Page 81 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

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Linear Elastic Fracture Mechanics 61

is assumed to remain planar and maintain a constant shape as it grows. Such is usually not the case
for mixed-mode fractures. See Section 2.11 for further discussion of energy release rate in mixed-
mode problems.

2.8 CRACK-TIP PLASTICITY

Linear elastic stress analysis of sharp cracks predicts infinite stresses at the crack tip. In real
materials, however, stresses at the crack tip are finite because the crack-tip radius must be finite
(Section 2.2). Inelastic material deformation, such as plasticity in metals and crazing in polymers,
leads to further relaxation of crack-tip stresses.
The elastic stress analysis becomes increasingly inaccurate as the inelastic region at the crack
tip grows. Simple corrections to linear elastic fracture mechanics (LEFM) are available when
moderate crack-tip yielding occurs. For more extensive yielding, one must apply alternative crack-
tip parameters that take nonlinear material behavior into account (see Chapter 3).
The size of the crack-tip-yielding zone can be estimated by two methods: the Irwin approach,
where the elastic stress analysis is used to estimate the elastic-plastic boundary, and the strip-yield
model. Both approaches lead to simple corrections for crack-tip yielding. The term plastic zone usually
applies to metals, but will be adopted here to describe inelastic crack-tip behavior in a more general
sense. Differences in the yielding behavior between metals and polymers are discussed in Chapter 6.

2.8.1 THE IRWIN APPROACH
On the crack plane (θ = 0), the normal stress σ in a linear elastic material is given by Equation
yy
(2.39). As a first approximation, we can assume that the boundary between elastic and plastic
behavior occurs when the stresses given by Equation (2.39) satisfy a yield criterion. For plane
stress conditions, yielding occurs when σ = σ , the uniaxial yield strength of the material.
yy
YS
Substituting yield strength into the left side of Equation (2.39) and solving for r gives a first-order
estimate of the plastic zone size:
K 

1
I
r = 2πσ YS   2 (2.64)

y

If we neglect strain hardening, the stress distribution for r = r can be represented by a horizontal
y
line at σ = σ , as Figure 2.29 illustrates; the stress singularity is truncated by yielding at the
yy
YS
crack tip.
The simple analysis in the preceding paragraph is not strictly correct because it was based on
an elastic crack-tip solution. When yielding occurs, stresses must redistribute in order to satisfy
equilibrium. The cross-hatched region in Figure 2.29 represents forces that would be present in an
elastic material but cannot be carried in the elastic-plastic material because the stress cannot exceed
the yield. The plastic zone must increase in size in order to accommodate these forces. A simple
force balance leads to a second-order estimate of the plastic zone size r :
p
r y r y K
σ YS p ∫ σ r yy d r= = ∫ I dr (2.65)
0 0 2 π r
Integrating and solving for r gives
p
K 

1
I
r = πσ YS   2 (2.66)

p

which is twice as large as r , the first-order estimate.
y