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

Problem 1: Part 7 edit

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




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