/MAT/LAW112 (PAPER or XIA)
Block Format Keyword The Paperboard law models an orthotropic and dissymmetric elastoplastic material from proposed by Xia, 2002.
The basic principle is to fully uncouple the behavior in the plane of the paper sheet and the behavior out of the plane. A yield stress is defined for each directions of loading, in tension and compression.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW112/mat_ID/unit_ID or /MAT/PAPER/mat_ID/unit_ID or /MAT/XIA/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
E_{1}  E_{2}  E_{3}  Ires  Itab  I_{smooth}  
${\nu}_{21}$  G_{12}  G_{23}  G_{13}  
K  E3C  CC  
${\nu}_{1p}$  ${\nu}_{2p}$  ${\nu}_{4p}$  ${\nu}_{5p}$ 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

S01  A01  B01  C01  
S02  A02  B02  C02  
S03  A03  B03  C03  
S04  A04  B04  C04  
S05  A05  B05  C05  
ASIG  BSIG  CSIG  
TAU0  ATAU  BTAU 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

TAB_YLD1  MAT_Xscale1  MAT_Yscale1  
TAB_YLD2  MAT_Xscale2  MAT_Yscale2  
TAB_YLD3  MAT_Xscale3  MAT_Yscale3  
TAB_YLD4  MAT_Xscale4  MAT_Yscale4  
TAB_YLD5  MAT_Xscale5  MAT_Yscale5  
TAB_YLDC  MAT_XscaleC  MAT_YscaleC  
TAB_YLDS  MAT_XscaleS  MAT_YscaleS 
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) 

${\rho}_{i}$  Initial
density. (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
E_{i}  Young’s modulus in the
i^{th} orthotropic direction. (Real) 
$\left[\text{Pa}\right]$ 
${\nu}_{ij}$  Poisson's ratio related to the
i^{th} and j^{th} orthotropic
direction. (Real) 

${G}_{ij}$  Shear modulus related to the
i^{th} and j^{th} orthotropic
direction. (Real) 
$\left[\text{Pa}\right]$ 
Ires  Resolution method for plasticity.
(Integer) 

Itab  Yield stresses computation.
(Integer) 

I_{smooth}  Interpolation type (in case of
tabulated yield function).
(Integer) 

K  Inplane yield surface
exponent. Default = 1.0 (Real) 

E3C  First elastic compression
parameter. Default = E_{3} (Real) 
$\left[\text{Pa}\right]$ 
CC  Second elastic compression
parameter. Default = 1.0 (Real) 

${\nu}_{1p}$  Tensile plastic Poisson’s ratio in
direction 1. (Real) 

${\nu}_{2p}$  Tensile plastic Poisson’s ratio in
direction 2. (Real) 

${\nu}_{4p}$  Compressive plastic Poisson’s ratio
in direction 1. (Real) 

${\nu}_{5p}$  Compressive plastic Poisson’s ratio
in direction 2. (Real) 

S0i  Initial yield stress in the
i^{th} direction of loading. Each direction is
associated to a given loading direction following the order:
Default = 1.0e20 (Real) 
$\left[\text{Pa}\right]$ 
A0i  First hardening parameter in the
i^{th} direction of loading. (Real) 
$\left[\text{Pa}\right]$ 
B0i  Second hardening parameter in the
i^{th} direction of loading. (Real) 

C0i  Third hardening parameter in the
i^{th} direction of loading. (Real) 
$\left[\text{Pa}\right]$ 
ASIG  Initial outofplane yield stress
in compression. Default = 1.0e20 (Real) 
$\left[\text{Pa}\right]$ 
BSIG  First outofplane hardening
parameter in compression. (Real) 
$\left[\text{Pa}\right]$ 
CSIG  Second outofplane hardening
parameter in compression. (Real) 

TAU0  Initial transverse shear yield
stress. Default = 1.0e20 (Real) 
$\left[\text{Pa}\right]$ 
ATAU  First transverse shear hardening
parameter. (Real) 
$\left[\text{Pa}\right]$ 
BTAU  Second transverse shear hardening
parameter. (Real) 

TAB_YLDi  Tabulated yield stress – plastic
strain  strain rate function identifier in the i^{th}
direction of loading. (Integer) 

MAT_Xscalei  X scale factor of the tabulated
yield – plastic strain  strain rate function in the
i^{th} direction of loading. Default = 1.0 (Real) 
$\text{[Hz]}$ 
MAT_Yscalei  Y scale factor of the tabulated
yield – plastic strain  strain rate function in the
i^{th} direction of loading. Default = 1.0 (Real) 
$\left[\text{Pa}\right]$ 
Example (Paper)
#RADIOSS STARTER
/UNIT/1
unit for mat
Mg mm s
#12345678910
/MAT/LAW112/1/1
Xia
# RHO_I
7.83E10
# E1 E2 E3 Ires Itab Ismooth
4193 1554 1554 2 0 0
# nu21 G12 G23 G13
0.1011 988 76 76
# K E3C CC
2.0 47.2 24.46
# nu1p nu2p nu4p nu5p
0.555 0.1537 0.18 0.145
# S01 A01 B01 C01
12.0 19.0 260.0 800.0
# S02 A02 B02 C02
6.5 40.0 160.0 250.0
# S03 A03 B03 C03
6.0 11.0 100.0 125.0
# S04 A04 B04 C04
7.3 6.0 160.0 300.0
# S05 A05 B05 C05
6.3 9.0 310.0 225.0
# ASIG BSIG CSIG
16.55 16.55 3.16
# TAU0 ATAU BTAU
2.1 9.0 2.0
#12345678910
#ENDDATA
Example (Tabulated)
#RADIOSS STARTER
/UNIT/1
unit for mat
Mg mm s
#12345678910
#12345678910
/MAT/LAW112/1/1
Xia_tab
# RHO_I
7.83E10
# E1 E2 E3 Ires Itab Ismooth
4193 1554 1554 1 1 1
# nu21 G12 G23 G13
0.1011 988 76 76
# K E3C CC
2.0 47.2 24.46
# nu1p nu2p nu4p nu5p
0.555 0.1537 0.18 0.145
# TAB_YLD1 MAT_Xscale1 MAT_Yscale1
25 1.0 1.0
# TAB_YLD2 MAT_Xscale2 MAT_Yscale2
25 1.0 0.35
# TAB_YLD3 MAT_Xscale3 MAT_Yscale3
25 1.0 0.75
# TAB_YLD4 MAT_Xscale4 MAT_Yscale4
25 1.0 0.6341
# TAB_YLD5 MAT_Xscale5 MAT_Yscale5
25 1.0 0.5
# TAB_YLDC MAT_XscaleC MAT_YscaleC
25 1.0 0.5
# TAB_YLDS MAT_XscaleS MAT_YscaleS
25 1.0 0.5
/FUNCT/46
ecoulement2
# plastic strain stress
0.0 12.00
0.012 32.979020979021
0.025 50.4615384615385
0.05 74.5
0.075 90.9473684210526
0.1 102.909090909091
0.125 112.00
0.15 119.142857142857
0.175 124.903225806452
0.2 129.647058823529
0.25 137.00
0.3 142.434782608696
0.4 149.931034482759
0.5 154.857142857143
1.0 165.846153846154
/TABLE/1/25
Yld Functions : plastic strain + strain rate dependency
#DIMENSION
2
# FCT_ID strain rate Scale_y
46 0.0 1.00
46 1.0 1.10
46 5.0 1.15
46 10.0 1.20
46 100.0 1.25
46 100000.0 1.35
#12345678910
#enddata
Comments
 To describe the behavior of the paperboard material law, the following orthotropic direction is considered.
 The elastic behavior
of this material law is orthotropic.
The inplane behavior should be fully uncoupled with the outofplane behavior, computed as:
$$\{\begin{array}{l}{\sigma}_{xx}={C}_{11}{\epsilon}_{xx}+{C}_{12}{\epsilon}_{yy}\\ {\sigma}_{yy}={C}_{21}{\epsilon}_{xx}+{C}_{22}{\epsilon}_{yy}\\ {\sigma}_{xy}={G}_{12}{\gamma}_{xy}\end{array}$$With $C=\frac{1}{1{\nu}_{12}{\nu}_{21}}\left[\begin{array}{cc}{E}_{1}& {\nu}_{12}{E}_{2}\\ {\nu}_{21}{E}_{1}& {E}_{2}\end{array}\right]$
The transverse shear components are computed as:
$$\{\begin{array}{l}{\sigma}_{yz}={G}_{23}{\epsilon}_{yz}\\ {\sigma}_{zx}={G}_{21}{\epsilon}_{zx}\end{array}$$The outofplane elastic behavior (for solids only) is treated as a uniaxial equivalent problem. However, the computation of the stress may differ between tension and compression. The elasticity becomes nonlinear for compressive loadings.
$$\begin{array}{ccc}{\sigma}_{zz}={E}_{3}{\epsilon}_{zz}^{e}& \text{if}& {\epsilon}_{zz}^{e}\ge 0\\ {\sigma}_{zz}={E}_{3C}(1{\mathrm{e}}^{{C}_{c}{\epsilon}_{zz}^{e}})& \text{if}& {\epsilon}_{zz}^{e}<0\end{array}$$  In the Xia 2002
formulation, the inplane yield criterion, denoted as
$f$
, is defined as:$$f={{\displaystyle \sum _{I=1}^{6}{\chi}_{I}\left(\frac{\sigma :{N}_{I}}{{\sigma}_{Y}^{I}}\right)}}^{2k}1$$
Where,
${\chi}_{I}=\{\begin{array}{ccc}1& \text{if}& \sigma :{N}_{I}>0\\ 0& \text{otherwise}& \end{array}$ ${\chi}_{I}$
 Switching parameters.
 $\sigma $
 Cauchy stress tensor.
 ${N}_{I}$
 Normal direction of the yield planes.
 ${\sigma}_{Y}^{I}$
 Yield stresses.
 $k$
 Positive integer.
Each direction is associated to a given loading direction following the order defined below: 1
 Tension in orthotropic direction 1.
 2
 Tension in orthotropic direction 2.
 3
 Positive inplane shear.
 4
 Compression in orthotropic direction 1.
 5
 Compression in orthotropic direction 2.
 6
 Negative inplane shear (same input as positive inplane shear ${\sigma}_{Y}^{6}={\sigma}_{Y}^{3}$ ).
The normal direction vector to the yield planes are:
$\begin{array}{l}{N}_{1}=\left[\begin{array}{cccccc}\frac{1}{\sqrt{1+{\nu}_{1p}^{2}}}& \frac{{\nu}_{1p}}{\sqrt{1+{\nu}_{1p}^{2}}}& 0& 0& 0& 0\end{array}\right]\\ {N}_{2}=\left[\begin{array}{cccccc}\frac{{\nu}_{2p}}{\sqrt{1+{\nu}_{2p}^{2}}}& \frac{1}{\sqrt{1+{\nu}_{2p}^{2}}}& 0& 0& 0& 0\end{array}\right]\\ {N}_{3}=\left[\begin{array}{cccccc}0& 0& 0& 1& 0& 0\end{array}\right]\\ {N}_{4}=\left[\begin{array}{cccccc}\frac{1}{\sqrt{1+{\nu}_{4p}^{2}}}& \frac{{\nu}_{4p}}{\sqrt{1+{\nu}_{4p}^{2}}}& 0& 0& 0& 0\end{array}\right]\\ {N}_{5}=\left[\begin{array}{cccccc}\frac{{\nu}_{5p}}{\sqrt{1+{\nu}_{5p}^{2}}}& \frac{1}{\sqrt{1+{\nu}_{5p}^{2}}}& 0& 0& 0& 0\end{array}\right]\\ {N}_{6}=\left[\begin{array}{cccccc}0& 0& 0& 1& 0& 0\end{array}\right]\end{array}$
Each direction $I$ is then associated to a specific yield stress whose expression is:
$\begin{array}{cc}{\sigma}_{Y}^{I}={S}_{I}^{0}+{A}_{I}^{0}\mathrm{tanh}\left({B}_{I}^{0}{\epsilon}_{p}^{f}\right)+{C}_{I}^{0}{\epsilon}_{p}^{f}& \begin{array}{cc}\text{with}& I\in \left[1,6\right]\end{array}\end{array}$
Where, ${\epsilon}_{p}^{f}$ is the inplane equivalent plastic strain (associated to the yield function $f$ ).
The outofplane yield function is denoted as $g$ is defined as:
$g={\sigma}_{zz}{\sigma}_{Y}^{C}$ with ${\sigma}_{Y}^{C}={A}_{\sigma}+{B}_{\sigma}\mathrm{exp}({C}_{\sigma}{\epsilon}_{p}^{g})$
Where, ${\epsilon}_{p}^{g}$ is the outofplane equivalent plastic strain (associated to the yield function $g$ ).
The transverse shear yield function is:
$$h=\frac{\sqrt{{\sigma}_{yz}^{2}+{\sigma}_{zx}^{2}}}{{\sigma}_{Y}^{S}}1$$Where, ${\sigma}_{Y}^{S}={\tau}_{0}+\left[{A}_{\tau}\mathrm{min}(0,{\sigma}_{zz}){B}_{\tau}\right]{\epsilon}_{p}^{h}$
 ${\epsilon}_{p}^{h}$
 Outofplane equivalent plastic strain (associated to the yield function $h$ ).
If the tabulated yield stress option is selected (Itab = 1), each yield stress is associated to a table (TAB_YLDi) to define the stress evolution with the plastic strain, at several plastic strainrate. Two scale factors can be also defined in the X and Y direction for each table. In this case, the hardening parameters $S0i$ , $A0i$ , $B0i$ , $C0i$ , ${A}_{\sigma}$ , ${B}_{\sigma}$ , ${C}_{\sigma}$ , ${\tau}_{0}$ , ${A}_{\tau}$ , and ${B}_{\tau}$ are ignored, and the yield stress becomes:
$\begin{array}{l}\begin{array}{cc}{\sigma}_{Y}^{I}={f}_{Y}^{tab\_YLDI}({\epsilon}_{p}^{f},{\dot{\epsilon}}_{p}^{f})& I\in \left[1,6\right]\end{array}\\ {\sigma}_{Y}^{C}={f}_{Y}^{tab\_YLDC}({\epsilon}_{p}^{g},{\dot{\epsilon}}_{p}^{g})\\ {\sigma}_{Y}^{S}={f}_{Y}^{tab\_YLDS}({\epsilon}_{p}^{h},{\dot{\epsilon}}_{p}^{h})\end{array}$
For output field, an equivalent “global” plastic strain is computed as:
$${\epsilon}_{p}=\sqrt{{({\epsilon}_{p}^{f})}^{2}+{({\epsilon}_{p}^{g})}^{2}+{({\epsilon}_{p}^{h})}^{2}}$$