Computational modeling of size effects in concrete specimens under

Int J Fract (2008) 154:27–49DOI 10.1007/s10704-009-9316-9

ORIGINAL PAPER

Computational modeling of size effects in concretespecimens under uniaxial tension

Miroslav Vorechovský · Václav Sadílek

Received: 14 April 2008 / Accepted: 14 January 2009 / Published online: 10 February 2009© Springer Science+Business Media B.V. 2009

Abstract The paper presents a follow-up study ofnumerical modeling of complicated interplay of sizeeffects in concrete structures. The major motivationis to identify and study interplay of several scalinglengths stemming from the material, boundary condi-tions and geometry. Methods of stochastic nonlinearfracture mechanics are used to model the well pub-lished results of direct tensile tests of dog-bone spec-imens with rotating boundary conditions. Firstly, thespecimens are modeled using microplane material andalso fracture-plastic material laws to show that a por-tion of the dependence of nominal strength on structuralsize can be explained deterministically. However, it isclear that more sources of size effect play a part, and weconsider two of them. Namely, we model local materialstrength using an autocorrelated random field attempt-ing to capture a statistical part of the combined sizeeffect, scatter inclusive. In addition, the strength dropnoticeable with small specimens which was obtainedin the experiments could be explained either by thepresence of a weak surface layer of constant thickness(caused e.g. by drying, surface damage, aggregate sizelimitation at the boundary, or other irregularities) orthree dimensional effects incorporated by out-of-planeflexure of specimens. The latter effect is examined bycomparison of 2D and 3D models withthe same material laws. All three named sources

M. Vorechovský (B) · V. SadílekInstitute of Structural Mechanics, Brno Universityof Technology, Veverí 95, 602 00 Brno, Czech Republice-mail: [email protected]

(deterministic-energetic, statistical size effects and theweak layer effect) are believed to be the sources mostcontributing to the observed strength size effect; themodel combining all of them is capable of reproduc-ing the measured data. The computational approachrepresents a marriage of advanced computational non-linear fracture mechanics with simulation techniquesfor random fields representing spatially varying mate-rial properties. Using a numerical example, we docu-ment how different sources of size effects detrimentalto strength can interact and result in relatively com-plicated quasibrittle failure processes. The presentedstudy documents the well known fact that the experi-mental determination of material parameters (neededfor the rational and safe design of structures) is verycomplicated for quasibrittle materials such as concrete.

Keywords Size effect · Scaling · Random field ·Weak boundary · Crack band · Dog-bone specimens ·Quasibrittle failure · Crack initiation · Stochasticsimulation · Characteristic length · Weibull integral ·Microplane model · Fracture-plastic model

1 Introduction

This paper studies interacting size effects on the nomi-nal strength of concrete structures using a combinationof finite element software enabling nonlinear analy-ses and probabilistic methods. The target is to identifypossible sources of size effect, study them and model

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28 M. Vorechovský, V. Sadílek

them together in one complex model. We want to showhow the different sources interact with each other. Weare particularly interested in the interaction of differentmaterial length scales and the effect of such interactionon strength size effect.

The work is related to previous papers by otherauthors among which the most influential is the workby Gutiérrez and de Borst (2002), dealing with deter-ministic and statistical lengths and their role in sizeeffect. Several very influential works were produced inthe 1990s; Carmeliet and Hens (1994) and Carmelietand de Borst (1995). They combined a simple non-local damage model and simulation of a bi-variate ran-dom field of material properties (damage threshold andstrain softening) within a single finite element com-putational model, and studied the two different lengthparameters: the characteristic length of the nonlocaldamage model, and the correlation distance for the ran-dom field. The illustrated example presenting finite-element analyses of direct-tension tests has shown thatthe specimen exhibits structural behavior that is repre-sentative of nonsymmetrical deformation, with a non-linear stress-displacement curve. It has also shown thatthe two sources of size effect can be modeled satis-factorily well. Their model utilizes experience gainedfrom a paper by Mazars et al. (1991), who also studiedthe two sources of size effect in cementitious materialsusing a continuous damage model, and compared theresults with experiments on both notched and unnot-ched bending beams. Unfortunately, they did not con-sider more than one random property, and ignored itsspatial correlation. The interplay of deterministic andstatistical size effects is one of the central topics in PhDthesis by Vorechovský (2004b). Some analytical resultssupported by a large computational case study of the

Malpasset dam failure are published by Vorechovskýet al. (2005), Bažant et al. (2007b).

Even though we have the ambition to study the sizeeffect phenomena in general terms, we have decidedto illustrate the problem using a particular example forthe sake of easier comprehension and transparency. Inparticular, we numerically study the well publishedexperimental results of direct tensile tests on dog-bonespecimens with rotating boundary conditions of vary-ing size (size range 1:32) performed by van Vliet andvan Mier and summarized in the PhD thesis by van Vliet(2000) and in papers by van Vliet and van Mier (1998,1999, 2000a,b), van Mier and van Vliet (2003), Dyskinet al. (2001). We are interested in the series of “dry”concrete specimens A to F (dimension D varying from50 to 1,600 mm, see Fig. 1 and Table 1); a series accom-panied by tensile splitting verification tests. The paperattempts an explanation of the interacting size effectson the mean and variance of nominal strength by a com-bination of random field simulation of local materialproperties and “weak boundary” effects, and a nonlin-ear fracture mechanics simulation based on a cohesivecrack model. There has been much effort expended ondifferent explanations of the experimentally obtainedsize effects on strength from several different points ofview. Firstly, the effect of a non-uniform distribution ofstrains in the smallest cross-section was studied usingsimple linear constitutive law (van Vliet and van Mier1999, 2000a), and a separation of structural and mate-rial size effects was discussed. Van Vliet and van Mier(1999) argue that most of the experimentally observedsize effect could be explained by strain/stress gradientsthat develop due to several reasons. The results werealso compared to the Weibull theory (Weibull 1939)based on the weakest-link model which was found to

r

D

D

D/4

D/4

D/5

rotating stiff

monitoredverticaldisplacements

andu uupp low

0.6 D

weakenedlayerthickness

F,u1600

2400

FD ECBA

A

F,u

e=D/50F,u

(b)(a) (c)

Fig. 1 (a) Dog-bone series (specimens A to F) tested by vanVliet and van Mier (1998); (b) elastic principal stress field;(c) 2D model in ATENA software with a surface layer. Strain

ε is calculated using the separation �u = uupp − ulow of twopoints over the control length lc = 0.6D

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Computational modeling of size effects 29

Table 1 Experimental data.Specimens’ dimensions,nominal strengths, samplesize, c.o.v. andcorresponding Weibullmodulus m

D r = 0.725D σN mean Specimens (#) c.o.v [%](mm) (mm) (std. dev.) (MPa) (shape par.)

A 50 36.25 2.54 (0.41) 10 16.2 (7.27)

B 100 72.5 2.97 (0.19) 4 6.28 (19.7)

C 200 145 2.75 (0.21) 7 7.67 (16.0)

D 400 290 2.30 (0.09) 5 4.02 (31.1)

E 800 580 2.07 (0.12) 4 5.91 (21.0)

F 1600 1160 1.86 (0.16) 4 8.67 (14.1)

fit the mean nominal strength of sizes B to F (vanVliet and van Mier 1998, 1999, 2000a,b, van Mierand van Vliet 2003). The slope of the mean size effectcurve corresponds to a Weibull modulus of 12, whichdoes not coincide exactly with the measured scatterof strengths at each size. However, this is required inthe Weibull type of size effect. Secondly, the effect ofGaussian stress fluctuation with non-uniform loadingwas studied by Dyskin et al. (2001), and the developedmodel, employing a limiting distribution of indepen-dent Gaussian variables with linear trend, agrees withthe experimental data very well. Van Mier and van Vli-et also compared the data to the “Delft lattice model”using a simple local elastic-brittle material with bothregular and random lattices, and they obtained goodresults. The statistical part of experimentally obtainedsize effect has also been modeled by Lehký and Novák(2002), employing a limiting distribution of indepen-dent Weibull variables describing the distribution ofstrength.

The present work extends a previous work presentedby Vorechovský (2007) in several ways. In this paper,we firstly try to explain the mean size effect curvewith deterministic effects (not taking into account thelocal material strength with a random field). To do so,two material laws are compared, namely the micro-plane material model and fracture-plastic model imple-mented as NLCEM model in ATENA software(Cervenka and Pukl 2005). There is a partial expla-nation of the decreasing slope of the mean size effectcurve (MSEC) in a double-logarithmic plot (nominalstrength versus characteristic size). However, the strongdecrease in the mean strength of the smallest specimenA is believed to be sufficiently captured by a modeledweak surface layer with a thickness of about 2 mm.A parametric study of the influence of “weak layer”thickness and the percentage reduction in the layer’s

strength compared to the bulk strength was presentedby Vorechovský (2007). Next, we approximate the localmaterial strength via an autocorrelated random field,attempting to capture the statistical size effect, scatterinclusive, and finally combine all sources together. Allabove named studies are performed in two dimensions.In order to quantify possible effects of out-of-planeflexure we have performed three dimensional model-ing taking into account the reported non-uniformity ofstiffness over the specimens’ width.

2 Experiments

The experiments by van Vliet and van Mier are welldocumented in the seven references cited in the intro-duction. We will briefly mention only those necessarydata needed to explain the computational model: allother details can be found in the cited publications.Dog-bone shaped specimens were loaded in uniaxialtension with geometrically scaled eccentricity from thevertical axis of symmetry e = D/50. The loading plat-ens were allowed to rotate freely in all directions aroundthe loading points at the top and bottom concrete faces.The loading platens were glued to the concrete. Six dif-ferent sizes were tested; all specimens were geometri-cally similar (see Fig. 1a). The specimen thickness waskept constant (b = 0.1 m), implying a transition fromplane strain like conditions at the smallest size to planestress conditions for the large sizes. The concrete mix-ture was reported to have an average cube compres-sive strength of 50 MPa and a maximum aggregate sizedmax = 8 mm.

For comparative purposes, it is necessary to define anominal strength σN. Since the eccentricity of the load-ing points has been geometrically scaled in both experi-ments and numerical models, we can ignore its effect on

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the linear state stress field and define the nominal stressσ simply as a function of the characteristic dimensionD (maximum specimen width), instantaneous tensileforce F applied at the concrete faces at the eccentric-ity e and the cross sectional area in the middle of thespecimen A (= 0.6Db = 0.06D m2):

σ = F

A(1)

Having defined the nominal stress, we define the nom-inal strength σN as the nominal stress attained at max-imum loading force (σN = Fmax/A).

Note that the smallest specimen size A has a width inthe ‘neck-area’ of only 30 mm. Compared to the max-imum aggregate size of 8 mm, it must be questionedwhether such a small specimen (being too small in sizeto be considered a representative volume element) canstill be treated identically to the rest of the series.

The authors of experiments have reported that spec-imens were casted such that, during manufacturing ofspecimens, the back surfaces were at bottom (mouldside) and that casting took place in three layersvan Vliet and van Mier (1999). This procedure is likelyto induce stiffness differences in the direction of cast-ing which could be more pronounced in relatively thicksmall specimens rather than in large specimens, seespecimen A in Fig. 1a and Fig. 6 right.

3 The deterministic models

3.1 Two-dimensional modeling

Most of the present studies were performed with 2Dmodels. We start with microplane modeling and com-pare the results to fracture-plastic models later on.

A strong contribution to the non-uniformity of thenominal strength is the “energetic-deterministic” sizeeffect caused by an approximately constant fractureprocess zone (FPZ) size with stress redistributionin specimens of all sizes; see e.g. Bažant and Planas(1998). This effect can be modeled e.g. by the finiteelement method provided that the fracture energy andthe whole shape of pre- and post-peak behavior is cor-rectly introduced. We created the deterministic modelin the ATENA software package (Cervenka and Pukl2005), using Bažant’s microplane material model (ver-sion 4) (Bažant et al. 2000) and the crack band model(Bažant and Oh 1983) as a simple regularization. The

basic idea of the crack band model for strain-soften-ing in tension (and also of the model of Pietruszczakand Mróz (1981) for strain-softening in shear) is tomodify the material parameters controlling the smearedcracking such that the energies dissipated by large andsmall elements per unit area of the crack band wouldbe identical. The choice of the microplane constitutivemodel is supported by the fact that M4 seems to bethe best model able to capture the complex behaviorof concrete under general conditions. The crack bandmodel has been chosen as the only technique widelyused and incorporated in commercial codes due to itssimplicity. The M4 microplane model does not explic-itly work with strain decomposition into elastic andinelastic parts and therefore the so-called equivalentlocalization element (Cervenka et al. 2005) has beenimplemented into ATENA. This technique removes theproblem of the spurious mesh size dependence of theresults, while a certain dependence on the mesh orien-tation still remains (for a concise overview of variousnumerical methods and their ability to analyse locali-zation and failure in engineering materials see de Borstet al. (2004)).

Specimens were loaded by deformation incrementsand the force F was monitored, see Fig. 1c. We ignoredthe transition from plane strain to plane stress condi-tions with growing specimen size in two-dimensionalmodels where we have modeled the whole series ofsizes with a plane stress model. Verifications usingthree dimensional models follow. Based on the infor-mation about the average cube compressive strength of50 MPa, ATENA generated a set of consistent micro-plane parameters: K1 = 1.5644 · 10−4, K2 = 500,K3 = 15, K4 = 150 (Caner and Bažant 2000), crackband cb = 30 mm, number of microplanes 21 (an effi-cient formula that still yields acceptable accuracyinvolves 21 microplanes to integrate over a sphere(Bažant and Oh 1986)). The parameters K1 throughK4 are phenomenological microplane model param-eters and they do not have a physical meaning; theycan be understood as scaling parameters of given curveshapes (criteria) describing the so-called “boundaries”.Briefly, K1 plays a role in relations for the tensile nor-mal boundary (needed for tensile cracking, fragmentpullout and crack closing), and also compressive devi-atoric and tensile deviatoric boundaries (spreading andsplitting); K1 and K2 affect shear boundary (friction);K1, K3 and K4 are present in the relations for bothtensile and compressive volumetric boundaries (pore

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0

1

2

3

[MPa

]

size Z size A size B

0

1

2

3

[MPa

]

size C size D

0 0.1 0.2 0.3

size E

0

1

2

0 0.1 0.2 0.3

N[M

P a]

size F

0 0.1 0.2

( 10 ) [-]x3

size H

1.5

2.0

2.5

3.03.5

1 10 100 1000 10000

1.0

1.21.4

specimen size [mm]D

A B CDE F

MicroplaneNLCEM

[MPa

]

GHZ

N

( 10 ) [-]x3

N,

Fig. 2 Comparison of deterministic nominal stress-strain dia-grams obtained with microplane model and fracture-plasticmodel for a range of structure sizes. Bottom right: nominalstrength dependence on structure size obtained with the twomaterial models compared to experiment average strengths witherrorbars

collapse, expansive breakup); for a full description seeBažant et al. (2000).

We changed the crack band to 8 mm, a value thatenables us to explain most of the experimentallyobtained size effect, see Fig. 2 bottom right. The crackband size is related to the fracture energy of the mate-rial and controls the size at which the continuum com-putational model undergoes transition from relativelyductile to elastic-brittle failure (transition between twohorizontal asymptotes in the size effect plot, see Fig. 2bottom right). By varying the strength related param-eter K1 (while keeping the characteristic length �ch orcb constant), the whole curve moves up and down asa rigid body. Another noticeable fact is that in the sizeeffect plot the curve can be shifted right or left as arigid body just by changing cb. More specifically, thedeterministic nominal strength computed for a certainsize of D using a cb value is also the nominal strengthof size s D computed with crack band width s cb (s is apositive scaling parameter):

for ∀s > 0: σ detN (D, cb) = σ det

N (s D, s cb) (2)

This has a direct implication noticed already by otherauthors before for the case of a fictitious crack (Bažantand Planas 1998): the nominal structural strengthdepends, when all other parameters are fixed, on thedimensionless ratio of D and �ch. The fictitious crackcan be shown to be equivalent to the smeared crack

band model for mode I fracture under consideration(see Bažant and Planas 1998, p. 255). The character-istic length �ch has a linear relationship with the crackband width cb (Bažant and Planas 1998). In our case,therefore, we can write that the nominal strength isproportional to the tensile strength times a functionof the ratio between D and �ch; a fact exploited inEq. 4 and discussed later on in this paper. It should notremain unnoticed that Eq. 2 has a direct relation to theVashy-Buckingham�-theorem (Vashy 1892, Bucking-ham 1914) on dimensional analysis (see e.g. Barenblatt1996). It turns out that, when the ratio between D andcb (or �ch) is close to unity, the structure is in transitionbetween two important asymptotes: plastic and elas-tic solutions. More precisely, if the structure is muchsmaller than �ch (D/cb → 0), the behavior is fullyelastic-plastic and can be simply predicted based onthe knowledge of direct tensile strength ft (the yield-ing point in this case). On the other hand, if the struc-tural size D is much larger compared to cb (or �ch), thebehavior is linear elastic with a sudden failure at theonset of reaching the direct tensile strength ft at anypoint in the material, see e.g. Vorechovský et al. (2005),Bažant et al. (2007b). In this case, what matters is theprofile of principal tensile stresses over the structure,see Fig. 1b.

From this we can also deduce the value ofσN(∞, cb) ≡ σN,∞, it being the large size asymp-tote in Fig. 2 bottom right. Simply, it is the nominalstress when the extreme principal tension reaches thedirect tensile strength ft . Note that in the definitionof σN the eccentricity of loading and possible stressconcentration in the specimen’s neck are not reflectedand therefore ft �= σN,∞. The ratio of ft/σN,∞ canbe deduced by considering two effects: (i) stress con-centration due to curved sides of the specimens and,(ii) eccentricity of the loading force. The first factorequals 1.24 (ratio between the maximum stress attainedat the curved boundary and a remote uniform stress).The second factor of 1.2 can be computed from normalstress due to centric normal force plus bending moment= F/A(1 + 6e/0.6D) = F/A(1 + 0.2). The multipleof these two factors 1.24 × 1.2 = 1.49 agrees wellwith FEM simulations of the eccentrically loaded largesized specimens giving ft/σN,∞ = 1.49.

Regarding the asymptotic strength of small speci-mens σN(0, cb) ≡ σN,0 one can argue that a speci-men made of ideal elastic-plastic material reaches itsmaximum force when the whole neck cross section is

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yielding (at stress equal to ft). Therefore the theoreticallimit nominal strength equals ft , no matter the eccen-tricity and the curved shape. Our FEM simulations withboth microplane and NLCEM models predict the max-imum nominal stress of small specimens at σN,0 ≈0.95 ft = 1.42 σN,∞. This value is, at the same time,the maximum size effect that can be captured determin-istically by considering stress redistribution (≈ 42%reserve), see the right hand ordinate in Fig. 2.

In order to compare the way microplane model cap-tures the energetic-deterministic size effect due to stressdistribution, we performed a parallel study with thefracture-plastic model named ‘NLCEM’ (nonlinearcementitious) in ATENA software. In this model theconventional parameters are used to define a mate-rial law. The most relevant are: the cube compressivestrength of 50 MPa (we have set it equal to the one in mi-croplane model and used it to generate the implicit val-ues for the rest of the material parameters applied in theintegration points mentioned next), uniaxial compres-sive strength fc = 42.5 MPa, modulus of elasticity E(set to 36.95 GPa such that the stiffness initialof microplane model and NLCEM models were equal),uniaxial tensile strength ft = 3.2 MPa, fracture energyGF = 200 N/m (exponential crack opening obtainedexperimentally by Hordijk 1991). Using this set ofmaterial parameters we have performed determinis-tic computations with a wide range of structure sizes(overlapping the tested range). Comparison of stress-strain diagrams obtained with the two material modelsis presented in Fig. 2 together with the size effect plotin the transition zone. It can be seen that the overallideal plastic behavior is the limiting behavior for thesmallest specimens while elastic-brittle-like curves areobtained for the large sizes. Note, that the transition isdifferent with NLCEM and microplane models. Whenthe structural size is very small, microplane model pre-dicts strong pre-peak stiffness reduction even thoughthe initial stiffness is equal to the one in NLCEM mate-rial model. The large size asymptotic strength is equalfor both models and so does the small size asymptoticstrength. The transition though is slightly different anddepends on the boundaries in microplane model andcrack-opening curve with other material parameters inNLCEM model.

The role of fracture energy in NLCEM model forscaling of structural strength is very similar to the roleof crack band in the microplane model in Eq. 2. It canbe easily checked that, for fixed E and ft:

for ∀s > 0: σ detN (D, GF) = σ det

N (s D, s GF) (3)

It is because the Irwin’s characteristic length�ch = E GF/ f 2

t scales linearly with GF and thereforevarying GF is equivalent to varying �ch. In other words,the size effect plot in Fig. 2 bottom right can be shiftedright or left as a rigid body just by changing GF: σ det

N(D, GF/s) = σ det

N (s D, GF), see Fig. 4. Not only thenominal strength is equal for the scaled structure. Ifboth the structure size and fracture energy in NLCEMmodel are scaled s times, the stress and displacementfields take the same values over the scaled coordinates.The same was true in the case of microplane modeling:if both the structure and crack band width is scaled stimes, the stress and displacement fields take the samevalues over the scaled coordinates. This fact simplifiesthe preprocessing of numerical models of a size effectseries: simply create a model of one size only and varyGF (or cb) instead of D.

3.2 Effect of varied GF and cb at element andstructural levels

We have performed simple numerical experiments withATENA software using which we document the effectsof varied GF and cb on tensile response of (i) one ele-ment and (ii) dog-bone specimens.

Fig. 3 presents stress-strain (force-displacement)diagrams of one square finite element of unit size loadedin uniaxial tension. The top row shows the situationwith NLCEM model, in which the original fractureenergy GF = 2, 000 N/m is multiplied by several fac-tors s ranging from 1/20 to 10 ( ft and E were keptconstant at values mentioned above). The bottom rowshows a similar numerical experiment in microplanemodel with original crack band width cb = 30 mm mul-tiplied by factors s ranging from 1/10 to 32 (again, theother parameters were kept as before). It is known (seee.g. Bažant and Planas 1998) that when using the crackband technology, the finite element can be imaginedto consist of an inelastic part with softening behav-ior and a perfectly elastic spring coupled in a series(see Fig. 3 top left). It can be seen that both the initial(spring) stiffness E and tensile strength are not affectedby variations of GF or cb. The area below the curvesis almost perfectly proportional to the scaling factor s(see the right hand side plots in Fig. 3 of stress againstscaled inelastic strains, that collapse into one curve).There is one condition, though, for the scalability of

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fracture energy, that is thoroughly described in sect.8.6.4 of Bažant and Planas (1998): the finite elementcan not be arbitrarily large compared to the character-istic length (or cb). Or, equivalently in NLCEM model,the fracture energy of a single finite element must begreater than the elastic strain energy accumulated in thespring at the peak stress: G(E)

F > f 2t /(2E). The prob-

lem is that snap-back behavior can not be captured bythe nodal displacement controlled computation. There-fore, one can see that when s is small in the two materialmodels, the finite elements dissipate more energy thanthey should. If no other criterion (such as those recom-mended in Bažant and Planas (1998), sect. 8.6) can beimplemented in the finite element program used, cau-tion must be paid that the element fracture energy ofthe crack band is greater than the elastic energy of thespring. The finite element fracture energy in our caseis simply G(E)

F = GF/L(E), where L(E) is the widthof elements perpendicular to the direction of cracking.We have checked that this criterion was fulfilled in alldog bone models studied in this paper. This was oneimportant limitation when exploiting Eqs. 2 and 3 forthe simplified modeling of size effect tests and appliesto modeling of very large structures using the referencesized model with an identical mesh (and with reducedGF [cb] in NLCEM [microplane] model respectively).

0

1

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3

0 0.001 0.0020

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enalporciM

0

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NL

CE

M

0.5ss

s

s= 0.5s

s

1/20s = 1/10s =

s0.5s =

2s =8s =

1/10s =s = 0.51s =

2s =

8s =

s = 32

16s =

L(E)

L(E)

σ , ε = εinel + εel σ , εinel

εel = σ / Eσ

σ

σ

σ

εεinel

s=

ε −σ /Es

Fig. 3 Effect of varying: fracture energy GF in NLCEM model(top); and crack band width cb in M4 model (bottom) in a singlefinite element under tension. Left: stress vs total strain curves;Right: stress vs scaled inelastic (fracturing) strain

Another limitation when using the reference size tomimic a very large size is that if the numerical mesh iskept in the whole series, the resolution of stresses (e.g.in the fracture process zone) might become insufficient.

Since the dog bone specimens are not loaded just byuniform tension, stress redistribution can take place. Itwas concluded that varying fracture energy (or crackband width) is in fact equivalent to varying the propor-tion between structural size and characteristic length(characterizing the material heterogeneity scale). Inorder to document this numerically using the dog bonespecimens models, we have varied the fracture energyGF by multiplying it with parameter s (s = 2, 4, 8, 16,

32 and their inverses, the largest to the smallest ratioequals 1:1,024) and plotted the nominal strength ofspecimens as a representative parameter against thestructural size (which was kept constant). If we, how-ever, shift the points towards the size D/s, the pointsfall exactly on the size effect curve computed with aconstant GF and varied size D, see Fig. 4. The nominalstrength dependence on size in the studied case of dog-bone specimens for instance, can be fitted very wellwith the following formula (Bažant and Planas 1998):

σ detN (D) = σN,∞

(1 + Db

D + l p

)(4)

where, aside from Db ≈ 300 mm, l p is a second deter-ministic characteristic length controlling the center oftransition to the left ‘plastic’ horizontal asymptote. Thevalue of l p can be deduced from the ratio of ‘ideal-plastic’ limiting strength and ‘elastic-brittle’ limitingstrength ηp = (1 + Db/ l p) ≈ 1.42; therefore l p ≈714 mm (which happens to be quite close to the Irwin’scharacteristic length �ch = E GF/ f 2

t ≈ 720 mm).Formula (4) gives the transition from perfectly plasticbehavior for D/ l p → 0 (corresponding to an elasticbody in which the crack is filled with a perfectly plasticglue), through quasibrittle behavior, to perfectly brit-tle behavior for D/Db → ∞. For the meaning of theparameters and justification of the formula, the readeris referred e.g. to Bažant et al. (2007b) and referencestherein.

It is an occasional practise to study a random modelresponse of structures with varied (randomized) frac-ture energy (keeping an identical crack opening lawcurve shape). If the size effect relation such as the one inEq. 4 is known, the probabilistic distribution of randomstrength σN for a given size D can be written analyti-cally just using a random variable transformation. The

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2.0

2.5

3.0

1 10 100 1000 10000 100000

N]aP

M[

D [mm]

G

1.01.11.21.31.4

htgnerts

sF

x

N,

N,0

N,

N

Fig. 4 Strength scaling using varying fracture energy GF (or cb)in a logarithmic size effect plot

situation is more complicated when randomizing thetensile strength and material toughness simultaneously.

3.3 Three-dimensional modeling

We have mentioned that the two dimensional modelswere created assuming plane stress conditions. Onemight get an impression that this simplification couldbe a source of an error, because the smallest specimenwidth is not negligible compared to other dimensions,see Fig. 1 left. Also, one can argue that a 2D model canbe insufficient, because in the experiment, the hinges(pendulum system) could freely rotate in all directionsenabling also out of plane rotation.

We have modeled the dog-bone specimens of allsizes in the three dimensional version of ATENAprogram using the same material law (fracture-plastic model NLCEM). The study was performed (i)with uniform stiffness distribution and also (ii) with athree-layer material of three different Young’smoduli E .

In the homogeneous case we have used the sameelastic modulus as in the two-dimensional models.There is nearly no difference between the 2-D and3-D model responses. The maximum forces and theσ–ε diagrams are almost identical, see Fig. 5. The onlymarginal difference observed with small specimens wasthat the strains obtained in the core of the neck crosssection was slightly greater than the strains obtainedin the front and back surfaces. This might be a resultof inducing tension in point hinges placed in the bot-tom and top loading platens that were not infinitelystiff. The reason for using the point hinges was to allowrotations of the platens in all directions. The diagramsof the medium sized and large specimens did not dif-fer in the pre-peak branches, see Fig. 5. Large speci-mens in 3D showed different shapes of the post-peakbranches obtained in the front and back faces. The

A C F

2D

redye

0

1

2

3

0 0.1 0.2 0.3

]aPM[

x3

0 0.1 0.2 0.3 0 0.1 0.2 0.3

x3

x3

Fig. 5 Comparison of σ–ε diagrams obtained with 2D and 3Dmodels

reason is that numerically the symmetry of the modelwas lost and the specimen started to flex at the onset ofstrain localization.

As for the nonhomogeneous case we have used thevalues and followed the concept of a previous study byvan Vliet and van Mier (1999). It has been speculatedalso by Vorechovský (2007) that the effect of out-of-plane flexure might be an additional reason for averagestrength drop noticeable with small specimens. Onemight expect that in the smallest specimens the crackfront is not initiated over the whole specimen width.Rather, due to stiffness inhomogeneity induced by thecasting procedure (causing, in fact, internal eccentric-ity) the crack might initiate from the front face andthe specimen flex out of the 2-D modeling plane. Theauthors of experiments have reported that due to thecasting of the specimens, the front layers have dif-ferent material properties than the back layers. VanVliet and van Mier (1999) have shown that the nominalstrength drop for the smallest size can nearly entirelybe explained by strain/stress gradients that can developdue to the specimen’s shape, eccentricity of the externalload, material inhomogeneity and eigen-stresses dueto differential shrinkage. They performed a thoroughstudy using a linear model in which they considerednormal stresses due to (i) tension (with a stress concen-tration factor corresponding to the dog-bone shape), (ii)bending moment due to the in-plane eccentricity and(iii) the out-of-plane bending moment caused by dif-ferent stiffness in the casting and mould sides. Theyshowed that most of the observed size effect could beexplained with such a model.

We have modified the homogeneous 3D model bydividing the width of 100 mm into three layers of differ-ent thicknesses depending on the manufacturing pro-cess (see Fig. 6 right). The weighted average of the threemoduli was equal to the modulus used in the homoge-neous case. The three values 35.13, 30.59 and 24.93GPa were set such that their ratios are equal to theratios used by van Vliet and van Mier (1999). Their

123


0.05

0.1

0.15

0.2

0.25

1 2 3 4 1

01niartS

( x3

1 23

4 1

eF,u

Fig. 6 Left: Strain distribution along the edges of the smallestcross section of an eccentrically loaded inhomogeneous speci-men. Comparison of our ATENA results with analyses by vanVliet and van Mier (1999). Right: Three-layered inhomogeneousmodel in ATENA 3D software

reasoning was as follows: the front face is less stiff andthis causes an internal eccentricity. Therefore, micro-cracking starts to initiate from the front face leading toa further decrease of local stiffness and increasing themicro-cracking even more. However, our computationswith three dimensional models employing nonlinearmaterial law did not support this idea. For large sizesthe layered material stiffness makes no difference dueto negligible specimen width compared to other dimen-sions. In small size specimens the response is relativelyductile, see Figs. 2 and 5. Even though the cracking didinitiate from the front surface rather than from the backsurface (see Fig. 1a) the inelastic response of materialpoints is nearly perfectly elastic-plastic and the overallbehavior is also ideally elastic-plastic.

The greatest difference predicted by the models wasobtained with medium sized specimens C where theresponse is between the brittle and plastic limits and the100 mm width is still comparable to other specimensdimensions. For specimen size C we present the com-puted strain profile in the linear pre-peak phase alongthe edges of the smallest cross-section (neck). Figure 6compares these computations with previously obtainedresults of van Vliet and van Mier (1999). These compu-tations confirm that the strain is far from uniformly dis-tributed over the cross section and microcracking doesnot initiate simultaneously (even if the material washomogeneous). This fact supports the conclusion of vanVliet and van Mier (1999) that the nominal strengthσN increases from A to C and then decreases as thesize goes to F and that this can be partly explainedby the effect of strain gradients. On the other hand,when the effect of nonlinear material response is takeninto account, one must consider that strain gradientsobtained in linear analysis do not hold in fully micro-cracked small specimens. Fig. 17 in (van Vliet and van

Mier 1999) shows that linear analysis with strain gra-dients qualitatively overestimates the strength drop ofsmall specimens. Our linear elastic strain profile pre-sented in Fig. 6 has slightly less peaking thus suggestingthat there is a strong sensitivity of the peak strain onthe way the stiffness is distributed. The conclusion isthat possible strength drop obtained with the smallestspecimens might still be explained partly by a strongnon-homogeneity (presence of aggregates—grains upto 8 mm size) combined with out-of-plane eccentricitydue to casting procedure, but the homogeneous modelbased on cohesive stresses (perfectly plastic glue in thelimit) is not able to reproduce it.

Another possible cause was described and numer-ically studied in the preceding paper by Vorechovský(2007). The hypothesis was that the smallest specimensuffers the most from having a surface layer of a mate-rial with lower stiffness and strength. This can explainthe strength decrease in small specimens. A parametricstudy with varied layer thickness and material strengthreduction in that layer is presented in that paper, Sect. 4.The deterministic size effect studied until this point wasautomatically included in the weak layer computationbecause the same material model and parameters wereused. However, the most important effect of strengthreduction for large specimens cannot be modeled bydeterministic effects studied so far. Neither is the modelable to represent the strength scatter because random-ness has not been considered in the model yet. Section 4is focused on modeling local material strength param-eters as a random property. Before proceeding to thoseresults, it is important to see how variations in ft ofNLCEM model [K1 in microplane model] influencesthe response of a single finite element.

3.4 Effect of varied ft and K1 at a finite element level

Let us now study what happens if ft is randomized onlyin NLCEM model of one finite element under uniaxialtension. Figure 7 right studies such a situation, wherewe have multiplied the original tensile strength bys = 1/2, 1 and 2. Since the fracture energy is notscaled, the initial softening slope of σ–ε diagramdepends on the peak stress ft to keep the same areaunder the curve. We can write that if ft ∝ s thenGF = const. Therefore, if s > 1 the same elementbecomes stronger but ‘more brittle’ and dissipates theoriginal amount of energy. The initial softening slope

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0

0.5

1

1.5

2

0 2 4 6 8 10 12 14

Microplane

0 100 200 300 400 500 600

(s f× K1NLCEM

s fK1× ( × ) × ( × )s ft

(s f× t

s

s

s

s

s

s

Fig. 7 Effect of varying strength K1 [ ft ] in microplane (left)[NLCEM (right)] models, respectively, on uniaxial tensileresponse of a single finite element

is in perfect negative dependence on ft , while GF isindependent of ft . Therefore, the size effect curve ofthe whole structure in Fig. 2 bottom right (or Fig. 4)computed with NLCEM model cannot be just movedup and down by changing ft only (GF is not propor-tional to ft). One would have to multiply GF by s2 toachieve it because the characteristic length �ch ∝ s−2.

A somewhat different situation is when strengthparameter K1 is varied in the microplane model. Fig. 7left shows results when a single element has K1 mul-tiplied by s = 1/2, 1 and 2. The tensile strength ofone element scales linearly with s, but the whole σ–ε

is scaled radially, keeping the instantaneous softeningslopes equal at corresponding loading stages. In otherwords, if ft ∝ s then GF ∝ s2 and the characteristiclength �ch = const. This can be viewed as a perfectpositive dependence between ft and GF.

Note that, there can be imagined another alternativein which, with ft ∝ s the energy GF ∝ s. The soften-ing curve would have to decrease towards an identicalstrain value irrespective the peak stress ft . The char-acteristic length �ch ∝ s−1. This would also imply aperfect positive dependence between ft and GF.

These illustrations are important when randomizingboth peak stress and fracture energy simultaneously.The most frequent combination in academic studiesis the simultaneous randomization of GF and tensilestrength ft . For example, it was shown previously byVorechovský (2004b), Vorechovský and Novák (2004)that a strong positive correlation between these twoparameters, when they are both randomly varying spa-tially, increases the slope of size effect curve in thetransitional region between the two asymptotic limits(positive correlation, in fact, speeds up the convergencetowards the classical Weibull statistical size effect).

4 The stochastic model

We believe that the strong size effect on strength in theexperimental data is predominantly caused by the spa-tial variability/randomness of local material strength.Therefore, in the previous study (Vorechovský 2007),we considered the strength related parameter in the mi-croplane model denoted K1 in ATENA to be random,and performed Monte Carlo type simulations for eachsize of specimen. The same strategy was performedhere also with the NLCEM fracture-plastic materialmodel in which we randomized the tensile strengthft . In particular, we sampled 64 random field realiza-tions of the parameter K1 [ ft] for each size and com-puted the responses (complete σ–�u diagrams, stressfields, crack patterns, etc.). The reason for sampling thelocal material strength by a random field instead of byindependent random variables is that we believe thatin reality the strength of any two close locations mustbe strongly related (correlated) and that such a rela-tionship can be suitably modeled by an autocorrelatedrandom field, see Fig. 8 right.

The distribution of local tensile strength at eachmaterial point is assumed to be identical and Weibulldistributed, see Fig. 8 top-left. The reason for selectingWeibull distribution is that the strength of large sizedstructures obeys this form of extreme value distribution.It can be argued that small to medium structures haveGaussian strength distribution with Weibullian left tail(Bažant and Pang 2007). A simple argument to supportsuch a distribution comes from the fact that a random

0

800 0

1200

80.1.21.62.02.4

K1( )10

x [mm]y [mm]

-5x

0

1

0 50 100 150 200

0 1 2 3

0

1

0 1

d [mm]

l

noitale rrocotuA

R

r

d

FDP

2

0

1

0 1 2 3-4

3.2

f [MPa]t

x

FDP

)(

f t

lr

f [MPa]t

Fig. 8 Top-left: Weibull probability distribution function of therandomized parameters K1 [ ft ] (Eq. 5) compared to Gaussiandistribution with equal mean and variance (dashed line). Bot-tom-left: Autocorrelation function (Eq. 6). Right: Realization ofa Weibull random field of K1 compared with dog-bone speci-mens type A – E. The dashed lines correspond to the mean andmean ± one standard deviation of K1 [ ft ]

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strength of Daniels’s fiber bundle model has this form(see e.g. Vorechovský and Chudoba 2006, Bažant andPang 2007 for more details on the distribution). Morefundamental arguments are presented in another paperin this issue (Pang et al. 2009). However, the differencebetween the Gaussian and Weibullian core with equalmean value and variance (see Fig. 8) has a negligibleeffect on the first few statistical moments of a randomstrength of small to medium structures. More impor-tantly, the strength of large structures depends solely onthe Weibull left tail, which is modeled correctly whenassuming the whole distribution Weibullian.

The local probability of failure pf (cumulative Wei-bull distribution function Fσ ) depending on stress levelσ reads:

pf = Fσ (σ ) = 1 − exp

[−

(σ

σ0

)m](5)

where σ0 = scale parameter of Weibull distribution, avalue of 1.6621·10−4 MPa is used for K1 and 3.4 MPafor ft; m = shape parameter of Weibull distribution(dimensionless, depends solely on cov = coefficient ofvariation), m = 7.91 being used for random K1 and ft

in the two parallel studies.To obtain results consistent with the previous deter-

ministic analyses, we kept the value of parameters ft

[K1] as the mean values, i.e. 3.2 MPa [1.5644 · 10−4].The second parameter of Weibull distribution has beenset with regard to the cov of the nominal strength ofthe smallest specimen A (in experiments the cov ofstrengths of size A was 0.16). This choice is supportedby the fact that size A has the largest sample size (10replications, see Table 1). Therefore, the estimation ofvariance has a higher statistical significance than forother sizes. We will explain later on why this choice isnot quite correct. For simplicity, we used the value ofcov = 0.15 (15% variability of local material strength).This is a relatively high value implying the unusuallylow Weibull modulus mentioned above. Note that adifferent choice of Weibull modulus based e.g. on thescatter of nominal strengths for size C would lead to agreater m (≈ 16, see the rightmost column of Table 1)and therefore less scattered results (cov ≈ 0.08) anda milder slope in the asymptotic size effect curve forD → ∞. On the other hand, the scatter of experimen-tally obtained peak forces is much higher for size Asuggesting that there was a strong influence of addi-tional imperfections in shape, geometry and boundary

conditions (eccentricity, etc). As will be seen later, theasymptotic slope of the mean size effect −2/7.91 doesnot equal the value of −2/12 suggested by averagesof all sizes except size A (and used in a simple Wei-bull slope fit by van Vliet and van Mier (2000a,b)). Theissue of the correct choice of statistical scatter of thematerial strength is further discussed in Sect. 5.

A discretized random field can be viewed as a set of(auto)correlated random variables. The most importantparameter (in a given form of autocorrelation function)is the autocorrelation length controlling the distanceover which the random material strengths are corre-lated. We used the squared exponential autocorrelationfunction (Fig. 8 bottom-left):

R = exp

[−

(d

lr

)2]

(6)

where d = distance between two points; lr = correla-tion length, a value of 80 mm used for a random fieldsof K1 [ ft].

The correlation length lr is here assumed to be amaterial (and possibly structural) constant related toboth the microstructure (grain size and defect distri-bution and their frequency, i.e. on their distance fromeach other), and also on the production technique (com-pacting, etc.). The autocorrelation function takes valuesclose to unity for any two close points in the specimen(unit correlation is the upper limit for two coincidingpoints). For a pair of remote points the autocorrelationdecays to zero implying no statistical correlation forthe material properties of those two points. It can beshown that for specimens much smaller than one auto-correlation length, the realization of a random field ofthe local strength K1 is a constant function over thewhole region (see Figs. 8 right and 9), and all localstrengths of the whole specimen can be representedby just one random variable (instead of a number ofspatially correlated variables). Since the specimen’snominal strength is just a simple transformation of theinput strength parameter K1 [ ft] (no spatial variability,allowing cracks to localize in other locations than indeterministic analysis), we knew that the mean nomi-nal strength of the smallest specimen will be the sameas that obtained by deterministic analysis. That is whywe used the K1 [ ft] from deterministic analysis as themean value of the random field of K1 [ ft]. We setthe correlation length lr such that the computed sizeeffect curve ’bends’ between a constant and Weibull

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A 60

50[mm] 75

[mm]

0 1 2 3 4 5

[MPa]B 10

100150

0 1 2 3 4

C 22

200300

0 1 2 3 4

D 1

400600

0 1 2 3 4

E 45

8001 200

0 1 2 3 4

F 5

1 6002 400

0 1 2 3 4

Fig. 9 Stress/strength fields corresponding to the peak load forselected realizations and specimen sizes. Results are computedwith randomized NLCEM material model. Fields from top: ran-dom strength field (threshold), principal stress of a brittle material

scaled to correspond to the peak load (nominal strength), actualprincipal stress at peak load, cracking strain at the bottom plane.See also selected realizations in Fig. 10

asymptotes at a crossover specimen size C, which bet-ter fits the experiments, see also Sect. 4.1.

The samples of random fields evaluated at the loca-tions of integration points were simulated by methodsdescribed in Vorechovský (2008, 2004b). In themethod, the support of the field is discretized (nodesof the random field mesh may coincide directly withthe integration points of the FEM mesh). Based onthe discretization and a given autocorrelation function(Eq. 6) an autocorrelation matrix C is assembled. Sucha matrix is symmetric and positive definite and hasorthogonal eigenvectors � and associated eigenvalues� such that C = ��T . The (discretized) Gaussianrandom field X is expanded using a Gaussian randomvector ξ and the computed eigenmodes asX = � (�)1/2 ξ . If non-Gaussian fields are to be sim-ulated, the Nataf model is usually employed (Liu andDer Kiureghian 1986). The simulated random fieldsare stationary, isotropic and homogeneous. Briefly, thedescribed orthogonal transformation of the covariancematrix has been used in combination with Latin Hyper-cube Sampling of the random part of field expansion(Novák et al. 2000). Such a combination proved itself tobe very effective in providing samples of random fieldsleading to high accuracy in estimated response statisticscompared to classical Monte Carlo sampling. Numer-ical studies documenting this efficiency are published

in (Vorechovský 2008). This is an extremely importantproperty in cases when the evaluation of each responseis very time consuming. In our case the evaluation isrepresented by one computation of response by the non-linear finite element method with the microplane orNLCEM material model inside. Obviously, this is veryexpensive and we must keep the number of simulationsas low as possible. The number of 64 simulations wastested to be high enough and to provide stable and accu-rate statistical estimates of fields’ statistics (averages,sample standard deviations, autocorrelation structure)as well as reproducible estimates of structural responsestatistics (nominal strength etc.).

The automatic simulation of all structural responseswas done by SARA software integrating (i) ATENAsoftware (evaluation of response) and (ii) FREET soft-ware (Vorechovský 2004b, Novák et al. 2003b, 2006)(simulation of samples of random parameters, statisti-cal assessment).

In Fig. 10 we plot selected realizations of the randomstrength field in NLCEM model for all sizes A – F,some of which are better visualized in Fig. 9. Similarly,Fig. 11 presents selected plots for randomized micro-plane model. We note that a similar scaling rule as inEq. 2 can be written for the role of statistical length(here in the form of autocorrelation length lr ). For agiven random strength field (statistical distribution and

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Fig. 10 Simulated randomstrength field realizationsand corresponding crackpatterns in deformedspecimens right afterattaining the maximumforce Fmax. Fields weresimulated and crack widthswere computed at theintegration points of finiteelements using the NLCEMmaterial model

Fig. 11 Selected crackpatterns from the series withrandomized microplanemodels. Intended for directcomparison with Fig. 10

autocorrelation structure) only the dimensionless pro-portion D/ lr matters (recall the dimensional analysis):

for ∀s > 0 : σN (D, lr ) = σN (s D, s lr ) (7)

Again, this can be used to simplify modeling becauseone size can be used with varying lr instead of varyingD with a constant lr . Similarly to Eq. 2, this property

illustrates the scaling properties with lr standing for aprobabilistic (or statistical) scaling length. Similarly aswith the deterministic size effect caused by stress redis-tribution in FPZ, the probabilistic size effect curve rep-resents a transition between two asymptotes (horizontalfor D → 0 and an inclined straight line for D → ∞).The transition happens around a cross-over size ls

123


(discussed below in Sect. 4.1), i.e. when the non-dimen-sional size D/ ls takes values approximately between0.1 and 10.

It can be seen that as the ratio of autocorrelationlength and specimen size D decreases, the rate of spatialfluctuation of random field realizations grows. There-fore, there are an increasing number of locations withlow material strength (locations prone to failure). Or,in other words, with increasing specimen size there isan increased probability that there will be a weak spotin highly stressed regions. This effect has long beenreferred to as the statistical size effect. The classicalstatistical size effect is modeled by the simple weakestlink model and is usually approximated by the Wei-bull power law (Weibull 1939). However, as explainedin (Vorechovský 2004b,a, Vorechovský and Chudoba2006), the classical Weibull model is not able to accountfor spatial correlation between local material strengths.Rather, the Weibull model is based on IID (independentand identically distributed) random variables linked inseries. The effect of such a consideration is that thestrength of an infinitely small specimen is infinite. Inthe Weibull model every structure is equivalent to achain under uniaxial tension, a chain of independentmembers having an identical statistical distribution ofstress. If the local strength is modeled by an autocor-related random field (and we consider the autocorre-lation length to be a material property), the small sizeasymptote of strength is equivalent to the distributionof local material strength. On the other hand, the largesize asymptote is exactly identical to that of the Wei-bull model (for a proper choice of reference length andthe corresponding scale parameter of Weibull distribu-tion in the Weibull model). The autocorrelation lengthplays an important role as a statistical scaling length ina material controlling the transition from a one strengthrandom variable model (full correlation in small struc-tures) to many independent local strengths (large struc-tures, Weibull model); see (Vorechovský 2004b) fordetails.

In Fig. 12 we plot computed sets of ‘nominal stress–strain’ (σ–ε) diagrams obtained with the NLCEMmodel and sketch the definition of strain (the separationof two measuring points �u over the control length).The corresponding diagrams obtained with microplanelaw were published in (Vorechovský 2007). In there,several selected load displacement curves were high-lighted and the corresponding realizations of randomstrength fields of microplane K1 parameter together

0

1

2

3

4

]aPM[

0

1

2

3

0 0.1 0.2

]aPM[

3

0 0.1 0.2 0 0.1 0.2

u low

upp

x3

x3

x

A B C

D E Fu

Fig. 12 Nominal stress-strain diagrams (64 realizations)obtained with randomized NLCEM material model

with crack patterns were presented. In the same paper,besides the most frequent simple σ–�u functions, wehave purposely highlighted several curves with unusualshapes (snap-back type or “a loop”). When testingconcrete structures in routine practice such specialshapes can only occasionally be experimentally mea-sured. They would indicate that the control length wasnot properly designed (with respect to the specimenshape and material strength variability) and that local-ized strains occur outside the control length. As dis-cussed there, some of the unusual or unexpected curveswere obtained partly due to the definition of displace-ment �u, and mainly due to the spatial randomnesswith (too) high variability. In the analyses with NLCEMmaterial law, these loops nearly did not occur. For com-parison purposes of the peak strength of the determin-istic σ–ε diagrams with the mean values of nominalstrength are added into Fig. 12. The difference betweenthem grows with specimen size. While for size C themean strength still nearly coincides with the peak of thedeterministic diagram, for specimen size F the deter-ministic curve is above all 64 random realizations of thediagram, see Fig. 12. This feature is related to the tran-sition from the central limit theorem applicable to smallsizes (nominal strength is a result of a sum of randomstrength of many links arranged in parallel) to extremesof independent variables (the smallest local strengthcompared to stress dictate the structural strength).

The crack patterns of two randomly chosen speci-mens A 60 and B 10 (see Fig. 10) show the most fre-quent location of strain localization. Fig. 9 shows howthe maximum principal stress field would look at thepeak load if no redistribution takes place and whenthe stress could exceed the local strength. This figurealso shows the actual (redistributed) stress field that

123


can be described as a ‘deformable ball pressed towardssealing (strength realization) from bottom’. Samplesof random fields in both cases (A, B) are nearly con-stant functions and therefore there is no space left forthe weakest link principle. The small eccentricity ofload and relatively narrow neck of dog bone specimensguarantee that cracking will initiate on the right side ofthe neck. Pattern C 22 in the same figure documentsthat the local strength can be, in some locations, sosmall that the relatively low stresses in that locationcan initiate fracturing. In specimen C 22 the rotationof platens was opposite to the usual direction. Since thedamage localized out of the distance (control length)over which we measured the separation �u, the corre-sponding σ–ε has the snap-back-like shape. The factthat the specimen breaks in the relatively low stressedregion is associated with the relatively high variabilityof local material strength. Simply, the realization of thestrength field in the cracked region was the closest to theprincipal tensile stress profile, see Fig. 9. If a differentstrength distribution was chosen (especially lower var-iability), the occurrence of fracturing outside the neckarea would be suppressed. The selection of materialparameter variability was in this numerical exampletoo high as will be discussed in Sect. 5. Pattern C 51 isalso quite unusual. Similar features can occasionally befound in series D. We illustrate the random samplingof crack initiation in the same figure. In series F theautocorrelation length becomes so small compared tospecimen dimension that again cracks mostly initiateon the right side of the neck in nearly all cases, seeFig. 9. The specimen F 5 in Fig. 9 illustrates the casewhen the strength realization is hit by the stress fieldat two points simultaneously. In such cases, two majorparallel cracks can develop at the peak load even in arelatively large structure with small amount of stressredistribution prior reaching the peak load.

In series A, we never reported a snap-back-like curvedue to cracking outside the measuring distance, becausethe random field is nearly a constant function over thespecimen. We can conclude that the most interestingprocesses happen in specimens with a dimension com-parable to one or two correlation lengths (region oftransition from one random strength variable to a set ofindependent strength variables).

From comparisons in Sect. 3.4 it transpires thatcomparisons between results with randomized ft in

NLCEM and K1 in microplane models cannot be donedirectly. Rather, it was expected to shed some lighton the effect of having a perfect positive dependencebetween material peak stress and fracture energy. It canbe seen that the post-peak curves in the bundles of dia-grams in Fig. 12 computed with NLCEM are steeperwhen they reach a higher peak (and opposite). On thecontrary, the softening slopes of diagrams in each bun-dle obtained with the microplane model (Fig. 4 in(Vorechovský 2007)) have the same post-peak slope,if the snap back did not occur.

A question appears: which of the three alternativesfor randomization of tensile strength, fracture energyand characteristic length described in Sect. 3.4 is morerealistic for real concrete? We do not give an answer,because the two compared material models behave dif-ferently clouding the picture. One would have to per-form simulations with a single material model in whichthe parameters are varied simultaneously according tothe three alternatives.

It might be interesting to compare the crack patternsobtained with microplane model and NLCEM modelwhen the realization of a random field of K1 [ ft] isidentical (one is just a multiple of the other). We haveselected five crack patterns from our previous study(Vorechovský 2007) and plot them here for compari-son. An interested reader can find the same size andnumber in Figs. 10 and 11. One can see that NLCEMmodel predicts much more localized cracks comparedto the microplane model. In D03, for example, micro-plane model predicts quite diffused cracking far fromthe neck while NLCEM model just predicts somemicrocracking there and the final magistral crack passesthrough the neck. The relatively diffused cracking pre-dicted by microplane model corresponds to milderslopes of the pre-peak branches of diagrams in Fig. 2.We note that the first guess might be different: when aweak element starts to soften in NLCEM, it is an ele-ment with a milder softening slope. This would supportsomewhat tougher behavior of NLCEM models whichis not found here.

Finally we note that, in contrast to the experiments,we did not control loading by displacement increments�u. Instead, we loaded the specimens by displace-ment at the ends, and therefore we were able to mon-itor the snap-back type of curves without anydifficulty.

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4.1 The Weibull integral, extremes of random fields,reference and representative volumes

We were able to simulate the random responses of spec-imens smaller than A with random fields of K1, andmoreover we could simply use random variable sam-pling to represent randomness in the small specimens(each realization becomes a random constant functionover the specimen). On the other hand, it becomes veryexpensive to simulate samples of random fields of spec-imens much larger than F due to the need of a densediscretization. Approaches already exist to overcomethe computational difficulties with the stochastic finiteelement computation of large structures (Vorechovskýet al. 2006) but we will present another technique here.Fortunately, only strength is random in our analysis andwe can use the classical Weibull integral for large struc-tures. As explained in (Vorechovský 2004b,a, Vorecho-vský and Chudoba 2006), if the structure is sufficientlylarge, the spatial correlation of local strengths becomesunimportant and the Weibull integral yields a solutionequivalent to a full stochastic finite element simula-tion. We will briefly sketch the computational proce-dure of evaluating the Weibull integral for structuralfailure probability: details can be found e.g. in (Bažantand Planas 1998). The Weibull integral has the form:

− ln(1 − Pf) =∫V

c [σ (x) ; m, σ0] dV (x) (8)

where Pf = probability (the cumulative probabilitydensity) of failure load of the structure; c [•] = stressconcentration function.

There are several possible definitions of the stressconcentration function, see Bažant and Planas (1998).In the studied specimens the major contributor to thestress tensor is the normal stress σyy . The field of stressσyy nearly coincides with the principal tensionσI . Sinceonly tensile stresses are assumed to cause a failure, wedefined the stress concentration function simply as:

c [σ (x) ; m, σ0] = 1

V0

⟨σI (x)

σ0

⟩m

(9)

where V0 = ln0 = reference volume associated with m

and σ0.In Fig. 1b, we plot the computed field of the maxi-

mum principal stress (tension) over a specimen in anelastic stress state. Numerical integration of this stressfield for different specimen sizes and failure probabili-ties can be suitably rewritten in dimensionless coordi-nates so that the computation becomes extremely easy.

In particular, let ξ = x/D, consider unit thicknessb and set the maximum elastic principal stress fieldσI (x) = σN S (ξ) where σN is the nominal stress andS (ξ) the dimensionless stress distribution independentof D. If, in accordance with Bažant et al. (2007a), wesubstitute these and dV (x) = Dn dV (ξ) into Eq. 8, weget − ln(1 − Pf) = (σN/σ0)

m Neq or

Pf(σN) = 1 − exp

[−Neq

(σN

σ0

)m](10)

where the equivalent number of identical links in achain

Neq =(

D

l0

)n

� (11)

depends on a geometry parameter

� =∫V

Sm(ξ) dV (ξ) (12)

This geometry parameter characterizes the dimensionalstress field that depends only on the structure geome-try and boundary conditions. As defined in a recentwork by Bažant et al. (2007a), Neq can be interpretedas the equivalent number of equally stressed materialelements of a size for which the reference material sta-tistical properties has been measured. At this place,we mention that asymptotically, the structure becomesa chain of Neq equally stressed RVEs in a series, seeFig. 13 right. Note that the number of the chain elementswith a random strength is proportional to the scalingdimension n (two in our case). It is an occasional prac-tise to place a fiber bundle model (FBM) inside a regionin which the crack is expected to be propagating, seeFig. 13. Note however, that for the purpose of asymp-totic strength prediction, this approach is inadequate. It

l 0

oneRVE

N Deq

F

2

f

NF

FF

l 0

ls0

s l0l0l0

Fig. 13 Illustration of a random strength representation. Refer-ence size and s-times scaled structure. Left: incorrect scaling ofstrength using FBM. Right: concept of RVEs in a solution usingWeibull integral

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is known that increasing the number of parallel fibersnf (or microbonds) in the FBM asymptotically doesnot decrease the mean strength per fiber (although thevariance decreases inversely proportionally to nf , seee.g. Vorechovský and Chudoba 2006). So, scaling thestructure size s times brings about only length effect,which is a one-dimensional effect, not two-dimensionaleffect as it should be (Neq ∝ D2). The prediction ofan asymptotic random strength distribution would thanbe incorrect, see (Vorechovský and Chudoba 2006) formore details.

The present derivation is fully complying with therecent work by Bažant et al. (2007a) in which the repre-sentative volume element (RVE) is defined as the small-est material volume whose failure causes the failure ofthe whole structure (this definition is valid for positivegeometry structures, i.e. structures that fail, under loadcontrol, as soon as the first RVE fails). The conceptof equivalent number Neq of equally stressed RVEs isintroduced to simplify the problem in cases when thestress state is not uniform and the actual number identi-cal volume elements are subjected to different stresses.In both cases, the probabilities of failure of the structurePf are identical.

Consider now a case when all the RVEs are indepen-dent of each other (as is the case of asymptotically largestructures). Since the structure survives as a whole ifand only if all the RVEs survive, one can write the sur-vival probability as a product 1− Pf = (1 − P1)

Neq or,equivalently (Bažant and Pang 2007):

Pf(σN) = 1 − [1 − P1(σN)]Neq (13)

This failure probability tend to

limNeq→∞ Pf(σN) = 1 − exp

[−Neq P1(σN)]

(14)

where P1(σN) is the cumulative distribution functionof the strength of one RVE. Clearly, as Neq → ∞,the structural strength distribution converges to Wei-bull form.

Using Eq. 10 we can easily relate the parameters ofa random material strength to the mean value of thestructural strength:

σN = σ0

N 1/meq

(1 + 1/m) = µ0

N 1/meq

(15)

where (·) is the Gamma function. The materialstrength is represented by the parameters of a randomRVE strength considered to follow Weibull distributionor a Gaussian distribution with a Weibull tail described

by the shape parameter m and corresponding scaleparameter σ0, a pair yielding the mean strength valueµ0 of one RVE of size l0.

In the particular case of studied dog-bone speci-mens, the Weibull solution gives the following results.First, the thickness b = 100 mm is not scaled and there-fore it does not contribute to the strength scaling. There-fore we ignore the thickness and define volumes asareas. When m = 7.91, as studied before, the geometryparameter defined in Eq. 12 can be computed to equal� ≈ 0.574. If one selects the length l0 to equal theautocorrelation length lr = 0.08 m (see below for rea-sons), each RVE has the mean strength µ0 = 3.2 MPaand scale parameter σ0 = 3.4 MPa. The number ofequivalent RVEs in specimens of various sizes can becalculated using Eq. 11, for size F the formula givesNeq ≈ 230. Therefore, the average nominal strength ofsize F is approximately 1.61 MPa. The resulting meansize effect for other sizes is plotted in Fig. 14 (asymp-totic mean size effect curve). Let us also mention thatanother way of simulating the random strength of largestructures can be done utilizing the stability postulateof extreme values (Fisher and Tippett 1928). Such acomputational procedure is an elegant trick using therecursive property of the distribution function and isdescribed in Bažant et al. (2007b), Novák et al. (2003a),Vorechovský (2004b) together with applications. Theresults of such an approach (and also the Weibull inte-gral as presented here) are valid only for extremelylarge sizes where the effects of structural nonlinear-ity (causing stress redistribution) disappear. There isno characteristic material length in the classical (local)Weibull theory, because the Weibull size effect is self-similar—a power law with no characteristic length andno upper bound. Rather, lr (or V0) in Weibull theory issimply a chosen unit of measurement to which the spa-tial density of failure probability is referred. For smallsizes there are two problems: (i) the spatial correlationof local strengths and (ii) the effect of stress redistri-bution. The result of the Weibull integral must be astraight line in a double logarithmic plot of size ver-sus strength (the size effect plot is a power law). Thesetwo issues will be discussed next. Note that there existalso a nonlocal alternative to the Weibull integral thatis commented on and compared with the classical localWeibull integral in Sect. 5.2 of (Vorechovský 2007).

Because the statistical and energetic physical causesof size effect are different and independent, the sta-tistical length lr cannot be affected by changes in the

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deterministic length cb (or similarly changes in GF).The mean value of a random nominal strength σN mustbe bounded when D → 0, (i.e., the statistical size effectcannot cause a strength increase when the structure istoo small as in the classical Weibull theory). The upperbound on the mean statistical size effect can be eas-ily calculated as the deterministic strength of a struc-ture with no stress redistribution allowed (cb/D → 0or GF/D → 0), see the bottom horizontal asymp-tote in Fig. 14. The same bound is also obtained as themean value of the distribution of extremes (minima)of random fields representing local material strength(Vorechovský 2004b, Vorechovský and Chudoba 2006).

To study the statistical size effect of structures withno redistribution, one has to select the size of the RVE inthe case when a random material strength is describedby random fields. Note that in the case of uncorre-lated Weibull strengths the choice is arbitrary; the ref-erence length is related to the strength scale parameterthrough a power law. In the autocorrelated case thechoice depends on the autocorrelation length — thelength l0 must be equal to a length over which the localstrengths are nearly uncorrelated. Therefore, we con-sider the equality between the autocorrelation length(Eq. 6) and the length l0 from here on:

lr = l0 (16)

An area A0 = l20 or a volume V0 = l3

0 has now themean strength of µ0.

To show the difference between the statistical sizeeffect in the Weibull sense and when autocorrelatedstrength is assumed, one must isolate the statisticaleffects from the deterministic effects. The pure sta-tistical size effect (i.e. the size effect of the structurewith no stress redistribution) can be numerically sim-ulated by replacing the crack band or fracture energywith zero and using the same realizations of a randomstrength fields. Numerical results are plotted in Fig. 14using a line with solid boxes and errorbars. One can seethat the calculated mean size effect curve is a smoothtransition between two asymptotic cases: the constantupper bound for small sizes and the Weibull asymptotefor large sizes. The cross-over size ls can be calcu-lated from the equality of deterministic strength of alarge structure σN,∞ ≡ σ det

N (∞, cb) ≡ σ detN (D, 0) =

2.15 MPa and the mean Weibull strength of 3.2 MPa inEq. 15. This equality yields

ls = l0 �−1/n[

µ0

σN,∞

]m/n

(17)

1.3

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.03.23.43.6

10 100 1000 10000

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.6

]aPM[

N

m

±

A B C D E F

asymptote

experiments ±

m 7.91

G H I J

7.91

2

12

2

D

Z

lsDb

N,

N,0

Fig. 14 Comparison of results in a size effect plot

123


which gives, in our numerical example ls ≈ 510 mm,see the abscissa scale in Fig. 14. The transition canbe approximated using the formulas proposed earlierby Vorechovský (2004b), Vorechovský and Chudoba(2006) as approximations to extremes of random pro-cesses:

σN (D) = σN,∞(

D

ls+ ls

ls + D

)−n/m

(18)

or

σN (D) = σN,∞(

lsls + D

)n/m

(19)

The numerically obtained mean of statistical solutionlies in between these two approximations. These for-mulas represent extension of extremes of stationaryand ergodic Weibull random processes (Vorechovský2004a,b, Vorechovský and Chudoba 2006) to n-dimen-sional random fields. The formulas were originallyapplied to predictions of the strength of thin fibers(Vorechovský and Chudoba 2006) and later in thecontext of the energetic-statistical size effect in quasi-brittle structures failing after crack initiation, see(Vorechovský et al. 2005, Bažant et al. 2007b).

5 Analysis of the results

By introducing three different scaling lengths we areable to independently incorporate three different effectsin the model resulting in three size effects on nominalstrength. The crack band width cb (deterministic scal-ing length, linearly related to the fracture energy GF)controls at which size the transition from ductile to elas-tic-brittle microplane model behavior takes place, andtherefore it controls the transition between two hor-izontal asymptotes in the size effect plot (see Fig. 2bottom right). The second introduced length (weakboundary thickness tw) together with material strengthreduction controls at which size there will be a sig-nificant reduction in nominal strength. The reductionbecomes amplified with decreasing specimen size andcauses an opposite slope of size effect than withthe deterministic and statistical ones (see Fig. 2 of(Vorechovský 2007)). The last introduced length is theautocorrelation length lr controlling the transition fromrandomness caused by overall material strength scatter(one random variable for material strength) to a set ofindependent identically distributed random variables oflocal material strengths via an autocorrelated random

field. In other words, it controls the convergence to theWeibull statistical size effect based on the weakest linkprinciple.

In the case of a random field description of localstrength we can see the following behavior. If the possi-bility to redistribute stresses (nonlinear phenomena) issuppressed (vanishing characteristic length representedby cb or GF compared to specimen size), the Weibullasymptote is reached from bottom, see the solid boxesin Fig. 14. If the deterministic length is comparable tothe statistical one, the transition to Weibull solution isfrom above, because small sizes exceed the nominalstrength σN,∞ (solid circles in the figure).

Numerical simulations show that the autocorrelationlength may, with its role as a measure of the rate of fluc-tuation of local material parameters, significantly influ-ence the damage process and the global response of thestructure. Especially when local strength is varied ran-domly and the relative variability is high, the spread ofdamaged zones will be dependent on both: crack bandsize (or similarly nonlocal length in nonlocal damagemodels) and also on the autocorrelation length/struc-ture. When the correlation length is much larger thanthe specimen, the realizations of a random strengthare nearly constant functions and the damage spreadis governed solely by the deterministic effects (stressfields and concentrations influenced by redistributionsdue to the nonzero nonlocal or crack band length), seeA 60 in Fig. 9. Here, with reference to the discussionon the concepts of nonlocal averaging in Sect. 5.2 of(Vorechovský 2007), we note that the so-called fractureprocess zone size depends on actual stress fields, whichare also influenced by presence of specimen bound-aries, apart from the usually attached physical meaningof the function of the maximum aggregate size in con-crete. All these can be put on the side of deterministiceffects. On the contrary, when the spatial distance ofserious local flaws or defects (the correlation length) isvery small compared to the nonlocal length, the damageprocess will depend on interactions of zones in whichthe damaged material softens locally. The zones with ahigh local strength adjacent to the weaker zones act as‘barriers’ for further spreading-out of damage, see e.g.F 5 in Fig. 9. Generally, the size of the damaged zonedepends probably strongly (and nearly linearly) on thedeterministic length and only weakly on the autocor-relation length of local strength field in our continuummodel. If the autocorrelation length is smaller than thedamage zone size in the deterministic case, it probably

123


slightly decreases the damaged zone size in the proba-bilistic case. Both lengths can be understood to rely onthe same physical background—microstructural prop-erties below the continuum level. However, a randomspatial strength variability can also occur at the macro-scopic level due to variable manufacturing process ofconcrete, etc.

The interplay of three independent material/struc-tural lengths as sketched in this paper is very compli-cated. It would be nearly impossible to determine allthese parameters from the available experimental dataeven if the model featuring the three effects was per-fectly correct.

In Fig. 6 of (Vorechovský 2007), the estimated distri-bution function of nominal strength for all tested sizesas we obtained them from the full stochastic finite ele-ment analysis with are plotted. The discussion thereexplains that models of large specimens suffer froman insufficient discretization of the random field withrespect to the autocorrelation length, see also F 5 inFig. 9. This is a clear limitation of the random fieldapproach. We sketch the size regions for different com-putational techniques used for the modeling of randomstrength above the plots in Fig. 14.

The resulting nominal strengths for all sizes obtainedby nonlinear stochastic FEM are plotted and comparedto experiments in Fig. 14. The figure is devoted mostlyto microplane modeling because at this graph theNLCEM modeling yields comparable results (even ifthere were differences in the stress–strain curves andcrack patterns and randomization of ft and K1 are notdirectly comparable). We see that starting from size Cthe dependence of mean nominal strength on size is pre-dominantly statistical, and we were not able to model itby deterministic model alone. We also included meannominal strengths for sizes D, E, F and G obtained byWeibull integral.

The asymptotic slope of statistical size effect sug-gested by the trend of the means of the Weibull solu-tion (−n/m = −2/7.91) is not in a good agreementwith the scatter of measured nominal strengths for sizesgreater than A. If we ignore the smallest size A wherethe variability of strength data is questionable, C is thesize with most of replications (seven). Therefore, thestandard deviation obtained from experiments is mostsignificant for size C. The very last column in Table 1presents the effective Weibull moduli computed fromexperimentally obtained averages and sample standarddeviations. Mostly these m values are greater than that

of C meaning that the estimated scatter is less, whichcorresponds to the fact that there were less realizations(four or five only). The reason for A yielding muchgreater variability (and thus smaller effective m) mustbe explained by several other effects. Numerical sim-ulations with a lattice model performed by van Vliet(2000) revealed that if a large grain is present in the sur-face layer of the specimen, the peak load can decreaseconsiderably. Such a mechanism certainly contributesto the increased statistical scatter of nominal strengthsin specimen type A. Even though the number of realiza-tions for size A was ten, the information on scatter doesnot seem to be relevant. A better choice in this studywould be m ≈ 14 suggested mainly by the slope ofthe size effect curve for the two largest specimen sizesE, F. Note that the steeper slope at medium sizes C, Dcan be attributed to a combination of both deterministicand statistical effects.

The tendency of bundles of diagrams computed withNLCEM (Fig. 12) corresponds better with the experi-mental curves: the greater the size the steeper the post-peak slope. This was not true for the microplanemodels, see Fig. 4 of (Vorechovský 2007). With regardto the above mentioned comparison to randomizationwith a constant �ch (NLCEM) versus constant GF (mi-croplane), the comparison with experimental curvessuggests that keeping the GF constant is somewhatcloser to reality than keeping a constant �ch.

We have also performed a study with NLCEM modelin which we divided the fracture energy GF by 16, thusshifting the deterministic transition to smaller speci-mens. In this way, the deterministic behavior previ-ously obtained for size E becomes behavior of size A.The stress–strain diagrams then correspond much bet-ter with those published by van Vliet (2000), van Vlietand van Mier (2000a). The resulting deterministic sizeeffect curve is just the same curve as before shiftedto the left (recall Eq. 3). The average curve obtainedwith random strength field with the same autocorrela-tion length then unsticks to join the Weibull integralsolution. Based on various parametric studies, it seemsthat a very good fit of the experimental results (stress–strain diagrams, the mean size effect curve shape andscatter at all sizes but A) would be obtained ifGF = 200/16 N/m, the mean material strength µ0 =1.5 × 3.2 = 4.8 MPa and m ≈ 14, a realistic valueof strength scatter for concrete. The horizontal deter-ministic asymptotes would appear higher in Fig. 14,the asymptotic slope of statistical part would be milder

123


and strength scatter would be less than using the pres-ent values. Transitions would happen at smaller sizesthan now.

It seems that the mean size effect as well as thestatistical scatter at the same time could be explainedby a combination of the ‘weak layer’, ‘deterministic-energetic’, ‘strain gradient’ and ‘statistical’ size effectstogether. The very thick curve in Fig. 7 of (Vorecho-vský 2007) (a curve denoted as 3) is the curve resultingfrom the combination of all three effects described here.With NLCEM modeling here, the curve would be verysimilar.

Finally, it can be questioned whether the crack bandwidth is the correct parameter to represent the determin-istic scaling length. Similarly the softening adjustedmodulus (driven by adjusted fracture energy GF) of amaterial point was designed so that a crack band occu-pying one band of an element’s width always dissi-pates the same amount of energy irrespective of theband width. Unfortunately, we are not able to model asituation where the deterministic characteristic lengthis greater than the statistical one (represented by theautocorrelation length in our model). This is becausewe cannot represent a real crack below the level of res-olution of FEM discretization.

Moreover, the crack band model is not suitable whenmore that one crack appears in parallel, because it wasprimarily designed to correctly represent a single crackpassing through a specimen without mesh size depen-dency. In our simulations it sometimes happened thatat the onset of cracking the crack pattern was diffused,and the consumed energy was then probably higherthan what was thought to be correct. Fortunately, local-ization always started soon (in terms of position on theσ–�u diagram) and the peak force we recorded was notinfluenced much. In our constitutive model the crackreturns energy during unloading, and this supports thehypothesis that the virtual error was not high.

Both the aforementioned issues can probably besolved by using a better regularization technique to pre-vent spurious mesh localization; the nonlocal contin-uum model proposed by Pijaudier-Cabot and Bažant(1987). In this model the deterministic length is welldefined by the averaging length la (internal length ofthe nonlocal continuum) over which a certain variableis averaged (based on the weight function α) (Jirásek1998). In our eyes such a model would better representthe effect of interaction of the two lengths: determinis-tic la and statistical lr . Another very promising option

seems to be the cohesive segments method (Remmerset al. 2003) in which the cohesive segments are insertedinto finite elements as discontinuities in the displace-ment field by exploiting the partition-of-unity propertyof the shape functions.

6 Conclusions

We present a combination of nonlinear computationalmechanics tools with a simulation of random fieldsof spatially correlated material properties in a singleplatform as an approach to the modeling of failure inquasibrittle materials. The performed numerical sim-ulations of the random responses of tensile tests withdog-bone specimens with rotating boundary conditionsperformed by van Vliet and van Mier are in good agree-ment with the published data. Based on the comparisonof trends of nominal strength dependency on structuralsize we conclude that the suggested numerical modelfeaturing three scaling lengths is capable of capturingthe most important mechanisms of failure.

In particular, we have shown that a portion of theexperimentally obtained size effect can be capturedat a deterministic level with the help of deterministiclength represented by crack band width in our models(the smeared cracking with the crack band models arecompared for a fracture-plastic and microplane mate-rial models).

Secondly, further strength dependence on size inlarge specimens is modeled by an autocorrelated ran-dom strength field. The important statistical lengthscale is introduced in the form of the autocorrelationlength of the field. It is shown that the inhomogeneityof material properties over the structure in the form ofan autocorrelated random strength field gives rise toimperfections that trigger fracturing in highly stressedregions of a structure. We believe that the statisticaldescription of standard continuum helps to resolve theill-posedness of the continuum model after the onsetof localization, it being a fundamental problem. Theasymptotic size effect form caused by local strengthrandomness is the classical Weibull power law. By ran-dom sampling of the local strength field we were alsoable to model the random scatter of resulting nomi-nal strengths. Also, simple scaling rules, anchored intheoretical dimensional analysis, are suggested.

In the presented model, the complex interplay ofseveral scaling lengths is captured at a time. Numeri-cal simulations of localization phenomena demonstrate

123


that the introduction of the stochastic distribution ofmaterial properties reveal phenomena that would oth-erwise remain unnoticed.

Acknowledgements Authors thank to anonymous reviewersfor their valuable comments that lead to many improvements ofthe paper. This outcome has been achieved with the financialsupport of the Ministry of Education, Youth and Sports, projectNo. MSMT 0021630519. In this undertaking, theoretical resultsgained in the project KJB201720902 from the Grant Agency ofthe Academy of Sciences of the Czech Republic were partiallyexploited.

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Computational modeling of size effects in concrete specimens under