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An illustration of the variation of stress intensity factor with crack length for prescribed load & prescribed displacement Using results from slide on DCB specimen 4 Pa 3 ∆= Eb3 Pa K = 2 3 3/ 2 , b 3Eb3/ 2 K= ∆ 2a 2 (P is the force/thickness) For fixed P, K increases as a increases. For fixed ∆, K decreases as a increases. K versus crack advance With initial crack length, a0 , load up to P0 4 2 3P0 a0 . 3/ 2 b If we fix P = P0 and increase the crack length, 3 with ∆ 0 = 4 P0 a03 /( Eb3 ) and K 0 = K a = K 0 a0 Prescribed P=P 0 2 However, if we fix ∆ = ∆ 0 and increase the crack length, K ⎛ a0 ⎞ =⎜ ⎟ K0 ⎝ a ⎠ 1 Prescribed ∆=∆ 2 0 1 See plot for very different trends 0 1.5 2 a/a 2.5 0 3 The concept of small scale yielding for cracks in metallic materials σ Y ∼ tensile yield stress L ∼ crack length (a ) or other relevant length region in which K-field holds rp Mode I small scale yielding plastic zones Details of these shapes will be discussed later 1 rp ≅ 3π ⎛K ⎞ ⎜ ⎟ σ ⎝ Y⎠ 2 1⎛ K ⎞ rp ≅ ⎜ ⎟ π ⎝ σY ⎠ 2 π⎛K ⎞ rp = ⎜ ⎟ 8 ⎝ σY ⎠ Small scale yielding requires that rp be sufficiently small For the crack of length 2a in an infinite compared to L such that there exists a region surrounding the plastic zone in which the K -field is accurate. sheetor slab, this condition requires rp << a / 4. 2 Griffith-Irwin Criterion for Mode I Crack Advance in an Ideally Brittle Material K (pg. 14 of notes) K IC Quasi-static advance occurs when K = K IC K IC is called the fracture toughness (units Pa m ) (1 −ν 2 ) K IC 2 GIC = is also called the fracture toughness (units J / m 2 ) E crack advance ∆a We will discuss experimental methods to measure fracture toughness later. To use LEFM, the test must satisfy small scale yielding conditions in plane strain. In the table Below we list representative values of yield strength, fracture toughness and plastic zone size for four metals ranging from relatively low toughness to high toughness. material T (o C ) σ Y (MN / m 2 ) K IC ( MPa m ) GIC ( J / m 2 ) rp (mm) 6061 − T 651( Al ) 20 269 33 14, 000 5 7075 − T 651( Al ) 20 AISI 4340 (Steel) 0 620 1500 36 33 16, 600 4900 0.35 0.05 A533B (Steel) Typical ceramic 620 ?? 200 0.3 − 0.6 93 20 180, 000 5 − 20 260 ?? From Material Selection in Mechanical Design, M.F. Ashby, Pergamon Press Crack Advance in Ideally in a Brittle 4340 Steel Compact Tension Specimen KI = ASTM compact tensile specimen P ⎛a⎞ a F1 ⎜ ⎟ bt ⎝b⎠ ⎛a⎞ ⎛ 0.6 − a / b ⎞ ⎛ a / b − 0.4 ⎞ F1 ⎜ ⎟ ≅ 11.7 ⎜ ⎟ + 17.6 ⎜ ⎟ ⎝b⎠ ⎝ 0.2 ⎠ ⎝ 0.2 ⎠ (see earlier slide) And from Tada, pg. 62 δ choose : b = 2in = 0.0508m h = 0.6b h1 = 0.275b c = D = 0.25b thickness ≡ t = b / 2 K IC = 33MPam −1/ 2 , E = 200GPa, ν = 0.29 P ⎛a⎞ ⎛1+ x ⎞ 2 3 4 5 δ = V2 ⎜ ⎟ , V2 ( x) = ⎜ ⎟ ( 2.163 + 12.219 x − 20.065 x − 0.9925 x + 20.609 x − 9.9314 x ) Et ⎝ b ⎠ ⎝1− x ⎠ 2 -6 3 10 4 4 10 2.5 10 4 3.5 10 2 10 4 3 10 1.5 10 4 2.5 10 1 10 4 2 10 -6 -6 -6 -6 0.4 0.45 0.5 a/b 0.55 0.6 0.4 0.45 0.5 a/b 0.55 0.6 Thickness and Temperature Dependence of Fracture Toughness (pg. 16 & 17 of notes) Schematic of Experimental Thickness dependence KC Schematic of Experimental Temperature dependence for a steel K IC Temperature T thickness t Lower temperature range governed by cleavage “Upper shelf” range governed by ductile void nucleation, growth an coalescence mechanism ASTM requirement for valid K IC test : The mechanism transition is referred to as the brittle to ductile toughness transition. 2 2 ⎛K ⎞ ⎛K ⎞ t > 2.5 ⎜ IC ⎟ ≅ 25rp & t > 2.5 ⎜ IC ⎟ ≅ 25rp ⎝ σY ⎠ ⎝ σY ⎠ Compact tension specimens for valid KIC testing. The smallest specimen is about 2 inches wide. The largest specimen was used to obtain the toughness of a very tough pressure vessel steel (A533B), as required by LEFM.