The Best Cars Gallery: Introduction to Elasticity

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

$\epsilon_{xx} = \frac{\partial u}{\partial x};~ \epsilon_{yy} = \frac{\partial v}{\partial y};~ \epsilon_{zz} = \frac{\partial w}{\partial z};~ \gamma_{xy} = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x};~ \gamma_{yz} = \frac{\partial v}{\partial z} + \frac{\partial w}{\partial y};~ \gamma_{zx} = \frac{\partial w}{\partial x} + \frac{\partial u}{\partial z}$

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);

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

g a m x y : = 0

From the above Maple calculation, and noting that there is no motion in the

z

direction, the strains in the cube are

$\epsilon_{xx} = -1;~\epsilon_{yy} = -1; \epsilon_{zz} = 0; \gamma_{xy} = 0; \gamma_{yz} = 0; \gamma_{zx} = 0$

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.

The Best Cars Gallery

Blogroll

Tattoo vs body painting

MY COLLECTION BLOGS

Introduction to Elasticity

0 komentar:

Post a Comment

Followers

Archive