Page 84 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

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64 Fracture Mechanics: Fundamentals and Applications

FIGURE 2.31 The strip-yield model. The plastic zone is modeled by yield magnitude compressive stresses
at each crack tip (b).

2.8.2 THE STRIP-YIELD MODEL
The strip-yield model, which is illustrated in Figure 2.31, was first proposed by Dugdale [24] and
Barenblatt [25]. They assumed a long, slender plastic zone at the crack tip in a nonhardening
material in plane stress. These early analyses considered only a through crack in an infinite plate.
The strip-yield plastic zone is modeled by assuming a crack of length 2a + 2ρ, where ρ is the
length of the plastic zone, with a closure stress equal to σ applied at each crack tip (Figure 2.31(b)).
YS
This model approximates elastic-plastic behavior by superimposing two elastic solutions: a
through crack under remote tension and a through crack with closure stresses at the tip. Thus the
strip-yield model is a classical application of the principle of superposition.
Since the stresses are finite in the strip-yield zone, there cannot be a stress singularity at the
crack tip. Therefore, the leading term in the crack-tip field that varies with 1 r (Equation (2.36))
must be zero. The plastic zone length ρ must be chosen such that the stress intensity factors from
the remote tension and closure stress cancel one another.
The stress intensity due to the closure stress can be estimated by considering a normal force
P applied to the crack at a distance x from the centerline of the crack (Figure 2.32). The stress
intensities for the two crack tips are given by

P a x +
K I a(+ ) = π a a x − (2.73a)

P a x −
K I a(− ) = π a a x + (2.73b)

assuming the plate is of unit thickness. The closure force at a point within the strip-yield zone is
equal to

P YS d =−σ x (2.74)

FIGURE 2.32 Crack-opening force applied at a dis-
tance x from the center-line.