Nonlinear finite elements/Kinematics - motion and displacement

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Continuum Mechanics

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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.


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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,


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The displacement of a material point is given by

In index notation,

where is the Kronecker delta.


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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.


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


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