TALAT Lecture 2403: Applied Fracture Mechanics

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This lecture demonstrates how to teach the principles and concepts of fracture mechanics as well as provide recommendations for practical applications; it provides necessary information for fatigue life estimations on the basis of fracture mechanics as a complementary method to the S-N concept. Background in engineering, materials and fatigue is required.

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TALAT Lecture 2403: Applied Fracture Mechanics

  1. 1. TALAT Lecture 2403 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. 2. 2403 Applied Fracture Mechanics Contents 2403.01 Historical Context ..................................................................................... 3 Fracture and Fatigue in Structures ...........................................................................3 2403.02 Notch Toughness and Brittle Fracture ................................................... 5 Notch-Toughness Performance Level as a Function of Temperature and Loading Rates.........................................................................................................................5 Brittle Fracture .........................................................................................................8 2403.03 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 ................................................................................21 Superposition of Stress Intensity Factors...............................................................22 2403.04 Experimental Determination of Limit Values according to Various Recommendations ................................................................................................... 23 Linear-Elastic Fracture Mechanics ........................................................................24 Experimental Determination of KIc - ASTM-E399........................................... 24 Test procedure: ................................................................................................. 25 Elastic-Plastic Fracture Mechanics ........................................................................27 Crack opening displacement (COD) - BS 5762 ................................................ 27 Determination of R-Curves - ASTM-E561........................................................ 28 Determination of JIc - ASTM-E813 .................................................................. 30 Determination of J-R Curves - ASTM-E1152 .................................................. 33 Crack Opening Displacement (COD) Measurements - BS 5762 ...................... 34 2403.05 Fracture Mechanics Instruments for Structural Detail Evaluation... 36 Free Surface Correction Fs ....................................................................................37 Crack Shape Correction Fe ....................................................................................37 Finite Plate Dimension Correction Fw ..................................................................38 Correction Factors for Stress Gradient Fg .............................................................38 Remarks on Crack Geometry.................................................................................39 2403.06 Calculation of a Practical Example: Evaluation of Cracks Forming at a Welded Coverplate and a Web Stiffener .............................................................. 41 Coverplate ..............................................................................................................42 Web Stiffener .........................................................................................................43 2403.07 Literature/References ............................................................................... 47 2403.08 List of Figures.......................................................................................... 48 TALAT 2403 2
  3. 3. 2403.01 Historical Context • Fracture and fatigue in structures In his first treatise on "Mathematical Theory of Elasticity" Love, 100 years ago, discus- sed several topics of engineering importance for which linear elastic treatment appeared inadequate. One of these was rupture. Nowadays structural materials have been im- proved 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 disconti- nuities. Fracture mechanics is the primary tool (characteristic material values, test procedures, failure analysis procedures) in controlling brittle fracture and fatigue failures in struc- tures. 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 econo- mical 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 prop- erty 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 1920 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 19th and early 20th 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. 1962 the Kings Bridge in Melbourne failed by brittle fracture at low tempera- tures 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, 1965. This was the turning point initia- ting 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 2403 3
  4. 4. (a) fracture appeared in the eye of an eyebar caused by the growth of a flaw to a criti- cal size under normal working stresses, (b) the initial flaw was caused through stress-corrosion cracking from the surface of the hole, hydrogen sulphide was probably the reagent responsible, (c) the chemical composition and heat treatment of the eyebar produced a steel with very low fracture toughness at the failure temperature, and (d) 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 2403 4
  5. 5. 2403.02 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 de- scribe 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 elasti- city 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 KIc, 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 2403.02.01 shows the var- ious 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 value 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 re- quired for satisfactory performance in a particular structure at a specific service tem- perature, see Figure 2403.02.02. In this example and for impact loading the three different steels exhibit either plane-strain behaviour (steel 1) or elastic-plastic behaviour (steel 2), or fully plastic behaviour (steel 3) at the indicated service temperature. TALAT 2403 5
  6. 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. Notch-Thoughness Performance Levels vs. Temperature Impact loading by absorbed energy in notched specimens Plastic Static loading Levels of performance as measured Elastic - Intermediate Plastic loading rate NDT (Nil-Ductility Transition) Plane Strain (Macro linear) Elastic Temperature D. Kosteas, TUM alu Notch-Thoughness Performance Levels 2403.02.01 Training in Aluminium Application Technologies vs. Temperature Service Temperature Plastic Steel 3 Steel 2 Steel 1 Levels of Performance Elastic- Plastic NDT NDT Plane NDT (Steel 2) (Steel 3) (Steel 1) Strain Temperature D. Kosteas, TUM alu Relation between Performance and Training in Aluminium Application Technologies Transition Temperature for three different Steel Qualities 2403.02.02 Not all structural materials exhibit a ductile-brittle transition. For example, aluminium 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 2403.02.03. TALAT 2403 6
  7. 7. Notch-Toughness vs. Temperature 6 Alloy 5083-H113 + + + + + Notch toughness kpm/ cm² + 4 x x x x x 2 x ISO - Probe x longitudinal } transverse parent metal + HAZ x filler 0 -200 -100 0 20 D. Kosteas, TUM Temperature in °C alu Notch-Toughness of Welded Aluminium Alloy 5083 Training in Aluminium Application Technologies vs. Temperature 2403.02.03 Notch toughness measurements express the behaviour and respective laboratory test re- sults 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 2403.02.01 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 mech- anics, 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 ef- fect 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 0.02 mm to several cm long. The significant point is that TALAT 2403 7
  8. 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 values and existing toughness test results. In many cases, because of the current limitations on test requirements for measuring fracture toughness KIc exist- ing notch-toughness tests must be used to help the designer estimate KIc values. Brittle Fracture The number of catastrophic brittle fractures have been very small generally. Especially for aluminium, exhibiting rather high fracture toughness values 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 2403 8
  9. 9. 2403.03 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 2) 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 Kc or plane strain KIc for slow loading and linear elastic behaviour or KId under con- ditions 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, JIc, 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 2403 9
  10. 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 2403.03.01 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 σx σx Crack σy (θ = 0) σ r tip σy Nominal θ stress x KI θ θ 3θ σx = cos (1 - sin sin ) (2 π r)1/2 2 2 2 KI θ θ 3θ σy = cos (1 + sin sin ) (2 π r)1/2 2 2 2 Source: Irwin and Williams, 1957 alu Training in Aluminium Application Technologies Elastic Stress Field Distribution near a Crack 2403.03.01 The equations describing the crack tip stress field distribution were formulated by Irwin and Williams (1957) as follows: KI θ θ 3θ  σx = cos  1 − sin sin  2πr 2 2 2 KI θ θ 3θ  σy = cos  1 + sin sin  2πr 2 2 2 KI θ θ 3θ τ xy = sin cos cos 2 πr 2 2 2 τ xz = τ yz = 0 σz =0 Plane Stress (thin sheet) σ z = µ ⋅ (σ x + σ y ) Plane Strain (thick sheet) 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, KI, designated the stress in- tensity factor. The applied stress, the crack shape, size, and orientation, and the struc- tural configuration of structural components subjected to this type of deformation affect the value 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 in- tensity values for some typical cases are given in Figure 2403.03.02. TALAT 2403 10
  11. 11. KI values for various crack geometries a σ σ σ 2a 2c a σ σ σ Through thickness crack Surface crack Edge crack KI = σ ⋅ π ⋅ a a KI = 112 ⋅ σ ⋅ π ⋅ a KI = 112 ⋅ σ ⋅ π ⋅ Q where Q = f(a/2c, σ) alu Training in Aluminium Application Technologies KI Values for Various Crack Geometries 2403.03.02 It is a principle of fracture mechanics that unstable fracture occurs when the stress in- tensity factor at the crack tip reaches a critical value Kc. For mode I deformation and for small crack tip plastic deformation, i.e. plane strain conditions, the critical stress in- tensity factor for fracture for fracture instability is KIc. The value KIc represents the fracture toughness of the material (the resistance to progressive tensile crack extension under plane strain conditions) and has units of MN/m3/2 (or MPa/mm1/2 or ksi√in). 1ksi√in = 34.7597 N/mm3/2 or MPa/mm1/2 {1ksi = 6.89714 N/mm2 or MPa} This material toughness property depends on the particular material, loading rate, and constraint: Kc critical stress intensity factor for static loading and plane stress conditions of variable constraint. This value depends on specimen thickness and geometry, as well as on crack size. KIc critical stress intensity factor for static loading and plane strain conditions of maximum constraint. This value is a minimum value for thick plates. KId critical stress intensity factor for dynamic (impact) loading and plane strain conditions of maximum constraint. TALAT 2403 11
  12. 12. where K c , K Ic , K Id = C ⋅ σ ⋅ a C = constant, function of specimen and crack geometry σ = nominal stress a = flaw size Through knowledge of the critical value 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 2403.03.03. Relation between material toughness, flaw size and stress σ 2a Increasing material toughness σ (COD, JIc, R) Increasing stress, σ (Fracture zone) KC KC of !f tough er ste el !0 KI = f(! , a) KC = critical value of KI a0 af Increasing flaw size, 2a alu Relation between Material Toughness, Flaw Training in Aluminium Application Technologies Size and Stress 2403.03.03 In an unflawed structural member, as the load is increased the nominal stress increases until an instability (yielding at σys) 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 KI increases until an instability, fracture at KIc, occurs. Another ana- logy that helps to understand the fundamental aspects of fracture mechanics is the com- parison with the Euler column instability, Figure 2403.03.04. To prevent buckling the actual stress and L/r values must be below the Euler curve. To prevent fracture the act- ual stress and flaw size must be below the Kc level. TALAT 2403 12
  13. 13. COLUMN RESEARCH COUNCIL COLUMN STRENGTH CURVE P L ! YS EULER CURVE P π 2⋅ E σc = ( L / r) 2 ! YIELDING ! = ! YS ! ! YS L/r (a) COLUMN INSTABILITY KC σc = 2a C ⋅ π ⋅a ! ! YIELDING ! = ! YS a (b) CRACK INSTABILITY Source: Madison and Irwin alu Analogy: Column Instability and Crack Instability 2403.03.04 Training in Aluminium Application Technologies The critical stress intensity factor KIc represents the terminal conditions in the life of a structural component. The total useful life NT of the component is determined by the time necessary to initiate a crack NI and by the time to propagate the crack NP from subcritical dimensions a0 to the critical size ac. 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 propaga- tion depends on the stress intensity factor KI which serves as a single term parameter re- presentative 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 KIscc- value is used as the failure criterion rather than KIc. This threshold value KIscc is the value 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 2403.03.03, KIscc replaces Kc as the critical value of KI. 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 2403 13
  14. 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 dif- ferent 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 underta- ken 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 (where- by the stress intensity is related to the applied nominal stress or stress fluctuation) de- monstrates 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 fati- gue. 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 struc- tures. Because fracture mechanics is based on a stress analysis, a quantitative evaluation 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 2403.03.05, represent the local deformation in an infinitesimal element containing a crack form. Figure 2403.03.06 shows the coordinate system and stress components ahead of a crack tip TALAT 2403 14
  15. 15. Basic cracking modes y x Mode I z y x Mode II z y x Mode III z alu Training in Aluminium Application Technologies Basic Modes of Cracking 2403.03.05 σy τ xy y τ yz τ xz σx σz x r θ Leading edge of the crack z Coordinate System and Stress Components ahead alu of a Crack Tip 2403.03.06 Training in Aluminium Application Technologies 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 2403.03.07. 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 2403 15
  16. 16. Stress Field Equations in the Vicinity of Crack Tips Mode I Mode II Mode III KI θ θ 3θ  K II θ θ 3θ  KIII θ σx = cos  1 − sin sin  σx =− sin  2 + cos cos  τ xz = − sin 2π ⋅ r 2 2 2 2π ⋅ r 2 2 2 2π ⋅ r 2 KI θ θ 3θ  KII θ θ 3θ KIII θ σy = cos 1 + sin sin  σy = sin cos cos τ yz = cos 2π ⋅ r 2 2 2 2π ⋅ r 2 2 2 2π ⋅ r 2 KI θ θ 3θ KII θ θ 3θ  τ xy = sin cos cos τ xy = cos  1 − sin sin  2π ⋅ r 2 2 2 2π ⋅ r 2 2 2 σ z = ν (σ x + σ y ) σ z = ν (σ x + σ y ) σ x = σ y = σ z = τ xy = 0 τ xz = τ yz = 0 τ xz = τ yz = 0 θ θ θ θ θ 1 2 1 2 1 2 KI  r  K II  r  K III  r  u= cos 1 − 2ν + sin 2  u= sin  2 − 2ν + cos2  w= sin G  2π    2 2 G  2π    2 2 G  2π    2 θ θ θ θ 1 2 1 2 KI  r  K II  r  v= sin  2 − 2ν − cos2  v= cos  −1 + 2ν − sin 2  G  2π    2 2 G  2π    2 2 w=0 w=0 u=v=0 alu Training in Aluminium Application Technologies Stress Field Equations in the Vicinity of Crack Tips 2403.03.07 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 confi- gurations, 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 [2] H. Tada, P.C. Paris and G.R. Irwing, ed., "Stress analysis of cracks handbook", Del Research corporation, Hellertown, Pa. 1973 [3] G.C. Sih, "Handbook of stress intensity factors for researchers and engi-neers", Institute of fracture and solid mechanics, Lehigh University, Bethlehem, Pa. 1973 Stress intensity factor for a through-thickness crack:Figure 2403.03.08 TALAT 2403 16
  17. 17. Through-Thickness Crack Finite Width Plate Infinite Width Plate σ σ Tangent correction for finite width a b [sec((π ⋅ a) / (2 ⋅ b))] 1 2 0.074 1.00 0.207 1.03 0.275 1.05 2a 0.337 1.08 2a 0.410 1.12 2b 0.466 1.16 0.535 1.22 0.592 1.29 σ Nominal Stress σ [ KI = σ π ⋅ a ⋅ sec( (π ⋅ a) / ( 2 ⋅ b)) ] 1 2 K = σ π ⋅a alu SIF for Through-Thickness Crack 2403.03.08 Training in Aluminium Application Technologies Stress intensity factor for a double-edge crack: Figure 2403.03.09 Double-Edge Crack σ Tangent correction for finite width a b [sec((π ⋅ a) / (2 ⋅ b))] 1 2 0.074 1.00 0.207 1.03 a a 0.275 1.05 0.337 1.08 0.410 1.12 2b 0.466 1.16 0.535 1.22 0.592 1.29 σ . [ K I = 112σ π ⋅ a ⋅ sec( (π ⋅ a ) / ( 2 ⋅ b)) ] 1 2 alu SIF for Double-Edge Crack 2403.03.09 Training in Aluminium Application Technologies TALAT 2403 17
  18. 18. Stress intensity factor for a single-edge crack: Figure 2403.03.10 Single-Edge Crack For a semi-finite edge-cracked specimen: σ KI =1.12 σ(πa)1/2 For a finite width edge-cracked specimen: KI = σ(πa)½ f(a/b) Correction factor for a single-edge crack in a finite width plate a/b f(a/b) a 0.10 1.15 0.20 1.20 2b 0.30 1.29 0.40 1.37 0.50 1.51 0.60 1.68 0.70 1.89 σ 0.80 2.14 0.90 2.46 1.00 2.86 alu SIF for Single-Edge Crack 2403.03.10 Training in Aluminium Application Technologies Stress intensity factor for cracks emanating from circular or elliptical holes: Figure 2403.03.11 1.1 Cracks Emanating from 1.0 Circular or Elliptical Holes 0.9 0.8 ! 0.7 The stress intensity factor 0.6 in the case of a finite plate is F( λ , δ ) 0.5 bN ρ K I = F( λ , δ ) ⋅ σ π ⋅ a 0.4 aN aF  a b  0.3 a where λ =   and δ =  N   aN   aN  0.2 0.1 ! For a circular hole δ = 1.0 0 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40  a λ =   aN  alu Training in Aluminium Application Technologies SIF for Cracks Emanating from Circular or Elliptical Holes 2403.03.11 TALAT 2403 18
  19. 19. Stress intensity factor for a single-edge crack in beam in bending: Figure 2403.03.12 Single-Edge Crack in Beam in Bending a M M w 6M a Correction factor for notched beams KI = f  3 W (W − a ) 2 a/W f(a/W) 0.05 0.36 0.10 0.49 0.20 0.60 0.30 0.66 0.40 0.69 0.50 0.72 W: Depth of the Beam >0.60 0.73 alu SIF for Single-Edge Crack in Beam in Bending 2403.03.12 Training in Aluminium Application Technologies Stress intensity factor for an elliptical or circular crack in an infinite plate: Figure 2403.03.13 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 ! 3 1 σ (πa ) 2   a 2 4 KI = sin2 β +   cos2 β  θ0    b   The point on the perimeter of the crack a is defined by the angle ß a and the elliptic integral #o π 3 c " 2  c2 − a 2 2 c θ 0 = ∫ 1 − sin 2 θ  ⋅ dθ 0  c2  For circular cracks, c = a ⇒ K I = 2 ⋅ σ (a π ) 12 When c → ∞ and β = π 2 , this case is reduced ! to a through - thickness crack. alu Training in Aluminium Application Technologies SIF: Elliptical or Circular Crack in an Infinite Plate 2403.03.13 TALAT 2403 19
  20. 20. Stress intensity factor for a surface crack: Figure 2403.03.14 Surface Crack ! "Thumbnail Crack" The stress intensity factor can be calculated from the equations of elliptical crack using a free surface correction factor of 1.12 and for the position ß=π/2 π ⋅ a  1 2 K I = 112 ⋅ σ  .  Q   with Q = θ 0 and the elliptic integral a 2 2c 3 π 2 2  c2 − a 2  θ 0 = ∫ 1 − sin 2 θ  ⋅ dθ 0 c 2  Q is regarded as a shape factor because its values depend on a and c. For values of Q see Figure 2402.03.15. ! alu Training in Aluminium Application Technologies SIF: Surface Crack 2403.03.14 Flaw shape parameter: Figure 2403.03.15 Flaw Shape Parameter, Q 0.5 2c a 0.4 σ σ ys = 0 Ratio a/2c 0.3 σ σ ys = 0.60 σ σ ys = 0.80 σ σ ys = 10 . 0.2 0.1 0 0.5 1.0 1.5 2.0 2.5 Flaw Shape Parameter, Q alu Flaw Shape Parameter, Q 2403.03.15 Training in Aluminium Application Technologies TALAT 2403 20
  21. 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 2403.03.16. 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 2rY Plastic Zone alu Training in Aluminium Application Technologies Distribution of Stress in the Crack Tip Region 2403.03.16 The size of the plastic zone ry can be estimated from the stress field equations for plane stress conditions and setting σy = σys 2 1  K  ry =  σ   2π  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 2 1  K  ry =   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 condi- tion near the surface of the specimen where σz = 0. That means that the plastic zone in the center of a thick specimen is smaller than at the surface of the specimen, Figure 2403.03.17. TALAT 2403 21
  22. 22. Midsection Plane stress A - Overall view B - Edge view mode I Plane Surface strain mode I y Cracktip z Crack tip x KI² 2 $! y² rIp (plane strain) Machine notch Fatigue crack z Plastic zone rp (Plane stress) Plane strain region Specimen cross-section alu Training in Aluminium Application Technologies Plastic Zone Dimensions 2403.03.17 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 distribu- tions in the vicinity of the crack tip according to the equations given in Figure 2403.03.07. Consequently the total stress intensity factor can be obtained by algebraically adding the individual stress intensity factors corresponding to each load. TALAT 2403 22
  23. 23. 2403.04 Experimental Determination of Limit Values according to Various Recommendations • Linear-Elastic Fracture Mechanics − Experimental determination of KIc - ASTM-E399 • Elastic-Plastic Fracture Mechanics − Crack opening displacement (COD) - BS 5762 − Determination of R-Curves - ASTM-E561 − Determination of JIc - ASTM-E813 − Determination of J-R Curves - ASTM-E1152 − Crack opening displacement (COD) measurements - BS 5762 Structural materials have certain limiting characteristics, yielding in ductile materials or fracture in brittle materials. The yield strength σys is the limiting value for loading stresses, the critical stress intensity factors KIc, KId or Kc are the limiting values for the stress intensity factor KI. The critical stress intensity factor Kc at which unstable crack growth occurs for conditions of static loading at a particular temperature depends on specimen thickness or constraint, Figure 2403.04.01. Effect of Thickness on Kc 240 Plane stress Plane strain 200 in. 160 Kc, ksi 120 K Ic 80 Thickness at the beginning of plane strain is: 2  K Ic  40 t = 2.5   σy 0 0.25 0.30 0.50 0.75 1.0 2.0 3.0 5.0 Thickness, in. alu Training in Aluminium Application Technologies Effect of Thickness on Kc 2403.04.01 The limiting value of Kc for plane strain (maximum constraint) conditions is KIc (slow loading rate) or KId (dynamic or impact load). The KIc-value is the minimum value for plane strain conditions. The KIc-value would also be a minimum value for conditions of maximum structural constraint (for example, stiffeners, intersecting plates, etc.) that TALAT 2403 23
  24. 24. might lead to plain-strain conditions even though the individual structural members might be relatively thin. Design procedures based on the KIc-value provide a "conservative" approach to the fracture problem. The following chapter gives information on experimental require- ments and procedures for the measurement of KIc-values. 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 in- side 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 KI 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 KIc ma- terial values is the ASTM-E399 test method. W a SE(B) Bend Specimen 2.1W B 2.1W =W/2 Compact Tension 0.25 W Dia. 0.6 W Specimen 0.275 W 0.275 W 0.6 W a W B 1.25 W =W/2 after ASTM-E399 alu SE Bend Specimen and CT Tension Specimen 2403.04.02 Training in Aluminium Application Technologies TALAT 2403 24
  25. 25. Several different test specimen forms have been proposed in the course of the de- velopment 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 2403.04.02 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 2403.04.03. Surface cracks grow initially in the direction of the plate thickness and propagate further on as through-thickness cracks. according to ASTM alu Specimen Orientation 2403.04.03 Training in Aluminium Application Technologies 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 > 2.5 ⋅ (KIc/σys)2 B = specimen thickness > 2.5 ⋅ (KIc/σys)2 W = specimen depth > 5.0 ⋅ (KIc/σys)2 This leads for instance to a specimen thickness of approximately 50 times the radius of the plane strain plastic zone. Even before a KIc test specimen can be machined, the KIc value to be obtained must already be known or at least estimated. Three general rules may be used TALAT 2403 25
  26. 26. − overestimate the KIc value 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 aluminium alloys having yield strength of at least 350 MPa. σys/E Minimum recommended thickness and crack length [mm] 0.0050-0.0057 75,0 0.0057-0.0062 63,0 0.0062-0.0065 50,0 0.0065-0.0068 44,0 0.0068-0.0071 38,0 0.0071-0.0075 32,0 0.0075-0.0080 25,0 0.0080-0.0085 20,0 0.0085-0.0100 12,5 0.0100 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 main- tain plane strain conditions under slow loading and at normal service temperatures. Thus, the linear elastic analysis to calculate KIc values 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 values are given as an example for common aluminium alloys KI σ0,2 KI/σ0,2 2.5(KIc/σ0,2)2 MPa MPa AlMgSi1 50 245 0,0416 104 AlZn4,5Mg1 73 370 0,0389 97 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 2403 26
  27. 27. Analyse P-∆v record, calculate conditional KIc (=KQ) values, perform validation check for KIc: If the KQ values meet the above stated requirements, like a or B ≥ 2.5(KQ/σ0.2) and W ≥ 5.0(KQ/σ0.2)2, 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, invali- dating the calculation of KIc values. There are three possible approaches into the elastic- plastic region, through − crack opening displacement (COD) − R-curve analysis − J-integral Crack opening displacement (COD) - BS 5762 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 displace- ment. He also showed that the concept of crack opening displacement was analogous to the concept of critical crack extension force and thus the COD values could be related to the plane strain fracture toughness KIc. 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 KI analysis the application of the COD approach to engineering structures requires the measurement of a fracture toughness parameter δc, the critical value 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 δc-test is regarded as an extension of the KIc testing the british standardized test method after BS 5762 is very similar to the ASTM-E399 test method for KIc. Similar specimen preparation, fatigue-cracking procedures, instrumentation, and test procedures are followed. The displacement gage is similar to the one used in KIc 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 value δc to the actual crack tip opening displacement of the component analyzed and characterized by geometrical dimensions of the component TALAT 2403 27
  28. 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 (εz=0) with small scale crack tip plasticity. Kc is governed by conditions of plane stress (σz=0) with large scale crack tip plasticity. Kc values are generally 2-10 times larger than KIc. KIc values depend on only two variables, temperature and strain rate. Kc values 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 evaluations 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 2403.04.04 the solid lines represent the R-curves for different initial crack lengths. The dashed lines represent the variation in KI with crack length for different constant loads P1<P2<P3. Each line is a function of the crack length, KI = f(P,√a). R-Curves KR = Kc P1<P2<P3 P3 for a0 = a1 KR Applied K I = f (P, a ) KI Levels P2 KR = Kc for a0 = a2 KR < Kc for a0 = a1 ∆aactual ∆aactual a1 a2 a alu R-Curves and Critical Fracture Toughness Values KR 2403.04.04 Training in Aluminium Application Technologies for Different Initial Cracks a0 The two points of tangency represent points of instability, or the critical plane stress intensity factor Kc = KR, at the particular crack length and, of course, for the given conditions of temperature and loading rate. The KR value is always calculated by using the effective crack length aeff and is plotted against the actual crack extension aact, that takes place physically in the material during the test. TALAT 2403 28
  29. 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 value where complete unstable fracture occurs. The displacement control technique can be used to obtain the entire R-curve. The evaluation of R-curves for relatively low- strength, high-toughness alloys exhibiting large scale crack tip plasticity σy 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 value. 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 values δ/P from these second lines are used to determine the associated aeff/W-values that reflect the effective crack length aeff = a0 + ∆a + ry using also an appropriate relationship for the given specimen between crack opening and effective crack length. Each aeff/W-value is then used to determine a respective Keff-value, the latter is plotted against aeff producing the R-curve for the given material. The significance of the critical fracture toughness values obtained from an R-curve ana- lysis is in the calculation of the critical flaw size acr required to cause fracture instability. A normalised plot showing the general relationship of acr 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 2403.04.05. The plot can be used for any material for which valid fracture mechanics results (KIc, KId, Kc, KIscc) are available under a given loading rate, temperature and state of stress. TALAT 2403 29

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