Revision Notes on Deformable Bodies and Elastic Deformation (Mechanical Properties of Solids):-

• Matter:- Anything which possesses mass and occupies space is called matter.

(a) Solid:- It is the type of matter which has a definite shape and a definite volume.

(b) Liquid:- It is the type of matter which has a definite volume but not definite shape.

(c) Gas:- It is the type of matter which has neither definite shape nor definite volume.

• Elasticity:-The property by virtue of which a body tends to recover its original configuration (shape and size) on the removal of the deforming forces, is  elasticity.

• Plastic bodies:- Bodies which do not show a tendency to recover their original configuration on the removal of deforming forces are called plastic bodies.

• Stress:- It is defined as the restoring force per unit area.

Stress = F/A

(a) Normal stress:- Stress is called a normal stress if the restoring force acts at right angles to the surface.

(Stress) N = Fsinθ/A

(i) Compressional stress:- This stress produces a decrease in lemgth per volume of the body.

(ii) Tensile stress:- This stress results in increase in length per volume of the body.

(b) Tangential stress:- Stress is said to be tangential if it acts in direction parallel to the surface.

(c) Unit of stress:- S.I- N/m²,   C.G.S-dyn/cm²

(d) Dimension of stress:- Stress = F/A = [M¹L^-1T^-2]

• Strain:- Relative change in configurationdue to the applicationof deforming forces is called strain.

(a) Longitudinal strain:- It is the ratio between change in length to its original length.

Longitudinal strain = l/L

(b) Lateral strain:- Lateral strain is the ratio between change in diameter to its original diameter when the cylinder is subjected to a force along its axis.

Lateral strain = change in diameter /original diameter

(c) Volume strain:- It is defined as the ratio between change in volume to its original volume.

Volume strain = v/V

(d) Shear strain:- Shear strain is measured by angle turned by a line originally perpendicular to the fixed face.

Shear strain = ?,

For small angles,

? = tan?

= DD₁/AD

= displacement in plane CD/distance of plane CD from fixed plane

So, shear strain is also defined as the ratio between displacements in one plane to its distance from the fixed plane. It has no unit.

(e) Unit of strain:- No unit

(f) Dimension of strain:- [M⁰L⁰T⁰]

• Hooke’s Law:-

It states that with in elastic limits, stress is proportional to strain.

Within elastic limits, tension is proportional to extension.

So, Stress ∝Strain

/A∝l/L

(a) For stretching: Stress = Y×strain  or Y = Fl/A(Δl)

(b) For shear: Stress = η×strain    or η = F/A?

(c) For volume elasticity: Stress = B×strain  or B = - P/(ΔV/V)

• Plasticity:- This property of body by virtue of which, it loses property of elasticity and acquires a permanent deformation on the removal of deforming force is called plasticity.

• Coefficient of elasticity:- It is the ratio between stress and strain.

Unit - S.I- N/m², C.G.S - dyn/cm²

• Young’s modulus of elasticity (Y):- It is defined as theratio betweennormal stress to the longitudinal strain.

Y = normal stress/longitudinal strain  = (F/A)/(l/L) = (Mg×L)/(πr²×L)

• Bulk modulus of elasticity (B):- It is defined as the ratio between normal stress to the volumetric strain.

B = normal stress/volumetric strain  = (F/A)/(v/V) = pV/v

• Compressibility:- Compressibility of a material is defined as the reciprocal of its bulk modulus of elasticity.

Compressibility = 1/B

• Modulus of rigidity(η):- It is defined as the ratio betweentangential stressto the shear strain.

η = tangential stress/ shear strain = (F/A)/θ = T/θ

• Unit of coefficient of elasticity:- S.I- N/m², C.G.S - dyn/cm²

• Dimension of coefficient of elasticity = stress/strain = [M¹L^-1T^-2]

• Effect of temperature on co-efficient of elasticity:-

(a) Y_t = Y₁₅[1-α (t-15)]

(b) η_t = η₁₅[1-α' (t-15)]

Here Y₁₅ and η₁₅ are the Young’s modulus and the modulus of rigidity at 15ºC whileY_t and  η_t are the corresponding values at tºC, α and α' are the temperature coefficient for Y and η.

• Work done in stretching:-

(a) W = ½ ×(stress)×(strain)×(volume) = ½ Y(strain)²×volume = ½ [(stress)²/Y]×volume

(b) Potential energy stored, U = W = ½ ×(stress)×(strain)×(volume)

(c) Potential energy stored per unit volume, U = ½ ×(stress)×(strain)

• Work done during extension (Energy density):-

W =½ F×l = ½ tension ×extension

• Energy density:-The amount of energy, storedin the wire, per unit volumeis calledits energy density.

Energy density = work done/volume

          = ½ [(F×l)/(A×L)]

          = ½ [stress×strain]

• Steel is more elastic than rubber:- A substancewhich has greater co-efficient of elasticity is said to be more elastic.

For steel, Y_S = F_L/al_s

For rubber, Y_R = F_L/al_r

Since, l_s<l_r       So, Y_S> Y_R

Therefore, steel is more elastic than rubber.

• Elastomer:- A substance in which a large strain can be produced due to a small stress is called an elastomer.

• Elastic fatigue:- The phenomenon by virtue of which a substance exhibits a delay in recovering its original configuration, if it had been subjected to a stress for a longer time, is called elastic fatigue.

• Elasticity of a gas:-

(a) Isothermal change:- If the temperature of the gas remains constant, change is said to be isothermal.

P =-(stress/volumetric strain) = B_i

Here B_i is the Bulk modulus of elasticity under isothermal conditions.

(b) Adiabatic Change:-If the temperature of the gasdoes not remainconstant, the change is said to be adiabatic.

γ_P =(stress/volumetric strain) = B_a

Here B_a is the Bulk modulus of elasticity under adiabatic conditions.

As γ is always greater than one, therefore B_a> B_i.

• Breaking stress:- Breaking stress = Breaking weight /πr²

• Poisson’s ratio (σ):- Poisson’s ratio of the material of a wire is defined as the ratio between lateral strains per unit stress to the longitudinal strain per unit stress.

σ = lateral strain/longitudinal strain = β/α = (ΔD/D)/(ΔL/L)

Values of σ lies between -1 and 0.5.

• Relations among elastic constants:-

(a) B= Y/[3(1-2σ)]

(b) η = Y/[2(1+ σ)]

(c)  9/Y = 3/η + 1/B

(d) σ = [3B-2η]/[6B+2η]

• Loaded beam:-

(a) Depression, δ = Wl³/4Ybd³ (rectangular)

(b) Depression, δ = Wl³/12Yπr² (cylindrical)

• Torsion of a cylinder:-

(a) rθ = l?   (Where θ is the angle of twist and ? is the angle of shear)

(b) Restoring torque, = cθ

(c) Restoring couple per unit twist, c = πηr⁴/2l   (for solid cylinder)

(d) Restoring couple per unit twist, c = πηr₂⁴-r₁⁴/2l   (for hollow cylinder)

• Bulk modulus of an ideal gas:-

(a) Bisothermal = P

(b) Badiabatic = γ_P

Here γ = C_p/C_v

• Elongation of a wire due to its own weight:- Δl = ½ [Mgl/YA] = ½ [l²ρg/Y]