Nonlinear finite elements/Homework 6/Solutions/Problem 1/Part 3
Problem 1: Part 3
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.
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
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, 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.
The angular velocity of the normal is given by