Given:
A monoclinic crystal has inversion symmetric about the
e
^
1
{\displaystyle {\widehat {\mathbf {e} }}_{1}}
-
e
^
2
{\displaystyle {\widehat {\mathbf {e} }}_{2}}
plane. Therefore, the material properties do not change for a mirror-reflection through this plane. The stress-strain relations must therefore remain unchanged under this transformation. The transformation matrix
[
L
]
{\displaystyle \left[L\right]\,}
for this
for the mirror inversion is given by
[
L
]
=
[
1
0
0
0
1
0
0
0
−
1
]
{\displaystyle \left[L\right]={\begin{bmatrix}1&0&0\\0&1&0\\0&0&-1\end{bmatrix}}}
Show:
If we apply this transformation to the stress and strain tensors, then the stiffness matrix of the material (in Voigt notation) is
[
C
]
=
[
C
11
C
12
C
13
0
0
C
16
C
21
C
22
C
23
0
0
C
26
C
31
C
32
C
33
0
0
C
36
0
0
0
C
44
C
45
0
0
0
0
C
54
C
55
0
C
61
C
62
C
63
0
0
C
66
]
{\displaystyle \left[C\right]={\begin{bmatrix}C_{11}&C_{12}&C_{13}&0&0&C_{16}\\C_{21}&C_{22}&C_{23}&0&0&C_{26}\\C_{31}&C_{32}&C_{33}&0&0&C_{36}\\0&0&0&C_{44}&C_{45}&0\\0&0&0&C_{54}&C_{55}&0\\C_{61}&C_{62}&C_{63}&0&0&C_{66}\\\end{bmatrix}}}
In 3
×
{\displaystyle \times }
3 matrix form, the strain tensor is given by
ε
=
[
ε
11
ε
12
ε
13
ε
21
ε
22
ε
23
ε
31
ε
32
ε
33
]
{\displaystyle {\boldsymbol {\varepsilon }}={\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}\end{bmatrix}}}
The transformation rule for a second order tensor
A
{\displaystyle {\boldsymbol {A}}\,}
is
[
A
]
′
=
[
L
]
[
A
]
[
L
]
T
{\displaystyle \left[A\right]^{'}=\left[L\right]\left[A\right]\left[L\right]^{T}}
Applying this transformation to the strain tensor, we have
[
ε
11
′
ε
12
′
ε
13
′
ε
21
′
ε
22
′
ε
23
′
ε
31
′
ε
32
′
ε
33
′
]
=
[
1
0
0
0
1
0
0
0
−
1
]
[
ε
11
ε
12
ε
13
ε
21
ε
22
ε
23
ε
31
ε
32
ε
33
]
[
1
0
0
0
1
0
0
0
−
1
]
=
[
1
0
0
0
1
0
0
0
−
1
]
[
ε
11
ε
12
−
ε
13
ε
21
ε
22
−
ε
23
ε
31
ε
32
−
ε
33
]
=
[
ε
11
ε
12
−
ε
13
ε
21
ε
22
−
ε
23
−
ε
31
−
ε
32
ε
33
]
{\displaystyle {\begin{aligned}{\begin{bmatrix}\varepsilon _{11}^{'}&\varepsilon _{12}^{'}&\varepsilon _{13}^{'}\\\varepsilon _{21}^{'}&\varepsilon _{22}^{'}&\varepsilon _{23}^{'}\\\varepsilon _{31}^{'}&\varepsilon _{32}^{'}&\varepsilon _{33}^{'}\end{bmatrix}}&={\begin{bmatrix}1&0&0\\0&1&0\\0&0&-1\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}\end{bmatrix}}{\begin{bmatrix}1&0&0\\0&1&0\\0&0&-1\end{bmatrix}}\\&={\begin{bmatrix}1&0&0\\0&1&0\\0&0&-1\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&-\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&-\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&-\varepsilon _{33}\end{bmatrix}}\\&={\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&-\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&-\varepsilon _{23}\\-\varepsilon _{31}&-\varepsilon _{32}&\varepsilon _{33}\end{bmatrix}}\end{aligned}}}
In engineering notation (Voigt notation),
[
ε
]
=
[
ε
11
ε
22
ε
33
2
ε
23
2
ε
31
2
ε
12
]
T
=
[
ε
1
ε
2
ε
3
ε
4
ε
5
ε
6
]
T
{\displaystyle {\begin{aligned}\left[{\boldsymbol {\varepsilon }}\right]&={\begin{bmatrix}\varepsilon _{11}&\varepsilon _{22}&\varepsilon _{33}&2\varepsilon _{23}&2\varepsilon _{31}&2\varepsilon _{12}\end{bmatrix}}^{T}\\&={\begin{bmatrix}\varepsilon _{1}&\varepsilon _{2}&\varepsilon _{3}&\varepsilon _{4}&\varepsilon _{5}&\varepsilon _{6}\end{bmatrix}}^{T}\end{aligned}}}
Therefore, the transformed strain tensor can be written as
[
ε
]
′
=
[
ε
1
′
ε
2
′
ε
3
′
ε
4
′
ε
5
′
ε
6
′
]
T
=
[
ε
1
ε
2
ε
3
−
ε
4
−
ε
5
ε
6
]
T
{\displaystyle {\begin{aligned}\left[{\boldsymbol {\varepsilon }}\right]^{'}&={\begin{bmatrix}\varepsilon _{1}^{'}&\varepsilon _{2}^{'}&\varepsilon _{3}^{'}&\varepsilon _{4}^{'}&\varepsilon _{5}^{'}&\varepsilon _{6}^{'}\end{bmatrix}}^{T}\\&={\begin{bmatrix}\varepsilon _{1}&\varepsilon _{2}&\varepsilon _{3}&-\varepsilon _{4}&-\varepsilon _{5}&\varepsilon _{6}\end{bmatrix}}^{T}\\\end{aligned}}}
The expression for the strain energy density of a linear elastic material imposes a constraint on the components of the stiffness tensor in the presence of planes of material symmetry. This constraint is
C
i
j
(
ε
i
ε
j
−
ε
i
′
ε
j
′
)
=
0
(
i
,
j
=
1
…
6
)
{\displaystyle C_{ij}(\varepsilon _{i}\varepsilon _{j}-\varepsilon _{i}^{'}\varepsilon _{j}^{'})=0~~~~(i,j=1\dots 6)}
where
C
i
j
{\displaystyle C_{ij}}
are the components of the 6
×
{\displaystyle \times }
6 matrix that contains the independent components of the stiffness tensor.
Therefore,
C
11
(
ε
1
ε
1
−
ε
1
′
ε
1
′
)
+
C
12
(
ε
1
ε
2
−
ε
1
′
ε
2
′
)
+
C
13
(
ε
1
ε
3
−
ε
1
′
ε
3
′
)
+
C
14
(
ε
1
ε
4
−
ε
1
′
ε
4
′
)
+
C
15
(
ε
1
ε
5
−
ε
1
′
ε
5
′
)
+
C
16
(
ε
1
ε
6
−
ε
1
′
ε
6
′
)
+
C
21
(
ε
2
ε
1
−
ε
2
′
ε
1
′
)
+
C
22
(
ε
2
ε
2
−
ε
2
′
ε
2
′
)
+
C
23
(
ε
2
ε
3
−
ε
2
′
ε
3
′
)
+
C
24
(
ε
2
ε
4
−
ε
2
′
ε
4
′
)
+
C
25
(
ε
2
ε
5
−
ε
2
′
ε
5
′
)
+
C
26
(
ε
2
ε
6
−
ε
2
′
ε
6
′
)
+
C
31
(
ε
3
ε
1
−
ε
3
′
ε
1
′
)
+
C
32
(
ε
3
ε
2
−
ε
3
′
ε
2
′
)
+
C
33
(
ε
3
ε
3
−
ε
3
′
ε
3
′
)
+
C
34
(
ε
3
ε
4
−
ε
3
′
ε
4
′
)
+
C
35
(
ε
3
ε
5
−
ε
3
′
ε
5
′
)
+
C
36
(
ε
3
ε
6
−
ε
3
′
ε
6
′
)
+
C
41
(
ε
4
ε
1
−
ε
4
′
ε
1
′
)
+
C
42
(
ε
4
ε
2
−
ε
4
′
ε
2
′
)
+
C
43
(
ε
4
ε
3
−
ε
4
′
ε
3
′
)
+
C
44
(
ε
4
ε
4
−
ε
4
′
ε
4
′
)
+
C
45
(
ε
4
ε
5
−
ε
4
′
ε
5
′
)
+
C
46
(
ε
4
ε
6
−
ε
4
′
ε
6
′
)
+
C
51
(
ε
5
ε
1
−
ε
5
′
ε
1
′
)
+
C
52
(
ε
5
ε
2
−
ε
5
′
ε
2
′
)
+
C
53
(
ε
5
ε
3
−
ε
5
′
ε
3
′
)
+
C
54
(
ε
5
ε
4
−
ε
5
′
ε
4
′
)
+
C
55
(
ε
5
ε
5
−
ε
5
′
ε
5
′
)
+
C
56
(
ε
5
ε
6
−
ε
5
′
ε
6
′
)
+
C
61
(
ε
6
ε
1
−
ε
6
′
ε
1
′
)
+
C
62
(
ε
6
ε
2
−
ε
6
′
ε
2
′
)
+
C
63
(
ε
6
ε
3
−
ε
6
′
ε
3
′
)
+
C
64
(
ε
6
ε
4
−
ε
6
′
ε
4
′
)
+
C
65
(
ε
6
ε
5
−
ε
6
′
ε
5
′
)
+
C
66
(
ε
6
ε
6
−
ε
6
′
ε
6
′
)
=
0
{\displaystyle {\begin{aligned}C_{11}(\varepsilon _{1}\varepsilon _{1}-\varepsilon _{1}^{'}\varepsilon _{1}^{'})+C_{12}(\varepsilon _{1}\varepsilon _{2}-\varepsilon _{1}^{'}\varepsilon _{2}^{'})+C_{13}(\varepsilon _{1}\varepsilon _{3}-\varepsilon _{1}^{'}\varepsilon _{3}^{'})&+\\C_{14}(\varepsilon _{1}\varepsilon _{4}-\varepsilon _{1}^{'}\varepsilon _{4}^{'})+C_{15}(\varepsilon _{1}\varepsilon _{5}-\varepsilon _{1}^{'}\varepsilon _{5}^{'})+C_{16}(\varepsilon _{1}\varepsilon _{6}-\varepsilon _{1}^{'}\varepsilon _{6}^{'})&+\\C_{21}(\varepsilon _{2}\varepsilon _{1}-\varepsilon _{2}^{'}\varepsilon _{1}^{'})+C_{22}(\varepsilon _{2}\varepsilon _{2}-\varepsilon _{2}^{'}\varepsilon _{2}^{'})+C_{23}(\varepsilon _{2}\varepsilon _{3}-\varepsilon _{2}^{'}\varepsilon _{3}^{'})&+\\C_{24}(\varepsilon _{2}\varepsilon _{4}-\varepsilon _{2}^{'}\varepsilon _{4}^{'})+C_{25}(\varepsilon _{2}\varepsilon _{5}-\varepsilon _{2}^{'}\varepsilon _{5}^{'})+C_{26}(\varepsilon _{2}\varepsilon _{6}-\varepsilon _{2}^{'}\varepsilon _{6}^{'})&+\\C_{31}(\varepsilon _{3}\varepsilon _{1}-\varepsilon _{3}^{'}\varepsilon _{1}^{'})+C_{32}(\varepsilon _{3}\varepsilon _{2}-\varepsilon _{3}^{'}\varepsilon _{2}^{'})+C_{33}(\varepsilon _{3}\varepsilon _{3}-\varepsilon _{3}^{'}\varepsilon _{3}^{'})&+\\C_{34}(\varepsilon _{3}\varepsilon _{4}-\varepsilon _{3}^{'}\varepsilon _{4}^{'})+C_{35}(\varepsilon _{3}\varepsilon _{5}-\varepsilon _{3}^{'}\varepsilon _{5}^{'})+C_{36}(\varepsilon _{3}\varepsilon _{6}-\varepsilon _{3}^{'}\varepsilon _{6}^{'})&+\\C_{41}(\varepsilon _{4}\varepsilon _{1}-\varepsilon _{4}^{'}\varepsilon _{1}^{'})+C_{42}(\varepsilon _{4}\varepsilon _{2}-\varepsilon _{4}^{'}\varepsilon _{2}^{'})+C_{43}(\varepsilon _{4}\varepsilon _{3}-\varepsilon _{4}^{'}\varepsilon _{3}^{'})&+\\C_{44}(\varepsilon _{4}\varepsilon _{4}-\varepsilon _{4}^{'}\varepsilon _{4}^{'})+C_{45}(\varepsilon _{4}\varepsilon _{5}-\varepsilon _{4}^{'}\varepsilon _{5}^{'})+C_{46}(\varepsilon _{4}\varepsilon _{6}-\varepsilon _{4}^{'}\varepsilon _{6}^{'})&+\\C_{51}(\varepsilon _{5}\varepsilon _{1}-\varepsilon _{5}^{'}\varepsilon _{1}^{'})+C_{52}(\varepsilon _{5}\varepsilon _{2}-\varepsilon _{5}^{'}\varepsilon _{2}^{'})+C_{53}(\varepsilon _{5}\varepsilon _{3}-\varepsilon _{5}^{'}\varepsilon _{3}^{'})&+\\C_{54}(\varepsilon _{5}\varepsilon _{4}-\varepsilon _{5}^{'}\varepsilon _{4}^{'})+C_{55}(\varepsilon _{5}\varepsilon _{5}-\varepsilon _{5}^{'}\varepsilon _{5}^{'})+C_{56}(\varepsilon _{5}\varepsilon _{6}-\varepsilon _{5}^{'}\varepsilon _{6}^{'})&+\\C_{61}(\varepsilon _{6}\varepsilon _{1}-\varepsilon _{6}^{'}\varepsilon _{1}^{'})+C_{62}(\varepsilon _{6}\varepsilon _{2}-\varepsilon _{6}^{'}\varepsilon _{2}^{'})+C_{63}(\varepsilon _{6}\varepsilon _{3}-\varepsilon _{6}^{'}\varepsilon _{3}^{'})&+\\C_{64}(\varepsilon _{6}\varepsilon _{4}-\varepsilon _{6}^{'}\varepsilon _{4}^{'})+C_{65}(\varepsilon _{6}\varepsilon _{5}-\varepsilon _{6}^{'}\varepsilon _{5}^{'})+C_{66}(\varepsilon _{6}\varepsilon _{6}-\varepsilon _{6}^{'}\varepsilon _{6}^{'})&=0\end{aligned}}}
For a monoclinic material, replacing the transformed strain components by the equivalent original strain components, we get
C
14
(
ε
1
ε
4
+
ε
1
ε
4
)
+
C
15
(
ε
1
ε
5
+
ε
1
ε
5
)
+
C
24
(
ε
2
ε
4
+
ε
2
ε
4
)
+
C
25
(
ε
2
ε
5
+
ε
2
ε
5
)
+
C
34
(
ε
3
ε
4
+
ε
3
ε
4
)
+
C
35
(
ε
3
ε
5
+
ε
3
ε
5
)
+
C
41
(
ε
4
ε
1
+
ε
4
ε
1
)
+
C
42
(
ε
4
ε
2
+
ε
4
ε
2
)
+
C
43
(
ε
4
ε
3
+
ε
4
ε
3
)
+
C
46
(
ε
4
ε
6
+
ε
4
ε
6
)
+
C
51
(
ε
5
ε
1
+
ε
5
ε
1
)
+
C
52
(
ε
5
ε
2
+
ε
5
ε
2
)
+
C
53
(
ε
5
ε
3
+
ε
5
ε
3
)
+
C
56
(
ε
5
ε
6
+
ε
5
ε
6
)
+
C
64
(
ε
6
ε
4
+
ε
6
ε
4
)
+
C
65
(
ε
6
ε
5
+
ε
6
ε
5
)
=
0
{\displaystyle {\begin{aligned}C_{14}(\varepsilon _{1}\varepsilon _{4}+\varepsilon _{1}\varepsilon _{4})+C_{15}(\varepsilon _{1}\varepsilon _{5}+\varepsilon _{1}\varepsilon _{5})+C_{24}(\varepsilon _{2}\varepsilon _{4}+\varepsilon _{2}\varepsilon _{4})&+\\C_{25}(\varepsilon _{2}\varepsilon _{5}+\varepsilon _{2}\varepsilon _{5})+C_{34}(\varepsilon _{3}\varepsilon _{4}+\varepsilon _{3}\varepsilon _{4})+C_{35}(\varepsilon _{3}\varepsilon _{5}+\varepsilon _{3}\varepsilon _{5})&+\\C_{41}(\varepsilon _{4}\varepsilon _{1}+\varepsilon _{4}\varepsilon _{1})+C_{42}(\varepsilon _{4}\varepsilon _{2}+\varepsilon _{4}\varepsilon _{2})+C_{43}(\varepsilon _{4}\varepsilon _{3}+\varepsilon _{4}\varepsilon _{3})&+\\C_{46}(\varepsilon _{4}\varepsilon _{6}+\varepsilon _{4}\varepsilon _{6})+C_{51}(\varepsilon _{5}\varepsilon _{1}+\varepsilon _{5}\varepsilon _{1})+C_{52}(\varepsilon _{5}\varepsilon _{2}+\varepsilon _{5}\varepsilon _{2})&+\\C_{53}(\varepsilon _{5}\varepsilon _{3}+\varepsilon _{5}\varepsilon _{3})+C_{56}(\varepsilon _{5}\varepsilon _{6}+\varepsilon _{5}\varepsilon _{6})+C_{64}(\varepsilon _{6}\varepsilon _{4}+\varepsilon _{6}\varepsilon _{4})&+\\C_{65}(\varepsilon _{6}\varepsilon _{5}+\varepsilon _{6}\varepsilon _{5})&=0\end{aligned}}}
or,
2
C
14
ε
1
ε
4
+
2
C
15
ε
1
ε
5
+
2
C
24
ε
2
ε
4
+
2
C
25
ε
2
ε
5
+
2
C
34
ε
3
ε
4
+
2
C
35
ε
3
ε
5
+
2
C
41
ε
4
ε
1
+
2
C
42
ε
4
ε
2
+
2
C
43
ε
4
ε
3
+
2
C
46
ε
4
ε
6
+
2
C
51
ε
5
ε
1
+
2
C
52
ε
5
ε
2
+
2
C
53
ε
5
ε
3
+
2
C
56
ε
5
ε
6
+
2
C
64
ε
6
ε
4
+
2
C
65
ε
6
ε
5
=
0
{\displaystyle {\begin{aligned}2~C_{14}\varepsilon _{1}\varepsilon _{4}+2~C_{15}\varepsilon _{1}\varepsilon _{5}+2~C_{24}\varepsilon _{2}\varepsilon _{4}+2~C_{25}\varepsilon _{2}\varepsilon _{5}+2~C_{34}\varepsilon _{3}\varepsilon _{4}+2~C_{35}\varepsilon _{3}\varepsilon _{5}&+\\2~C_{41}\varepsilon _{4}\varepsilon _{1}+2~C_{42}\varepsilon _{4}\varepsilon _{2}+2~C_{43}\varepsilon _{4}\varepsilon _{3}+2~C_{46}\varepsilon _{4}\varepsilon _{6}+2~C_{51}\varepsilon _{5}\varepsilon _{1}+2~C_{52}\varepsilon _{5}\varepsilon _{2}&+\\2~C_{53}\varepsilon _{5}\varepsilon _{3}+2~C_{56}\varepsilon _{5}\varepsilon _{6}+2~C_{64}\varepsilon _{6}\varepsilon _{4}+2~C_{65}\varepsilon _{6}\varepsilon _{5}&=0\end{aligned}}}
Using the symmetry of the stiffness matrix, we have
4
C
14
ε
1
ε
4
+
4
C
15
ε
1
ε
5
+
4
C
24
ε
2
ε
4
+
4
C
25
ε
2
ε
5
+
4
C
34
ε
3
ε
4
+
4
C
35
ε
3
ε
5
+
4
C
46
ε
4
ε
6
+
4
C
56
ε
5
ε
6
=
0
{\displaystyle {\begin{aligned}4~C_{14}\varepsilon _{1}\varepsilon _{4}+4~C_{15}\varepsilon _{1}\varepsilon _{5}+4~C_{24}\varepsilon _{2}\varepsilon _{4}+4~C_{25}\varepsilon _{2}\varepsilon _{5}+4~C_{34}\varepsilon _{3}\varepsilon _{4}+4~C_{35}\varepsilon _{3}\varepsilon _{5}&+\\4~C_{46}\varepsilon _{4}\varepsilon _{6}+4~C_{56}\varepsilon _{5}\varepsilon _{6}&=0\end{aligned}}}
Since the strains can be arbitrary, the above condition is satisfied only if
C
14
=
C
15
=
C
24
=
C
25
=
C
34
=
C
35
=
C
46
=
C
56
=
0
{\displaystyle C_{14}=C_{15}=C_{24}=C_{25}=C_{34}=C_{35}=C_{46}=C_{56}=0}
Therefore, the stiffness matrix is given by
[
C
]
=
[
C
11
C
12
C
13
0
0
C
16
C
21
C
22
C
23
0
0
C
26
C
31
C
32
C
33
0
0
C
36
0
0
0
C
44
C
45
0
0
0
0
C
54
C
55
0
C
61
C
62
C
63
0
0
C
66
]
{\displaystyle \left[C\right]={\begin{bmatrix}C_{11}&C_{12}&C_{13}&0&0&C_{16}\\C_{21}&C_{22}&C_{23}&0&0&C_{26}\\C_{31}&C_{32}&C_{33}&0&0&C_{36}\\0&0&0&C_{44}&C_{45}&0\\0&0&0&C_{54}&C_{55}&0\\C_{61}&C_{62}&C_{63}&0&0&C_{66}\\\end{bmatrix}}}
Hence shown.