Bulk modulus and poisson's ratio

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SUMMARY

The discussion centers on deriving the equation for the bulk modulus, K = E/3(1 - 2v), where v represents Poisson's ratio. The participant initially struggles with the derivation, particularly with a factor of 1/3 discrepancy. Key equations include E = stress/e, where e is strain, and the relationship e_v = e_x + e_y + e_z, with e_y and e_z expressed in terms of v and e_x. The resolution highlights that the total hydrostatic pressure leads to the factor of 3, emphasizing the negative sign due to compression.

PREREQUISITES
  • Understanding of bulk modulus and its significance in material science.
  • Familiarity with Poisson's ratio and its role in elasticity.
  • Knowledge of stress and strain relationships in solid mechanics.
  • Basic grasp of hydrostatic pressure effects on materials.
NEXT STEPS
  • Study the derivation of the bulk modulus in detail, focusing on the role of hydrostatic pressure.
  • Explore the implications of Poisson's ratio in different materials and its effect on mechanical properties.
  • Learn about the relationship between stress, strain, and elastic modulus in solid mechanics.
  • Investigate the applications of bulk modulus in engineering and material selection.
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Students and professionals in materials science, mechanical engineering, and civil engineering who are involved in understanding material properties and their behavior under stress.

ABoul
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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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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.
 
Perhaps this will bring the answer out.

A body subjected to a uniform hydrostatic pressure all three stress components are equal to a -p.
 
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?
 
Yes that is right. But it is a negative 3. You are in compression.
 

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