Chapter 8. Fracture of Cracked Members Mechanical Strengths and Behavior of Solids 2015/8/9 Seoul National University -1- Contents 1 Introduction 2 Preliminary Discussion 3 4 Relationship between πΊπΊ and πΎπΎ 5 πΎπΎ for various cases Safety Factors 6 Additional topics on K 7 Trends of Fracture Toughness π²π²π°π°ππ 8 2015/8/9 Fracture mechanics under Plasticity Seoul National University -2- 8.1 Introduction • Unexpected failures below the material’s yield strength initial cracks Figure 8.1 A propane tank truck explosion due to fracture from initial cracks in welds • Fracture mechanics <Notch-impact test> Rough guide for choosing materials 2015/8/9 <Fracture mechanics> Specific analysis of strength and life for various cracks Seoul National University - 3/35 - 8.2 Preliminary Discussion • Cracks as Stress Raisers (stress concentration factor) πππ¦π¦ = ππ 1 + 2 2015/8/9 ππ ππ = ππ 1 + 2 ππ ππ Seoul National University - 4/35 - 8.2 Preliminary Discussion • Behavior at Crack Tips in Real Materials – Large plastic deformations near the crack tip (plastic zone) – High stress is spread over a region (stress redistribution) 2015/8/9 Seoul National University - 5/35 - 8.2 Preliminary Discussion • Behavior at Crack Tips in Real Materials <Polymer> <Ceramic> craze zone Plastic Zone *Measurement of Cohesive Parameters of Crazes in Polystyrene Films (Experimental and Applied Mechanics) http://what-when-how.com/ **Enhancement mechanisms of graphene in nano-58S bioactive glass scaffold: mechanical and biological performance http://www.nature.com/srep/2014/140416/srep04712/full/srep04712.html 2015/8/9 Seoul National University - 6/35 - 8.2 Preliminary Discussion • Effects of Cracks on Strength Stress intensity factor K - a measure of crack severity - affected by size, stress, and geom. - linear-elastic assumption (LEFM) Fracture toughness Kc - criteria for brittle fracture - affected by material, temperature, loading rate, thickness Plane strain fracture toughness KIc - thicker plate: a lower value of Kc - a worst-case value of Kc - material-dependent property 2015/8/9 πΎπΎ = ππ ππππ (ππ βͺ ππ) πΎπΎ < πΎπΎππ βΆ elastic deformation πΎπΎ > πΎπΎππ βΆ brittle fracure Material Toughness Kic MPa m Seoul National University Steel Polymer Ceramic AISI 4130 ABS Concrete 110 3.0 1.19 - 7/35 - 8.2 Preliminary Discussion • Effects of Cracks on Strength Area loss ( ) : area loss due to crack ππ = ππ 2π‘π‘(ππ−ππ) = ππ0 1 − ππ/ππ Critical stress ( ) : fracture due to stress intensity ππππ = πΎπΎππ / ππππ Stress deviation ( ) : due to plastic deformation violating LEFM assumption Figure 8.5 2015/8/9 Failure data for cracked plates of 2014-T6 Al at −195°C. Seoul National University - 8/35 - 8.2 Preliminary Discussion • Effects of Cracks on Brittle vs. Ductile behavior Transition crack length ππππ : critical size b/w yielding and brittle fracture <Ductile material> 2 <Brittle material> Ductile Yield strength ππ0 ↓ Fracture toughness πΎπΎππ ↑ Transition crack length πππ‘π‘ Brittle ↑ ↓ ↓ Material 2015/8/9 1 πΎπΎππ πππ‘π‘ = ππ ππ0 ↑ *above πππ‘π‘ is valid for wide and center-cracked plate Seoul National University - 9/35 - 8.3 Relationship between πΊπΊ and πΎπΎ (1) • Strain Energy Release Rate πΊπΊ : Energy per unit crack are to extend the crack πΊπΊ = − 1 ππππ π‘π‘ ππππ • Stress Intensity Factor πΎπΎπ°π° : stresses near the ideal sharp crack (linear-elastic & isotropic) πΎπΎπΌπΌ ππ ππ 3ππ πππ¦π¦ ≅ ππππππ 1 + sin sin 2 2 2 2ππππ πΎπΎπΌπΌ = lim πππ¦π¦ 2ππππ ππ,ππ→0 πΎπΎπΌπΌ = πΉπΉπΉπΉ ππππ 2015/8/9 πΉπΉ βΆ a dimensionless function that depends on the geometry and loading configuration Seoul National University - 10 /35- 8.3 Relationship between πΊπΊ and πΎπΎ (1) 2015/8/9 Seoul National University - 11/35 - 8.3 Relationship between πΊπΊ and πΎπΎ (2) • Energy-balance approach Fracture energy = released strain energy ππ + bond-breaking energy ππ Released strain energy πΌπΌ 1 πΈπΈππ 2 ππ 2 = ππ = οΏ½ ππ ππππ = ππ 2 2πΈπΈ ∗ π½π½π½π½ ∗ ππ ∗ ππ 2 2 ππ = − ππππ ππ = −2 2 2πΈπΈ for plane stress loading π½π½ = ππ Bond-breaking energy πΊπΊ ππ = πΊπΊππ ∗ ππ Energy-balance ππππ2 ππ(ππ + ππ) = − ππππππ + πΊπΊππ = 0 For ππ = ππππ , ππππ πΈπΈ πΈπΈπΊπΊππ ∴ ππππ = ππππππ β ππ = πΊπΊππ ππ *Introduction to Fracture Mechanics, David Roylance, 2001 http://ocw.mit.edu/courses/materials-science-and-engineering/3-11-mechanics-of-materials-fall-1999/modules/frac.pdf 2015/8/9 Seoul National University - 12/35 - 8.3 Relationship between πΊπΊ and πΎπΎ(3) • Stress criteria for Rupture πΎπΎππ = ππππ ππππππ β‘ • Relationship b/w πΊπΊ and πΎπΎ From β & β‘, πΎπΎππ = πΈπΈπΊπΊππ ∗ ππππππ = ππππππ πΎπΎππ2 = πΊπΊππ πΈπΈ πΈπΈπΊπΊππ for plane stress (πππ§π§ = 0) πΎπΎππ2 = πΊπΊππ πΈπΈ/(1 − ππ 2 ) for plane strain (πππ§π§ = 0) *Introduction to Fracture Mechanics, David Roylance, 2001 http://ocw.mit.edu/courses/materials-science-and-engineering/3-11-mechanics-of-materials-fall-1999/modules/frac.pdf 2015/8/9 Seoul National University - 13/35 - 8.4 πΎπΎ for various cases (1) • Cracked plates under tension 2015/8/9 Seoul National University - 14/35 - 8.4 πΎπΎ for various cases (2) • Cracked plates under bending 2015/8/9 Seoul National University - 15/35 - 8.4 πΎπΎ for various cases (3) • Round shaft with circumferential crack 2015/8/9 Seoul National University - 16/35 - 8.4 πΎπΎ for various cases (4) • Plate w/ forces to the crack faces • ASTM standard compact specimen πΉπΉππ = 2015/8/9 2 + πΌπΌ (0.886 + 4.64πΌπΌ 1 − πΌπΌ 3/2 −13.32πΌπΌ 2 + 14.72πΌπΌ 3 − 5.6πΌπΌ 4 ) Seoul National University - 17/35 - 8.4 πΎπΎ for various cases (5) • Safety factors against brittle fracture πΎπΎπΌπΌπΌπΌ πΎπΎπΌπΌπΌπΌ = πππΎπΎ = πΎπΎ πΉπΉππππ ππππ πΎπΎπΌπΌπΌπΌ = πΉπΉππ ππππ ππππππ ππππ = 2015/8/9 ππππ πΉπΉ = ππ ππ πΉπΉππ πΎπΎ 2 πππΎπΎ βΆ safety factor for fracture toughness ππππ βΆ safety factor for crack size ππππ βΆ applied stress ππ βΆ crack size ππππ βΆ critical crack length πΎπΎ βΆ fracture toughness πΎπΎπΌπΌπΌπΌ βΆ plane strain fracture toughness πΉπΉππ βΆ fracture toughness at ππππ – Elements for assigning safety factors 1) statistical information of crack shape, stress, material prop. 2) safety factor set by design code, company policy, government regulation – Crack size ππ should be quite smaller than ππππ to satisfy reasonable πππΎπΎ – In general, πππΎπΎ is set to be large due to great variance of πΎπΎπΌπΌπΌπΌ – Safety factors on crack length must be rather large to achieve reasonable safety factors on πΎπΎ and stress Seoul National University - 18/35 - 8.5 Additional topics on K (1) • Practical applications: Complex 3-D crack cases – Useful cases include cracked plates, shafts, cracked tubes, discs, stiffened panels, etc., including three-dimensional cases – πΉπΉ values are elevated for points where the crack front intersects the surface and max. πΎπΎ (b)-(d). (a) (b) πΎπΎ = πΉπΉπΉπΉ ππππ (c) (d) πππ‘π‘ βΆ tension, P ππππ βΆ bending, M Figure 8.17 Stress intensity factors for (a) an embedded circular crack, (b) half-circular surface crack, (c) quarter-circular corner crack, and (d) half-circular surface crack in a shaft 2015/8/9 Seoul National University - 19/35 - 8.5 Additional topics on K (2) • Half-circular surface crack 2015/8/9 Seoul National University - 20/35 - 8.5 Additional topics on K (3) • Elliptical crack (a) ππππ ππ , πΎπΎ = ππ ππ ππ ππ/2 ππ = οΏ½ 0 1− ππππ = ππ ππ 2 2 2 cos ππ + sin ππ ππ 2 sin2 π½π½ ππππ, 1/4 ππ ππ = 1 − ππ 2 2 ππ/ππ ≤ 1 ππ: flow shape factor (b) ππππ , πΎπΎπ·π· = πΉπΉπ·π· ππ ππ 2015/8/9 ππ ππ ≈ 1 + 1.464 ππ Seoul National University 1.65 (ππ/ππ ≤ 1) - 21/35 - 8.5 Additional topics on K (4) • Crack growing from notches (holes, fillets, rivets, etc.) (by numerical analysis) <Short crack> <Long crack> For small ππ/ππ, For large ππ/ππ, π€π€hole ≈ ππcrack . πΎπΎπ΄π΄ = 1.12πππ ππππ ππ ′ = πππ‘π‘ ππ 2015/8/9 Seoul National University πΎπΎπ΅π΅ = πΉπΉπΉπΉ ππππ - 22/35 - 8.5 Additional topics on K (5) • Superposition for combined loading ππ ππππ 6ππ 6ππππ πΎπΎ2 = πΉπΉ2 ππ2 ππππ, ππ2 = 2 = 2 ππ π‘π‘ ππ π‘π‘ ππ 6πΉπΉ2 ππ πΎπΎ = πΎπΎ1 + πΎπΎ2 = πΉπΉ + ππππ ππππ 1 ππ πΎπΎ1 = πΉπΉ1 ππ1 ππππ, <Eccentric loading of a plate> πΎπΎ1 = πΉπΉπππ ππ1 = ππ π‘π‘ ππ πΎπΎ3 = πΉπΉ3 ππ ππππ πΎπΎ2 = πΎπΎ1 = 1 ππ πΉπΉ3 ππππ πΎπΎ1 + πΎπΎ3 = πΉπΉπππ + 2 2 2π‘π‘ ππ ππ 1 π‘π‘ ππ sin ππππ πΎπΎ3 = ππ 2ππ tan <Single/row of a cracks (bolt, rivet)> 2015/8/9 Seoul National University πΎπΎ2 = ππππ 2 1 ππ πΎπΎ1 + πΎπΎ3 = 2 2π‘π‘ ππ 1 sin ππππ + 1 ππππ tan 2 2 - 23/35 - 8.5 Additional topics on K (6) • Inclined or parallel cracks to an stress – Alternating crack direction : It does not grow in its original plane. Toughness for mixed-mode – Interactive stresses are generally unknown. : Fracture modes are not independent. – Possible approach : Projection of crack normal to the stress direction Opening πΎπΎπΌπΌ = ππ cos 2 ππ ππππ πΎπΎπΌπΌπΌπΌ = ππ cos ππ sin ππ 2015/8/9 Seoul National University Sliding ππππ Tearing ≈ πΎπΎ = ππ ππ a cos ππ - 24/35 - 8.5 Additional topics on K (7) • Leak-Before-Break (LBB) design of pressure vessels – Pressure vessels should be designed to leak before fracture. – A through-wall crack length 2ππ ≈ 2π‘π‘ – Critical crack size ππππ πΎπΎπΌπΌππ = πΉπΉπΉπΉ ππππππ πΉπΉ = 1 (β΅ wide plate) 2015/8/9 1 πΎπΎπΌπΌππ ππππ = ππ πππ‘π‘ 2 Seoul National University π‘π‘ < ππππ : leak before break π‘π‘ > ππππ : brittle fracture - 25/35 - 8.6 Trends of Fracture Toughness π²π²π°π°ππ (1) • Fixtures for a fracture toughness with crack growth bend specimen Figure 8.27 Fixtures for a fracture toughness test on a bend specimen. The dimension W corresponds to our b. (Adapted from [ASTM 97] Std. E399; copyright © ASTM; reprinted with permission.) 2015/8/9 Seoul National University - 26/35 - 8.6 Trends of Fracture Toughness π²π²π°π°ππ (2) • Fracture Toughness – A deviation from linearity on the P-v plot, or a sudden drop in force due to rapid cracking, identifies a point ππππ corresponding to an early stage of cracking – The value of πΎπΎ, denoted πΎπΎππ , is the stress intensity factor corresponding to ππππ – πΎπΎππ may be somewhat lower than the value πΎπΎππ corresponding to the final fracture of the specimen. – Fracture toughness testing of metals based on LEFM principles governed by several ASTM standards, notably Standard Nos. E399 and E1820. – Standards No. D5045 (polymers) and No. C1421 (ceramics) 95% of slope πΎπΎππ decreases with thickness 2015/8/9 Seoul National University - 27/35 - 8.6 Trends of Fracture Toughness π²π²π°π°ππ (3) • Material – Material-dependent πΎπΎπΌπΌππ Material Toughness [MPa m] Metal Polymer Ceramic 20~200 1~5 1~5 – Large CoV of πΎπΎπΌπΌππ : 10~20% • Microstructural influences – Chemical composition : sulfide inclusions facilitate fracture. – Processing (forging, rolling, extruding) : anisotropy and planes of the flattened grains – Neutron radiation (radiation embrittlement) : large numbers of point defects 2015/8/9 Seoul National University *CoV: coefficient of variation (CoV = ππ/ππ) - 28/35 - 8.6 Trends of Fracture Toughness π²π²π°π°ππ (4) • Temperature – Cleavage @ low temp. : fracture with little plastic deform. along the crystal planes of low resistance – Dimples rupture @ high temp. : fracture with plasticity-induced formation, growth, and joining of tiny voids Upper shelf Cleavage Dimpled rupture (microvoid coalescence) Lower shelf (πΎπΎπΌπΌππ ) <Fracture mechanics shift> 2015/8/9 Seoul National University - 29/35 - 8.6 Trends of Fracture Toughness π²π²π°π°ππ (5) • Temperature and Loading rate – A higher rate of lading lowers the fracture toughness πΎπΎπΌπΌππ . (temperature shift) – Statistical variation of πΎπΎπΌπΌππ is especially large within the temperature transition. 2015/8/9 Seoul National University - 30/35 - 8.7 Fracture mechanics under Plasticity (1) • The size of plastic zone ππππ – – Yielding at crack tips (plastic zone) will be studied. Plastic zone may not be large if the LEFM theory is to be applied. For plane strain (πππ§π§ = 0, πππ§π§ = 2πππππ¦π¦ ), For plane stress (πππ§π§ = 0), πππ₯π₯ = πππ¦π¦ = ππππππ = πππ¦π¦ ≅ πΎπΎ 2ππππ lim πππ¦π¦ = ππ→0 2015/8/9 1 πΎπΎ 2ππ ππππ ππππππ πΎπΎ πΎπΎ 2 2ππππ ο 2ππππππ = ππ ππ 3ππ 1 + sin sin 2 2 2 1 πΎπΎ ππ ππππ 2 For octa. or max. shear stress yield criterion, ππ0 ≈ 2.5ππ0 → 3ππ0 πππ₯π₯ = πππ¦π¦ = 1 − 2ππ 1 πΎπΎ 2ππππππ = 3ππ ππππ 2 2ππππ Seoul National University - 31/35 - 8.7 Fracture mechanics under Plasticity (2) • The size of plastic zone ππππ – For thin specimen, Poisson contraction in the thickness occurs. – It results in yielding on shear planes inclined through the thickness. Plane stress 2015/8/9 Seoul National University Plane strain - 32/35 - 8.7 Fracture mechanics under Plasticity (3) • Plasticity limitations on LEFM – LEFM is valid for small plastic zone compared with crack tip-to-boundary dist. – 8ππππ is generally considered to be sufficient ο 4 times of crack zone size – Since ππππππ is larger than ππππππ , an overall limit of the use of LEFM is ππ, ππ − ππ , β ≥ 8ππππππ 4 πΎπΎ = ππ ππππ 2 (LEFM applicable) or Region of K-dominance <Crack specimen geometry> 2015/8/9 <LEFM applicable region, K-field> Seoul National University - 33/35 - 8.7 Fracture mechanics under Plasticity (3) • Plasticity limitations on LEFM – LEFM is valid for small plastic zone compared with crack tip-to-boundary dist. – 8ππππ is generally considered to be sufficient ο 4 times of crack zone size – Since ππππππ is larger than ππππππ , an overall limit of the use of LEFM is ππ, ππ − ππ , β ≥ 8ππππππ 2015/8/9 4 πΎπΎ = ππ ππππ 2 (LEFM applicable) Seoul National University - 34/35 - 8.7 Fracture mechanics under Plasticity (4) • Fracture mechanics beyond linear elasticity – Condition of "ππ, ππ − ππ , β ≥ 8ππππππ " is not satisfied, (under excessive yielding) – LEFM and πΎπΎ are not applicable due to excessive yielding. – Following three approaches are available. (1) Plastic zone adjustment – The stress outside of the plastic zone is similar to elastic stress. – Hypothetical crack (ππππ =ππ + ππππππ ) with its tip near the center of the plastic zone. – Not applicable for large stress to cause yielding on whole section; 80% of the fully plastic force or moment. πΎπΎ = πΉπΉπΉπΉ ππππ πΎπΎππ = πΉπΉππ ππ ππππππ = πΉπΉππ ππ ππ ππ + ππππππ 2015/8/9 πΉπΉππ = πΉπΉ ππππ /ππ where 1 πΎπΎππ ππππππ = 2ππ ππππ Seoul National University 2 - 35/35 - 8.7 Fracture mechanics under Plasticity (5) (2) J-Integral – π½π½ is the generalization of the strain energy release rate, πΊπΊ, to nonlinear-elastic. – It reains significance as a measure of the intensity of the elasto-plastic stress and strain fields around the crack tip. – Two different and independent P-v curves are required. – Basis of fracture toughness tests, ASTM Standard No. E1820. πΎπΎπΌπΌ2ππ = πΊπΊπΌπΌππ πΈπΈ′ Eq. (8.10) πΎπΎπΌπΌππ π½π½ = π½π½πΌπΌππ πΈπΈ ′ for plane stress (πππ§π§ = 0) πΈπΈ ′ = πΈπΈ πΈπΈ ′ = πΈπΈ/(1 − ππ 2 ) for plane strain (πππ§π§ = 0) πΎπΎπ½π½ = π½π½π½π½ ππ πΎπΎπ½π½ ≈ πΎπΎ 1 + ππππππππ 2015/8/9 Seoul National University - 36/35 - 8.7 Fracture mechanics under Plasticity (6) (2) J-Integral: Fracture toughness tests for π½π½πΌπΌππ – Complexity encountered in π½π½πΌπΌππ testing is that nonlinearity in P-v behavior is due to a combination of crack growth and plastic deformation – π½π½ calculation for the standard bend and compact specimens π½π½ = π½π½ππππ + π½π½ππππ πππ΄π΄ππππ πΎπΎ 2 1 − ππ 2 + = πΈπΈ π‘π‘ ππ − ππ ππ = 1.9 for bend specimen ππ = 2 + 0.522(1 − ππ/ππ) for compactspecimen <Bend specimen> <Unloading compliance method> 2015/8/9 Seoul National University <Compact specimen> <Potential drop test> - 37/35 - 8.7 Fracture mechanics under Plasticity (7) (3) Crack-Tip Opening Displacement (CTOD; πΉπΉ) – πΎπΎ can be used to estimate the displacement separating the crack faces. – CTOD is also used as the basis of fracture toughness tests; ASTM Standards E1290 and E1820 πΎπΎ 2 π½π½ πΏπΏ ≈ ≈ πΈπΈππππ ππππ • Summary 2015/8/9 Seoul National University - 38/35 -
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