1. Ratings#
\(\sigma\) refers to the stress tensor, arranged as a vector according to the convention:
\(\left\{\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{33}\\ {\sigma }_{12}\\ {\sigma }_{13}\\ {\sigma }_{23}\end{array}\right\}\)
We note:
\({I}_{1}=\text{Trace}(\sigma )\)
\({\sigma }_{H}=\frac{1}{3}\text{tr}(\sigma )\) |
hydrostatic stress |
\(s=\sigma -\frac{1}{3}\text{tr}(\sigma )I\) |
the constraint deviator |
\({\varepsilon }_{H}=\frac{1}{3}\text{tr}(\varepsilon )\) |
volume deformation |
\(\tilde{\varepsilon }=\varepsilon -\frac{1}{3}\text{tr}(\varepsilon )I\) |
the deflection deflector |
\({\dot{\tilde{\varepsilon }}}_{\mathrm{eq}}=\sqrt{\frac{3}{2}\text{trace}(\dot{\tilde{\varepsilon }}\mathrm{.}\dot{\tilde{\varepsilon }})}\) |
the equivalent deformation rate |
\({J}_{2}=\frac{1}{2}\text{trace}({s}^{2})\) |
the second invariant of constraints |
\({\sigma }^{\text{eq}}=\sqrt{{\mathrm{3J}}_{2}}=\sqrt{\frac{3}{2}\text{trace}({s}^{2})}\) |
the equivalent stress |
\({\tau }_{\text{oct}}=\sqrt{\frac{2}{3}{J}_{2}}=\sqrt{\frac{\text{trace}({s}^{2})}{3}}\) |
|
\({\sigma }_{\text{oct}}={\sigma }_{H}=\frac{{I}_{1}}{3}=\frac{\text{trace}(\sigma )}{3}\) |
\({f}_{c}^{\text{'}}\) |
initial break limit in simple compression |
\({f}_{\text{cc}}^{\text{'}}\) |
initial break limit in bi compression |
\(\varphi {f}_{c}^{\text{'}}\) |
elastic limit in compression |
\({f}_{t}^{\text{'}}\) |
initial tensile failure limit |
\(\alpha =\frac{{f}_{t}^{\text{'}}}{{f}_{c}^{\text{'}}}\) |
relationship between tensile and compression failure limit |
\(\beta =\frac{{f}_{\text{cc}}^{\text{'}}}{{f}_{c}^{\text{'}}}\) |
relationship between rupture limit in bi-compression and simple compression |
\({\kappa }_{{}_{t}}^{p}\) |
plastic deformation under tension |
\({\lambda }_{t}\) |
plastic multiplier in traction |
\({\kappa }_{{}_{c}}^{p}\) |
plastic deformation under compression |
\({\lambda }_{c}\) |
plastic multiplier in compression |
\({f}_{c}({\kappa }_{{}_{c}}^{p})\) |
compression work hardening curve |
\({f}_{t}({\kappa }_{{}_{t}}^{p})\) |
tensile work hardening curve |
\({\kappa }_{t}^{u}\) |
ultimate plastic deformation under traction |
\({k}_{c}^{u}\) |
ultimate plastic deformation under compression |
\({G}_{c}^{f}\) |
compression rupture energy (material characteristic) |
\({G}_{t}^{f}\) |
tensile failure energy (material characteristic) |
\(\theta\) |
the maximum temperature during the load history |