Formula sheet: Mechanical Properties of Solids

Mechanical Properties of Solids12 min readLesson 1 of 5

1.1 Stress and strain Stress is the internal restoring force per unit area set up inside a deformed solid. It is measured in N m^-2, which is the pascal (Pa). Strain is the fractional change of shape or size and has no unit. The three pairs that appear in this chapter are longitudinal, shearing and hydraulic (volumetric) stress with their matching strains. A wire under a load F has a uniform tensile stress F/A far from the clamps, where A is the cross-sectional area. Near a clamp the stress is not uniform, so exam questions usually ask for the stress at a section well away from the ends.

σ = \frac{F}{A}, ϵ = \frac{Δ L}{L}, σ_{s h e a r} = \frac{F}{A}, θ = \frac{Δ x}{L}

Longitudinal stress and strain, and the shear stress and shear strain used for a deforming layer

1.2 The three elastic moduli Young's modulus Y applies to a wire or rod pulled along its length. The shear modulus G applies when one face of a body is pushed sideways relative to the opposite face. The bulk modulus B applies when a body is squeezed from all sides, and its reciprocal is the compressibility.

Y = \frac{σ}{ϵ} = \frac{F L}{A Δ L}, G = \frac{σ _{s h e a r}}{θ} = \frac{F L}{A Δ x}

Young's modulus of a stretched wire and shear modulus of a layer of thickness L displaced by x

B = - \frac{P}{Δ V / V}, \frac{1}{B} = κ, \frac{Δ V}{V} = - \frac{P}{B}

Bulk modulus and compressibility; a positive pressure P produces a negative fractional volume change

1.3 Energy, strength and the common combinations The elastic potential energy stored in a stretched wire is the work done by the stretching force, so it equals one half of the final force times the extension. The energy per unit volume is one half of stress times strain, which is the area under the straight part of the stress-strain curve. Breaking stress, or tensile strength, is the largest stress the material survives. For a given material, Y and G do not depend on the size or shape of the sample; only the material matters.

U = \frac{1}{2} F Δ L, u = \frac{U}{V} = \frac{1}{2} σ ϵ = \frac{σ ^{2}}{2 Y} = \frac{1}{2} Y ϵ^{2}

Elastic potential energy of a stretched wire and the energy density in the elastic region

1.4 Units and values to remember A useful check: Y for steel is about 2 x 10^11 Pa, for copper about 1.1 x 10^11 Pa, and for rubber it is small (about 10^6 to 10^7 Pa). G for steel is about 8 x 10^10 Pa and for aluminium about 2.5 x 10^10 Pa. B for water is about 2.2 x 10^9 Pa, and for air at constant temperature it is about 10^5 Pa. A large Y means a stiff material; a large B means a liquid or solid resists being squeezed.

depth for a 0.1% volume change: h = \frac{P}{ρ g}, P = B \frac{Δ V}{V}

Pressure needed to compress a volume by a given fraction, and the depth of liquid that supplies it

1.5 When each relation is used Use Y when the shape is a wire or rod and the force acts along the length. Use G when a book is pushed sideways on a table or a block is sheared. Use B when the pressure on all faces changes, as with a ball taken deep into the sea. Use the energy relation in questions about work done in stretching, and use the strain formula twice when a composite wire has several sections.

Exam

Write the units of stress (Pa), strain (no unit) and the three moduli (Pa) before starting a numerical. A dimensional check catches most careless slips: Y, G and B all have the dimension of pressure.

in hinglish

Is chapter ka core ek hi idea hai: deformation doosri cheezon se compare karke measure karna. Stress bolta hai ki kitna force per area hai, strain bolta hai ki shape kitni badli. Dono ko divide karo to modulus milta hai. Wire ke liye Young's modulus Y = FL/(A delta L), sideways push ke liye shear modulus G, aur chaar taraf se dabane ke liye bulk modulus B . Sab ki unit pascal hai, strain ki koi unit nahi. Elastic energy ka formula ek second year ke numerical jaisa simple hai: half of force times extension, ya energy density half stress times strain. Exam me pehle unit check karo, phir formula me values daalo. Steel ka Y rubber se bahut bada hota hai, isliye steel ka wire kam khinchta hai. Compressibility bulk modulus ka ulta hota hai, so paani ka B bada hai to uska volume change chhota hota hai. Aur ek baat yaad rakho: Y material ki property hai, wire ki size ya length badalne se nahi badalta, sirf extension badalta hai.