Stress triaxiality information

In continuum mechanics, stress triaxiality is the relative degree of hydrostatic stress in a given stress state.^[1] It is often used as a triaxiality factor, T.F, which is the ratio of the hydrostatic stress, ${\displaystyle \sigma _{m}}$, to the Von Mises equivalent stress, ${\displaystyle \sigma _{eq}}$.^[2][3][4]

${\displaystyle T.F.={\frac {\sigma _{m}}{\sigma _{eq}}}={\frac {{\frac {1}{3}}(\sigma _{1}+\sigma _{2}+\sigma _{3})}{\sqrt {\frac {(\sigma _{1}-\sigma _{2})^{2}+(\sigma _{2}-\sigma _{3})^{2}+(\sigma _{3}-\sigma _{1})^{2}}{2}}}}={\frac {{\frac {1}{3}}(\sigma _{11}+\sigma _{22}+\sigma _{33})}{\sqrt {\frac {(\sigma _{11}-\sigma _{22})^{2}+(\sigma _{22}-\sigma _{33})^{2}+(\sigma _{33}-\sigma _{11})^{2}+6(\sigma _{12}^{2}+\sigma _{23}^{2}+\sigma _{31}^{2})}{2}}}}}$

Stress triaxiality has important applications in fracture mechanics and can often be used to predict the type of fracture (i.e. ductile or brittle) within the region defined by that stress state. A higher stress triaxiality corresponds to a stress state which is primarily hydrostatic rather than deviatoric. High stress triaxiality (> 2–3) promotes brittle cleavage fracture^[3] as well as dimple formation within an otherwise ductile fracture.^[1][5] Low stress triaxiality corresponds with shear slip and therefore larger ductility,^[5] as well as typically resulting in greater toughness.^[6] Ductile crack propagation is also influenced by stress triaxiality, with lower values producing steeper crack resistance curves.^[7] Several failure models such as the Johnson-Cook (J-C) fracture criterion (often used for high strain rate behavior),^[8] Rice-Tracey model, and J-Q large scale yielding model incorporate stress triaxiality.

History

In 1959 Davies and Connelly introduced so called triaxiality factor, defined as the ratio of Cauchy stress first principal invariant divided by effective stress ${\displaystyle {{\eta }_{DC}}\equiv 3\,{{\sigma }_{m}}/{{\sigma }_{ef}}={{I}_{1}}/{\sqrt {3{{J}_{2}}}}}$, cf. formula (35) in Davies and Conelly (1959).^[9] The ${\displaystyle {{I}_{1}}\equiv {{\sigma }_{I}}+{{\sigma }_{II}}+{{\sigma }_{III}}}$ denotes first invariant of Cauchy stress tensor, ${\displaystyle {{\sigma }_{I}},{{\sigma }_{II}},{{\sigma }_{III}}}$ denote principal values of Cauchy stress, ${\displaystyle {{\sigma }_{m}}={\tfrac {1}{3}}{{I}_{\,1}}}$ denotes mean stress, ${\displaystyle {{J}_{2}}\equiv {\tfrac {1}{2}}{{s}_{ij}}{{s}_{ij}}={\tfrac {1}{2}}({{s}_{I}}^{2}+{{s}_{II}}^{2}+{{s}_{III}}^{2})}$ is second invariant of Cauchy stress deviator, ${\displaystyle {{s}_{I}},{{s}_{II}},{{s}_{III}}}$ denote principal values of Cauchy stress deviator, ${\displaystyle {{\sigma }_{ef}}\equiv {\sqrt {3{{J}_{2}}}}}$ denotes effective stress.

Davies and Conelly were motivated in this proposal by supposition, correct in view of their own and later research, that negative pressure (spherical tension) ${\displaystyle -p\equiv {{\sigma }_{m}}}$ called by them rather exotically triaxial tension, has a strong influence on the loss of ductility of metals, and the need to have some parameter to describe this effect.

Wierzbicki and collaborators adopted a slightly modified definition of triaxiality factor than the original one ${\displaystyle \eta \equiv \,{{\sigma }_{m}}/{{\sigma }_{ef}}\in <-\infty ,\ \infty >}$, ${\displaystyle \eta ={{\eta }_{DC}}/3\ }$, cf. e.g. Wierzbicki et al (2005).^[10]

The name triaxiality factor is rather unfortunate, inadequate, because in physical terms the triaxiality factor determines the calibrated ratio of pressure forces relative to shearing forces or the ratio of isotropic (spherical) part of stress tensor in relation to its anisotropic (deviatoric) part both expressed in terms of their moduli, ${\displaystyle \eta =({\sqrt {2}}/3)||{{\boldsymbol {\sigma }}^{\,sph}}||/||\mathbf {s} ||}$; ${\displaystyle ||{{\boldsymbol {\sigma }}^{\,sph}}||\ ={\sqrt {3}}\,{{\sigma }_{m}}}$, ${\displaystyle ||\mathbf {s} ||\ ={\sqrt {2{{J}_{\,2}}}}}$.

The triaxiality factor does not discern triaxial stress states from states of lower dimension.

Ziółkowski proposed to use as a measure of pressure towards shearing forces another modification of the index ${\displaystyle \eta }$, not burdened with whatever strength effort hypothesis, in the form ${\displaystyle {{\eta }_{\,i}}\equiv \ ||{{\boldsymbol {\sigma }}^{\,sph}}||/||\mathbf {s} ||\ \in <-\infty ,\ \infty >}$, cf. formula (8.2) in Ziółkowski (2022).^[11] In the context of material testing a reasonable mnemonic name for ${\displaystyle {{\eta }_{\,i}}}$ could be, e.g. pressure index or pressure factor.