Bulk properties of solids: Elastic strain energy, breaking stress.

The idea

When a material is stretched elastically, work is done against the interatomic forces and stored as elastic strain energy; on unloading, this energy is recovered. For a wire obeying Hooke's law, the stored energy equals the area under the force–extension graph, which for a straight-line region is ½FΔL, or equivalently ½k(ΔL)². Beyond the elastic limit, energy is dissipated as heat and permanent deformation. Breaking stress is the stress at which a material fractures; it depends on the material and on flaws, and is not the same as the ultimate tensile stress, which is the maximum stress the material can withstand before necking. A brittle material such as glass breaks at its elastic limit with little plastic deformation, whereas a ductile material such as copper extends and necks before fracture.

What earns the marks

• Elastic strain energy is the energy stored in a stretched or compressed material when it is deformed elastically and is recovered when the deforming force is removed.
• For a material obeying Hooke's law, the stored energy equals the area under the force–extension graph, which is ½FΔL or ½k(ΔL)².
• Work done in stretching a wire is stored as elastic strain energy; beyond the elastic limit, some energy is dissipated as thermal energy and permanent deformation remains.
• Breaking stress is the stress at which a material fractures; it is a property of the material and is affected by surface flaws and internal defects.
• Breaking stress is distinct from ultimate tensile stress: the ultimate tensile stress is the maximum stress reached before necking, while breaking stress is the stress at fracture.
• A brittle material fractures with little plastic deformation, so its force–extension graph is approximately linear up to a sudden break.
• A ductile material shows plastic deformation and necking before fracture, so its force–extension graph curves and extends well beyond the elastic limit.

Common mistakes and how to put them right

• Confusing elastic strain energy with the total work done when the material is stretched beyond its elastic limit.

  Put it right: Only the elastic part is recoverable, while plastic deformation dissipates energy.

• Assuming breaking stress always equals ultimate tensile stress.

  Put it right: The ultimate tensile stress is the maximum stress before necking, whereas breaking stress is the stress at fracture, which can be lower.

• Using ½FΔL for a non-linear force–extension graph.

  Put it right: The area under the graph must be found, and ½FΔL applies only to the straight-line Hooke's law region.

Exam tips

• Sketch the force–extension graph and shade the area representing stored energy before substituting values, so the correct region is used.
• State clearly whether the material is brittle or ductile and link this to the shape of the force–extension graph and the fracture behaviour.
• When comparing breaking stress with ultimate tensile stress, refer to the stress–strain graph and identify the fracture point and the maximum point separately.