Hydrostatic stress
= Hydrostatic stress
{title2=$\boldsymbol{\sigma}=-p\mathbf I$}
With tension taken positive, <hydrostatic stress> is the <stress tensor> $\boldsymbol{\sigma}=-p\mathbf I$, where $\mathbf I$ is the <identity matrix>. Every plane has the same compressive normal <stress>, and no tangential traction. More generally,
$$
p=-\frac13\operatorname{tr}\boldsymbol{\sigma},\qquad
\boldsymbol{\sigma}=-p\mathbf I+\mathbf s,\qquad
\operatorname{tr}\mathbf s=0,
$$
where $\mathbf s$ is <deviatoric stress>. Two samples can share the same mean <pressure> while experiencing different <deviatoric stresses>. A stationary <liquid> cannot sustain equilibrium shear <stress>; a <solid> transmitting medium can support it. Consequently, a load divided by <surface area> does not by itself establish hydrostatic sample conditions. A transmitting medium's state at the actual <pressure> and <temperature>, and contact between sample and walls, matter.
Bridgman's Figure 4 compares mild <steel> fractured in tension at <atmospheric pressure> with <steel> stretched further in a <liquid> at $25000\,\mathrm{kgf\,cm^{-2}}$.<General Survey of Certain Results in the Field of High-Pressure Physics by Percy Williams Bridgman>[p. 56, Figure 4 caption: atmospheric-pressure tensile fracture versus greater reduction of area in pressurized liquid; historical kg/cm² rendered explicitly as force per area]{ref} The imposed tension remains a directional load: the experiment combines confinement with deformation, rather than measuring a <hydrostatic stress> alone. Its comparison motivates recording the loading path and the surrounding medium as well as <pressure>; it does not establish a universal fracture threshold for all <steels>.