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