= Hydraulic opening of a mineral vein
{title2=$P_f>\sigma_n+T_0$}
A fluid-filled fracture can open perpendicular to the least compressive <principal stress>. In a compression-positive contact approximation, <effective stress> is $\sigma_n'=\sigma_n-P_f$. Opening an initially intact plane of <tensile strength> $T_0$ requires
$$
\sigma_n-P_f<-T_0.
$$
A sufficiently weak pre-existing crack can open near zero effective compression. Increasing <isotropic> <pore fluid pressure> translates all effective <principal stresses> without rotating their directions, so an opening fracture and a frictional <normal fault> can record compatible stress axes at different <pressure> episodes. Subsequent mineral precipitation creates a <mineral vein>; white appearance alone does not identify its filling mineral.
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