Nonlinear finite elements/Homework 6/Solutions/Problem 1/Part 3

Problem 1: Part 3

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The rate of deformation is defined as

 

where   is the velocity. In index notation, we write

 

Given the above definition, derive equations (9.2.1) through (9.2.7) of the book chapter.

The motion of a point   on the beam with respect to a point on the reference line   is shown in Figure 2.

 
Figure 2. Motion of continuum-based beam.

Since the normal ( ) rotates as a rigid body, the velocity of point   with respect to   is given by

 

where   is the angular velocity of the normal, and   is the vector from   to  .

Expressed in terms of the local basis vectors  ,  , and  , the angular velocity and the radial vector are

 

Therefore,

 

Let   be the velocity of the point   at time  . Then the actual velocity of point   is

 

Now, in terms of the local basis vectors

 

Therefore,

 

Therefore, the velocity of any point   in terms of the local basis at its orthogonal projection at the reference line is

 

The components of the rate of deformation tensor are

 

In terms of the local basis, these components are

 

For the Euler-Bernoulli beam theory, the normals remain normal to the reference line. Let   be the rotation of the normal. Then, the rotation is given by (see Figure 3)

 

where   is the displacement in the local  -direction at a point on the reference line.

 
Figure 3. Euler-Bernoulli beam kinematics.

The angular velocity of the normal is given by

 

Hence,