Question

A cylindrical specimen of some metal alloy 7.3 mm in diameter is stressed in tension. A force of 7810 N produces an elastic reduction in specimen diameter of 0.0031 mm. Calculate the elastic modulus (in GPa) of this material if its Poisson's ratio is 0.34.

Answer 1

To calculate the elastic modulus, one needs the stress, longitudinal strain, and the Poisson's ratio. By using the formulae for stress and strain in conjunction with the given Poisson's ratio, Young's modulus of the material can be derived and then converted to GPa.

Step-by-step explanation:

To find the elastic modulus of the material in question, we first need to understand that it is a measure of a material's ability to resist deformation in response to an applied force. Given the diameter reduction under stress and Poisson's ratio, we can employ the relationship between the lateral strain (transverse contraction) and longitudinal strain provided by Poisson's ratio (ν). The formula for Poisson's ratio is ν = - ε_transverse/ε_longitudinal, where ε_transverse is the transverse strain, and ε_longitudinal is the longitudinal strain.

In this case, ε_longitudinal is not directly given, but we can use the concept that stress (σ) is equal to force (F) divided by the cross-sectional area (A), σ = F/A, and that strain (ε) is the change in length (ΔL) over the original length (L), ε = ΔL/L. Young's modulus (E) is the ratio of stress to strain, E = σ/ε. To find the reduction in diameter due to the applied force, transverse strain must first be calculated using ε_transverse = Δd/d_original, where Δd is the change in diameter and d_original is the original diameter.

From the transverse strain and the given Poisson's ratio, we can find the longitudinal strain. Then, calculate stress using the force and original cross-sectional area of the cylindrical specimen. Finally, with both stress and longitudinal strain, we can solve for Young's modulus. Since the unit of Young's modulus is Pascal (Pa) and 1 GPa = 109 Pa, we can convert the final answer to GPa as requested.

Let's perform the calculations:

1. Calculate the original cross-sectional area, A = π(d_original/2)2.
2. Calculate the stress using σ = F/A.
3. Calculate the transverse strain, ε_transverse = Δd/d_original.
4. Calculate the longitudinal strain using Poisson's ratio, ε_longitudinal = -ε_transverse/ν.
5. Finally, calculate Young's modulus, E = σ/ε_longitudinal and convert the result to GPa.

Answer 2

To calculate the elastic modulus of a metal alloy from the given tension test data, we use stress and lateral strain equations derived from the tension force, the specimen's diameter, the reduction in diameter, and Poisson's ratio. The formula considering Poisson's ratio gives us Young's modulus, which is an essential mechanical property representing the material's stiffness.

Step-by-step explanation:

To calculate the elastic modulus (in GPa) of the metal alloy, we will use the relationship between stress, strain, and Poisson's ratio. Stress (σ) is the internal force per unit area within materials that arises from externally applied forces, defined as force (F) divided by the area (A). Strain (ε) on the other hand is the measure of deformation representing the displacement between particles in the material body. Due to the cylindrical geometry, we have to consider both axial and lateral strains and their relationship determined by Poisson's ratio (ν).

The formula for Young's modulus (E) considering Poisson's ratio is E = σ/(1-2ν)ε. However, since we don't have direct measures of axial strain, we will use lateral strain, calculated through reduction in diameter, to find strain. The lateral strain is ε₂ = Δd/d where d is the original diameter and Δd is the diameter change. The cross-sectional area of the specimen is A = π(d/2)^2. This allows us to calculate stress σ = F/A. Using the given data, let's work through the calculation:

• Original diameter (d) = 7.3 mm
• Force (F) = 7810 N
• Diameter reduction (Δd) = 0.0031 mm
• Poisson's ratio (ν) = 0.34

Firstly, calculate the stress using the force and the original cross-sectional area. Then, calculate the lateral strain based on the diameter reduction and the original diameter. Finally, use the stress and strain values along with Poisson's ratio to calculate Young's modulus (E), converting the final result to GPa as required.

The elastic modulus of the material under test provides important information about its stiffness and is a critical property for engineering and material science applications.