II.1.10.20 Hashin criteria

This criterion is meant to be used for uni-directional materials. Direction 1 is assumed to be direction of fibers. One first presents the way the reserve factor is calculated:

• 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:

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

  one has simply:

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

  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 one keeps the value of safety factor.

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