Deformation: Strain and Rotation

In Chap. 3 , we introduced infinitesimal and finite strain tensors. The Eulerian displacement gradients $$ \frac{\partial {u}_i}{\partial {x}_j} $$ and the Lagrangian displacement gradients $$ \frac{\partial {u}_i}{\partial {X}_j} $$ occur in the strain tensor definitions. This Chapter is concerned with the mathematical description of finite strain deformations. We will consider two configurations of a continuous body: the reference configuration and the deformed configuration. The reference configuration may be the undeformed state or any intermediate state in the history of deformation. The deformed configuration is any other state. The description presented here is also called a two-state description of deformation. Generally, the body may have undergone some finite strain and rotation from the reference to the deformed configuration. In the limit where the strain increment between the two states is infinitesimal, the two-state description converges to the small-strain approximation.