9 -- annotated -- Fatigue and ceramics

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U.C. Berkeley Department of Mechanical Engineering ME 108: Mechanical Behavior of Engineering Materials Week 9: fatigue and ceramics Hayden Taylor hkt@berkeley.edu
Deciding whether crack propagation will be stable or unstable
Importance of the relative magnitudes of ideal and yield strengths
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Recap: what makes materials tough?
Two possible criteria for designing pressure vessels to avoid fast fracture 6 Smaller vessel Larger vessel Crack detection Easier ( e.g. X-ray, ultrasound, proof testing) May be harder to image entire vessel or proof- test it Gradual crack propagation Less likely: cyclic loading less common More likely: corrosion, cyclic loading etc Design principle Yield before fracture (yield easily detectable) Leak before fracture (leak easily detectable)
If an upper limit on crack length cannot be guaranteed, make leaking the failsafe Thickness, t Pressure, p Crack, length 2 a = t Spherical pressure vessel As before, stress σ in vessel wall should be below yield stress: 𝑝𝑝𝑝𝑝 2𝑡𝑡 < 𝜎𝜎 𝑦𝑦 𝑡𝑡 > 𝑝𝑝𝑝𝑝 2𝜎𝜎 𝑦𝑦 Substitute into ( ) setting 𝜎𝜎 = 𝜎𝜎 𝑦𝑦 (since operation can guarantee stress below yield stress): 𝑝𝑝 < 4 𝑌𝑌 2 𝜋𝜋𝜋 𝐾𝐾 1𝑐𝑐 2 𝜎𝜎 𝑦𝑦 Maximize M 2 : maximize operating pressure Radius, R If the crack must grow to span the vessel wall ( i.e. 2 a = t ) before propagating unstably, the pressure inside the vessel can be safely relieved by leaking: 𝜎𝜎 = 𝑌𝑌𝐾𝐾 1𝑐𝑐 𝜋𝜋𝜋𝜋 = 𝑌𝑌𝐾𝐾 1𝑐𝑐 𝜋𝜋𝑡𝑡 /2 ( ) M 2
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8 𝑝𝑝 < 4 𝑌𝑌 2 𝜋𝜋𝜋 𝐾𝐾 1𝑐𝑐 2 𝜎𝜎 𝑦𝑦 To maximise safe operating pressure, pick materials with high K 2 / σ y ratio If an upper limit on crack length cannot be guaranteed, make leaking the failsafe To minimize thickness (and thus mass), maximize σ y 𝑡𝑡 > 𝑝𝑝𝑝𝑝 2 𝜎𝜎 𝑦𝑦
Example: leak before break Coolant pipes in pressurized water reactors (1960s-70s) (Petroski ch 10) Selected a stainless steel where cracks grow through wall faster than along, so could achieve leak-before- break criterion Used purified cooling water to eliminate salts from the pipes which would increase rate of corrosion Fracture toughness falls with temperature. The ‘reference temperature’ below which material is brittle increases with time. Cracks were discovered in some of the pipes. Some did leak and were replaced. Intergranular stress corrosion cracking was cause of propagation. Pipe Crack Study Group was formed. Byron Nuclear Plant, Illinois, 2007. Source: Nuclear Regulatory Commission via https://blog.ucsusa.org/dlochbaum/nuclear- pipe-nightmares/ https://www.nrc.gov/docs/ML0037/ML003719804.pdf
https://www.flickr.com/photos/boston_public_library/4901555337/
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https://www.linkedin.com/pulse/great-molasses-flood-origin-professional-engineers-stamp-shevchenko/
Case study: The Great Molasses Flood Large tank for storing molasses suddenly failed, releasing a wave of liquid Tank was 50 feet tall and 90 feet in diameter Steel is thought to have been too thin for the specified load: 0.67 inches at bottom; 0.31 inches at top Density of molasses: ~1.5 times water There were large pre-existing cracks before the disaster (large enough to fill a cup) Concentrated stresses around rivet holes: also punched , not reamed Manganese content of the steel was too low: enabled a ductile-to-brittle transition at winter temperature (40 °F on day of disaster) https://en.wikipedia.org/wiki/Great_Molasses_Flood https://www.history.com/news/great-molasses-flood-science https://csengineermag.com/the-great-boston-molasses-tank-failure-of-1919/
Starting an analysis of the cause of rupture in Great Molasses Flood
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Fatigue
Fatigue Crack growth over multiple stress cycles, leading to fracture
Categories of fatigue Key question: how many loading cycles does it take a crack to initiate and/or grow large enough that on the next cycle there will be catastrophic fracture?
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Initially uncracked samples: two regimes Depending on whether yield stress is exceeded during cycling
Initially uncracked samples: high-cycle fatigue Basquin’s Law
Initially uncracked samples: low-cycle fatigue Coffin-Manson Law e.g. paperclip
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Dealing with non-zero average stress A non-zero mean stress 𝜎𝜎 𝑚𝑚 reduces the cycles to failure for a given Δ𝜎𝜎 Compensate for this with Goodman’s rule:
Dealing with stress amplitude changes over time Miner’s rule
Pre-cracked components The larger the structure, the harder to avoid initial cracks/imperfections If stress intensity is great enough to grow the crack: Stress intensity too low to grow crack below threshold
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The idea of proof testing Load occasionally up to a higher-than- service load in a safe way (e.g. hydraulic pressure) If no fast fracture results, the proof test guarantees that the largest crack length is no larger than a certain value We can then calculate how many service cycles would be needed for that guaranteed maximum-length crack to grow to the critical crack length. Repeat proof test before that number of service cycles has been completed.
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Summary of fatigue mechanisms Pre-cracked Non-cracked, low cycle Non-cracked, high cycle
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How fatigue cracks grow
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Crack initiation and growth in low- vs high-cycle fatigue
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Examples of each type of fatigue? Comet airliner disasters (1950s) Early days of Al alloys: fatigue resistance was not yet very high Fatigue failure at a small antenna opening in the fuselage ( not passenger windows) Analysis of the causes: https://www.youtube.com/watch?v=- DjnG74DDno Failure analysis case study: doi.org/ 10.1016/B978-0-08-043959-4.50018-3 Initial defect size estimated at ~ 100 𝜇𝜇 m . Failures occurred after ~1000 pressurization cycles
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Ceramics
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Classes of ceramic material Glasses Vitreous ceramics High-performance (engineering) ceramics Concrete and cement Rocks and minerals
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Molecular structures of ceramics: ionic
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Molecular structures of ceramics: covalent
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Amorphous covalent ceramics
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Ceramic alloys
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Typical ceramic microstructure
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Mechanical properties of ceramics Lattice resistance (i.e. monocrystalline shear yield strength) is ~E/30 for ceramics vs ~E/1000 for metals Typical ceramic yield strengths ~ 5 GPa
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Fracture strength of ceramics Very little plasticity because 𝜎𝜎 𝑦𝑦 is much closer to �𝜎𝜎 than in metals Thus 𝐾𝐾 𝐼𝐼𝐼𝐼 ~ 50 times lower than ductile metals Also: cracks and flaws are common in ceramics: From processing (typically sintering) Thermal cycling Abrasion, corrosion Cyclic loading – elastic anisotropy of grains or slip causes crack nucleation ‘Weakest link’ model for ceramic failure:
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How to increase 𝜎𝜎 𝑇𝑇𝑇𝑇 of a ceramic? Examples of ceramic composites Carbon fiber-reinforced polymer Glass fiber-reinforced polymer ‘Cermets’ e.g. WC-Co Bone Glass, metal and polymer fibers in concrete
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Testing of ceramics; modulus of rupture 𝜎𝜎 𝑟𝑟 ~1.7 𝜎𝜎 𝑇𝑇𝑇𝑇
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Cracks in compression
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Statistics of brittle fracture Manufacturing creates a random distribution of cracks Hence there is a finite probability of a component having a given tensile strength There is also a volume dependence: smaller samples have a lower chance of exceeding any given crack size Survival probability : Lower m means a higher degree of strength variability
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Weibull plot of fracture results
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Volume dependence of tensile strength in ceramics Consider n samples each of volume V 0 and with survival probability 𝑃𝑃 𝑠𝑠 𝑉𝑉 0
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Computed axial lithography (CAL), a volumetric additive manufacturing technique SPIE Advanced Lithography 2021, 22-26 Feb. 2021 46 Projections Sinogram 10 mm 1 Projection computation and optimization 2 Printing 3 Development Kelly et al., Science. (2019)
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Sintering process in micro-CAL Toombs et al., Science 376, 308-312 (2022)
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MicroCAL structures Scale bars: A-E 1 mm; F-H 0.2 mm; I-J 0.25 mm; K 0.5 mm; L 0.1 mm Toombs et al., Science 376, 308-312 (2022)
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49 VAM2021 Surface roughness evaluation of microCAL-printed components Toombs et al., Science 376, 308-312 (2022)
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Toombs et al., Science 376, 308-312 (2022) Bending strength of microCAL-printed silica glass structures
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Gradual crack propagation in ceramics ‘Static fatigue’
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