Correlation between fracture and damage for quasi-brittle bi-material interface cracks

$: Correlation between fracture and damage for quasi-brittle bi-material interface cracks$
Available online at www.sciencedirect.com

Engineering Fracture Mechanics 75 (2008) 2208–2224

www.elsevier.com/locate/engfracmech

Correlation between fracture and damage for quasi-brittlebi-material interface cracks

Vikas Garhwal, J.M. Chandra Kishen *

Department of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India

Received 9 May 2007; received in revised form 18 September 2007; accepted 8 October 2007Available online 13 October 2007

Abstract

Fracture at a bi-material interface is essentially mixed-mode, even when the geometry is symmetric with respect to thecrack and loading is of pure Mode I, due to the differences in the elastic properties across an interface which disrupts thesymmetry. The linear elastic solutions of the crack tip stress and displacement fields show an oscillatory type of singularity.This poses numerical difficulties while modeling discrete interface cracks. Alternatively, the discrete cracks may be modeledusing a distributed band of micro-cracks or damage such that energy equivalence is maintained between the two systems.In this work, an approach is developed to correlate fracture and damage mechanics through energy equivalence conceptsand to predict the damage scenario in quasi-brittle bi-material interface beams. The study is aimed at large size structuresmade of quasi-brittle materials failing at concrete–concrete interfaces. The objective is to smoothly move from fracturemechanics theory to damage mechanics theory or vice versa in order to characterize damage. It is concluded, that throughthe energy approach a discrete crack may be modeled as an equivalent damage zone, wherein both correspond to the sameenergy loss. Finally, it is shown that by knowing the critical damage zone dimension, the critical fracture property such asthe fracture energy can be obtained.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Fracture; Damage; Bi-material interface; Energy equivalence

1. Introduction

The application of interfacial fracture mechanics has increased due to the ever escalating use of adhesivejoints, composite laminates, multilayered electronic devices, etc. In the field of civil engineering, interfacesare formed whenever a repair material is applied to an infrastructure system after rehabilitation. Interfacesbetween different materials are found in dams wherein the concrete superstructure rests on rock foundation.Usually, the interface is relatively weaker than the material on either side of it. In a patch repaired system,wherein a new layer of cementitious material is applied on the parent concrete, the chances of failure by crack-ing along the interface is higher because of the stress concentration and rapid change of stress levels along the

0013-7944/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engfracmech.2007.10.001

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

mailto:[email protected]

Nomenclature

B width of beamD damage indexE modulus of elasticity of homogeneous materialE1 modulus of elasticity of material 1 in a bi-material beamE2 modulus of elasticity of material 2 in a bi-material beamGc critical fracture energyGf fracture energyGq energy release rate for crack in a quasi-brittle materialGIc energy dissipation due to Griffith–Irwin mechanismGr energy dissipation due to Dugdale–Barrenblatt mechanismH depth of beam[K] global stiffness matrixK complex stress intensity factor (K = K1 + iK2)KIc fracture toughnessLD total depth of the damage zoneN load cyclePmax maximum load{Q} load vectorS span of beamSe entropyT temperatureTb nominal bending stress at mid-span of beamU strain energy or internal energyUD strain energy due to damageV volume of the specimenf(a) geometry factora crack lengthlc width of damage zone{u} global displacement vectorW free energyP potential energya relative crack lengthdr maximum mid-span deflection

V. Garhwal, J.M. Chandra Kishen / Engineering Fracture Mechanics 75 (2008) 2208–2224 2209

interface. Thus, an improvement in the understanding of the behavior of an interface is an important issue inthe selection of a repair material in the design of rehabilitation of concrete structures.

Damage in plain concrete during its initial loading phase appears in the form of distributed micro-cracks.These micro-cracks are primarily due to the shrinkage of cement paste around the aggregates. During its ser-vice life, under sustained loading, the diffused damage expands and forms a distinct large crack due to coales-cence of micro-cracks which would propagate and cause final failure of the structural member. There are twomain category of models that describe this failure process: one that uses fracture mechanics concepts and theother which uses continuum damage mechanics concepts. Fracture mechanics describes the separation due todecohesion of two parts of the continuum [1,2]. It can be applied when a crack has been initiated, or assumingthat 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: elas-tic stiffness degradation, inelastic strains, etc. When the location of an expected crack and the direction alongwhich it propagates are unknown, fracture mechanics can hardly be used, because the critical flaw from which

2210 V. Garhwal, J.M. Chandra Kishen / Engineering Fracture Mechanics 75 (2008) 2208–2224

cracking initiates needs to be determined first. On the contrary, damage mechanics offers the essential advan-tage of predicting the location of this critical flaw. The bridge between fracture and damage mechanics can beconsidered to be the situation wherein damage is equal to unity at a material point, or in a small region defin-ing the size of an initial flaw in the theories of fracture. In most cases, this situation corresponds to the local-ization of strains and damage due to strain softening. Strain softening yields, however, several problems whichneed to be solved in order to bridge the two theories.

As explained in the paper by Mazars and Cabot [2], the major inconsistency which is faced to correlate frac-ture and classical damage theories, are that the failure predicted through local damage models occurs withoutdissipation of energy; whereas, most criteria for discrete crack propagation are based on the quantity of energywhich must be released in order to propagate a crack. To circumvent this problem, non-local damage modelsor gradient dependent damage models are proposed. The key idea of non-local damage models is to assumethat the condition of growth of damage is non-local, i.e., it depends at each material point on a weighted aver-age of the strains in a neighborhood. This neighborhood is scaled by an internal length parameter related tothe size of heterogeneities [3]. As a promising mark along the line of correlating damage and fracture theory,works available in the literature are by Oliver et al. [4], Mazars and Cabot [2], Jirasek [5], etc. Mazars [6] hasdescribed a micro-scale damage model based on thermodynamics framework and proposed a correlationbetween micro- and macro-scale damage in concrete. In the similar direction, a work by Mazars and Cabot[2] establishes an energetic equivalence between micro-scale damage and macro-scale fracture. They havereported that the equivalence from fracture to damage necessitates the transformation of a crack into a vol-ume distribution of damage on a band around this crack. For the general case of specimens with finite dimen-sions, the energy consumed during the crack formation is Gf, the fracture energy, generally deduced from thearea under the load–displacement curve. The classical approach uses non-linear fracture theory (e.g., fictitiouscrack model by Hillerborg et al. [7]), to determine the fracture energy. The others use mechanics of continuumthrough a non-local damage model able to describe the overall process: the cracked zone (D = 1) and, aheadand around, the process zone (0 < D < 1). Planas et al. [8] have derived the relationship between non-localmodels 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 dis-crete crack for a bi-material interface formed is developed through energy-based equivalence concept. Thisstudy is aimed at large size structures made of quasi-brittle materials failing at concrete–concrete interfaces.The equivalent damage zone dimension is obtained as a function of increasing crack length until failure. Itis shown that by knowing the critical crack length at failure, the fracture energy of the interface could beobtained from the energy equivalence approach and vice versa. The validation of the proposed procedureis done using the experimental results of concrete–concrete transversely cold jointed interface beams of differ-ent sizes that has been already published [9].

2. Fracture behavior of concrete interface

In an intact concrete specimen, a fracture process zone containing a number of micro-cracks exists in frontof a true crack and the fracture behavior is greatly influenced by the presence of this process zone. Since con-crete is a heterogeneous material consisting of different phases, the process zone is dominated by complicatedtoughening mechanisms such as micro-cracking, crack deflection, aggregate bridging, crack-face friction,crack tip blunting by voids and crack branching. These toughening mechanisms in the fracture process zoneare modeled by a cohesive pressure acting on the crack surfaces. It is well known that linear elastic fracturemechanics (LEFM) cannot be directly applied to concrete due to the presence of sizeable fracture process zone[10]. Further, the crack path in concrete is tortuous and it is difficult to determine the crack tip due to particlesbridging and variation in fracture process zone along the thickness direction [11]. Currently available fracturemodels for concrete attempt to simulate Mode I cracking with an effective line crack. The inelastic responsedue to the fracture process zone is taken into account by a cohesive pressure acting on the crack faces.

When a concrete structure with a quasi-brittle crack is subjected to loading, the applied load results in anenergy release rate Gq at the tip of the effective quasi-brittle crack. The energy release rate Gq may be dividedinto two portions:


(1) the energy rate consumed during material fracturing in creating two surfaces, GIc, which is equivalent tothe material surface energy, and

(2) the energy rate to overcome the cohesive pressure in separating the surfaces, Gr.

As a result, the energy release rate for a Mode I quasi-brittle crack, Gq, may be expressed as

Fig.

Gq ¼ GIc þ Gr ð1Þ

Although propagation of a quasi-brittle crack can be described by Eq. (1), we may approximately use only asingle fraction energy dissipation mechanism, either the Griffith–Irwin mechanism by assuming Gr = 0 or theDugdale–Barrenblatt mechanism by assuming GIc = 0. Based on different energy dissipation mechanismsused, non-linear fracture mechanics models for quasi-brittle materials like concrete may be classified as a fic-titious crack approach and an equivalent-elastic crack approach (or an effective-elastic crack approach). Frac-ture mechanics models using only the Dugdale–Barrenblatt energy dissipation mechanism are usually referredto as the fictitious crack approach, whereas the ones using only the Griffith–Irwin energy dissipation mecha-nism are usually referred to as the effective-elastic crack approach [11].

The behavior of a concrete–concrete joint or interface is much different when compared to an intact con-crete as explained below:

(1) Fig. 1 shows a typical load versus crack mouth opening displacement (CMOD) plot for an intact con-crete and concrete–concrete interface three-point bend beam [12]. It is seen from this plot that the basicdifference in the behavior of an interface beam when compared with an intact beam appears in the post-peak response. The intact concrete beam requires higher energy for complete fracture as seen from thearea under the load–CMOD curve when compared to an interface beam. This is due to the presence ofsizeable fracture process zone in intact concrete which is negligibly small in an interface beam.

(2) The size effect is one of the compelling reasons for adopting fracture mechanics in concrete. For concretestructures, the curve obtained by plotting strength versus size on log–log plot approaches a horizontalline for the strength criterion if the structure is very small and an inclined straight line of slope �1/2 fol-lowing LEFM if the structure is very large [13]. In large concrete specimens, the characteristic sizebecomes negligible compared to the size of the specimen. In small sized concrete specimens, the charac-teristic size is considerable when compared to the specimen dimensions. The effect of size on concrete–concrete interface beams have been studied experimentally by Chandra Kishen and Subba Rao [9]. Theresults have shown that the slope of the strength versus size plot are �1/2 for all the interface specimensconsidered and they followed LEFM theory. This implies that the failure of concrete–concrete interfacespecimens is governed by linear elastic fracture mechanics theory.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

Crack Mouth Opening Displacement (mm)

Load

(kN

)

Intact Concrete A

Concrete — Concrete Interface AA

1. Typical load versus crack mouth opening displacement plot for intact concrete and concrete–concrete interface beams [12].


From the above two differences in behavior of intact concrete with an interface, it is clear that the existenceof the fracture process zone having toughening mechanisms as in intact concrete cannot be visualized for inter-face specimens. Neither can the existence of micro-cracking zone ahead of the true crack be neglected. Hence,in this study, it is assumed that only the Griffith–Irwin energy dissipation mechanism is present for an interfacecrack with the Dugdale–Barrenblatt type energy dissipation mechanism being negligible and an effective-elas-tic crack approach is used. The effective-elastic crack is governed by the criterion from LEFM theory and theequivalence between the actual and corresponding effective crack is proposed.

Fracture at a bi-material interface is essentially mixed-mode, even when the geometry is symmetric withrespect to a crack and loading is pure Mode I. This is due to the differences in the elastic properties acrossan interface which would disrupt the symmetry [14]. Consequently, both tensile and shear stresses act onthe interface ahead of the crack and opening and sliding displacements of the crack flanks occur behindthe crack tip. The linear elastic solutions of the crack tip stress and displacement fields show that the stressesand displacements ahead of the crack front behave in an oscillatory manner. Due to this oscillatory behavior,the definition of the stress intensity factors needs special consideration, and in addition crack-face contact mayoccur for a small distance near the crack tip. The Modes I and II stress intensity factors cannot be decoupledto represent tension and shear stress fields as seen in the case of homogeneous materials. Appendix A gives thelinear elastic stress and displacement fields for a crack lying between the interface of two different materials.

3. Thermodynamics basis for energy equivalence between damage and fracture

The following assumptions are made in the following analysis of energy equivalence:

(1) Material behavior is linear elastic coupled with damage or fracture and the evolutions are at fixedtemperature.

(2) Analysis is restricted to Mode I fracture with the local state of stress in uniaxial tension.(3) Damage is isotropic.(4) The distribution of damage is constant along lines parallel to the crack.

Using the concepts of thermodynamics, it is easy to relate the local and global damage variables [2]. Thefree (reversible) energy stored in the material during straining is defined as

W ¼ U � TSe ð2Þ

where U is the internal energy, T the temperature and Se the entropy. For an elementary volume at a givenstate of damage, the internal energy is given by

U ¼ 1

2Eijklð1� DÞ�ij�kl ð3Þ

where Eijkl is the elasticity tensor of the undamaged material, D is the scalar damage parameter and �ij is thelocal strain component. For the overall body, damaged or partially cracked, the internal energy may also bewritten in the form

U ¼ 1

2fugt½K�fug ð4Þ

where [K] is the global stiffness matrix and {u} is the global displacement vector when a load denoted by {Q} isapplied on the structure. For the damaged material, using Eqs. (2) and (3), we can write [2]

rij ¼oWo�ij¼ Eijklð1� DÞ�kl ð5Þ

Y ¼ oWoD¼ � 1

2Eijkl�ij�kl ð6Þ

where rij is the local stress tensor and Y is the damage energy release rate. For the cracked structure, with theactual area of crack being A, using Eqs. (2) and (4), we can write


Q ¼ oWofqg ¼ ½K�fqg ð7Þ

G ¼ oWoA¼ 1

2fqgt o½K�

oAfqg ð8Þ

where G is the fracture energy release rate.The relationship between fracture and damage can be obtained on the basis of an energy equivalence theory

that must be thermodynamically acceptable. In this work, an attempt has been made to replace an existingdiscrete interface crack with an equivalent volume distribution of damage using the energy equivalence prin-ciple. The energy required for unit crack propagation is equated with the energy loss due to the formation of adamage band of micro-cracks. The objective of this exercise is to offer the possibility of passing from one the-ory to the other (fracture to damage or vice versa) or to obtain from one theory, the information to be used inthe other.

According to fracture mechanics theory, energy is required for an existing crack to propagate by an amountda. 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 toincrease in damage by an amount dD is referred to as damage strain energy release rate. The idea of energeticequivalence is based on equating the energy loss due to damage, with the energy required for an equivalentcrack propagation within the member. In a more explanatory sense, energetic equivalence correlates two struc-tures having the same geometry and loading condition, but different damage definitions. In a global sense theybehave in the same manner, when the energy dissipation corresponding to two different damage conditionsbecome 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 K by [15]

1

BdUda¼ � 1

BdPda¼ ð1� b2ÞðK2

1 þ K22Þ

Eeff

ð9Þ

where K1 and K2 are the components of complex stress intensity factor K = K1 + iK2 as detailed in AppendixA. Considering a three-point bend specimen with a crack at the bottom of mid-span, the stress intensity factoris given by [16]

K ¼ T b

ffiffiffiap

a�ieeiwf ðaÞ ð10Þ

where Tb is the nominal bending stress at mid-span region (Tb = 6PS/4H2), a is the crack length, Eeff is theeffective modulus of elasticity for the bi-material beam with an interface, f is the geometry factor, a is the rel-ative crack length = a/H, w is the mode mixity angle, P is the applied load at the mid-span, S is the span of thebeam and H is the overall depth of the beam.

The effective modulus of elasticity of a beam with bi-material interface is derived using the compatibility ofdisplacements between a homogeneous beam and a bi-material beam as shown in Appendix B and can be writ-ten as

Eeff ¼cS bþ c

3

� �� c

3þ 2b2

3Sc

SE1bþ c

3

� �� c

3E1þ 2b2

3SE2

ð11Þ

where b, c and S are as shown in Fig. 2, and Ei is the modulus of elasticity of material i. For the symmetricalcase, that is when the interface is at the mid-span, then b = c and the above equation reduces to

Eeff ¼2E1E2

E1 þ E2

ð12Þ

The geometry factor f(a) for S/H = 2.5 is given by [17]

f ðaÞ ¼ ½ð1� 2:5aþ 4:49a2 � 3:98a3 þ 1:33a4Þffiffiffipp�

ðBH :5Þð1� aÞ1:5ð13Þ

Fig. 2. Geometry of bi-material interface beam.


The mode mixity angle w which gives the relative proportions of normal traction and in-plane shear near thebi-material interface is defined as [18]

w ¼ tan�1 rxy

rxx

� �r¼l¼a

ð14Þ

Substituting Eq. (10) into Eq. (9), and integrating over a from the instant when there is no crack in the beam tothe current crack length a, the strain energy as a function of the relative crack depth becomes

UðaÞ ¼ 9

4

P 2S2B

H 2

ð1� b2ÞE

F ðaÞ ð15Þ

where

F ðaÞ ¼Z a

0

af 2ðaÞda ð16Þ

Using Eq. (15), the strain energy as a function of the 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

U D ¼Z

Dð�Y ÞdDdV ð17Þ

where D is the scalar damage parameter and Y is the damage strain energy release rate which for an uniaxialloading is given by

Y ¼ 1

2r�� ¼ 1

2r

rEeffð1� DÞ ¼

r2

2Eeffð1� DÞ ð18Þ

Here, r is the uniaxial stress and �* is the damage strain. For the damage to increase from 0! D

DUD ¼ VZ D

0

ð�Y ÞdD ¼ VZ D

0

�r2

2Eeffð1� xÞ dx ¼ r2V2Eeff

lnð1� DÞ ð19Þ

The main objective of the present analysis is to replace a discrete interface crack with an equivalent damagezone by equating the energy loss for both the systems. The idea is to obtain an equivalent damage modelreplacing a discrete discontinuity without altering the global structural response. At the location of discretediscontinuity, strain localization takes place, and the adjacent material undergoes strain softening. In conven-tional fracture mechanics theory this discontinuity is modeled using finite element method, as a zero widthelement. In the literature available, the local damage based approach considers the growth of damage as afunction of the previously described localized strain. But localization into an arbitrarily small regions leadsto unacceptable behavior both physically and computationally: the softening zone has a zero width and vol-ume and resulting into zero energy loss, which seems to be impractical. This localization of damage is incon-sistent with the definition of damage variable as an average representation of microstructural damage andmakes the finite element analysis mesh-unobjective [6]. 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 fi-nite band, the width of which is related to the internal scale provided by the non-locality. Since the volumeaffected by damage is in reality, strongly related to the scale of micro-structure of a material, the internallength must be related to the scale of the micro-structure. This length was assessed experimentally on the basisof an energy equivalence between a specimen where damage is constrained to remain diffuse and another onein which damage localizes in order to obtain a single crack [2].

The behavior of bi-material interface beams subjected to three-point bending shows a stable crack growthfollowed by softening type of response finally leading to unstable crack propagation as shown in Fig. 1. In thepresent analysis, a discrete interface crack has been replaced by an equivalent damage zone. As per the wellknown convention of the damage mechanics, D = 0 corresponds to no damage (zero crack opening) andD = 1 corresponds to full crack opening resulting in a zero stress transfer along the crack surface. A beamwith bi-material interface crack as shown in Fig. 2 has been replaced with an equivalent damage zone of sizelc · LD · B in Fig. 3, where lc and LD represent the width and length of damage zone. According to damageanalysis, when D becomes unity for the fully shaded zone, it corresponds to the crack propagation of lengthLD = (a + lD), where lD is the depth of the shaded triangular damage portion which represents the equivalentdamage of the small fracture process zone that forms in front of a true crack.

In three dimensional form, the energy dissipated through damage is expressed as

UD ¼Z

V

ZDðyÞð�Y ÞdDdV ð20Þ

and for this three dimensional stress state, using Eq. (6), Y can be expressed as

Y ¼ 1

2Eð1� DÞ2ð1þ mÞr : r� mðtrrÞ2h i

ð21Þ

where r is the state of stress on an infinitesimally small element and m is the Poisson’s ratio. For a two dimen-sional plane stress problem, the state of stress may be written in matrix form as

r ¼rxx rxy

rxy ryy

� �ð22Þ

The state of stress on a small element of a beam under three-point bending can be obtained using theory ofelasticity as

rxx ¼�6Pxðy � H=2Þ

BH 3; ryy ¼ 0; rxy ¼

3Pðy � H=2Þ2

BH 3� 3P

4BH

!ð23Þ

where P is the central point load on the beam, B and H are the width and height of the beam, respectively; y isthe distance measured along the depth from the bottommost fiber of the cross-section, and x is the distancemeasured along the length of the beam.

Fig. 3. Beam showing an equivalent damage zone.


Referring to Fig. 3, the energy dissipated due to progressive damage in the shaded zone near the location ofthe crack results in the gradual change of damage variable D from 0! 1 and is given by

DUD ¼ Blc

Z a

0

Z 1

0

ð�Y dDÞdy þZ aþlD

0

Z 1�y�alD

� �0

ð�Y ÞdDdy

8<:

9=; ð24Þ

Equating the two energy terms in Eqs. (15) and (24), we can solve for the unknown width of the damage zonelc through a trial and error procedure, after defining the length of the damage zone LD as the critical cracklength at which unstable crack propagation takes place. The detailed procedure is as follows:

• For a given interface beam of known dimensions and known elastic constants for the two materials oneither side of the interface, the interface constant b is computed using Eq. (A.4).

• For an assumed crack length, the geometry factor f(a) is computed using Eq. (13), where a is the relativecrack length defined as the ratio of crack length to the total beam depth.

• The strain energy U(a) is computed for the assumed crack length using Eq. (15).• Assuming the shape of the process zone to be an equilateral triangle of base width lc, as shown in Fig. 3, the

depth lD can be written in terms of lc. Thus, the energy dissipated by progressive damage for the assumedcrack length is computed using Eq. (24) by numerical integration.

4. Case studies

To validate the aforementioned theory of energetic equivalence, three-point bend beams having a trans-verse interface between two different mixes of concrete similar to the ones used by Chandra Kishen and SubbaRao [9] in their experimental studies, have been considered. Geometrically similar notched beams of differentdepths, with span to depth ratio of 2.5 and notch to depth ratio of 0.1 have been used. Fig. 4 shows the geom-etry of the beams used. The cross-sectional dimensions and the span of the beams are shown in Table 1. Thethickness of all the beams are kept constant at 50 mm. Table 2 shows the mechanical properties of differentgrades of concrete used in the experiments. Four different concrete mixes designated as A, B, C and D havingproperties as shown in Table 2 have been used in the experiments. The specimens have been prepared by cast-ing the first half of beam with concrete mix A, while the second half of the beam has been cast with concretemix A, B, C or D after two days. This creates an interface between the two mixes of concrete. In the case stud-ies, the geometry of damage zone is predicted as a function of increasing crack length for all three sizes of

Fig. 4. Geometry of the interface beam used in experiments.

Table 1Dimensions of interface beams [9]

No. Beam designation Depth b (mm) Span s (mm) Length L (mm)

1 Small 76 190 2412 Medium 152 380 4313 Big 304 760 810

Table 2Mechanical properties for different grades of concrete for three-point bend specimens

Mix designation Compressive strength (MPa) Modulus of elasticity (GPa) Poisson’s ratio

A* 27 25.3 0.20A 28 25.5 0.20B 39 29.3 0.19C 51 36.0 0.18D 58 38.5 0.18


specimens considered and all four combination of interfaces, namely AA, AB, AC and AD. In addition, it isalso shown that the fracture energy can be estimated by knowing the critical damage zone size.

4.1. Size of the equivalent damage zone

In this case study, the dimensions of an equivalent damage zone is determined for the three-point bendspecimen of all the sizes and interface combinations considered. As shown in Fig. 3, the depth of the damagezone LD is considered to be (a + lD), where a is the size of discrete interface crack and lD is the size of theprocess zone that forms ahead of the discrete crack. In this study, the shape of the process zone is assumedto be an equilateral triangle of base width lc. Hence, the depth lD can be written in terms of lc thereby reducingthe two unknowns to one. Hence, using Eq. (24), the width of the damage zone lc can be computed for a givencrack length. Figs. 5–7 show the variation of the width of the damage zone with the crack length for small,medium and large sized beams, respectively, and for different interface combinations. It is seen that there isalmost no variation in the damage zone width between the different material combinations considered inthe study since all the curves overlap on each other. The plots show that the size of the damage zone increasesslowly with increase in crack length during the initial stage of crack propagation. The rate of increase in thesize of the damage zone increases considerably as the critical or failure crack length is approached.

10 20 30 40 50 60 700

50

100

150

200

Crack length (mm)

Wid

th o

f dam

age

zone

(m

m)

A (Intact)

AB

AC

AD

Fig. 5. Width of damage zone versus crack length for small interface beams.

20 40 60 80 100 1200

50

100

150

200

Crack length (mm)

Wid

th o

f dam

age

zone

(m

m)

A (Intact)

AB

AC

AD

Fig. 6. Width of damage zone versus crack length for medium interface beams.

30 60 90 120 150 180 210 2400

50

100

150

200

250

300

350

400

Crack length (mm)

Wid

th o

f dam

age

zone

(m

m)

A(Intact)

AB

AC

AD

Fig. 7. Width of damage zone versus crack length for large interface beams.


4.2. Determination of fracture energy

Based on the energy equivalence principle, the critical fracture parameter such as the fracture energy can beobtained by knowing the critical damage zone size. From experiments, it is convenient to obtain the criticalcrack length. Using this critical crack length, we can determine the size of the critical damage zone asexplained in the previous section. Using the critical damage zone dimension LDc · lc · B corresponding to fail-ure of the specimen, the critical damage energy release (UDc) can be computed from Eq. (24). The critical frac-ture energy can be written as Gcda; where Gc is the critical fracture energy release rate. Hence, based on energyequivalence, we can write

Gcda ¼ U Dc ð25Þ

The critical fracture energy is computed for the bi-material three-point bend beams and for all sizes andinterface combinations using this procedure. Figs. 8–10 show the variation of fracture energy with cracklength for small, medium and large beams, respectively, and for different material combinations considered.

10 20 30 40 50 600

5

10

15

20

25

30

35

40

Crack length (mm)

Fra

ctur

e en

ergy

(N

/mm

)

A (Intact)

AA

AB

AC

AD

Fig. 8. Fracture energy versus crack length for small interface beams.

20 30 40 50 60 70 80 90 100 1100

5

10

15

20

25

30

35

40

Crack length (mm)

Fra

ctur

e en

ergy

(N

/mm

)

A (Intact)

AA

AB

AC

AD

Fig. 9. Fracture energy versus crack length for medium interface beams.


It is seen that the curves initially remain constant and later on become asymptotic as the critical crack length isreached. The initial constant portion of the curves indicate stable crack propagation within the interface.From these figures, we can obtain the critical fracture energy of the interface by knowing the critical cracklength. In the experiments performed by Chandra Kishen and Subba Rao [9], loading and unloading weredone at constant intervals during the test. From these loading unloading load–deflection plots, the complianceis measured at different stages of crack propagation. Through numerical calibration using the compliancemethod [19], the critical crack length is determined. Tables 3–5 show the computed critical crack length fordifferent interfaces used in the test for small, medium and large specimens, respectively. Using these criticalcrack lengths, the critical fracture energy is determined from Figs. 8–10 for small, medium and large speci-mens, respectively. In the same tables, the computed values of fracture energy together with those reportedby Chandra Kishen and Subba Rao [9] from their experimental studies are shown. It is seen that there is avery good match between the computed values and the experimental ones for all sizes of specimens and allcombination of interfaces. It may be noted that the fracture energy being a material property has the sameexperimental values for all specimen sizes.

Table 3Critical fracture energy of interfaces for small specimen

Specimen designation Critical crack length (mm) Predicted GIc (N/m) Experimental GIc (N/m)

A (intact) 45 31.92 34.37AA 46 23.13 24.51AB 48 19.89 21.57AC 49 18.5 20.01AD 54 17.77 18.81

30 80 130 180 2300

5

10

15

20

25

30

35

40

Crack length (mm)

Fra

ctur

e en

ergy

(N

/mm

)

A (Intact)

AA

AB

AC

AD

Fig. 10. Fracture energy versus crack length for large interface beams.

Table 4Critical fracture energy of interfaces for medium specimen



Table 5Critical fracture energy of interfaces for large specimen




5. Discussions

In the present work, it is shown through case studies on three-point bend bi-material beams that the twotheories, 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 micro-cracks. The damage zone dimensions can be obtained as a function of increasing crack length. Hence, basedon analyst’s choice, either of the theories may be used to study the behavior of beams having interfaces. There-fore, it can be concluded that either fracture mechanics or damage mechanics theory could be used to predictthe residual strength of bi-material interface beams. The response/output from one of the analysis (say frac-ture mechanics based) could be used to determine the behavior using the other method (say damage mechanicsbased), once the energy-based equivalence is made between the two models. Finally, knowing the equivalentdamage zone dimension corresponding to a specific crack length (or critical crack length) it is shown that thecritical fracture energy may be computed. A close agreement is obtained between the computed values and theexperimental 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 theform of a distributed damage zone for bi-material interface beams. Through the case studies of three-pointbending beam specimen, it has been shown, that the progressive cracking phenomenon can be modeled, asan equivalent damage zone, without altering its global structural response. Knowing the damage zone dimen-sions at critical crack length, the critical fracture energy can be computed with good accuracy. Thus, by deter-mining only the critical crack length from experiments, the fracture energy could be conveniently obtainedfrom the theory of energy equivalence. The main benefits of the energy equivalence approach for interfacecracks are:

(1) It is possible to model a discrete interface crack by an equivalent damage zone. This zone may be mod-eled by finite element discretization for investigating the residual strength of any structure wherein theinterface crack length is known without any further information on subsequent cracking.

(2) It is shown that the fracture energy can be obtained from the damage model. This would be helpful, espe-cially for large specimens wherein the critical value of fracture energy can be determined directly fromsize effect experiments [2].

The approach followed in this work is a general one and applicable as long as the materials on either side ofthe interface behaves in a quasi-brittle manner. Further, the proposed method may result in unrealistic pre-dictions if the materials used lack a macroscopic characteristic size, in which case, a multiscale analysis needto be performed where at a lower scale a characteristic size is encountered.

Appendix A. Elastic fields for an interface cracks

Williams [20] performed an asymptotic analysis of the elastic fields at the tip of an open interface crack andfound that the stresses and displacements behaved in an oscillatory manner as

r � r�1=2ðsin; cosÞðe log rÞ ðA:1Þu � r1=2ðsin; cosÞðe log rÞ ðA:2Þ

e ¼ 1

2plog

1� b1þ b

ðA:3Þ

where b is a dimensionless composite parameter depending on the material properties and introduced by Dun-ders [21] and given by

b ¼ l1ð1� 2m2Þ � l2ð1� 2m1Þ2½l1ð1� m2Þ þ l2ð1� m1Þ�

ðA:4Þ

in which l and m are the shear modulus and Poisson’s ratio, respectively, and subscripts 1 and 2 refer to thematerials above and below the interface, respectively. b varies from �1/2 to 1/2 and vanishes for identicalmaterials. It also vanishes for two incompressible materials, or one incompressible and the other rigid under

δy

δx

y

x

θ

μ ν E r

μ ν E1 1 1

2 2 2

Fig. 11. Geometry and conventions of an interface crack [16].


conditions of plane strain, among other combinations [22]. Under plane strain conditions, b gives a measurefor the mismatch in bulk moduli.

Considering the bi-material interface crack in Fig. 11, for traction-free plane problems the near-tip normaland shear stresses ryy and sxy, may conveniently be expressed in terms of complex stress intensity factors [14]as

1 No

ryy þ isxy ¼ðK1 þ iK2Þrieffiffiffiffiffiffiffiffiffiffiffi

ð2prÞp ðA:5Þ

where i ¼ffiffiffiffiffiffiffi�1p

, K1 and K2 are components of the complex stress intensity factor K = K1 + iK2.1 From Eq.(A.5) it is seen that the usual definition of the stress intensity factors as limr!0{(2p r)1/2r} will not workand it produces logarithmically infinite factors. Furthermore, any attempt to define the stress intensity factorswithout reference to a characteristic length, such as the crack length, will produce dimensionally meaninglessresults [22]. However, the strain energy release rate G computed by the dimensionally wrong factors will still bedimensionally valid. The expressions for these bi-material stress intensity factors, derived by Rice [23] are

K1 ¼r½cosðe log 2aÞ þ 2e sinðe log 2aÞ� þ ½s½sinðe log 2aÞ � 2e cosðe log 2aÞ�

cosh pe

ffiffiffiap

ðA:6Þ

K2 ¼s½cosðe log 2aÞ þ 2e sinðe log 2aÞ� � ½r½sinðe log 2aÞ � 2e cosðe log 2aÞ�

cosh pe

ffiffiffiap

ðA:7Þ

In the above equations, e is the oscillation index given by Eq. (A.3) and b is Dunders elastic mismatch param-eters defined by Eq. (A.4), r and s are the normal and shear stresses along the interface and a is the cracklength.

When e 5 0, Eq. (A.5) shows that the stresses oscillate heavily as the crack tip is approached (r! 0). Fur-thermore, the relative proportion of interfacial normal and shear stresses varies slowly with distance from thecrack tip because of the factor rie. Thus, K1 and K2 cannot be decoupled to represent the intensities of inter-facial normal and shear stresses as in homogeneous fracture.

The crack flank displacements, for plane strain are given by [14,16]

dy þ idx ¼4ffiffiffiffiffiffi2pp ð1=E1 þ 1=E2ÞðK1 þ iK2Þ

ð1þ 2ieÞ coshðpeÞffiffirp

rie ðA:8Þ

where dy and dx are the opening and sliding displacements of two initially coincident points on the crack sur-faces behind the crack tip, as shown in Fig. 11, r is the distance from the crack tip and Ei is the plane strainmodulus of elasticity. It may be noted from Eq. (A.8) that crack-face interpenetration is implied when e 5 0,as e is an oscillatory term as shown in Eq. (A.3). The zone of contact, however, is generally exceedingly smallcompared to the crack tip plastic zone and may therefore be neglected [16].

te that for interface cracks, it is customary to replace KI and KII by K1 and K2.


The energy release rate, G, for extension of the crack along the interface, for plane strain is given by [14]

G ¼ ð1=E1 þ 1=E2ÞðK21 þ K2

2Þ2cosh2ðpeÞ

ðA:9Þ

Appendix B. Effective modulus of elasticity of interface beams

The effective modulus of elasticity of an interface beam is defined as the modulus of elasticity of an equiv-alent homogeneous beam that would give the same deflection at the load point as that of the interface beam.The load point deflection of the interface beam shown in Fig. 2 computed using the moment–area theoremwould be

d1 ¼Pcb2

2SIb

SE1

cþ b3

þ 2c2

3SE2

� b3E1

� �ðB:1Þ

The load point deflection of an homogeneous beam having modulus of elasticity E of the same span l as theinterface beam is

d2 ¼Pcb2

2SEIbS

cþ b3

þ 2c2

3S� b

3

� �ðB:2Þ

Equating the load point deflections of Eqs. (B.1) and (B.1), we obtain

Eeff ¼cS bþ c

3

� �� c

3þ 2b2

3Sc

SE1bþ c

3

� �� c

3E1þ 2b2

3SE2

ðB:3Þ

where b, c and S are as shown in Fig. 2, and Ei is the modulus of elasticity of material i. For the symmetricalcase, that is when the interface is at the mid-span, then b = c and the above equation reduces to

Eeff ¼2E1E2

E1 þ E2

ðB:4Þ

References

[1] Kaplan M. Crack propagation and the fracture of concrete. ACI J 1961;58:591–610.[2] Mazars J, Cabot GP. From damage to fracture mechanics and conversely: a combined approach. Int J Solids Struct 1996;33:3327–42.[3] Mazars J, Cabot GP. Continuum damage theory – application to concrete. J Engng Mech ASCE 1989;115:345–65.[4] Oliver J, Huespe AE, Pulido MDG, Chaves E. From continuum damage mechanics to fracture mechanics: the strong discontinuity

approach. Engng Fract Mech 2002;69:113–36.[5] Jirasek M. Nonlocal models for damage and fracture: comparison of approaches. Int J Solids Struct 1998;35:4143–5.[6] Mazars J. A description of micro- and macroscale damage of concrete structures. Engng Fract Mech 1986;25(5/6):729–37.[7] Hillerborg X, Modeer M, Petersson PE. Analysis of crack formation and crack growth in concrete by means of fracture mechanics

and finite elements. Cement Concr Res 1976;6:773–82.[8] Planas J, Elices M, Guinea G. Cohesive cracks versus non-local models-closing the gap. Int J Fract 1993;63:173–87.[9] Chandra Kishen JM, Subba Rao P. Fracture of cold jointed concrete interfaces. Engng Fract Mech 2007;74:122–31.

[10] Shah SP, McGarry FJ. Griffith fracture criterion and concrete. J Engng Mech ASCE 1971;97:1663–76.[11] Shah SP, Swartz SE, Ouyang C. Fracture mechanics of concrete: applications to concrete, rock and other quasi-brittle

materials. John Wiley and Sons; 1995.[12] Subba Rao P. Fracture behavior of jointed concrete interfaces. Ph.D. Thesis, Department of Civil Engineering, Indian Institute of

Science, Bangalore, India, June 2006.[13] Bazant Z, Planas J. Fracture and size effect in concrete and other quasi-brittle materials. CRC Press; 1997.[14] Carlsson LA, Prasad S. Interface fracture of sandwich beams. Engng Fract Mech 1993;44:581–90.[15] Malyshew BM, Salganik RL. The strength of adhesive joints using the theory of cracks. Int J Fract 1965;1:114–8.[16] Hutchinson JW, Suo Z. Mixed mode cracking in layered materials. Adv Appl Mech 1992;29:62–191.[17] Mobasher B, Yu Li C. Effect of interfacial properties on the crack propagation in cementitious composites. Adv Cement Based Mater

1996;4:93–105.[18] Shih C. Cracks on bi-material interfaces: elasticity aspects. Mater Sci Engng A 1991;143:77–84.


[19] Bruhwiler E, Saouma VE. Fracture testing of rock by the wedge splitting test. In: Proceedings of the 31st US symposium on rockmechanics. Golden, CO: Balkema; 1990. p. 287–94.

[20] Williams ML. The stresses around a fault or crack in dissimilar media. Bull Seismolog Soc Am 1959;49(2):199–204.[21] Dunders J. Edge bonded dissimilar orthogonal elastic wedge under normal and shear loading. J Appl Mech 1969(36):270–8.[22] Comninou M. An overview of interface cracks. Engng Fract Mech 1990;37:197–208.[23] Rice JR, Sih GC. Plane problem of cracks in dissimilar media. J Appl Mech 1965;32:418–23.

Correlation between fracture and damage for quasi-brittle bi-material interface cracks

Documents

Transcript of Correlation between fracture and damage for quasi-brittle bi-material interface cracks