statistics of ductile fracture surfaces: the effect of...

$: Statistics of ductile fracture surfaces: the effect of ...laurentponson.com/assets/statistics-of-ductile... · Statistics of ductile fracture surfaces 139 principle of virtual work$
Int J Fract (2013) 184:137–149DOI 10.1007/s10704-013-9846-z

Statistics of ductile fracture surfaces: the effect of materialparameters

Laurent Ponson · Yuanyuan Cao ·Elisabeth Bouchaud · Viggo Tvergaard ·Alan Needleman

Received: 21 January 2013 / Accepted: 26 March 2013 / Published online: 9 April 2013© Springer Science+Business Media Dordrecht 2013

Abstract The effect of material parameters on thestatistics of fracture surfaces is analyzed under smallscale yielding conditions. Three dimensional calcu-lations of ductile crack growth under mode I planestrain, small scale yielding conditions are carried outusing an elastic-viscoplastic constitutive relation for aprogressively cavitating plastic solid with two popula-tions of void nucleating second phase particles repre-sented. Large particles that result in void nucleation atan early stage are modeled discretely while small par-ticles that require large strains to nucleate are homo-geneously distributed. The three dimensional analy-sis permits modeling of a three dimensional materialmicrostructure and of the resulting three dimensionalstress and deformation states that develop in the frac-ture process region. Material parameters characteriz-ing void nucleation are varied and the statistics of theresulting fracture surfaces is investigated. All the frac-

L. Ponson · Y. CaoInstitut Jean le Rond d’Alembert (UMR 7190), CNRS,Université Pierre et Marie Curie, Paris, France

E. BouchaudCEA-Saclay and ESPCI, Paris Tech, Paris, France

V. TvergaardDepartment of Mechanical Engineering, The TechnicalUniversity of Denmark, Lyngby, Denmark

A. Needleman (B)Department of Materials Science and Engineering,University of North Texas, Denton, TX, USAe-mail:[email protected]

ture surfaces are found to be self-affine over a size rangeof about two orders of magnitude with a very similarroughness exponent of 0.56 ± 0.03. In contrast, the fullstatistics of the fracture surfaces is found to be moresensitive to the material microscopic fracture proper-ties: height fluctuations are shown to crossover froma Student’s distribution with power law tails at smallscales to a Gaussian behavior at large scales, but thistransition occurs at a material dependent length scale.Using the family of Student’s distributions, this transi-tion can be described introducing an additional expo-nent μ = 0.15 ± 0.02, the value of which compareswell with recent experimental findings. The descriptionof the roughness distribution used here gives a morecomplete quantitative characterization of the fracturesurface morphology which allows a better compari-son with experimental data and an easier interpretationof the roughness properties in terms of microscopicfailure mechanisms.

Keywords Fracture surfaces · Roughness statistics ·Ductile fracture · Crack growth · Scaling behavior ·Finite elements

1 Introduction

Mandelbrot et al. (1984) first characterized the self-affine scaling properties of fracture surfaces. A hopein this study was to provide a quantitative correla-tion between fracture surface roughness and a mate-rial’s crack growth resistance. This hope has yet to be

123

138 L. Ponson et al.

realized. Nevertheless, Mandelbrot et al. (1984) stimu-lated a large body of experimental and theoretical workaimed at characterizing the scaling properties of frac-ture surfaces. Experimentally, the self-affine nature ofthe roughness of fracture surfaces has been observed ina wide variety of materials (metals, ceramics, glasses,rocks…) under a wide variety of loading conditions(quasi-static, dynamic, fatigue) and is typically char-acterized by two exponents: one measured parallel tothe crack front, denoted by ζ , and one measured alongthe crack propagation direction, denoted by β, see Pon-son (2007), Alava et al. (2006), Bonamy and Bouchaud(2011) for overviews. In some studies, e.g. Bouchaudet al. (1990), Måløy et al. (1992), it was suggested thatthe values of these exponents are universal.

Others have maintained that the characterization ofthe roughness of fracture surfaces is more complex. Forexample, in Bonamy and Bouchaud (2011), Bonamyet al. (2006) it is argued that there are two roughnessregimes: one pertaining to length scales smaller thanthe fracture process zone and the other to length scaleslarger than the fracture process zone, with each regimecharacterized by different values of the fracture sur-face roughness exponents. Alternatively, a multifractalcharacterization of fracture surface roughness has beensuggested as discussed in Cherepanov et al. (1995).The extent to which the appropriate characterizationof fracture surface roughness depends on the materialmicrostructure and/or the fracture mechanism remainsan open question.

A variety of theoretical analyses of fracture surfaceroughness have been carried out using a linear elasticfracture mechanics framework, e.g. Ramanathan et al.1997, or a molecular dynamics framework, e.g. Nakanoet al. (1995). For ductile fracture of structural metalsat room temperature the governing mechanism is thenucleation, growth and coalescence of micron scalevoids, which involves large plastic deformations, andoccurs over size and time scales much larger than cur-rently accessible by molecular dynamics. This processgoverns the evolution of ductile fracture surface rough-ness. The importance of accounting for porosity evo-lution and the accompanying plastic deformation inmodeling fracture surface roughness is emphasized inBouchaud (2003), Bouchbinder et al. (2004).

Calculations of fracture surface roughness using afinite deformation continuum mechanics formulationfor a progressively cavitating solid were carried outin Needleman et al. (2012). The analyses in

Needleman et al. (2012) were based on a modified Gur-son (1975) constitutive relation for a porous viscoplas-tic solid with two types of void nucleating particlesmodeled: large particles that nucleate voids at an earlystage and smaller particles that nucleate voids at a laterstage. The larger particles were represented as uniformsized discrete ”islands” of void nucleation, thus intro-ducing a material length scale, while the smaller par-ticles were uniformly distributed. This framework nat-urally accounts for the effects of damage evolution onthe stress and deformation state in the fracture processzone.

In Needleman et al. (2012), the calculations werecarried out for fixed material properties but for fourrandom distributions of the larger particles. Here, weemploy the same theoretical framework as in Needle-man et al. (2012) and solve the same small scale yield-ing problem. We focus attention on one of the fourspatial distributions of the larger particles in Needle-man et al. (2012) and vary two material parameters thatcharacterize void nucleation. We also use a more robustprocedure to define the fracture surface geometry fromthe output of our simulations and investigate the rough-ness statistical properties for the cases calculated inNeedleman et al. (2012) as well as for the simulationspresented here. We go beyond the characterization offracture surfaces by their correlation function and thevalue of the roughness exponent, and investigate thescaling of the full distribution of roughness. This scal-ing can be described by a second exponent character-izing the transition from power law tail at small scalesto Gaussian roughness statistics at large scales. Thisquantitative characterization of fracture surface geom-etry allows a better comparison with experimental dataand an easier interpretation of the roughness propertiesin terms of microscopic failure mechanisms.

2 Problem formulation

The boundary value problem analyzed is identical tothat in Needleman et al. (2012). Only parameters char-acterizing void nucleation differ. For completeness, theformulation and constitutive relation are briefly statedhere.

The focus is on quasi-static crack growth butdynamic analyses are carried out for numerical reasons.The calculations are carried out using a Lagrangian,convected coordinate formulation and the dynamic

123123

Statistics of ductile fracture surfaces 139

principle of virtual work written as∫

V

τ i jδEi j dV =∫

S

T iδui d S−∫

V

ρ∂2ui

∂t2 δui dV (1)

All field quantities are taken to be functions of theconvected coordinates, yi , and time, t ; Ei j are thecovariant components of the Lagrangian strain ten-sor, T i are the contravariant components of the trac-tion vector, τ i j are the contravariant components ofKirchhoff stress on the deformed convected coordinatenet (τ = Jσ , with σ being the Cauchy or true stressand J the ratio of current to reference volume), ν j andu j are the covariant components of the reference sur-face normal and displacement vectors, respectively, ρ

is the mass density, V and S are the volume and sur-face of the body in the reference configuration, and( ),i denotes covariant differentiation in the reference(y1, y2, y3) Cartesian frame. In presenting the resultswe will use the notations x , y and z for y1, y2 and y3,respectively.

A slice of material orthogonal to the initial crackplane is analyzed and the quasi-static mode I isotropicelastic singular displacement field is imposed on theremote boundaries. Also imposed are symmetry con-ditions corresponding to an overall plane strain con-straint.

As in Needleman et al. (2012), the in-plane blockdimensions are hx = hy = 0.4 m with an initialcrack tip with an opening of b0 = 1.875 × 10−4 m.The finite element mesh consists of 428, 256 twentynode brick elements giving 1, 868, 230 nodes and5, 604, 690 degrees of freedom. Ten uniformly spacedelements are used through the thickness hz of 0.005 m,with 10 elements through the thickness, and a uniform208 × 64 in-plane mesh is used in a 0.02 m × 0.006 mregion immediately in front of the initial crack tipgiving an in-plane element size of 9.62 × 10−5 m by9.38 × 10−5 m.

2.1 Constitutive relation

The constitutive framework is the modified Gursonconstitutive relation with

d = de + dΘ + dp (2)

and

de = L−1 : σ (3)

dΘ = αΘI (4)

dp =[

(1 − f )σ ˙εσ : ∂

∂σ

]∂

∂σ(5)

Small elastic strains are assumed, L is the tensor ofisotropic elastic moduli, σ is the Jaumann rate ofCauchy stress and Θ is the temperature.

Adiabatic conditions are assumed so that

ρcp∂Θ

∂t= χτ : dp (6)

with ρ = 7, 600 kg/m3 = 7.6×10−3 MPa/(m/sec)2,

cp = 465 J/(kg ◦K), χ = 0.9 and α in Eq. (4) is 1 ×10−5/K.

The flow potential is ( Gurson 1975),

= σ 2e

σ 2 +2q1 f ∗ cosh

(3q2σh

2σ

)−1−(

q1 f ∗)2 =0

(7)

where q1 = 1.25, q2 = 1.0 are parameters introducedin Tvergaard (1981), Tvergaard (1982a), f is the voidvolume fraction, σ is the matrix flow strength, and

σ 2e = 3

2σ ′ : σ ′, σh = 1

3σ : I, σ ′ = σ − σhI (8)

The function f ∗, introduced in Tvergaard andNeedleman (1984), is given by

f ∗ ={

f f < fc

fc+(1/q1− fc)( f − fc)/( f f − fc) f ≥ f

(9)

where the values fc = 0.12 and f f = 0.25 are used.The matrix plastic strain rate, ˙ε, is taken as

˙ε = ε0

[σ

g(ε, Θ)

]1/m

,

g(ε, Θ) = σ0G(Θ) [1 + ε/ε0]N (10)

with ε = ∫ ˙εdt , E = 70 GPa, ν = 0.3, σ0 = 300 MPa(ε0 = σ0/E = 0.00429), N = 0.1, m = 0.01 andε0 = 103/s.

The function defining the temperature-dependenceof the flow strength is

G(Θ) = 1 + bG exp(−c[Θ0 − 273])× [

exp(−c[Θ − Θ0]) − 1]

(11)

with bG = 0.1406 and c = 0.00793/K. In (11), Θ

and Θ0 are in K and Θ0 = 293K. Also, the initialtemperature is taken to be uniform and 293K.

123


The initial void volume fraction is taken to be zeroand the evolution of the void volume fraction is gov-erned by

f = (1 − f )dp : I + fnucl (12)

where the first term on the right hand side of Eq. (12)accounts for void growth and the second term for voidnucleation.

Eight point Gaussian integration is used in eachtwenty node element for integrating the internal forcecontributions. Lumped masses are used and twenty-seven point Gaussian integration is used for the ele-ment mass matrix. The discretized equations are inte-grated using the explicit Newmark β-method (β = 0),Belytschko et al. (1976). The constitutive updating isbased on the rate tangent modulus method of Peirceet al. (1984) while material failure is implemented viathe element vanish technique in Tvergaard (1982b).

2.2 Inclusions and fracture properties

The calculations model a material with two popula-tions of void nucleating second phase particles: (1) uni-formly distributed small particles that are modeled byplastic strain controlled nucleation and (2) large, lowstrength inclusions that are modeled as “islands” ofstress controlled nucleation. In each case, void nucle-ation is assumed to be described by a normal distribu-tion as in Chu and Needleman (1980).

For plastic strain nucleation,

fnucl = D ˙ε, D = fN

sN√

2πexp

[−1

2

( ε − εN

sN

)2]

(13)

The parameters fN = 0.04 and sN = 0.1 are fixed.For stress controlled nucleation

fnucl = A[ ˙σ + σh

](14)

with

A = fN

sN√

2πexp

[−1

2

( σ + σh − σN

sN

)2](15)

if (σ + σh) is at its maximum over the deformationhistory. Otherwise A = 0.

We confine attention to a single inclusion distrib-ution that is the distribution labeled 411 in Needle-man et al. (2012). There are 2016 possible inclusions(mean spacing about 6.7×10−4 m) in the uniform meshregion. Each inclusion radius is r0 = 1.5×10−4 m. For

an inclusion governed by stress nucleation centered at(y1

0 , y20 , y3

0), the value of fN in Eq. (15) at the point(y1, y2, y3) is

fN =⎧⎨⎩

fN for√

(y1 − y10 )2 + (y2 − y2

0 )2 + (y3 − y30 )2 ≤ r0;

0 for√

(y1 − y10 )2 + (y2 − y2

0 )2 + (y3 − y30 )2 > r0

(16)

The values fN = 0.04 and sN /σ0 = 0.2 are fixed.In the following, materials having various values of

parameters characterizing void nucleation are investi-gated (Fig. 1). For all materials analyzed the inclusiondistribution is that termed case411 in Needleman et al.(2012):

– Material # 1 has the parameter values analyzed inNeedleman et al. (2012) which are εN = 0.3 andσN /σ0 = 1.5. The fracture surfaces for this case arecalculated using the procedure introduced here fordefining the roughness geometry and the statisticalgeometry of this case is investigated in more detailthan in Needleman et al. (2012).

– Material # 2 has the nucleation strain of the smallparticles changed from the reference value to εN =0.4 with the nucleation stress of the inclusions fixedat σN /σ0 = 1.5.

– Material #2x has the nucleation strain of the smallparticles changed from the reference value to εN =0.2 with the nucleation stress of the inclusions fixedat σN /σ0 = 1.5. This calculation terminated afteran amount of crack growth that was too small toprovide a fracture surface that could be character-ized statistically in the same manner as for the othercases.

– Material # 3 has the nucleation stress of the inclu-sions changed from the reference value to σN /σ0 =2.0 with the nucleation strain of the small particlesfixed at εN = 0.3.

The results of the calculations are regarded as mod-eling quasi-static response via dynamic relaxation. Theresults are reported as for a quasi-static solution; forexample, the absolute magnitude of geometric dimen-sions do not matter; only geometric ratios matter. In thefollowing, the results are presented as for a quasi-staticsolution.

123123


Fig. 1 Initial inclusiondistribution. a On z = 0.b On z = hz. Note that thepositive y−axis is inopposite directions in aand b

Fig. 2 Void volumefraction distributionshowing the mode of crackgrowth for Material#2(εN = 0.4, σN /σ0 = 1.5).a On z = 0. b On z = hz.Note that the positivey−axis is in oppositedirections in a and b

Fig. 3 Void volumefraction distributionshowing the mode of crackgrowth for Material #2x(εN = 0.2, σN /σ0 = 1.5).a On z = 0. b On z = hz.Note that the positivey−axis is in oppositedirections in a and b

3 Results

3.1 Crack growth

Figures 2, 3 and 4 show distributions of void volumefraction f that indicate the crack growth path on theplanes z = 0 and z = hz for Material #2, Material #2x

and Material #3, respectively. In each case, the stage ofloading corresponds to the last stage of loading com-puted for that case. Note that in (a) for all three figuresthe y−axis is positive to the left whereas in (b) for allthree figures the y−axis is positive to the right. The darkgray regions show where f ≥ 0.1 so that the mater-ial within those regions has essentially lost all stress

123


Fig. 4 Void volumefraction distributionshowing the mode of crackgrowth for Material #3(εN = 0.3, σN /σ0 = 2.0).a On z = 0. b On z = hz.Note that the positivey−axis is in oppositedirections in a and b

Fig. 5 Distributions of void volume fraction f in a plane z =constant for material # 3 illustrating the definition of fracturesurfaces based on void volume fraction threshold values of ft =0.03, 0.04 and 0.10

carrying capacity. The lighter gray regions which cor-respond to 0.1 ≥ f ≥ 0.01 show the inclusions on thetwo planes that have nucleated voids. Shear localiza-tion can play a significant role in linking voids to themain crack. For Material #2x with εN = 0.2, the cal-culation was terminated after a relatively small amountof crack growth because intense localized deformationled to the stable time step becoming very small.

3.2 Fracture surface definition

The first step in defining the fracture surface is toplot the void volume fraction f distribution on a cross

section z = constant. Next a threshold value ft is cho-sen so that f ≥ ft corresponds to a connected crackwith ft being the largest value that gives a connectedcrack over a predefined region. Figure 5 shows a typicalcross section with contour values of f = 0.03, 0.04and 0.10. The fracture surfaces for these three con-tour values are rather close. Changing the size of thecracked region modifies the value of ft needed for aconnected crack. Interestingly, the scaling behavior,and especially the value of the roughness exponent,was found to be largely independent of this value. Onthe contrary, the roughness amplitude was observed toslightly increase while the value of ft was decreased.

For the fracture surfaces analyzed in the follow-ing, ft is taken close to 0.10 but sufficiently small toavoid any uncracked ligaments behind the main crack.Regions with f ≥ ft , but that are not connected to themain crack, were found—these are not shown in Fig. 5.They do not contribute to the roughness of the fracturesurfaces.

The roughness is computed for the two fracture sur-faces produced by each calculation. One fracture sur-face, termed as the lower fracture surface, is obtainedfrom the smallest value of z, for a given position (x , y)in the mean fracture plane, for which f ≥ ft on the con-nected crack while the other fracture surface, termedthe upper fracture surface, corresponds to the largestvalue of z, for a given position (x , y) in the mean frac-ture plane, for which f ≥ ft on the connected crack.The crack profiles obtained from the lower and upperfracture surfaces are denoted by hbot(x) and htop(x),respectively. These are shown on Fig. 6 for the voidvolume fraction distribution of Fig. 5. The jumps in theprofiles reflect the overhangs in the connected crack

123123


0 20 40 60 80 100 120 140 160−30

−20

−10

0

10

20

30

x/ex

h/e x

hbot(x)

htop(x)

Fig. 6 Fracture profiles extracted along the crack propagationdirection x obtained from the void volume fraction distributionof Fig. 5 using ft = 0.04 for material # 3

as seen in the porosity distribution. Our definition ofthe crack profiles mimics a profilometer that measuresonly the highest point on the fracture surface and so pre-vents the presence of overhang on the crack surfacesanalyzed subsequently.

4 Statistical analysis of fracture surfaces

4.1 Correlation functions of fracture surfaces androughness exponent

We first characterize the fluctuations of heights ofthe fracture surface using the correlation function, �h,defined as

�h(δx) =√⟨[h(x + δx, y) − h(x, y)]2

⟩x, y (17)

Here, 〈〉x denotes the average over x and y. The quan-tity �h(δx) can be interpreted as the typical differenceof height between two points separated by a distanceδx along the mean fracture plane. We focus on the cor-relation of heights in the direction of propagation only(x-axis), the width of the specimen in the perpendic-ular direction z being too small to allow a statisticalanalysis of the roughness in that direction.

The correlation function is computed for both theupper and lower fracture surfaces. The final correla-tion function is then obtained by averaging over thesestatistically equivalent surfaces. Indeed, from symme-try arguments, we expect the upper and lower fracturesurfaces in each calculation to share similar statisticalproperties. The correlation function �h(δx) is shownin Fig. 7 on a logarithmic scale, after normalization ofthe axis by the element spacing ex along the x-axis inthe fine mesh region. Figure 7a shows plots for the fourinclusion distributions analyzed in Needleman et al.

(2012) corresponding to material #1 (εN = 0.3 andσN /σ0 = 1.5) using the same notation as in Needle-man et al. (2012) to identify the inclusion distributions.The correlation functions exhibit a power law behavior

�h(δx) ∝ δxβ (18)

over more than two decades, characterized by theroughness exponent β. In this logarithmic represen-tation, β is the slope of the solid straight lines, powerlaw fits of the correlation functions computed from thefracture surfaces. Although the roughness amplitude—the position of the lines along the ordinates—may varyfrom one sample to another, the slope remains ratherindependent of the inclusion spatial distribution, asillustrated by the values of β given in the legend ofthe figure and obtained from a power law fit of thesecurves. These results give an average roughness expo-nent β = 0.55 ± 0.02, independent of the inclusiondistribution.

Our calculations capture the self-affine nature ofductile fracture surfaces, as observed experimentallyin Bonamy and Bouchaud (2011), and as also foundnumerically in Needleman et al. (2012), but here overa larger range of length scales than in Needleman et al.(2012). Since quadratic elements are used the meshspacing is ex/2 but the scaling in Fig. 7 holds for evensmaller δx values. This indicates that the interpolationprocedure between mesh points when defining the frac-ture surface geometry still preserves the roughness scal-ing. In Needleman et al. (2012) values 0.4 < β < 0.6were reported for the simulations in Fig. 7a. The pro-cedure used in this study to define the fracture surfacesfrom the porosity field gives a more precise character-ization of the crack roughness, a more uniform valueof the roughness exponent and the value of the rough-ness exponent β exhibits a smaller variation with theinclusion distribution than in Needleman et al. (2012).There are variations in the amplitude of the roughnessthat reflect variations in the threshold value ft of theporosity used to define the fracture surfaces. Neverthe-less, despite these fluctuations, the value of the rough-ness exponent is very robust, and changes very weaklyfrom one inclusion distribution to another. We have alsochecked that the value of the roughness exponent doesnot depend much on the value of the threshold ft usedto define the fracture surfaces using both ft = 0.04 andft = 0.10, supporting the value β � 0.55 reported inthis study.

123


10−1

100

101

102

10−1

100

101

δx/ex

Δh/e

x

case411 Fit: β = 0.56 case419 Fit: β = 0.57 case421 Fit: β = 0.56

Fit: β = 0.53 case447

10−1

100

101

102

10−1

100

101

δx/ex

Δh/e

x

Fit: β = 0.56

Fit: β = 0.55

Fit: β = 0.57

Material #1

Material #2

Material #3

(a) (b)

Fig. 7 Correlation function of the roughness of simulated frac-ture surfaces. a Material # 1: scaling behavior of the crack rough-ness observed for four different spatial distributions of the inclu-sions (see Needleman et al. 2012) characterized by a similarroughness exponentsβ = 0.55 ± 0.02. b Comparisons of materi-

als #1, #2 and #3: Scaling behavior of the crack roughness for thesame spatial distribution of inclusions, but with different materialproperties. regardless of the value of the nucleation threshold, theroughness follows power law behavior with a roughness expo-nent β = 0.56 ± 0.01

Figure 7b shows the effect of varying a material para-meter for a fixed spatial distribution of inclusions. Theinclusion distribution chosen here is case411 (Needle-man et al. 2012). In one calculation, the void nucleationstrain is taken to be εN = 0.4 (material #2) while in theother calculation σN /σ0 = 2.0 (material #3). All othermaterial and void nucleation parameters remain fixed.Also, for comparison purposes, the data for case411with the reference void nucleation parameters (material# 1) is also plotted. The simulated fracture surfaces hereshow a power law behavior, with a roughness exponent,β � 0.56, rather independent of the material proper-ties in the range investigated. This result is consistentwith a large body of experimental work that reports auniversal value of the roughness exponent, independentto a large extent of the microstructure of the materialstudied, its mechanical properties or the loading condi-tions used during the fracture test as seen in Bonamyand Bouchaud (2011), Bouchaud et al. (1990), Ponsonet al. (2006).

4.2 Non-Gaussian statistics of height fluctuations

As seen in Fig. 7, the Hurst exponents calculated for thevarious inclusion distributions and material parametervariations considered here and in Needleman et al.

(2012) are nearly the same. In Vernède et al. (2013)the full statistics of fracture surface height fluctuationswere obtained for cracks in a variety of materials. It wasfound that the height fluctuations could be described bya distribution that differed from a Gaussian by havingpower law tails. The deviation from Gaussian statisticswas found to be material dependent. Therefore, in orderto explore possible effects of material characteristics onthe predicted fracture surface morphology, we inves-tigate the full statistics of height variations δh(x, y)

defined by

δh(x, y) = h(x + δx, y) − h(x, y) (19)

Following Ponson (2007), Santucci et al. (2007), theprocedure used to compute the histogram P(δh) at agiven scale δx is the following.

1. We fix first a value of δx .2. For each location (x, y) on both the upper and

lower fracture surface, the corresponding heightvariations δh are computed. This procedure resultsin a large set {δh}δx of height variations at a givenscale δx .

3. The histogram of this set of values is computed.The histogram of δh is calculated by placing into‘boxes’ [bminb2], [b2b3],…, [bn−1bmax] the valuesof δh where the side b1, b2 of the boxes are distrib-

123123


−5 0 510

−3

10−2

10−1

100

δh/ex

p

100

102

100

101

δx/ex

Δh/e

x

δx/ex = 0.8

δx/ex = 1.6

δx/ex = 3.2

δx/ex = 6.4

Fig. 8 Histograms P(δh|δx) of height variations δh (see Eq.(19)) for various values of δx for material #1 and the spatial dis-tribution of particles labeled case421 in Needleman et al. (2012).The solid lines are fits based on the Student’s distribution of Eq.(20) using various values of the parameters k and δhc.The insetrepresents the correlation function of the surface. The points (orsecond moment) corresponding to the distributions representedin the main panel are plotted using the same color code

uted homogeneously between bmin = min[δh] andbmax = max[δh].

4. The histogram or probability density

Pi

(δhi = bi + bi+1

2|δx

)

is calculated as the fraction of values of δh con-tained in the i th box.

In order to study the scaling behavior of the frac-ture surface roughness, this procedure is repeated forvarious scales δx , leading to a family of histogramsP(δh|δx). An important property is that the standarddeviation of the distributions P(δh|δx) corresponds tothe correlation function �h(δx) of the fracture sur-faces, as can be observed from its definition in Eq. (17).

The distribution P(δh|δx) is shown in Fig. 8 forfour values of δx for the simulation labeled case421 inNeedleman et al. (2012). The larger the value of δx , thebroader the distribution, as expected from the scaling ofthe correlation function �h(δx) ∝ δxβ . The standarddeviation of these histograms is plotted in the inset as afunction of δx using the same color code as in the mainpanel. As expected, the standard deviation evolves as apower law with exponent β = 0.56, in agreement withthe plot of �h(δx) shown in Fig. 7.

The distribution of height variations computed onnumerical fracture surfaces are not Gaussian, but

δhc/e

x

δx/ex

[k/(

k−2)

]1/2

Fit: γ = 0.71

Fit: μ = 0.15

101

100

100

101

[k/(k−2)]1/2

δhc/e

x

2.100

3.100

100

Fig. 9 Variations of the parameters√

k/(k − 2) and δhc/exobtained from the fit of the roughness distributions P(δh) shownin Fig. 8 for case 421 in Needleman et al. (2012) using Student’sdistribution (see Eq. (20)). The transition from power law tails(√

k/(k − 2) � 1) to Gaussian (√

k/(k − 2) � 1) statistics asthe length scale δx increases is characterized by the exponentμ = 0.15. The typical value of δhc of the roughness extractedusing fits by Student’s distribution also evolves as a power lawof the length scale δx with exponent γ = 0.71

exhibit fat tails with power law behavior P(δh) ∝δh(k+1)/2 for large values of δh. This means that largeheight fluctuations are not exponentially rare on duc-tile fracture surfaces as is the case for brittle fracturesurfaces (Ponson et al. 2007). Indeed, the crack profilesin the calculations here display a non-negligible num-ber of abnormally large fluctuations δh, as qualitativelyseen in Fig. 6.

To describe this effect more quantitatively, the distri-butions P(δh) are described using a family of probabil-ity distributions referred to as Student’s t distributions

tk,δhc(δh) ∝ 1

δhc

(1 + 1

k

(δh

δhc

)2)−(k+1)/2

(20)

with parameters k and δhc, and represented by solidlines in Fig. 8. For reasons discussed in the next sec-tion, it is more appropriate to consider the parameter√

k/(k − 2) instead of k.These parameters, obtained from the fit of the distri-

butions P(δh|δx), are represented in Fig. 9 as a functionof the scale δx . The first parameter that characterizesthe shape of the distribution decreases as a power lawwith the scale δx√

k

k − 2∝ δx−μ (21)

with μ = 0.15 as illustrated by the straight line varia-tions in the logarithmic representation of Fig. 9.

123


The second fitting parameter used to describe theroughness distributions P(δh|δx) is also representedin Fig. 9 as a function of the scale δx . It here increasesas a power law

δhc ∝ δxγ (22)

with exponent γ = 0.71.

The ductile crack profiles studied here exhibit a morecomplex behavior than brittle fracture surfaces. Indeed,in brittle materials, distributions of height variationsfollow a Gaussian behavior at all scales (Ponson 2007;Ponson et al. 2007). In our description based on Stu-dent’s t distribution, this corresponds to k → ∞, orequivalently, to

√k/(k − 2) = 1, for any value of δx .

In other words, only one exponent is needed to describethe scaling of the roughness distribution, since μ = 0.For ductile fracture surfaces, not one, but three scal-ing exponents β, μ and γ are required. However, thiscan be reduced to two independent exponents basedon a simple relationship. Using the definition (20) ofStudent’s t distribution, one can show that its secondmoment, and so the correlation function, varies as

�h(δx) ∝ δhc

√k/(k − 2) (23)

As a result, from the scaling relations in Eqs. (18), (21)

and (22), one obtains

β = γ − μ. (24)

This relation is satisfied by the exponents extractedhere for which β = 0.56 (case421) was measured (seeFig. 7). This supports our description of the roughnessstatistics based on Student’s t distribution.

The relation between exponents suggests an inter-pretation of the roughness exponent β that character-izes the scaling behavior of the correlation functionwith the scale δx as the combined effect of the varia-tions of the typical roughness δhc with the scale δx andthe variations in the actual shape of the distribution ofheight fluctuations. It also suggests that the universalvalue of the roughness exponent β relies on the value ofμ that describes the change in the shape of roughnessdistribution. Experimental results in Vernède et al.(2013) are consistent with these findings: the statisticalanalysis of both aluminum alloy and mortar fracturesurfaces indicates a similar behavior, with a measuredvalue μexp � 0.15. However, the experimental valueof the second exponent, γ exp � 0.9, is larger than thevalue obtained for the calculated fracture surfaces here.

In the following, we use the above analysis of thedistribution of height variations to compare the fracturesurfaces obtained for materials #1, #2 and #3.

4.3 Roughness statistics: comparison between thematerials

The analysis in Sect. 4.2 revealed that the scaling ofthe roughness distribution of ductile fracture surfacescould be described using Student’s tk,δhc distributionintroduced in Eq. (20), with the adjustable parameters√

k/(k − 2) and δhc that follow power laws with thescale δx . We focus in this section on the parameter√

k(k − 2) that characterizes the distribution shape. Fora finite value of the parameter k, Student’s t distributiondisplays fat tails with

tk,δhc(δh) ∝ δh(k+1)/2 for δh � δhc

while tk,δhc(δh) approaches a Gaussian distributionwhen k tends to infinity. As a result, this family of distri-butions is suited to describe a transition from power lawtails to Gaussian statistics. In particular

√k/(k − 2) =

1 for a Gaussian distribution and√

k/(k − 2) > 1 oth-erwise. We expect the value of

√k/(k − 2) to indicate

the distance from Gaussian behavior.The analysis of the fracture surface of material

#1 has revealed that√

k/(k − 2) remains larger thanone, indicating fat tails statistics and deviation froma Gaussian distribution at the length scales investi-gated (see Fig. 9). However, extrapolating the powerlaw behavior

√k/(k − 2) ∼ δx−μ towards larger val-

ues of δx , predicts that it will reach√

k/(k − 2)(δx =ξ) = 1 at the crossover length δx = ξ . Interestingly,it means that for δx > ξ , the fracture surface mightrecover Gaussian statistics. Such an extrapolation leadsto ξ = 100ex for material #1.

The transition from fat tail statistics for δx < ξ toGaussian behavior for δx > ξ calls for an interpreta-tion in terms of fracture mechanisms. In the scenarioproposed in Vernède et al. (2013), Gaussian statisticsof the roughness would be reminiscent of brittle frac-ture surfaces while distributions with fat tails wouldbe signature of damage mechanisms. This behavior isfairly consistent with the scaling behavior of the cor-relation function that gives roughness exponents of theorder of β = 0.5 for brittle failure and larger val-ues around β = 0.6 at a smaller scale when fail-ure is accompanied by damage mechanisms ( Bonamy

123123


10−1

100

101

100

δx/ex

[k/(

k−2)

]1/2

Material #1 Fit: μ = 0.16 Material #2 Fit: μ = 0.14 Material #3 Fit: μ = 0.14

2.100

3.100

Fig. 10 Variations of the parameter√

k/(k − 2) used in the fitby Student’s distribution, Eq. (20), of the histrograms P(δh) ofheight fluctuations as a function of δx

et al. 2006; Morel et al. 2008). The crossover lengthbetween the two regimes - large exponents and fattail statistics at small scale to small exponents withGaussian statistics at large scale - may be a reflectionof the size scale associated with the micro mechanismof fracture. If so, the value of ξ extracted from theanalysis of the roughness statistics could serve as alength scale associated with the underlying damageprocess.

Figure 10 compares the variations of the parame-ter

√k/(k − 2) for the materials analyzed. The power

law decay differs from one material to another. Theexponent μ � 0.15 —the slope of the straight linevariations in this logarithmic representation—remainsrather constant. But the variations in the position alongthe ordinates indicate different values of ξ , and moregenerally, different level of deviations from Gaussian(and so brittle) behavior.

The deviation from Gaussian behavior is more pro-nounced for material #3 than for material #2 for whichthe values of

√k/(k − 2) is smaller at all length scales

δx . This may indicate that material #3 is more ductilethan material #2 which is expected due to the largervalue of the nucleation stress for the large inclusions.Extrapolation of

√k/(k − 2) to large values of δx indi-

cates that the crossover value ξ = 300ex for material#3 which is much larger than the value estimated forthe materials #1 and #2.

The comparison of material #1 with material #2is more complex. The relative position of the curves

√k/(k − 2) indicates stronger deviations from Gaussian

behavior for material #1. However, the extrapolatedvalue of ξ � 100ex gives a similar value for bothmaterials.

5 Discussion

Calculations have been carried out for a fixed size, den-sity and distribution of discretely modeled void nucle-ating inclusions. The statistics of the fracture surfaceswas investigated. In addition, the statistics of the frac-ture surfaces obtained in Needleman et al. (2012) wererecalculated. In Needleman et al. (2012), the fracturesurface was defined by a threshold value ft = 0.1,somewhat smaller than the material parameter ft usedin the constitutive model to define final failure. In thepresent studies, in some cases, an even smaller value,ft = 0.03, is chosen to define a well connected cracksurface. Fig. 5 shows that the crack surfaces obtainedby these definitions are rather close, as the value of fincreases steeply in the material near the fully opencrack. In Needleman et al. (2012), a thickness averageof the fracture surface height was calculated and thisaverage profile was used to obtain the correlation func-tion �h(δx). Here, instead, �h(δx) is calculated foreach cross section, and is subsequently averaged, andit turns out that this procedure gives power law scal-ing over a larger range, with less difference betweenthe various inclusion distributions in Needleman et al.(2012). Also in Needleman et al. (2012) only the rough-ness exponent was calculated whereas here we calcu-late more complete fracture surface statistics.

As noted in Needleman et al. (2012) the calcula-tions contain several length scales: (1) the mean spac-ing between inclusion centers; (2) the inclusion radius;(3) the slice thickness; and (4) the finite element meshspacing. A significant difference between these lengthscales is that the first three are physically relevant lengthscales whereas any dependence of the results on thefinite element mesh length scale is a numerical arti-fact. Whether or not there is a significant effect of thefinite element mesh spacing on the predicted statisticsof fracture roughness remains to be determined.

Our results give rise to a value of β = 0.56 ± 0.03that is essentially independent of the inclusion distribu-tions and fracture properties considered and that is alsoclose to the value β3D � 0.6 reported for 3D fracturesurfaces of a wide range of ductile and quasi-brittle

123


materials in the crack propagation direction Ponsonet al. (2006), Ponson et al. (2006). While the proba-bility distribution of heights is a Gaussian for brittlefracture, it is more complex for ductile fracture sur-faces. We showed that Student’s t distribution can fitall the histograms in Fig. 8. This distribution is equiva-lent to a Gaussian at sufficiently small values of δh/ex

but crosses over to a power-law at larger values. Also,as seen in Fig. 8 for small values of δx/ex the com-puted distributions display fat power law tails whereasfor sufficiently large values of δx/ex the distributionis well-fit by a Gaussian. The transition from fat tailto Gaussian statistics is described by introducing a sec-ond exponent μ � 0.15. In the experiments in Vernèdeet al. (2013) where the roughness statistics were calcu-lated as an average over all directions in the fracture sur-face, the statistics were also observed to deviate froma Gaussian and the deviation was well-described bytwo exponents. The value of μ was close to the oneobtained here but the value of γ was larger than inour calculations. Hence, while for brittle fracture sur-faces, knowledge of the roughness index β is sufficientto describe completely the probability distribution, forductile fracture surfaces independent exponents areneeded.

Our results indicate that for ductile fracture sur-faces there is a length characterizing the crossover frompower law to Gaussian statistics. The results also sug-gest that this length may correlate with the size of thefracture process zone. What remains to be determinedis the extent to which this crossover length as well asother aspects of the non-Gaussian statistics correlatewith measures of fracture toughness such as KIc andcrack growth resistance curves. For the type of materi-als considered here, this requires more extensive calcu-lations with increased crack growth, larger variationsof inclusion density and distribution, and greater varia-tion in material properties. Such calculations are beingundertaken and the results here suggest the possibil-ity of a connection being made between post-mortemfracture surface statistics and crack growth resistance.

6 Conclusions

1. A central result of this paper is that, for the materialparameter variations analyzed, the computed duc-tile fracture surfaces exhibit self-affine propertieswith similar values of the fracture surface rough-

ness exponent β but their full statistical propertiesdiffer.

2. For the ductile fracture surfaces analyzed:

– The computed fracture surfaces are self-affine overa range of length scales of about two orders of mag-nitude.

– The computed values of the fracture surface rough-ness exponents along the crack growth direction arenot sensitive to the larger particle distributions norto the fracture parameter variations investigated.

– The computed fracture surface roughness distrib-utions are not Gaussian but they are all well fittedby Student’s distribution. Both the roughness expo-nent β = 0.56 ± 0.03 and the exponent μ � 0.15characterizing the transition with δx/ex from fattail to Gaussian statistics are found to be ratherindependent of the spatial distributions of the largerinclusions and of material parameters in the rangeinvestigated.

– The computed full fracture surface roughness sta-tistics vary with the fracture parameters investi-gated.

Acknowledgments The financial support provided by the U.S.National Science Foundation through Grant CMMI-1200203 andby the European Union through the ToughBridge Marie CurieGrant (LP) is gratefully acknowledged. We also thank Jean-Philippe Bouchaud and Stéphane Vernède for fruitful discussions

References

Alava MJ, Nukala PKVV, Zapperi S (2006) Statististical modelsof fracture. Adv Phys 55:349–476

Belytschko T, Chiapetta RL, Bartel HD (1976) Efficient largescale non-linear transient analysis by finite elements. Int JNumer Methods Eng 10:579–596

Bonamy D, Ponson L, Prades S, Bouchaud E, Guillot C (2006)Scaling exponents for fracture surfaces in homogeneous glassand glassy ceramics. Phys Rev Lett 97:135504

Bonamy D, Bouchaud E (2011) Failure of heterogeneous mate-rials: a dynamic phase transition? Phys Rep 498:1–44

Bouchaud E, Lapasset G, Planès J (1990) Fractal dimension offractured surfaces: a universal value? Europhys Lett 13:73–79

Bouchaud E (2003) The morphology of fracture surfaces: a toolfor understanding crack propagation in complex materials.Surf Rev Lett 10:797–814

Bouchbinder E, Mathiesen J, Procaccia I (2004) Roughening offracture surfaces: the role of plastic deformation. Phys RevLett 92:245505

Cherepanov GP, Balankin AS, Ivanova VS (1995) Fractal frac-ture mechanics–a review. Eng Fract Mech 51:997–1033

Chu CC, Needleman A (1980) Void nucleation effects in biaxiallystretched sheets. J Eng Mater Technol 102:249–256

123123


Gurson AL (1975) Plastic flow and fracture behavior of ductilematerials incorporating void nucleation, growth and interac-tion. Ph.D. Thesis, Brown University.

Måløy KJ, Hansen A, Hinrichsen EL, Roux S (1992) Experi-mental measurements of the roughness of brittle cracks. PhysRev Lett 68:213–215

Mandelbrot BB, Passoja DE, Paullay AJ (1984) Fractal characterof fracture surfaces of metals. Nature 308:721–722

Morel S, Bonamy D, Ponson L, Bouchaud E (2008) Transientdamage spreading and anomalous scaling in mortar crack sur-faces. Phys Rev E 78:016112

Nakano A, Kalia RK, Vashista P (1995) Dynamics and morphol-ogy of brittle cracks: a molecular-dynamics study of siliconnitride. Phys Rev Lett 75:3138–3141

Needleman A, Tvergaard V, Bouchaud E (2012) Predictionof ductile fracture surface roughness scaling. J Appl Mech79:031015

Peirce D, Shih CF, Needleman A (1984) A tangent modulusmethod for rate dependent solids. Comput Struct 18:875–887

Ponson L, Auradou H, Vié P, Hulin JP (2006) Low self-affineexponents of fractured glass ceramics surfaces. Phys Rev Lett97:125501

Ponson L, Bonamy D, Bouchaud E (2006) Two dimensional scal-ing properties of experimental fracture surfaces. Phys Rev Lett96:035506

Ponson L, Bonamy D, Auradou H, Mourot G, Morel S, BouchaudE, Guillot C, Hulin J-P (2006) Anisotropic self-affine proper-ties of experimental fracture surfaces. Int Fract 140:27–36

Ponson L (2007) Crack propagation in disordered materials: howto decipher fracture surfaces. Ann Phys 32:1–120

Ponson L, Auradou H, Pessel M, Lazarus V, Hulin J-P (2007)Failure mechanisms and surface roughness statistics of frac-tured Fontainebleau sandstone. Phys Rev E 76:036108

Ramanathan S, Ertas D, Fisher DS (1997) Quasistatic crack prop-agation in heterogeneous media. Phys Rev Lett 79:873–876

Santucci S, Måløy KJ, Delaplace A, Mathiesen J, Hansen A,Bakke J, Schmittbuhl J, Vanel L, Ray P (2007) Statistics offracture surfaces. Phys Rev E 75:016104

Tvergaard V (1981) Influence of voids on shear band instabilitiesunder plane strain conditions. Int J Fract 17:389–407

Tvergaard V (1982a) On localization in ductile materials con-taining spherical voids. Int J Fract 18:237–252

Tvergaard V (1982b) Influence of void nucleation on ductileshear fracture at a free surface. J Mech Phys Solids 30:399–425

Tvergaard V, Needleman A (1984) Analysis of the cup-cone frac-ture in a round tensile bar. Acta Metall 32:157–169

Vernède S, Cao Y, Bouchaud JP, Ponson L (2013) Extreme eventsand non-Gaussian statistics of experimental fracture surfaces(submitted for publication)

123

statistics of ductile fracture surfaces: the effect of...

Documents