Numerical model for time-dependent fracturing of concrete structures and itsapplications

G. Di Luzio & L. CedolinDepartment of Structural Engineering, Politecnico di Milano, Milano, Italy

ABSTRACT: A constitutive law which incorporates creep and fracture behavior for the analysis of time-dependent cracking of concrete structures is briefly presented. The constitutive model is composed of twoparts acting in series: a Maxwell chain for the creep of uncracked concrete and a recent version of the mi-croplane model (M4), which includes the rate-dependence of fracturing associated with the activation energyof bond ruptures, for the cracking. The proposed formulation is applied to the simulation of the size effect ofsingle notched specimens in direct tension for quasi-static fracture in the time domain with different loadingrates. The numerical results reveal clearly the link between the size of the fracture process zone (FPZ) and therate loading.

1 INTRODUCTION

The load carrying capacity of a concrete structureunder sustained constant load can be considerablysmaller with respect to that of the same structure in-stantaneously loaded. This can be explained by theinteraction between cracking and creep, which maylead to an increase of damage with a reduction ofthe ultimate load capacity. Moreover, it is well-knownthat fracture propagation is rate-sensitive, due to thefact that the rupture of interatomic or intermolecularbonds is a thermally activated process (Bazant 1995).Wu & Bazant (1993) used the activation energy pro-cess of bond ruptures in order to capture the strengthincrease with loading rate in a wide range of quasi-static loading rates.

In the last years a significant progress has beenmade in the modeling of fracture and damage of ce-mentitious materials. To understand and model thecreep rupture of concrete different approaches can befound in the literature (among others Bazant 1992,Bazant & Jirasek 1993, van Zijl et al. 2001, Zhou &Hillerborg 1992, Zhang & Karihaloo 1992, Carpinteriet al. 1995).

In the present paper two different sources of time-dependence in concrete have been considered, arate dependence of crack propagation, introduced ina three-dimensional constitutive model for concretesuch as the microplane model M4, and a linear creepmodel for the prediction of the viscoelastic behav-ior of uncracked concrete. In the present approach

the non-linear creep of concrete may be regarded asa consequence of the redistribution of stresses dueto creep and, related to it, the increase of damage.Both sources of time-dependence (creep and rate-dependence of crack growth) play an important rolein the lifetime of the concrete structures. The predic-tion capability of this model is demonstrated throughthe experimental data available in the literature (Zhou1992). Moreover, the effectiveness of the proposedmodel is demonstrated through the analysis of singlenotched specimens in direct tension under differentloading rates, which reveals clearly the link betweenthe size of the fracture process zone (FPZ) and therate loading.

2 COMPUTATIONAL MODEL FOR LONG-TERM BEHAVIOR OF CONCRETE

The stress increment over a time step t, =(t + t) (t), in a linear visco-elastic materialcan be written as (Bazant & Wu 1974)

= Dveve + (1)

with

Dve =

[E0(t

) +N

n=1

En(t)n

t

(1 etn

)]D (2)

and = N

n=1

(1 etn

)n(t) (3)

where D is a dimensionless matrix obtained from thelinear elastic stiffness matrix divided by the Youngmodulus, ve is the visco-elastic strain increment inthe time step t, En(t) is the stiffness modulus ofthe element n, constant in the range t t t+t,n is the relaxation time of the element n and n(t)is the initial stress in the element n at the time t. Themodel is equivalent to a Maxwell chain model withtime-dependent element stiffnesses which, for exam-ple, can be determined through the Solidification The-ory (Carol & Bazant 1993).

Assuming the additivity of strains, the strain ratecan be expressed as follows

= ve + cr + sh + t (4)

where is the total strain rate, ve is the visco-elasticstrain rate, cr is the cracking strain rate, sh is theshrinkage strain rate, and t is the thermal strain rate.Substituting the strain rate with the strain incrementover a finite time step and recalling Equation 1 onegets

= Dveve + =

Dve(cr sh t)+ (5)

In the following, shrinkage and thermal strain, whichare not essential in the present application, will be dis-regarded (sh = t = 0).

The evolution of the cracking strain is assumed tobe governed by the Microplane Model M4 (Bazant etal, 2000a). This constitutive model has an explicit for-mulation that gives the stress tensor for a given straintensor

= fM4(elcr

)(6)

in which elcr is the elastic and cracking deforma-tion and fM4 is a function which represents the mi-croplane model formulation. According to Equation6, the stress increment for a given strain incrementelcr in the time step can be written as

= fM4(elcr +elcr

)initial (7)where initial is the initial stress at the beginning ofthe time step. Since the stress increment can be writ-ten as = Eel, the cracking strain increment is

cr = elcr E1[fM4(elcr +elcr

)+

initial] (8)where E is the linear elastic stiffness matrix. Substi-tuting Equation 8 into Eq. 5 one obtains

= Dve[elcr+E1 (fM4 (elcr +elcr)initial)] + (9)

Since the viscoelastic model and the MicroplaneModel M4 are coupled in series, the viscoelastic stressincrement and the Microplane model M4 stress incre-ment must be the same. After some mathematical ma-nipulations Equations 5 and 9 become an equation inthe only unknown elcr

fM4(elcr +elcr

) DveE1fM4

(elcr +elcr

)+ Dveelcr = b (10)

where b = Dve( E1initial) + + initial is a

vector of constants in each time step. The previousset of nonlinear equations must be solved iteratively.At a certain iteration i of the solution algorithm theresidual is

r(elcr) = fM4(elcr +elcr

) DveE1fM4

(elcr +elcr

)+ Dveelcr b (11)

Taking a Taylor expansion of the residual about thecurrent value of the elastic-cracking strain, elcri ,dropping the terms which are of higher order thanlinear in elcr, and setting the resulting residualequal to zero, one obtains a linear system of equationsfor elcr which represents the linearized model ofthe nonlinear equation (11). The proposed linearizedmodel is quite powerful, but it still has the disadvan-tage of requiring the calculation of the tangent stiff-ness matrix (Jacobian matrix). To overcome this prob-lem, the initial (elastic) stiffness matrix for the mi-croplane model M4 is employed and the Jacobian ma-trix becomes equal to the elastic stiffness matrix E.

In the Newton procedure, the solution of the non-linear equation is obtained by iteratively solving a se-quence of linear models. The new value for the un-known in each step of the iteration is obtained aselcri+1 =

elcr + elcri and it is continued until theconvergence criterion is met. The convergence crite-rion used to terminate the iteration is based on themagnitude of the unknown increments elcr.

Using the initial stiffness matrix the convergenceto the solution of the iterative procedure is alwaysassured, but it is very slow in the nonlinear regime.Therefore, to accelerate the convergence Broydensmethod has been adopted (Press et al. 1996). In theproposed iterative procedure the first three iterationsare performed using the Newton procedure with theinitial linear stiffness matrix. If the convergence cri-terion is not met after these three iterations, startingfrom that point Broydens method is adopted and theconvergence to the solution is obtained in one to twoiterations. Once elcr is known, one can calculatethe other unknowns cr, ve and

= fM4(elcr +elcr

)initial (12)

cr = elcr E1 andve = cr (13)

At convergence the visco-elastic strain tensor is ob-tained according to Equation 13, which also enablesthe calculation of the internal stresses of the Maxwellchain elements. For the first element one finds

0(t+t) = 0(t) + DE0(t)ve (14)

while for the others

n(t+t) = n(t)+[DEn(t

)nt

ve n(t)](

1 et/n) (15)More details on the proposed formulation can befound in Di Luzio (in prep.).

3 RATE DEPENDENT MICROPLANE MODELM4

The microplane model is a three-dimensional macro-scopic constitutive law for modelling quasi-brittlematerials which exhibit softening. In the microplanemodel the material law is characterized by simple re-lations between the stress and strain components onplanes of various orientations. In the present papera recent version of the microplane model for con-crete, proposed and tested by Bazant and coworkers(Bazant et al. 2000a; Caner & Bazant 2000) calledM4, is used. The microplane model M4 has the in-elastic behavior characterized on the microplane levelby the so-called stress-strain boundaries, which maybe regarded as strain-dependent yield limits and ex-hibit strain softening. The original formulation of themicroplane model M4 (Bazant et al. 2000a) has beenslightly modified as reported in Bazant et al. (2004)and Di Luzio (in press).

The formulation proposed by Bazant (Bazant 1993,Bazant 1995), in which the rate dependence of a co-hesive crack is described by the activation energy the-ory, is adopted. According to this theory the rate ef-fect dependence is nonlinear. The rate-dependence isintroduced in the microplane model M4 as proposedby Bazant et al. (2000b). The macroscopic strain soft-ening function = F () may be written as

F () = F 0() [1 +C2asinh(/C1)] (16)

F () represents the stress-strain boundary (tensilenormal stress; compressive volumetric stresses; ten-sile and compressive deviatoric stresses; and the cohe-sion in the friction boundary) on the microplane cor-responding to strain rate and F0() has the meaningof the static stress-strain boundary that corresponds

Figure 1: Maxwell chain fit to relaxation data.

to a vanishing strain rate. Equation 16, which trans-forms the static boundary F0() to a rate-dependentboundary, represents a vertical scaling of the bound-ary curve. The rate effect on the frictional yield sur-face is also applied as a vertical shift of the horizontalasymptote.

It is known that the concept of softening continuumleads to serious problems, i. e. the boundary valueproblem becomes ill-posed and the numerical calcu-lations cease to be objective, exhibiting pathologicalspurious mesh sensitivity and unrealistic damage lo-calization. To suppress it and to prevent the damagefrom localizing into a zone of zero volume, the sim-plest remedy is to adjust the post-peak slope of thestress-strain diagram as a function of the element size(this was done in the crack band model of Bazant &Oh 1983). This type of localization limiter is adoptedin the microplane model M4, in which the microplanelimit curves are assumed as functions of the crackband width h. For solid eight node elements the crackband width is assumed to be equal to the average el-ement size h = V 1/3, where V is the volume of thefinite element.

4 NUMERICAL SIMULATIONS OF BENDINGTESTS

The experimental investigation of Zhou (1992) hasbeen considered for validating the proposed compu-tational framework. The beams were sealed to avoidshrinkage and and subjected to three-point bendingtests. Initially the parameters of the microplane modelwere calibrated on the basis of the mechanical prop-erties of the concrete: Young modulus of 36GPaand tensile strength of 2.8MPa. The parameters ofa Maxwell chain model with 7 elements were cali-brated through the least square method, by fitting theexperimental relaxation curve obtained from cylindri-cal notched tensile specimens (Fig. 1). Since the re-laxation tests were performed over a short time (1hour) and all of the other tests have a short duration, anon-aging material behavior has been assumed. Zhou

Figure 2: Three-point-bending tests by Zhou (1992):a) load vs deflection curves for different deflectionrates; b) time to failure under sustained loads.

(1992) also studied the influence of loading rate on thepeak strength through three-point-bending tests underdisplacement control on smaller beams (cross sectionof 50x50mm2 and length of 600mm) with differentdeflection rates (from 0.05ms1) to 50ms1). Tofit these test data the rate parameters were varied in atrial-and-error fashion until a satisfactory agreementwas found.

After the calibration of the model parameters, adisplacement-controlled three-point-bending test wassimulated with a deflection rate of 5 ms1, afterwhich the creep tests were conducted on the sametype of specimen by increasing the central force toa predefined level and then sustaining it. Four loadlevels of 76%, 80%, 85% and 92% of the maxi-mum load capacity of the displacement-controlledcase have been considered. The failure in the creeptests occurred in a such a way that the displacement-controlled curve describes very well the failure en-velope characterizing the deformation at failure un-der sustained load. The numerical procedure goes onuntil equilibrium is no longer ensured for the sus-tained load level, i.e. the load bearing capacity ofthe specimen is exceeded. When this happens, the

Figure 3: a) Sample geometry of the tensile directtest, b) distribution of the maximum principal crack-ing strain at the peak for different loading rates andsizes.

Newton-Raphson iterative procedures are no longerable to reach convergence. The numerical results arepresented in Figure 2, showing a good agreement withexperiments.

5 NUMERICAL ANALYSIS OF SIZE EFFECTTESTS AT VARIOUS LOADING RATES

The proposed computational framework, calibratedon the basis of the experimental data studied in theprevious section, has been used to predict the influ-ence of loading rate on geometrically similar speci-mens. As already pointed out from experimental in-vestigations (Bazant & Gettu 1992), the lower theloading rate the more brittle the fracture propagation,which means that the size effect curve has a signif-icant shift toward the right as the loading rate de-creases. This physical phenomenon can be explainedby a reduction of the damage distribution (FPZ) as theloading rate decreases.

The direct tensile test has been adopted for thesesimulations, because a constant crack mouth open-ing displacement rate can be imposed. Figure 3ashows the specimen geometry. The finite element

Figure 4: Load versus time for fast (a) and slow (b)loading rates for different sizes (d).

meshes for five different specimen sizes, with size ra-tio 1:2:4:8:16, are shown in Figure 3b. The smallestspecimen depth is d = 38mm, and the largest one isd= 608mm. The thickness is b= 38mm for each size.The specimens were loaded through a constant dis-placement velocity of 5103 mm/s and of 5108 mm/sfor the faster and the slower loading rates, respec-tively. The maximum principal cracking strain distri-bution at the peak for different loading rates and sizesis depicted in Figure 3b, which reveals clearly that thedamage and the energy dissipation for each size aremore spread out for high loading rates. The volumein which the energy is dissipated (FPZ) is reduced forvery slow loading rates by the effect of creep. For bothfast and slow loading rates, Figures 4a, b report thenumerical load versus time curves.

Let us apply now to the results of the numericalsimulations (expressed in terms of nominal stress atmaximum load N = Pmax/db) Bazants size effectlaw (Bazant & Planas 1998):

N =0

1 + d/d0(17)

in which the constants 0 and d0 can be determinedfrom the numerical (or experimental) results. Rewrit-ing Equation 17 in the plot Y = (1/N)2 versusX = d, the size-effect-law appears as a straight lineY = AX + C in which 0 = 1/

C and d0 = C/A.

Figure 5: Size effect law (S.E.L.) for fast and slowloading rates. Comparison of the two size effect re-sults with the shift toward the right as the loading be-comes slower.

The coefficients A and C have been determined per-forming a linear regression of Y versus X, gettingA = 4.525104 MPa2/mm and C = 0.157 MPa2

(from which 0 = 2.525 MPa and d0 = 346.66 mm)for the fast series and A = 1.476103 MPa2/mm andC = 0.282 MPa2 (from which 0 = 1.884 MPa andd0 = 191.0 mm) for the slow series. Figure 5 clearlyshows that by reducing the loading rate the size effectcurve has a shift toward the right with a change in themode of failure, i.e. the fracture becomes more brittleas the loading becomes slower. This is confirmationthat fracture propagation and the size effect are timedependent processes for concrete.

6 CONCLUSIONS

The microplane model M4 with rate dependence, cou-pled with a linear viscoelastic element, is capable ofpredicting the interaction between fracture and creepin concrete. The proposed time-dependent model hasbeen verified for experimental tests of up to 3 hours induration (Zhou, 1992). It has been shown that the in-clusion of both sources of time-dependence, i.e. creepand rate-dependence of crack growth, must be consid-ered to obtain the correct prediction of life expectancyunder a constant loads.

The proposed constitutive model is also able to cap-ture the increase of concrete brittleness as the loadingrate becomes slower, which appears with a significantshift toward the right (toward LEFM in the size effectplot) as the loading rate decreases. This phenomenonis explained in the numerical simulations with a re-duction of the size of the FPZ as the loading rate be-comes slower.

ACKNOWLEDGMENTThe authors gratefully acknowledge the financial sup-port of the Italian Department of Education, Univer-sity and Scientific Research (MIUR) to the projectCalcestruzzi Fibro-rinforzati per Strutture ed Infras-trutture Resistenti, Durevoli ed Economiche.

REFERENCES[1] Bazant, Z.P. 1992. Rate effects, size effects and

nonlocal concept for fracture of concrete andother quasi-brittle material. Mechanisms ofQuasi-brittle Material, S. Shah (ed.), Klu-ver, Dordrecht, 131-153.

[2] Bazant, Z.P. 1995. Creep and damage in con-crete. Proc., Materials science of concreteIV, J. Skalny and S. Mindess, Eds., Am. Ce-ramic. Soc., Westrville, OH, 355-389.

[3] Bazant, Z.P., Caner, F., Carol, I., Adley, M.& Akers, S.A. 2000a. Microplane ModelM4 for Concrete. I: formulation with work-conjugate deviatoric stress. J. Engrg. Mech.,ASCE 126, 944-953.

[4] Bazant, Z.P., Caner, F., Adley, M. & Akers, S.A.2000b. Fracturing rate effect and creep inmicroplane model for dynamics. J. Engrg.Mech., ASCE 126, 962-970.

[5] Bazant, Z.P., Caner, F.C., Cedolin, L., Cusatis,G. & Di Luzio, G. 2004. Fracturing materialmodels based on micromechanical concepts:recent advances. In V.C. Li, C.K.Y. Leung,K.J. Willam, S.L. Billington (eds.), FractureMechanics of Concrete and Concrete Struc-tures - FraMCoS-5, Proc., Vail Colorado,83-89.

[6] Bazant, Z. P. & Oh, B.-H. 1983. Crack band the-ory for fracture of concrete. Mat. and Struct.,RILEM 16(93), 155-177, Paris.

[7] Bazant, Z. P. & Gettu, R. 1992. Rate effect andload relaxation in static fracture of concrete.ACI Mat. J., 89(5), 456-468.

[8] Bazant, Z. P. & Wu, S.T. 1974. Rate-type creeplaw of aging concrete based on Maxwellchain. Mat. and Struct., RILEM 7, 45-60.

[9] Bazant, Z.P. & Jirasek, M. 1993. R-curve mod-eling of rate and size effects in quasi-brittlematerial. Int. J. of Fract. 62, 355-373.

[10] Caner, F., and Bazant, Z. P. 2000. MicroplaneModel M4 for Concrete. II: algorithm andcalibration. J. Engrg. Mech., ASCE 126,954-961.

[11] Carol, I. & Bazant, Z.P. 1993. Viscoelasticitywith aging caused by solidification of nonag-

ing constituent. J. Engrg. Mech., ASCE119(11), 2252-2269.

[12] Carpinteri, A., Valente, S., Zhou, F.P., Ferrara,G. & Melchiorri, G. 1995. Crack propaga-tion in concrete specimens subjected to sus-tained loads. Fracture Mechanics of Con-crete Structures, Wittmann, F.H. (Ed.), Aed-icatio, Germany, 13151328.

[13] Di Luzio, G. 2007a. A numerical model for con-crete time-dependent fracturing. submittedto J. Engrg. Mech., ASCE.

[14] Di Luzio, G. 2007b. A symmetric over-nonlocalmicroplane model M4 for fracture in con-crete. Int. J. Solids and Struct. in press.

[15] Press, W.H., Teukolsky, W.A., Vetterling, W.T.& Flennery, B.P. 1996. Numerical Recipesin Fortran 90: the Art of Parallel ScientificComputing, Cambridge University Press.

[16] van Zijl, G., de Borst, R. & Rots, J. 2001. A nu-merical model for the time-dependent crack-ing of cementitious materials. Int. J. Num.Met. in Engrg. 38, 5063-5092.

[17] Wu, Z.S. & Bazant, Z. P. 1993. Finite elementmodeling of rate effect in concrete fracturewith influence of creep. Z.P. Bazant, I. Carol(eds.), 5th International RILEM Symposiumon Creep and Shrinkage of Concrete, Proc.,London, U.K., 427-432.

[18] Zhang, C. & Karihaloo, B. 1992. Stability ofcrack in linear visco-elastic tension soften-ing material. Proc., Fracture Mechanics ofConcrete Structures, Z.P. Bazant (ed.), Else-vier Applied Science, The Netherlands, 75-81.

[19] Zhou, F.P. & Hillerborg, A. 1992. Time-dependent fracture of concrete: testing andmodelling. Proc., Fracture Mechanics ofConcrete Structures, Z.P. Bazant (ed.), Else-vier Applied Science, The Netherlands, 75-81.

[20] Zhou. F.P. 1992. Time-dependent crack growthand fracture in concrete, PhD thesis, LundUniversity of Technology, Sweden.