Nonlinear finite elements/Lagrangian finite elements

Lagrangian Finite Elements

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

Total Lagrangian:
- The stress and strain measures are Lagrangian, i.e.,they are defined with respect to the original configuration.
- Derivatives and integrals are computed with respect to the Lagrangian (or material) coordinates $(\mathbf{X})$ .
Updated Lagrangian:
- The stress and strain measures are Eulerian, i.e.,they are defined with respect to the current configuration.
- 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.

Figure 1. Axially loaded bar

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

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

For the reference configuration,

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

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