The Physics
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Opus in profectus

# 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 (Δℓ/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 = Δℓ/ℓ0) due to a tensile stress.
• Compressive strain is the fractional decrease in length of an object (ε = Δℓ/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 Δℓ 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 (Δy/y0 or Δz/z0) to axial strain (Δx/x0) is called Poisson's ratio.
• The symbol for Possion's ratio is ν (nu).  ν = − Δy/y0 = − Δz/z0 Δx/x0 Δx/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 Δx 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 ΔV 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].