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A cube subjected to three mutually perpendicular stress of equal intensity p expenses a volumetric strain
- a)3p/ E × (2/m - 1)
- b)3p/ E × (2 - m)
- c)3p/ E × (1 - 2/m)
- d)E/ 3p × (2/m - 1)
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
A cube subjected to three mutually perpendicular stress of equal inten...
Stress p in x-direction causes tensile strain p/E in x-direction, while the ‘p’ in ‘y’ and ‘z’ direction causes compressive strains μ(p/E) in x-direction.
Therefore, ex = p/E - μ(p/E) - μ(p/E) = p/E x (1- 2μ)
So, ey =p/E x (1- 2μ),
ez = p/E x (1- 2μ)
Therefore, Volumetric Strain, ev = ex + ey + ez = 3p/E x (1- 2μ) = 3p/E x (1 –2/m)
Most Upvoted Answer
A cube subjected to three mutually perpendicular stress of equal inten...
Given: Poisson's ratio (ν) = 0.4
We need to find the ratio of Shear modulus (G) to Modulus of elasticity (E).
Formulae:
Shear modulus (G) = (3E(1 - 2ν)) / (2(1 + ν))
Modulus of elasticity (E) = (3K(1 - 2ν)) / (1 + ν)
Where K = Bulk modulus
Calculation:
Let's assume K = 1 for simplicity.
Modulus of elasticity (E) = (3(1)(1 - 2(0.4))) / (1 + 0.4) = 0.9
Shear modulus (G) = (3(0.9)(1 - 2(0.4))) / (2(1 + 0.4)) = 0.225
Ratio of Shear modulus to Modulus of elasticity = G / E = 0.225 / 0.9 = 5/14
Therefore, the correct answer is option C, 5/14.
We need to find the ratio of Shear modulus (G) to Modulus of elasticity (E).
Formulae:
Shear modulus (G) = (3E(1 - 2ν)) / (2(1 + ν))
Modulus of elasticity (E) = (3K(1 - 2ν)) / (1 + ν)
Where K = Bulk modulus
Calculation:
Let's assume K = 1 for simplicity.
Modulus of elasticity (E) = (3(1)(1 - 2(0.4))) / (1 + 0.4) = 0.9
Shear modulus (G) = (3(0.9)(1 - 2(0.4))) / (2(1 + 0.4)) = 0.225
Ratio of Shear modulus to Modulus of elasticity = G / E = 0.225 / 0.9 = 5/14
Therefore, the correct answer is option C, 5/14.
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Community Answer
A cube subjected to three mutually perpendicular stress of equal inten...
Given P1 ,P2 & P3 be the 3 mutually perpendicular stresses acting in X , Y and Z directions having equal intensity P.
E be the modulus of elasticity
and 1/m be Poisson's ratio
P1 = P2 = P3 = p eq 1
Longitudinal strain in X direction
E1 = P1/E - P2/mE - P3/mE
from eq. 1
or E1 = P/E (1- 2/m) [A]
Lateral strain in Y direction
E2 = P2/E - P1/E - P3/E
from eq. 1
or E2 = p/E (1-2/m) [B]
Lateral strain in Z direction
E3 = P3/E - P2/E - P1/E
from eq.1
or E3 = p/E (1-2/m) [C]
We know Volumetric strain = E1 +E2 + E 3
Adding all A +B +C
we get the answer.
E be the modulus of elasticity
and 1/m be Poisson's ratio
P1 = P2 = P3 = p eq 1
Longitudinal strain in X direction
E1 = P1/E - P2/mE - P3/mE
from eq. 1
or E1 = P/E (1- 2/m) [A]
Lateral strain in Y direction
E2 = P2/E - P1/E - P3/E
from eq. 1
or E2 = p/E (1-2/m) [B]
Lateral strain in Z direction
E3 = P3/E - P2/E - P1/E
from eq.1
or E3 = p/E (1-2/m) [C]
We know Volumetric strain = E1 +E2 + E 3
Adding all A +B +C
we get the answer.
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