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

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

 

[piercas]

The bulk modulus is a measure of a material's resistance to compression when subjected to external pressure. It is a fundamental property of a material and is related to its elastic properties. It is defined as the ratio of the change in pressure (delta p) to the resulting change in volume (delta V).

In order to understand how the bulk modulus is related to pressure and volume, we can look at the equation provided in your textbook: delta V = [-3V delta p (1 - 2c)] /E. This equation shows that the change in volume (delta V) is directly proportional to the change in pressure (delta p) and inversely proportional to the material's Young's modulus (E).

When we take the limit as delta p tends to zero, we are essentially looking at what happens to the bulk modulus when the change in pressure becomes infinitely small. In this case, the change in volume would also become infinitely small and the ratio of delta p to delta V would approach a constant value, which is the bulk modulus (K).

In other words, by taking the limit as delta p goes to zero, we are essentially looking at the bulk modulus at a specific point, or in this case, at a point where the change in pressure is infinitesimally small. This gives us a more accurate and precise value for the bulk modulus, as it eliminates any errors that may arise from larger changes in pressure.

In summary, the bulk modulus is defined as the ratio of the change in pressure to the resulting change in volume. By taking the limit as delta p tends to zero, we are able to obtain a more precise value for the bulk modulus, as it eliminates any errors that may arise from larger changes in pressure.