Nonlinear finite elements/Lagrangian finite elements

Lagrangian finite elements
Two types of approaches are usually taken when formulating Lagrangian finite elements:


 * 1)  Total Lagrangian:
 * 2) *The stress and strain measures are Lagrangian, i.e.,they are defined with respect to the original configuration.
 * 3) *Derivatives and integrals are computed with respect to the Lagrangian (or material) coordinates $$(\mathbf{X})$$.
 * 4)  Updated Lagrangian:
 * 5) *The stress and strain measures are Eulerian, i.e.,they are defined with respect to the current configuration.
 * 6) *Derivatives and integrals are computed with respect to the Eulerian (or spatial) coordinates $$(\mathbf{x})$$.

The following 1-D examples illustrate what these approaches entail.

Consider the axially loaded bar shown in Figure 1. In the reference (or initial) configuration, the bar has a length $$L_0$$, an area $$A_0(X)$$, and density $$\rho_0(X)$$. A tensile force $$T$$ is applied at the free end. In the current (or deformed) configuration at time $$t$$, the length of the bar increases to $$L$$, the area decreases to $$A(X, t)$$, and the density changes to $$\rho(X,t)$$.

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 (regarding this step, read the Discuss page)



{ J = \cfrac{A}{A_0} F ~. } $$

For the reference configuration,



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