A round bar 40 mm dia meter is subjected to an axial pull of 80 kn and reduction in diameter was found to be 0.00775mm.find poisson's ratio and young's modules for the material of the bar. take value of shear modules as 40kn/mm square.
Assuming an isotropic and elastic material,
εy=-0.00775/40
σx=80 kN
G=40 N/mm^2 (40 kN was probably a typo)
Use
εy=(σy-ν(σx+σz))/E ....(1)
and
G=E/[2(1+ν)] ...(2)
Solve for ν by substituting
E=2G(1+ν) from (2)
into (1)
εy=(σy-ν(σx+σz))/[2G(1+ν)]
and solve for ν.
Substitute ν into (2) to find E.
I get ν=0.24, E=99.2 MPa
Give the answer
To find Poisson's ratio (ν) and Young's modulus (E), we can use the formulas that relate these parameters to the applied axial load and the change in diameter of the bar.
Poisson's ratio (ν) is given by:
ν = (change in diameter/original diameter) / (change in length/original length)
Young's modulus (E) is given by:
E = (axial load * original length) / (change in length * original area)
Given data:
- Diameter (D) = 40 mm = 40/1000 = 0.04 m
- Axial load (P) = 80 kN = 80*1000 N
- Change in diameter (ΔD) = 0.00775 mm = 0.00775/1000 = 0.00000775 m
- Shear modulus (G) = 40 kN/mm^2 = 40*1000 N/mm^2
Step 1: Calculate the original length of the bar (L0).
To calculate L0, we need to find the original cross-sectional area (A0) of the bar.
Area (A) = π*(D/2)^2
A0 = π*(0.04/2)^2 = π*(0.02)^2
Step 2: Calculate the change in length (ΔL).
We can find ΔL using the formula:
ΔL = (ΔD * L0) / D
Step 3: Calculate Poisson's ratio (ν).
ν = ΔD / D
Step 4: Calculate Young's modulus (E).
E = (P * L0) / (ΔL * A0)
Step 5: Calculate the value of Young's modulus in MPa.
To convert N/mm^2 to MPa:
E (MPa) = E (N/mm^2) / 1000
Now, let's calculate the values:
Step 1: Find original length (L0).
A0 = π*(0.02)^2
Step 2: Find change in length (ΔL).
ΔL = (0.00000775 * L0) / 0.04
Step 3: Find Poisson's ratio (ν).
ν = 0.00000775 / 0.04
Step 4: Find Young's modulus (E).
E = (80 * 1000 * L0) / (ΔL * A0)
Step 5: Convert Young's modulus to MPa.
E (MPa) = E / 1000
Now, you can substitute the values into the equations to find the results.
To find Poisson's ratio and Young's modulus for the material of the bar, we need to use the equations relating these parameters with the given information.
Poisson's ratio (ν) is defined as the ratio of lateral strain to the axial strain. It can be calculated using the formula:
ν = Δd / d_0 * ΔL / L_0
where Δd is the reduction in diameter, d_0 is the initial diameter, ΔL is the axial deformation, and L_0 is the initial length.
In this case, Δd = 0.00775 mm, d_0 = 40 mm, and ΔL = 80 kN (force) applied to the bar. However, we need to convert the force into deformation using the relation:
ΔL = F / (A * E)
where F is the force applied, A is the cross-sectional area, and E is the Young's modulus.
The cross-sectional area can be calculated using the formula:
A = π * (d^2) / 4
where d is the diameter.
Given that the shear modulus (G) is 40 kN/mm², we can use the relation between Young's modulus (E) and shear modulus (G):
E = 2 * G * (1 + ν)
Now, let's calculate the values step by step:
1. Calculate the cross-sectional area (A):
A = π * (40 mm)^2 / 4
= π * (1600 mm²) / 4
= 1256.637 mm²
2. Convert the force into deformation (ΔL):
ΔL = 80 kN / (1256.637 mm² * 40 kN/mm²)
= 0.001 mm (or 1 µm)
3. Calculate Poisson's ratio (ν):
ν = Δd / d_0 * ΔL / L_0
= (0.00775 mm) / (40 mm) * (0.001 mm) / (0 mm)
= 0.0019375
4. Calculate Young's modulus (E):
E = 2 * G * (1 + ν)
= 2 * 40 kN/mm² * (1 + 0.0019375)
≈ 80.775 kN/mm²
Therefore, the Poisson's ratio (ν) is approximately 0.0019375 and the Young's modulus (E) is approximately 80.775 kN/mm² for the material of the bar.