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:

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

  The following quantities are then computed successively:

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

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

  $$c=-1,$$

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

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 the actual value of the safety factor is used.

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