/FAIL/BIQUAD

By default, /FAIL/BIQUAD (S-Flag=1) uses two parabolic curves to describe the plastic failure strain ${\epsilon }_f$ , as a function of stress triaxiality ${\sigma }^{*}$ . The two parabolic curves use:(1)

$$f_1\left(x\right)=a{x}^2+bx+c$$

(2)

$$f_2\left(x\right)=d{x}^2+ex+f$$

Where,

$a$ , $b$ , $c$ , $d$ , $e$ , and $f$

Parabolic coefficients

$x$

Stress triaxiality

$f_1\left(x\right)$ and $f_2\left(x\right)$

Plastic failure strain

Figure 1. /FAIL/BIQUAD Failure Strain Curve Made of 2 Parabolic

The parabolic coefficients $a$ , $b$ , $c$ , $d$ , $e$ , and $f$ are computed by Radioss using a curve fit based on the plastic failure strain c1-c5 input 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. The results of the curve fit are in the Starter *0000.out file.

      Bi-Quadratic FAILURE 
       -------------------- 
     c1. . . . . . . . . . . . . . . . . . .=  0.2419E+00
     c2. . . . . . . . . . . . . . . . . . .=  0.1900E+00
     c3. . . . . . . . . . . . . . . . . . .=  0.1585E+00
     c4. . . . . . . . . . . . . . . . . . .=  0.1437E+00
     c5. . . . . . . . . . . . . . . . . . .=  0.1394E+00

      COEFFICIENTS OF FIRST PARABOLA 
      ----------------------------- 
     a . . . . . . . . . . . . . . . . . . .=  0.9180E-01
     b . . . . . . . . . . . . . . . . . . .= -0.1251E+00
     c . . . . . . . . . . . . . . . . . . .=  0.1900E+00


      COEFFICIENTS OF SECOND PARABOLA 
      ----------------------------- 
     d . . . . . . . . . . . . . . . . . . .=  0.3753E-01
     e . . . . . . . . . . . . . . . . . . .= -0.9483E-01
     f . . . . . . . . . . . . . . . . . . .=  0.1859E+00

The c1–c5 plastic failure strains definitions are:

c1

Plastic failure strain in uniaxial compression

c2

Plastic failure strain in shear

c3

Plastic failure strain in uniaxial tension

c4

Plastic failure strain in plane strain tension

c5

Plastic failure strain in biaxial tension

Depending on the M-Flag input option, there are three different ways to define the c1-c5 values.

• M-Flag=0, User-defined Test Data

  For this case, you must enter c1-c5 which represents the plastic failure strain for the 5 different stress states. Ideally this data would be obtained from test or the material supplier.

• M-Flag=1-7, Predefined Material Data
  If failure strain data is not available, you can pick from 7 predefined materials. Figure Figure 2 shows the plastic strain at failure curves for the 7 materials.

  Note: The predefined values are supplied for early design exploration and it is your responsibility to verify that their material has the same properties.

  
  
  Figure 2. Predefined Material Failure Curves

• M-Flag=99, Plastic Failure Strain Ratio Input, r1-r5
  The last input method is to enter the plastic failure strain in uniaxial tension, c3, and plastic failure strain ratios for the other four stress states. These ratios are defined as:

  r1

  Failure plastic strain ratio, Uniaxial Compression (c1) to Uniaxial Tension (c3), so $c1=r1\cdot c3$

  r2

  Failure plastic strain ratio, Pure Shear (c2) to Uniaxial Tension (c3), so $c2=r2\cdot c3$

  r4

  Failure plastic strain ratio, Plane Strain Tension (c4) to Uniaxial Tension (c3), so $c4=r4\cdot c3$

  r5

  Failure plastic strain ratio, Biaxial Tension (c5) to Uniaxial Tension (c3), so $c5=r5\cdot c3$

  Using this method, it is easy to change the failure curve by adjusting the single plastic failure strain in uniaxial tension value, c3.

  
  
  Figure 3. Changes in Plastic Failure Strain Curve. by increasing the uniaxial tension failure, c3, with the same failure plastic strain ratios

By default, values different than 0 need to be entered for c1 to c5. However, specific default behaviors exists, in case failure information are missing.

• 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 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.

A cumulative damage method is used to sum the amount of plastic strain that has occurred at each integration point in the element using:(3)

$$D={\displaystyle \sum \frac{\text{Δ}{\epsilon }_p}{{\epsilon }_f}\ge 1}$$

Where,

$D$

Damage

$\text{Δ}{\epsilon }_p$

The change in plastic strain of the integration point

${\epsilon }_f$

Plastic failure strain for the current stress triaxiality

In materials such as sheet metal, thickness thinning and diffuse necking may appear during tensile loading of the material. This is called localized necking and normally occurs in the stress triaxiality range of $\frac{1}{3}\le {\sigma }^{*}\le \frac{2}{3}$ :

Figure 4.

In /FAIL/BIQUAD it is possible to simulate this localized necking using the option, S-Flag=3 and Inst_start. This option uses the same plastic failure strain curve as S-Flag=2 and adds 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 a user-defined value in the Inst_start field and occurs at plane strain tension ${\sigma }^{*}=\frac{1}{\sqrt{3}}$ . Using this localized necking curve, a second localized necking damage value is calculated and failure due to necking only occurs when all integration points reach $D=1$ . The localized necking criteria is based on the Marciniak-Kuczynski analysis. ¹

Figure 5. Default Failure Strain Curve . with additional localized necking curve (blue)

If S-Flag=3is used to describe the localized necking, damage accumulation begins once the plastic strain reaches in the localized necking curve (blue curve in Figure 6). For localized necking, the element is deleted when all integration points reach damage, $D=1$ , whereas element deletion not due to localized necking is defined by P_thickfail.

Figure 6.

Due to a materials imperfection or production process, a material’s failure strain may not be exactly the same everywhere and thus very small perturbations of failure limit may exist. When using M-Flag>0 with /PERTURB/FAIL/BIQUAD, a statistical distribution of the failure limit is applied to each element assigned the failure model. This is accomplished by calculating the normal or random distribution of a failure scale factor which is applied to /FAIL/BIQUAD, c3. The two different distributions methods are shown in Figure 7 and Figure 8.

Figure 7. Random Distribution. Idistri=1, of the failure limit in the Starter *0.out file

Figure 8. Normal (Gaussian) Distribution. Idistri=2, of the failure limit in the Starter *0.out file

/FAIL/BQUAD uses small perturbations of c3 generated by /PERTURB/FAIL/BIQUAD to scale the entire failure curve using the strain ratio values for the predefined materials or user-defined ratios, r1, r2, r4 and r5, depending on the value of M-Flag.

Figure 9.