Bulk modulus and poisson's ratio

In summary, the bulk modulus equation, K = E/3(1-2v), is derived by considering the equations for stress, strain, and Poisson's ratio. The factor of 3 comes from the sum of all stress components in a body subjected to uniform hydrostatic pressure.
  • #1
ABoul
28
0

Homework Statement


derive the equation for the bulk modulus, K = E/3(1 - 2v), where v is poisson's ratio.


Homework Equations


E = stress/e, where e is strain


The Attempt at a Solution


e_v = e_x + e_y + e_z
e_y = e_z = -v*e_x
e_v = (1 - 2v)*e_x

K = stress/e_v
therefore K = stress/[(1 - 2v)*e_x]

i am out by a factor of 1/3. where have i gone wrong?
 
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  • #2
Here is a hint. e = epsilon sub x + epsilon sub y + epsilon sub z. Look at the equations for epsilon sub x, epsilon sub y, and epsilon sub z.

For instance, epsilon sub x = (sigma sub x) / E - (v*sigma sub y) / E - (v*sigma sub z) / E.
 
  • #3
Perhaps this will bring the answer out.

A body subjected to a uniform hydrostatic pressure all three stress components are equal to a -p.
 
  • #4
CFDFEAGURU said:
Perhaps this will bring the answer out.

A body subjected to a uniform hydrostatic pressure all three stress components are equal to a -p.

i see. so the total hydrostatic pressure is the sum of all components, and that's where the factor of 3 comes in, right?
 
  • #5
Yes that is right. But it is a negative 3. You are in compression.
 

1. What is the definition of bulk modulus?

Bulk modulus is a measure of a material's resistance to changes in volume under external pressure.

2. How is bulk modulus related to compressibility?

Bulk modulus is the inverse of compressibility, meaning that materials with high bulk modulus are less compressible and vice versa.

3. What is the formula for calculating bulk modulus?

The formula for bulk modulus is K = -V(dP/dV), where K is the bulk modulus, V is the volume of the material, and (dP/dV) is the change in pressure over the change in volume.

4. What is Poisson's ratio?

Poisson's ratio is a measure of the ratio of lateral strain (strain in the direction perpendicular to applied force) to longitudinal strain (strain in the direction of the applied force) in a material.

5. How are bulk modulus and Poisson's ratio related?

There is a direct relationship between bulk modulus and Poisson's ratio, as they both measure the elastic properties of a material. Materials with high bulk modulus tend to have a low Poisson's ratio, and vice versa.

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