Homogeneous material
Homogeneous material is a type of material with the same properties along any mutually orthogonal axes. To simulate this type of material, the solid in section 4.5 will need to be rotated 90° clockwise about the zaxis, so the material properties along x and у axes can be the same. Since the material properties along the zaxis are same as уaxis, the properties along the .vaxis will also be similar to those along the zaxis, and thus the simulation matches the characteristics of a homogeneous material, as shown in Fig. 4.11.
Table 4.4 show's the direction cosines between mutually orthogonal axes before and after rotation:
The following equations are obtained after substituting the values in Table 4.4 to Eqs. (2.31) and (3.36):
FIGURE 4.11 Solid before and after 90° clockwise rotation about the zaxis.
TABLE 4.4
Transformation of axis for a homogeneous material
cos 90° = 0 
cos 0° = 1 
cos 90° = 0 

cos 180° = 1 
cos 90° = 0 
cos 90° = 0 

cos 90° = 0 
cos 90° = 0 
cos 0° = 1 
With the comparison of stress and strain components before and after rotation in Eqs. (4.55) and (4.56), the following transformation relationships were obtained:
q_{v} before rotation can be expressed as below by referring to Eq. (4.54):
The expression for ct_{x}, w'hich denotes the normal stress in positive лdirection after rotation, can be written based on the general form as show'n in Eq. (4.58):
By equating the normal stress in the positive .vdirection before and after rotation, we apply the condition where the stresses to be developed in both the positive and negative directions along the original global лaxis are the same. This simulates the
FIGURE 4.12 Transformation from orthotropic material with symmetrical properties along у and z axes to homogeneous material.
case where the material properties are the same for the faces normal to the .vaxis, as shown in Fig. 4.12.
Eq. (4.59) needs to be expressed as cr, in the first place. This can be obtained by applying the relationships as shown in Eq. (4.57):
The expression for oy before rotation is required for further derivation. It can be expressed as follows by referring to Eq. (4.54):
The expression for ay can be written based on the general form as shown in Eq. (4.61):
By applying relationship in Eq. (4.57) to the equation above, we can obtain:
Eqs. (4.58) and (4.62) are comparable as they both express oy component. By equalling the terms on the righthand side in Eqs. (4.58) and (4.62), then comparing the coefficients of the common terms leads to the following:
Same equations will be obtained by repeating the process above for Eqs. (4.60) and (4.61).
g_{:} before rotation can be expressed as below by referring to Eq. (4.54):
The expression for oj can be written based on the general form as shown in Eq. (4.63):
After the relationship in Eq. (4.57) is applied to the equation above, the following is obtained:
Equalization the terms on the righthand side in Eqs. (4.63) and (4.64), follows with the comparison of the coefficients of the common terms results in the following:
The expression for r_{vv} before rotation can be derived as below by referring to Eq. (4.54):
The expression for r_{v}y ^{can} be written based on the general form as shown in Eq. (4.65):
By taking the relationship as per Eq. (4.57) into account, the following can be obtained from the equation above:
By equating the terms on the righthand side in Eqs. (4.65) and (4.66), then comparing the coefficients of the common terms gives us the following:
The expression for before rotation can be developed by referring to Eq. (4.54):
Based on the general form as shown in Eq. (4.67), the expression for r_{V}'_{Z}' can be written as follows:
Application of relationship in Eq. (4.57) to the equation above results in the following:
By referring to Eq. (4.54), t_{xz} before rotation can be expressed as:
The expression for r,/. can be written by referring to the general form as shown in Eq. (4.69):
By considering the relationship in Eq. (4.57), the equation above gives us:
After equalizing both Eqs. (4.68) and (4.69), the following is obtained by comparing the coefficients of the common terms:
Same equations will be obtained by repeat the process above for Eqs. (4.67) and (4.70).
With all the derivation above, the expression in Eq. (4.54) is simplified as follows for a homogeneous material:
FIGURE 4.13 Solid before and after 45° clockwise rotation about zaxis.
TABLE 4.5
Transformation of axis for isotropic material