observed experimentally on Figure 1).

${\bar{z}}_{2}$

Internal strain-like scalar associated to with chain alignment and coiling at large strains, that causes material hardening.

$β$

Internal strain-like tensor associated to with chain orientation and chain stretching at large strains, causing material hardening.

Figure 1. Decomposition of the total deformation gradient

The description of the kinematics of the problem is based on the decomposition of the deformation gradient into elastic, viscoplastic, and isotropic thermal components.(1)
$F = F^{e} F^{p} F^{θ}$
The model considers the incompressible plastic flow, that brings $J^{p} = 1$ . Considering that $F^{e} = R^{e} U^{e}$ and that the elastic logarithmic strain tensor $E^{e} = \ln U^{e}$ , the mandel stress tensor $M$ can be calculated as:(2)
$M = 2 μ (θ) E^{e} + (K (θ) - \frac{2}{3} μ (θ)) t r (E^{e}) I$

With,(3)
$μ (θ) = \frac{E (θ)}{2 + 2 ν}$

And,(4)
$K (θ) = 2 μ (θ) \frac{1 + ν}{3 (1 - 2 ν)}$

Where,

$E (θ)$

Young's modulus.

$ν$

Poisson coefficient.

$K (θ)$

Bulk moduli.

$μ (θ)$

Shear moduli.
The Cauchy stress tensor $σ$ can be obtained from the Mandel stress:(5)
$σ = J^{e}^{- 1} R^{e} M R^{e}^{T}$

The viscoplastic flow rule, which describes the evolution equation for $F^{p}$ is:(6)
${\dot{F}}^{p} = D^{p} F^{p}$

With,(7)
$D^{p} = \frac{1}{\sqrt{2}} {\dot{γ}}^{p} N^{p}$

Where,

$N^{p}$

Viscous flow direction.

$d e v$

Deviatoric part of the tensor.

(8)
$N^{p} = \frac{d e v (M - α)}{‖ d e v (M - α) ‖}$

The $α$ tensor is the corresponding stress-like tensor to the strain-like tensor $β$ , representing the chain stretching at large deformation. The viscous flow is given by:(9)
${\dot{γ}}^{p} = {\dot{γ}}_{0}^{p} \times e^{- \frac{Δ H}{R θ}} {[\sinh (\frac{τ_{e q} V}{k_{B} θ})]}^{m}$

Where,

${\dot{γ}}^{p}$

Viscous shear strain rate.

$τ_{e q}$

Amount of plastic shear stress.

Finally, the internal state variables evolution rules are:(10)
${\dot{\bar{z}}}_{1} = h_{0} (1 - \frac{{\bar{z}}_{1}}{z *}) {\dot{γ}}^{p}$
(11)
${\dot{z}}^{*} = ({\bar{z}}_{s a t}^{*} - g_{0} (θ) z^{*}) {\dot{γ}}^{p}$
(12)
${\dot{\bar{z}}}_{2} = h_{1} (λ^{p} - 1) (1 - \frac{{\bar{z}}_{2}}{z_{2 s a t} (θ)}) {\dot{γ}}^{p}$

And(13)
$\dot{β} = R_{s} (θ) (D^{p}_{} β + β D^{p}_{})$
Young's modulus as a function of temperature is:(14)
$E (θ) = [E_{r e f} + E_{1} (θ - θ_{r e f})] [1 + \frac{V E_{1}}{1 + \exp (- \frac{\log \dot{ε} - \log {\dot{ε}}_{r e f}}{V E_{2}})}]$

Internal State Variables (ISVs):

ISV 1	ISV 2	ISV 3
$k_{1} = α_{k 1} μ (θ) {\bar{z}}_{1}$	$k_{2} = α_{k 2} μ (θ) {\bar{z}}_{2}$	$α = μ_{B} (θ) β$
${\dot{\bar{z}}}_{1} = h_{0} (1 - \frac{{\bar{z}}_{1}}{z *}) {\dot{γ}}^{p}$	${\dot{\bar{z}}}_{2} = h_{1} (λ^{p} - 1) (1 - \frac{{\bar{z}}_{2}}{z_{2 s a t} (θ)}) {\dot{γ}}^{p}$	$\dot{β} = R_{s} (θ) (D^{p}_{} β + β D^{p}_{})$
${\dot{z}}^{} = ({\bar{z}}_{s a t}^{} - g_{0} (θ) z^{*}) {\dot{γ}}^{p}$	${\bar{z}}_{2 s a t} (θ) = C_{11} (θ - θ_{r e f}) + C_{12}$	$μ_{B} (θ) = μ_{R} (θ) {(1 - \frac{t r (β) - 3}{λ_{L}})}^{- 1}$
${\bar{z}}_{s a t}^{*} (θ) = C_{7} (θ - θ_{r e f}) + C_{8}$		$μ_{R} (θ) = C_{1} (θ - θ_{r e f}) + C_{2}$
$g_{0} (θ) = C_{9} (θ - θ_{r e f}) + C_{10}$		$R_{s} (θ) = C_{13} (θ - θ_{r e f}) + C_{14}$
$z_{0}^{*} (θ) = C_{5} (θ - θ_{r e f}) + C_{6}$

Flow rule:
- ${\dot{F}}^{p} = D^{p} F^{p}$
- $D^{p} = \frac{1}{\sqrt{2}} {\dot{γ}}^{p} N^{p}$
- $N^{p} = \frac{d e v (M - α)}{‖ d e v (M - α) ‖}$
- ${\dot{γ}}^{p} = {\dot{γ}}_{0}^{p} e^{- \frac{Δ H}{k θ}} {[\sinh (\frac{τ_{e q} V}{k_{B} θ})]}^{m}$
- $τ_{e q} = τ - (Y (θ) + κ_{1} + κ_{2} + α_{p} \bar{π})$
- $Y (θ) = C_{3} (θ - θ_{r e f}) + C_{4}$
- $τ = \frac{1}{\sqrt{2}} ‖ d e v (M - α) ‖$

Heat generation (in adiabatic conditions):

$c_{v} \dot{θ} = ω M : D^{p}$
$ρ (θ) = ρ (θ_{r e f}) \frac{1.42 θ_{g} + 44.7}{1.42 θ_{g} + 0.15 θ}$
$c_{v} = c_{v} (θ_{r e f}) (0.106 + 3 \times 10^{- 3} θ)$

Where,

Notation	Description	Notation	Description
$α_{k 1}$	Material parameter	$D^{p}$	Plastic component of the velocity gradient
$α_{k 2}$	Material parameter	$E$	Logarithmic strain tensor
$α$	Internal shear stress	$E_{r e f}$	Young's modulus at reference temperature
$α_{p}$	Pressure sensitivity parameter	$E_{1}$	Material parameter
$β$	ISV 3 tensor	$F$	Deformation gradient
${\dot{γ}}^{p}$	Viscous flow	$F^{e}$	Elastic component of $F$
$ε^{n o m}$	Nominal strain	$F^{p}$	Plastic component of $F$
$ε^{t r u e}$	Hencky (true) strain	$F^{θ}$	Thermal component of $F$
$η$	Viscosity	$g_{0}$	Material parameter
$θ$	Temperature	$Δ H$	Activation energy
$κ_{1}, κ_{2}$	Internal stress fields induced by entanglement points	$h_{0}$	Hardening modulus
$λ$	Stretch	$h_{1}$	Hardening modulus
$λ_{L}$	Network locking stretch	$J$	Determinant of $F$
$λ_{p}$	Equivalent plastic stretch	$K$	Elastic bulk modulus
$μ$	Elastic shear modulus	$k_{B}$	Boltzmann constant
$μ_{B}$	Internal shear stress modulus	$M$	Mandel stress
$μ_{R}$	Rubbery modulus	$m$	Viscous flow exponent
$ν$	Poisson coefficient	$N^{p}$	Viscous flow direction
$\bar{π}$	Effective pressure	$R$	Gas constant
$σ$	Cauchy (true) stress	$R_{s}$	Material parameter
$τ$	Equivalent shear stress	$R, U$	Antisymmetric, symmetric part of $F$
$ψ$	Free energy	$V$	Activation volume
$ω$	Conversion factor	$Y$	Yield surface
$C$	Cauchy-Green tensor	${\bar{z}}_{1}$	ISV 1
$c_{v}$	Heat capacity	${\bar{z}}_{2}$	ISV 2
$C_{1}, C_{2}, ..., C_{14}$	Material parameters	${\bar{z}}^{*}$	Strain criteria for chain slip
$D$	Velocity gradient	${\bar{z}}_{s a t}^{*}$	Saturation value for ${\bar{z}}^{*}$

¹ Bouvard JL, Francis DK, Tschopp MA, Marin EB, Bammann DJ, Horstemeyer MF (2013) An internal state variable material model for predicting the time, thermomechanical, and stress state dependence of amorphous glassy polymers under large deformation. Int J Plast 42:168–193