<CRITERION> bron

Description

The Bron criterion allows modeling of anisotropic behavior in the criterion and flow directions.

The proposed yield function is defined by an equivalent stress:

(347)\[\begin{split}\begin{aligned} \bar{\sigma} &= \left( \alpha (\bar{\sigma}^1)^a + (1-\alpha)(\bar{\sigma}^2)^a \right)^{1/a}\\ \bar{\sigma}^k &= \left( \psi^k \right)^{1/b^k}\\ \psi^1 & = \frac{1}{2}\left( \left| S_2^1-S_3^1 \right|^{b^1} + \left| S_3^1-S_1^1 \right|^{b^1} + \left| S_1^1-S_2^1 \right|^{b^1} \right)\\ \psi^2 & = \frac{3^{b^2}}{2^{b^2}+2}\left( \left| S_1^2 \right|^{b^2} + \left| S_2^2 \right|^{b^2} + \left| S_3^2 \right|^{b^2} \right) \end{aligned}\end{split}\]

where \(S_{i=1-3}^k\) are the principal values of a modified stress deviator \(\overline{s}^k\) defined as follows:

(348)\[\begin{split}\begin{gathered} \ten{s}^k = \tenf{L}^k\!:\!\ten{\sigma}\\ \ten{L}^k = \begin{pmatrix} (c_2^k+c_3^k)/3 & -c_3^k/3 & -c_2^k/3 & 0 & 0 & 0 \\ -c_3^k/3 & (c_3^k+c_1^k)/3 & -c_1^k/3 & 0 & 0 & 0 \\ -c_2^k/3 & -c_1^k/3 & (c_1^k+c_2^k)/3 & 0 & 0 & 0 \\ 0 & 0 & 0 & c_4^k & 0 & 0 \\ 0 & 0 & 0 & 0 & c_5^k & 0 \\ 0 & 0 & 0 & 0 & 0 & c_6^k \end{pmatrix} \end{gathered}\end{split}\]

\(a\), \(b^1\), \(b^2\) and \(\alpha\) are four material parameters that influence the shape of the yield surface but not its anisotropy which is only controlled by \(c_{i=1-6}^{k=1-2}\). Thereby, the yield function has 16 parameters. To ensure convexity and derivability the following conditions are required: \(a\geq 1\) and \(b^k \geq 2\). No restriction applies to the \(c_i^k\) coefficients; in particular they can be negative. The particular case where \(\tenf{L}^1 = \tenf{L}^2\) and \(a=b^1=b^2\) corresponds to the yield function of Karafillis and Boyce (1993) and the case where \(\alpha = 1\) corresponds to the yield function of Barlat et al. (1991). Finally, when \(\alpha=1\) and \(c_i^1=1\), it amounts to von Mises yield function if \(b^1=2\text{ or }4\) and to Tresca yield function if \(b^1=1 \text{ or }+\infty\). If \(c_i^k=1\), the resulting yield function is isotropic as it only depends on the eigenvalues of \(\overline{\sigma}\).

Example

**porous_potential
   *porous_criterion modified_rousselier ...
   *isotropic function ...
   *flow plasticity
   *shear_anisotropy bron
      a 2.2 alpha 0.60
      b1 10.3
      c11 0.58 c12 1.35 c13 1.14 c14 1.23 c15 1.35 c16 1.57
      b2 13.1
      c21 2.07 c22 0.20 c23 0.33 c24 0.85 c25 1.31 c26 0.59