4.2 Theory of Elasticity

4.2 Theory of Elasticity

1. Stress and Strain

1. Stress ($\sigma$):

   • Internal resistance per unit area.

   • Normal Stress: Perpendicular to surface.

     • Tensile (+) or Compressive (-).

   • Shear Stress ($\tau$): Parallel to surface.

   • Units: Pa (N/m²) or psi.

   • $\sigma = \frac{F}{A}$ (average stress).

2. Strain ($\epsilon$):

   • Measure of deformation.

   • Normal Strain: Change in length per original length.

     • $\epsilon = \frac{\Delta L}{L_0}$

   • Shear Strain ($\gamma$): Angular distortion.

     • $\gamma = \tan \theta \approx \theta$ (small angles).

   • Dimensionless quantity.

2. Hooke's Law

1. Fundamental Relation:

   • Within elastic limit: stress ∝ strain.

   • For normal stress: $\sigma = E\epsilon$

   • For shear stress: $\tau = G\gamma$

2. Elastic Limit:

   • Maximum stress without permanent deformation.

   • Below this: material returns to original shape.

3. Proportional Limit:

   • Maximum stress where linear relationship holds.

3. Elastic Moduli

1. Young's Modulus ($E$):

   • Ratio of normal stress to normal strain.

   • $E = \frac{\sigma}{\epsilon}$ (tension/compression).

   • Measures stiffness (larger $E$ = stiffer material).

2. Shear Modulus ($G$):

   • Ratio of shear stress to shear strain.

   • $G = \frac{\tau}{\gamma}$

   • Also called modulus of rigidity.

3. Bulk Modulus ($K$):

   • Ratio of volumetric stress to volumetric strain.

   • $K = \frac{-p}{\Delta V/V}$

   • Measures compressibility resistance.

4. Relationships:

   • $G = \frac{E}{2(1+\nu)}$

   • $K = \frac{E}{3(1-2\nu)}$

4. Thermal Stress

1. Thermal Expansion:

   • Length change: $\Delta L = \alpha L_0 \Delta T$

   • $\alpha$ = coefficient of thermal expansion.

2. Thermal Strain:

   • $\epsilon_{thermal} = \alpha \Delta T$

3. Thermal Stress:

   • When expansion/contraction is constrained:

     • $\sigma_{thermal} = E\epsilon_{thermal} = E\alpha \Delta T$

   • Compressive for temperature increase (if constrained).

   • Tensile for temperature decrease (if constrained).

5. Poisson's Ratio ($\nu$)

1. Definition:

   • Ratio of lateral strain to axial strain.

   • $\nu = -\frac{\epsilon_{lateral}}{\epsilon_{axial}}$

2. Range:

   • Theoretical: $-1 \leq \nu \leq 0.5$

   • Most materials: $0.2 \leq \nu \leq 0.4$

   • Rubber: $\nu \approx 0.5$ (nearly incompressible).

   • Cork: $\nu \approx 0$ (negligible lateral contraction).

3. Physical Meaning:

   • Material expands laterally when compressed axially.

   • Material contracts laterally when stretched axially.

4. Volume Change:

   • For uniaxial stress: $\frac{\Delta V}{V} = \epsilon(1-2\nu)$

6. Strain Energy and Impact Loading

1. Strain Energy ($U$):

   • Energy stored in deformed elastic body.

   • $U = \frac{1}{2} \times \text{Force} \times \text{Deformation}$

2. Strain Energy Density:

   • Energy per unit volume.

   • $u = \frac{1}{2}\sigma\epsilon = \frac{\sigma^2}{2E} = \frac{E\epsilon^2}{2}$

3. Impact Loading:

   • Sudden application of load (e.g., falling weight).

   • Stress and deflection are higher than static case.

4. Impact Factor:

   • Ratio of dynamic to static deformation/stress.

   • For sudden load (no initial velocity): factor = 2.

   • For falling mass from height $h$:

     • $\delta_{dynamic} = \delta_{static} \times \left(1 + \sqrt{1 + \frac{2h}{\delta_{static}}}\right)$

     • $\sigma_{dynamic} = \sigma_{static} \times \text{same factor}$

5. Assumptions:

   • Material remains elastic.

   • No energy loss during impact.

   • Mass of striking body >> struck body.