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FRACTURE MECHANICS II. recap:. Sara Ferry NSE|H.H.Uhlig Corrosion Lab 22.71| October 23, 2012 . Griffiths vs. Irwin & Orowan. (but much larger!). Energy Release Rate. arbitrary body:. Precracked Purely elastic Loaded with weight P Δ describes displacement of weight
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FRACTURE MECHANICS II recap: Sara Ferry NSE|H.H.Uhlig Corrosion Lab 22.71|October 23, 2012
Griffiths vs. Irwin & Orowan (but much larger!)
Energy Release Rate arbitrary body: • Precracked • Purely elastic • Loaded with weight P • Δ describes displacement of weight • A describes crack area Elastic energy in the body: P How to find the form of U? 1. Solve boundary value problems using elasticity theory 2. Measure experimentally
Energy release rate dU 1. • Only the weight does work on the body • This work is stored as elastic energy • Crack doesn’t change • Obtain expression for change in U and integrate 2. • Crack increases in area, reducing elastic energy • Weight does no work • Displacement held fixed 1+2. dU= P dΔ – G dA
Fracture Energy • Most of the work done by the weight is stored as U: only some of the work goes toward inelastic processes (fracture, plastic deformation) • If small scale yielding condition applies, can still find U as if the body was purely elastic PdΔ = dU + ΓdA Elastic processes Inelastic processes …and fracture criterion: The crack grows if energy release rateG = fracture energy Γ. (fracture energy resists crack growth)
How can we measure fracture energy? • Look it up: results from previous experiments • Perform a fracture test yourself • Use a computer simulation (not standard) For common materials, you can expect to measure a fracture energy of … 10 J/m2 in glass 50 J/m2 in ceramics 103 J/m2 in polymers 104 J/m2 in aluminum 105 J/m2 in steel* *heat treating the steel can drastically change its fracture energy!
Potential Energy Consider elastic body + weight P as a combined system Π = U – PΔ • Analogous to Gibbs free energy. Subbing in expression for dU: dΠ = -ΔdP – GdA, such that Π = Π(P, A)
Linear Elasticity There is a linear relationship between applied force P and displacement Δ. Elastic energyU =PΔ/2 Potential energyΠ = -U Energy release rate load fixed • Finding G: • Look it up; there are handbooks for elasticity solutions for many situations (G is specific to the configuration of the system) • Measure C using multiple identical specimens that have different crack areas. Obtain C(A), find U, and then find G (expression on next slide) • Elasticity boundary-value problem (finite element program)
Compliance • Linearly elastic body • Linear relationship between load and displacement • C = compliance, is a function of crack area • Opposite of stiffness (stiffness: rigidity; resistance to deformation when force is applied) Δ = C(A)P When A increases, compliance C increases. More compliance = more elastic energy stored when load is fixed More compliance = less elastic energy stored when displacement is fixed
Applications of Fracture Mechanics 1. Measure fracture energy • If other constants known except σc and Γ • Load a precracked sample, record critical stress, solve for Γ • Compare fracture energies of materials and study ways to improve fracture resistance 2. Predict critical load • Solve for σc if other values known (may have to experimentally determine Γ and measure afirst) without carrying out fracture experiment • Use to compare the critical load for various crack sizes (what is maximum allowable crack size) 3. Estimate flaw size • Measure σc, Γ, and β • a = ΓE/πσ2 4. Knowing the material that will be used, expected stresses, and typical flaw sizes, design a structure to minimize likelihood of fracture
