Int J Fract (2013) 181:241–255DOI 10.1007/s10704-013-9838-z

ORIGINAL PAPER

Spacing and failure mechanism of edge fracturein two-layered materials

Chunyan Bao · C. A. Tang · Ming Cai ·Shibin Tang

Received: 15 August 2012 / Accepted: 12 March 2013 / Published online: 26 March 2013© Springer Science+Business Media Dordrecht 2013

Abstract The phenomenon of edge fractures is ana-lyzed using a fracturing process approach. Such frac-tures often initiated from the free surface of layeredmaterials, and often terminate at the interface thatdivides the fractured layer and the matrix layer, or con-tinue to expand along the interface, or create a peel-crack. To understand better the pattern of fractures withdifferent spacing and the fracture mechanism, a double-layer elastic model with the same material propertieswith a fractured overlying layer subjected to uniax-ial tension is simulated firstly. The stress distributionbetween two adjacent fractures is periodically distrib-uted as a function of the ratio of the fracture spacingto the thickness of the fractured layer (S/t ratio), andhomogeneous and heterogeneous material propertiesare considered in the simulation. The simulation resultsshow that both the stress distribution and the criticalvalue of fracture spacing to layer thickness ratio areaffected by material heterogeneity; the S/t ratio of het-erogeneous material is much smaller and it is mucheasier to crack compared with the homogeneous mate-rials. In particular, the fracture saturation mechanismis analyzed by stress state change. Then a numerical

C. Bao · C. A. Tang (B) · S. TangState Key Laboratory of Coastal and Offshore Engineering,Dalian University of Technology,Dalian 116024, Liaoning, Chinae-mail:tang_chunan@yahoo.com

M. CaiBharti School of Engineering, Laurentian University,Sudbury, ON, Canada

simulation is carried out to reveal the fracturing processfrom micro-fracture formation, propagation, coales-cence, nucleation, fracture infilling, fracture saturation,termination, to interface delamination. A fitting curveof the relationship between strain and spacing to layerthickness radio is obtained. It is found that infillingfractures may grow near the bottom or the free sur-face of the fractured layer by coalescence of micro-fractures and flaws. We also find that fracture spacingin the case of interface delamination is greater than thatwithout interface delamination. A study of stress tran-sition between the two layers on fracture spacing inthe fracture process is also carried out, with a focus onstress transfer mode.

Keywords Edge fracture · Heterogeneity ·Progressive fracture · Numerical modeling ·Layered materials

1 Introduction

Fractures initiated from a free surface in layeredmaterials, such as surface fractures in asphalt concretepavements, desiccation shrinkage fractures in mud andsurface coating films, fractures on the convex side offlexure etc., often terminate at the interface that dividesthe fractured layer and the matrix layer. Examples ofsurface fractures are presented in Fig. 1. This is a verycommon phenomenon observed both in natural andin man-made materials, and it has being studied in

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Fig. 1 Examples of surface fractures formed on city roads.a Road surface fractures in Northern Ontario, Canada. b Roadsurface fractures in Liaoning Province, China

geological and earth sciences (Iyer and Podladchikov2009; Ji and Saruwatari 1998; Mandl 2005), materi-als science, civil engineering (Hong and Li 1997) andother fields. These types of fractures, which are alsocalled edge fractures, consist of arrays of parallel sur-face cracks or polygonal fractures (e.g., mud cracks) inthe layer perpendicular to their respective tensile stressdirections. A good understanding of the surface frac-ture mechanism is vital for many engineering designssuch as asphalt concrete pavement design. If these sur-face fractures can be reduced or eliminated, less roadmaintenance would be required, leading to less impactto commuters and cost saving.

Several studies have been devoted to the understand-ing of the mechanism of this phenomenon. One ofthe earliest theoretical treatments of fracture spacingof parallel edge fractures is based on the concept ofstress shadow provided by the opening of a fracture

(Lachenbruch 1961). Another study focused on thedevelopment of a methodology that related the satu-rated fracture spacing to interface fracture toughness(Nanta and Moran 1996), and it is revealed that the min-imum fracture spacing is a function of applied stressfor different values of interface fracture toughness.Bai et al. (2000a) using a two-layer elastic numeri-cal model, obtained the stress distribution between twoadjacent edge fractures as a function of the ratio offracture spacing to the thickness of the fractured toplayer. They found that the critical fracture spacing tolayer thickness ratio increases with increasing Pois-son’s ratio of the fractured layer, but it decreases withincreasing Poisson’s ratio of the underlying layer. Tanget al. (2008) reproduced the phenomenon of networkpolygonal fractures as observed in shrinkage mud usingRFPA2D (two-dimensional realistic fracture processanalysis), a numerical code originally developed byTang and Zhao (1997). The same numerical code wasused to study thermo-hydro-cracking in concrete (Tang2009). When a layered concrete, bounded on both sidesin the direction parallel to the layers, is subjected toshrinkage in the drying or cooling process, the mater-ial is subjected to an extensional strain in the directionparallel to the layers. When the contraction-inducedtensile stresses are high enough, fracturing of the con-crete may result.

Most fractures found in materials are formed fromcontraction due to cooling or drying, and in manyinstances the extent of the fractures are confined bylayer boundaries with their height equal to the layerthickness (Fischer et al. 1995; Gross and Engelder1995; Helgeson and Aydin 1991). The contraction, cou-pled with adhesion to neighboring layers, leads to abuildup of strain or stress in the material. It is recog-nized that if the stress state in the fractured layer beforefracturing is uniaxial, or biaxial with one of the prin-cipal stresses greater than the other, or if the mater-ial of the fractured layer has a preferred orientationfor fracturing, a set of parallel fractures will form inthe material (Thouless 1990; Thouless et al. 1992; Wuand Pollard 1995). Experimental results show that frac-ture spacing decreases approximately as the inverseof the applied strain in the direction perpendicular tothe fractures, by fractures forming between previouslyformed fractures. This process is known as sequen-tial infilling (Gross 1993). In the fracturing process,the spacings between the fractures become so smallthat additional strain will not induce additional new

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Spacing and failure mechanism of edge fracture 243

fractures but just further open the existing fractures,which is called fracture saturation (Wu and Pollard1995).

Although the aforementioned possible explanationfor fracture saturation has been widely accepted, fewof the existing theoretical and numerical models canadequately reproduce the complete process of frac-ture initiation, propagation, infilling, and saturation, asobserved both in the laboratory and in the field. Forexample, the numerical simulation by Bai et al. (2000a)did not consider the progressive fracturing of the mate-rial and this may limit the interpretation of the mech-anism of fracture saturation. We found in our currentstudy that in addition to the reasons explained by otherresearchers, there are other reasons that could drive thesystem to reach fracture saturation. Two of the possiblereasons that catch our attention are the development ofinterface delamination and fractures passing throughthe underlying layer.

In this paper, the phenomenon of edge fracturesin dissimilar layers on elastic and elasto-plastic sub-strates is analyzed. We document the stress state changebetween adjacent edge fractures in a row of paral-lel, equally spaced fractures in a two-layered modelwith a fractured top layer. We consider uniaxial load-ing and analyze the dependence of the length scaleon the strain, for both homogeneous and heteroge-neous fractured layer and underlying layer with dif-ferent elastic constants and substrate thickness. Theobjective is to demonstrate how to predict fracture for-mation, a phenomenon representing an exception tothe widely accepted fracture-spacing-theory-based-onmodels (Bai et al. 2000a,b; Wu and Pollard 1995).However, neither of the previous models considersmaterial heterogeneity and gradual fracture propa-gation. In the present study, particular interest andcomplexity are homogeneous and heterogeneous mate-rial properties and gradual fracture propagation, andthe results show that the micro-fractures or flaws tendto propagate a shorter distance, and macro-fracturesare formed by the coalescence of micro-fractures andflaws.

In the following discussion, the term “fracture” isused to represent any edge fracture that cuts through thefractured layer, and the term “flaw” is used to representany crack that is very short in length compared withthe thickness of the fractured layer. Flaws are used todetermine whether the sequential infilling will occur ornot by fracture propagation.

2 Numerical method and verification

A two-dimensional finite element code named RFPA2D

is used in the current investigation. This code is basedon the theory of elastic-damage mechanics, and wasoriginally developed by Tang and Zhao (1997) and fur-ther developed at Mechsoft, China. In RFPA2D, thesolid or material is assumed to be comprised of manyelements with the same size, and the mechanical prop-erties of the elements are assumed to conform to a givenWeibull distribution presented in “Appendix”. RFPA2D

has been successfully applied to the study of variousfracture problems, ranging from tensile fracture for-mation in Brazilian test (Zhu et al. 2005), fracture pat-tern transition from parallel to polygonal fractures inlayered materials induced by a quasi-statistical biaxialloading (Tang et al. 2006) to thermo-hydro cracking inconcrete (Tang 2009).

Of particular interest to the current study is the abil-ity of RFPA2D to model fracture spacing in layeredmaterials. A previous study conducted by Tang et al.(2008) on a three-layered model verified that the codecould simulate fractures in layered materials reason-ably well. Figure 2 presents the normalized horizontalstress variation with the S/t ratio, where S is fracturespacing and t is layer thickness. The results obtained byTang et al. (2008) agree well with the numerical resultsobtained by Bai et al. (2000a). The materials are homo-geneous in both models and fractures occurred in themiddle layer. In the current investigation, we focus onfracture formation on the surface layer and the phe-nomenon of surface fracture saturation and factors thatinfluence fracture saturation.

(a) (b)

Fig. 2 Variation of σx/σxmax at the middle point of the middleline between two fractures with the fracture spacing to thicknessratio S/t; Tensile and compressive stresses are positive and neg-ative, respectively. a Results obtained from by RFPA. b Resultsof Bai and Pollard (2000a, 2000b)

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3 Numerical simulation of edge fracture sequentialinfilling and saturation

In this section, we simulate the process of edgefracturing, fracture-induced acoustic events and stressredistribution in a two-layered model; furthermore, weinvestigate the influence of the interface debonding onthe fracturing process.

As shown in Fig. 3, the material properties of eachlayer are different, and the material heterogeneities arereflected by parameter m in Table 1. A large and asmall m value indicate relatively homogeneous andheterogeneous material property, respectively. Hence,the top layer is more heterogeneous than the bottomlayer. Other model parameters such as Young’s mod-ulus, Poisson’s ratio, tensile and compressive strengthetc. are also listed in Table 1. Because the Young’s mod-ulus of the top layer is larger than that of the bottomlayer, the top layer will have a larger local tensile stresswhen subjected to the same strain. In addition, the ten-sile strength of the top layer is small; therefore, thetop layer will fracture initially. The model boundary isfixed in the y direction at the bottom edge and in the xdirection at the right side as shown in Fig. 3. Strains areapplied to the model through the application of lateraldisplacement to the left end in the negative x direction.The total strain is applied in 230 increments, from aninitial value of 0.006 % to a final value of 0.464 %.

Because there are hundreds of loading steps in thenumerical simulation and fracture formation is a grad-ual process, it is difficult to show all the images thatindicate the whole fracture process. In Fig. 4 we justpresent the main fracture formation steps to show thefracture sequential infilling process and fracture satu-ration at different strain levels. The order of fractureformation is shown by the sequential numbers shown

in Fig. 4. The brightness in the images indicates theminimum principal stress (tensile stress) and it is seenthat stress distribution is affected by the evolution offracturing in the model.

First, let us analyze the stress transfer in the fractureformation process. We divide the fracture process intofour stages. In the first stage (Fig. 4a, b), because thestrain is <0.78×10−3, there are randomly distributedflaws or short fractures formed in the upper layer atweak locations, where the local tensile stress reachesthe local tensile strength. In the second stage (Fig. 4c,d), more flaws are generated and three long fracturesare formed consecutively. In the third stage (Fig. 4e–i), six additional fractures are embedded in the seg-ment between the previously formed fractures. Finally,because the stress in the bottom layer reaches its ulti-mate tensile strength, a fracture propagates across theunderlying layer and catastrophic failure occurs. Sub-sequent strain increase does not result in the formationof additional infilling fractures in the top layer; instead,it only leads to further propagation of the through-goingfracture in the bottom layer (Fig. 4j). Hence, sequentialfracture infilling in the top layer can be stopped by thefracturing of the underlying layer.

As is seen from Fig. 4d–j, delamination between thetwo layers occurs, resulting in partial disconnection ofthe fractured (top) layer from the underlying layer. It isobserved that for fractures numbered 2 and 7, delami-nation occurs and the fracture spacing between the twofractures is much wider than that without delamination.For example, the fracture spacing between fractures 2and 7, 7 and 4 are wider than that between 4 and 6.When layer interface delamination occurs, the defor-mation in the fractured layer is less affected by theunderlying layer. The fracture infilling is affected andthe critical spacing to layer thickness ratio is higher

Fig. 3 Model dimensionand boundary conditions(unit: mm)

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Spacing and failure mechanism of edge fracture 245

Table 1 Model parameters

Refer to Layer level E (GPa) σc (MPa) σt (MPa) Poisson’sratio

Homogeneityfactor m

Layer size(mm2)

Numberof elements

Rupture process Top 30 150 15 0.25 3 5,000 × 300 15,000

Bottom 10 300 30 0.35 100 5,000 × 1,500 75,000

Fig. 4 Selectednumerically obtainedsequence of imagesshowing fracture formationin the weak layer (m = 3,relatively heterogeneousmaterial). This sequenceillustrates the process fromfracture initiation, fractureinfilling, fracture saturation,to propagation to the bottomlayer. Numbers a to jindicate the order of fractureinfilling. Note: the darkelements represent thenucleated flaws. Fracturesare formed from flawsconnection. The shadingintensity indicates therelative magnitude of thehorizontal stress in theelements

(f) ε = 2.00×10-3

1

2 1 2 1 3

2 4 1 3 2 4 1 5 3

2 4 6 1 5 3

2 7 4 6 8 1 5 3 2 7 4 6 8 1 9 5 3

2 7 4 6 1 5 3

(a) ε = 0.06×10-3 (b) ε = 0.78×10-3

(c) ε = 0.98×10-3 (d) ε = 1.42×10-3

(e) ε = 1.56×10-3

(g) ε = 3.16×10-3 (h) ε = 3.72×10-3

(i) ε = 3.92×10-3 (j) ε = 4.64×10-3

than that for completely bonded layers. The mecha-nism of layer delamination on fracture generation willbe further investigated in a separate publication.

Numerically obtained mean fracture spacing S, frac-ture spacing to layer thickness ratio S/t and the associ-ated average strain at each loading step are presented inTable 2. The S/t ratios are plotted against the averagestrain in Fig. 5 (the numbers shown in the bracket are thenumbers of fractures generated at different strain lev-els). It is seen that theS/t ratio decreases with increas-ing strain; the decrease is rapid in the strain rangebetween 0 and 0.002 and slow afterwards. The resultsinclude the influence of interface debonding, which isnoticed for the first time in the present study. A curvefitting is conducted and the equation that best describesthe relationship between the S/t ratio and strain ε is:

Table 2 Numerically obtained mean S/t ratio and the corre-sponding average strain

No S (mm) S/t Strain

(0) 500 16.67 0.00006

(1) 250 8.33 0.00078

(2) 274 9.13 0.00098

(3) 218 7.27 0.00142

(4) 145 4.83 0.00156

(5) 109 3.63 0.002

(6) 87 2.90 0.00316

(7) 72.5 2.42 0.00372

(8) 62.3 2.08 0.00392

(9) 62.3 2.08 0.00464

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0 1 2 3 4 50

2

4

6

8

10

12

14

16

18

S/t Asymptotic1 Fit of Sheet1 S/t

S/t

Strain(×0.001)

Model Asymptotic1

Equation y = a-b*c^x

Reduced Chi-Sqr 0.67

Adj. R-Square 0.9684

Value Standard Error

S/t a 1.80903 0.55725

S/t b -15.61032 0.94056

S/t c 0.39953 0.04826

(0)

(1)

(2)

(3)

(4)(5)

(6) (7)(8) (9)

Fig. 5 Fitting curve of numerically obtained mean fracture spac-ing to layer thickness ratio S/t as a function of average strain. Thenumber shown in the bracket is the total number of fractures gen-erated at each strain level

S/t = 1.809 + 15.610 × 0.4001,000ε (1)

with a correlation coefficient value of R2 = 0.97, indi-cating a good correlation between the two parameters.From Eq. (1), we notice that the horizontal line repre-sented by S/t = 1.809 is the asymptotic line when thestrain becomes infinite. This means that as the strainincreases continuously to infinite, the fracture spacingwill not reduce accordingly and this is the phenom-enon described as fracture saturation. We also noticethat there is a limit for the S/t ratio at zero strainingbased on Eq. (1). For this particular case, the limit is17.42 at zero strain. For our model at an initial strainof 10−6, there is no cracking in the layer and we plotS/t as L/t (= 16.67) in the figure.

Figure 6 presents the temporal distribution ofacoustic events (AE) that occurred during the load-ing process. The failure of each cell element is con-sidered as an AE event. A flaw or a short fracture maybe indicated by a small number of AE events. Whenmany small fractures are combined, a macro-fracturemay result. The curve correlates the AE event countswith the fracture infilling phenomenon well. It is seenfrom Fig. 6 that AE events are generated in the entiredeformation process. In stage 1, an increasing num-ber of small fractures occur, as randomly distributedflaws or short fractures form in the top layer at its weaklocations when the local tensile stress reaches the localtensile strength. During this stage, tensile stress results

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00

5

10

15

20

25

30

35

40

45

50

Stage 4Stage 3Stage 2Stage 1

Strain ∗0.001

Mea

n S

tres

s(M

Pa)

0

40

80

120

160

AE

Eve

nt C

ount

s

Fig. 6 Plots of numerically obtained mean horizontal stress (σx)

and AE event counts (N) as a function of average strain (ε)

mainly from the tensile strain applied to the model’sboundary. However, during stages 2 and 3, the numbersof flaw nucleation events no longer increase systemati-cally with increasing strain; instead, as more local ele-ments between two fractures fail, a drastic decrease inlocal tensile stress between adjacent fractures occurs.Consequently, additional fractures can be generatedonly after further straining is applied. In other words,once a macro-fracture is nucleated, the layer gener-ally remains stable during gradual increases in exten-sional strain to the point when the stress reaches thestrength. A comparison between Figs. 4 and 6 revealsthat each abrupt increase in AE event counts corre-sponds to an event of fracture infilling, and each turn-ing point on the stress-strain curve can be correlated tothe formation of a macro-fracture through the top layer.For the through going fracture that occurs in the bot-tom layer in the final stage, the curve shows a suddenstress drop to zero and an abrupt increase of AE eventcounts.

4 Stress state change and critical fracture spacingto layer thickness ratio in a homogeneoustwo-layered material model

4.1 Numerical model and boundary condition

The model used by Bai et al. (2000a) is adopted inthe current study to investigate the stress distributionsbetween adjacent equally spaced fractures. As shown inFig. 7, there are four equally spaced fractures in the toplayer. Similar to the work by Bai et al. (2000a), we alsocarried out numerical simulations with more than four

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Spacing and failure mechanism of edge fracture 247

Fig. 7 The two-layeredmodel and its boundarycondition with four equallyspaced edge fractures in thefractured layer

fractures using RFPA2D, and the results show that onlythe stress distributions in areas near the boundary endare significantly different from that in areas betweenthe fractures. In the equally spaced four-fracture model,the middle two fractures are used to represent any twoadjacent fractures in a row composed of many equallyspaced fractures. For example, when the stress distri-bution in the middle two fractures of a four-fracturemodel is compared with that from a 20-fracture model,we found that the difference in stress was only 2 %,but the computation time was 90 % less for the smallermodel.

The model boundary and loading condition areshown in Fig. 7. The thickness of the top fractur-ing layer is t = 20 mm, which is also the height ofthe pre-existing fractures (H). The model thicknessis t + T = 120 mm. The model width (W) variesaccording to the fracture spacing (S) such that thedistance from the left (or right) boundary to the leftmost (or right most) fracture is set as three timesof the fracture spacing S. Hence, the total modelwidth is W = 9S. The size of each element is 1 ×1 mm2. As for the sign convention, tensile stress ispositive, and compressive stress is negative. Both lay-ers have the same elastic constants (Young’s mod-uli E f = Eu = 30 GPa, Poisson’s ratios υ f =υu = 0.25). To study the variation of horizontal stress(σx) along line A–A′ in Fig. 7 in a homogeneousmedium, a very large homogeneity index (m = 106)is used.

The x-axis is parallel to the layer boundary andperpendicular to the fracture orientation, and the y-axis is perpendicular to the layer boundary as seen inFig. 7. The right boundary is fixed in x-direction. A con-stant displacement Ux boundary condition is appliedin the negative x-direction on the left boundary, andthe model is subjected to an average tensile strainof εx = Ux/W. The top and bottom boundaries arefree.

4.2 Fringe contours of horizontal stress betweenedge fractures

Photoelasticity is one of the most widely used tech-niques to visualize stress fields. To show how stressvaries as the fracture spacing changes, we plot thefringe contours of horizontal stress (σx) in Fig. 8 forS/t = 3.4, 3.2, 3.0, and 2.8. As is seen from Fig. 8a–d,the tensile stress zone (black zone) shrinks from theedge, which means that as the S/t ratio decreases, thehorizontal stress along the fractured layer edge changesfrom tensile stress to compressive stress. The ranges ofσx values between fractures are illustrated in the Fig. 9and it is seen that as the S/t ratio increases, tensilestress field is developed. On the other hand, as morefractures are generated, the S/t ratio decreases and theoverall stress field is more inclined to compression.

4.3 Effect of loading on the critical fracture spacingto layer thickness ratio

To develop a clear understanding of the horizontalstress (σx) distribution along line A–A′ in Fig. 7 in ahomogeneous medium, we plot the stress versus thenormalized position (x − 0.5W)/t for S/t ratios inFig. 10a, using the sign convention that tensile stressis positive. This stress state varies with the S/t ratio,and our result agrees well with that Bai et al. (2000a),shown in Fig. 10b.

As show in Fig. 10a, as the S/t ratio changes, thereexists a threshold S/t value at the middle point thatgoverns whether the stress state is in tension or incompression. The plots show that the critical S/t ratiois between 3.0 and 3.2. When the S/t ratio is equalto 3.0 or smaller, the stress is compressive (negative)everywhere along line A–A′. When the S/t ratio is 3.2or greater, the stress along the central portion of lineA–A′ changes to tensile (positive). The sign of the

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Fig. 8 Numerical fringecontours of σx for differentfracture spacing to thicknessratios S/t; a S/t = 3.4, σx =−1.30E − 3 ∼ 15.3 MPa,b S/t = 3.2, σx =−1.40E − 3 ∼ 14.0 MPa,c S/t = 3.0, σx =−4.63E − 3 ∼ 13.5 MPa,d S/t = 2.8, σx =−9.79E − 3 ∼ 13.1 MPa.Tensile stresses are shownas black, and compressivestresses as white

(a) (b)

(c) (d)

2.6 2.8 3.0 3.2 3.4 3.6

-0.009

-0.006

-0.003

13

14

15

16

Range of σx with S/t=2.8

Range of σx

with S/t=3.0

Range of σx

with S/t=3.2

Range of σx with S/t=3.6

S/tRan

ge o

f σx

Fig. 9 Range of horizontal σx between fractures for various S/tratios. Tensile and compressive stresses are positive and negative,respectively

stress at the central point of line A–A′ is used to definethe critical S/t ratio (S/t)cr . The (S/t)cr is definedwhen the horizontal stress is zero at the central pointbetween two adjacent fractures. When the sign of thestress at the central point is negative, the correspond-ing spacing to layer thickness ratio is below the criticalvalue; otherwise this ratio is above the critical value.

Figure 11 presents the results by plotting the distri-bution of the horizontal stress at the middle point online A–A′ against the S/t ratio. In the case of homo-geneous top and bottom layers with the same elasticconstants, the critical S/t ratio lies between 3.0 and3.2. By linear interpolation, we find the critical S/tratio is 3.142, which is slightly different from the resultof (S/t)cr = 3.114 reported in Bai et al. (2000a) The

difference is attributed to the element sizes used in thesetwo models. In our model, smaller element sizes areused, leading to an increased accuracy in stress analy-sis.

5 Stress state change and critical fracture spacingto layer thickness ratio in a heterogeneoustwo-layered material model

Most natural materials, especially rocks, are not homo-geneous but contain defects at various length scales.Hence, it is necessary to consider local scale hetero-geneity in fracture modeling. In the following simu-lation, we used the same model shown in Fig. 7 butwith heterogeneous material properties (m = 2). Thestress σx distributions along the line A–A′ (indicatedin Fig. 7) for various S/t ratios are plotted in Fig. 12.For comparison, the stress distributions for the homo-geneous model are also shown in Fig. 12 as smoothlines. The mean values of the material properties inboth the homogeneous and the heterogeneous modelsare the same. It is seen that for heterogeneous materi-als, the stress distributions fluctuate around the lines ofthe homogenous cases, and the local stress fluctuationis governed by the heterogeneity parameter m. This isobviously caused by the local variation in stress andstrength. At locations where the local stress is high andthe strength is weak, fractures are more likely to ini-tiate, grow, and nucleate, which in turn will affect thefracture patterns.

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Spacing and failure mechanism of edge fracture 249

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

-60

-40

-20

0

20

40

60

σ x:M

Pa

S/t=2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

3.8Ef=Eu=30,000 MPa

υυυυf=υυυυu=0.25

εεεεxx(ave)=0.01

Position along line A-A' in Fig.7 nomalized by the thickness of the fractured layer( (x-0.5W)/t)

(a) (b)

-60

-40

-20

0

20

40

60

σ x:M

Pa

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

S/tr=2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

3.8Ef=Eu= 30.000 MPa

Vf = Vu = 0.25

εεεεxx(ave) = 0.01

Fig. 10 Distributions of the horizontal stress σx along line A–A′ (refer to Fig. 7 for its location) for various S/t ratios. Tensile stressesare positive, and compressive stresses are negative. a Results from RFPA. b Results of Bai and Pollard (2000a)

0 1 2 3 4 5 6

-100

-50

0

50

100

150

S/t

σ x(M

Pa)

Ef=Eu=30,000 MPa

υf=υu=0.25

εxx(ave)=0.01

Fig. 11 Variation of stress σx at the middle point of the lineA–A′ with the S/t ratio

The peak tensile stress near the center of A–A′ ofthe heterogeneous material case is always less than thepeak tensile stress of the homogenous material case.This means that for heterogeneous materials, a fracturemay not occur in the middle between two existing frac-tures as the tensile stress there is not necessarily themaximal. Hence, in the following discussion on frac-ture pattern in heterogeneous materials, we will acceptthe fact that in the process of fracture infilling, a newfracture does not always initiate at the middle pointbetween two previously generated fractures; instead,the new fracture will initiate at a point where the local

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

-70

-60

-50

-40

-30

-20

-10

0

10

20

30

40

50

60

Position along line A-A' in Fig.7 nomalized by the thickness of the fractured layer

σσ σσ x:M

Pa

S/t =2.2 S/t =2.8 S/t =3.0 S/t =3.2 S/t =3.6 S/t =3.8

Ef=Eu=30,000 MPa

υυυυf=υυυυu=0.25

εεεεx(ave)=0.01

Fig. 12 Comparison of the horizontal stress distributions alongline A–A′ from the heterogeneous model with that from thehomogeneous model, for various S/t ratios. Stresses of heteroge-neous materials (m = 2) are shown to fluctuate around the stressesof homogeneous materials (m = 106)

tensile stress reaches its strength. Figure 13 presentsthe distribution of the horizontal stress at the middlepoint on line A–A′ against the S/t ratio. For compar-ison, the result from the homogenous material case isalso shown in the same figure. By linear interpolation,we obtain the (S/t)cr as 3.154, which is slightly largerthan that obtained from the homogeneous model.

For an elastic system, the tensile stress magnitudedepends on the Young’s modulus. In a model consisting

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0 1 2 3 4 5 6

-100

-50

0

50

100

150

200

m=10E6

m=2

σ x(M

Pa)

S/t

Ef=Eu=30,000 MPa

υf=υu=0.25

εxx(ave)=0.01

Fig. 13 Variation of stress σx at the middle point of line A–A′with the S/t ratio for the heterogeneous material model (m = 2).Result from the homogeneous model (m = 106) is also shown forcomparison

of elements with different Young’s moduli, stresseswill be higher in elements with larger moduli. Hence,the tensile stress at the middle point in a heteroge-neous model is not necessarily the maximum. Whenthe tensile stress in the layer reaches the tensile strengthof a certain element, flaws will form thereby reliev-ing the stress. By comparing the two curves shown inFig. 13, we notice that the peak stress in the hetero-geneous model is less than that in the homogeneousmodel.

The stress state change analyzed above indicates thatin the process fracturing in the top layer, the horizontalstress in the adjacent domain of fractures is reducedwhen a new fracture is introduced (fracture infilling),thereby inhibiting the formation of additional new frac-tures unless the strain level is increased. A new edgefracture cannot insert between two existing fractureswhen the fracture spacing to layer thickness ratio lessthan the critical value, unless a surface flaw exists. Inthe next section, we will examine stress state changeduring the fracture infilling process.

6 The mechanism of stress state change in theprocess of edge fracture infilling

In Sects. 4 and 5, we examined the stress state changeunder extension in a model with four equally spacedfractures. The change from tensile stress to compres-sive stress was not explained by any of the known the-ories and conceptual models until Bai et al. (2000a),

who studied the spacing of fractures formed during theextension of a three-layered model using the finite ele-ment method.

To further study the mechanism of stress statechange in the process of edge fracture infilling, weuse a model slightly different from the one shownin Fig. 7. As shown in Fig. 14, the model thicknessis t + T = 120 mm, the model width is unchanged[W = 360 mm + (N −1)× S + 360 mm = 840 mm],and the size of each element is 1×1 mm2. The lengths ofthe end sections are kept as 360 mm, and fractures withvarious spacings are introduced in the middle sectionwith a length of (N − 1) × Smm. Fractures are spacedequally and the total number of fractures is N. In thisstudy, we construct models with N = 0, 2, 3, 4, 5, 6, 7,9, 11, and 21, which correspond to S/t ratios of ∞, 6, 3,2, 1.5, 1.2, 1, 0.75, 0.6, and 0.3, respectively. The modelis subjected to a lateral tensile pressure (σx = 3 MPa),and the material is homogeneous.

The distributions of the horizontal stress for the ele-ments along line 3–3′ are presented in Fig. 15a, forvarious S/t ratios. Elements next to the left most frac-ture are considered (Fig. 14). The stress distributionfor the case where there is no fracture is representedby the dash dot line (σx = 3 MPa), which is equal tothe applied stress. As fracture infilling continues in thefractured layer, the S/t ratio decreases from 6 to 3,and then to 2, 1.5 and 0.75 accordingly. At the locationwhere line 3–3′ intersects the layer boundary (repre-sented by the element whose number is 100), a sharpstress change can be seen. In most part of the bottomlayer, the stress is more or less constant. Near the layerinterface, the stress increases drastically and reachesits maximum at the interface, and then decrease rapidlyto a constant value in the top layer. The drastic stressincrease and decrease happen only in a very narrowband around the layer interface, in a range covered byapproximately 30 elements or 30 mm.

As shown in Fig. 15b, the horizontal stress alongline 3–3′ in the bottom layer decreases from top to bot-tom and approaches a constant value which is greaterthan 3 MPa. As the S/t ratio decreases, the constantvalue increases, which means that the bottom layerwill take more load as the top layer is heavily frac-tured. As shown in Fig. 15c, with the decrease of theS/t ratio, the stress value which corresponds to thesharp stress change point decreases as well. It meansthat the top layer will take fewer loads as it is heavilyfractured.

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Spacing and failure mechanism of edge fracture 251

Fig. 14 Model size andboundary conditions

Fig. 15 Distribution ofhorizontal stress σx alongline 3–3′ for various S/tratios. a Stress along line3–3′. b Line 3–3′ in thebottom layer. c Line 3–3′ inthe top (fractured) layer

-1 0 1 2 3 4 5 6 7 8 9 10 11 12

0

20

40

60

80

100

120

Horizontal stress σx (MPa)

Ele

men

t n

um

ber

no fracture

S/t=6

S/t=3

S/t=2

S/t=1.5

S/t=0.75

(a)

3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55

0

20

40

60

80

0.75

1.5

3

23

6

Horizontal stress σx (MPa)

S/t=6S/t=3S/t=2S/t=1.5S/t=0.75

Ele

men

t n

um

ber

alo

ng

lin

e 3-

3'

0 1 2 3 4 5 6 7 8 9 10 11

97

98

99

100

101

102

0.75

1.5 2 6

Horizontal stress σx (MPa)

Ele

men

t n

um

ber

alo

ng

lin

e 3-

3'

S/t=6 S/t=3 S/t=2 S/t=1.5 S/t=0.75

3

(b) (c)

The horizontal stress distributions along lines B–B′and C–C′ (see Fig. 14) are presented in Fig. 16a, b. Thearea enclosed by the curve and the x-axis is the totalload acting on the cross section (the model thickness is1). When the S/t ratio decreases, the enclosed area inFig. 16a decreases, which means that the load carriedby the layer interface decreases. For all S/t ratios, thestresses along line B–B′ are all tensile. Line C–C′ isright under the model surface, and its stress distributionis very different from that along line B–B′. As the S/tratio decreases, the stress state changes from tension to

compression. The only fact that affects the stress statechange is fracture spacing.

The vertical stresses σy along lines B–B′ and C–C′are plotted in Fig. 16c, d. As shown in Fig. 16c, whenthe S/t ratio is 6, the middle part is in compression andboth ends are in tension. With the S/t ratio decreasesto the critical value (around 3), the compressive stressin the middle part increases. With further decrease ofS/t , the compressive stress decreases and the stressstate changes to tension when the S/t ratio is less than1. As the S/t ratio decreases, the tensile stress in the end

123

252 C. Bao et al.

0.0 0.2 0.4 0.6 0.8 1.0-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

S/t=6S/t=3S/t=2S/t=1.5S/t=0.75σσ σσ

x al

ong

line

B-B

'(M

Pa)

Position along line B-B' in Fig.14 0.0 0.2 0.4 0.6 0.8 1.0

-0.5

0.0

0.5

1.0

1.5

2.0S/t=6S/t=3S/t=2S/t=1.5S/t=0.75

Position along line C-C' in Fig.14

σσ σσ x a

long

line

C-C

'(M

Pa)

(a) (b)

0.0 0.2 0.4 0.6 0.8 1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

σ σσy

of t

he e

lem

ent

Position along line B-B' in Fig.14

S/t=6S/t=3S/t=2S/t=1.5S/t=0.75

0.0 0.2 0.4 0.6 0.8 1.0

-0.006

-0.004

-0.002

0.000

0.002

0.004

S/t=6S/t=3S/t=2S/t=1.5S/t=0.75

σσ σσ y o

f the

ele

men

t

Position along line C-C' in Fig.14

(c) (d)

Fig. 16 σx (a, b) and σy (c, d) distributions along lines B–B and C–C for various S/t ratios

areas decreases and stress state changes to compressionwhen the S/t ratio is <1.5. The stress change in theelements 1, 2, 3, 4 shown in Fig. 17 is summarized inTable 3.

Similar to the mechanism of stress generation in thethree-layered model of Bai et al. (2000a), the horizontalstress (σx) between two fractures is the summation ofthe transferred tensile stress (σT

x ) and the compressivestress (σC

x ) due to fracture shortening and horizontalconstraint, i.e., σx = σT

x + σCx . The first mechanism is

stress transfer from the neighboring layer, and it pro-duces tensile stress in the fractured layer (Table 3, σT

3x,represented by hollow arrows). The second mechanisminvolves contraction of the fractured layer in the verti-cal direction, and it causes vertical compressive stress

in the fractured layer. The stress magnitude reduces asthe fracture spacing increases (Table 3, σC

3y and σC4y,

shown as solid arrows). The third mechanism involvesthe transferred stress cancellation by the stress gener-ated by the combined stress due to lateral constraint andthe near-fracture vertical stress (Table 3, σC

3x and σC4x,

represented by solid arrows). It becomes clear fromTable 3 that Element 1 is subjected to tension in thex-direction, and the tensile stress comes mainly fromthe transferred stress σT

x . Element 4 is subjected tocompression in both x and y directions, caused mainlyby the constraint in both directions.

A horizontal constraint produces a compressivestress. Because the existence of the traction-free frac-ture surface, the compressive stress decreases with

123123

Spacing and failure mechanism of edge fracture 253

Table 3 Stress state change in elements 1, 2, 3, 4 in Fig. 17 for S/t equals to 6, 3, 2, 1.5, and 0.75

decreasing S/t ratio. When the S/t ratio is smallenough (e.g., S/t = 0.75, Table 3), the horizontalconstraint becomes weak, and the compressive stressis reduced to zero. Although there are compressivestresses in the vertical direction in Elements 1 and 2,the horizontal compressive stresses in Elements 3 and4 are close to zero. This is because that as the fracturespacing decreases, very little stress can be transferredinto the narrow region between fractures so that thetotal stress is close to zero.

7 Discussion and conclusions

In all previous models, the materials are assumedhomogeneous and continuous, consequently, the frac-ture patterns are formed by fractures that propagate instraight lines and interact to form an array of polygons.Few of the existing theoretical and numerical modelscan adequately reproduce the complete process of frac-ture initiation, propagation, infilling, and saturation, asobserved both in the laboratory and in the field. There-fore, investigation based on material heterogeneity is animportant means to understand the failure mechanismof edge fractures. We consider material heterogene-ity and gradual fracture propagation in our numericalmodel, by using a 2D finite element tool, we reproducedthe process of fracture initiation, propagation, nucle-ation, infilling, and saturation in layered materials, and

the results are comparable to what occurred in natureand observed in laboratory experiment. We find thatmany factors control when the system will reach frac-ture saturation. In addition to fracture spacing, the twoother important factors identified are interface delam-ination and fracture propagation into the underlyinglayer. The results are in good consistency with the nat-ural phenomena.

We find that fracture spacing is influenced by inter-face debonding and base layer fracturing. When delam-ination occurs along the interface, the critical fracturespacing to layer thickness ratio S/t is much greaterthan the S/t ratio for completely bonded layers. This isbecause that the interface debonding consumes energy,reduces the contact area between the layers and hencereduces the stress transferred from the bottom layer tothe top layer. As a result, fewer new fractures will begenerated in the top layer.

We find also that the S/t ratio can be expressed asa function of the applied strain using an exponentialfunction. A curve which correlates the AE event countswith the fracture infilling phenomenon is obtained. Wedivide the fracture infilling process into four stages—microfracture generation, fracture propagation, nucle-ation, and fracture propagation into the underlyinglayer.

Numerical fringe contours of horizontal stress withdifferent S/t ratios are obtained and the results illus-trate that the horizontal stress along the fractured edge

123

254 C. Bao et al.

Fig. 17 Locations of fourelements in the model

changes from tensile to compressive stress as the S/tratio decreases. Stress variations with respect to the S/tratio for both homogeneous and heterogeneous mate-rials are presented. The critical S/t ratios obtained inthis paper are slightly larger than the values from pre-vious studies because we used finer element meshes inour simulation. The result of the homogeneous modelprovides an explanation for fracture saturation withoutconsidering the effect of material heterogeneity.

The modeling results show that fracture patternsstrongly depend on material heterogeneity, i.e., the dis-tribution of elastic modulus and tensile strength. In het-erogeneous models, the critical fracture length is longerthan that in homogeneous models with the same meanmaterial properties. Fracture patterns obtained from theheterogeneous models are closer to the ones observed inthe laboratory and in the field. In the study of the stressstate change between two adjacent edge fractures, wedefined a critical fracture spacing to layer thicknessratio (S/t)cr . When the S/t ratio changes from a valuegreater than the critical value to a value smaller thanthe critical value, the normal stress (in the direction per-pendicular to the fractures in the region along the freesurface) changes from tensile to compressive stress.

Stress distribution along one section near the endarea next to the fracture is plotted to illustrate the mech-anism of stress state change in the two-layered model.

Part of the top layer is subjected to tensile stress and thestress magnitude decreases as the S/t ratio decreases.

The horizontal stress between two fractures is gov-erned by level of the transferred tensile stress from thebottom layer, and the compressive stress from fractureshortening and the horizontal constraint. The overallstress distribution is controlled by the S/t ratio. Whenthe S/t ratio is greater than the critical value, the overallhorizontal stress is tensile because more tensile straincan be transferred to the fractured layer. When the frac-ture spacing increases, the vertical compressive stressdecreases and changes to tensile stress at locations farfrom the fracture. If the S/t ratio is less than the criticalvalue, instead of an edge crack propagating downwardsto the interface, an internal crack of the same size withits lower tip at the interface is more likely to propa-gate upwards toward the free surface. This is becausethat the overall stress state is in compression and fewerstresses are transferred from the bottom layer to the toplayer.

Being able to understand the mechanism of stresschange during fracturing of layered materials and pre-dict the minimum fracture spacing is important toexplain many natural phenomena such as joint spac-ing in rocks and fracture spacing in asphalt road sur-face. The numerical tool used in this study is provento be applicable to the studying of fracture spacing and

123123

Spacing and failure mechanism of edge fracture 255

failure mechanism of edge fractures in layered materi-als. Future study is planned to investigate surface crack-ing on road in cold regions due to contraction and findeffective means to limit or eliminate surface fractures.

Acknowledgments This research in this paper was supportedby the CNSF (National Natural Science Foundation of China)under contract numbers 50820125405 and 51121005, the 973Program under contract number 2011CB013503, and the CNSFunder contract number 51004020. We appreciate the valuablecomments and suggestions provided by two external reviewers.

Appendix: Weibull distribution function

In RFPA2D, the solid or material is assumed to be com-posed of many elements with the same size, and themechanical properties of these elements are assumedto conform to a given Weibull distribution defined asfollowing:

f (u) = m

u0

(u

u0

)m−1

exp

[−

(u

u0

)m](2)

where u is the parameter of the element (such asstrength or elastic modulus); the scale parameter, u0, isrelated to the average of the element parameter and theshape parameter, m, which defines the shape of the dis-tribution function. According to the definition, a largem implies a more homogeneous material property.

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