# Nonlinear finite elements/Lagrangian finite elements

## Lagrangian finite elements

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

1. 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 ${\displaystyle (\mathbf {X} )}$ .
2. 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 ${\displaystyle (\mathbf {x} )}$ .

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 ${\displaystyle L_{0}}$ , an area ${\displaystyle A_{0}(X)}$ , and density ${\displaystyle \rho _{0}(X)}$ . A tensile force ${\displaystyle T}$  is applied at the free end. In the current (or deformed) configuration at time ${\displaystyle t}$ , the length of the bar increases to ${\displaystyle L}$ , the area decreases to ${\displaystyle A(X,t)}$ , and the density changes to ${\displaystyle \rho (X,t)}$ .

### Motion in Lagrangian form

The motion is given by

${\displaystyle {x=\varphi (X,t)=x(X,t)~,~~\qquad X\in [0,L_{0}]~.}}$

For the reference configuration,

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

The displacement is

${\displaystyle {u(X,t)=\varphi (X,t)-X=x-X~.}}$

For the reference configuration,

${\displaystyle u_{0}=u(X,0)=\varphi (X,0)-X=X-X=0~.}$

${\displaystyle {F(X,t)={\frac {\partial }{\partial X}}[\varphi (X,t)]={\frac {\partial x}{\partial X}}~.}}$

For the reference configuration,

${\displaystyle F_{0}=F(X,0)={\frac {\partial }{\partial X}}[\varphi (X,0)]={\frac {\partial X}{\partial X}}=1~.}$

The Jacobian determinant of the motion is (regarding this step, read the Discuss page)

${\displaystyle {J={\cfrac {A}{A_{0}}}F~.}}$

For the reference configuration,

${\displaystyle J_{0}={\cfrac {A_{0}}{A_{0}}}F_{0}=1~.}$