# Nonlinear finite elements/Lagrangian finite elements

## Lagrangian finite elements[edit | edit source]

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 .

**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 .

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 , an area , and density . A tensile force is applied at the free end. In the current (or deformed) configuration at time , the length of the bar increases to , the area decreases to , and the density changes to .

### Motion in Lagrangian form[edit | edit source]

The ** motion** is given by

For the reference configuration,

The ** displacement** is

For the reference configuration,

The ** deformation gradient** is

For the reference configuration,

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

For the reference configuration,