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The Stress Intensity Factor

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1 The Stress Intensity Factor
The Elastic Stress Field Approach The Stress Intensity Factor The three basic modes of crack surface displacements

2 Derivation of the Elastic Stress Field Equations
Concepts of plane stress and plane strain Equilibrium equations Compatibility equations for strains Airy stress function Biharmonic equation Complex stress functions: Westergaard function for biaxially loaded plate (Mode I) Mode I stress / displacement fields Mode I stress intensity factor Westergard (1939), Irwin (1957)

3 Linear Elastic Crack-tip Fields
(general case) Mode I:

4 Mode II: Mode III:

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)1/2 But 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 plate Semi infinite edge notched specimens Finite width centre cracked specimens Finite width edge notched specimens edge notched finite width Crack-line loading Elliptical / Semielliptical cracks

7 Finite-width centre-craked specimens:
f(a/W) Irwin: Isida: 36 term power series approx. 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 stress Single edge notched (SEN) Double edge notched (DEN) SEN: Finite-width edge-notched specimens: 0.5% accurate for a/W < 0.6 DEN: 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 plates Internal Pressure problems

10 * Elliptical Cracks Example: corner crack in a longitudinal section of a pipe-vessel intersection in a pressure vessel actual cracks often initiate at surface discontinuities or corners in structural components !!! We start considering idealised situations: from embeded elliptical crack to semielliptical surface cracks

11 KI varies along the elliptical crack front
The embeded (infinite plate) elliptical crack under Mode I loading Irwin solution for Mode I: Where F: elliptic integral of the second type with: circular crack: a/c 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 F 1.000 1.016 1.051 1.097 1.151 1.211 1.277 1.345 1.418 1.493 1.571 KI varies along the elliptical crack front max. 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 cracks Best 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).

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 crack where P is the force per unit thickness

15 selected shape: better size estimation
CRACK TIP PLASTICITY First approximation: Better approaches selected shape: better size estimation Irwin Dudgale Better shape but first order approximation for the size Irwin approach: - stress redistribution; elastic – plastic; plane stress

16 Plastic zone shape from Von Mises yield criterion
First Order Aproximations of Plastic Zone Shapes Through-thickness plastic zone in a plate of intermediate thickness Plastic zone shape from Von Mises yield criterion Empirical Rules to estimating Plane Stress vs. Plane Strain conditions: Plane Stress: 2.ry ≈ B Plane 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 strain Kc = KIC Variation in KC with specimen thickness in a high strength maraging steel KIC 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 strain small scale plasticity Remember: 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 Fracture Fatigue pre-cracked specimens ! where ASTM Standard Single Edge notched Bend (SENB) Specimen ASTM Standard Compact Tension (CT) Specimen where

22 Clip gauge and ist attachment to the specimen
Principal types of load-displacement plots obtained during KIC testing ANALYSIS Line at 5% offset (95 % of tg OA equivalent to 2 % crack extension Ps: intersection 5 % offset with P-v record if there is a P value > Ps before Ps, then PQ = Ps check if Pmax / PQ < 1.10, then go to K(PQ): calculate KQ (conditional KIC) check if for KQ the specimen size requirements are satisfied, then check if crack front is symmetric, then KQ = 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 1971 S l 2c 2a L* : transitional crack length Example: 2c = 5 mm, L* = 0.25 mm c = 25 mm, L* = 1.21 mm For 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 window Crack in groove of a pressurized cylinder plane window Lager 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)

Fatigue Sustained load crack growth behaviour - stress corrosion cracking cracking due to embrittlement by internal or external gaseous hydrogen liquid metal embrittlement creep and creep crack growth *

27 Fatigue crack growth rate curve da/dN - DK
Fatigue Crack Propagation DKmax = KIC (1-R) !! Fatigue crack growth rate curve da/dN - DK

28 How to describe crack growth rate curves: crack growth “laws“
Paris Law: only Region II, no R effects Forman: also Region III complete crack growth rate curve n1, n2, n3, empirically adjusted parameters Complete curve McEvily: the three regions

29 EXAMPLES of Crack Growth rate Behaviour
Fatigue crack growth rate vs. DK for various structural materials at low R values Fatigue 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 sheet CRACK CLOSURE

31 Crack closure effects Elber: Actually:
Measuring the crack opening stress by means of a stress-displacement curve Elber: Actually:

32 Elber obtained the empirical relationship

33 Closure Mechanisms

34 Sustained load crack growth behaviour
Time to Failure Tests: preferred technique in the past Initial KI !!!

35 Generalised sustained load crack growth behaviour
KISCC KIC or KQ Modern 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 specimen Self stressed and portable Clear steady state and arrest Corrosion 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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