Strain Transformation

Say the direction cosines for two different coordinate systems are as shown in Table 3.1: Corresponding to stress components in the form of Eq. (2.31), the transformation of strain components in Eq. (3.14) can be derived as follows:

TABLE 3.1

Direction cosines between new and old coordinate systems

Principal Strain and Maximum Shear Strain

Corresponding to stress components in the form of Eq. (2.31), the corresponding strain components in Eq. (3.14) are applied for the strain invariants. By replacing the stress components with corresponding strain components as per Eq. (3.14) into Eq. (2.35) leads to the follows:

By replacing the stress components with corresponding strain components as per Eq. (3.14) into Eq. (2.36) results in the following:

The following yielded after expansion of equation above:

Through simplification we can get the following equation:

By replacing the stress components with corresponding strain components as per Eq. (3.14) into Eq. (2.37) yields the follows:

Expand the equation above results in the following expression:

Simplify the equation above and we obtain:

The maximum shear strain can be expressed by replacing the stress components with corresponding strain components as per Eq. (3.14) into relationships in Eq. (2.44). Say, for n_Amax:

Simplify the equation above yields the follow:

Similarly, shear strain for other planes can be written in the following form:

Deviatoric Strain

Like stresses, the deviatoric strain can be written with the corresponding strain components in Eq. (3.14). Replacing the stress components with those corresponding strain components in Eq. (2.45) yields the following:

Substitution of Eq. (3.37) into equation above provides us the definition below:

The deviatoric strain invariants in the other hand, remains the same as they are for the stress components:

Octahedral Strain

The octahedral strains are no different from the octahedral stresses in term of the structure of expression. The following expression is obtained by replacing the stress components with the corresponding strain components as per Eq. (3.14) in Eq. (2.52):

Similarly, octahedral shear strain can be derived by replacing the stress components with the corresponding strain components as per Eq. (3.14) in Eq. (2.54):

The following equation is resulted after simplification of equation above:

Plane Strain

Consider the length of a body (say, along z-axis) is much greater than its width and height (along ,v and у axes). When force exerted in the direction normal to z-axis, strains are developed in any direction but along z-axis. Since there is no longitudinal strain, such scenario can be simplified into plane strain scenario, as shown in Fig. 3.5.

Under plane strain scenario, the component of strain as per Eq. (3.13) can be simplified as follows:

In expression above, it is noteworthy that y_vv and y_n are only due to corresponding shear stress components, r_yx and z_xy.

FIGURE 3.5 Plane strain.