/MAT/LAW78
Block Format Keyword This law is the Yoshida-Uemori model for describing the large-strain 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 plastic-strain dependency of the Young's modulus and the work-hardening 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) |
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/MAT/LAW78/mat_ID/unit_ID | |||||||||
mat_title | |||||||||
E | |||||||||
Y | b | h | B0 | ||||||
m | Rsat | OptR | C1 | C2 | |||||
r00 | r45 | r90 | Mexp | Icrit | |||||
fct_IDE | Einf | CE |
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) |
|
Initial density. (Real) |
||
E | Young's modulus. (Real) |
|
Poisson's ratio. (Real) |
||
Y | Yield stress. (Real) |
|
b | Center of the bounding
surface. (Real) |
|
Parameter for kinematic hardening rule
of yield surface (or used in calculation of
if
). 5 (Real) |
||
h | Material parameter for controlling work
hardening stagnation. (Real) |
|
B0 | Initial size of the bounding
surface. (Real) |
|
m | Parameter for isotropic and kinematic
hardening of the bounding surface. (Real) |
|
Rsat | Saturated value of the isotropic
hardening stress. (Real) |
|
OptR | Modified isotropic hardening rule flag
(available for shells only):
(Integer) |
|
C1, C2 | Constant used in the modified
formulation of the isotropic hardening of bounding surface (available for shells
only). (Real) |
|
r00 | Lankford parameter (0 degree) used for
shell elements. Default = 1.0 (Real) |
|
r45 | Lankford parameter (45 degree) used for
shell elements. Default = 1.0 (Real) |
|
r90 | Lankford parameter (90 degree) used for
shell elements. Default = 1.0 (Real) |
|
Mexp | Exponent
in Barlat's 1989 Yield Criterion for shell elements.
8.
(Real) |
|
Icrit | Plastic criterion selection flag.
|
|
fct_IDE | ID of the function defining the scale
factor of Young's modulus evolution versus effective plastic strain. 9 (Integer) |
|
Einf | Asymptotic value of Young's
modulus. (Real) |
|
CE | Parameter controlling the dependency of
Young's modulus on the effective plastic strain. (Real) |
|
Parameter
for kinematic hardening rule of yield surface. 5
Ignore if . (Real) |
Example
#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/LAW78/1/1
DP600-HDG
# RHO_I
7.8E-9
# 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
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
Comments
- For solid elements, von Mises yield
criterion is used, so the yield function is expressed as: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:Where,
- Yield stress.
- Total back stress. 4
If , then is expressed as:
- The Barlat’s criterion is expressed as:
Where, is the exponent in Barlat's yield criterion.
is expressed as:
With,
and .
Parameters , , and are computed from the Lankford coefficients.
, ,
Parameter is obtained by solving:
- The Hill’s criterion is expressed as:
- Yield stress, Poisson ratio and Young's modulus should be strictly positive. The other parameters should be non-negative value.
- The schematic illustration of the
two-surface model is shown in Figure 1.
Where, 0 is the original center of the yield surface, the yield surface with its center and its radii Y, is moving kinematically, within a bounding surface that has a size indicated by B+R and tensor indicating its center position.
- The yield surface is subjected to a
kinematic hardening. The kinematic motion is described by
that has the following evolution:
- for shell elements
- for solid elements
Where,- Equivalent plastic strain rate
- C and a
- Material parameters and
- Total back stress
Note: In case of a quadratic (using Hill or =2.0 with Barlat) and isotropic criterion ( ), the shell kinematic motion equation equals the solid one. - for shell elements
- The parameter
for kinematic hardening rule can take two different values
to describe a change of work hardening rate for forward and reverse deformations.
- If ,
- If , the change of work hardening rate is deactivated and .
- The bounding surface is subjected to an
isotropic-kinematic hardening. The evolution equation for isotropic hardening is:
- Default (if OptR = 0) Yoshida expression
- Available for shell elements, if OptR = 1
The evolution equation for kinematic hardening of bounding surface is:- for shell elements:
- for solid elements:
Note: In case of a quadratic (using Hill or Mexp=2 with Barlat) and isotropic criterion ( ), the shell kinematic hardening equation equals the solid one. - The work-hardening stagnation during
unloading is described using a J2-type surface
with a radius r and a center
q:Where,
- Denotes a material parameter that determines the rate of expansion of surface .
- Should be either inside or on the surface .
- 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_IDE > 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
:
The initial value of the scale factor should be equal to 1 and it decreases.
- If fct_IDE = 0, the Young's modulus is calculated as:Where,
- E and Einf
- Initial and asymptotic value of Young's modulus, respectively.
- Accumulated equivalent plastic strain.
Note: If fct_IDE = 0 and CE = 0, Young's modulus E is kept constant.
- If fct_IDE > 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
: