Engineering Fracture Mechanics 74 (2007) 2320–2333 www.elsevier.com/locate/engfracmech

Energy-based equivalence between damage and fracture in concrete under fatigue Trisha Sain, J.M. Chandra Kishen

*

Department of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India Received 18 May 2006; received in revised form 26 October 2006; accepted 18 November 2006 Available online 26 January 2007

Abstract Damage in concrete members, occur in a distributed manner due to the formation and coalescence of micro-cracks, and this can easily be described through a local damage approach. During subsequent loading cycles, this distributed zone of micro-cracks get transformed into a major crack, introducing a discrete discontinuity in the member. At this stage, concepts of fracture mechanics could be used to describe the behavior of the structural member. In this work, an approach is developed to correlate fracture and damage mechanics through energy equivalence concepts and to predict the damage scenario in concrete under fatigue loading. The objective is to smoothly move from fracture mechanics theory to damage mechanics theory or vice versa in order to characterize damage. The analytical methods developed here have been exemplified with some already available data in the literature. The strength and stiffness reduction due to progressive cracking or increase in damage distribution, has been characterized using the available indices such as the strength reduction and stiffness reduction factors. It is seen through numerical examples, that the strength and stiffness drop indices using fracture and damage mechanics theory agree well with each other. Hence, it is concluded, that through the energy approach a discrete crack may be modeled as an equivalent damage zone, wherein both correspond to the same energy loss. Finally, it is shown that by knowing the critical damage zone dimension, the critical fracture property such as the fracture energy can be obtained.  2006 Elsevier Ltd. All rights reserved. Keywords: Fatigue; Fracture; Damage index; Energy equivalence

1. Introduction Damage in concrete during its initial loading phase appears in the form of distributed micro-cracks. These micro-cracks are primarily due to the shrinkage of the cement paste around the aggregates. In due course of time, under sustained loading, the diffused damage expands and forms a distinct large crack due to coalescence of microcracks which would propagate and cause final failure of the structural member. There are two main category of models that describe this failure process: one that uses fracture mechanics concepts and the other which uses continuum damage mechanics concepts. Fracture mechanics is well suited to describe the *

Corresponding author. Tel.: +91 80 2293 3117; fax: +91 80 2360 0404. E-mail address: [email protected] (J.M. Chandra Kishen).

0013-7944/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2006.11.014

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Nomenclature B C D DR E Gf H K0 Kr K KIC L Mmax N Pmax SD U UD V Y(a) a da m, n, p f lch lc P Dr a rH req dr / /y /f

width of beam fatigue law parameter damage index stiffness reduction factor elastic modulus fracture energy depth of beam initial secant stiffness reduced secant stiffness stress intensity factor fracture toughness ligament length maximum bending moment load cycle maximum load strength reduction factor strain energy strain energy due to damage volume of the specimen geometry factor crack length maximum aggregate size fatigue law constants loading frequency characteristic length localization limiter potential energy stress range relative crack length hydrostatic stress Von Mises equivalent stress maximum mid-span deflection maximum curvature corresponding to damage curvature at yield load final curvature at failure

separation due to decohesion of two parts of the continuum [1,2]. It can be applied when a crack has been initiated, or assuming that there are initial flaws of known sizes and at known locations within the continuum. On the other hand, damage mechanics, which includes smeared (or distributed) crack models, describes the local effects of micro-cracking, that is the evolution of the mechanical properties of the continuum as micro-cracking develops: elastic stiffness degradation, inelastic strains, etc. When the location of an expected crack and the direction along which it propagates are unknown, fracture mechanics can hardly be used, because the critical flaw from which cracking initiates needs to be determined first. On the contrary, damage mechanics offers the essential advantage of predicting the location of this critical flaw. The bridge between fracture and damage mechanics can be considered to be the situation wherein damage is equal to unity at a material point, or in a small region defining the size of an initial flaw in the theories of fracture. In most cases, this situation corresponds to the localization of strains and damage due to strain softening. Strain softening yields, however, several problems which need to be solved in order to bridge the two theories.

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As explained in the paper by Mazars [2] the major inconsistency which is faced to correlate fracture and classical damage theories, are that the failure predicted through local damage models occurs without dissipation of energy; whereas, most criteria for discrete crack propagation are based on the quantity of energy which must be released in order to propagate a crack. This paradox has been recognized in the literature 20 years ago [3]. To circumvent this problem, non-local damage models or gradient dependent damage models are proposed. The key idea of non-local damage models is to assume that the condition of growth of damage is non-local, i.e. it depends at each material point on a weighted average of the strains in a neighborhood. This neighborhood is scaled by an internal length parameter related to the size of heterogeneities [4]. As a promising mark along the line of correlating damage and fracture theory, works available in the literature are by Oliver et al. [5], Mazars and Cabot [2], Jirasek [6], etc. Mazars [7] has described a micro-scale damage model based on thermodynamics framework and proposed a correlation between micro- and macroscale damage in concrete. In the similar direction, a work by Mazars and Cabot [2] establishes an energetic equivalence between micro-scale damage and macroscale fracture. They have reported that the equivalence from fracture to damage necessitates the transformation of a crack into a volume distribution of damage on a band around this crack. For the general case of specimens with finite dimensions, the energy consumed during the crack formation is Gf, the fracture energy, generally deduced from the area under the load–displacement curve. The classical approach uses non-linear fracture theory (e.g. fictitious crack model by Hillerborg [8]), to determine the fracture energy. The others use mechanics of continuum through a non-local damage model able to describe the overall process: the cracked zone ðD ¼ 1Þ and, ahead and around, the process zone ð0 < D < 1Þ. Planas et al. [9] have derived the relationship between non-local models for concrete and the fictitious crack model. Using a uniaxial formulation and a Rankine type model, they have shown that the cohesive crack may be obtained as a particular case of a fully non-local formulation. In the present work, an approach to correlate the state of diffused micro-cracking with an equivalent discrete crack is developed through energy-based equivalence concept. The equivalent damage zone dimension is obtained as a function of increasing crack length upto failure. A finite element analysis is performed to assess the strength and stiffness degradation through modeling of the damage zone with a reduced elastic moduli. The advantage of this finite element model is that it is much simpler, as one can avoid the modeling of stress concentrations occurring at the crack tip and, thus computationally efficient. Finally, it has been shown that by having the critical damage parameters known, fracture parameters such as the fracture energy can be obtained with reasonable accuracy. 2. Fracture mechanics based fatigue law for concrete Structures such as airport/highway pavements and bridge decks are subjected to repetitive loads of high stress amplitude due to moving vehicles. According to classical theory, applied loads result in in-plane tensile stresses at the bottom of the pavement structure. The stress state in such structures is often simulated with three-point bending tests. Therefore, flexural fatigue is a common phenomenon in case of concrete members. Plain concrete subjected to flexural loading fails due to crack propagation. Repeated loading results in a steady decrease in the stiffness of the structure, eventually leading to failure. It is of interest to characterize the material behavior subjected to such loading and study the crack propagation resulting from such loading. Based on linear elastic fracture mechanics principle Slowik et al. [10] have suggested for modifications of the well known Paris law [11], to describe fatigue cracking in concrete members. The proposed law is given by: K mImax DK nI da ¼C þ F ða; DrÞ dN ðK IC  K Isup Þp

ð1Þ

where C is a parameter which gives a measure of crack growth per load cycle, K Isup is the maximum stress intensity factor ever reached by the structure in its past loading history, KIC the fracture toughness, K Imax is the maximum stress intensity factor in a cycle, N is the number of load cycles, a is the crack length, DK is the stress intensity factor range, and m, n, p, are constants. These constant co-efficients are determined by Slowik et al. through an optimization process using the experimental data and are 2.0, 1.1, and 0.7, respectively. The parameter C in Eq. (1) basically gives a measure of crack growth per load cycle. In concrete

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members this parameter indicates the crack growth rate for a particular grade of concrete and is also size dependent. Slowik et al. [10] have determined the value of C to be equal to 9.5 · 103 and 3.2 · 102 mm/cycle for small and large size specimen, respectively. It should be noted here that the stress intensity factor be expressed in MN m3/2. These values were determined for a particular loading frequency. Further, the fatigue crack propagation takes place primarily within the fracture process zone and hence C should be related to the relative size of the fracture process zone, which itself is related to characteristic length. Therefore, C should depend on the characteristic length lch and ligament length L, where lch ¼

EGf ft2

ð2Þ

Here, E is the elastic modulus, ft the tensile strength of concrete and Gf is the specific fracture energy. According to Slowik et al. [10] a linear relationship exists between parameter C and the ratio of ligament length to characteristic length, and is expressed as   L C ¼ 2 þ 25 ð3Þ  103 mm=cycle lch This equation does not account for the frequency of fatigue loading. The frequency or rate of loading influences the strain accumulation rate per cycle in the member, thereby controlling the crack propagation. Hence, the authors in their previous work [12], have proposed a modification to this equation to include the effect of, Table 1 Geometry and loading details Specimen

Depth (mm)

Initial notch (mm)

KIC MN m3/2

Peak load (N)

Cyclic load Pmax

Pmin

Large Medium Small

152.4 76.2 38.1

25.4 12.7 6.35

1.41 1.51 1.66

1815.6 2986 5184

1452.5 2389 4147

0 0 0

0.45

Relative Crack Depth (a/D)

0.4

Small (proposed model) Small (experimental) Medium (proposed model) Medium (experimental) Large (proposed model) Large (experimental)

0.35

0.3

0.25

0.2

0.15

1

10

2

10

Number of Fatigue Load Cycle (N) Fig. 1. Fatigue crack propagation (LEFM law).

3

10

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loading frequency. This is done through a regression analysis, using experimental results of Slowik et al. [10] and Bazant and Xu [13]. Normal strength concrete with load frequency of 3 Hz and 0.033 Hz have been used by Slowik et al. [10] and Bazant and Xu [13], respectively, for three different sizes of specimens. The resulting best fit curve represents a quadratic polynomial given by:  2   L L Cf ¼ 0:0193 þ 0:0809 þ 0:0209 mm=s ð4Þ lch lch From this equation one can obtain the value of parameter C for any given load frequency, grade of concrete and specimen size. The modified fatigue law for concrete is validated using the experimental results of Bazant and Xu [13]. Three point bending beams of three different sizes under fatigue loading were used by Bazant and Xu [13]. Geometrically similar specimens, each having width of 38.1 mm have been considered. Table 1 shows the details of specimen geometry and loads used. Young’s modulus for the grade of concrete has been considered to be as 27,120 MPa, for all the three specimens. Fig. 1 shows the fatigue crack propagation curves obtained using Eqs. (1) and (4) together with the experimental results as a function of loading cycles. It is seen that the analytical predictions using the above formulation is in good agreement with the experimental ones. 3. Energy-based equivalence between damage and fracture The relationship between fracture and damage can be obtained on the basis of an energy equivalence theory. In the present work, an equivalent volume distribution of damage has been obtained by replacing a discrete crack based on the energy equivalence principle. The energy required for unit crack propagation is equated with the energy loss due to the formation of a damage band of micro-cracks. The progressive crack propagation under fatigue loading has been replaced as a progressive formation of equivalent damage zone. The objective of this exercise is to offer the possibility of passing from one theory to the other or to obtain from one theory, information to be used in the other. According to fracture mechanics theory, energy is required for an existing crack to propagate by an amount da. This energy is commonly expressed as strain energy release rate per unit crack extension and denoted by G. Similarly, in case of damage based analysis, the strain energy loss per unit volume of the material due to increase in damage by an amount dD is referred to as damage strain energy release rate. The idea of energetic equivalence is based on equating the energy loss due to damage, with the energy required for an equivalent crack propagation within the member. In a more explanatory sense, energetic equivalence correlates two structures having the same geometry and loading condition, but different damage definitions. In a global sense they behave in the same manner, when the energy dissipation corresponding to two different damage conditions become equal for the two structures [2]. The energy release rate dU per unit crack extension da (which in turn is equal to the potential energy lost (dP) by the applied load) is related to the stress intensity factor KI by [14], 1 dU 1 dP K 2I ¼ ¼ B da B da E

ð5Þ

Considering a three point bend specimen with a crack at the bottom of mid-span, the stress intensity factor is given by pffiffiffi 6Y ðaÞM max a KI ¼ ð6Þ BH 2 where Mmax is the maximum bending moment at mid-span region ð¼ P max L=4Þ; a is the crack length; E is the modulus of elasticity and Y(a) is the geometry factor given by, Y ðaÞ ¼

½1:99  að1  aÞð2:15  3:93a þ 2:7a2 Þ ð1 þ 2aÞð1  aÞ1:5

a is the relative crack depth (¼ a=H ).

ð7Þ

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Substituting Eq. (6) into Eq. (5), and integrating over a from the instant when there is no crack in the beam to the current crack length a, we obtain: U ðaÞ ¼ where F ðaÞ ¼

9 P 2 L2 1 F ðaÞ 4 BH 2 E Z

ð8Þ

a

aY 2 ðaÞda

ð9Þ

0

Using Eq. (8), the strain energy as a function of relative crack depth a can be evaluated. In order to compute the energy dissipation due to increase in the degree of damage by an amount dD, the energy dissipated in the elemental volume dV is required and this quantity can be obtained using, Z U D ¼ ðY ÞdD dV ð10Þ D

where D is the scalar damage parameter and Y is the damage strain energy release rate which for an uniaxial loading is given by, 1 1 r r2 ¼ Y ¼ r ¼ r 2 2 Eð1  DÞ 2Eð1  DÞ Here, r is the uniaxial stress and * is the damage strain. For the damage to increase from 0 ! D, Z D Z D r2 r2 V lnð1  DÞ dx ¼ DU D ¼ V ðY ÞdD ¼ V 2E 2Eð1  xÞ 0 0

ð11Þ

ð12Þ

The main objective of the present analysis is to replace a discrete crack with an equivalent damage zone by equating the energy loss for both the systems. The idea is to obtain an equivalent damage model replacing a discrete discontinuity without altering the global structural response. At the location of discrete discontinuity, strain localization takes place, and the adjacent material undergoes strain softening. In conventional fracture mechanics theory this discontinuity is modeled using finite element method, as a zero width element. In the literature available, the local damage based approach considers the growth of damage as a function of the previously described localized strain. But localization into an arbitrarily small regions leads to unacceptable behavior both physically and computationally: the softening zone has a zero width and volume and resulting into zero energy loss, which seems to be impractical. This localization of damage is inconsistent with the definition of damage variable as an average representation of microstructural damage and makes the finite element analysis mesh-unobjective [7]. An effective method to avoid pathological localization of damage is to add non-local terms to the constitutive model. The spatial interactions resulting from the non-locality prevent the damage growth from localizing into a surface. Instead, the damage growth occupies a finite band, the width of which is related to the internal scale provided by the non-locality. Since the volume affected by damage is in reality, strongly related to the scale of micro-structure of a material, the internal length must be related to the scale of the micro-structure. This length was assessed experimentally on the basis of an energy equivalence between a specimen where damage is constrained to remain diffuse and another one in which damage localizes in order to obtain a single crack [2]. According to Bazant’s approach [15], the width of the band cannot be less than a certain characteristic value lc. For concrete, it can be approximately considered to be between 3da and 5da, where da is the maximum size of aggregates used. In concrete structures subjected to fatigue loading, the fracture process can be divided into three stages: crack initiation, stable crack growth and final fracture. In the first stage, repeated straining of the component results in an accumulation of microstructural damage, culminating in the formation of a macroscopic crack. In the stable crack growth stage, this crack propagates by a small amount per load cycle, until the remaining cross-section becomes too small to bear the load which is transferred by the component. In the present analysis, the progressive fracture phenomenon has been replaced by an equivalent continuously growing damage process. As per the well known convention of the damage mechanics, D ¼ 0 corresponds to no damage (zero crack opening) and D ¼ 1 corresponds to full crack opening resulting in a zero stress transfer along the crack

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surface. Fig. 2 shows the sequence of the progress of damage at different loading cycles. In Fig. 2a a discrete crack has been shown. In Fig. 2b the crack has been replaced with an equivalent damage zone of size lc  LD  B, where lc represents the localization limiter. Fig. 2c gives an idea about how damage progresses within this zone. In this figure, N indicates the number of fatigue load cycles. According to damage analysis, when D becomes unity for the fully shaded zone, it corresponds to the crack propagation of length LD. In three-dimensional form, the energy dissipated through damage is expressed as Z Z UD ¼ ðY ÞdD dV ð13Þ V

DðyÞ

and for this three-dimensional stress state, Y can be expressed as "  2 # r2eq 2 rH ð1 þ mÞ þ 3ð1  2mÞ Y ¼ 2 3 r eq 2Eð1  DÞ

ð14Þ

where req is Von Mises equivalent stress. For a two-dimensional plane stress problem, this equation simplifies to  1=2 ð15Þ req ¼ r21 þ r22  r1 r2 and rH ¼ 13 ðr1 þ r2 Þ is the hydrostatic stress at a point, where r1 and r2 are the principal stresses. The state of stress on a small element of a beam under three-point bending can be obtained using theory of elasticity as ! 2 6Pxðy  H =2Þ 3P ðy  H =2Þ 3P rxx ¼ ; ryy ¼ 0; sxy ¼  : ð16Þ 4BH BH 3 BH 3 where P, B and H are the load, width and height of the specimen, respectively; y is the distance measured from the bottom of the cross-section, along the depth of the beam; x is the distance along the length of the beam. Referring back to Fig. 2, the energy dissipated due to progressive damage in the shaded zone near the location of the crack results in the gradual change of damage variable D from 0 ! 1 and is given by Z LD Z ðy=LD Þþ1 DU D ¼ blc ðY ÞdD dy ð17Þ 0

0

Equating the two energy terms in Eqs. (8) and (17), we can solve for the unknown length of damage zone, LD through a trial and error procedure, after defining lc, the internal length of the continuum as mentioned earlier.

a

b

d a

LD D=1

L

lc N=0

c D=1

00

D=1 N=Nf

Fig. 2. Details of fracture based crack model and progressive damage zone.

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4. Case studies To implement the aforementioned theory of energetic equivalence, three point bend specimens under constant amplitude fatigue loading similar to the ones used by Bazant and Xu [13] in their experimental studies, have been considered. The details of the specimen geometry and loading are given in Table 1. In the case studies, the geometry of damage zone is predicted as a function of increasing crack length for all three sizes of specimens considered. Further, the stiffness and strength reduction factors as functions of crack and damage zone length are obtained. In addition, it is also shown that the fracture parameter such as the fracture energy can be estimated by knowing the critical damage zone size. 4.1. Size of the damage zone In this case study, the length LD and width (lc) of an equivalent damage zone is determined for the three point bending specimen of small size whose geometry and loading condition is shown in Table 1. The width (lc) of the damage zone is related to the internal length parameter and depends on the characteristics of the microstructure of the material, as discussed in the earlier section. Following Bazant’s approach [15], in the present study, the dimension of lc has been fixed by relating it to the maximum size of the aggregates used. A parametric study is done by considering three different widths of the damage zone: lc ¼ 3d a ; 4d a ; 5d a . The maximum size of the aggregate used in this study is 15 mm, same as the one used by Bazant and Xu [13] in their experimental study. Fig. 3, shows the fatigue damage propagation curve for the small beam specimen for different lc values together with the experimentally obtained plot. It is seen that there is a close match between the experimental and analytical curves for lc value of 4da. Finally, considering lc ¼ 4d a , the length of the damage zone LD is computed for small, medium and large specimens as a function of increasing crack length. Fig. 4 shows the equivalent damage zone length as a function of the relative crack depth (a/D) for all the three specimens. Thus, by knowing the discrete crack length, an equivalent damage zone size could be obtained by using the energy equivalence concept. 4.2. Stiffness and strength reduction factors It is a well known fact that concrete members subjected to cyclic loading undergo both stiffness and strength degradation that accompanies the damage process. Fracture mechanics based analysis quantifies this

Damage zone length (mm)

20

damage zone (lc=3da) damage zone (lc=5da) damage zone (lc=4da) Experimental Crack length

15

10

5 1 10

2

10

3

10

4

10

Number of fatigue load cycle (N) Fig. 3. Equivalent damage zone length as a function of fatigue load cycles (small specimen).

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Equivalent Damage Zone Length (mm)

90 80

Small Medium Large

70 60 50 40 30 20 10 0

0.2

0.25

0.3

0.35

0.4

0.45

Relative Crack Depth (a/H) Fig. 4. Equivalent damage zone length as a function of relative crack depth.

degradation as an effect of progressive cracking phenomenon within the member. On the other hand, damage mechanics theory assumes that the strain accumulation within the material volume is attributed to the degradation of elastic modulus, which eventually results into stiffness degradation. In this study, the strength and stiffness reduction in concrete beams are determined using both, fracture mechanics and damage mechanics approach. The same three-point bend specimens that were used in the previous case study are modeled using the finite element program FRANC [16]. 4.2.1. Stiffness reduction factor The stiffness reduction factor, is defined by Lybas and Sozen [17] as DR ¼

K0 Kr

ð18Þ

where K0 is the initial stiffness corresponding to no damage and Kr is the reduced secant stiffness due to damage. In this study, the reduced secant stiffness is computed using the maximum displacement (dr) as a function of increasing crack length for fracture based analysis and as a function of increasing damage zone length for damage based analysis. In the fracture mechanics based analysis, a rosette of singular crack tip quarter point elements are used to model the crack tip. The damage zone, in the damage mechanics based analysis is modeled using reduced value of modulus of elasticity. According to the theoretical definition of damage, the modulus of elasticity of the elements in the damage zone should approach zero. In this finite element study, two trial values, (1/100 of undamaged E) and (1/1000 of undamaged E) are considered in order to avoid the numerical difficulties arising from using a zero value. The width of the damage zone considered is lc ¼ 4d a with d a ¼ 15 mm, since this gave closer prediction with the experimental results as shown in Fig. 3. Thus, by computing the maximum displacement (dr) from the finite element analysis, the reduced secant stiffness Kr can be obtained using K r ¼ P =dr . Figs. 5–7 show the plots of stiffness reduction factor as a function of increasing crack length or damage zone length, for small, medium and large specimens, respectively. The maximum variation in the stiffness reduction factor between the fracture and damage based analysis is found to be in the range of 5–15%. As expected, the lower the value of modulus of elasticity of the elements in the damaged zone, the closer is the match between finite element and damage mechanics theory. Hence, the residual strength of a damaged member can be computed using either the fracture mechanics or damage mechanics based analysis since both approaches yield similar stiffness reduction factors. One may use a combined analysis by switching from one method to the other.

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Discrete crack Equivalent damage zone, Ed=E/100

1.6

Stiffness Drop Index (DR)

2329

Equivalent damage zone, E =E/1000 d

1.5 1.4 1.3 1.2 1.1 1 0

2

4

6

8

10

12

14

16

Crack/Damage zone Length (mm) Fig. 5. Stiffness degradation factor as a function of crack depth/damage zone length (small).

1.4

Discrete crack Equivalent damage zone: Ed=E/100

R

Stiffness Drop Index (D )

1.35

Equivalent damage zone: Ed=E/1000 1.3

1.25 1.2 1.15 1.1 1.05 1 0

5

10

15

20

25

30

35

40

Crack/Damage zone Length (mm) Fig. 6. Stiffness degradation factor as a function of crack depth/damage zone length (medium).

4.2.2. Strength reduction factor It is well known that concrete members suffer from strength deterioration when subjected to cyclic loading. Experimental studies of Hwang [18] have shown that the strength deterioration starts at considerably lower load levels when the concrete member is subjected to large numbers of load cycles. Young et al. [19] have defined a strength reduction index as !w /  /y SD ¼ ð19Þ /f  /y where /y is the curvature at the peak load (monotonic) or yield level corresponding to undamaged situation; /f is the final curvature at failure and / is the curvature at peak load corresponding to a particular damage level. The authors, through calibration studies have suggested a value of 1.5 for the parameter w.

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Discrete crack Equivalent damage zone: E =E/100 d

Equivalent damage zone: Ed=E/1000

R

Stiffness Drop Index (D )

1.35 1.3 1.25 1.2 1.15 1.1 1.05 1 0

10

20

30

40

50

60

70

80

Crack/Damage zone Length (mm) Fig. 7. Stiffness degradation factor as a function of crack depth/damage zone length (large).

In the present study, finite element analysis is performed using the program FRANC [16] to obtain yield curvature / for different damage zone dimension with a reduced elastic modulus, as well as with different discrete crack lengths. To compute the yield curvature, the monotonic peak load is used in the finite element analysis, as reported in Table 1. The curvature (/) values corresponding to different damage zone dimensions as well as for different discrete crack lengths, are obtained from the computed strains. Finally, the strength drop index has been computed using Eq. (19), and plotted in Figs. 8–10, for small, medium and large specimens, respectively. For the analysis of distributed damage zone, two different values of reduced elastic modulus have been used, as mentioned earlier. It is seen that there is not much difference between the strength reduction factors computed for the two values of reduced modulus considered in the study. Further, a close match is obtained for the strength reduction factor between the damage theory and fracture theory. Hence, from this analysis it can be concluded, that knowing the equivalent damage zone dimension or the discrete crack length, one can easily find out the reduced secant stiffness and the residual strength of the member, without much difference in the structural response.

0.7

Discrete crack Equivalent damage zone: Ed=E/1000 Equivalent damage zone Ed=E/100

D

Strength Drop Index (S )

0.6 0.5 0.4 0.3 0.2 0.1 0 6

8

10

12

14

16

Crack/ Equivalent Damage Zone Length (mm) Fig. 8. Strength drop index as a function of crack depth/damage zone length (small).

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D

Strength Drop Index (S )

1 0.9

Discrete crack Equivalent damage zone: E =E/1000

0.8

Equivalent damage zone: E =E/100

2331

d d

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 5

10

15

20

25

30

35

40

Crack/ Equivalent Damage Zone Length (mm) Fig. 9. Strength drop index as a function of crack depth/damage zone length (medium).

Strength Drop Index (SD)

0.25

Discrete carck Equivalent damage zone, Ed=E/1000 Equivalent damage zone, E =E/100

0.2

d

0.15

0.1

0.05

0 20

30

40

50

60

70

Crack/ Equivalent Damage Zone Length (mm) Fig. 10. Strength drop index as a function of crack depth/damage zone length (large).

4.3. Determination of fracture parameter Based on the energy equivalence principle, the critical fracture parameter such as the fracture energy can be obtained by knowing the critical damage zone size. From experiments, it is convenient to obtain the critical crack length. Using this critical crack length, we can determine the size of the critical damage zone as explained in the previous section. Using the critical damage zone dimension LDc  Lc  B corresponding to failure of the specimen, the critical damage energy release (UDc) can be computed from Eq. (17). The critical fracture energy can be written as Gc da; where Gc is the critical fracture energy release rate. Hence, based on energy equivalence, we can write, Gc da ¼ U Dc

ð20Þ

The critical fracture energy is computed for the three-point bending beams used earlier for all the three sizes using this procedure. Table 2 shows the computed values of GC together with those reported by Bazant and Xu

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Table 2 Determination of GIC Specimen depth

Critical LD

Predicted GIC

Reported GIC

% Error

38.1 76.2 152.4

15.02 37.425 78.25

98.01 89.5 71.5

101.8 84.01 73.19

3.7 6.5 2.3

[13] from their experimental studies. It is seen that there is a very good match between the computed values and the experimental ones. 5. Discussions It has been mentioned earlier, that there was a major inconsistency between fracture and continuum damage theory [2], while describing the structural response based on energy consideration. In the present work, it has been assumed that damage appears in a distributed manner within a zone of certain dimensions, which avoids the pathological difficulty of computing the energy dissipation in the damage process. Following this assumption, it is shown through case studies on three point bend beams that the two theories, namely, the fracture mechanics and damage mechanics, can be related by equating the energy required for unit crack propagation with the energy loss due to the formation of a damage band of microcracks. It is shown that the damage zone dimensions can be obtained as a function of increasing crack length. Further, from the parametric study on the width of damage zone, a value corresponding to four times the maximum aggregate size gave close prediction with experimental results. Knowing the equivalent damage zone dimension corresponding to a specific crack length (or critical crack length), finite element analysis may be performed to assess the residual stiffness and strength. It is seen that there is a close match between the stiffness and strength reduction factors computed using the fracture mechanics and damage mechanics theories. Hence, based on analyst’s choice, either of the theories may be used to assess the residual stiffness and strength of concrete beams. Finally, it is shown that the critical fracture energy may be computed by knowing the critical crack length and thus the critical damage zone dimensions. A close agreement is obtained between the computed values and the experimental values for the fracture energy using these two theories in conjunction. 6. Conclusions In the present study, an energy-based equivalence approach is proposed to model a discrete crack in the form of a distributed damage zone. Through the case studies of three-point bending beam under fatigue loading, it has been shown, that the progressive cracking phenomenon can be modeled, as an equivalent damage zone, without altering its global structural response. Knowing the damage zone dimension, the strength and stiffness drop index have been computed and compared with those values obtained through discrete crack analysis. The variation between two series of values was within acceptable limit. Therefore, it can be concluded that both fracture mechanics or damage mechanics theory can be used to predict the residual strength and flexibility of the cracked/damaged member, depending on the complexity involved in modeling of the problem. The response/output from one of the analyses (say fracture mechanics based) could be used to determine the behavior using the other method (say damage mechanics based), once the energy-based equivalence is made between the two models. Finally, it has been shown that by knowing the critical damage zone dimensions, the critical fracture property, such as fracture energy can be obtained with reasonable accuracy. References [1] [2] [3] [4]

Kaplan M. Crack propagation and the fracture of concrete. ACI J 1961;58:591–610. Mazars J, Cabot G. From damage to fracture mechanics and conversely: a combined approach. Int J Solids Struct 1996;33:3327–42. Bazant Z. Instability, ductility and size effect in strain-softening concrete. J Engng Mech ASCE 1976;102:331–44. Mazars J, Cabot G. Continuum damage theory-application to concrete. J Engng Mech ASCE 1989;115:345–65.

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