CONTINUUM MECHANICS for ENGINEERS

x 1x 2 plane to be one of elastic symmetry. Accordingly, the elastic constant
matrix for this case has the form
C C C C
C C C C
C C C C
[ Cαβ
] C C
C C
C C C C
=
⎡ 11 12 13 0 0 16 ⎤
⎢
⎥
⎢ 21 22 23 0 0 26 ⎥
⎢
⎥
31 32 33 0 0 36
⎢
⎥
⎢ 0 0 0 44 45 0 ⎥
⎢ 0 0 0 ⎥
54 55 0
⎢
⎥
⎣⎢
61 62 63 0 0 66 ⎦⎥
(6.3-6)
and the original 36 constants are reduced to 20. Also, if a strain energy
functions exists, C αβ = C βα and these 20 nonzero constants would be further
reduced to 13.
If the x 2x 3 plane is also one of elastic symmetry at the same time as the
x 1x 2 plane at a point and we repeat the procedure outlined above, we find
that C 16, C 26, C 36, C 45, C 54, C 61, C 62, and C 63 must also be zero, and the C matrix
is further reduced to
C C C
C C C
C C C
[ Cαβ
] C
C
C
=
⎡ 11 12 13 0 0 0 ⎤
⎢
⎥
⎢ 21 22 23 0 0 0
⎥
⎢
⎥
31 32 33 0 0 0
⎢
⎥
⎢ 0 0 0 44 0 0 ⎥
⎢ 0 0 0 0 ⎥
55 0
⎢
⎥
⎣⎢
0 0 0 0 0
66 ⎦⎥
(6.3-7)
having 12 nonzero coefficients, or 9 if a strain energy function exists.
Interestingly enough, if x 1x 3 is also a plane of elastic symmetry along with
the two considered above, no further reduction in the C αβ matrix occurs. A
material possessing three mutually perpendicular planes of elastic symmetry
is called an orthotropic material, and its elastic constants matrix is that given
in Eq 6.3-7.
The reduction of the orthotropic elastic matrix to that of the isotropic
matrix may be completed by successive consideration of the three axes of
elastic symmetry shown in Figure 6-4 as well as their respective transformation
matrices. By the rotation of 90° about the x 1 axis (Figure 6-4a), we find
that C 12 = C 13, C 21 = C 31, C 22 = C 33, C 23 = C 32, and C 55 = C 66. For the rotation of
90° about the x 3 axis (Figure 6.4b), we see that C 12 = C 21, C 11 = C 22, C 12 = C 23,
C 31 = C 32, and C 44 = C 55. Finally, by a rotation of 45° about the x 3 axis
(Figure 6-4c), we obtain 2C 44 = C 11 – C 12. Therefore, by setting C 44 = µ and
C 12 = λ, we may identify these remaining three C αβ’s with the Lamé constants
and write the elastic coefficient matrix for isotropic behavior as