2
$\begingroup$

A cube made of an isotropic material is subjected to a tensile stress as shown in Figure 2. What is the relationship between the strain in different directions ?
enter image description here

There is an elongation in $z$ direction and contraction in $x, y$ directions. So $\epsilon_x, \epsilon_y < \epsilon_z$. Is my reasoning correct ? Since they say material is isotropic, can I say anything else about $\epsilon_x, \epsilon_y$ ? (Like $\epsilon_x = \epsilon_y$). Please explain how someone can arrive at a conclusion about $\epsilon_x, \epsilon_y, \epsilon_z$.

$\endgroup$
3
$\begingroup$

In this case, we say that $\epsilon_{axial}$ ($z$-direction in your diagram) is positive by convention. Then, for a normal material with a Poisson's ratio $\nu \ge 0$, $\epsilon_{transverse}$ ($x$- and $y$-directions in your diagram) will be negative. The fact that the material is isotropic means we have no way of distinctly labeling one direction $x$ and one direction $y$, so it must be the case that $\epsilon_x = \epsilon_y$.

So in summary, yes, assuming $\nu \ge 0$ you are justified in saying $\epsilon_x = \epsilon_y < \epsilon_z$.

Note, however, that even though almost all materials will have $\nu \ge 0$, there does exist a class of materials known as auxetics which have $\nu \lt 0$. A common naturally occurring example is $\alpha$-cristobalite silica. In this case, the relationship between the strains is not so immediately obvious, and would depend on the exact value of $\nu$ (since the strain would be positive in all directions). These materials are not usually isotropic though, so $\nu \ge 0$ is probably a safe assumption.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.