Generalised stress-strain relationship

From Eq. (4.1), stress can be expressed in terms of the following relationship:

Since a and e are stress and strain components at a point in matrix form, E would be expressed in matrix form as well to generalise the stress-strain relationship. In this case, D matrix is introduced.

Based on the relationships as shown in Eqs. (2.8), (2.9) and (2.10), we know that the following shear stress components are complementary:

Therefore, the corresponding strain components have the same relationship:

Therefore, the generalised stress-strain relationship for anisotropic material for stress and strain components as stated in Eq. (2.11) and Eq. (3.36), respectively, can be expressed in the form of {a} = [£>]{e}:

FIGURE 4.5 Solid before and after 180° rotation about z-axis.

D matrix can be derived from Eq. (4.5) for any type of material by taking an anisotropic material as the reference material. Sections 4.3 to 4.7 illustrate progressive derivation of D matrix for various materials sorted in a descending order of the level of anisotropy.

Material with symmetrical properties about z axis

Say there is a material with properties symmetrical about the z axis, which means the material properties along the positive л-direction are the same as those in the negative л-direction, and so on, for у-axis. By rotating the solid by 180° about the z-axis, the face of the solid that carries the properties of material in the positive л-direction now coincides with the face that carries properties in the negative л-direction as per the global coordinate system, which is the coordinate system before rotation, as shown in Fig. 4.5.

The direction cosines between mutually orthogonal axes before and after rotation are shown in Table 4.1.

The following stress and strain expressions are obtained by substituting the values in Table 4.1 to Eqs. (2.31) and (3.36):

TABLE 4.1

Transformation of axis for material with symmetrical properties about z axis

	cos 180° = -1	cos 90° = 0	cos 90° = 0
	cos 90° = 0	cos 180° = — 1	cos 90° = 0
	cos 90° = 0	cos 90° = 0	cos 0° = 1

By comparing the stress and strain components before and after transformation in Eqs. (4.6) and (4.7), respectively, we get the following relationship:

By referring to Eq. (4.5), ct_v before rotation can be expressed as follows:

The expression for ct_x<, which denotes the normal stress in the positive л-direction after rotation, can be written in the following form based on the general form as shown in Eq. (4.9):

By equating the normal stress in the positive л-direction before and after rotation, we apply the condition where the stresses to be developed in both positive and negative directions along the original global л-axis are the same. This simulates the case where the material properties are the same for the faces normal to л-axis, as shown in Fig. 4.6.

However, Eq. (4.10) needs to be expressed as cr_v in order to do so. This can be achieved by applying the relationships as shown in Eq. (4.8):

FIGURE 4.6 Transformation from anisotropic material to material with symmetrical properties about z-axis.

By equating the terms on the right-hand side in Eqs. (4.9) and (4.11) the following is obtained:

By comparing the coefficients of the common terms in the equation above, we know that

The equations above are valid if and only if both D/s and Dj6 are zero. The other D components remain the same as they are for anisotropic material.

By referring to Eq. (4.5), oy before rotation can be expressed as follows:

The expression for ay can be written as below, based on the general form as shown in Eq. (4.12):

The following is produced after applying relationship in Eq. (4.8) to the equation above:

Similarly, by equating the terms on the right-hand side in Eqs. (4.12) and (4.14) and then comparing the coefficients of the common terms, we know' that:

The equations above are valid if and only if both D25 and D26 are zero. The other D components remain the same as they are for anisotropic material.

From the derivation above, we can summarise that the coefficients of a* and cr_v (e.g. D/5 and D/в) are zero w'hen stresses before are the same as after rotation (e.g. a_x = ct'._v), while the strains associated with those coefficients are equal to their counterparts in opposite sign (e.g. y_v,_z, = —у and y_x,_z, = —y_xz). Therefore, D35, D_}6, D45 and D₄(, are zero because of the following expressions:

By referring to Eq. (4.5), r_yz before rotation can be expressed as:

The expression for Ty_z- can be written based on the general form as showrn in Eq. (4.15):

Applying the relationship in Eq. (4.8) to the equation above results in the following equations:

By equating the terms on the right-hand side in Eqs. (4.15) and (4.17) we obtain the following:

The following is obtained by comparing the coefficients of the common terms in equation above:

The equations above are valid if and only if D_y, D52. D53 and D54 are zero. The other D components remain the same as they are for anisotropic material.

By referring to Eq. (4.5), t_xz before rotation can be expressed as below:

Based on the general form as shown in Eq. (4.18), the expression for t_x>_z> can be written as below:

Application of relationship in Eq. (4.8) to the equation above produces the following:

Equating the terms on the right-hand side in Eqs. (4.18) and (4.20), and comparing the coefficients of the common terms leads to the following:

The equations above are valid if and only if D_6h Dg₂. D₆₃ and D₆₄ are zero. The other D components remain the same as they are for anisotropic material.

With all the derivation above, the expression in Eq. (4.5) is simplified as follow for material with symmetrical properties along the z axis: