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RESEARCH PAPER
Numerical modelling of a field soil desiccation test using a cohesivefracture model with Voronoi tessellations
Y. L. Gui1,2 • W. Hu2 • Z. Y. Zhao3 • X. Zhu2
Received: 19 September 2016 / Accepted: 2 May 2017 / Published online: 30 May 2017
� Springer-Verlag Berlin Heidelberg 2017
Abstract Numerical modelling of a field soil desiccation
test is performed using a hybrid continuum-discrete ele-
ment method with a mix-mode cohesive fracture model and
Voronoi tessellation grain assemblages. The fracture model
considers material strength and contact stiffness degrada-
tion in both normal and tangential directions of an inter-
face. It is found that the model can reasonably reproduce
the special features of the field soil desiccation, such as
curling and sub-horizontal crack. In addition, three signif-
icant factors controlling field desiccation cracking, fracture
energy, grain heterogeneity and grain size are identified.
Keywords Cohesive fracture model � Elastic–plastic-fracture model � Fracture energy � Numerical modelling �Soil desiccation test
1 Introduction
Desiccation cracking is a commonly occurring phe-
nomenon in unsaturated soil with a high degree of satura-
tion. The occurrence of soil desiccation cracking has a
significant effect on the engineering properties of soils as it
can considerably increase the hydraulic conductivity and
decreases the shear strength of a soil [17]. Thus, it can pose
a significant threat to the hydraulic and structural integrity
of earthworks [32], dykes [34] and embankments
[14, 41, 46]. Indeed, the research on the internal and
external factors relating to the phenomenon has been a
subject of great interest in environmental geotechnical
practices dealing with waste disposal, earth embankment
protection and climate change impact on earthworks and so
on. This research has also been of great interest in other
fields studying general porous media, such as material
science and engineering, chemical engineering and agri-
cultural engineering, to name a few.
Several numerical techniques have been developed over
the last few years to simulate cracking/fracturing phe-
nomenon in both soft and brittle materials. Among the
notable contributions have included the numerical methods
of finite difference method (FDM) [48], finite element
method (FEM) [43] and mesh-free method (MM) [29].
These methods either model the crack as strong disconti-
nuity or ‘smear’ the crack over a certain width [20]. The
first category can be categorised into ‘enriched’ and ‘non-
enriched’ methods. Enriched methods are based on parti-
tion of unity [8, 27] and introduce additional degrees of
freedom in the variational formulation and commonly
allow for crack propagation without re-meshing. Such
approaches include the embedded finite element method
[10], generalised or extended finite element method
(XFEM) [9, 28], extended mesh-free method [38, 39],
phantom node method (PNM) [47, 50], numerical manifold
method (NMM) [12, 45], cracking particles method (CPM)
[36, 37, 40], or extended IGA [11, 16, 30], peridynamics
(PD) and dual-horizon peridynamics (DH-PD) [42],
screened-Poisson models [6, 7] and efficient re-meshing
techniques [2–5]. Element deletion techniques, cohesive
& Y. L. Gui
1 School of Civil Engineering and Geosciences, Newcastle
University, Newcastle upon Tyne NE1 7RU, United
Kingdom
2 State Key Laboratory of Geohazard Prevention and
Geoenvironmental Protection, Chengdu University of
Technology, Chengdu 610059, China
3 School of Civil and Environmental Engineering, Nanyang
Technological University, Singapore 639798, Singapore
123
Acta Geotechnica (2018) 13:87–102
https://doi.org/10.1007/s11440-017-0558-9
elements, certain meshless methods based on the visibility/
diffraction/transparency method and FEM based re-mesh-
ing are belonging to the category of non-enriched discrete
crack methods. A comprehensive description of these
methods is given, for example, in [35].
The conventional approaches to investigating soil des-
iccation cracking are primarily on laboratory testing (e.g.
[13, 18, 32, 49, 51]) and rarely on field testing (e.g. [25]).
The experimental testing can provide intuitive and phe-
nomenological observations, but the measurement of
parameters is problematic and the intrinsic mechanism
cannot be explored easily in an experimental test. To tackle
the shortcoming of the experiments, some numerical
investigations are conducted in the literature to simulate the
soil desiccation cracking with the help of various numerical
methods, such as FEM (e.g. [43, 52]), FDM (e.g. [24]) and
discrete element method (DEM) (e.g. [33]). The current
continuum computational mechanic-based methods, such
as FEM and FDM, generally predefine a few cracks [31] or
just use model boundary as the potential crack [44], which
inevitably undermines the capability of prediction on des-
iccation crack initiation and propagation. As an alternative
approach, DEM is considered to be relatively promising
[33] due to its capability to capture the highly discrete
nature of soil grains [17]. Although there is much literature
on numerical modelling of soil desiccation, there have been
very few numerical simulations carried out for field soil
desiccation. This is possibly due to the complex physics
involved in a field soil desiccation test. In the laboratory
tests, the thickness of the soil samples is thin, ranging from
a few millimetres to a few centimetres, and the shrinkage
can be deemed to be homogeneous throughout the depth of
the sample. Therefore, a simple shrinkage mechanism, for
example homogeneous shrinkage, of the sample can be
used (e.g. [43]). However, in the field soil desiccation test,
due to the nature of the half-infinite domain, the soil
material properties and the shrinkage are no more homo-
geneous. Therefore, the whole depth of the soil has to be
dealt with layer by layer. The other distinct feature of field
soil desiccation cracking is that there are sub-horizontal
cracks occurring during desiccation. Therefore, the pure
tensile failure model may not be physically sufficient to
characterise the failure of the soil. Rather, a failure model
considering the tension and shear combined effect and
strength softening should be adopted for a field soil des-
iccation cracking problem.
Based on the above consideration, the main objective in
this paper is to apply a mix-mode cohesive fracture model
to simulate a field desiccation test in the literature using a
hybrid continuum-discrete element method (i.e. UDEC). In
the mix-mode cohesive fracture model, the elastic, plastic
and damage mechanical behaviour are all considered for
both normal and shear directions of each fracture/potential
fracture. The fracturing behaviour is governed by a
parameter identified as effective norm of inelastic fracture
displacement (i.e. plastic and fracture displacement) and a
failure model. Implementation of the cohesive fracture
model in the hybrid continuum-discrete element method
makes it capable of handling multiple fracture as well as
deformation problems in materials, which is a considerable
advantage over other methods such as FEM and DEM. This
is because the conventional FEM has the problem of han-
dling multiple fractures, and the grain deformation is not
possible to be considered in conventional DEM [19]. In the
current simulations, the experimental suction profile is used
as the shrinkage driver from which the effective stress is
calculated. Thus, the shrinkage throughout the whole depth
of the model is not uniform or linear, so that it is more
physically rationale in terms of field soil desiccation pro-
cess. Additionally, the potential factors affecting the sim-
ulation of field soil desiccation cracking are identified.
2 The field test and numerical model
2.1 Field soil desiccation test
Konrad and Ayad [25] conducted a field test by excavation
to three different levels from top to bottom to investigate
desiccation cracking in top soil, weathered and intact clay
under restrained conditions, respectively, at the experi-
mental site of Saint-Alban, Quebec, Canada. In their
investigation, the moisture content and suction profiles
were recorded. It was found that the gravimetric water
content decreased significantly in the soil close to the
surface, while the deeper soil had a much smaller decrease
in water content. This effect was most pronounced in the
upper 40 cm. The suction profile was similar to the water
content. The suction profile monitored in the whole des-
iccation test is shown in Fig. 1. It was also found that
0
10
20
30
40
50
60
700 10 20 30 40 50
Dep
th (c
m)
Suction (kPa)
0 (h) 18 (h)24 (h) 42 (h)48 (h) 65 (h)73 (h) 97 (h)145 (h) 193 (h)241 (h)
Fig. 1 The measured suction profile during the field desiccation test
[27]
88 Acta Geotechnica (2018) 13:87–102
123
desiccation cracking occurred in less than 17 h after the
start of the test for both intact and weathered clays with
average crack spacing of 20–24 cm for the intact clay. In
addition, horizontal crack occurred and propagated hori-
zontally at certain depth in the intact clay, thereby soil
wedges forming and being easily removed.
2.2 Numerical model
A 2D model with length of 1.2 m and depth of 0.7 m is
built, as shown in Fig. 3. The model is discretised using
Voronoi tessellation which can reduce mesh bias yet
facilitate the crack initiation and propagation. The average
size of the Voronoi grains is 0.01 m, with total of 8528
grains in the model. The selection of the grain size is
determined by the computational capacity and the observed
crack spacing in the field desiccation test. In addition, the
Voronoi grain itself is meshed though finite difference
triangles (Fig. 2a) which can help the grain to be deform-
able, hence to model the deformation of soil skeleton.
Thus, the Voronoi grains in the simulations can be deemed
to represent the soil aggregates instead of soil single par-
ticles as illustrated in Fig. 2e. In real soil desiccation
shrinkage, the deformation is primarily from the defor-
mation of soil skeleton in the form of soil particle packing
and rearrangement, rather than soil particle itself whose
compressibility is negligible in calculations compared to
the skeleton [23]. The interface of the grains is treated
using the mix-mode cohesive fracture model described in
Appendix. The validity and applicability of the fracture
model were verified by simulating a number of preliminary
cases including direct shear, direct tension and mixed
grainVoronoi
Bond
grainVoronoi
contactNormal
contactTangential
Gridpoint
Contact
enu
pnufnu
esu
psu f
su
Spring
Slider
Divider
SpringSlider
Divider
ieffuσw
tσ
0tσ
ieffucw
c
0c
(a) (b) (c)
Zone
IfG
IIafG
0=D
1=D
10 << Dτ
nσ0tσ
0c
(d)
particleSoil
(e)
aggregateSoil
Fig. 2 Illustration of the hybrid continuum-discrete element method and the mix-mode cohesive fracture model [15]: a hybrid continuum-
discrete element method, b mechanical element representation of the mix-mode cohesive fracture model, c constitutive model of the contact,
d evolution of failure surface for the mix-mode cohesive fracture model (solid line representing the initial failure surface of the contact, the long
dash line standing for the final yield surface with the contact fully propagated, and the square dot line is the failure surface for the partially
propagated contact. Tensile stress is denoted as positive and compressive stress is negative), and e physical implication of the Voronoi grain in
the hybrid continuum-discrete element method
Acta Geotechnica (2018) 13:87–102 89
123
shear-tension tests, and they are presented in Ref. [19]. The
left, bottom and right boundaries are fixed using rollers.
The model is divided into six layers segregated by the
levels where the suction history was measured by ten-
siometers in the field desiccation test, i.e. 2, 7, 16, 33 and
57 cm below the ground surface. In each layer, the suction
is interpolated linearly. Since the soil becomes harder and
harder during desiccation, the elastic modulus of the soil is
increased during the desiccation process and it is calculated
based on the initial suction value (289 Pa) measured in the
field and the suction at each time point (i.e. 18, 24, 42, 48,
65, 73, 97, 145, 193 and 241 h) through the following
equation [1]
E ¼ E0 þ 24ð1� 2tÞðs� s0Þ ð1Þ
where E,E0 and t are the updated elastic modulus, initial
elastic modulus and the Poisson’s ratio, respectively. s and
s0 are the updated and initial suction, respectively. The
increase in the effective stress due to desiccation follows
the effective stress principle as
Dr0 ¼ Drnet þ vDs ð2Þ
where r0, rnet and v are the effective stress, net stress and
effective stress parameter accounting the contribution of
suction to the value of effective stress. During the soil
desiccation shrinkage, the soil aggregate (i.e. Voronoi grain
in the model) is almost saturated, even after cracking.
Therefore, the effective stress parameter is set to v ¼ 1
[22]. As the value of the net stress has no change, the
effective stress change is only contributed to by the
increase in the suction. The initial parameters used in the
simulations are listed in Table 1, except otherwise men-
tioned. The parameters are primarily determined from the
reported literature about the Saint-Alban clay [1, 25].
During the simulations, the settlements at various depths
are monitored, and the crack initiation time and crack
propagation patterns are recorded.
In the numerical simulations, a pseudo-time is used to
increase the suction following the field measurement as
shown in Fig. 1. Since the suction was monitored at five
levels in the field test, the whole soil layer is divided into
six sub-layers in the simulations. The suction value
between adjacent monitoring levels is interpolated linearly
along the depth axis in Fig. 1. For each sub-layer of soil,
the suction is also varied linearly between two consecutive
time points in Fig. 1 by 500 steps to make the calculation
stable. Thus, for each sub-layer of soil at each time instant,
the suction value can be obtained. Therefore, the Young’s
modulus and stress increment can be updated using
Eqs. (1) and (2), respectively.
3 Simulation of a field desiccation cracking usingthe proposed approach
Figure 4 shows the modelled micro-crack initiation and
propagation of the field desiccation. It can be seen that the
desiccation cracks start from the ground surface and are
initiated at some locations prior to other locations. These first
occurred cracks are referred to as primary crack in soil des-
iccation (Fig. 4a). The average spacing of the primary cracks
is 24 cm as there are four primary cracks in Fig. 4a. With
desiccation progressing, more cracks (secondary crack) are
Table 1 Summary of soil parameters used in the desiccation
simulation
Density [q (kg/m3)] 1440
Elastic modulus [E0 (MPa)] 5.0
Poisson’s ratio (t) 0.3
Friction angle [u (degree)] 19
Cohesion [c0 (kPa)] 30.0
Tensile strength [rt0 (kPa)] 15.0
Normal stiffness [kn0 (Pa/m)] 4 9 1010
Shear stiffness [ks0 (Pa/m)] 2 9 1010
wr (m) 1.875 9 10-6
wc (m) 7.5 9 10-6
m7.0
m2.1
(a)
(b)
Fig. 3 a Physical model and b numerical model of the soil sample
90 Acta Geotechnica (2018) 13:87–102
123
initiated within the aggregates bounded by the primary
cracks, and the cracks propagate primarily into the deeper
level vertically. Due to the discrete nature of the numerical
method, the crack propagating path is not straight which
agrees with the observations from both laboratory and field
testing. Some of the cracks bifurcate when they propagate
(Fig. 4d, e). The branches of the bifurcated cracks propagate
nearly horizontally and meet each other, resulting in the
Fig. 4 Simulated micro-crack pattern evolution during the field soil desiccation
Fig. 5 The simulated final macro-crack pattern at 241 h
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 50 100 150 200 250 300 350
Settl
emen
t (m
m)
Time (h)
2 cm4 cm7 cm10 cm16 cm25 cm40 cm2 cm (model)4 cm (model)7 cm (model)10 cm(model)16 cm (model)25 cm (model)40 cm (model)
Fig. 6 Comparison of experimental measured and numerically
modelled settlements
Acta Geotechnica (2018) 13:87–102 91
123
Table 2 Summary of the seven cases simulated
Parameter Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7
Density [q (kg/m3)] 1440 1440 1440 1440 1440 1440 1440
Elastic modulus [E0 (MPa)] 5.0 5.0 5.0 5.0 5.0 5.0 5.0
Poisson’s ratio (t) 0.3 0.3 0.3 0.3 0.3 0.3 0.3
Friction angle [u (degree)] 19 19 19 19 19 19 19
Cohesion [c0 (kPa)] 30.0 30.0 30.0 30.0 30.0 30.0 30.0
Tensile strength [rt0 (kPa)] 15.0 15.0 15.0 15.0 15.0 15.0 15.0
Normal stiffness [kn0 (Pa/m)] 4 9 1010 4 9 1010 4 9 1010 4 9 1010 4 9 1010 4 9 1010 4 9 1010
Shear stiffness [ks0 (Pa/m)] 2 9 1010 2 9 1010 2 9 1010 2 9 1010 2 9 1010 2 9 1010 2 9 1010
Ultimate norm displacement
[wr(m)]
1.875 9 10-6 3.75 9 10-6 7.5 9 10-6 1.875 9 10-6 1.875 9 10-6 1.875 9 10-6 1.875 9 10-6
Ultimate shear displacement [wc
(m)]
7.5 9 10-6 1.5 9 10-5 3 9 10-5 7.5 9 10-6 7.5 9 10-6 7.5 9 10-6 7.5 9 10-6
Table 3 Statistics of the seven cases simulated
Time (h) Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7
18
T 93,998 94,012 93,995 96,223 101,410 45,504 31,275
F 18,051 22,132 22,796 19,048 22,870 9348 8069
R 0.1920 0.2354 0.2425 0.1980 0.2255 0.2054 0.2580
L 27.882 34.798 36.128 29.127 33.736 15.262 13.125
24
T 93,960 93,966 93,960 96,189 101,351 45,483 31,264
F 24,882 27,956 28,594 25,184 28,640 13,060 11,168
R 0.2648 0.2975 0.3043 0.2618 0.2826 0.2871 0.3572
L 38.517 43.779 44.989 38.398 42.157 21.393 18.007
42
T 93,943 93,957 93,942 96,178 101,339 45,476 31,264
F 25,663 28,463 29,226 25,660 29,172 13,483 11,639
R 0.27318 0.3029 0.3111 0.2668 0.2879 0.2965 0.3723
L 39.64 44.457 45.856 38.986 42.807 22.08 18.718
48
T 93,941 93,951 93,935 96,172 101,334 45,474 31,260
F 26,036 28,807 29,512 25,966 29,503 13,695 11,920
R 0.2772 0.3066 0.3142 0.2600 0.2911 0.3012 0.3813
L 40.166 44.924 46.236 39.382 43.223 22.398 19.18
65
T 93,927 93,932 93,918 96,164 101,308 45,461 31,254
F 26,788 29,324 30,107 26,626 30,275 14,323 12,484
R 0.2852 0.3122 0.3206 0.2769 0.2988 0.3151 0.3994
L 41.16 45.522 46.974 40.194 44.208 23.375 20.063
73
T 93,872 93,868 93,851 96,100 101,220 45,437 31,231
F 29,379 31,101 32,015 29,039 32,019 15,679 13,529
R 0.3130 0.3313 0.3411 0.3022 0.3163 0.3451 0.4332
L 44.659 47.713 49.346 43.333 46.237 25.441 21.536
97
T 93,833 93,846 93,829 96,061 101,179 45,425 31,225
92 Acta Geotechnica (2018) 13:87–102
123
Table 3 continued
Time (h) Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7
F 29,978 31,596 32,400 29,606 32,391 15,956 13,879
R 0.3195 0.3367 0.3453 0.3082 0.3201 0.3513 0.4445
L 45.342 48.212 49.632 43.951 46.537 25.808 22.041
145
T 93,829 93,832 93,822 96,050 101,168 45,420 31,221
F 30,120 31,661 32,457 29,672 32,503 16,081 14,000
R 0.3210 0.3374 0.3459 0.3089 0.3213 0.3541 0.4484
L 45.48 48.249 49.642 43.978 46.634 25.971 22.221
193
T 93,808 93,808 93,791 96,029 101,138 45,408 31,210
F 30,540 31,994 32,800 30,089 32,906 16,432 14,256
R 0.3256 0.3411 0.3497 0.3133 0.3254 0.3619 0.4568
L 45.971 48.598 50.011 44.405 47.052 26.483 22.581
241
T 93,778 93,713 93,769 95,994 101,108 45,390 31,199
F 31093 32,518 33,293 30,456 33,427 16,951 14,628
R 0.3316 0.3470 0.3551 0.3173 0.3306 0.3735 0.4689
L 46.653 49.35 50.544 44.783 47.622 27.245 23.115
Grain no. 8528 8528 8528 8543 8592 2157 986
T—total number of contacts; F—number of failed contacts; R—contact failure rate; L—accumulated length of failed contact; grain no.—number
of grains in a model
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0
20
40
60
80
100
0 50 100 150 200 250 0 50 100 150 200 250
0 50 100 150 200 250 0 50 100 150 200 250
Con
tact
failu
re ra
te
Con
tact
num
ber,
×100
0
Time, h
(a)
Total contactFailed contactFailure rate
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0
20
40
60
80
100
Con
tact
failu
re ra
te
Con
tact
num
ber,
×100
0
Time, h
(b)
Total contactFailed contactFailure rate
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0
20
40
60
80
100
Con
tact
failu
re ra
te
Con
tact
num
ber,
×100
0
Time, h
(c)
Total contactFailed contactFailure rate
20
25
30
35
40
45
50
Tota
l fai
led
cont
act l
engt
h, m
Time, h
(d)
Case 1
Case 2
Case 3
Fig. 7 Contact statistic for the three models with different fracture energy: a Case 1, b Case 2, c Case 3 and d the total length of the failed
contacts in the simulations
Acta Geotechnica (2018) 13:87–102 93
123
formation of horizontal cracks. Therefore, soil wedges are
formed close to the surface (Fig. 4f). It is also observed that
the crack can reach as deep as more than 60 cm from the
surface at the end of the simulation.
Figure 5 shows the final macro-crack pattern. Twelve
cracks can be seen from the top surface, resulting in an
average crack spacing of 9 cm, which is within the range of
field observation. A few soil wedges can be counted from the
ground surface boundary. The tortuosity of themacro-cracks
can be clearly seen. However, the straight part of each crack
(i.e. the depth can be seen from the surface) is about 4–6 cm,
measured from the surface, which agrees with the field
investigation. In the field test, the sub-horizontal cracks
appeared progressively at depth ranging between 5 and 7 cm,
and this sub-horizontal cracks are also occurred in this depth
range in the numerical simulation. For the surface soil, cur-
ling effect is also observed in the simulation (Fig. 5).
Figure 6 shows the comparison of experimental and
numerical simulation settlements. Generally, the modelled
settlement is within the range of the measured settlement,
indicating the validity of the numerical scheme. However, it
is noted that the predicted settlement for each layer is slightly
larger than the measured settlement. The discrepancy can be
explained as follows: (1) the measured field settlement val-
ues from the field test might be underestimated due to the
unloading from the excavation, because the field test was
carried out after excavation to a certain depthwhichmade the
soil overconsolidated and the elastic deformation be recov-
ered; (2) cracking and sub-horizontal cracking could
decrease the settlement; and (3) the soil parameters (e.g.
friction angle) varied spatially, which were not considered in
the simulations, except for the Young’s modulus. Although
there is difference between the simulation and the experi-
ment, the simulated soil settlement can still reflect the gen-
eral trend: settlement value is larger at shallower level. In the
simulation, the settlement at the depth of 25 cm is larger than
the one at the depth of 16 cm. This can be explained by the
formation of the sub-horizontal crack above which the set-
tlement is decreased by shrinkage, while below which the
settlement is increased also due to the shrinkage.
Fig. 8 Comparison of fracture energy effect on simulated desiccation macro- and micro-crack patterns: a Case 1, b Case 2 and c Case 3
94 Acta Geotechnica (2018) 13:87–102
123
4 Effect of material properties and grain sizeon desiccation cracking
In this section, the potential factors affecting the simulation
results of field desiccation cracking are investigated, such
as grain contact fracture energy, heterogeneity of the grain
size distribution, and grain size. In all the following sim-
ulations, the soil parameters listed in Table 1 are still
applied, except specially indicated. There are total seven
cases simulated, and the parameters used are listed in
Table 2.
4.1 Effect of the fracture energy
In the mix-mode cohesive fracture model, the fracture
energy can be denoted as the triangle area in Fig. 2c. In the
above simulation, wr ¼ 5ðrt0=kn0Þ, wc ¼ 5ðc0=ks0Þ (this
case is Case 1 in the following statement). Here two more
fracture energy cases are used, i.e. wr ¼ 10ðrt0=kn0Þ, wc ¼10ðc0=ks0Þ (Case 2) and wr ¼ 20ðrt0=kn0Þ, wc ¼20ðc0=ks0Þ (Case 3). For the ease of the readability of the
paper, the statistics of all the cases considered in this paper
are listed in Table 3. Figure 7 shows the evolution of total
contact, failed contact and failed contact rate (number of
failed contact/number of total contact) in the three sce-
narios. It can be seen that with desiccation progressing,
more and more contacts start to fail and the failure rate
increases. However, the change of failed contacts and
failure rate is only notable for the first 73 h. After that, the
change becomes less and less obvious. This is because the
suction changes are mainly occurred in the first 73 h as
shown in Fig. 1. More specifically, at the time of 18 h, the
number of total contact and failed contact and failure rate
were 93,998, 18,051 and 19.20%, respectively, for Case 1.
(a)
(b)
(c)
Fig. 9 The models with three different meshes: a Case 1 b Case 4 and c Case 5
Table 4 Statistic of the grain assemblages in Cases 1, 4 and 5
Case no. 1 4 5
No of grains 8528 8543 8592
A: m2 9.8499 9 10-5 9.8326 9 10-5 9.7765 9 10-5
SD 1.6885 9 10-5 1.8189 9 10-5 3.2872 9 10-5
A average area of the grains; SD standard deviation of grains area
Acta Geotechnica (2018) 13:87–102 95
123
With desiccation progressing, the three values changed to
93,778, 31,093 and 33.16%, respectively, at 241 h. For
Case 2, the three values increased from 94,012, 22,132 and
23.54% at 18 h to 93,713, 32,518 and 34.70% at 241 h,
respectively. However, for Case 3, they ascended from
93,995, 22,796 and 24.25% at 18 h to 93,769, 33,293 and
35.51%, separately, at 241 h. Observing the failure rate, it
can be seen that the failure rate increases both with des-
iccation progressing and with increase in contact fracture
energy. Figure 7d counts the total length of failed contact
in the three simulations. It is interesting to note that the
higher contact fracture energy induces longer accumulated
length of failed contact during the entire numerical desic-
cation process. Figure 8 shows the final micro- and macro-
crack patterns from the three simulations. Although the
macro-crack patterns are almost the same, the higher
fracture energy results in thicker micro-crack band.
4.2 Effect of grain distribution
In this section, the grain heterogeneity effect on desiccation
cracking simulation is discussed. Three cases, including
Case 1, Case 4 and Case 5, are considered as shown in
Fig. 9. All the parameters used in each of these simulations
are the same and listed in Table 1. The models are acquired
through adopting various iteration numbers in UDEC. The
grain size in the three models is similar and is 0.01 m.
There are 8528, 8543 and 8592 Voronoi grains in the three
models, respectively, for Case 1, Case 4 and Case 5. The
statistic of the three models is listed in Table 4. It can be
seen that Case 5 has the largest number of grains, and the
largest standard deviation of grain area distribution
demonstrating the most discrete nature of the grain distri-
bution in Case 5.
Figure 10 depicts the evolution of the total contact,
failed contact and failed contact rate in the three scenarios.
Similarly to the study in Sect. 4.1, it can be seen that with
desiccation progressing, more and more contacts start to
fail and the failure rate increases. This is particularly
notable for the first 73 h due to the dramatic change of
suction during the first 73 h. After that, the change
becomes less and less obvious. More specifically, at the
time of 18 h, the total contact, failed contact and failure
rate are 93,998, 18,051 and 19.20%, respectively, for Case
1. With desiccation progressing, the three values changed
to 93,778, 31,093 and 33.16%, respectively, at 241 h. For
Case 4, the three values increase from 96,223, 19,048 and
19.80% at 18 h to 95,994, 30,456 and 31.73% at 241 h,
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0
20
40
60
80
100
120
0 50 100 150 200 250
Con
tact
failu
re ra
te
Con
tact
num
ber,
×100
0
Time, h
(a)
Total contactFailed contactFailure rate
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0
20
40
60
80
100
120
0 50 100 150 200 250
Con
tact
failu
re ra
te
Con
tact
num
ber,
×100
0
Time, h
(b)
Total contact
Failed contact
Failure rate
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0
20
40
60
80
100
120
0 50 100 150 200 250
Con
tact
failu
re ra
te
Con
tact
num
ber,
×100
0
Time, h
(c)
Total contactFailed contactFailure rate
20
25
30
35
40
45
50
0 50 100 150 200 250
Tota
l fai
led
cont
act l
engt
h
Time, h
(d)
Case 1
Case 4
Case 5
Fig. 10 Contact statistic for the three models with different meshes: a Case 1, b Case 4, c Case 5 and d the failed contact length in the
simulations
96 Acta Geotechnica (2018) 13:87–102
123
respectively. However, for Case 5, they rise from 101,410,
22,870 and 22.55% at 18 h to 101,108, 33,427 and 33.06%
at 241 h, separately. It can be seen that the failure rate
increases with desiccation progressing but fluctuate with
the grain distribution heterogeneity.
Overall, the three values are almost changed similarly
for the three cases. The difference between the three
cases, in terms of number of total contact, number of
failed contact and the failure rate, is relatively small
compared to the absolute value of them. Despite the
similarity, the failure rate at 241 h is 1.727, 1.603 and
1.466 times as the failure rate at 18 h for Case 1, Case 4
and Case 5, respectively, demonstrating that higher dis-
crete nature of the grain distribution can cause smaller
increase of the failure rate. Figure 10d counts the total
length of the failed contact in the three simulations.
Generally, Case 5 has the largest failure contact length
among the three scenarios. Figure 11 presents the final
micro- and macro-crack patterns from the three simula-
tions. There is no significant difference both for micro-
and macro-crack patterns.
4.3 Effect of grain size
Now the effect of grain size on the simulated desiccation
crack pattern is investigated. As shown in Fig. 12, three
grain sizes (i.e. 0.01, 0.02 and 0.03 m denoted as Case 1,
Case 6 and Case 7) are chosen to study the effect of grain
size on the desiccation crack patterns. All the parameters
used in these simulations are the same and listed in
Table 1. There are 8528, 2175 and 986 Voronoi grains in
the three models, respectively.
Figure 13 illustrates the contact statistic evolution in the
three grain-sized models. Apparently, the model with
smaller grain size has more contact numbers. Specifically,
there are 93,998, 45,504 and 31,275 contacts for the Case
1, Case 6 and Case 7, respectively. After 18-h desiccation,
there are 18,051, 9348 and 8069 contacts failed due to
desiccation shrinkage and the failure rate is 19.20, 20.54
and 25.80%, respectively. The number of the failed con-
tacts and the failure rate ascend rapidly for the first 73 h in
all the three cases. The final failed contact number is
31,093, 16,951 and 14,628, respectively. The failure rate
increases to 33.16, 37.35 and 46.89%, respectively, which
Fig. 11 Comparison of mesh on simulated desiccation crack pattern: a Case 1, b Case 4, and, c Case 5
Acta Geotechnica (2018) 13:87–102 97
123
are almost as twice as the failure rate at 18 h after desic-
cation start. Figure 13d depicts the evolution of the total
failed crack length. As anticipated, the total failure crack
length for the model with the smallest grains is the largest.
The final desiccation crack patterns are shown in
Fig. 14. It can be seen that the crack spacing is significantly
affected by the grain size and the model with the larger
grain size can cause larger macro-crack spacing under
desiccation shrinkage. The micro-crack width is generally
higher for the model with larger grain size. In terms of the
cracking depth, all the three grain-sized models are similar.
5 Conclusions
The paper presents the numerical simulations of a field soil
desiccation test using a hybrid continuum-discrete element
method with a mix-mode cohesive fracture model and
Voronoi grains. In the simulation, the desiccation-induced
soil stiffness hardening is considered by adopting an
empirical equation for soil elastic modulus. It is found that
the desiccation phenomena can be reasonably replicated;
especially, the sub-horizontal crack is successfully repro-
duced. It is demonstrated that the mix-mode cohesive frac-
ture model is able to handle desiccation-induced multi-
cracks in the field soil desiccation. In addition, the influence
factors affecting the desiccation cracking behaviour are
studied. The results demonstrate that the considered factors
such as fracture energy, grain distribution heterogeneity and
grain size have significant impact on the simulation micro-
cracks. However, the observed macro-crack is less sensitive
regarding to these factors, except for the grain size.
Acknowledgements Funding support from China State Key Labo-
ratory of Geohazard Prevention and Geoenvironmental Protection,
Chengdu University of Technology, via project SKLGP2016K003 is
gratefully acknowledged.
Appendix: Mix-mode cohesive fracture model
The cohesive fracture model used here is an extension and
application of the elastic–plastic-damage interface consti-
tutive framework originally presented in Ref. [21], and the
model has been successfully applied in static [19, 26] and
dynamic [20] problem of geomaterials, i.e. rock and soil.
To facilitate the readability, the model is described.
The cohesive fracture model takes into account the
cohesive effect on both tension and shear. In the model, the
fracture interface is idealised with zero thickness; in other
words, there is no layer of element embedded at the shared
boundary of the two adjacent grains. Figure 2 illustrates the
fracture model used in this paper. As shown in Fig. 2b,
when the bond is undergoing loading, its displacement can
be partitioned into elastic displacement represented by a
spring and inelastic displacement represented by a slider
and a divider in series. Therefore, the total bond dis-
placement is expressed as
u ¼ ue þ ui ð3Þ
where u is the total displacement tensor of the bond, ue is
the elastic displacement tensor and ui is the tensor of the
inelastic displacement occurring in the bond. As the
inelastic displacement is contributed from the slider and
divider in series, it can be further decomposed into plastic
displacement described by sliders, which is irreversible and
fracture displacement by dividers (reversible) as
ui ¼ up þ uf ð4Þ
where up is the plastic displacement tensor from the
deformation of the sliders and uf is the fracture
Fig. 12 The models with three different grain sizes: a 0.01 m (Case
1), b 0.02 m (Case 6) and c 0.03 m (Case 7)
98 Acta Geotechnica (2018) 13:87–102
123
displacement tensor measured from the dividers. The norm
of the inelastic displacement is computed as
uieff ¼ jjuijj ¼ jjup þ uf jj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðui2n þ ui2s Þq
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðupn þ ufnÞ2 þ ðups þ u
fsÞ2
q
ð5Þ
where uin and uis are the inelastic displacement scalar along
the normal and tangential direction of the contact,
respectively. In tensile loading, the governing variables
are the tensile strength (rt) and the norm of the inelastic
displacement (uieff ). The tensile strength evolves as a
linearly decreasing function of uieff as shown in Fig. 2c. It
is given as
rt uieff
� �
¼rt0 1� uieff
wr
� �
0
uieff\wr
uieff �wr
8
>
<
>
:
ð6Þ
and
wr ¼2GI
f
rt0: ð7Þ
In Eqs. (6) and (7), wr, rt0 and GIf are the ultimate norm
of the inelastic displacement corresponding to zero tensile
strength, the initial tensile strength and the mode I fracture
energy, respectively. The mode I fracture energy can be
obtained through a mode I test. The ultimate norm of the
inelastic displacement corresponds to the threshold
condition where the fracture is fully developed and the
contact is no longer capable of transferring stress. The
initial tensile strength is the stress at which the cohesive
zone starts to develop and the crack starts to undergo
softening.
A micro-damage variable is introduced as the per-
centage of fracture surface to the overall interface area to
degrade the stiffness. This definition can reflect the
physical behaviour when the contact is undergoing
fracturing. In addition, it also complies with the classical
definition of damage parameter in damage mechanics
(e.g. [26]). The micro-damage variable can be calculated
as
D ¼ Af
A0
¼ 1� kns
kn0ð8Þ
where D is the micro-damage variable and Af and A0 are
the fracture surface area and the overall contact area,
respectively. kns and kn0 are, respectively, the degraded and
initial normal stiffness. kns can be computed as
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0
20
40
60
80
100
0 50 100 150 200 250
Con
tact
failu
re ra
te
Con
tact
num
ber,
×100
0
Time, h
(a)
Total contact
Failed contact
Failure rate
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0
10
20
30
40
50
Con
tact
failu
re ra
te
Con
tact
num
ber,
×100
0
Time, h
(b)
Total contact
Failed contact
Failure rate
00.050.10.150.20.250.30.350.40.450.5
0
5
10
15
20
25
30
35
Con
tact
failu
re ra
te
Con
tact
num
ber,
×100
0
Time, h
(c)
Total contactFailed contactFailure rate
10
20
30
40
50
0 50 100 150 200 250
0 50 100 150 200 250 0 50 100 150 200 250
Tota
l fai
led
cont
act l
engt
h, m
Time, h
(d)
0.01 m0.02 m0.03 m
Fig. 13 Contact statistic for the three models with different grain sizes: a 0.01 m (Case 1), b 0.02 m (Case 6), c 0.03 m (Case 7) and d the failed
contact length in the simulations
Acta Geotechnica (2018) 13:87–102 99
123
kns ¼rn
un � upn¼ rtðuieffÞ
uen þ upn þ u
fn � u
pn
¼ rtðuieffÞrtðuieffÞ=kn0 þ ð1� gÞuieff
ð9Þ
where g is the ratio of plastic displacement to the total
value of inelastic displacement (i.e. g ¼ up=ui with ui the
norm of inelastic displacement before unloading) and it can
be determined experimentally using the pure mode I test.
Substituting Eq. (9) into Eq. (8), the micro-damage
variable D is expressed based on the norm of the inelastic
displacement as
D ¼ 1� rtðuieffÞrtðuieffÞ þ ð1� gÞuieffkn0
ð10Þ
Accordingly, the normal stress–displacement
relationship, i.e. the relationship between rn and ðun �upnÞ of the interface, is presented as
rn ¼ knsðun � upnÞ ¼ aknoðun � upnÞ ð11Þ
where parameter a is the integrity parameter defining the
relative active area of the fracture. The integrity parameters
is defined as
a ¼ 1� rnj j þ rn2 rnj j D ð12Þ
This integrity parameter is used to simulate the tensile
unloading–reloading behaviour in the cohesive fracture
model. It can be seen from Eq. (12) that the activation of
the micro-damage variable is controlled by the fraction
normal stress, which is activated in tension (i.e. rn [ 0)
and deactivated in compression (i.e. rn\0). Thus, the
normal stiffness in tension can be degraded, while it is kept
unchanged in compression.
The governing variables for the shear loading are
cohesion c (the contribution of normal stress to shear
strength can be neglected if the friction angle is taken to be
zero) and the norm of the inelastic displacement uieff . As
the crack propagates, the cohesion degrades and can be
expressed as a linear function of the norm of the inelastic
displacement uieff as shown in Fig. 2c. It is expressed as
c uieff� �
¼c0 1� uieff
wc
� �
0
uieff\wc
uieff �wc
8
>
<
>
:
ð13Þ
and
Fig. 14 Comparison of grain size on simulated desiccation macro- and micro-crack pattern: a 0.01 m (Case 1), b 0.02 m (Case 6) and c 0.03 m
(Case 7)
100 Acta Geotechnica (2018) 13:87–102
123
wc ¼2GIIa
f
c0ð14Þ
In Eqs. (13) and (14), wc is the ultimate norm of the
inelastic displacement corresponding to zero cohesion, c0 is
the initial cohesion, and GIIaf is the mode II fracture energy
dissipated during shear at high confining normal stress (i.e.
without the influence of the tensile loading regime). The
ultimate norm of the inelastic displacement corresponds to
zero cohesion. It gives the threshold condition at which the
shear crack is fully developed. It also indicates that the
material is no longer capable of transferring cohesion.
Similar to the softening treatment under tensile loading, the
degraded shear stiffness can be written as
kss ¼ aks0 ð15Þ
where ks0 and kss are the initial shear stiffness and the
degraded shear stiffness, respectively. The definition of ain Eq. (15) is same as in Eq. (11). The shear stress (s) iscomputed by
s ¼ kss us � ups� �
¼ akso us � ups� �
ð16Þ
Now the failure function is described. The normal stress
can be either compressive or tensile. As described earlier,
with the variation of the norm of the inelastic displacement,
thematerial tensile strength and cohesion are changed. In this
paper, the inter-block failure criterion can be described by a
failure envelope as shown in Fig. 2d. The failure envelope
represented by a solid line is the initial failure envelope. rt0and c0 represent the initial tensile strength and cohesion of
the fracture, respectively. According to Eqs. (6) and (13), the
two parameters decrease with increasingDwhen the fracture
develops. Therefore, the failure envelope shrinks when the
fracture is developing, i.e. the square dot line in Fig. 2d. If the
fracture is totally destroyed, the failure envelope will
become a line which is the conventional Mohr–Coulomb
failure envelopewith zero cohesion and zero tensile strength,
i.e. the long dash line in Fig. 2d. Mathematically, the failure
surface function can be expressed as
F ¼ s2 � 2c tan uð Þ rt � rnð Þ � tan2 uð Þ r2n � r2t� �
¼ 0
ð17Þ
where u is the friction angle which is kept unchanged
during the calculation, while rt and c are evolved as per
Eqs. (6) and (13), respectively.
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