21 Fracture and Fatigue Revision

Transcription

1 21 Fracture and Fatigue Revision EG2101 / EG2401 March 2015 Dr Rob Thornton Lecturer in Mechanics of Materials

2 Fracture concepts Fracture: Initiation and propagation of cracks within a material Structure no longer sustains any applied loading Often occurs at nominal stresses/strains below those material is expected to sustain (why?) Design against fracture: Crack-free materials are difficult to produce Therefore, we design for non-propagation or controlled propagation of cracks 2

3 Fracture mechanisms (1) Transgranular (Cleavage) Rupture Intergranular Brittle (fast) fracture: Failure occurs without plastic deformation Crystalline materials split along defined planes Ductile fracture: Failure occurs following necking or shearing Substantial plastic deformation 3

4 Fracture mechanisms (2) Type of fracture dependent on: Material bonding and structure Type of stress applied (e.g. tensile, shear, torsion) Rate of stress application Temperature (creep) Operating environment (corrosion) Component geometry (stress concentrators) Internal flaws (e.g. vacancies, precipitates, cracks etc.) External flaws (e.g. cracks, oxides etc.) 4

5 Brittle fracture by cleavage c So what do brittle fractures look like? Occurs in materials where the yield strength is high relative to the bonding strength: e.g. ceramics, glasses As p.d. does not occur, stress at sharp crack tips becomes extremely high: Sufficient to break interatomic bonds Brittle or fast fracture can propagate at the speed of sound Crack propagation by cleavage Jones and Ashby, (2011), Engineering Materials 1 5

6 Brittle fracture Zn at -120 C Very ductile at room temperature Not so much at -120 C when Zn has gone through its ductile-brittle transition phase Smooth surfaces are characteristic of brittle fractures 6

7 Brittle fracture Al-Li 8090 Aluminium alloys are usually ductile but not when inclusions form along grain boundaries this is an example of an intergranular fracture 7

8 Ductile fracture by tearing Nucleation Growth Coalescence In ductile materials, yield strength is low relative to bonding strength, hence p.d. occurs near crack tip: e.g. metals, polymers Within the plastic zone, voids nucleate and grow near defects Voids grow, coalesce (merge); allowing crack to propagate Crack propagation by ductile tearing Jones and Ashby, (2011), Engineering Materials 1 8

9 Ductile fracture (Almost) pure Cu Cu is a very ductile material and can display stable necking if processed correctly Ductile tearing can occur as voids nucleate and grow around precipitates or inclusions and then coalesce Rough surfaces are characteristic of ductile fracture 9

10 Fracture mechanics approaches Energy balance criterion (Griffith, brittle materials): An existing crack will propagate if crack growth releases more stored energy than is absorbed by the creation of the new crack surface U = U e + U s du da > 0 for crack stability Stress intensity factors (Irwin, ductile materials): The elastic stress distribution in a loaded material is distorted near crack tips and can be characterised if we assume standard crack geometries σ ij r, θ K 2πr f ij θ where K = Yσ πa Objective: To determine the critical size of defect necessary for fast fracture to occur 10

11 Energy balance criterion (1) σ Assume a plate contains an edge-crack (grossly enlarged!) Under an applied stress, crack grows by δa a δa For crack growth, work must be done (energy must be input): δw represents work required to enlarge crack by δa Crack growth releases elastic energy but requires energy for crack surface: σ δw = δu e + δu s 11

12 Energy balance criterion (2) σ δw = δu e + δu s a δa Energy absorbed by new crack tip: δu s = G c tδa G c is a material property: Toughness or critical strain energy release rate Tough materials have high G c : Copper, G c 106 Jm -2 Glass, G c 10 Jm -2 σ In tough materials it is difficult for cracks to propagate 12

13 Energy balance criterion (6) U U s = G c ta a c du da = 0 a σ πa c = EG c or σ c πa = EG c Total energy, U = U e + U s Critical crack length a c : Unstable equilibrium If a < a c : Growth requires additional energy (stress) U e = σ2 2E πa 2 t 2 If a > a c : Growth reduces energy Spontaneous and catastrophic 13

14 Stress intensity factor (1) From our fast fracture condition, σ πa = EG c, we can also see that: A critical combination of stress and crack length exist when fast fracture will commence Defining LHS as the stress intensity factor: K = σ πa (MN m -3/2 ) Defining RHS (only material properties) as the fracture toughness: K c = EG c (MN m -3/2 ) Fast fracture occurs when: K = K c 14

15 a σ W σ Stress intensity factor (2) Strictly the result K = σ πa is only valid for wide plates (thin, semifinite materials) A correction factor must be applied: For an edge-crack in a semi-finite plate (W >> a) K = 1.12σ πa In general, for other geometries: K = Yσ πa Values for Y can be found in data books 15

16 Stress intensity factor (3) σ σ c σ y = K c πa y σ y Failure by fast fracture Failure by yielding σ c = K c πa a y a 16

17 What is fatigue? Fatigue: Slow crack growth at loads less than that described by the fast fracture criterion, K = K c Occurs due to cyclic loading Why is understanding fatigue important? Estimated that 75% of all failures in engineering components due to fatigue e.g. De Havilland Comet failures Design against fatigue: Minimise both initial size and rate of growth of cracks Use of S-N curves to ensure stress cycling does not exceed the fatigue limit of the material 17

18 Fatigue loading (2) Linear-elastic Linear-elastic / yielding σ a Δσ Δσ σ y ε a ε a σ a Δε el Δε pl Δε tot 18

19 ε tot 2 (log scale) ε f σ m, ε m = 0 True fracture strain True fracture stress Fatigue loading (4) Fatigue curve results from elastic and plastic strain amplitudes: ε tot 2 σ f 2N f b E + ε f 2N f c σ f E N f (log scale) Constants b and c determined from fitting test data; typically: < b < < c <

20 Fatigue failure surfaces Characteristics: No necking prior to failure elastic strain Flat fracture surface Beach marks visible Brittle or ductile fracture follows fatigue surface 20

21 Fatigue failure rotating steel shaft Initiation at stress concentrating feature Beach marks formed during each loading cycle Fracture occurs once component can no longer sustain applied stress 21

22 Fatigue of cracked components (3) Δσ Initial crack Tension Crack widens by δ Unloading Crack grows by δ a δ Δσ Tension Crack grows by da/dn δ Fatigue crack growth Jones and Ashby, (2011), Engineering Materials 1 22

23 σ a / MPa %C steel Stress-cycle (S-N) curves Wöhler curve: Stress amplitude (S) against logarithmic cycles to failure (N f ) Fatigue limit series Al-Cu Endurance limit for N cycles N N f 23

24 Fatigue of cracked components (5) log da dn Threshold Crack growth per cycle (da/dn): Steady-state crack growth rate, da dn = A K m If initial (a 0 ) and failure lengths (a f ) are known: N f = N f = N f 0 dn a f a 0 da A K m K 0 Paris Law da dn log K = A K m Fast fracture K max = K c 24

25 Variable loading Typically, Δσ changes during life of components (e.g. due to wind loading, haulage loads, fuel load): How do we calculate total cycles to failure? Miner s Rule: The sum of the fractions of the cycles to fracture under each loading regime equals 1 N i i = 1 = N 1 N fi + N 2 + N 3 N f1 N f2 N f3 Procedure: 1) For each stress range, divide number of cycles by calculated cycles to failure 2) Add fractions until the sum exceeds 1 at which point fracture should have occurred 25

26 Exam reminder Two and a half hours: 6 questions in two sections; answer 4 in total Part A Answer one question out of two Part B Answer three questions out of four Questions in Part A do not follow style of previous years examples: New lecturer, new content new example slides! 26

27 Good luck!