crack patterns in directional drying

8/13/2019 Crack Patterns in Directional Drying

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approach to fracture mechanics proposed byFrancfort & Marigo(1998)appears as a promis-

ing framework to address these issues. It extends the energetic theory of Griffith and treats

the crack geometry as a genuine unknown, making no a prioriassumptions on its geometry

or temporal evolution. Instead, it postulates that the deformation and crack configuration ofa body at a given time is obtained by minimising the sum of its elastic and fracture ener-

gies, among all admissible crack sets and (discontinuous) displacement fields. In the simplest

model, the fracture energy is proportional to its surface in two space dimensions and its area

in three dimensions (Griffith fracture energy).

Of course such a minimisation problem is challenging in particular because it is techni-

cally not possible to test all crack configurations. One possibility is to suppose that the crack

shape is known and to restrict the minimisation on the subset formed by these morphologies.

But how can one be sure that the optimum solution belongs to this subset? Alternatively, to

answer this question, numerical regularized approaches can be used. Roughly, they consist

in replacing the minimisation on all possible crack configurations corresponding to a discon-

tinuity surface by a minimisation on a continuous scalar field more suitable for numerical

purposes.

More precisely, their analysis borrows tools from the calculus of variations and free-discontinuity problems (Ambrosio et al. 2000). Their numerical implementation relies on

the concept of variational approximation by elliptic functional (Braides 1998), where approx-

imated functionals depending on a regularization parameters are constructed. From a technical

standpoint, the approximation takes place in the sense ofconvergence (Braides 2002) i.e.one can prove that as the regularization parameter goes to 0, the minimisers of the regularized

functionals approaches that of the total energy. The specific regularized functional we focus

on here resembles gradient damage laws or phase field approximations of sharp interfaces

models(Hakim & Karma 2009,Corson et al. 2009). It is very similar to the one proposed

byAmbrosio & Tortorelli(1992)where the crack set is represented by a secondary smooth

variable and the displacement field is also approximated by a smooth function. The main ad-

vantages of this approach is that it eliminates the issue of representing discontinuous fields

when their discontinuity set is not known a priori. It also reduces energy minimisation withrespect to any admissible crack geometry to minimisation with respect to a smooth field, a

much simpler problem. In addition, it can be discretized numerically using standard continu-

ous finite elements. The first numerical implementation of the variational fracture mechanics

is reported inBourdin et al. (2000). Further developments and applications may be found

inBourdin et al. (2008),Chambolle et al.(2009),Del Piero et al.(2007),Lancioni & Royer-

Carfagni(2009),Amor et al.(2009),Freddi & Royer-Carfagni(2010). To our knowledge, no

precise quantitative comparison between this approach and experiments has been done so far.

One aim of this paper is to fill this gap.

For this, we focus on the complex cracking phenomena encountered in directional drying

experiments of colloidal suspensions confined in capillary tubes. During drying, the suspen-

sion gradually transforms into a drained porous solid matrix. Further drying induces a natural

shrinking of the solid matrix. In capillary cells, the shrinking is prevented by the strong adhe-

sion to the wall of the cell, and gives rise to high tensile stresses in the matrix. These stressesare at the origin of complex crack patterns, whose shapes depend on the geometry of the cell

and the drying conditions. This kind of experiments has been proposed first in flat rectangular

capillary tubes (Allain & Limat 1995,Dufresne et al. 2003, 2006) and later on, in circular

ones (Gauthier et al. 2007, 2010). They are extended here to squared cell shapes. Crack prop-

agation in flat specimens has been extensively studied by the traditional approach for instance

byBazant et al.(1979),Nemat-Nasser et al.(1980),Hofmann et al. (2006),Bahr et al.(2010).

Gauthier et al. (2007, 2010) recently showed that the observed crack patterns can be correctly

predicted by energy minimisation amongst a given family of cracks, namely arrays of paral-

lel cracks or star shaped cracks. Here, our aim is to compare the fracture patterns observed

experimentally to those found by numerical energy minimisation according to the variational

approach to fracture introduced above.

We show that the regularized form of the variational approach is able to predict the shapeof the crack patterns as a function of the cell shape, without any a priorihypotheses.

The outline of the paper is as follows. In Section2, the experimental setup is described.

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Various crack shapes are obtained by changing the suspension, the cell geometry and the

drying velocities. In the aim to predict the crack shape, we model (Section 3) the experiments

by a 2D linear elasticity problem in the cross-section, the drying loading being given by a

tensile isotropic inelastic strain. Then we suppose that the fracture configuration for a givenloading, material and cell geometry minimises the sum of the elastic and fracture energies.

This problem is first solved by restricting the crack shapes to radial cracks and searching their

number that minimises the total energy (Section4). The question then is to show if star-shape

cracks are energetically optimal. To answer it, the minimisation is performed numerically

using the regularizated form of the variational approach mentioned above. In Section5, the

principle of this method is described and the simulation results are given and shown to be in

agreement with the direct minimisation for not too high loadings. For higher loadings more

complex crack shapes seems to be preferred to radial cracks. The close agreement with the

experiments is shown in Section 7. This demonstrates that (i) this simple 2D elastic model

captures the physics of the fracture in suspension drying phenomenon and (ii) the efficiency

of the variational approach to predict complex crack morphologies without any preliminar

assumption on the shape.

2 Experiments

Fractured gel

Air satured with water

Suspension

Evaporation surfaceCrackinganddryingdire

ction

Crosssection view

(a) Experimental setup. (b) Pictures of some

cross section cuts (the

colors depend on the

light used).

Figure 1: A vertical glass capillary is filled with a colloidal suspension; the single bottom open

edge allows for evaporation of the water in a surrounding maintained at a constant relative humidity

(RH) and temperature (T). The cross-sectional shape of the cracks depend on the cell shape andsize and on the drying conditions.

The experiments are similar to those ofGauthier et al. (2007, 2010). They are carried

out using aqueous suspensions of mono disperse silica spherical particles (Ludox SM30 of

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radiusr 3.5 nm or Ludox HS40 of radius r 6 nm) and volume fraction 0.3.To investigate unidirectional drying, vertical circular or square glass capillary tubes are used

here. The radius of the cross-section of the circular cells areR = 0.05mm, R = 0.15mm,

R= 0.5mm or R= 0.75mm; the edge length of the square cells is 2R= 1mm. The lengthsare given with a precision of 10%.The tube is filled with the colloidal suspension (Figure 1(a)) and placed in a controlled

environment maintained at a constant relative humidity (RH) and temperature T. Experimentsare performed either:

1. at room temperatureT 20C and relative humidity maintained below 10% usingdesiccant;

2. at room temperature and RH kept over 90% by introducing water in the chamber;

3. at fridge temperature ofT 3C and relative humidity kept below 10% using desiccant.The top is closed and the bottom is open allowing the water to evaporate. The tube is only par-

tially filled with the suspension, so that the air and water vapor located above the suspension

can expand to compensate the loss of water during desiccation. More technical details about

the experiments can be found inGauthier et al. (2007, 2010).

As the sample loses water, particles aggregating at the open edge form a growing drained

solid porous medium (Figure 1(a)). High negative capillary pressure in the draining fluid

generates high tensile stresses in the gel (Dufresne et al. 2003). This causes crack formation

along the drying direction. Their sectional shape can be visualized either by transparency with

a camera or by cutting a tube to see the cross-section. The later manipulation is delicate and

doesnt work for very small tubes. Some pictures are given in Figure1(b). We observe that

the crack shape for a given suspension and tube geometry depends on the drying conditions

through the drying velocity only. Indeed drying at T 3 C, RH 10 % or at T 20C, RH90 % gives the same crack tip velocities and the same crack patterns. Thus in thesequel, experiments performed at T 3 C, RH10 % or atT 20C and RH90 %will be gathered together under slow velocity (SV) experiments and those performed at T

20C and RH 10 % under higher velocity (HV) experiments.A summary of the shapes obtained can be found in Figure 1(b). For square cells, two

perpendicular cracks appear along the diagonals of the square. For circular cells, the cross-

section forms mostly a star-shape crack breaking up the circular cross section in n circularsectors (with a central angle2/n). The number of sectorsn is observed to increase with thecell sizeR and the drying velocities. A summary of the panel of experiments made can befound in table3(section7).

3 Model

3.1 Basic hypotheses

Our aim is to predict for a given tube geometry, suspension and drying condition, the shape ofthe cracks appearing in the tube. For this matter, we make the following simplifying hypothe-

ses:

H1. The solid adheres to the cell walls, and cracks cannot grow along the walls. This is

required to create the tensile stresses that lead to crack formation.

H2. The solid medium behaves as a linear isotropic homogeneous material. This is reason-

able since the porous matrix is formed by relatively hard silica particles.

H3. Changes in material properties induced by the drying phenomenon are negligible.

H4. The drying phenomenon induces isotropic inelastic strains 0 =0 1, where 1 denotesthe3 3identity matrix and 0 < 0. The inelastic strains are taken as independent ofthe deformation in the material, according to a simplified view of the full poroelastic

problem (Wang 2000).

Concerning the fractures, we make the following hypotheses:

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H5. The material is perfectly brittle.

H6. Crack surfaces are stress-free. This is reasonable since it has been shown (Dufresne

et al. 2003)that the crack surfaces are dry.

H7. Growing cracks have no impact on the value of the inelastic strain 0. This allows usto first determine 0 independently of the crack configuration, then to solve the crack

propagation problem.

We focus on situations where the cracks grow only along the z-axis and their cross-sectional geometry remains unchanged (Figure 1(a)). Instead of solving the full three-dimensional

evolution problem, for any given loading and geometry we look for the cross-sectional crack

morphology as a two-dimensional static problem. Since the solid is perfectly bounded to

the wall, the reduced two-dimensional model is derived under the plane strain assumption.

Following the variational approach to brittle fracture (Francfort & Marigo 1998):

H8. For a given loading, the deformation state and fracture configuration correspond to a

minimum of a total energy defined by the sum of the bulk elastic energy and a fracture

energy.

H9. The fracture energy is of Griffiths type. The energy S() per height unit, associated toa crack setis proportional to its length given by:

S() :=GcL(), (1)whereGcis the fracture energy of the material, and L denotes the length of the crack.

We invite the reader more interested in the physics of the drying and fracturation of col-

loidal suspensions than in the technicalities of the model and its numerical implementation to

switch directly to Section6 for a presentation of the numerical simulations and to Section 7

for a comparison with experiments.

3.2 Variational fracture modelLet us introduce the following equivalent 2d inelastic strain defined by 2d0 = (1 +)0 12,where 12 is the2 2identity matrix. With this notation and the aforementioned hypotheses,the strain energy can be written under the following the form:

w(, 0) := E

2(1 + )

(1 2) tr2( 2d0 ) + ( 2d0 ) ( 2d0 )

(2)

whereEand are the Young modulus and the Poisson ratio of the material, is the symmet-ric second-order2 2matrix representing the linearized plane strain and tr denote the traceoperator, and the dot is used for the scalar product. In linear elasticity, kinematical compati-

bility implies that (u) = 12

(u+ uT), whereu is the displacement field,the gradientoperator, and the superscriptT denotes the transpose operator. We parametrise the inelastic

strain 2d0 representing the drying loading by a non dimensional drying intensity, defined by(see the first remark at the end of this section)

2d0 :=

GcER

12 (3)

where R is a characteristic length associated with the cross-section, typically its radius. Hence,the potential energyP of the cross-section occupying the open setC and associated to adisplacement fieldu and a crack setfor a loading parameteris given by

P(u, ) :=C/

w((u))dS, with w() :=w

,

Gc/ER 12

. (4)

The total energy is defined as the sum of the potential energy and the surface energy requiredto create the cracks:

E(u, ) :=P(u, ) + S(). (5)

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For any given loading parameter, we seek to find the crack set and displacement field uas the global minimiser of(5) amongstanyadmissible crack set and kinematically admissible

displacement fields. The admissible crack sets consist of all possible curves or sets of curves

inside the boundary ofC. For any given crack set , the space of the admissible displacementsisU() :={uH1(C \ ;R2), u= 0onC}, (6)

i.e. it consists of all vector valued fields satisfying the adhesion boundary condition and suf-

ficiently smooth (square integrable with square integrable first derivatives) on the uncracked

domain.

More precisely, the global minimality condition can be expressed as:

Find C, u U() : E(u, ) E(u, ), C, u U(). (7)

3.3 Remarks

The model presented above deserves several remarks:

The scaling factor

Gc/ER in(3) renders all the results, presented in terms ofin therest of the paper, independent of the material constants and cross-sectional dimension.

Other choices for the relevant non dimensional parameter are possible. In particular, as

inGauthier et al. (2010), one could also chose to parameterize the loading in terms of

the Griffith length Lc := EGc/20, where0 is a prestress. After some calculations,one can relateandLc by

Lc= R

2(1 2)2(1 + )2. (8)

This relation will be useful in Section7as it will allow us to estimate the value offorvarious experiments.

We consider the problem of finding the optimal displacement field and crack pattern in a

cross section of the tube of Figure1for a given value of the loading parameter, indepen-

dently of the previous history or irreversibility conditions on preexistent crack patterns.

We refer to this problem as a static formulation of the fracture mechanics problem, in

opposition to thequasi-static setting, where one need to account for the previous his-

tory and the irreversible nature of crack propagation through unilateral minimisationas

inFrancfort & Marigo (1998).

Perfect bonding to the walls of the cell is accounted for in the minimisation principle (7)

by imposing null-displacement boundary conditions on u, and that the admissible cracksbe included in the open set C. This in particular proscribes cracks along C.

The admissible displacement fields are potentially discontinuous across cracks, butthe location of the potential discontinuities themselves is not known a priori. This ren-

ders the numerical minimisation of (5) challenging as most numerical methods suchas cohesive, discontinuous or extended finite element methods require at least some a

priori knowledge of the crack path or of its topology. Indeed, this problem falls into

the broader class offree discontinuity problemsfor which a wealth of mathematical and

numerical literature now exists. In the following, we solely focus on the numerical im-

plementation using an extension of that proposed in Bourdin et al.(2000) andBourdin

(2007), and inspired from Ambrosio and Tortorellis results on the approximation of

the Mumford-Shah functional by means of elliptic functionalsAmbrosio & Tortorelli

(1990, 1992). We refer the reader interested in the analysis of the model to Francfort &

Larsen(2003),Dal Maso et al.(2005),Bourdin et al. (2008), and references within.

4 Simple illustration: star-shaped cracks in circular cells

Although the main strength of the variational approach to fracture is that it does not require any

a priorihypotheses on crack geometry, the following basic computation provides a valuable

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we do not see our regularized formulation as a fracture model, but merely as a numerical

approximation of the total energy of the static or quasi-static variational approach ofFranc-

fort & Marigo(1998). This approximation is deeply rooted in the mathematical literature on

free-discontinuity problems (Braides 1998). In particular the minimisation principle for theregularized energy is derived from that of the variational model in the static case.

5.2 Numerical implementation

The numerical minimisation of (12) is implemented in a way similar to that described in Bour-

din (2007). We discretize the regularized energy by means of linear Lagrange finite ele-

ments over an unstructured mesh. As long as the mesh sizeh is such that h = o(), theconvergence property of (12) to (5) is also true for the discretization of the regularized en-ergy (seeBellettini & Coscia(1994),Bourdin(1999),Burke et al.(2010) for instance). This

compatibility condition leads to fine meshes, which are better dealt with using parallel super-

computers. We use PETSc(Balay et al. 1997, 2010, 2011) for data distribution, parallel linear

algebra, and TAO (Benson et al. 2010) for the constrained optimization. In order to avoid

prefered directions in the mesh, we use the Delaunay-Voronoi mesh algorithm implementedin Cubit, from Sandia National Laboratories.

Due to the size of the problems, global minimisation algorithms are not practical. Instead,

we notice that although (12) is non convex, it is convex with respect to each variable indi-

vidually. We alternate minimisations with respect tou and , an algorithm akin to a blockNewton method or a segregated solver. Note that minimisation with respect to uis equivalentto solving a simple linear elasticity problem, but that minimisation with respect to[0, 1]requires an actual box-constrained minimisation algorithm. Of course, as the total energy is

non convex, one cannot expect convergence to a global minimiser. However, one can prove

that the alternate minimisation process is unconditionally stable and globally decreasing and

that it leads to a stationary point of ( 12) which may be a local (or global) minimiser or a

saddle point of the energy. From a practical standpoint we observe that the algorithm is quite

robust with respect to the mesh discretization, provided that the regularization length is largeenough compared to the mesh size. However it can be quite sensitive with respect to the initialvalue ofu and. Different choices of the initial guess or of the regularization parametercan lead to convergence to different solutions

In the following section, we present numerical experiments performed using the method

as described above, highlight its shortcomings and illustrate how to decrease the sensitivity

with respect to the choice of initial configuration and regularization parameter.

6 Numerical simulations

6.1 Selection of crack shapes

We first illustrate our numerical approach on circular cells. Experimental evidence suggeststhat for small values of , the actual fracture pattern resembles the star-shaped cracks fromSection4. We can use this feature to perform partial verificationof our numerical approach.

As dimensional analysis highlights the dependency of the fracture energy (5) on a single

loading parameter,, it is natural to replace the regularization length with a non-dimensionalparameter= /R.

Figure 3 presents the field obtained by numerical minimisation of (12) for variouschoices of the parameters and with a cell of radius R = 1. The material parameters (E,Gc) were set to 1 without loss of generality, and the Poisson ratio to 0.3. In each computation,the mesh size was h = 0.025and the residual stiffness was set to k = 106. The alternateminimisation algorithm was initialized with = 0, u = 0. The value 1 (corresponding tocracks) ofis encoded in red and the value 0 (the un-cracked material) in blue. A first glanceat the table highlights the wide variety of crack geometries obtained, and that the complexity

of the fracture pattern increases with the loading parameter. Again we stress that no hypothe-sis on this geometry is made in the model and that the shape of the crack patterns is purely an

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0.2 0.1 0.05

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.5

3.0

4.5

5.0

(a) Direct numerical simulations. Each prob-

lem was solved independently initializing the

alternate minimizations algorithm with the un-

cracked solution = 0.

0.2 0.1 0.05

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.5

3.0

4.5

5.0

(b) Numerical results obtained using

continuation. Each row corresponds to a set of

computations, each taking the one at its left as a

first guess foruand .

Figure 3: Numerical results by minimisation of functional (12) for circular cross-sections. The

results are obtained using uniform Delaunay-Voronoi unstructured meshes with size h= 0.025ondisk of radiusR = 1. The material properties areE= 1,Gc= 1,= 0.3.

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0.0 0.5 1.0 1.5 2.0 2.5 3.00

2

4

6

8

10

2.652.011.510.960.71

RG

c

(a) Energies associated with the crack geometries identified in Figure3(b)as a func-

tion of the loading parameter. Thick black lines distinguishes energy curves corre-

sponding to configurations attaining the minimal energy for some value of the loading

parameters. The gridlines marks the critical loading for the passage of one optimal

curve to the next. Note that for large the identification of critical loadings becomes

difficult.

0.0 0.5 1.0 1.5 2.0 2.5 3.00

2

4

6

8

102.652.011.510.960.71

RG

c

(b) Comparison of the optimal energy obtained using minimisation over star shaped

cracks (dashed line) and numerical simulation (continuous line). The continuous line

is the lower envelope of the curves in figure4(a). The pictures represent the optimal

crack shapes in each range of the loading parameter delimited by the gridlines.

Figure 4: Minimisation over star-shaped cracks vs. minimisation of functional (12). As the loading

increases, our numerical method identifies crack configurations with much lower energetically than

star-shaped cracks.

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Table 1: Energies of the numerical solutions in Figure3for = 0.05without (left) or with (right)continuation.

Elastic Surface Total0.6 2.2 0 2.20.8 3.9 0 3.91.0 6.0 0 6.01.2 8.7 0 8.71.4 1.6 3.9 5.51.6 2.0 4.0 6.01.8 1.9 4.9 6.82.0 2.0 5.3 7.32.2 2.1 5.7 7.82.5 2.1 6.4 8.5

3.0 2.1 7.3 9.44.5 2.3 9.6 11.85.0 2.4 9.8 12.2

Elastic Surface Total0.6 2.2 0 2.20.8 1.5 1.8 3.41.0 1.3 2.8 4.11.2 1.7 2.9 4.61.4 2.0 3.3 5.31.6 2.0 3.9 5.91.8 2.0 4.4 6.42.0 2.0 4.9 6.92.2 2.0 5.4 7.42.5 2.1 5.9 8.0

3.0 2.1 6.9 9.14.5 2.3 9.2 11.55.0 2.4 9.9 12.3

6.2 Comparison with star-shaped cracks

Figure 4(b) compares the total energy of these configurations with the energy of the star-

shaped cracks taken from Figure2 (dashed line). For small values of the loading parameter,

we obtain similar geometries and critical loading. The surface energy obtained is close to the

number of branches, and the critical loading upon which we obtain a single straight crack is

0.71 (vs. a theoretical value of2= 0.73). Bifurcation between straight and Y-shaped cracks

take place at0.94(vs. a theoretical value of2= 0.97).More interestingly, for larger values of, our numerical simulations have identified mul-

tiple configurations that are energetically close to each other but always less expensive than

star-shaped cracks. In particular, we show that perfect 5-branch stars are never optimal and

that configurations consisting of either two triple junctions very close to each others (see for

instance = 1.6 in Figure3(b)), a 4-branch star whose branches split in two near the cellboundary (see for instance = 1.8, 2.0, 2.2 in Figure3(b)), or a more complicated patternslike the stick figure looking 5 cracks configuration that we obtain for = 2.5 have lesserenergy. Of course, that the local geometry near the crack crossing resembles 2 triple junctions

near each others rather that an X does not really come up as a surprise. As mentioned earlier,

the fracture energy (5) resembles the Mumford-Shah energy for edge segmentation (Mumford

& Shah 1989). Therefore, it seems natural to expect that if they posses some form of regu-

larity, optimal crack geometries satisfy the Mumford-Shah conjecture which rules out crackcrossing, kinks and only allows cracks to meet at 120 triple junctions, locally.

These quantitative and qualitative properties of the numerical solution provide a powerful

illustration of the strength of the variational approach and our numerical methods by high-

lighting that global minimisation over unrestricted crack shapes can lead to configurations

that are energetically favored over simple ones. In this context, the choice of the family of

cracks one has to minimize the total energy over in classical approaches is neither trivial nor

innocent.

6.3 Square cells

Finally, we performed another set of numerical simulations on unit square cells. The results

obtained by the same -continuation method, as in Figure3(b)for circular cells, are depicted

on Figure5. The materials parameters and mesh size are unchanged (E=1, Gc=1, = 1,h= 0.025). Whereas for circular cells, starshaped cracks are natural candidates, there were

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0.2 0.1 0.05

0.6

0.8

1.0

1.2

1.4

1.6

0.2 0.1 0.05

1.8

2.0

2.2

2.5

3.0

4.5

Figure 5: Numerical results for square cells by minimisation of functional (12) using the -continuation method.

no obvious family of cracks in this case. This geometry also leads to a rich variety of crackpatterns and highlights the strength of the proposed method in identifying complex crack pat-

terns without a priori hypothesis. Some of the quantitative properties of the optimal cracks

highlighted in the case of circular cells are still observed. Again, cracks seem to split near

the edges of the cells. Triple junctions seem to be favored over crack crossing, although in

the case of two diagonal or longitudinal cracks, the resolution of our numerical experiments

does not allow us to clearly identify the configuration. As for the circular cell, one can further

post process the numerical result in order to identify the range of loadings for which each

of the identified configuration is optimal. This is presented in Figure6. Again, for small

values of the loading parameters, simple and somewhat predictable crack geometries are ob-

tained. For larger values of, more complex and less intuitive patterns are energetically moreadvantageous.

7 Comparison between experimental and numerical results

7.1 Identification of the loading parameter

Dimensional analysis shows that the model relies on a single parameter, the value of whichneeds to be estimated in order to perform quantitative comparison between experiments and

numerical simulation. As depends on experimental conditions, colloidal suspension type,and tube geometry, one solution is to try to measure separately 0,E, andGcappearing in thedefinition (3) of. One may obtain the material constants E,Gc by indentation (Malzbenderet al. 2002)and the mismatch strain0 by beam deflection technics (Tirumkudulu & Russel2004,Chekchaki et al. 2011) from a thin film drying experiments, for instance. However,

such direct measurements are difficult, and transposing the values obtained from one typeof experiments (thin film drying) to another (directional drying) is questionable. Indeed, the

parameters may depend on the type of experiments and even evolve in time. For example,

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0.0 0.5 1.0 1.5 2.0 2.5 3.0

0

2

4

6

8

10

2.181.61.381.00.68

RG

c

Figure 6: Range of parameters in which each of the configuration identified in Figure5is optimal

the material constants E andGc of the porous medium may depend on the microstructure,influenced by formation dynamics.

Instead of performing such difficult measurements, whose relevance to our problem may

be questioned, we used the method presented in Gauthier et al. (2010), which we briefly

summarize. The basis of the method is to consider a directional drying experiment in thin

rectangular cells(Allain & Limat 1995). In this geometry, an array of parallel tunneling cracks

is obtained and the cracks spacing can be correlated to the Griffith lengthLc := EGc/20

(0being the prestress induced by the films drying). Using an energy minimisation principlesimilar to the one in Section4,one can show that the spacing is proportional to Lct, tbeing the cells thickness and in particular, for = 0.3, one obtains 3.1Lct. For a givenmaterial and drying parameter, the value ofLc can therefore be deduced from measurementsof. Table2presents the value ofLc for Ludox

SM30 (r 3.5 nm) and LudoxHS40(r6 nm) under high velocity and slow velocity conditions.

Lc LudoxSM30 (r3.5nm) LudoxHS40 (r6 nm)

HV 34 10 40 10SV 60 18 45 15

Table 2: Values of Griffiths length Lc (inm) for several Ludoxsuspensions and drying rates

(SV=[T 3C and RH 10% or atT 20C and RH 90%] and HV=[T 20C and RH 10%])

We assume that the Griffith length Lc is a well-defined material parameter for a givensuspension and drying condition, and that it is independent of the cross-sectional geometry of

the directional drying experiment. Hence, from the values ofLc in Table2, we estimate thevalue ofin the directional drying of circular and square cells of different diameters using therelation(8), which gives= 0.52

R/Lc for = 0.3.

7.2 Results and analysis

Table3 reports on the series of experiments on circular cells described in Section 2. From

a qualitative standpoint we observe that star shaped appear above a critical load, and thatthe number of branches increases with the loading, which is consistent with the analysis in

Section4and the numerical simulations of Section 6.

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Ludox HV/SV Lc(m) R (m) n

a SM30 HV 34 10 50 5 1 0.6 0.12b SM30 SV 60 18 150 15 2 0.8 0.16c SM30 SV 60 18 500 50 3 1.5 0.30d HS40 SV 45 15 500 50 4 1.7 0.37e HS40 HV 40 10 500 50 4 1.8 0.27f SM30 HV 34 10 500 50 4 2.0 0.39g HS40 HV 40 10 750 75 5 2.3 0.41h SM30 HV 34 10 750 75 5 2.4 0.48

Table 3: Experimental results on circular cells. The value ofn corresponds to the number ofsectors delimited by the cracks as in Section4.

In order to perform a quantitative comparison, we summarize all the results obtained inthe case of circular cells in Figure 7. The first row corresponds to the outcome of the semi-

analytical minimization over star shaped cracks. The critical values of the loading parameters

computed in Section 4 upon which bifurcation between different morphologies take place

is printed in red letters, and represented by red dashed vertical lines. The second row cor-

responds to the numerical experiments without a priori hypotheses on the crack path. The

critical loads extracted from Figure 4(a)are printed in black letters, and represented by ver-

tical solid black lines. As highlighted in Figure4(b), the critical loads obtained in the case

of the bifurcation from1 to2, then3 and 4 are very close. This part of the table can beseen as a verification of the numerical implementation, i.e. as evidences that the computed

solutions are indeed solution of the variational fracture model. The third row summarizes the

outcome of the experiments from Table3. For each experiment, the value of the loading pa-

rameter is shown together with the accuracy of the measurement. When available, photos of

the cross sections are also displayed. We observe that for every single choice of, the crackgeometry predicted by our approach matches the one observed in the experiment. This acts

as a validationof the variational fracture model as a predictive tool in the setting of drying of

colloidal suspension.

an1

bn2 cn3

dn4

en4

fn4

gn5

hn5

0.71 0.96 1.51 2.01

0.73 0.97 1.55 2.15

Figure 7: Comparison between semi-analytical, numerical and experimental results for circular

cells.

We also did a single experiment on a square cell, for an estimated value 1.8 of the

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loading parameter for which we obtained two diagonal cracks (see the bottom image in Fig-

ure1b). Again, the numerical simulation in this case matches the experiment (see Figures5

and6).

Despite the modeling simplifying assumption, the complexity of the numerical technique,

and the uncertainty of the measurement of the parameters, the agreement between analysis,

simulation, and experiments is excellent. Our model correctly captures the essential physics

of the crack formation giving credit to the idea that crack growth can be predicted by minimi-

sation of the sum of elastic and surface energy over all possible crack path. In order to further

justify this idea, one will need to compare experiments and simulations for higher loading

parameters in which case numerical simulations identify complex crack patterns with signif-icantly lower energy than classical star-shaped solutions. For instance, better quality imaging

will be required to unambiguously determine if the the 5 cracks configurations we observe ex-

perimentally (see Figures1(b)or7)resembles a stick figure as in our numerical simulation

(cf. Figure3for= 2.5), a regular 5-branch star, or something completely different.

8 Conclusions and future work

In this paper, we have presented some directional drying experiments of colloidal suspensions

realized in capillary cells where solvent evaporation leads to the formation of a growing porous

solid media. Due to shrinkage prevented by adhesion on the wall of the cells, high tensile

stresses appear that give rise to cracks of various morphologies depending on the cell geometry

and the drying velocities.

We proposed a simple model based on the assumption that when the crack cross sec-

tional geometry does not evolve, this problem can be reduced to a 2d static one. We showed

that changes in crack geometry due to different materials and experimental condition can be

accounted for by a single dimensionless parameter which can represent the intensity of the

tensile strain or stress induced by drying or the energetic cost of growing cracks. We used

the variational approach to fracture to account for crack propagation in the porous medium as

a function of this parameter without any underlying assumptions on crack geometry. Under

additional assumptions on the crack geometry, we computed the range of loading parameters

for which optimality may be achieved by star-shaped cracks. We then presented a numeri-

cal method and some simulations allowing us to predict crack patterns without anya priori

knowledge. Finally, we achieve qualitative and quantitative agreement between numerical

simulation, semi-analytical solutions and experiments.

At this point, though, we were not able to perform qualitative or quantitative comparisons

for higher values of the loading parameters, where the virtue of the variational approach to

fracture over more conventional ones requiring at least some a prioriknowledge of the crack

path becomes more striking. Such experiments will require additional work to deal with larger

cells for instance. In these situations, the main difficulty is thepost-mortem analysis of the

crack geometry. Microphotography though the sides of the cells becomes hard to interpret,and cutting the tubes without perturbing the cracks geometry is difficult. Perhaps the solution

lies in full 3d imaging of the cells and post-processing in order to highlight the location of

the cracks. From the modeling perspective, for larger cell size, a full three-dimensional linear

poroelasticity model initiated byBiot(1941)will become necessary. Finally, from a physico-

chemical point of view, the link between the drying velocity and the macroscopic signaturewill have to be explored.

Acknowledgements

The work of V. Lazarus and G. Gauthier was partially supported by the ANR Program JC-

JC ANR-05-JCJC-0029 Morphologies. C. Maurini gratefully acknowledges the funding of

the French National Research Council (CNRS) for a PICS bilateral exchange program withB. Bourdin and a grant of the University Pierre et Marie Curie EMERGENCES-UPMC.

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B. Bourdins work was supported in part by the National Science Foundation under the grant

DMS-0909267. Some of the numerical experiments were performed using the National Sci-

ence Foundation TeraGrid resources(Grandinetti 2007)provided by TACC at the University

of Texas under the Teragrid Resource Allocation TG-DMS060014N and the resources of theInstitut du Developpement et des Ressources en Informatique Scientifique (IDRIS) under the

DARI 2011 allocation 100064.

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