Application of the Model

As with the J-Q approach, the implementation of the scaling model requires detailed elastic-plastic finite element analysis of the configuration of interest. The principal stress contours must be con-structed and the areas compared with the T = 0 reference solution obtained from a modified boundary layer analysis. The effective driving force Jo is then plotted against the applied J, as Figure 3.41 schematically illustrates. At low deformation levels, the Jo-J curves follow the 1:1 line, but deviate from the line with further deformation. When J Jo, the crack-tip stress fields are close to the Q = 0 limit, and fracture toughness is not significantly influenced by specimen boundaries. At high defor-mation levels J > Jo and the fracture toughness is artificially elevated by constraint loss. Constraint loss occurs more rapidly in specimens with shallow cracks, as Figure 3.28 illustrates. A specimen with a/W = 0.15 would tend to fail at a higher Jc value than a specimen with a/W = 0.5. Given the Jo-J curve, however, the Jc values for both specimens can be corrected to Jo, as Figure 3.41 illustrates.

Figure 3.42 is a nondimensional plot of Jo at the midplane vs. the average J through the thickness of SENB specimens with various W/B ratios [36]. These curves were inferred from a

FIGURE 3.41 Schematic illustration of the scaling model. A specimen with a/W = 0.15 will fail at a higher Jc value than a specimen with a/W = 0.5, but both Jc values can be corrected down to the same critical Jo value.

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three-dimensional elastic-plastic analysis. The corresponding curve from a two-dimensional plane strain analysis is shown for comparison. Note that for W/B = 1 and 2, Jo at the midplane lies well above the plane-strain curve. For W/B = 4, Jo at the midplane follows the plane-strain curve initially, but falls below the plane-strain results at high deformation levels. The three-dimensional nature of the plastic deformation apparently results in a high level of constraint at the midplane when the uncracked ligament length is ≤ the specimen thickness.

Figure 3.43 is a plot of effective thickness Beff as a function of deformation. The trends in this plot are consistent with Figure 3.42; namely, the constraint increases with decreasing W/B. Note that all three curves reach a plateau. Recall that Beff is defined in such a way as to be a measure of the through-thickness relaxation of constraint, relative to the in-plane constraint at the midplane. At low deformation levels there is negligible relation at the midplane and J ≈ Jo, but a through-thickness constraint relation occurs, resulting in a falling Beff/B ratio. At high deformation levels, the Beff/B ratio is essentially constant, indicating that the constraint relaxation is proportional in three dimen-sions. Figure 3.44 and Figure 3.45 show data that have been corrected with the scaling model.

3.6.4 LIMITATIONSOF TWO-PARAMETER FRACTURE MECHANICS

The T stress approach, J-Q theory, and the cleavage scaling model are examples of two-parameter fracture theories, where a second quantity (e.g., T, Q, or Jo) has been introduced to characterize the crack-tip environment. Thus these approaches assume that the crack-tip fields contain two degrees of freedom. When single-parameter fracture mechanics is valid, the crack-tip fields have only one degree of freedom. In such cases, any one of several parameters (e.g., J, K, or CTOD) will suffice to characterize the crack-tip conditions, provided the parameter can be defined unambiguously; K is a suitable characterizing parameter only when an elastic singularity zone exists ahead of the crack tip.⁸ Similarly, the choice of a second parameter in FIGURE 3.42 Effective driving force for cleavage Jo for deeply notched SENB specimens.

8 An effective K can be inferred from J through Equation (3.18). Such a parameter has units of K, but it loses its meaning as the amplitude of the elastic singularity when such a singularity no longer exists.

150 Fracture Mechanics: Fundamentals and Applications

FIGURE 3.43 Effective thickness for deeply notched SENB specimens.

FIGURE 3.44 Fracture toughness data for a mild steel, corrected for constraint loss. Taken from Anderson, T.L. and Dodds, R.H., Jr., ‘‘Specimen Size Requirements for Fracture Toughness Testing in the Ductile-Brittle Transition Region.’’Journal of Testing and Evaluation, Vol. 19, 1991, pp. 123–134; Sorem, W.A., ‘‘The Effect of Specimen Size and Crack Depth on the Elastic-Plastic Fracture Toughness of a Low-Strength High-Strain Hardening Steel.’’ Ph.D. Dissertation, The University of Kansas, Lawrence, KS, 1989.

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the case of two-parameter theory is mostly arbitrary, but the T stress has no physical meaning under large-scale yielding conditions.

Just as plastic flow invalidates single-parameter fracture mechanics in many geometries, two-parameter theories eventually break down with extensive deformation. If we look at the structure of the crack-tip fields in the plastic zone, we can evaluate the range of validity of both single- and two-parameter methodologies.

A number of investigators [39–43] have performed asymptotic analyses of the crack-tip fields for elastic-plastic materials. These analyses utilize deformation plasticity and small-strain theory.

In the case of plane strain, these analyses assume incompressible strain. Consequently, asymptotic analyses are not valid close to the crack tip (in the large-strain zone) nor remote from the crack tip, where elastic strains are a significant fraction of the total strain. Despite these limitations, asymptotic analysis provides insights into the range of validity of both single- and two-parameter fracture theories.

In the case of a plane strain crack in a power-law-hardening material, asymptotic analysis leads to the following power series:

(3.87) FIGURE 3.45 Experimental data from Figure 3.28 corrected for constraint loss. Taken from Anderson, T.L., Stienstra, D.I.A., and Dodds, R.H., Jr., ‘‘A Theoretical Framework for Addressing Fracture in the Ductile-Brittle Transition Region.’’Fracture Mechanics, Vol. 24, ASTM STP 1207, American Society for Testing and Materials, Philadelphia, PA (in press).

σ_{ij k} α ε σ σ θ

k o o

k ijk

A J

r s

=  n





= 

∑

ˆ^{( )}( , )

152 Fracture Mechanics: Fundamentals and Applications The exponents sk and the angular functions for each term in the series can be determined from the asymptotic analysis. The amplitudes for the first five terms are as follows:

The two unspecified coefficients A₂ and A₄ are governed by the far-field boundary conditions. The first five terms of the series have three degrees of freedom, where J, A₂, and A₄ are independent parameters. For low and moderate strain hardening materials, Crane [43] showed that a fourth independent parameter does not appear in the series for many terms. For example, when n = 10, the fourth independent coefficient appears in approximately the 100th term. Thus for all practical purposes, the crack-tip stress field inside the plastic zone has three degrees of freedom.

Since two-parameter theories assume two degrees of freedom, they cannot be rigorously correct in general. There are, however, situations where two-parameter approaches provide a good engineering approximation.

Consider the modified boundary layer model in Figure 3.32. Since the boundary conditions have only two degrees of freedom (K and T), the resulting stresses and strains inside the plastic zone must be two-parameter fields. Thus there must be a unique relationship between A₂ and A₄ in this case. That is

(3.88) The two-parameter theory is approximately valid for other geometries to the extent that the crack-tip fields obey Equation (3.88). Figure 3.46 schematically illustrates the A₂-A₄ relationship.

This relationship can be established by varying the boundary conditions on the modified boundary layer model. When a given cracked geometry is loaded, A₂ and A₄ initially will evolve in accordance with Equation (3.88) because the crack-tip conditions in the geometry of interest can be represented by the modified boundary layer model when the plastic zone is relatively small. Under large-scale yielding conditions, however, the A₂-A₄ relationship may deviate from the modified boundary layer solution, in which case the two-parameter theory is no longer valid.

Figure 3.47 is a schematic three-dimensional plot of J, A₂, and A₄. Single-parameter fracture mechanics can be represented by a vertical line, since A₂ and A₄ must be constant in this case.

The two-parameter theory, where Equation (3.88) applies, can be represented by a surface in this three-dimensional space. The loading path for a cracked body initially follows the vertical

FIGURE 3.46 Schematic relationship between the two independent amplitudes in the asymptotic power series.

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single-parameter line. When it deviates from this line, it may remain in the two-parameter surface for a time before diverging from the surface.

The loading path in J-A2-A4 space depends on geometry [43]. Low-constraint configurations such as the center-cracked panel and shallow notched bend specimens diverge from the single-parameter theory almost immediately, but follow Equation (3.88) to fairly high deformation levels.

Deeply notched bend specimens maintain a high constraint to relatively high J values, but they do not follow Equation (3.88) when constraint loss eventually occurs. Thus low-constraint geometries should be treated with the two-parameter theory, and high-constraint geometries can be treated with the parameter theory in many cases. When high-constraint geometries violate the single-parameter assumption, however, the two-single-parameter theory is of little value.

APPENDIX 3: MATHEMATICAL FOUNDATIONS OF ELASTIC-PLASTIC