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 z-axis, so the material properties along x and у axes can be the same. Since the material properties along the z-axis are same as у-axis, the properties along the .v-axis will also be similar to those along the z-axis, 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 z-axis.

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 л-direction 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 .v-direction 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 л-axis 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 .v-axis, 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 right-hand 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 o-j 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 right-hand 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_vy ^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 right-hand 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 z-axis.

TABLE 4.5

Transformation of axis for isotropic material