Elastic-Plastic Fracture Mechanics
3.1 CRACK-TIP-OPENING DISPLACEMENT
When Wells [1] attempted to measure KIcvalues in a number of structural steels, he found that these materials were too tough to be characterized by LEFM. This discovery brought both good news and bad news: High toughness is obviously desirable to designers and fabricators, but Wells’
experiments indicated that the existing fracture mechanics theory was not applicable to an important class of materials. While examining fractured test specimens, Wells noticed that the crack faces had moved apart prior to fracture; plastic deformation had blunted an initially sharp crack, as illustrated in Figure 3.1. The degree of crack blunting increased in proportion to the toughness of the material. This observation led Wells to propose the opening at the crack tip as a measure of fracture toughness. Today, this parameter is known as CTOD.
In his original paper, Wells [1] performed an approximate analysis that related CTODto the stress intensity factor in the limit of small-scale yielding. Consider a crack with a small plastic zone, as illustrated in Figure 3.2. Irwin [2] postulated that crack-tip plasticity makes the crack behave as if it were slightly longer (Section 2.8.1). Thus, we can estimate the CTODby solving for the displacement at the physical crack tip, assuming an effective crack length of a + ry. From Table 2.2, the displacement ry behind the effective crack tip is given by
(3.1)
where E′ is the effective Young’s modulus, as defined in Section 2.7. The Irwin plastic zone correction for plane stress is
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Substituting Equation (3.2) into Equation (3.1) gives
(3.3) where δ is the CTOD. Alternatively, CTODcan be related to the energy release rate by applying Equation (2.54):
(3.4) Thus, in the limit of small-scale yielding, CTODis related to G and KI. Wells postulated that CTOD is an appropriate crack-tip-characterizing parameter when LEFM is no longer valid. This assumption was shown to be correct several years later when a unique relationship between CTODand the J integral was established (Section 3.3).
The strip-yield model provides an alternate means for analyzing CTOD [3]. Recall Section 2.8.2, where the plastic zone was modeled by yield magnitude closure stresses. The size of the strip-yield zone was defined by the requirement of finite stresses at the crack tip. The CTODcan be defined as the crack-opening displacement at the end of the strip-yield zone, as Figure 3.3 illustrates. According to this definition, CTODin a through crack in an infinite plate subject to a remote tensile stress (Figure 2.3) is given by [3]
(3.5) FIGURE 3.1 Crack-tip-opening displacement (CTOD).
An initially sharp crack blunts with plastic deforma-tion, resulting in a finite displacement (d) at the crack tip.
FIGURE 3.2 Estimation of CTOD from the displace-ment of the effective crack in the Irwin plastic zone correction.
δ =2u =π σ4 K2
y E
I YS
δ=π σ4 G
YS
δ σ
π π σ
= σ
8
2
YS
YS
a E ln sec
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Elastic-Plastic Fracture Mechanics 105
Equation (3.5) is derived in Appendix 3.1. Series expansion of the “ln sec” term gives
(3.6)
Therefore, as
(3.7) which differs slightly from Equation (3.3).
The strip-yield model assumes plane stress conditions and a nonhardening material. The actual relationship between CTODand KI and G depends on stress state and strain hardening. The more general form of this relationship can be expressed as follows:
(3.8)
where m is a dimensionless constant that is approximately 1.0 for plane stress and 2.0 for plane strain.
There are a number of alternative definitions of CTOD. The two most common definitions, which are illustrated in Figure 3.4, are the displacement at the original crack tip and the 90° intercept. The latter definition was suggested by Rice [4] and is commonly used to infer CTODin finite element measurements. Note that these two definitions are equivalent if the crack blunts in a semicircle.
Most laboratory measurements of CTODhave been made on edge-cracked specimens loaded in three-point bending (see Table 2.4). Early experiments used a flat paddle-shaped gage that was inserted into the crack; as the crack opened, the paddle gage rotated, and an electronic signal was sent to an x-y plotter. This method was inaccurate, however, because it was difficult to reach the crack tip with the paddle gage. Today, the displacement V at the crack mouth is measured, and the CTODis inferred by assuming the specimen halves are rigid and rotate about a hinge point, as illustrated in Figure 3.5.
Referring to this figure, we can estimate CTODfrom a similar triangles construction:
FIGURE 3.3 Estimation of CTOD from the strip-yield model. Taken from Burdekin, F.M. and Stone, D.E.W., ‘‘The Crack Opening Displacement Approach to Fracture Mechanics in Yielding Materials.’’Journal of Strain Analysis, Vol. 1, 1966, pp. 145–153.
δ σ
106 Fracture Mechanics: Fundamentals and Applications
Therefore
(3.9) where r is the rotational factor, a dimensionless constant between 0 and 1.
The hinge model is inaccurate when displacements are primarily elastic. Consequently, standard methods for CTODtesting [5, 6] typically adopt a modified hinge model, in which displacements are separated into elastic and plastic components; the hinge assumption is applied only to plastic dis-placements. Figure 3.6 illustrates a typical load (P) vs. displacement (V) curve from a CTODtest.
The shape of the load-displacement curve is similar to a stress-strain curve: It is initially linear but deviates from linearity with plastic deformation. At a given point on the curve, the displacement is separated into elastic and plastic components by constructing a line parallel to the elastic loading line.
FIGURE 3.4 Alternative definitions of CTOD: (a) displacement at the original crack tip and (b) displacement at the intersection of a 90° vertex with the crack flanks.
FIGURE 3.5 The hinge model for estimating CTOD from three-point bend specimens.
δ = −
− + r W a V r W a a
( )
( )
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Elastic-Plastic Fracture Mechanics 107
The dashed line represents the path of unloading for this specimen, assuming the crack does not grow during the test. The CTODin this specimen is estimated by
(3.10)
The subscripts el and p denote elastic and plastic components, respectively. The elastic stress intensity factor is computed by inserting the load and specimen dimensions into the appropriate expression in Table 2.4. The plastic rotational factor rp is approximately 0.44 for typical materials and test specimens.
Note that Equation (3.10) reduces to the small-scale yielding result (Equation (3.8)) for linear elastic conditions, but the hinge model dominates when ..
Further details of CTODtesting are given in Chapter 7. Chapter 9 outlines how CTODis used in design.