Elasticity of Materials - Interactive Visualization

Interactive simulation demonstrating Hooke's Law, stress-strain relationships, and material properties

Spring-Mass System

Original Length (L₀): 100 mm
Current Length (L): 100 mm
Elongation (ΔL): 0 mm
Elastic Potential Energy: 0 J

Applied Force (F)

Spring Constant (k)

Material Type

Material Properties

Young's Modulus (E): 200 GPa
Elastic Limit: 250 MPa
Yield Strength: 250 MPa

Force vs Displacement (F-x) Curve

Elastic Region (Hooke's Law)
Plastic Region
Fracture Point
Current Position

Material Comparison: Stress-Strain Curves

Steel - High Modulus, Large Linear Range
Rubber - Low Modulus, Good Elasticity
Concrete - Brittle, Small Elastic Range

Mathematical Foundation

Hooke's Law

F = k · x

Stress

σ = F / A

Strain

ε = ΔL / L₀

Young's Modulus

E = σ / ε

Elastic Potential Energy

Eₚ = ½kx²

Stress-Strain Relation

σ = E · ε

What is Elasticity?

Elasticity is the ability of a material to return to its original shape after being deformed by an external force. When you stretch a spring within its elastic limit, it will return to its original length when the force is removed. This behavior is described by Hooke's Law: the force is directly proportional to the displacement.

Stress-Strain Curve Regions

Elastic Region (Hooke's Law)

In this region, the material follows Hooke's Law (σ = E·ε) with a linear relationship between stress and strain. When the load is removed, the material returns to its original shape completely. The slope of this line is Young's Modulus (E), representing the material's stiffness.

Plastic Region

Beyond the elastic limit, the material undergoes permanent deformation. The stress-strain relationship becomes non-linear as the material's structure begins to change. When the load is removed, the material will not return to its original length.

Fracture Point

At this point, the material breaks or fractures. The stress at fracture is called the ultimate tensile strength. Different materials have very different behaviors at this point.

Material Properties Comparison

Steel

  • High Young's Modulus (~200 GPa)
  • Large elastic region
  • Ductile behavior
  • Used in construction, tools

Rubber

  • Low Young's Modulus (~0.01-0.1 GPa)
  • Very elastic (large strain)
  • Hyperelastic behavior
  • Used in seals, tires

Concrete

  • Medium Young's Modulus (~30 GPa)
  • Small elastic region
  • Brittle (sudden fracture)
  • Weak in tension, strong in compression

Real-World Applications

  • Mechanical Springs: Used in vehicle suspensions, watches, and various mechanisms. The spring constant determines how much force is needed for a given displacement.
  • Structural Engineering: Buildings and bridges must operate within the elastic range of their materials. Steel reinforcement provides tensile strength while concrete handles compression.
  • Material Testing: Tensile tests measure stress-strain curves to determine material properties like Young's Modulus, yield strength, and ultimate tensile strength.
  • Biomaterials: Understanding elasticity is crucial for implants and prosthetics. Materials must match the mechanical properties of biological tissues.
  • Nanotechnology: Carbon nanotubes have exceptionally high Young's Modulus (~1 TPa), making them ideal for reinforcing composite materials.

Experiment Guide

Start with Small Forces

Begin with forces under 20N to observe Hooke's Law clearly. Notice the linear relationship between force and displacement in the elastic region.

Compare Materials

Try different materials to see how stiffness affects elongation. Steel stretches very little, while rubber stretches significantly under the same force.

Observe Energy Storage

Watch how elastic potential energy increases quadratically with displacement (Eₚ = ½kx²). The area under the F-x curve represents the stored energy.