Mechanical Properties of Solids Notes
Study Notes
Topics
4π 1. Chapter Overview
Overview
Mechanical Properties of Solids explains how solids respond when external forces stretch, compress, twist, shear or bend them. A solid resists deformation because internal restoring forces develop between its particles. Stress measures restoring force per unit area, while strain measures fractional deformation. If a body regains its original shape after the deforming force is removed, it shows elastic behaviour; if permanent deformation remains, it shows plastic behaviour. Hooke's law states that stress is proportional to strain within the proportional limit. The proportionality constants are elastic moduli: Young's modulus, bulk modulus and shear modulus. The stress-strain curve helps identify elastic limit, yield point, ultimate stress and breaking point, which are very important for NEET.
- 1Elasticity is the property by which a body regains its original shape and size after removal of deforming force.
- 2Greater elastic modulus means greater stiffness and smaller strain for the same stress.
- 3The slope of the linear part of stress-strain curve gives Young's modulus.
- 4Area under stress-strain graph gives elastic energy stored per unit volume.
- 5Safety factor is used in construction so that working stress stays much below breaking stress.
- 6NEET commonly asks formulas, units, graph regions and comparisons of ductile and brittle materials.
Stress vs Strain
Stress is the force effect; strain is the shape effect.
Three Moduli
Young changes length, Bulk changes volume, Shear changes shape.
Real-Life Example
Steel cables in cranes stretch slightly under heavy loads and return to original length if the elastic limit is not crossed.
NEET Quick Example
If stress is 2 Γ 10^8 Pa and strain is 10^-3, Young's modulus is 2 Γ 10^11 Pa.
Confusing Elasticity with Rigidity
Elasticity means ability to regain original shape; rigidity means resistance to deformation.
Treating Strain as Having Unit
Strain is dimensionless because it is a ratio of similar quantities.
Using Hooke's Law Beyond Limit
Hooke's law is valid only up to proportional limit, not for all deformations.
Stress is the internal restoring force per unit area developed in a body under deforming force.
Variables
F=Restoring or applied force
A=Area of cross-section
Strain measures relative deformation and has no unit.
Variables
change in dimension=Change in length, volume or shape
original dimension=Initial length, volume or relevant dimension
ποΈ 2. Stress & Strain
Overview
When a deforming force is applied to a solid, the solid develops internal restoring forces that oppose deformation. Stress is defined as restoring force per unit area. It may be tensile, compressive, hydraulic or shear depending on the way force acts. Strain is the relative change in shape or size caused by stress. Longitudinal strain measures change in length, volumetric strain measures change in volume and shear strain measures change in shape. For small deformations, stress is proportional to strain, giving the stress-strain relationship. If deformation disappears after removing the force, it is elastic deformation. If a permanent change remains, it is plastic deformation.
- 1Stress is calculated using force per area but represents internal restoring response.
- 2The direction of applied force decides the type of stress.
- 3Strain has no dimensions because it is a ratio.
- 4For a thin wire under tension, longitudinal stress and strain are most important.
- 5Shear strain is approximately equal to angular displacement in radians for small angles.
- 6Elastic and plastic deformation are separated by the elastic limit.
Stress Types
Pull stretches, push compresses, side force shears.
Strain Rule
Strain is always change divided by original.
Solved Example: Stress
A force of 200 N acts on a wire of area 2 Γ 10^-6 mΒ². Stress = F/A = 200/(2 Γ 10^-6) = 1 Γ 10^8 Pa.
Solved Example: Strain
A 2 m wire extends by 1 mm. Strain = 0.001/2 = 5 Γ 10^-4.
Using External Force Direction Incorrectly
Stress type depends on whether force is normal, tangential, tensile or compressive.
Forgetting Area in Stress
Same force produces larger stress in a thinner wire because area is smaller.
Confusing Elastic and Plastic Behaviour
Elastic recovery is complete; plastic deformation remains after unloading.
Stress due to force normal to the cross-sectional area.
Variables
Ο=Normal stress
F=Normal force
A=Area of cross-section
Fractional change in length due to tensile or compressive stress.
Variables
Ξ΅=Longitudinal strain
ΞL=Change in length
L=Original length
Fractional change in volume due to hydraulic stress.
Variables
ΞV=Change in volume
V=Original volume
πͺ 3. Hooke's Law & Elastic Moduli
Overview
Hooke's law states that stress is directly proportional to strain within the proportional limit. This simple law allows us to define elastic moduli, which measure stiffness of a material. Young's modulus describes resistance to change in length, bulk modulus describes resistance to change in volume and shear modulus describes resistance to change in shape. These constants are material properties and have the same unit as stress because strain is dimensionless. Elastic constants are related through Poisson's ratio for isotropic materials. When a body is elastically deformed, work done by external force is stored as elastic potential energy. Elastic energy density equals the area under the stress-strain graph in the linear region.
- 1Hooke's law does not apply beyond proportional limit.
- 2Young's modulus is high for steel, so steel is difficult to stretch.
- 3Bulk modulus is high for nearly incompressible materials.
- 4Shear modulus is related to shape rigidity.
- 5Fluids cannot sustain static shear, so shear modulus is effectively zero for fluids.
- 6Energy stored in a stretched wire equals area under force-extension graph.
Y-B-G Shortcut
Y changes length, B changes bulk volume, G changes geometry or shape.
Elastic Energy
Stretching energy has a half because force rises gradually from zero to F.
Numerical Problem: Young's Modulus
A wire of length 2 m and area 1 Γ 10^-6 mΒ² extends by 1 mm under 100 N. Y = FL/(AΞL) = 100Γ2/(10^-6Γ10^-3) = 2 Γ 10^11 Pa.
Numerical Problem: Elastic Energy
A wire stretches by 2 mm under final force 50 N. U = 1/2 FΞL = 1/2Γ50Γ0.002 = 0.05 J.
Missing Minus Sign in Bulk Modulus
Compression makes ΞV negative, so K = -P/(ΞV/V) is positive.
Confusing Energy and Energy Density
Energy is in joule, while energy density is energy per unit volume.
Assuming Higher Modulus Means More Stretching
Higher modulus means less strain for the same stress, so the material is stiffer.
Within proportional limit, stress varies linearly with strain.
Variables
constant=Elastic modulus for the deformation type
strain=Relative deformation
Young's modulus measures resistance to longitudinal stretching or compression.
Variables
Y=Young's modulus
F=Longitudinal force
L=Original length
A=Area of cross-section
ΞL=Change in length
Bulk modulus measures resistance to volume compression.
Variables
K=Bulk modulus
P=Applied pressure or hydraulic stress
ΞV/V=Volumetric strain
Shear modulus measures resistance to change in shape.
Variables
G=Shear modulus or modulus of rigidity
F/A=Shear stress
ΞΈ=Shear strain in radians
π 4. Stress-Strain Curve
Overview
The stress-strain curve shows how a material responds as stress is gradually increased. Initially, the graph is a straight line and Hooke's law is obeyed up to the proportional limit. The elastic limit is the maximum stress up to which the material completely regains its original shape on unloading. Beyond this, plastic deformation begins. The yield point marks the stage where strain increases greatly with little increase in stress. Ultimate stress is the maximum stress the material can bear. After this, necking or weakening may occur, and the specimen finally breaks at the breaking point. Ductile materials show large plastic deformation, while brittle materials break with little strain.
- 1Proportional limit and elastic limit may be close but are conceptually different.
- 2Beyond elastic limit, unloading leaves permanent strain.
- 3Ultimate stress can be higher than breaking stress for ductile materials.
- 4Brittle materials do not show a well-defined yield region.
- 5Ductile materials can be drawn into wires because they undergo large plastic deformation.
- 6NEET frequently tests graph labeling and comparison of ductile versus brittle materials.
Curve Order
Remember P-E-Y-U-B: Proportional, Elastic, Yield, Ultimate, Breaking.
Ductile vs Brittle
Ductile draws into wire; brittle breaks like glass.
Previous NEET Concept
If a stress-strain graph is initially a straight line, its slope represents Young's modulus.
Material Example
Copper is ductile because it can undergo large plastic deformation before breaking, while glass is brittle because it fractures suddenly.
Confusing Proportional Limit and Elastic Limit
Hooke's law ends at proportional limit, but elastic recovery may continue up to elastic limit.
Thinking Ultimate Stress Is Breaking Stress
Ultimate stress is maximum stress; breaking stress is stress at fracture and may be lower in ductile materials.
Ignoring Permanent Set
Beyond elastic limit, unloading does not bring the material back to original length.
The initial linear portion of the stress-strain graph gives Young's modulus.
Variables
Y=Young's modulus
stress=Longitudinal stress
strain=Longitudinal strain
Stress corresponding to final fracture of the material.
Variables
breaking force=Force at which body breaks
original area=Initial area of cross-section
ποΈ 5. Applications of Elasticity
Overview
Elasticity is essential in designing safe buildings, bridges, cranes, cables, springs and machines. Materials in engineering structures must withstand stress without exceeding elastic or breaking limits. A safety factor is used so that working stress remains much smaller than breaking stress. Suspension bridges use steel cables because steel has high Young's modulus, high tensile strength and reliable elastic behaviour. Elasticity also appears in daily life through springs, rubber bands, vehicle suspension, mattresses, shock absorbers and sports equipment. Practical NEET applications often test why certain materials are chosen, how safe load is calculated and how extension changes with length, area and Young's modulus.
- 1Engineering structures are designed to stay within elastic limit under normal working loads.
- 2Safety factor accounts for sudden loads, fatigue, defects and environmental effects.
- 3A high breaking stress means the material can withstand large stress before failure.
- 4A high Young's modulus means the material does not deform much under stress.
- 5Elasticity in daily life helps absorb shocks and restore shape.
- 6NEET often combines application-based theory with Young's modulus formula.
Safety Factor
Break divided by work: safety factor tells how far working stress is from failure.
Extension Control
To reduce stretch: make it shorter, thicker or use a stiffer material.
Real-World Example
Suspension bridge cables are made thick and strong so that actual working stress remains much less than breaking stress.
Solved Practice
A material has breaking stress 900 MPa and safety factor 3. Working stress = 900/3 = 300 MPa.
Practical NEET Application
If two wires of the same material and length support the same load, the wire with larger area has smaller stress and smaller extension.
Confusing Strength and Stiffness
Strength relates to breaking stress, while stiffness relates to Young's modulus.
Ignoring Safety Factor
Structures are never designed to work near breaking stress because sudden loads and defects can cause failure.
Assuming Elastic Materials Cannot Break
Elastic materials break if stress exceeds breaking stress.
Safety factor tells how many times stronger a material is compared with the stress allowed in use.
Variables
breaking stress=Stress at which material breaks
working stress=Maximum stress permitted during normal operation
Maximum safe force a cable or rod can support under working stress.
Variables
F_safe=Safe load
A=Cross-sectional area
Formula Sheet
10Stress is the internal restoring force per unit area developed in a body under deforming force.
Variables
F=Restoring or applied force
A=Area of cross-section
Strain measures relative deformation and has no unit.
Variables
change in dimension=Change in length, volume or shape
original dimension=Initial length, volume or relevant dimension
Within proportional limit, stress is directly proportional to strain.
Variables
Stress=Restoring force per unit area
Strain=Fractional deformation
Energy stored per unit volume in a linearly elastic body.
Variables
u=Elastic energy per unit volume
stress=Force per unit area
strain=Relative deformation
Stress due to force normal to the cross-sectional area.
Variables
Ο=Normal stress
F=Normal force
A=Area of cross-section
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