The Young’s modulus of steel is 2 × 1011 N/m2
and its rigidity modulus is 8 × 1010 N/m2
Find the Poisson’s ratio & bulk modulus.
To find the Poisson's ratio and bulk modulus of steel, we can use the formulas that relate these properties to Young's modulus and rigidity modulus.
The Poisson's ratio (ν) is a measure of the lateral strain (deformation perpendicular to the applied force) to the longitudinal strain (deformation in the direction of the applied force). It is given by the formula:
ν = (3K - 2G) / (2(3K + G))
where K is the bulk modulus and G is the rigidity modulus.
The bulk modulus (K) is a measure of a material's resistance to uniform compression. It is related to Young's modulus (E) and the Poisson's ratio (ν) by the formula:
K = E / (3(1 - 2ν))
Let's calculate the Poisson's ratio first:
Given:
Young's modulus (E) = 2 × 10^11 N/m^2
Rigidity modulus (G) = 8 × 10^10 N/m^2
Using the formula:
ν = (3G - 2E) / (2(3G + E))
Substituting the known values:
ν = (3(8 × 10^10) - 2(2 × 10^11)) / (2(3(8 × 10^10) + (2 × 10^11)))
ν = (-4 × 10^11) / (2(24 × 10^10 + 2 × 10^11))
ν = -4 × 10^11 / (2(24 × 10^10 + 2 × 10^11))
ν = -4 × 10^11 / (2(26 × 10^10))
ν = -4 × 10^11 / (52 × 10^10)
ν = -4 / 52
ν ≈ -0.0769
The Poisson's ratio (ν) for steel is approximately -0.0769.
Now, let's calculate the bulk modulus (K) using the formula:
K = E / (3(1 - 2ν))
Substituting the known values:
K = (2 × 10^11) / (3(1 - 2(-0.0769)))
K = (2 × 10^11) / (3(1 + 0.1538))
K = (2 × 10^11) / (3(1.1538))
K = (2 × 10^11) / (3.4614)
K ≈ 5.77 × 10^10 N/m^2
The bulk modulus (K) for steel is approximately 5.77 × 10^10 N/m^2.