fatigue crack propagation model for plain concrete – an analogy with population growth

Engineering Fracture Mechanics 77 (2010) 3418–3433

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Engineering Fracture Mechanics

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Fatigue crack propagation model for plain concrete – An analogywith population growth

Sonalisa Ray, J.M. Chandra Kishen ⇑Department of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India

a r t i c l e i n f o

Article history:Received 26 November 2009Received in revised form 28 July 2010Accepted 14 September 2010Available online 29 September 2010

Keywords:FatiguePlain concretePopulation growth modelDimensional analysisSelf-similaritySize-effect

0013-7944/$ - see front matter � 2010 Elsevier Ltddoi:10.1016/j.engfracmech.2010.09.008

⇑ Corresponding author. Tel.: +91 80 2293 3117;E-mail address: [email protected] (J.M.

a b s t r a c t

In this work, an analytical model is proposed for fatigue crack propagation in plain concretebased on population growth exponential law and in conjunction with principles of dimen-sional analysis and self-similarity. This model takes into account parameters such as load-ing history, fracture toughness, crack length, loading ratio and structural size. Thepredicted results are compared with experimental crack growth data for constant and var-iable amplitude loading and are found to capture the size effect apart from showing a goodagreement. Using this model, a sensitivity analysis is carried out to study the effect of var-ious parameters that influence fatigue failure.

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1. Introduction

Fatigue phenomenon is a gradual, permanent, micro structural change that takes place in a material due to the applicationof repeated loading. Although, the nominal stress may be well below the yield limit of the material, the local stress may behigh enough near the crack tip causing further propagation of crack, eventually leading to failure. Fatigue problems may arisewhen concrete structures such as bridges, airport pavements and highway pavements are subjected to repetitive kind ofloading during their service lives. The fatigue behavior for metals and ceramics is well explained in the literature basedon dislocation theory. However, fatigue study on concrete is rather limited and not well understood.

In case of concrete, fatigue failure is attributed to inhomogeneity and pre-existing structural defects. Fatigue failuremechanism has three distinct phases, starting from the weak regions called flaw initiation followed by progressive growthof initial flaw to critical size and leading to unstable crack propagation and failure of the structure [1]. The complexity inconcrete increases due to the presence of a large process zone ahead of the crack tip. Within the fracture process zonethe distribution of normal tensile stress gradually increases from the initial crack tip and reaches the tensile strength ofthe material (ft), at the end of FPZ [2], as shown in Fig. 1.

Fatigue behavior of concrete is generally characterized by the well known S–N curve and the mean fatigue life of themember is predicted under a given fatigue stress level. According to Oh [3], probabilistic distributions of fatigue life of con-crete depend on the level of applied stress and hence, conventional S–N curve approach requires time consuming experi-mental data collection followed by statistical analysis. Each curve is valid for a constant ratio between maximum andminimum stress and not applicable to different design and loading cases. Most importantly, this approach does not includefundamental material parameters [4].

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Nomenclature

B width of beamC Paris law parameterD depth of beamE elastic modulusF overload functionGf fracture energyDGI change in mode I energy release rateKImax maximum stress intensity factorDKI mode I stress intensity factor rangeKIC mode I fracture toughnessKIf size independent fracture toughnessL ligament lengtha0 initial notch lengtha crack lengthd0 transitional sizef frequency of external loadingrt tensile strengthn power-law exponentN no. of cyclesNf computed number of cycles at failureNc number of cycles at failure obtained from experimentsR loading ratiob brittleness numbera relative crack depth (a/D)pi coefficient of variationvyi coefficient of variation with ith parameter as randomvy coefficient of variation with all input quantities as randomri Spearman rank coefficients

S. Ray, J.M. Chandra Kishen / Engineering Fracture Mechanics 77 (2010) 3418–3433 3419

Development of mechanistic approaches using the concepts of fracture mechanics, for the study of crack propagation un-der fatigue loading began with the well known Paris law [5] wherein crack growth increment per load cycle is a function ofapplied stress intensity factor amplitude. Attempts have been made by many researchers [6–8] to model crack growth ofconcrete by applying Paris law [5]. However, one important aspect, namely the size effect has been scarcely reported inthe literature concerned with fatigue of concrete.

Fig. 1. Fracture process zone.

3420 S. Ray, J.M. Chandra Kishen / Engineering Fracture Mechanics 77 (2010) 3418–3433

The first attempt was made by Bazant and Xu [9] wherein they combined the size effect law to the existing Paris law andproposed a size adjusted Paris law applicable for plain concrete as

dadN¼ C

DKI

KIC

� �m

ð1Þ

KIC ¼ KIfb

1þ b

� �12

ð2Þ

where KIC is defined as the size dependent fracture toughness, KIf is the fracture toughness which has been extrapolated forinfinitely large size structure and b = D/d0 with D being the beam depth and d0 the transitional size. This law gives the cracklength increment per cycle as a power function of the amplitude of a size adjusted stress intensity factor. The size adjustmentwas done by adjusting the transition size d0, which is considered as a material constant. The numerical value of the transi-tional size is different for fatigue loading than the monotonic one. In their model, to compute fatigue law coefficients, tran-sitional size has been adjusted and the adjustment factor used for d0 happens to be 10 times the monotonic one whichaccording to the proposers is obtained by chance. The law has been validated with the experimental data on three-pointbend beams. A similar law has been proposed describing the fatigue fracture of high strength concrete for predicting crackgrowth [10]. However, concrete exhibits typically nonlinear fracture processes due to the large size process zone and makesthis linear elastic fracture mechanics approach (LEFM) questionable.

Slowik et al. [11] proposed a linear elastic fracture mechanics based fatigue crack propagation law which included param-eters such as the fracture toughness, loading history, and specimen size. This law applicable to variable amplitude fatigueloading is described by

dadN¼ C

KmImaxDKn

I

KIC � KpIsup

� �þ Fða;DrÞ ð3Þ

where KImax is defined as the maximum stress intensity factor for a particular load cycle, KIsup is the maximum stress intensityfactor ever reached by the structure in its loading history and F(a,Dr) is a function which takes into account the effect ofoverloads. KIC is the material fracture toughness and the coefficients m, n and p are constants for all structural sizes andare determined through an optimization process.

Sain and Chandra Kishen [16] have modified the above law to include the effects of loading frequency and overload func-tion, F(a,Da). The major limitation of the Slowik’s model and the modification made by Sain and Chandra Kishen [16] is that,they are not dimensionally homogeneous due to empirical nature of the proposed equations. Similar models based on thevariation of Paris law have been proposed by Kolluru et al. [4]. They have observed that the crack growth due to fatigue load-ing comprises of a deceleration stage followed by an acceleration stage and they developed analytical expressions for thecrack growth in both the stages. Zhang et al. [12] have proposed a semi analytical method for modeling fatigue performancein flexure of plain concrete. Through S–N curve approach it has been shown that the fatigue performance in bending isstrongly dependent on the structural size.

Many researchers have used the concepts of dimensional analysis and self-similarity to study fatigue behavior of plainconcrete. Carpinteri and Spagnoli [13] have proposed a size dependent fatigue crack propagation law for concrete that ex-presses the crack growth rate against the stress intensity factor range. The concept of fractal geometry was used togetherwith a new definition of fracture energy and SIF based on physical dimensions that are different from the classical ones.Spagnoli [14] has derived a crack size dependent Paris law using similarity methods and fractal concepts. The form of thefatigue law proposed is based upon the assumption of an incomplete self-similarity. The author has only shown the depen-dency of the fatigue parameters on growth rate and has not obtained any closed form expression for the fatigue crack prop-agation model. Carpinteri and Paggi [15] have proposed an approximate relationship between the Paris law coefficients Cand m. Two independent approaches, self-similarity concepts and the condition that the Paris law instability correspondsto the Griffith–Irwin instability at the onset of rapid crack growth have been used. Further, Ciavarella et al. [17] have general-ized the Barenblatt and Botvina dimensional analysis approach to study the functional dependencies of m and C on addi-tional dimensionless parameters using experimental data for both steel and concrete. A correlation between m and C isdeveloped which is observed to be quite different for ductile and quasi-brittle materials. Similar studies using dimensionalanalysis are reported [18,19] to derive generalized Wohler’s and Paris equations. An analytical approach was proposed tostudy the relationships between Wohler’s and Paris representations of fatigue. It is observed that the parameters, micro-structural size and crack size in both the relationships are size-scale dependent.

2. Research significance

In order to throw further light on the dependence of various parameters on the fatigue behavior of concrete, in this study,a fatigue crack propagation law is proposed. This law is based on the population growth model and is developed in conjunc-tion with the principles of dimensional analysis and self-similarity. The specific growth rate parameter in population growthmodel is defined in an analogous manner for the crack growth rate and it includes the crack growth influencing parameterssuch as loading history, fracture toughness, loading ratio, crack length and size effect. The effect of crack size on growth rate


and the relationship with Paris law constants are discussed. A sensitivity study is carried out to determine the dominantparameters that affect the fatigue life.

3. Theory of growth

To model the growth of biological system various models have been proposed in the literature. The history of growth the-ory started around 200 years ago and is explained by Malthus [20] as ‘Law of geometric growth’. According to this model,population grows at a rate proportional to its size at any given time and described by

dxdt¼ kx ð4Þ

The proportionality constant k relates to the size of the population at any time x(t), and its rate of growth, dx/dt. The solutionto Eq. (4) is given by

xðtÞ ¼ mekt ð5Þ

where m = x(0). In short, unconstrained natural growth is exponential in nature. Verhulst [21] has developed a logistic equa-tion which states that the population competes with itself, that is, as population increases, its growth rate decreases linearly.The differential equation representing this growth rate is given by

dxdt¼ k 1� x

l

� �x ð6Þ

The parameter l is the carrying capacity. As x(t) approaches l, the growth rate approaches zero and the growth ultimatelystops. According to Turner et al. [22] all classic growth models are interrelated and can be derived from a single parent gen-eric equation written in the form:

dxdt¼ m

kn x1�npðkn � xnÞ1þp ð7Þ

where x is the population size at time t. x is determined by integrating Eq. (7) as:

x ¼ k

1þ ½1þmnpðt � sÞ��1p

� �1n

ð8Þ

where m, k, n, and p are the parameters in the rate equation which determine the shape of the curve. The parameter m is themaximum specific growth rate which the population could attain in the absence of any limitations to growth. Similarly, theparameter k is the maximum growth in the stationary phase under limited growth. The parameter s is related to the time atwhich the maximum specific growth rate is attained. The parameters n and p are related to metabolic efficiency.

Classical growth models can be derived by placing appropriate limits and conditions on the selected parameters in thegeneric model. For example, in the case wherein the growth rate is proportional to the size of population under no limitingfactors, the generic curve reduces to an exponential curve. Bacterial growth in liquid is triphasic in nature. In the initial per-iod, the growth rate will be small followed by rapid growth which eventually ceases and reaches stationary phase, as shown

Fig. 2. Microbial growth curve.


in Fig. 2. A crack growth curve for plain concrete shows a trend similar to microbial growth during the first two phases. Dur-ing the initial stage of fatigue loading, a crack grows at a slower rate and after sufficient number of cycles have elapsed, thecrack begins to propagate in an unstable manner, thereby leading to failure of the component. In this study, an attempt hasbeen made to model crack growth rate using the theory of microbial growth. The parameters in the growth model are deter-mined using the concepts of dimensional analysis and self-similarity.

4. Self-similar solutions as intermediate asymptotic

Scaling laws or power-laws which describe the power-law relationship between different quantities give an evidence of avery important property of self-similarity. In case of a self-similar phenomenon, the spatial distribution of its properties atvarious different moments of time can be obtained from one another by a similarity transformation [23]. In construction ofan analytical model, it is impossible to take into account all the factors which influence the phenomenon. So, every model isbased on certain idealization of the phenomenon. In constructing these idealizations, the phenomenon under study shouldbe considered at intermediate times and distances. Therefore, every mathematical model is based on intermediate asymp-totic. In fact, self-similar solutions not only describe the behavior of the physical systems under some special conditions butalso describe the ‘intermediate asymptotic’ behavior of the solution to broader classes of problems, i.e. the behavior in theregions where these solutions have ceased to depend on the details of the initial conditions and boundary conditions [24]. Inother words, the intermediate asymptotic is a time-space dependent solution of an evolution equation that already forgot itsinitial conditions, but still does not feel the limitations imposed by the system boundary. Fig. 3 shows a typical demonstra-tion of intermediate asymptotic where, surface waves are generated by vibration of triangular element on surface of origi-nally still liquid confined in rectangular container (top view). As the waves propagate outwards, they gradually obtain thenatural circular shape, being undisturbed by either boundary. This happens at the intermediate distance between containercenter and the container wall. The solution describing the circles is an intermediate asymptotic (IA) of the system [25]. Fig. 4represents, all the phenomena in the universe through nesting of similarity concepts.

5. Complete and incomplete similarity

In any physical study (theoretical or experimental), we attempt to obtain the relationships among the quantities thatcharacterize the phenomenon being studied. Thus, the problem always reduces to determining one or several relationshipof the form

a ¼ f ða1; . . . ; ak; akþ1; . . . ; anÞ ð9Þ

where a is the quantity being determined in the study and is known as governed parameter, and (a1, . . . ,ak,ak+1 ,. . . ,an) are thequantities that are assumed to be given, called as governing parameters. On applying Buckingham P theorem to Eq. (9), ittakes the form

a ¼ ap1; . . . ; ar

kUakþ1

apkþ11 ; . . . ; arkþ1

k

; . . . ;an

apn1 ; . . . ; arkþ1

k

!ð10Þ

Fig. 3. Demonstration of intermediate asymptotic.

Fig. 4. Nesting of similarity concepts. DA: dimensional analysis; CS: complete similarity; IS: incomplete self-similarity; IA: intermediate asymptotic; AI:absence of intermediateness.


Introducing the dimensionless parameters as:

Pi ¼akþi

apkþi1 ; . . . ; arkþi

k

ð11Þ

Let P ¼ aap

1; . . . ; ark

ð12Þ

Eq. (10) may be rewritten as

P ¼ UðP1; . . . ;Pn�kÞ ð13Þ

where U is a function containing nondimensional terms. On applying P theorem, U turns out to be a function of (n � k) vari-ables only. We shall now discuss two important terms associated with dimensional analysis.

(1) Self-similarity of first kind(2) Self-similarity of second kind

Let us consider the parameter a1. This parameter is considered as non-essential if the corresponding dimensionlessparameter P1 is too large or too small (tend to zero or infinity), giving rise to a finite non-zero value of the function U withthe other similarity parameters remaining constant. The number of arguments can now be reduced by one and we can writeEq. (13) as:

P ¼ U1ðP2; . . . ;Pn�kÞ ð14Þ

where U1 is the limit of the function U as P1 ? 0 or P1 ?1. This is called complete self-similarity or self-similarity of firstkind [23]. On the other hand, for P1 ? 0 or P1 ?1, if U also tends to zero or infinity, then the quantity P1 becomes essen-tial, no matter how large or small it becomes. However, in some cases, the limit of the function U tends to zero or infinity, butthe function U has power type asymptotic representation which can be written as,

U ffi Pc1U1ðP2; . . . ;Pn�kÞ ð15Þ

Substituting into Eq. (13), we obtain

P� ¼ PPc

1

¼ U1ðP2; . . . ;Pn�kÞ ð16Þ

where the constant c and the nondimensional parameter U1 cannot be obtained from the dimensional analysis alone. This isthe case of incomplete self-similarity or self-similarity of second kind. It can be noted here that, the parameter c can only beobtained either from a best fitting procedure on experimental results or according to numerical simulations.

6. Formulation of fatigue crack propagation model and discussions

On adopting the population growth exponential model to the study of crack propagation due to fatigue loading, the rate ofcrack propagation takes the form


dadN¼ ak ð17Þ

where a is the crack length, N is the number of load cycles and k is the proportionality constant analogous to the specificgrowth rate and is derived here using the concepts of dimensional analysis. This simplest model states that the growth rateis proportional to the crack length and the proportionality constant k relates the crack length a(N) to its rate of growth da/dNthrough other governing parameters. It should be noted at this point that the effect of loading history is of paramount impor-tance in the fatigue behavior of concrete. It can be seen from Eq. (17) that, the crack length at a particular instance is theconsequence of loads that have been already applied. So, computation of crack growth due to each load cycle considersthe crack length due to previous cycle and thereby includes history effect.

We consider k as the quantity to be determined in the phenomenon which is governed by the quantities such as change inenergy release rate (DG) and the brittleness number (b). Brittleness number is defined as the ratio of structural size to tran-sitional size (D/d0) and can be used to indicate the closeness of the structural behavior to linear elastic fracture mechanics orlimit analysis [26]. Also, the rate of fatigue crack growth is governed by the loading ratio, R (ratio of minimum applied stressto maximum applied stress) and loading frequency x. The material properties considered in the model development are sizeindependent fracture energy Gf and tensile strength rt. Now we can write the above dependence as follows

k ¼ UðDG;Gf ;rt;R; b;x; tÞ: ð18Þ

where governing variables are summarized with their physical dimensions expressed in the Length–Force–Time class (LFT)in Table 1. Considering a state of no explicit time dependence and Gf, rt as nondimensional parameters, dimensional analysisgives

k ¼ UDGGf

;R; b� �

ð19Þ

where the nondimensional quantities are

P1 ¼DGGf

; P2 ¼ R; P3 ¼ b ð20Þ

Considering P1, it is usually small in the intermediate range of fatigue crack growth and consideration of complete self-sim-ilarity would make the crack growth independent of DG. Hence, assumption of incomplete self-similarity in P1, gives

k ¼ DGGf

� �c

U1ðR;bÞ ð21Þ

Now, the crack growth rate can be expressed in terms of k in the following way:

dadN¼ ak ð22Þ

dadN¼ a

DGGf

� �c

U1 ð23Þ

The above equation may be rewritten as

dadN¼ a

DGGf

� �p

U ð24Þ

where c = p, U1 = U, where the exponent p and the nondimensional parameter U, cannot be determined from the consider-ation of dimensional analysis alone. These parameters can only be obtained either from a best fitting procedure on experi-mental results or according to numerical simulations. In the next section, these parameters are obtained by a best fit methodfollowing the principles of least square using the experimental data of Bazant and Xu [9].

Table 1Governing variables of the fatigue crack growth phenomenon in plain concrete.

Variables Definitions Dimensions

DG Energy release rate range FL�1

Gf Fracture energy FL�1

rt Tensile strength FL�2

R Loading ratio –

b Dd0

� �Brittleness number –

x Loading frequency T�1

t Time T


Further, the aforementioned assumptions of incomplete self-similarity conditions are validated with the available exper-imental data. Using the experimental results of Bazant and Xu [9] and Shah [27] for normal strength concrete, the nondimen-sional quantity P1 is computed to be between 0.35–0.70 and 0.19–0.40 respectively. Similarly, for high strength concrete[10] the value of P1 falls in the range of 0.25–0.56. From these values, it can be observed that the assumption of incompleteself-similarity in P1 is approached asymptotically in the experimental study. Hence, on observation of these range of exper-imental values we may assume that an incomplete self-similarity is achieved for 0 < P1 < 1. Now considering the parameterP3, for normal strength concrete it varies between 0.52–2.00 and 0.54–2.18 for both the set of experimental data. On theother hand, the experimental values fall between 1.19 and 9.58 for high strength concrete. It is to be noted here that, theconsideration of incomplete self-similarity in P3 applies for P3 ? 0. However, for relatively large size specimen, P3 ?1,resulting in a complete self-similarity. For the intermediate range of structural sizes a transition between these two kindsof self-similarity may occur which can be categorized as non self-similarity. In the present study, for normal strength con-crete [9,27] the structural sizes are small hence, P3 ? 0 and a power law relationship exists between C and P3 (P3 ? 0). Butfor relatively large size specimen [10] scaling law does not hold between P3 and Paris’ coefficient, C. To verify this further,available experimental data for normal and high strength concrete is used to plot the LogC versus P3 and LogC versus LogP3

relationship as shown in Fig. 5. Based on the regression coefficient values (R2), those having higher coefficients give the bestcorrelation. For normal strength concrete it is observed that, the assumption of incomplete self-similarity gives better cor-relation. But on the other hand, for high strength concrete an opposite trend is observed indicating that scaling law does nothold for this case.

7. Calibration of the proposed model

Experimental results of Bazant and Xu [9] are used to calibrate the proposed model. In their experimental study, Bazantand Xu [9] have tested a series of geometrically similar three-point beams under fatigue loading. The details of dimensionsand physical properties of these specimens are given in Table 2. Sinusoidal loading was applied with a frequency varyingbetween 0.033 and 0.04 Hz. The maximum loads were kept constant and equal to 80% of the monotonic peak load andthe minimum loads were kept at zero. Hence, the loading ratio R turns out to be zero in this case and therefore a closed formexpression for U(b,R) is obtained only in terms of b. The coefficients p and U are determined through an optimization pro-cess using the principle of least squares. The best suited value of p is found to be 4.6663 and indeed considered as a materialconstant. Using the experimental results of Bazant and Xu [9] for small, medium and large sized specimens, a relationshipbetween U and the nondimensional parameter b is obtained in Fig. 6 and the best fit quadratic relationship is given by

LogðUÞ ¼ �0:5202fLogbg2 � 4:9447fLogbg � 1:9046 ð25Þ

Since the nondimensional parameter b can reflect the differences in structural size as well as the geometry, the above equa-tion can be used to obtain the value of U for any given size and geometry of specimen.

8. Validation of the proposed model and discussions

The proposed fatigue crack propagation law represented by Eq. (24) contains two constants p and U which have beenobtained from the experimental results of Bazant and Xu [9] in the previous section. In this section, the proposed law ischecked for its validity on different loading ratios (R) and other specimen geometry. Since the loading ratio in the experimen-tal data used for the calibration of the model was zero, its presence could not be reflected in Eq. (25). Hence, this is achievedby using the fatigue test data and results of Shah [27]. The proposed fatigue model is validated using the experimental resultsof Shah [27] for three-point geometrically similar beams of different sizes and R ratios, Toumi et al. [28] for beams with vary-ing R ratios and Slowik et al. [11] for compact tension specimens.

In the fatigue test performed by Shah [27], three-point geometrically similar bend beams of small, medium and large sizeswere subjected to variable amplitude sinusoidal loading of frequency 1 Hz. The specimen details and material properties aretabulated in Table 3. The fatigue loading was applied in such a way that the maximum load was increased by 0.5 kN afterevery 500 cycles as shown in Fig. 7. A minimum load of 0.2 kN is applied for all cycles in order to have some contact betweenloading device and the specimen and also to avoid any impact loading during application of the fatigue load.

As reported in the literature [29], the crack growth is a function of mean stress level and hence the loading ratio, whichleads to a decrease in the number of cycles to failure with increasing stress range or loading ratio. Therefore, the effect ofloading ratio is included into Eq. (25) through an optimization process using the principle of least squares. This principleis based on the concept of minimization of sum of squared residuals, a residual being the difference between an experimen-tally observed value and the value obtained from the model. The modified expression for U takes the form

LogðUÞ ¼ �0:5202fLogbg2 � 4:9447fLogbg � 1:9046þ Rf1:8bg ð26Þ

It may be noted here that the form of Eq. (25) is retained in Eq. (26). Furthermore, the transition size d0 used in size effect law[2] corresponds to peak load states at monotonic loading and is different for fatigue loading. In this validation study, an opti-mum fit with the experimental data has been obtained with d0 = 230 mm whereas, the measured d0 value for the static test is138.4 mm. Using the modified model described by Eq. (24) and (26), the relative crack growth is computed and is plotted as a

Fig. 5. Power law assessment in P3. (a and b) Normal concrete [9]. (c and d) Normal concrete [27]. (e and f) High strength concrete [10].


function of number of cycles together with the experimental data, as shown in Fig. 8. It is observed that, the proposed modelfollows the experimental trend quite well.

To verify the accuracy of the proposed model after including the effect of loading ratio (R), the experimental data of Toumiet al. [28] was used. The dimensions and the material properties are tabulated in Table 2. They have tested three-point bendbeams at different values of upper fatigue loading (0.87Fu,0.81Fu,0.76Fu,0.71Fu) keeping lower limit as constant at 0.23Fu,where Fu is the peak load in the static tests. Fig. 9 shows the plot of crack growth rate versus the stress intensity factor rangefor different stress ratios predicted by the proposed model together with the experimental ones. In Table 4, a comparative

Table 2Geometry and material properties of specimens.

Specimen designation Depth D (mm) Span S (mm) Thickness B (mm) Notch size a (mm) Gf (N/mm) E (MPa)

Beam (small) [9] 38.1 95 38.1 6.35 0.038 27,120Beam (medium)[9] 76.2 191 38.1 12.7 0.038 27,120Beam (large) [9] 152.4 381 38.1 25.4 0.038 27,120Beam [28] 80 420 50 40 0.07 40,000Compact tension (small) [11] 300 – 100 90 0.05 16,000Compact tension (large) [11] 900 – 400 270 0.05 17,000

Fig. 6. Relationship between U and the nondimensional parameter.

Table 3Geometry and material properties of beam specimens [27].

Depth D (mm) Span S (mm) Thickness B (mm) Notch size a (mm) Gf (N/mm) d0 (mm)

76 190 50 15.2 0.07 138.84152 380 50 30.4 0.07 138.84304 760 50 60.8 0.07 138.84

Fig. 7. Pattern of fatigue loading.


Fig. 8. Experimental and computed growth of relative crack length with the number of load cycles.

Fig. 9. Crack growth rate versus stress intensity factor amplitude.

Table 4Validation of proposed model with experimental results of Toumi et al. [28].

Stress ratio Log DKIKIC

� �Log da

dN

� �(Proposed model) Log da

dN

� �(Experiment)

Fmax/Fu = 0.87 �0.136 �4.02 �3.97�0.104 �3.71 �3.74�0.072 �3.39 �3.57�0.008 �2.96 �3.34

Fmax/Fu = 0.81 �0.16 �4.21 �3.98�0.136 �3.99 �3.81�0.123 �3.86 �3.75�0.104 �3.67 �3.65

Fmax/Fu = 0.76 �0.216 �4.72 �4.24�0.208 �4.64 �4.08�0.2 �4.57 �3.97�0.123 �3.85 �3.82�0.048 �3.1 �3.34


study of numerical values is made between the crack growth rates obtained from the proposed model and the experimentalstudy. A reasonably good agreement is seen between the two.

The proposed model is also validated for a different specimen geometry, other than the one used for calibration. For this,the experimental results of Slowik et al. [11] are used. They have performed experiments on compact tension specimens for

Table 5Validation of proposed model with compact tension specimen results [11].

Specimen Designation Pmax (kN) Pmin (kN) dN (Cycles) da (mm) dadN (Experiment) da

dN (Proposed model)

Large G13 50 0 900 3.2 3.5 � 10�3 2.7 � 10�3

G10 50 0 1440 1.0 6.94 � 10�4 2.30 � 10�3

G07 50 0 900 2.0 2.22 � 10�3 2.9 � 10�3

G06 50 0 1800 0.5 5.5 � 10�4 8.17 � 10�4

Small K13 2.1 1.1 540 0.5 9.26 � 10�4 7.83 � 10�4

K15 1.5 0.2 540 2.8 1.85 � 10�3 5.0 � 10�3

K06 1.65 0.55 900 1.3 1.44 � 10�3 1.10 � 10�3


both large and small size under variable amplitude loading interrupted by spikes. The specimen details are given in Table 2.The validation study was done only on a few experimental data points containing no spikes. The crack growth results fromthe model predictions and experimental study are tabulated in Table 5. Once again, a fairly good agreement is seen by theproposed model and the experimental results.

9. The effect of crack size on growth rate

In the experimental study [4], it has been shown that the crack growth comprises of two phases: a deceleration followedby an acceleration phase. For cracks shorter than the crack length at peak load in quasi-static monotonic loading, da/dN is adecreasing function of a and later on the acceleration phase begins. To model the crack growth, Spagnoli [14] proposed afatigue crack propagation law based on dimensional analysis concept, wherein the crack growth rate is a power type func-tion of crack length. In their work, it may be noted that for small cracks, the nondimensional ratio (P5) associated with atends to zero, which means that an incomplete self-similarity occurs in P5 and giving rise to a power type function involvingthe crack length. However, for relatively long cracks, the nondimensional parameter tends to infinity and a complete self-similarity occurs which corresponds to a crack growth independent of crack size. Further, using experimental data it hasbeen observed that the coefficient (c2) in P5 takes a negative value for short cracks indicating a deceleration occurring inthe crack growth. During the acceleration phase of crack growth, this nondimensional parameter takes a positive value.In the present study, the proposed model is the direct consequence of the analogy with the population growth model wherecrack growth is a linear function of crack size. This can be interpreted as a particular case of the fatigue law proposed bySpagnoli [14] wherein the parameter c2 ? 2. Hence, the proposed model does not capture the deceleration phase of the crackpropagation. Furthermore, to verify the dependence of growth rate on crack length, the experimental data of Shah [27] andthe model predictions are plotted together in Fig. 10 wherein Log da

dN

� �is plotted against the crack length for the small, med-

ium and large size specimen. It is seen that the proposed model can capture the acceleration phase reasonably well, espe-cially in the region close to failure.

10. Paris law coefficients

In this section, the proposed model is compared with the Paris law and its coefficients are determined. The experimentalresults of Bazant and Xu [9] that were used for calibration of the model are used to compute the Paris law coefficients. Small,

Fig. 10. Fatigue crack growth (deceleration and acceleration).

Fig. 11. Relative crack length with number of load cycles.

Table 6Number of cycles to failure.

Specimen No. of cycles

Proposed model Experiment

Beam (small) 914 974Beam (medium) 846 850Beam (large) 910 882


medium and large sized three-point beam specimens whose details are given in Table 2 are analyzed. Fig. 11 shows the plotof relative effective crack length a/D, as a function of number of load cycles for the proposed fatigue law. The number of cy-cles at failure which is computed using the proposed fatigue law are listed in Table 6 together with the experimentally ob-served ones. It is seen that the agreement between the two is reasonably good. Fig. 12 shows logarithmic plot of crack growthrate versus stress intensity factor amplitude normalized with size independent stress intensity factor (KIf). In this figure, theexperimental results reported by Bazant and Xu [9] are plotted together with the proposed fatigue law considering the caseof constant amplitude loading. From this plot, the values of the Paris law coefficients LogC (vertical axis intercepts) and m(slope of the straight line) are computed and compared with the experimentally reported ones in Table 7. A good agreementis seen between the experimental results and the proposed fatigue model. If Paris law has to be valid for concrete in general,the plot in Fig. 13 should have to be a single straight line for all specimen sizes. Although the slopes are not very much dif-ferent from each other, there are great differences among the C values, implying that Paris law is not valid for concrete ingeneral. Hence, the Paris law was modified by Bazant and Xu as described by Eqs. (1) and (2) wherein a size dependent frac-ture toughness KIC is used instead of KIf.

Fig. 12. Crack length increment per cycle versus the stress intensity factor amplitude.

Table 7Paris law parameters.

LogC m

Specimen Proposed model Experiment Proposed model Experiment

Beam (small) �16.284 �16.7 11.43 11.78Beam (medium) �16.517 �18.2 10.5 9.97Beam (large) �17.831 �19.6 10.46 9.27

Fig. 13. Distribution curves of fatigue life Nf for different random variables.


11. Sensitivity analysis

In the previous section, it is observed that the fatigue crack propagation law that has been proposed for plain concretetakes into consideration several parameters such as fracture toughness, loading ratio, structural size and loading history.Many of these parameters vary randomly and cannot be obtained in a deterministic sense. Hence, it becomes necessaryto determine the parameters which play a dominant role on the fatigue life. To this end, a sensitivity study is carried outto determine which of these parameters are more sensitive to be considered as a random quantity rather than deterministic.To achieve this, the influence in the variation of the maximum number of load cycles Nf corresponding to the critical cracklength, on the variability of input random quantities is studied. Two statistical methods are used for computing the sensi-tivity coefficients by considering depth of beam (D), applied loading range (DP), transition size (d0) and fracture toughness(Gf) as random variables.

11.1. Sensitivity analysis using coefficient of variation

This method is based on the comparisons of the sensitivity coefficients pi, which is defined by

pi ¼ 100v2

yi

v2y

ð27Þ

where vyi is the coefficient of variation of the output quantity keeping the ith input parameter as random and others deter-ministic; where i = 1,2, . . . ,N and N is the number of input quantities examined in the sensitivity analysis. vy is the coefficientof variation of the output quantity, considering all the input quantities as random ones. The concept is based on the assump-tion that higher value of sensitive coefficient pi indicates higher degree of correlation and therefore higher influence of inputvariable on the output [30].

11.2. Sensitivity analysis using Spearman rank-order correlation

Spearman rank-order correlation ri is a non-parametric measure of correlation that is, it gives an assessment of how wellan arbitrary monotonic function could describe the relationship between two variables. The calculation of Spearman rankcoefficient is carried out based on the ranks of the data. Each parameter is ranked separately by arranging the values in orderand ranking is achieved by giving the rank ‘1’ to the biggest number in a column, ‘2’ to the second biggest value and so on.The smallest value in the column will get the lowest rank. Tied scores are given the mean (average) rank. The null hypothesisin this method is that the rank of one variable does not covary with the rank of other variables; in other words, as the ranks ofone variable increases, the rank of other variables are not more likely to increase (or decrease). Spearman coefficient gives a

Table 8Statistical parameters used in the sensitivity studies.

Variables Distribution Mean Standard deviation Coefficient of variation (pi) % Spearman rank correlation coefficient (ri)

D (mm) Normal 38.1 5 61.52 0.68DP (N) Normal 1452 95 16.5 �0.28d0 (mm) Normal 72.6 7.26 14.63 �0.26Gf (N/mm) Normal 0.04 0.004 11.30 0.223


measure of the linear relationship between two sets of ranked data, i.e. it measures how tightly the ranked data clustersaround a straight line [31]. The correlation coefficient always takes a value between �1 and +1. A positive correlation means,the rank of both the variables increase together whereas a negative correlation means that, as the rank of one variable in-creases, the rank of the other decreases. A correlation of �1 or +1 indicates that the relationship between the two variables isexactly linear. The closer is the value of ri to +1 or �1, the stronger the likely correlation. A perfect positive correlation is +1and a perfect negative correlation is �1 with zero correlation at exactly zero. Spearman rank-order correlation ri is expressedas

ri ¼ 1�6P

jkji � lj� �2

NðN2 � 1Þð28Þ

where ri is the order representing the value of random variable Xi in an ordered sample among N simulated values applied inthe jth simulation, lj is the order of an ordered sample of the resulting variable for the jth run of the simulation process,(kji � lj) is the difference between the rank of two samples.

The sensitivity analysis is done using the two methods discussed above. The fatigue life Nf of the concrete beam is com-puted using the proposed fatigue law defined by Eq. (24) as
Z Nf
0dN ¼

Z ac

a0

da

a DGGf

� �pU

ð29Þ

or

Nf ¼Z ac

a0

da

a DGGf

� �pU

ð30Þ

where ac is the critical crack length which is obtained from the experimental results and U is defined by Eq. (26).By knowing the mean, standard deviation and the type of distribution of a particular input quantity, random variables are

generated using the Latin hypercube Sampling (LHS) method. These randomly generated variables are used to compute coef-ficient of variation as well as Spearman rank coefficient.

Sensitivity study is performed on small sized specimen detailed in Table 2. The statistical distributions of the randomquantities along with the results of the coefficient of variation and the Spearman rank correlation coefficient are tabulatedin Table 8. From these results, it is observed that the structural size is the most sensitive parameter followed by the loadingparameter. The quantities Gf and d0 fall in the same range with lowest coefficient. Similar trends are observed by both themethods.

In order to verify the correctness of the sensitivity coefficients, that is, whether the coefficients yield correct influence ofdifferent parameters, the results of the sensitivity analysis are plotted in terms of probability distribution functions. Fig. 13shows the computed probability distribution function wherein the ultimate fatigue life in terms of number of load cycles tofailure ‘NC’ is plotted against the chance that the computed life ‘Nf’ is less than a specific NC value. The Nf values are computedthrough Monte Carlo simulation considering one parameter as random at a time and keeping other parameters as determin-istic. If the probability distribution function depicts a steeper (more or less vertical) curve, then it implies that the parameteris not much sensitive and thus can be considered as deterministic. It is seen from this figure that the structural size is moresensitive followed by the loading parameter, the fracture energy and the transition size parameter which is similar to thoseobserved in Table 8.

12. Conclusions

In this study, a fatigue crack propagation law for plain concrete is developed based on the population growth model and aclosed form expression for the specific growth rate parameter in the fatigue model is derived using concepts of dimensionalanalysis. The model takes into account a number of parameters, such as, loading history, fracture toughness, stress ratio,crack length and most importantly the structural size. The nondimensional ratio b used in the formulation takes into accountthe different sizes as well as specimen geometry. The proposed law has been validated for constant and variable amplitude


loading cases. In addition, the law is found to agree well with the experimental results of three-point bend beams as well ascompact tension specimens. A sensitivity analysis is performed using two different methods to study the influence of each ofdifferent input parameters such as the structural size, loading ratio, fracture energy and transition size parameter on the fa-tigue crack propagation. It is observed that, the structural size is the most sensitive quantity followed by the loadingparameter.

References

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fatigue crack propagation model for plain concrete – an analogy with population growth

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