Mechanical Properties of Solids · Lesson 2 of 7
Hooke’s Law and Stress–Strain Curves
“Use the stress–strain graph to distinguish linear elasticity, yielding, plastic deformation and fracture, and compare material responses.”
• State Hooke’s law and distinguish proportionality from recoverability. • Interpret the labelled regions and characteristic stresses of a metal curve. • Read Young’s modulus from an initial stress–strain slope. • Compare stiffness, strength, ductility and the nonlinear response of elastomers.
Hooke’s law is a material response law
Stress and strain describe what is applied internally and what deformation results. They do not by themselves predict the response. Experiment supplies the relation. For many solids under sufficiently small deformation, doubling the stress doubles the strain. Stress is then proportional to strain. This empirical observation is Hooke’s law; the proportionality constant is an elastic modulus.
The word empirical means established by observation, not a mathematical identity that every material must satisfy. A modulus has the units of stress because strain is dimensionless. Its numerical value depends on the material and physical conditions such as temperature. A large modulus means that a given small strain requires a large stress. Hooke’s law should never be extended automatically to a heavily deformed specimen.
How a stress–strain curve is measured
Start with a test wire or rod of known initial length L and area A. Increase the tensile load in small steps and measure the extension. Calculate stress from F/A and strain from ΔL/L, then plot stress vertically and strain horizontally. The graph separates a material property from specimen dimensions more effectively than a raw force–extension graph. Keep the convention for stress fixed; the source’s typical metal curve uses the original area when interpreting the fall after ultimate strength.
From O to A the curve is straight: stress/strain is constant and Hooke’s law holds. Remove the load and the specimen returns to its original dimensions. Between A and B the curve bends, so one constant no longer describes stress divided by strain; in the source’s idealized curve the deformation is still recoverable. Thus linear response and elastic response are different tests. Proportionality is a stricter condition than recovery.
Yielding and permanent set
The source identifies B as the yield point or elastic limit, with yield strength σy. Real engineering definitions may distinguish these more precisely, but the role here is clear: beyond the recoverable range, unloading leaves a permanent strain. At a point C in the plastic region, reducing the load brings stress to zero at a nonzero strain. That residual extension is the permanent set. The dashed unloading line ends to the right of the origin, rather than returning to O.
Between B and D, a comparatively small extra stress can cause a substantial increase in strain. At D the engineering tensile stress reaches its greatest value, the ultimate tensile strength σu. Beyond D, deformation can continue while the applied force falls, and the specimen eventually fractures at E. A declining engineering-stress curve does not mean the material has become an ordinary spring with negative stiffness; the deformation and changing specimen geometry near failure need a different interpretation.
A ductile material can undergo appreciable plastic deformation before breaking. A brittle material fractures with little plastic deformation. The source illustrates the contrast through the separation of the ultimate-strength and fracture portions of the curve. Use the whole curve and the amount of plastic strain; a high Young’s modulus by itself does not say that the material is strong or ductile.
Read stiffness and strength from different graph features
Problem
Find the Young’s modulus and approximate yield strength from the displayed source-based graph.
- 1.Use the initial straight portion: at strain 0.002 the stress is 150 MPa = 150 × 10⁶ Pa.
- 2.Y = (150 × 10⁶)/0.002 = 7.5 × 10¹⁰ Pa = 75 GPa. Strain has no unit, so the slope has units Pa.
- 3.The curve departs from the straight response near a stress of 300 MPa. Its approximate yield strength on this graph is 3 × 10⁸ Pa.
- 4.The first calculation concerns stiffness; the second identifies the stress scale for yielding. They are different quantities with the same unit.
When two graphs use the same axis scales, a steeper initial stress-versus-strain line means greater Young’s modulus. In the displayed A/B comparison, A has the greater initial slope and reaches fracture at the greater stress, so A is both stiffer and stronger in this tensile test. These two outcomes happen to agree here; they need not agree for every pair of materials. If graph axes were interchanged, the initial strain-versus-stress slope would be 1/Y. Always read the axis labels rather than memorizing “steeper means larger Y” without conditions.
Elastomers: large recovery without a straight line
Rubber and the elastic tissue of the aorta can be stretched through large strains and recover their shape. Such materials are called elastomers. The aorta curve becomes progressively steeper: later deformation requires a larger extra stress for the same extra strain. It does not have the broad, clearly separated plastic region of the idealized metal curve. A single constant Y cannot describe its entire response.
In ordinary speech, “more elastic” often means “easier to stretch”. In the school comparison made here, a material with a larger modulus resists a given small strain more strongly; steel has a much larger Young’s modulus than rubber. Keep that stiffness comparison separate from recoverability and from the amount of deformation possible before failure. A rubber band can stretch much farther and recover while still being far less stiff than steel.
Quick Quiz
Quiz
A specimen returns to its original length after unloading, but its stress–strain curve is curved. Which conclusion is valid?
Recoverability defines elastic response. Hooke’s law additionally requires a linear relation.
What does permanent set mean?
The plastic unloading path reaches zero applied stress at nonzero strain.
On a stress-versus-strain graph, which feature determines Young’s modulus in the Hookean range?
Y is the increase in stress divided by the corresponding increase in strain in the linear region.
Which statement about rubber and steel is correct?
Large extensibility and recoverability are different from a high stress/strain ratio.
A material withstands a high fracture stress but has a shallow initial stress–strain slope. What follows?
Strength concerns failure stress; stiffness concerns the initial slope. Neither alone establishes ductility.
Practice Problems
- In the initial straight region of a stress–strain test, strain 0.00020 corresponds to stress 40 MPa, and strain 0.00040 corresponds to stress 80 MPa. Determine Young’s modulus. Predict the stress at strain 0.00070 if that strain is still within the same proportional region.
- A metal’s loading curve is straight from O to A, curved but completely recoverable from A to B, and leaves a permanent extension when loaded beyond B. Identify the proportional limit and the boundary of recoverable behaviour in this model. Explain why the curved A-to-B response can be elastic without obeying Hooke’s law.
- Material P has an initial Young’s modulus of 200 GPa and fractures at 220 MPa. Material Q has an initial Young’s modulus of 80 GPa and fractures at 420 MPa. Which is stiffer, and which has the higher tensile fracture strength? Explain why these data alone do not establish which is more ductile.
- A 0.50 m specimen extends by 4.0 mm during loading and retains 1.0 mm extension after complete unloading. Calculate the recovered extension and the residual strain as a fraction and percentage. Explain what the retained extension says about the loading range reached.
- A designer compares a steep, nearly straight initial stress–strain graph with a broad curved graph showing large recoverable strains. Explain which response is more suitable for a part that should deform very little and which could suit a flexible elastic seal. Distinguish stiffness, elastic recovery and strength in your reasoning.
Key Takeaways
• Hooke’s law describes the linear part of the response; elastic recovery can extend beyond linearity. • Yield strength, ultimate tensile strength and fracture identify different features of the loading curve. • Young’s modulus comes from the initial stress-versus-strain slope, not a breaking point. • Elastomers can sustain large recoverable strains without obeying one linear stress–strain law over the whole range.