THE OVERVIEW OF FRACTURE MECHANICS MODELS FOR …

1. INTRODUCTIONCracking of concrete is one of the most common flawsand problems in practical engineering. It can serve asan important signal that, for example, the constructionis not working properly. It may also be a direct causeof structures’ catastrophes even, if ignored.It is not an easy task to describe fracture process in con-crete, as there are many factors affecting crack develop-ment and propagation. In literature usually three mainstages of the crack propagation are distinguished: crackinitiation, stable crack development and unstable crackpropagation. The third stage is especially problematicbecause it can cause a quick failure of a structural mem-ber. As a consequence, the need to research crackingprocesses in more depth has arisen in a form of a sepa-rate scientific field under the name of fracture mechan-ics. The start of its development is owed to A. A.Griffith [1]. In 1920 he formulated rules of an energeticmodel describing an element with an artificial flaw, sim-

ulating a naturally occurring phenomenon. The conclu-sion derived from this research was that a crack needs acertain amount of energy to lengthen, under a specificvalue of stress. Once the stress reaches a critical value,an unstable propagation occurs, followed by destructionof a structural member. His idea was a general one,using a mathematical model based on linear elasticityassumptions, and not focusing only on concrete. Evenso the base of fracture mechanics was created. Next,after Griffith’s theory, G.R. Irwin expanded fracturemechanics’ research and introduced new parameterswhich are sometimes referred to as fracture criteria [2].The most important one was a critical stress intensityfactor KIC, which was also the first concept of fracturetoughness. Over fifty years after Griffith’s first fracturetheory a breakthrough for concrete structures was madewhen A. Hilleborg introduced his own model. It tookinto account properties of concrete as a material verydifferent from steel, which was mainly researched mate-rial until then. After that many other theoretical models

THE OVERVIEW OF FRACTURE MECHANICS MODELS FOR CONCRETE

Amanda AKRAM *

*MSc Eng.; Department of Structural Engineering; Faculty of Civil Engineering and Architecture;Lublin University of Technology; 40 Nadbystrzycka Street, 20-618 Lublin, PolandE-mail address: [email protected]

Received: 24.09.2020; Revised: 16.02.2021; Accepted: 22.02.2021

A b s t r a c tFracture mechanics of concrete is a complex matter still thoroughly researched from different angles. It is not an easy taskto describe fracture process in concrete, as there are many factors affecting crack development and propagation. Practicalapplications of fracture mechanics could allow engineers to design concrete structures more effectively and safely. At theminimum, it could help estimate the “safe” period of time left before the unstable, dangerous crack propagation. This util-itarian goal was the reason for many researchers to invent numerous theoretical models in order to describe the crackoccurrence better. However, dealing with various analytical problems was not a simple matter and thus existing models offracture mechanics for concrete have different limitations. Over the years first fracture theories for concrete were reviewedrepeatedly. All of these investigations lead to modifications of older models in order to overcome found drawbacks, whichproved not to be an easy task. Recently, new approaches to fracture analyses seemed to produce promising results, like uni-versal size effect law (USEL) or modified two parameter fracture model (MTPM) with alternative ways for evaluating frac-ture parameters. In the paper some of them will be discussed together with other fracture models, starting from some of thevery first ones introduced for concrete, like fictitious crack model (FCM) and crack band model (CBM).

K e y w o r d s : Fracturemechanics; Fracture of concrete; Fracturemodels; Cracking of concrete; Fracturemechanics of concrete.

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were proposed, the field of concrete fracture mechan-ics grew and new development is being conducted still,for example [3, 4]. Despite developing new solutions,some of the older models are still valid and applied innumerical and experimental research over recent years[5, 6, 7]. In this paper an analysis of main, selectedmodels used to describe crack propagation in concretewill be carried out.

2. OVERVIEW OF CHOSEN THEORETI-CAL FRACTURE MECHANICS MODELS2.1. Fictitious Crack Model (FCM) and crack bandmodel (CBM)In the following section two models will be discussed:the fictitious crack model and crack band model asthey are mathematically and ideologically similar.Furthermore they struggle with coinciding problems.Differences between the models will be also pointedout.The research of cracking begun with using knowledgederived from theory of linear elasticity. For concretethough it proved not to be the best approach, as it isnot a perfectly brittle material. As a consequence,mathematical results showed stress singularity at thecrack tip. That lead to the necessity of introducingmodifications in a form of turning to nonlinear solu-tions. In case of fracture mechanics, the nonlinearityis related to fracture process zone (FPZ) whichdevelops in front of a crack tip and replaces the erro-neous idea of stress singularity. One of the first mod-els for concrete was the fictitious crack model (FCM)introduced by Hilleborg et al. in 1976 [8]. It wasbased on the earlier work concerning metals whichalso fall under the category of materials behavingnonlinearly. The FPZ in the fictitious crack modelwas modeled in a linear form and the fracture processis characterized by a stress (in normal direction to thecrack, σ) – displacement (δ) relationship in a soften-ing range shown in the Fig. 1.That is the first of three main elements essential tothe fictitious crack model (FCM). As the tensilestress decreases in the fracture process zone, thestrain, such as the crack tip opening displacement,increases. This strain-softening occurrence for con-crete in tension is presented in the Fig. 2 [9].Two other elements that make up the core of FCM,labeled as the fracture mechanics parameters: the ten-sile strength limit fct and the fracture energy GF, arealso shown in Fig. 1. The concept of fct as one of thematerial properties is a well-known idea and its exis-

tence explains why there is no brittle fracture in con-crete. On the other hand, GF was a new value strictlyconnected to fracture mechanisms and it is describedas the area under the curve shown in Fig. 1. The shapeof the curve itself is also often called a fracture para-meter and depends on the material. For example, forhigh strength concretes the initial part of the softeningcurve would be noticeably steeper compared to nor-mal strength concretes. Fracture energy can be takenfrom Eq. 1 and it is understood as the area under thedescending curve of σ-δ relationship.

Another model that took into account the existenceof FPZ was the crack band model (CBM) introducedby Bažant and Oh [10]. The main idea of CBM wasfor the deformations of the fracture process zoneFPZ to be smeared over a band in front of the maincrack’s tip. That produced an additional materialparameter – a fixed value of crack band width (wc). Itseemed as a more realistic approach from the ficti-tious crack model, seeing as in concrete there usuallywas a lot of microcracks distributed along some part

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Figure 1.Illustration of fracture parameters: a descending branch ofstress-displacement curve (a) and a generalized bilinearsoftening function (b)

Figure 2.The strain-softening phenomenon in the post-peak rangeand the fictitious crack model for concrete in tension [9]

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of the material in front of the crack tip that later coa-lesced into bigger cracks. Mathematically though thefictitious crack model was a special case of a crackband model, if an assumption was made that wc wasinfinitely small (wc→0). It was also proven that thesetwo models were basically equivalent in terms of theresults they gave.Even if the fracture process zone (FPZ) and its sig-nificant size in concrete was taken into considerationin the models, there were still a few points at issue.The first one would be the model of the crack itself inthe fictitious crack model (FCM). There is a possibil-ity of a singular crack existing and propagating butthe linear shape of it is quite a significant simplifica-tion. The more natural path of propagation of such adefect is a torturous one. The cause of that is the het-erogeneity of concrete. Because the aggregate andthe cement matrix have different strength properties,the matrix being the weaker one, it’s only natural thatthe crack will develop avoiding the stronger aggre-gate particles. However, there are exceptions to thisrule when considering high-strength and ultra high-performance concretes in which the matrix’s strengthtends to be higher in comparison with the aggregate’sone. In such cases there is a possibility of a morestraight crack propagation, cutting also through thegrains. The matter of the fictitious crack model beingan one-axial is also an issue to point out. In realitymost common constructions work in a three-axialstress state. On the other hand, it is possible to takeinto account the three-dimensional stresses in termsof fracture, for example if the linear crack model is

not used [11]. Also the crack analysis becomes morecomplicated in reinforced concrete, where the pres-ence of steel can cause cracks to appear parallely. Insuch case, the fictitious crack model (FCM) does notpresent realistic results.Looking at the crack band model, in which theoreti-cally the crack form is shaped more realistically in asmeared manner (Fig. 3), the question about wc

appears.The width of a crack band, also known as a FPZwidth, should be fixed for numerical analysis’ pur-poses but various research results could not seem toagree on the definitive value of this parameter. Themost popular thesis stated that the width of FPZ wasdependent on the maximum aggregate’s size (Dmax).This dependence was researched over the years [12-14] and varies for different reasons, for example thetype of concrete or the type of a tested member, it’sshape and size. There are also some findings thatdid not confirm wc – Dmax dependency at all [15].Another concept to consider would be the stability ofcrack band during the crack propagation. The gener-al idea in crack band model is that of a fixed valuebut if it is really not dynamic at all during the pro-gressing of fracture and the course of loading. Onething which is quite certain though is that crackband’s width has a considerable impact on numericalresults. The problem is discussed wider for instancein [16]. One undeniable conclusion is that the crackband width is a relevant parameter of the crack bandmodel (CBM) used commonly in numerical simula-tions and as such it requires a finite and consistentrules of applicability.The final issue with discussed models is the size effectwhich could be observed for the nominal stress atmaximum load. Fracture energy GF was proven todepend on the geometry of tested concrete members,in terms of both size and shape, as well as the aggre-gate size. Recent research shows that there are otherfactors impacting the fracture energy [17], such as:water-cement ratio or type of aggregate.That fact effectively restrained the possibilities ofthese first crack models for concrete to be utilitarianin their design applications. The need for appropriatesize effect law to be used, in order to avoid this prob-lem and extrapolate the value of fracture energy forinfinite size was a continuous struggle and an elementof research, e.g. [18, 19]. This issue will be discussedin more depth in paragraph 3 of the paper.

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Figure 3.The visualization of the crack morphology (a), the model ofstrain distribution in the fracture process zone and its vicin-ity (b), the crack band model (CBM) concept with the crackband width wc (c) and the fictitious crack model (FCM) con-cept for comparison (d)

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2.2. Two Parameter Fracture Model of Jenq andShah (TPFM)One of the first special fracture models for concretewith no direct usage of stress – displacement curva-ture was the two-parameter fracture model (TPFM)proposed by Jenq and Shah in 1985 [20]. Instead, lin-ear elastic fracture mechanics’ (LEFM) elementswere modified in order to take into considerationslow crack growth which is a nonlinear phenomenonin concrete. Fracture parameters were introduced askey elements to this model: the critical stress intensi-ty factor KIC and the critical value of the crack tipopening displacement CTODc. Both of themrequired to be measured experimentally in Mode Iconditions, in a three – point bending test of anotched concrete member.

However, an issue appears to emerge in relations tothese fracture parameters such as the dependency oncomposition of concrete mix. It has been proven thatwater-cement ratio has an impact on fracture para-meters [21, 22]. Fly ash and slag were also researchedin relation to that matter [23].Other values crucial to the two-parameter fracturemodel TPFM, connected to aforementioned maintwo parameters, need to be considered as well(Fig. 4): the effective crack length ae and the lengthof an existing, artificial notch a0. At the tip of theeffective crack, corresponding to the maximum load,KIC should be measured. The width of the notchintroduced for the experimental purposes gives thevalue of CTODc. However, the calculations for allthose parameters are based on the LEFM equationsand were proven to be geometry – dependent.Sundara Raya Iyengar et al. [24], based on an earlierattempts of other researchers, proposed an alterna-tive, graphical method of determining critical stressintensity factor, as well as the critical value of thecrack tip opening displacement. Graphical solutionsthat could produce fracture mechanics’ parameters in

a simpler way would be a meaningful aid. On theother hand, it should be remembered they are alsoderived from LEFM equations which in case of con-crete can lead to various mathematical issues. Finally,Jenq – Shah model was based on mode I only, leavingroom and necessity for further analysis of mode II,mode III and mixed modes, which would presumablylead to an improvement of understanding fracture inconcrete structure on a more realistic and practicallevel.The experimental method of measuring itself is bur-dened with some difficulties, such as possibility ofinaccurate calculating of KIC and CTODc because ofcrack propagation after the peak load. This topic wasraised for example by D. C. Jansen et al. [25].Therefore the authors proposed a different, modifiedtesting procedure for the two-parameter fracturemodel – the focal point method. It was divided intotwo sub-methods: the focal point method I and II.Both of them were graphic – based, similarly to thesolution proposed in [24]. Focal Point methods serveas a simulation of the closed – loop testing in a wayfor determining unloading compliances, which arelater used in a standard two-parameter fracturemodel (TPFM) to calculate KIC and CTODc. Inmethod I the idea is to find a centroid of a triangleconstructed from three lines going through points ofthree, initial peak loads. The centroid then acts as afoundation for determining unloading complianceCuc. In the second method two lines from the firstloading and first unloading are required and theirintersection point provides also a solution for defin-ing Cuc. These methods seem to give more realisticvalues of two basic fracture parameters but still onlyfor testing under mode I conditions.

2.3. Modified Two Parameter Fracture Model(MTPM)With regard to the crack model itself there wereattempts made to take into consideration the tortu-ous path that the crack may follow. In that case thereason generally may be twofold. The kinked crack-ing may occur as a result of a difference in strength ofcement matrix and aggregate or as a consequence ofmixed mode loading. A. Carpinteri et al. [26] carriedout a series of test focusing on the second aspect con-cerning the deflection of a crack from its supposed,linear path (Fig. 5). The modified two parametermodel’s core consists of mathematical changes in for-mulae in a form of introducing a deflection angle θ,for computing the value of fracture parameters.


Figure 4.A notched beam in a three-point bending test with parame-ters of Jenq – Shah model


The theoretical background for the tests was imple-mentation of additional virtual forces perpendicularto the notch. These forces along with the basic verti-cal, loading force in a three-point bending togethercreated conditions of mix mode loading. For obtain-ing the critical stress intensity factor K(I+II)c the effec-tive crack length is necessary [27]:

Where: ao – initial crack length, a1 = 0.3ao and a2 canbe calculated through an iterative procedure. In casethe a2 proves to be negative, the effective crack lengthconsists only of ao and a1.

Although the starting point of described modifiedtwo parameter fracture model’s (MTPM) analysisinvolved bovine cortical bones, quite recent researchadapted and implemented it into fracture of con-crete. Some similarities may be observed with regardto the general quasi – brittle nature of both materialsbut the structure of bone tissue with many elementsof various shape and mechanical characteristics isfairly different from concrete structure, which con-sists of only two basic parts – the matrix and theaggregate. However, the aim to take account of thecrack behavior under mixed mode loading would be avaluable input in concrete fracture mechanics. Theinvestigation in [27] shows promising results based onfiber-reinforced concrete members.

2.4. Double – K Fracture Model (DKFM)Double-K model was based on a criterion proposed byXu and Reinhardt [28]. As the name suggests, it con-sists of two separate stress intensity factors. The ideabehind this combined form of a stress intensity factor,which in itself was known and considered before[2, 29], was to propose a tool for estimating crack prop-agation in a structure in a twofold way. The first onerepresents the initial cracking toughness KIc

ini and the

second one – unstable fracture toughness KIcun. The

initial stress intensity factor is especially significant, asdescribing the phase of initial crack propagation wasnot attempted before. Other models already men-tioned are related to the last phase only – unstablecracking. For three-point bending tests the values ofKIc

ini and KIcun can be determined from the following

formula based on the test data [28]:

where: P – the applied load, S – length of the beam,B – thickness of the beam, D – depth of the beam,a – the crack length, F(α) – function dependent onthe a/D proportion.Depending on which values of P and a are put intothe above formula, the KIc

ini and KIcun can be

obtained.

For evaluating initial stress cracking toughness, thevalue of initial cracking load Pini is needed. One of thepossible ways of determining Pini is through the use ofP-CMOD curve (Fig. 6) but it is not an easy task [30].Hence the analytical way of evaluating initial stressintensity factor was proposed:

The unstable fracture toughness KIcun can be simply

obtained from Eq. 3. KIcc is the stress intensity factor

due to cohesive force and can be found through theuse of the distribution function of the cohesive stress

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Figure 5.Theoretical basics of modified two parameter fracturemodel: the deflection (kinking) angle θθ, effective criticalcrack length ae, a1 and a2 together form the kinked crackbranch [27]

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Figure 6.An example of a P – CMOD curve for the purposes of calcu-lating the double – K fracture parameters, where: Cu – theunloading compliance, Pini – initial cracking load, CMODp –plastic crack mouth opening, CMODe – elastic crack mouthopening. From point 0 to 1 the fracture process can be char-acterized as elastic and there is no further propagation of theinitial crack, at point 2 maximum load Pmax is achieved

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along the fictitious crack zone at the peak load(Fig. 7). It is important to note that the shape ofcohesive stress distribution is assumed to be linearbetween the initial notch tip to the tip of effectivecrack extension.

However, a problem can occur with the analyticalsolution, as it is a simplified method of evaluatingfracture toughness and the possibility of discrepancybetween mathematically obtained results and practi-cal use exists. Next to the mentioned two types offracture toughness gives another, subsidiary fractureparameter – critical crack tip opening displacementCTODc [30], as in the two parameter model of Jenqand Shah. Unlike the KIc

ini and KIcun parameters,

experimental research showed the size-dependenceof CTODc.

In order to investigate the evaluating procedure ofdouble – K parameters in wider spectrum, besidesthree-point bending tests, Xu and Reinhardt carriedout a series of experiments on compact tension spec-imens, as well as wedge splitting specimens [31]. Theresults revealed that for large CT members KIc

ini andKIc

un show no dependency on size. These conclusionsare logically somewhat similar to the idea of “extrap-olation to infinity” about which Bažant et al. talks inhis research of size effect. Further discussion aboutthis topic will be carried out in the next part of thepaper. On the other hand, for the small WS speci-mens KIc

ini and KIcun were found to increase in value

as the size of the tested members grew. Wang et al. [32] tested specimens of different com-pressive strengths for initial and unstable crackingtoughness with consistent results. They proposed dif-ferent, size-independent empirical formulas from Xuand Reinhardt, for obtaining KIc

ini and KIcun:

in which fc is the cube compressive strength of con-crete.Further research is still ongoing and throughout therecent years was added to the investigation of dou-ble-K fracture parameters [33–36].

3. SIZE EFFECT IN FRACTURE OF CON-CRETESize effect is a problem associated with many theoriesand models of fracture mechanics (Fig. 8) and at thesame time it presents an obstacle in universal andpractical use of fracture parameters. Because of theimportance of this challenge, this part of the paperwill be dedicated to it as a separate paragraph.

This problem was observed very early and the idea offinding a golden mean surfaced a few years afterintroducing first fracture models for concrete. Manylaboratory tests and analytical investigations werecarried out [38–40]. A lot of research was carried outby Bažant [41–44]. He analyzed size-effect occurringin RILEM recommendations and observed that thefracture energy obtained through these recommen-dations was strongly dependant on size and also onthe notch length of the members [41]. Experimentalstudies have been carried out in order to find size-independent Gf by other scientists and compare itsvalues obtained through various methods [45].Similar studies from various angles are continuouslycarried out. Bažant based his first investigations on the size effect


Figure 7.The cohesive stress distribution σσ(x) in the fictitious crackzone at the peak load; ao – initial crack length, ac – criticalcrack length, ft – tensile strength of concrete

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Figure 8.Size effects for geometrically similar structures on a bi-loga-rithmic scale with strength criterion and LEFM asymptotes(dashed lines). Solid arched line shows the transitionalbehavior of concrete between different types of size effects[37]


law which was proposed for nominal stress. The firstevaluation of the nominal strength, however, was alsosize-dependant:

where: cN – after Bažant, a general coefficient forconvenience [41], Pmax – ultimate load, b – thicknessof the specimen, d – characteristic dimension of thespecimen. Another form of formula used for fracture of con-crete, devised also by Bažant [42], for nominal stress,was as shown below:

or alternatively with the use of fracture characteris-tics:

where: ft – tensile strength, B and �λo – empirical con-stants, da – the maximum aggregate size, d – charac-teristic dimension of the specimen, E’ – Young’smodulus for plane stress or plane strain, Gf – fractureenergy, g(αo) – dimensionless energy release func-tion, αo = ao/d, ao – initial notch depth, cf – the frac-ture zone length at maximum load. In literature the λo is also replaced by do and namedas the transitional structure size [43] or the brittle-ness number β [41]. Other researchers also madeattempts to calculate and assign brittleness numbersto investigated materials, e.g. Carpinteri [44] orGustafsson and Hilleborg [46], in order to predictpossible fracture behaviour. The empirical constant �could be characterized as another fracture parameterand in [44] the role of it is compared to the characterof Reynolds number. In [46] the measure of brittle-ness took a form of a characteristic length lch which isproportional to the fracture energy. One of the main theory principles dealing with sizedependency is the existence of two types of sizeeffects for fracture occurring in quasi-brittle materi-als such as concrete: type I (deterministic) and type II(energetic). The formula mentioned above (Eq. 8) isconnected with type II size effect or in general it iscalled the size effect law of Bažant. This occurs whenthe specimens are geometrically similar, with a con-siderably large depth of the notch in comparison tothe characteristic dimension D. Type I size effect for-mula is as follows [43]:

Where: fr�– nominal strength extrapolated to infinityfor very large specimens, db – boundary layer length,r – empirical constant, lp – characteristic length of thematerial. This type of size effect is caused by the randomnessof material strength and it originated from Weibull’stheory [47]. To simplify briefly, type I size effectoccurs when the crack length is equal to 0 and type IIsize effect is observed for deep notches. However, there are cases where the fracture behav-iour in a concrete member falls neither strictly undertype I, nor type II size effect and instead it is a tran-sitional variation of the two. That was a direct causeof introducing a combined law under the name ofuniversal size effect law (USEL). In [43] Hoover et al.raise an additional point of statistical element andthus the USEL has two forms. Other research on ageneral size effect law was carried out also [48, 49]. Asignificant analysis was performed by Hoover andBažant [50], in which the authors gathered enoughexperimental material to properly verify the applica-bility of USEL. These comprehensive tests lead todefining an empirical parameter � which describessize effect law in terms of transition between type Iand II. The question of geometry independence ofthis parameter remains still unanswered and needs tobe verified in further tests. Other researchers also focused on the problem ofsize effect in fracture of concrete over the course ofmany years. Carpinteri et al. considered the fractalnature of cracking in concrete and its input to the sizeeffect [51, 52]. It resulted in introducing the multi-fractal scaling law (MFSL) which analytically is givenby the following equation:

where: �σN – the nominal tensile strength, A and B –the constants related to the physical dimensions ofthe square of a stress and the square of a stress-inten-sity factor respectively, both to be determined from anon-linear least-squares numerical algorithm, d –characteristic structural size.The idea behind this approach is to consider renor-malized fracture parameters (such as the tensilestrength and the fracture energy) given by the analy-sis of the observation scales. At the normally used,macroscopic scale, the fracture energy is dissipatedover the nominal area. In the fractal sense, the scalewould be progressively enlarged, leading to the

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decrease of the impact that microcracks have on themechanical behaviour of concrete. At the same timethe fracture energy becomes a renormalized, scale-invariant fracture parameter as a part of a moreorderly multifractal scaling concept, in a form of alinear relationship (Fig. 9a). Analogical reasoningcan be applied to the tensile stress (Fig. 9b).

Another approach to the size effect was taken byDuan and Hu in a form of the boundary effect con-cept [53]. The idea was to associate the size-depen-dant fracture energy and strength with the memberboundary – crack tip relation (Fig. 10).

In short, the localization of the fracture process zonewill directly affect the fracture parameters. If thecrack length is short (a→0), the maximum stress willreach the tensile strength of concrete as the crackitself will remain shielded before specimen failure.On the other hand, if the crack length is significantand the fracture process zone is localized in the mid-dle of a specimen and far away from the boundarieslayers, the strength criterion will follow the rules oflinear elastic fracture mechanics LEFM. Further investigations lead to the development of an

asymptotic boundary effect model [54] which pro-duced the asymptotic solution as follows:

where: ft – the tensile stress, a – the crack length, a∞ – the reference crack length depending on thestress intensity factor, at the intersection of twoasymptotic limits.However, the equation mentioned above is applica-ble only for the large specimens so the LEFM condi-tions are possible.

4. CONCLUSIONSThe broad scale analysis of the professional literaturepresented in the paper has proven that fracturemechanics of concrete is worth dealing with. Each ofthe discussed models has its limitations. The two-parameter fracture model and size-effect model pro-duce very similar results but at the same time thereare other parameters which differ from one fracturemodel to another. For example, the fracture energyGf – Hilleborg’s model and the size-effect model pro-duce disparate values of Gf. Additionally, other sev-eral unknowns and points are still being discussed byscientists.When describing fracture processes in concrete, agreat deal depends on concrete strength, type ofaggregate, concrete age or geometry of a structuralmember. Proven size-dependency of many fractureparameters derived from the discussed models in thepaper is one of the main problems concerning thetheory of fracture mechanics, as well as the key tofind universal laws for evaluating fracture parametersfor all types of structural members. Even in case ofthe size effects models there are still questions aboutthe range of their applicability, especially in terms ofnon-linear conditions. Initial notches which are artifi-cially introduced into the researched member inthree-point bending tests, used by all the models, arenot always a good imitation of the actual cracks thatcould exist internally in concrete structures. Majorityof theoretical range in all the discussed modelsdepends on the linear elastic fracture mechanics,whereas the lack of linear behavior of fracture in con-crete is very apparent. Another point is the utilizationof beams in the three-point bending tests mainly inthe models. That kind of laboratory environment willnot always depict realistically the actual concretebehaviour in terms of cracking. It could be worth-


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Figure 9.The effects of multiscaling fractal law on a bi-logarithmicscale: a) for the fracture energy, b) for the tensile stress; b – the characteristic reference size, dG – the fractional incre-ment of the energy dissipation, dσσ – the fractional decrementcaused by the analyzed disorder

Figure 10.The member boundary – crack tip relation in the boundaryeffect concept: a) the case of a “short crack”, when thecrack’s tip is close to the edge, in other words front boundary,b) the crack tip is remote from both front and back bound-aries and the linear elasticity’s rules applies, c) the case of a“short ligament” when the crack tip is close to the backboundary. The hatched area corresponds to the fractureprocess zone


while to conduct more research on cylinders, whichare the type of sample that can be extracted from theactual existing concrete structures. In spite of a variety of tests which are still being con-ducted in the field of fracture mechanics, it can beconcluded that both fracture parameters of concreteas well as fracture models need further investigations.In particular the influence of the depth of the notchand the maximum aggregate size on fracture parame-ters are worth to deal with.

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