Modeling Material Symmetry - Continuum Mechanics of Anisotropic Materials - page 96

coefficients appearing in C must be unchanged by one reflective symmetry
transformation. Let e 1 be the normal to the plane of reflective symmetry so that
the reflective symmetry transformation is R ðe 1 Þ , given by the first of ( 4.5 ). The tensor
C must be invariant under the transformation

T

C ðLÞ

¼ R ðe 1 Þ C ðGÞ ½R ðe 1 Þ ; (4.16)

¼ R ðe 1 Þ . The pattern of this
calculation follows the pattern of calculation in ( 4.9 ). That pattern is the substitu-
tion for C and R ðe 1 Þ into this equation and the execution of the matrix multiplication.
The resulting matrices are not documented here. As mentioned above, they may be
easily obtained with any symbolic algebra program. The result is that the tensor C is
unchanged by the reflection only if

Q

which follows from the first of (A162) by setting

^ c 15 ¼ ^ c 51 ¼ ^ c 16 ¼ ^ c 61 ¼ ^ c 25 ¼ ^ c 52 ¼ ^ c 26 ¼ ^ c 62

¼ ^ 0. It follows then that the form

of the tensor C consistent with monoclinic symmetry characterized by a plane
of reflective symmetry normal to the e 1 base vector must satisfy the conditions
^

c 35 ¼ ^ c 53 ¼ ^ c 36 ¼ ^ c 63 ¼ ^ c 45 ¼ ^ c 54 ¼ ^ c 46 ¼ ^ c 64 ¼

c 15 ¼ ^ c 51 ¼ ^ c 16 ¼ ^ c 61 ¼ ^ c 25 ¼ ^ c 52 ¼ ^ c 26 ¼ ^ c 62 ¼ ^ c 35 ¼ ^ c 53 ¼ ^ c 36 ¼ ^ c 63 ¼ ^ c 45 ¼

c 54 ¼ ^ 0. This result for monoclinic symmetry is recorded in Table 4.4 .

Three mutually perpendicular planes of reflective symmetry characterize
orthotropic symmetry, but the third plane is implied by the first two. This means
that the material coefficients appearing in the representation of C for monoclinic
symmetry must be unchanged by another perpendicular reflective symmetry
transformation. Let the e 2 be the normal to the plane of reflective symmetry so
the reflective symmetry transformation is R ðe 2 Þ as given by the second of ( 4.5 ).
The monoclinic form of the tensor C must be invariant under the transformation

^ c 46 ¼ ^ c 64 ¼

T

C ðLÞ

¼ R ðe 2 Þ C ðGÞ ½R ðe 2 Þ ; (4.17)

which follows from the first of (A162) by setting Q ¼ R ðe 2 Þ . The pattern of this
calculation follows the pattern of calculation in ( 4.11 ). That pattern is the substitu-
tion of the monoclinic form for C and R ðe 2 Þ into this equation and the execution of
the matrix multiplication. The resulting matrices are not documented here; they
may be easily obtained with any symbolic algebra program. The result is that the
tensor C is unchanged by the reflection only if c 14 ¼ c 41 ¼ c 24 ¼ c 42 ¼ c 34 ¼ c 43
¼ ^ 0 . It follows then that the form of the tensor C consistent with

orthotropic symmetry characterized by planes of reflective symmetry normal to the
e 1 and e 2 base vectors must satisfy the conditions

c 56 ¼ ^ c 65 ¼

^ c 14 ¼ ^ c 41 ¼ ^ c 24 ¼ ^ c 42 ¼ ^ c 34 ¼

c 43 ¼ ^ 0. This result for orthotropic symmetry is recorded in Table 4.4 .

A transversely isotropic material is one with a plane of isotropy. This means that
the material coefficients appearing in the representation of

^ c 56 ¼ ^ c 65 ¼

C for orthotropic