δσij is 2nd order tensor represented by a 3x3 matrix of form:
with six independent stress parameters (σ12 = σ21, σ13 = σ31, σ23 = σ32) and
σm = (σ11 + σ22 + σ33)/3
In a triaxial test it is possible to arrange the stress components in a way that the diagonal stress components σ11, σ22 and σ33 are oriented parallel to the applied Cartesian coordinate system. Consequently, these three stress components become the principal stresses σ1, σ2 and σ3 and the off-diagonal components equal zero. Therefore, in a laboratory situation the deviatoric stress tensor [δσij] can be simplified to:
According to Engelder (1994) deviatoric stress is very often confused with other types of stresses such as differential stress and effective stress. Engelder (1994) recommends not to use the term “deviatoric stress” unless for fault slip problems.
ΔS is the difference between the maximum principal stress S1 and the minimum principal stress S3.
ΔS = S1 - S3
is also the diameter of a Mohr circle.
σij (with i=j and the main principle stresses σ1 ≥ σ2 ≥ σ3) is the difference between the applied external stress Sij and the pore pressure Pp. In its simple form (Terzaghi 1923):
σij = Sij - δijPp
it denies the importance of pore volume and its compressibility by disregarding the Biot coefficient α. This equaition is applied generally in soil mechanics where full efficacy of pore pressure (i.e. α = 1) can be assumed.
In case of solid rock things appear more complicated and pore pressure efficacy never is at 100%. Consequently, the Biot coefficient has to be considered and the exact form of the equation according to Nur & Byerlee 1971 is:
σij = Sij - δijαPp.
See "Threshold pressure"