Table 2 3D Hashin-type and Maximum stress failure criteria
From: 3D Progressive Damage Based Macro-Mechanical FE Simulation of Machining Unidirectional FRP Composite
Failure mode | 3D Hashin-type | 3D maximum stress |
|---|---|---|
Fiber breakage \((\sigma_{11} \ge 0)\) | \(f_{{ 1 {\text{T}}}} \, = \,\left( {\frac{{\sigma_{11} }}{{X_{\text{T}} }}} \right)^{2}\) | \(f_{{ 1 {\text{T}}}} \, = \,\left( {\frac{{\sigma_{11} }}{{X_{\text{T}} }}} \right)^{2}\) |
Fiber crushing \((\sigma_{11} \, < \,0)\) | \(f_{{1{\text{C}}}} = \left( {\frac{{\sigma_{11} }}{{X_{\text{C}} }}} \right)^{2}\) | \(f_{{1{\text{C}}}} = \left( {\frac{{\sigma_{11} }}{{X_{\text{C}} }}} \right)^{2}\) |
Matrix cracking in Dir. 2 \((\sigma_{22} \, \ge \,0)\) | \(f_{{2{\text{T}}}} \, = \,\left( {\frac{{\sigma_{22} }}{{Y_{\text{T}} }}} \right)^{2} \, + \,\left( {\frac{{\sigma_{12} }}{{S_{12} }}} \right)^{2} \, + \,\left( {\frac{{\sigma_{23} }}{{S_{23} }}} \right)^{2}\) | \(f_{{ 2 {\text{T}}}} \, = \,\left( {\frac{{\sigma_{22} }}{{Y_{\text{T}} }}} \right)^{2}\) |
Matrix crushing in Dir. 2 \((\sigma_{22} \, < \,0)\) | \(f_{{ 2 {\text{C}}}} \, = \,\left( {\frac{{\sigma_{22} }}{{Y_{\text{C}} }}} \right)^{2} \, + \,\left( {\frac{{\sigma_{12} }}{{S_{12} }}} \right)^{2} \, + \,\left( {\frac{{\sigma_{23} }}{{S_{23} }}} \right)^{2}\) | \(f_{{ 2 {\text{C}}}} \, = \,\left( {\frac{{\sigma_{22} }}{{Y_{\text{C}} }}} \right)^{2}\) |
Matrix cracking in Dir. 3 \((\sigma_{33} \, \ge \,0)\) | \(f_{{ 3 {\text{T}}}} \, = \,\left( {\frac{{\sigma_{33} }}{{Z_{\text{T}} }}} \right)^{2} \, + \,\left( {\frac{{\sigma_{13} }}{{S_{13} }}} \right)^{2} \, + \,\left( {\frac{{\sigma_{23} }}{{S_{23} }}} \right)^{2}\) | \(f_{{ 3 {\text{T}}}} \, = \,\left( {\frac{{\sigma_{33} }}{{Z_{\text{T}} }}} \right)^{2}\) |
Matrix crushing in Dir. 3 \((\sigma_{33} \, < \,0)\) | \(f_{{ 3 {\text{C}}}} \, = \,\left( {\frac{{\sigma_{33} }}{{Z_{\text{C}} }}} \right)^{2} \, + \,\left( {\frac{{\sigma_{13} }}{{S_{13} }}} \right)^{2} \, + \,\left( {\frac{{\sigma_{23} }}{{S_{23} }}} \right)^{2}\) | \(f_{{ 3 {\text{C}}}} \, = \,\left( {\frac{{\sigma_{33} }}{{Z_{\text{C}} }}} \right)^{2}\) |
Shear failure in 1–2 plane | N/A | \(f_{12} \, = \,\left( {\frac{{\sigma_{12} }}{{S_{12} }}} \right)^{2}\) |
Shear failure in 1–3 plane | N/A | \(f_{13} \, = \,\left( {\frac{{\sigma_{13} }}{{S_{13} }}} \right)^{2}\) |
Shear failure in 2–3 plane | N/A | \(f_{23} \, = \,\left( {\frac{{\sigma_{23} }}{{S_{23} }}} \right)^{2}\) |