Compendium

Mechanical Behavior of Materials — Elastic, Plastic, Fracture, Fatigue, Creep

24 min read

Mechanical Behavior of Materials

The study of how solids deform, yield, and fracture under load. This note covers the full deformation hierarchy from infinitesimal elastic response through dislocation-mediated plasticity, brittle and ductile fracture, fatigue, creep, viscoelasticity, hardness, wear, and environmentally assisted cracking. The framework is continuum mechanics on the macroscale, dislocation theory on the microscale, and fracture mechanics for defect-tolerant design. Units are SI primary; US customary in parentheses where idiomatic in engineering practice.

1. Stress and Strain

1.1 The Cauchy stress tensor

The Cauchy stress σ is a 3×3 second-order symmetric tensor (six independent components) relating a unit outward normal n on an internal surface to the traction vector t via t = σ·n. Symmetry σ_ij = σ_ji follows from balance of angular momentum (Cauchy 1827) in the absence of body couples.

Components in Cartesian coordinates:

  • Normal stresses: σ_xx, σ_yy, σ_zz (units Pa = N/m²; engineering MPa or ksi)
  • Shear stresses: σ_xy = τ_xy, σ_yz, σ_xz

Hydrostatic (mean) stress: σ_h = (σ_xx + σ_yy + σ_zz)/3 = (1/3) tr(σ). The deviatoric stress is s = σ − σ_h I (controls shape change; the hydrostatic part controls volume change).

Engineering stress σ_e = F/A_0 uses the initial cross-section area; true stress σ_t = F/A_i uses the instantaneous area. For incompressible plastic flow with no necking, σ_t = σ_e·(1 + ε_e) and ε_t = ln(1 + ε_e). The two diverge sharply past 5–10% strain.

1.2 Strain measures

Infinitesimal (small) strain tensor: ε_ij = (1/2)(∂u_i/∂x_j + ∂u_j/∂x_i) valid when |∇u| ≪ 1 (engineering metals at room temp, polymers below glass transition for small loads).

Finite strain measures:

  • Green–Lagrange E = (1/2)(F^T F − I), reference configuration (Lagrangian)
  • Almansi e = (1/2)(I − F^-T F^-1), current configuration (Eulerian)
  • Logarithmic (Hencky, true) ε = ln(λ), preferred for large plastic strains because successive strains add linearly

Deformation gradient F = ∂x/∂X relates reference X to current x. Polar decomposition F = R·U = V·R splits rotation from stretch.

1.3 Principal stresses and invariants

Diagonalizing σ yields the three principal stresses σ_1 ≥ σ_2 ≥ σ_3 along mutually orthogonal principal directions. The three invariants:

  • I_1 = σ_1 + σ_2 + σ_3 (trace)
  • I_2 = σ_1σ_2 + σ_2σ_3 + σ_3σ_1
  • I_3 = σ_1σ_2σ_3 (det σ)

Deviatoric invariants used in plasticity:

  • J_1 = 0
  • J_2 = [1/6]((σ_1−σ_2)² + (σ_2−σ_3)² + (σ_3−σ_1)²)
  • J_3 = det(s)

1.4 Equivalent stress measures

Von Mises equivalent stress: σ_VM = √(3 J_2) = √([1/2]((σ_1−σ_2)² + (σ_2−σ_3)² + (σ_3−σ_1)²)) = √((3/2) s:s)

The von Mises stress reduces to the uniaxial tensile stress in a pure tension test (σ_VM = σ_1 when σ_2 = σ_3 = 0). It is the workhorse scalar for ductile-metal stress assessment.

Tresca (maximum shear stress) equivalent stress: σ_Tresca = σ_1 − σ_3 = 2 τ_max More conservative than von Mises by up to 15.5% (factor 2/√3); historically preferred for hand calculations because of its simplicity.

2. Elasticity

2.1 Hooke’s law (generalized)

Linear elasticity σ = C : ε where C is the 4th-order stiffness tensor (81 components in 3D; 21 independent for the most general anisotropy thanks to symmetries of σ, ε, and the strain-energy density).

Compliance form: ε = S : σ, S = C⁻¹.

Symmetry reductions for the number of independent elastic constants:

  • Triclinic: 21
  • Monoclinic: 13
  • Orthotropic (3 mutually perpendicular planes of symmetry — wood, rolled sheet, many composites): 9
  • Transversely isotropic (one axis of symmetry — unidirectional fiber composites, drawn polymers): 5
  • Cubic (Fe, Cu, NaCl): 3 (C_11, C_12, C_44)
  • Isotropic (random polycrystals, glass): 2

2.2 Isotropic elastic constants

For isotropic solids only two independent constants are needed. The common pairs and conversions:

PairConversion
(E, ν)G = E/[2(1+ν)], K = E/[3(1−2ν)]
(G, ν)E = 2G(1+ν)
(K, G)E = 9KG/(3K+G), ν = (3K−2G)/(6K+2G)
(λ, μ) LaméE = μ(3λ+2μ)/(λ+μ), ν = λ/[2(λ+μ)]

Constraints: −1 < ν < 0.5; E, G, K > 0. Auxetic materials (ν < 0) exist (Lakes 1987 reentrant foams, some crystalline phases).

Representative room-temperature isotropic values:

MaterialE (GPa)νG (GPa)K (GPa)
Diamond10500.07478442
Tungsten carbide6000.22245350
Steel (mild)2070.3080170
Cast iron100–1600.25——
Titanium (Grade 5)1100.3441105
Aluminum (6061)690.332667
Glass (soda lime)700.222840
Concrete300.20——
PMMA30.401.15.6
Polyethylene HDPE1.00.46——
Natural rubber0.01–0.1≈0.4990.0003–0.03≈2
Cortical bone170.306—

2.3 Anisotropic and crystal elasticity

For a cubic single crystal the 3 independent constants (Voigt notation) C_11, C_12, C_44 determine direction-dependent moduli:

  • E_<100> = (C_11 + 2 C_12)(C_11 − C_12)/(C_11 + C_12)
  • E_<111> involves C_44

Anisotropy ratio A = 2 C_44/(C_11 − C_12). Tungsten is nearly isotropic (A ≈ 1); copper A ≈ 3.2; iron A ≈ 2.4.

2.4 Stored elastic strain energy

Density U = (1/2) σ : ε; uniaxial U = σ²/(2E). At yielding of a 1 GPa-strength steel, U ≈ 2.4 MJ/m³ — close to the kinetic energy density of a bullet, illustrating why elastic snap-back of pressurized vessels is dangerous.

3. Yield Criteria

A yield criterion specifies the surface in stress space enclosing the elastic region.

3.1 Von Mises (J_2) — ductile metals

f(σ) = √(3 J_2) − σ_y = 0 A right circular cylinder centered on the hydrostatic axis. Derived from distortional strain energy (Maxwell 1865, Huber 1904, Hencky 1924). Most accurate for FCC metals.

3.2 Tresca — conservative metal design

τ_max = (σ_1 − σ_3)/2 = σ_y/2 A regular hexagonal prism inscribed in the von Mises cylinder. Used in ASME Boiler & Pressure Vessel Code Section VIII Div. 1.

3.3 Drucker–Prager — pressure-sensitive solids (granular, polymers, geomaterials)

√J_2 + α I_1 = k A cone in principal-stress space. Reduces to von Mises at α = 0.

3.4 Mohr–Coulomb — soils and rocks

τ = c + σ_n tan φ where c is cohesion and φ is internal friction angle. A pyramid in stress space.

3.5 Hill-48 anisotropic — rolled sheet

F(σ_y−σ_z)² + G(σ_z−σ_x)² + H(σ_x−σ_y)² + 2L τ_yz² + 2M τ_zx² + 2N τ_xy² = 1 Six anisotropy parameters fitted from r-values (Lankford coefficients) measured in 0°/45°/90° tensile tests on rolled sheet.

3.6 Tsai–Hill and Tsai–Wu — composites

Tsai–Hill (1968): a Hill-type extension treating fiber-matrix anisotropy. Tsai–Wu (1971): polynomial form with explicit tension/compression asymmetry, used in laminate failure analysis under ASTM D3039 / D3410 / D3518 protocols.

4. Plasticity

4.1 The tensile stress–strain curve

Standard tensile test (ASTM E8 / ISO 6892):

  • Elastic region: linear up to proportional limit
  • Yield: 0.2% offset proof stress σ_0.2 by convention (use upper/lower yield for low-carbon steel that shows a yield drop)
  • Strain hardening: σ rises with ε
  • Necking onset at ultimate tensile strength (UTS): Considère criterion dσ/dε = σ
  • Fracture: at the engineering strain ε_f

4.2 Hardening laws

Hollomon: σ_t = K · ε_t^n (n = strain-hardening exponent, K = strength coefficient) Ludwik: σ_t = σ_0 + K · ε_t^n Swift: σ_t = K(ε_0 + ε_t)^n Voce: σ_t = σ_s − (σ_s − σ_0) exp(−ε_t/ε_c) (saturating)

Typical Hollomon n values: 0.2–0.3 for annealed low-carbon steel, 0.1–0.2 for stainless, 0.05–0.15 for Al alloys, 0.5 for fully annealed Cu.

4.3 Strain-rate sensitivity

σ ∝ ε̇^m m = ∂ ln σ / ∂ ln ε̇ at fixed ε, T. For superplastic alloys (Pb-Sn eutectic, fine-grain Ti-6Al-4V) m ≈ 0.5, enabling tensile elongations of >500% (basis for Boeing 787 titanium part forming). Standard steels at room T: m ≈ 0.01.

4.4 Considère criterion (necking)

A tensile specimen necks when the rate of geometric softening equals the rate of strain hardening: dσ_t/dε_t = σ_t Equivalently for Hollomon: ε_t at necking = n.

4.5 Flow rule

Associated (von Mises): plastic strain increment is normal to the yield surface, dε^p = dλ ∂f/∂σ. For J_2 plasticity this gives the Lévy–Mises equations and incompressibility tr(dε^p) = 0.

5. Cold Work, Recovery, Recrystallization, Grain Growth

Plastic deformation stores roughly 5% of the plastic work as dislocation strain energy (the rest dissipates as heat).

Sequence on heating after cold work (e.g., 70%-rolled Cu):

  • Recovery (0.2–0.4 T_m on the absolute scale): dislocation rearrangement, subgrain formation, residual-stress relief. Resistivity drops; strength almost unchanged.
  • Recrystallization (~0.4 T_m for pure metals, higher for alloys): new strain-free grains nucleate and grow. Strength drops to annealed level. Recrystallization temperature is suppressed by impurities and prior strain.
  • Grain growth: large grains consume small ones to reduce grain-boundary area. d² − d_0² = k t (normal grain growth, n = 2 idealization).

Hot working = deformation above the recrystallization temperature, restores ductility continuously; cold working below it builds strength but reduces ductility.

6. Strengthening Mechanisms

All mechanisms work by impeding dislocation motion. Total strength is roughly additive (linear sum or root-sum-square depending on obstacle strength).

6.1 Solid-solution strengthening (Mott–Nabarro, Fleischer)

Δτ ∝ √c (atomic concentration) for weak obstacles; ∝ c^(2/3) for strong. Size misfit ε_b and modulus misfit ε_G drive interactions. Substitutional (Cu in Ni) vs interstitial (C, N in Fe — especially strong because they distort the bcc lattice tetragonally).

6.2 Precipitation hardening (Orowan looping vs particle shearing)

Discovered by Alfred Wilm 1906 (Duralumin, accidentally aged at room T over a weekend). For coherent shearable particles τ ∝ √r (Friedel); for incoherent bypassed particles (Orowan loops) τ = (Gb)/(L − 2r), so strength peaks at the cross-over (peak-aged condition). Examples: Al 2024 (Al-Cu-Mg, θ’ precipitates), 7075 (Al-Zn-Mg, η’/η), 6061 (Mg₂Si), Ni-base superalloys (γ’ Ni₃(Al,Ti) coherent L1₂).

6.3 Grain refinement (Hall–Petch)

σ_y = σ_0 + k_HP / √d σ_0 = friction stress; k_HP = Hall–Petch slope (MPa·m^0.5):

  • Low-carbon steel ≈ 0.7
  • Cu ≈ 0.11
  • Al ≈ 0.07
  • Mg ≈ 0.28

Numerical example: low-carbon steel with d = 100 μm gives σ_0 + k/√d = 70 + 0.7/√(10⁻⁴) = 70 + 70 = 140 MPa. Refining to 10 μm: 70 + 221 = 291 MPa. Refining to 100 nm (severe plastic deformation, ECAP — Valiev 1990s) approaches ~2 GPa but the law breaks down at nanocrystalline sizes (inverse Hall–Petch below ~20 nm, GB sliding dominates).

6.4 Dislocation forest hardening (Taylor)

τ = α G b √ρ α ≈ 0.3 (FCC), ρ = dislocation density (m⁻²). Annealed metals ρ ≈ 10¹⁰ m⁻²; heavily cold-worked ρ ≈ 10¹⁶ m⁻². Numerical: Cu G = 48 GPa, b = 0.256 nm, ρ = 10¹⁵ → τ = 0.3·48e9·0.256e-9·√1e15 = 117 MPa.

6.5 Work hardening

Stage I (easy glide, single slip), Stage II (linear hardening, multi-slip, dislocation tangles), Stage III (dynamic recovery, parabolic). Dislocation generation by Frank–Read sources.

7. Dislocations

The carriers of metallic plasticity, predicted by Orowan, Polanyi, Taylor independently in 1934.

7.1 Geometry

Burgers vector b: closure failure of a circuit around the dislocation line. Edge: b ⊥ line. Screw: b ∥ line. Mixed otherwise. Magnitude |b| = a/√2 for FCC ½<110>, a√3/2 for BCC ½<111>, a for HCP on basal.

Line tension T_l ≈ G b² /2 (energy per unit length ~ 5 eV/atom-length). Strain field 1/r — slowly decaying.

7.2 Slip systems

7.3 Peierls stress

Lattice-friction stress σ_P = (2G/(1−ν)) exp[−2π w/b], where w is dislocation core width. Low for FCC (10⁻⁵ G), high for BCC at low T (10⁻³ G) and covalent crystals (Si, diamond). Explains the FCC/BCC DBTT contrast.

7.4 Sources and multiplication

Frank–Read source: a dislocation pinned at two points bows under shear stress until it becomes unstable at semicircle (τ = Gb/L), then loops off, regenerating the original segment. This is how plastic strains of order 0.01 are produced despite finite initial dislocation density.

8. Fracture Mechanics

8.1 Griffith energy balance (1921)

For a center crack of length 2a in an infinite plate under uniaxial σ, the energy release rate balances surface energy creation when the crack extends: σ_c = √(2 E γ_s / (π a)) A. A. Griffith, working on glass at the Royal Aircraft Establishment, explained why bulk glass fails far below theoretical strength (atomic bond strength ~E/10): pre-existing flaws.

For ductile materials Irwin and Orowan extended this with plastic-work term γ_p ≫ γ_s.

8.2 Irwin’s stress intensity factor K (1957)

Near-tip stress field σ_ij = (K/√(2πr)) f_ij(θ) + … K characterizes the singularity. For Mode I: K_I = Y σ √(π a) where Y is a geometry factor (Y = 1 for center crack in infinite plate, ≈1.12 for edge crack, tabulated in handbooks — Tada/Paris/Irwin, Rooke/Cartwright).

Three loading modes:

  • Mode I (opening, tension perpendicular to crack)
  • Mode II (in-plane shear, sliding)
  • Mode III (out-of-plane shear, tearing)

8.3 Fracture toughness K_IC

Critical value of K_I at which crack growth becomes unstable. Measured per ASTM E399 with a fatigue-precracked compact-tension (CT) or single-edge-notched-bend (SENB) specimen. Plane-strain conditions require thickness B ≥ 2.5 (K_IC/σ_y)² so that the plastic zone is small compared to specimen dimensions.

Representative K_IC (MPa·m^0.5):

  • Glass (soda lime): 0.7
  • Pyroceram, soda-lime glass: 1
  • Alumina (99.9%): 3–5
  • Silicon carbide: 3–6
  • Silicon nitride: 5–10
  • Zirconia (Y-TZP, transformation toughened): 9–15
  • Cast iron (gray): 6–20
  • Aluminum 7075-T6: 24
  • Aluminum 2024-T351: 26–37
  • Aluminum 6061-T6: 29
  • Titanium Ti-6Al-4V (annealed): 55–115
  • 4340 steel (quenched & tempered): 50–100
  • HY-130 steel: 150
  • A533B nuclear pressure vessel steel: 200+
  • Maraging 250 steel: 90–110

Plastic zone size (Irwin): r_p = (1/(2π)) (K/σ_y)² plane stress, (1/(6π)) (K/σ_y)² plane strain.

8.4 CTOD (crack tip opening displacement)

δ = K² / (m σ_y E’) where m = 1 (plane stress), 2 (plane strain), E’ = E or E/(1−ν²). Critical CTOD δ_c is used in BS 7448 and API 579-1 for weldments where K_IC measurement is impractical.

8.5 J-integral (Rice 1968)

J = ∫_Γ (W dy − T_i ∂u_i/∂x ds) Path-independent contour integral that generalizes K to elastoplastic fracture. For LEFM J = G = K²/E’. Critical J_IC is measured per ASTM E1820 from a resistance (J-R) curve of stable tearing. Used universally for ductile metals where LEFM is invalid.

8.6 R-curve and stable tearing

Crack-growth resistance R rises with Δa in ductile materials (more plastic-zone development); intersection of crack-driving-force curve with R-curve defines instability.

8.7 Master Curve (ASTM E1921)

Wallin (VTT, Finland) showed BCC ferritic steel cleavage toughness follows a single-parameter master curve in the transition region: K_JC(median) = 30 + 70 exp[0.019(T − T_0)] MPa·m^0.5 T_0 = reference temperature where median K_JC = 100 MPa·m^0.5. Adopted by ASME Code Case N-629 for reactor pressure vessel surveillance.

9. Fatigue

Failure under cyclic loading at stresses well below the monotonic yield. Wöhler’s 1860s railway-axle tests founded the field.

9.1 S-N curve and endurance limit

Plot of stress amplitude S_a vs cycles to failure N_f on log-log axes (or semi-log). Ferritic steels exhibit a true fatigue limit (knee around 10⁶–10⁷ cycles, often σ_e ≈ 0.5 σ_UTS). Non-ferrous metals (Al, Cu, Mg) do not — they continue to degrade indefinitely (specify endurance at 10⁸ or 10⁹ cycles).

9.2 Basquin (HCF) and Coffin–Manson (LCF)

High-cycle fatigue: σ_a = σ_f’·(2 N_f)^b (Basquin), b ≈ −0.05 to −0.12 Low-cycle fatigue (plastic strain dominant): Δε^p/2 = ε_f’·(2 N_f)^c (Coffin–Manson, 1954), c ≈ −0.5 to −0.7 Total strain (Morrow): Δε/2 = (σ_f’/E)·(2 N_f)^b + ε_f’·(2 N_f)

9.3 Mean stress corrections

Goodman line: σ_a/σ_e + σ_m/σ_UTS = 1 Gerber parabola: σ_a/σ_e + (σ_m/σ_UTS)² = 1 (less conservative) Soderberg: σ_a/σ_e + σ_m/σ_y = 1 (most conservative) Smith–Watson–Topper: σ_max · ε_a = const (handles ratio R effects with one parameter)

9.4 Cumulative damage (Palmgren–Miner 1924/1945)

Σ (n_i / N_fi) = 1 at failure Linear damage rule, ignores load sequence. Sequence effects (high-low vs low-high) can produce Σ between 0.3 and 3 — known limitation, but the rule is enshrined in ASME, DNV, IIW design codes.

9.5 Crack-growth approach (Paris law 1963)

Once a crack is detectable (NDT limit), use: da/dN = C (ΔK)^m with ΔK = K_max − K_min = Y Δσ √(π a) (R-dependent in detailed formulations).

Typical m values:

  • Ferritic steels: 3
  • Aluminum alloys: 4
  • Stainless steels: 2.5–3.5
  • Ti alloys: 3.5–5
  • Cast iron: 5–10

Threshold ΔK_th below which growth is negligible (~5 MPa·m^0.5 for steel R=0.1, lower at high R).

Forman model adds K_IC asymptote: da/dN = C (ΔK)^m / [(1−R) K_IC − ΔK]. NASGRO equation (NASA, Forman/Newman) is the modern industry standard, capturing threshold and R effects.

9.6 Damage-tolerant design

Origin: 1969 USAF Aircraft Structural Integrity Program (ASIP) after F-111 wing failure. Assume an initial flaw of detectable size, compute remaining life by integrating Paris, schedule inspections at intervals leaving at least 2× life margin. Required for FAA Part 25 commercial aircraft structure and for nuclear (ASME Section XI).

9.7 Variable amplitude and load sequence

Rainflow counting (Matsuishi & Endo 1968) decomposes random spectra into closed hysteresis loops; combined with Miner gives an estimate. Retardation effects after overloads (Wheeler, Willenborg models).

10. Creep

Time-dependent plastic deformation at T/T_m ≳ 0.3–0.4. Strain ε grows under constant load even below σ_y.

10.1 Creep curve stages

  • Primary (Stage I, transient): ε̇ decreasing (Andrade ε ∝ t^(1/3))
  • Secondary (Stage II, steady-state): ε̇ = constant — the engineering design regime
  • Tertiary (Stage III): ε̇ accelerating, voids/cracks → rupture

10.2 Steady-state creep equation

ε̇_ss = A σ^n exp(−Q/RT) n = stress exponent, Q = activation energy.

Mechanism map (Frost & Ashby 1982):

  • Dislocation creep (climb-controlled), n ≈ 3–8, Q ≈ Q_self-diffusion
  • Diffusional creep:
    • Nabarro–Herring (bulk diffusion through grain interior), n = 1, ε̇ ∝ σ Ω D_v / (k_B T d²)
    • Coble (grain-boundary diffusion), n = 1, ε̇ ∝ σ Ω δ D_gb / (k_B T d³)
  • Grain boundary sliding (GBS, n ≈ 2)
  • Power-law breakdown at high stress
  • Harper–Dorn (n = 1, low stress, large grains)

Coble dominates at low T/d, NH at high T/large d, dislocation at high σ.

10.3 Larson–Miller parameter (1952)

P_LM = T (C + log t_r), C ≈ 20 for many steels. Allows extrapolation of short-duration creep-rupture tests to design lifetimes (10⁵ h ≈ 11 years), heavily used by power-plant alloy designers (P91, 9Cr-1Mo, IN-718, IN-738LC for gas-turbine blades).

10.4 Stress rupture

Failure under prolonged creep. Monkman–Grant: ε̇_ss · t_r = constant — empirically tight across many alloys.

11. Viscoelasticity

Time-dependent response intermediate between elastic and viscous. Critical for polymers, biological tissue, hot metals, asphalt, concrete.

11.1 Simple mechanical analogues

  • Maxwell (spring + dashpot in series): σ + (η/E) σ̇ = η ε̇. Stress relaxes exponentially (τ = η/E); strain unbounded under constant stress.
  • Kelvin–Voigt (spring ∥ dashpot): σ = E ε + η ε̇. Bounded creep ε(t) = [σ/E](1 − exp(−t/τ)); no instantaneous elastic strain.
  • Standard linear solid (SLS, spring in series with K-V): captures both instantaneous elasticity and finite relaxation/creep.

11.2 Boltzmann superposition

For linear viscoelastic materials with relaxation modulus E(t): σ(t) = ∫₀^t E(t − τ) dε/dτ dτ This is the basis of dynamic mechanical analysis (DMA) interpretation (storage E’, loss E”, tan δ).

11.3 Time–temperature superposition (WLF, Williams–Landel–Ferry 1955)

log a_T = −C_1 (T − T_g) / (C_2 + T − T_g) Universal constants C_1 ≈ 17.4, C_2 ≈ 51.6 K when reference is T_g. Master curves at one T cover decades of frequency. Foundational tool for tire compound design and polymer-component lifetime extrapolation.

12. Hardness

A penetration resistance probe — quick, semi-nondestructive, and a stand-in for tensile data in field inspection.

12.1 Brinell HB (1900, Johan A. Brinell, Sweden)

10 mm steel/carbide ball, 3000 kgf typically. HB = F / surface-area-of-indent in kgf/mm². Now ASTM E10.

12.2 Vickers HV (1924, Smith & Sandland at Vickers, UK)

136° diamond pyramid. HV = 1.8544 F / d² (kgf, mm). Useful from 1 gf (microhardness) to 100 kgf. ASTM E384, E92.

12.3 Rockwell (1914, Hugh & Stanley Rockwell)

Differential-depth method. Scales: HRC (150 kgf, diamond cone, hardened steel — automotive bearings 58–62 HRC), HRB (100 kgf, 1/16” ball, brass-steel range), HRA, HR15N etc. ASTM E18.

12.4 Nanoindentation (Oliver–Pharr 1992)

Berkovich pyramid, continuous load-depth recording. Extract reduced modulus E_r and hardness H from unloading-curve slope and contact area. Used for thin films, ion-implanted layers, single-grain mapping.

12.5 Hardness-strength correlations

σ_UTS (MPa) ≈ 3.45 · HV (Tabor 1948) σ_UTS (MPa) ≈ 3.45 · HB for steel σ_UTS (ksi) ≈ 0.5 · HB

These hold for non-cold-worked steels in the 100–600 HV range. Useful for quick field assessment but not for design.

13. Impact Toughness

13.1 Charpy V-notch (CVN)

ASTM E23. 10×10×55 mm bar, 2 mm-deep 45°-V notch (root radius 0.25 mm), 2 m pendulum. Absorbed energy in J. Cheap, fast, used for material specification (e.g., 27 J at −40 °C for pressure-vessel steels).

13.2 Izod

Cantilever variant. Common for polymers (ASTM D256) with notched/unnotched, including reverse-notch.

13.3 Ductile–brittle transition temperature (DBTT)

BCC and HCP metals show a temperature below which Charpy energy drops sharply. Liberty ships (WWII) — at least 1,031 ships built, 19 broke completely in two and ~200 suffered serious cracks, several in cold Atlantic. Root cause: low-toughness sulfur-rich rimming steel + welded (rather than riveted) construction + stress concentrators (square hatch corners). Constance Tipper at Cambridge ran the metallurgical investigation that established DBTT as a design driver, leading to modern toughness specifications.

13.4 Drop-weight (Pellini NDT)

NRL drop-weight test (Pellini, 1950s) finds nil-ductility transition temperature for ship and pressure-vessel plate (ASTM E208).

14. Wear and Tribology

14.1 Archard’s law (1953)

V = k · W · L / H V = worn volume, W = normal load, L = sliding distance, H = hardness, k = wear coefficient (10⁻² severe adhesive to 10⁻⁸ mild lubricated).

Wear modes:

  • Adhesive (cold welding & shearing)
  • Abrasive (two-body / three-body, hard particles ploughing)
  • Surface fatigue (rolling-contact spalling — bearings)
  • Fretting (small-amplitude oscillation, surprisingly aggressive, common at bolt heads)
  • Erosive (impinging particles or droplets — turbine blade trailing edges)
  • Corrosive/oxidative (passive film disrupted, repassivated, removed)

Friction (Amontons–Coulomb): F = μ N, with μ independent of nominal contact area because real contact area ∝ N at asperities (Bowden & Tabor 1939–50).

See tribology for lubrication regimes (boundary / mixed / hydrodynamic Stribeck curve) and bearing design.

15. Environmentally Assisted Cracking

15.1 Stress corrosion cracking (SCC)

Sustained tensile stress + susceptible alloy + specific environment → brittle cracking far below the air toughness. Notorious combinations:

  • Austenitic stainless (304/316) + chloride (>60 °C) → transgranular SCC
  • Brass + ammonia (season cracking — first identified in cartridges in 1900s)
  • High-strength steels + H₂S (sour-gas, sulfide stress cracking) — NACE MR0175 / ISO 15156
  • Aluminum 7000 series + chloride
  • Sensitized stainless (Cr-depleted near grain boundaries) + oxidizing aqueous → IGSCC

15.2 Hydrogen embrittlement (HE)

Atomic H from cathodic protection, pickling, plating, or service in H₂ infiltrates BCC steel, lowers cohesion at trapping sites. Major mechanisms: HEDE (hydrogen-enhanced decohesion), HELP (hydrogen-enhanced local plasticity), AIDE. Catastrophic in high-strength steels (>1000 MPa). Mitigation: bake-out after plating (190–220 °C, 4–24 h per AMS 2759/9), Cd-replacement coatings, low-hydrogen welding electrodes.

15.3 Liquid metal embrittlement (LME)

Solid metal contacted by a specific liquid metal cracks under tension (Hg on Al, Zn on austenitic stainless during galvanized-bolt heating — caused fatalities at refinery hot bolting). Theoretical: surface-energy reduction (Rebinder effect) + diffusion-mediated decohesion.

15.4 Corrosion fatigue

Cyclic loading in corrosive medium — no fatigue limit, da/dN-ΔK shifted upward, sometimes with frequency-dependent plateau region.

16. High Strain Rate Behavior

16.1 Strain-rate regimes

  • Quasi-static: ε̇ < 1 s⁻¹ (universal testing machine)
  • Intermediate: 10⁰–10² s⁻¹ (servohydraulic, drop tower)
  • High: 10²–10⁴ s⁻¹ (Split Hopkinson pressure bar)
  • Very high: 10⁴–10⁷ s⁻¹ (plate impact, Taylor cylinder)

16.2 Split Hopkinson Pressure Bar (SHPB)

Kolsky bar arrangement: striker → input bar → specimen → output bar; strain gauges record incident/reflected/transmitted elastic pulses; sample stress and strain rate inferred from 1-D wave theory. Tension and torsion variants exist (Kolsky 1949 monograph; Davies 1948 modification).

16.3 Johnson–Cook constitutive (1983)

σ = (A + B ε^n)(1 + C ln ε̇*)(1 − T*^m) T*= (T − T_room)/(T_melt − T_room), ε̇* = ε̇/ε̇_0. Five-parameter, empirical, widely used in LS-DYNA / Abaqus Explicit for ballistic, crash, blast simulations.

Zerilli–Armstrong (1987) is physics-based (Peierls-stress + thermal activation), separating BCC and FCC forms. Steinberg–Guinan and Mechanical Threshold Stress (MTS) models cover extreme regimes.

17. Composites Mechanics

17.1 Rule of mixtures

Longitudinal (iso-strain): E_∥ = V_f E_f + V_m E_m, σ_∥ = V_f σ_f + V_m σ_m Transverse (iso-stress): 1/E_⊥ = V_f/E_f + V_m/E_m Volume fractions V_f + V_m = 1.

17.2 Halpin–Tsai (1969)

Semi-empirical correction for transverse and shear moduli with reinforcement aspect ratio ξ (2 for transverse modulus of long fibers, ≥ 10 for shear). Reduces to ROM in limits.

17.3 Classical Laminate Theory (CLT)

For thin laminates: [N; M] = [A B; B D] [ε^0; κ], where A is in-plane, D is bending, B is coupling stiffness matrix. Lamina stiffness Q_ij rotated by ply angle to get global Q̄_ij, then summed through thickness. Predicts ply-by-ply stresses, basis for failure analysis with Tsai–Wu or Hashin criteria.

17.4 Interlaminar shear strength (ILSS)

Out-of-plane (Mode II) strength controls delamination. Tested by short-beam shear ASTM D2344 (~50–100 MPa for typical CFRP). Drives use of toughened thermoplastic interlayers in modern aerospace prepregs (Hexcel HexPly M91, Toray T800/3900-2B for 787).

17.5 BVID (barely visible impact damage)

Low-velocity impact (tool drop, hail) creates extensive sub-surface delamination invisible from the surface — drives the “no-growth” compression-after-impact (CAI) design philosophy for primary aircraft structure, requirement encoded in FAA AC 20-107B.

18. Famous Fracture Case Histories

  • Versailles rail disaster (1842): brittle fracture of wrought-iron axle, ~55 deaths. First systematic study of metal fatigue (Rankine’s 1843 report).
  • Liberty ships (1942–46): cleavage fracture of welded hulls in cold seas. Catalyst for the field of fracture mechanics (Constance Tipper, Cambridge).
  • De Havilland Comet I (1954): two cabin-pressurization fatigue failures (G-ALYP off Elba; G-ALYY off Naples) from corner stress concentration at square ADF aerial cutout. Square windows myth — actually square ADF aerial windows. Drove fail-safe design philosophy.
  • Tacoma Narrows Bridge collapse (1940): aeroelastic flutter; not a strength failure per se but aeroelastic stability. The “Galloping Gertie” case study founded modern bridge wind engineering.
  • Hyatt Regency walkway collapse (Kansas City 1981): design change doubled load on connecting rods; 114 deaths. ASCE ethics case study.
  • Alexander L. Kielland (1980): fatigue crack from fillet weld on a hydrophone bracket initiated catastrophic capsize of semi-submersible drilling rig, 123 deaths. Drove NORSOK fatigue design rules for offshore.
  • Aloha Airlines Flight 243 (1988): multi-site fatigue damage (MSD) along lap-joint rivet rows on 19-year-old Boeing 737-200, top fuselage skin separated in flight, 1 flight attendant lost. Founded the Aging Aircraft Program and widespread fatigue damage research.
  • Eschede ICE rail disaster (1998): fatigue of a rubber-sprung wheel rim; 101 dead. German railway moved away from compound wheel design.
  • Space Shuttle Columbia (2003): foam impact-induced delamination of RCC carbon-carbon leading-edge panels; not a metal fatigue failure but illustrates BVID-class composite damage tolerance.

19. Key ASTM / ISO Standards

StandardTopic
ASTM E8 / ISO 6892Tensile testing of metals
ASTM E18Rockwell hardness
ASTM E10Brinell hardness
ASTM E384 / E92Vickers / microhardness
ASTM E23Charpy / Izod impact
ASTM E208Drop-weight NDT
ASTM E399K_IC plane-strain fracture toughness
ASTM E1820J-integral / CTOD
ASTM E647Fatigue crack growth da/dN
ASTM E466 / E468 / E739Stress-life fatigue testing and analysis
ASTM E606Strain-life (LCF) testing
ASTM E1921Master curve (T_0) reference temperature
ASTM E139Conventional creep / stress rupture
ASTM E328Stress relaxation
ASTM E1681Threshold K_IEAC for environmentally assisted cracking
ASTM D3039 / D3410 / D3518Composite tensile / compressive / shear
ASTM D2344Short-beam (ILSS) for composites
ASTM D7136/D7137Composite impact and CAI
ISO 12108Fatigue crack growth (international)
ISO 11782Corrosion fatigue
BS 7910Fracture assessment of metallic structures (FAD)
API 579-1 / ASME FFS-1Fitness-for-service

20. Quick Engineering Formulas

  • Hoop stress in thin pressure vessel: σ_h = p r / t (axial σ_a = p r / 2 t)
  • Stress concentration factor at elliptical hole (Inglis 1913): K_t = 1 + 2 a/b (= 3 for a circular hole)
  • Beam bending: σ = M c / I; curvature 1/ρ = M/(E I)
  • Torsion of round bar: τ = T r / J
  • Buckling (Euler): P_cr = π² E I /(K L)²
  • Slenderness ratio L/r: column transitions to Johnson formula below critical slenderness
  • Fatigue strength reduction K_f = 1 + q (K_t − 1), notch sensitivity q (Peterson curves)

21. Adjacent

Graph View

Backlinks