EG4022 Engineering Materials

University of East London | Malvern University Partnerships

INTERACTIVE LEARNING HUB

Lecturer: Vismay Loliyaniya | Module Lead

Tensile Strength & Fracture Behaviour

Understand how materials behave under load, yield, fracture, and fatigue — and why these properties are critical to HS2 Phase One's structural integrity.

HS2 Phase One — Core Assessment Context

📚 LECTURE CONTENT — 2 HOURS

12 topics · Use the section headers to navigate · HS2 context woven throughout

PART A: TENSILE BEHAVIOUR PART B: FRACTURE MECHANICS PART C: FATIGUE & FAILURE

🎯 Learning Objectives

By the end of this session you should be able to:

  • Define engineering stress and strain and apply their formulae correctly
  • Interpret all regions of a stress–strain curve for both ductile and brittle materials
  • Distinguish between elastic and plastic deformation at the atomic scale
  • Explain yield strength, UTS, proof stress, and ductility with correct units
  • Describe the mechanisms of ductile and brittle fracture
  • Understand the Griffith crack theory and its engineering significance
  • Apply the stress intensity factor K and fracture toughness KIC to crack problems
  • Explain fatigue mechanisms, S-N curves, and the fatigue limit concept
  • Use Paris Law to describe crack growth under cyclic loading
  • Recognise the role of the Charpy impact test in material selection
  • Connect all concepts to HS2 Phase One material specifications
  • Critically evaluate material selection decisions using mechanical property data

PART A — TENSILE BEHAVIOUR

Topics 1–5 · Approximately 50 minutes

🧪 Topic 1 — The Tensile Test

The tensile test (or uniaxial tension test) is the most fundamental mechanical test in engineering. A standardised specimen is gripped at both ends and pulled apart at a controlled rate while a load cell measures the applied force and an extensometer measures elongation.

Standard Specimen Geometry

  • Gauge length (L₀): the measured portion, typically 50 mm or 80 mm (BS EN ISO 6892-1)
  • Parallel length: uniform cross-section where deformation occurs
  • Grip ends: wider, threaded or shouldered to prevent grip failure
  • Cross-section: circular (round bar) or rectangular (flat plate)
  • Standard: BS EN ISO 6892-1 (metals) at room temperature

What the Test Measures

  • Force vs. extension — converted to stress vs. strain
  • Elastic modulus (slope of elastic region)
  • Yield strength (onset of plastic deformation)
  • Ultimate tensile strength (maximum load ÷ original area)
  • % Elongation at fracture (ductility measure)
  • % Reduction in area (another ductility measure)

🚄 HS2 Quality Control

Every batch of S355 steel delivered to HS2 construction sites (e.g. the Colne Valley Viaduct steelwork) requires tensile test certificates conforming to EN 10025-2. Test pieces are taken from each cast of steel and must demonstrate yield strength ≥355 MPa, UTS 470–630 MPa, and elongation ≥22% at fracture. Failure to meet these values triggers batch rejection.

📐 Topic 2 — Stress, Strain and Hooke's Law

2.1 Engineering Stress (σ)

Engineering stress is defined as the applied force divided by the original cross-sectional area. It is a nominal stress — it does not account for the change in cross-section during loading.

σ = F / A₀

F = applied tensile force (N) · A₀ = original cross-sectional area (m²) · Units: N/m² = Pa, or more commonly MPa (N/mm²)

Worked Example: A steel rod of diameter 20 mm carries a tensile load of 90 kN. Calculate the engineering stress.

A₀ = π(0.020)²/4 = 3.14 × 10⁻⁴ m²
σ = 90,000 / 3.14 × 10⁻⁴ = 286.6 MPa

2.2 Engineering Strain (ε)

Engineering strain is the ratio of the change in gauge length to the original gauge length. It is dimensionless and often expressed as a percentage.

ε = ΔL / L₀ = (L − L₀) / L₀

ΔL = extension (mm or m) · L₀ = original gauge length · Dimensionless, or expressed as %

Worked Example: A 50 mm gauge length specimen extends by 0.15 mm under elastic loading. Calculate the strain.

ε = 0.15 / 50 = 0.003 = 0.3%

2.3 Hooke's Law and Young's Modulus

Within the elastic region, stress and strain are proportional. This linear relationship is Hooke's Law, and the proportionality constant is Young's Modulus (E) — a fundamental measure of material stiffness.

σ = E · ε    ∴    E = σ / ε

Units of E: GPa (gigapascals) = 10⁹ Pa

210 GPa

Structural steel (S355)

68 GPa

Aluminium alloy 6061

35 GPa

C40 concrete

2.4 True Stress vs Engineering Stress

Engineering stress uses the original area A₀. True stress uses the instantaneous area A, which decreases as the specimen elongates. The difference becomes important after necking begins:

Engineering: σeng = F / A₀
True: σtrue = F / Ainstantaneous

After necking, engineering stress appears to drop (because A₀ is fixed) while true stress continues to rise until fracture. For structural design we use engineering stress since components are designed not to neck.

2.5 Poisson's Ratio (ν)

When a material is stretched longitudinally, it contracts laterally. The ratio of lateral strain to longitudinal strain is Poisson's ratio — a key constant in structural analysis:

ν = −εlateral / εlongitudinal

Typical values: steel ν ≈ 0.30 · aluminium ν ≈ 0.33 · concrete ν ≈ 0.20 · rubber ν ≈ 0.50

🚄 HS2 Stiffness Design

Young's Modulus governs deflection under service loads. HS2 viaduct beams (E = 210 GPa) must limit mid-span deflection to L/800 under train loading to prevent passenger discomfort and maintain OLE (overhead line equipment) geometry. A lower-E material like aluminium (E = 68 GPa) would require three times the cross-sectional area to achieve equivalent stiffness.

📈 Topic 3 — The Stress–Strain Curve in Depth

The stress–strain curve is the complete mechanical fingerprint of a material. It encodes stiffness, strength, toughness, and ductility in a single diagram. Understanding every region is essential for structural design.

Region 1 — Proportional Limit

The initial straight-line portion where stress is exactly proportional to strain (Hooke's Law). The slope of this line is Young's Modulus. All deformation here is fully elastic — removing the load returns the specimen to its original dimensions with zero permanent change. At the atomic scale, interatomic bond lengths are stretching reversibly — no bonds are broken.

Region 2 — Elastic Limit

Slightly beyond the proportional limit, stress and strain are no longer strictly proportional but deformation is still elastic. In practice for most metals the elastic limit and proportional limit are very close and often treated as the same point.

Region 3 — Yield Point (Upper and Lower)

In low-carbon (mild) structural steels — including S355 used in HS2 — there is a distinct upper yield point followed by a sudden drop to the lower yield point. This drop occurs because dislocations (line defects in the crystal lattice) are suddenly released from their carbon atom "pins" and begin to move freely through the crystal.

Why does the yield drop occur? Carbon and nitrogen atoms in mild steel cluster around dislocations (Cottrell atmospheres), locking them in place. The upper yield point is the stress required to unpin these dislocations. Once free, they glide easily — causing the sudden load drop to the lower yield point. This is unique to low-carbon steels and is why S355 has such a well-defined design yield strength of exactly 355 MPa.

For materials without a clear yield point (e.g. aluminium alloys, high-strength steels), a 0.2% proof stress is used — the stress that causes 0.2% permanent strain, determined by drawing a line parallel to the elastic slope offset by 0.2% strain.

0.2% Proof Stress: offset strain = 0.002 → draw parallel line → intersection with curve

Region 4 — Strain Hardening (Work Hardening)

After yielding, continued plastic deformation requires increasing stress. This is strain hardening (also called work hardening). As dislocations move through the crystal lattice, they encounter grain boundaries, other dislocations, and precipitates — becoming tangled and unable to move freely. The material becomes progressively stronger and harder.

Region 5 — Necking (UTS to Fracture)

At the UTS, localised deformation (necking) begins at the weakest cross-section. The engineering stress now falls because the cross-sectional area is reducing rapidly — even though the true stress is still rising. Necking signals imminent failure. The engineering stress at fracture is called the fracture stress, which is lower than the UTS on an engineering curve.

Region 6 — Fracture

Final separation of the specimen. In ductile materials this produces a characteristic cup-and-cone fracture surface. The elongation at fracture and reduction in area are measured and reported as ductility values.

% Elongation = [(Lf − L₀) / L₀] × 100%
% Reduction in Area = [(A₀ − Af) / A₀] × 100%

🚄 Design Significance for HS2

Structural design to Eurocodes uses the characteristic yield strength (fyk) as the fundamental design parameter. For S355 steel, fyk = 355 MPa for t ≤ 16 mm, reducing to 335 MPa for 16 mm < t ≤ 40 mm (EN 10025-2). Safety factors (γM0 = 1.0 for yielding; γM2 = 1.25 for net section fracture) are then applied in Eurocode 3 calculations. The design yield strength used in calculations is fyd = fyk / γM0 = 355 / 1.0 = 355 MPa.

⚛️ Topic 4 — Deformation at the Atomic Scale

Understanding why materials deform the way they do requires looking at what happens to atoms and crystal structure during loading.

Elastic Deformation — Bond Stretching

In the elastic region, atoms are pulled slightly apart from their equilibrium positions. The interatomic forces act like springs — when the load is removed, atoms return to equilibrium. No bonds are broken, no atoms move permanently. This is why Young's Modulus is a material constant that depends only on the interatomic bond type (e.g. metallic, covalent, ionic) — not on heat treatment or microstructure.

  • Covalent bonds (diamond): extremely stiff → very high E
  • Metallic bonds (steel): moderately stiff → E ≈ 200 GPa
  • van der Waals (polymers): very flexible → E ≈ 1–3 GPa

Plastic Deformation — Dislocation Motion

Plastic deformation in metals occurs through the movement of dislocations — line defects in the crystal lattice. Dislocations allow slip to occur at stresses far below the theoretical strength of a perfect crystal. The theoretical shear strength of iron is ~32 GPa, but its actual yield stress is ~200 MPa — 160 times lower — because dislocations provide an easy path for atomic movement.

  • Edge dislocations: extra half-plane of atoms
  • Screw dislocations: helical distortion of lattice
  • Slip occurs on close-packed planes in close-packed directions (slip systems)

Strengthening Mechanisms

Engineers can increase yield strength by making dislocation movement more difficult. Four main mechanisms are used:

① Solid Solution Strengthening

Foreign atoms (e.g. carbon in steel, copper in aluminium) distort the lattice and impede dislocation movement. Manganese in S355 steel contributes to its 355 MPa yield strength.

② Work Hardening

Increasing dislocation density through cold working. Each dislocation impedes others. Cold-drawn rebar (B500B) achieves 500 MPa yield through controlled cold work during wire drawing.

③ Grain Boundary Strengthening

Smaller grains = more grain boundaries = more obstacles for dislocations. The Hall-Petch equation: σy = σ₀ + k·d where d = grain diameter. Normalised (fine-grained) steels are stronger than coarse-grained equivalents.

④ Precipitation Hardening

Fine particles (precipitates) block dislocation movement. Used in high-strength aluminium alloys (7075-T6 achieves 503 MPa UTS). Not commonly used in structural steels due to weldability concerns.

💪 Topic 5 — Yield Strength, UTS, and Ductility

5.1 Yield Strength (σy or fyk)

The yield strength is the most critical design parameter for structural components loaded in tension or bending. It defines the boundary between safe elastic service behaviour and unacceptable permanent deformation.

Steel GradeYield StrengthUTSTypical HS2 Application
S275275 MPa430–580 MPaSecondary steelwork, handrails
S355355 MPa470–630 MPaPrimary viaduct structure, portal frames
S460460 MPa550–720 MPaHigh-load pin connections, splice plates
Grade 900A Rail≥680 MPa≥900 MPaHigh-speed rail track
B500B Rebar500 MPa≥540 MPaReinforced concrete elements

5.2 Ultimate Tensile Strength (UTS)

The UTS is the maximum engineering stress the material can sustain. It corresponds to the maximum load on the load-extension curve divided by the original cross-sectional area. Beyond this point, necking begins. The UTS is used in fracture-based failure criteria and in determining safety margins.

Design Note: Structural connections (bolts, welds) are often designed to the net section fracture limit, which is based on UTS (with γM2 = 1.25) rather than yield (γM0 = 1.0). This larger safety factor on fracture reflects its catastrophic, unwarned nature compared to ductile yielding.

5.3 Ductility

Ductility is the ability to undergo substantial plastic deformation before fracture. It is critical for structural safety because it provides:

High Ductility Materials

  • Mild steel S275: ~28% elongation
  • S355 structural steel: ≥22% elongation
  • B500B rebar: ≥5% elongation (ductility class B)
  • 6061-T6 aluminium: ~17% elongation

Low Ductility (Brittle) Materials

  • Cast iron: ~0.5% elongation
  • Glass: ~0% elongation
  • Plain concrete: ~0.01% strain at fracture
  • Ceramics: typically <1% strain

🚄 HS2 Ductility Requirements

Eurocode 8 (seismic design) and Eurocode 3 require structural steel to meet minimum ductility requirements. For HS2 connections, ductility class S355 J2 is specified — the J2 suffix meaning the steel must achieve a minimum Charpy impact value of 27 J at −20°C, ensuring ductile fracture behaviour even in winter service conditions. Rebar used in HS2 tunnel linings must meet ductility class B (B500B) to ensure ductile reinforced concrete failure modes, giving occupants time to evacuate in any seismic or overload event.

PART B — FRACTURE MECHANICS

Topics 6–9 · Approximately 40 minutes

🔩 Topic 6 — Fracture: Ductile vs Brittle

6.1 Ductile Fracture — Mechanisms

Ductile fracture progresses slowly through three stages:

Stage 1 — Void Nucleation

After the UTS, stress concentrations at inclusions, second-phase particles, and grain boundary triple junctions cause tiny voids to nucleate (typically at MnS or Fe₃C particles in steel).

Stage 2 — Void Growth

Under continued loading, voids grow as the surrounding material deforms plastically. The rate of void growth depends on stress triaxiality — higher triaxiality (as in a thick specimen) accelerates growth.

Stage 3 — Void Coalescence and Fracture

Adjacent voids link up (coalescence) to form the fracture surface. The flat central region (fibrous zone) forms by void coalescence; the outer shear lip (conical part of the cup-and-cone) forms by shear stress. Final fracture is typically at 45° to the tensile axis.

6.2 Brittle Fracture — Mechanisms

Brittle fracture occurs with negligible plastic deformation. It propagates by cleavage — the breaking of atomic bonds along specific crystallographic planes (the cleavage planes, typically the {100} planes in BCC metals like iron).

KEY FEATURES

  • Crack propagates at very high speed (≈ 2000 m/s, ~40% of speed of sound in steel)
  • No necking, no warning, sudden catastrophic failure
  • Flat, reflective, crystalline fracture surface ("chevron" or "herringbone" markings point to origin)
  • Promoted by: low temperature, high strain rate (impact), thick sections, stress concentrations, high-strength steels
  • BCC metals (iron, steel) can undergo ductile-to-brittle transition (DBT); FCC metals (aluminium, copper) do not

HISTORICAL EXAMPLES

  • Liberty Ships (1940s): brittle fracture of hull welds in cold North Atlantic waters caused catastrophic structural failures — up to 3 ships broke in two
  • Aloha Airlines 737 (1988): fatigue crack growth led to brittle separation of upper fuselage
  • King's Bridge, Melbourne (1962): brittle fracture due to notch sensitivity in high-constraint weld zones
  • HS2 Prevention: all critical steelwork tested to EN 10025 J2 sub-grade (27 J at −20°C)

6.3 Ductile-to-Brittle Transition (DBT)

BCC metals (including structural steel) can behave in a ductile manner at room temperature but transition to brittle behaviour at lower temperatures. The temperature at which this transition occurs is called the DBTT (Ductile-to-Brittle Transition Temperature).

The Charpy Impact Test measures the energy absorbed (in Joules) when a standard notched specimen is fractured by a pendulum hammer. By testing at different temperatures, a transition curve is produced. The DBTT is taken at the temperature giving 27 J absorbed energy (the European Charpy criterion). S355 J2 steel must give ≥27 J at −20°C — well below the minimum UK winter air temperature (typically −15°C).

🚄 HS2 Charpy Requirements

HS2 Phase One construction spans from London to Birmingham across varying exposure conditions. Bridge steelwork within the M25 and Chiltern tunnel portal zones is specified as S355 J2+N (normalised, 27 J at −20°C) to guarantee ductile fracture behaviour in all UK service temperatures. The "+N" suffix indicates normalised rolling — a heat treatment producing fine, uniform grain structure that lowers the DBTT and improves weldability.

🧬 Topic 7 — Griffith Crack Theory

In 1921, A.A. Griffith showed that real materials fracture at stresses far below the theoretical atomic bond strength because of pre-existing cracks or flaws. These flaws concentrate stress and provide the energy needed for crack growth.

7.1 Stress Concentration at a Crack Tip

At the tip of an elliptical crack of half-length a and tip radius ρ, the local stress is magnified above the applied stress σ by the stress concentration factor Kt:

σmax = 2σ√(a/ρ)

A sharp crack has ρ → 0, meaning σmax → ∞ theoretically. In reality, plastic deformation at the crack tip blunts it slightly.

Worked Example: A plate of glass contains a surface crack of depth a = 0.5 mm with a tip radius ρ = 0.001 mm. If σ = 10 MPa applied:

σmax = 2 × 10 × √(0.5/0.001) = 2 × 10 × 22.4 = 448 MPa — enough to fracture glass!

7.2 Griffith Energy Balance

Griffith's energy criterion states that a crack will propagate when the strain energy released by crack growth equals or exceeds the energy required to create new crack surfaces:

σf = √(2Eγs / πa)

σf = fracture stress · E = Young's Modulus · γs = surface energy per unit area · a = half crack length

This equation shows that fracture stress decreases as crack size increases — a longer crack requires less applied stress to propagate. This is the fundamental reason why Non-Destructive Testing (NDT) — finding cracks before they reach critical size — is central to structural safety management on HS2.

7.3 Modification for Metals (Irwin/Orowan)

Griffith's theory was derived for perfectly brittle materials (glass). Irwin and Orowan independently modified it for metals by replacing surface energy γs with the plastic work term Gc (critical strain energy release rate), which is orders of magnitude larger:

σf = √(EGc / πa)

Gc = critical strain energy release rate (J/m²) · For steel ≈ 10,000–100,000 J/m² vs glass ≈ 10 J/m²

🚄 Why NDT is Mandatory on HS2

The Griffith equation shows that a 10 mm crack in S355 steel at 355 MPa applied stress corresponds to K = 1.12 × 355 × √(π × 0.01) = 70.5 MPa√m — approaching the KIC of 50–100 MPa√m. All HS2 welds are inspected by ultrasonic testing (UT) and magnetic particle inspection (MPI) to ensure no crack-like flaws exceed the critical length for the design stress state.

🔬 Topic 8 — Fracture Toughness KIC

8.1 The Stress Intensity Factor K

The stress intensity factor K characterises the severity of the stress field at a crack tip. It combines applied stress, crack geometry, and specimen geometry into a single parameter:

K = Yσ√(πa)

K = stress intensity factor (MPa√m) · Y = dimensionless geometry factor (≈1.0–1.2 for common cases) · σ = applied stress (MPa) · a = crack half-length (m)

8.2 The Three Modes of Crack Opening

↕️

Mode I — Opening

Crack opens perpendicular to the applied tensile stress. Most common in structural failures. KIC is the toughness for Mode I.

↔️

Mode II — Sliding

In-plane shear — crack faces slide parallel to each other in the crack plane. Common in contact fatigue (e.g. rail-wheel contact).

🔄

Mode III — Tearing

Out-of-plane shear — crack faces twist relative to each other. Occurs in torsion-loaded components.

8.3 Fracture Toughness KIC

KIC (pronounced "K-one-C") is the critical stress intensity factor for Mode I loading — the value of K at which unstable crack propagation begins. It is a material property, measured by standardised tests (BS EN ISO 12737) on pre-cracked specimens.

MaterialKIC (MPa√m)
S355 Structural Steel50–100
Grade 900A Rail Steel25–35
Austenitic Stainless Steel200+
7075-T6 Aluminium24
6061-T6 Aluminium29
Cast Iron (grey)6–20
C40 Concrete1.0–1.5
Soda-lime glass0.7–0.8
Nylon PA62.5–3.0

Design Application: If K < KIC → safe (crack stable). If K ≥ KIC → fracture (crack unstable). The critical crack length for a given stress:

ac = (1/π)(KIC/Yσ)²

Worked Example: S355 steel weld, σ = 355 MPa, Y = 1.12, KIC = 50 MPa√m. Find critical crack half-length ac:

ac = (1/π)(50/(1.12×355))² = (1/π)(0.1258)² = 5.0 mm

Any crack ≥5 mm in this weld at yield stress would cause fracture → NDT detection limit must be <5 mm.

8.4 Plane Strain vs Plane Stress Fracture

Fracture toughness depends on specimen thickness. Thin specimens (plane stress) give higher apparent toughness due to shear lip formation; thick specimens (plane strain) give the conservative KIC value used in design. Plane strain conditions exist when:

B ≥ 2.5 (KICy

B = specimen thickness · This explains why thick structural sections (flanges >40 mm) have reduced yield strength specifications in EN 10025.

🚄 Fracture Mechanics in HS2 Weld Inspection

HS2 applies Engineering Critical Assessment (ECA) per BS 7910 to all Category 1 welds (primary load-carrying connections). ECA uses KIC values, design stress states, and NDT detection limits to calculate the maximum tolerable flaw size. For a viaduct main beam end connection with σ = 200 MPa and KIC = 60 MPa√m, the critical crack size is approximately 14 mm — so UT inspection must reliably detect cracks ≥5 mm (with a 3× safety factor). Phased array UT (PAUT) is specified to achieve this.

PART C — FATIGUE & FAILURE

Topics 9–12 · Approximately 30 minutes

Topic 9 — Fatigue: Introduction and Significance

Fatigue is failure under repeated cyclic loading, at stress levels well below the static yield strength or UTS. It is the most common cause of mechanical failure in engineering — responsible for approximately 80–90% of all in-service metallic failures.

Why Fatigue is So Dangerous

  • Occurs at stresses far below yield — no visible yielding or warning deformation
  • Initiates at microscopic surface defects, notches, or stress concentrations
  • Progresses slowly and invisibly until the crack reaches critical size
  • Final fracture is sudden, brittle, and catastrophic
  • Cannot be detected by visual inspection during stable propagation
  • Fracture surface shows characteristic "beach marks" (arrest lines) and a smooth zone (fatigue crack) followed by a rougher, final fracture zone
  • Affected by surface finish, corrosive environment, temperature, and residual stresses
  • Weld toe is the highest-risk location in welded structures

Three Stages of Fatigue Failure

Stage I — Crack Initiation

Begins at the surface, typically at a stress concentration (notch, corrosion pit, weld toe, machining mark). Persistent slip bands form as dislocations accumulate along a few slip planes, creating intrusions and extrusions on the surface. The microcrack is on the order of grain size.

Stage II — Stable Crack Propagation

The crack re-orients to grow perpendicular to the maximum principal stress (Mode I). Each load cycle advances the crack by a small increment (nanometres to micrometres per cycle). "Striations" visible on the fracture surface represent individual load cycles. This stage can last millions of cycles and occupies most of the fatigue life.

Stage III — Rapid Crack Growth and Final Fracture

When K reaches KIC, unstable brittle fracture occurs suddenly. This stage occupies very few cycles relative to the total fatigue life. The fracture surface shows a rougher, more irregular texture compared to the smooth Stage II striations.

📉 Topic 10 — S-N Curves and the Fatigue Limit

10.1 The S-N (Wöhler) Curve

The S-N curve plots stress amplitude (S) against the number of cycles to failure (N) on a log scale. It is determined by testing many identical specimens at different stress levels and recording the number of cycles to failure at each.

Key Features of the S-N Curve:

  • Low Cycle Fatigue (LCF): N < 10⁴ cycles, high stress (approaching yield). Relevant to structures experiencing infrequent but large overloads.
  • High Cycle Fatigue (HCF): N = 10⁴ to 10⁷ cycles, stress below yield. Most structural applications.
  • Fatigue Limit (Endurance Limit): For plain carbon steels, the S-N curve becomes horizontal at ~10⁶–10⁷ cycles, defining a stress below which the material theoretically has infinite life. σe ≈ 0.4–0.5 × UTS for steel in air.
  • No Fatigue Limit: Aluminium alloys, high-strength steels in corrosive environments, and most non-ferrous metals show continuously decreasing curves with no flat region. A fatigue strength at 10⁸ cycles is reported instead.

10.2 Stress Ratio R

Fatigue behaviour depends not just on stress amplitude but on the mean stress and the stress ratio R:

R = σmin / σmax

R = −1

Fully reversed loading (σmin = −σmax). Most severe fatigue condition. Used in standard S-N tests.

R = 0

Pulsating tension (load goes from zero to maximum). Typical of rail loading (tension only when train passes).

R = +0.5

High mean stress. Positive mean stress (tension) reduces fatigue life; compressive mean stress increases it — basis of shot peening.

10.3 Factors Affecting Fatigue Life

LIFE-REDUCING FACTORS

  • Stress concentrations (notches, holes, welds)
  • High tensile residual stresses (from welding)
  • Corrosive environment (corrosion fatigue — no fatigue limit)
  • High surface roughness
  • Large section size (higher likelihood of defects)
  • Elevated temperature

LIFE-IMPROVING FACTORS

  • Compressive residual stresses (shot peening, cold rolling)
  • Smooth surface finish (ground, polished)
  • Weld toe grinding or peening
  • High-frequency peening (HiFIT) on weld toes
  • Protective coating (prevents corrosion fatigue)
  • Reducing stress concentration factor Kt

🚄 HS2 Fatigue Design — EN 1993-1-9

HS2 steel structures are designed for fatigue to Eurocode 3 Part 1-9. Each type of weld/connection detail is assigned a fatigue detail category (e.g. Detail 71, 90, 125) representing the stress range (in MPa) causing failure at 2 × 10⁶ cycles. A transverse butt weld (Category 71) in the main viaduct beams must sustain the calculated stress range from train loading multiplied by λ (damage factor) over the 120-year design life. Weld toe grinding upgrades details by 1–2 categories, and is specified at critical HS2 connections to extend fatigue life.

📊 Topic 11 — Paris Law and Crack Growth Rate

11.1 Paris Law

For Stage II stable crack propagation, the crack growth rate per cycle is related to the stress intensity factor range ΔK by the Paris-Erdogan Law:

da/dN = C · (ΔK)m

da/dN = crack growth rate (m/cycle) · ΔK = Kmax − Kmin = stress intensity factor range (MPa√m) · C and m = material constants (experimentally determined)

Typical values for structural steel:

C ≈ 6.9 × 10⁻¹² (m/cycle when ΔK in MPa√m)
m ≈ 3.0

11.2 The Full da/dN vs ΔK Curve

The Paris Law describes only the middle linear region of the log(da/dN) vs log(ΔK) curve. The full curve has three regions:

Region I — Threshold

Below the threshold stress intensity range ΔKth, cracks do not propagate (da/dN → 0). For structural steel in air, ΔKth ≈ 6–8 MPa√m. Keeping ΔK < ΔKth ensures infinite fatigue life — the design philosophy used for aircraft turbine discs.

Region II — Paris Law (Stable Growth)

Linear on log-log plot, described by da/dN = C·(ΔK)m. This is the main engineering design region. Most of the useful fatigue life is spent here. The slope m ≈ 2–4 for metals.

Region III — Rapid Growth and Fracture

As Kmax approaches KIC, crack growth accelerates rapidly per cycle and the specimen fractures within a few more cycles. This region contributes very little to total life.

11.3 Calculating Fatigue Life from Paris Law

By integrating Paris Law between initial crack size a₀ and critical crack size ac, the number of cycles to failure Nf can be predicted:

Nf = ∫(a₀ → ac) da / [C·(Yσ√πa)m]

Worked Example: Steel component (C = 6.9×10⁻¹², m=3, Y=1.0, σ=200 MPa). Initial crack a₀ = 1 mm, critical crack ac = 10 mm.

ΔK₀ = 1.0×200×√(π×0.001) = 11.2 MPa√m
ΔKc = 1.0×200×√(π×0.01) = 35.4 MPa√m
Nf ≈ 2/(C·m·σ²·π·Y²) × [a₀1-m/2 − ac1-m/2]
Nf1.4 × 10⁶ cycles

🚄 Paris Law in HS2 Maintenance Planning

HS2 uses fracture mechanics-based inspection intervals calculated from Paris Law. For a critical weld with an assumed initial defect of 2 mm (conservative NDT detection limit), and a calculated critical crack size of 12 mm, Paris Law integration gives Nf ≈ 1.5 × 10⁷ cycles. At 300 trains/day, this is 137 years — exceeding the 120-year design life. However, with a safety factor of 3, the inspection interval is set at Nf/3 ÷ annual load cycles = ~46 years, with intermediate visual checks every 6 years.

📊 Topic 12 — Material Properties Summary & HS2 Comparison

The table below summarises the key tensile and fracture properties for all primary materials used in HS2 Phase One construction.

MaterialE (GPa)σy (MPa)UTS (MPa)Elongation (%)KIC (MPa√m)Primary HS2 Use
S355 J2+N Structural Steel210355470–630≥2250–100Viaduct beams, portal frames, connections
S460 M/ML210460550–720≥1740–80High-load pin plates, splice plates
Grade 900A Rail Steel210≥680≥900≥1025–35High-speed mainline rail track
B500B High-Ductility Rebar200500≥540≥5 (class B)RC tunnel linings, viaduct decks
C40/50 Concrete (compressive)3540–50 (comp)~01.0–1.5Tunnel segments, pile caps, deck slab
Grade 8.8 Bolts (HSFG)210640800≥12Structural connections (slip-resistant)
7075-T6 Aluminium715035721124Overhead line equipment masts
6061-T6 Aluminium682763101729Non-structural cladding, fitments
HDPE (pipe lining)0.820–3030–40>5001.5–4Drainage pipe, cable conduit

Why S355 Dominates HS2 Structural Steel Specification

✅ Strength

355 MPa yield strength is sufficient for most primary structural elements with economical section sizes. Higher grades (S460) give thinner sections but introduce weldability and DBTT concerns.

✅ Ductility

≥22% elongation ensures plastic hinge formation in overload scenarios, allowing moment redistribution and preventing progressive collapse — critical for public infrastructure.

✅ Weldability

Carbon equivalent CE ≤ 0.43 for S355 ensures good weldability without preheat on sections up to 25 mm. Higher-strength steels require preheat, slowing fabrication and increasing cost.

🚄 Integrated Summary — Why These Properties Matter for HS2

Every mechanical property covered in this lecture directly governs an HS2 design decision. Young's Modulus governs viaduct deflection limits. Yield strength determines beam section sizes. UTS controls bolt and net section failure checks. Ductility ensures seismic and overload robustness. KIC sets inspection intervals and NDT detection limits. Fatigue detail categories determine weld treatment requirements. Paris Law defines maintenance inspection intervals. This is not abstract theory — it is live engineering being built today.

⚙️ Stress–Strain Curve Simulator

Select a material and apply tensile load to watch how the specimen behaves. Observe elastic region, yield point, UTS, and fracture.

STRESS–STRAIN CURVE

SPECIMEN VISUALISATION

Stress
0 MPa
Strain
0.000
Phase
ELASTIC
Status
INTACT

Fatigue & Crack Propagation Simulator

Simulate cyclic tensile loading on a steel specimen. Watch the crack initiate and grow using Paris Law: da/dN = C(ΔK)ᵐ. Higher stress = faster crack growth. Critical crack length = 20 mm.

CRACK GROWTH CURVE (Paris Law)

SPECIMEN CROSS-SECTION

Cycles (N)0
Crack Length (a)0.000 mm
ΔK (Stress Intensity) MPa√m
PhaseNO CRACK

🚄 Paris Law in HS2 Context

HS2 rail fixings experience ~1.3 × 10¹⁰ load cycles over 120 years. Engineering design keeps stress intensity range (ΔK) well below the threshold where crack growth becomes significant, typically ΔK < 5 MPa√m for welded steel details (EN 1993-1-9 Category F2).

🧮 Fracture Toughness Calculator

Calculate the critical stress for crack propagation using KIC = Yσ√(πa). Enter values below:

🎯 Game 1: Property Sort

Drag each property card into the correct category — Elastic Behaviour or Plastic Behaviour.

Hooke's Law applies
Permanent deformation
Stress ∝ Strain
Dislocation motion
Reversible shape change
Strain hardening
E = σ/ε
Necking begins

ELASTIC

PLASTIC

Game 2: Rapid Fire Challenge

Answer as many questions as you can in 60 seconds. No second chances!

🖊️ Game 3: Label the Curve

Click on the correct region of the stress-strain curve when prompted. Match all 5 labels to pass!

🧩 Game 4: Match the Definition

Match each term to its correct definition using the dropdowns.

Ultimate Tensile Strength
Yield Strength
Young's Modulus
Fracture Toughness KIC

✏️ Game 5: Complete the Equation

Click a word from the bank to fill each blank. Complete all equations correctly.

Force
Cross-sectional Area
Strain
Young's Modulus
Original Length
Change in Length
Stress

① Stress (σ) = ÷

② Strain (ε) = ÷

(E) = ÷

🟢 Game 6: True or False Blitz

Is the statement TRUE or FALSE? Answer quickly to score bonus points!

Question 1/10 Score: 0/10

🚄 HS2 Phase One — Core Assessment Context

HS2 Phase One is your primary research context for this module's assessment. Use the structured tasks below to investigate how tensile and fracture properties govern material selection and structural safety in this landmark project.

LONDON EUSTON → BIRMINGHAM CURZON STREET 225 KM ROUTE MAX 360 KM/H DESIGN SPEED 120-YEAR DESIGN LIFE

🔍 Research Task 1 — Rail Steel Selection

HS2 uses Grade 900A pearlitic rail steel. Research and explain why this grade was selected over alternatives such as Grade 700 or bainitic steel for the high-speed mainline.

📌 Research Guidance

Consider: UTS requirements, rolling contact fatigue (RCF) resistance, wear behaviour, weldability, and the mechanical demands of 360 km/h train operations. Refer to EN 13674-1 (Railway Applications — Track — Rail).

⚠️ No hints provided — this is your own research and analysis.

🔍 Research Task 2 — Viaduct Structural Steel

The Colne Valley Viaduct (3.4 km) uses S355 structural steel. Analyse why S355 was specified over S275 or higher-grade S460 in terms of tensile properties, fracture toughness, and whole-life cost.

📌 Research Guidance

Explore: Charpy impact test requirements (EN 10025), weld hydrogen cracking risk at higher grades, fatigue detail category (EN 1993-1-9), and the implications of using steels with yield strengths above 355 MPa for connection design.

⚠️ No hints provided — this is your own research and analysis.

🔍 Research Task 3 — Tunnel Lining Concrete

HS2 tunnel bores (including the 10 km Long Itchington Wood tunnel) use C50/60 precast concrete segments reinforced with steel fibres and rebar. Explain how this composite system addresses concrete's inherently low fracture toughness.

📌 Research Guidance

Investigate: crack bridging mechanisms in steel-fibre-reinforced concrete (SFRC), how prestress reduces net tensile stress, Eurocode 2 durability requirements for tunnel environments (XS3 exposure class), and consequences of KIC being ~1.5 MPa√m in plain concrete.

⚠️ No hints provided — this is your own research and analysis.

🔍 Research Task 4 — Fatigue in High-Speed Rail Connections

Identify a critical fatigue-sensitive connection in HS2 (e.g. rail clips, viaduct beam-to-column joints, overhead line mast bases). Explain how fatigue life was assessed and what design decisions were made to meet the 120-year design life.

📌 Research Guidance

Use Paris Law principles (da/dN = C·ΔKᵐ), EN 1993-1-9 fatigue detail categories, and concept of fatigue limit vs safe-life design philosophy. Consider that 300 trains/day × 365 × 120 years ≈ 1.3 × 10¹⁰ load cycles at a joint.

⚠️ No hints provided — this is your own research and analysis.

🔍 Research Task 5 — Critical Analysis

HS2 Ltd has faced significant public and professional criticism regarding material cost overruns and specification changes during construction. Critically evaluate whether any materials substitutions (e.g. specification downgrading) could have compromised the long-term mechanical integrity of Phase One infrastructure.

📌 Research Guidance

Draw on: PAC (Public Accounts Committee) reports on HS2 cost overruns, IStructE guidance on material specification, the implications of reducing steel grade or concrete class on tensile margins, fracture toughness, and fatigue life. Be balanced — include both risk and mitigation perspectives.

⚠️ No hints provided — this is your own research and analysis.

📝 End-of-Topic Quiz — Tensile Strength & Fracture

15 questions covering all learning objectives. Answer each question — no hints are provided. You must achieve 68% (10/15) to pass. Answers are locked once submitted.

📖 Key Terms Glossary

Click any term to reveal its definition. All definitions are relevant to tensile testing, fracture mechanics, and HS2 applications.