/FAIL/SAHRAEI

Block Format Keyword This orthotropic strain-based failure model can be used to predict failure and shortcut in battery cells. It is available for solid elements only.

Format

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
/FAIL/SAHRAEI/`mat_ID`/`unit_ID`

Card 1 - Damage accumulation parameters

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
`Fct_ratio`	`NUM`	`DENOM`	`ORDIN`	`VOL_STRAIN`			`Fct_ID`_el	`El_ref`

Card 2 – Biaxial tension failure strain and strain-rate dependency

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
`COMP_DIR`	`IDEL`	`MAX_COMP_STRAIN`		`RATIO`

Optional line

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
`fail_ID`

Definition

Field	Contents	SI Unit Example
`mat_ID`	Material identifier (Integer, maximum 10 digits)
`unit_ID`	(Optional) Unit identifier. (Integer, maximum 10 digits)
`Fct_ratio`	Strain ratio function identifier. (Integer)
`NUM`	Numerator strain flag. = 1 Strain along X axis = 2 Strain along Y axis = 3 Strain along Z axis = 4 1^st principal strain = 5 2^nd principal strain = 6 3^rd principal strain (Integer)
`DENOM`	Denominator strain flag. = 1 Equivalent strain along X-Z plane = 2 Equivalent strain in X-Y plane = 3 Equivalent strain in Y-Z plane = 4 1^st principal strain = 5 2^nd principal strain = 6 3^rd principal strain (Integer)
`ORDIN`	Maximum strain at failure flag. = 1 max(eps_xx, eps_yy, eps_zz) = 2 eps_xx = 3 eps_yy = 4 eps_zz =5 1^st principal strain = 6 Equivalent strain in x – z plane = 7 Equivalent strain in x – y plane = 8 Equivalent strain in y – z plane (Integer)
`VOL_STRAIN`	Trigger volumetric strain for damage. (Real)
`Fct_ID`_el	Element size regularization function identifier. (Integer)
`El_ref`	Reference element size. (Real)	$[m]$
`COMP_DIR`	Direction component for compression failure. (Integer)
`IDEL`	Element deletion activation in compression flag. = 0 (Default) No element deletion in compression. = 1 Element deletion in compression. (Integer)
`MAX_COMP_STRAIN`	Maximum strain value in compression for element failure. (Real)
`RATIO`	Strain ratio for failure in compression. (Real)
`fail_ID`	(Optional) Failure criteria identifier. (Integer, maximum 10 digits)

Example

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
#              MUNIT               LUNIT               TUNIT
                  kg                  mm                  ms
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  1. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW28/1
MAIN_1
#        Init. dens.          Ref. dens.
              2.5E-6                   0
#               E_11                E_22                E_33
                  10                   8                   8
#               G_12                G_23                G_31
                   5                   5                   5
#      Y11       Y22       Y33    Iflag1            Fscale11            Fscale22            Fscale33
        10        11        11         1                   0                   0                   0
#         Eps_max_11          Eps_max_22          Eps_max_33
                   0                   0                   0
#      Y12       Y23       Y31    Iflag2            Fscale12            Fscale23            Fscale31
        12        12        12         1                   0                   0                   0
#         Eps_max_12          Eps_max_23          Eps_max_31
                   0                   0                   0
/FAIL/SAHRAEI/1
#Fct_ratio       NUM     DENOM     ORDIN          VOL_STRAIN            Fct_IDEL              EL_REF
      3000         6         4         1                  .5                3001                   5                 
# COMP_DIR     MAX_COMP_STRAIN               RATIO
         0                   1                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. FUNCTIONS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/10
Load 1st direction
                  -1                   5
                   0                  .1
                   1                  .1
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/11
Load 2nd and 3rd direction
                  -1                   4
                   0                 .08
                   1                 .08
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/12
Shear
                  -1                 .05
                   0                 .05
                   1                 .05
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/3000
fail strain as ratio of E33/E11 vs. failure strain
#                  X                   Y
                   0         0.335081967
         0.141129032         0.330491803
         0.181451613         0.312131148
                0.27         0.271967213
         0.403225807         0.222622951
         0.483870968         0.203114754
         0.705645161         0.149180328
         0.826612903         0.110163934
         1.008064516         0.082622951
         1.411290323         0.059672131
         1.975806452         0.055081967
         2.661290323         0.061967213
         3.286290323         0.063114754
         4.032258065         0.064262295
         4.677419355         0.064262295
         5.705645161         0.061967213
         6.693548387         0.061967213
         7.540322581         0.050491803
                  9.         0.032131148
                 10.         0.032131148
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/3001
fail strain as ratio of E33/E11 vs. failure strain
#                  X                   Y
                   0                   1
                   1                   1
                   5                  .5
                  10                  .5
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#enddata
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

The SAHRAEI failure criterion considers the evolution of a strain at failure with a strain ratio. This strain ratio is user-defined by a numerator (NUM) and a denominator (DENOM):(1)
$ε_{M A X} = f (\frac{ε_{n u m}}{ε_{d e n o m}})$
The strain used as a numerator $ε_{n u m}$ can be:
- Strain along X axis $ε_{x}$
- Strain along Y axis $ε_{y}$
- Strain along Z axis $ε_{z}$
- First principal strain $ε_{1}$
- Second principal strain $ε_{2}$
- Third principal strain $ε_{3}$
The strain used as the denominator $ε_{d e n o m}$ can be:
- The equivalent strain in X-Z plane defined as:(2)
  $ε_{e q}^{X - Z} = \frac{ε_{x} + ε_{z}}{2} + \sqrt{{(\frac{ε_{x} - ε_{z}}{2})}^{2} + ε_{x z}^{2}}$
- The equivalent strain in X-Y plane defined as:(3)
  $ε_{e q}^{X - Y} = \frac{ε_{x} + ε_{y}}{2} + \sqrt{{(\frac{ε_{x} - ε_{y}}{2})}^{2} + ε_{x y}^{2}}$
- The equivalent strain in Y-Z plane defined as:(4)
  $ε_{e q}^{Y - Z} = \frac{ε_{y} + ε_{z}}{2} + \sqrt{{(\frac{ε_{y} - ε_{z}}{2})}^{2} + ε_{y z}^{2}}$
- First principal strain $ε_{1}$
- Second principal strain $ε_{2}$
- Third principal strain $ε_{3}$
The strain computation for maximum value $ε_{M A X}$ can be:
- Maximum value between normal strains $\max (ε_{x}, ε_{y}, ε_{z})$
- Strain along X axis $ε_{x}$
- Strain along Y axis $ε_{y}$
- Strain along Z axis $ε_{z}$
- First principal strain $ε_{1}$
- Equivalent strain in X-Z plane $ε_{e q}^{X - Z}$
- Equivalent strain in X-Y plane $ε_{e q}^{X - Y}$
- Equivalent strain in Y-Z plane $ε_{e q}^{Y - Z}$
The evolution $ε_{M A X} = f (\frac{ε_{n u m}}{ε_{d e n o m}})$ is given by the tabulated function ID, Fct_ratio. An example of the recommended shape is:

Figure 1.
The damage variable computation starts when the absolute value of volumetric strain is higher than the limit you defined. The damage evolution is:(5)
$D = \frac{ε_{O R D I N}}{ε_{M A X}}$

Where, $ε_{O R D I N}$ is computed the same way as $ε_{M A X}$ .
A failure in compression can also be defined using COMP_DIR, RATIO and MAX_COMP_STRAIN (denoted $ε_{M A X_C O M P} < 0$ ).
- If COMP_DIR = 1, failure is reached when:
  $\begin{matrix} ε_{y} < ε_{M A X_C O M P} & or & ε_{z} < ε_{M A X_C O M P} \times r a t i o \end{matrix}$
- If COMP_DIR = 2, failure is reached when:
  $\begin{matrix} ε_{z} < ε_{M A X_C O M P} & or & ε_{x} < ε_{M A X_C O M P} \times r a t i o \end{matrix}$
- If COMP_DIR = 3, failure is reached when:
  $\begin{matrix} ε_{x} < ε_{M A X_C O M P} & or & ε_{y} < ε_{M A X_C O M P} \times r a t i o \end{matrix}$
It is possible to consider element size in material failure by function fct_ID_el to scale the failure strain:(6)
$f a c t o r_{e l} = f c t_I D_{e l} (\frac{S i z e_{e l}}{E l_r e f})$

Where, $S i z e_{e l}$ is the reference element mesh size.
The failure in compression only sets the damage variable to the value 1 without any element deletion (it is used as an indicator). If you want to activate the element deletion in compression, set the flag IDEL to 1.

¹ Elham Sahraei, Emanuela Bosco, Brandy Dixon, Benjamin Lai. Microscale failure mechanisms leading to internal short circuit in Li-ion batteries under complex loading scenarios