# FCP Short Course. Ductile and Brittle Fracture. Stephen D. Downing. Mechanical Science and Engineering

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1 FCP Short Course Ductile and Brittle Fracture Stephen D. Downing Mechanical Science and Engineering University of Illinois Board of Trustees, All Rights Reserved

2 Agenda Limit theorems Plane Stress Plane Strain Notched plates Notched bars Brittle Faracture FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 1 of 4

3 Beam in Bending M M M FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved of 4

4 Stress Distribution Elastic Elastic-Plastic Fully Plastic M FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 4

5 Plastic Moment y s Elastic: M e ys c I ys bh 6 y s Fully Plastic: M dy y o ys 4 M M o e h b ydy ys bh FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 4

6 Fully Plastic Moment F M o Beam can be loaded until M o is reached after which a plastic hinge is formed and the beam is free to rotate. FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 5 of 4

7 Beam With a Support L F L M o M o FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 4

8 Collapse F right end support prevents collapse M = M o F M = M o F left end support prevents collapse collapse M = M o M = M o FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 4

9 Lower Bound Theorem If an equilibrium distribution of stress can be found which balances the applied load and is everywhere below or at yield, the structure will not collapse or just be at the point of collapse. Guaranteed load capacity where system will not have large deformations FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 8 of 4

10 Upper Bound Theorem The structure must collapse if there is any compatible pattern of plastic deformation for which the rate at which external work is equal to or greater than the rate of internal dissipation. Guaranteed load capacity where the system will have large deformations. FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 9 of 4

11 Application to a Cracked Plate Assume a stress distribution that satisfied equilibrium P LB y b z x a W y = ys x = 0 z = 0 P LB = ys ( W a ) b xy = 0 FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 10 of 4

12 Upper Bound Solution t Rigid blocks allowed to slide Shear band of thickness t FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 11 of 4

13 Upper Bound Solution (continued) F s P UB du Slip plane area: A s W a b cos Force on shear plane: F s ys 3 W a b cos FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 1 of 4

14 Upper Bound Solution (continued) P UB Internal work: F s du duf du ys External work: s du 3 W a b cos du dw P du sin UB FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 13 of 4

15 Upper Bound Solution (continued) Upper Bound Theorem: du = dw P UB ys W a b du PUB dusin 3 cos ys W ab ys W ab 3 sin cos 3 sin Lowest upper bound for sin = 1 or = 45 P UB = 1.15 ys ( W a ) b P LB = ys ( W a ) b FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 14 of 4

16 Agenda Limit theorems Plane Stress Plane Strain Notched plates Notched bars Brittle Faracture FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 15 of 4

17 Plane Stress Plane Strain thick plate plane strain z = 0 z = 0 thin plate plane stress z = 0 z = 0 FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 16 of 4

18 Thick or Thin? Plane strain Plane stress FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 17 of 4

19 Transverse Strains Longitudinal Tensile Strain Transverse Compression Strain Thickness mm 0 mm 10 mm = = mm y z x 5 D 100 FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 18 of 4

20 Notch Stresses y z x t x z x z FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 19 of 4

21 Fracture Surfaces FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 0 of 4

22 Fracture Surfaces FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 1 of 4

23 Yielding plane stress z = 0 z = 0 x x y y 3 3 k plane strain z = 0 z = k x y FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved of 4

24 Flow Stress Ultimate strength Stress (MPa) Yield strength Strain Flow Stress: flow ys uts k flow 3 FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 4

25 Notches Edge Center Same net section, same K T FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 4

26 Plane Stress n = flow n = flow flow 0 0 FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 5 of 4

27 Edge Notch Plane Strain From the punch problem: n = k P =.96 flow By analogy: n =.96 flow.96 flow 1.81 flow.38 flow FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 4

28 Center Notch Plane Strain Slip Line Field x = 1.15 flow flow FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 4

29 Agenda Limit theorems Plane Stress Plane Strain Notched plates Notched bars Brittle Faracture FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 8 of 4

30 What does all this tell us? A B C D Lower Bound Theorem: P A = P B = P C = P D Upper Bound Theorem: P A = P B = P C < P D Slip Line Theory: P A = P B = P C < P D No effect of stress concentration on static strength! Some stress concentrations can increase the strength! FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 9 of 4

31 Failure Loads Edge Center Plane Stress P A net 1.15 flow P A net flow Plane Strain P A net.96 flow P A net 1.15 flow FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 30 of 4

32 Conclusion Net section area, state of stress and material strength control the failure load in a structure. FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 31 of 4

33 Stress or Strain Control? elastic material plastic zone Elastic material surrounding the plastic zone forces the displacements to be compatible, I.e. no gaps form in the structure. Boundary conditions acting on the plastic zone boundary are displacements. Strains are the first derivative of displacement FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 4

34 Define K and K S, e, Define: nominal stress, S nominal strain, e notch stress, notch strain, Stress concentration Strain concentration K K S e FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 33 of 4

35 K and K K T S Stress (MPa) K K S e K T e Strain FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 34 of 4

36 Stress and Strain Concentration Stress/Strain Concentration First yielding K K T K T K 1 Nominal Stress FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 35 of 4

37 Failure of a Notched Plate FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 36 of 4

38 Real Data Materials: 1018 Hot Rolled Steel 7075-T6 Aluminum /4 thick 1 1 FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 37 of 4

39 1018 Stress-Strain Curve true fracture strain, f = 0.86 true fracture stress, f = 770 MPa Strain FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 38 of 4

40 7075-T6 Stress-Strain Curve true fracture strain, f = 0.3 true fracture stress, f = 70 MPa Strain FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 39 of 4

41 7075-T6 Test Data 100 load, kn edge diamond slot hole displacement, mm FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 40 of 4

42 1018 Steel Test Data load, kn edge diamond slot hole displacement, mm FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 41 of 4

43 Agenda Limit theorems Plane Stress Plane Strain Notched plates Notched bars Brittle Faracture FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 4

44 Stress Concentration o Axial, z Tangential, Radial, r r FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 43 of 4

45 Bridgeman Analysis (1943) ys z ys r z r a r a r Elastic stress distribution Plastic stress distribution z r cons tan t FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 44 of 4

46 FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 45 of 4 Stresses a r a a ln 1 o z a 0 z z dr r P a 1 ln a 1 a P flow max P max = A net flow CF CF constraint factor

47 Constraint Factors a / CF P max = A net flow CF FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 46 of 4

48 Effect of Constraint Stress, z Increasing constraint Strain, z Higher strength and lower ductility FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 47 of 4

49 Failures from Stress Concentrations Net section stresses must be below the flow stress Notch strains must be below the fracture strain FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 48 of 4

50 Typical Stress Concentration K T ductile brittle K T brittle ductile Stress distribution Strain distribution A ductile material has the capacity for very large strains and it reaches the strength limit first A brittle material reaches the strain limit first FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 49 of 4

51 Strain Distribution Nominal strain will reach the yield strain when the strength limit is reached Strain distribution K T K T yield yield ductile transition K T f K yield T brittle yield f K f T yield K T yield FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 50 of 4

52 Notched Bars 1 8 R 1 16 R 1 R V notch FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 51 of 4

53 7075-T6 Test Data V displacement, mm FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 5 of 4

54 1018 Steel Test Data V displacement, mm FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 53 of 4

55 Conclusion Net section area, state of stress and material strength control the failure load in a structure only in ductile materials. In brittle materials, cracks will form before the maximum load capacity of the structure is reached. FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 54 of 4

56 Agenda Limit theorems Plane Stress Plane Strain Notched plates Notched bars Brittle Fracture FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 55 of 4

57 Brittle Fracture Failure of structures by the rapid growth of cracks (or crack-like defects) until loss of structural integrity. FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 56 of 4

58 Fracture FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 57 of 4

59 Stress Concentration applied Inglis Equation local a local 1 applied K t applied Crack dimensions a = 10-3 = 10-9 local Does this make sense? applied a Kt 000 FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 58 of 4

60 Griffith Approach l applied a Uncracked plate elastic energy 1 Ue w*l*t w*l*t E Introduce a crack which Reduces elastic energy by: a Ue a *t E Increases total surface energy by: w at s applied thickness, t For a crack to grow, the energy provided by new surfaces must equal that lost by elastic relief a U e FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 59 of 4

61 Griffith Approach applied Minimum criterion for stable crack growth: Strain energy goes into surface energy a U l a a a a a E t a s a 4 a t w a E s applied thickness, t s E a FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 60 of 4

62 Plastic Energy Term l a a w applied Previous derivation is for purely elastic material Most metals and polymers experience some plastic deformation Strain energy (U) goes into surface energy () & plastic energy (P) U P a a a Orowan introduced plastic deformation energy, γ p E ( s a p ) p s applied thickness, t Materials which exhibit plastic deformation absorb much more energy, removing it from the crack tip FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 61 of 4

63 Energy Release Rate, G l a a w applied applied thickness, t Irwin chose to define a term, G, that characterizes the energy per unit crack area required to extend the crack: G G 1 U t a a E Comparing to the previous expression, we see that: G s p Works well when plastic zone is small ( fraction of crack dimensions ) EG a Change in crack area FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 4

64 GExperimentally Measuring G a F F x x t Load ICa Ea U a + a Displacement atfracture Load cracked sample in elastic range Fix grips at given displacement Allow crack to grow length a Unload sample Compare energy under the curves G is the energy per unit crack area needed to extend a crack Experimentally measure combinations of stress and crack size at fracture to determine G IC FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 63 of 4

65 Stress Based Crack Analysis Westergaard, Irwin analyzed fracture of cracked components using elastic-based stress theory Three modes for crack loading Mode I opening Mode II in-plane shear Mode III out-of-plane shear From: Socie FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 64 of 4

66 Stress Intensity Factor, K af astress intensity factor flaw size operating stresses b Kspecimen geometry Critical parameter is now based on: Stress Flaw Size Specimen geometry included using correction factor crack shape specimen size and shape type of loading (i.e. tensile, bending, etc) FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 65 of 4

67 Stress Intensity Factor, K Stress Fields (near crack tip) x K 3 cos 1 sin sin r y K 3 cos 1 sin sin r xy K 3 cos sin cos r Note: as r 0, stress fields Plane stress - too thin to support stress through thickness z 0 z 0 Plane strain - so thick that constrains strain through thickness z 0 x y z FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 66 of 4

68 FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 67 of 4 Stress Intensity Factor, K Displacement Fields sin cos r G K u cos sin r G K v x u x x v y x v y u xy Displacement in x-direction Displacement in y-direction

69 K and G G K G E 1 E K plane stress plane strain FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 68 of 4

70 Linear Elastic Fracture Mechanics How can we use an elastic concept (K) when there is plastic deformation? Small scale yielding Blunts the crack tip At elastic-plastic boundary, y r r* K 3 1 sin sin y cos r* Note: zone of yielding will vary with plastic zone r p Consider stress at = 0 K y r* FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 69 of 4

71 Design Philosophy K IC a f a w fracture toughness flaw shape flaw size operating stresses FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 70 of 4

72 Critical Crack Sizes a critical 1 K IC f FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 71 of 4

73 Fracture Toughness From M F Ashby, Materials Selection in Mechanical Design, 1999, pg 431 FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 4

74 Thickness Effects fracture toughness, MPa m thickness, mm FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 73 of 4

75 FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 74 of 4 Plastic Zone Size plane stress plane strain ys p K 1 r ys p K 6 1 r

76 3D Plastic Zone plane stress plane strain FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 75 of 4

77 Fracture Surfaces FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 76 of 4

78 Size Requirements t,w a,a.5 K ys 50r ys K Ic t, mm 04- T T Ti-6Al-4V Ti-6Al-4V PH p FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 77 of 4

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