keph201
Chapter Eight: Mechanical Properties of Solids
8.1 Introduction
Examines how mass distribution influences the motion of bodies, particularly rigid bodies.
Rigid body: A solid object with a definite shape and size.
Real-world bodies can be stretched, compressed, or bent — not perfectly rigid.
External force is required to change a body’s shape or size (e.g., stretching a spring).
Elasticity: Ability to return to original shape after deforming force removed (elastic deformation).
Plasticity: Permanent deformation where the body doesn't regain its shape (e.g., putty).
Importance of elastic behavior in engineering design (e.g., buildings, bridges, vehicles).
Exploring material properties can aid in designing lighter and stronger structures, like aeroplanes or artificial limbs.
8.2 Stress and Strain
Deformation of a solid body under external force leads to a restoring force.
Stress: Restoring force per unit area.
Formula: [ \text{Stress} = \frac{F}{A} ] where F = applied force, A = cross-sectional area.
Units: N/m² or Pascal (Pa) [ [M L^{-1} T^{-2}] ]
Types of stress:
Tensile Stress: Deformation due to stretching (length increases).
Compressive Stress: Deformation due to compression (length decreases).
Strain: Ratio of change in length to original length (( \Delta L/L )).
Longitudinal strain: Under tensile/compressive stress.
Shearing stress: Relative displacement between opposite faces.
Formula for shearing strain: [ \text{Shearing strain} = \frac{\Delta x}{L} \text{ or } \tan(\theta) ]
8.3 Hooke's Law
States that stress is proportional to strain for small deformations.
Formula: ( \text{Stress} = k \times \text{Strain} )
k: Modulus of elasticity (proportionality constant).
Valid for most materials, exceptions exist.
8.4 Stress-Strain Curve
Experimental data showing the relationship between stress and strain for materials.
Key points in the curve:
Region (O to A): Linear, obeying Hooke’s law (elastic behavior).
Yield Point (B): Stress level where permanent deformation begins (yield strength).
Beyond yield point, material undergoes plastic deformation (permanent shape change).
Ultimate tensile strength (D): Maximum stress before fracture (E).
If D and E are close, material is brittle; if far apart, it is ductile.
8.5 Elastic Moduli
Measures material's response to deformation:
Young's Modulus (Y): Ratio of tensile/compressive stress to longitudinal strain.
Formula: [ Y = \frac{\sigma}{\epsilon} = \frac{F/A}{\Delta L/L} ]
Shear Modulus (G): Ratio of shearing stress to shearing strain.
Bulk Modulus (B): Measure of a material's response to uniform pressure (hydraulic stress).
Formula: [ B = -\frac{p}{\Delta V/V}]
8.6 Applications of Elastic Behavior of Materials
Engineering designs (e.g., buildings, cranes): knowledge of material strength is crucial.
Designing ropes and beams to withstand applied loads without permanent deformation.
Importance of material choice affects the safety and functionality of structures.
Summary
Stress: restoring force per unit area.
Strain: fractional change in dimension (
Types: tensile, compressive, shearing, hydraulic.
Hooke’s Law: Relationship between stress and strain in the elastic limit.
Elastic moduli highlight material characteristics: Young’s modulus, shear modulus, bulk modulus.
Points to Ponder
Stress defined clearly in relation to tension and net forces in a wire.
Hooke’s law is relevant only in elastic (linear) regions of the stress-strain curve.
Elastic moduli specific to solids, liquids, and gases.
Real-world applications emphasize the importance of understanding material properties in design.