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Materials Science · Chapter 7 of 10 · Advanced

Failure: Fracture, Fatigue, and Creep

Parts rarely fail because the load exceeded the strength. They fail from a tiny crack, from millions of small cycles, or from slow stretching in the heat. Three modes, three defences.

Readiness check

This chapter builds on stress, strength, and a little fracture geometry. Tick only what you can do closed-notes.

Compute stress from force and area.
Recall yield and tensile strength.
Take a square root and work with √(πa).
Read a log-scale (S-N) plot.
Understand a stress that cycles between two values.

0 or 1 weak itemsContinue with this chapter.

2 weak itemsReview mechanical properties in Chapter 5.

3 or more weak itemsRevisit stress and strength before continuing.

The core idea

Real parts fail from flaws, cycles, or time, not from a single overload. Fracture mechanics, fatigue limits, and creep rules each guard against one of these.

K = Yσ√(πa)fast fracture when K = K_ICσ_a/σ_e + σ_m/σ_UTS = 1

A crack concentrates stress at its tip; the stress intensity K = Yσ√(πa) measures how severe that is, and fast fracture occurs when K reaches the material's fracture toughness K_IC. Under repeated loading, parts fail by fatigue far below the yield strength. At high temperature and long times, they slowly stretch and rupture by creep. Each mode has its own design rule.

The skill works when: you identify which mode threatens the part and apply its specific criterion (K_IC, endurance limit, or creep limit).

The skill breaks down when: a part is checked only against yield, ignoring flaws, cyclic loading, or temperature.

The concept. A crack of length 2a concentrates the remote stress at its tips. The stress intensity K gathers stress, crack size, and geometry into one number; when it reaches the toughness K_IC, the crack runs.

The skills, taught in order

Failure comes in three kinds, each with its own analysis. Five skills cover fracture, fracture mechanics, the brittle transition, fatigue, and creep.

7.1 Ductile versus brittle fracture

Ductile fracture absorbs much energy and warns through necking; brittle fracture is sudden and energy-poor. The fracture surface tells which occurred: cup-and-cone and dimples for ductile, flat and faceted for brittle.

Aspect	Ductile	Brittle
Plastic deformation	extensive	little or none
Warning	necking, slow	sudden, catastrophic
Energy absorbed	high	low
Crack behaviour	stable, needs more load	unstable, self-running

7.2 Fracture mechanics

A sharp flaw raises the local stress far above the nominal value. The stress intensity K = Yσ√(πa) combines applied stress σ, crack length a, and a geometry factor Y. Fast fracture occurs when K reaches the fracture toughness K_IC, a material property. This lets you find a critical crack size a_c for a given stress, or a safe stress for a known flaw.

7.3 Impact and the ductile-brittle transition

Impact (Charpy) tests show that many steels turn brittle below a ductile-brittle transition temperature. BCC metals show this transition; FCC metals largely do not. Ignoring it sank welded ships in cold seas, a historic lesson in why the transition temperature must sit below the service temperature.

7.4 Fatigue

Cyclic stress nucleates and grows cracks until the part fails, often far below the yield strength. The S-N curve plots stress amplitude against cycles to failure; many steels show an endurance limit below which life is effectively infinite. A mean (non-zero) stress is handled with the Goodman relation, σ_a/σ_e + σ_m/σ_UTS = 1 at the limit.

7.5 Creep

Under stress at high temperature (above roughly 0.4 T_m), materials slowly and permanently stretch. The creep curve has primary, steady-state (minimum rate), and tertiary stages ending in rupture. The steady rate follows an Arrhenius law, and the Larson-Miller parameter trades temperature for time so short hot tests predict long service.

Mode	Cause	Key parameter	Design rule
Fast fracture	flaw plus stress	K_IC	keep K below K_IC
Fatigue	cyclic stress	endurance limit, S-N	stay below the limit (Goodman)
Creep	stress at high T over time	creep rate, Larson-Miller	limit T, stress, and time

Engineering connection: fracture toughness governs pressure vessels and aircraft; fatigue dominates rotating and vibrating parts; creep limits turbine blades and boilers.

Worked example 1: critical crack size

A steel plate has fracture toughness K_IC = 50 MPa√m and geometry factor Y = 1.0. Find the stress that would fracture it with a 2 mm internal crack (a = 2 mm), and the critical crack size if it operates at 300 MPa.

Figure 1. A centre-cracked plate under tension. The same equation gives a fracture stress for a known crack, or a critical crack size for a known operating stress.

ProblemFind the fracture stress for a 2 mm crack and the critical crack size at 300 MPa, for the plate in Figure 1.
Given / findK_IC = 50 MPa√m, Y = 1.0, a = 2 mm = 0.002 m, operating σ = 300 MPa. Find σ_c and a_c.
AssumptionsLinear-elastic fracture mechanics, the given Y, plane-strain toughness.
ModelSet K = K_IC in K = Yσ√(πa) and solve for stress (with a fixed) or for crack size (with σ fixed).
Equationsσ_c = K_IC/(Y√(πa)) a_c = (1/π)(K_IC/(Yσ))²
Solveσ_c = 50/(1.0 × √(π × 0.002)) = 50/0.0793 = 631 MPa. At 300 MPa, a_c = (1/π)(50/300)² = (1/π)(0.0278) = 0.0088 m = 8.8 mm.
CheckThe 2 mm crack fractures at 631 MPa, above the 300 MPa service stress, so the plate is safe with that flaw. The critical size at 300 MPa is 8.8 mm, so inspection must reliably catch cracks smaller than that.
ConclusionFracture mechanics turns toughness into an inspectable crack size. A tougher steel (higher K_IC) tolerates larger flaws, which is why toughness, not just strength, governs safety-critical structures.

Result. Fracture stress 631 MPa for a 2 mm crack; critical crack size 8.8 mm at 300 MPa.

Worked example 2: fatigue with a mean stress

A component cycles between σ_min = 50 MPa and σ_max = 300 MPa. The material has an endurance limit σ_e = 250 MPa and tensile strength σ_UTS = 600 MPa. Use the Goodman criterion to judge whether it has infinite life.

Figure 2. The Goodman diagram. The line runs from the endurance limit to the tensile strength; an operating point below it has infinite fatigue life.

ProblemDecide whether the component in Figure 2 survives indefinitely under its cyclic load.
Given / findσ_min = 50, σ_max = 300 MPa; σ_e = 250, σ_UTS = 600 MPa. Find σ_a, σ_m, and the Goodman result.
AssumptionsConstant-amplitude loading, Goodman mean-stress correction, well-finished part.
ModelSplit the cycle into amplitude and mean, then evaluate the Goodman sum (below 1 means infinite life).
Equationsσ_a = (σ_max − σ_min)/2, σ_m = (σ_max + σ_min)/2 σ_a/σ_e + σ_m/σ_UTS ≤ 1
Solveσ_a = (300 − 50)/2 = 125 MPa, σ_m = (300 + 50)/2 = 175 MPa. Goodman sum = 125/250 + 175/600 = 0.50 + 0.29 = 0.79, below 1, so infinite life with a safety factor of 1/0.79 = 1.26.
CheckThe peak stress (300 MPa) is below yield, so static failure is not the concern; fatigue is, and the operating point sits safely below the Goodman line. A safety factor of 1.26 is modest but positive.
ConclusionFatigue must be checked separately from static strength, and the mean stress matters: a higher mean would push the point past the line even at the same amplitude. Surface finish and stress concentrations would erode the margin further.

Result. σ_a = 125 MPa, σ_m = 175 MPa; Goodman sum 0.79, so infinite life (safety factor 1.26).

Misconceptions and diagnostics

Mistake	Symptom	Diagnostic question	Correction
Checking only against yield	Cracked or cyclic part deemed safe	"Are there flaws, cycles, or heat?"	Add the fracture, fatigue, or creep check the situation demands.
Strength means toughness	High-strength steel assumed crack-tolerant	"What is K_IC?"	Toughness is separate; strong steels can be brittle (low K_IC).
Ignoring mean stress in fatigue	Only amplitude checked	"Is the mean stress non-zero?"	Use Goodman: a tensile mean stress cuts the allowable amplitude.
Creep ignored below yield	Hot part sized only for short-term strength	"Is T above ~0.4 T_m for a long time?"	At high temperature, creep limits stress even well below yield.

Practice ladder

Level 1 · Direct skill

A part of toughness K_IC = 30 MPa√m (Y = 1) carries 200 MPa. Find the critical crack size.

Show answer

a_c = (1/π)(30/200)² = (1/π)(0.0225) = 0.00716 m = 7.2 mm. Lower toughness shrinks the tolerable flaw, demanding finer inspection.

Level 2 · Mixed concept

Two steels carry the same 300 MPa with the same 2 mm crack, but one has K_IC = 50 and the other 30 MPa√m. Which is safe?

Show answer

K = 1.0 × 300 × √(π × 0.002) = 23.8 MPa√m. The 50-toughness steel (K < K_IC) is safe; the 30-toughness steel is also safe here (23.8 < 30) but with far less margin and a smaller critical crack, so it is the riskier choice.

Level 3 · Independent problem

The Worked Example 2 component is redesigned so the load is fully reversed (σ_m = 0) with the same 125 MPa amplitude. How does the fatigue margin change?

Show answer

Goodman sum = 125/250 + 0/600 = 0.50, safety factor 2.0. Removing the tensile mean stress nearly doubles the margin, showing why mean stress is a key fatigue lever.

Level 4 · Transfer to real engineering

Pick a real failure or critical part (a cracked bracket, an aircraft fuselage, a turbine blade). Identify the dominant failure mode and the property or criterion that governs its design.

What good work looks like

The mode named (fracture, fatigue, or creep), the controlling parameter (K_IC, endurance limit, or creep limit) identified, and a quantitative criterion applied.

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that I picked the failure mode that actually threatens this part."

"Give me five components; I will name the dominant failure mode for each."

"Compute the critical crack." Setting K = K_IC yourself is the skill.

"Is this safe?" Choosing and applying the right criterion is the point.

Portfolio task

Analyse one part for its most likely failure mode: apply the matching criterion (K_IC, Goodman, or creep) and report a critical flaw, life, or safe stress.

Must include: the mode justified, the governing equation applied, and a safety margin.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. Write the stress-intensity relation and the fast-fracture condition.

K = Yσ√(πa); fast fracture when K = K_IC.

2. How does ductile differ from brittle fracture?

Ductile absorbs much energy and warns by necking; brittle is sudden and energy-poor.

3. What is the endurance limit?

A stress amplitude below which (for many steels) fatigue life is effectively infinite.

4. State the Goodman relation.

σ_a/σ_e + σ_m/σ_UTS = 1 at the fatigue limit; below 1 is safe.

5. When does creep matter, and what are its stages?

Above about 0.4 T_m over time; primary, steady-state, and tertiary (to rupture).

TodayFinish this quiz and Levels 1 and 2 of the ladder.

+1 dayRe-derive a critical crack size from a blank page.

+3 daysOne fatigue (Goodman) problem.

+7 daysConnect toughness back to microstructure and grain size.

+30 daysCarry failure thinking into selection, Chapter 10.

Textbook mapping

Item	Mapping
Primary source	Callister and Rethwisch, Materials Science and Engineering: An Introduction, Chapter 8 (Failure)
Cross-reference	Askeland, Ch. 7 and 23 · Shackelford, Ch. 8 · Mechanics of Materials
Core topics	7.1 Ductile vs brittle · 7.2 Fracture mechanics · 7.3 Ductile-brittle transition · 7.4 Fatigue · 7.5 Creep
Engineering connection	Pressure vessels and aircraft (toughness), rotating parts (fatigue), turbines (creep).
Read next	Chapter 8: Phase Diagrams.