Elasticity/Kinematics example 1

Example 1

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

The strains are related to displacements by

${ϵ}_{xx}=\frac{\partial u}{\partial x};{ϵ}_{yy}=\frac{\partial v}{\partial y};{ϵ}_{zz}=\frac{\partial w}{\partial z};{\gamma }_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x};{\gamma }_{yz}=\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y};{\gamma }_{zx}=\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}$

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);

$r:=\sqrt{x^2+y^2}$

theta := arctan(y/x);

$\theta :=arctan(\frac{y}{x})$

x1 := r*cos(theta);

$x1:=\frac{\sqrt{x^2+y^2}}{\sqrt{1+\frac{y^2}{x^2}}}$

y1 := r*sin(theta);

$y1:=\frac{\sqrt{x^2+y^2}y}{x\sqrt{1+\frac{y^2}{x^2}}}$

x2 := r*cos(theta+Pi/2);

$x2:=-\frac{\sqrt{x^2+y^2}y}{x\sqrt{1+\frac{y^2}{x^2}}}$

y2 := r*sin(theta+Pi/2);

$y2:=\frac{\sqrt{x^2+y^2}}{\sqrt{1+\frac{y^2}{x^2}}}$

u := x2 - x1;

$u:=-\frac{\sqrt{x^2+y^2}y}{x\sqrt{1+\frac{y^2}{x^2}}}-\frac{\sqrt{x^2+y^2}}{\sqrt{1+\frac{y^2}{x^2}}}$

v := y2 - y1;

$v:=\frac{\sqrt{x^2+y^2}}{\sqrt{1+\frac{y^2}{x^2}}}-\frac{\sqrt{x^2+y^2}y}{x\sqrt{1+\frac{y^2}{x^2}}}$

epsx := simplify(diff(u,x));

$epsx:=-\frac{\sqrt{x^2+y^2}}{x\sqrt{\frac{x^2+y^2}{x^2}}}$

epsy := simplify(diff(v,y));

$epsy:=-\frac{\sqrt{x^2+y^2}}{x\sqrt{\frac{x^2+y^2}{x^2}}}$

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

$gamxy:=0$

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

${ϵ}_{xx}=-1;{ϵ}_{yy}=-1;{ϵ}_{zz}=0;{\gamma }_{xy}=0;{\gamma }_{yz}=0;{\gamma }_{zx}=0$

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.