II.1.10.20 Hashin criteria

This criterion is meant to be used for unidirectional materials. Direction 1 is assumed to be the fiber direction. The reserve factor is calculated as follows:

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

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

  Reserve factor is given by:

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

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

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

  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 ${\sigma }_{22}>0$, ply passes the failure criterion if the following condition is satisfied:

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

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

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

  A quadratic equation is solved to obtain the corresponding reserve factor.

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

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

  $$c=-1,$$

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

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 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 “Hashin” criterion in FeResPost. Correspondingly, version “Hashin_b” is defined, in which only the fiber failure is checked, and version “Hashin_c”, in which only the matrix failure is checked.