Natural Sciences Tripos Part II MATERIALS SCIENCE C15: Fracture, Fatigue and Creep Deformation Dr C. Rae Lent Term 2011-2012 I IPart II Materials C15 Lent 2012 1 C15:FRACTURE FATIGUE AND CREEP DEFORMATION Catherine Rae 12 LecturesSynopsis Introduction:This course examines the use of fracture mechanics in the prediction of mechanical failure.We explore the range of macroscopic failure modes; brittle and ductile behaviour. We take a closer look at fast fracture in brittle and ductile materials characteristics of fracture surfaces; inter-granular and intra-granular failure, cleavage and micro-ductility.We describe the range of fatigue failure and apply fracture mechanics to the growth of fatigue cracks.Finally we look at the processes of creep and how it combines with fatigue. Griffiths analysis:Revision of concept of energy release rate, G, and fracture energy,R.Obreimoffs experiment.Timeline for developments. Linear Elastic Fracture Mechanics, (LEFM).We look at the three loading modes and hence the state of stress ahead of the crack tip.This leads to the definition of the stress concentration factor, stress intensity factor and the material parameter the critical stress intensity factor. Superposition principle, Energy release rate, prediction of crack growth direction. Plasticity at the crack tip and the principles behind the approximate derivation of plastic zone shape and size.Limits on the applicability of LEFM. The effect of Constraint, definition of plane stress and plane strain and the effect of component thickness. Concept of G - R curves, measuring G and K. Elastic-Plastic Fracture Mechanics; (EPFM).The definition of alternative failure prediction parameters, Crack Tip Opening Displacement, and the J integral.Measurement of parameters and examples of use. The effect of Microstructure on fracture mechanism and path, cleavage and ductile failure, factors improving toughness, Fatigue: definition of terms used to describe fatigue cycles, High Cycle Fatigue, Low Cycle Fatigue, mean stress R ratio, strain and load control.S-N curves. Adapting data to real conditions: Goodmans rule and Miners rule.Micro-mechanisms of fatigue damage, fatigue limits and initiation and propagation control, leading to a consideration of factors enhancing fatigue resistance. Total life and damage tolerant approaches to life prediction, Paris law. Creep deformation: the evolution of creep damage, primary, secondary and tertiary creep. The use of Larson-Miller parameters.Micro-mechanisms of creep in materials and the role of diffusion. Ashby creep deformation maps.Stress dependence of creep power law dependence. Comparison of creep performance under different conditions extrapolation and. Creep-fatigue interactions. Part II Materials C15 Lent 2012 2 Booklist: T.L. Anderson, Fracture Mechanics Fundamentals and Applications, 2nd Ed. CRC press, (1995) (Fracture mechanics and its application to fatigue, very thorough and readable) B. Lawn, Fracture of Brittle Solids, Cambridge Solid State Science Series 2nd ed 1993. (Exactly as it says on the label very good on LEFM) J.F. Knott, P Withey, Worked examples in Fracture Mechanics, Institute of Materials.(Excellent short summary of fracture mechanics and good worked examples) H.L. Ewald and R.J.H. Wanhill Fracture Mechanics, Edward Arnold, (1984).(Provides very clear explanations different perspective from Anderson) S. Suresh, Fatigue of Materials, Cambridge University Press, (1998)(Excellent on fatigue but not very readable) G. E. Dieter, Mechanical Metallurgy, McGraw Hill, (1988) (Good entry-level text on mechanical properties) D.C. Stouffer and L.T. Dame,Inelastic Deformation of Metals, Wiley (1996) (Particularly chapters 2 and 3 for creep and fatigue) R.C Reed, The Superalloys, CUP (2006).Particularly Chapters 2 and 3 for creep and fatigue in superalloys and Chapter 4 for lifing strategies. Part II Materials C15 Lent 2012 3 FRACTURE, FATIGUE AND CREEP DEFORMATION SYNOPSIS This course examines the use of fracture mechanics in the prediction of mechanical failure.We explore macroscopic failure modes; brittle and ductile behaviour, and take a closer look at fast fracture in brittle and ductile materials characteristics of fracture surfaces; inter-granular and intra-granular failure, cleavage and micro-ductility. Fatigue causes 90% of engineering failures: we examine how we characterise the susceptibility of materials to fatigue and estimate lifetimes. At high temperatures time-dependent plastic deformation occurs: we describe the mechanisms of creep and how it can both exacerbate and mitigate the effects of fatigue. GRIFFITHS THEORY, REVISION FROM 1B COURSE. Griffiths Theory provides the thermodynamic or energetic criterion for failure: it does not consider the mechanism by which failure occurs. The basic premise is that a crack will propagate in a material when the elastic energy released as a result of that propagation exceeds the energy required to propagate the crack.In the first instance just the surface energy needed to create two new surfaces was considered, but this applies only to ideal brittle solids i.e. those where fracture occurs without any plastic deformation.Subsequently this was widened to include the work required to perform the plastic deformation associated with ductile failure and, in principle, can include any work necessary such as de-cohesion on composites phase changes etc. If we introduce a crack of length 2a into an infinite plate of thickness B under a uniform stress , the elastic stresses relax around the crack and reduce the elastic potential energy UE stored in the plate. Extra surface is created at the crack, US, and, if the grips are fixed, no external work, UF,is done by the applied force, UF = 0.Part II Materials C15 Lent 2012 4 ( )s E FU U U a U + + =At equilibrium:

! dUda=dUEda+dUSda= 0 The change in the potential energy is estimated from an elastic analysis of the stresses around the crack:

! UE " #$%2a2BE And the work done to propagate the crack is: s SaB 4 U ! =Where the area of the crack is 2aB,the surface area is 4aB and the surface energy is s. Thus:

! d(UE)da=2B"#2aE and

! dUSda= 4B"s : hence:Ea22s!"= # Rearranging:Griffiths Equation This is for an ideal brittle solid; for a ductile material the plastic work of deformation p , is introduced: aE ) 2 (p s!" + "= # Modification of the fracture criterion to include plastic work leads to the more general definition of the energy release rate or the crack extension force:G.This is the change in the potential energy, U, of the system per unit increase in crack area, A, and has the dimensions of force/length. Energy Release Rate:

! G = "dUdA= "dU2Bda= " #$2aE According to Griffiths crack extension occurs when this equals the work to fracture, 2s + p . p s c2 G G ! + ! = =Gc is a material constant and a measure of the fracture toughness. The RHS is the resistance to crack growth termed R where R = 2s + p. Part II Materials C15 Lent 2012 5 OBREIMOFFS EXPERIMENT A real example illustrates two important points: firstly that brittle fracture is reversible under the right circumstances and secondly, that whether it occurs or not is governed by balancing stored elastic energy with the work of fracture. In 1930 Obreimoff split a thin sheet of mica off a larger piece by inserting a wedge of thickness h beween the layers. The crystal cleaves along the weak interfaces between the layers to give a thin upper fillet and a thick lower section. As the wedge is driven into the crack the crack grows to keep the length constant.The elastic energy stored as the wedge is forced into the open crack is principally in the thin upper fillet, and is balanced by the cohesive forces at the crack tip.The crack opens until these are balanced.The energy is calculated easily from the elastic properties of the mica, and the geometry of the set-up. The elastic strain in the cantilever is given by beam theory:

! U= UE=Ed3h28a3 where the constants are given in the diagram. The surface energy needed to grow the crack is

! US= 2a"where is the surface energy. Equating the elastic energy to the surface energy gives an equilibrium crack length ao of:

! ao= 3Ed3h2/16"4 As the wedge is withdrawn the crack closes and the damage is pretty much repaired if the process is done in vacuum.This can be shown by reopening the crack and noting that the value of ao for the re-opened crack is almost the same.As air and moisture are introduced, the quality of the repair deteriorates and the equilibrium length ao increases.Part II Materials C15 Lent 2012 6 TIME LINE FatigueFracture~1500- Leonardo da Vinci failure stress of iron wires depends on length i.e. on probability of flaw 1842- Railway accident Versailles - failure of axle 1843- significance of fatigue striations recognized WJM Rankin 1852-1869- Wohler systematic experiments on bending and torsion development of S-N curves 1874& 1899Gerber and Goodman life prediction methodologies 1886Baushinger effect noted 1900Ewing and Rosenberg recognition of persistent slip bands extrusions and intrusions 1913Inglis elastic stress field around elliptical hole 1920Griffiths equation for brittle materials 1930 1938 Obreimoffs experimentWestergaarde elastic solution of the stress distribution at a sharp crack 1945Constance Tipper and the Liberty ships - Recognition of the Ductile Brittle transition Tipper test and the role of crystal structure in failure 1945Minor accumulation of fatigue damage 1953 -54Comet airliner losses due to fatigue failure1954Coffin Manson empirical laws for HCF and LCF 19561956Wells applies fracture mechanics to fatigue to explain the Comet fatigue fractures 1956Irwin development of the concept of energy release rate based on Westergardes work 1956Demonstration of the role of PSB in initiating fatigue failure 1957Fracture mechanics predicts disc failures for GE 19601960Paris law relating the crack growth rate to the stress intensity factor 1960-61Irwin/Dugdale/Wells development of LEFM and effect of plastic zone size and shape 1968Proposal of the J integral by rice and the CTOD by Wells to cope with the failure of ductile materials 1976Shih and Hutchinson establish the theoretical basis of the J-Integral and link it to the CTOD 1980 Chaboche Development of time dependant fracture interactions between creep and fatigue. Part II Materials C15 Lent 2012 7 WHAT IS A BRITTLE FRACTURE? Very few fractures are truly brittle i.e. have no permanent deformation. But fracture is still determined by the energy balance and the energy driving the cracking process is still the elastic energy stored in the cracked body. Fast fracture is a more accurate term than brittle fracture to use for rapid failure. Where local deformation occurs the cracking process is not reversible as it was in the case of Mica. Can deal with a great many materials and situations using simple elastic assumptions.This is known as linear elastic fracture mechanics. [There is a fundamental flaw inherent in LEFM the calculations assume elastic behaviour but we know that for the crack to have any chance of growing the stresses at the tip must vastly exceed the yield stress: yet we carry on anyway!The point is that in many materials the contribution to the energy balance from the non-elastic part is a tiny fraction of the total equation. We can put this to one side for the time being, but will examine this later.] Brittle Brittle Ductile Ductile Part II Materials C15 Lent 2012 8 LINEAR ELASTIC FRACTURE MECHANICS When a crack occurs in a material the local stress around the crack is raised. LEFM relies on the sufficient of the specimen/component being elastic such that the energy release rate can be calculated from the elastic displacements around the crack tip.Hence if you can solve for the elastic stress in any configuration you can (in principle) calculate G from dUE/da. STRESS CONCENTRATION AT FEATURES In some simple situations the equations governing elastic deformation can be solved analytically: i.Expressing the stresses in terms of complex potentials ii.Specifying the boundary conditions iii.Finding functions to satisfy the above Or, more generally, solving the problem using finite element analysis.One problem for which there is a solution is that of a circular hole in an infinite thin plate subject to a stress o. In polar co-ordinates the stresses are given by:

! "rr= "o21 +ro2r2+ 1 + 3ro4r4 # 4ro2r2$ % & & ' ( ) ) cos2*+ , - . - / 0 - 1 -

! "##= "o21 +ro2r2 $ 1 + 3ro4r4% & ' ' ( ) * * cos2#+ , - . - / 0 - 1 -

! "r#= $ "o21 $ 3ro4r4+ 2ro2r2% & ' ' ( ) * * sin2#+ , - . - / 0 - 1 - Substituting r = ro and = 90 and 0: gives the maximum and minimum hoop stresses , at the edge of the notch as 3o and -o.Thus the presence of a round hole in the plate increases the tensile stress by a factor of three in one direction and introduces a compressive stress at the top of the hole equal to the distant tensile stress. Part II Materials C15 Lent 2012 9 Because all the stresses are elastic and therefore small, the imposed stress fields, and the solutions for those stress fields, can be added: this is known as the PRINCIPLE OF SUPERPOSITION. Hence, adding two stresses o at right angles to each other to produce a 2D hydrostatic tension and the stresses around the hole in the plate are now: 3o- o = 2o. Another important situation for which an exact solution exists is that of an elliptical hole, semi-axes a and b, in a plate, subject to a distant stress o.In this case the maximum stress is at the tip of the ellipse: 2a 2b 2! "o ! "max= "o1+2ab# $ % & ' ( or ! "max= "o1+2a#$ % & & ' ( ) )

where ab2= !the radius tangential at the tip. Hence for a long thin crack where a >>b, ! "max= "o2a#$ % & & ' ( ) ) This is slightly modified for a half crack at the edge of a plate by the factor 1.12 because the free surface (zero stress) allows the ellipse to open rather wider than for the embedded crack. The factor max/o by which the elastic stress is raised by a feature such as a crack or a hole is the stress concentration factor kt.This is dimensionless. Part II Materials C15 Lent 2012 10 SHARP CRACKS The above is very useful for finding the effect of features (intended or unintended) in the structure, but most cracks are long and have sharp tips.These can be of atomic dimensions in brittle materials. In 1939 Westergaard solved the stress field for an infinitely sharp crack in an infinite plate.The elastic stresses were given by the equations; yy= oa2rcos 2| | | | 1+ sin 2| | | | sin32| | | | | | | | | | xx= oa2rcos 2| | | | 1sin 2| | | | sin32| | | | | | | | | | xy= oa2rsin 2 cos 2 cos32 + similar expressions for displacements u [Equations for the polar stresses as a function of r and are in the data-book.] All the equations separate into a geometrical factor and the stress intensity factor:

! K = "o#a Kdetermines the amplitude of the additional stress due to the crack over the whole specimen, but particularly at the crack tip where growth has to occur. When = 0 the stress opening the crack has the value : yy= oa2r=K2r The value of K at which fracture occurs is the material-dependant Fracture Toughness:

KIc= fa For a fixed stress this defines the maximum stable crack length or for a fixed crack length the maximum stress. r Part II Materials C15 Lent 2012 11 You have come across K in 1A and 1B:Be careful, there are a number of parameters K: kt= maxostress concentration factor (dimensionless)

K = astress intensity factor Pa m

KIc= fa critical stress intensity factor Pa m or Fracture Toughness The equations indicate an infinite stress at the crack tip when r = 0.This is not a problem as the stored elastic energy forms a finite interval.A small volume at the crack tip will be above the yield stress and thus in a plastic state. OTHER MODES OF FAILURE PRINCIPLE OF SUPERPOSITION The above equations considered only a stress normal to the crack surface but much more complex states of stress will exist at cracks.These can be resolved in to three distinct crack opening modes, termed with extraordinary imagination, modes I II and III. Combinations of these can describe any state of stress and the stresses are additive as they remain elastic.For example the mode II stress equations include the factor: = KII2r , for any location around the crack tip since the stresses are additive, the values of K from the separate crack modes are also additive. Crack opening modes I, II and III. The energy release rate is given by integrating stress strain with respect to r, and has the value: G =K2EFor plain stress,or G =K2E(12) for plane strain. Hence, because the values of K for each opening mode can be assessed independently and then added, it is possible to assess complex multimode cracking modes. The total change in energy in the body as a whole can be expressed directly in terms of the individual stress intensities which characterise the crack tip stress and displacement fields.The total energy release rate is given by the expression: Part II Materials C15 Lent 2012 12 ! EG = KI2+KII2+(1+")KIII2 For plane stress or: ! EG = (1"#2)KI2+(1"#2)KII2+(1+#)KIII2For plane strain. Note: These equations do not include the background stress which must be added. !ys !o K dominated Overall stress r Plastic zone Diagram showing the net stress resulting from the remote stress and the stress intensity .For o