Some Mechanical Properties of Matter

Poisson ratio (σ):

• Ratio of transverse strain to longitudinal strain with opposite sign
• $\sigma = -\frac{\Delta d/d}{\Delta L/L}$
• Describes how a material contracts perpendicular to the direction it's stretched
• When a rod is stretched longitudinally, its diameter decreases
• Typical values range from 0.16 to 0.32 for common metals
• Physical significance: characterizes volume conservation during deformation
• Example: For steel (σ = 0.19), if stretched 1% longitudinally, diameter decreases by 0.19%

Two fundamental types of stress:

1. Normal (Longitudinal) Stress (Γₙ):

   • Defined as force component normal to the cross-sectional area divided by that area
   • Formula: Γₙ = Fₙ/ΔS
   • Can be tensile (pulling) or compressive (pushing)

2. Tangential (Shearing) Stress (Γₜ):

   • Defined as force component tangential to the cross-sectional area divided by that area
   • Formula: Γₜ = Fₜ/ΔS
   • Causes layers of the material to slide relative to each other

Each stress type is measured in N/m² or Pascal (Pa).

For small deformations, shearing stress and strain follow Hooke's Law:

• Shearing stress (Γt) is proportional to shearing strain (x/h)
• The constant of proportionality is called the shear modulus (η), also known as modulus of rigidity or torsional modulus

Mathematically:

• Γt = η × (x/h)
• η = (F/A)/(x/h) = Fh/Ax

Where:

• F = applied tangential force
• A = area parallel to force
• x = displacement
• h = distance from fixed layer

The shear modulus is a material property that indicates resistance to shearing deformation. Materials with higher shear modulus (like steel: 0.84×10¹¹ N/m²) resist shearing better than those with lower values (like aluminum: 0.30×10¹¹ N/m²).

The stress-strain relationship of vulcanized rubber has several unique characteristics:

• Remains elastic even when stretched to 8+ times its original length
• No region of proportionality between stress and strain
• Follows a non-linear curve throughout its elastic range
• Displays elastic hysteresis - the loading curve differs from the unloading curve
• Returns to original length after releasing stress despite extreme elongation
• Absorbs energy during deformation cycles, converting it to heat
• Lower magnitude of stress for a given strain compared to metals

This behavior makes vulcanized rubber excellent for shock absorbers, as it can absorb vibration energy through its hysteresis loop.

The U-shaped wire frame demonstration illustrates the relationship between surface tension and surface energy through:

• When the sliding wire is pulled outward by distance x:

• The film's surface area increases by 2·l·x (two surfaces)
• Work done against surface tension = F·x = 2·S·l·x
• This work is stored as increased surface energy = S·(2·l·x)

• This shows that surface tension S equals surface energy per unit area:

• Surface energy = work done / area created
• Surface energy/area = (2·S·l·x)/(2·l·x) = S

• The soap film behavior reveals that:

• Surfaces behave like stretched elastic membranes
• They store energy proportional to their area
• They exert tangential forces to minimize their area

This demonstrates why S has equivalent units of N/m (force/length) and J/m² (energy/area).

The transformation occurs due to surface tension:

• Initially: The thread experiences balanced forces from both sides of the soap film

  • Forces from inner and outer surfaces cancel out
  • Thread can maintain any arbitrary shape

• After breaking inner film: Force equilibrium is disrupted

  • Only the outer surface remains intact
  • Surface tension creates uniform pulling forces acting radially outward
  • Each point on the thread experiences equal tension in the direction perpendicular to the thread

• Circular shape forms because:

  • For a fixed perimeter length, a circle encloses maximum area
  • This represents the minimum energy configuration
  • Surface energy is minimized when surface area is minimized

This phenomenon demonstrates how surfaces under tension behave like elastic membranes seeking their lowest energy state.

The resultant pressure force on a hemispherical surface can be simplified using:

1. For any axis (e.g., X-axis), the resultant force equals: F = P₁ × (Projection of entire hemispherical surface onto plane perpendicular to axis)

2. For a hemisphere of radius R, this projection is simply the circular disc with area πR²

3. Therefore: F = P₁πR²

This relationship works because:

• Each surface element contributes P₁ΔS·cosθ along the axis
• The sum of all projections equals the area of the circular cross-section
• This method avoids complex integration over the curved surface

This principle applies to finding resultant pressure forces on any curved surface where the projection can be determined.

Area projection is crucial in fluid mechanics because:

1. It simplifies force calculations:

   • Instead of integrating force components over complex curved surfaces
   • We can multiply pressure by the projected area on a plane

2. It reveals that the resultant force depends on the silhouette, not the actual surface area:

   • Two different shaped objects with identical projections experience identical resultant forces along that axis
   • This explains why a hemisphere and cone with the same base experience the same horizontal pressure force

3. It enables practical solutions for engineering problems:

   • Dam design calculations
   • Pressure vessel analysis
   • Fluid thrust on curved surfaces like turbine blades

4. It demonstrates the vector nature of pressure forces, showing how their components can be resolved along different coordinate axes for analysis.

This concept is foundational for analyzing hydrostatic forces on submerged curved surfaces in fluid mechanics.

Viscous force-velocity proportionality:

The viscous force is proportional to velocity because:

• Fluid layers move at different speeds in the presence of an object
• The velocity gradient (shear rate) between layers causes internal friction
• This friction is proportional to how quickly the layers slide past each other
• The mathematical expression is: $F = 6\pi\eta rv$ for spheres

Effect on terminal velocity development:

1. Initial state: Object has zero velocity, so viscous force = 0
2. Net force = Weight - Buoyancy > 0, causing acceleration
3. As velocity increases, viscous force increases proportionally
4. The increasing viscous force gradually reduces the net acceleration
5. Terminal velocity is reached when: Net force = 0
   • Weight - Buoyancy - Viscous force = 0
   • Viscous force = Weight - Buoyancy = constant

This creates a self-regulating system:

• If v > terminal velocity: viscous force exceeds Weight - Buoyancy, causing deceleration
• If v < terminal velocity: viscous force is less than Weight - Buoyancy, causing acceleration

Buoyancy complications in Stokes' law applications:

The buoyancy force creates several complications:

1. Force balance complexity:

   • Net force = Weight - Buoyancy - Viscous force
   • If buoyancy is neglected, viscosity calculations will be incorrect

2. Density difference dependence:

   • Terminal velocity depends on (ρ-σ), not just object density
   • Objects with density close to fluid may move very slowly
   • Air bubbles rise rather than fall due to negative (ρ-σ)

3. Experimental considerations:

   • For accurate viscosity measurements: $\eta = \frac{2r^2(\rho-σ)g}{9v_0}$
   • Both object and fluid density must be precisely known
   • Temperature affects both viscosity and density

4. Practical solutions:

   • Choose objects with significantly higher density than fluid
   • Measure densities at the same temperature as the experiment
   • For rising objects (like air bubbles), use the same equation but account for direction
   • Use multiple objects of different densities to verify consistency

5. Special case - air bubbles:

   • When measuring with air bubbles, ρ ≈ 0, so: $\eta = \frac{2r^2σg}{9v_0}$
   • This simplifies density measurements to just the fluid