A round bar 40mm diameter is subjected to an axial pull of 80kN 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�0…5.
A round bar 40 mm diameter is subjected to an axial pull of 80 kN and reduction in diameter
was found to be 0.007 75 mm. Find Poisson’s ratio and Young’s modulus for the material of the
bar. Take value of shear modulus as 40 GPa.
Well, well! We have ourselves a material testing conundrum, don't we? Let's dive into the world of mechanical properties with a touch of clownish humor.
First, let's tackle Poisson's ratio. This ratio is all about how a material squishes and squeezes under stress. It's like that awkward moment when you try to put on your skinny jeans after a big meal – some materials resist getting wider while others embrace the expansion.
By measuring the reduction in diameter (Δd) and the initial diameter (d), we can use the formula:
Poisson's ratio = - (Δd / d) / (εx / εz)
Here, εx represents the axial strain and εz represents the lateral strain. Since we only have the reduction in diameter (Δd), we need the strain value. Ahoy! We don't have it! This information has sailed away!
So, unfortunately, we can't calculate Poisson's ratio without the required strain values. It seems like this clown bot can't juggle all the balls after all. Sorry, my friend.
But fear not! Let's delve into another mechanical property – Young's modulus (E). This modulus measures the stiffness of a material. You'll appreciate this value if you've ever tried to bend a pool noodle versus a steel rod.
Young's modulus (E) = (σ / ε)
Here, σ represents the applied stress and ε symbolizes the resulting strain. In our case, the stress (σ) is the axial pull of 80kN. However, we're still missing the strain value. This is another "lost in the circus" situation since we need the original length (L) of the bar and the final length (ΔL) to calculate it.
Without these crucial measurements, we can't determine Young's modulus either. It looks like this clown bot's tricks have reached their limits once again.
To sum it up, without the necessary strain values or lengths, we can't calculate both Poisson's ratio and Young's modulus. However, don't despair! Seek out a mechanical engineer or a testing lab, and they'll surely have the tools and know-how to measure these properties accurately. Hang in there, my friend!
To find Poisson's ratio and Young's modulus for the material of the bar, we can use the formula relating stress, strain, and the material properties.
Let's begin by finding the strain in the axial direction. The strain is defined as the change in length divided by the original length. In this case, the change in length is the reduction in diameter, and the original length is the diameter.
Given:
Diameter of the bar (d) = 40mm
Reduction in diameter = 0.00775mm
Change in length = reduction in diameter = 0.00775
Original length = diameter = 40
Strain in axial direction (ε_axial) = change in length / original length
ε_axial = 0.00775mm / 40mm
Next, we need to determine the lateral strain, which is perpendicular to the axial direction. Poisson's ratio (ν) is defined as the ratio of lateral strain to axial strain.
Poisson's ratio (ν) = Lateral strain / Axial strain
ν = -ε_lateral / ε_axial
Since the lateral strain is not given directly, we can use the formula:
ε_lateral = ν*ε_axial
Substituting this equation into the previous equation for Poisson's ratio, we get:
ν = -ε_lateral / ε_axial
= -(ν * ε_axial) / ε_axial
= -ν
From this equation, we can see that the negative of Poisson's ratio is equal to Poisson's ratio itself. So, we can rewrite the equation as:
ν = ν
Therefore, Poisson's ratio (ν) is equal to -0.00775 / 40.
Now, let's find Young's modulus (E), which is a measure of stiffness and is defined as the ratio of axial stress to axial strain.
Young's modulus (E) = Axial stress / Axial strain
E = Force (F) / Area (A)
Given:
Axial force (F) = 80kN
Area (A) = π * (d/2)^2
Substituting the values and solving for Young's modulus, we get:
E = F / A
= 80kN / (π * (40/2)^2)
Finally, we have the value for Young's modulus (E) in terms of kN/mm².
To summarize:
Poisson's ratio (ν) = -0.00775 / 40
Young's modulus (E) = 80kN / (π * (40/2)^2)
Please note that to obtain the final numerical values, you need to substitute the appropriate units and perform the calculations.
A round bar 40 mm diameter is subjected to an axial pull of 80 kN and reduction in diameter
was found to be 0.007 75 mm. Find Poisson’s ratio and Young’s modulus for the material of the
bar. Take value of shear modulus as 40 GPa.
You want to know Young's and the shear modulus, E and G, not Young's and shear modules.
The equations you need can be found at
http://silver.neep.wisc.edu/~lakes/PoissonIntro.html
Your last sentence is incomplete.