Elasticity

Summary

• Generalized Hooke's law
  • stress
    • A stress is a force or combination of forces distributed throughout the whole of an object that acts to deform it.
    • Stresses take the general form of force divided by area (F/A).
    • The SI unit of stress is the pascal or newton per meter sqared [Pa = N/m²]
  • strain
    • A strain is any deformation of an object measured geometrically as a fraction of some original value.
    • Strains take the general form of a change in one geometric quantity divided by the original value of that quantity or a similar quantity with the same unit (∆ℓ/ℓ₀, ∆V/V₀, etc.).
    • Strains are always dimensionless or unitless ratios [m/m, m³/m³, etc.]
  • modulus (plural, moduli)
    • Stress is directly proportional to strain.
    • An elastic modulus is the ratio of some stress to the strain caused by that stress.
    • The SI unit of all elastic moduli is the pascal or newton per meter squared [Pa = N/m²], but gigapascals [1 GPa = 10⁹ Pa] are more commonly used.
    • Elastic moduli are properties of materials, not the objects made from those materials.
• Tension and compression
  • stress
    • Tensile stress is the normal force per area (σ = F/A) that causes an object to increase in length.
    • Compressive stress is the normal force per area (σ = F/A) that causes an object to decrease in length.
  • strain
  • Tensile strain is the fractional increase in length of an object (ε = ∆ℓ/ℓ₀) due to a tensile stress.
  • Compressive strain is the fractional decrease in length of an object (ε = ∆ℓ/ℓ₀) due to a compressive stress.
  • modulus
    • Young's modulus or elastic modulus is the ratio of tensile stress to tensile strain or compressive stress to compressive strain.
    • The symbol for Young's modulus is E (for élasticité) or Y (for Young).

      F  = E  ∆ℓ A ℓ0

      σ = Eε

  • Poisson's ratio
    • Axial strain in one sense is usually accompanied by transverse strain in the opposite sense.
      • Tensile stress makes objects longer and thinner.
      • Compressive stress makes objects shorter and fatter.
    • The negative ratio of transverse strain (∆y/y₀ or ∆z/z₀) to axial strain (∆x/x₀) is called Poisson's ratio.
      • The symbol for Possion's ratio is ν (nu).

        ν = −	∆y/y0	= −	∆z/z0
        	∆x/x0		∆x/x0

• Shear
  • stress
    • Shear stress is the tangential force per area (τ = F/A) that causes one face of an object to become displaced parallel to the opposite face.
      • Shear stress changes rectangles into parallelograms.
  • strain
    • Shear strain is the fractional tangential displacement relative to the normal distance between opposite parallel faces (γ = ∆x/y) caused by a shear stress.
      • Shear strain is the tangent of the shear angle.
  • modulus
    • The shear modulus or rigidity modulus is the ratio of shear stress to shear strain.
      • The symbol for shear modulus is G (for glissement) or S (for shear)

        F  = G  ∆x A y

        τ = Gγ

• Bulk
  • stress
    • Pressure is the compressive stress (P = F/A) applied uniformly to all surfaces of an object.
      • Uniform compression or decompression changes the volume of objects but not their shape.
  • strain
    • Volume strain is the fractional change in volume of an object (θ = ∆V/V₀) due to a change in pressure.
  • modulus
    • The bulk modulus or compression modulus is the ratio of the increase in pressure to the relative decrease in volume.
      • The symbol for bulk modulus is K (for kompression) or B (for bulk).

        F  = K  ∆V A V0

        P = Κθ

    • The reciprocal of bulk modulus is called compressibility.
      • The symbol for compressibility is β (beta) or κ (kappa).

        β =	1
        	K

      • The SI unit of compressibility is the inverse pascal [Pa⁻¹].