The amount of displacement of each particle along the line element varies from left to right. It can be shown that the displacement of each particle increases linearly with respect to x as plotted below. This is how normal strains relate to displacements. The relationship between shear strain and displacement is more complex mathematically. Consider the following deformation.
Shear deformation is related to the rotations of the sides of a control element.
Even though the two sides of the original cubic element are rotated with different amounts, the final configuration can be accomplished in two steps with a symmetrical operation. This rotation, however, does not contribute to any shear strain because the original configuration does not change. After this operation, the final configuration is obtained. This is the only operation that contributes to the production of shear strains. Hence, Because the deformation angles are infinitesimal in magnitude, Hence, In matrix form, a strain tensor becomes Or, it can be written in tensor form as follows.