Question

If stress-strain relation for volumetric change is in the form $\dfrac{{\Delta V}}{{{V_0}}} = KP$ where $P$ is applied uniform pressure, then $K$ stands for
A) shear modulus
B) compressibility
C) Young's modulus
D) bulk modulus

Answer 1

Hint: When a uniform pressure is applied on a body, you can see a deformation of the body. The deformation in terms of volume causes a volume strain or bulk strain on the body. This bulk strain is related to the pressure by an elastic modulus.

Formulae Used:
If a uniform pressure $P$ causes a deformation of volume $\Delta V$ where the total volume of the body was initially ${V_0}$, then we have the relation
Stress $ = $ Elastic Modulus $ \times $ Strain
$P = B \times \dfrac{{\Delta V}}{{{V_0}}}$
where, $B$ is the bulk modulus.
The compressibility $K$ is defined as the reciprocal of the bulk modulus $B$
$K = \dfrac{1}{B}$

Complete step by step solution:
Given, The volumetric change is in the form $\dfrac{{\Delta V}}{{{V_0}}} = KP$.

Step 1:
There is no mention of shear happening in the given problem.
Hence you can discard the elastic modulus of shearing that is the shear modulus from the consideration.
So, the option (A) is incorrect

Step 2:
The Young modulus is the elastic modulus which acts in the process of linear deformation.
But here, a volumetric change is happening hence the $K$ in the equation can not be Young’s modulus.
So, the option (C) is incorrect

Step 3:
From eq (1), you can see that a volumetric change is expressed, where the elastic modulus is the bulk modulus $B$ and the stress applied is the pressure $P$.
Express eq (1) in terms of the given form of the volumetric change.
$
  P = B \times \dfrac{{\Delta V}}{{{V_0}}} \\
   \Rightarrow \dfrac{{\Delta V}}{{{V_0}}} = \dfrac{1}{B}P \\
 $
So, evidently comparing this with the given form you can get
$K = \dfrac{1}{B}$
Hence this is not the bulk modulus but its reciprocal.
So, option (D) is also not correct.
By definition from eq (2), you can realize that the correct option is (B) which is the compressibility.

If stress-strain relation for volumetric change is in the form $\dfrac{{\Delta V}}{{{V_0}}} = KP$ where $P$ is applied uniform pressure, then $K$ stands for compressibility. Hence, option (B) is correction.

Note:
The bulk modulus is the ratio of the volumetric stress by volumetric strain. So, the bulk modulus will be proportional to the volumetric stress that is the uniform pressure rather than the volumetric strain. This problem also can be argued as the increase in volumetric strain means a change in the volume rather than the elastic property to keep it unchanged. So, the bulk modulus which represents the elastic property of the body that tries to resist the deforming stress can not be proportional to the strain. Hence compressibility would be the more correct option which represents the ability of the body to be compressed with certain stress applied.