Solids maintain a definite shape and size because their constituent particles are held together by strong interatomic forces. When external forces act on a solid, it deforms. The study of how solids respond to forces — and how they return to their original shape — is called the mechanical properties of solids.
Key Concepts
Elasticity is the property of a body to regain its original size and shape when the deforming force is removed. Plasticity is the opposite — a material that does not return to its original form after deformation (e.g., clay).
Stress is the internal restoring force per unit cross-sectional area developed when a body is deformed.
Stress = F / A (unit: N/m2 or Pascal)
Strain is the ratio of the change in dimension to the original dimension. It has no unit.
- Three types of stress and corresponding strain:
- Longitudinal (tensile/compressive) stress — change in length
- Shearing stress — change in shape at constant volume
- Hydraulic (volumetric) stress — change in volume due to pressure from all sides
Hooke's Law and Elastic Moduli
Hooke's Law: Within the elastic limit, stress is directly proportional to strain.
Stress / Strain = constant = Modulus of Elasticity
Young's Modulus (Y): Y = (F/A) / (deltaL / L) = F L / (A deltaL)
Measures resistance to longitudinal deformation.
Shear Modulus (G) (also called Modulus of Rigidity):
G = Shearing Stress / Shearing Strain = (F/A) / (deltax / L)
Bulk Modulus (B):
B = -p / (deltaV / V)
The negative sign indicates volume decreases when pressure increases.
Compressibility = 1/B
Stress-Strain Curve
- The stress-strain graph for a metallic wire shows:
- Proportional limit: Hooke's law obeyed up to this point.
- Elastic limit: Maximum stress up to which the body returns to original shape.
- Yield point: Beyond this, permanent deformation begins.
- Ultimate tensile strength: Maximum stress before fracture.
- Fracture point: Where the wire breaks.
Elastic potential energy stored = (1/2) x Stress x Strain x Volume = (1/2) x Y x (strain)2 x Volume
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A steel wire of length 2 m and cross-section 2 x 10-6 m2 is stretched by 0.001 m when a force of 200 N is applied. Find Young's modulus.
Y = F L / (A deltaL) = (200 x 2) / (2 x 10-6 x 0.001) = 400 / (2 x 10-9) = 2 x 1011 N/m2
A copper rod 1 m long, radius 1 cm, is compressed by 104 N. Y = 1.2 x 1011 Pa. Find compression.
A = pi x (0.01)2 = 3.14 x 10-4 m2
deltaL = F L / (A Y) = (104 x 1) / (3.14 x 10-4 x 1.2 x 1011) = 104 / (3.77 x 107) = 2.65 x 10-4 m
Find the bulk modulus of water if 1000 L decreases by 0.5 L under pressure of 107 Pa.
B = -p / (deltaV / V) = 107 / (0.5/1000) = 107 / (5 x 10-4) = 2 x 1010 Pa
A square lead slab (side 10 cm, thickness 1 cm) is subjected to a shear force of 9 x 104 N on the upper face. G = 5.6 x 109 Pa. Find shear strain.
Shear Stress = F/A = 9 x 104 / (0.1 x 0.1) = 9 x 106 Pa
Shear Strain = Stress / G = 9 x 106 / 5.6 x 109 = 1.6 x 10-3
A wire stretches by 0.1 mm under load. If the same load is applied to a wire of same material but double the length and half the radius, find the extension.
New extension: deltaL' = F L' / (A' Y). L' = 2L, A' = pi(r/2)2 = A/4
deltaL' = F(2L) / (A/4 x Y) = 8 FL/AY = 8 x 0.1 = 0.8 mm
Calculate elastic potential energy stored per unit volume in a steel wire, strain = 10-3, Y = 2 x 1011 N/m2.
Energy per unit volume = (1/2) Y x strain2 = 0.5 x 2 x 1011 x (10-3)2 = 0.5 x 2 x 1011 x 10-6 = 105 J/m3
Ratio of radii of two wires is 1:2, same material and length. Same force is applied. Find ratio of Young's moduli if elongation is same.
Same elongation, same F, same L: Y = FL/(A deltaL). Y is same (same material). Ratio is 1:1.
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Common mistakes
- Strain is dimensionless — students sometimes assign units to strain. Never add units to strain.
- Do not confuse elastic limit with proportional limit — elastic limit is slightly beyond proportional limit.
- Bulk modulus formula has a negative sign — always include it in derivation.
Summary
Stress and strain quantify deformation. Elastic moduli (Y, G, B) connect them. Hooke's law holds up to the elastic limit. The stress-strain curve reveals the mechanical behaviour of materials from elasticity to fracture.