Nonlinear finite elements/Motion in Lagrangian form

Motion in Lagrangian form
The  motion is given by



{ x = \varphi(X, t) = x(X, t) ~, \qquad X \in [0, L_0]~. } $$ For the reference configuration,

X = \varphi(X, 0) = x(X, 0) ~. $$

The  displacement is

{ u(X, t) = \varphi(X, t) - X = x - X ~. } $$ For the reference configuration,

u_0 = u(X, 0) = \varphi(X, 0) - X = X - X = 0 ~. $$

The  deformation gradient is

{ F(X, t) = \frac{\partial }{\partial X}[\varphi(X,t)] = \frac{\partial x}{\partial X} ~. } $$ For the reference configuration,

F_0 = F(X, 0) = \frac{\partial }{\partial X}[\varphi(X,0)] = \frac{\partial X}{\partial X} = 1 ~. $$ The  Jacobian determinant of the motion is

{ J = \cfrac{A}{A_0} F ~. } $$ For the reference configuration,

J_0 = \cfrac{A_0}{A_0} F_0 = 1 ~. $$