1、ACI 446.1 R-91Fracture Mechanics of Concrete: (Reapproved 1999)Concepts, Models and Determinationof Material PropertiesReported by ACI Committee 446, Fracture Mechanics*Zdenek P. Bazant1,2(Chairman)MembersVellore S. Gopalaratnaml1,3(Secretary)Oral Buyukozturk4Luigi Cedolin4David Darwin3Manuel Elices
2、1,3Shu-Jin FangWalter GerstlesNeil M. HawkinsHideyuki HoriilJeremy IsenbergVictor C. Li1Feng-Bao LinlSteven L. McCabeSheng-Taur Mau3Jacky Mazars4Sidney MindessAntoine E. NaamanC. Dean NormanPhillip A. PfcifferGilles Pijaudier-Cabot3Victor Saouma1,3Surendra P. Shah1,3Robert L. SicrakowskiWimal Suaris
3、Stuart E. Swartz1,2Tatsuya TsubakiC. VipulanandanlMethi WecharatanalThe committee wishes to recognize the contributions of the following non-voting members:Farhad AnsarilArne Hillerborg1Ravindra Gettu4B. L. KarihaloolPere C. Prat1Hans W. Reinhardt11Members of Subcommittee I (chaired by Bazant) which
4、 prepared the report2Principal Authors3Contributing Authors*This report was approved by a vote of the full ACI Committee 446 in December 1989. It.does not reflect research after that date.ACI Committe e Reports, Guides, Standard Practices, andCommentaries are intended for guidance in designing,plann
5、ing, executing, or inspecting construction and inpreparing specifications. Reference to these documents shallnot be made in the Project Documents. If items found in thesedocuments are desired to be part of the Project Documentsthey should be phrased in mandatory language andincorporated into the Pro
6、ject Documents.Copyright 0 1991, American Concrete Institute.All rights re served including the rights of reproduction and use in any form orby any means, including the making of copies by any photo process or by anyelectronic or mechanical device, printed, written, o r oral or recording for soundor
7、 visual reproduction or for use in any knowledge or retrieval system or device,unless permission in writin g is obtained from the copyright proprietors.446.1 R-l-446.1 R-2 ACI COMMITTEE REPORTCONTENTSSynopsis, p. 446.1R-4Introduction, p. 446.1R-5Chapter 1 - Why fracture mechanics?, p . 44 6.1 R- 61.
8、1 - Five reasons for fracture mechanics approach1.2 - Is Weibulls statistical theory of size effect applicable?1.3 - Simple energy explanation of size effect1.4 - Experimental evidence for size effect in structures1.5 - Explanation of size effect on ductilityChapter 2 - Essential results from linear
9、 elastic fracture mechanics, p . 446.1R-242. 1 - - Stress singularity2. 2 - - Energy criterion2. 3 - - Limits of applicabilityChapter 3 - Nonlinear fractu r e models with softening zone, p . 446.1R-283. 1 - - Softening stress-displacement relations3. 2 - - Softening stress-strain relations3. 3 - - S
10、tress-displacement vs. stress-strain softenin g relations3. 4 - - Nonlinear triaxial models for strain-softening3. 5 - - Random particle simulation of microstructureChapter 4 - Special nonlinear fracture models based on adaptation of LEF M , 446.lR-534.1 - Effective crack models4.2 - Two-parameter m
11、odel of Jenq and Shah4.3 - Geometry-dependent R-curve determined from size effect lawChapter 5 - Size effect and brittleness o f structures p. 446.1R-585.1 - Size effect law for maximum nominal stress5. 2 - - Brittlenes s number5. 3 - - Other siz e effects an d limitationsChapter 6 - Experimental or
12、 analytical determination of material fracture parameters,p. 446.1R-686. 1 - - Notched beam tests6. 2 - - Wedge-splitting test6. 3 - - Work-of-fracture method (RILEM , Hillerborg)6. 4 - - Size effect in work-of-fracture method6.5 - Two-parameter fracture model of Jenq and Shah6.6 - Effective crack m
13、odel of Karihaloo and Nallathambi446.1 R- 36.7 - Determination of material parameters by size effect method6.8 - Size required for applicability of LEFM6.9 - Identification of nonlocal characteristics length6.10 - Identification of tensile post-peak softening stress-strain curve6.11 - Material param
14、eters for Mode II and planar mixed mode fracture6.12 - Material parameters for Mode III fractureChapter 7 - Factors influencing fracture parameters, p . 446.1R-1047.1 - - Effect of loading rate and creep7.2 - Effect of temperature and humidity on fracture energy7.3 - Effect of cyclic loadingChapter
15、8 - - Effect o f reinforcement, 446.1R-1098. 1 - - Effect o f reinforcing steel bars8. 2 - - Fractur e in fiber-reinforced concreteChapter 9 - - Crack systems, p . 446.1R-1139.1 - Response of structures with interacting growing cracks9. 2 - - Interacting parallel cracks9.3 - Crack spacing and width
16、in beams9. 4 - - Interacting microcracksConcluding remarks, p . 446.1R-119Acknowledgement, p . 446.1R-119References, p . 446.1R-120Appendix - Derivations of some formulas, p . 446.1R-140Extended summary, p. 446.1R-141Basic notations, p . 446.1R-146446.1 R-4 ACI CO MMITT EE REPORTSYNOPSISIn the first
17、 of its series of four state-of-the-art reports under preparation, the Committee describesthe basic concepts of fracture mechanics of concrete, the existing theoretical models, and the methodsfor determining the material fracture parameters. Chapter 1 offers five reasons for introducing fracturemech
18、anics into certain aspects of design of concrete structures, including some code provisions: (1) atheoretical energy argument; (2) the need to achieve objectivity of finite element solutions, i.e., eliminatespurious mesh sensitivity; (3) the progressive (propagating) nature of failure, implied whene
19、ver th e load-deflection diagram lacks a yield plateau; (4) the need to rationally predict ductility and energy absorptioncapability; and most importantly, (5) the effect of structure size on the nominal strength (i.e., nominalstress at maximum or ultimate load) as well as on ductility and energy ab
20、sorption capability. Thesize effect is due to stored energy release into the fracture front, and is not governed by Weibull-typestatistical theory. Experimental evidence on the existence of the size effect, hitherto ignored in designpractice and code provisions, is documented.Chapter 2 gives a brief
21、 review of the necessary basic results of linear elastic fracture mechanics(LEFM). In concrete, departures from this classical theory are caused by the existence of distributedcracking (or damage) in a progressively softening fracture process zone which surrounds the tip of a con-tinuous crack. In C
22、hapter 3 nonlinear fracture models characterizing the softening stress-displacementor stress-strain relations (such as those of Hillerborgs fictitious crack model, crack band model, nonlocalstrain-softening models, etc.) are described and random particle simulation of aggregate microstruc-ture is di
23、scussed. The principles of implementation of these models in finite element programs are alsooutlined. Chapter 4 presents simpler nonlinear fracture models which represent adaptations of linearelastic fracture mechanics, such as Jenq and Shahs model and the R-curve, along with determinationof geomet
24、ry-dependent R-curves from the size effect law proposed b y Ba zant . This law, describingthe approximate dependence of the nominal stress at maximum load on structure size, is discussed inChapter 5 , and structural response is characterized by the brittleness number.Chapter 6 presents in considerab
25、le detail the current methods for experimental and analytical deter-mination of material fracture parameters, including the quasi-LEFM methods , RILEM (work-of-fracture)method, the Jenq-Shah and Karihaloo-Nallathambi methods, and the size-effect method. Experimen-tal determination of the characteris
26、tic length for nonlocal continuum models and the strain-softeningproperties is then examined, and material parameters for modes II and III, shear fractures and mixedmode fracture are also discussed. Chapter 7 then proceeds to describe various influencing factors, suchas the loading rate, humidity an
27、d temperature, as well as the effect of cyclic loading. Chapter 8 isdevoted to the effect of reinforcing bars and their bond slip on fracture propagation, and to fractureof fiber-reinforced concrete. Chapter 9 deals with more theoretical problems of modeling systems ofinteracting cracks. Attention i
28、s focused on systems of parallel growing cracks. Their stability decidesthe spacing and width of the cracks from the mechanics viewpoint.It is concluded that, after a decade of rapid progress in research, the time appears ripe for introducingfracture mechanics into design practice. This should not o
29、nly bring about more uniform safety margins,thus improving safety and economy of design, but also pave the way for safer and more efficient use ofhigh-performance concretes and permit design extrapolations beyond the range of previous experimentsand design.KEYWORDS: Brittleness, concrete, concrete s
30、tructures, crack spacing and width, cracking, damagemechanics, design codes, ductility, failure, fiber-reinforced concrete, nonlocal continuum models,reinforced concrete, size effect, strain softening, structural design, testing methods, ultimate loads.FR AC TUR E MECHANICS446.1R-5IntroductionConcre
31、te structures are full of cracks. Failure of concrete structures typically involvesstable growth of large cracking zones and the formation of large fractures before the maximumload is reached. Yet design is not based on fracture mechanics, even though the basic fracturemechanics theory has been avai
32、lable since the middle of this century. So why has not fracturemechanics been introduced into concretedesign? Have concrete engineers been guilty ofignorance? Not at all. The forms of fracture mechanics which were available until recentlywere applicable only to homogeneous brittle materials such as
33、glass, or to homogeneousbrittle-ductile metals. The question of applicability of these classical theories to concretewas explored long ago - the idea of using the stress intensity factor appeared already in theearly 1950s (e.g., Bresler and Wollack, 1952)and serious investigations started in the 196
34、0s(e.g., Kaplan, 1961, and others). But the answer was, at that time, negative (e.g., Kesler,Naus and Lott, 1971). As is now understood, the reason was that in concrete structures onemust take into account strain-softening due to distributed cracking, localization of crackinginto larger fractures pr
35、ior to failure, and bridging stresses at the fracture front. A form offracture mechanics that can be applied to such structures has been developed only duringthe last decade.Concrete design has already seen two revolutions. The first, which made the technologyof concrete structures possible, was the
36、 developrnent of the elastic no-tension analysis during1900-1930. The second revolution, based on a theory conceived chiefly during the 1930s, wasthe introduction of plastic limit analysis, which occurred during 1940-1970. There are goodreasons to believe that the introduction of fracture mechanics
37、into the design of concretestructures, both reinforced and unreinforced, might be the third major revolution. Thetheory, formulated mostly during the last dozen years, finally appears to be ripe.Fracture researchers have at the present no doubt that the introduction of fracture me-chanics into the d
38、esign criteria for all brittle failures of reinforced concrete structures (suchas diagonal shear, punching shear, torsion or pull out, or for concrete dams), can bringabout significant benefits. It will make it possible to achieve more uniform safety margins,especially for structures of different si
39、zes.This, in turn, will improve economy as well asstructural reliability. It will make it possible to introduce new designs and utilize new con-crete materials. Fracture mechanics will be particularly important for high strength concretestructures, fiber-reinforced concrete structures, concrete stru
40、ctures of unusually large sizes,and for prestressed structures.The application of fracture mechanics is most urgent forstructures such as concrete dams and nuclear reactor vessels or containments, for which thesafety concerns are particularly high and the consequences of a potential disaster enormou
41、s.Surveys of concrete fracture mechanics have recently been prepared by various commit-tees (Wittmann, 1983, and Elfgren, 1989). However, due to the rapidly advancing research,the contents of the present state-of-the-art report are quite different. A unified, systematicpresentation, rather than a co
42、mpilation of all the contributions by various authors, is at-tempted in the present state-of-art report. The report is aimed primarily at researchers, notnecessarily specialists in fracture mechanics.However, it should also be of interest to design446.1R-6 ACI COMMITTEE REPORTengineers because it de
43、scribes a theory that is likely to profoundly influence the design prac-tice in the near future. Subsequent reports dealing with applications in design, finite elementanalysis of fracture, and dynamic fracture analysis, are in preparation by ACI Committee446.Chapter 1. WHY FRACTURE MECHANICS?Fractur
44、e mechanics, in a broad sense, is a failure theory which (1) uses energy crite-ria, possibly in conjunction with strength criteria, and (2) which takes into account failurepropagation through the structure.1.1 Five Reasons for Fracture Mechanics ApproachSince concrete structures have been designed a
45、nd successfully built according to codeswhich totally ignore fracture mechanics theory, it might seem unnecessary to change thecurrent practice. Nevertheless, there are five compelling reasons for doing so.Reason 1: Energy Required for Crack FormationFrom the strictly physical viewpoint, it must be
46、recognized that while crack initiationmay depend on stress, the actual formation of cracks requires a certain energy - the fractureenergy -which represents the surface energy of a solid.Hence, energy criteria should beused. This argument might suffice to a physicist but not a designer. But there are
47、 otherreasons.Reason 2: Objectivity of CalculationsAny physical theory must be objective in the sense that the result of calculations madewith it must not depend on subjective aspects such as the choice of coordinates, the choiceof mesh, etc. If a theory is found to be unobjective, it must be reject
48、ed. There is no needto even compare it to experiments. Objectivity comes ahead of experimental verification.A powerful approach to finite element analysis of concrete cracking is the concept ofsmeared cracking, introduced by Rashid (1968).According to this approach, the stress in afinite element is
49、limited by the tensile strength of the material, f t, and after reaching thisstrength limit, the stress in the finite element must decrease. As initially practiced, the stresswas assumed to decrease suddenly to zero, in a vertical drop; but soon it was realized thatbetter and more realistic results are usually obtained if the stress is reduced gradually, i.e.,the material is assumed to exhibit strain-softening (Scanlon, 1971; Lin and Scordelis, 1975);see Fig. l.la. The concept of strain-softening, though, proved to be a mixed blessing. Afterstrain-softeni
