# /FAIL/JOHNSON

Block Format Keyword This failure model uses a nonlinear, plastic strain-based, failure criteria with linear damage accumulation. It describes the failure criteria by Johnson-Cook failure model.

## Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/FAIL/JOHNSON/mat_ID/unit_ID
D1 D2 D3 D4 D5
${\stackrel{˙}{\epsilon }}_{0}$ Ifail_sh Ifail_so     Dadv   Ixfem
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 Unit Identifier

(Integer, maximum 10 digits)

D1 1st parameter.

(Real)

D2 2nd parameter.

(Real)

D3 3rd parameter.

(Real)

D4 4th parameter.

(Real)

D5 5th parameter.

(Real)

${\stackrel{˙}{\epsilon }}_{0}$ Reference strain rate.

(Real)

$\left[\frac{\text{1}}{\text{s}}\right]$
Ifail_sh Shell failure flag.

(Integer)

If Ixfem =0: failure - element deleted.

If Ixfem =1: failure - element cracked. 2

= 1: (Default) Shell is deleted or cracked when $D=\sum \frac{\Delta {\epsilon }_{p}}{{\epsilon }_{f}}\ge 1$ for one integration point or layer.

= 2: For each integration point, the stress tensor is set to zero when $D=\sum \frac{\Delta {\epsilon }_{p}}{{\epsilon }_{f}}\ge 1$ . The shell is deleted or cracked when $D=\sum \frac{\Delta {\epsilon }_{p}}{{\epsilon }_{f}}\ge 1$ for all integration points or layers.

Ifail_so Solid failure flag.
= 1 (Default)
Solid element is deleted, when $D=\sum \frac{\Delta {\epsilon }_{p}}{{\epsilon }_{f}}\ge 1$ for one integration point.
= 2
For each integration point, the deviatoric stress tensor vanishes, when $D=\sum \frac{\Delta {\epsilon }_{p}}{{\epsilon }_{f}}\ge 1$ .

(Integer)

Dadv Criterion for the crack advancement (Only active if with Ixfem =1). 4
= 1 (Default)
> 1
Then Dadv = 1 (max accumulation damage value D, used for crack initiation).

(Real between 0 and 1)

Ixfem XFEM flag (for /PROP/SHELL, /PROP/SH_SANDW, and /PROP/TYPE51 properties only).
= 0 (Default)
Without XFEM.
= 1
XFEM formulation. 2

(Integer)

fail_ID Failure criteria identifier. 3

(Integer, maximum 10 digits)

## Examples

In these two simple examples, strain rate and temperature are not considered. The stress-strain relationship could be simplified:(1)
${\epsilon }_{f}=\left[{D}_{1}+{D}_{2}\text{exp}\left({D}_{3}{\sigma }^{*}\right)\right]$
With three typical loadings (example: pure tension, pure shear or pure compression) material parameter D1 D2 and D3 could be determined using the above simplified stress-stain relationship.

## Example 1 (Steel)

Use Ixfem=0, if the element deleted reaches the rupture criteria.
#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
#              MUNIT               LUNIT               TUNIT
Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  1. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/PLAS_JOHNS/1/1
Steel
#              RHO_I
7.8E-9                   0
#                  E                  Nu
210000                  .3
#                  a                   b                   n           EPS_p_max            SIG_max0
270                 450                  .6                   0                   0
#                  c           EPS_DOT_0       ICC   Fsmooth               F_cut               Chard
0                   0         0         0                   0                   0
#                  m              T_melt              rhoC_p                 T_r
0                   0                   0                   0
/FAIL/JOHNSON/1/1
#                 D1                  D2                  D3                  D4                  D5
0.11                0.08                -1.5                   0                   0
#              EPS_0  Ifail_sh  Ifail_so                                    Dadv               Ixfem
1         1         1                                       0                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#enddata
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

## Example 2 (Steel)

Use Ixfem=1, if the element cracked reaches the crack criteria.
#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
#              MUNIT               LUNIT               TUNIT
Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  1. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/PLAS_JOHNS/1/1
Steel
#              RHO_I
7.8E-9                   0
#                  E                  Nu
210000                  .3
#                  a                   b                   n           EPS_p_max            SIG_max0
270                 450                  .6                   0                   0
#                  c           EPS_DOT_0       ICC   Fsmooth               F_cut               Chard
0                   0         0         0                   0                   0
#                  m              T_melt              rhoC_p                 T_r
0                   0                   0                   0
/FAIL/JOHNSON/1/1
#                 D1                  D2                  D3                  D4                  D5
0.11                0.08                -1.5                   0                   0
#              EPS_0  Ifail_sh  Ifail_so                                    Dadv               Ixfem
1         1         1                                     0.5                   1
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#enddata
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

1. The parameters are used in stress-strain relationship:(2)
${\epsilon }_{f}=\left[{D}_{1}+{D}_{2}\text{exp}\left({D}_{3}{\sigma }^{*}\right)\right] \left[1+{D}_{4}\mathit{ln}\left({\stackrel{˙}{\epsilon }}^{*}\right)\right] \left[1+{D}_{5}{T}^{*}\right]$
Where, ${\sigma }^{*}=\frac{{\sigma }_{m}}{{\sigma }_{VM}}$ ( ${\sigma }^{*}$ is the stress triaxiality).(3)
${\stackrel{˙}{\epsilon }}^{*}=\frac{\stackrel{˙}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{0}}$
$T*$ is computed for all material laws, as:(4)
${T}^{*}=\frac{T-{T}_{r}}{{T}_{melt}-{T}_{r}}$
Where,
Tr
Initial temperature
Tmelt
Melting temperature for materials LAW2 and LAW4

When /HEAT/MAT (with Iform =1) references this material model, the values of Tr and Tmelt defined in this card will be overwritten by the corresponding T0 and Tmelt defined in /HEAT/MAT.

2. XFEM formulation (Ixfem=1) is only compatible with BT Q4 (Ishell=1, 2, 3 or 4), and QEPH (Ishell=24) shell elements. If XFEM is activated (Ixfem=1), the failure criteria will lead to element cracking instead of element or layer deletion.
Two XFEM options are available: mono-layer and multi-layer. The XFEM option depends on the property type associated to the failure criterion applied to the material identifier:
• If /PROP/SHELL (TYPE1) is used, then mono-layer XFEM will be applied.

In this case, the whole element thickness is considered as a single layer. The failure criterion is calculated in each integration point, but only one single crack can appear in this element. This approach is compatible with all values of the shell flag (Ifail_sh=1 or 2). The crack direction is determined by the principal constraints in the last failed integration point.

• If /PROP/SH_SANDW (TYPE11) is used, then multi-layer XFEM will be applied.
In this case, each integration point over thickness is considered as a distinct layer. The failure criterion is calculated separately and the crack direction may be different in each layer. Crack direction in each layer will independently propagate from one element to another. Multi-layer XFEM is not compatible with Ifail_sh=1. Its value will be automatically set to Ifail_sh=2 in this case.
• If /PROP/TYPE51 is used, then multi-layer XFEM will be applied, and the separate cracks may appear in each layer and propagate independently from one element to another. Thus, crack directions and patterns will be different in each layer. The failure criterion is calculated separately in each integration point and crack will propagate when all the integration points fail within a layer. Multi-layer XFEM is not compatible with Ifail_sh=1. Its value will be automatically set to Ifail_sh=2.
Warning: Mono-layer and multi-layer XFEM formulations cannot be mixed in the same model, yet. The choice between them must be made for the whole model.
3. The fail_ID is used with /STATE/BRICK/FAIL and /INIBRI/FAIL. There is no default value. If the line is blank, no value will be output for failure model variables in the /INIBRI/FAIL (written in the .sta file with /STATE/BRICK/FAIL option).
4. Two different failure (rupture or crack) are introduced in this failure model. The failure criteria is calculated, as:
• Element rupture (Ixfem=0):

Element rupture (deleted) if D > 1

• Element crack (Ixfem=1):

Element cracked, if:

this element has no failed neighbors and D > 1, then in this case, new crack initialization in element.

this element has failed neighbors and D > Dadv, then in this case, crack advanced and Dadv is used for crack advancement. Dadv will be used if existing crack arrives to a boundary of an element.

Element deleted, if a second crack arrives to the same element.

Where,(5)
$D=\sum \frac{\text{Δ}{\epsilon }_{p}}{{\epsilon }_{f}}\ge 1$

Dadv should always be less than 1.