Modeling Material Symmetry - Continuum Mechanics of Anisotropic Materials - page 99

4.7.2 Show that the representation for isotropic symmetry in Table 4.4 is also
invariant under the transformation R ðy 13 Þ constructed in Problem 4.4.3.
4.7.3 Construct a representation for C that is invariant under the seven reflective
transformations formed from the set of normals to the planes of reflective
symmetry given in Problem 4.5.2: k , p , m , n , e 1 , e 2 , and e 3 . Does the result
coincide with one of the representations already in Table 4.4 ? If it does,
please explain.
4.8 Curvilinear Anisotropy
In the case where the type of textured material symmetry is the same at all points in an
object, it is still possible for the normals to the planes of mirror symmetry to rotate as a
path is traversed in the material. This type of anisotropy is referred to as curvilinear
anisotropy . The cross-section of a tree illustrated in Fig. 4.5 and the nasturtium
petiole in Fig. 4.6 have curvilinear anisotropy. At any point the tree has orthotropic
symmetry, but as a path across a cross-section of the tree is followed, the normals to
the planes of symmetry rotate. In the cross-section the normals to the planes of
symmetry are perpendicular and tangent to the growth rings. Curvilinear anisotropy,
particularly curvilinear orthotropy, and curvilinear transverse isotropy are found in
many man-made materials and in biological materials. Wood and plane tissue are
generally curvilinear orthotropic, as are fiber wound composites. Only textured
symmetries can be curvilinear. Crystalline symmetries are rectilinear, that is to say
the planes of symmetry cannot rotate as a linear path is traversed in the material.
Curvilinear anisotropies such as those based on the ideal cylindrical and spherical
coordinate systems may have mathematical singularities. For example, a curvilinear
orthotropy characterized by the ideal cylindrical coordinate system has a singularity
at the origin (Tarn 2002 ). This is due to the fact that the modulus associated with the
radial direction is different from that associated with the circumferential or hoop
direction (c.f., Fig. 4.5 ). The singularity at the origin arises due to the fact that the
radial direction and the circumferential or hoop direction are indistinguishable at the
origin yet they have different moduli. A simple resolution of the mathematical
singularity in the model is possible with the proper physical interpretation of its
significance in the real material. One such proper physical interpretation of such
singular points is to note that a volume element containing such a singular point is
not a typical RVE and must be treated in a special manner. This is basically the
approach of Tarn ( 2002 ) who constructs a special volume element, with transversely
isotropic symmetry, enclosing the singularity in the cylindrical coordinate system.
The mathematical singularity in the model is, in this way, removed and the model
corresponds more closely to the real material.
Problems
4.8.1 Sketch the curvilinear nature of the set of three normals to the planes of
reflective symmetry that characterize the wood tissue of a tree.