/MAT/LAW78
Block Format Keyword This law is the YoshidaUemori model for describing the largestrain cyclic plasticity of metals. The law is based on the framework of two surfaces theory: the yielding surface and the bounding surface.
During the plastic deformation, a yield surface will move within the bounding surface and will never change its size, and the bounding surface can change both in size and location. The plasticstrain dependency of the Young's modulus and the workhardening stagnation effect are also taken into account. Concerning SPH, it is compatible with solid only, this can be verified with the /SPH/WavesCompression test. The solid version is only isotropic. The shell version is anisotropic based on Hill criterion.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW78/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
E  $\nu $  
Y  b  ${C}_{2}^{KH}$  h  B_{0}  
m  R_{sat}  OptR  C_{1}  C_{2}  
r_{00}  r_{45}  r_{90}  Mexp  Icrit  
fct_ID_{E}  E_{inf}  C_{E}  ${C}_{1}^{KH}$ 
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) 

${\rho}_{i}$  Initial density. (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
E  Young's modulus. (Real) 
$\left[\text{Pa}\right]$ 
$\nu $  Poisson's ratio. (Real) 

Y  Yield stress. (Real) 
$\left[\text{Pa}\right]$ 
b  Center of the bounding
surface. (Real) 
$\left[\text{Pa}\right]$ 
${C}_{2}^{KH}$  Parameter for kinematic hardening rule
of yield surface (or used in calculation of
$C$
if
${C}_{1}^{KH}>{C}_{2}^{KH}$
). 5 (Real) 

h  Material parameter for controlling work
hardening stagnation. (Real) 

B_{0}  Initial size of the bounding
surface. (Real) 
$\left[\text{Pa}\right]$ 
m  Parameter for isotropic and kinematic
hardening of the bounding surface. (Real) 

R_{sat}  Saturated value of the isotropic
hardening stress. (Real) 
$\left[\text{Pa}\right]$ 
OptR  Modified isotropic hardening rule flag
(available for shells only):
(Integer) 

C_{1}, C_{2}  Constant used in the modified
formulation of the isotropic hardening of bounding surface (available for shells
only). (Real) 

r_{00}  Lankford parameter (0 degree) used for
shell elements. Default = 1.0 (Real) 

r_{45}  Lankford parameter (45 degree) used for
shell elements. Default = 1.0 (Real) 

r_{90}  Lankford parameter (90 degree) used for
shell elements. Default = 1.0 (Real) 

Mexp  Exponent
$M$
in Barlat's 1989 Yield Criterion for shell elements.
See Comment 8.
(Real) 

Icrit  Plastic criterion selection flag.


fct_ID_{E}  ID of the function defining the scale
factor of Young's modulus evolution versus effective plastic strain. 9 (Integer) 

E_{inf}  Asymptotic value of Young's
modulus. (Real) 
$\left[\text{Pa}\right]$ 
C_{E}  Parameter controlling the dependency of
Young's modulus on the effective plastic strain. (Real) 

${C}_{1}^{KH}$  Parameter
$C$
for kinematic hardening rule of yield surface. 5
Ignore if ${C}_{1}^{KH}\le {C}_{2}^{KH}$ . (Real) 
Example
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
Mg mm s
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/LAW78/1/1
DP600HDG
# RHO_I
7.8E9
# E NU
206000 .3
# Y B C2KH H B0
420 112 200 0 555
# m RSAT OPTR C1 C2
12 190 0 1 1
# R0 R45 R90 Mexp Icrit
1 1 1
# Fct_IDE EINF CE C1KH
0 163000 50 0
#12345678910
#ENDDATA
/END
#12345678910
Comments
 For solid elements, von Mises yield criterion is
used, so the yield function is expressed as:
(1) $$f=\frac{3}{2}\left(\mathbf{s}\mathbf{\alpha}\right):\left(\mathbf{s}\mathbf{\alpha}\right){Y}^{2}$$Whereas for shell elements, Hill’s (1948) or Barlat’s (1989) yield criterion are used, which allows for the modeling of anisotropic materials: The Hill’s criterion is expressed as:
(2) $$f=\phi (\mathbf{\sigma}\mathbf{\alpha}){Y}^{2}$$Where, $Y$
 Yield stress.
 $\alpha $
 Total back stress.
If $A=\sigma \alpha $ , then $\phi $ is expressed as:(3) $$\phi (A)={A}_{xx}^{2}\frac{2{r}_{0}}{1+{r}_{0}}{A}_{xx}{A}_{yy}+\frac{{r}_{0}\left(1+{r}_{90}\right)}{{r}_{90}\left(1+{r}_{0}\right)}{A}_{yy}^{2}+\frac{{r}_{0}+{r}_{90}}{{r}_{90}\left(1+{r}_{0}\right)}\left(2{r}_{45}+1\right){A}_{xy}^{2}$$  The Barlat’s criterion is expressed as:
(4) $$f=\varphi \left(\sigma \alpha \right)2{Y}^{M}$$Where, $M$ is the exponent in Barlat's yield criterion.
$\varphi $ is expressed as:(5) $$\varphi \left(A\right)=a{\left{K}_{1}+{K}_{2}\right}^{M}+a{\left{K}_{1}{K}_{2}\right}^{M}+c{\left2{K}_{2}\right}^{M}$$With,
${K}_{1}=\frac{{A}_{xx}+h{A}_{yy}}{2}$ and ${K}_{2}=\sqrt{{\left(\frac{{A}_{xx}h{A}_{yy}}{2}\right)}^{2}+{p}^{2}{A}_{xy}^{2}}$ .
Parameters $a$ , $c$ , and $h$ are computed from the Lankford coefficients.
$a=22\sqrt{\frac{{r}_{00}}{1+{r}_{00}}\frac{{r}_{90}}{1+{r}_{90}}}$ , $c=2a$ , $h=\sqrt{\frac{{r}_{00}}{1+{r}_{00}}\frac{1+{r}_{90}}{{r}_{90}}}$
Parameter $p$ is obtained by solving:(6) $$\frac{2M{Y}^{M}}{\left(\frac{\partial f}{\partial {A}_{xx}}+\frac{\partial f}{\partial {A}_{yy}}\right){\sigma}_{45}}1{r}_{45}=0$$
 The Hill’s criterion is expressed as:
 Yield stress, Poisson ratio and Young's modulus should be strictly positive. The other parameters should be nonnegative value.
 The schematic illustration of the
twosurface model is shown in Figure 1.
Where, 0 is the original center of the yield surface, the yield surface with its center $\alpha $ and its radii Y, is moving kinematically, within a bounding surface that has a size indicated by B+R and tensor $\mathbf{\beta}$ indicating its center position.
 The yield surface is subjected to a kinematic
hardening. The kinematic motion is described by
${\mathbf{\alpha}}^{*}$
that has the following evolution:
 for shell elements
(7) $${\dot{\alpha}}^{*}=C\left[\left(\frac{a}{Y}\right)\left(\mathbf{\sigma}\mathbf{\alpha}\right)\sqrt{\frac{a}{\Vert {\mathbf{\alpha}}^{\text{*}}\Vert}}{\mathbf{\alpha}}^{*}\right]{\dot{\overline{\epsilon}}}_{p}$$  for solid elements
(8) $${\dot{\alpha}}^{*}=C\left[\left(\frac{2}{3}\right)a{\dot{\epsilon}}_{{}^{\mathbf{p}}}\sqrt{\frac{a}{\Vert {\mathbf{\alpha}}^{*}\Vert}}{\mathbf{\alpha}}^{*}{\dot{\overline{\epsilon}}}_{{}^{P}}\right]$$
Where, ${\dot{\overline{\epsilon}}}_{p}$
 Equivalent plastic strain rate
 C and a
 Material parameters and $a=B+RY$
 $\mathbf{\alpha}={\mathbf{\alpha}}^{*}+\mathbf{\beta}$
 Total back stress
Note: In case of a quadratic (using Hill or $M$ =2.0 with Barlat) and isotropic criterion ( ${r}_{00}={r}_{45}={r}_{90}=1$ ), the shell kinematic motion equation equals the solid one.  for shell elements
 The parameter
$C$
for kinematic hardening rule can take two different values
to describe a change of work hardening rate for forward and reverse deformations.
 If ${C}_{1}^{KH}>{C}_{2}^{KH}$ , $\{\begin{array}{c}\begin{array}{ccc}C={C}_{1}^{KH}& \text{if}& Max\left(\Vert {\alpha}^{*}\Vert \right)<BY\end{array}\\ \begin{array}{ccc}C={C}_{2}^{KH}& \text{otherwise}& \begin{array}{cc}\begin{array}{cc}\begin{array}{cc}& \end{array}& \end{array}& \end{array}\end{array}\end{array}$
 If ${C}_{1}^{KH}\le {C}_{2}^{KH}$ , the change of work hardening rate is deactivated and $C={C}_{2}^{KH}$ .
 The bounding surface is subjected to an
isotropickinematic hardening. The evolution equation for isotropic hardening is:
 $\dot{R}=m\left({R}_{\mathit{sat}}R\right){\dot{\overline{\epsilon}}}_{{}^{p}}$
 Default (if OptR = 0) Yoshida expression
 $R={R}_{\mathit{sat}}\left[{\left({C}_{1}+{\overline{\epsilon}}_{{}^{p}}\right)}^{{C}_{2}}\text{\hspace{0.17em}}{C}_{1}^{{C}_{2}}\right]$
 Available for shell elements, if OptR = 1
The evolution equation for kinematic hardening of bounding surface is: for shell elements:
(9) $$\dot{\beta}=m\left[\left(\frac{b}{Y}\right)\left(\sigma \alpha \right)\beta \right]{\dot{\overline{\epsilon}}}_{{}^{p}}$$  for solid elements:
(10) $$\dot{\beta}=m\left(\frac{2}{3}b{\dot{\epsilon}}_{{}^{p}}\beta {\dot{\overline{\epsilon}}}_{{}^{p}}\right)$$
Note: In case of a quadratic (using Hill or Mexp=2 with Barlat) and isotropic criterion ( ${r}_{00}={r}_{45}={r}_{90}=1$ ), the shell kinematic hardening equation equals the solid one.  The workhardening stagnation during unloading is
described using a J_{2}type surface
${g}_{\sigma}$
with a radius r and a center
q:
(11) $$\begin{array}{l}{g}_{\sigma}(\mathbf{\beta},{\mathbf{q}}^{\prime},r)=\frac{3}{2}(\mathbf{\beta}{\mathbf{q}}^{\prime}):(\mathbf{\beta}{\mathbf{q}}^{\prime}){r}^{2}\\ {\dot{\mathbf{q}}}^{\prime}=\text{\mu}\left(\mathbf{\beta}{\mathbf{q}}^{\prime}\right)\\ r=h\dot{\Gamma}\hspace{0.17em},\dot{\Gamma}\frac{3\left(\mathbf{\beta}{\mathbf{q}}^{\prime}\right):\dot{\beta}}{2r}\end{array}$$Where, $h\text{\hspace{0.17em}}\left(0<h\le 1\right)$
 Denotes a material parameter that determines the rate of expansion of surface ${g}_{\sigma}$ .
 $\mathbf{\beta}$
 Should be either inside or on the surface ${g}_{\sigma}$ .
 The exponent in Barlat’s (1989) yield
criterion can be set by considering the microstructure of the material. Any value greater
than 2.0 is valid, but typically:
 Mexp = 6.0 (Default) for a Body Centered Cubic (BCC) material
 Mexp = 8.0 for a Face Centered Cubic (FCC) material
 The evolution of Young's modulus:
 If fct_ID_{E} > 0, the curve defines a scale factor for Young's modulus
evolution with equivalent plastic strain, which means the Young's modulus is scaled by
the function
$\mathrm{f}\left({\overline{\epsilon}}_{p}\right)$
:
 $E\left(t\right)=E\cdot \mathrm{f}\left({\overline{\epsilon}}_{p}\right)$
The initial value of the scale factor should be equal to 1 and it decreases.

If fct_ID_{E} = 0, the Young's modulus is calculated as:
(12) $$E\left(t\right)=E\left(E{E}_{\mathit{inf}}\right)\left[1\mathrm{exp}\left({C}_{E}{\overline{\epsilon}}_{p}\right)\right]$$Where, E and E_{inf}
 Initial and asymptotic value of Young's modulus, respectively.
 ${\overline{\epsilon}}_{p}$
 Accumulated equivalent plastic strain.
Note: If fct_ID_{E} = 0 and C_{E} = 0, Young's modulus E is kept constant.
 If fct_ID_{E} > 0, the curve defines a scale factor for Young's modulus
evolution with equivalent plastic strain, which means the Young's modulus is scaled by
the function
$\mathrm{f}\left({\overline{\epsilon}}_{p}\right)$
:
 This material law is not available for implicit analysis.