Nonlinear finite elements/Lagrangian and Eulerian descriptions

In our study of the axially loaded bar, we have assumed that the area of cross section remains constant during the deformation. Therefore, the physical stress is not affected by the area of cross section of the rod. However, in most materials, the area of cross section changes with deformation. This change is most prominent during  large deformations. In this handout, we will discuss a  Lagrangian finite element formulation for large deformations.

There are two main ways of approaching problems that involve the motion of deformable materials - the  Lagrangian way and the  Eulerian way. These approaches are distinguished by three important aspects:


 * 1) The mesh description.
 * 2) The stress tensor and momentum equation (kinetics).
 * 3) The strain measure (kinematics).

Definitions
Some essential definitions are given below:


 *  Spatial or Eulerian coordinates ($$\mathbf{x}$$):
 * These coordinates are used to locate a point in space with respect to a fixed basis. You can think of these coordinates as the ones you are familiar with.


 *  Material or Lagrangian coordinates ($$\boldsymbol{X}$$):
 * These coordinates are used to  label material points. If we sit on a material point, the label does not change with time. We do start with a  reference label which we usually choose as the initial spatial coordinates of a material point.


 *  Motion or Deformation  $$(\boldsymbol{\varphi}(\boldsymbol{X},t))$$:
 * A motion or deformation is defined as a mapping between the initial and the current configuration. We usually write this relationship as

\mathbf{x} = \boldsymbol{\varphi}(\boldsymbol{X}, t) ~. $$
 * The initial configuraton is

\boldsymbol{X} = \boldsymbol{\varphi}(\boldsymbol{X}, 0) ~. $$
 *  Displacement  $$(\mathbf{u}(\boldsymbol{X},t))$$:
 * The displacement is defined as the difference between the reference and the current configuration. We write

\mathbf{u}(\boldsymbol{X}, t) = \boldsymbol{\varphi}(\boldsymbol{X}, t) - \boldsymbol{X} \equiv \mathbf{x} - \boldsymbol{X} ~. $$
 *  Velocity  $$(\mathbf{v}(\boldsymbol{X},t))$$:
 * The velocity of a material point is the derivative of the motion with $$\boldsymbol{X}$$ fixed.

\mathbf{v}(\boldsymbol{X}, t) = \frac{\partial }{\partial t}(\boldsymbol{\varphi}(\boldsymbol{X}, t)) ~. $$
 *  Acceleration  $$(\mathbf{a}(\boldsymbol{X},t))$$:
 * The acceleration of a material point is the derivative of the velocity with $$\boldsymbol{X}$$ fixed.

\mathbf{a}(\boldsymbol{X}, t) = \frac{\partial }{\partial t}(\mathbf{v}(\boldsymbol{X}, t)) ~. $$

An example
Let us consider a one-dimensional problem. Assume that the motion is

x = \varphi(X, t) = 1 + X(1 + 2t + t^2) ~. $$ The inverse of the map $$\varphi$$ gives us $$X$$ in terms of $$x$$, i.e.,

X = \varphi^{-1}(x, t) = \cfrac{x - 1}{1 + 2t + t^2} ~. $$ Then the displacement of the material point $$X$$ is

u(X, t) = x - X = 1 + X(2t + t^2) ~. $$ The velocity of the material point is

v(X, t) = \frac{\partial x}{\partial t} = 2X(1 + t) \qquad \leftarrow \qquad \text{ Lagrangian description of velocity.} $$ We could alternatively wish to express the velocity in terms of the spatial coordinate $$x$$. In that case, we have

\bar{v}(x, t) = v(\varphi^{-1}(x,t),t) = \cfrac{2(x-1)(1+t)}{1+ 2t + t^2} \qquad \leftarrow \qquad \text{ Eulerian description of velocity.} $$ Note that $$v(X,t)$$ and $$\bar{v}(x,t)$$ give us different ways of expressing the same quantity. To keep the number of symbols reasonably small, we usually represent both functions with the same symbol ($$v$$) and determine which form we are talking about on the basis of the arguments.

We can find the  Lagrangian and  Eulerian versions of the acceleration in a similar manner. Thus,

a(X, t) = \frac{\partial v}{\partial t} = 2X \qquad\leftarrow \qquad \text{Lagrangian description of acceleration,} $$ and

a(x, t) = a(\varphi^{-1}(x,t),t) = \cfrac{2(x-1)}{1+ 2t + t^2} \qquad\leftarrow \qquad \text{Eulerian description of acceleration.} $$

Lagrangian and Eulerian Meshes
The Lagrangian and Eulerian descriptions can be visualized in terms of the corresponding meshes (see Figure 1).

We can think of the Lagrangian mesh as being drawn on the body. The mesh deforms with the body. Both the nodes and the material points change position as the body deforms. However, the position of the material points relative to the nodes remains fixed.

On the other hand, the Eulerian mesh is a background mesh. The body flows through the mesh as it deforms. The nodes remain fixed and the materials points move through the mesh. The position of a material point relative to the nodes varies with the motion.

Some features, advantages, and disadvantages of the two descriptions are given below.



 Lagrangian mesh 
 * 1) Lagrangian coordinates of nodes move with the material. Material coordinates of material points are time invariant.
 * 2) No material passes between elements.
 * 3) Element quadrature points remain coincident with material points.
 * 4) Boundary nodes remain on the boundary. Therefore, boundary conditions and interface conditions are easily applied.
 * 5) Severe mesh distortion can occur because the mesh deforms with the material.

 Eulerian mesh
 * 1) Eulerian coordinates of nodes are fixed and coincide with spatial points. Spatial coordinates of material points vary with time.
 * 2) Material flows through the mesh.
 * 3) The material point at a given element quadrature point changes with time. This makes dealing with history-dependent materials difficult.
 * 4) Boundary nodes and the material boundary may not coincide. Therefore, boundary conditions and interface conditions are hard to apply.
 * 5) There is no mesh distortion because the mesh is fixed in space. However, the domain that needs to be modeled is larger because we do not want the body to leave the domain.