Subjective Type

The relationship between the impressibility $$\beta $$ and the elastic constants $$E$$ and $$\mu$$. Show that Poisson's ratio $$\mu$$ cannot exceed $$1/2$$.

Solution

Let us consider a cube under an equal compressive stress $$\sigma$$, acting on all its faces.
Then,
Volume strain $$=-\dfrac{\Delta V}{V}=\dfrac{\sigma}{k'}$$...............$$1$$
where $$k$$ is the bulk modulus of elasticity.
So
$$\dfrac{\sigma}{k}=\dfrac{3\sigma }{E}(1-2\mu)$$
or,
$$E=3k(1-2\mu)=\dfrac{3}{\beta}(1-2\mu)\left(as\ k=\dfrac{1}{\beta}\right)$$
$$\mu \le \dfrac{1}{2}$$ if $$E$$ and $$\beta$$ are both to remain positive.

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