Mechanical Properties of Solids: NEET notes

1. Elasticity and plasticity

NCERT §8.1

• A rigid body is an idealisation. Every real solid, even a thick steel bar, changes length, shape or volume when a large enough force acts on it.
• Elasticity: the property by which a body tries to get back its original size and shape once the deforming force is taken away. The change it undergoes meanwhile is an elastic deformation, as in a gently stretched helical spring.
• Plasticity: the property of bodies such as putty or mud, which show no real tendency to recover and stay deformed for good. Putty and mud come close to ideal plastic bodies.
• Engineers need the elastic behaviour of steel, concrete and other materials to design buildings, bridges, automobiles and ropeways, and to make aircraft or artificial limbs that are light yet strong.
• Questions such as why a rail has an I-shaped section, or why glass is brittle while brass is not, start from how simple loads deform solids.

2. Stress and strain

NCERT §8.2

• When a body is deformed but kept in equilibrium, a restoring force appears inside it, equal in size and opposite in direction to the applied force. Stress is this restoring force per unit area: magnitude F/A.
• Stress has the SI unit N m⁻², called the pascal (Pa), and the dimensional formula [ML⁻¹T⁻²], the same as pressure.
• Tensile stress: equal and opposite pulls normal to the cross-section stretch a cylinder. Compressive stress: pushes shorten it. Both are longitudinal stress, and the resulting longitudinal strain is ΔL/L.
• Shearing (tangential) stress: equal and opposite forces act parallel to the cross-section, so opposite faces slide by Δx relative to each other. Shearing strain = Δx/L = tan θ, where θ is the tilt from the vertical.
• θ is usually tiny, so tan θ ≈ θ (in radians). Even at θ = 10°, θ and tan θ differ by only about 1%. Pushing the top cover of a thick book sideways shows shear.
• Hydraulic stress: a solid held in a fluid under high pressure is squeezed normal to its surface at every point. Its volume falls with no change of shape; the internal restoring force per unit area equals the hydraulic pressure.
• Volume strain = ΔV/V. Every strain is a ratio of a change in a dimension to the original dimension, so strain has no unit and no dimensions.

3. Hooke's law and the stress-strain curve

NCERT §8.3, §8.4

• Hooke's law: for small deformations, stress is proportional to strain. Stress = k × strain, where the constant k is the modulus of elasticity.
• Hooke's law is empirical. It holds for most materials over small strains, but some materials never show a straight-line relation.
• In a tensile test a wire or test cylinder is loaded in steps and its strain recorded; stress (applied force per unit area) is then plotted against strain.
• O to A: a straight line. Hooke's law holds and the body recovers fully when unloaded, so it behaves elastically.
• A to B: stress and strain are no longer proportional, yet the body still recovers on unloading. B is the yield point, also called the elastic limit, and the stress there is the yield strength σy.
• Beyond B the strain grows fast for small increases in stress. If the load is removed at a point C between B and D, the strain does not return to zero: the body has a permanent set, a plastic deformation.
• D marks the ultimate tensile strength σu. Past D the strain keeps growing even under a smaller force, and the wire fractures at E.
• D and E close together: the material is brittle. D and E far apart: the material is ductile.
• Elastomers such as rubber and the elastic tissue of the aorta can be stretched to large strains and still recover. Their elastic region is huge, yet Hooke's law fails over most of it and there is no well-defined plastic region.

4. Young's modulus

NCERT §8.5.1

• The ratio of stress to strain within the proportional region OA is a property of the material, called a modulus of elasticity.
• Young's modulus Y = tensile (or compressive) stress σ ÷ longitudinal strain ε = (F/A)/(ΔL/L) = FL/(AΔL). Experiment shows the same strain for equal tensile and compressive stress.
• Strain has no unit, so Y has the unit of stress, N m⁻² or Pa. Metals have large Y; wood, bone, concrete and glass have rather small Y.
• To stretch a wire of cross-section 0.1 cm² by 0.1% takes 2000 N for steel, and 690 N, 900 N and 1100 N for aluminium, brass and copper. Steel is therefore the most elastic of the four, which is why heavy machines and structures use it.
• Steel rod, radius 10 mm, length 1.0 m, pulled with 100 kN, Y = 2.0 × 10¹¹ N m⁻²: stress = 3.18 × 10⁸ N m⁻², elongation = 1.59 mm, strain = 1.59 × 10⁻³ ≈ 0.16%.
• Copper wire 2.2 m and steel wire 1.6 m, both 3.0 mm in diameter, joined end to end: same tension and area, so ΔLc/ΔLs = (Ys/Yc)(Lc/Ls) = 2.5 with Yc = 1.1 × 10¹¹ and Ys = 2.0 × 10¹¹ N m⁻². A total stretch of 0.70 mm splits as 0.50 mm + 0.20 mm, and the load is about 1.8 × 10² N.
• Human pyramid: 280 kg in all, the bottom performer 60 kg, so his legs carry 220 kg, or 2156 N, 1078 N per thighbone. With L = 0.5 m, radius 2.0 cm (A = 1.26 × 10⁻³ m²) and Y(bone) = 9.4 × 10⁹ N m⁻², each bone shortens by 4.55 × 10⁻⁵ m, a fractional change of 0.0091%.

5. Shear modulus

NCERT §8.5.2

• Shear modulus (modulus of rigidity) G = shearing stress σs ÷ shearing strain = (F/A)/(Δx/L) = FL/(AΔx).
• Because shearing strain ≈ θ, G = (F/A)/θ = F/(Aθ), and so σs = Gθ.
• G is in N m⁻² or Pa. It is generally smaller than Young's modulus; for most materials G ≈ Y/3.
• Typical G values: tungsten has the stiffest at 150 GPa, steel 84, nickel 77 and iron 70; copper 42, brass 36, aluminium 25 and glass 23; wood 10 and lead only 5.6 GPa.
• Lead slab 50 cm square and 10 cm thick, lower edge riveted to the floor, sheared by 9.0 × 10⁴ N on its narrow face: A = 0.5 m × 0.1 m = 0.05 m², stress = 1.8 × 10⁶ N m⁻², and with G = 5.6 × 10⁹ N m⁻² the upper edge moves Δx = stress × L/G = 1.6 × 10⁻⁴ m = 0.16 mm.
• Shear, like stretching, needs a body with a definite shape, so Young's modulus and shear modulus apply only to solids.

6. Bulk modulus and compressibility

NCERT §8.5.3

• Bulk modulus B = −p/(ΔV/V): hydraulic stress over volume strain. The minus sign records that raising the pressure lowers the volume, so B itself is positive for a body in equilibrium.
• B has the unit of pressure, N m⁻² or Pa. Bulk modulus applies to solids, liquids and gases alike.
• Compressibility k = 1/B = −(1/Δp)(ΔV/V): the fractional change in volume per unit rise in pressure.
• Typical B values (GPa). Solids: nickel 260, steel 160, copper 140, iron 100, aluminium 72, brass 61, glass 37. Liquids: mercury 25, glycerine 4.76, water 2.2, carbon disulphide 1.56, ethanol 0.9. Air at STP: just 1.0 × 10⁻⁴.
• Solids have much larger B than liquids, and liquids much larger than gases. Gases are roughly a million times more compressible than solids, and their compressibility changes with pressure and temperature.
• The reason is coupling between neighbours: atoms in a solid are tightly bound, molecules in a liquid less so, and molecules in a gas hardly at all.
• Indian Ocean, average depth about 3000 m, g taken as 10 m s⁻²: p = hρg = 3 × 10⁷ N m⁻², so ΔV/V = p/B = 3 × 10⁷/2.2 × 10⁹ = 1.36 × 10⁻² or 1.36% for water at the bottom.
• Summary of moduli: Y for tensile or compressive stress (shape changes, solids only); G for shear (shape changes, solids only); B for hydraulic stress (volume changes, shape does not; solids, liquids and gases).

7. Poisson's ratio and elastic energy

NCERT §8.5.4, §8.5.5

• Stretching a wire also thins it. The strain at right angles to the applied force is the lateral strain, Δd/d for a wire of diameter d.
• Within the elastic limit lateral strain is proportional to longitudinal strain (Simon Poisson). Poisson's ratio = (Δd/d)/(ΔL/L) = (Δd/ΔL)(L/d).
• Poisson's ratio is a ratio of two strains, so it is a pure number with no unit or dimensions, and it depends only on the material: 0.28 to 0.30 for steels and about 0.33 for aluminium alloys.
• Stretching a wire does work against the interatomic forces, and that work is stored as elastic potential energy.
• For an extension l of a wire of length L and area A, the force is F = YA(l/L). Integrating F dl from 0 to l gives W = ½ YA l²/L.
• So U = ½ × Y × strain² × volume = ½ × stress × strain × volume, and the energy per unit volume is u = ½ σε.

8. Applications of elastic behaviour

NCERT §8.6

• Designing columns, beams and supports needs the strength and elastic behaviour of the materials used. Structural engineering explains why bridge beams have an I-shaped section and why a sand heap or a hill takes a pyramid shape.
• Crane rope for 10 tonnes (1 metric ton = 1000 kg): the rope must stay within its elastic limit, so A ≥ Mg/σy. With mild steel σy ≈ 300 × 10⁶ N m⁻², A ≥ (10⁴ × 9.8)/(300 × 10⁶) = 3.3 × 10⁻⁴ m², a radius of about 1 cm.
• A safety margin of about ten times the load raises the recommended radius to about 3 cm. A single wire that thick would be a rigid rod, so crane ropes are many thin wires braided together, for easier manufacture, flexibility and strength.
• Rest a bar (span l, breadth b, depth d) on supports near its two ends and hang a load W from its middle: the middle drops by δ = Wl³/(4bd³Y).
• To reduce sagging use a material of large Y, keep the span short, and increase the depth rather than the breadth: δ ∝ d⁻³ but only ∝ b⁻¹.
• A deep, thin bar can buckle sideways when the load is not exactly in place, as with moving traffic. The I-section is the compromise: a large load-bearing surface and enough depth to resist bending, with less weight and cost for the same strength.
• A pillar with rounded ends supports less load than one whose ends are spread out (distributed).
• Maximum height of a mountain: the base feels a shear stress of about hρg. Setting it equal to a typical rock's elastic limit, 30 × 10⁷ N m⁻², with ρ = 3 × 10³ kg m⁻³ and g = 10 m s⁻², gives h = 10 km, more than the height of Mt. Everest.

Must-know facts

1. Stress = restoring force per unit area = F/A; unit N m⁻² = Pa; dimensions [ML⁻¹T⁻²].
2. Strain is a ratio of lengths or volumes: no unit, no dimensions.
3. Three strains: longitudinal ΔL/L, shearing Δx/L = tan θ ≈ θ, volume ΔV/V.
4. Hooke's law: stress ∝ strain for small deformations; the constant is the modulus of elasticity.
5. Stress-strain curve: O-A proportional (Hooke's law), B yield point or elastic limit (σy), C permanent set if unloaded, D ultimate tensile strength (σu), E fracture.
6. D and E close: brittle. D and E far apart: ductile.
7. Elastomers (rubber, aorta tissue): large elastic region, no Hooke's law over most of it, no well-defined plastic region.
8. Y = FL/(AΔL); G = F/(Aθ); B = −p/(ΔV/V); all three have the unit Pa.
9. Y and G apply only to solids; B applies to solids, liquids and gases.
10. For most materials G ≈ Y/3.
11. Y: steel 2.0 × 10¹¹, copper 1.1 × 10¹¹, bone 9.4 × 10⁹ N m⁻² (values used in NCERT's examples).
12. Same 0.1% strain in a 0.1 cm² wire: steel 2000 N, copper 1100 N, brass 900 N, aluminium 690 N; steel is the most elastic.
13. Compressibility k = 1/B; a gas squeezes roughly a million times more easily than a solid.
14. Bulk modulus of water 2.2 × 10⁹ N m⁻²; air at STP 1.0 × 10⁵ N m⁻² (1.0 × 10⁻⁴ GPa).
15. Water at 3000 m ocean depth (g = 10) is compressed by 1.36%.
16. Poisson's ratio = lateral strain/longitudinal strain; unitless; steels 0.28-0.30, aluminium alloys about 0.33.
17. Elastic energy U = ½ × stress × strain × volume; per unit volume u = ½σε = ½Y(strain)².
18. Beam sag δ = Wl³/(4bd³Y): depth counts as d³, breadth only as b.
19. Crane rope for 10 t: A ≥ Mg/σy = 3.3 × 10⁻⁴ m² (radius about 1 cm); with a safety factor of 10, radius about 3 cm, made of braided thin wires.
20. Mountain height limit: hρg = 30 × 10⁷ N m⁻² with ρ = 3 × 10³ kg m⁻³ gives about 10 km.
21. A wire hung from a ceiling with weight F at its end has tension F, not 2F, at every section, so the stress is F/A.
22. Stress is not a vector: no single direction can be assigned to it.

Common traps

Formulas

Key terms

Elasticity

The tendency of a body to recover its original size and shape once the deforming force is removed.

Plasticity

The behaviour of a body that stays deformed after the force is removed, as putty and mud do.

Stress

Internal restoring force per unit area of a deformed body; equal in size to the applied force per unit area.

Strain

Change in a dimension divided by the original dimension; a pure number.

Hydraulic stress

Uniform normal stress from a surrounding fluid; equal to the fluid pressure and changing only volume.

Elastic limit (yield point)

Point B of the stress-strain curve, the largest stress after which the body still recovers fully.

Permanent set

The strain left in a body after unloading from beyond its yield point.

Ultimate tensile strength

The highest stress on the stress-strain curve, point D; past it the wire keeps stretching under a smaller force until it breaks.

Brittle

A material whose fracture point lies close to its ultimate strength.

Ductile

A material that strains a long way between its ultimate strength and fracture.

Elastomer

A substance such as rubber or aorta tissue that can take large strains and recover, without following Hooke's law.

Modulus of rigidity

Another name for the shear modulus G.

Compressibility

Reciprocal of bulk modulus; fractional volume change per unit rise in pressure.

Poisson's ratio

Lateral strain divided by longitudinal strain for a stretched wire.

Buckling

Sideways bending of a deep, thin bar under a load that is not exactly placed.