I /1-12
M o d i f i e d R e s u l t / / .
~ ~ . . . - " " '" f i .
5Q
. . . . . . .
0 , i , 0
0 100 200 300 400
T e m p e r a t u r e -- ~
FIG. 4--The minimum driving force G~ needed to drive the crack dynamically as a function of temperature. The solid line corresponds to Fig. 3, and the dashed line from a modified'result in Ref 21.
modification of the basic model was proposed by Mataga, Freund, and Hutchinson [21]
which provides a description in better agreement with full-field numerical solutions than the model outlined above. In the original development [20], it was assumed that the plastic dissipation was completely controlled by the near tip stress-intensity factor field, say Kt,p.
However, this stress-intensity factor, which is asymptotically correct, must give way to the far-field stress stress-intensity factor K with increasing distance from the crack edge. It was observed in Ref 21 that the estimate of plastic dissipation was improved significantly, at least in comparison to finite element simulations, if the plastic dissipation was estimated on the basis of a "mean" stress-intensity factor V ~ , p K . The graph of arrest toughness versus temperature for parameters corresponding to mild steel is also shown in Fig. 4.
Important experiments on crack propagation and arrest in steel specimens are currently being carried out by deWit and Fields [22]. Their specimens are enormous, single-edge notched plates loaded in tension. The growing crack thus experiences an increasing driving force as it advances through the plate. A temperature gradient is also established in the specimen so that the crack grows from the cold side of the specimen toward the warm side.
Based on the presumption that the material becomes tougher as the temperature is increased, the crack also experiences increasing resistance as it advances through the plate. The spec- imen material is A533B pressure vessel steel, which is both very ductile and strain-rate sensitive. In the experiments, the fracture initiates as a cleavage fracture and propagates at high speed through the specimen into material of increasing toughness. The crack then arrests abruptly in material whose temperature is above the nil-ductility temperature for the material based on Charpy tests. A large plastic zone grows from the arrested crack edge, and cleavage crack growth is occasionally reinitiated. The essential features of the experiment appear to be consistent with the model of high strain rate crack growth outlined in Ref 19, and this model appears to provide a conceptual framework for interpretation of the phe- nomenon. An analysis of rapid crack growth in a rate dependent plastic solid has also been carried out by Brickstad [15] in order to interpret some experiments on rapid crack growth in a high-strength steel. It is noted that the work of deWit and Fields [22] is part of the Nuclear Regulatory Commission's Heavy Section Steel Technology Program, which sustains an integrated dynamic fracture mechanics effort involving experiments, computation, and material characterization.
FREUND ON CRACK-TIP PLASTICITY 95 Concluding Remarks
The results described in the preceding sections reflect some progress toward discovery of the role played by crack-tip plastic fields in establishing conditions for rapid advance of a crack in an elastic-plastic material. Understanding of this issue is far from complete, and a few of the open questions that could be profitably pursued are identified in this concluding section. For example, much of the modeling that has resulted in a detailed description of crack-tip elastic-plastic fields has been based on the assumption that the fields are steady as seen by a crack-tip observer. This approach overlooks all transient aspects of the process.
The picture of the way in which a crack-tip plastic zone develops in a cracked, rate-sensitive structural material under the action of stress wave loading is not clear, but the question is important in the sense that these fields determine whether or not the crack will advance.
The same issue appears to be at the heart of the cleavage initiation process in steels, but on a microstructural scale. Here, the sudden cracking of carbides or other brittle phases due to incompatible plastic strains provides a nucleation mechanism, and the question is whether or not these dynamic microcracks penetrate into the adjacent ferrite as sharp cracks.
The answer seems to hinge on the way in which plastic strains develop near the carbide- ferrite interface due to the appearance of the microcracks in the brittle phases.
The transients of the arrest of a cleavage crack in a structural material are also unclear at this point. A running crack appears to arrest because conditions for the continuous reinitiation of cleavage cannot be maintained [23]. In terms of the model discussed in the previous section, arrest occurs because conditions for elastic rate dominance of the local field cannot be maintained. However, the model does not provide information on the process thereafter. It appears from the experiments reported by deWit and Fields [22] that arrest is quite abrupt, that a large plastic zone grows from the crack edge following arrest of the cleavage crack, that the crack may grow subsequently in a ductile mode, and that cleavage may be reinitiated at a later stage. It is not clear if the cleavage reinitiation is due to a rate effect or to a combination of strain hardening and constraint in the interior portions of the specimen.
Modeling of plasticity effects in dynamic crack growth has been restricted to two-dimen- sional systems, for the most part. It is likely that a number of three-dimensional effects are of sufficient importance to warrant further investigation. For example, crack propagation studies are often carried out with plate specimens. For such specimens, the transition from plane-stress conditions in regions far from the crack tip compared with plate thickness to plane-strain or generalized plane-strain conditions near the crack edge is not d e a r . Yang and Freund [24] suggest that plane-stress conditions prevail only for points beyond about one-half the plate thickness from the crack edge for elastic deformations. Out-of-plane inertia is of potential importance in these three-dimensional fields, but this effect has not been investigated to date. Furthermore, the roles of ductile shear lips at the free surfaces or of ductile ligaments left behind a cleavage crack as it advances through a structural metal are not clear at this time. In a study of fracture initiation in dynamically loaded specimens of a ductile material by Nakamura, Shih, and Freund [25], it was shown that these three- dimensional effects are potentially very significant.
Virtually all of the foregoing discussion has focused on issues from the traditional fracture mechanics point of view, that is, the use of a single parameter to characterize the mechanical state of a dominant crack in a stressed body. This section is concluded, however, with mention of somewhat speculative interpretations of recent work that suggest a departure from the traditional fracture mechanics viewpoint. Some exciting new data on fracture initiation and crack growth in a 4340 steel in a very hard condition were recently reported by Ravichandran and Clifton [26]. Through a modification of the plate impact apparatus,
96 FRACTURE MECHANICS: TWENTIETH SYMPOSIUM
they were able to examine fracture initiation and crack growth of a few millimetres for the plane-strain situation of a semi-infinite crack in an unbounded body subject to plane wave loading, at least for a microsecond or two. A t the lowest testing temperature reported, the cracks grew as cleavage cracks. Based on optical measurements of the surface motion of the specimen and comparison with detailed elastic-viscoplastic calculations, it appeared that the cracks grew more nearly at constant velocity crack than with a fixed level of energy release rate or stress intensity factor. This observation is similar to that made by Ravi- Chandar and Knauss [27], who studied crack growth in the brittle polymer Homalite-100.
In both cases, this observation was made in situations where the load was suddenly applied and the load level was very intense compared with the minimum load necessary to induce fracture in the same situation. The results suggest that the one-parameter characterization of the crack-tip conditions may not be adequate to describe fracture response under such conditions.
Acknowledgments
It is a pleasure to acknowledge ongoing collaboration on various aspects of plasticity effects in dynamic fracture with J. W. Hutchinson of Harvard University. Research on plasticity effects in dynamic fracture is supported at Brown University by the Office of Naval Research through contracts N00014-87-K-0481, the A r m y Research Office through contract DAAG29-85-K-0003, and the Brown University NSF Materials Research Group through grant DMR-8714665. This support is gratefully acknowledged.
References
[1] Rice, J. R., "Mathematical Analysis in the Mechanics of Fracture," Fracture--Vol. 2, H. Liebowitz, Ed., Academic Press, 1968, pp. 191-311.
[2] Rice, J. R., Drugan, W. J., and Sham, T. L:, "Elastic-Plastic Analysis of Growing Cracks," Fracture Mechanics, (Twelfth Conference), ASTM STP 700, American Society for Testing and Materials, Philadelphia, 1980, pp. 189-219.
[3] Freund, L. B. and Douglas, A. S., "The Influence of Inertia on Elastic-Plastic Antiplane Shear Crack Growth," Journal of the Mechanics and Physics of Solids, Vol. 30, 1982, pp. 59-74.
[4] Dunayevsky, V. and Achenbach, J. D., "Boundary Layer Phenomenon in the Plastic Zone Near a Rapidly Propagating Crack Tip," International Journal of Solids and Structures, Vol. 18, 1982, pp. 1-12.
[5] Slepyan, L. I., "Crack Dynamics in an Elastic-Plastic Body," Mechanics of Solids, Vol. 11, (English translation of Mechanika Tverdogo Tela, 1976, pp. 126-134.
[6] McClintock, F. A. and Irwin, G. R., "Plasticity Aspects of Fracture Mechanics," Fracture Tough- ness Testing and Its Applications, ASTM STP 381, American Society for Testing and Materials, Philadelphia, 1965, pp. 84-113.
[7] Lam, P. S. and Freund, L. B., "Analysis of Dynamic Growth of a Tensile Crack in an Elastic- Plastic Material," Journal of the Mechanics and Physics of Solids, Vol. 33, 1985, pp. 153-167.
[8] Leighton, J. T., Champion, C. R., and Freund, L. B., "Asymptotic Analysis of Steady Dynamic Crack Growth in an Elastic-Plastic Material," Journal of the Mechanics and Physics of Solids, Vol.
35, 1987, pp. 541-563.
[9] Achenbach, J. D. and Dunayevsky, V., "Fields Near a Rapidly Propagating Crack-Tip in an Elastic-Plastic Material," Journal of the Mechanics and Physics of Solids, Vol. 29, 1981, pp. 283- 303.
[10] Gao, Y. C. and Nemat-Nasser, S., "Dynamic Fields Near a Crack Tip in an Elastic Perfectly Plastic Solid," Mechanics of Materials, Vol. 2, 1983, pp. 47-60.
[11] Rosakis, A. J., Duffy, J., and Freund, L. B., "The Determination of Dynamic Fracture Toughness of AISI 4340 Steel by the Shadow Spot Method," Journal of the Mechanics and Physics of Solids, Vol. 32, 1984, pp. 443-460.
[12] Kobayashi, T. and Dally, J. W., "Dynamic Photoelastic Determination of the a-K Relation for
FREUND ON CRACK-TIP PLASTICITY 97 4340 Steel," Crack Arrest Methodology and Applications, ASTM STP 711, G. T. Hahn and M. E Kanninen, Eds., American Society for Testing and Materials, Philadelphia, 1979, pp. 189- 210.
[13] Dahlberg, L., Nilsson, E, and Brickstad, B., "Influence of Specimen Geometry on Crack Prop- agation and Arrest Toughness," Crack Arrest Methodology and Applications, ASTM STP 711, G. T. Hahn and M. E Kanninen, Eds., American Society for Testing and Materials, Philadelphia, 1980, pp. 89-108.
[14] Lo, K. K., "Dynamic Crack-Tip Fields in Rate Sensitive Solids," Journal of the Mechanics and Physics of Solids, Vol. 31, 1983, pp. 287-305.
[15] Brickstad• B. • " A Visc•p•astic Ana•ysis •f Rapid Crack Pr•pagati•n Experiments in Stee•••• J•urnal of the Mechanics and Physics of Solids, Vol. 31, 1983, pp. 307-327.
[16] Frost, H. J. and Ashby, M. F., Deformation-Mechanism Maps, Pergamon Press, Oxford, 1982.
[17] Hui, C. Y. and Riedel, H., "The Asymptotic Stress and Strain Field Near the Tip of a Growing Crack Under Creep Conditions," International Journal of Fracture, Vol. 17, 1981, pp. 409-425.
[18] Freund, L. B. and Douglas, A. S., "Dynamic Growth of an Antiplane Shear Crack in a Rate- Sensitive Elastic-Plastic Material," Elastic-Plastic Fracture: Second Symposium. Volume 2, ASTM STP 803, C. E Shih and J. Gudas, Eds., American Society for Testing and Materials, Philadelphia, 1983, pp. 5-20.
[19] Freund, L. B. and Hutchinson, J. W., "High Strain-Rate Crack Growth in Rate-Dependent Plastic Solids," Journal of the Mechanics and Physics of Solids, Vol. 33, 1985, pp. 169-191.
[20] Freund, L. B., Hutchinson, J. W., and Lam, P. S., "Analysis of High Strain Rate Elastic-Plastic Crack Growth," Engineering Fracture Mechanics, Vol. 23, 1986, pp. 119-129.
[21] Mataga, P. A., Freund, L. B., and Hutchinson, J. W., "Crack Tip Plasticity in Dynamic Fracture,"
Journal of the Physics and Chemistry of Solids, Vol. 48, 1987, pp. 985-1005.
[22] deWit, R. and Fields, R., "Wide Plate Crack Arrest Testing," Nuclear Engineering and Design, Vol. 98, 1987, pp. 149-155.
[23] Irwin, G. R., "Brittle-Ductile Transition Behavior in Reactor Vessel Steels," Proceedings, WRSI Meeting, Gaithersburg, MD, Oct. 1986, NUREG/CP-0082, Vol. 2, U.S. Nuclear Regulatory Commission, Feb. 1987, pp. 251-272.
[24] Yang, W. and Freund, L. B., "Transverse Shear Effects for Through-Cracks in an Elastic Plate,"
International Journal of Solids and Structures, Vol. 21, 1985, pp. 977-994.
[25] Nakamura, T., Shih, C. F., and Freund, L. B., "Three Dimensional Transient Analysis of ~a Dynamically Loaded Three Point Bend Ductile Fracture Specimen," Nonlinear Fracture Mechan- ics, ASTM STP 995, American Society for Testing and Materials, Philadelphia, 1988, pp. 217- 241.
[26] Ravichandran, G. and Clifton, R. J., "Dynamic Fracture Under Plane Wave Loading," Brown University Report, 1986.
[27] Ravi-Chandar, K. and Knauss, W. G., "An Experimental Investigation into Dynamic Fracture:
III. On Steady State Crack Propagation and Crack Branching," International Journal of Fracture, Vol. 26, 1984, pp. 141-154.
Microstructure and
Micromechanical Modeling
H e r m a n n R i e d e l 1
Creep Crack Growth
REFERENCE: Riedel, H., "Creep Crack Growth," Fracture Mechanics: Perspectives and Directions (Twentieth Symposium), ASTM STP 1020, R. P. Wei and R. P. Gangloff, Eds., Amer- ican Society for Testing and Materials, Philadelphia, 1989, pp. 101-126.
ABSTRACT: On the background of the historical evolution of the subject area, the current knowledge on creep crack growth is reviewed. The discussion is grouped around the C* integral, which is the appropriate load parameter for describing creep crack growth in ductile materials.
Crack growth by coalescence with grain boundary cavities is modeled and the lifetimes of cracked specimens are calculated by integrating the resulting laws for crack growth. Limitations to C* arise from the initial elastic-plastic material response, from primary creep, from crack- tip blunting and from widespread cavitation damage. All these bounds are shown on a load parameter map. This is a diagram in the stress-time plane indicating the regimes in which different load parameters are applicable: KI for predominantly elastic deformation; J if plas- ticity plays a role; C* for primary creep; and C* for steady-state creep. Tertiary creep is included in a damage-mechanics analysis of creep crack growth. The final section discusses a few three-dimensional aspects of fracture mechanics in general and of creep crack growth in particular.
KEY WORDS: creep crack growth, high temperatures, grain boundary cavities, damage mechanics, three-dimensional aspects
A component operating in the creep regime may fail by the slow extension of a macroscopic crack. Slow stable cracking at elevated temperatures is called "creep crack growth" if it occurs under more or less constant load, and "creep-fatigue crack growth" under cyclic loading conditions. Since corrosive effects often play a role, it may be justified to speak of high-temperature stress corrosion cracking in some cases. The present paper focuses the attention on the mechanics and the mechanisms of creep crack growth.
Usually, but not necessarily, creep crack growth occurs along grain boundaries by the formation and coalescence of grain boundary cavities. Cavitation may be confined to a small zone near the tip of the growing main crack ("small-scale damage"). Then, the lifetime will be determined by crack growth and a fracture-mechanics approach appears to be appropriate to deal with the problem. In other cases, the whole specimen may have developed cavities before the crack grows markedly. Then failure occurs by more or less homogeneous cavi- tation and the crack plays no particular role for the lifetime.
Because creep crack growth has been investigated for almost 20 years, only a fraction of the published literature can be referenced here. More complete reference lists can be found in previous reviews [1-8].
The Early Years of Creep Crack Growth Testing
The history of research on creep crack growth could have started a few years earlier than it actually did. In the 1960s, major advances had been made in the theory of elastic-plastic
1 Head of the Department Applications of Materials at High Temperatures, Fraunhofer-Institut fiir Werkstoffmechanik, W6hlerstraSe 11, D-7800 Freiburg, Federal Republic of Germany.
101
102 FRACTURE MECHANICS: TWENTIETH SYMPOSIUM
fracture mechanics. In particular, the relevance of the J-integral for crack analysis had been recognized [9,10], the Hutchinson, Rice, and Rosengren (HRR) crack-tip fields had been derived [11,12], and the role of crack-tip blunting had been explored [13]. For theoreticians, who were aware of Hoff's analogy [14], it was obvious that these discoveries could be applied directly to viscously creeping bodies. Hoff's analogy states that a nonlinear elastic body obeying a material law 9 = / ( ( r ) , and a nonlinear viscous body characterized by ~ = f(~), develop the same stress field when subjected to the same loads. The elastic strain field corresponds to the viscous strain-rate field, and a viscous analogue to the J-integral exists, which has become known as C* [15-17]. But in the 1960s, these ideas were not directly pursued, since creep crack growth had only started to be recognized as a problem of practical significance, and this knowledge had not yet spread within the scientific community.
In the early 1970s, the first papers appeared which reported slow stable cracking under elevated-temperature creep conditions [18-22]. To analyze the measured crack growth rates, some of the early workers tried to use the linear-elastic stress intensity factor, KI. They were aware of the fact that KI could only be the appropriate load parameter under pre- dominantly elastic, or small-scale yielding, conditions. This is illustrated by Fig. 1, which shows two specimens after crack growth tests under sustained load. It cannot be expected that the linear-elastic stress intensity factor can be applied to the heavily deformed specimen of the chromium-molybdenum steel. But crack growth in the Nimonic alloy was accompanied by no detectable nonelastic deformation, and hence, K~ should be applicable. Similarly, Siverns and Price [22] obtained a good correlation between KI and the crack growth rate in an embrittled 21/4 Cr-lMo steel (heat treated to simulate the coarse-grained, heat-affected zone of a weld), which exhibited little nonelastic deformation during crack growth. In more ductile materials, crack growth is accompanied by extensive creep of the specimen. Before the C* integral had become known among experimentalists, many workers tried to correlate crack growth rates in ductile materials with the net section stress, tr,e,, or with a reference stress, ~ref- Harrison and Sandor [19], for example, found a good correlation of crack growth rate with crref for a small set of data on a turbine rotor steel.
However, it should be noted that there is no theory which suggests using (r,r or (r,~ as a parameter correlating crack growth rates in different specimens. It may be useful to compare rupture lifetimes (but not crack growth rates) of cracked, notched, and smooth specimens on the basis of the net section stress. In this connection, the terms notch strengthening and notch weakening have been introduced to indicate that a cracked or notched specimen
FIG. 1--Two specimens after creep crack growth at 535~ ( CrMo steel) and 650~ (Nimonic 80 A ).
RIEDEL ON CREEP CRACK GROWTH 103 sustains a given net section stress for a longer or shorter time, respectively, than a smooth specimen. (In the case of notch strengthening, the triaxial constraint, which retards creep flow, dominates, whereas notch weakening occurs if the stress concentration at the crack or notch causes premature crack growth and failure.) However, these terms are vague and specimen-geometry dependent, and should be used with care as an empirical guideline for characterizing the creep ductility or brittleness of materials.
Although ~r,e, and cr,e~ have no theoretical basis in relation to creep crack growth rates, many data were evaluated using these parameters (for a summary, see Ref 23). The question of which parameter should be used has important practical consequences, since, as Riedel and Rice [24] point out, the lifetime of a cracked comPonent may be under- or overestimated by orders of magnitude if an inappropriate load parameter is employed to transfer crack growth rates from laboratory test specimens to the component. However, the theoretical basis for discussing that question was lacking in the early 1970s.
The C* Integral
An important step towards a well-founded theory of creep crack growth was the intro- duction of C* by Landes and Begley [15], by Ohji, Ogura, and Kubo [16], and by Nikbin, Webster, and Turner [17]. Other notations for C* are J, J*, and J'.
The General Idea
The use of C*, like that of other fracture mechanics parameters, relies on the following set of arguments:
1. In nonlinear viscous materials characterized by an arbitrary stress/strain-rate relation,
= f(cr), the contour integral C* defined in Fig. 2 is independent of the choice of the path.
The proof of the path independence of C* is completely analogous to that of J [9,10].
2. The numerical value of C* depends on the applied loading, the specimen geometry and the specific form of the law describing the material's nonlinear-viscous behavior. Practical guidelines on how to determine C* from the second formula in Fig. 2 are given by Landes and Begley [15]. For the special case of power-law viscous materials, more convenient expressions for C* will be presented in the next subsection.
3. The same C* which is measured far away from the crack tip at the load line must characterize the deformation field near the crack tip, since C* is path independent.
4. The asymptotic field near the crack tip has a unique form independent of the specimen geometry and is unequivocally determined once C* is specified. A power-law viscous material is a convenient example to illustrate this (see the next subsection).
FIG. 2IDefinition of the contour integral C*.
C* = " J (W dy-nidij-"~- * 8flj ds)
r-
1 a /pd c* =--6 ~- E