Mechanical Properties of Solids — Complete Formula

Stress

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Stress	[\text{Stress} = \dfrac{F}{A}]	[F] = force, [A] = area	Pa	Force per unit area ⭐
Longitudinal Stress	[\dfrac{F}{A}]	[F] = normal force, [A] = cross-sectional area	Pa	Acts along length
Shear Stress	[\dfrac{F}{A}]	[F] = tangential force	Pa	Causes change in shape ⚠️
Bulk Stress	[\dfrac{\Delta P}{1}]	[\Delta P] = change in pressure	Pa	Equal in all directions

💡 Memory Hint:
Stress = internal restoring force per unit area

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Longitudinal Strain	[\dfrac{\Delta L}{L}]	[\Delta L] = change in length, [L] = original length	—	Dimensionless ⭐
Shear Strain	[\theta]	[\theta] = angular deformation	rad	Small angle approx ⚠️
Volume Strain	[\dfrac{\Delta V}{V}]	[\Delta V] = change in volume	—	Used in bulk modulus

💡 Memory Hint:
Strain = relative deformation

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Hooke’s Law	[\text{Stress} \propto \text{Strain}]	—	—	Valid within elastic limit ⭐
Linear Form	[\text{Stress} = Y \times \text{Strain}]	[Y] = Young’s modulus	Pa	Straight-line graph

💡 Memory Hint:
Valid only in elastic region ⚠️

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Young’s Modulus	[Y = \dfrac{F L}{A \Delta L}]	[L] = original length	Pa	For length change ⭐
Bulk Modulus	[K = -\dfrac{\Delta P}{\Delta V / V}]	[\Delta P] = pressure change	Pa	Negative sign important ⚠️
Shear Modulus	[G = \dfrac{\text{Shear Stress}}{\text{Shear Strain}}]	—	Pa	Shape change

💡 Memory Hint:

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Relation	[Y = 2G(1 + \nu)]	[\nu] = Poisson’s ratio	Pa	Important ⭐
Alternate	[Y = 3K(1 – 2\nu)]	—	Pa	Used in numericals ⭐

💡 Memory Hint:
All constants are interrelated

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Poisson’s Ratio	[\nu = -\dfrac{\text{Lateral Strain}}{\text{Longitudinal Strain}}]	[\nu] = Poisson’s ratio	—	Usually between 0 and 0.5 ⭐

💡 Memory Hint:
Stretch → width decreases

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Elastic Potential Energy	[U = \dfrac{1}{2} \times \text{Stress} \times \text{Strain} \times \text{Volume}]	—	J	Energy stored ⭐
Alternate Form	[U = \dfrac{1}{2} \dfrac{F \Delta L}{1}]	[\Delta L] = extension	J	Work done in stretching

💡 Memory Hint:
Energy = area under stress–strain graph

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Breaking Stress	[\sigma = \dfrac{F_{max}}{A}]	[F_{max}] = breaking force	Pa	Maximum stress before breaking ⭐

💡 Memory Hint:
Material fails beyond breaking stress

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