/MAT/LAW120 (TAPO)

Block Format Keyword This is a non-associated elasto-plastic model for polymer adhesives. The constitutive model is based on a I1-J2 criterion that can be reduced either to a von Mises or Drucker-Prager type in compression.

It can be used to represent the mechanical behavior of adhesives under complex loading paths with combined shear and tension. The material model includes a nonlinear damage model depending on plastic strain, triaxiality and strain rate. This material is applicable only to solid hexahedron elements (/BRICK).

Format

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(9)
/MAT/LAW120/`mat_ID`/`unit_ID` or /MAT/TAPO/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$
`E`		$ν$		`I`_form	`Itrx`	`Idam`
`Table_ID`	`Xscale`		`Yscale`
$τ_{0}$		`Q`		$β$		`H`
`A`_1F		`A`_2F		`A`_1H		`A`_2H	`A`_S
`C`		${\dot{ε}}_{r e f}$		${\dot{ε}}_{m a x}$
`D`_1c		`D`_2c		`D`_1f		`D`_2f
`D`_trx		`D`_JC		`Exp_n`

Definition

Field	Contents	SI Unit Example
`mat_ID`	Material identifier. (Integer, maximum 10 digits)
`unit_ID`	(Optional) 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 (stiffness) modulus. (Real)	$[Pa]$
$ν$	Poisson's coefficient. (Real)
`I`_form	Yield criterion formulation flag. = 1 (Default) Drucker-Prager model in compression. = 2 von Mises in compression. (Integer)
`Itrx`	Damage dependency on triaxiality in compression flag. = 1 Damage depends on triaxiality in tension and compression. = 2 (Default) Damage depends on triaxiality in tension only. (Integer)
`Idam`	Strain rate definition in damage model flag. = 1 Damage factor defined with undamaged plastic strain rate. = 2 (Default) Damage factor defined with damaged plastic strain rate. (Integer)
`Table_ID`	Table identifier to define yield stress as a function of plastic strain, strain rate and temperature. (Integer)
`Xscale`	Scale factor for strain rate variable in `Table_ID`. (Real)	$[Hz]$
`Yscale`	Scale factor for yield stress value defined by `Table_ID`. (Real)	$[Pa]$
$τ_{0}$	Initial shear yield stress. (Real)	$[Pa]$
`Q`	Voce hardening modulus. (Real)	$[Pa]$
$β$	Voce nonlinear hardening exponent. Default = 1.0 (Real)
`H`	Linear hardening exponent. Default = 1.0 (Real)	$[Pa]$
`A`_1F	Yield function parameter. (Real)
`A`_2F	Yield function parameter. (Real)
`A`_1H	Yield function distortional hardening parameter. (Real)
`A`_2H	Yield function distortional hardening parameter. (Real)
`A`_S	Plastic flow function parameter for hydrostatic term. (Real)
`C`	Johnson-Cook strain rate coefficient for hardening. (Real)
${\dot{ε}}_{r e f}$	Quasi-static threshold strain rate in Johnson-Cook term. (Real)	$[Hz]$
${\dot{ε}}_{m a x}$	Maximum dynamic threshold strain rate in Johnson-Cook term. (Real)	$[Hz]$
`D`_1c	Johnson-Cook parameter for damage initiation. (Real)
`D`_2c	Johnson-Cook parameter for damage initiation. (Real)
`D`_1f	Johnson-Cook parameter for failure strain. (Real)
`D`_2f	Johnson-Cook parameter for failure strain. (Real)
`D`_trx	Johnson-Cook damage parameter for triaxiality term. (Real)
`D`_JC	Johnson-Cook strain rate parameter for damage. (Real)
`Exp_n`	Exponential coefficient for damage strain rate dependency. (Real)

Example (Adhesive Polymer)

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/20
Material model units
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/TAPO/1/20
Adhesive polymer
#              RHO_I
              1.2E-9                   0
#                  E                  Nu     Iform      Itrx      Idam
                1588                 .34         1         0         0          
#               TAU0                   Q                beta                   H
               19.66               2.746               24.98               13.35
#                A1F                 A2F                 A1H                 A2H                  AS
               0.446               0.218                0.24                 0.1               0.338
#                 CC            Epsp_ref            Epsp_max
                 0.1               0.002                1726
#                D1c                 D2c                 D1f                 D2f
               0.345               1.094               6.935                0.00 
#              D_trx                D_JC               Exp_n
               0.001               1.044                   0

Comments

The yield function is described depending on the I_form flag:
- I_form = 1: Drucker-Prager formulation:(1)
  $f = J_{2} + \frac{a_{1}}{\sqrt{3}} τ_{0} I_{1} + \frac{a_{2}}{3} I_{1}^{2} - τ_{y}^{2}$
  
  $a_{1} = A_{1 F} + A_{1 H} ε_{p l}$ and $a_{2} = A_{2 F} + A_{2 H} ε_{p l}$
- I_form = 2: von Mises formulation:(2)
  $f = J_{2} + \frac{A_{2 F}}{3} I_{1} + \frac{\sqrt{3}}{2} \frac{A_{1 F}}{A_{2 F}} τ_{0}^{2} - (τ_{y}^{2} + \frac{A_{1 F}^{2}}{A_{2 F}} \frac{τ_{0}^{2}}{4})$
These 2 functions are written in terms of the damaged stress tensor: $σ_{d} = σ / (1 - D)$

Where, $D$ represents the isotropic damage.
Plastic potential is expressed as:(3)
$f^{*} = J_{2} + \frac{A_{S}}{3} I_{1}^{2}$
Yield stress is rate dependent:
- Table_ID ≠ 0, the yield stress is tabulated.
- Table_ID = 0, it is analytic.
(4)
$τ_{y} = (τ_{0} + R) g (\dot{ε})$

Where, $R = Q (1 - exp (- β ε_{p l})) + H ε_{p l}$ .(5)
$g (\dot{ε}) = 1 + C [l n (\frac{\dot{ε}}{{\dot{ε}}_{r e f}}) - l n (\frac{\dot{ε}}{{\dot{ε}}_{m a x}})]$
Damage initiation and rupture are function of triaxiality $σ^{*} = \frac{σ_{m}}{\bar{σ}}$ with $σ_{m} = \frac{I_{1}}{3}$ and ${\bar{σ}}_{e q} = \sqrt{3 J_{2}}$ .(6)
$\dot{D} = n {\frac{ε_{p l} - ε_{c}}{ε_{f} - ε_{c}}}^{n - 1} \frac{{\dot{ε}}_{p l}}{ε_{f} - ε_{c}}$
(7)
$ε_{c} = [D_{1 c} + D_{2 c} exp (D_{t r x} σ^{*})] (1 + D_{J C} l n (\frac{\dot{ε}}{{\dot{ε}}_{r e f}}))$
(8)
$ε_{f} = [D_{1 f} + D_{2 f} exp (D_{t r x} σ^{*})] (1 + D_{J C} l n (\frac{\dot{ε}}{{\dot{ε}}_{r e f}}))$