This cheat sheet covers the engineering basics of stress, strain, and Hooke's law for materials under tension or compression. Students need these ideas to predict how rods, wires, beams, and structural parts deform when forces act on them. It helps connect physical measurements like force and extension to material properties used in design.
The reference is useful for physics, engineering, robotics, and materials science coursework.
The core ideas are that stress measures force per area, strain measures fractional change in length, and Hooke's law describes elastic behavior. Young's modulus links stress and strain with the formula E = stress / strain. A material behaves elastically when it returns to its original shape after the load is removed.
Engineers must also understand limits, including the proportional limit, elastic limit, yield point, and breaking point.
Key Facts
- Normal stress is calculated with stress = force / cross-sectional area, or sigma = F / A.
- Tensile strain is calculated with strain = change in length / original length, or epsilon = delta L / L0.
- Hooke's law for a spring is F = kx, where F is force, k is spring constant, and x is extension or compression.
- Young's modulus is E = stress / strain, so E = (F / A) / (delta L / L0).
- For a uniform rod or wire in the elastic region, extension is delta L = FL0 / AE.
- Stress is measured in pascals, where 1 Pa = 1 N/m^2, while strain has no units.
- A larger Young's modulus means a material is stiffer and stretches less for the same stress.
- Hooke's law applies only in the linear elastic region before the proportional limit is reached.
Vocabulary
- Stress
- Stress is the internal force per unit area in a material caused by an external load.
- Strain
- Strain is the fractional deformation of a material, calculated as change in length divided by original length.
- Hooke's Law
- Hooke's law states that force is proportional to extension for an elastic object within its proportional limit.
- Young's Modulus
- Young's modulus is a measure of stiffness equal to stress divided by strain in the elastic region.
- Elastic Limit
- The elastic limit is the maximum stress a material can experience and still return to its original shape.
- Yield Point
- The yield point is the stress at which a material begins to deform permanently.
Common Mistakes to Avoid
- Using diameter instead of cross-sectional area, which is wrong because stress = F / A requires area in square meters, not a length.
- Forgetting that strain has no units, which is wrong because strain is a ratio of two lengths with the same units.
- Applying Hooke's law after permanent deformation begins, which is wrong because F = kx and stress = E strain only apply in the elastic linear region.
- Mixing millimeters and meters in the same calculation, which is wrong because SI formulas require consistent units such as meters, newtons, and pascals.
- Assuming a stronger material is always stiffer, which is wrong because strength relates to failure stress while stiffness is described by Young's modulus.
Practice Questions
- 1 A wire has a force of 200 N applied to a cross-sectional area of 0.00004 m^2. What is the stress in the wire?
- 2 A 2.0 m metal rod stretches by 0.001 m under load. What is the strain?
- 3 A spring stretches 0.05 m when a 15 N force is applied. What is the spring constant k?
- 4 Explain why engineers must know both the elastic limit and Young's modulus when choosing a material for a bridge cable.
Understanding Stress, Strain & Hooke's Law Reference
A force does not tell the full story of how likely an object is to deform. The same pulling force can have a small effect on a thick steel cable but a large effect on a thin wire. This happens because the force is spread through the material’s cross section.
Small changes in diameter matter greatly because cross sectional area depends on diameter squared. A wire with twice the diameter has four times the area.
In classroom problems, check whether a diameter or a radius is given before finding area. Mixing millimetres with metres is another common source of large errors.
A stress strain graph shows the changing response of a sample as load increases. Its straight section is especially useful because equal increases in stress produce equal increases in strain. The gradient in this region represents stiffness through Young’s modulus.
Past this region, the graph begins to curve. The internal structure of the material is no longer responding in a simple proportional way. Near the yield point, permanent deformation can begin.
After unloading, the sample may become shorter or longer than its starting shape. This is called plastic deformation. Engineers avoid this region when a part must keep its exact dimensions.
The area under a force extension graph represents energy transferred to the object. For a spring behaving linearly, the graph is a triangle, so the stored elastic energy is one half times force times extension. This energy can return when the spring or elastic object recovers its shape.
Vehicle suspension, bows, trampolines, weighing scales, and door closers use this effect. Some materials do not return all the stored energy.
Rubber, foam, and many plastics lose part of it as heating during repeated stretching. Their loading and unloading graphs follow different paths, a behaviour called hysteresis.
Real components rarely experience one simple pull. A bridge member may be compressed, a bolt may be stretched, and a ruler may bend. Bending places one side in tension while the other side is in compression.
Repeated loading can cause fatigue even when each load is below the breaking force. Tiny cracks may grow over thousands of cycles, especially near scratches, holes, sharp corners, or screw threads. Engineers use smooth shapes, safety factors, and suitable materials to reduce this risk.
When solving problems, state the assumptions clearly. A formula for a uniform wire works best when the material has a constant area, the deformation is small, the load is axial, and the material remains within its elastic linear range.