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Unformatted text preview: ME 382 Lecture 23 1 S TATISTICS OF F RACTURE • Very brittle materials have very small critical flaw sizes E.g., Consider a ceramic with K Ic ≈ 1 MPa √ m ( σ Y ≈ 14 GPa) Crack length for a “useful” strength of 150 MPa? Assume loading parameter for fracture: K I ≈ σ π a ∴ Critical crack size for fracture when σ f = 150 MPa a cr ≈ K Ic 2 πσ f 2 = 10 12 π × 2.25 × 10 16 ≈ 14 μ m ∴ Critical crack length ≈ grain size & random natural defects • Difficult to measure size of strengthlimiting flaw (problem is both small size, and the fact that there are lots of flaws of similar sizes) ∴ Use statistical methods to determine probability of failure Weakestlink statistics • Consider a chain made up of many brittle links Probability of failure of each link: P f = 0.3 / link Probability of survival of each link: P s = 1  P f = 0.7 ∴ For 1 link: P f = 0.3, P s = 0.7 For 2 links: P s = 0.7 x 0.7 = 0.49 ( P f = 0.51) For n links: P s = 0.7 n ⇒ P f = 1  (0.7) n ∴ P f ↑ as n ↑ • A general feature of nonredundant systems: P f ↑ as size ↑ • Brittle materials are full of flaws ME 382 Lecture 23 2 • If one flaw is critical entire material fails: ∴ Weakestlink statistics apply • If volume flaws responsible for fracture: P f ↑ as volume ↑ • If surface flaws responsible for fracture: P f ↑ as surface area ↑ • Strength of glass limited by surface flaws Fresh glass fibers are very strong because they are very small ( ≈ 1 GPa) Contact forms surface flaws ⇒ strength falls Weibull statistics • If the maximum normal stress in element of material δ V is σ (...
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This note was uploaded on 09/08/2011 for the course MECHENG 382 taught by Professor Thouless during the Fall '08 term at University of Michigan.
 Fall '08
 Thouless

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