Nonlinear finite elements/Homework 6/Solutions/Problem 1/Part 7

Problem 1: Part 7

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Derive the equation (9.3.13) of the book chapter starting from equation (9.3.3).

Figure 4 shows the notation used to represent the motion of the master and slave nodes and the directors at the nodes.

 
Figure 4. Notation for motion of the continuum-based beam.

Since the fibers remain straight and do not change length, we have

 

where   is the location of master node  ,   is the unit director vector at master node  , and   is the initial thickness of the beam.

Taking the material time derivatives of equations (5), we get

 

where   is the angular velocity of the director.

From equations (5) we have

 

where   is the global basis.

In terms of the global basis, the angular velocity is given by

 

Therefore,

 

Substituting equation (8) into equations (6), we get

 

Let the velocity vectors be expressed in terms of the global basis as

 

Then equations (9) can be written as

 

Therefore, the components of the velocity vectors are

 

Then, the matrix form of equations (11) is

 

or