2s33 S11 S22

The deviatoric stress tensor represents a state of pure shear. It is obtained by subtracting the mean stress from the three normal stresses (cn, ff22, and a33) in a stress tensor. It is important from the viewpoint of inelastic analysis because experiments have shown that inelastic behavior of most ductile materials is independent of the mean normal stress, but is related primarily to the deviatoric stress.

If the indicial notation s,j is used to represent the nine deviatoric stress components given in Equation 1.19, the maximum deviatoric stress acting on each of the three orthogonal planes (which are the same as the principal planes) can be computed from the cubic equation s3 — /1s2 — /2s — J = 0 where /j, /2, and /3 are the first, second, and third deviatoric stress invariants given by

J2 = 22s;jsj; = 1(s21 + s22 + s33 + 2s22 + 2s23 + 2s23)

J2 = 22s;jsj; = 1(s21 + s22 + s33 + 2s22 + 2s23 + 2s23)

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