Tag: elastic energy

$Y=2\eta(1+\sigma)\Rightarrow \sigma=\displaystyle\frac{0.5Y-\eta}{\eta}$

Hookes law states that stress is proportional to strain up to elastic limit. If p is the stress induced in material and e the corresponding strain, then according to Hooke's law, 
$\dfrac{P}{E}$ = E, a constant.

The Poisson's ratio of a stable, isotropic, linear elastic material cannot be less than $1.0$ nor greater than $0.5$. So only possible value among the options is $0.4.$

Experimental value of poisson's ratio is always between $0$ to $1/2$ .

1/ Young's modulis is property of a metal independent of its dimensions
2/Definition of Poisson's ratio 
Poisson's ratio is the ratio of transverse contraction strain to longitudinal extension strain in the direction of stretching force. Tensile deformation is considered positive and compressive deformation is considered negative. The definition of Poisson's ratio contains a minus sign so that normal materials have a positive ratio.Virtually all common materials, such as the blue rubber band on the right, become narrower in cross section when they are stretched. 

Relation  between  Young's  modulus,  bulk  modulus  and  poisson's  ratio  is  given  below :
$Y = 3B (1-2\sigma)$
So,  according  to  problem
$ 6.6 \times  10^{10} = 3 \times 11 \times 10^{10} (1-2\sigma)$
$ \sigma = 0.4$

$\epsilon x=0.05$  (given)
$\sigma =0.25$
$\dfrac{\epsilon y}{0.05}=-0.25$ (standard result)
$=-0.0125$

$\epsilon x=\dfrac{0.3}{3}$$=0.1$ (standard result)

Let the material of length $l$ and side $s$ 

$Y=2\eta(1+\sigma)$
$\Rightarrow 2.4\eta =2\eta(1+\sigma)$
$\Rightarrow 1.2=1+\sigma$
$\Rightarrow \sigma=0.2$