1、ACI 446.1R-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 Elices1
2、,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 SuarisS
3、tuart 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 Committee Reports, Guides, Standard Practices, andCommentaries are intended for guidance in designing,plannin
5、g, 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 Proje
6、ct Documents.Copyright 0 1991, American Concrete Institute.All rights reserved 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, or oral or recording for soundor vis
7、ual reproduction or for use in any knowledge or retrieval system or device,unless permission in writing is obtained from the copyright proprietors.446.1 R-l-446.1R-2 ACI COMMITTEE REPORTCONTENTSSynopsis, p. 446.1R-4Introduction, p. 446.1R-5Chapter 1 - Why fracture mechanics?, p. 446.1R-61.1 - Five r
8、easons 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 elastic f
9、racture mechanics, p. 446.1R-242.1 - Stress singularity2.2 - Energy criterion2.3 - Limits of applicabilityChapter 3 - Nonlinear fracture models with softening zone, p. 446.1R-283.1- Softening stress-displacement relations3.2 - Softening stress-strain relations3.3 - Stress-displacement vs. stress-str
10、ain softeningrelations3.4 - Nonlinear triaxial models for strain-softening3.5- Random particle simulation of microstructureChapter 4 - Special nonlinear fracture models based on adaptation of LEFM, 446.lR-534.1 - Effective crack models4.2 - Two-parameter model of Jenq and Shah4.3 - Geometry-dependen
11、t R-curve determined from size effect lawChapter 5 - Size effect and brittleness o f str uctures p. 446.1R-585.1 - Size effect law for maximum nominal stress5.2 - Brittlenessnumber5.3- Other sizeeffects andlimitationsChapter 6 - Experimental or analytical determination of material fracture parameter
12、s,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 model of Karihaloo and Nallathambi446.1 R-36.7 - Determination of mater
13、ial 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 parameters for Mode II and planar mixed mode fracture6.12 - Material paramet
14、ers 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 8 - Effect ofreinforcement, 446.1R-1098.1- Effect ofreinforcing steel bars
15、8.2 - Fracture 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 in beams9.4 - Interacting microcracksConcluding remarks, p. 446.1R-119Acknowledgement, p. 446.
16、1R-119References, p. 446.1R-120Appendix - Derivations of some formulas, p. 446.1R-140Extended summary, p. 446.1R-141Basic notations, p. 446.1R-146446.1R-4 ACI COMMITTEE REPORTSYNOPSISIn the first of its series of four state-of-the-art reports under preparation, the Committee describesthe basic conce
17、pts of fracture mechanics of concrete, the existing theoretical models, and the methodsfor determining the material fracture parameters. Chapter 1 offers five reasons for introducing fracturemechanics into certain aspects of design of concrete structures, including some code provisions: (1) atheoret
18、ical 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 whenever th e load-deflection diagram lacks a yield plateau; (4) the need to rationally predict ductility and
19、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 absorption capability. Thesize effect is due to stored energy release into the fracture front, and is not g
20、overned 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 review of the necessary basic results of linear elastic fracture mechanics(LEFM). In concrete, departure
21、s 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 Chapter 3 nonlinear fracture models characterizing the softening stress-displacementor stress-strain relat
22、ions (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 discussed. The principles of implementation of these models in finite element programs are alsooutlined. Ch
23、apter 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 geometry-dependent R-curves from the size effect law proposed b y Bazant. This law, describingthe approximate d
24、ependence 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 considerable detail the current methods for experimental and analytical deter-mination of material fracture parameters
25、, 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 characteristic length for nonlocal continuum models and the strain-softeningproperties is then examined, and material p
26、arameters 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 and temperature, as well as the effect of cyclic loading. Chapter 8 isdevoted to the effect of reinforcing bar
27、s 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 is focused on systems of parallel growing cracks. Their stability decidesthe spacing and width of the cracks
28、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 only bring about more uniform safety margins,thus improving safety and economy of design, but also pave the w
29、ay for safer and more efficient use ofhigh-performance concretes and permit design extrapolations beyond the range of previous experimentsand design.KEYWORDS: Brittleness, concrete, concrete structures, crack spacing and width, cracking, damagemechanics, design codes, ductility, failure, fiber-reinf
30、orced concrete, nonlocal continuum models,reinforced concrete, size effect, strain softening, structural design, testing methods, ultimate loads.FRACTURE MECHANICS446.1R-5IntroductionConcrete structures are full of cracks. Failure of concrete structures typically involvesstable growth of large crack
31、ing 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 available since the middle of this century. So why has not fracturemechanics been introduced into concretedesign? H
32、ave 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 glass, or to homogeneousbrittle-ductile metals. The question of applicability of these classical theories to co
33、ncretewas 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 1960s(e.g., Kaplan, 1961, and others). But the answer was, at that time, negative (e.g., Kesler,Naus and Lott, 197
34、1). 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 prior to failure, and bridging stresses at the fracture front. A form offracture mechanics that can be applied to
35、 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 developrnent of the elastic no-tension analysis during1900-1930. The second revolution, based on a theory conc
36、eived 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 into the design of concretestructures, both reinforced and unreinforced, might be the third major revolution. T
37、hetheory, 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 design criteria for all brittle failures of reinforced concrete structures (suchas diagonal shear, punching shea
38、r, 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 sizes.This, in turn, will improve economy as well asstructural reliability. It will make it possible to introduce
39、 new designs and utilize new con-crete materials. Fracture mechanics will be particularly important for high strength concretestructures, fiber-reinforced concrete structures, concrete structures of unusually large sizes,and for prestressed structures.The application of fracture mechanics is most ur
40、gent 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 enormous.Surveys of concrete fracture mechanics have recently been prepared by various commit-tees (Wittmann, 1983, an
41、d 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 compilation of all the contributions by various authors, is at-tempted in the present state-of-art report. The re
42、port 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 describes a theory that is likely to profoundly influence the design prac-tice in the near future. Subsequent rep
43、orts dealing with applications in design, finite elementanalysis of fracture, and dynamic fracture analysis, are in preparation by ACI Committee446.Chapter 1. WHY FRACTURE MECHANICS?Fracture mechanics, in a broad sense, is a failure theory which (1) uses energy crite-ria, possibly in conjunction wit
44、h 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 and successfully built according to codeswhich totally ignore fracture mechanics theory, it might seem unnecessa
45、ry to change thecurrent practice. Nevertheless, there are five compelling reasons for doing so.Reason 1: En ergy Required for Crack FormationFrom the strictly physical viewpoint, it must be recognized that while crack initiationmay depend on stress, the actual formation of cracks requires a certain
46、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 otherreasons.Reason 2: Objectivity of CalculationsAny physical theory must be objective in the sense that the
47、 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 rejected. There is no needto even compare it to experiments. Objectivity comes ahead of experimental verification.A
48、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 limited by the tensile strength of the material, f t, and after reaching thisstrength limit, the stress in the
49、 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-softening had been implemented in large finite element programs and widely applied,it was discovered that the conver
