Mechanical Properties of Solids

The One Thing

The mechanical behaviour of a solid is determined by how its deformation responds to applied stress. Within the elastic limit, stress is proportional to strain, while elastic moduli quantify a material’s resistance to changes in length, volume, or shape.

Reactions, Processes and Experiments

Key Terms

• Elasticity: The property of a material by which it regains its original shape and size after the deforming force is removed.
• Plasticity: The property of a material by which it undergoes permanent deformation and does not completely regain its original shape after the force is removed.
• Deformation: A change in the shape, size, or volume of a body due to an external force.
• Restoring force: An internal force developed in a deformed body that opposes the deformation and tends to restore the original shape.
• Stress: The internal restoring force acting normally or tangentially per unit area of cross-section. Its SI unit is pascal (Pa).
• Normal stress: Stress acting perpendicular to the cross-sectional area. Tensile stress pulls a body apart, while compressive stress pushes it together.
• Tangential or shearing stress: Stress acting parallel to the surface or cross-sectional area, tending to change the shape of a body.
• Strain: The ratio of the change in a physical dimension to its original dimension. Strain is dimensionless.
• Longitudinal strain: The ratio of change in length to original length: longitudinal strain = ΔL/L.
• Volumetric strain: The ratio of change in volume to original volume: volumetric strain = ΔV/V.
• Shearing strain: The angular deformation produced by tangential stress; for small deformation, it is approximately equal to the lateral displacement divided by the perpendicular height.
• Elastic limit: The greatest value of stress for which a material completely regains its original shape after the stress is removed.
• Hooke's law: Within the elastic limit, stress is directly proportional to strain: stress ∝ strain.
• Young's modulus: The ratio of longitudinal stress to longitudinal strain: Y = (F/A)/(ΔL/L) = FL/(AΔL). It measures resistance to change in length.
• Bulk modulus: The ratio of hydraulic or volumetric stress to volumetric strain: K = -ΔP/(ΔV/V). The negative sign indicates that pressure increase generally causes a decrease in volume.
• Shear modulus: The ratio of tangential stress to shearing strain: G = (F/A)/(Δx/L) = FL/(AΔx). It measures resistance to change in shape.
• Poisson's ratio: The negative ratio of lateral strain to longitudinal strain: σ = - lateral strain/longitudinal strain. It has no unit.
• Elastic fatigue: The weakening of a material's elastic behaviour after repeated loading and unloading.
• Elastic hysteresis: The phenomenon in which strain does not immediately return to zero when stress is removed, causing energy loss as heat during a loading cycle.
• Ductility: The ability of a material to undergo large plastic deformation and be drawn into wires.
• Malleability: The ability of a material to be hammered or rolled into thin sheets.
• Brittleness: The tendency of a material to fracture with little or no plastic deformation.
• Elastic energy: The energy stored in a body due to elastic deformation. For a stretched wire, energy stored = 1/2 × force × extension.

Easily Confused

• Elasticity and plasticity: Elasticity involves recovery of the original shape after unloading; plasticity involves permanent deformation.
• Elasticity and flexibility: A material may undergo large deformation yet be less elastic; steel is more elastic than rubber because it has a larger Young’s modulus and produces less strain for the same stress.
• Stress and strain: Stress is restoring force per unit area, whereas strain is the ratio of deformation to the original dimension.
• Normal stress and tangential stress: Normal stress acts perpendicular to an area and changes length, whereas tangential stress acts parallel to an area and changes shape.
• Young’s modulus, bulk modulus and shear modulus: Young’s modulus measures resistance to change in length, bulk modulus to change in volume, and shear modulus to change in shape.
• Ductility and malleability: Ductility concerns drawing a material into wires; malleability concerns hammering or rolling it into thin sheets.
• Elastic fatigue and elastic hysteresis: Elastic fatigue is weakening after repeated loading and unloading, whereas elastic hysteresis is delayed strain recovery and energy loss as heat during a loading cycle.
• Elastic limit and proportional limit: The proportional region is the straight-line region in which stress is proportional to strain; the elastic limit is the greatest stress for complete recovery of shape.
• Elastic modulus and stress or strain: Elastic moduli are properties of materials, whereas stress and strain describe the current state of loading and deformation.
• Stress and force: Stress is force per unit area, so equal forces can produce different stresses on different cross-sectional areas.

What Gets Asked

• Define stress, strain, restoring force, elastic limit, or one of the elastic moduli; marks are lost by confusing stress, which is measured in pascals, with strain, which is dimensionless.
• Calculate extension, Young’s modulus, bulk modulus, shear modulus, or Poisson’s ratio using the stated equations; marks are lost by using the wrong type of deformation or omitting the negative sign in $K=-\Delta P/(\Delta V/V)$.
• Derive or apply the wire-extension relation $\Delta L=FL/(AY)$; marks are lost by treating $Y$ as a force rather than a material property.
• Interpret a stress-strain curve, including the proportional limit, elastic limit, yield region, plastic region, ultimate tensile stress, and fracture point; marks are lost by identifying the proportional limit and elastic limit as the same point.
• Compare materials such as steel and rubber; marks are lost by stating that rubber is more elastic merely because it is more flexible or undergoes greater deformation.
• Explain elastic energy and energy density using $U=1/2F\Delta L=1/2\times (stress)\times (strain)\times volume$; marks are lost by confusing total stored energy with energy stored per unit volume.