Page 385 - Mechanical Behavior of Materials

386 Chapter 8 Fracture of Cracked Members

σ y , and thus relatively small Poisson contraction in the z-direction. This makes it difficult for the
material inside the plastic zone to deform in the z-direction, as its length in the z-direction is held
nearly constant by the surrounding material. Hence, the behavior is said to approximate plane strain,
defined by ε z = 0. As a result, a tensile stress develops in the z-direction, which elevates the value
of σ x = σ y necessary to cause yielding, in turn decreasing the plastic zone size relative to that for
plane stress.
To explore this in more detail, note that ε z = 0, when substituted with σ x = σ y into Hooke’s
law, Eq. 5.26, gives a stress in the z-direction of σ z = 2νσ y . Substituting these stresses into either
the octahedral shear stress yield criterion or the maximum shear stress yield criterion gives a stress at
yielding of σ x = σ y = σ o /(1 − 2ν); that is, σ x = σ y = 2.5σ o for a typical value of Poisson’s ratio
of ν = 0.3. This situation has already been analyzed as Ex. 7.5, except that the stresses there were
compressive, but the same result as just stated is obtained for tensile stresses. Hence, the constrained
deformation creates a tensile hydrostatic stress that, in effect, subtracts from the ability of the applied
stresses σ x = σ y to cause yielding, resulting in an apparent elevation of the yield strength.
The more refined estimate by G. R. Irwin suggests that the effect is somewhat smaller, with
√
yielding around σ y = 3σ o . Proceeding as for the plane stress estimate, except for using the latter
value of σ y , we obtain
1 K 2
2r oε = (8.38)
3π σ o
This is noted to be one-third as large as the plane-stress value.
The plastic zone size equations given are based on simple assumptions and should be considered
to be rough estimates only. The particular estimates given follow the early work of G. R. Irwin.

8.7.3 Plasticity Limitations on LEFM
If the plastic zone is sufficiently small, there will be a region outside of it where the elastic stress field
equations (Eq. 8.7) still apply, called the region of K-dominance,orthe K-field. This is illustrated
in Fig. 8.45. The existence of such a region is necessary for LEFM theory to be applicable. The
K-field surrounds and controls the behavior of the plastic zone and crack tip area, which can
be thought of as an incompletely understood “black box.” Thus, K continues to characterize the
severity of the crack situation, despite the occurrence of some limited plasticity. However, if the
plastic zone is so large that it eliminates the K-field, then K no longer applies.
As a practical matter, it is necessary that the plastic zone be small compared with the distance
from the crack tip to any boundary of the member, such as distances a,(b − a), and h for a cracked
plate, as in Fig. 8.46(a). A distance of 8r o is generally considered to be sufficient. Note from
Eqs. 8.37 and 8.38 that 8r o is four times the plastic zone size, which can be either 2r oσ or 2r oε ,
depending on which applies. Since 2r oσ is larger than 2r oε , an overall limit on the use of LEFM is
2

4 K

a,(b − a), h ≥ (LEFM applicable) (8.39)
π σ o
This must be satisfied for all three of a,(b − a), and h. Otherwise, the situation too closely
approaches gross yielding with a plastic zone extending to one of the boundaries, as shown in