Material Failure Criterion

Material failure criterion can be defined using the MATF Bulk Data Entry or the MATS1 Bulk Data Entry (for damage initiation/evolution criteria only). Failure of materials is strongly influenced by the loading conditions and thus, the stress state. Hence, several criteria available refer to the notions of stress triaxiality and optionally to the Lode parameter to describe the loading conditions (uniaxial tension, pure shear, plane strain etc).

To describe a failure criterion based on plasticity and stress states, the value stress triaxiality, $η$ , and the lode parameter, $ξ$ , are needed. For shells only, stress triaxiality is needed.

Stress Triaxiality

Stress triaxiality (

η

) is used to differentiate between compressive and tensile loadings and depends on the trace of the stress tensor. It can determine the position of the stress state on the hydrostatic axis shown in Figure 1.

Figure 1. Description of the stress state on hydrostatic axis and deviatoric plane

It is computed as follows:

η = \frac{\frac{1}{3} (σ_{x x} + σ_{y y} + σ_{z z})}{σ_{V M}}

Where

σ_{V M}

is the equivalent von Mises stress.

The values of stress triaxiality vary from

- \infty

to

+ \infty

for solids (in practice bounded to -1 and 1) and -2/3 to 2/3 for shells.

Table 1. Stress triaxiality values for some common stress states
Loading condition	Solids	Shells
Confined compression	-1
Biaxial compression	-2/3	-2/3
Uniaxial compression	-1/3	-1/3
Pure shear	0.0	0.0
Uniaxial tension	1/3	1/3
Plane strain	0.5751	0.5751
Biaxial tension	2/3	2/3
Confined tension	1

Lode Angle

To describe 3D loading conditions, another important quantity is the lode angle ( $θ$ ) given by:

\cos (3 θ) = \frac{27}{2} \frac{J_{3}}{σ_{V M}^{3}}

Where

J_{3}

is the third deviatoric invariant.

The lode angle determines the position of the stress state in the deviatoric section. Its value is between 0 (for tension) and

π / 3

(for compression).

Figure 2. Stress state position on the deviatoric plane depending on the lode angle value

Shear and plane strain condition takes a lode angle value of

π / 6

.

Under plane stress hypothesis (for shell elements), the lode angle and the stress triaxiality are linked and thus one for them can be used to recover the other:

\cos (3 θ) = - \frac{27}{2} η (η^{2} - \frac{1}{3})

As it is much easier to deal with normalized value instead of radians, the lode angle is usually switched by the Lode parameter denoted $ξ$ , given by:

ξ = 1 - \frac{6}{π} θ

The lode parameter's values are:

-1.0 in compression
In pure shear or plane strain
In tension

Supported Failure Criteria

Currently, four failure criteria are supported for Explicit Dynamic Analysis namely, BIQUAD, TSTRN, tabulated failure criteria and INIEVO.

BIQUAD

The BIQUAD criterion is a stress triaxiality based failure criterion mostly used for ductile metals. Its double quadratic curve shape describes the evolution of plastic strain,

ε_{p}^{f}

, at failure with respect to stress triaxiality,

η

, as shown in the below image.

Figure 3. Failure plastic strain evolution with stress triaxiality for BIQUAD criterion

It then requires five parameters called c1, c2, c3, c4 and c5 respectively corresponding to V1, V2, V3, V4 and V5 value in the MATF Bulk Data Entry. These five values correspond to plastic strain at failure for five different stress states:

Uniaxial compression
Pure shear
Uniaxial tension
Plane strain
Biaxial tension

Note: The parabolic curve computation at high stress triaxiality is made so that c4 is always the minimum value.

For shell elements, strain localization and necking occurring at high strain rate might not be correctly detected as the thickness variation is purely numerical. Thus, failure can be delayed in comparison to an equivalent sized solid element. To avoid that, an additional curve (see the blue curve in the below figure) can be defined for shells using INST parameter (V6), replacing c4 in the high stress triaxiality parabolic curve computation.

Figure 4. Additional failure quadratic curve (in blue) at high stress triaxiality for shells

If enough experimental data is unavailable to identify all the c1, c2, c3, c4 and c5 parameters, a material selector input is also available for BIQUAD criterion. Depending on the keyword MATER value chosen in the list presented above, the c1, c2, c4 and c5 parameters will be automatically computed with respect to c3 value, as shown below.

c 1 = r 1 \cdot c 3

c 2 = r 2 \cdot c 3

c 3 = c 3

c 4 = r 4 \cdot c 3

c 5 = r 5 \cdot c 3

The value c3 is then the only expected parameter when using material input for BIQUAD criterion. However if no c3 value is specified, a default value of c3 will automatically be set.

Table 2. Automatic parameters settings for MATER keyword
Keyword	c3 (Default)	r1	r2	r4	r5
MILD	0.60	3.5	1.6	0.6	1.5
HSS	0.50	4.3	1.4	0.6	1.6
UHSS	0.12	5.2	3.1	0.8	3.5
AA5182	0.30	5.0	1.0	0.4	0.8
AA6082	0.17	7.8	3.5	0.6	2.8
PA6GF30	0.10	3.6	0.6	0.5	0.6
PP T40	0.11	10.0	2.7	0.6	0.7

For each timestep, the plastic strain at failure,

ε_{p}^{f} (η)

, is estimated according to the stress triaxiality and the parabolic curves. This allows increases to the damage variable accounting for the stress state history:

D = \sum_{t = 0}^{\infty} \frac{Δ ε_{p}}{ε_{p}^{f} (η)}

TSTRN

The TSTRN failure criterion is a strain based damage model and is supposed to be fully coupled (DAMAGE keyword activated and

D_{C} = 0

). However, you have the freedom to use it as a failure criterion or a pure output damage variable. It considers a linear evolution of the damage variable between two starting and ending strain values, in tensile loading conditions (

η > 0

):

D = \frac{ε - ε_{s t a r t}}{ε_{e n d} - ε_{s t a r t}}

A couple of values

ε_{s t a r t}

and

ε_{e n d}

are then needed in the card. V1 and V2 values corresponds to starting and ending von Mises equivalent strain. The von Mises equivalent strain is computed as follows:

ε = \sqrt{\frac{2}{3} (ε_{x x}^{2}^{'} + ε_{y y}^{2}^{'} + ε_{z z}^{2}^{'} + 2 ε_{x y}^{2} + 2 ε_{y z}^{2} + 2 ε_{z x}^{2})}

Where

ε^{'}

is the deviatoric strain tensor.

If V3 and V4 values are specified, they correspond to starting and ending major principal strain.

ε = \max (ε_{1}, ε_{2}, ε_{3}) > 0

Note: V3 and V4 values are always prioritized when both V1/V2 and V3/V4 pairs are specified.

Tabulated failure criteria

The TAB failure criterion is used to give as much freedom as possible to describe a plastic strain based tabulated criterion. The TABLEMD entry defined by EPS_TID describes the map showing the evolution of plastic strain at failure,

ε_{p}^{f}

, with respect to stress triaxiality and, optionally for solid elements, with lode parameter,

ξ

, as shown in Figure 5.

Figure 5. Tabulated failure criterion map showing the evolution of plastic strain at failure with respect to stress triaxiality and lode parameter

For solid elements, the entire map with all possible couple of values,

(η, ξ)

, is considered. However, for shells stress triaxiality and lode parameter are linked due to plane stress conditions. Hence, only the plane stress (blue line in Figure 5) is considered.

The V1 value is a scale factor that allows you to quickly increase or decrease in entire map.

The damage variable evolution is given by a specific formula using the parameter in defined in V2 value:

D = \sum_{t = 0}^{\infty} \frac{Δ ε_{p}}{ε_{p}^{f} (η, ξ)} \cdot n \cdot D^{(1 - \frac{1}{n})}

Thus, including its own current value, the damage variable evolution is taking into account the stress state history but also the damage history. The exponent

n

allows to indirectly change the shape of the damage evolution with respect to plastic strain as presented in Figure 6. The increase of the

n

exponent parameter tends to delay the stress softening effect as shown.

Figure 6. Effect of n parameter on the damage versus plastic strain evolution (left picture) and effect of n parameter on a single element uniaxial tension behavior (right picture)

You can use the TAB criterion defining only the first line of parameters (EPS_TID, V1 and V2). In this case, like any other criterion available, you can activate the element deletion using DAMAGE, chose the beginning of stress softening with the constant value for critical damage DC and the shape of the stress softening using EXP.

Another approach of stress softening approach with TAB criterion is called the necking-controlled approach.

To use this new approach, the two first parameters of the second line INST_TID and V6 must be defined. INST_TID defines the ID of a TABLEMD entry defining a map showing the evolution of the plastic strain value (denoted $ε_{p}^{l o c}$ ) for which necking instability and thus strain localization starts, with respect to stress triaxiality and, optionally, lode parameter. It is an instability limit curve or map mostly defined at high stress triaxiality as the one described above for BIQUAD criterion in Figure 3 and is supposed to be lower than the failure curve/map to have an effect. It can be used with solids or shells.

This INST second map allows to compute the evolution of a new variable called necking-triggering variable and denoted $f$ . Its evolution is very similar to the damage variable one:

f = \sum_{t = 0}^{\infty} \frac{Δ ε_{p}}{ε_{p}^{l o c} (η, ξ)} \cdot n \cdot f^{(1 - \frac{1}{n})}

Once this variable reaches the value 1, a stress softening is triggered (defined by Comment 12 in the MATF Bulk Data Entry). However, instead of using the constant value, $D_{C}$ , in the MATF entry, the parameter, $D_{C}$ , becomes an integration point. Thus $D_{C}$ can be very different from one element to another depending on the history of the element stress state.

Thus, when INST_TID is used, the $D_{C}$ value corresponds to the value taken by the damage variable $D$ at the exact moment when $f$ reaches or overtakes the value 1. In other words, $D_{C}$ is the $D$ value when the necking criterion is reached the first time. Then, $D_{C}$ remains untouched until the end of the simulation.

\begin{array}{l} D = \int Δ D \\ f = \int Δ f \\ D_{C} = \{\begin{matrix} \begin{matrix} 1 & while & f < 1 \end{matrix} \\ \begin{matrix} D & when & f \geq 1 \end{matrix} \end{matrix} \\ σ = σ_{e f f} (1 - {(\frac{D - D_{c}}{1 - D_{c}})}^{\exp}) \end{array}

Unlike the

D_{C}

parameter, the exponent (EXP) is a constant parameter over all elements.

This necking-controlled approach can offer a higher predictivity for a large range of stress state but needs to define an instability map especially at high stress triaxiality when necking is more likely to happen.

Finally, parameters V7 and V8 values are stress triaxiality boundaries for element size scaling defined below. If this pair of values are defined, the size scaling only occurs when:

V 7 < η < V 8

Damage initiation and evolution (INIEVO)

INIEVO failure criterion is very specific and provides the ability to define a failure approach based on the use of a DMGINI Bulk Data Entry and, optionally a DMGEVO Bulk Data Entry.

For the DMGINI Bulk Data Entry, only DUCTILE criterion is available. For the DMGEVO Bulk Data Entry, only DISP and ENERGY evolution are available.

This criterion can be defined using two methods:

The DAMAGE continuation line in the MATS1 Bulk Data Entry. This method is supported both for Implicit and Explicit Dynamic Analysis.
CRI=INIEVO in the MATF Bulk Data Entry. This method is supported only for Explicit Dynamic Analysis.

Note: For INIEVO, strain rate dependency and element size dependency are not available.