X.K. Zhu

J-A2 Characterization of Crack-Tip Fields: Extent of J-A2 Dominance and Size Requirements

The requirements for J-dominance, limits of the single-parameter criterion to characterize the fracture of engineering structures, and two-parameter fracture analyses are first reviewed. Through comparison, it is argued that the two-parameter fracture methodology based on the J-A2 theory is a reasonable extension of the single parameter (J-integral) fracture methodology. Consequently the extent of J-A2 dominance in various specimens under either tension or bending is investigated in detail in this paper. Using the J2 flow theory of plasticity and within the small-strain framework, full field finite element solutions are obtained for both deep and shallow crack geometries of single edge notch bar under pure bending [SEN(B)] and central cracked panel in uniform tension [CC(T)]. These crack-tip stresses are compared with those in the HRR singularity fields and the J-A2 asymptotic fields at the same level of applied J. The comparison indicates that the size R of the region dominated by the J-A2 field is much larger than that of the HRR field around the crack tip. Except for deeply-cracked SEN(B) in low hardening material (n=10) under fully plastic conditions, the numerical results near the crack tip in both SEN(B) and CC(T) match very well with the J-A2 asymptotic solutions in the area of interest 1<r/(J/σ0)<5 from well-contained to large scale plasticity. The implications of these results on the minimal specimen size requirements essential to a two-parameter fracture criterion based on the J-A2 asymptotic solution are then discussed.

Quantification of constraint on elastic–plastic 3D crack front by the J–A2 three-term solution

Abstract Three-dimensional (3D) modified boundary layer analyses are performed using the finite element method to study the crack-front constraint for an elastic–plastic thin plate. Far field loading is applied through the plane-stress displacement fields based on the elastic K I -field, and specimen constraint levels are varied through the T -stress applied on the far field boundary. Numerical stresses at the crack front at different planes of the plate are compared with those determined by the J–A 2 three-term solution. Results show that the in-plane-stress fields at the crack front for various K I – T loads possess the plane-strain nature throughout the thickness except for the region near the free surface, and can be characterized by the J–A 2 three-term solution under the small scale yielding condition. The transition of the stress field from the far field, in the plane-stress state, to the near crack-front field, dominated by the plane-strain state, is explained by the iso-contours of effective stress and the plane-strain constraint parameter. At the plane near the free surface, the crack-front field is close to the plane-stress state provided that the applied load is high and the plastic zone size is relatively large. Additional aspects of the 3D fields at the crack front through the thickness are also analyzed. In particular, a linear relationship between the constraint parameter A 2 and the hydrostatic stress is found in the 3D crack-front fields. It is also found that unlike the two-dimensional case where the J at the crack tip is not affected by the T -stress, the J -integrals at the crack front in the 3D case vary with the applied far field T -stress.

Constraint-modified J−R curves and its application to ductile crack growth

The concept of J-controlled crack growth is extended to J−A2 controlled crack growth using J as the loading level and A2 as the constraint parameter. It is shown that during crack extension, the parameter A2 is an appropriate constraint parameter due to its independence of applied loads under fully plastic conditions or large-scale yielding. A wide range of constraint level is considered using five different types of specimen geometry and loading configuration; namely, compact tension (CT), three-point bend (TPB), single edge-notched tension (SENT), double edge-notched tension (DENT) and centre-cracked panel (CCP). The upper shelf initiation toughness JIC, tearing resistance TR and J−R curves tested by Joyce and Link (1995) for A533B steels using the first four specimens are analysed. Through finite element analysis at the applied load of JIC, the values of A2 for all specimens are determined. The framework and construction of constraint-modified J−R curves using A2 as the constraint parameter are developed and demonstrated. A procedure of transferring the J−R curves determined from standard ASTM procedure to non-standard specimens or practical cracked structures is outlined. Based on the test data, the constraint-modified J−R curves are presented for the test material of A533B steel. Comparison shows the experimental J−R curves can be reproduced or predicted accurately by the constraint-modified J−R curves for all specimens tested. Finally, the variation of J−R curves with the size of test specimens is produced. The results show that larger specimens tend to have lower crack growth resistance curves.

Effect of specimen size and crack depth on 3D crack-front constraint for SENB specimens

Abstract Three-dimensional (3D) elastic–plastic finite element analyses (FEA) are performed to study constraint effect on the crack-front stress fields for single-edge notched bend (SENB) specimens. Both rectangular and square cross-section of the specimens with a deep crack of a / W =0.5 are considered to investigate the effect of specimen size. A square-cross-section specimen with a shallow crack of a / W =0.15 is also considered to examine the effect of crack depth. Stresses from FEA at the crack front on different planes of the specimen are compared with those determined by the J – A 2 three-term solution. Results show that in-plane stress fields can be characterized by the three-term solution throughout the thickness even in the region near the free surface. Cleavage fracture toughness data is compared to predict the effects of specimen size and crack depth on fracture behavior. It is found that the distributions of crack opening stress are nearly the same for the SENB specimens at the critical J which is consistent with the RKR model. Furthermore our results indicate that there is a distinct relationship between the crack-front constraint and the cleavage fracture toughness. By introducing the failure curves, the minimum fracture toughness and scatter band can be well captured using the J – A 2 approach.

A Modification of J-Q Theory and Its Applications

To more effectively quantify constraint effects on crack-tip fields and fracture resistances in ductile materials, a modification of the J-Q theory (ODowd and Shih, 1991) is proposed by introducing a parameter Q*. Results show that Q* is a load-independent constraint parameter under LSY or fully plastic deformation. The validity of J-Q* description has been demonstrated by three applications to: (a) predict crack-tip fields without further FEA calculations; (b) rank constraint levels for different fracture specimens; (c) predict constraint-corrected J-R curves.

Three-dimensional stress and displacement fields near an elliptical crack front

Local stress and deformation fields for an elliptical crack embedded in an infinite elastic body subjected to normal, shear and mixed loads are considered. Particular emphasis is placed on the direction of propagation of points along the crack border. A confocal curvilinear coordinate system related to a fundamental ellipsoid, and a local spherical coordinate system attached to the crack border are adopted. Using asymptotic analysis, this paper obtains the stress and displacement fields in a plane inclined to the 3D crack front. Results show that the present solutions are independent of the curvature of the ellipse, and different from those given by Sih (1991). Based on two different fracture criteria, crack growth analysis shows that a 3D crack would propagate in the direction of the normal plane along the crack front. As a result, the fracture initiation and propagation of a 3D flat crack can be analyzed in the plane normal to the crack front, and the local fields in the normal plane are the linear superposition of the plane strain mode-I, mode-II, and mode-III crack-tip fields.

Dynamic crack-tip field for tensile cracks propagating in power-law hardening materials

The near-tip field of a dynamically propagating crack in an incompressible power-law hardening material is studied using the asymptotic method as an extension of Leighton etxa0al. (Journal of the Mechanics and Physics of Solids, 1987, Vol.35, pp.541–563). The crack is subjected to tensile loads and propagates steadily under plane strain conditions. The material deformation is described by the J2 flow theory of plasticity with infinitesimal displacement gradients. Results show that, in the present crack-tip field, (a)xa0the angular variations of stresses, strains and particle velocities around the crack tip are fully continuous; (b)xa0the stresses and strains at the crack tip are bounded; (c)xa0there is a free parameter σeq0 which cannot be determined in the asymptotic analysis. Comparisons indicate that the present asymptotic solution matches well with full-field numerical results, and the parameter σeq0 can characterize the effects of the far field on the crack-tip field. Furthermore, the present solution approaches that of Leighton etxa0al. (1987) in the limit as the material hardening exponent goes to infinity, but does not reduce to the accepted solution for quasi-statically growing cracks in the limit as the crack speed goes to zero.

Constraint effects on plastic crack-tip fields for plane strain mode-I, II and III cracks in non-hardening materials

To explore constraint effects on fully plastic crakc-tip fields, analytical solutions are examined for mode-I, II and III loading in non-hardening materials under plane strain conditions. The results reveal that under mode-II and III loading the crack-tip stress fields are unique, and thus can be characterized by a `single parameter. Under mode-I loading, however, the crack-tip stress field is non-unique but can be characterized by two sets of solutions or `two parameters. One set of the solutions is the well-known Prandtl field and the other is a plastic T-stress field. This conclusion corroborates the observation of McClintock (1971) that the slip-line field is non-unique for plane strain tensile cracks. A two-term plastic solution which combines the Prandtl field and the plastic T-stress field with two parameters B1 and B2 can then characterize the crack-tip stress field of plane strain mode-I crack over the plastic region and quantify the magnitude of crack-tip constraints. These characters are similar to those for hardening materials. Analyses and examples show that the two-term plastic solution can match well with the slip-line field or finite element results over plastic region. Thus the parameters B1 and B2 can be used to characterize the constraint level for mode-I finite-sized crack specimens in non-hardening materials under plane strain conditions.

Determination of Constraint-Modified J-R Curves for Carbon Steel Storage Tanks

Mechanical testing of A285 carbon steel, a storage tank material, was performed to develop fracture properties based on the constraint theory of fracture mechanics. A series of single edge-notched bend (SENB) specimen designs with various levels of crack tip constraint were used. The variation of crack tip constraint was achieved by changing the ratio of the initial crack length to the specimen depth. The test data show that the J-R curves are specimen-design-dependent, which is known as the constraint effect. A two-parameter fracture methodology is adopted to construct a constraint-modified J-R curve, which is a function of the constraint parameter, A2 , while J remains the loading parameter. This additional fracture parameter is derived from a closed form solution and can be extracted from the finite element analysis for a specific crack configuration. Using this set of SENB test data, a mathematical expression representing a family of the J-R curves for A285 carbon steel can be developed. It is shown that the predicted J-R curves match well with the SENB data over an extensive amount of crack growth. In addition, this expression is used to predict the J-R curve of a compact tension specimen (CT), and reasonable agreement to the actual test data is achieved. To demonstrate its application in a flaw stability evaluation, a generic A285 storage tank with a postulated axial flaw is used. For a flaw length of 10% of the tank height, the predicted J-R curve is found to be similar to that for a SENB specimen with a short notch, which is in a state of low constraint. This implies that the use of a J-R curve from the ASTM (American Society for Testing and Materials) standard designs, which typically are high constraint specimens, may be overly conservative for analysis of fracture resistance of large structures.Copyright