(5.3.12) (5.3.13) (5.3.14) and (5.3.15) With these stress invariants and stress state the gradient along the boundary for stress region #2 is (5.3.16) Utilizing equations (5.3.9) and (5.3.16) in equation (5.3.3) then at the boundary between stress region #1 and stress region #2 (5.3.17) The following three expressions can be extracted from equation (5.3.19), i.e., (5.3.18) (5.3.19) and (5.3.20) From equations (5.3.18) and (5.3.20) it is apparent that (5.3.21) From equations (5.3.19) and (5.3.18) it can be shown that (5.3.22) For the stress state in region #3 given above the unit principle stress vector is (5.3.23) Thus
(5.3.12) (5.3.13) (5.3.14) and (5.3.15) With these stress invariants and stress state the gradient along the boundary for stress region #2 is (5.3.16) Utilizing equations (5.3.9) and (5.3.16) in equation (5.3.3) then at the boundary between stress region #1 and stress region #2 (5.3.17) The following three expressions can be extracted from equation (5.3.19), i.e., (5.3.18) (5.3.19) and (5.3.20) From equations (5.3.18) and (5.3.20) it is apparent that (5.3.21) From equations (5.3.19) and (5.3.18) it can be shown that (5.3.22) For the stress state in region #3 given above the unit principle stress vector is (5.3.23) Thus