Mechanical Properties of Solids: common doubts, answered
The questions students ask most often about Mechanical Properties of Solids, each with a short answer. For the full chapter, read the Mechanical Properties of Solids notes.
Elasticity and plasticity
Read this section in the notes →Is rubber more elastic than steel?
No, steel is more elastic in the physics sense. Elasticity measures how strongly a material resists deformation and returns to shape, given by its modulus. Steel needs a far larger stress than rubber to produce the same strain, so its Young's modulus is much larger. Rubber stretches easily precisely because it is less elastic in this sense.
What is the difference between elasticity and plasticity?
An elastic body returns fully to its original shape and size when the deforming force is removed, while a plastic body stays deformed. Putty and clay are close to ideal plastics. Real materials are elastic only up to a limit; loaded beyond it, they take a permanent set and behave plastically.
Stress and strain
Read this section in the notes →What is the difference between stress and pressure?
Both are force per unit area with the unit N m⁻², but stress is the internal restoring force per unit area that develops inside a deformed body, and it can be tensile, compressive or shearing. Pressure is the normal force per unit area exerted by a fluid, and acts equally in all directions at a point. Hydraulic stress in a solid is numerically equal to the applied pressure.
Why does strain have no unit?
Because strain is a ratio of two quantities of the same kind: change in length divided by original length, or change in volume divided by original volume. The units cancel, so strain is a pure number. Shearing strain is an angle in radians, tan θ ≈ θ, which is also dimensionless.
What are the three types of strain?
Longitudinal strain is the change in length per unit length, ΔL/L, produced by tensile or compressive stress. Shearing strain is the sideways displacement per unit height, Δx/L, produced by a tangential force. Volume strain is the change in volume per unit volume, ΔV/V, produced by uniform pressure from all sides.
Hooke's law and the stress-strain curve
Read this section in the notes →What does Hooke's law state and when does it hold?
Hooke's law says stress is proportional to strain, stress = k × strain, where k is the modulus of elasticity. It holds only for small deformations, up to the proportional limit of the material. Beyond that point the stress-strain graph curves and the simple proportionality no longer applies.
What is the difference between the proportional limit and the elastic limit?
The proportional limit is where stress stops being proportional to strain, ending the straight part of the graph. The elastic limit, or yield point, is slightly beyond it: up to there the body still recovers fully when unloaded, even though Hooke's law no longer holds exactly. Loading past the elastic limit leaves a permanent deformation.
What is the difference between ductile and brittle materials?
A ductile material, such as copper, has a long plastic region between its elastic limit and fracture, so it can be drawn into wires before breaking. A brittle material, such as glass or cast iron, breaks soon after the elastic limit with very little plastic deformation. The shape of the stress-strain curve shows which kind a material is.
Young's modulus
Read this section in the notes →How do you calculate Young's modulus from a wire experiment?
Use Y = FL/(AΔL), where F is the load, L the original length, A the cross-sectional area and ΔL the extension. Find A from the radius, πr², remembering to halve the measured diameter. Convert every quantity to SI units before substituting, since areas in mm² are a common source of errors.
Does Young's modulus depend on the length or thickness of a wire?
No. Young's modulus is a property of the material alone, at a given temperature. A longer or thinner wire stretches more under the same load, but its ratio of stress to strain stays the same. Changing dimensions changes the extension, not the modulus.
What is the tension in a wire held at both ends by forces F?
The tension is F, not 2F. When a wire hangs from a ceiling with a load F, the ceiling pulls up with F only to keep the wire in equilibrium. The same is true for a wire pulled by F at each end. The stress is therefore F/A, and adding the two end forces would double the answer wrongly.
Shear modulus
Read this section in the notes →What is shear modulus?
Shear modulus, or modulus of rigidity, is the ratio of shearing stress to shearing strain: G = F/(Aθ). It measures how strongly a material resists changes of shape when a force acts parallel to a surface. For most solids it is roughly a third of Young's modulus.
Why don't liquids have Young's modulus or shear modulus?
Because liquids cannot hold a fixed shape and simply flow when a tangential or stretching force is applied. They cannot sustain shearing or longitudinal stress, so those moduli have no meaning for them. Liquids and gases do resist compression, so they have a bulk modulus only.
Bulk modulus and compressibility
Read this section in the notes →Why is there a minus sign in the formula for bulk modulus?
Because an increase in pressure causes a decrease in volume, so ΔV is negative. The minus sign in B = −p/(ΔV/V) makes the bulk modulus come out positive. Bulk modulus is never negative; a negative answer means a sign was missed.
What is compressibility?
Compressibility is the reciprocal of bulk modulus, k = 1/B. It tells how much a material's volume changes per unit increase in pressure. Gases have very high compressibility and low bulk modulus; solids are the least compressible, with liquids in between.
Poisson's ratio and elastic energy
Read this section in the notes →What is Poisson's ratio?
Poisson's ratio is the ratio of lateral strain to longitudinal strain when a wire is stretched: (Δd/d)/(ΔL/L). A wire gets slightly thinner as it gets longer, and this ratio measures by how much. It has no unit, since both strains are dimensionless.
What is the elastic potential energy stored in a stretched wire?
It equals the work done in stretching it, U = ½ × stress × strain × volume. Per unit volume, the energy stored is ½ × stress × strain. The factor of a half appears because the stretching force builds up from zero to its final value, just as for a spring.
Applications of elastic behaviour
Read this section in the notes →Why are beams used in buildings made with an I-shaped section?
Because a beam's sag falls sharply with depth, δ = Wl³/(4bd³Y), varying as one over the cube of the depth d but only as one over the width b. An I-section puts most of the material in the top and bottom flanges, where it best resists bending, giving a stiff beam with less weight and material.
Why can't mountains be taller than about 10 km?
The pressure at the base of a mountain, hρg, rises with its height. If it exceeds the elastic limit of the rock, the rock at the base flows and the mountain cannot grow taller. Using typical rock values gives a limiting height of roughly 10 km, comparable to the tallest mountains on Earth.
How is the thickness of a crane's steel rope chosen?
The rope must carry the load without reaching its yield strength, so its area must satisfy A ≥ Mg/σy. Engineers then add a large safety factor and use many thin strands wound together, which makes the rope flexible and strong. A single thick solid rod would be too stiff to handle.
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