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1 Politecnico di Torino Porto Institutional Repository [Article] Coesive Crack Model Description of Ductile to Brittle Size-scale Transition: Dimensional Analysis vs. Renormalization Group Teory Original Citation: Carpinteri A.; Cornetti P.; Barpi F.; Valente S. (23). Coesive Crack Model Description of Ductile to Brittle Size-scale Transition: Dimensional Analysis vs. Renormalization Group Teory. In: ENGINEERING FRACTURE MECHANICS, vol. 7, pp ISSN Availability: Tis version is available at : ttp://porto.polito.it/ / since: October 26 Publiser: Elsevier Terms of use: Tis article is made available under terms and conditions applicable to Open Access Policy Article ("Public - All rigts reserved"), as described at ttp://porto.polito.it/terms_and_conditions. tml Porto, te institutional repository of te Politecnico di Torino, is provided by te University Library and te IT-Services. Te aim is to enable open access to all te world. Please sare wit us ow tis access benefits you. Your story matters. Publiser copyrigt claim: NOTICE: tis is te autor s version of a work tat was accepted for publication in "ENGINEERING FRACTURE MECHANICS". Canges resulting from te publising process, suc as peer review, editing, corrections, structural formatting, and oter quality control mecanisms may not be reflected in tis document. Canges may ave been made to tis work since it was submitted for publication. A definitive version was subsequently publised in ENGINEERING FRACTURE MECHANICS, [ vol:7, issue:unspecified, date:23] DOI: (Article begins on next page)

2 Post print (i.e. final draft post-refereeing) version of an article publised on Engineering Fracture Mecanics. Beyond te journal formatting, please note tat tere could be minor canges from tis document to te final publised version. Te final publised version is accessible from ere: ttp://dx.doi.org/1.116/s (3)126-7 Tis document as made accessible troug PORTO, te Open Access Repository of Politecnico di Torino (ttp://porto.polito.it), in compliance wit te Publiser s copyrigt policy as reported in te SHERPA-ROMEO website: ttp:// Coesive Crack Model Description of Ductile to Brittle Size-scale Transition: Dimensional Analysis vs. Renormalization Group Teory A. Carpinteri 1, P. Cornetti 2, F. Barpi 3 and S. Valente 4 1 Dipartimento di Ingegneria Strutturale, Edile e Geotecnica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 1129 Torino, Italy. alberto.carpinteri@polito.it 2 Dipartimento di Ingegneria Strutturale, Edile e Geotecnica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 1129 Torino, Italy. pietro.cornetti@polito.it 3 Dipartimento di Ingegneria Strutturale, Edile e Geotecnica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 1129 Torino, Italy. fabrizio.barpi@polito.it 4 Dipartimento di Ingegneria Strutturale, Edile e Geotecnica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 1129 Torino, Italy. silvio.valente@polito.it Keywords Coesive crack; Concrete; Fractal geometry; Size-effect Abstract Abstract Te present paper is a review of te researc works carried out on te coesive crack model and its applications at te Politecnico di Torino during te last decade. Te topic encompasses experimental, numerical and teoretical aspects of te coesive crack model. Te researc work followed two main directions. Te early work concerns te development and te implementation of te coesive crack model, wic as been sown to be able to simulate experiments on concrete specimens and structures. It is referred to as te dimensional analysis approac, since it succeeds in capturing te ductile-tobrittle transition by increasing te structural size owing to te different pysical dimensions of two material parameters: te tensile strengt and te fracture energy. On te oter and, te later researc direction aims at extending te classical coesive model to quasibrittle materials sowing (as tey often do) fractal patterns in te failure process. Tis approac is referred to as te renormalization group (or fractal) approac and leads to a scale-invariant coesive crack model. Tis model is able to predict te size effects even in tests were te classical approac fails, e.g. te direct tension test. Te two researc pats, terefore, complete eac oter, allowing a deeper insigt into te ductile-tobrittle transition usually detected wen testing quasi-brittle material specimens or structures at different size-scales.

3 1 Introduction Concrete in tension exibits strain softening, i.e., a negative slope in te stress-deformation diagram, due to microcracking and localization of te deformation in a narrow band, were energy dissipation occurs. Te beaviour of te material outside tis band is still linear and elastic. Tis penomenon, observed experimentally by L Hermite [1], Rusc and Hilsdorf [2], Huges and Capman [3], Evans and Marate [4], among oters, must be taken into account in order to provide a good explanation of te beaviour of te material. From te Continuum Mecanics viewpoint, strain softening represents a violation of Drucker s Postulate [5], as was pointed out by Maier [6, 7] and Maier et al. [8]. Tese autors sowed tat, even in te absence of geometrical instability effects, te following penomena may occur: loss of stability in te controlled load condition (snap-troug); loss of stability in te controlled displacement condition (snap-back); bifurcation of te equilibrium pat; loss of uniqueness of te solution in te incremental elasto-plastic response; dependence of te results on te type of mes used in te numerical analysis. A continuum described by strain-softening is also caracterized by an imaginary wave speed or by te cange of te equation of motion from yperbolic to elliptic, as pointed out by Hadamard [9]. Tis confirms te difficulties involved in tis constitutive relationsip, as compared to te classical strain-ardening one. Te Finite Element Metod (F.E.M.) was first applied to te problem of concrete cracking by Rasid [1], wo adopted te so-called Smeared Crack Model. In tis approac to te cracking process, te stress in te element was limited by te tensile strengt of te material, beyond wic point te stress registered a vertical drop to zero. Scanlon [11], among oters, used a constitutive model in wic te stress is reduced gradually to zero in a sequence of small stress drops, i.e., by introducing a strain-softening constitutive law. Softening was added in Finite Element codes and applied to a large number of problems, but it was discovered tat te convergence properties were incorrect and te results strongly dependent on te mes size. Te problem was tat te energy dissipated by cracking decreases wit te refinement of te mes and converges to zero. Tis trend, referred to as spurious mes sensitivity, is pysically unacceptable. By specifying te energy dissipated over cracking surfaces, it is possible to eliminate te mes sensitivity. A relationsip describing te softening damage must be introduced in te model; tis can be done basically in two ways: by introducing a softening stress-crack opening displacement constitutive law, or by using a softening stress-strain relation for te material included in a band around te crack (te bandwidt is an additional parameter to be determined). Te first approac represents te basis of te Coesive Crack Model; it involves te separation of te nodes of two adjacent elements belonging to te crack. Te second refers to two models, not described ere, referred to as te Crack Band Model and te Non-Local Model (see, for instance, [12] and [13]). Tese models are attractive because tey involve only te modification of te classical constitutive relationsip witout requiring remesing procedures. Te Coesive Crack Model is able to describe materials tat exibit a strain-softening type beaviour. Te area under te closing stress versus crack opening displacement curve represents te fracture energy G F assumed as a material property. Tis approac eliminates te mes sensitivity. Mode I problems are caracterized by te a priori knowledge of te crack trajectory, wilst in Mixed-Mode problems tis is an additional unknown. In eiter case, te Finite Element Metod represents an effective way to address crack propagation problems and penomena suc as size effect and ductileto-brittle transition; it will be used later in tis paper.

4 Real crack tip, R.C.T. (s = r ) Process zone Fictitious crack tip, F.C.T. (s = f ) s u u Figure 1: Process zone (witout searing stresses). Te Coesive Crack Model was initially proposed by Barenblatt [14, 15] and Dugdale [16]. Subsequently, Dugdale s model was reconsidered by Bilby et al. [17], Willis [18], Rice [19], and utilized by Wnuk, wo referred to it as te Final Stretc criterion [2]. Hillerborg et al. [21] proposed te Fictitious Crack Model in order to study crack propagation in concrete. Te crack is assumed to propagate wen te stress at te crack tip reaces te tensile strengt σ u. Wen te crack opens, te stress is not assumed to fall to zero at once, but to decrease wit increasing crack widt w. Te amount of energy absorbed per unit crack area (denoted ere by G F ) is: G F = wc σdw (1) and represents te area under te curve σ-w (were w c is te critical displacement, i.e., te distance between te crack surfaces, at wic te interaction vanises). Concrete is assumed to be linear elastic until σ u is reaced. Tere are some similarities among Barenblatt s, Dugdale s, Rice s and Hillerborg s formulations: te crack tip faces close smootly (te stress intensity factor K I vanises at te crack tip in Mode I propagation) and te fracture process zone is of negligible tickness. On te oter and, te closing stresses in te fracture process zone are constant only in Dugdale s model, wile te size of tis zone is constant and small in comparison wit te lengt of te main crack only in Barenblatt s model. Again, wit some modifications, te model was furter applied by Wecaratana and Sa [22], Bažant and O [23] and Ingraffea and Gerstle [24]. More recently, te former terminology of Coesive Crack Model as been reproposed by Carpinteri [25 28], Carpinteri et al. [29 31] and te model as been used wit tis name by a number of researcers (for instance, Carpinteri and Valente [32], Cen and Maier [33], Elices et al. [34], among oters). Later on, in order to explain te size effects upon te parameters of te coesive crack model, Carpinteri [35, 36] applied fractal geometry concepts and described te influence of te microstructural disorder typical of most of quasi-brittle materials. Te fractal approac was futer developed by Carpinteri et al. [37, 38]. Recently, an improvement of te coesive crack model, te so-called (scale-invariant) fractal coesive crack model [39], as been proposed and applied to interpret te most extensive experimental tensile data from concrete specimens tested over a broad range of scales [4, 41]. Finally, te reader is referred to te following books [42 49] autored or edited by te first autor of te present paper. 2 Basic Concepts of te Coesive Crack Model Te basic assumption is te formation, as an extension of te real crack, of a fictitious crack, referred to as te process zone, were te material, albeit damaged, is still able to transfer stresses (Fig. 1).

5 (a) u Stress, 1 E u (b) u Stress, c G F G F 1 2 u w c w c Strain, Opening, w Figure 2: Consitutive laws: (a) undamaged material, (b) process zone. Te point separating te stress-free area, i.e., te real crack, from te process zone, is called Real Crack Tip (RCT), wilst te point separating te process zone from te uncracked material is referred to as Fictitious Crack Tip (FCT). Te process zone represents te area in wic energy dissipation takes place: it begins to form wen te principal tensile stress reaces te material ultimate tensile strengt, σ u, in te direction perpendicular to te direction of te principal tensile stress. Furtermore, in te process zone, te stresses transferred by te material are decreasing functions of te displacement discontinuity, according to a proper coesive law (linear in Fig. 2b), wilst in te uncracked zone te beaviour of te material is linear-elastic, as sown in Fig. 2a. At te end of te fictitious crack, te stress will always be equal to te value of ultimate tensile strengt. Tus, no singularities arise in te state of stress. In te model described so far, searing stresses in te process zone are disregarded. Te area under te σ-w segment represents te fracture energy, G F, wic, liketeultimatetensilestrengt, isusuallyconsideredasapropertyoftematerial. Tis assumption will be removed in Section Elementary Beam Model Te linear elastic beaviour of an initially uncracked beam in Tree-Point Bending may be represented by te following dimensionless equation [27, 5, 51]: P = 4 λ 3 δ (2) were te dimensionless load and central deflection are given respectively by: P = P l σ u t 2 (3a) δ = δl ε u 2 wit l=beam span, =beam dept, t=beam tickness and λ = l/=beam slenderness. Once te ultimate tensile strengt is reaced at te lower edge of te beam, a fracturing process in te central cross section is assumed to set in. Suc a process admits of a limit-situation like tat in Fig. 3. Te limit stage of te fracturing and deformation process may be considered as tat of two rigid parts connected by te inge A at te upper edge of te beam. Te equilibrium of eac part is ensured by te external load, te support reaction and te closing coesive forces (Fig. 3). Te latter depend on te distance between te two interacting surfaces: as te distance w increases, te coesive forces decrease until tey vanis for w > w c. Te geometrical similarity of te triangles ABC and AB C in Fig. 3 yields: δ l/2 = w c/2 x were x is te extension of te triangular distribution of te coesive forces. Equation (4) can be rearranged as follows: (3b) (4) x = w cl 4δ (5)

6 P B A u C x w c P/2 P/2 B' C' Figure 3: Elementary beam model wit coesive forces. Te rotational equilibrium around point A is possible for eac beam part only if te moments of support reaction and coesive forces are equal: P 2 l 2 = σ uxt x 2 3 Recalling Eq. (5), te relation between load and deflection may be obtained: (6) P = σ utwc 2 24 Equation (7) can be cast in dimensionless form: P = 1 6 ( GF σ u λ 2 ε u δ l δ 2 (7) ) 2 = 1 λ (s 2 ) 2 E (8) 6 ε u δ were s E = G F /(σ u ) represents te energy brittleness number [25]. Wile te linear Eq. (2) describes te linear elastic beaviour of te initially uncracked beam, te yperbolic Eq. (8) represents te asymptotic beaviour of te same beam wen it is totally cracked. Equation (2) is valid only for load values lower tan tat producing te ultimate tensile strengt σ u at te lower beam edge: P 2 3 (9) On te oter and, Eq. (8) is valid only for deflection values iger tan tat producing a coesive zone of extension x equal to te beam dept : From Eqs. (5) and (1) it follows tat: x (1) δ δ 2 = s E λ 2 2ε u (11) Te bounds (9) and (11), upper for load and lower for deflection respectively, can be transformed into two equivalent bounds, bot upper for deflection and load. Equations (3a) and (9) yield: wereas Eqs. (8) and (11) yield: δ δ 1 = λ3 6 P 2 3 (12) (13)

7 P 2/3 1 < P 2 1 > 2 2/3 (a) (b) Figure4: Loadvs.deflectiondiagrams: (a)ductile,(b)brittlecomdition(δ 1 = λ 3 /6, δ 2 = s E λ 2 /(2ε u )). Conditions (9) and (13) are identical. Terefore, a stability criterion for elastic-softening beams may be obtained by comparing Eqs. (11) and (12). Wen te two domains are separated, te two loaddeflection brances (linear and yperbolic) may be assumed to be connected by a regular curve, as in Fig. 4a. Onteoterand, wentetwodomainsarepartiallyoverlapped, itisreasonabletoassumetattey are connected by a curve wit igly negative or even positive slope (Fig. 4b). Unstable beaviour and catastropic events (snap-back) may be possible for: s E λ 2 λ3 2ε u 6 and te brittleness condition for te Tree-Point Bending geometry becomes: (14) s E ε u λ 1 3 Terefore, te system is brittle for low brittleness numbers s E, ig ultimate tensile strain ε u, and ig slenderness λ. It is terefore evident tat te relative brittleness for a structure depends not only on material properties but also on te structural size and slenderness. Te global brittleness of te beam can be defined as te ratio of te ultimate elastic energy contained in te body to te energy dissipated by fracture (see also [52]): Brittleness = Suc a ratio is iger tan unity wen: (15) 1 2 P 1 uδ u G F Area = 18 σ uε u tl = 1 ( ) 1 se (16) G F t 18 ε u λ s E ε u λ 1 18 Equation (17) represents a stricter condition for global structural brittleness compared wit Eq. (15). 2.2 Beam Model Interpretation Based on Ultimate Strengt Criterion and Linear Elastic Fracture Mecanics Owing to te different pysical dimensions of ultimate tensile strengt σ u and fracture tougness K IC, under te usual conditions of engineering materials and fracture testing, scale effects are practically always present [28, 5, 53 55]. Te key point is tat te collapse can be governed by ultimate strengt or by crack propagation: suc a competition between types of collapse of a different nature becomes readily evident if we consider te ASTM formula for te Tree-Point Bending Test(or TPBT) evaluation of fracture tougness [56]: wit: f ( a ) = 2.9 ( a ) 1/2 4.6 ( a K I = P l ( t 3/2 f a ) ) 3/ ( a ) 5/ ( a (17) (18) ) 7/2 ( a ) 9/ (19)

8 1..8 P u t 2 P u u a s = a Figure 5: Dimensionless load of crack instability vs. relative crack dept a /. For incipient crack propagation, K I = K IC ; Eq. (18) becomes: K IC σ u 1/2 = s = P LEFM l σ u t 2 f ( a ) were P LEFM is te external load of brittle fracture and s = K IC /(σ u 1/2 ) is te static brittleness number [53]. Rearranging of Eq. (2) yields: P LEFM l σ u t 2 = K IC 1 σ u 1/2 f ( a ) = s f ( a ) (21) On te oter and, it is possible to consider te non-dimensional load of ultimate strengt in a beam of dept ( a ): P US l σ u t 2 = 2 3 ( 1 a (2) ) 2 (22) Equations (21) and (22) are plotted in Fig. 5 as functions of te relative crack dept a /. Wereas te former produces a set of curves as te brittleness number s is varied, te latter is represented by a single curve. It is evident tat te ultimate strengt collapse at te ligament precedes crack propagation for eac initial crack dept wen te brittleness number s is iger tan te limit value s =.5. For lower brittleness numbers, ultimate strengt collapse precedes crack propagation only for crack depts outside a certain range. Tis means tat a true LEFM collapse occurs only for comparatively low fracture tougnesses, ig tensile strengts and/or large structural sizes. It is not te individual values of K IC, σ u and tat determine te nature of te collapse, but only teir functions brittleness number - see Eq. (2). It is also possible to study te beaviour of te beam in terms of load-deflection curves. Te deflection due to te elastic compliance of te uncracked beam is: δ e = P l3 (23) 48EI were I is te inertial moment of te cross-section. On te oter and, te deflection due to te local crack compliance is [57]: δ c = 3 2 P l 2 ( t 2 E g a ) (24)

9 .8 P.6.5 u t 2 Virtual crack propagation (K I = K IC ) Virtual progressive failure ( = u) s =.4 a / =. a / =.1 a / =.2 a / = s =.3 a / =.4 s =.6 s =.2 a / =.5 a / =.6 s =.5 a / =.7 a / =.8 u Figure 6: Dimensionless load of crack instability vs. dimensionless deflection. wit: g ( a ) ( a ) [ = 1 a Te superposition principle yields: ( a ) ( a ) ( a ) 3 ( a ) ] (25) and, in non-dimensional form: δl ε u 2 = P l σ u t 2 δ = δ e +δ c (26) [ 1 4 ( ) l ( ) l 2 g ( a ) ] (27) were ε u = σ u /E. Te term in square brackets is te dimensionless compliance, wic is a function of beam slenderness l/, as well as of crack dept, a /. Some linear load-deflection diagrams are represented in Fig. 6, for varying crack dept a / and for te fixed ratio l/ = 4. By means of Eqs. (21) and (22) it is possible to determine te point of crack propagation as well as te point of ultimate strengt on eac linear plot of Fig. 6. Wereas te former depends on te brittleness number s, te latter is unique. Te set of crack propagation points wit constant s and varying crack dept represents a virtual load-deflection pat, were point by point te load is always tat producing crack instability. Wen te crack grows, te instability load decreases and te compliance increases, so tat te product on te rigt-and side of Eq. (27) may be eiter decreasing or increasing. Te diagram of Fig. 6 sows te deflection decreasing (wit te load) up to te crack dept a /.3 and ten increasing (against te load). Terefore, wereas for a / >.3 te load-deflection curve presents te usual softening trend wit negative derivative, for a / <.3 it presents a positive derivative. Suc a branc could not be revealed by deflection-controlled testing, and te representative point would jump from te positive to te negative branc wit a beaviour discontinuity. Te set of ultimate tensile strengt points, wit varying crack dept, is represented by te tick line in Fig. 6. Tis line intersects te virtual crack propagation curves wit s s =.5, wic is analogous to wat is sown in Fig. 5, and presents a sligt indentation wit dp/dδ >.

10 .48 P R u 2.36 N O M.24 L H I.12 G F A B E A: s E = B: s E = C: s E = D: s E = E: s E = F: s E = G: s E = H: s E = I: s E = L: s E = M: s E = N: s E = O: s E = P: s E = Q: s E = R: s E = Figure 7: Load vs. deflection curves, a / =.. 3 Mode I Problems Numerical Simulations Te analyses discussed in tis section concern te beaviour of concrete elements in Mode I conditions (TPBT). For reasons of symmetry, te crack trajectory is known a priori. Te numerical results, based on te coesive model, were obtained using te Finite Element Code FR.ANA. (FRacture ANAlysis) [25 31]. An extensive series of analyses was carried out from 1984 to 1989 by A. Carpinteri and co-workers, and some of te results obtained are mentioned later [25 31]. Te experimental results can be found in [58 6]. Te cases described in te reference papers regard tree l/ ratios (4, 8 and 16), and four a / ratios (.,.1,.3,.5), see, for instance, Figs For eac ratio, te response was analyzed for different values of te brittleness number, s E, ranging from to Te concrete properties are listed in Table 1. As can be seen from te diagrams, te brittleness number s E as a decisive effect on te structural response of te element: by increasing s E, te beaviour of te element canges from brittle to ductile, as anticipated at te end of te previous section. Hence, te structural response is not described by te parameters σ u, G F and independently, but rater by a combination of tese parameters, as expressed by te brittleness number s E. A comparison between te results obtained wit te Finite Element Metod and tose provided by te Boundary Element Metod can be found in [33] and [61]. Te agreement between te results is excellent: bot implementations of te coesive model are able to describe snap-back penomena. 3.1 Strain Localization and Apparent Strengt of Initially Uncracked Beams Some dimensionless load-deflection diagrams for a concrete-like material are plotted in Figs Te specimen beaviour is brittle (snap-back) for low s E numbers, i.e., for low fracture tougnesses G F, ig tensile strengts σ u, and/or large sizes. In Fig. 7, for s E , te P δ curve presents a positive slope in te softening branc, and a catastropic event occurs if te loading process is deflection-controlled. Suc an indenting branc is not virtual only if te te loading process Table 1: Material properties (tree point bending test). E ν σ u ε u (MPa) (-) (MPa) (-)

11 P u 2 B I L A: s E = B: s E = C: s E = D: s E = E: s E = F: s E = G: s E = H: s E = I: s E = L: s E = F Figure 8: Load vs. deflection curves, a / = P u 2 F R M H I G E D C B A A: s E = B: s E = C: s E = D: s E = E: s E = F: s E = G: s E = H: s E = I: s E = L: s E = M: s E = N: s E = O: s E = P: s E = Q: s E = R: s E = Figure 9: Load vs. deflection curves, a / =.5. is controlled by a monotonically increasing function of time, suc as te displacement discontinuity across te crack [5, 51]. On te oter and, Eq. (15) yields s E Suc a condition reproduces te one sown in Fig. 7 very accurately, wereas Eq. (17) appears too severe. Wen te post-peak beaviour is kept under control up to complete structural separation, te area delimited by te load-deflection curve and deflection axis represents te product of fracture energy, G F, and initial cross-section area, t. Te maximum loading capacity P Coes of te initially uncracked specimen wit l = 4 is obtained from Fig. 7. On te oter and, te maximum load P US of ultimate strengt is given by [5, 51]: P US = 2 σ u t 2 (28) 3 l Te values of te ratio P Coes /P US may also be regarded as te ratio of te apparent tensile strengt σ f (given by te maximum load P Coes and applying Eq. (28)) to te true tensile strengt (considered as a material constant). It is evident from Fig. 11 tat te results of te Coesive Crack Model tend to tose of ultimate strengt analysis for low s E values: lim P Coes = P US (29) s E Terefore, only for comparatively large specimen sizes can te tensile strengt σ u be obtained as

12 .36 P u A C L E F G H I 1 4 A: s E = B: s E = C: s E = D: s E = E: s E = F: s E = G: s E = H: s E = I: s E = L: s E = Figure 1: Load vs. deflection curves, a / =.5. 3 P Coes P U.S., f 2 1 u a. 1/s E Dimensionless size, u/g F (1 3 ) Figure 11: Decrease in apparent strengt by increasing te specimen size (λ = 4, ε u = ). σ u =σ f. Wit te usual laboratory specimens, an apparent strengt iger tan te true one is always found. As a limit case, for te size or fracture energy G F (elastic-perfectly plastic material intension), i.e., fors E, teapparenttensilestrengtσ f 3σ u. Infact, intecentreoftebeam, te uniform stress distribution (Fig. 12) produces a plastic inge wit a resistant moment M max wic is twice te classical moment of te bi-rectangular limit stress distribution (elastic-perfectly plastic material in tension and compression). Te fictitious crack dept at te maximum load is plotted as a function of 1/s E in Fig. 13. Te brittleness increase for s E is evident also from tis diagram, te process zone at dp/dδ = tending to disappear (brittle collapse), wereas it tends to cover te wole ligament for s E (ductile collapse). On te oter and, te real (or stress-free) crack dept at te maximum load is always zero for eac value of s E. Tis means tat te slow crack growt does not start before te softening stage. Consequently, neiter is slow crack growt found to occur nor does te coesive zone develop before te peak, wen s E. Recalling once again Figs. 11 and 13, it is possible to M max = u t (/2) M max M max u M P = M max /2 /2 u t M u = M max /3 Figure 12: Constant distribution of coesive stresses.

13 1. Fictitious crack dept at maximum load a. 1/s E Dimensionless size, u/g F (1 3 ) Figure 13: Fictitious crack dept at maximum load as a function of specimen size (λ = 4, ε u = ). 1 % P Coes, P L.E.F.M. 75 % 5 % P U.S. P L.E.F.M. Linear Elastic Fracture Mecanics Coesive model 25 % Limit analysis a.5 1/s E Dimensionless size, u/g F (1 3 ) Figure 14: Increase of fictitious tougness wit increasing specimen size (λ = 4, ε u = ). state tat, te smaller te brittleness number s E, te more accurate te snap-back in reproducing te perfectly-brittle ultimate strengt instability (a / = ). 3.2 Fracture Tougness of Initially Cracked Slabs Te mecanical beaviour of Tree-Point Bending slabs wit initial cracks is investigated on te basis of te coesive numerical model presented previously. Te initial crack renders te specimen beaviour more ductile tan in te case of te initially uncracked specimen [5, 51] (see Figs. 9-1). Te area delimited by te load-deflection curve and te deflection axis represents te product of fracture energy and initial ligament area, ( a ) t. Te areas under te nondimensional P δ curves are tus proportional to te respective s E numbers (Figs. 9-1) 1. Tis result is based upon te assumption tat te energy dissipation occurs only on te fracture surface, wereas in reality energy is also dissipated in a damage volume around te crack tip as assumed in [62] and sown in [63]. Te maximum loading capacity P Coes according to te Coesive Crack Model, is obtained from Figs 9-1. On te oter and, te maximum load P LEFM of brittle fracture can be obtained from Linear Elastic Fracture Mecanics (Eq. (2)), wit K IC = (G F E) 1/2 (plane stress condition). Te values of te ratio P Coes /P LEFM are given as functions of te inverse of s E in Fig. 14. Tis ratio may also be regarded as te ratio of te fictitious fracture tougness (given by te maximum load P Coes ) to te true fracture tougness (considered as a material constant). It is evident tat for low s E numbers te results of te Coesive Crack Model tend to tose of Linear Elastic Fracture Mecanics [5, 54]: lim P Coes = P LEFM (3) s E ( 1 t In te nondimensional diagrams of Figs. 9-1 te area delimited is equal to s E 1 a ).

14 2.2 F 2, 2 F 1, 1 a = t Figure 15: Four-Point Sear Test wit one notc. and terefore, te maximum loading capacity can be predicted if te condition K I = K IC is applied. It appears tat te true fracture tougness K IC of te material can be obtained only wit very large specimens. In fact, wit laboratory specimens, a fictitious fracture tougness lower tan te true one is always measured. 4 Mixed-Mode Problems Numerical Simulations Te analyses discussed in tis section concern te beaviour of concrete elements in Mixed-Mode conditions (Four-Point Sear Test wit one or two notces and Dam Models). Te crack trajectory is not known a priori, so tat te Finite Element mes must be modified at eac step of crack propagation. All numerical simulations were performed wit te aid of te Coesive CRAck Program (C.CRA.P.) devised at te Politecnico di Torino by Valente and developed by Barpi in te last few years. More information related to different Mixed-Mode conditions, namely te Pull-out test, can be found in [64 68]. 4.1 Four-Point Sear Test Wit One Notc Te testing set-up for te so-called Four-Point Sear Test (FPST or Single-Edge Notced Beam Test, SENBT) is sown in Fig. 15. A detailed description of tese tests is provided in [69 72]. Concrete properties are illustrated in Table 2. Te load vs. crack mout opening displacement diagrams are sown in Fig. 16. Figures 17, 18 and 19 sow te load vs. displacement diagrams. Te quantities F 1, δ 1, F 2, δ 2 are defined in Fig. 15. As can be seen, numerical and experimental results are in good agreement. On te oter and, te experimental results sow a more rigid response, wic can be explained by taking into account tat te displacements were measured relative to a bar wic neutralizes local strains in te constraints. Finally, Fig. 2 sows te mes used in te numerical simulation wen =.2m [73]. It can be seen tat te smallest specimen broke into tree parts, wereas te oters into two. Tis trend was observed not only in te tests described in [7], but also in tose illustrated in [69, 74] and [71, 72, 75]. Tis penomenon is taken into accout by te coesive model: Fig. 21 sows te mes in te case of te propagation of te secondary crack, wile Fig. 22 sows te corrisponding load vs. displacement curve. For more details, te reader is referred to [76]. Table 2: Material properties (single-edge notced beam). E ν G F σ u (MPa) (-) (N/m) (MPa)

15 F 1 + F 2 (kn) Numerical results Experimental results =.3 m 2 =.2 m =.1 m C.M.O.D. (1 6 m) Figure 16: Load vs. C.M.O.D. ( =.1,.2,.3m). F (kn) 3 F Numerical results 1 Experimental results F 2-2 (1 6 m) Figure 17: Loads F 1,F 2 vs. displacements δ 1,δ 2 for =.1m. A brief overview of te different metods proposed to study tis penomenon is now given. Te metod proposed in [77] is based on a remesing tecnique similar to te one illustrated in Fig. 2 and 21. Tis is a discrete-type model, and eac crack is idealized by means of a line. Anoter approac [78], is represented by te so-called Smeared Crack Model. Wilst in te model proposed in [77] te crack trajectories are not connected wit te direction of te mes elements, in te Smeared Crack Model tey are affected by te arrangement of te elements. An approac wic is similar to te foregoing one is tat proposed in [79]: te crack trajectories are still affected by te type of mes. In [8] te approac is still of te discrete type, wit te crack trajectories correlated to te interface between adjacent elements. 6 F (kn) F Numerical results 2 Experimental results F m) Figure 18: Loads F 1,F 2 vs. displacements δ 1,δ 2 for =.2m.

16 8 F (kn) 6 F 1-1 Numerical results 4 Experimental results 2 F m) Figure 19: Loads F 1,F 2 vs. displacements δ 1,δ 2 for =.3m. Figure 2: Mes used in te numerical simulation ( = 2cm). 4.2 Gravity Dam Models We sall now examine te beaviour of a gravity dam model, te dimensions of wic are as sown in Fig. 23 [81]. Te properties of te material used are listed in Table 3. In bot cases, specimen tickness t was 3 cm, wereas te notc dept was taken to be 15 or 3 cm. Two constitutive laws were considered for te process zone: linear and bilinear. Figure 24 presents te numerical and experimental results obtained for te specimens wit 15-cm long notces. Wen te state of stress at te tip of te fictitious crack is isotropic, i.e., caracterized by te condition σ xx σ yy, tismakestecrackpropagationcriterion(basedontedirectionofprincipaltensilestress) unapplicable. For tis reason, te analyses discussed in tis section were stopped at tis condition and are necessarily incomplete. An alternative criterion is proposed in [82, 83]. Numerical and experimental crack trajectories are sown in Fig. 25. Te study presented in [84] proposes an interesting comparison wit te Finite Element Metod and te Diffused Cracking Model, wile, in [85] an Anisotropic Damaging Model is proposed. A furter and interesting comparison is based on an investigation [86] tat sets fort te results obtained using different bilinear constitutive laws and different models of te searing stresses in te process zone. Concerning experimental tests on dam models, we mention te ones carried out by means of centrifugal equipment in order to simulate te beaviour of a 96-m ig prototype. For more information see [87 Primary crack F 2 = F 1 /1 F 1 Secondary crack Figure 21: Mes for te case =.1m (specimen broken into tree parts).

17 1.5 F 1-1 Dimensionless load, F/( u H t) 1..5 One crack Two cracks Experimental results (average values) F Dimensionaless deflection, /H Figure 22: Dimensionless loads vs. dimensionless displacements for te case =.1m (specimen broken into tree parts). 9]. Te coesive crack model as been extended to te study of crack propagation under constant load, coupling creep and fracture, in te case of Mode I problems, see [91 95]. and Mixed Mode problems, see[96,97]. Itasbeenfurtergeneralizedbymeansofaviscousmodelbasedonafractional order rate law to overcome te difficulties encontered using long cains of reological elements, wose properties are difficult to determine (see [98 1]). Due to te scatter in te material parameters, te same autors also examined te influence of uncertainity (or imprecision) in te material parameters by means of a fuzzy-set approac [11, 12]. Given a value of vagueness in te material parameters, te fuzzy set teory makes it possible to evaluate te vagueness in te results avoiding te difficulties of a stocastic analysis. 5 Direct tension tests: te effect of te microstructural disorder Dealing wit te failure of unnotced specimens in tension, te coesive crack model fails in predicting te size effects, wic, neverteless, rise. Te explanation of te size effects upon tensile fracture properties of concrete specimens, expecially in direct tension tests, is an ongoing matter of discussion inside te scientific community. A sound approac to tis problem as been proposed by Carpinteri since 1994 by means of fractal geometry. 5.1 Te fractal approac Te fundamental reason of size effects rising in quasi-brittle material structures is damage localization. In te previous sections we sowed ow te coesive crack model is able to catc tis peculiar beaviour in several configurations. More generally, te coesive crack model is able to simulate tose tests were ig stress gradients are present, i.e., tests on pre-notced specimens or in bending. In tese cases, te Table 3: Material properties (dam model). E ν G F σ u σ u (linear law) (bilinear law) (MPa) (-) (N/m) (MPa) (MPa)

18 y % P % P % P 3% 7 % P 43.75% P x 2 3 Figure 23: Gravity dam model wit notc lengt of 15cm (dimensions in centimeters). P (kn) 75 Experimental results 5 25 Linear c-w law Bilinear c-w law C.M.O.D. (cm) Figure 24: Load vs. C.M.O.D. curves (notc lengt 15cm). coesive crack model captures te ductile-to-brittle transition occurring by increasing te size of te structure. On te oter and, smaller but neverteless relevant size effects are encountered in uniaxial tension tests on dog-bone saped specimens, were muc smaller stress gradients are present. In tis case, size effects sould be inerent to te material beaviour rater tan to te stress gradient. Apart from te tests carried out by Bažant & Pfeiffer [13] in a limited scale range (1:4), uniaxial tensile tests on dog-bone saped specimens were performed by Carpinteri & Ferro [4], wit controlled boundary conditions, in a scale range 1:16, and by van Mier & van Vliet [41], wit rotating boundary conditions, in a scale range 1:32. Te tests, in bot cases, proved tat te pysical parameters caracterizing te coesive law are scale-dependent, tus sowing te limits of Hillerborg s model. By increasing te size of te specimen, te peak of te coesive law decreases wile te tail rises. In oter words, te tensile strengt decreases wile te fracture energy as well as te critical displacement increase. A consistent explanation of te size effects affecting te coesive law parameters in direct tension test was provided by Carpinteri [35, 36] and by Carpinteri et al. [39] assuming fractal damage domains. Tis ypotesis is motivated by te disorder caracterising te microstructure of most quasi-brittle materials in a broad range of scales. Size effects in uniaxial tensile tests can terefore be seen as a consequence of te eterogeneous microstructure of concrete and rocks. Since te flaw distribution in quasi-brittle materials is often self-similar (i.e., it looks te same at different magnification levels), te

19 Linear c-w law Bilinear c-w law Experimental results Figure 25: Crack trajectories (notc lengt of 15cm). microstructure can be correctly modelled by fractal sets. Fractal sets are caracterized by non-integer dimensions [14, 15]. For instance, te dimension α of a fractal set in te plane can vary between and 2. Accordingly, increasing te measure resolution, its lengt tends to zero if its dimension is smaller tan 1 or tends to infinity if it is larger. In tese cases, te lengt is a nominal, useless quantity, since it vanises or diverges as te measure resolution increases. A finite measure can be acieved only using non-integer units, suc as meters raised to α. Analogously, if te stress and strain localization occurs in a fractal damaged zone, te nominal quantities (ultimate strengt, critical strain, fracture energy) sould depend on te resolution used to measure te set were stress, strain and energy dissipation take place. In te limit of a very ig measure resolution, te stress and te strain sould be infinite, wile te dissipated energy sould be zero. Finite values can be obtained only introducing fractal quantities, i.e., mecanical quantities wit non-integer pysical dimensions. On te oter and, if te measure resolution is fixed, te nominal quantities undergo size effects. More specifically, te fractal strain localization explains te observed increasing tail of te coesive law as te specimen size increases (see [16]), i.e., it clarifies te scaling of te critical displacement w c. Similarly, te fractal stress localization explains te experimentally observed decreasing peak in te coesive law wile increasing te specimen size, i.e., it clarifies te scaling of te tensile strengt σ u. Finally, te scaling of te fracture energy is a consequence of te invasive fractality of te set were energy dissipates (i.e. a fracture surface or a damaged band). According to te fractal approac, te scaling of te coesive law parameters is represented by power laws, wose exponents are linked eac oter by a relation (as sown later). Witout entering te details, we wis to empasize ow te ypotesis of te fractal damage domain in quasi-brittle material failure is not a matematical abstraction, since fractal patterns ave been detected in several experiments(see, for instance, [17, 18]). Furtermore, for wat concerns concrete, anoter explanation of te fractal features of te damaged zones as been recently derived from te analysis of te aggregate size distribution [19, 11]. Tis stereological analysis confirmed te values of tensile strengt and fracture energy power law exponents previously conjectured by Carpinteri based on dimensional analysis arguments [36]. Te analyses of te fracture surfaces ave sown tat te fractal beaviour is more evident at te smaller scales. At te larger scales, te disorder and its influence onto te mecanical properties seem to diminis. Wile classical (i.e. self-similar) fractal sets cannot catc tis trend, te self-affine fractals can. Te scaling laws previously derived ave been terefore extended to te self-affine case, leading to te definition of te so-called multifractal scaling laws [111, 112] for tensile strengt [37, 113], for fracture energy [38, 114], and, more recently, for critical displacement [11]. In tis researc field, tese concepts ave been applied not only to tensile tests, but also to explain te R-curve material beaviour [115] and to interpret te results of bending [116] and compression [117] tests. Finally, Carpinteri et al. [118, 119] compared te size effect predictions provided by te fractal approac wit te ones given by te gradient teory approac. Anoter important researc field as been recently opened by Carpinteri & Cornetti [12]. Tey ave tried to generalize te classical differential equations of continuum mecanics to fractal media. Since te fractal functions, because of teir irregularity, cannot be solutions of any differential equations, Carpinteri & Cornetti argued tat suitable matematical operators sould replace te integro-differential operators of classical calculus [121]. Te attention was drawn to te local frac-

20 F b b * A res * A dis (a) F (b) (c) Figure 26: A concrete specimen subjected to tension (a). Fractal localization of te stress upon te resistant cross section (b) and of te energy dissipation upon crack surface (c). tional calculus operators recently introduced by Kolwankar [122] stemming from fractional calculus 2. Here we just wis to point out tat te order of differentiation is linked to te fractal dimension of te domain were te differential equations old and tat, by local fractional calculus, te autors succeeded in proving te Principle of Virtual Work for fractal media [ ]. We ave previously seen tat te coesive model parameters are size-dependent. In order to get true material parameters, we are forced to introduce quantities wit anomalous (non-integer) pysical dimensions: te fractal tensile strengt, te fractal critical strain and te fractal fracture energy. Tanks to teir non-integer pysical dimensions, tey intrinsecally introduce te fractal dimensions of te sets were stress, strain and energy dissipation localize. In te next section, we will focus our attention upon tese fractal mecanical quantities, in terms of wic it is possible to define a scale-invariant (or fractal) coesive law tat represents a true material property. Togeter wit te linear elastic constitutive law valid for te undamaged part of te material, te fractal coesive law defines a material model tat we call te (size-independent) fractal coesive crack model. Te model will be applied to te results of te tests carried out by Carpinteri & Ferro [127] and by van Mier & van Vliet [41], in order to prove te soundness of te fractal approac to te size effect prediction. 5.2 Scale-independent coesive crack model In order to introduce te fractal coesive crack model, we ave to consider separately te size effects on te tree parameters caracterizing te coesive law. Let us start analyzing te size effect on te tensile strengt by considering a concrete specimen subjected to tension (Fig. 26a). Recent experimental results about te porous concrete microstructure [18] as well as a stereological analysis of concrete flaws [19] led us to believe tat a consistent modelling of concrete damage can be acieved by assuming tat te rarefied resisting sections A res in correspondence of te peak load can be represented by stocastic lacunar fractal sets wit dimension 2 d σ (d σ ). From fractal geometry, we know tat te area of lacunar sets is scale-dependent and tends to zero as te resolution increases: te corresponding tensile strengt sould be infinite, wic is pysically meaningless. Finite measures can be obtained only wit non-integer (fractal) dimensions. For te sake of simplicity, let us represent te specimen resistant cross-section as a Sierpinski carpet built on te square of side b (Fig. 26b). Te fractal dimension of tis lacunar domain is (d σ =.17). Te assumption of Euclidean domain caracterizing classical continuum mecanics 2 Fractional calculus is te branc of te calculus dealing wit integrals and derivatives of any order. See, for instance, te treatise [123].

21 ε c * b 1 d ε w F b b (b) (a) b z F F F Figure 27: Fractal localization of te strain (a) and of te energy dissipation inside te damaged band (b). states tat te maximum load F is given by te product of te strengt σ u times te nominal area A = b 2, wereas, in te present model, F equals te product of te (Hausdorff) fractal measure [15] A res b 2 dσ of te Sierpinski carpet times te fractal tensile strengt σ u [35]: F = σ u A = σ ua res (31) were σ u presents te anomalous pysical dimensions [F][L] (2 dσ). Fractal tensile strengt is te true material constant, i.e., it is a scale-invariant. From Eq. (31) we obtain te scaling law for tensile strengt: σ u = σ ub dσ (32) i.e. a power law wit negative exponent d σ. Equation (32) represents te negative scale effect on tensile strengt, experimentally revealed by several Autors. Experimental and teoretical results allow us to affirm tat d σ can vary between te lower limit canonical dimensions for σ u and absence of size effect on tensile strengt and te upper limit 1/2 σ u wit te pysical dimensions of a stress-intensity factor and maximum size effect on tensile strengt (as in te case of LEFM). Secondly, let us consider te work W necessary to break a concrete specimen of cross section b 2 (Fig. 26a). It is equal to te product of te fracture energy G F times te nominal fracture area A = b 2. On te oter and, te surface were energy is dissipated is not a flat cross-section: it is a crack surface, wose area A dis diverges as te measure resolution tends to infinity because of its rougness at any scale. Terefore, te fracture energy sould be zero, wic is pysically meaningless. Finite values of te measure of te set were energy is dissipated can be acieved only via non-integer fractal dimensions. For te sake of simplicity, let us represent te crack surface as a von Kock surface builtontesquareofsideb(fig.26c). Tefractaldimensionoftisinvasivedomainis2.262, i.e.2+d G (d G =.262). Te classical coesive crack model states tat te failure work W is given by te product of te fracture energy G F times te nominal area A = b 2, wereas, in te present model, W equals te product of te fractal (Hausdorff) measure [15] A dis b2+d G of te von Kock surface times te fractal fracture energy G F [35]: W = G F A = G FA dis (33) G F = G Fb d G (34) G F is te true scale invariant material parameter, wereas te nominal value G F is subjected to a scale effect described by a positive power law.

22 σ (a) σ u σ u * σ* (b) ε ε* ε u ε c * Figure 28: Fractal coesive model. Now we turn our attention to te deformation inside te zone were damage localizes (te so-called damaged band). We assume tat te strain field presents fractal patterns. Tis could appear strange at a first glance; on te contrary, fractal strain distributions are rater common in material science. For instance, in some metals, te slip-lines develop wit typical fractal patterns [128]. Fractal crack networks develop also in dry clay [129] or in old paintings under tensile stresses due to srinkage. Tus, as representative of te damaged band, consider now te simplest structure, a bar subjected to tension (Fig. 27a), were, at te maximum load, dilation strain tends to concentrate into different softening regions, wile te rest of te body undergoes elastic unloading. Assume, for instance, tat te strain is localized at cross-sections wose projections onto te longitudinal axis are provided by te triadic Cantor set, wose dimension is ln 2/ ln 3 =.639. Te displacement function at rupture can be represented by a Cantor staircase grap, sometimes also called devil s staircase (Fig. 27a). Te strain defined in te classical manner is meaningless in te singular points, were it diverges. Tis drawback can be overcome by introducing a fractal strain. Let us indicate wit 1 d ε (d ε ) te fractal dimension of te lacunar projection of te cracked sections (in tis case d ε =.361). According to te fractal measure of te damage line projection, te total elongation w c of te band at rupture must be given by te product of te Hausdorff measure b b (1 dε) oftecantor settimestecritical fractal strain ε c, wileinclassicalcontinuum mecanics it equals te product of te lengt b times te critical strain ε c : w c = ε c b = ε cb (1 dε) (35) ε c = ε cb dε (36) were ε c as te anomalous pysical dimension [L] dε. Te fractal critical strain is te true material constant, i.e. it is te only scale-invariant parameter governing te kinematics of te crack band. On te oter and, Eq. (35) states tat te scaling of te critical displacement is described by a power law wit positive exponent (1 d ε ). Te fractional exponent d ε is intimately related to te degree of disorder in te mesoscopic damage process. Wen d ε varies from to 1, te kinematical control parameter ε c moves from te canonical critical strain ε c dimensionless [L] to te critical crack opening displacement w c of dimension [L] 1. Terefore, wen d ε = (diffused damage, ductile beaviour), one obtains te classical response, i.e. collapse governed by te strain ε c, independently of te bar lengt. In tis case, continuum damage mecanics olds, and te critical displacement w c is subjected to te maximum size effect (w c b). On te oter and, wen d ε = 1 (localization of damage onto isolated sections, brittle beaviour) traditional fracture mecanics olds and te collapse is governed by te critical opening displacement w c, wic is size-independent as in te usual coesive model. Te tree scaling laws (32), (34), (35) of te coesive parameters are not completely independent of eac oter. In fact, tere is a relation among te scaling exponents tat must be always satisfied. Tis means tat, wen two exponents are given, te tird follows from te first two. In order to get tis relation, suppose, for instance, to know d σ and d ε. Generalizing Eqs. (32) and (35) to te wole