Stress, Strain and the Young Modulus
Stress is axial force divided by cross-sectional area. Strain is extension divided by original length. In the linear elastic region, the gradient of stress against strain gives the Young modulus, a material property under stated test conditions.
What a learner can do afterwards
- Calculates stress in pascals and strain as a ratio for a wire under a stated load
- Finds the Young modulus as a gradient and quotes it with its unit
- Explains why two wires of the same material but different thickness give the same modulus
1 · Read
You have compared force and extension. For a uniform wire pulled along its length, tensile stress = force / cross-sectional area. Use newtons and square metres to get pascals: 1 Pa = 1 N/m squared. For circular diameter d, area = pi × (d/2) squared. Convert d to metres before squaring; 1 mm = 0.001 m. Use the wire's tension, not the sum of the opposite end forces.
Tensile strain = extension / original length. Extension is loaded length minus original length. Use matching length units so they cancel. A 1.5 m wire extending by 0.003 m has strain 0.003 / 1.5 = 0.002, with no units. Strain is the fractional increase in length, not the extension itself.
In the linear elastic region through the origin, Young modulus Y = stress / strain = (force/area) / (extension/original length). Plot stress vertically and strain horizontally. Y is the gradient: change in stress / change in strain. Its unit is Pa because strain has no units. For the bars, 100 MPa / 0.001 = 100000000000 Pa. A larger Y means less strain at the same stress.
Compare wires of the same material, processing and direction at the same temperature. In the linear elastic region, Y does not depend on length or area. At fixed force, doubling area halves stress and strain, leaving their ratio unchanged. A sample's force-extension stiffness still depends on its dimensions. The material modulus and the sample's stiffness are different quantities.
Elastic deformation disappears after unloading. The proportionality limit ends the straight-line region; the elastic limit marks the onset of permanent deformation. A curved region can still be elastic. Do not use one constant stress/strain ratio outside the linear region. Working below yield with a suitable safety factor reduces the risk of permanent deformation. Young modulus alone cannot establish a safe working load.
Stress = force/area. Strain = extension/original length. Their ratio gives Young modulus in the linear elastic region, in pascals.
Lesson, questions and flashcards written for this dataset. Mechanics references were consulted for facts and calculation methods.
Readings written for this dataset. Flashcards draw on mechanics fact records. Video reference credited to Khan Academy: Intro to springs and Hooke's law.
More to read
Measurements needed for a wire test
Record a wire's unloaded length and its original cross-sectional area. Under tension, measure the increase in length. Dividing that increase by the original length gives strain. Dividing the tension by area gives stress.
For a circular wire, halve the diameter to obtain radius, then calculate pi times radius squared. Convert the diameter to metres first if the area must be in square metres. Mixing millimetres and metres before squaring produces a large area error.
Repeat the measurements at several small loads. Plot strain horizontally and stress vertically. The straight-line gradient in the linear elastic region is the Young modulus. If the graph curves, a single constant stress/strain ratio no longer describes that region.
A material property and a sample property
Two wires made from the same material can extend by different amounts under the same force. A longer wire extends more, while a larger cross-sectional area reduces extension. In the linear elastic region, extension = force × original length / (area × Young modulus).
The force-extension stiffness belongs to the particular sample. Young modulus describes the material response after accounting for length and area. Compare material, processing, temperature and loading direction before expecting samples to share a modulus.
Young modulus applies to axial stretching or compression. Sliding layers sideways involves shear deformation and a shear modulus. Pressure that changes a body's volume involves a bulk modulus. These moduli describe different responses and cannot be substituted for one another.
What unloading reveals
A loading graph alone does not tell you whether deformation is permanent. Remove the load and check whether the sample returns to its original dimensions. A curved loading region may still be elastic. The proportionality limit and elastic limit describe different changes.
Recovery of shape also does not require complete recovery of energy. Loading and unloading can follow different paths and dissipate energy, as in the earlier rubber-band example.
Materials do not all show the same yield plateau, necking or fracture pattern. Some break with little permanent deformation. Working stress should remain below yield with a suitable safety factor. Young modulus measures stiffness; it does not by itself give a safe load for a bridge, cable or other structure.
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