Understanding Bulk Modulus: Explaining the Relationship with Pressure and Volume

[discombobulated]

Homework Statement

My textbook says, delta V = [-3V delta p (1 - 2c)] /E
where,
delta V = change in volume
delta p = change in pressure
c = poisson's ratio
E = young's modulus

Taking the limit as delta p tends to zero, we can write the bulk modulus K as,
K = -V dp/dV

but I'm not clear with how they got K by taking the limit as delta p goes to zero...could someone explain that please. Thanks!

 

[Mapes]

The second equation isn't meant to follow from the first. The second equation is the definition of the bulk modulus. The first equation comes from using the generalized Hooke's equation for an isotropic material (in your notation):

$$\epsilon=\frac{1}{E}\sigma_1-\frac{c}{E}\sigma_2-\frac{c}{E}\sigma_3$$

where we plug in the pressure p for all three stresses. Also, the change in volume

$$\Delta V=(1+\epsilon)^3-1\approx3\epsilon$$

for small strains. The bulk modulus for an isotropic material would therefore be

$$K=\frac{E}{3(1-2c)}$$