Presentation on theme: "The Stress Intensity Factor"— Presentation transcript:
1 The Stress Intensity Factor The Elastic Stress Field ApproachThe Stress Intensity FactorThe three basic modes of crack surface displacements
2 Derivation of the Elastic Stress Field Equations Concepts of plane stress and plane strainEquilibrium equationsCompatibility equations for strainsAiry stress functionBiharmonic equationComplex stress functions: Westergaard function for biaxially loaded plate (Mode I)Mode I stress / displacement fieldsMode I stress intensity factorWestergard (1939), Irwin (1957)
3 Linear Elastic Crack-tip Fields (general case)Mode I:
5 CHARACTERISTICS OF THE STRESS FIELDS Angular distributions of crack-tip stresses for the three modes (rectangular: left; polar:right)CHARACTERISTICS OF THE STRESS FIELDS- The stress and displacement formulas may reduced to particularly simple forms:Details of the applied loading enters only through K !!!for the infinite plate: K = s(p*a)1/2But for a given Mode there is a characteristic shape of the field !!!- Principle of Superposition: for a given Mode, K terms from superposed loadings are additive
6 Semi infinite edge notched specimens We consider next some other cases apart from the cracked infinite plateSemi infinite edge notched specimensFinite width centre cracked specimensFinite width edge notched specimensedge notchedfinite widthCrack-line loadingElliptical / Semielliptical cracks
7 Finite-width centre-craked specimens: f(a/W)Irwin:Isida: 36 term power seriesapprox.Brown:Feddersen:
8 Finite-width edge-notched specimens: Semi infinite edge-notched specimens:Free edges: crack opens more than in the infinite plate resulting in 12% increase in stressSingle edge notched (SEN)Double edge notched (DEN)SEN:Finite-width edge-notched specimens:0.5% accurate for a/W < 0.6DEN:0.5% accurate for any a/W
9 KI decrease when crack length increases ! TWO IMPORTANT SOLUTIONS FOR PRACTICAL USE* Crack-line Loading(P: force per unit thickness)for centrally located force:Crack under internal pressure (force per unit thickness is now P.dx, where P is the internal pressure)same result by end loading with s !KI decrease when crack length increases !Very useful solution:Riveted, bolted platesInternal Pressure problems
10 * Elliptical CracksExample: corner crack in a longitudinal section of a pipe-vessel intersection in a pressure vesselactual cracks often initiate at surface discontinuities or corners in structural components !!!We start considering idealised situations:from embeded elliptical crack tosemielliptical surface cracks
11 KI varies along the elliptical crack front The embeded (infinite plate) elliptical crack under Mode I loadingIrwin solution for Mode I:Where F: elliptic integral of the second typewith:circular crack:a/c0.00.10.20.220.127.116.11.18.104.22.168F1.0001.0161.0511.0971.1511.2111.2771.3451.4181.4931.571KI varies along the elliptical crack frontmax. at =p/2:min. at = 0:During crack growth an elliptical crack will tend to become circular: important in fatigue problems
12 The semi- elliptical surface crack in a plate of finite dimensions under Mode I loading In practice elliptical cracks will generally occur as semi-elliptical surface cracks or quarter-elliptical corner cracksBest solutions for semielliptical:FEM calculations from Raju-Newman:Raju I.S.,Newman J.C. Jr. Stress Intensity Factors for Two Symmetric Corner Cracks, Fracture Mechanics, ASTM STP 677, pp (1979).
13 SUPERPOSITION OF STRESS INTENSITY FACTORS 1) Crack under Internal Pressure:2) Semi-elliptical Surface Crack in a Cylindrical Pressure Vessel:
14 where P is the force per unit thickness 3) Cracks growing from both sidesof a loaded hole where the hole is small with respect to the crackwhere P is the force per unit thickness
15 selected shape: better size estimation CRACK TIP PLASTICITYFirst approximation:Better approachesselected shape: better size estimationIrwinDudgaleBetter shape but first order approximation for the sizeIrwin approach:- stress redistribution; elastic – plastic; plane stress
16 Plastic zone shape from Von Mises yield criterion First Order Aproximations of Plastic Zone ShapesThrough-thickness plastic zone in a plate of intermediate thicknessPlastic zone shape from Von Mises yield criterionEmpirical Rules to estimating Plane Stress vs. Plane Strain conditions:Plane Stress: 2.ry ≈ BPlane Strain: 2.ry < 1/10 B
17 Planes of maximum shear stress: location of the planes of maximum shear stress at the tip of the crack for plane stress (a) and plane strain (b) conditions
18 Deformation Modes: plane strain (a) and plane stress (b)
19 Is K a useful parameter to characterise fracture toughness? Under conditions of:- small scale plasticity- plane strainKc = KICVariation in KC with specimen thickness in a high strength maraging steelKIC is a material property: fracture toughness of linear elastic materials
20 Effect of Specimen Thickness on Mode I Fracture Toughness Limits to the Validity of LEFM:After considerable experimental work the following minimum specimen size requirements were established to be in a condition of :plane strainsmall scale plasticityRemember: Empirical Rules to estimate Plane Strain conditions: 2*ry < 1/10 B
21 Fatigue pre-cracked specimens ! LEFM Testing: ASTM E-399, committee E8 Fatigue and FractureFatigue pre-cracked specimens !whereASTM Standard Single Edge notched Bend (SENB) SpecimenASTM Standard Compact Tension (CT) Specimenwhere
22 Clip gauge and ist attachment to the specimen Principal types of load-displacement plots obtained during KIC testingANALYSISLine at 5% offset (95 % of tg OA equivalent to 2 % crack extensionPs: intersection 5 % offset with P-v recordif there is a P value > Ps before Ps, then PQ = Pscheck if Pmax / PQ < 1.10, thengo to K(PQ): calculate KQ (conditional KIC)check if for KQ the specimen size requirements are satisfied, thencheck if crack front is symmetric, thenKQ = KIC (valid test)
23 Material Toughness Anisotropy To provide a common scheme for describing material anisotropy, ASTM standardized the following six orientations:L-S, L-T, S-L, S-T, T-L, and T-S.The first letter denotes the direction of the applied load; the second letter denotes the direction of crack growth. In designing for fracture toughness, consideration of anisotropy is very important, as different orientations can result in widely differing fracture-toughness values.When the crack plane is parallel to the rolling direction, segregated impurities and intermetallics that lie in these planes represent easy fracture paths, and the toughness is low. When the crack plane is perpendicular to these weak planes, decohesion and crack tip blunting or stress reduction occur, effectively toughening the material. On the other hand, when the crack plane is parallel to the plane of these defects, toughness is reduced because the crack can propagate very easily.
24 Applications of Fracture Mechanics to Crack Growth at Notches Numerical Solution: Newman 1971Sl2c2aL* : transitional crack lengthExample:2c = 5 mm, L* = 0.25 mm c = 25 mm, L* = 1.21 mmFor crack length l ≥ 10% c: crack effective length is from tip to tip!!! (including notch)
25 Lager effective crack length by a contribution of a notch ! Consequences:Edge crack at windowCrack in groove of a pressurized cylinderplane windowLager effective crack length by a contribution of a notch !For relatively small (5-10 % notch size) cracks at a hole or at a notch, the stress intensity factor K is approximately the same as for a much larger crack with a length that includes the hole diameter / notch depth.Reading: Fatige and the Comet Airplane (taken from S. Suresh, Fatigue of Materials)
26 SUBCRITICAL CRACK PROPAGATION IN COMPONENTS WITH PREXISTING FLAWS FatigueSustained load crack growth behaviour- stress corrosion crackingcracking due to embrittlement by internal or external gaseous hydrogenliquid metal embrittlementcreep and creep crack growth*
28 How to describe crack growth rate curves: crack growth “laws“ Paris Law:only Region II, no R effectsForman:also Region IIIcomplete crack growth rate curven1, n2, n3, empirically adjusted parametersComplete curveMcEvily:the three regions
29 EXAMPLES of Crack Growth rate Behaviour Fatigue crack growth rate vs. DK for various structural materials at low R valuesFatigue crack growth rate for Structural steel (BS4360) at room temperature and with cycling frequencies 1-10 Hz.EXAMPLES of Crack Growth rate Behaviour
30 Influence of R on fatigue crack growth in Al 2024-T3 Alclad sheet Effect of R:Influence of R on fatigue crack growth in Al 2024-T3 Alclad sheetCRACK CLOSURE
31 Crack closure effects Elber: Actually: Measuring the crack opening stress by means of a stress-displacement curveElber:Actually:
32 Elber obtained the empirical relationship Schijve:
34 Sustained load crack growth behaviour Time to Failure Tests:preferred technique in the pastInitial KI !!!
35 Generalised sustained load crack growth behaviour KISCCKIC or KQModern techniques: based on fracture mechanics parameter K !
36 Increasing or decreasing K specimens crack- line wedge-loaded specimen (CLWL)tapered double cantilever beam-specimen (TDCB):constant K !!!bolt loaded cantilever beam-specimen (DCB)
37 Difference in crack growth behaviour for increasing K (cantilever beam) and decreasing K (modified CLWL or DCB specimens)Decreasing K specimens:Advantages:Entire crack growth with one specimenSelf stressed and portableClear steady state and arrestCorrosion product wedging:gives higher crack growth rate at a given nominal KI.Disadvantages:
38 Example:Outdoor exposure stress corrosion cracking propagation in 7000 series Al-alloy plate
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