Nonlinear finite elements/Kinematics - motion and displacement

Continuum MechanicsEdit

To understand the updated Lagrangian formulation and nonlinear finite elements of solids, we have to know continuum mechanics. A brief introduction to continuum mechanics is given in the following. If you find this handout difficult to follow, please read Chapter 2 from Belytschko's book and Chapter 9 from Reddy's book. You should also read an introductory text on continuum mechanics such as Nonlinear continuum mechanics for finite element analysis by Bonet and Wood.

MotionEdit

Let the undeformed (or reference) configuration of the body be   and let the undeformed boundary be  . Let the deformed (or current) configuration be   with boundary  . Let   be the motion that takes the body from the reference to the current configuration (see Figure 1).

 
Figure 1. The motion of a body.

We write

 

where   is the position of material point   at time  .

In index notation,

 

DisplacementEdit

The displacement of a material point is given by

 

In index notation,

 

where   is the Kronecker delta.

VelocityEdit

The velocity is the material time derivative of the motion (i.e., the time derivative with   held constant). This type of derivative is also called the total derivative.

 

Now,

 

Therefore, the material time derivative of   is

 

Alternatively, we could have expressed the velocity in terms of the spatial coordinates  . Let

 

Then the material time derivative of   is

 

AccelerationEdit

The acceleration is the material time derivative of the velocity of a material point.