Biomedical Engineering Reference
In-Depth Information
Table 4.2 The normals to the planes of symmetry of the indicated symmetry group
Type of material symmetry Normals to the planes of symmetry of the indicated symmetry group
Triclinic
None
Monoclinic
e 1
Orthotropic
e 1 , e 2 , e 3
Tetragonal
e 1 , e 2 , e 3 , (1/
2)( e 1
þ
e 2 ) and (1/
2)( e 1
e 2 )
Cubic
e 1 , e 2 , e 3 , (1/
2)( e 1
þ
e 2 ), (1/
2)( e 1
e 2 ), (1/
2)( e 1
þ
e 3 )
(1/
2)( e 1
e 3 ), (1/
2)( e 2
þ
e 3 ), (1/
2)( e 2
e 3 )
Trigonal
e 1 , (1/2)( e 1
þ
3 e 2 ) and (1/2)( e 1
3 e 2 )
Hexagonal
e 1 , e 2 , e 3 , (1/2)(
3 e 1
þ
e 2 ), (1/2)(
3 e 1
e 2 ), (1/2)( e 1
þ
3 e 2 )
and (1/2)( e 1
3 e 2 ).
Transverse isotropy
e 3 and any vector lying in the e 1 , e 2 plane
Isotropy
Any vector
π /4
π
/4
π /4
π /4
a 5
a 1
a 4
a 2
a 3
Fig. 4.11 An illustration of the five planes of symmetry characterizing tetragonal symmetry.
The normals to the five planes are denoted by
4 . Four of the five planes of symmetry have
normals that all lie in the same plane and make angles of 45 with each other. The fourfold
character of the symmetry stems from the relative orientation of its planes of symmetry. The fifth
plane of symmetry is the plane containing the normals to the other four planes of symmetry.
Modified from Rovati and Taliercio ( 2003 )
1 -
a
a
If every vector in a plane is normal to a plane of reflective symmetry, the plane is
called a plane of isotropy . It can be shown that a plane of isotropy is itself a plane of
reflective symmetry. The material symmetry characterized by a single plane of
isotropy is said to be transverse isotropy . In the case of linear elasticity the C matrix
for transversely isotropic symmetry is the same as the C matrix for hexagonal
symmetry and so a distinction is not made between these two symmetries. Isotropic
symmetry is characterized by every direction being the normal to a plane of
reflective symmetry, or equivalently, every plane being a plane of isotropy.
In this presentation the reference coordinate system for the elastic symmetries is
selected so that there are only 18 distinct components of C for triclinic symmetry, 12
for monoclinic and so that the 7 constant tetragonal and trigonal symmetries are the
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