The Generalised Definition of Strain

Mechanical Properties of Solid Polymers, Third Edition. I. M. Ward and J. Sweeney. 2013 John Wiley Sons, Ltd. Published 2013 by John WUey Sons, Ltd. 

This leaves one point, the origin, unchanged, and thus rigid-body translations have been automatically eliminated by this approach. We adopt the Lagrangian approach to strain, in which the reference state is the undeformed state of the material, in contrast to the alternative Eulerian model, in which reference is made to the deformed state. We can rewrite Equation (3.1) in matrix form as 

As a matter mainly of notation. Equation (3.1) can be differentiated to give 

The matrix F and its components are referred to as deformation gradients, F is also termed the deformation gradient tensor. It defines a transformation of the undeformed state onto the deformed state. 

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