Mechanical Properties of Metals & Alloys – Part 1

Learning Outcomes

• By the end of the chapter a student should be able to:
  • Define engineering stress and engineering strain.
  • Given an engineering stress–strain diagram, determine:
  • Modulus of elasticity $E$ (Young’s modulus).
  • Tensile (ultimate) strength $\sigma_\text{UTS}$.
  • Estimate percent elongation $\%EL$ at fracture.
  • Compute ductility in terms of:
  • Percent elongation $\%EL$.
  • Percent reduction of area $\%RA$.

Why Study Mechanical Properties?

• Mechanical engineers
  • Design machines/structures that experience forces & deformations.
  • Must choose materials so that unacceptable deformation or failure does not occur.
• Structural engineers
  • Determine & predict stress distributions in loaded members (experimentally or analytically).
• Materials & metallurgical engineers
  • Produce / fabricate materials to meet the above service requirements.
  • Study relationships between micro-structure and mechanical properties.

Definition of Mechanical Properties

• Describe a material’s response/deformation under applied load.
• Behaviour shows both elastic (recoverable) and plastic (permanent) deformation before failure.
• Measured by mechanical tests; response depends on bonding & atomic arrangement.
• Two basic deformation types for load-carrying materials:
  • Elastic deformation – instantaneous, fully recoverable.
  • Plastic deformation – non-recoverable; permanent shape change.

Common Mechanical Properties & Associated Tests

• Properties: hardness, ductility, yield strength, toughness, brittleness, tensile/ultimate strength, modulus of elasticity, malleability, fracture strength, stiffness, flexural strength.
• Tests: tension, compression, hardness (micro/macro), impact, flexural/bending, torsion/shear, fatigue (S–N, fatigue crack growth), creep (elevated-temperature), fracture toughness.

Material Property Assessment Summary

• Hardness → Micro/Macro hardness tests.
• Strength/yield/ultimate → Tension tests, torsion tests.
• Ductility (elongation, area reduction) → Tension tests.
• Creep (high-T strength) → Creep tests.
• Toughness (resistance to failure) → Impact tests, fracture toughness tests.
• Fatigue strength → S–N tests, fatigue crack growth tests.
• Modulus of elasticity/stiffness/flexural strength → Tension, flexural tests.

Elastic Deformation ("Elastic means reversible")

• Sequence for small load $F$ on bonds:
  1. Initial, undeformed.
  2. Bonds stretch by distance $d$.
  3. Upon unloading, material returns to original shape.
• Two elastic categories on stress–strain curve:
  • Linear-elastic (Hookean).
  • Non-linear elastic.

Plastic Deformation ("Plastic means permanent")

• After yield, atoms/glide planes shear; even after unloading only elastic part recovers, plastic offset remains $d_{plastic}$.
• For metals, plasticity is accommodated by slip of planes & dislocation motion.

Modes of Loading / Types of Stress

• Tension, compression, shear, torsion, bending, buckling.
• Visualised deformations:
  • (a) Tension → elongation (+ve strain).
  • (b) Compression → contraction (−ve strain).
  • (c) Shear → $\gamma = \tan\theta$.
  • (d) Torsion → angle of twist $\phi$ under torque $T$.

Concept of Stress & Strain – General States

• Tensile stress: $\sigma = \dfrac{F}{A_0}$.
• Shear stress (torsion): $\tau = \dfrac{F<em>s}{A</em>0}$ or for a circular shaft $\tau = \dfrac{M}{A_c R}$.
• Compression identical formula with \sigma<0.
• Engineering strains (dimensionless):
  • Axial tensile/compressive: $\varepsilon = \dfrac{\Delta L}{L_0}$.
  • Lateral: $\varepsilon<em>L = \dfrac{\Delta w}{w</em>0}$.
  • Shear: $\gamma = \tan\theta = \dfrac{\Delta x}{y}$.

Poisson’s Ratio $\nu$

• $\nu = -\dfrac{\varepsilon_L}{\varepsilon}$ (negative sign gives positive $\nu$ because lateral strain is opposite sign to axial strain).
• Typical values: metals ≈ 0.33; ceramics ≈ 0.25; polymers ≈ 0.40.
• Theoretical isotropic: $0.25$; maximum $0.50$ (incompressible). \nu>0.50 → density increases; \nu<0.50 → density decreases (voids form).

Tensile Test: Equipment & Procedure

• Universal test machine with hydraulic actuator.
• Specimen clamped, elongated at constant rate.
• Load cell measures instantaneous force $F$, extensometer/gauge marks measure elongation $\Delta L$.
• Results converted to engineering stress $\sigma=F/A<em>0$ and engineering strain $\varepsilon=\Delta L/L</em>0$ to compare specimens of differing size.

Engineering Stress–Strain Curve (Typical)

• Regions:
  • Elastic (linear) with slope $E$.
  • Yield point / proportional limit $P$.
  • Strain-hardening (uniform plastic deformation).
  • Ultimate tensile strength (maximum $\sigma_{UTS}$).
  • Necking (non-uniform plastic deformation).
  • Fracture (stress at break $\sigma_f$).
• Key ordinates:
  • $E$: modulus of elasticity $E = \dfrac{\Delta\sigma}{\Delta\varepsilon}$ within linear segment.
  • Yield strength $\sigma_y$ defined by $0.2\%$ offset ( $\varepsilon=0.002$ ).
  • Ultimate strength $\sigma_{UTS}$ – peak engineering stress.
  • Fracture strength $\sigma_f$ – stress at specimen failure.

Mechanical Properties Obtained from Tensile Test

1. Modulus of elasticity $E$ (GPa).
2. Proportional limit & $0.2\%$ offset yield strength $\sigma_y$.
3. Ultimate tensile strength $\sigma_{UTS}$.
4. Fracture strength $\sigma_f$.
5. Ductility metrics: $\%EL$, $\%RA$.

• Example property table (room-temperature, typical):
  • Aluminium: $\sigma<em>y≈35\,\text{MPa},\;\sigma</em>{UTS}≈90\,\text{MPa},\;\%EL≈40$.
  • Steel 1020: $\sigma<em>y≈180\,\text{MPa},\;\sigma</em>{UTS}≈380\,\text{MPa},\;\%EL≈25$.
    (Full table provided in transcript.)

Calculating Ductility

• Percent elongation:
  • $\%EL = \dfrac{L<em>f - L</em>0}{L_0}\times 100$.
• Percent reduction in area:
  • $\%RA = \dfrac{A<em>0 - A</em>f}{A_0}\times 100$.
• Requires measurements after fracture (final gauge length & necked diameter).

Linear Elasticity & Hooke’s Law

• For tension/compression: $\sigma = E\varepsilon$.
• Rearrangements:
  • $E = \dfrac{\sigma}{\varepsilon}$.
  • $\varepsilon = \dfrac{\sigma}{E}$.
• Elastic elongation of prismatic bar:
  • $\Delta L = \dfrac{F L<em>0}{A</em>0 E}$.

Example Problems (Highlights)

Example 1: Elastic Elongation of Copper Bar

• Given: $L<em>0=305\,\text{mm}$, $\sigma=276\,\text{MPa}$, $E</em>{Cu}=110\,\text{GPa}$.
• $\Delta L = \dfrac{\sigma L_0}{E} = \dfrac{(276)(305)}{110\times10^3}=0.77\,\text{mm}$.

Example 2: Load Causing Specified Diameter Change in Brass Rod

• Diameter $d_0=10\,\text{mm}$, desired $\Delta d = -2.5\times10^{-3}\,\text{mm}$ (negative = shrink).
• Lateral strain $\varepsilon<em>x = \dfrac{\Delta d}{d</em>0} = -2.5\times10^{-4}$.
• Axial strain $\varepsilon<em>z = -\dfrac{\varepsilon</em>x}{\nu} = 7.35\times10^{-4}$ with $\nu=0.34$.
• Axial stress $\sigma = E\varepsilon<em>z = 71.3\,\text{MPa}$ $\big(E</em>{Brass}=97\,\text{GPa}\big)$.
• Load: $F = \sigma A_0 = (71.3\times10^6)\dfrac{\pi (10\times10^{-3})^2}{4}=5.6\,\text{kN}$.

Example 3: Brass Specimen – Properties from Stress–Strain Curve

• (a) $E \approx 94\,\text{GPa}$ (from linear slope).
• (b) $\sigma_y \approx 250\,\text{MPa}$ at $0.2\%$ offset.
• (c) Max load for $d<em>0=12.8\,\text{mm}$: $F</em>{max}=57.9\,\text{kN}$ using $\sigma_{UTS}=450\,\text{MPa}$.
• (d) For $\sigma=345\,\text{MPa}$, read $\varepsilon≈0.06$ → $\Delta L=(0.06)(250)=15\,\text{mm}$.

Example 4: Steel Specimen Ductility & True Stress at Fracture

• $d<em>0=12.8\,\text{mm},\;d</em>f=10.7\,\text{mm},\;\sigma_f=460\,\text{MPa}$.
• $A<em>0=128.7\,\text{mm}^2,\;A</em>f=89.9\,\text{mm}^2$.
• $\%RA = \dfrac{128.7-89.9}{128.7}\times100=30\%$.
• Load at fracture: $F=\sigma<em>f A</em>0 = 59.2\,\text{kN}$.
• True stress at fracture: $\sigma<em>T = \dfrac{F}{A</em>f}=660\,\text{MPa}$.

Key Equations (Summary)

• Engineering stress: $\sigma = \dfrac{F}{A_0}$.
• Engineering strain: $\varepsilon = \dfrac{\Delta L}{L_0}$.
• Hooke’s law (linear elastic): $\sigma = E \varepsilon$.
• Poisson’s ratio: $\nu = -\dfrac{\varepsilon_L}{\varepsilon}$.
• Percent elongation: $\%EL = \dfrac{L<em>f-L</em>0}{L_0}\times100$.
• Percent reduction in area: $\%RA = \dfrac{A<em>0-A</em>f}{A_0}\times100$.
• True stress (at any instant): $\sigma<em>T = \dfrac{F}{A</em>{inst}}$; at fracture use $A_f$.

Practical / Philosophical Implications

• Choice of material & cross-section is guided by property limits such as $\sigma<em>y$, $\sigma</em>{UTS}$, $E$, $\%EL$ ensuring safety factors.
• Ethical responsibility for engineers to avoid catastrophic failure through accurate testing & interpretation of mechanical properties.
• Awareness of test limitations (destructive, room-temperature, static) is crucial; actual service may involve high T, cyclic or impact loads → additional tests (creep, fatigue, impact) required.

Connections & Real-World Relevance

• Design codes (ASME, AISC, ISO) embed yield & tensile requirements; test data feed into Finite-Element stress analyses.
• Ductility metrics inform forming processes (rolling, drawing) and failure modes (ductile vs brittle fracture).
• Poisson’s ratio enters vibration, stability and multi-axial stress calculations (e.g., pressure vessels, biomechanics).

Reference Values (Room-Temperature)

• $E<em>{Steel}\approx207\,\text{GPa}$; $E</em>{Al}\approx69\,\text{GPa}$; $E<em>{Brass}\approx97\,\text{GPa}$; $E</em>{Cu}\approx110\,\text{GPa}$.
• Shear modulus $G$ linked: $G = \dfrac{E}{2(1+\nu)}$ for isotropic materials.

Testing Ethics & Disclaimer

• Tests are destructive; specimens permanently deformed/fractured.
• Data compiled for educational, non-commercial purposes; reliability depends on cited sources (Callister, Smith & Hashemi, etc.).