Stress intensity factors

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008II. Stress Intensity FactorsIt is instructive to examine the concentrated stresses at theedge of an elliptical hole in a thin plate, when the plate issubjected to tensile stressesThe maximum stress at the long end of the elliptical hole isσa= σ (1+2 ) bmax 0K t= SCFNote that when a=b (circular hole), the stress concentrationfactor is 3, i.e. a well-known result.

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008The radius of curvature at the end of the major axis isρ =2ba⇒ K = 1+2Thus, as ρ →0, K t →∞ and σ max →∞. This marks thetransition from a smooth defect to a sharp crack.tThe formal definition of the stress intensity factor is:aρK ≡ lim σ ( r,0) 2πrI r→0yy• The SIF depends on the intensity of the stress near thecrack tip. It will depend only on the geometry andloading (material also if a composite)

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 20081.) Stress Analysis of Cracks Handbook• Center-cracked plate• Double-edge crack plate• Single edge crack plate• Edge-cracked beam• Edge-cracked simply supported beam• Cracks emanating from a thru-holes in a plate• Crack emanating from an elliptical notch• Buried penny-shaped crack• Edge-cracked solid shaft• Semi-elliptical surface crack• Circumferentially cracked thin-walled pressure vessel• Longitudinally cracked thin-walled pressure vessel

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J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 20082.) Practical applicationsDesign of a high strength steel pressure vessel-Data: p 0 =5000psi, D=2R=30in, t>0.5in, γ steel =489lb/ft 3σ ys K Ic $/lbA 260ksi 80ksi-in 1/2 $1.40B 220 110 1.40C 180 140 1.00D 180 220 1.20E 140 260 0.50F 110 170 0.15Problem: Design for satisfactory performance (SF=2),minimum cost and weight are also important

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Stress based design:σσmaxmaxminallowhoopσysσallow=2pR σ0 ys=t 2t= σ= σ =pDpR0t5000(30) 150,0000⇒min= = =σys σys σysσ allow t min Cost-$/ft Weight-lb/ftA 130ksi 0.58in 255$/ft 182lb/ftB 110 0.68 298 213C 90 0.83 258 258D 90 0.83 310 258E 70 1.07 165 330F 55 1.36 62 416

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Fracture mechanics based design:The leak-before-break (LBB) failure mode assumes a thruwallcrack with length equal to twice the wall thickness.This crack is designed to leak before causing a failure byfractureKIcKI=2pR0πttminminK=20⇒ tmin= =⎜ K ⎟IcIc2 2⎛ pD π ⎞ ⎛265,868⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ KIc⎠

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Κ Ιc /2 t min Cost-$/ft Weight-lb/ftA 40ksi-in 1/2 11.04in 3126 $/ft 2233 lb/ftB 55 5.84 2107 1505C 70 3.61 1016 1016D 110 1.46 533 444E 130 1.05 162 324F 85 2.45 108 720

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Hot Isostatic Press failure-HIP Specs:• Pressure= 15,000psi• Temperature=2400F• Up to 24hr process times• Q&T steel, σ uts =185ksi, σ ys =165ksi• OD=97in, ID=76in, L=14ft (huge!)• Failed at cycle 1898, 507 cycles after change incooling jacket

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Failure Analysis• Failed pieces reassembled- crack map points tomultiple origin sites, presence of corrosion pits onvessel OD• Final fracture origin sites indicate semi-circularsurface flaws in the r-θ plane on vessel OD neardiameter transition• Possible temper embrittlement of steel during heattreatment at manufacture• Problems with water treatment chemistry

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Stress Analysis:• Force on lid:F = pπr = 68,000,000lbs• Axial stress (due to pressure only):2i2σ = prizz23,700 psi2 2r − r=• Axial stress (thermal, 300F ΔT):oiσzzEαΔT=± =+ 42,000 psi (OD)2(1 −υ)• Total stress: σ zz =65,700psi• Stress concentration at diameter taper neglected sofar, FEA only way to calculate• Concentrated total stress with FEA σ zz =118,000psi• Various heat flux models yield a range 90,600< σ zz

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Fracture Mechanics Analysis:• Flaw model:rIDOD2aaθ• SIFKI= 1.12σzzπ aQ• Critical crack size measurements 0.857