Young's Modulus: Stress, Strain, and Material Properties

• A solid has definite shape and seize and requires an external force to change it

Elasticity

• Elasticity is the property of a body to return to its original shape when deforming forces are removed
• Substances exhibiting this property are called elastic
• Steel and rubber are examples of elastic materials
• Steel is more elastic than rubber

Plasticity

• Plasticity describes substances that do not regain their original shape after deformation
• Plastic substances remain permanently deformed after the removal of the deforming force
• Putty and mud are examples of plastic materials

Stress and Strain

• Applying a force to a body causes deformation, which can be small or large
• A restoring force develops within the body that is equal in magnitude but opposite in direction to the applied force

Stress

• Stress is the restoring force acting per unit area
• If F is the applied force and A is the cross-sectional area, then Stress = F/A
• The SI unit of stress is N/m² or Pascal (Pa)
• The dimensional formula for stress is [ML⁻¹T⁻²]

Strain

• Strain represents a change in dimension
• Strain is the fractional change in dimension
• Strain = (Change in dimension) / (Original dimension)
• Strain has no unit and is dimensionless

Types of Stress and Strain

• External forces can alter the dimensions of a solid, resulting in three types of stress and strain
• They include Longitudinal, Shearing and Hydraulic

Longitudinal Stress and Longitudinal Strain

• Longitudinal stress is the restoring force per unit area, where the force is applied normal to the cross-sectional area
• The formula is: Longitudinal stress = F/A
• Tensile stress occurs when a cylinder is stretched
• Compressive stress occurs when a cylinder is compressed
• Longitudinal strain is the ratio of the change in length (ΔL) to the original length (L)
• Longitudinal strain = ΔL / L

Shearing Stress and Shearing Strain

• Shearing stress is the restoring force per unit area when a tangential force is applied to the cylinder
• Shearing stress = F/A
• Shearing strain is the ratio of the relative displacement of the faces Δx to the length of the cylinder L
• Shearing strain = Δx / L = tanθ; where θ is very small, tan θ ≈ θ
• Shearing strain = θ

Hydraulic Stress and Hydraulic Strain (Volume Strain)

• Hydraulic stress is the restoring force per unit area on a solid sphere placed in a fluid, with force applied perpendicularly
• Hydraulic stress = -F/A = -P (pressure)
• The negative sign indicates that increasing pressure decreases volume
• Volume strain (hydraulic strain) is the ratio of the change in volume (ΔV) to the original volume (V)
• Volume strain = ΔV / V

Hooke's Law

• For small deformations, stress is directly proportional to strain
• This relationship is known as Hooke’s Law
• Stress ∝ Strain can be written as stress = k × strain or Stress / Strain = k
• 'k' is a constant called Modulus of Elasticity
• The SI unit of modulus of elasticity is N/m² or Pascal (Pa), the same as stress, since strain is unitless
• The dimensional formula is [ML⁻¹T⁻²]

Stress-Strain Curve

• The region from O to A shows a linear curve, indicating stress is proportional to strain
• Hooke's law is obeyed within this region
• In the region from A to B, stress and strain are not proportional, and Hooke's law is not obeyed, the body remains elastic
• Point B on the curve marks the yield point or elastic limit
• Yield strength (Sy) is the stress corresponding to the yield point
• Beyond point B, in the region from B to D, strain increases rapidly with small changes in stress
• If the load is removed at some point C between B and D, the body does not regain its original dimensions, resulting in a permanent set
• The material exhibits plastic behavior in this region
• Point D on the graph represents the ultimate tensile strength (Su) of the material
• In the region from D to E, additional strain occurs even with reduced applied force, leading to fracture at point E, called the Fracture Point
• If points D and E are close together, the material is brittle
• If points D and E are far apart, the material is ductile

Elastomers

• Elastomers are substances like tissue of the aorta or rubber that can be stretched to cause large strains
• Even though the elastic region is extensive, elastomers do not obey Hooke's law for most regions
• Elastomers lack a well-defined plastic region

Elastic Moduli

• The modulus of elasticity is the ratio of stress to strain
• There are three moduli of elasticity, depending on the type of stress and strain

1. Young's Modulus (Y)
2. Shear Modulus or Rigidity Modulus (G)
3. Bulk Modulus (B)

Young's Modulus (Y)

• Young's modulus defines 'the ratio of longitudinal stress to longitudinal strain'
• Y = (longitudinal stress) / (longitudinal strain)
• Formula: Y=(F/A)/(ΔL/L) = FL / AΔL
• If force F = mg and area A = πr², then Y = mgL / πr²ΔL
• The SI unit of Young's modulus is N/m² or Pa
• Metals have large Young's moduli
• Steel is more elastic than rubber because steel has a larger Young's modulus
• Wood, bone, concrete, and glass have relatively small Young's moduli

Steel in Heavy-Duty Machines and Structural Designs

• Steel is more elastic than copper, brass, and aluminum due to its higher Young's modulus, making it preferable in heavy-duty machines and structural designs

Shear Modulus or Rigidity Modulus (G)

• Shear modulus or rigidity modulus is the ratio of shearing stress to shearing strain
• The formula for shear modulus is G = (Shearing stress)/(Shearing strain)
• Which can be denoted by: G = (F/A) / θ
• Resulting in: G = F / Aθ
• The SI unit of shear modulus is N/m² or Pa
• Shear modulus is typically less than Young's modulus
• For most materials, G ≈ Y/3

Bulk Modulus (B)

• Bulk modulus is the ratio of hydraulic stress to hydraulic strain
• B = (Hydraulic stress)/(Hydraulic strain)
• Can be written as: B = (F/A) / (ΔV/ V)
• Resulting in: B = -P / (ΔV/ V), or B = -PV /ΔV
• The SI unit of the bulk modulus is N/m² or Pa
• The negative sign indicates that when pressure increases, volume decreases
• The bulk modulus B is always positive for a system in equilibrium

Compressibility (k)

• Compressibility is the reciprocal of the bulk modulus
• k = 1/B or k = -1/V * (ΔV/ ΔP)
• The bulk moduli for solids are much larger than for liquids, which are larger than for gases (air)
• Solids are the least compressible, while gases are the most compressible

Poisson's Ratio

• Stretching a material in one direction leads to compression in the perpendicular direction, and vice versa
• Longitudinal strain is the strain in the direction of the applied force
• Lateral strain is the strain perpendicular to the direction of the applied force
• Poisson's ratio (σ) is the ratio of lateral strain to longitudinal strain
• Written as: σ = (Lateral Strain) / (Longitudinal Strain)
• Which can be denoted: σ = (Δd / d) / (ΔL / L)
• Poisson's ratio is dimensionless

Elastic Potential Energy in a Stretched String

• The work done to deform a body is stored as Elastic Potential Energy (PE)
• For a string with small elongation dl, the work done dW = F dl, therefore W = ∫F dl
• Substituting F, it simplifies to: U = 1/2 x stress x strain x volume

Applications of Elastic Behavior of Materials

• Cranes use thick metal ropes made of metals with higher Young's modulus to lift heavy loads
• Ropes are made of multiple thin wires braided together for ease of manufacturing, flexibility, and strength
• The maximum height of mountains on Earth is limited to ~10 km due to the elastic properties of rocks
• In construction, beams should not bend or break; materials with a large Young's modulus Y are used to minimize bending
• Buckling a bending can be reduced by increasing the depth, but deep bars may bend sideways, hence avoiding buckling uses beams with cross-sectional shapes of I