Chapter 7:Fracture: 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.)

Tensile 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.

Griffith Criterion on Fracture

(a) Stress distribution ina large plate containing a circularhole. (b) Stress concentrationfactor Kt as a function of the radiusof a circular hole in a large plate intension.

Maximum Stress

Stress concentration atan elliptical hole for a = 3b.

Stress Distribution at Tip of Crack

Griffith Criterion

Crack Propagation

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 at Crack Tip

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

Plane Stress and Plane Strain

Linear Elastic Fracture Mechanics

Inherent material σ resistance to crack growth 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 (see Figure 7.1).

Stress Field

Crack Tip Stress Field

Crack Configuration

Some common load and crack configurations and the corresponding expressions for the stress intensity factor, K.

Plastic Zone Correction

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

Dugdale–Bilby–Cottrell–Swinden model of a crack.

Plastic Zone at the Crack Tip

Variation of Fracture Toughness with Thickness

(a) Variation infracture toughness (Kc) with platethickness (B) for Al 7075-T6 andH-11 Steel. (Reprinted withpermission from J. E. Srawley andW. F. Brown, ASTM STP 381(Philadelphia: ASTM, 1965), p 133,and G. R. Irwin, in Encyclopaedia ofPhysics, Vol. VI (Heidelberg:Springer Verlag, 1958); see also J.Basic Eng., Trans. ASME, 82 (1960)417.) (b) Schematic variation offracture toughness Kc andpercentage of flat fracture P withthe plate thickness B.

Elastic Body with a Crack

(a) Elastic body containing a crack of length 2aunder load P. (b) Diagram of load P versus displacement e.

Fracture Toughness Parameters

Crack Opening Displacement

Crack Opening Displacement

A body subjected toexternal forces F1, F2, . . ., Fn andwith a closed contour .

Integral of External Forces

J Integral: Contours Around Cracks

J Integral

R Curves for Brittle and Ductile Material

Growth of a Crack-Equivalence between Equations

Fracture Toughness and Yield Stress

Variation of fracturetoughness KIc with tensile strengthand 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

Fracture Toughness for Different Alloys

Different Measures of Crack Tip Opening

Strength Distribution for a Brittle and Ductile Solid

Weibull Distribution

A Weibull Plot

A Weibull plot for asteel, a conventional alumina, and acontrolled-particle-size (CPS)alumina. Note that the slope(Weibull modulus m)→∞forsteel. For CPS alumina, m is doublethat of conventional alumina.(After E. J. Kubel, Adv. Mater. Proc.,Aug (1988) 25.)

Flexural Strengths for Ceramics

Flexural strengths(4-point bend test with inner andouter spans 20 and 40 mm,respectively, and cross section of 3× 4 mm) for three ceramics.(Courtesy of C. J. Shih.)