# Mechanical Properties of Metals

## Key Concepts

- **Mechanical properties** discussed:
- Stiffness (Elastic modulus)
- Strength (Yield \& tensile strength)
- Ductility
- Hardness
- Toughness
- **Mechanical failure** often occurs due to fatigue (~90% of metallic failures).
**Examples**: Computer chips (thermal cycling), hip implants (walking), bridges (cyclic loading).

***

## Fundamental Parameters

- **Stress** ($\sigma$)
- Measures force per unit area ($\frac{F}{A}$)
- Types: Tensile (pulling), compressive (pushing), and shear (sliding)
- **Strain** ($\epsilon$)
- Measures deformation per unit length ($\frac{\Delta L}{L_0}$)
- Dimensionless quantity

***

## Elastic and Plastic Deformation

- **Elastic deformation**: Reversible, material returns to original shape when load removed.
- Follows Hooke's Law: $\sigma = E \times\epsilon$
- $E$: Young’s modulus (stiffness)
- **Plastic deformation**: Permanent, atoms are displaced and material does not revert.
- Associated with dislocation motion in crystals

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## Testing Methods

- **Tensile Test**: Measures stress-strain curve, determines yield strength, tensile strength, ductility.
- **Hardness Test**: Rockwell, Brinell, Mohs scales—quantify resistance to surface indentation.

***

## Important Equations

- **Normal Stress**:

$$
\sigma = \frac{F}{A}
$$
- **Strain**:

$$
\epsilon = \frac{\Delta L}{L_0}
$$
- **Shear Stress**:

$$
\tau = \frac{F}{A_S}
$$
- **Torsion for Shear**:

$$
\tau = \frac{M}{A_C R}
$$
- **Modulus of Resilience**:

$$
U_r = \frac{y_y^2}{2E}
$$

***

## Material Property Comparisons

| Material | Young’s Modulus (GPa) |
| :-- | :-- |
| Aluminum | ~62 |
| Iron | ~208 |
| Magnesium | ~44 |
| Glass | ~70 |
| Polystyrene | ~2.8 |
| Nylon | ~2.8 |

***

## Ductility \& Toughness

- **Ductility**: Ability to deform without breaking (measured by % elongation/reduction in area).
- **Toughness**: Energy absorbed before fracture (area under stress-strain curve).
- **Brittle vs. Ductile Behavior**: Brittle materials fail with little deformation; ductile show significant plastic deformation.

***

## Hardness Correlations

- Hardness ↑ ⇒ Yield/Tensile strength ↑
- Conversion between scales approximate for steels: $TS (MPa) \approx 3.45 \times HB$ (Brinell)

***

## Design \& Safety

- **Factor of Safety** ($N$): Margin to account for uncertainties. Working stress $w = \frac{y}{N}$.
- Example: For yield strength $y$ and applied force $F$, determine minimum required diameter for a rod.

***

## Summary Table

| Property | Resistance Type | Measured By |
| :-- | :-- | :-- |
| Stiffness | Elastic deformation | Young’s modulus |
| Strength | Plastic deformation | Yield/Tensile strength |
| Ductility | Failure deformation | % elongation/reduction |
| Hardness | Surface deformation | Hardness scales (HR, HB) |
| Toughness | Energy absorption | Area under curve |