A proposed relation between the norm ratio and the anisotropy degree

It is known that the anisotropy of the material, i. e., the symmetry group of the material and the anisotropy of the measured property depicted in the same material may be quite different. Obviously, the property tensor must show, at least, the symmetry of the material. For example, a property which is measured in a material can almost be isotropic but the material symmetry group itself may have very few symmetry elements.

In the elastic stiffness tensors, the isotropic symmetry material is decomposed into two parts, the decomposition of the cubic symmetry material is consisted of the same two isotropic decomposed parts and a third part, and the decomposition of the hexagonal symmetry material is consisted of the same two isotropic decomposed parts and another three parts. Consequently, the Norm Ratio Criteria (NRC) proposed in this chapter is close to that proposed in (Gaith & Alhayek, 2009; Gaith & Akgoz, 2005). For isotropic materials, the elastic stiffness tensor has two parts, so the norm of the elastic stiffness tensor for isotropic materials is equal to the norm of these two parts, i. e., N = N. so. Hence, the ratio

N.

(—1^° = 1) for isotropic materials. For cubic symmetry materials the elastic stiffness tensor

has the same two parts that consisting the isotropic symmetry materials and a third, will be designated as the other than isotropic or the anisotropic part, so two ratios are defined:

Niso Nanis

—for the isotropic parts and —for the anisotropic part. For more anisotropic

Nanis

materials, the elastic stiffness tensor additionally contains more anisotropic parts, so — is defined for all the anisotropic parts.

Although the norm ratios of different parts represent the anisotropy of that particular part, they can also be used to asses the anisotropy degree of a material property as a whole, in this chapter the following criteria are proposed:

1. When N. is dominating among norms of the decomposed parts, the closer the norm

Niso

ratio —is to one, the closer the material property is isotropic.

2. When Niso is not dominating or not present, norms of the other parts can be used as a

criterion. But in this case the situation is reverse; the larger the norm ratio value, the more anisotropic the material property is.