Universal Size Effect Law and Effect of Crack Depth on Quasi-Brittle Structure Strength

Transcription

1 Universal Size Eect Law and Eect o Crack epth on Quasi-Brittle Structure Strength Zdeněk P. Bažant 1 and Qiang Yu 2 Astract: In cohesive racture o quasi-rittle materials such as concrete, rock, ier composites, tough ceramics, rigid oams, sea ice, and wood, one can distinguish six simple and easily modeled asymptotic cases: the asymptotic ehaviors o very small and very large structures, structures ailing at crack initiation rom a smooth surace and those with a deep notch or preexisting deep crack, the purely statistical Weiull-type size eect, and the purely energetic deterministic size eect. Size eect laws governing the transition etween some o these asymptotic cases have already een ormulated. However, a general and smooth description o the complex transition etween all o them has een lacking. Here, a smooth universal law ridging all o these asymptotic cases is derived and discussed. A special case o this law is a ormula or the eect o notch or crack depth at ixed specimen size, which overcomes the limitations o a recently proposed empirical ormula y uan et al., 2003, 2004, OI: /ASCE :278 CE ataase suject headings: Size eect; Fracture; Cracking; Notches; Structural reliaility; Concrete. Introduction In plastic limit analysis or elasticity with strength limit, the nominal strength o structure is size independent. However, or quasirittle materials such as concrete, rock, ier composites, tough ceramics, rigid oams, sea ice, sti soils, and wood i.e., heterogeneous rittle materials with a racture process zone FPZ that is not negligile compared to structural dimensions, the nominal strength depends on structure size. This phenomenon is called the size eect. There are two kinds o size eect: a statistical, descried y the Weiull 1939a,, 1951 theory oandom local material strength, and energetic deterministic. In the latter, one discerns principally the Type I size eect, occurring in structures that ail at crack initiation rom a smooth surace, and the Type II size eect, occuring in structures with a deep notch or deep stress-ree atigued crack ormed staly eore reaching the maximum load. The large size and small size asymptotic ehaviors have already een ridged y closed-orm size eect laws, oth or the purely energetic size eect Type II and energetic-statistical size eect Type I Bažant 1984, 1997, 2001, 2002, 2004; Bažant and Chen 1997; Bažant and Planas However, ridging the size eects or the cases o no notch and a deep notch or crack is a more diicult prolem and is important or predicting the ehavior o structures with shallow ut nonzero cracks. This prolem was tackled y Bažant 1997 and a kind o universal size eect law was derived; however, with two serious limitations: 1 it was 1 McCormick Institute Proessor and W.P. Murphy Proessor o Civil Engineering and Materials Science, Northwestern Univ., CEE, 2145 Sheridan Rd., Evanston, IL z-azant@northwestern.edu 2 Postdoctoral Research Associate, Northwestern Univ., CEE, 2145 Sheridan Rd., Evanston, IL qiangyu@northwestern.edu Note. iscussion open until July 1, Separate discussions must e sumitted or individual papers. The manuscript or this paper was sumitted or review and possile pulication on May 18, 2006; approved on July 29, This paper is part o the Journal o Engineering Mechanics, Vol. 135, No. 2, Feruary 1, ASCE, ISSN /2009/ /$ purely energetic and deterministic i.e., the Weiull statistical asymptote or Type I size eect was not captured, and 2 the dependence o nominal strength on the notch or crack depth was not smooth. The ojective o the present study is to derive an improved universal size eect law that is ree o these two limitations and smoothly descries the ehavior transitional among all the six simple asymptotic cases. As a yproduct, the dependence o the nominal strength on the notch depth over the entire range o depth will also e otained, and compared to previous work o uan et al Review o Energetic Types I and II Size Eect Laws When a quasi-rittle structure has a deep notch or a large tractionree i.e., atigued crack that has ormed eore reaching the maximum load, the size eect on the mean nominal strength o structure is essentially energetic, with the negligile statistical component Bažant and Xi This size eect, termed Type II size eect, may approximately e descried y the size eect law proposed y Bažant1984 N = B t / 0 1/2 where N = P/ or P/ 2 =nominal strength or scaling in two or three dimensions; P = maximum applied load or load parameter; =characteristic size o structure; =thickness in the third dimension o a structure scaled in two dimensions; and B and 0 = parameters depending on structural geometry. Eq. 1 was later rederived more generally y Bažant and Kazemi 1990 and Bažant 1997 using asymptotic approximations o the energy release unction o a propagating crack ased on equivalent linear elastic racture mechanics LEFM; see also Bažant and Planas Truncating the expansion ater the second term, one can express the size eect law in terms o racture characteristics 1 78 / JOURNAL OF ENGINEERING MECHANICS ASCE / FEBRUARY 2009

2 N = EG g 0 c + g 0 where the parameters are expressed as 0 = c g 0 g 0 B t = EG c g 0 Here 0 =a 0 /=relative initial crack length; c =eective length o racture process zone considered as a material parameter; g=k I / P 2 =dimensionless energy release unction o equivalent LEFM characterizing the specimen geometry K I =stress intensity actor; g 0 =dg/d at = 0, with the prime denoting derivatives; E=E or E/1 2 or plane stress or plane strain =Poisson s ratio; E=Young s modulus; and G =racture energy. The Type II size eect law in Eq. 1 Bažant 1984, 2001, 2002 applies to most notched racture specimens and also to most ailure types oeinorced concrete structures. Structures exhiiting Type II size eect are also the ojective o good design ecause large stale crack growth prior to maximum load endows the structure with large energy dissipation capaility and signiicant ductility. Many quasi-rittle structures, however, ail at crack initiation rom a smooth surace, as soon as the racture process zone or oundary layer o cracking ully develops. In that case, the size eect is o Type I Bažant 2001, It was analyzed y Bažant and Li 1995 ased on stress redistriution caused y a oundary layer o densely distriuted microcracking that has, at the peak load, a size-independent critical thickness. In a more general way, related to racture mechanics, the same Type I size eect was deduced y Bažant 1997 rom the limiting case o energy release and dissipation or crack length approaching zero. Because unction g vanishes or 0 while its irst and second derivatives do not, the third term o the large-size asymptotic series expansion o unction g aout the point =0 or must e retained i the size eect should e captured; this gives 1/2 EG N = g0c + g0c 2 = /2 r 1 2 1/2 r 1/r 4 in which the asymptotic approximations 1 2x 1/2 1+x rx 1/r or x= /1 having an error o the order o x 2 have een used and the ollowing notations have een made: = EG g0c = 2 3 g0 4g0 c c = c 5 Here,, and r=positive constants or geometrically similar specimens; has the meaning o nominal strength or a very large structure; has the meaning o eective thickness o the oundary layer; c =eective length depth o racture process zone or racture initiation rom a smooth surace; the operator. Macauley racket means the positive part, i.e., x=maxx,0; and =constant1 ut close to 1, which characterizes the ratio o the eective sizes o the cracking zones or FPZ at a smooth surace and at the tip o a deep notch or crack. From oservations, it appears that the FPZ or crack initiation is generally larger than the FPZ or a crack starting rom a deep notch. The empirical coeicient r had to e introduced ecause the case r=1 gives only one among ininitely many cases that give the same irst two terms o the asymptotic expansion and are equally plausile rom the mathematical viewpoint. Considering only r = 1 would, thus, e an aritrary, unreasonale, restriction. According to experimental data, the optimum r-value generally lies etween 1/2 and 1 depending on the coeicient o variation oandom material strength Bažant and Pang 2006, Note that the second expression in Eq. 4 would give complex N -values when is not large enough. This eature is, o course, unrealistic though not surprising ecause this expression was otained as a large-size asymptotic approximation. However, this eature is eliminated y the last expression in Eq. 4, which is equally justiied as Eq. 4 ecause it has the same irst two terms o the large-size asymptotic expansion in powers o a/, yet is realistic or a road range o practically all the cross sections larger than the representative volume o material except or 0. According to the cohesive crack model Barenlatt 1959, which is a continuum model, the limit o N or 0 should e inite. This condition may e satisied y the ollowing modiication having no eect on the large size asymptotic expansion Bažant 1997: r 1/r + l p where l p =material characteristic length that represents the size o the representative volume element o quasi-rittle material. This length equals aout two to three aggregate sizes in concrete, and is aout the same as the minimum possile spacing o parallel cohesive cracks, or as the eective width o the racture process zone across the direction o propagation Bažant and Pang 2006, Although the value o l p is empirical, its introduction is necessary or mathematical reasons, as a means to satisy the asymptotic requirement or 0 while ensuring the eect o l p to e negligile or l p. Note that l p diers rom the Irwin 1958 characteristic length l 0 =EG / t 2, which characterizes the length o the racture process zone in the direction o propagation. Review o Energetic-Statistical Type I Size Eect Law Since the material strength is random, a macrocrack can initiate at many dierent points in the structure. Thereore, the size eect o Type I must, or /l p, approach the Weiull statistical size eect. Based on the nonlocal Weiull theory, which comines the energetic and statistical size eects as conceived y Bažant and Xi 1991 and extended y Bažant and Novák 2000a,, the ollowing generalization o Eq. 4 was derived y Bažant and Novák 2000a,c: rn/m + r 1/r For small, this ormula converges to Eq. 4, and or large it converges to Weiull size eect N n/m. A similar statistical generalization o the extended energetic ormula in Eq. 6 reads Bažant 2004; Bazǎnt et al ls + + rn/m r 1/r l p + where l s =second statistical characteristic length. Although its value is empirical, l s must e introduced or the same mathematical reasons as already explained or l p elow Eq JOURNAL OF ENGINEERING MECHANICS ASCE / FEBRUARY 2009 / 79

3 In the size eect o Type II, the randomness o material strength has no signiicant eect on the mean nominal strength N o the structure, which is the ojective o ormulating the present universal size eect law. The randomness controls only the statistical distriution o N. There also exists a Type III size eect Bažant 2001, 2004, which, however, is so close to Type II that it is hardly distinguishale experimentally, and will not e considered here. Asymptotic Conditions Required in ierent Types o Size Eect The cohesive crack model has emerged as the most realistic among simple models or quasi-rittle racture. According to this deterministic model, cohesive racture must have the ollowing asymptotic properties or vanishing and ininite structure sizes Bažant 2001, 2002: For 0: N 1 k 0 + O 2 all types 9 For and 0 0: N 1+k O 2 Type I 10 For and large 0 : N 1/2 1 k O 2 Type II 11 where k 0,k 1,k 2 =positive constants Type III is omitted, since it is similar to Type II. The deterministic size eect laws in Eqs. 1 and 6 or Eqs. 2 and 6 satisy these asymptotic properties. This may e checked y the ollowing second-order approximations derived y inomial power series expansions: 0 1+/ 0 1/2 0 1+/ 0 1/2 r 1/r l p + /l p 0 / 0 / 0 0 r l p r 1/r l p / /1 0 /2 / 1/r1 1+l p /rl p where L p =constant. These equations show that the ormulas or Types I and II size eects have very dierent asymptotic properties or 0 and ; see Fig. 1. In Type II, the asymptote or is 1/2 a straight line o slope 1/2 in doule logarithmic scale, which is the size eect o similar cracks or perectly rittle ehavior, governed y LEFM. For 0, the size eect curve o N versus approaches a horizontal asymptote, i.e., the size eect disappears, which is typical o plasticity. In Type I size eect, the racture process zone, represented y the oundary layer o distriuted cracking, ecomes negligile compared to the specimen size when. There is then negligile stress redistriution and ailure occurs when the maximum elastically calculated stress attains the material strength value r. So, or, the size eect asymptotically vanishes; see the horizontal asymptote in Fig. 1. σ N 6 2 Type II d 2 1 For vanishing structure size 0, the cohesive crack model implies that the material strength is moilized at all the points o the ailure surace or crack. This is equivalent to a crack illed y a perectly plastic glue, in which case the size eect also vanishes. Ater analyzing the asymptotic expansion and determining the asymptotic properties or dierent size eects, Bažant 1995, 1997 derived a kind o universal size eect ormula, which has oth Type I crack initiation and Type II large notch as its limit cases; it reads Nu = B t r 1/s 2l 0 0 1/2r1+s 2l in which r,s,=empirical parameters, the values o which can e set approximately as r=s=1 see their discussion in Bažant 1995, The constant oundary layer thickness rom Eq. 5 is here replaced y parameter l depending on the initial crack or notch length l = g 0 c 4g0 17 In this previous attempt or a universal size eect ormula, the irst term contains the size eect law or notched specimen, while the second term contains the law or crack initiation. The threedimensional plot o this ormula is given in Fig in Bažant and Planas 1998, and also in Fig. 6 o Bažant There are, however, two shortcomings o Eq. 16, which need to e remedied. First, Eq. 16 at =0 crack initiation converges or to a horizontal asymptote while correctly it should converge to the power law n/m o the Weiull statistical size eect, i.e., to a straight line o slope n/m in the logarithmic scale m=weiull modulus or concrete m24, and or most materials m=10 to 50. For racture specimens and many structures, the racture geometry scales in two dimensions, i.e., n=2, whether or not the structure is scaled in two or three dimensions i.e., the eam width has no eect on the nominal strength. Second, Eq. 16 is not smooth, which is evident in its threedimensional picture shown in Bažant 1995, 1997 and in Bažant and Planas The surace has a sudden change o slope at 0.1, which is caused y the Macauley racket in Eq. 17. Fig. 2 shows the plot o the energy release unction and its irst and second derivatives or a three-point end eam with the spandepth ratio o S/=4. This is a standard ASTM test specimen geometry, or which the approximate energy release unction is given in handooks and textooks Tada et al. 1985; Murakami 6 2 Type I n d Fig. 1. Two types o size eect ehavior; let: Type II; right: Type I m 80 / JOURNAL OF ENGINEERING MECHANICS ASCE / FEBRUARY 2009

4 1987; Broeck 1988; Kanninen and Popelar Here we use the ollowing more accurate approximation, derived y Pastor et al in which g, g'/8, g''/ g'(α)/8 g = k 2 p 4 k = /2 p 4 = It is typical or this and other geometries that g 0 changes its sign rom negative to positive. This occurs at 0 0.1, which is where the slope o the surace is discontinuous, ecause o discontinuity o g 0. Universal Size Eect Law S An improved universal size eect law, which has already een reported without derivation at a recent conerence Bažant and Yu 2004, will now e derived in detail, ased on asymptotic arguments. For close to 0, the dimensionless energy release unction g may e approximated y its irst three terms o the Taylor series expansion at 0 N = EG 1/2 g EG g 0 + g 0 c + gc 2 or a 0 / where, or revity, g 0 =g 0, g 0 =g 0, g=g 0 0. For ailure at crack initiation Type I, g 0 =g 0 =g0=0. To separate this case rom Type II, or which g 0 0 and the third term with g 0 must e separated, or else the opposite asymptotic properties o cohesive crack model or 0 could not e matched; Bažant 2001, the last expression, applicale only or large enough, may e rearranged as ollows: 2 1/2 EG N = g 0 c g 0 + gc 0 1/21+ 2g 0 + gc 0 EG 2 1/r rg 0 c 21 g 0 + gc 0 1 4g 0 + g 0 c α g(α) g''(α)/80 Fig. 2. Energy release unction and its irst and second order derivatives or a three-point end eam a 0 Here we inserted an aritrary coeicient r. This insertion is permitted, and in act required or generality, ecause the asymptotic expansions o the last two expressions in terms o powers o a 0 / are independent o up to the quadratic terms, and ecause there is no reason or r to e 1. However, note that although oth previous expressions coincide exactly or r = 2, a negative r could not e used ecause typically g 0 0, which would make N imaginary. When the specimen has a deep notch i.e., when a 0 is not negligile compared to the cross section dimension, the preceding ormula must e made identical to Eq. 2. So, g 0 must e replaced y a unction smoothly approaching 0 when 0 ecomes large enough i.e., when a 0 ecomes non-negligile compared to the cross section dimension. To this end, we can replace g 0 y g 0 e k 0 q, where k and q=positive empirical constants controlling the transition. Eq. 21 may, thus, e rewritten as N = EG c 2 ge k 0 q 1/r 0 22 g 0 + g 0 c 1 4g 0 + g 0 c To check the general applicaility o Eq. 22, note the ollowing two opposite asymptotic cases o crack initiation and o deep notch: For large 0 : For 0 0: e k 0 q 0 and N = EG g 0 + gc 0 N = EG 1+ g 1/r 0 rc gc 0 4g For 0, Eq. 22 gives an ininite nominal strength N, this would violate the small-size asymptotic limit o the cohesive crack model, which is always inite. We can circumvent this prolem y replacing 4 y 4+l p in Eq. 22, where l p gives the center o transition to a horizontal asymptote and represents a material characteristic length, which should e approximately equal to the maximum aggregate size Bažant and Pang 2006, For /l p, Eq. 22 is approached asymptotically. With this modiication, the oregoing asymptotic cases satisy the asymptotic conditions o the cohesive crack model Eqs. 23 and 24; hence, Eq. 22 satisies them too. Finally, y comparison with test data, it seems possile to set q = 2. So, i only the deterministic-energetic size eect is considered, and i g 0 0, the new deterministic universal size eect law or mean N may e expressed as ollows: N = EG g 0 c + g 0 1 rc 2 ge k 2 1/r 0 0 4l p + g 0 + g 0 c 25 Fig. 3 shows a three-dimensional plot o Eq. 25 or the previously considered three-point end eam with S/ = 4, and or the ollowing typical material parameters o concrete: c =200 mm, l p =100 mm, E=28.0 GPa, G =70 N/m, and t =3.0 MPa with empirical constants r=1 and k=115. The cross sections o this surace or constant 0 represent the size eect curves, Type I or 0 =0 and Type II or large 0, and it can e clearly seen that a smooth transition etween these two types is achieved thanks to the suppression o discontinuous Macauley rackets with the multiplier e k2. JOURNAL OF ENGINEERING MECHANICS ASCE / FEBRUARY 2009 / 81

5 Fig. 3. Improved universal size eect law; let: without Weiull statistics; right: with Weiull statistics To capture the Weiull statistical size eect, which ecomes signiicant or very large unnotched structures, a statistical part, analogous to Eq. 7, may e superposed on Eq. 25 N = EG g 0 c + g 0 l s l s + e 0 rc 2 ge k l p + g 0 + g 0 c 1/r rn/m 2 26 i g 0 0. This inal ormula, representing a general universal size eect law, satisies three asymptotic conditions or size eect at crack initiation 0 =0: For no notch, 0 0 g 0 0, and or small enough sizes, l s, Eq. 26 asymptotically approaches the deterministicenergetic ormula r 1/r l p + 27 For no notch, 0 0 g 0 0, and or large enough sizes, Maxl s,l p, Eq. 26 asymptotically approaches the Weiull type size eect Way o Experimental Identiication o Material Parameters In Eq. 26, there are seven ree parameters: m, r, c, l p, l s, k, and. First, c as well as EG may e identiied rom Type II size eect tests on scaled notched specimens. Knowing c, parameters, r, and may then e identiied y itting Eq. 27 to the test results or modulus oupture or lexural strength at dierent sizes, with a suicient size range they may also e otained y discrete particle simulations, ater that c may e solved rom Eq. 4, and then =c /c according to Eq. 5. Parameter l p matters only or extrapolation to zero size and can e otained only y calculating the zero size limit with the cohesive crack model, although estimating it as equal to the maximum aggregate size seems adequate. Knowing, parameters m Weiull modulus and l s Weiull scaling parameter can e identiied y itting Eq. 28 to test data on the statistical size eect, which can e otained directly only on very large unnotched specimens. Parameters and k can e experimentally identiied only y testing the Type I-Type II transition, although approximately one can proaly assume that, or relative notch depth 0 =0.1, the values o k 0 2 and 0 2 are 1, which gives k100. l s / n/m 28 For m and any size, Eq. 26 coincides with deterministic-energetic ormula 25. The three-dimensional surace o the universal size eect law in Eq. 26 is shown in Fig. 3. The surace is seen to e smooth. Special Case o Crack Length Eect, Contrasted with uan-hu Formula A semiempirical size eect ormula or the maximum load dependence on the crack length at constant size was proposed y 82 / JOURNAL OF ENGINEERING MECHANICS ASCE / FEBRUARY 2009

6 Fig. 4. Proiles otained rom improved universal size eect ormulas Eqs. 25 and 26 compared to uan et al approximation uan et al. 2002, 2003, 2004, 2006 and y Hu and Wittmann 2000 n = 0 1+a/a * 1/2 29 Here 0 = t is assumed or small three-point end specimens, and a * is a certain constant representing the maximum tensile stress in the ligament ased on a linear stress distriution over the ligament, and in uan and Hu s notation n = N /A 0 where A 0 =1 0 2 or three-point end specimens. uan and Hu s ormula ought to e equivalent to the proile o the universal size eect law Fig. 4 at constant size, scaled y the ratio n / N =1/A 0. The curve o that ormula Eq. 29 approaches the asymptotic case or 0 0 with a horizontal asymptote in log 0 scale. However, the asymptotic limit or 0 0 is independent o the structure size. So we conclude that, according to uan and Hu s ormula, there is no size eect or ailure at crack initiation rom a smooth surace Type I, e.g., in the tests or lexural strength or modulus oupture. This is an unrealistic eature o their ormula, conlicting with extensive experimental evidence or unnotched eams Bažant and Li 1995; Bažant 1998, 2001, 2002; Bažant and Novák 2000,c. Except within a small portion o the ranges o crack length and structure size, the proiles at constant and given y the present universal size eect law are not matched y uan and Hu s ormula ater its conversion rom n to N. So, even though Eq. 29 seems to work or a narrow range o typical deep-notched racture specimens o concrete, it does not have road applicaility. This is not surprising ecause Eq. 29 is not ased on the energy release unction g in the sense o equivalent LEFM, and ecause the material strength t in the cohesive crack model, which is a constant, is not applicale to very short or vanishing notches, or which the tensile strength must exhiit the Type I size eect. For very short cracks or notches 0 0.1, the tensile strength must e treated in the way o either the Guinea et al. 1994a, method, or a similar method y Bažant et al called the zero-rittleness method. Conclusions There are two simple asymptotic types o size eect in quasirittle racture: Type I, which occurs in ailures at crack initiation rom a smooth surace, and Type II, which occurs in ailures starting rom a deep notch or crack. To descrie the continuous transition etween these two types o size eect, an improved universal size eect law is required. The improved universal size eect law can e derived y matching asymptotic series expansions or six asic limit cases: 1 the asymptotic ehaviors or very small and very large sizes which can e captured y the irst two nonzero terms o the expansion or each case; 2 the large notch and vanishing notch ehaviors; and 3 the energetic and statistical parts o size eect. In contrast to a previous ormulation Eq. 16, the present universal size eect law achieves a smooth transition etween the Types I and II size eects. In contrast also to the previous ormulation, the classical Weiull statistical size eect is captured as a limiting case o proposed universal size eect law. The dependence o the nominal strength o structure on the notch depth at constant specimen size is a special case o the present universal size eect law. This dependence is more realistic than an empirical ormula previously proposed y uan and Hu. That ormula does not have realistic asymptotics and conlicts with the Type I size eect law, which must e the limit case or a vanishing notch depth. Reerences Barenlatt, G. I The ormation o equilirium cracks during rittle racture. General ideas and hypothesis, axially symmetric cracks. Prikl. Mat. Mekh., 23, Bažant, Z. P Size eect in lunt racture: Concrete, rock, metal. J. Eng. Mech., 1104, Bažant, Z. P Scaling theories or quasiritttle racture: Recent advances and new directions. Proc., 2nd Int. Con. on Fracture Mechanics o Concrete and Concrete Struct., FraMCoS2, F. H. Wittmann ed., Aediicatio Pulishers, Freiurg, Germany, Bažant, Z. P Scaling o quasirittle racture: Asymptotic analysis. Int. J. Fract., 83, Bažant, Z. P Size eect in tensile and compression racture o concrete structures: Computational modeling and design. Fracture mechanics o concrete structures, Proc., 3rd Int. Con., FraMCoS-3, H. Mihashi and K. Rokugo, eds., Aediicatio Pulishers, Freiurg, Germany, Bažant, Z. P Size eects in quasirittle racture: Apercu o recent results. Fracture mechanics o concrete structures, Proc., FraMCoS-4 Int. Con., R. de Borst, J. Mazars, G. Pijaudier-Caot, and J. G. M. van Mier, eds., Swets and Zeitlinger, Balkema, Lisse, The Netherlands, JOURNAL OF ENGINEERING MECHANICS ASCE / FEBRUARY 2009 / 83

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