Principal Stresses in Two and Three Dimensions
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The principal stresses are the components of the stress tensor when the basis is changed in such a way that the shear stress components become zero. To find the principal stresses in two dimensions, we have to find the angle
θ
{\displaystyle \textstyle \theta }
σ
12
′
=
0
{\displaystyle \textstyle \sigma _{12}^{'}=0}
θ
=
1
2
tan
−
1
(
2
σ
12
σ
11
−
σ
22
)
{\displaystyle \theta ={\cfrac {1}{2}}\tan ^{-1}\left({\frac {2\sigma _{12}}{\sigma _{11}-\sigma _{22}}}\right)}
Plugging
θ
{\displaystyle \textstyle \theta }
transformation equations for stress we get,
σ
1
=
σ
11
+
σ
22
2
+
(
σ
11
−
σ
22
2
)
2
+
σ
12
2
{\displaystyle {\begin{aligned}\sigma _{1}&={\frac {\sigma _{11}+\sigma _{22}}{2}}+{\sqrt {\left({\frac {\sigma _{11}-\sigma _{22}}{2}}\right)^{2}+\sigma _{12}^{2}}}\end{aligned}}}
Where are the shear tractions usually zero in a body?
The principal stresses in three dimensions are a bit more tedious to calculate. They are given by,
σ
1
=
I
1
3
+
2
3
(
I
1
2
−
3
I
2
)
cos
ϕ
σ
2
=
I
1
3
+
2
3
(
I
1
2
−
3
I
2
)
cos
(
ϕ
−
2
π
3
)
σ
3
=
I
1
3
+
2
3
(
I
1
2
−
3
I
2
)
cos
(
ϕ
−
4
π
3
)
{\displaystyle {\begin{aligned}\sigma _{1}&={\frac {I_{1}}{3}}+{\frac {2}{3}}\left({\sqrt {I_{1}^{2}-3I_{2}}}\right)\cos \phi \\\sigma _{2}&={\frac {I_{1}}{3}}+{\frac {2}{3}}\left({\sqrt {I_{1}^{2}-3I_{2}}}\right)\cos \left(\phi -{\frac {2\pi }{3}}\right)\\\sigma _{3}&={\frac {I_{1}}{3}}+{\frac {2}{3}}\left({\sqrt {I_{1}^{2}-3I_{2}}}\right)\cos \left(\phi -{\frac {4\pi }{3}}\right)\end{aligned}}}
where,
ϕ
=
1
3
cos
−
1
(
2
I
1
3
−
9
I
1
I
2
+
27
I
3
2
(
I
1
2
−
3
I
2
)
3
/
2
)
I
1
=
σ
11
+
σ
22
+
σ
33
I
2
=
σ
11
σ
22
+
σ
22
σ
33
+
σ
33
σ
11
−
σ
12
2
−
σ
23
2
−
σ
31
2
I
3
=
σ
11
σ
22
σ
33
−
σ
11
σ
23
2
−
σ
22
σ
31
2
−
σ
33
σ
12
2
+
2
σ
12
σ
23
σ
31
{\displaystyle {\begin{aligned}\phi &={\cfrac {1}{3}}\cos ^{-1}\left({\frac {2I_{1}^{3}-9I_{1}I_{2}+27I_{3}}{2(I_{1}^{2}-3I_{2})^{3/2}}}\right)\\I_{1}&=\sigma _{11}+\sigma _{22}+\sigma _{33}\\I_{2}&=\sigma _{11}\sigma _{22}+\sigma _{22}\sigma _{33}+\sigma _{33}\sigma _{11}-\sigma _{12}^{2}-\sigma _{23}^{2}-\sigma _{31}^{2}\\I_{3}&=\sigma _{11}\sigma _{22}\sigma _{33}-\sigma _{11}\sigma _{23}^{2}-\sigma _{22}\sigma _{31}^{2}-\sigma _{33}\sigma _{12}^{2}+2\sigma _{12}\sigma _{23}\sigma _{31}\end{aligned}}}
The quantities
I
1
,
I
2
,
I
3
{\displaystyle \textstyle I_{1},I_{2},I_{3}}
stress invariants .
Note: Be careful while implementing above relations in a solver, as the value of:
2
I
1
3
−
9
I
1
I
2
+
27
I
3
2
(
I
1
2
−
3
I
2
)
3
/
2
{\displaystyle {\frac {2I_{1}^{3}-9I_{1}I_{2}+27I_{3}}{2(I_{1}^{2}-3I_{2})^{3/2}}}}
can be out of range of
cos
−
1
{\displaystyle \cos ^{-1}}
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