RILEM TC QFS Quasibrittle fracture scaling and size effect - Final report 1

Transcription

1 Materials and Structures / Matériaux et Constructions, Vol. 37, October 4, pp RILEM TC QFS Quasibrittle racture scaling and size eect - Final report 1 TC Membership Chairman: Z.P. Bažant, USA; Secretary: R. Gettu, Spain; Report editor: M. Jirásek, Switzerland; Members: B.I.G. Barr, UK; I. Carol, Spain; A. Carpinteri, Italy; M. Elices, Spain; C. Huet, Switzerland; H. Mihashi, Japan; K.M. emati, USA; J. Planas, Spain; F.-J. Ulm, USA; J.G.M. van Mier, Switzerland; M.R.A. van Vliet, The etherlands. Other contributions rom: S. Burtscher, B. Chiaia, J.P. Dempsey, G. Ferro, V.S. Gopalaratnam, P. Prat, K. Rokugo, V.E. Saouma, V. Slowik, L. Vítek and K. Willam. ABSTRACT The report attempts a broad review o the problem o size eect or scaling o ailure, which has recently come to the oreront o attention because o its importance or concrete and geotechnical engineering, geomechanics, arctic ice engineering, as well as in designing large loadbearing parts made o advanced ceramics and composites, e.g. or aircrat or ships. First the main results o Weibull statistical theory o random strength are briely summarized and its applicability and limitations described. In this theory as well as plasticity, elasticity with a strength limit, and linear elastic racture mechanics (LEFM), the size eect is a simple power law because no characteristic size or length is present. Attention is then ocused on the deterministic size eect in quasibrittle materials which, because o the existence o a non-negligible material length characterizing the size o the racture process zone, represents the bridging between the simple power-law size eects o plasticity and o LEFM. The energetic theory o quasibrittle size eect in the bridging region is explained and then a host o recent reinements, extensions and ramiications are discussed. Comments on other types o size eect, including that which might be associated with the ractal geometry o racture, are also made. The historical development o the size eect theories is outlined and the recent trends o research are emphasized. RÉSUMÉ Ce rapport tente de passer en revue le problème de l eet de taille ou d échelle sur la rupture, qui est récemment devenu un point ocal d attention à cause de son importance pour les structures en béton, la géotechnique, la géoméchanique, l étude des glaces arctiques ainsi que dans la conception de grandes pièces structurelles en céramiques ou composites modernes, pour l industrie navale ou aéronautique par exemple. Tout d abord, les résultats principaux de la théorie statistique de résistance aléatoire de Weibull, son domaine d application et ses limitations sont décrits. Dans cette théorie, ainsi qu en plasticité, élasticité avec une résistance limite et mécanique de la rupture linéaire élastique, l eet de taille s exprime seulement par une loi puissance puisqu aucune longueur ou taille caractéristique n est présente. L attention se tourne ensuite sur l eet de taille déterministe dans les matériaux quasi-ragiles qui, à cause de l existence d une longueur matérielle non-négligeable caractérisant la taille de la zone de rupture, représente le lien entre les simples lois puissance de la plasticité et de la mécanique de la rupture linéaire. On explique la théorie énergétique des eets de taille quasi-ragiles dans la région de transition. On discute ensuite une multitude de rainement récents, extensions et ramiications. On ait également des commentaires sur d autres types d eets de taille, y compris ceux qui pourraient être associés avec la géométrie ractale des issures. L histoire du développement des théories de l eet de taille est résumée et les tendances récentes de la recherche sont mises en relie. 1 Signiicant portions o this article are reprinted rom [1] with the kind permission o the Editor o the Archives o Applied Mechanics and o Springer Verlag /4 RILEM 547

2 RILEM TC QFS 1. ITRODUCTIO The size eect is a problem o scaling, which is central to every physical theory. In luid mechanics research, the problem o scaling continuously played a prominent role or over a hundred years. In solid mechanics research, though, the attention to scaling had many interruptions and became intense only during the last decade. ot surprisingly, the modern studies o non-classical size eect, begun in the 197s, were stimulated by the problems o concrete structures, or which there inevitably is a large gap between the scales o large structures (e.g. dams, reactor containments, bridges) and o laboratory tests. This gap involves in such structures about one order o magnitude (even in the rare cases when a ull scale test is carried out, it is impossible to acquire a suicient statistical basis on the ull scale). The question o size eect recently became a crucial consideration in the eorts to use advanced iber composites and sandwiches or large ship hulls, bulkheads, decks, stacks and masts, as well as or large load-bearing uselage panels. The scaling problems are even greater in geotechnical engineering, arctic engineering, and geomechanics. In analyzing the saety o an excavation wall or a tunnel, the risk o a mountain slide, the risk o slip o a ault in the earth crust, or the orce exerted on an oil platorm in the Arctic by a moving mile-size ice loe, the scale jump rom the laboratory spans many orders o magnitude. In most o mechanical and aerospace engineering, on the other hand, the problem o scaling has been less pressing because the structures or structural components can usually be tested at ull size. It must be recognized, however, that even in that case the scaling implied by the theory must be correct. Scaling is the most undamental characteristic o any physical theory. I the scaling properties o a theory are incorrect, the theory itsel is incorrect. The size eect on structural strength is understood as the eect o the characteristic structure size (dimension) D on the nominal strength o structure when geometrically similar structures are compared. The nominal stress (or strength, in case o maximum load) is deined as cp bd or c P D or two- or three-dimensional similarity, respectively; P = load (or load parameter), b structure thickness, and c arbitrary coeicient chosen or convenience (normally c = 1). So is not real stress but a load parameter having the dimension o stress. The deinition o D can be arbitrary (e.g. the beam depth or hal-depth, the beam span, the diagonal dimension, etc.) because it does not matter or comparing geometrically similar structures. The basic scaling laws in physics are power laws in terms o D, or which no characteristic size (or length) exists. The classical Weibull theory o statistical size eect caused by randomness o material strength [] is o this type. During the 197s it was ound that a major deterministic size eect, overwhelming the statistical size eect, can be caused by stress redistributions due to stable propagation o racture or damage and the inherent energy release. The law o the deterministic size eect provides a way o bridging two dierent power laws applicable in two adjacent size ranges. The structure size at which this bridging transition occurs represents a characteristic size. The material or which this new kind o size eect was identiied irst, and studied in the greatest depth and with the largest experimental eort by ar, is concrete. In general, a size eect that bridges the small-scale power law or nonbrittle (plastic, ductile) behavior and the large-scale power law or brittle behavior signals the presence o a certain non-negligible characteristic length o the material. This length, which represents the quintessential property o quasibrittle materials, characterizes the typical size o material inhomogeneities or the racture process zone (FPZ). Aside rom concrete, other quasibrittle materials include rocks, cement mortars, ice (especially sea ice), consolidated snow, tough iber composites and particulate composites, toughened ceramics, iber-reinorced concretes, dental cements, bone and cartilage, biological shells, sti clays, cemented sands, grouted soils, coal, paper, wood, wood particle board, various reractories and illed elastomers, as well as some special tough metal alloys. Keen interest in the size eect and scaling is now emerging or various high-tech applications o these materials. Quasibrittle behavior can be attained by creating or enhancing material inhomogeneities. Such behavior is desirable because it endows the structure made rom a material incapable o plastic yielding with a signiicant energy absorption capability. Long ago, civil engineers subconsciously but cleverly engineered concrete structures to achieve and enhance quasibrittle characteristics. Most modern high-tech materials achieve quasibrittle characteristics in much the same way by means o inclusions, embedded reinorcement, and intentional microcracking (as in transormation toughening o ceramics, analogous to shrinkage microcracking o concrete). In eect, they emulate concrete. In materials science, an inverse size eect spanning several orders o magnitude must be tackled in passing rom normal laboratory tests o material strength to microelectronic components and micromechanisms. A material that ollows linear elastic racture mechanics (LEFM) on the scale o laboratory specimens o sizes rom 1 to 1 cm may exhibit quasibrittle or even ductile (plastic) ailure on the scale o.1 to 1 microns. The purpose o this report is to present a brie review o the basic results and their history. For an in-depth review with several hundred literature reerences, the recent article by Bažant and Chen [3] and Bažant s book [4] may be consulted. A ull exposition o most o the material reviewed here is ound in the recent book by Bažant and Planas [4]. The problem o scale bridging in the micromechanics o materials, e.g., the relation o dislocation theory to continuum plasticity, is beyond the scope o this review.. HISTORY OF SIZE EFFECT UP TO WEIBULL Speculations about the size eect can be traced back to Leonardo da Vinci (15s); see [5] and page 546 in [6]. He observed that among cords o equal thickness the longest is the least strong, and proposed that a cord is so much 548

3 Materials and Structures / Matériaux et Constructions, Vol. 37, October 4 stronger... as it is shorter, implying inverse proportionality. A century later, Galileo Galilei [7], the inventor o the concept o stress, argued that Leonardo s size eect cannot be true. He urther discussed the eect o the size o an animal on the shape o its bones, remarking that bulkiness o bones is the weakness o the giants. A major idea was spawned by Mariotte [8]. Based on his extensive experiments, he observed that a long rope and a short one always support the same weight unless that in a long rope there may happen to be some aulty place in which it will break sooner than in a shorter, and proposed the principle o the inequality o matter whose absolute resistance is less in one place than another. In other words, the larger the structure, the greater is the probability o encountering in it an element o low strength. This is the basic idea o the statistical theory o size eect. Despite no lack o attention, not much progress was achieved or two and hal centuries, until the remarkable work o Griith [9], the ounder o racture mechanics. He showed experimentally that the nominal strength o glass ibers was raised rom 9 MPa to 3.39 GPa when the diameter decreased rom 17 µm to 3.3 µm, and concluded that the weakness o isotropic solids...is due to the presence o discontinuities or laws... The eective strength o technical materials could be increased 1 or times at least i these laws could be eliminated. In Griith s view, however, the laws or cracks at the moment o ailure were still only microscopic; their random distribution controlled the macroscopic strength o the material but did not invalidate the concept o strength. Thus, Griith discovered the physical basis o Mariotte s statistical idea but not a new kind o size eect. The statistical theory o size eect began to emerge ater Peirce [1] ormulated the weakest-link model or a chain and introduced the extreme value statistics which was originated by Tippett [11], Fischer and Tippett [1], and Fréchet [13], and reined by von Mises [14] and others [15-18]. The capstone o the statistical theory was laid by Weibull [, 19-1]. On a heuristic and experimental basis, he concluded that the tail distribution o low strength values with an extremely small probability could not be adequately represented by any o the previously known distributions. He introduced what came to be known as the Weibull distribution, although this distribution has been mathematically derived in [1] 11 years earlier in a dierent context. The Weibull distribution gives the probability o a small material element as a power law o the strength dierence rom a inite or zero threshold. Others later oered a theoretical justiication by means o a statistical distribution o microscopic laws or microcracks [15, 17]. Reinement o applications to metals and ceramics (atigue embrittlement, cleavage toughness o steels at low and brittle-ductile transition temperatures, evaluation o scatter o racture toughness data) has continued until today; see e.g. [18, -4]. Most subsequent studies o the statistical theory o size eect dealt basically with reinements and applications o Weibull s theory to atigue embrittled metals and to ceramics [5, 6]. Applications to concrete, where the size eect were o the greatest concern, have been studied e.g. in [7-34]. Until about 1985, most mechanicians paid almost no attention to the possibility o a deterministic size eect. Whenever a size eect was detected in tests, it was automatically assumed to be statistical, and thus its study was supposed to belong to statisticians rather than mechanicians. The reason probably was that no size eect is exhibited by the classical continuum mechanics in which the ailure criterion is written in terms o stresses and strains (elasticity with strength limit, plasticity and viscoplasticity, as well as racture mechanics o bodies containing only microscopic cracks or laws); see [35]. The subject was not even mentioned by Timoshenko in 1953 in his monumental History o the Strength o Materials. The attitude, however, changed drastically in the 198s. In consequence o the well-unded research in concrete structures or nuclear power plants, theories exhibiting a deterministic size eect have been developed. We will discuss it later. 3. POWER SCALIG AD THE CASE OF O SIZE EFFECT It is proper to explain irst the simple scaling applicable to all physical systems that involve no characteristic length. Let us consider geometrically similar systems, or example the beams shown in Fig. 1a, and seek to deduce the response Y (e.g., the maximum stress or the maximum delection) as a unction o the characteristic size (dimension) D o the structure. We choose a certain reerence size D and denote the corresponding response as Y. For a geometrically similar structure o an arbitrary size D, the response can be expressed as Y Y( D D) where is a dimensionless unction o a dimensionless argument, describing the scaling law. For example, or sizes D and 1 D we have Y 1 Y ( D 1 D ) and Y Y( D D). However, since there is no characteristic length, we can also take D as the reerence size 1 and write Y Y1 ( D D1). Consequently, the equation D D D 1 D1 D D must hold or any combination o sizes D, D and 1 D. This is a unctional equation or the unknown scaling unction. Any possible solution must have the orm o a power law D D D D s where s is an arbitrary but ixed exponent. On the other hand, when or instance ( DD ) log( DD ), Equation (1) is not satisied. So, the logarithmic scaling could be possible only i the system possessed a characteristic length and a change o the reerence size implied a change o the scaling unction. The power scaling must apply or every physical theory in which there is no characteristic length. In solid mechanics such ailure theories include elasticity with a strength limit, (1) () 549

4 RILEM TC QFS Fig. - (a) Chain with many links o random strength, (b) Failure probability o a small element, (c) Structure with many microcracks o dierent probabilities to become critical. Fig. 1 - (a) Geometrically similar structures o dierent sizes, (b) power scaling laws, (c) size eect law or quasibrittle ailures bridging the power law o plasticity (horizontal asymptote) and the power law o LEFM (inclined asymptote). elasto-plasticity, viscoplasticity as well as LEFM (or which the FPZ is assumed shrunken into a point). To determine exponent s, the ailure criterion o the material must be taken into account. For elasticity with a strength limit (strength theory), or plasticity (or elastoplasticity) with a yield surace expressed in terms o stresses or strains, or both, one inds that s = when response Y represents the stress or strain (or example the maximum stress, or the stress at certain homologous points, or the nominal stress at ailure); see [35]. Thus, i there is no characteristic dimension, all geometrically similar structures o dierent sizes must ail at the same nominal stress. By convention, this came to be known as the case o no size eect. In LEFM, on the other hand, s = 1/, provided that the geometrically similar structures with geometrically similar cracks or notches are considered. This may be generally demonstrated with the help o Rice s J-integral [35]. I log is plotted versus log D, the power law is a straight line (Fig. 1b). For plasticity, or elasticity with a strength limit, the exponent o the power law vanishes, i.e., the slope o this line is For LEFM, the slope is 1/. An emerging hot subject is the behavior o quasibrittle materials and structures, or which the size eect bridges these two power laws. 4. WEIBULL STATISTICAL SIZE EFFECT The classical theory o size eect has been statistical. Three-dimensional continuous generalization o the weakest link model or the ailure o a chain o links o random strength (Fig. a) leads to the distribution P ( ) 1exp c[ ( x) )] dv( x) (3) V which represents the probability that a structure that ails as soon as macroscopic racture initiates rom a microcrack (or a some law) somewhere in the structure; = stress tensor ield induced by the load that corresponds to the nominal stress, x = coordinate vector, V = volume o structure, and c( ) = unction giving the spatial concentration o ailure probability 1 o the material (= V r ailure probability o material representative volume V ) [15]; c( ) r P1( i ) V i where = principal stresses (i = 1,,3) and i P 1 ( ) = ailure probability (cumulative) o the smallest possible test specimen o volume V (or representative volume o the material) subjected to uniaxial tensile stress []; or suiciently small P 1 it holds u P1 ( ) (4) s m where ms 1 = material constants ( m = Weibull modulus, usually between 5 and 5; s = scale parameter; u = strength threshold, which may usually be taken as ) and V = reerence volume understood as the volume o specimens on which c( ) was measured. For specimens under uniorm uniaxial stress (and u = ), (3) and (4) lead to the ollowing simple expressions or the mean and coeicient o variation o the nominal strength: 1 1 (1 )( ) m s m V V (5) 1 (1 m ) 1 1 (1 m ) where is the gamma unction. Since depends only on m, it is oten used or determining m rom the observed statistical scatter o strength o identical test specimens. The expression or includes the eect o volume V which depends on size D. In general, or structures with nonuniorm multidimensional stress, the size eect o Weibull theory (or ) is o the type: u (6) 55

5 Materials and Structures / Matériaux et Constructions, Vol. 37, October 4 nd m (7) D where n d = 1, or 3 or uni-, two- or three-dimensional similarity. ote that, to validate this size eect, one must check that the coeicient o variation o measured loads P agrees with Equation (6) i the m value identiied rom size eect tests is used. 1 In view o (5), the value ( VV ) m W or a uniormly stressed specimen can be adopted as a sizeindependent stress measure called the Weibull stress. Taking this viewpoint, Beremin [] proposed taking into account the nonuniorm stress in a large crack-tip plastic zone by the so-called Weibull stress: W 1m m V i I i (8) i V where V i ( i 1 ) are elements o the plastic zone W having maximum principal stress. Ruggieri and Dodds I i [3] replaced the sum in (5) by an integral; see also [4]. Equation (8), however, considers only the crack-tip plastic zone whose size is almost independent o D. Consequently, Equation (8) is applicable only i the crack at the moment o ailure is not yet macroscopic, still being negligible compared to structural dimensions. As ar as quasibrittle structures are concerned, applications o the classical Weibull theory ace a number o serious objections: 1. The act that in (7) the size eect is a power law implies the absence o any characteristic length. But this cannot be true i the material contains sizable inhomogeneities.. The energy release due to stress redistributions caused by macroscopic FPZ or stable crack growth beore the peak load, P max, gives rise to a deterministic size eect which is ignored. Thus the Weibull theory is valid only i the structure ails as soon as a microscopic crack becomes macroscopic. 3. According to the classical Weibull theory, every structure would be mathematically equivalent to a uniaxially stressed bar (or chain, Fig. ), which means that no inormation on the structural geometry and ailure mechanism is taken into account. 4. The size eect dierences between two- and threedimensional similarity ( n d = or 3) are predicted much too large. 5. Many tests o quasibrittle materials (e.g., diagonal shear ailure o reinorced concrete beams) show a much stronger size eect than predicted by Weibull theory; see [4] and the review in [36]. 6. The classical theory neglects the spatial correlations o material ailure probabilities o neighboring elements caused by nonlocal properties o damage evolution (while generalizations based on some phenomenological loadsharing hypotheses have been divorced rom mechanics). 7. When (5) is it to the test data on statistical scatter or specimens o one size (V = const.), and when (7) is it to the mean test data on the eect o size or V (o unnotched plain concrete specimens), the optimum values o Weibull exponent m are very dierent, namely m = 1 and m = 4, respectively [37]. I the theory were applicable, these values would have to coincide. The contribution o Weibull-type material randomness to mean size eect is signiicant only or initially positive geometries, such as three-point or our-point bending or pure tension o a strip, and only i the specimen is suiciently large. Asymptotically, or D, the Weibull size eect, or such geometries, dominates [38], although or concrete only extremely large structures such as dams are large enough to approach this behavior. For small enough sizes, the mean size eect is, even or such geometries, almost purely energetic, associated with stress redistribution due to a inite racture process zone [39]. For bending, the Weibull size eect at such small sizes is strong, while or centrically tensioned strip this is true only or a short specimen ixed at ends. Otherwise, the strips starts to lex sideway beore the ull racture process zone develops, and this engenders at least some energetic size eect, superposed on Weibull size eect. The tail o the probability distribution o the nominal strength is always governed by extreme value statistics, and is o Weibull type. In view o these limitations, among concrete structures pure Weibull theory appears applicable to some extremely thick plain (unreinorced) structures, e.g., the lexure o an arch dam acting as a horizontal beam (but not or vertical bending o arch dams nor gravity dams because large vertical compressive stresses cause long cracks to grow stably beore the maximum load). Most other plain concrete structures are not thick enough to prevent the deterministic size eect rom dominating. Steel or iber reinorcement prevents it as well. 5. QUASIBRITTLE SIZE EFFECT BRIDGIG PLASTICITY AD LEFM, AD ITS HISTORY Quasibrittle materials are materials that (1) are incapable o purely plastic deormations and (), in normal use, have an FPZ which is not negligible compared to structure size D. The concept o quasi-brittleness is not absolute but relative, depending on D. For a large enough D, every quasibrittle structure becomes brittle, i.e., ollows LEFM, except that crack initiation is governed by material strength (which itsel is determined by racture behavior o microscopic laws in the FPZ, as in brittle ceramics or atigue-embrittled steel). For small enough D, every quasibrittle structure is equivalent to an elastic body with a perectly plastic crack (as proven in [4]) and ollows the theory o plasticity, although the size D or which such plastic behavior is attained may represent an abstract theoretical extrapolation in which D is smaller than the inhomogeneity size o the material. All brittle materials (i.e., materials in which the crack growth is governed by LEFM) become quasibrittle on a small enough scale (e.g., a ine-grained ceramic, brittle or D > 1 mm, may be quasibrittle or D 1 µm), and all quasibrittle materials become perectly brittle on a large enough scale (e.g., concrete with normal-size aggregate on the scale o a large gravity dam, sea ice on the scale o 1 m, or jointed rock mass, with joints at 1 m separation, on the scale o a whole mountain, exceeding 1 km). 551

6 RILEM TC QFS While plasticity alone, as well as LEFM alone, possesses no characteristic length, the combination o both, which must be considered or the bridging o plasticity and LEFM, does. Irwin [41] studied the size p o the plastic zone that orms ahead o a crack tip. He derived a rough estimate p KI, where K is the mode-i stress intensity actor I and is the material strength or yield limit. At incipient crack propagation under plane stress, K I is equal to the racture toughness, K IC EG F, where E is Young s modulus and G F is the racture energy. This motivates the deinition o a characteristic length (material length) EG F (9) which approximately characterizes the size o the FPZ in quasibrittle materials. So the key to the deterministic quasibrittle size eect is a combination o the concept o strength or yield with racture mechanics. In dynamics, this urther implies the existence o a characteristic time (material time): (1) v representing the time a wave o velocity v takes to propagate by the distance. Ater LEFM was irst applied to concrete [4], it was ound to disagree with test results [43-46]. Leicester [44] tested geometrically similar notched beams o dierent n sizes, it the results by a power law, D, and observed that the optimum n was less than 1/, the value required by LEFM. The power law with a reduced exponent o course its the test data in the central part o the transitional size range well but does not provide the bridging o the ductile and LEFM size eects. It was tried to explain the reduced exponent value by notches o a inite angle, which however is objectionable or two reasons: (i) notches o a inite angle cannot propagate (rather, a crack must emanate rom the notch tip), (ii) the singular stress ield o inite-angle notches gives a zero lux o energy into the notch tip. Same as Weibull theory, Leicester s power law also implied nonexistence o a characteristic length (see [3], Equations (1)-(3)), which cannot be the case or concrete due to the large size o its inhomogeneities. More extensive tests o notched geometrically similar concrete beams o dierent sizes were carried out by Walsh [45, 46]. Although he did not attempt a mathematical ormulation, he was irst to make the doubly logarithmic plot o nominal strength versus size and observe that it was transitional between plasticity and LEFM. An important advance was made by Hillerborg et al. [47]; see also [48]. Inspired by the sotening and plastic FPZ models o Barenblatt [49, 5] and Dugdale [51], they ormulated the cohesive (or ictitious) crack model characterized by a sotening stress-displacement law or the crack opening and showed by inite element calculations that the ailures o unnotched plain concrete beams in bending exhibit a deterministic size eect, in agreement with tests o the modulus o rupture. Analyzing distributed (smeared) cracking damage, Bažant [5] demonstrated that its localization into a crack band engenders a deterministic size eect on the postpeak delections and energy dissipation o structures. The eect o the crack band is approximately equivalent to that o a long racture with a sizable FPZ at the tip. Subsequently, using an approximate energy release analysis, Bažant [53] derived or the quasibrittle size eect in structures ailing ater large stable crack growth the ollowing approximate size eect law: 1 D B 1 R D or more generally: 1 r r D B 1 R D (11) (1) in which r, B = positive dimensionless constants; D = constant representing the transitional size (at which the power laws o plasticity and LEFM intersect); both D and B depend on the structure geometry (shape). Usually constant R, except when there is a residual crackbridging stress r outside the FPZ (as in iber composites), or when at large sizes some plastic mechanism acting in parallel emerges and becomes dominant (as in the Brazilian split-cylinder test). Equation (11) was shown to be closely ollowed by the numerical results or the crack band model [5, 54], as well as or the nonlocal continuum damage models, which are capable o realistically simulating the localization o strainsotening damage and avoiding spurious mesh sensitivity. Beginning in the mid 198s, the interest in the quasibrittle size eect o concrete structures surged enormously and many researchers made noteworthy contributions; to name but a ew: Petersson [48], Carpinteri [55, 33], and Planas and Elices [56-58]. The size eect has recently become a major theme at conerences on concrete racture [59-64]. Measurements o the size eect on P max were shown to oer a simple way to determine the racture characteristics o quasibrittle materials, including the racture energy, the eective FPZ length, and the (geometry dependent) R-curve. 6. SIZE EFFECT MECHAISM: STRESS REDISTRIBUTIO AD EERGY RELEASE Let us now describe the gist o the deterministic quasibrittle size eect. LEFM applies when the FPZ is negligibly small compared to structural dimension D and can be considered as a point. Thus the LEFM solutions can be obtained by methods o elasticity. The salient characteristic o quasibrittle materials is that there exists a sizable FPZ with distributed cracking or other sotening damage that is not negligibly small compared to structural dimension D. This makes the problem nonlinear, although 55

7 Materials and Structures / Matériaux et Constructions, Vol. 37, October 4 approximately equivalent LEFM solutions can be applied unless FPZ reaches near the structure boundaries. The existence o a large FPZ means that the distance between the tip o the actual (traction-ree) crack and the tip o the equivalent LEFM crack at P max is equal to a certain characteristic length c (roughly one hal o the FPZ size) that is not negligible compared to D. An equivalent LEFM solution may be rigorously deined as the solution or which the load-point stiness is the same as the actual stiness [4]; this occurs when the tip o the LEFM crack is placed approximately in the center o FPZ. The large FPZ size causes a non-negligible macroscopic stress redistribution with energy release rom the structure. With respect to the racture length a (distance rom the mouth o notch or crack to the beginning o the FPZ), two basic cases may now be distinguished: (i) a =, which means that P max occurs at the initiation o macroscopic racture propagation, and (ii) a is inite and not negligible compared to D, which means that P max occurs ater large stable racture growth. 6.1 Scaling or ailure at crack initiation An example o the irst case is the modulus o rupture test, which consists in the bending o a simply supported beam o span L with a rectangular cross section o depth D and width b, subjected to concentrated load P. The maximum load is not decided by the stress 1 3PL bd (3L D) at the tensile ace, but by the stress value roughly at distance c rom the tensile ace (which is roughly at the middle o FPZ). Approximately, 1 1c where 1 D = stress gradient. Setting = tensile strength o the material, t we have (3L D) (1 cd) t, which gives (1 ) Db D, in which (D 3 L) and t Db c (= thickness o the boundary layer o cracking) are constants because the ratio D L is constant or geometrically similar structures. This expression or, however, is unacceptable or D D. But since the b derivation is valid only or small enough c D (i.e., up to the irst-order term o the asymptotic expansion o in terms o 1 D ), one may replace it by the ollowing asymptotically equivalent size eect ormula: 1r 1 rdb D (13) which happens to be acceptable or the entire range o D (including the smallest specimens); r and are positive constants (or the oregoing deinition o, = 1, but not in general). The values r = 1 or have been used or concrete [65], while r 1.47 is optimum according to Bažant and ovák s latest analysis o test data [66]. 6. Scaling or ailures with a long crack or notch Let us now give a simple explanation o the second case o structures ailing only ater stable ormation o large Fig. 3 - Approximate zones o stress relie due to racture. cracks, or notched racture specimens. Failures o this type, exhibiting a strong size eect [4, 67-7], are typical o reinorced concrete structures or iber composites [73, 74], and are also exhibited by some unreinorced structures (e.g., dams, due to the eect o vertical compression, or loating ice plates in the Arctic). Consider the rectangular panel in Fig. 3, which is initially under a uniorm stress equal to. Introduction o a crack o length a with a FPZ o a certain length and width h may be approximately imagined to relieve the stress, and thus release the strain energy, rom the shaded triangles on the lanks o the crack band shown in Fig. 3. The slope k o the eective boundary o the stress relie zone need not be determined; what is important is that k is independent o the size D. For the usual ranges o interest, the length o the crack at maximum load may normally be assumed approximately proportional to the structure size D while the size h o the FPZ is essentially a constant, related to the inhomogeneity size in the material. This has been veriied or many cases by experiments (showing similar ailure modes or small and large specimens) and inite element solutions based on crack band, cohesive or nonlocal models. The stress reduction in the triangular zones o areas ka (Fig. 3) causes (or the case b 1) the energy release Ua ( ka ) E. The stress drop within the crack band o width h causes urther energy release U b ha E. The total energy dissipated by the racture is W = ag F, where G F is the racture energy, a material property representing the energy dissipated per unit area o the racture surace. Energy balance during static ailure requires that ( Ua Ub) a dw da. Setting a D( a D) where ad is approximately a constant i the ailures or dierent structure sizes are geometrically similar, the solution o the last equation or yields Bažant s [53] approximate size eect law in (11) with R = (Fig. 1c). More rigorous derivations o this law, applicable to arbitrary structure geometry, have been given in terms o asymptotic analysis based on equivalent LEFM [75] or on Rice s path-independent J-integral [4] or on the integral equation o the smeared-tip method [4]. This law has also been veriied by nonlocal inite element analysis, and by random particle (or discrete element) models. The experimental veriications, among which the earliest is ound in the amous Walsh s tests o notched concrete beams [45, 46], have by now become abundant (e.g. Fig. 4). For very large sizes ( D D ), the size eect law in 1 (11) reduces to the power law D, which 553

8 RILEM TC QFS Fig. 4 - (a) Comparisons o size eect law with Mode I test data obtained by various investigators using notched specimens o dierent materials; (b) size eect in compression kink-band ailures o geometrically similar notched carbon-peek specimens (ater [76]). represents the size eect o LEFM (or geometrically similar large cracks) and corresponds to the inclined asymptote o slope 1/ in Fig. 1c. For very small sizes ( D D ), this law reduces to = const, which corresponds to the horizontal asymptote and means that there is no size eect, as in plastic limit analysis. The ratio D D is called the brittleness number o a structure. For the structure is perectly brittle (i.e. ollows LEFM), in which case the size eect is the strongest possible, while or the structure is nonbrittle (or ductile, plastic), in which case there is no size eect. Regardsless o geometry, quasibrittle structures are those or which 1 1, in which case the size eect represents a smooth transition (or interpolation) that bridges the power law size eects or the two asymptotic cases. The law (11) has the character o asymptotic matching and serves to provide the bridging o scales. In the quasibrittle range, the stress analysis is o course nonlinear, calling or the cohesive crack model or the crack band model (which are mutually almost equivalent), or some o the nonlocal damage models. The meaning o the term quasibrittle is relative. I the size o a quasibrittle structure becomes suiciently large compared to material inhomogeneities, the structure becomes perectly brittle (or concrete structures, only the global racture o a large dam is describable by LEFM), and i the size becomes suiciently small, the structure becomes non-brittle (plastic, ductile) because the FPZ extends over the whole cross section o the structure (thus a micromachine or a miniature electronic device made o silicone or ine-grained ceramic may be quasibrittle or non-brittle). 6.3 Size eect on postpeak sotening and ductility The plots o nominal stress versus the relative structure delection (normalized so as to make the initial slope in Fig. 5 size independent) have, or small and large structures, the shapes indicated in Fig. 5. Apart rom the size eect on P max, there is also a size eect on the shape o the postpeak descending load-delection curve. For small structures the postpeak curves descend slowly, or larger structures they are steeper, and or suiciently large structures they may exhibit a snapback, that is, a change o slope rom negative to positive. These structural size eects were analyzed or concrete structures by Carpinteri and coworkers [77-79] as well as others; see [4, 4]. I a structure is loaded under displacement control through an elastic device with spring constant C, it loses stability and ails at the s point where the load-delection diagram irst attains the slope C (i ever); Fig. 5. The ratio o the delection at these points to the elastic delection characterizes the ductility o the structure. As apparent rom the igure, small quasibrittle structures have a large ductility while large quasibrittle structures have small ductility. The areas under the load-delection curves in Fig. 5 characterize the energy absorption. The capability o a quasibrittle structure to absorb energy decreases, in relative terms, as the structure size increases. The size eect on energy absorption capability is important or blast loads and impact. The progressive steepening o the postpeak curves in Fig. 5 with increasing size and the development o a snapback can be most simply described by the series coupling model, which assumes that the response o a structure may be at least approximately modeled by the series coupling o the cohesive crack or damage zone with a spring characterizing the elastic unloading o the rest o the structure; see [8] or Section 13. in [81]. One possible exception to the behavior described above is in the racture o iber-reinorced concrete (FRC), where the larger crack opening that occurs in bigger s 554

9 Materials and Structures / Matériaux et Constructions, Vol. 37, October 4 LEFM may be used. In this approach the tip o the equivalent LEFM (sharp) crack is assumed to lie at distance c ahead o the tip o the traction-ree crack or notch, c being a constant (representing roughly one hal o the length o the FPZ ahead o the tip. Two cases are relatively simple: (i) I a large crack grows stably prior to P max or i there is a long notch, EG ( ) c ( ) D Y g( ) c g( ) D (14) and (ii) i P max occurs at racture initiation rom a smooth surace EG c c D Y () ()( ) g c g c D () ()( ) (15) Fig. 5 - Load-delection curves o quasibrittle structures o dierent sizes, scaled to the same initial slope. [75, 65] where the primes denote derivatives; g( ) = KI P D and ( ) = K I Y D are dimensionless energy release unctions o LEFM o a D where a = length o notch or crack up to the beginning o the FPZ; K, K I P I = stress intensity actors or load P and or loading by uniorm residual crack-bridging stress Y, respectively; Y or tensile racture, but Y in the cases o compression racture in concrete, kink band propagation in iber composites, and tensile racture o composites reinorced by ibers short enough to undergo rictional pullout rather than breakage. The asymptotic behavior o (14) or D is o the LEFM type, 1 Y D where Y Y ( ) g( ). Formula (15) approaches or D a inite asymptotic value. So does ormula (14) i Y. ote that parameter G in (14) (15) is related to but dierent rom the racture energy G, as will be explained in the next subsection. F 6.5 Size-eect method or measuring material racture parameters and R-curve Comparison o (14) with (11) yields the relations: D c g( ) g( ) (16) B EG c g( ) (17) Fig. 6 - Load-delection curves o plain concrete and iberreinorced concrete beams o dierent sizes (ater [8]). specimens/structures mobilizes the ibers more eectively. This results in more ductile response ater cracking in larger specimens o similar geometry as seen in Fig. 6, where experimental results rom tests o two sizes o plain concrete (PC) and FRC beams [8] are compared. 6.4 Asymptotic analysis o size eect by equivalent LEFM To obtain simple approximate size eect ormulae that give a complete prediction o the ailure load, including the eect o geometrical shape o the structure, equivalent Thereore, by itting ormula (11) with R to the values o measured on test specimens o dierent sizes with a suiciently broad range o brittleness numbers 1 D Dg( ) D c g( ) (18) the values o G and c can be identiied [87, 88]. The itting can best be done by using the Levenberg-Marquardt 1 Other deinitions o brittleness numbers were used in [55, 83-85]. The advantage o deinition (18), proposed by Bažant [86] on the basis o the size eect law, is that it is independent o structure geometry. 555

10 RILEM TC QFS nonlinear optimization algorithm, but it can also be accomplished by a (properly weighted) linear regression o versus D. The specimens do not have to be geometrically similar, although when they are the evaluation is simpler and the error smaller. The lower the scatter o test results, the narrower is the minimum necessary range o (or concrete and iber composites, the size range 1 : 4 is the minimum). The size eect method o measuring racture characteristics has been adopted or an international standard recommendation or concrete ([89], or Section 6.3 in [4]), and has also been veriied and used or various rocks, ceramics, orthotropic iber-polymer composites, sea ice, wood, tough metals and other quasibrittle materials. The advantage o the size eect method is that the tests, requiring only the maximum loads, are oolproo and easy to carry out. With regard to the cohesive crack model, note that the size eect method gives the energy value corresponding to the area under the initial tangent o the sotening stressdisplacement curve, rather than the total area under the curve. The stress-displacement curves used by cohesive crack models or concrete typically start by a relatively steep descending part ollowed by a long tail. The area under the entire stress-displacement curve corresponds to the racture energy G F that would be consumed per unit area o the crack advance in an ininitely large specimen. Laboratory specimens used by the size-eect method are not large enough to activate the long tail o the curve already beore peak, and the peak load is usually attained with only a partially developed process zone. Consequenly, the shape o the tail has no inluence on the peak load and the corresponding part o the racture energy cannot be captured by the size-eect method [91]. This is why, or the usual range o sizes tested in the laboratory, the racture energy G identiied rom the size eect law is smaller than the racture energy or an ininite specimen, G F, which can be approximately determined by the work-oracture method [9]. Parameter G can be roughly understood as the area under the initial tangent o the sotening stress-displacement curve, and the typical ratio G GF is about 1 :.5 [91]. The size eect method also permits determining the R- curve (resistance curve) o the quasibrittle material a curve that represents the apparent variation o racture energy with crack extension or which LEFM becomes approximately equivalent to the actual material with a large FPZ. The R-curve, which (in contrast to the classical R- curve deinition) depends on the specimen geometry, can be obtained as the envelope o the curves o the energy release rate at P = P max (or each size) versus the crack extension or specimens o various sizes. In general, this can easily be done numerically, and i the size eect law has the orm in (11) with =, a parametric analytical expression or the R R-curve exists; see [88] or Section 6.4 in [4]. The racture model implied by the size eect law in (11) or (14) has one independent characteristic length, c, representing about one hal o the FPZ length. From c, using the relation 1 B g( ) c, one can determine the characteristic length 1 EG t where G represents the area under the initial tangent o the sotening stressseparation curve o the cohesive crack model and must be distinguished rom racture energy G F, which represents the area under the entire sotening curve and is measured by the work-o-racture method (the ratio G F G exhibits very high scatter and on the average is about.5, which means that 1 is on the average about.5); see [91]. The value o c controls the size D at the center o the bridging region (intersection o the power-law asymptotes in Fig. 1c), and or G controls a vertical shit o the size eect curve at constant D. Aside rom geometry actors expressed in terms o unction g( ), the locations o the large-size and small-size asymptotes depend only on Kc EG and EG c, respectively. A very eective method or measuring G has been the notched-unnotched method, conceived by Guinea, Planas and Elices [9] without any reerence to size eect. Bažant, Yu and Zi [91] recently improved this method by exploiting the exact dimensionless size eect curve o the cohesive crack model which is calculated in advance or given specimen geometry. This is possible because only the initial downward slope o the sotening curve matters or the maximum load. The reason is that, in normal-size notched specimens, the crack stress proile at maximum load terminates at notch tip with a inite stress so large that the tail portion o the sotening curve is not reached. The improved method, as well as Guinea et al. s, makes it possible to determine G (or the initial slope o the sotening curve) simply by measuring solely the maximum loads o notched specimens o only one size (and one geometry), supplemented by direct measurement o t (or which the Brazilian split-cylinder test has been recommended). I the cohesive crack model is assumed to hold or the entire size range D ( ), then the strength data correspond to the zero-size limit o the size eect plot (i.e., to zero brittleness number [93]) because lim or D depends only on the tensile strength (being independent o the sotening curve). The Bažant-Yu-Zi method uses the regression equation Y AX C ( X) (19) with X D 1 Y ( t ) () Function ( X ), which must be accurately computed in advance, gives the deviations o the exact size eect curve o the cohesive crack model rom the size eect law (11) (with R ); ( X ) vanishes or D and its asymptotic expansion begins with the term 1 D. For normal-size notched three-point bend specimens, the correction ( X ) is insigniicant (error o a ew percent only), but or zero size the correction by () is important [91]. Knowing unction ( X ), including the limit (), Equation (19) can be itted to the measured values or notched specimens o one size and the values corresponding to at zero-size limit. The itting yields t 556

11 Materials and Structures / Matériaux et Constructions, Vol. 37, October 4 the values o A and C, rom which G, c and 1 ollow [91]. The improved size eect method o Bažant, Yu and Zi is equivalent to Guinea et al. s method except or the statistical evaluation. The act that the ormer permits identiication o material parameters by simultaneous statistical regression o both the notched specimen data and the strength data is an advantage. 6.6 Critical crack-tip opening displacement, CTOD The quasibrittle size eect, bridging plasticity and LEFM, can also be simulated by the racture models characterized by the critical stress intensity actor K c (racture toughness) and CTOD ; or metals see [94, 95] and or concrete [96]. Jenq and Shah s model [96], called the two-parameter racture model, has been shown to give similar results as the R-curve derived rom the size eect law in (11) with R =. The approximate relationship o size eect law and Jenq-Shah model is given by K c EG (1) 1 8Gc CTOD E () However, Jenq-Shah model suers rom a dependence o its results on the slope o the unloading curve o the cohesive crack model. This dependence can change the measured G within a range o about 15% and is, in principle, inadmissible because the racture energy is deined by the sotening curve independently o the unloading properties o the cohesive crack model [91]. Using (1) and (), the values o K c and CTOD can be easily identiied by itting the size eect law (11) to measured values o the peak load P max. Like the size eect law in (11) with R =, the twoparameter model has only one independent characteristic length, CTOD, which is related to 1 or c. 6.7 Material heterogeneity and representative volume element The smallest specimens in size eect tests have oten been only about ive aggregate sizes in cross section dimension. One may wonder whether this is enough in view o the concept o the representative volume element (RVE). The answer depends on statistical considerations. The RVE is deined as the smallest element which, when translated through the heterogeneous material, does not change its statistical properties. But what statistical properties? And with what accuracy? There is much conusion in the literature stemming rom ignorance o the statistical aspect. I one considers the moments o the probability distributions o the strength and stiness parameters up to ininite order, and demands complete accuracy, an RVE would have to be ininitely large (and continuum mechanics inapplicable at any size) unless that material has an artiicial perectly periodic structure. For the irst two moments o the distribution (i.e., including the invariance o the standard deviation during RVE shits), the RVE must be much larger than or the irst moment only, i.e., or the mean response (as long as the shiting o the RVE through a single specimen is considered). The size o a useul RVE also depends on whether or not the tests or microstructural simulations o randomly heterogeneous material are averaged over several realizations, and on how many realizations are averaged (i many are averaged, the RVE can be much smaller than i none are). What is the minimum cross section dimension D that min is meaningul or a concrete racture specimen? Based on critical analysis o the new experimental data, in particular o eects due to environmental constraints (see Fig. 8 and the discussion in Section 7.6) and large scale heterogeneity at the level o the considered specimen that usually go unnoticed, van Mier and van Vliet [97-99] opined that the minimum RVE size is Dmin 8d ( d = maximum a a aggregate size). However, according to the size eect tests in [87], beam depth Dmin 3d suices or approximating a the mean behavior, provided that only the averages over suiciently many tests are considered. This is conirmed by the act that, in those tests, the average or beams 3d a in depth lied, or each o three dierent geometries, very close to the size eect curve obtained by regression o test data or sizes D da = 6,1,4, and the G value obtained by the size eect method was nearly the same whether or not the smallest specimens were included or excluded. This indicates that, i only the average behavior is needed, the specimens can be as small as can be cast. There is, however, another limitation on D min or the size eect method described in Section 5: Approximation (11) should not deviate rom the exact size eect curve o the cohesive crack model by more than deemed allowable; i 3% is allowable, then D 4d a or the three-point bent notched beams; or details see [91]. 7. EXTESIOS, RAMIFICATIOS AD APPLICATIOS 7.1 Size eects in compression racture Loading by high compressive stress without suicient lateral conining stresses leads to damage in the orm o axial splitting microcracks engendered by pores, inclusions or inclined slip planes. This damage localizes into a band that propagates either axially or laterally. For axial propagation, the energy release rom the band drives the ormation o the axial splitting racture, and since this energy release is proportional to the length o the band, there is no size eect. For lateral propagation, the stress in the zones on the sides o the damage band gets reduced, which causes an energy release that grows in proportion to D, while the energy consumed and dissipated in the band grows in proportion to D. The mismatch o energy release rates inevitably engenders a deterministic size eect o the quasibrittle type, analogous to the size eect associated with tensile racture. In consequence o the size eect, 557