Applied Fracture Mechanics

Transcription

1 TALAT Lecture 403 Applied Fracture Mechanics 49 pages and 40 figures Advanced Level prepared by Dimitris Kosteas, Technische Universität München Objectives: Teach the principles and concepts of fracture mechanics as well as provide recommendations for practical applications Provide necessary information for fatigue life estimations on the basis of fracture mechanics as a complementary method to the S-N concept Prerequisites: Background in engineering, materials and fatigue required Date of Issue: 1994 EAA - European Aluminium Association

2 403 Applied Fracture Mechanics Contents Historical Context... 3 Fracture and Fatigue in Structures Notch Toughness and Brittle Fracture... 5 Notch-Toughness Performance Level as a Function of Temperature and Loading Rates...5 Brittle Fracture Principles of Fracture Mechanics... 9 Basic Parameters...9 Material Toughness... 9 Crack Size... 9 Stress Level... 9 Fracture Criteria...13 Members with Cracks...14 Stress Intensity Factors...16 Deformation at the Crack Tip...1 Superposition of Stress Intensity Factors Experimental Determination of Limit Ves according to Various Recommendations... 3 Linear-Elastic Fracture Mechanics...4 Experimental Determination of K Ic - ASTM-E Test procedure:... 5 Elastic-Plastic Fracture Mechanics...7 Crack opening displacement (COD) - BS Determination of R-Curves - ASTM-E Determination of J Ic - ASTM-E Determination of J-R Curves - ASTM-E Crack Opening Displacement (COD) Measurements - BS Fracture Mechanics Instruments for Structural Detail Evation Free Surface Correction F s...37 Crack Shape Correction F e...37 Finite Plate Dimension Correction F w...38 Correction Factors for Stress Gradient F g...38 Remarks on Crack Geometry Calculation of a Practical Example: Evation of Cracks Forming at a Welded Coverplate and a Web Stiffener Coverplate...4 Web Stiffener Literature/References List of Figures TALAT 403

3 Historical Context Fracture and fatigue in structures In his first treatise on "Mathematical Theory of Elasticity" Love, 100 years ago, discussed several topics of engineering importance for which linear elastic treatment appeared inadequate. One of these was rupture. Nowadays structural materials have been improved with a corresponding decrease in the size of safety factors and the principles of modern fracture mechanics have been developed, mainly in the 1946 to 1966 period. Fracture mechanics is the science studying the behaviour of progressive crack extension in structures. This goes along with the recognition that real structures contain discontinuities. Fracture mechanics is the primary tool (characteristic material ves, test procedures, failure analysis procedures) in controlling brittle fracture and fatigue failures in structures. The desire for increased safety and reliability of structures, after some spectacular failures, has led to the development of various fracture criteria. Fracture criteria and fracture control are a function of engineering contemplation taking into account economical and practical aspects as well. Fracture and Fatigue in Structures Brittle fracture is a type of catastrophic failure that usually occurs without prior plastic deformation and at extremely high speeds. Brittle fractures are not so common as fatigue (the latter characterised by progressive crack development), yielding, or buckling failures, but when they occur they may be more costly in terms of human life and property damage. Fatigue failures according to statistics is responsible for approx. 7% of failures. Aristotle talked about hooks on molecules, breaking them meant fracture. Da Vinci and Gallileo talked about fracture, too. The big break in fracture mechanics came in 190 with the Griffith theory, applicable mostly to brittle materials, as well as Orowan and Irwing and Williams in the 1940's. Catastrophic brittle failures were recorded in the 19 th and early 0 th century. There were several failures in welded Vierendeel-truss bridges in Europe shortly after being put into service before World War II. However, it was not until the large number of World War II ship failures that the problem of brittle fracture was fully appreciated by engineers. 196 the Kings Bridge in Melbourne failed by brittle fracture at low temperatures due to poor details and fabrication resulting in cracks which were nearly through the flange prior to any service loading. Although this failure was studied extensively, bridge-builders did not pay particular attention until the failure of the Point Pleasent Bridge in West Virginia, USA on December 15, This was the turning point initiating the possibilities of fracture mechanics in civil engineering. This failure was unique in several ways, it was investigated extensively and its results were characteristic for the procedures and possibilities of fracture mechanics analysis. Therefore they are mentioned briefly here: TALAT 403 3

4 (a) (b) (c) (d) fracture appeared in the eye of an eyebar caused by the growth of a flaw to a critical size under normal working stresses, the initial flaw was caused through stress-corrosion cracking from the surface of the hole, hydrogen sulphide was probably the reagent responsible, the chemical composition and heat treatment of the eyebar produced a steel with very low fracture toughness at the failure temperature, and fracture resulted from a combination of factors and it would not have occurred in the absence of anyone of these - the high hardness of the material made it susceptible to stress corrosion cracking - close spacing of joint components made it impossible to apply paint to high stressed regions yet provided a crevice where water could collect - the high design load of the eyebar resulted in high local stresses at the inside of the eye greater than the yield strength of the steel. - the low fracture toughness of the steel led to complete fracture from the slowly propagating stress corrosion crack when it had reached a depth of only 3.0 mm It has been shown that an interrelation exists between material, design, fabrication and loading as well as maintenance. Fractures cannot be eliminated in structures by merely using materials with improved notch toughness. The designer still has the fundamental responsibility for the overall safety and reliability of the structure. TALAT 403 4

5 403.0 Notch Toughness and Brittle Fracture Notch-toughness performance level as a function of temperature and loading rates Brittle fracture In the following chapters it will be shown how fracture mechanics can be used to describe quantitatively the trade-offs among stress, material toughness, and flaw size so that the designer can determine the relative importance of each of them during design rather than during failure analysis. Notch-Toughness Performance Level as a Function of Temperature and Loading Rates The traditional mechanical property tests measure strength, ductility, modulus of elasticity etc. There are also tests available to measure some form of notch toughness. Notch toughness is defined as the ability of a material to absorb energy (usually when loaded dynamically) in the presence of a flaw. Toughness is defined as the ability of a smooth, unnotched member to absorb energy (usually when loaded slowly). Notch toughness is measured with a variety of specimens such as the Charpy-V-notch impact specimen, dynamic tear test specimen K Ic, pre-cracked Charpy, etc. Toughness is usually characterised by the area under a stress-strain curve in a slow tension test. Notches or other forms of stress raisers make structural materials susceptible to brittle fracture under certain conditions. The ductile or brittle behaviour of some structural materials like steels is well known, depending on several conditions such as temperature, loading rate, and constraint (the latter arising often in welded components among other reasons due to residual stresses and the complexity of welds). Ductile fractures are generally preceded by large amount of plastic deformation occurring usually at 45 to the direction of the applied stress. Brittle or cleavage fractures generally occur with little plastic deformation and are usually normal to the direction of principal stresses. Figure shows the various fracture states and the transition from one to another depending on environmental conditions. Plane-strain behaviour refers to fracture under elastic stresses and is essentially brittle. Plastic behaviour refers to ductile failure under general yielding conditions accompanied usually, but not necessarily, with large shear lips. The transition between these two is the elastic-plastic region or the mixed-mode region. Higher loading rates move the characteristic transition curve to higher temperatures. A particular notch toughness ve called the nil-ductility transition (NDT) temperature generally defines the upper limits of plane-strain behaviour under conditions of impact loading. In practice the question has to be answered regarding the level of material performance which should be required for satisfactory performance in a particular structure at a specific service temperature, see Figure In this example and for impact loading the three different steels exhibit either plane-strain behaviour (steel 1) or elastic-plastic behaviour (steel ), or fully plastic behaviour (steel 3) at the indicated service temperature. TALAT 403 5

6 Although fully plastic behaviour would be a very desirable level of performance, it may not be necessary or even economically feasible for many structures. Levels of performance as measured by absorbed energy in notched specimens D. Kosteas, TUM Notch-Thoughness Performance Levels vs. Temperature Plastic Elastic - Plastic Plane Strain (Macro linear) Elastic Intermediate loading rate Temperature Static loading Impact loading NDT (Nil-Ductility Transition) Notch-Thoughness Performance Levels vs. Temperature Service Temperature Plastic Steel 3 Steel Steel 1 Levels of Performance Elastic- Plastic Plane Strain NDT (Steel 3) NDT (Steel ) NDT (Steel 1) Temperature D. Kosteas, TUM Relation between Performance and Transition Temperature for three different Steel Qualities Not all structural materials exhibit a ductile-brittle transition. For example, minium as well as very high strength structural steels or titanium do not undergo a ductile-brittle transition. For these materials temperature has a rather small effect on toughness, see Figure TALAT 403 6

7 D. Kosteas, TUM Notch-Toughness vs. Temperature Notch toughness kpm/ cm² x 0-00 Alloy 5083-H x + x x x x x ISO - Probe longitudinal transverse } parent metal HAZ filler Temperature in C + 0 x 0 Notch-Toughness of Welded Aluminium Alloy 5083 vs. Temperature Notch toughness measurements express the behaviour and respective laboratory test results can be used to predict service performance. Many different tests have been used to measure the notch toughness of structural materials. These include Charpy-V notch (CVN) impact, drop weight NDT dynamic tear (DT) wide plate Battelle drop weight tear test (DWTT) pre-cracked Charpy, etc. Notch toughness tests produce fracture under carefully controlled laboratory conditions. Hopefully, test results can be correlated with service performance to establish curves like in Figure for various materials and specific applications. However, even if correlations are developed for existing structures, they do not necessarily hold for certain designs or new operating conditions or new materials. Test results are expressed in terms of energy, fracture appearance, or deformation, and cannot always be translated to engineering parameters. A much better way to measure notch toughness is with the principles of fracture mechanics, a method characterising the fracture behaviour in structural parameters readily recognised and utilised by the engineer, namely stress and flaw size. Fracture mechanics is based on a stress analysis as described in the next chapters and can account for the effect of temperature and loading rate on the behaviour of structural members that have sharp cracks. Large and complex structures always have discontinuities of some kind. Dolan has made the flat statement that "every structure contains small flaws" whose size and distribution are dependent upon the material and its processing. These may range from non-metallic inclusions and microvoids to weld defects, grinding cracks, quench cracks, surface laps, etc. Fisher and Yen have shown that discontinuities exist in practically all structural members, ranging from below 0.0 mm to several cm long. The significant point is that TALAT 403 7

8 discontinuities are present in fabricated structures even though the structure may have been inspected. The problem of establishing acceptable discontinuities, for instance in welded structures, is becoming an economic problem since techniques that minimize the size and distribution of discontinuities are available if the engineer chooses to use them. Whether a given defect is permissible or not depends on the extent to which the defect increases the risk of failure of the structure. It is quite clear that this will vary with the type of structure, its service conditions, and the material from which it is constructed. The results of a fracture-mechanics analysis for a particular application (specimen size, service temperature, and loading rate) will establish the combinations of stress level and flaw size that would be required to cause fracture. The engineer can then quantitatively establish allowable stress levels and inspection requirements so that fracture cannot occur. Fracture mechanics can also be used to analyse the growth of small cracks, as for example by fatigue loading, to critical size. Fracture mechanics have definite advantages compared with traditional notch-toughness tests. The latter are still useful though, as there are many empirical correlations between fracture-mechanics ves and existing toughness test results. In many cases, because of the current limitations on test requirements for measuring fracture toughness KIc existing notch-toughness tests must be used to help the designer estimate KIc ves. Brittle Fracture The number of catastrophic brittle fractures have been very small generally. Especially for minium, exhibiting rather high fracture toughness ves over the whole temperature regime, probability of failure is low. Nonetheless, when the design becomes complex, thick welded plates from high strength materials are used cost minimisation for the structure becomes more significant the magnitude of loading increases, and actual factors of safety decrease because of more precise computer designs, the possibility of brittle fracture in large complex structures must be considered. TALAT 403 8

9 Principles of Fracture Mechanics Basic parameters Fracture criteria Members with cracks Stress intensity factors Deformation at the crack tip Superposition of stress intensity factors Basic Parameters Numerous factors like service temperature, material toughness, design, welding, residual stresses, fatigue, constraint, etc., can contribute to brittle fractures in large structures. However, there are three primary factors in fracture mechanics that control the susceptibility of a structure to brittle fracture: 1) Material toughness Kc, KIc, KId ) Crack size a 3) Stress level σ Material Toughness is the ability to carry load or deform plastically in the presence of a notch and can be described in terms of the critical stress-intensity factor under conditions of plane stress K c or plane strain K Ic for slow loading and linear elastic behaviour or K Id under conditions of plane strain and impact or dynamic loading, also for linear elastic behaviour. For elastic-plastic behaviour, i.e. materials with higher levels of notch toughness than linear elastic behaviour, the material toughness is measured in terms of parameters such as R-curve resistance, J Ic, and COD as described later on. Crack Size Fracture initiates from discontinuities or flaws. These can vary from extremely small cracks within a weld arc strike to much larger or fatigue cracks, or imperfections of welded structures like porosity, lack of fusion, lack of penetration, toe or root cracks, mismatch, overfill angle, etc. Such discontinuities, though even small initially, can grow by fatigue or stress corrosion to a critical size. Stress Level Tensile stresses (nominal, residual, or both) are necessary for brittle fractures to occur. They are determined by conventional stress analysis techniques for particular structures. Further factors such as temperature, loading rate, stress concentrations, residual stresses, etc., merely affect the above three primary factors. Engineers have known these facts for many years and controlled the above factors qualitatively through good design (adequate sections, minimum stress concentrations) and fabrication practices (decreased imperfection or discontinuity size through proper welding and inspection), as well as the use of materials with sufficient notch toughness levels. TALAT 403 9

10 A linear elastic fracture mechanics technology is based upon an analytical procedure that relates the stress field magnitude and distribution in the vicinity of a crack tip to the nominal stress applied to the structure, to the size, shape, and orientation of the crack, and to the material properties. In Figure are the equations that describe the elastic stress field in the vicinity of a crack tip for tensile stresses normal to the plane of the crack (Mode I deformation). Elastic Stress Field Distribution near a Crack σ y Magnitude of stress along x axis, σy σy σ Nominal stress Crack tip σx r K I θ σy σx θ θ 3 θ σx = cos (1 - sin sin ) ( π r) 1/ σy (θ = 0) x K I θ θ 3 θ σy = cos (1 + sin sin ) ( π r) 1/ Source: Irwin and Williams, 1957 Elastic Stress Field Distribution near a Crack The equations describing the crack tip stress field distribution were formulated by Irwin and Williams (1957) as follows: σ = K I θ θ θ x πr 1 3 cos sin sin σ = K I θ θ θ y πr cos sin sin K θ θ θ τ I 3 xy = sin cos cos πr τ xz = τ yz = 0 σ z = 0 Plane Stress (thin sheet) σ = µ σ + σ Plane Strain (thick sheet) ( ) z x y The distribution of the elastic stress field in the vicinity of the crack tip is invariant in all structural components subjected to this type of deformation. The magnitude of the elastic stress field can be described by a single parameter, K I, designated the stress intensity factor. The applied stress, the crack shape, size, and orientation, and the structural configuration of structural components subjected to this type of deformation affect the ve of the stress intensity factor but do not alter the stress field distribution. This allows to translate laboratory results directly into practical design information. Stress intensity ves for some typical cases are given in Figure TALAT

11 K I ves for various crack geometries a σ σ σ a c a σ σ σ Through thickness crack Surface crack Edge crack KI = σ π a a KI = 11 σ π Q KI = 11 σ π a where Q = f(a/c, σ) K I Ves for Various Crack Geometries It is a principle of fracture mechanics that unstable fracture occurs when the stress intensity factor at the crack tip reaches a critical ve K c. For mode I deformation and for small crack tip plastic deformation, i.e. plane strain conditions, the critical stress intensity factor for fracture for fracture instability is K Ic. The ve K Ic represents the fracture toughness of the material (the resistance to progressive tensile crack extension under plane strain conditions) and has units of MN/m 3/ (or MPa/mm 1/ or ksi in). 1ksi in = N/mm 3/ or MPa/mm 1/ {1ksi = N/mm or MPa} This material toughness property depends on the particular material, loading rate, and constraint: K c critical stress intensity factor for static loading and plane stress conditions of variable constraint. This ve depends on specimen thickness and geometry, as well as on crack size. K Ic critical stress intensity factor for static loading and plane strain conditions of maximum constraint. This ve is a minimum ve for thick plates. K Id critical stress intensity factor for dynamic (impact) loading and plane strain conditions of maximum constraint. TALAT

12 where K, K, K = C σ a c Ic Id C = constant, function of specimen and crack geometry σ = nominal stress a = flaw size Through knowledge of the critical ve at failure for a given material of a particular thickness and at a specific temperature and loading rate, tolerable flaw sizes for a given design stress level can be determined. Or design stress levels may be determined that can be safely used for an existing crack that may be present in a structure, see Figure Relation between material toughness, flaw size and stress σ a Increasing stress, σ! f! 0 Increasing material toughness (COD, J Ic, R) K I = f(!, a) K C σ (Fracture zone) K C of tougher steel K C = critical ve of K I a 0 a f Increasing flaw size, a Relation between Material Toughness, Flaw Size and Stress In an unflawed structural member, as the load is increased the nominal stress increases until an instability (yielding at σ ys ) occurs. Similarly in a structural member with a flaw as the load is increased ( or as the size of the flaw grows by fatigue or stress corrosion) the stress intensity K I increases until an instability, fracture at K Ic, occurs. Another analogy that helps to understand the fundamental aspects of fracture mechanics is the comparison with the Euler column instability, Figure To prevent buckling the actual stress and L/r ves must be below the Euler curve. To prevent fracture the actual stress and flaw size must be below the K c level. TALAT 403 1

13 COLUMN RESEARCH COUNCIL COLUMN STRENGTH CURVE P L! YS EULER CURVE! YIELDING π E σ = c ( L/ r) P! =! YS L/r (a) COLUMN INSTABILITY! YS K C σ c = C π a! a! YIELDING!! =! YS a (b) CRACK INSTABILITY Source: Madison and Irwin Analogy: Column Instability and Crack Instability The critical stress intensity factor K Ic represents the terminal conditions in the life of a structural component. The total useful life N T of the component is determined by the time necessary to initiate a crack N I and by the time to propagate the crack N P from subcritical dimensions a 0 to the critical size a c. Crack initiation and subcritical crack propagation are localised phenomena that depend on the boundary conditions at the crack tip. Subsequently it is logical to expect that the rate of subcritical crack propagation depends on the stress intensity factor K I which serves as a single term parameter representative of the stress conditions in the vicinity of the crack tip. Fracture mechanics theory can be used to analyse the behaviour of a structure throughout its entire life. For materials that are susceptible to crack growth in a particular environment the K Iscc - ve is used as the failure criterion rather than K Ic. This threshold ve K Iscc is the ve below which subcritical crack propagation does not occur under static loads in specific environment. In the relationship between material toughness, design stress and flaw size, Figure , K Iscc replaces K c as the critical ve of K I. Fracture Criteria A careful study of the particular requirements of a structure is the basis in developing a fracture control plan, i.e. the determination of how much toughness is necessary and adequate. Developing a criterion one should consider service conditions such as temperature, loading, loading rate, etc. the level of performance (plane strain, elastic-plastic, plastic) consequences of failure TALAT

14 Developing a fracture control plan for a complex structure is very difficult. All factors that may contribute to the fracture of a structural detail or failure of the entire structure have to be identified. The contribution of each factor and the combination effect of different factors have to be assessed. Methods minimising the probability of fracture have to be determined. Responsibility has to be assigned for each task that must be undertaken to ensure the safety and reliability of a structure. A fracture control plan can be defined for a given application and cannot be extended indiscriminately to other applications. Certain general guidelines pertaining to classes of structures (such as bridges, ships, vehicles, pressure vessels, etc.) can be formulated. The fact that crack initiation, crack propagation, and fracture toughness are functions of the stress intensity fluctuation KI and of the critical stress intensity factor KIc (whereby the stress intensity is related to the applied nominal stress or stress fluctuation) demonstrates that a fracture control plan depends on fracture toughness KIc or Kc of the material at the temperature and loading rate of the application. The fracture toughness can be modified by changing the material. the applied stress, stress rate, stress concentration, and stress fluctuation. They can be altered by design changes and fabrication. the initial size of the discontinuity and the size and shape of the critical crack. These can be controlled by design changes, fabrication, and inspection. Members with Cracks Failure of structural members is caused by the propagation of cracks, especially in fatigue. Therefore an understanding of the magnitude and distribution of the stress field in the vicinity of the crack front is essential to determine the safety and reliability of structures. Because fracture mechanics is based on a stress analysis, a quantitative evation of the safety and reliability of a structure is possible. Fracture mechanics can be subdivided into two general categories namely linear-elastic and elastic-plastic. The following relationships and equations for stress intensity factors are based on linear elastic fracture mechanics (LEFM). It is convenient to define three types of relative movements of two crack surfaces. These displacement modes, Figure , represent the local deformation in an infinitesimal element containing a crack form. Figure shows the coordinate system and stress components ahead of a crack tip TALAT

15 Basic cracking modes y x z Mode I y x z Mode II y x z Mode III Basic Modes of Cracking σ y τ xy y τ yz σ x r θ σ z τ xz x Leading edge of the crack z Coordinate System and Stress Components ahead of a Crack Tip Mode I is characterized by displacement of the two fracture surfaces perpendicular to each other in opposite directions. Local displacements in the sliding or shear Mode II and Mode III, the tearing mode, are the other basic types corresponding to respective stress fields in the vicinity of the crack tips, Figure In any problem the deformations can be treated as one or combination of these local displacement modes. Respectively the stress field at the crack tip can be treated as one or a combination of the three basic types of stress fields. For practical applications Mode I is the most important since according to Jaccard even if a crack starts as a combination of different modes it is soon transformed and continues its propagation to a critical crack under Mode I. TALAT

16 Stress Field Equations in the Vicinity of Crack Tips Mode I Mode II Mode III σ = x K I θ θ θ π r 1 3 cos sin sin σ = x K II θ θ θ + π r 3 sin cos cos τ xz = K III θ sin π r σ = y K I θ θ θ + π r 1 3 cos sin sin σ = y K II θ θ 3θ sin cos cos π r τ yz = K III θ cos π r τ τ xy xz = K I θ θ 3θ sin cos cos π r = τ =0 yz KI r θ u = G + 1 θ cos ν sin π 1 KI r θ v = G θ sin ν cos π w = 0 ( ) σ = ν σ + σ z x y 1 τ xy = K II θ θ θ π r 1 3 cos sin sin ( ) σ = ν σ + σ τ z x y xz = τ =0 yz KII r θ u = G + θ sin ν cos π 1 KII r θ θ v = G cos 1+ ν sin π w = 0 1 σ x = σ y = σ z = τ xy = 0 w K III r θ = G π sin u= v =0 1 Stress Field Equations in the Vicinity of Crack Tips Dimensional analysis of the equations shows that the stress intensity factor must be linearly related to stress and directly related to the square root of a characteristic length, the crack length in a structural member. In all cases the general form the stress intensity factor is given by K = f ( g) σ a where f(g) is a parameter that depends on the specimen and crack geometry. Stress Intensity Factors Various relationships between the stress intensity factor and structural component configurations, crack sizes, orientations, and shapes, and loading conditions can be taken out of respective literature [1] C.P. Paris and G.C. Sih, "Stress analysis of cracks" in "Fracture toughness testing and its applications", ASTM STP No381, ASTM, Philadelphia 1965 [] H. Tada, P.C. Paris and G.R. Irwing, ed., "Stress analysis of cracks handbook", Del Research corporation, Hellertown, Pa [3] G.C. Sih, "Handbook of stress intensity factors for researchers and engi-neers", Institute of fracture and solid mechanics, Lehigh University, Bethlehem, Pa Stress intensity factor for a through-thickness crack:figure TALAT

17 Through-Thickness Crack Finite Width Plate σ a b Tangent correction for finite width a [ sec (( π 1 a) / ( b) )] b Infinite Width Plate σ a σ Nominal Stress σ [ sec (( π )/ ( ))] KI = σ π a a b 1 K = σ π a SIF for Through-Thickness Crack Stress intensity factor for a double-edge crack: Figure Double-Edge Crack σ a b a Tangent correction for finite width a [ sec (( π 1 a) / ( b) )] b σ KI = 11. σ π a sec π a / b [ (( ) ( ))] 1 SIF for Double-Edge Crack TALAT

18 Stress intensity factor for a single-edge crack: Figure Single-Edge Crack σ b σ a For a semi-finite edge-cracked specimen: K I =1.1 σ(πa) 1/ For a finite width edge-cracked specimen: K I = σ(πa) ½ f(a/b) Correction factor for a single-edge crack in a finite width plate a/b f(a/b) SIF for Single-Edge Crack Stress intensity factor for cracks emanating from circular or elliptical holes: Figure Cracks Emanating from Circular or Elliptical Holes 0.9 F ( λ, δ) b N! a N a ρ a F The stress intensity factor in the case of a finite plate is K F( ) = λ, δ σ π a I where λ= a a N and = b N δ a N 0.1! For a circular hole δ = λ = a a N SIF for Cracks Emanating from Circular or Elliptical Holes TALAT

19 Stress intensity factor for a single-edge crack in beam in bending: Figure Single-Edge Crack in Beam in Bending a M w M K I = 6M ( W a) 3 a f W W: Depth of the Beam Correction factor for notched beams a/w f(a/w) > SIF for Single-Edge Crack in Beam in Bending Stress intensity factor for an elliptical or circular crack in an infinite plate: Figure Elliptical or Circular Crack in an Infinite Plate The stress intensity factor at any point along the perimeter of elliptical or circular cracks in an infinite body subjected to uniform tensile stress is! ( a) 3 σπ a KI = + sin β θ b cos β a a The point on the perimeter of the crack is defined by the angle ß and the elliptic integral # o c c " π c a θ 0 = 1 sin θ dθ c 0 3 For circular cracks, c = a K = σ I ( a π) 1! When c and β = π, this case is reduced to a through - thickness crack. SIF: Elliptical or Circular Crack in an Infinite Plate TALAT

20 Stress intensity factor for a surface crack: Figure ! Surface Crack "Thumbnail Crack" c! a The stress intensity factor can be calculated from the equations of elliptical crack using a free surface correction factor of 1.1 and for the position ß=π/ with 1 K I = 11. σ π a Q Q= θ 0 and the elliptic integral π c a θ 0 = 1 θ θ sin d c 0 Q is regarded as a shape factor because its ves depend on a and c. For ves of Q see Figure SIF: Surface Crack Flaw shape parameter: Figure Flaw Shape Parameter, Q 0.5 c a Ratio a/c σ σ ys = 0 σ σ ys = 060. σ σ ys = 080. σ σ ys = Flaw Shape Parameter, Q Flaw Shape Parameter, Q TALAT 403 0

21 Deformation at the Crack Tip The stress field equations show that the elastic stress at a distance r from the crack tip (where r<<a) can be very large, see Figure In reality the material in this region deforms plastically. A plastic zone surrounds the crack tip. Elastic Stress Distribution! YS Stress Distribution after Local Yielding! x Crack Tip r Y Plastic Zone Distribution of Stress in the Crack Tip Region The size of the plastic zone r y can be estimated from the stress field equations for plane stress conditions and setting σ y = σ ys 1 K ry = π σ ys Following a suggestion by Irwin that the increase in tensile stress for plastic yielding caused by plane strain elastic constraint is of the order of 3, we can estimate the size of the plastic zone under plane strain conditions as r y 1 K = 6π σ ys The plastic zone along the crack front in a thick specimen is subjected to plane strain conditions in the center portion of the crack front where w = 0 and to plane strain condition near the surface of the specimen where σ z = 0. That means that the plastic zone in the center of a thick specimen is smaller than at the surface of the specimen, Figure TALAT 403 1

22 A - Overall view Midsection B - Edge view Plane stress mode I Surface Crack tip y z x Cracktip Plane strain mode I K I ² $! y ² Machine notch Fatigue crack Plastic zone r Ip (plane strain) z r p(plane stress) Specimen cross-section Plane strain region Plastic Zone Dimensions Superposition of Stress Intensity Factors Stress components from such loads as uniform tensile loads, concentrated tensile loads, or bending loads, all belonging to Mode I type loads, have the same stress field distributions in the vicinity of the crack tip according to the equations given in Figure Consequently the total stress intensity factor can be obtained by algebraically adding the individual stress intensity factors corresponding to each load. TALAT 403

23 Experimental Determination of Limit Ves according to Various Recommendations Linear-Elastic Fracture Mechanics Experimental determination of K Ic - ASTM-E399 Elastic-Plastic Fracture Mechanics Crack opening displacement (COD) - BS 576 Determination of R-Curves - ASTM-E561 Determination of J Ic - ASTM-E813 Determination of J-R Curves - ASTM-E115 Crack opening displacement (COD) measurements - BS 576 Structural materials have certain limiting characteristics, yielding in ductile materials or fracture in brittle materials. The yield strength σys is the limiting ve for loading stresses, the critical stress intensity factors KIc, KId or Kc are the limiting ves for the stress intensity factor KI. The critical stress intensity factor K c at which unstable crack growth occurs for conditions of static loading at a particular temperature depends on specimen thickness or constraint, Figure Effect of Thickness on K c Plane stress Plane strain Kc, ksi in K Ic Thickness at the beginning of plane strain is: t=.5 K Ic σ y Thickness, in Effect of Thickness on K c The limiting ve of K c for plane strain (maximum constraint) conditions is K Ic (slow loading rate) or K Id (dynamic or impact load). The K Ic -ve is the minimum ve for plane strain conditions. The K Ic -ve would also be a minimum ve for conditions of maximum structural constraint (for example, stiffeners, intersecting plates, etc.) that TALAT 403 3

24 might lead to plain-strain conditions even though the individual structural members might be relatively thin. Design procedures based on the KIc-ve provide a "conservative" approach to the fracture problem. The following chapter gives information on experimental requirements and procedures for the measurement of KIc-ves. For thin section problems further procedures based on a Kc or R-curve analysis for elastic-plastic behaviour problems an analysis on the basis of JIc- or COD-procedures will be described in further chapters. Linear-Elastic Fracture Mechanics Experimental Determination of KIc - ASTM-E399 The accuracy with which KIc describes the fracture behaviour of real materials depends on how well the stress intensity factor represents the conditions of stress and strain inside the actual fracture process zone. This is the extremely small region just ahead of the tip of a crack where crack extension would originate. In this sense K I is exact only in the case of zero plastic strain as in brittle materials. For most structural materials, a sufficient degree of accuracy may be obtained if the plastic zone ahead of a crack tip is small in comparison with the region around the crack in which the stress intensity factor yields a satisfactory approximation of the exact elastic stress field. The decision of what is sufficient accuracy depends on the particular application. A standardised test method must be reproducible, specimen size requirements are chosen so that there is essentially no question regarding this point. The standardised test method for determining K Ic material ves is the ASTM-E399 test method..1w.1w a W B =W/ SE(B) Bend Specimen Compact Tension Specimen 0.5 W Dia W 0.6 W 0.75 W 0.6 W after ASTM-E399 a W 1.5 W B =W/ SE Bend Specimen and CT Tension Specimen TALAT 403 4

25 Several different test specimen forms have been proposed in the course of the development and standardization. Two of the most common for engineering applications, the bend specimen or SE(B) specimen and the compact tension or C(T) specimen are reproduced in Figure according to ASTM-E399. Test procedure: Determine location and orientation of test specimen in respect to component to be analysed: Specimen orientation L-S and L-T cover the most common crack cases in structural engineering components, such as welded structures, see Figure Surface cracks grow initially in the direction of the plate thickness and propagate further on as through-thickness cracks. according to ASTM Specimen Orientation Determine critical specimen size dimension: To fulfil requirements of maximum constraint and small plastic zone in relation to specimen dimensions the following relations must be observed. a = crack depth >.5 (K Ic /σ ys ) B = specimen thickness >.5 (K Ic /σ ys ) W = specimen depth > 5.0 (K Ic /σ ys ) This leads for instance to a specimen thickness of approximately 50 times the radius of the plane strain plastic zone. Even before a K Ic test specimen can be machined, the K Ic ve to be obtained must already be known or at least estimated. Three general rules may be used TALAT 403 5

26 overestimate the K Ic ve on the basis of experience with similar materials and judgement based on other types of notch-toughness tests use specimens that have as large as thickness as possible, namely a thickness equal to that of the plates to be used in service use the following ratio of yield strength to modulus of elasticity to select a specimen size. These estimates are valid for very high strength structural materials, steels having yield strength of at least 1000 MPa and minium alloys having yield strength of at least 350 MPa. σ ys /E Minimum recommended thickness and crack length [mm] , , , , , , , , , or greater 6,5 Many low- to medium-strength structural materials in section sizes of interest for most large structures (ships, bridges, pressure vessels) are of insufficient thickness to maintain plane strain conditions under slow loading and at normal service temperatures. Thus, the linear elastic analysis to calculate K Ic ves is invalidated by general yielding and the formation of large plastic zones. Alternative methods must be used for fracture analysis as described in further chapters. The following ves are given as an example for common minium alloys K I σ 0, K I /σ 0,.5(K Ic /σ 0, ) MPa MPa AlMgSi , AlZn4,5Mg , Select and prepare a test specimen: Most probably one of the two standard specimen shapes will be selected, slow-bend test specimen or compact-tension specimen. The initial machined crack length 'a' should be 0.45 W so that the crack can be extended by fatigue to approximately 0.5 W. Usually the selection of the specimen thickness B is made first. Perform test following requirements of ASTM-E399 procedure: This includes initial fatigue cracking of the test specimen. Measure and plot crack opening displacement v against load P. TALAT 403 6

27 Analyse P- v record, calculate conditional KIc (=KQ) ves, perform validation check for K Ic : If the KQ ves meet the above stated requirements, like a or B.5(KQ/σ 0. ) and W 5.0(KQ/σ 0. ), the KQ=KIc. If not the test is invalid, the results may be used to estimate the material toughness only. Elastic-Plastic Fracture Mechanics Elastic-plastic fracture mechanics analysis is an extension of the linear elastic analysis. As already mentioned low- to medium-strength structural materials used in the section sizes of interest for large complex structures are of insufficient thickness to maintain plane strain conditions under slow loading conditions at normal service temperatures. Large plastic zones from ahead of the crack tip, the behaviour is elastic-plastic, invalidating the calculation of KIc ves. There are three possible approaches into the elasticplastic region, through crack opening displacement (COD) R-curve analysis J-integral Crack opening displacement (COD) - BS 576 Proposed by Wells in 1961 the fracture behaviour in the vicinity of a sharp crack could be characterized by the opening of the notch faces, namely the crack opening displacement. He also showed that the concept of crack opening displacement was analogous to the concept of critical crack extension force and thus the COD ves could be related to the plane strain fracture toughness K Ic. COD measurements can be made even when there is considerable plastic flow ahead of a crack. Using a crack tip plasticity model proposed by Dugdale it is possible to relate the COD to the applied stress and crack length. As with the K I analysis the application of the COD approach to engineering structures requires the measurement of a fracture toughness parameter δ c, the critical ve of the crack tip displacement, which is a material property as a function of temperature, loading rate, specimen thickness, and possibly specimen geometry, i.e. notch acuity, crack length and overall specimen size. Since the δ c -test is regarded as an extension of the K Ic testing the british standardized test method after BS 576 is very similar to the ASTM-E399 test method for K Ic. Similar specimen preparation, fatigue-cracking procedures, instrumentation, and test procedures are followed. The displacement gage is similar to the one used in K Ic testing (Clip-Gage) and a continous load-displacement record is obtained during the test. On the basis of the British Standard PD 6493 an analysis results is based on the comparison of the critical COD ve δ c to the actual crack tip opening displacement of the component analyzed and characterized by geometrical dimensions of the component TALAT 403 7

28 and the existing flaw and its location, as well as the material used - for a specific service temperature and loading rate. Determination of R-Curves - ASTM-E561 KIc is governed by conditions of plane strain (ε z =0) with small scale crack tip plasticity. Kc is governed by conditions of plane stress (σ z =0) with large scale crack tip plasticity. Kc ves are generally -10 times larger than KIc. KIc ves depend on only two variables, temperature and strain rate. Kc ves depend on 4 variables, temperature, loading rate, plate thickness and initial crack length. Plane stress conditions rather than plane strain conditions actually exist in service. Plane stress fracture toughness evations using an R-curve or resistance curve analysis as one of several extensions of linear elastic fracture mechanics into elastic-plastic fracture mechanics is envisaged. An R-curve characterizes the resistance to fracture of a material during incremental slow stable crack extension. An R-curve is a plot of crack growth resistance as a function of actual or effective crack extension. KR, also in MPa m units is the crack growth resistance at a particular instability condition during the R-curve test, i.e. the limit prior to unstable crack growth. In Figure the solid lines represent the R-curves for different initial crack lengths. The dashed lines represent the variation in K I with crack length for different constant loads P1<P<P3. Each line is a function of the crack length, K I = f(p, a). R-Curves K R = Kc for a 0 = a 1 P 3 P 1 <P <P 3 K R KI = f ( P, a) P Applied K I Levels K R = K c for a 0 = a K R < K c for a 0 = a 1 a actual a 1 a a a actual R-Curves and Critical Fracture Toughness Ves K R for Different Initial Cracks a The two points of tangency represent points of instability, or the critical plane stress intensity factor K c = K R, at the particular crack length and, of course, for the given conditions of temperature and loading rate. The K R ve is always calculated by using the effective crack length a eff and is plotted against the actual crack extension a act, that takes place physically in the material during the test. TALAT 403 8

29 R-curves can be determined either by load control or displacement control tests. The load control technique can be used to obtain only that portion of the R-curve up to the Kc ve where complete unstable fracture occurs. The displacement control technique can be used to obtain the entire R-curve. The evation of R-curves for relatively lowstrength, high-toughness alloys exhibiting large scale crack tip plasticity σ y at fracture, relative to the test specimen in plane dimensions W and a, requires an elastic-plastic approach. Here the crack-opening displacement δ at the physical crack tip is measured and used in calculating the equivalent elastic K ve. This elastic-plastic crack model is designated the crack-opening-stretch (COS) method, where δ and COS are equivalent terms. This method can be used with either a load-control or displacement control test. A standardized testing procedure is available after ASTM-E561. Generally the thickness of a test specimen is equal to the plate thickness considered for actual service. The other dimensions are made considerably larger. The advantage of the displacement control technique is partially offset by the necessity for new or unique loading facilities and sophisticated instrumentation, whereas the load-control method in conjunction with relatively simple measuring devices can be used with a conventional tension machine. The limits of application of this technique are for materials with high strength with low toughness, small plastic zone ahead of the crack tip. For materials with high levels of toughness this analysis becomes increasingly less accurate. The R-curve is determined by graphical means. A series of secant lines are constructed on the load displacement record from a test sample. The compliance ves δ/p from these second lines are used to determine the associated a eff /W-ves that reflect the effective crack length a eff = a 0 + a + r y using also an appropriate relationship for the given specimen between crack opening and effective crack length. Each a eff /W-ve is then used to determine a respective K eff -ve, the latter is plotted against a eff producing the R-curve for the given material. The significance of the critical fracture toughness ves obtained from an R-curve analysis is in the calculation of the critical flaw size a cr required to cause fracture instability. A normalised plot showing the general relationship of a cr to such design parameters, nominal design stress and yield stress, for a large center-cracked tension specimen subjected to uniform tension is given in Figure The plot can be used for any material for which valid fracture mechanics results (K Ic, K Id, K c, K Iscc ) are available under a given loading rate, temperature and state of stress. TALAT 403 9

30 acr, critical flaw size, inches Normalized Plot for Critical Flaw Size K ( C ) = 0.10 σys a cr a cr σd K I = σd π a a cr = 1 π σd K ( C σd) σd ( σys) Normalized Plot for Critical Flaw Size Determination of J Ic - ASTM-E813 Another means of directly extending fracture mechanics concepts from the linear-elastic behaviour to the elastic-plastic behaviour is the path independent J-integral proposed by Rice as a method of characterizing the stress strain field at the tip of a crack by an integration path taken sufficiently far from the crack tip to be substituted for a path close to the crack tip region. For linear elastic behaviour the J-interal is identical to the energy release rate G per unit crack extension. Therefore a J-failure criterion for the linear elastic case is identical to the K Ic -failure criterion, under linear-elastic plane strain conditions. J Ic = G = Ic ( υ ) 1 K E Ic The energy line integral J is defined for either elastic or elastic-plastic behaviour J = W dy T U x dx R where R=any contour around the crack tip. Figure shows the crack-tip coordinate system and arbitrary line integral contour. Note the counterclockwise evation starting from the lower flat notch surface. TALAT