In the Holzapfel's book "Nonlinear solid mechanics - a continuum approach for engineering", at page 64, equation 2.20 shows that the material time derivative of F (scalar, vector or tensor field) is equal to the directional derivative of F in the direction of the velocity vector, v:

DF(X, t)/Dt = DvF(X, t)

The equation was widely used in the book, but Holzapfel did not provide its derivation and I cannot derive it.

First, according to the definition of the directional derivative (page 47, equation 1.266), with the implementation of Taylor's expansion, it seems that we can obtain:

DvF(X, t) = dF(X + sv, t)/ds | s=0 = GradF · v

However, the directional derivative should not be applied to the velocity vector, because it is not a unit vector.

Now, let us assume that DvF(X, t) works for a velocity field. We still cannot reach the conclusion that DvF(X, t) is equal to the material time derivative of F (considering a scalar field here):

DvF(X, t) = GradF · v = ∂F/∂X · v = ∂F/∂X · ∂x(X, t)/∂t ≠ DF(X, t)/Dt

I would greatly appreciate if anyone can help me!