Page 47 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C02.fm Page 27 Thursday, April 14, 2005 6:28 PM

Linear Elastic Fracture Mechanics 27

The surface energy can be estimated as follows:

γ 1 ∫ λ σ = sin  πx  dx = σ λ (2.6)
s
2 0 c  λ  c π
The surface energy per unit area, γ , is equal to one-half of the fracture energy because two surfaces
s
are created when a material fractures. Substituting Equation (2.4) into Equation (2.6) and solving
for σ gives
c
E γ
σ = x o s (2.7)
c

2.2 STRESS CONCENTRATION EFFECT OF FLAWS

The derivation in the previous section showed that the theoretical cohesive strength of a material
is approximately E/π, but experimental fracture strengths for brittle materials are typically three
or four orders of magnitude below this value. As discussed in Chapter 1, experiments by Leonardo
da Vinci, Griffith, and others indicated that the discrepancy between the actual strengths of brittle
materials and theoretical estimates was due to flaws in these materials. Fracture cannot occur unless
the stress at the atomic level exceeds the cohesive strength of the material. Thus, the flaws must
lower the global strength by magnifying the stress locally.
The first quantitative evidence for the stress concentration effect of flaws was provided by Inglis
[1], who analyzed elliptical holes in flat plates. His analyses included an elliptical hole 2a long by
2b wide with an applied stress perpendicular to the major axis of the ellipse (see Figure 2.2). He
assumed that the hole was not influenced by the plate boundary, i.e., the plate width >> 2a and the
plate height >> 2b. The stress at the tip of the major axis (Point A) is given by

a 
σ A σ =   1 + 2 b  (2.8)

The ratio σσ/ is defined as the stress concentration factor k . When a = b, the hole is circular
t
A
and k = 3.0, a well-known result that can be found in most strength-of-materials textbooks.
t
As the major axis, a, increases relative to b, the elliptical hole begins to take on the appearance
of a sharp crack. For this case, Inglis found it more convenient to express Equation (2.8) in terms
of the radius of curvature ρ:
 a 
σ σ =  12  (2.9)
+
A
 ρ 
where

b 2
ρ = (2.10)
a
When a >> b, Equation (2.9) becomes

a
σ A σ = 2 ρ (2.11)