Free Pearson Edexcel International A Level Physics lesson
Stress, strain and deformation energy
Separate material behaviour from sample size.
What you’ll learn
- Calculate tensile and compressive stress and strain with correct units.
- Interpret deformation graphs, elastic limits, yielding and breaking stress.
- Estimate deformation work and distinguish stored and dissipated energy.
Two ratios remove geometry from the comparison
Tensile stress σ (sigma) is the tensile force divided by the original cross-sectional area perpendicular to that force: σ = F/A. Compressive stress uses the same force-per-area idea for squeezing. A sample held in tension has equal and opposite forces at its ends; use the magnitude transmitted through a cross-section, not their zero external resultant and not twice one end force. Stress is measured in pascals, where 1 Pa = 1 N m⁻² and 1 MPa = 10⁶ Pa.
Longitudinal strain ε (epsilon) is change in length divided by original gauge length: ε = ΔL/L₀. It is dimensionless because the two lengths use the same unit. A strain of 0.002 is 0.2%, not 0.002%. Tensile strain lengthens a specimen and compressive strain shortens it. We normally quote positive magnitudes and identify tension or compression in words here; a signed convention is also valid when consistently defined.
Stress and strain allow samples with different dimensions to be compared. Doubling area halves stress at the same force. Two samples that extend by the same number of millimetres need not have the same strain if their original lengths differ. For a round wire, A = πd²/4 with d converted to metres before squaring. These engineering ratios use original dimensions; large deformation and necking can make the actual local area different, so do not interpret them as a detailed local stress model after severe deformation.
σ = F/A ε = ΔL/L₀ A = πd²/4 for a circular wire
Symbols and units
- σ is tensile stress in Pa = N m⁻²; F is tensile force in N; A is original cross-sectional area in m².
- ε is strain, the dimensionless extension ratio ΔL/L₀; ΔL and L₀ are in the same length unit.
How the equation explains the idea
- Divide force by area to compare loading intensity: σ = F/A. Divide extension by original length to compare relative deformation: ε = ΔL/L₀.
- For a circular wire A = π(d/2)² = πd²/4. Doubling diameter quadruples area and quarters stress under the same force.
- The ratios remove specimen geometry from a material comparison; strain has no unit but is not the same numerical quantity as extension.
Conditions. Two ideal linear-elastic models obey σ = Eε with E = 100 GPa and 200 GPa across the plotted small-strain range. Stress is displayed in MPa; no yield or failure behaviour is asserted.
Strain / dimensionless; Tensile stress / MPa.
At strain 0.001, stresses are 100 MPa and 200 MPa. The higher gradient means a larger Young modulus. These material plots remove specimen length and area, unlike a raw force–extension plot.
Open the interactive lesson to view the plotted curves and their coordinate tables.
Worked example
One wire, two ratios
- A wire has original cross-sectional area 0.20 mm² = 2.0 × 10⁻⁷ m² and gauge length 1.50 m.
- A 30 N tensile force produces an extension of 1.20 mm = 0.00120 m.
- Stress = 30/(2.0 × 10⁻⁷) = 1.5 × 10⁸ Pa; strain = 0.00120/1.50 = 8.0 × 10⁻⁴.
Stress is 150 MPa and strain is 0.00080, or 0.080%.
Keep this idea. Use original area and gauge length, and distinguish stress in pascals from dimensionless strain.
Read shape, recovery and failure separately
On a force–extension graph with force vertically and extension horizontally, the initial straight-line gradient is sample stiffness. A force–compression graph can use positive shortening on the horizontal axis and compressive force magnitude vertically; label this convention so that increasing compression is clear. On stress–strain axes, the initial straight gradient describes the material’s Young modulus. Dividing the force coordinates by area and the extension coordinates by original length changes the scales; the physical events refer to the same loading experiment.
The limit of proportionality is where the straight-line relationship ends. Elastic deformation is recovered after unloading, while plastic deformation leaves a permanent change. The elastic limit is the greatest loading level that still allows full recovery within measurement resolution; it need not coincide exactly with the proportional limit. A yield region is substantial additional plastic strain for little additional stress. Some materials do not have a sharp visible yield point. A loading curve alone cannot establish complete recovery: unloading evidence is needed.
A ductile material can undergo appreciable plastic extension before fracture; a brittle material fractures with little plastic deformation. Brittle does not mean low stiffness, and stiffness does not specify strength. Breaking stress is the engineering stress at fracture: breaking force divided by original area. For a ductile specimen that necks, the maximum engineering stress may occur before the final breaking stress. Different tension and compression behaviour matters in applications; for example a short specimen may withstand compression well but crack in tension, while a long slender specimen may buckle before the material’s compressive failure limit is reached.
Initial force–extension slope = stiffness initial stress–strain slope = Young modulus breaking stress = Fbreak/A₀
Symbols and units
- Stiffness k is force–extension gradient in N m⁻¹; Young modulus E is stress–strain gradient in Pa.
- Fbreak is breaking force in N and A₀ is original cross-sectional area in m².
How the equation explains the idea
- In a linear elastic region, F = kx while σ = Eε. The two slopes have different units and cannot be interchanged.
- Breaking engineering stress is Fbreak/A₀. Compare a failure point separately from the initial slope: a stiff specimen need not have a large breaking strain.
- Recovery is determined by unloading behaviour; departure from proportionality alone does not establish permanent deformation.
Worked example
Interpret a loading and unloading record
- A wire’s loading curve becomes nonlinear above 80 MPa; unloading from 90 MPa leaves no measurable residual strain.
- Unloading after a later 130 MPa load leaves strain 0.006, so that later deformation included plastic strain.
- The wire fractures at force 44 N with original area 2.0 × 10⁻⁷ m²: breaking stress = 2.2 × 10⁸ Pa.
Nonlinearity alone did not prove plastic deformation; the later residual strain does. Breaking stress is 220 MPa.
Keep this idea. Read the axes and separate proportionality, recovery, plastic deformation and fracture; use unloading data when judging elasticity.
Area records work, with a condition on recovery
During a small extension increment, work transferred to a slowly loaded sample is approximately force times that increment. Adding these contributions gives the area under the force–extension loading curve. For linear loading from zero to force F at extension x, the triangular area is ½Fx = ½kx². If a sample is already under a preload, the additional area includes the preload rectangle as well as the increase in force: it is not generally ½ΔFΔx alone.
For a nonlinear curve, estimate the actual area using narrow trapezia or a suitably fine grid. Work put in is not automatically all recoverable elastic energy. The area under a quasistatic unloading curve is the work returned as the specimen shortens. The difference between loading work and returned work is dissipated, for example through internal friction or plastic deformation. A loading–unloading loop encloses that loss. A reversible nonlinear specimen can return its loading work even though the loading area is not triangular.
Energy per original volume follows from dividing by A₀L₀. In the uniform small-strain model, force becomes stress and extension becomes strain, so area under stress–strain gives work per volume in J m⁻³. A linear elastic sample stores u = ½σε = ½Eε². This derived relationship is useful for comparing sample sizes; multiplying u by original volume gives total energy. Use the whole curve or unloading evidence outside the linear reversible region, and remember that a large area to fracture measures absorbed work rather than wholly recoverable storage.
W = area under F–x linear elastic: Eelastic = ½Fx u = Eelastic/(A₀L₀) = ½σε
Symbols and units
- W and Eelastic are energies in J; F is force in N; x is extension in m.
- u is energy per original volume in J m⁻³; A₀L₀ is original volume in m³; σ is stress in Pa and ε is dimensionless strain.
How the equation explains the idea
- For linear reversible loading, area under F against x is Eelastic = Fx/2.
- Divide by A₀L₀: u = (F/A₀)(x/L₀)/2 = σε/2. Pa and J m⁻³ are equivalent units.
- For a nonlinear curve, use its actual area; for unloading with dissipation, recoverable energy follows the unloading curve rather than all the input work.
Conditions. Computed u = ½Eε² for a linear elastic material with E = 200 GPa. Energy density in J m⁻³ is divided by 1000 for the plotted kJ m⁻³ scale.
Strain / dimensionless; Elastic energy density / kJ m⁻³.
At strain 0.001, energy density is 100 kJ m⁻³; at strain 0.002 it is 400 kJ m⁻³. Doubling strain quadruples energy density. This is recoverable energy only for the stated reversible model.
Open the interactive lesson to view the plotted curves and their coordinate tables.
Worked example
Nonlinear loading and incomplete recovery
- A loading record gives (extension m, force N): (0, 0), (0.002, 4), (0.004, 10).
- Trapezium work estimate = ½(0 + 4) × 0.002 + ½(4 + 10) × 0.002 = 0.018 J.
- An independently measured unloading area is 0.014 J, so about 0.004 J was not returned.
About 0.018 J is put in, 0.014 J is returned and 0.004 J is dissipated in this cycle.
Keep this idea. Use the area under the correct curve and distinguish total work, recoverable energy and energy per volume.
Open the interactive lesson for free practice and reviewed three-question samples when available. New full question sets and timed unit checks are in preparation. Existing complete Physics revision packs retain paid access; existing free teaching PDFs remain free.
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