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Chapter 7 Fracture: Macroscopic Aspects

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Goofy Duck Analog for Modes of Crack Loading “Goofy duck” analog for three modes of crack loading. (a) Crack/beak closed. (b) Opening mode. (c) Sliding mode. (d) Tearing mode. (Courtesy of M. H. Meyers.)

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Theoretical Tensile Strength

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Theoretical Cleavage Strength

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Stress Concentration “ Lines of force” in a bar with a side notch. The direction and density of the lines indicate the direction and magnitude of stress in the bar under a uniform stress σ away from the notch. There is a concentration of the lines of force at the tip of the notch.

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Inglis: Stress Concentration

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(a)Stress distribution in a large plate containing a circular hole. (b) Stress concentration factor K t as a function of the radius of a circular hole in a large plate in tension. Stress Concentration due to a Circular Hole

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Stress concentration at an elliptical hole for a = 3b. Stress Concentration due to an Elliptical Hole

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Griffith Criterion of Crack Propagation

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Crack in Thin and Thick Plates Crack in (a) thin (t 1 ) and (b) thick (t 2 ) plates. Note the plane-stress state in (a) and the plane-strain state in (b).

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Dislocation Emission at Crack Tip Dislocations emitted from a crack tip in copper. TEM. (Courtesy of S. M. Ohr.)

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Plane Stress and Plane Strain

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Linear Elastic Fracture Mechanics Inherent material resistance to crack growth, K R and its relationship to the applied stress σ and crack size a.

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Three Modes of Fracture The three modes of fracture. (a) Mode I: opening mode. (b) Mode II: sliding mode. (c) Mode III: tearing mode.

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Stress Field at a Crack Tip

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Crack Tip Stress Field

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Some Crack and Loading Configurations

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Plastic Zone Correction Plastic-zone correction. The effective crack length is (a + r y ).

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Dugdale–Bilby–Cottrell–Swinden Model of a Crack.

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Plastic Zone at Crack Tip Plane Stress and Plane Strain

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Variation of Fracture Toughness with Thickness (a) Variation infracture toughness (K c ) with plate thickness (B) for Al 7075-T6 and H-11 Steel. (Reprinted with permission from J. E. Srawley and W. F. Brown, ASTM STP 381 (Philadelphia: ASTM, 1965), p 133, and G. R. Irwin, in Encyclopaedia of Physics, Vol. VI (Heidelberg: Springer Verlag, 1958). (b) Schematic variation of fracture toughness K c and percentage of flat fracture P with the plate thickness B.

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Elastic Body with a Crack (a)Elastic body containing a crack of length 2a under load P. (b) Diagram of load P versus displacement e.

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Crack Extension Force

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Crack Opening Displacement

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A body subjected to external forces F1, F2,..., Fn and with a closed contour. Body under External Forces

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J Integral

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R Curves for Brittle and Ductile Material

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Different Parameters for Fracture Toughness

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Fracture Toughness vs. Yield Stress

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Variation of fracture toughness K Ic with tensile strength and sulfur content in a steel. (Adapted from A. J. Birkle, R. P. Wei, and G. E. Pellissier, Trans. ASM, 59 (1966) 981.) Fracture Toughness: Effect of Impurities

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Plane Strain Fracture Toughness

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Fracture Toughness vs. Yield Strength for Different Alloys

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Measures of Crack Tip Opening Displacement

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Strength Distribution for a Brittle and Ductile Solid

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Weibull Distribution

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Typical Values of Weibull Modulus

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Weibull Plots for Steel and Two Alumina samples Weibull plots for a steel, a conventional alumina, and a controlled-particle-size (CPS) alumina. Note that the slope (Weibull modulus m)→∞ for steel. For CPS alumina, m is double that of conventional alumina. (After E. J. Kubel, Adv. Mater. Proc., Aug (1988) 25.)

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Probability of Failure for Three Ceramics Probability of failure of flexural strength (4-point bend test with inner and outer spans 20 and 40 mm, respectively, and cross section of 3 × 4 mm) for three ceramics. (Courtesy of C. J. Shih.)

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