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 $(\mathbf {X} )$.
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 $(\mathbf {x} )$.

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 (regarding this step, read the Discuss page)

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

For the reference configuration,

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

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Nonlinear finite elements

Motion in Lagrangian form →