Octahedral stresses Cauchy stress tensor
figure 6. octahedral stress planes
considering principal directions coordinate axes, plane normal vector makes equal angles each of principal axes (i.e. having direction cosines equal
|
1
/
3
|
{\displaystyle |1/{\sqrt {3}}|}
) called octahedral plane. there total of 8 octahedral planes (figure 6). normal , shear components of stress tensor on these planes called octahedral normal stress
σ
o
c
t
{\displaystyle \sigma _{\mathrm {oct} }}
, octahedral shear stress
τ
o
c
t
{\displaystyle \tau _{\mathrm {oct} }}
, respectively. octahedral plane passing through origin known π-plane (π not confused mean stress denoted π in above section) . on π-plane,
s
i
j
=
i
/
3
{\displaystyle s_{ij}=i/3}
.
knowing stress tensor of point o (figure 6) in principal axes is
σ
i
j
=
[
σ
1
0
0
0
σ
2
0
0
0
σ
3
]
{\displaystyle \sigma _{ij}={\begin{bmatrix}\sigma _{1}&0&0\\0&\sigma _{2}&0\\0&0&\sigma _{3}\end{bmatrix}}}
the stress vector on octahedral plane given by:
t
o
c
t
(
n
)
=
σ
i
j
n
i
e
j
=
σ
1
n
1
e
1
+
σ
2
n
2
e
2
+
σ
3
n
3
e
3
=
1
3
(
σ
1
e
1
+
σ
2
e
2
+
σ
3
e
3
)
{\displaystyle {\begin{aligned}\mathbf {t} _{\mathrm {oct} }^{(\mathbf {n} )}&=\sigma _{ij}n_{i}\mathbf {e} _{j}\\&=\sigma _{1}n_{1}\mathbf {e} _{1}+\sigma _{2}n_{2}\mathbf {e} _{2}+\sigma _{3}n_{3}\mathbf {e} _{3}\\&={\tfrac {1}{\sqrt {3}}}(\sigma _{1}\mathbf {e} _{1}+\sigma _{2}\mathbf {e} _{2}+\sigma _{3}\mathbf {e} _{3})\end{aligned}}}
the normal component of stress vector @ point o associated octahedral plane is
σ
o
c
t
=
t
i
(
n
)
n
i
=
σ
i
j
n
i
n
j
=
σ
1
n
1
n
1
+
σ
2
n
2
n
2
+
σ
3
n
3
n
3
=
1
3
(
σ
1
+
σ
2
+
σ
3
)
=
1
3
i
1
{\displaystyle {\begin{aligned}\sigma _{\mathrm {oct} }&=t_{i}^{(n)}n_{i}\\&=\sigma _{ij}n_{i}n_{j}\\&=\sigma _{1}n_{1}n_{1}+\sigma _{2}n_{2}n_{2}+\sigma _{3}n_{3}n_{3}\\&={\tfrac {1}{3}}(\sigma _{1}+\sigma _{2}+\sigma _{3})={\tfrac {1}{3}}i_{1}\end{aligned}}}
which mean normal stress or hydrostatic stress. value same in 8 octahedral planes. shear stress on octahedral plane then
τ
o
c
t
=
t
i
(
n
)
t
i
(
n
)
−
σ
n
2
=
[
1
3
(
σ
1
2
+
σ
2
2
+
σ
3
2
)
−
1
9
(
σ
1
+
σ
2
+
σ
3
)
2
]
1
/
2
=
1
3
[
(
σ
1
−
σ
2
)
2
+
(
σ
2
−
σ
3
)
2
+
(
σ
3
−
σ
1
)
2
]
1
/
2
=
1
3
2
i
1
2
−
6
i
2
=
2
3
j
2
{\displaystyle {\begin{aligned}\tau _{\mathrm {oct} }&={\sqrt {t_{i}^{(n)}t_{i}^{(n)}-\sigma _{\mathrm {n} }^{2}}}\\&=\left[{\tfrac {1}{3}}(\sigma _{1}^{2}+\sigma _{2}^{2}+\sigma _{3}^{2})-{\tfrac {1}{9}}(\sigma _{1}+\sigma _{2}+\sigma _{3})^{2}\right]^{1/2}\\&={\tfrac {1}{3}}\left[(\sigma _{1}-\sigma _{2})^{2}+(\sigma _{2}-\sigma _{3})^{2}+(\sigma _{3}-\sigma _{1})^{2}\right]^{1/2}={\tfrac {1}{3}}{\sqrt {2i_{1}^{2}-6i_{2}}}={\sqrt {{\tfrac {2}{3}}j_{2}}}\end{aligned}}}
Comments
Post a Comment