Transformation rule of the stress tensor Cauchy stress tensor
figure 2.4 transformation of stress tensor
expanding matrix operation, , simplifying terms using symmetry of stress tensor, gives
σ
11
′
=
a
11
2
σ
11
+
a
12
2
σ
22
+
a
13
2
σ
33
+
2
a
11
a
12
σ
12
+
2
a
11
a
13
σ
13
+
2
a
12
a
13
σ
23
,
{\displaystyle \sigma _{11} =a_{11}^{2}\sigma _{11}+a_{12}^{2}\sigma _{22}+a_{13}^{2}\sigma _{33}+2a_{11}a_{12}\sigma _{12}+2a_{11}a_{13}\sigma _{13}+2a_{12}a_{13}\sigma _{23},}
σ
22
′
=
a
21
2
σ
11
+
a
22
2
σ
22
+
a
23
2
σ
33
+
2
a
21
a
22
σ
12
+
2
a
21
a
23
σ
13
+
2
a
22
a
23
σ
23
,
{\displaystyle \sigma _{22} =a_{21}^{2}\sigma _{11}+a_{22}^{2}\sigma _{22}+a_{23}^{2}\sigma _{33}+2a_{21}a_{22}\sigma _{12}+2a_{21}a_{23}\sigma _{13}+2a_{22}a_{23}\sigma _{23},}
σ
33
′
=
a
31
2
σ
11
+
a
32
2
σ
22
+
a
33
2
σ
33
+
2
a
31
a
32
σ
12
+
2
a
31
a
33
σ
13
+
2
a
32
a
33
σ
23
,
{\displaystyle \sigma _{33} =a_{31}^{2}\sigma _{11}+a_{32}^{2}\sigma _{22}+a_{33}^{2}\sigma _{33}+2a_{31}a_{32}\sigma _{12}+2a_{31}a_{33}\sigma _{13}+2a_{32}a_{33}\sigma _{23},}
σ
12
′
=
a
11
a
21
σ
11
+
a
12
a
22
σ
22
+
a
13
a
23
σ
33
+
(
a
11
a
22
+
a
12
a
21
)
σ
12
+
(
a
12
a
23
+
a
13
a
22
)
σ
23
+
(
a
11
a
23
+
a
13
a
21
)
σ
13
,
{\displaystyle {\begin{aligned}\sigma _{12} =&a_{11}a_{21}\sigma _{11}+a_{12}a_{22}\sigma _{22}+a_{13}a_{23}\sigma _{33}\\&+(a_{11}a_{22}+a_{12}a_{21})\sigma _{12}+(a_{12}a_{23}+a_{13}a_{22})\sigma _{23}+(a_{11}a_{23}+a_{13}a_{21})\sigma _{13},\end{aligned}}}
σ
23
′
=
a
21
a
31
σ
11
+
a
22
a
32
σ
22
+
a
23
a
33
σ
33
+
(
a
21
a
32
+
a
22
a
31
)
σ
12
+
(
a
22
a
33
+
a
23
a
32
)
σ
23
+
(
a
21
a
33
+
a
23
a
31
)
σ
13
,
{\displaystyle {\begin{aligned}\sigma _{23} =&a_{21}a_{31}\sigma _{11}+a_{22}a_{32}\sigma _{22}+a_{23}a_{33}\sigma _{33}\\&+(a_{21}a_{32}+a_{22}a_{31})\sigma _{12}+(a_{22}a_{33}+a_{23}a_{32})\sigma _{23}+(a_{21}a_{33}+a_{23}a_{31})\sigma _{13},\end{aligned}}}
σ
13
′
=
a
11
a
31
σ
11
+
a
12
a
32
σ
22
+
a
13
a
33
σ
33
+
(
a
11
a
32
+
a
12
a
31
)
σ
12
+
(
a
12
a
33
+
a
13
a
32
)
σ
23
+
(
a
11
a
33
+
a
13
a
31
)
σ
13
.
{\displaystyle {\begin{aligned}\sigma _{13} =&a_{11}a_{31}\sigma _{11}+a_{12}a_{32}\sigma _{22}+a_{13}a_{33}\sigma _{33}\\&+(a_{11}a_{32}+a_{12}a_{31})\sigma _{12}+(a_{12}a_{33}+a_{13}a_{32})\sigma _{23}+(a_{11}a_{33}+a_{13}a_{31})\sigma _{13}.\end{aligned}}}
the mohr circle stress graphical representation of transformation of stresses.
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