A loading rate dependent cohesive model for concrete fracture

Engineering Fracture Mechanics 82 (2012) 195–208

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Engineering Fracture Mechanics

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A loading rate dependent cohesive model for concrete fracture

A.L. Rosa a,b, R.C. Yu a, G. Ruiz a, L. Saucedo a, J.L.A.O. Sousa b,⇑a Universidad de Castilla-La Mancha, 13071 Ciudad Real, CLM, Spainb Universidade Estadual de Campinas, 13083-852 Campinas, SP, Brazil

a r t i c l e i n f o a b s t r a c t

Article history:Received 14 March 2011Received in revised form 30 September 2011Accepted 29 December 2011

Keywords:Viscous cohesive crack modelLoading rate effectFracture processesHigh strength concrete

0013-7944/$ - see front matter � 2012 Elsevier Ltddoi:10.1016/j.engfracmech.2011.12.013

⇑ Corresponding author.E-mail address: [email protected] (J.L.A.O. Sousa).

A time dependent cohesive model to account for the influence of loading rate on concretefracture is addressed. Viscous-cohesive parameters are introduced in the fictitious crackmodel to represent a time dependent stress-crack opening zone in an elastic solid. A com-putational implementation has been performed, simulating the cohesive stress-crackopening and rate dependence in a smeared crack tip framework. The model has been val-idated with results from three-point-bend notched specimens of high strength concrete,tested at five different loading rates. Numerical results show that the approach reflectsthe experimentally documented dependence of crack propagation in cementitious materi-als on the loading rate.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Cohesive fracture models for quasi-brittle materials like concrete – characterized by the development of a large nonlinearcrack tip fracture process zone (FPZ) prior to failure in tension – require a constitutive relationship (named cohesive model,stress versus crack opening or softening curve), that relates the stresses developed at the crack front, i.e., in the fracture pro-cess zone (FPZ), with the crack opening as the basic ingredient to represent crack nucleation and propagation, either in animplicit or in an explicit manner (e.g., in the smeared crack [1–3], discrete crack [4,5], enriched and strong discontinuityapproaches [6–10] and fragmentation methods [11–14]). However, neglecting the influence of factors related to the exper-imental techniques, the specific fracture energy measurement is performed without taking into account the influence of timeand environmental conditions on the fracture process. Even though a test is a simplification of reality, experimentallydocumented evidences indicate that the load rate prescribed in fracture tests set-up exerts significant influence on thefailure process of concrete samples, which is macroscopically reflected on the load versus deflection or load versus crackmouth opening displacement (CMOD) responses ([15–19]). The peak load and the apparent fracture energy increase mono-tonically with increasing loading rate, while the displacement corresponding to the peak load remains virtually the same[15,16,18,19].

Although the physical micro-mechanisms involved in the fracture process due to loading rate and/or load duration are notyet clarified, experimental evidences indicate that, on the micro-level, the quasi-brittle failure is characterized, in a firststage, by a delay (for a particular level of strain) of micro-cracking growth at high strain rates [20]. In a second stage, thephenomenon of failure is followed by the propagation of many micro-cracks at the same time [21]. The fact that for qua-si-brittle materials a discrete crack is essentially due to nucleation and growth of micro-voids and micro-cracks inducesthe understanding that a reduction of micro-cracking with increasing strain rate leads to a reduction of the macroscopic non-linear behavior and to an increase of the nominal failure stress (material strength increase). It is likely that, for this second

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Nomenclature

a0 notch depthcf effective fracture process zone size defined by Bazantf (w) static cohesive law (stress vs. crack opening relation)f ⁄(w⁄) nondimensional static cohesive lawfc compressive strength (cylinder)ft material tensile strengthf st material tensile strength (static)

gð _wÞ viscous function, dependent on the crack-opening rateg�ð _w�Þ nondimensional viscous functionn exponent of rate dependencew crack opening displacementwc critical crack opening displacement_w crack opening displacement rate_w0 normalization parameter‘ch characteristic lengthwch ¼ Gs

F=f st characteristic crack opening displacement

B beam widthD beam depthFð _w;wÞ time dependent (viscous) cohesive modelF�ð _w�;w�Þ nondimensional time dependent (viscous) cohesive modelGF fracture energyGs

F fracture energy (static)L beam lengthP loadS beam spanbH Hillerborg’s brittleness number_d loading displacement rater cohesive stressrN nominal tensile stressCMOD crack mouth opening displacementFPZ fracture process zoneHSC high strength concreteTPBT three-point-bend test

196 A.L. Rosa et al. / Engineering Fracture Mechanics 82 (2012) 195–208

stage, micro-cracking does not have sufficient time to search for paths of minimum energy or minimum resistance. Thus,micro-cracks are forced to propagate along the shortest path with higher resistance. This suggests that the fracture energyand the peak load should increase as a function of the loading rate [15,16]. However, at low loading rates, the tendency of thefracture energy and the nominal failure stress with increasing loading rate is different. The fracture energy is almost constantalthough the peak load continues to increase as the loading rate increases. At slower loading rates more water-related phe-nomena, like water migration through the porous network and mainly the cohesive induced forces due to the cohesive forcesinduced by the so-called Stefan’s effect (see [22–24]), are activated and can explain the exhibited rate sensitivity.

Computer codes simulating time-dependent concrete crack growth at the macroscopic level use rate dependent modelswith relative success. In these models the time-dependent concrete fracture incorporates the analysis of the bulk materialviscous behavior [25–27]. Such can simulate the experimentally observed rate effect phenomena to a certain degree of lim-itations and accuracy, and cannot be used as a general model for arbitrary loading histories. Therefore, an attempt to a moregeneral and fundamental approach to adequately simulate the loading rate effect is an open question. More research in thisfield is required to shed light on the matters of rate sensitivity of the quasi-brittle fracture and on the causes that generatesuch sensitivity. The main goal of this research is a better understanding of the loading rate dependence of macroscopic crackgrowth through numerical simulation. The findings on time dependence of a macroscopic crack are also relevant, since thegrowth of micro-cracks must abide by the same law [25]. Although the physical micro-mechanisms involved in these effectshave not yet been clarified, an important incursion in this direction is presented by Denarié et al. [28], Denarié [29], whosimulated the time-dependent crack growth in concrete considering its microstructure and viscoelastic effects.

This paper describes an implementation of a macroscopically time-dependent model according to Ruiz and co-workers[15,16], which accounts for loading rate effects by coupling a viscous parameter, as a function of the crack-opening rate, withthe classical Hillerborg’s fictitious crack approach [30]. The model is implemented in a finite element framework by means ofthe commonly denominated smeared crack tip method, developed by Planas and Elices [31], except for the introduction of aviscous cohesive zone, in which the behavior is represented by a time-dependent stress-crack opening.

A.L. Rosa et al. / Engineering Fracture Mechanics 82 (2012) 195–208 197

The macroscopically time-dependent model is validated by comparison with results from fracture tests on notched highstrength concrete (HSC) beams loaded in three-point-bend (TPBT) configuration, at five distinct loading rates [15]. The re-sults numerically obtained with the current viscous cohesive crack representation coincide accordingly with the experimen-tal ones, particularly the maximum load for the several loading rates used in the tests. A parametric study has also beenperformed to demonstrate that the dependence of the fracture energy on the size of the specimen and on the length ofthe ligament is mainly attributable to rate effects. The numerical results show that the model, regardless of its simplicity,provides a general approach to replicate the experimental findings which document that crack propagation, and so theparameters governing the fracture process, depend on the loading rate.

The rest of the paper is structured as follows. In Section 2 the assumptions and the computational aspects are presentedand discussed. In Sections 3 and 4, the experimental program designed to investigate the loading rate effect on the fracturebehavior and material characterization are outlined. Numerical treatment of data from HSC specimens tested in TPBT con-figuration and a parametric analysis of the model sensitivity relating the work of fracture to the size of the specimen and tothe length of the ligament is presented in Section 5. Section 6 ends the paper with a summary and conclusions.

2. The time dependent cohesive model

2.1. Governing equation

To take into account the rate effect on the quasi-brittle fracture process of the hardened HSC, a time dependent (viscous)cohesive model, stated as a function of the crack opening (w) and the crack opening velocity ( _w), is proposed as follows:

r ¼ Fð _w;wÞ ¼ gð _wÞf ðwÞ ð1Þ

and

Fð _w;wÞ ¼ f ðwÞ for _w ¼ 0Fð _w;wÞP 0

�ð2Þ

where the function f(w) defines a static cohesive law (stress vs. crack opening relation), according to the classical Hillerborg’sfictitious crack model [30] and g _ðwÞ is a viscous function dependent on the crack-opening rate, _w, such as:

gð _wÞ ¼ 1þ_w_w0

� �n

ð3Þ

In the above equation _w0 is a normalization parameter, with dimension [L][T]�1, and n, exponent of rate dependence, is anondimensional constant describing the degree of viscosity of the material. The function r ¼ Fð _w;wÞ completely character-izes the mechanical behavior of the crack. Fig. 1 presents the boundaries (projection on the r�w plane) of the viscous-cohe-sive function r ¼ Fð _w;wÞ. For this picture a viscous-cohesive crack initiates when the stress at the crack tip reaches thematerial tensile strength, f s

t , referred as static, evolving until a critical crack opening value, wc, is reached, in such a way thatthere are no stresses bridging from one crack face to the other, i.e., no stress transfer across the crack occurs for w P wc. InFig. 1 the shaded area below the dashed line quantifies the fracture energy, Gs

F , considered here as static (herein, unlessotherwise stated, superscript s denotes a material property under static conditions). The areas below the continuous linesrepresent the apparent fracture energy GF measured as a function of loading rate.

Fig. 1. Viscous cohesive function.

198 A.L. Rosa et al. / Engineering Fracture Mechanics 82 (2012) 195–208

2.2. Numerical solution

As aforementioned the viscous-cohesive model is implemented in a finite-element framework based on the smearedcrack tip method devised by Planas and Elices [31] from the classical influence method of Petersson [32]. The virtue ofthe method is that it leads to an improvement of the solution algorithm that describes the crack propagation problem inthe Petersson’s method by a triangulation of the system of equations, enabling a direct solution by forward substitution.So, the solution is sought from a triangular system of equations obtained by superposing linear elastic fracture mechanics(LEFM) cases.

In the following section, the essential aspects of the numerical formulation are presented. For a detailed description, thereader is referred to the work of Ruiz [33], where the method is applied to study the propagation of a cohesive crack througha reinforcement layer.

2.2.1. Numerical formulationThis time dependent cohesive model sees discrete crack propagation as a superposition of linear elastic fracture mechan-

ics (LEFM) problems, in which a cohesive model (in our case a viscous-cohesive model) describes the stress transferringcapability of the fracture process zone (FPZ) and, thus, the conditions to the crack formation and advance. In this way thenonlinear fracture response of HSC specimens is written as the sum of N elastic cases corresponding to N different cracklengths, each one with a stress-free crack with tip at node j (j = 1, 2, . . . , N) and an external incremental load DPj (Fig. 2).Then, for each node i, the distributions of nodal stresses ri, nodal crack openings wi, the total external load P and the totaldisplacement d of the load-point can be evaluated by the following equations:

ri ¼XN

i;j¼1

RijDPj for i ¼ 1;2; . . . ;N ð4Þ

wi ¼XN

i;j¼1

DijDPj ð5Þ

P ¼XN

j¼1

DPj ð6Þ

d ¼XN

j¼1

CjDPj ð7Þ

where Cj and Rij and Dij are coefficients denoting, respectively, the load–point displacement, the nodal stress and the crackopening at a node i when an external unit force is applied with a crack reaching node j (see: Fig. 2). These coefficients arestored in a coefficient matrix M, known a priori through finite element method (FEM). By imposing the boundary conditionsof the problem (Eqs. (8)–(10)) and considering the viscous-cohesive model given in Eq. (1) the nodal stresses ri, the nodalcrack openings wi, the total external load P and the total displacement d of the load-point (Eqs. (4)–(7)) can be computed bysolving iteratively the nonlinear system of m � e + 1 equations (in which m and e come out from Fig. 2) given by Eq. (11) asfollows: starting from an estimate of the incremental load DPj ¼ DPa

j in the right-hand side and solving for a better estimateDPaþ1

j from the left-hand side at the iterative step a.

ri ¼ 0 for i ¼ 1; . . . ; e� 1 ð8Þ

ri ¼ Fð _wi;wiÞ for i ¼ e; . . . ;m ð9Þ

Fig. 2. Schematic illustration of the calculation method.

A.L. Rosa et al. / Engineering Fracture Mechanics 82 (2012) 195–208 199

wi ¼ for i P m ð10Þ

wi ¼Xi

j¼e

RijDPaþ1j ¼ F _wi;

Xm

j¼iþ1

DijDPaj

!for i ¼ e; eþ 1; . . . ;m ð11Þ

Considering a TPBT configuration under displacement control, the nodal crack opening velocity _wi is obtained through therate relation:

_wðjÞi ¼wðjÞi �wðj�1Þ

i

dðjÞp � dðj�1Þp

_dp ð12Þ

where a dot superscript denotes a time derivative; index j corresponds to variables at loading step j; p denotes a monitoringpoint and _dp is the displacement rate of the monitoring point. Please note that _wðjÞi is the crack opening rate at node i in thestep j, while _dp represents the loading rate measured at the monitoring point, assumed constant in our tests. The time inter-val between steps j � 1 and j can be computed as ðdðjÞp � dðj�1Þ

p Þ= _dp.In this work, the computations were performed with the involving equations (based on a three-point bend beam config-

uration) reduced to a nondimensional form. To write the equations in a nondimensional form (identified by an asterisk), thefollowing nondimensional stress, crack opening, load and displacement have been defined:

r� ¼ rf st

ð13Þ

w� ¼ wwch

ð14Þ

P� ¼ rN

f st¼ r�N ¼

32

� �PS

BD2f st

ð15Þ

d� ¼ dwch

ð16Þ

where wch ¼ GsF=f s

t is the characteristic crack opening displacement; rN is the nominal tensile stress, S, B and D are, respec-tively, the beam span, width and depth. When all variables are reduced to a nondimensional form and the analysis is restrictedto geometrically similar structures, it so happens that the size effect is controlled by the Hillerborg’s brittleness number, bH ,which is the essential characteristic of the fracture processes in plain cohesive specimens ([30,32]), and is defined as:

bH ¼D‘ch

ð17Þ

where ‘ch ¼ EcGsF=ðf s

t Þ2 is the characteristic length; Ec is the secant elastic modulus; Gs

F and f st are, respectively, the fracture

energy and the material tensile strength, which is considered as static, as stated before.According to Eqs. (13) and (14), the nondimensional form of the viscous cohesive model can be set as:

r� ¼ F�ð _w�;w�Þ ¼ g�ð _w�Þf �ðw�Þ ð18Þ

Note that _w� ¼ 0 implies the static case: F�ð _w�;w�Þ ¼ f �ðw�Þ and _dp ¼ 0. The function f �ðw�Þ is such that:

f �ð0Þ ¼ 1; f �ðw�cÞ ¼ 0 andZ w�c

0f �ðw�Þdw� ¼ 1; where w�c ¼ wc=wch ð19Þ

3. Experimental program

To validate the findings, the proposed methodology was applied to data from an experimental program, designed to inves-tigate the loading rate effect on the fracture behavior of high strength concrete (HSC) [15]. Only the relevant aspects related tothis study, such as the materials parameters characterizing this time dependent (viscous) cohesive crack model, are reproducedherein. To obtain the experimental material parameters required for the numerical model subject of this study, independenttests were performed for a complete material characterization. The experimental program was conducted on notched beamsin three-point bend (TPBT) test to characterize the apparent fracture energy, GF, from the load versus deflection curve; splittingcylinders (Brazilian tests) for the tensile strength, f s

t ; and axial compression tests on cylinders for the compressive strength, fc,and the secant elastic modulus, Ec, of the material. The apparent fracture energy and the tensile strength characterize the staticsoftening curve. The secant elastic modulus characterizes the bulk material surrounding the discrete crack and the FPZ.

Quantification of other parameters required to completely describe our time dependent (viscous) cohesive model, such asthe exponent of rate dependence, n, and the normalization parameter, _w0, are obtained indirectly from experimental results,as described in (Section 5).

Fig. 3. TPBT configuration and geometric dimensions.

200 A.L. Rosa et al. / Engineering Fracture Mechanics 82 (2012) 195–208

3.1. Specimens testing and apparatus

3.1.1. Three-point bending test (TPBT)The geometry dimensions of the tested specimens in TPBT configuration is shown in Fig. 3. HSC notched beams were tested

following, essentially, the RILEM recommendation [34] for the measurement of GF and improvements devised by Elices, Guineaand Planas [35–40]. Tests were performed under displacement-control by a closed-loop servo-hydraulic test machine, with theload-point displacement (P–d curve) as the feedback data acquisition signal. With this purpose, two linear variable differentialtransducers (LVDTs), fixed on a stiff steel beam of the set-up apparatus, were used to measure the load-point displacement ofthe central loading rod, in such a way that the displacement was taken as the average deflection measured by the two LVDTs. Thesupports rest and roll on the upper face of a stiff steel beam (fastened to the machine actuator), allowing rotation out of the planeof the beam longitudinal axis with negligible friction. In all the tests a weight compensation strategy was adopted by using aspring to hold the specimen during the applied loading history. Five different loading rates were applied in the experimentalsetup from a quasi-static1/6 level (1.74 � 10�5 mm/s and 5.50 � 10�4 mm/s) to a dynamic level (1.74 � 10�2 mm/s, 5.50 �10�1 mm/s and 1.74 � 10+1 mm/s). In addition, four specimens for each loading rate were employed during the tests. All thespecimens were cast in steel molds, compacted with a vibrating table, wrap-cured for 24 h and demolded. Before testing, thespecimens were stored for 28 days, under 20 �C and 98% relative humidity, sawn through the central cross section to a nominalnotch depth a0

D ¼ 0:5 and 2.5 mm thickness.

3.1.2. Splitting cylinder (Brazilian) and axial compression testsBrazilian tests were carried out on specimens following the procedures recommended by ASTM C496. The specimen was

loaded through plywood strips with a width of the specimen diameter. The velocity of the machine actuator was 0.43 mm/min. Axial compression tests were carried out according ASTM C39 and ASTM C469 to obtain the compressive strength, fc,and the secant elastic modulus, Ec, respectively. The tests were run under displacement control. The strain was measuredover a 50 mm gage length using two clip gage extensometers, Instron

�2630, centered on the specimen to measure the dis-

placement in the central part of the sample to obtain the secant modulus.A total of 8 cylindrical specimens of 75 mm � 150 mm (diameter � height) were cast in metallic molds, 4 for axial com-

pression tests and 4 for splitting cylinder (Brazilian) tests. The bases of the cylinders tested in axial compression were sur-face-ground with a wet diamond wheel to guarantee perfectly-plane and parallel surfaces. All specimens were tested on acompressive setup performed on a universal Instron

�8805 closed-loop servo-hydraulic dynamic testing machine with a load

capacity of 1 MN and load platens of hardened steel and polished surfaces. To control the tests a pair of LVDTs, with a cal-ibrated range of ±2.5 mm, was used to measure the average axial platen-to-platen displacement. This average signal repre-sents the axial velocity at the center of the loading plate. This type of control prevents snap-back to take place in the r–ecurves. The axial strain rate is defined as _e ¼ _d=D/, where _d is the axial platen-to-platen displacement rate and D is the initialdistance between the top and the bottom plane surfaces over the height of the specimen. This axial strain rate was kept con-stant for all testing have a value 6:7�6=s.

4. Material characterization

4.1. HSC material properties

4.1.1. Mechanical properties of the hardened HSCThe main properties of the hardened HSC determined in the various characterization tests, in a static point of view, are

summarized in Table 1. In this table the mean and standard deviation values from the tests are shown. Note that the staticvalue of the apparent fracture energy, GF ¼ Gs

F , assumed to be a true material property, was obtained, by curve fitting, fromthe mean values of the specific fracture energy values computed for each set of specimens subject to the same applied load-ing rate.

Table 1Hardened HSC mechanical properties.

Eca (GPa) fc

a (MPa) f st

b (MPa) GsF

c (J/m2) ‘chd (mm) bh

e

Mean 33.9 97 5.2 128 160.47 0.6232Std 1.2 5.9 0.5 – – –

a Cylinder axial compression tests.b Splitting cylinder (Brazilian) tests.c Obtained by curve fit of experimental data.d ‘ch ¼ EcGs

F=ðf st Þ

2.e bH ¼ D=‘ch .

Table 2Fresh mix additive content by m3.

Micro-silica fume (kg) Metakaolin (kg) Expander (Denka CSA 20) (kg) Super-plasticizer (kg)

37.80 3.10 1.90 17.92

Fig. 4. Curves fitting and comparison between predicted and experimental results for the nondimensional load and the fracture energy at different loadingrates.

A.L. Rosa et al. / Engineering Fracture Mechanics 82 (2012) 195–208 201

4.1.2. Material constituentsA single high strength concrete mix was used to cast the tested specimens of the experimental program. The mix weight

proportion was 2.68:0.33:1 (aggregate:water:cement), with a ASTM type-I cement content of 577 kg/m3 and a maximumaggregate size of 12 mm. The granulometric curve of the fine aggregate followed the ASTM C33 and was obtained by appro-priate mixing of fractions of siliceous (R-0/4-S) and andesite (T-6/12-C). Also micro-silica slurry, super-plasticizer (ACE 325,B225) and additives, to permit the workability of the fresh mix, were used in the concrete composition (see: Table. 2). Therewas a strict control of the specimen-making process to minimize scatters in the test results.

5. Numerical results and discussion

The numerical calculations were validated against the experimental results performed on notched HSC concrete beams inTPB configuration, according Ruiz et al. [15,16]. All validations have been performed using the beam geometry and dimen-sions shown in Fig. 3. In the parametric study, presented in Section 5.2, the geometry dimensions are proportionally scaledby varying the Hillerborg’s brittleness number, bH, to take into account the size effect on the peak load. Notch sensitivityinfluencing the model response is also addressed.

The numerical simulation of experiments have been performed using a bilinear softening representation for the cohesivefunction f(w) (dashed line in Fig. 5a), which is referred a particular case (static case) of the viscous-cohesive model for therate effects. Such cohesive representation was computed according to Guinea, Planas and Elices [39]. In the parametric inves-tigation a linear-softening curve was employed as a constitutive relation in order to model the tensile failure of the material

Fig. 5. (a) Bi-linear viscous cohesive law used throughout the calculations of the concrete fracture and (b–f) comparison between the predicted and theexperimental load versus displacement curves at different loading rates.

202 A.L. Rosa et al. / Engineering Fracture Mechanics 82 (2012) 195–208

A.L. Rosa et al. / Engineering Fracture Mechanics 82 (2012) 195–208 203

as static, i.e. rate-independent (dashed line in Fig. 7b). The curves presented in the Figs. 5a and 6b are a normalized repre-sentation in which the cohesive function f(w) complies with the conditions stated in Eq. (19). Note that in the Figs. 5a and 6b,the continuous lines represent possible viscous-cohesive paths as a projection at the nondimensional r/ft � w/wch plane. Thereal viscous-cohesive paths depend on the crack opening history. The coefficients of the M matrix were computed using thefinite element method by six-noded triangular elements as piece-wise for the structural approximation. The middle sectionwas discretized into 100 equal divisions (i.e., N = 100 in Fig. 2).

5.1. Numerical simulation of experimental tests

The hardened HSC mechanical properties and the computed fracture parameters representing the prismatic notchedspecimens tested in TPB configuration, under different loading rates, are given in Table 1. For numerical simulation of exper-iments, the applied loading rate is an input argument from the tests. The value of the exponent of rate dependence, n, wasobtained by curve fitting of the nondimensional peak load as a function of the applied loading rate. This value is then used toobtain the normalization parameter _w0 by curve fitting, resulting _w0 ¼ 9:65� 10�2 mm=s. As shown in Fig. 4a, this pairn; _w0ð Þ ¼ ð0:15;9:65� 10�2 mm=sÞ is a reasonable solution for the problem of fitting a curve to the experimental data. An

approach that determines the parameters related to rate dependence, n and, and the softening parameters related to the sta-tic case, e.g., ft and GF, through inverse analysis on the data from the whole set of tests (four specimens for each of the fiveloading rates) is the objective of an ongoing investigation that extends the approach proposed by Sousa and Gettu [41], and isleft for a subsequent companion paper.

Fig. 4a and b show the predicted and the experimental results for the nondimensional load and fracture energy for dif-ferent applied loading rates. In these figures fitted curves from the proposed model and from experiments are also plotted. Asseen in Fig. 4b the measured fracture energy for high loading rates cannot be considered as a material property. Although, forlow loading rates the experimental value of fracture energy does not vary significantly, it is clear from Fig. 4a and b that thenumerical response follows the same trends shown by the experiments, so that the peak load and the fracture energy in-crease monotonically as the loading rate increases.

Fig. 5b–f show a comparison between the predicted and the experimental curves for the five different loading rates em-ployed in the tests. As depicted from these figures, the numerical results are quite close to their experimental counterparts. Itis shown that the model, despite its simplicity, provides a general approach to reflect the experimentally documented factthat the crack propagation, as well as the parameters governing the fracture process, are a function of the applied loadingrate.

In Fig. 6, we choose four snapshots of the crack propagation to demonstrate the cohesive-stress distribution within thedeveloped FPZ. Those snapshots correspond to advanced distances of the cohesive crack tip of 9, 28, 38 mm respectively fromthe initial notch tip. The five profiles at each snapshot represent for the five loading rates employed in the experimental tests.The cohesive crack tip is marked by the normalized stress equals to one and crack opening displacement equals to zero,whereas the stress-free crack tip is signaled by a null value of the normalized stress. Consequently, the developed FPZ sizeis the distance in between. It is remarkable that this fully developed FPZ size is around 20 mm, which coincides with therecent experimental measurements using strain gauges, by Zhang et al. [42]. In this reference we have shown that the mea-sured FPZ size for HSC does not vary significantly with the loading rate during the range of seven order of magnitude from5.5 � 10�4 mm/s to 2640 mm/s, i.e., the phenomena that creep will decrease the FPZ size was not observed in our measure-ments. In other words, the apparent contradiction with the work of Bazant [17] is not due to the limitation of not having

Fig. 6. Evolution of the cohesive stress distribution within the developed fracture process zone (FPZ) for the five loading rates employed in experimentaltests: _d ¼ 1:74� 10�5 mm

s ; _d ¼ 5:50� 10�4 mms ; _d ¼ 1:74� 10�2 mm

s ; _d ¼ 5:50� 10�1 mms ; _d ¼ 1:74� 10þ1 mm

s .

Fig. 7. Model description and parametric response of fracture parameters for bH ¼ 0:6232 (a) rigid-body kinematics assumption in three-point bend (TPB)configuration; (b) linear viscous cohesive model used throughout the parametric calculations; (c) and (d) nondimensional load vs. nondimensionaldisplacement curves for n = 0.15 and n = 0.30, respectively; (e) and (f) nondimensional peak load and nondimensional work of fracture per unit area versusthe nondimensional loading rate for different values of the exponent of rate dependence.

204 A.L. Rosa et al. / Engineering Fracture Mechanics 82 (2012) 195–208

A.L. Rosa et al. / Engineering Fracture Mechanics 82 (2012) 195–208 205

accounted for creep in the bulk material, but rather possibly the way of defining the FPZ. Here we define FPZ size as the ac-tual distance between the stress-free crack tip and the cohesive crack tip, whereas Bazant has defined the cf as an effectivelength of FPZ. The ratio of cf to the true length of the FPZ in concrete depends on the shape of the softening stress-displace-ment or stress–strain relations [17].

In Fig. 6, we can observe that the developed FPZ size decreases when the loading rate increases when the FPZ is allowed tofully develop (at snapshot 3 when the cohesive crack tip is at 28 mm), i.e. the material is more brittle at higher loading rate;whereas at snapshot 4, the FPZ size at all loading rates remains practically the same, this might due to the fact that the crackadvancing speed at post-peak stage does not vary much, see Ref. [43]. Even though further investigation needs to be done toclarify the reason behind.

5.2. Parametric analysis

In the preceding sections we proposed an expression to model the influence of the loading rate on the cohesive behaviorof concrete. An example application showed that, although only two additional parameters were introduced, the model iscapable of giving a representation of the phenomenon, fitting reasonably well with experimental observations. Further re-search should be developed in order to validate the model for other sets of specimens, varying dimensions, geometry andtesting setup, which influence the crack opening velocity at each point on the crack faces, even if the same loading rate isapplied. In order to deepen understanding of the phenomenon, we present in this section a parametric analysis to pointout how the viscous-cohesive model related parameters, as well as the structural dimensions, are capable of reflectingthe material’s fracture response. In particular we investigate the influence of the applied load rate on the peak load by chang-ing the value of the exponent of rate dependence, n; the influence of the loading rate on the fracture energy by changing theHillerborg’s brittleness number value bH and the notch sensitivity on the load versus displacement response.

In Fig. 7c and d we plot the nondimensional load vs. load–point displacement curves for n = 0.15 and n = 0.30 by varyingthe nondimensional load applied rate _d= _w0 from 0 to 103. Note that in these figures the computed load and its correspondingdisplacement are normalized in such a way that the region below the curve for _d= _w0 ¼ 0:0 (dashed line representing the sta-tic case) has unit area.

As seen in Fig. 7c and d, the plotted load vs. load–point displacement curves behave in a similar manner. A loading rateincrease is followed by an increase on the peak of the P–d curve. A comparison between these figures shows that for a non-dimensional applied load rate below _d= _w0 ¼ 102, the peak load increasing is more pronounced for n = 0.15 than for n = 0.30.

A plot of the nondimensional peak load 2Ps=BDft versus the nondimensional loading rate _d= _w0 (Fig. 7e shows that for dif-ferent values of n there is a transition domain at the peak load, close to the value _d= _w0 ¼ 102. Before this point, for a givenloading rate, the peak load experiences a decrease as the exponent of rate dependence n increases. Fig. 7e and f show that inthe range 0 < _d= _w0 < 102 the peak load and the fracture energy increase more rapidly for smaller values of n. After the point_d= _w0 ¼ 102 highest peak values occur for the highest n and the peak load and the fracture energy increase more rapidly forlarger values of n. Thus, one can suppose that for larger values of n a reduced material limit velocity wave propagation can beinduced dynamically. Additionally, for higher value of the index n (n = 0.60 in our example in Fig. 7d), the peak load and thework of fracture per unit area is almost constant in an extent range of values for _d= _w0. Since n is a nondimensional parameterthat describes the degree of viscosity of the bulk material it is probably related with the presence of water activity in thematerial porous network and fracture energy can be considered as a rate independent parameter only for a certain range

Fig. 8. Parametric response of fracture parameters: influence of bH .

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of applied loading rate and for a certain free water content acting in the material pores. The confirmation or not of theseevidences from experimental tests, apart from the analysis of the mechanisms linked to the micro-crack interactions inthe fracture process zone (FPZ), can bring more understanding about the rate sensitivity behavior on the quasi-brittle frac-ture response.

The same trends for the nondimensional peak load vs. the nondimensional loading rate are observed in the plot of thenondimensional fracture energy per unit area (GF=Gs

Fðd=bHwchÞ) vs. the nondimensional loading applied rate for different val-ues of n in Fig. 7f. Note that in the plots of Fig. 7c–f the value of bH is the same as given in Table 1.

It is noteworthy that similar trends for the peak load and for the work of fracture as a function of the nondimensionalloading rate, such as shown in Fig. 7c and Fig. 7d, was achieved by Ruiz et al. [15,16] from an analytical expression derivedfrom a beam in TPB configuration and under rigid-body kinematics assumptions, according to Petersson [32], so that tworigid-body sections are connected through a cohesive ligament at the central line, as depicted in Fig. 7a.

Fig. 8a and b show the nondimensional load (2Ps=BDft) vs. the nondimensional displacement obtained from the proposedmodel for different sizes (bH ranging from 0.25 to 4.00) and fixing n as 0.15. In both cases, as the structural size increases (bH

increases) the nondimensional peak load decreases, exhibiting a size effect.As expected, it is clear a registered increase in the peak load value of the nondimensional load vs. the nondimensional

displacement curves for dynamic cases (Fig. 8b) in comparison with the static values (Fig. 8a). By comparison between static

Fig. 9. Parametric response of fracture parameters: bH influence and notch sensitivity.

Fig. 10. Notch effect on nondimensional load–deflection curves.

A.L. Rosa et al. / Engineering Fracture Mechanics 82 (2012) 195–208 207

and dynamic cases we can conclude that the loading rate induces a slightly more brittle behavior, dictated by the slightlyhigher relative displacements ðd=bHwchÞ at peak. However, in the loading rate range studied, the size effect plays a dominantrole compared to the rate effect. Note that, for dynamic cases, the largest beam (bH = 4) was loaded with a rate 16 times thatof the smallest beam (bH = 0.25).

In the Fig. 9a and b the notch sensitivity on the work of fracture per unit area is depicted for various values of n and spec-imen size. For a fixed notch depth ratio the work of fracture per unit area is more pronounced for large specimens than forsmall ones. For dynamic cases, for a fixed notch-depth ratio a0=D and bH, a reduction in the exponent of rate dependence n isfollowed by an increase in GF=Gs

F .For high notch-depth ratio ða0=D > 0:7Þ the structural beam shows a high notch sensitivity (Fig. 9a and b). In this range

the beam experiences a ductility enhancement caused by the reduction of the ligament length and an adequate developmentof the FPZ on the crack front is not achieved. The notch-depth sensitivity on the structural ductility is more clearly shown inFig. 10, where the nondimensional load–deflection curve predicted by the current model is depicted. In addition, analyzingFig. 8, we can see that for any nondimensional applied loading rate _d= _w0 in the range (0, 0.7), the decrease of the value GF=Gs

F

is small and it is therefore difficult to detect this influence from experiments. Conversely the parametric analysis depicted inthe Fig. 9a and b indicate that the notch size should not be larger than 70% of the beam depth in order to avoid considerableerrors in measuring fracture energy for static and dynamic cases. It needs to be pointed out that, when the notched tip isbeyond 70% of the beam depth, the numerical calculations in Fig. 9a and b are less accurate due to the relatively fewer dis-cretization nodes employed.

Based on this parametric study, we can conclude that for dynamic cases, due to the rate effect, the fracture energy is sig-nificantly greater than the static fracture energy and therefore ceases to be a material parameter.

6. Summary and conclusions

This paper accounts for a numerical implementation of a time dependent cohesive model aimed at disclosing loading rateeffects in quasi-brittle fracture. We have assumed predefined crack (or discrete crack) so that the cohesive material proper-ties along the crack path are time-dependent and the bulk material follows a linear elastic, time-independent relation. At thisstage of the investigation we have shown that the phenomenon can be adequately described by a simple model, using twoadditional parameters to the usual softening model (two for Planas’ model, but possibly more for other models). The meth-odology has been successfully applied to experimental data obtained from a set of high strength concrete (HSC) beams testedin a three point bending test (TPBT) configuration. Further investigation should be conducted to take into account size andgeometry effects.

A parametric study was performed to demonstrate how the parameters related with the time dependent model reflect thefracture response of the material. The time dependent model implemented a viscous cohesive model according to Ruiz et al.[15,16] in a finite element based program using the smeared crack tip method developed by Planas and Elices [31], based onPetersson’s classical influence method [32]. The viscous cohesive model was expressed in a macroscopic point of view so thatthe parameters governing the model were obtained from the parameters governing Hillerborg’s fictitious crack model [30]and by regression data of experimental peak load versus applied loading rate. Inertial and wave effects are negligible for theloading rate range applied in the tests, which varied over six orders of magnitude, from 10�5 mm/s to 101 mm/s.

Based on the findings of this research, the following conclusions can be drawn:

� The numerically obtained results match reasonably well the experimental ones, particularly the maximum load for theseveral loading rates used in the tests.� The model can explicitly represent the rate dependence of the peak load and fracture energy such that the peak load and

the apparent fracture energy increase monotonically with increasing loading rate.� The developed FPZ size can also be extracted and coincides with that of experimental values measured through strain

gauge techniques.� Based on the parametric study we can conclude that the fracture energy is significantly affected by the applied loading

rate, and so, that rate effect is the main reason for the differences found in determinations reported in the literature.� Numerical results show that the model, despite its simplicity, provides a general approach to reflect the experimentally

documented fact that crack propagation, and so the parameters governing the fracture process, are dependent on theloading rate.� As pointed out by numerical and experimental comparisons, the model seems promising and further refinements to it are

likely to improve accuracy of its predictions of rate effects.

Acknowledgments

The authors gratefully acknowledges the financial support provided by the Brazilian institutions CAPES (Grant: CAPES/DGU 152/07, Process: BEX 1409/07-7) and FAPESP (Grant: 2006/01647 � 4), by the Spanish Ministry of Science and Innova-tion (Grant: MAT2009-12023) and by the UCLM Project (Ref.: 2008-BIN-2506).

208 A.L. Rosa et al. / Engineering Fracture Mechanics 82 (2012) 195–208

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