R-Curve Modelling with Constraint Effect

D.W. Zhou, W.G. Xu and S.D. Smith
Structural Integrity Technology Group, TWI Ltd, Cambridge, UK

Paper presented at 12th International Conference on Fracture, July 12 - 17, 2009, Ottawa, Canada.

This paper is dedicated in memoriam to Dr. W.G. Xu who passed away from cancer in 2008.

Abstract

A method for constructing R-curve has been developed. Three constraint theories (T-stress, Q-stress and A2) have been used to construct constraint-dependent R-curves. The predicted R-curves have been compared against test results taken from published literature. A comparison of the predicted constraint based R-curves against the testing result was given. It is shown that the developed method provides an efficient approach to obtain R-curves under low constraint and can potentially be used for engineering critical assessment (ECA) in various industry sectors.

1. Introduction

For a cracked component, the fracture initiation toughness J_1c determines when the crack extension initiates and the fracture resistance curve (i.e. R-curve) is concerned with how far the crack will grow in a stable manner under an applied load. From the view of single-parameter fracture mechanics, the J_1c and R-curve are considered sufficient to describe the crack tip condition in fracture analyses of engineering structures. The assumption behind the single-parameter theory is that the J_1c and R-curve obtained from laboratory are transferable to a full-scale structure. In practice, however, the fracture initiation toughness and fracture resistance curve are dependent on the test specimen geometry and loading mode. This dependence is attributed to the constraint effect at the crack tip. Standard laboratory testing specimens are typically of high constraint which appears to underestimate the fracture initiation toughness when the actual cracked structure is under low constraint. A typical example of the transferability problems is observed in performing ECA of pipe girth welds. Pisarski and Wignall^[1] suggested that use of specimen geometries and loading modes associated with lower constraint such as single edge notched tension (SENT) and shallow-notched single edge notched bending (SENB) specimens, allow improved estimates of fracture toughness to be obtained that are appropriate for the assessment of circumferential flaws in pipe girth welds.

Due to their simplicity, the T-stress and the Q-stress are frequently used for quantifying constraint at the crack tip. By combining the constraint parameter with a loading parameter such as stress intensity factor K or J-integral, two-parameter fracture mechanics (2PFM) is sufficient for describing the stress, strain or displacement field at a crack tip for different degrees of constraint.

Constraint-based approaches for determining the material fracture initiation toughness J_1c are described in industry codes such as R6^[2] where empirical formulas to calculate the T-stress and Q-stress have been given. However, these codes tend not to give a quantitative description of the effect of constraint on the R-curve. Various experiments have suggested that the R-curve depends on the level of constraint. Joyce and Link^[3] have shown that the constraint level can affect the prediction of crack initiation. In general, a specimen with a low constraint level tends to give a higher R-curve than a specimen associated with a high constraint level (Fig.1).

Δa =1mm as a function of Q can be linearly fitted as

Using a similar procedure to that adopted to obtain Eq.13, the two functions C₁(Q) and C₂(Q) are given by

Substituting Eq.16 into Eq.8, the constraint-modified R-curve for HY-100 using Q stress is given by,

The predicted R-curves using Eq.17 are also shown in Fig.5.

4.5 Use of A₂ to construct the constraint corrected R-curve

In the case of A₂, the J-integral at Δa =0.2 mm and Δa =1 mm as a function of A₂ can be linearly fitted as

Following the same approach, the functions C₁(A₂) and C₂(A₂) can be given by

The R-curve using A₂ is then given by,

5 Discussions

A comparison of the three predicted R-curves shows that all three parameters can be used to construct constraint-based R-curves. It has been argued that T-stress and Q-stress will fail under large scale yielding in bending-dominant specimens due to the presence of global bending stress which alters the crack tip stress field. Zhu and Leis^[13] developed a modified J-Q theory to construct constraint-dependent R-curves for this purpose. However, the present results show that R-curves predicted with T-stress and Q-stress are within up to about 20% of the test results (see Fig.5d for deeply-cracked SENB specimens). Further investigations are needed in order to clarify the validity of the J-Q theory in this region.

6 Conclusions

This paper has reviewed three constraint theories (T-stress, Q-stress and A₂) for characterizing the fracture initiation toughness and fracture resistance curve. The three constraint parameters were used to construct constraint-corrected R-curves. An example was used to demonstrate the validity of the theory by comparing the predicted R-curves with the test data at different constraint levels. The method can be potentially used for engineering critical assessment of flaws where standard tests associated with high constraint give conservative R-curves.

Acknowledgements

The authors wish to thank Professor Yuh J Chao, Department of Mechanical Engineering, University of South Carolina, USA for providing the report 'Tables of plane strain crack tip fields: HRR and higher order terms'.

References

1. Pisarski HG and Wignall CM, Fracture toughness estimation for pipeline girth welds, Proceeding of IPC2002 International pipeline conference, (2002), Calgary, Canada
2. R6: Assessment of the Integrity of Structures Containing Defects, revision 4. 2001, British Energy Generation Ltd., UK
3. Joyce JA and Link RE, Effect of constraint on upper shelf fracture toughness Fracture Mechanics, ASTM STP-1256, 26, (1995) 142-177
4. Yang S, Chao YJ and Sutton MA, Higher order asymptotic crack tip fields in a power-law hardening material, Engineering Fracture Mechanics 45, (1993) 1-20
5. Chao YJ and Zhu XK, Constraint-modified J-R curves and its application to ductile crack growth, Journal of Fracture 106,(2000) 135-160
6. Parks DM, Advances in characterization of elastic-plastic crack-tip fields in topics in Fracture and Fatigue, Edited by AS Argon, (1992)59-98
7. Kirk MT, The second ASTM/ESIS symposium on constraint effects in fracture; an overview, International Journal of Pressure Vessel & Piping 64,(1995)259-275
8. Chao YJ and Zhang L, Tables of plane strain crack tip fields: HRR and higher order terms ME-Report, Department of Mechanical Engineering, University of South Carolina, (1997) 97-1
9. Wei Y and Wang T, Characterization of elastic-plastic fields near stationary crack-tip and fracture criterion, Engineering Fracture Mechanics 51, (1995) 541-553
10. Karstensen AD, Nekkal A and Hancock, JW, Constraint estimation for edge cracked bending bars, IUTAM Symposium on Nonlinear Analysis of Fracture, Kluwer Academic Publishers, Netherlands, (1997) 23-32
11. Chao YJ, Zhu XK and Kim Y, et al, Characterization of crack-tip field and constraint for bending specimens under large-scale yielding, International Journal of Fracture 127, (2004) 283-302
12. Nyhus, B, Zhang, ZL, Thaulow, C, Normalisation of material crack resistance curves by the T-stress, In: Proceedings of 14th European Conference on Fracture, 8-13 Sept, 2002, Cracow, Poland
13. Zhu XK and Leis BN, Bending modified J-Q theory and crack-tip constraint quantification, International Journal of Fracture 141, (2006) 115-134