RD-V: 0700 Johnson-Cook Failure Criteria

Figure 1. Left: shell element; Right: hexahedron element

The subject of this analysis is to evaluate the Johnson-Cook failure criteria using a material LAW36 and LAW2.

The analysis is for 6 cases:

A square coupon modeled with shell elements /SHELL with /PROP/SHELL property and QEPH formulation (I_shell=24)
A square coupon modeled with sh3n elements /SH3N with /PROP/SHELL property and QEPH formulation (I_sh3n=0)
A square coupon modeled with hexahedron elements /BRIC with /PROP/SOLID property and HEPH formulation (I_solid =24)
A square coupon modeled with hexahedron elements /BRIC with /PROP/TSHELL property and HSEPH formulation (I_solid =15)
A square coupon modeled with hexahedron degenerated elements /BRIC with /PROP/SOLID property and HEPH formulation (I_solid =24)
A square coupon modeled with tetrahedron elements /BRIC with /PROP/SOLID property and HEPH formulation (I_tetra4 =3)

Options and Keywords

The following keywords are used in the models.

Material: /MAT/LAW2 (PLAS_JOHNS), /MAT/LAW36 (PLAS_TAB)
Failure: /FAIL/JOHNSON
Property: /PROP/TYPE14 (SOLID), /PROP/TYPE1 (SHELL), /PROP/TYPE20 (TSHELL)
Element: /BRICK, /SHELL
Boundary conditions: /BCS and /IMPVEL
Curves: /FUNCT
Local coordinate system: /SKEW and /FRAME
Sets: /GRNOD/NODE

Model Files

Before you begin, copy the file(s) used in this problem to your working directory.

RD-V-0700.zip

Model Description

Units: Kg, mm, ms, GPa

The boundary conditions are represented.

Material Law Characteristics

The material to be characterized is DP600 steel. The model is tested with different elements mentioned above with 10x10 mm side lengths.

Material Property: Values
Young's modulus: 210 GPa
Poisson ratio: 0.3
Density: 7.8e-06 kg/mm³

LAW36
The elasto-plastic behavior is defined using the tabulated material LAW36. The True Stress versus Plastic True Strain curve is used as an input of LAW36. For more information about the LAW36 material, refer to RD-E: 1101 Elasto-plastic Material Law Characterization in the Example Guide.
Figure 3. True stress versus True strain
LAW2
This law represents an isotropic elastoplastic material using the Johnson-Cook material model.
The materiel LAW2 parameters yield stress “ $a$ ”, Plastic hardening parameter “ $b$ ” and Plastic hardening exponent “ $n$ ” have been defined. The true stress is calculated using:
$σ = (a + b ε_{p}^{n})$
Where,

$a$

0.270 GPa

$b$

0.75 GPa

$n$

0.6

The true stress versus true strain curve is represented below:
Figure 4. Stress versus plastic strain curve LAW2

Johnson-Cook Failure

The Johnson-Cook failure model is defined using /FAIL/JOHNSON in the input. The model uses accumulative damages to compute failure:

$D = \sum \frac{Δ ε}{ε_{f}}$ with $ε_{f} = max ([D_{1} + D_{2} * \exp (D_{3} σ^{*})] [1 + D_{4} \ln (\frac{\dot{ε}}{\dot{ε_{0}}})] [1 + D_{5} T^{*}]; E P S F_{m i n})$

Where, $Δ ε$ is the increment of plastic strain during a loading increment, $σ^{*} = \frac{σ_{m}}{σ_{V M}} = - \frac{p}{σ_{V M}}$ the normalized mean stress and the parameters $D_{i}$ the material constants. Failure is assumed to occur when D=1. The strain rate and thermo-plastic effects are not considered in this example. Only three parameters are required $D_{1}$ , $D_{2}$ and $D_{3}$ .

The material constants can be calculated by using:

D_{1} = ε_{f} (0) - D_{2}

D_{2} = \frac{ε_{f} (1 / 3) - ε_{f} (0)}{e^{\frac{D_{3}}{3}} - 1}

D_{3} = 3 l n [\frac{ε_{f} (0) - ε_{f} (1 / 3)}{ε_{f} (- 1 / 3) - ε_{f} (0)}]

Where,

$ε_{f} (1 / 3) = 0.3$: Pure tensile failure
$ε_{f} (0) = 0.5$: Pure shear failure
$ε_{f} (- \frac{1}{3}) = 0.8$: Pure compression failure

The

ε_{f} - σ^{*}

curve describes the material failure model using

D_{1} = - 0.1, D_{2} = 0.6 a n d D_{3} = - 1.2

the Johnson-Cook failure model parameters calculated above.

Figure 5. $ε_{f} - σ^{*}$ curve describing the material failure

Results

The Johnson-Cook failure criteria was evaluated with LAW2 and LAW36. As the results are similar between both material laws, only material LAW36 results are represented below.

Square coupon with SHELL elements

The following element formulation is evaluated:

/PROP/TYPE1 (SHELL), QEPH shell formulation I_shell=24, 5 integration points over the thickness.
For each loading, the triaxiality and plastic strain curves are represented at failure.

Table 1. Results for SHELL QUAD element
Pure Tensile	Pure Shear	Pure Compression

As expected, the elements failed once they reached the plastic strain, then the stress decreases:

$ε_{f} = 0.3$: Pure tensile
$ε_{f} = 0.5$: Pure shear
$ε_{f} = 0.8$: Pure compression

Square coupon with SH3N elements

The following element formulation is evaluated:

/PROP/TYPE1 (SHELL), QEPH shell formulation I_sh3n=0, 5 integration points over the thickness.
For each loading, the triaxiality and plastic strain curves are represented at failure.

Table 2. Results for SH3N element
Pure Tensile	Pure Shear	Pure Compression

As expected, the elements failed once they reached the plastic strain, then the stress decrease:

$ε_{f} = 0.3$: Pure tensile
$ε_{f} = 0.5$: Pure shear
$ε_{f} = 0.8$: Pure compression

Cube coupon with Solid Hexahedron elements

The following element formulation is evaluated:

/PROP/TYPE14 (SOLID), I_solid =24.
For each loading, the triaxiality and plastic strain curves are represented at failure.

Table 3. Results for Hexahedron element
Pure Tensile	Pure Shear	Pure Compression

As expected, the elements failed once they reached the plastic strain, then the stress decrease:

$ε_{f} = 0.3$: Pure tensile
$ε_{f} = 0.5$: Pure shear
$ε_{f} = 0.8$: Pure compression

Cube coupon with TSHELL elements

The following element formulation is evaluated:

/PROP/TYPE20 (TSHELL), I_solid =15.
For each loading, the triaxiality and plastic strain curves are represented at failure.

Table 4. Results for TSHELL element
Pure Tensile	Pure Shear	Pure Compression

As expected, the elements failed once they reached the plastic strain, then the stress decrease:

$ε_{f} = 0.3$: Pure tensile
$ε_{f} = 0.5$: Pure shear
$ε_{f} = 0.8$: Pure compression

Cube coupon with Solid Hexahedron degenerated elements

The following element formulation is evaluated:

/PROP/TYPE14 (SOLID), I_solid =24.
For each loading, the triaxiality and plastic strain curves are represented at failure.

Table 5. Results for HEXAHEDRON degenerated element
Pure Tensile	Pure Shear	Pure Compression

As expected, the elements failed once they reached the plastic strain, then the stress decrease:

$ε_{f} = 0.3$: Pure tensile
$ε_{f} = 0.5$: Pure shear
$ε_{f} = 0.8$: Pure compression

Cube coupon with Solid Tetrahedron elements

The following element formulation is evaluated:

/PROP/TYPE14 (SOLID), I_tetra4 =3.
For each loading, the triaxiality and plastic strain curves are represented at failure.

Table 6. Results for TETRAHEDRON element
Pure Tensile	Pure Shear	Pure Compression

As expected, the elements failed once they reached the plastic strain, then the stress decrease:

$ε_{f} = 0.3$: Pure tensile
$ε_{f} = 0.5$: Pure shear
$ε_{f} = 0.8$: Pure compression

Conclusion

This study highlights that the /FAIL/JOHNSON failure criteria implemented in Radioss behaves as expected. As soon as the element reaches the targeted failure plastic strain for a given triaxiality level, the element is eroded. This study was carried out with material LAW2 and material LAW36.