Nonlinear finite elements/Motion in Lagrangian form

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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 ~.
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