Stress Strain Equations Calculator

Mechanics of Materials - Solid Formulas


Problem:

Solve for Young's modulus

Young's modulus

Enter Inputs:

stress
strain
unitless

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Solution:

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stress stress
force force
area area

strain strain
change in length change in length
original length original length

Young's modulusYoung's modulus
stressstress
strainstrain

References - Books

Tipler, Paul A.. 1995. Physics For Scientists and Engineers. Worth Publishers. 3rd ed.


Background

Young's modulus, te modulus of elasticity, measures how stiff a solid material is. It shows the relationship between stress (force per unit area) and strain, which is the proportional deformation of the object. It is a fundamental concept in engineering and materials science.


Equation

The formula to calculate Young's modulus ( E ) is given by:

E = σ / ε

Where:

  • E is Young's modulus in pascals (Pa)
  • σ is the stress in pascals (Pa)
  • ε is the strain (dimensionless)

How to Solve

To solve for Young's modulus, follow these steps:

  • Determine the stress (σ): This is calculated by dividing the force applied (F) by the area (A) over which the force is distributed. ( σ = F / A ).
  • Determine the strain (ε): Strain is calculated as the change in length (ΔL) divided by the original length (L). ( ε = ΔL / L ).
  • Apply the Formula: Substitute the values of stress and strain into Young's modulus equation ( E = σ / ε ).

Example

Consider a metallic rod subjected to a tensile force of 1000 N. Assume the original length of the rod is 2 meters, and under the load, it stretches by 1 mm. The cross-sectional area of the rod is 0.01 square meters.

Calculate the stress:

σ = 1000 N / 0.01 m2 = 100000 Pa

Calculate the strain:

ε = 0.001 m / 2 m = 0.0005

Calculate Young's modulus:

E = 100000 Pa / 0.0005 = 200 x 106 Pa

Therefore, Young's modulus for the material of the rod is 200 x 106 Pa.


Fields/Degrees

  • Mechanical Engineering
  • Civil Engineering
  • Aerospace Engineering
  • Materials Science
  • Biomechanics

Real Life Applications

  • Construction: Determining suitable materials for beams, columns, and slabs.
  • Automotive: Designing components such as springs, chassis, and engine parts.
  • Aerospace: Material selection for aircraft frames and other structural elements.
  • Consumer Electronics: In designing thin, robust cases and frames for smartphones and laptops.
  • Medical Devices: Design of prosthetics and other supportive equipment.

Common Mistakes

  • Mixing units (e.g., using mm versus meters for length).
  • Not accounting for uniform cross-sectional area along the length of the material.
  • Neglecting the effect of temperature and environment on material properties.
  • Overlooking existing microscopic stress while measuring applied stress.
  • Applying the modulus of elasticity beyond the elastic limit of the material.

Frequently Asked Questions

  • Is Young's modulus the same for all materials?
    No, Young's modulus varies from material to material, reflecting different stiffness levels.
  • Can Young's modulus be negative?
    No, Young's modulus is a measure of stiffness and cannot be negative.
  • Does temperature affect Young's modulus?
    Yes, with most materials, the increasing temperature tends to lower Young's modulus.
  • How does Young's modulus relate to strength?
    Young's modulus measures stiffness, not strength. High Young's modulus means a material is stiff but not necessarily strong.
  • Is it applicable only under tensile stress?
    Young's modulus is generally considered under both tensile and compressive stress as long as the material remains within the elastic limit.
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