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

Problem 1: Part 3 edit

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




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.

Figure 3. Euler-Bernoulli beam kinematics.

The angular velocity of the normal is given by