PLASTIC

PLASTIC option is a modular approach that can be used to add plasticity to an existing elastic material defined through MAT1. The yield function is a basic to define an elasto-plastic constitutive model through the equation:

From a numerical point of view, a trial value of the yield function is used to determine if the plastic return mapping is necessary. This return mapping procedure takes place when $f_{trial}\ge 0$ , and finish after several iterations until the yield function is brought back to a zero value. The default algorithm is the cutting plane method.

PLASTIC then offers the possibility to create a specific assembly of constitutive equations choosing a custom-made combination of yield criterion, hardening rule, and/or strain rate dependency, among an existing library.

The CRIT continuation line is used to define the yield criterion. Currently, only HILL criterion is supported. This criterion defines a surface in the stress state reference system that triggers the beginning of plastic behavior. Its shape is defined through the equation of the equivalent stress denoted as ${\sigma }_{eq}$ .

Hill 1948 (HILL):

The Hill yield criterion (1948) is an orthotropic version of the quadratic yield criterion which is described as:

The equivalent stress for Hill yield criterion is obtained with:

For shell elements, the plane stress assumption implies the following equivalent stress:

Where, $F,G,H,L,M,\text{and}N$ are the Hill orthotropy coefficients.

Some examples of the presented criterion shapes are given in Figure 1.

• The default input format of Hill criterion parameters describes a list of stress ratios which allows to compute the Hill coefficients as:

  $F=\frac{1}{2}\left(\frac{1}{R_{22}^2}+\frac{1}{R_{33}^2}-\frac{1}{R_{11}^2}\right)$	$L=\frac{3}{2R_{23}^2}$
  $G=\frac{1}{2}\left(\frac{1}{R_{33}^2}+\frac{1}{R_{11}^2}-\frac{1}{R_{22}^2}\right)$	$M=\frac{3}{2R_{31}^2}$
  $H=\frac{1}{2}\left(\frac{1}{R_{22}^2}+\frac{1}{R_{11}^2}-\frac{1}{R_{33}^2}\right)$	$N=\frac{3}{2R_{12}^2}$

  Where,

  $R_{ii}=\frac{{\sigma }_Y^{ii}}{{\sigma }_Y}$

  is for normal directions

  $R_{ij}=\frac{\sqrt{3}{\sigma }_Y^{ij}}{{\sigma }_Y}$

  is for shear planes

  Note: ${\sigma }_Y^{ii}$ is the initial elastic limit stress measured for the direction $i$ .
  These stress ratios can be input as:

  (1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
  	CRIT	HILL
  		R11_1	R22_1	R33_1	R12_1	R31_1	R23_1	TEMP_1
  		R11_2	R22_2	R33_2	R12_2	R31_2	R23_2	TEMP_2
  		etc	etc	etc	etc	etc	etc	etc

  A series of several lines of stress ratios can be defined for “n” specific temperature values (TEMP_n). If several lines are defined, the stress ratios are interpolated according to the current element temperature value.

• Another input format is available by using the keyword CLAS, for the classical Hill shape input. This allows to directly input the Hill coefficients instead of the stress ratios:

  (1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
  	CRIT	HILL	CLAS
  		F_1	G_1	H_1	L_1	M_1	N_1	TEMP_1
  		F_2	G_2	H_2	L_2	M_2	N_2	TEMP_2
  		etc	etc	etc	etc	etc	etc	etc

• The last input format is particularly useful when orthotropic sheets of material are considered (and thus, plane stress conditions). It allows the user to define the Lankford coefficient, considering the corresponding Hill criterion formula:
  $${\sigma }_{eq}=\sqrt{H\left(1+\frac{1}{r_{00}}\right){\sigma }_{xx}^2+H\left(1+\frac{1}{r_{90}}\right){\sigma }_{yy}^2-2H{\sigma }_{xx}{\sigma }_{yy}+2H\left(r_{45}+0.5\right)\left(\frac{1}{r_{00}}+\frac{1}{r_{90}}\right){\sigma }_{xy}^2}$$

  Where, $H=\frac{R}{1+R}$ and $R=\frac{r_{00}+2r_{45}+r_{90}}{4}$ .

Another keyword can be combined with LANK, allowing to select the reference direction for which the yield stress ${\sigma }_Y$ has been measured. Orthotropic directions 1 and 2 can be selected using respectively the keywords DIR1 or DIR2. If a keyword is not used, an average yield stress is considered by default.

In addition to work hardening, you may want to introduce a strain rate dependency. This allows to increase or decrease the yield stress depending on either the total strain rate (that may be filtered using the real value parameter FCUT), or the plastic strain rate (which corresponds to a full viscoplastic formulation activated by the keyword VPLAS).

• Johnson-Cook (JCOOK)

  The Johnson-Cook strain rate dependency is a well-known solution to add strain rate dependency to an elasto-plastic behavior. The hardening rule is then multiplied by:

  $${\sigma }_Y=R\left({\epsilon }_p\right)\left(1+C\ln \left(1+\frac{\dot{\epsilon }}{{\dot{\epsilon }}_0}\right)\right)$$
  Where,

  $C$

  Johnson-Cook parameter

  ${\dot{\epsilon }}_0$

  Reference strain rate delimiting the transition between the inviscid domain and the strain rate dependent domain

  The Johnson-Cook model thus describes a linear increase of the material strength with respect to the logarithm of the strain rate. (Figure 5). An example of yield stress increase shape obtained with the presented model is given in Figure 6.

  
  
  

  
  
  

• Cowper-Symonds (COWPER)

  The Cowper-Symonds rate dependency is another solution commonly found in the literature. It modifies the yield stress equation a:

  $${\sigma }_Y=R\left({\epsilon }_p\right)\left(1+{\left(\frac{\dot{\epsilon }}{c}\right)}^{\frac{1}{p}}\right)$$
  Where,

  $c$

  Strain rate dependency parameter

  $p$

  Strain rate exponent parameter

  The shape of the presented model is given in Figure 7.

  
  
  

  
  
  

• Nonlinear (NLINEAR)

  The nonlinear strain rate dependency is similar to the Johnson-Cook model described above. Nevertheless, it describes a nonlinear evolution of the strain rate dependency in the logarithmic scale (Figure 9). This is introduced in the yield stress formula as:

  $${\sigma }_Y=R\left({\epsilon }_p\right){\left(1+\frac{\dot{\epsilon }}{{\dot{\epsilon }}_0}\right)}^{CS}$$
  Where,

  $CS$

  Strain rate dependency exponent

  ${\dot{\epsilon }}_0$

  Reference strain rate delimiting the transition between the inviscid domain and the strain rate dependent domain

  An example of the presented model on the yield stress shape is given in Figure 10.

  
  
  

  
  
  

• A tabulated work hardening,
• A nonlinear strain rate dependency with full visco-plastic formulation