Chapter 7Fracture: Macroscopic Aspects

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.)

Theoretical Tensile Strength

Theoretical Cleavage Strength

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.

Inglis: Stress Concentration

(a) Stress distribution in a large plate containing a circular hole.

(b) Stress concentration factor Kt as a function of the radius of a circular hole in a large plate in tension.

Stress Concentration due to a Circular Hole

Stress concentration at an elliptical hole for a = 3b.

Stress Concentration due to an Elliptical Hole

Griffith Criterion of Crack Propagation

Crack in Thin and Thick Plates

Crack in (a) thin (t1) and (b) thick (t2) plates. Note the plane-stress state in (a) and the plane-strain state in (b).

Dislocation Emission at Crack Tip

Dislocations emitted from a crack tip in copper. TEM. (Courtesy of S. M. Ohr.)

Plane Stress and Plane Strain

Linear Elastic Fracture Mechanics

Inherent material resistance to crack growth, KR and its relationship to the applied stress σ and crack size a.

Three Modes of Fracture

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

Stress Field at a Crack Tip

Crack Tip Stress Field

Some Crack and Loading Configurations

Plastic Zone Correction

Plastic-zone correction. The effective crack length is (a + ry).

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

Plastic Zone at Crack Tip Plane Stress and Plane Strain

Variation of Fracture Toughness with Thickness

(a) Variation infracture toughness (Kc) 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 Kc and percentage of flat fracture P with the plate thickness B.

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.

Crack Extension Force

Crack Opening Displacement

Crack Opening Displacement

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

Body under External Forces

J Integral

J Integral

R Curves for Brittle and Ductile Material

Different Parameters for Fracture Toughness

Fracture Toughness vs. Yield Stress

Variation of fracture toughness KIc 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

Plane Strain Fracture Toughness

Fracture Toughness vs. Yield Strength for Different Alloys

Measures of Crack Tip Opening Displacement

Strength Distribution for a Brittle and Ductile Solid

Weibull Distribution

Typical Values of Weibull Modulus

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.)

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.)