/FAIL/ORTHBIQUAD

Figure 1. Biquad failure criterion shape

The parameters for the 2 parabolic failure strain curves versus the state of stress (stress triaxiality) are calculated iteratively by Radioss during the initialization phase using the input c1-c5 values.

If the calculated parabolic failure strain curves have negative failure strain values, these negative values will be replaced by a failure strain of 1E-6 which results in a very high damage accumulation and brittle behavior.

This failure criteria is usable for all elasto-plastic material models with shells and solids.

Where, $Q_m^i$ are the interpolation factors for parameter $i$ . These interpolation factors are automatically computed in the Radioss Starter.

For instance, if two sets of parameters are given for the direction 0 and 90 degrees, the Fourier series interpolation enables to determine the 45 degrees curve as:

Figure 3. Example of orthotropic biquad criterion shape

Loading directions not given in the input are interpolated using Fourier series. All directions have the same plastic strain at failure in biaxial tension.

• In case the material failure behavior is unknown, c1 to c5 are set to 0.0 and the mild steel behavior (M-Flag=1) is used.
• If only the tensile failure value is known, c3 is defined (c1=c2=c4=c5 = 0.0). The mild steel behavior is used and scaled by the user-defined c3 value.
• In case the material behavior is known, M-Flag is defined and c3 can be used to adjust the failure model according to the expected tensile failure. The selected material behavior is scaled by the user-defined c3 value.
• For all other cases, all c1 to c5 are intended to be defined and default value of 0.0 is used.
  Note: If c5 = 0.0, it is automatically computed depending on the value of M-Flag. In this case, the minimum value computed among all input directions will be retained.

• If M-Flag > 0, the entered c1, c2, c4 and c5 values will not be used, but will be calculated as follows, using the predefined ratio values from the table.

  $c1=r1\cdot c3$

  $c2=r2\cdot c3$

  $c3=c3$

  $c4=r4\cdot c3$

  $c5=r5\cdot c3$

• If c3=0, the default value shown in the table is used.

  M-Flag	Roughly Corresponds to Material	c3 (Default)	r1	r2	r4	r5
  1	Mild steel	0.60	3.5	1.6	0.6	1.5
  2	HSS steel	0.50	4.3	1.4	0.6	1.6
  3	UHSS steel	0.12	5.2	3.1	0.8	3.5
  4	Aluminum AA5182	0.30	5.0	1.0	0.4	0.8
  5	Aluminum AA6082-T6	0.17	7.8	3.5	0.6	2.8
  6	Plastic PA6GF30	0.10	3.6	0.6	0.5	0.6
  7	Plastic PP T40	0.11	10.0	2.7	0.6	0.7
  99	Self-defined values (optional line)	0.30	Optional input	Optional input	Optional input	Optional input

  Important: Neither Altair nor the authors assume any responsibility for the validity, accuracy or any results obtained from these values. You must verify your own values by reasonable test results. Usage is only recommended for early design exploration.
• If c3 > 0, the selected material behavior is scaled by c3 and the r1 to r5 predefined ratio values.
• If c5 = 0.0, the minimum value of among all input directions will be retained.
• If the M-Flag = 99, failure strain ratios r1, r2, r4 and r5 must be input in a following additional line.

For shell elements when an integration point reaches $D=1$ , the integration points stress tensor is set to 0. Shell elements are deleted based on the value of P_thick_fail.

If P_thick_fail is set blank or 0, the value of P_thick_fail from the shell property is used. If P_thick_fail > 0, any P_thick_fail value defined in the shell properties is ignored and the value entered in this failure model is used.

For values of P_thick_fail set > 0, the element fails and is deleted when the ratio of through thickness failed integration points equals or exceeds P_thick_fail.

In solid elements, the element is deleted when any integration point reaches $D=1$ .

• S-Flag = 1: the failure curves are created, as shown in Comment 1. In this case, the curves may not reach their minimum value for the same stress triaxiality (Figure 4).

  
  
  Figure 4. Orthotropic biquadratic failure criterion shape when S-Flag =1

• S-Flag = 2: is set by default. It ensures value c4 as global minimum for all directions. To achieve this for all curves, the second equation is split into 2 separate quadratic sub-functions where the minimum value of the curves is at c4; where, ${\sigma }^{*}=\frac{1}{\sqrt{3}}$ (Figure 5).

  
  
  Figure 5. Orthotropic biquadratic failure criterion shape when S-Flag =2

• S-Flag = 3: same as S-Flag = 2, plus a simplified localized necking criterion (only for shells). The localized necking criteria is based on the Marciniak-Kuczynski analysis. It uses two additional quadratic functions that define a curve that represents the start of localized necking between stress triaxiality $\frac{1}{3}$ and $\frac{2}{3}$ . The minimum value of this curve is user-defined in the Inst_start field and occurs at plane strain tension, ${\sigma }^{*}=\frac{1}{\sqrt{3}}$ (Figure 6).

  Using this curve, a second localized necking damage value is calculated, and failure only occurs when all integration points reach $D=1$ .

  The Inst_start value can be estimated as the (true plastic) strain at maximum stress, from the uniaxial tension test.

  
  
  Figure 6. Orthotropic biquadratic failure criterion shape when S-Flag =3

• ${\dot{\epsilon }}_0\ne 0$ and $C_{JC}\ne 0$ , the Johnson-Cook’s strain rate dependency is used. In this case, the plastic strain at failure value is multiplied by the strain rate dependency factor as:(3)
  $$D={\displaystyle \sum _{t=0}^{\infty }\frac{\text{Δ}{\epsilon }_p}{{\epsilon }_f^{\theta }(\eta )\left(1+C_{JC}{〈\ln \frac{\dot{\epsilon }}{{\dot{\epsilon }}_0}〉}_{+}\right)}}$$
  Where,

  ${\dot{\epsilon }}_0$

  Inviscid limit strain rate

  $C_{JC}$

  Strain rate dependency parameter

  ${\langle \rangle }_{+}$

  Macaulay brackets, which considers only positive values

  Note: If using a Johnson-Cook material law coupled to the /FAIL/ORTHBIQUAD criterion, the Johnson-Cook parameters used for the constitutive law might not be the same for the failure criterion.
• fct_ID_rate ≠ 0, a tabulated function of strain rate dependency factor is used. In this case, you have to define a function (/FUNCT) to describe the evolution of the strain rate factor (denoted $f_{rate}$ ) with the strain rate. You can also input a strain rate scale factor for the abscissa of the function: Xscale_rate (by default, this scale factor is set to 1.0). Using the tabulated strain rate dependency, the damage variable computation becomes:(4)
  $$D={\displaystyle \sum _{t=0}^{\infty }\frac{\text{Δ}{\epsilon }_p}{{\epsilon }_f^{\theta }(\eta )f_{rate}\left(\dot{\epsilon }\right)}}$$