Introduction to Fracture

Transcription

1 Introduction to Fracture

2 Introduction Design of a component Yielding Strength Deflection Stiffness Buckling critical load Fatigue Stress and Strain based Vibration Resonance Impact High strain rates Fracture -???

3 Introduction Design based on Strength of Materials / Mechanics of Solids Strength based design Check for allowable stress Stiffness based design Check for allowable deformation / deflection Presence of defects in the material ideal Imperfections Higher factor of safety 3

4 Introduction Fracture separation of a body in response to applied load Crack tip Crack Fracture Relieving stress and shed excess energy Main focus whether a known crack is likely to grow under a certain given loading condition 4

5 Introduction Fracture mechanics approach Implicit assumption Crack exists Severity of the existing crack when loads applied on the structure / component Application of Fracture Mechanics (FM) to Crack growth Fatigue failure 90s Griffith developed right ideas for growth of a crack Working parameter Modern FM 948 George Irwin devised a working parameter 5

6 Introduction Irwin s work Mainly for brittle materials Introduced Stress Intensity Factor (SIF) and Energy Release Rate (G) Linear Elastic Fracture Mechanics (LEFM) Plastic deformation negligible Irwin s theory application of FM to design problems Focus was on crack tip not on the crack 6

7 Introduction Crack Tip Opening Displacement (CTOD) 96 Wells J Integral 968 Rice CTOD and J integral Ductile materials large plastic zone at tip 7

8 Modes of fracture Modes of Fracture mode I mode II mode III K I K II K III Opening Sliding Tearing 8

9 Stresses at crack tip From Linear Elastic Fracture Mechanics (LEFM) the stresses near the crack tip in Mode I σ x σ y τ xy θ 3θ sin sin K I θ θ 3θ cos sin sin πr + θ 3θ sin sin The displacements near the crack tip θ cos ( κ + sin ) u KI r v G π θ θ sin ( κ + cos ) θ Plane strain Plane stress y r θ σ yy σ xy σ xx x κ 3 4υ 3 υ κ + υ 9

10 Griffith s theory Theory of unstable crack growth Theory establishes a relation unstable crack growth Basic underlying principle Energy balance Introducing a crack in a stressed / loaded component release of strain energy Reduction in stiffness What happens to change in strain energy??? or released strain energy 0

11 An infinite body Griffith s theory Linear elastic subjected to stress σ Initially there was no crack Strain energy stored σ U o E Vol σ βa a σ

12 Griffith s theory Crack size a the crack surfaces traction free no traction loads acting Material above and below stress free to some extent Stress relived portion assuming triangular distribution Height of triangle β Griffith carried out extensive calculations and experiments > β π

13 Griffith s theory Strain energy after introducing crack Strain energy before introducing crack Strain energy loss (relived) σ > U U ( ) o Volume of le E > U U > o U U σ σ E βa E t E R aβat Energy released because of presence of crack 3

14 Griffith s theory When a body is cracked breaking of material bonds Energy required for breaking bonds Source of energy??? Loss / Deficiency in strain energy > caters for formation of crack Formation of crack > Generation of new traction free surfaces Energy used for breaking bonds stored as surface energy on newly formed surfaces 4

15 Griffith s theory Two key aspects in crack growth How much energy is released strain energy when crack advances Minimum energy required for the crack advance in forming two new surfaces Some external work is done Increase in strain energy Surface energy crack growth 5

16 Griffith s theory Assume a crack of a size exists σ Initial stress free area, π(a+ a) a a σ πa A * ½ (a*πa) πa Crack grows by a on both sides Stress free area after crack growth, A */ ((a+ a)*π(a+ a) > A π(a+ a) 6

17 Griffith s theory Change in stress free area > A > A A A V 4 π a at Change in strain energy π σ > U E σ > U E ER πσ at > a E [( ) ] π a + a a ( a a) 4πa a V ( 4πa at ) () E R 7

18 Griffith s theory Surface energy required to create new area > W s γ ( 4 at) Ws > 4γ st () a Unstable crack growth takes place, if der dws da da πσ at > 4γ st E > Eγ σ πa s s 8

19 Griffith s theory Expression E γ s σ πa 'a' existingcrack length - Stress required to grow crack crack size- Damagetolerant design Eγ s c πσ Eγ s > σ c πa Given loading- Stress- maximumallowable a 9

20 Griffith s theory γ s Specific surface energy Surface energy per unit area of crack surface Area of crack (a*t) 4at Surface energy stored > 4at γ s Energy balance > > πσ E E a R t W s 4at γ s 0

21 Griffith s theory Plot E R and W s wrt crack length a πσ a t E R ER W s E W 4atγ E R s s W s Initial crack size a o W s Crack grows by a a o a E R a Strain energy released E R Surface energy required W s

22 Griffith s theory Crack growth takes place, if E R W s Not satisfying above inequality crack remains dormant Body acts as a strain energy reservoir Surface energy required can be obtained from external sources as well increase in applied stress No change in W s

23 Griffith s theory Strain energy used to break the bonds stored as surface energy Strain energy source & surface energy sink Irreversible thermodynamic process Crack growth energy conversion process When a crack propagates strain energy gets reduced and surface energy increases for a constant displacement case 3

24 Griffith s theory In the limiting case > > lt E E de da R R a R W lt a 0 a 0 > dw da s s W a Check the slope of E R and W S Satisfying above condition onset of crack growth s 4

25 Griffith s theory If strain energy release rate is higher than required rate of surface energy unstable crack growth de da R > dw da S Difference in energy rates > kinetic energy Higher difference > Faster crack growth 5

26 Griffith s theory Strain energy release rate per unit increase in area during crack growth > Griffiths G G de R da, E > R da πσ E a t tda, A at > G de R da de t da R Units of G > N.m/m or J/m πσ E a 6

27 Griffith s theory Crack area (A) and crack surface area (A s ) Crack area > Simply the area of crack A a*t at Crack surface area > Sum of surface areas of all the crack surfaces Crack surfaces Two (top and bottom) Each > at A s (at) 4at A a 7

28 Griffith s theory Surface energy required for crack to grow per unit area of extension crack resistance R S S ( increase in total surface surface ) W γ area > W S Crack area γ S > A ( ( at) ) > R dw da S at 4 atγ γ S S γ A Brittle fracture no plastic deformation R surface energy Elastic and plastic R caters for surface energy and energy for plastic deformation at crack tip S 8

29 Griffith s theory Crack start propagating if, G > R Strain energy release rate of a crack must be greater than the crack resistance for the crack to grow Crack propagation occurs G is sufficient to provide all the energy that is required for the crack formation Energy release rate is more than crack resistance, crack acquires KE the growth speed may be faster than speed of a supersonic flight 9

30 Griffith s theory Onset of crack growth σ πa Eγ RHS completely depends on material properties Constant From the above - a σ s Maximum allowable crack size depends on loading or Loading decides the allowable crack size Bigger cracks lower loads 30

31 Griffith s theory Crack growth at different load / stress conditions Strain energy - High stress. ER & Ws Strain energy - Low stress Surface energy Ramadas Crack Chennamsetti size 3

32 Griffith s theory Thin and thick brittle plates Thin plate Plane stress condition Thick plate Plane strain condition Brittle insignificant plastic zone at crack tip - LEFM Ductile materials considerable plastic deformation at crack tip EPFM Plane stress and planes strain Poisson ratio Allowable crack size of stress different 3

33 Griffith s theory Thin and thick plates Griffith s theory σ σ π π a a E γ s E υ γ s Plane stress Plane strain RHS higher in plane strain For a given load higher crack size in thick plate than in thin For a given crack size higher load in thick plate than in thin plate 33

34 Griffith s theory Taking square root > > s c E a E a c γ π σ γ υ π σ 34 Constant > > s c c IC s c c E a K E a γ υ π σ γ υ π σ K I Stress Intensity Factor (SIF) K c Stress Concentration Factor (SCF)

35 Griffith s theory Critical strain energy release rate G IC R Crack growth takes place, if G IC > K G E a c πσ E a G K E E G IC In the limiting case, πσ IC c IC IC γ S γ S E R 35

36 Griffith s theory Typical values of critical stress intensity factor K IC Glass 0.5 to MPa m 0.5 Alloy steel 50 MPa m 0.5 Aluminium alloy 5 to 40 MPa m

37 Griffith s theory Plate under constant displacement P o A P B O C U o Plate stiffness reduced OAB Strain energy released 37

38 Griffith s theory No crack > More slope high stiffness more force required to pull Presence of crack > lower slope reduced stiffness less force required to pull same length u o Difference in strain energy Area OAB used to form new surface / break the bonds Major portion of strain energy release takes place at above and below crack 38

39 Fracture toughness total potential A body with a crack subjected to external loading work done on the body Utilization of this energy Increase in strain energy Utilization of energy to create two new surfaces In this process Point of application of load may or may not move Force moves > work is done by the force Decrease in stiffness 39

40 Conservation of energy Work performed per unit time rate of change of internal energy + plastic energy + kinetic energy + surface energy (crack formation) W U + U + K E P + Γ Crack grows slowly inertia effects negligible > KE 0 Equation was wrt time. Crack grows with time change time to crack area t A A t A A Chain rule A at a 40

41 Conservation of energy Change variable to crack dw du du E P dγ A A + A + A da da da da dw du E du P d Γ > + + da da da da dw du E du P dγ > + da da da da du E dw du P dγ > + da da da da 4

42 Conservation of energy Ideal brittle material > Negligible plastic deformation > U p 0 > Total potential π du E da U E + V d π da dw da U dγ da E d Γ da W 4

43 Surface energy Surface energy per unit area > γ s Total surface area, A s *(at) 4at A Change in surface energy to form new crack of length a Γ γ s A s Γ > A γ γ s ( 4 at) s d Γ > da It was shown that, G I γ s γ s γ A s 43

44 G total potential Surface energy and total potential dπ d Γ da da d π > γ R da s G I Impending of crack growth G I R 44

45 G total potential Relation between G I and π > W > d > > π ext W G I d du π du ext E E du d π da GdA W d E + π ext G I G da I da External work done on the body > increase in elastic strain energy and formation of new surfaces 45

46 G total potential No movement of external forces, W ext 0 G I du da E When there is no external work done, energy required for crack growth is obtained from the strain energy stored in the body Conservation of energy > Decrease in strain energy increase in surface energy 46

47 Compliance approach Compliance > Inverse of stiffness C, P ku > u k CP Fracture toughness compliance approach Constant load Constant displacement Increase in compliance with crack length loss of stiffness Calculation of Fracture toughness, G I 47

48 P o Constant load Consider a body with some initial crack of length a - Load applied > P o P F a B a+da u u A C E π - Area ABF, π - Area ADF π π - π D u Write total potentials when crack sizes are a and a+da > > P u P u o o π π U U P u o P u o + V + V P u o P u o 48

49 Constant load Potentials, π > π π > π P u o π P o, π P u o P u o + P u o ( u u ) P u Strain energy release rate, G I π G ( P u) I d da d da o o P o du da 49

50 Constant load Use compliance relation u CP o G I du > da P o P o du da dc da P o dc da G Experimental measurements required to estimate change in compliance with crack area I P o dc da 50

51 Constant displacement Constant displacement Fixed grip condition a (a+ a) Since the grip is fixed no external work is done by the forces, W ext 0 5

52 Constant displacement P P P O Load-displacement curve D E a a+da A u o B C u π Change in potential, ( P P )uo > ve > π Puo dπ dp > GI uo da da dc dp uo CP > P C da da5

53 Constant displacement This gives, G I dp P dc da C da dp P dc u o uo da C da P G I dc da G I same in both constant load and displacement conditions 53

54 P o Constant load & displacement Comparison P P const a B a+da D P P P D E u const u u u C O u o A C E U > U Increase in strain energy Caters for crack growth a a+da A B No external work done U < U Decrease in strain energy used for crack growth u 54

55 Energy release rate P D A Energy release rate referred as rate of strain energy flux flowing towards a crack tip as the crack extends Linear and Nonlinear - π Linear U B C u P D A Nonlinear - π U B C u J integral contour integral J d π da Area ABD For linear elastic, G J 55

56 Measurement of G IC Expression for G IC Critical SERR P P o P G IC dc da A A A 3 A 4 Parameter to be measured experimentally > dc/da C dc/da k k k 3 k 4 u u u 3 u 4 u A o A 56

57 Measurement of G IC Size of crack > a o, corresponding area A o Load required to initiate crack growth, P c G IC P c dc da A o If G I is expressed in terms of stiffness, P dk G c IC k dk/da is -ve da o A o 57

58 Griffith s theory Crack in stiff and flexible components Same force applied on stiff and flexible components having same size of crack Energy stored U > U P C K P K ( C ) Flexible component stores more energy than stiffer component More store of energy availability of energy for crack propagation Stiff bodies do not store more energy less release 58

59 Surface energy Surface energy per unit area R material property In Fracture Mechanics R > total energy consumed in propagation of crack In ductile materials Plastic deformation takes place energy required to deform Energy required for a crack to grow is much larger than the surface energy Significant plastic deformation 59

60 Surface energy Crack growth accompanied with plastic deformation G ( ) w R γ γ + s p f γ p > Plastic work per unit area of surface created, γ s > Surface energy per unit area of surface created Most of metals surface energy is much smaller than (/000 th ) of the total energy consumed in crack propagation 60

61 Surface energy Crack resistance (R) depends on material and geometry thickness Plastic zone size at the crack tip depends on thickness Bigger plastic zone more dissipation of energy Higher crack resistance - Plane stress Smaller plastic zone less dissipation of energy Lower crack resistance Plane strain 6

62 Crack in a pipe Eg. Longitudinal crack of 0 mm is allowed in a pipe. What is the allowable pressure? P r a P a K IC 93 MPa m 0.5 Allowable stress 878 MPa 6

63 Crack in a pipe Hoop stress causes crack opening mode I Considering this as an infinite plate with a crack of length a p r σ max > t K IC σ > K π IC a c > p max t r Hoop stress Critical π a c SIF 63

64 Crack in a pipe Maximum pressure pmaxr K IC πac t 6 t K IC t 93 0 > pmax r πa r π 5 0 > p max c Design based on Fracture 6 t r 3 64

65 Crack in a pipe Same maximum pressure causes Hoop stress Check for yield failure σ allowble > Y p r max t p max If design is based on fracture criteria, it does not fail in yielding t r 65

66 66