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Task 6 - Safety Review and LicensingOn the Job Training on Stress AnalysisFracture Mechanics: Linear Elastic Fracture Mechanics 1/2Davide Mazzini – Ciro SantusPisa (Italy)June 15 – July 14, 2015

Table of content – Class VI.b.1Content Stress singularity-Notch degenerating into a crack-Multi-axial stress at notch root/ crack tip-The Williams problemLinear Elastic Fracture Mechanics (LEFM)-The Westergaard stress function-Definition and calculation of the Stress Intensity Factors (SIFs)-LEFM Validity limitationsPisa, June 15 – July 14, 20152

BooksBooks on Fracture MechanicsT.L. Anderson, Fracture Mechanics: Fundamentals and Applications,third edition. CRC Press 2005.D. Broek. The Practical Use of Fracture Mechanics. Kluwer 1989. and many many othersPisa, June 15 – July 14, 20153

History of “Strength of Materials” Experities and similitude (up to 1700) Elastic evaluations (nominal solutions) (Eulero, Cauchy, DeSaintVenant, 1800) Stress concentrations (Kirsch, Inglis, 1900) Theory of plasticity (Prandtl, 1920) Sharp tip defects (Griffith, 1922)Pisa, June 15 – July 14, 20154

History of “Fracture Mechanics” Griffith’s energy approach for brittle materials (1930)Practical relevance (1940-1950)Definition of K, extension to metallic materials, complete develpmentof the Linear Elastic Fracture Mechanics (LEFM) (Williams, Irwin,1950)Application of the LEFM to Fatigue (Paris, 1960)Extension to ductile materials (Elatic Plastic Fracture MechanicsEPFM) (Irwin, Dugdale, Baremblatt, Wells, Landes, Rice, 1960)Dynamics and crack arrest (DFM), viscous and (NLFM) (AA.VV. 1980)Engineering applications, standards for design and testing, NDT,corrosion, anisotropic materials, Damage Tolerant approaches, .(ASTM, ASME, ESIS, BS)Pisa, June 15 – July 14, 20155

History of “Fracture Mechanics”Pisa, June 15 – July 14, 20156

Liberty ships – World War II2500 liberty ships, hull assembled by the innovative process of weldingPisa, June 15 – July 14, 20157

Liberty ships – World War II700 experienced heavy structural damage, 145 completely destroyed,many lost (complete breakage of the hull)Pisa, June 15 – July 14, 20158

Liberty ships – World War IIPost-failure analysis Failure at low stress (sometimes with the ship in the arbor) Quite “brittle” fractures Failure more frequent in winter time (ductile to brittle transitiontemperature) Effect of the technological process (metallurgical, geometrical: weldcrack-like defects)Fracture mechanics was born to understand these failure!Pisa, June 15 – July 14, 20159

Kirsch 1898Circular hole in a flat plate 0Complete analytical solutionPlane stress solution if a BFar boundariesaPlane strain if a BExtension to other problemsB 0Pisa, June 15 – July 14, 201510

Kirsch 1898yCircular hole in a flat plater xFar boundaries 0 a2 a2 rr 1 2 1 1 3 2 cos 2 r 2 r r 0 a2 a4 1 1 3 4 cos 2 2 r 2 r 0 a 2 a2 1 1 3 2 sin 2 2 r 2 r Pisa, June 15 – July 14, 201511

Kirsch 1898Circular hole in a flat plate 1y/ay/a 0 rr 0-1 3x/ax/aWhy rr atthese points?Pisa, June 15 – July 14, 201512

Kirsch 1898Circular hole in a flat plate, bi-axial loading UniaxialKt 3EquibiaxialKt 2Pure shearKt 4Pisa, June 15 – July 14, 201513

Inglis 1913Elliptical hole in a flat plateFar boundariesProblem definition:Geometrya, bLoad, nominalstress (far field stress)Pisa, June 15 – July 14, 201514

Inglis 1913Elliptical hole in a flat plateStress concentration: 2a A 1 b 2aKt A 1 b Kirsch solution for central holeb aKt 3Far boundariesPisa, June 15 – July 14, 201515

Inglis 1913Elliptical hole in a flat plateMore significant, local radius:b2 aaa Far boundaries Pisa, June 15 – July 14, 201516

Inglis 1913Elliptical hole in a flat plateKt 1 2abb2being: athen:Far boundariesKt 1 2a , a are more properly definingthe local geometrywhen: aKt 2Pisa, June 15 – July 14, 2015a 17

Inglis 1913Elliptical hole in a flat platelim(b, ) 0 0Far boundariesLimit: a lim K t lim 2 0 0 and the power of singularity isthe square root of the local radiusPisa, June 15 – July 14, 201518

Multi-axial stress at notch rootStress components in-plane, plane stress 0y/a x yKt z 0aB aPisa, June 15 – July 14, 2015x/a19

Multi-axial stress at notch rootPlane stressAlmost zero stressat interior pointsTransversal stressfree surfacesPisa, June 15 – July 14, 201520

Multi-axial stress at notch rootStress components in-plane, plane strain (approx.) 0y/a x yKt z ( x y )ax/aB aPisa, June 15 – July 14, 201521

Multi-axial stress at notch rootPlane strain 11 E 1 x x y y 1 z z After imposing z 0Zero transversaldisplacement: z 01( x y z ) 0E z ( x y )Pisa, June 15 – July 14, 201522

Multi-axial stress at notch rootInglis notch-like, plane stressANSYS Wb 5 mm2a 40 mmKt 1 2a 1 2K t 5.5Why a different value here?20 55Pisa, June 15 – July 14, 201523

Multi-axial stress at notch rootInglis notch-like, plane stressANSYS WbStress components, MPaPath on the geometry600 500 400 yxz3002001000-10002Pisa, June 15 – July 14, 201546x coordinate, mm81024

Multi-axial stress at notch rootInglis notch-like, plane strainANSYS WbExercise:Calculate the Stress components, withANSYS Workbench, at the notch tip for thelarge thickness geometry, and then verifythe plain strain assumptionRepeat same calculation with imposed(exactly) plain strain constraintPisa, June 15 – July 14, 201525

Williams 1957The Williams problem local geometry :r 0 governing parameters: s s local polar coordinates:r , useful angular variable: sPisa, June 15 – July 14, 201526

Williams 1957Airy function: x, y 2 xx 2 x yy 2 2 y xy 2 x y 2 2 2 2 Governing equation: 2 2 2 2 0 x y x y 4 4 4 2 2 2 4 0 x 4 x y yPolar coordinates:22 2 1 1 2 2 1 1 2 2 2 2 02 22 r r r rr r r r1 3 1 4 1 3 4 1 2 1 4 2 4 2 2 2 2 2 2 3 042432r rr r rr r r r r1 2 1 1 2 1 2 ; 2 ; r 2 rr 22 rr r rr r r 22Pisa, June 15 – July 14, 2015Stresscomponents27

Williams 1957 Williams hypothesis for the Airy function: r 1 F r 1 c1 sin 1 c2 cos 1 c3 sin 1 c4 cos 1 1. General parameters: c1, c2, c3, c4 and exponent (a dimensionless real number)2. Airy equation fulfilled in the domain for any combination of c1, c2, c3, c4 and Corresponding stress field: rr r 1 F 1 F r 1 1 F r r 1 F r Strain and displacement: s ij r 1 ui r Pisa, June 15 – July 14, 201528

Williams 1957 In order to keep the displacements bounded:ui r 0 Local boundary conditions:r 0 2 s 0Traction free edges r 0 r 2 s 0 Boundary conditions in explicit form: c2 c4 0 s c1 1 c3 1 0 c1 sin 2 s 1 c2 cos 2 s 1 c3 sin 2 s 1 c4 cos 2 s 1 0 c1 1 cos 2 s 1 c2 1 sin 2 s 1 c3 1 cos 2 s 1 c4 1 cos 2 s 1 0 Homogeneous linear system with unknowns: c1, c2, c3, c4 and the parameter Typical outcome of several problems: instability, free vibrations, etc.We are interested in not trivial solutions (eigenvalue problem)Let’s put the determinant of the system matrix to zeroCharacteristic equation with as unknown (infinite solutions)Pisa, June 15 – July 14, 201529

Williams 1957Crack as the special case with ψ 0 For this case the eigensolutions aren n where n 1, 2,3,.2 and the corresponding Airy’s function becomes: rn 12n 1 s n n n n 2 n sin 1 c4 n cos 1 cos 1 c3n sin 1 2 2 n 2 2 2 The infinite couples c3n, c4n are determined by the other boundary conditions (remotegeometry of the body, applied loads, constraints) Final general expression for the stress components: ij rn 1n 12 ij , n, c3n , c4 n Aij r 1212 Bij Cij r .Square root singular term !Pisa, June 15 – July 14, 201530

Williams 1957General conclusions of the Williams analysis Among the (usually) infinite terms of the stress expansion at the notch tip, only thefirst is unbounded (it goes to infinite as r approaches zero) The other terms are bounded or tends to zero approaching the notch tip The power of the singular term is a function of the angle 2 of the notch The strength of the singularity is the highest when 0: the crack is the mostsevere notch The power of the leading singular term is universal (the same for any crack), theasymptotic terms of the elastic fields at the tip are:1 ij , ij randui rPisa, June 15 – July 14, 201531

Williams 1957Exercise – MATLAB:r Implement a parametric calculation for theWillams problem and find the λ solution inthe range of angles ψ 0 - 89 s c2 c4 0 c1 1 c3 1 0 c1 sin 2 s 1 c2 cos 2 s 1 c3 sin 2 s 1 c4 cos 2 s 1 0 c 1 cos 2 1 c 1 sin 2 1 c 1 cos 2 1 c 1 cos 2 1 0 s 2 s 3 s 4 s 1 Then the system can be put in matrix form:01 0 1 sin 2 s 1 cos 2 s 1 1 cos 2 s 1 1 sin 2 s 1 c 0 1 c 0 1 2 sin 2 s 1 cos 2 s 1 c3 0 1 cos 2 s 1 1 cos 2 s 1 c4 0 0Pisa, June 15 – July 14, 20151032

Williams 1957Exercise – MATLAB:r Write the determinant of the matrix,impose it to zero and solve to find s0 1 100 1 10sin 2 s 1 cos 2 s 1 sin 2 s 1 cos 2 s 1 1 cos 2 s 1 1 sin 2 s 1 1 cos 2 s 1 1 cos 2 s 1 0 .Pisa, June 15 – July 14, 201533

Williams 1957Exercise – MATLAB:r2 604 s20 0.51222-2-400.51 1.52Pisa, June 15 – July 14, 201534

Williams 1957Exercise – MATLAB:0.5rPower-law singularity exponent s0.4 ij r 1 1- 0.31r1 0.20.1002040 6080Pisa, June 15 – July 14, 201535

T.L. Anderson, Fracture Mechanics: Fundamentals and Applications, third edition. CRC Press 2005. D. Broek. The Practical Use of Fracture Mechanics. Kluwer 1989. and many many others Books Pisa, June 15 – July 14, 2015. 4 Experities and similitude (up to 1700) Elastic evaluations (nominal solutions) (Eulero, Cauchy, De SaintVenant, 1800) Stress concentrations (Kirsch, Inglis .