RD-E: 2602 Ductile Failure Model

Failure criteria defined with ductile failure model.

In Radioss, it is possible to simulate failure with a failure model. Johnson-Cook failure model is the most commonly used, which uses exponent function to describe the $ε_{f} - σ^{*}$ material failure behavior. /FAIL/TAB1 is the most sophisticated failure model to describe $ε_{f} - σ^{*}$ with simple curve input, which is more convenient. /FAIL/BIQUAD can describe $ε_{f} - σ^{*}$ with intuitive parameter input. Equivalent $ε_{f} - σ^{*}$ failure for these three failure models are discussed with one shell element and circular plate model.

Options and Keywords Used

Johnson-Cook failure model (/FAIL/JOHNSON) with parameters
Tabulated failure model (/FAIL/TAB1) with curve input
Simplified nonlinear strain based failure (/FAIL/BIQUAD) with parameter input
Imposed velocities (/IMPVEL)
Material LAW2 (/MAT/LAW2 (PLAS_JOHNS))
Rigid sphere (/RWALL)

Input Files

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

RD-E-2602_Ductile.zip

Model Description

LAW2 and Johnson-Cook Failure Model

The elasto-plastic behavior of the material is defined using the Johnson-Cook law (/MAT/LAW2 (PLAS_JOHNS)), with damage (maximum plastic strain, $ε_{p}^{m a x}$ ). The failure model /FAIL/JOHNSON is independent from the material law and the hardening model.

The Johnson-Cook failure model is defined using /FAIL/JOHNSON in the input deck. The model uses accumulative damage to compute failure.(1)

D = \sum \frac{Δ ε_{p}}{ε_{f}}

Where,

$D$: Current damage and is in range of $0 \leq D \leq 1$ .; The element is in failure, if $D = 1$ . This can be output to the animation file by defining /ANIM/SHELL/DAMA in the Engine.
$Δ ε_{p}$: Increment of plastic strain during the loading.

In /FAIL/JOHNSON, the failure strain,

ε_{f}

is computed as:(2)

ε_{f} = D_{1} + D_{2} \exp (D_{3} \cdot σ^{*})

Where,

$σ^{*}$: Stress triaxiality (or normalized mean stress) $σ^{*} = \frac{σ_{m}}{σ_{V M}}$ .
$D_{1}$ , $D_{2}$ and $D_{3}$: The first three parameters for Johnson-Cook failure model.

The strain rate and thermo-plastic effects are not taken into account in this example. Thus, only three parameters are required ( $D_{1}$ , $D_{2}$ and $D_{3}$ ).

For more information about this failure model, refer to Johnson-Cook Failure Model in the Theory Manual and /FAIL/JOHNSON in the User Guide.

Two cases are considered:

The failure plastic strain $ε_{p}^{m a x}$ in /MAT/LAW2 is not taken into account.
In addition to the Johnson-Cook failure model, the maximum stress and the failure plastic strain are activated.

Two failure approaches are also investigated:

Shell element is deleted, if damage $D = 1$ for one layer (I_{fail_sh} = 1).
The layer stress tensor is set to zero and the shell element is deleted, if damage $D = 1$ for all layers (I_{fail_sh} = 2).

The four simulations performed are:

	`I`_{fail_sh} = 1	`I`_{fail_sh} = 2
Johnson-Cook failure model	$D_{1}$ = 0.11 $D_{2}$ = 0.08 $D_{3}$ = -1.5 $ε_{p}^{m a x}$ = 0.151	$D_{1}$ = 0.09 $D_{2}$ = 0.08 $D_{3}$ = -1.5	$D_{1}$ = 0.11 $D_{2}$ = 0.08 $D_{3}$ = -1.5 $ε_{p}^{m a x}$ = 0.151	$D_{1}$ = 0.09 $D_{2}$ = 0.08 $D_{3}$ = -1.5

I_{fail_sh} = 1

I_{fail_sh} = 2

/FAIL +

$ε_{p}^{m a x}$ , $σ_{\max 0}$

/FAIL Only

/FAIL +

$ε_{p}^{m a x}$ , $σ_{\max 0}$

/FAIL Only

Johnson-Cook failure model

D_{1}

= 0.11

$D_{2}$ = 0.08

$D_{3}$ = -1.5

$ε_{p}^{m a x}$ = 0.151

D_{1}

= 0.09

$D_{2}$ = 0.08

$D_{3}$ = -1.5

D_{1}

= 0.11

$D_{2}$ = 0.08

$D_{3}$ = -1.5

$ε_{p}^{m a x}$ = 0.151

D_{1}

= 0.09

$D_{2}$ = 0.08

$D_{3}$ = -1.5

Figure 1 shows

ε_{f} - σ^{*}

curve with

D_{1}

= 0.11,

D_{2}

= 0.08 and

D_{3}

= -1.5 in the Johnson-Cook failure model.(3)

ε_{f} = 0.11 + 0.08 \exp (- 1.5 \cdot σ^{*})

Where,

ε_{f} - σ^{*}

curve describes the material failure. Above this curve, the material failed, below this curve, the material is safe (not failed).

For example, with a one shell element, from above curve, failure strain

ε_{f} = 0.1585

σ^{*} = 1 / 3

(for uniaxial tension). In results it shows element failed once reach the smallest failure strain either defined with

ε_{p}^{m a x}

in LAW2 or

D = 1

in /FAIL/JOHNSON.

Ductile Failure Models

In /FAIL/TAB1 and /FAIL/BIQUAD it is also possible to have equivalent $ε_{f} - σ^{*}$ , as in /FAIL/JOHNSON.

Curve input for

ε_{f} - σ^{*}

in /FAIL/TAB1 failure model is possible, which is very continent. Directly input

ε_{f} - σ^{*}

curve (created from above Johnson-Cook failure) in table1_ID of /FAIL/TAB1, and then the same failure, as in /FAIL/JOHNSON observed with /FAIL/TAB1.

/FAIL/TAB1 Example:
In the /FAIL/BIQUAD failure model, it is also possible to describe $ε_{f} - σ^{*}$ with intuitive input parameter c1, c2, c3, c4 and c5 as:

Parameter

Description

c1

Failure strain at uniaxial compression test, where $σ^{*} = - 1 / 3$

c2

Failure strain at pure shear test, where $σ^{*} = 0$

c3

Failure strain at uniaxial tension test, where $σ^{*} = 1 / 3$

c4

Failure strain at plainstrain tension test, where $σ^{*} = 1 / \sqrt{3}$

c5

Failure strain at biaxial tension test, where $σ^{*} = 2 / 3$

The parameter c1, c2, c3, c4 and c5 are from $ε_{f} - σ^{*}$ curve (created from above Johnson-Cook failure) in /FAIL/BIQUAD, and then the same failure as in /FAIL/JOHNSON, observed with /FAIL/BIQUAD.

Figure 4.
/FAIL/BIQUAD Example:
The same results show in /FAIL/JOHNSON, /FAIL/TAB1 and /FAIL/BIQUAD, with the equivalent $ε_{f} - σ^{*}$ failure curve.

Figure 5.

Figure 6.

In /FAIL/BIQUAD actually use parabolic function to reproduce the $ε_{f} - σ^{*}$ failure curve cording to input c1, c2, c3, c4 and c5. In fact, the $ε_{f} - σ^{*}$ for /FAIL/BIQUAD used in Radioss may slightly different than Johnson-Cook failure model or TAB1 model. As in this example, $ε_{f} - σ^{*}$ in /FAIL/BIQUAD is printed in start output *0000.out file.

Figure 7.

Use the above a, b, c, d, e, f coefficient from start output file in below parabolic functions from /FAIL/BIQUAD, the $ε_{f} - σ^{*}$ is:(4)
$f_{1} (x) = a x^{2} + b x + c$

$f_{2} (x) = d x^{2} + e x + f$

Figure 8.

At point c1, c2, c3, c4 and c5 are the same; but outside these points, there are some differences. In the simple one element example (uniaxial test), it does not involve other stress state during simulation. Therefore, it shows no difference between these three failure models.

Results

The influence of

ε_{p}^{m a x}

in material law and

ε_{f}

in failure model:

Model Setup	Failure Strain
$ε_{p}^{m a x}$ = 0.151 in LAW2 /FAIL/JOHNSON (with $ε_{f} = 0.1585$ )	By 0.151 ( $ε_{p}^{m a x}$ in LAW2)
$ε_{p}^{m a x}$ = 0.165 in LAW2 /FAIL/JOHNSON (with $ε_{f} = 0.1585$ )	By 0.1585 (failure $ε_{f}$ defined in /FAIL/JOHNSON, where $\sum Δ ε_{p} = ε_{f} \Rightarrow D = 1$ )
$ε_{p}^{m a x}$ = 0 (no failure strain) in LAW2 /FAIL/JOHNSON (with $ε_{f} = 0.1585$ )	By 0.1585 (failure $ε_{f}$ defined in /FAIL/JOHNSON, where $\sum Δ ε_{p} = ε_{f} \Rightarrow D = 1$ )

In the plate model test with above plane shows different damage behavior. The element failed with:

I_{fail_sh} =1: if strain in one integration point reached the criteria
I_{fail_sh} =2: if strain in all integration point reached the criteria

It shows less failure in plate with I_{fail_sh} =2, than in I_{fail_sh} =1.

Table 1. Johnson Failure
`I`_{fail _sh}=1 $ε_{p}^{m a x}$ in LAW2	`I`_{fail _sh}=1	`I`_{fail _sh}=2 $ε_{p}^{m a x}$ in LAW2	`I`_{fail _sh}=2

Influence

Use the above equivalent failure model (from above one element model). Here you can also observe the same failure behavior with these equivalent failure models.

Table 2. Plate Results with Different Failure Models
Johnson Failure `I`_{fail_sh}=1	TAB1 Failure `I`_{fail_sh}=1	BIQUAD Failure `P_thick`_fail=1/N

For /FAIL/JOHNSON and /FAIL/TAB1 the option I_{fail_sh} in these models is to control the element delete criteria. In this plate model, I_{fail_sh}=1 means the element is deleted once damage criteria is reached in just one integration point.

In /FAIL/BIQUAD the equivalent option Pthick_fail is to control the element deletion. It defines element deletion criteria with percentage of through thickness integration points. In this plate example, 5 integration points are defined through thickness (N=5 in /PROP/SHELL). Set Pthick_fail=1/5, which means element deleted if one of the 5 integration points reaches the damage criteria.

Conclusion

Failure model Johnson-Cook, TAB1 and BIQUAD (which use $ε_{f} - σ^{*}$ failure curve) are studied here with one shell element and also circular plate.

Failure could be defined in material model and in failure model at same time. Compare different combinations of failure definition in LAW2 and in Johnson-Cook failure model, it shows the element damaged once first reach any criteria defined in material model or failure model.

Equivalent failure criteria could be generated with Johnson-Cook, TAB1 and BIQUAD, but for BIQUAD there might be a slight difference since it uses two parabolic functions.

Pthick_fail in /FAIL/BIQUAD is similar to I_{fail_sh}, but is more flexible since it uses a percentage of thickness to define element deletion criteria.