Morphogenesis and Propagation of Complex Cracks Induced by Thermal Shocks

Blaise Bourdin,1,* Jean-Jacques Marigo,2 Corrado Maurini,3,4 and Paul Sicsic2,5

1Department of Mathematics and Center for Computation & Technology, Louisiana State University,Baton Rouge, Louisiana 70803, USA

2Laboratoire de Mécanique des Solides (UMR-CNRS 7649), École Polytechnique, 91128 Palaiseau Cedex, France3Institut Jean Le Rond d’Alembert (UMR-CNRS 7190), Université Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris, France

4Institut Jean Le Rond d’Alembert (UMR-CNRS 7190), CNRS, 4 Place Jussieu, 75252 Paris, France5Lafarge Centre de Recherche, 95 Rue de Montmurier 38290 St-Quentin-Fallavier, France

(Received 12 September 2013; revised manuscript received 22 November 2013; published 10 January 2014)

We study the genesis and the selective propagation of complex crack networks induced by thermal

shock or drying of brittle materials. We use a quasistatic gradient damage model to perform large-scale

numerical simulations showing that the propagation of fully developed cracks follows Griffith criterion

and depends only on the fracture toughness, while crack morphogenesis is driven by the material’s internal

length. Our numerical simulations feature networks of parallel cracks and selective arrest in two

dimensions and hexagonal columnar joints in three dimensions, without any hypotheses on cracks

geometry, and are in good agreement with available experimental results.

DOI: 10.1103/PhysRevLett.112.014301 PACS numbers: 46.15.Cc, 62.20.mt

Complex crack patterns are ubiquitous in nature and intechnology applications. Yet the theoretical understandingand predictive numerical simulation of how and whencomplex crack patterns arise (nucleation) and how theyevolve (crack propagation) are fraught with challenges.Although approaches based on phase fields [1] or varia-tional regularizations [2] have led to significant advance inthe numerical simulation of complex crack patterns, shortof introducing initial flaws at the structural scale [3],prescribing ad hoc stress criteria [4], or accepting globalenergy minimization arguments whose physical relevanceis debated [5–7], the predictive understanding of cracknucleation is still an elusive goal.

It is well accepted that while Griffith-like models areappropriate for crack propagation at the scale of a struc-ture, they are inadequate for the modeling of crack nuclea-tion in brittle materials. Arguably, finer models, where amicroscopic (material) length scale plays a fundamentalrole, are necessary to determine the critical load and crackgeometry at the onset, especially in situations where com-plex crack patterns arise straight from the nucleation. Theconsistent combined modeling and numerical simulationof crack nucleation and propagation from the material tothe structural length scale is a challenging and largely openissue.

In this Letter, we study the morphogenesis and theselective growth of complex crack patterns induced bymaterial shrinking under thermal shock. We report unpre-cedented quantitative agreement between numerical simu-lations, a theoretical model, and experiments at scalesspanning from the material internal length to the structurallength scale. Our numerical simulations predict key fea-tures of fracture patterns observed in experiments, such asthe formation of periodic patterns and the scaling laws

governing their selective propagation in two and threedimensions, and do not require any a priori hypotheseson cracks geometry. The method we use leverages recentprogress in the understanding of the links between damagemodels [8,9] and the variational approach to fracture[2,10]. It is based on a rate-independent gradient damagemodel with stress softening based on two material parame-ters: the fracture toughness, which rules the evolution offully developed cracks, and the material’s internal length,which controls the initial stages of crack nucleation.We investigate the thermal shock of brittle ceramics, a

now classical experimental setup [3,11,12] where a sampleinitially at a uniform temperature T0 is quenched in a coldbath at temperature T0 ��T. We consider a rectangularslab � exposed to the thermal shock through its thin facesonly. We focus first on very thin slabs, which we representby a two-dimensional body in plane stress. We assume thatwithin the range of temperatures involved, the materialproperties remain constant. Denoting by u the displace-ment field and " ¼ ðruþrTuÞ=2 the linear strain tensor,we consider for the sound material a linear elastic behaviorof energy density c tð"Þ ¼ A0ð"� "tht Þ � ð"� "tht Þ=2,where A0 is the isotropic elastic stiffness tensor. The in-elastic deformation induced by the time-dependent tem-perature field Tt is "

tht ¼ �ðTt � T0ÞI, where I is the

identity matrix. The index t is meant to highlight thedependence on time. We neglect the cracks’ influence onheat transfer so that the temperature field Tt solves the heatequation @tTt � kcr2Tt ¼ 0 on�. Phase changes, nonuni-form convection, and other nonlinear aspects of the heatexchange between the fluid and the sample are neglectedby assuming that the temperature of the domain boundaryexposed to the thermal shock is constant and equal to thatof the water bath, i.e., Tt ¼ T0 � �T on @�. Inertial

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effects are not considered because the diffusion velocity ofthe temperature field is much slower than the wave speed inthe material at the relevant scales in time and space. Thishypothesis is universally accepted in the literature onthermal shock problems [3,4,12–14]. We model materialfailure using a gradient damage model characterized by theenergy function

E tðu; �Þ ¼Z�

c tð"Þsð�Þ þ

Gc4cw

�wð�Þ‘

þ ‘jr�j2�dx; (1)

where � is a scalar damage field varying between 0 (soundmaterial) and 1 (fully damaged material), Gc is the mate-

rial’s fracture toughness, ‘ is an internal length, and cw ¼R10

ffiffiffiffiffiffiffiffiffiffiwðsÞp ds is a normalization constant. In a time-discrete

setting, the quasistatic evolution is obtained by solvingat time ti the following minimization problem:minu;���i�1Etiðu; �Þ, where the unilateral constraint on �enforces the irreversibility condition on the damage. Thecompliance function s and the energy dissipation functionw should be chosen such that (1) converges as ‘ ! 0 to aGriffith-like energy

R�n� c tð"ÞdxþGcSð�Þ, where S is

the surface measure of the crack � [2,15,16]. In this model,material interpenetration in the fully damaged area ispossible. In all the simulations presented here, it can bechecked a posteriori that this issue does not present itself.Here, we use sð�Þ ¼ 1=ð1� �Þ2 and wð�Þ ¼ �, a choicemotivated by the convenience of its numerical implemen-tation and specific analytical studies [9,17]. With thischoice the damage model has a stress-softening behaviorand remains purely elastic without damage until the stressreaches the critical value,

�c :¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiGcEw

0ð0Þ2cw‘s

0ð0Þ

s¼

ffiffiffiffiffiffiffiffiffiffiffiffi3GcE

8‘

s: (2)

The relation above may be used to determine the numericalvalue of the internal length for a specific material from theknowledge of its elastic limit �c, Young modulus E, andfracture toughness Gc [17]. The present model is in manyaspects similar to the phase-field models of fracturedeveloped independently [18]. Those with single-welldissipation potentials [1,19] are in the form of (1) withwð�Þ ¼ c½1� gð1� �Þ�, where gð�Þ ¼ 4�3 ��4. Onesignificant difference is that, while phase-field modelstypically involve some form of viscous regularizations,our formulation is rate independent. In addition, the currentliterature based on phase-field models is concerned onlywith the propagation of preexisting cracks and does notconsider the initiation problem.

The dimensional analysis of the energy (1) highlightsthree characteristic lengths: the geometric dimension of thedomain L, the internal length ‘, and the Griffith length‘0 ¼ Gc=ðE�2�T2Þ. Using the material’s internal lengthas the reference unit, the problem can be reformulated in

terms of two dimensionless parameters, the dimension ofthe structure L=‘ (a geometric parameter) and the intensityof the thermal shock ‘0=‘ (a loading parameter). This is asignificant departure from the classical Griffith settingwhere the only relevant parameter is L=‘0 [4,5,13].Figure 1 compares the experiment from Fig. 5(d) of

Ref. [20] (1� 9:8� 50 mm ceramic slab, �T ¼ 380 �C)with the damage field from a numerical solution of thegradient damage model. The material properties, commu-nicated by the authors of Ref. [8], are E ¼ 340 GPa, � ¼0:22, Gc ¼ 42:47 Jm�2, �c ¼ 342:2 MPa, and � ¼ 8�10�6 K�1, which using (2) gives ‘ ¼ 46 �m and ‘0 ¼14 �m. As our model is rate independent, its solution isindependent of kc, up to a change of time scale. Thenumerical results are obtained through a finite elementdiscretization and the approach of Refs. [2,17,21]. Themain technical difficulties are the constrained minimiza-tion of a nonconvex energy and the need for a spatialdiscretization adapted to the material length scale ‘.Cracks correspond to the localized bands where � goesfrom 0 to 1 and back to 0. The qualitative agreementbetween experiments and simulation is very good. In par-ticular, our simulations reproduce the key phenomenon:the emergence of a periodic array of parallel short cracks atthe initiation and their selective propagation toward theinterior of the slab. Figure 1(c) shows a quantitative com-parison between the numerical simulation of Fig. 1(a) and

FIG. 1 (color online). Full-scale numerical simulation of aceramic slab submitted to a thermal shock. (a) Damage fieldfrom the numerical simulation (blue, � ¼ 0; red, � ¼ 1).(b) Experimental results from Fig. 5(d) of Ref. [20].(c) Average crack spacing d as a function of their depth a for(a) and (b). The solid line is an approximate scaling law obtainedin Ref. [13] by imposing a period doubling condition on aGriffith model. Here, ‘ ¼ 46 �m is the material internal length,‘0 ¼ Gc=ðE�2�T2Þ ¼ 14 �m the Griffith length (loading pa-rameter), 2L ¼ 9:8 mm the total depth of the slab. See Fig. 2 forthe meaning of the distributed damage zone.

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experimental data from Ref. [20] by plotting the averagecrack spacing d as a function of the distance a to the edgeexposed to the thermal shock for the final configuration;the agreement is striking. Note that in the experimentalresults shorter cracks are probably filtered out by theadopted experimental crack detection methods [17]. In afirst regime, very short equidistributed cracks nucleate (theplateaus of the crack spacing for short depth in the numeri-cal experiments), followed by selective arrest and perioddoubling (see the Supplemental Material [22]). In thecentral region of the plot, we can compare experimentaland simulation data with a scaling law obtained in Ref. [13]through linear fracture mechanics calculations by impos-ing a bifurcation condition between crack propagationmodes with period doubling or not (solid line). For largervalues of a, we observe the final crack arrest caused by thefinite size of the sample, again in very good agreement withthe experiments. Whereas classical theories can be appliedin the second and third regimes consisting of fully devel-oped cracks, they cannot properly account for the nuclea-tion phenomenon observed here without preexisting flaws.Our simulations are initialized with a null damage field, anhomogenous material, and an unflawed geometry. Thecrack nucleation is due to the softening character of thematerial behavior.

The second series of simulations focuses precisely onthe crack nucleation process and, hence, on short times. Inthis setting, one can assume that the domain is semi-infinite, so that the geometric parameter L=‘ is infiniteand the only parameter is the intensity of the thermal shock‘0=‘. For an undamaged material, the stress is uniaxial andreaches its maximum value �max ¼ E��T at the surfaceof the thermal shock. Since ð�max=�cÞ2 ¼ 3‘=8‘0, formild-enough thermal shocks (‘0=‘ > 8=3), the criticalstress is never reached and the solution remains elastic atall time. If ‘0=‘ < 8=3, damage takes place at t ¼ 0, ishomogeneous in the horizontal direction, and non-null in aband of finite thickness D, which penetrates progressivelyinside the body until a critical time t�. At t ¼ t� thehorizontally homogeneous solution becomes unstable andthe damage field develops oscillations of periodicity ��[Fig. 2(a)]. The analytical solutions for the damage field inthe first stage of the evolution and its bifurcation andstability analysis are reported in Ref. [23], providing semi-analytical results for the periodicity ��, the damage pene-tration D�, and the time t� at the bifurcation. Here, weperform several simulations varying ‘0=‘ and detect thecritical parameter at the bifurcation. In Fig. 2(b) the nu-merical simulations (dots) are compared to Ref. [23] (solidlines). The good agreement provides an excellent verifica-tion of our numerical model. For severe shocks (‘0 � ‘),the results disclose a well-definite asymptotic behavior

with �� ffiffiffiffiffiffiffiffi‘0‘p , D� ‘, and t� ‘0‘=kc. In this regimewe observe numerically that all oscillations at the bifurca-tion develop in fully formed cracks (max� ¼ 1), which is

not the case for milder shocks (‘0 ‘). However, the fullpostbifurcation analysis remains an open problem at thistime.Experimental studies show that in three dimensions,

cracks delineate cells with coarsening polygonal crosssections [24]. Because of the complexity of the problem,the few available theoretical and numerical studies arebased either on simplified two-dimensional models[25,26] or on strong assumptions on the crack geometry[27]. The numerical simulation and analysis of the fullthree-dimensional problem is a major challenge for classi-cal fracture mechanics tools and remains, therefore, largelyunexplored.Figure 3 is a three-dimensional version of the simulation

from Fig. 1 on a plate of thickness 1 mm, for increasingvalues of �T. The fracture geometry is representedby the level surface � ¼ 0:95. In order to reduce the

(a)

(b)

FIG. 2 (color online). Crack nucleation. (a) Damage field nearthe shock surface before (left) and after (right) bifurcation timet� for ‘0=‘ ¼ 0:107 showing the bifurcation of a horizontallyhomogeneous damaged band of depth D� toward a periodicsolution with wavelength ��. (b) Wavelength, time, and damagepenetration in numerical simulations for several intensities of thethermal shock ‘0=‘ (dots), compared to the semianalyticalresults from Ref. [23] (solid lines).

(a)

(b)

(c)

(d)

FIG. 3 (color online). Three-dimensional version of the ex-periment from Fig. 1(b) showing the transition from two- tothree-dimensional crack patterns. The simulations are performedon a subdomain of dimension 5� 1� 1 mm and temperaturecontrast (a) 380 �C (‘0 ¼ 0:27‘), (b) 480 �C (‘0 ¼ 0:17‘),(c) 580 �C (‘0 ¼ 0:12‘), and (d) 680 �C (‘0 ¼ 0:08‘).

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computational cost, the computation is performed on afragment of width 5 mm and height 1 mm of the domain,and the temperature is assumed constant throughout thethickness of the sample. We observe a transition fromtransverse cracks to three-dimensional fracture patternsdelimiting polygonal cells between 480 �C and 580 �C,which is consistent with Fig. 5 of Ref. [20]. Another seriesof simulations performed at constant temperature forincreasing sample thickness (not shown here) highlightsthe same behavior: transverse cracks for thin domains,transitioning to three-dimensional cracks for thicknessesbetween 1 and 2 mm. This is also consistent with theobservations in Fig. 5 of Ref. [20] and justifies the use ofa two-dimensional model, a posteriori.

Figure 4 shows a fully three-dimensional crack patternobtained for a domain of dimension 150‘� 150‘� 20‘for ‘0 ¼ 0:05‘. During the simulation, a disorderedpattern of small cells nucleates in the first time steps andpropagates quasistatically inside the domain. A selectionmechanism leading to honeycomb patterns with increas-ingly large and regular cells arises from energy minimiza-tion. Tracking the propagation of the three-dimensionalcrack front of Fig. 4 using a classical Griffith-based modelrequiring an explicit description of the crack surface and itspropagation criterion would be prohibitively complex.Instead, our three-dimensional computations are per-formed through a straightforward extension of the discre-tization and minimization algorithm for the energy (1).Obtaining a three-dimensional equivalent of the scalinglaw from Fig. 1 giving fracture cell diameter as a functionof the depth would require simulations on larger domains.Such simulations are computationally prohibitive, hence,we perform a series of numerical experiments by fixing theloading ‘0 and by varying at the same time the internal

length ‘ and the domain size L so as to keep their ratioequal to L=‘ ¼ 40. The number of elements is kept con-stant with a mesh size h ¼ ‘=5. For each computation, wecompute the average cell diameter d as a function ofdistance from the bottom edge a using postprocessingsoftware. This process does not involve any adjustableparameter, shown in Fig. 5, yet our results match thetwo-dimensional scaling law of Ref. [13] over severalorders of magnitude, leading us to conclude that the scaleselection mechanisms in two and three dimensions are verysimilar. In addition, while the initial phase of the evolutiondepends strongly on ‘, later time evolution of fully devel-oped cracks at the structural scale is unaffected by thisparameter, matching the general scaling law for a Griffith-based model. This finding is consistent with the propertiesof the energy functional Et which is known to lead to aGriffith-type propagation criterion [16,19,28,29].Our simulations show that a purely quasistatic model

based on energy minimization can fully explain the for-mation of imperfect polygonal patterns and their selectivecoarsening as a ‘‘maturation’’ mechanism during propaga-tion, a phenomenon sometimes attributed to nonequilib-rium processes [24]. We show that a carefully chosengradient damage model can be used to account simulta-neously for the nucleation of complex crack patterns andtheir propagation following Griffith criterion. Further workwill be carried out to perform a careful statistical analysisof the geometry of the 3D crack patterns and furthercomparisons to experimental results. The present modelingframework has a general validity and can be applied toother domains including, for example, the formation ofbasalt columns with uniform cross-sectional diameters

FIG. 4 (color online). Complex fracture pattern for ‘0 ¼Gc=ðE�2�T2Þ ¼ 0:05‘ in a domain of size 150‘� 150‘�20‘ color coded by distance from the bottom surface wherethe thermal shock is applied. The problem was discretized withapproximately 4:4� 107 linear finite elements in space (meshsize h ¼ ‘=5) and 100 time steps. The computation was per-formed on 1536 cores of the NSF-XSEDE cluster Stampede atTexas Advanced Computing Center in 10 h.

(a) (b)

(c)

FIG. 5 (color online). Average size d (square root of theaverage cross-sectional area) of the fracture-delimited cells asa function of the depth a (distance to the exposed face) in cubicdomains with edge length L ranging from 2 mm to 2 m com-pared with the two-dimensional scaling law from Ref. [13] (solidline). All the simulations are with ‘0 ¼ 14 �m and ‘ ¼ L=40.Inset: Top view of the crack patterns for (a) L ¼ 2 mm,(b) L ¼ 63:2 mm, and (c) L ¼ 2 m.

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through the solidification of lava fronts [24] or shaping ofbiological systems as observed in the scales on the heads ofcrocodiles [30]. We are also considering stronger thermo-mechanical coupling including the effect of cracks on heattransfer as in Refs. [14,31].

The authors wish to thank Yingfeng Shao for providingthe experimental data used in Fig. 1. The work of B. B. wassupported in part by the National Science FoundationGrant No. DMS-0909267. Some numerical experimentswere performed using resources of the Extreme Scienceand Engineering Discovery Environment (XSEDE), whichis supported by National Science Foundation GrantNo. OCI-1053575 under the Resource Allocation TG-DMS060014. J.-J.M. and C.M. gratefully acknowledgethe funding of the ANR program T-Shock OTP J11R087.

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