Foundations of Elasticity Theory

tion of X, Ø is called the deformation at time t; regarded as a function of t, Ø describes a motion. See Figure 1. In the theory of elastic materials the deformation gradient F is important, (2:03) F = @Ø @X : (1.3) Remembering that bx and bX are vectors we see that F is the Jacobian matrix of the transformation X ! x. In fact, if we write X 1 , X 2 , X 3 , Figure 1 x 1 , x 2 , x 3 for the components of X, x relative to the fixed cartesian frame of reference, then (2:04) F<F