Nonlinear finite elements/Lagrangian finite elements

Lagrangian finite elementsEdit

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  .
  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  .


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  , 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 formEdit

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,