Page 52 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

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32 Fracture Mechanics: Fundamentals and Applications

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Consider a crack with ρ = 5 × 10 m. Such a crack would appear sharp under a light microscope,
but ρ would be four orders of magnitude larger than the atomic spacing in a typical crystalline
solid. Thus the local stress approach would predict a global fracture strength 100 times larger than
the Griffith equation. The actual material behavior is somewhere between these extremes; fracture
stress does depend on notch root radius, but not to the extent implied by the Inglis stress analysis.
The apparent discrepancy between the critical stress criterion and the energy criterion based
on thermodynamics can be resolved by viewing fracture as a nucleation and growth process. When
the global stress and crack size satisfy the Griffith energy criterion, there is sufficient thermodynamic
driving force to grow the crack, but fracture must first be nucleated. This situation is analogous to
the solidification of liquids. Water, for example, is in equilibrium with ice at 0°C, but the liquid-
solid reaction requires ice crystals to be nucleated, usually on the surface of another solid (e.g.,
your car windshield on a January morning). When nucleation is suppressed, liquid water can be
super cooled (at least momentarily) to as much as 30°C below the equilibrium freezing point.
Nucleation of fracture can come from a number of sources. For example, microscopic surface
roughness at the tip of the flaw could produce sufficient local stress concentration to nucleate
failure. Another possibility, illustrated in Figure 2.5, involves a sharp microcrack near the tip of a
macroscopic flaw with a finite notch radius. The macroscopic crack magnifies the stress in the
vicinity of the microcrack, which propagates when it satisfies the Griffith equation. The microcrack
links with the large flaw, which then propagates if the Griffith criterion is satisfied globally. This
type of mechanism controls cleavage fracture in ferritic steels, as discussed in Chapter 5.

2.3.2 MODIFIED GRIFFITH EQUATION
Equation (2.19) is valid only for ideally brittle solids. Griffith obtained a good agreement between
Equation (2.19) and the experimental fracture strength of glass, but the Griffith equation severely
underestimates the fracture strength of metals.
Irwin [4] and Orowan [5] independently modified the Griffith expression to account for materials
that are capable of plastic flow. The revised expression is given by
/
γ E( s  2 γ +  ) 12
σ =   πa p   (2.21)
f
where γ is the plastic work per unit area of surface created and is typically much larger than γ .
s
p

FIGURE 2.5 A sharp microcrack at the tip of a macroscopic crack.