Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 70 / 1 / b / Solution

A material fibre segment of reference length $L$ and unit direction $l$ has reference vector $Ll$. The uniform deformation gradient maps it to $LAl$, whose current length isTherefore its length is preserved if and only ifThis is the inextensible fibre constraint. The quadratic form uses the right Cauchy-Green deformation tensor; a subsequent rigid rotation does not change it, consistently with the polar factorization $A=RU$.