 
 
 
 
 
 
 
  
This model describes small deformations for elastically anisotropic materials with a von Mises type yield surface. Often, this model is used as a compromise for anisotropic materials with lack of data or detailed knowledge about the anisotropic behavior in the viscoplastic range.
The total strain is supposed to be the sum of the elastic strain and the plastic strain:
|       | (174) | 
An isotropic hardening variable  and a kinematic
hardening tensor
 and a kinematic
hardening tensor 
 are introduced representing the
isotropic and kinematic change of the yield surface,
respectively. The yield surface takes the form:
 are introduced representing the
isotropic and kinematic change of the yield surface,
respectively. The yield surface takes the form:
|  | (175) | 
where 
 is the deviatoric stress tensor, and
 is the deviatoric stress tensor, and  is the
size of the elastic range at zero yield. The constitutive equations for the
hardening variables satisfy:
 is the
size of the elastic range at zero yield. The constitutive equations for the
hardening variables satisfy:
|  | (176) | 
and
|    | (177) | 
where  and
 and 
 are the hardening variables
 in strain space. It can be shown that
 are the hardening variables
 in strain space. It can be shown that
|  | (178) | 
|  | (179) | 
where 
 is the equivalent plastic strain 
defined by
 is the equivalent plastic strain 
defined by
|  | (180) | 
and 
 is the equivalent value of the tensor
 is the equivalent value of the tensor
 defined in a similar way. Thus, the
constitutive equations amount to
 defined in a similar way. Thus, the
constitutive equations amount to
|  | (181) | 
and
|  | (182) | 
where
|  | (183) | 
has the meaning of an equivalent stress value or von Mises value. The
same applies to  . Consequently, the constitutive equations assume a
linear relationship between the hardening stress and the equivalent
plastic strain.
. Consequently, the constitutive equations assume a
linear relationship between the hardening stress and the equivalent
plastic strain.
The constitutive equation for the stress is Hooke's law:
|  | (184) | 
The evolution equations for the plastic strain and the hardening variables in strain space are given by:
|   | (185) | 
|  | (186) | 
and
|   | (187) | 
where
|  | (188) | 
The variable 
 is the consistency coefficient known
from the Kuhn-Tucker conditions in optimization theory
[44]. It can be proven to satisfy:
 is the consistency coefficient known
from the Kuhn-Tucker conditions in optimization theory
[44]. It can be proven to satisfy:
|  | (189) | 
Finally, the creep rate is modeled as a power law function of the yield exceedance and total time t:
|  | (190) | 
The brackets 
 reduce negative function values to zero
while leaving positive values unchanged, i.e.
 reduce negative function values to zero
while leaving positive values unchanged, i.e. 
 if
if  and
 and 
 if
 if  .
.
In the present implementation orthotropic elastic behavior is
assumed. Consequently, for each temperature 15
constants need to be defined: the elastic constants  ,
,
 ,
,  ,
, ,
,
 ,
,  ,
, ,
,
 ,
,  , and the viscoplastic constants
, and the viscoplastic constants 
 .
. 
The material definition consists of a *MATERIAL card defining the name of the material. This name HAS TO START WITH ''ANISO_PLAS'' but can be up to 80 characters long. Thus, the last 70 characters can be freely chosen by the user. Within the material definition a *USER MATERIAL card has to be used satisfying:
First line:
Following lines, in sets of 2:
First line of set:
 .
.
 .
.
 .
.
 .
.
 .
.
 .
.
 .
.
 .
.
 .
.
 .
.
 .
.
 .
.
 .
.
 .
.
 .
.
Repeat this set if needed to define complete temperature dependence.
The principal axes of the material are assumed to coincide with the global coordinate system. If this is not the case, use an *ORIENTATION card to define a local system.
For this model, there are 14 internal state variables:
 (1)
 (1)
 (6)
 (6) 
 (1)
 (1)
 (6)
 (6)
These variables are accessible through the *EL PRINT (.dat file) and *EL FILE (.frd file) keywords in exactly this order (label SDV). The *DEPVAR card must be included in the material definition with a value of 14.
Example: *MATERIAL,NAME=ANISO_PLAS *USER MATERIAL,CONSTANTS=15 500000.,157200.,500000.,157200.,157200.,500000.,126200.,126200., 126200.,0.,0.,0.,1.E-10,5,0. *DEPVAR 14
defines a single crystal with elastic constants
500000., 157200., 500000., 157200., 157200., 500000., 126200., 126200.,
126200., and viscoplastic parameters  ,
,  ,
,  ,
,
 ,
,  and
 and  . Thus, the yield surface has a zero radius
and there is no hardening. Only creep is activated.
. Thus, the yield surface has a zero radius
and there is no hardening. Only creep is activated.
 
 
 
 
 
 
