Introduction to Elasticity/Kinematics example 1

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Example 1[edit | edit source]

Take a unit cube of material. Rotate it 90 degrees in the clockwise direction around the z-axis. Calculate the strains. Discuss your results - their accuracy and the reasons for your conclusions.

Solution[edit | edit source]

The strains are related to displacements by

Let us consider rotation about the center of the cube. Since the problem concerns a pure rotation, a cylindrical co-ordinate system is appropriate. This problem also provides us a easy way of trying out Maple. Here are the steps that you can follow to find the strains at a point in the cube.

r := sqrt(x^2+y^2);

theta := arctan(y/x);

x1 := r*cos(theta);

y1 := r*sin(theta);

x2 := r*cos(theta+Pi/2);

y2 := r*sin(theta+Pi/2);

u := x2 - x1;

v := y2 - y1;

epsx := simplify(diff(u,x));

epsy := simplify(diff(v,y));

gamxy := simplify(diff(u,y) + diff(v,x));

From the above Maple calculation, and noting that there is no motion in the direction, the strains in the cube are

A pure rigid body rotation should not result in any non-zero strains.

Therefore, the measure of strain we have used is not appropriate for large rigid body motions.