Fundamentals of Materials Science and Engineering: An Integrated Approach, 3e

GTBL042-09 GTBL042-Callister-v3 October 4, 2007 11:53

2nd Revised Pages

9.5 Principles of Fracture Mechanics • 297

ρt

σ 0

σ 0

X X'

x'x

2 a

a

Position along X–X'

(b)

σ 0

σm

Stress

x x'

(a)

Figure 9.8 (a) The geometry of surface and internal cracks. (b) Schematic stress profile

along the lineX–X′in (a), demonstrating stress amplification at crack tip positions.

which is simply a measure of the degree to which an external stress is amplified at

the tip of a crack.

By way of comment, it should be said that stress amplification is not restricted to

these microscopic defects; it may occur at macroscopic internal discontinuities (e.g.,

voids), at sharp corners, and at notches in large structures.

Furthermore, the effect of a stress raiser is more significant in brittle than in duc-

tile materials. For a ductile material, plastic deformation ensues when the maximum

stress exceeds the yield strength. This leads to a more uniform distribution of stress in

the vicinity of the stress raiser and to the development of a maximum stress concen-

tration factor less than the theoretical value. Such yielding and stress redistribution

do not occur to any appreciable extent around flaws and discontinuities in brittle

materials; therefore, essentially the theoretical stress concentration will result.

Using principles of fracture mechanics, it is possible to show that the critical stress

σcrequired for crack propagation in a brittle material is described by the expression

σc=

(

2 Eγs

πa

) 1 / 2

(9.3)

Critical stress for

crack propagation in

a brittle material

where

E=modulus of elasticity

γs=specific surface energy

a=one half the length of an internal crack

All brittle materials contain a population of small cracks and flaws that have a

variety of sizes, geometries, and orientations. When the magnitude of a tensile stress

at the tip of one of these flaws exceeds the value of this critical stress, a crack forms