8 -- annotated -- Fracture

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Oct 30, 2023

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U.C. Berkeley Department of Mechanical Engineering ME 108: Mechanical Behavior of Engineering Materials Week 8: Fracture and fatigue Hayden Taylor hkt@berkeley.edu
Consider the effect of a flaw/crack on a material By what factor is the stress increased? Balloon demo
Harold Edgerton (MIT), 1959
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Stress concentrations around a crack https://www.doitpoms.ac.uk/tlplib/brittle_fracture/crack_tip_stress_popup.php
Fracture starts near stress concentrations A crack reduces breaking strength by a greater factor than the reduction in cross-sectional area: this is why cracks are dangerous Stress is highly concentrated around the tip of a crack: this is how the ideal stress is reached in those few atomic bonds around the tip Birefringence image of stresses around crack in polycarbonate How concentrated is the stress around a crack? Elliptical crack, 2c long, 2b wide: 𝜎𝜎 𝑡𝑡 = 𝜎𝜎 1 + 2 𝑐𝑐 𝑏𝑏 So: circular hole, stress concentration of 3 at edge of hole, regardless of diameter! Shape is what matters Concentrations can arise from design as well as from flaws Drilling holes at crack tips to arrest growth
What is the stress concentration factor for an atomically narrow crack?
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Comparison of strengths, from lecture 3… Note: slip applies to crystalline materials only: not amorphous materials like glass Ideal rupture strength (tensile) Slip without dislocations (hypothetical) Slip with dislocations �𝜎𝜎 ~ 𝐸𝐸 15 𝜏𝜏 crit ~ 𝐺𝐺 5 Roughly comparable Much lower 𝜏𝜏 p 𝐺𝐺 180 𝜏𝜏 p ∼ 3𝐺𝐺 exp 2𝜋𝜋𝜋𝜋 𝑏𝑏 𝐸𝐸 = 8 3 𝐺𝐺 for 𝜈𝜈 = 0.3
Consider brittle fracture first Separation of material into two or more pieces with clean, smooth surfaces Bonds break ( do reach ideal strength) a few at a time, at the tip of a crack, and the material ‘unzips’ as the crack propagates through the material For this to happen, reaching the ideal strength must be possible – i.e. the action of other deformation mechanisms like slip are not enough to relieve the stresses around the crack. Spring ‘unzipping’ model https://www.doitpoms.ac.uk/tlplib/brittle_fracture/bond_break.php
But creating fracture surfaces requires an input of energy Exceeding the ideal strength at the crack tip does not in itself mean that the crack will catastrophically propagate through the component and make it break. The energy to keep breaking bonds needs to come from somewhere: Mechanical work and/or release of elastic strain energy
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Case 1: negligible elastic strain energy Work done on material all goes into creating new surfaces
Case 2: crack propagation through a fixed body No external work provided All energy for crack growth comes from release of elastic strain energy If crack growth consumes more energy than is released, crack growth is stable If crack growth consumes less energy than it releases, crack growth is unstable: fast fracture
How does 𝑈𝑈 𝑒𝑒𝑒𝑒 depend on a ?
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Case 3: fast fracture at fixed load; moving boundary
The stress intensity factor K Geometry dependence Critical stress intensity factor is a material property Fast fracture occurs when:
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G and K c for engineering materials
Compare K c to strength, 𝜎𝜎 𝑦𝑦
Concept of the plastic zone Stress ahead of crack tip Note: derived using a continuum model – valid for 𝑟𝑟 ≫ 𝑏𝑏 where b is the atomic spacing. Consider the ratio of to
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Brittle or cleavage fracture Stress exceeds ideal cleavage strength of bonds at the crack tip before yield strength is reached ahead of the tip
Ductile tearing/fracture Zone in front of the crack tip yields Blunts the crack tip
Global vs local , and brittle vs ductile fracture Examples of fracture
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USS Schenectady “The Design and Methods of Construction of Welded Steel Merchant Vessels” U.S. GPO (1947) Ductile to brittle transitions
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Ways of increasing K c
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Adding ductility to concrete
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Modes of fracture
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Fracture testing Charpy test What mode of fracture does the Charpy test bring about?
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