📘 Elasticity & Fluid Motion

Stress and Strain

• Stress = Force applied per unit area $$\text{Stress} = \frac{F}{A}$$Stress=AF​
• Strain = Change in dimension / Original dimension
  (Unitless)

Hooke’s Law:$$\text{Stress} \propto \text{Strain}$$Stress∝Strain

Elastic Constants

Relation Between Elastic Constants

1️⃣ Relation between $Y$Y, $η$η, $σ$σ:$$Y = 2η(1 + σ)$$Y=2η(1+σ)

2️⃣ Relation between $Y$Y, $K$K, and $σ$σ:$$Y = 3K(1 – 2σ)$$Y=3K(1−2σ)

3️⃣ Relation between $K$K, $η$η, $σ$σ:$$K = \frac{2η(1+σ)}{3(1-2σ)}$$K=3(1−2σ)2η(1+σ)​

Twisting Torque on a Cylinder or Wire

When a wire is twisted, a restoring torque is produced:$$τ = \frac{π η r^4 θ}{2l}$$τ=2lπηr4θ​

Where:
$η$η = Shear modulus, $r$r = Radius, $l$l = Length, $θ$θ = Twist angle

Bending of Beams

When a beam with length $L$L is loaded at its end, it bends. Tensile stress occurs on top layer and compressive on bottom.

External Bending Moment

$$M = Y I \frac{1}{R}$$M=YIR1​

where $R$R is radius of curvature and $I$I is the geometric moment of inertia.

Flexural Rigidity

$$\text{Flexural Rigidity} = YI$$Flexural Rigidity=YI

Higher value → more resistance to bending.

Cantilever Beams

2️⃣ Fluid Motion

Kinematics of Moving Fluids

Flow of liquid involves velocity, pressure, and viscosity.
We use streamlines to show direction of flow.

Viscosity (η)

Internal friction between layers of a fluid.

Newton’s Law of Viscosity:$$F = η A \frac{dv}{dx}$$F=ηAdxdv​

Unit: Poise, SI: Pa·s

Poiseuille’s Equation (Flow through Capillary Tube)

For laminar flow through a tube of radius $r$r and length $l$l:$$V = \frac{\pi r^4 (P_1 – P_2)}{8ηl}$$V=8ηlπr4(P1​−P2​)​

Where $V$V = Volume of liquid flowing per second.

Corrections in Poiseuille’s Law

• End correction for extra pressure near entry/exit
• Effective length: $l + 2.4r$l+2.4r

Surface Tension (T)

Force acting along the surface of fluid causing it to minimize area.

Examples:

• Liquid drops are spherical
• Capillary rise in thin tubes

Formula:$$T = \frac{F}{2L}$$T=2LF​

Gravity Waves & Ripples

Summary Table

📌 Applications in Real Life

✔ Bridges, Buildings → Flexural rigidity
✔ Capillary rise → Ink pens, soil water transport
✔ Oil & blood flow → Viscosity
✔ Torsion springs → Mechanical clocks