II.1.10.21 Hashin criteria (3D)

A 3D version of the 2D criterion defined in section II.1.10.20 is defined as follows

• If ${\sigma }_{11}>0$, ply passes the failure criterion if the following condition is satisfied:

  $${\left(\frac{{\sigma }_{11}}{{\sigma }_{11}^{\text{t,all}}}\right)}^2+\frac{{\tau }_{12}^2+{\tau }_{31}^2}{{\left({\tau }_{12}^{\text{all}}\right)}^2}\le 1,$$

  and reserve factor is given by:

  $$R{F}_b=\frac{1}{FoS\cdot \sqrt{{\left(\frac{{\sigma }_{11}}{{\sigma }_{11}^{\text{t,all}}}\right)}^2+\frac{{\tau }_{12}^2+{\tau }_{31}^2}{{\left({\tau }_{12}^{\text{all}}\right)}^2}}}.$$

  If ${\sigma }_{11}<0$, ply passes the failure criterion if the following condition is satisfied:

  $$\left|{\sigma }_{11}\right|\le {\sigma }_{11}^{\text{c,all}}.$$

  The corresponding reserve factor is

  $$R{F}_b=\frac{{\sigma }_{11}^{\text{c,all}}}{FoS\cdot \left|{\sigma }_{11}\right|}.$$

• The calculation of matrix failure is slightly more complicated. If ${\sigma }_{22}+{\sigma }_{33}>0$ ply passes the failure criterion if the following condition is satisfied:

  $$\frac{{\tau }_{12}^2+{\tau }_{31}^2}{{\left({\tau }_{12}^{\text{all}}\right)}^2}+\frac{{\tau }_{23}^2-{\sigma }_{22}{\sigma }_{33}}{{\left({\tau }_{23}^{\text{all}}\right)}^2}+{\left(\frac{{\sigma }_{22}+{\sigma }_{33}}{{\sigma }_{22}^{\text{t,all}}}\right)}^2\le 1.$$

  Reserve factor is then given by:

  $$R{F}_c=\frac{1}{FoS\cdot \sqrt{\frac{{\tau }_{12}^2+{\tau }_{31}^2}{{\left({\tau }_{12}^{\text{all}}\right)}^2}+\frac{{\tau }_{23}^2-{\sigma }_{22}{\sigma }_{33}}{{\left({\tau }_{23}^{\text{all}}\right)}^2}+{\left(\frac{{\sigma }_{22}+{\sigma }_{33}}{{\sigma }_{22}^{\text{t,all}}}\right)}^2}}.$$

  Otherwise, ply passes the failure criterion if the following condition is satisfied:

  $$\frac{{\tau }_{12}^2+{\tau }_{31}^2}{{\left({\tau }_{12}^{\text{all}}\right)}^2}+\frac{{\tau }_{23}^2-{\sigma }_{22}{\sigma }_{33}}{{\left({\tau }_{23}^{\text{all}}\right)}^2}+{\left(\frac{{\sigma }_{22}+{\sigma }_{33}}{2{\tau }_{23}^{\text{all}}}\right)}^2+\frac{{\sigma }_{22}+{\sigma }_{33}}{{\sigma }_{22}^{\text{c,all}}}\left[{\left(\frac{{\sigma }_{22}^{\text{c,all}}}{2{\tau }_{23}^{\text{all}}}\right)}^2-1\right]\le 1.$$

  For that case, one successively calculates:

  $$a=\frac{{\tau }_{12}^2+{\tau }_{31}^2}{{\left({\tau }_{12}^{\text{all}}\right)}^2}+\frac{{\tau }_{23}^2-{\sigma }_{22}{\sigma }_{33}}{{\left({\tau }_{23}^{\text{all}}\right)}^2}+{\left(\frac{{\sigma }_{22}+{\sigma }_{33}}{2{\tau }_{23}^{\text{all}}}\right)}^2,$$

  $$b=\frac{{\sigma }_{22}+{\sigma }_{33}}{{\sigma }_{22}^{\text{c,all}}}\left[{\left(\frac{{\sigma }_{22}^{\text{c,all}}}{2{\tau }_{23}^{\text{all}}}\right)}^2-1\right],$$

  $$c=-1,$$

  $$R{F}_c=\frac{-b+\sqrt{b^2-4ac}}{2\cdot FoS\cdot a}.$$

Finally, the reserve factor is given by:

Here again, the calculation of failure index is based on the calculation of the reserve factor:

in which the safety factor used in the calculation of the RF is set to 1, and the strength ratio is calculated as

in which one keeps the value of safety factor.

The criterion presented above is referred to by “Hashin3D” criterion in FeResPost. Correspondingly, one defines version “Hashin3D_b” in which only the fiber failure is checked and “Hashin3D_c” in which only the matrix failure is checked. (These correspond to the values $R{F}_b$ and $R{F}_c$ calculated above.)