Elasticity and Young's Modulus

In summary, this conversation discusses deriving expressions for the atomic force constant k in a metal bar under tension. This involves calculating the number of chains of atoms, number of atoms in a single chain, microscopic extension between atoms, and tensile force between atoms. The value of k can be calculated using the equation f=kΔx, which is also equal to Ex. The modulus of elasticity in compression and tension are different, and the appropriate modulus to use in a finite element model depends on the specific material and situation.
  • #1
R2Zero
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Homework Statement



Consider a metal bar of length L, cross-sectional area A, equilibrium atomic separation x, and Young's modulus E. When a tension force F is applied to the bar, it causes an extension ΔL. Calculate the atomic force constant k by deriving expressions for (a) the number of chains of atoms in any cross section, (b) the number of atoms in a single chain of length L, (c) the microscopic extension Δx between atoms, and (d) the tensile force f between atoms. (e) Write f= kΔx and show that k=Ex. (f) Calculate the value of k for a typical metal for which E = 1.2 GN/m[itex]^{2}[/itex] and x=0.16 nm.

Homework Equations



f=kΔx

k=Ex

stress = modulus x strain

F/A = E ΔL/L

ΔL= FL/(EA)



The Attempt at a Solution



Part f is probably the only part of the problem I feel confident about doing. As far parts a through e, I can't make heads or tails of how to derive the expressions involving atomic separation x. This problem seems to somewhat relate Hooke's law with elastic materials and a picture on my book describes the interatomic forces in the material as spring-like.
 
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  • #2
What is the difference between the modulus of elasticity in compresão and modulus of elasticity in tension?
Which Modulus of elasticity should employ a finite element model?
 

1. What is elasticity and why is it important?

Elasticity is the property of a material that describes its ability to return to its original shape after being deformed. It is important because it allows materials to withstand stress and strain without permanent damage, making them useful for a variety of applications.

2. What is Young's Modulus and how is it related to elasticity?

Young's Modulus, also known as the modulus of elasticity, is a measure of a material's stiffness. It is defined as the ratio of stress to strain in a material under linear elastic deformation. In other words, it is a measure of how much a material will deform under a certain amount of stress. It is directly related to elasticity, as a higher Young's Modulus indicates a stiffer and more elastic material.

3. Can elasticity and Young's Modulus change with temperature?

Yes, both elasticity and Young's Modulus can change with temperature. As temperature increases, the atoms in a material vibrate more vigorously, which can cause them to move farther apart and decrease the material's stiffness. This is known as thermal expansion and can affect a material's elasticity and Young's Modulus.

4. How is Young's Modulus measured?

Young's Modulus is typically measured using a tensile test, where a sample of the material is stretched until it breaks. The stress and strain at various points during the test are recorded, and Young's Modulus can be calculated by dividing the stress by the strain.

5. Can different materials have the same Young's Modulus?

Yes, it is possible for different materials to have the same Young's Modulus. This is because Young's Modulus is a measure of a material's stiffness, not its strength. Two materials can have the same Young's Modulus but differ significantly in their strength and other properties.

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