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H. Mazighi et alii, Frattura ed Integrità Strutturale, 61 (2022) 154-175; DOI: 10.3221/IGF-ESIS.61.11                                                                
 

154 
 

 

 
 
 
 
 
Hybrid phase-field modeling of multi-level concrete gravity dam 
notched cracks  
  
 
H. Mazighi, M.K. Mihoubi  
Mobilisation et Valorisation des Ressources en Eau (MVRE), Ecole Nationale Supérieure d’Hydraulique (ENSH), Blida, 
Algeria. 
h.mazighi@ensh.dz, https://orcid.org/0000-0001-5983-5473  
mihkam@ensh.dz, https://orcid.org/0000-0002-7858-0127 
 
D. Santillán 
Dept. de Ingeniería Civil: Hidráulica, Energía y Medio Ambiente, Universidad Politécnica de Madrid, Madrid, 28040, Spain  
david.santillan@upm.es, https://orcid.org/0000-0002-9749-0522 
 
 
ABSTRACT. Phase-field models have become a powerful tool to simulate 
crack propagation. They regularize the fracture discontinuity and smooth the 
transition between the intact and the damaged regions. Based on the 
thermodynamic function and a diffusive field, they regularize the variational 
approach to fracture that generalizes Griffith’s theory for brittle fracture. 
Phase-field models are capable to simulate complex fracture patterns 
efficiently and straightforwardly. In this paper, we introduce a hybrid phase-
field approach to simulate the crack propagation in laboratory-scale and life-
scale structures. First, we apply our methodology to the three-point bending 
test on notched laboratory beams. Second, we simulate the fracture 
propagation in a life-size structure: the Koyna gravity dam. We account for 
the pressure load inside the fracture, and we study the effect of the position 
and number of initial fractures in the upstream face and the value of the 
Griffith critical energy release, on the fracture propagation under a flood 
event. The position of the fracture plays an important role in the final fracture 
pattern and crest displacements, whereas the value of the Griffith critical 
energy release alters the onset of the fracture propagation. We conclude that 
phase-field models are a promising computational tool that may be applied to 
real engineering problems. 
  
KEYWORDS. Concrete; Crack; Dam; Fluid pressure; Phase-field model. 
 

 

 
 

Citation: Mazighi, H., Mihoubi, M.K., 
Santillán, D., Hybrid phase-field modeling of 
multi-level concrete gravity dam notches 
crack, Frattura ed Integrità Strutturale, 61 
(2022) 154-175. 
 
Received: 24.01.2022 
Accepted: 21.03.2022 
Online first: 30.04.2022 
Published: 01.07.2022 
 
Copyright: © 2022 This is an open access 
article under the terms of the CC-BY 4.0, 
which permits unrestricted use, distribution, 
and reproduction in any medium, provided the 
original author and source are credited. 

 

 
 

https://youtu.be/TH9o8Mi3l0M


 

                                                              H. Mazighi et alii, Frattura ed Integrità Strutturale, 61 (2022) 154-175; DOI: 10.3221/IGF-ESIS.61.11 
 

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INTRODUCTION  
 

oncrete is one of the most common building materials in the world, and society demands proper safety levels in 
those infrastructures made of concrete. Safety assessment tasks require predicting the behavior of the structures 
until their ruin [1,2]. Engineers demand numerical models to simulate the fracture process. In this line, the two most 

employed approaches aiming at the initiation, localization, and propagation of cracks are the continuous [3] and 
discontinuous models [4]. 
The fracture is considered a discontinuity in the material due to internal or external stresses. The two most widely used 
theories in fracture mechanics are the Linear Elastic Facture Mechanics (LEFM) [1,5] and Nonlinear Fracture Mechanics 
(NLFM) [6–9]. The latter theory adopts the nonlinear behaviors which take into account micro-cracks located near the crack 
tip, the so-called fracture process zone [10]. This theory is suitable for large structures i.e. dams, bridges, etc. Contrary to 
the NLFM, the LEFM approach is simpler and assumes linear elastic behavior of materials [11]. 
In parallel, Continuum Damage Mechanics (CDM) [12,13] is a robust concept to model the degradation of materials, leading 
to the localization of fractures [14,15]. Many models have been proposed to simulate damage in concrete based on CDM 
[16–25]. Significant research efforts on modeling cracks have been conducted using the Finite Element Method (FEM) [26–
34], and the Extended Finite Element Method (XFEM) [35–43]. 
Recently, a new continuous model has appeared acknowledged as phase-field [44–47]. It models the discontinuities as a 
diffusive process and interpolates between both the broken and unbroken regions, the crack is determined by a scalar 
variable that takes two distinct values (0 inside the crack and 1 away) [48,49]. The damaged regions are determined by a 
coupled system of Partial Differential Equations (PDEs) based on the energy minimization; thus, additional computations 
as stress intensity factors are not necessary to calculate the crack initiation and propagation [50]. However, the method 
requires a regularization parameter called length scale [51], recently validated with experiments [52]. The great advantage of 
the phase-field approach is the ability to simulate complex crack patterns, such as twisting, kriging, joining [53,54]. It 
provides excellent results for brittle fracture [49,54–57], ductile fracture [58–61], cohesive fracture [62–64], and other 
complex applications [50]. 
In this paper, we focus on the ability of phase-field models to simulate the propagation of multi-level notched cracks in 
large structures, such as dams, and study its influence on the behavior of the structure compared to the load exerted by the 
fluid pressure inside the fracture. A recent and robust hybrid formulation of continuum damage mechanics is adopted to 
solve the governing PDEs due to its low computational cost. Our model encompasses all results previously found by other 
researchers considering the most influential basic parameters and possible crack levels. 
The paper is organized as follows: first, we present the governing equation of our model; afterward, we validate our model 
by comparing our numerical results with reported laboratory experiments on the literature; thereafter, we simulate the 
fracture propagation in a live-size gravity dam. Finally, we draw some overall conclusions. 
 
 
GOVERNING EQUATIONS 
 

his section introduces the coupled mathematical equations for fracture propagation in quasi-brittle materials. Once 
presented with the geometry of the domain, we derive the PDEs that govern our problem. 
 

 
Geometry 
Fig. 1 shows the proposed geometry, where the domain Ω  ℜδ has dimension δ{1,2,3}. Ω is composed of two 
subdomains, the elastic ΩE, and the fracture ΩF. The boundary conditions applied on the subdomain ΩE can be time-
dependent Dirichlet conditions on ∂DΩE or time-dependent Neumann conditions on ∂NΩE. We denote with f(x,t) the 
external traction force applied on ∂NΩE, b(x,t) is the body force, x is the position vector, and u(x,t)  ℜδ is the displacement 
field at time t. 
 
Constitutive relations 
We compute the crack propagation in quasi-brittle material through a quasi-static phase-field formulation, based on 
Griffith’s theory, an energy-based failure criterion propagation in brittle materials [1]. The crack propagates when the stored 
energy is higher than the fracture resistance of the material.  

C 

T 



 

H. Mazighi et alii, Frattura ed Integrità Strutturale, 61 (2022) 154-175; DOI: 10.3221/IGF-ESIS.61.11                                                                
 

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Figure 1: Scheme of the domains of the problem. 

 
The crack propagation states the minimization of the total potential energy stored ψ: 
 

d e sψ= ψ + ψ - ψ                                                            (1) 
 
where ψd is the critical energy release during the fracture process, ψe is the elastic energy, and ψS are the external sources of 
energy due to body and surface loads. The critical energy release during the fracture propagation equals: 
 

 
F

F

d
c c lΩ

Ω

ψ =  G dS=  G γ dΩ                                       (2) 

 
where Gc is the Griffith critical energy release rate for mode-I of fracture, and γl is the crack surface density per unit volume 
of the solid, given by Miehe et al. [48]: 
 

    
2

20
l

0

l
γ , = +

2l 2
                                                     (3) 

 
where l0 is the length scale parameter, and ϕ is the phase-field which satisfies the following condition: 
 

 




0,   if the material is intact    
x,t =   

1,   if the material is cracked
                                        (4) 

 
Therefore, ψd equals: 
 

 
F

2
2d 0

c
0Ω

l
ψ =  G ( + )dΩ

2l 2
                                                             (5) 

 
Borden et al. and Zhang et al. [65,66] proposed an analytical solution for computing the length scale parameter, l0, which 
depends on the resistance tensile strength, ft, the Young’s Modulus, E, and the Griffith critical energy release rate, Gc, as 
follows: 

 

c
0 2

t

27EG
l =

256f
                                                                       (6) 



 

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157 
 

In Eqn. (5), the critical energy release can be calculated through [67]: 
 

 





2
21c

c 2
1c

K
1-ν ,             plane strain

EG =  
K

,                       plane stress
E

                                             (7) 

 
where K1c is the fracture toughness and υ is the Poisson’s ratio. 
The strain tensor is decomposed into two parts [68]: 
 


δ

± a± a a
a=1

ε = ε n n                                                             (8) 

 
where ε+ and ε- are the tensile and compressive strain tensors, respectively, εa is the ath principal strain, and na is the principal 
direction of strain tensor εa. The elastic energies are expressed as: 
 

 ± 2 2
ε ± ±

λ
ψ ε = tr(ε) +μtr(ε )

2
                                                                (9) 

 
where λ and μ are Lamé constants, given by: 
 







νE E
λ=     ,   μ=            plane strain

(1+ν)(1-2ν) 2(1+ν)

E(1+2ν)νE
λ=     ,   μ=        plane stress

(1+ν)(1-ν) 3(1+ν)(1-ν)

                                             (10) 

 
The strain tensor is related to the displacement field u by: 
 

  T1
ε= ( u+ u )

2
                                                                      (11) 

 
We adopt the anisotropic formulation proposed by Miehe et al. [68], which states that fractures only propagate under 
tension, i.e., due to the positive part of the elastic energy. The total elastic energy is then expressed as:  
 

       ± + -
ε ε εψ ε =g  ψ ε +ψ ε                                                             (12) 

 
where g(ϕ) = (1-k)(1-ϕ2) + k is the degradation function, and k (0 < k << 1) is a parameter that avoids numerical 
singularities. 
The external energy functional, ψS accounts for the body forces and the applied loads on the boundaries as follows: 
 


 

E

s

Ω
Ω

ψ =  f.u dΩ+ t.u d                                                             (13) 

 
The fluid pressure inside the fracture, pf, exerts a force on the surface of the fracture. We include this load in the term of 
traction vector force as follows: 
 

  
    

E N N F
fΩ Ω Ω

t.u d = t.u d -  p n.u d                                               (14) 

 
We develop the second term in Eqn. (14) using the Divergence Theorem as follows: 



 

H. Mazighi et alii, Frattura ed Integrità Strutturale, 61 (2022) 154-175; DOI: 10.3221/IGF-ESIS.61.11                                                                
 

158 
 

 
    

N F E N E
f f fΩ Ω Ω

p n.u d = .(p u) dΩ- p n.u d                                    (15) 

 
The pressure load on the surface of the fracture is then formulated as a body force applied to the entire domain. To consider 
the crack propagation, we introduce the phase-field in Eqn. (15): 
 

 
        
E N E N

f f f fΩ Ω Ω Ω
.(p u) dΩ- p n.u d = g( ) .(p u) dΩ- g( )p n.u d                     (16) 

 
Thus, the external energy ψS becomes: 
 

 
 

       
N N

s
f fΩ Ω Ω

Ω

ψ =  f.u dΩ- g( ) .(p u) dΩ+ g p n.u d + t.u d           (17) 

 
By substituting Eqns. (5), (12), and (17) into Eqn. (1), the Fréchet derivative of the total potential energy ψ yields the 
following Euler equations [65]: 
 

             
2 2 +c

0 ε f f
0

G
-l =2 1-k 1- ψ ε +p .u+u. p

l
                                               (18) 

 
and: 
 

   fσ+p g +f=0                                                        (19) 

 
where the Cauchy stress tensor is given by: 
 

        ++ -
σ=g λtr ε I+2με +λtr ε I+2μ                                                         (20) 

 
The regime flow inside the porous media material is based on Darcy’s law, which is derived from the Navier-Stokes equation. 
It describes a linear relationship between the velocity v (m/s) and the gradient of pressure pf (Pa). Lomiz [69] and Louis 
[70] carried out that the penetration of the fluid through a rock fracture follows the Cubic law developed using the parallel-
plates approach [71]. 
 

fp v


 
k

                                                                                    (21) 

 
with   the dynamic viscosity (Pa.s) and k (m2) the permeability of the porous medium of the fluid respectively. 
Darcy’s law is only valid for low velocities when the regime is laminar [69,70]. 
We account for the irreversibility of the crack propagation process by adopting the following strain-history field H+(u, pf, t) 
[72]: 
 

 
 

   + +
f ε f f

x 0,t
H u,p ,t =max ψ ε +p .u+u. p


                                  (22) 

 
Eqn. (18) can then be rewritten as: 
 

       2 2 +c
0

0

G
-l =2 1-k 1- H

l
                                              (23) 

 
Thereby, the Euler-Lagrangian equation becomes: 
 



 

                                                              H. Mazighi et alii, Frattura ed Integrità Strutturale, 61 (2022) 154-175; DOI: 10.3221/IGF-ESIS.61.11 
 

159 
 

 
   

  

  



 
 
 

f

+ +
0 2 2 0

0
c c

σ+p g +f=0

2l 1-k H 2l 1-k H
+1 -l =

G G

                                              (24) 

 
Eqn. (24) are the strong form of the phase-field problem, subject to the following Neumann boundary conditions:  
 

  
 





f N

N

σ.n= t+g d p                         in  Ω

.n=0                                in  Ω
                                  (25) 

 
Eqn. (24) summarizes the nonlinear phase-field problem. The equations include the decomposition of the Cauchy stress 
tensor into its compression and tension parts. Since we assume fractures only propagate under tension, and not due to 
compression, the degradation function only affects the positive elastic energy. From a mathematical point of view, this 
assumption leads to a highly nonlinear and computationally expensive system of PDEs. Recently, these drawbacks have 
been sorted out with the so-called hybrid formulation [73]. The formulation linearizes the problem with a third constraint 
as follows: 
 

 
   

  

  

  



 
 
 
 

f

+ +
0 2 2 0

0
c c

+ -
ε ε

σ+p g +f=0

2l 1-k H 2l 1-k H
+1 -l =

G G

x:ψ <ψ : 0

                                         (26) 

 
In quasi-static calculation, the fractures propagate under a toughness-dominated regime. The energy expended during the 
fracture process is much higher than the viscous dissipation [74], i.e., the energy dissipated due to the water flow within the 
fracture. This assumption involves that the water pressure inside the crack is more or less constant and equal to the 
hydrostatic pressure at the crack level [50]. Consequently, the term u.∇pf in Eqn. (22) is null. 
We solve the system of PDEs given by Eqn. (26) with the staggered scheme proposed by Miehe et al. [72], successfully used 
in engineering problems [50,75]. The displacement field, u, the phase-field, ϕ, and the strain-history field, H+, are solved 
sequentially, as shown in Fig. 2. This approach requires small loading increments, i.e., small-time steps [50]. We adopt an 
implicit Backward Differentiation Formula [76] for the time integration. At the beginning of the j+1 time step, we take as 
the initial condition, the solution of the previous time step, j. The displacement field is first computed using H+,j, and ϕj. 
Afterward, the stain-history field is updated with uj+1 and ϕj. Finally, the phase-field is updated. In each time step, a minimum 
tolerance convergence is required. 
 
 
MODEL VALIDATION 
 

he purpose of the study proposed in this section is to validate the numerical model presented previously. 
The validation consists to compare the obtained results from the application of our numerical model on a 
benchmark widely used in fracture mechanics, which is the notched beam problem, where experimental data are 

available. 
 
Notched concrete beam 
In this example, we simulate the propagation of a fracture in the notched beam problem [77]. The geometry is depicted in 
Fig. 3(a), according to Meschke et al. [78]. The thickness of the beam is 127 mm, the height is 254 mm, the length is 1118 
mm, and the span is 1016 mm. The mechanical properties are: E = 4.36 x104 MPa, υ = 0.2, ft = 4.0 MPa, and Gc=119 N/m. 
We simulate the propagation of an initial vertical fracture parallel to the side of the beam. The fracture is 78 mm in length 
and 5 mm in width and is located in the bottom part of the beam. We apply an incremental vertical displacement u on the 
top central part of the beam, and we neglect the self-weight. We adopt as length scale parameter l0 = 10 mm. 
 

T 



 

H. Mazighi et alii, Frattura ed Integrità Strutturale, 61 (2022) 154-175; DOI: 10.3221/IGF-ESIS.61.11                                                                
 

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Figure 2: Staged scheme for the numerical resolution of the PDE system. 
 
 
We simulate the fracture propagation with a 2-dimensional model under plane strain conditions. We discretize the domain 
around the fracture with 8272 quadrilateral elements and the rest of the beam with 6427 triangular elements. The mesh size 
around the fracture is h0 = 1.0 mm = l0/10, small enough to accurately capture the transition from damaged to undamaged 
regions.  
We plot the fracture pattern in Fig. 3(b), where the red color represents the fracture, and the evolution of the applied force 
against the Crack Mouth Opening Distance (CMOD) in Fig. 3(c). We compare our simulations with those reported by 
Meschke et al. and Mandal et al. [78,79], and experimental results. Our simulations are quite similar to both reported results 
as well as with laboratory experiments. The applied load increases linearly until the critical load is reached. Afterward, the 
crack grows quickly, and the CMOD increases. 

 

 
Calculate ϕi

 j+1  

Using: ui
 j+1, Hi

+, j+1 

 Start 

 Next step

Yes

Se
gr

eg
at

ed
 s

te
ps

 

 

 

Calculate ui
 j+1 

Using: Hi
+, j=0, ϕi

j=0 

 

Calculate Hi
+, j+1  

Using: ui
 j+1, ϕi

j=0 

 
Initial conditions 

ui
j=0, Hi

+, j=0, ϕi
j=0 

 

  Convergence:  

u
i
j+1,H

i
+, j+1,ϕ

i

j+1 - 

u
i
j=0,H

i
+, j=0,ϕ

i

j=0 <Tolerance 

No



 

                                                              H. Mazighi et alii, Frattura ed Integrità Strutturale, 61 (2022) 154-175; DOI: 10.3221/IGF-ESIS.61.11 
 

161 
 

 
(a) 

 
(b) 

 
(c) 

Figure 3: (a) Geometry and boundary conditions of the notched beam test. Dimensions are in mm. (b) Fracture pattern. (c) Evolution 
of the applied force against CMOD. 
 
 
FULL-SCALE CONCRETE GRAVITY DAM 
 

ur second benchmark is a life-size structure, the Koyna gravity dam. The propagation of fracture along the dam 
body has been previously simulated by many researchers. We assume plane-strain conditions and homogenous 
materials. The structure is 103 m in height and is located in India. It has been widely used as a benchmark model O 



 

H. Mazighi et alii, Frattura ed Integrità Strutturale, 61 (2022) 154-175; DOI: 10.3221/IGF-ESIS.61.11                                                                
 

162 
 

for static analysis of fracture propagation with 2-dimensional mathematical models. Gioia et al. [80] compared fracture 
analyses of notched the Koyna dam using plastic and linear elastic fracture mechanics, the effect of three initial horizontal 
cracks on the upstream face on the failure process of the structure is studied as well. The authors concluded that the most 
critical position of the initial fracture is at the change of the downstream slope. Bhattacharjee and Léger [81] employed two 
smeared models based on nonlinear fracture mechanics to study the behavior of the dam under two critical energy 
dissipation parameters. The results showed that the critical energy affects considerably the initiation of the crack. Ghrib and 
Tinawi [82] proposed an anisotropic continuous damage model formulation and studied the influence of mesh size. The 
fracture becomes smoother as the mesh is refined, but no major changes in the crest displacement were detected. Roth et 
al. [83] adopted a coupled anisotropic damage mechanic model based on the continuous approach and XFEM to simulate 
the fracture propagation in the Koyna dam. The level sets functions used to define the fracture path, together the continuum 
damage mechanics provided a correct fracture path. Santillán et al. [50] implemented a hybrid-phase field model to simulate 
the failure of the Koyna dam. The authors disregarded the pressure load inside the fracture, yet they found a good agreement 
with the results of Bažant, Z.P., Planas [84]. 
In this study, we adopt our hybrid model and incorporate the effect of the pressure load inside the fracture. We study the 
influence of the initial crack level, the number, and length of initial fractures, as well as the value of the critical energy 
dissipation on the behavior of the dam under a flood event. We illustrate the Koyna dam geometry in Fig. 4(a), where we 
also depict the three initial fracture levels in the upstream face denoted as crack I, crack II, and crack III. The distance 
between crack levels is 22.17 m which corresponds to one-third of 66.51 m, and the depth of the fractures is 0.1Bi, where 
Bi is the width of the dam at the given height.  
The applied loads on the dam are the hydrostatic load plus the overflow on the upstream face, and the self-weight. The 
mechanical properties of the concrete are taken from the simulations of Bhattacharjee and Léger [81], and these are: Young 
Modulus E= 25000 MPa, υ = 0.2, ft = 1.0 MPa, ρ = 2450 kg/m3. Regarding the Griffith critical energy release rate, we adopt 
two plausible values: Gc = 100 N/m and Gc = 200 N/m, with l0 = 0.4 m. For each value of the critical energy release rate, 
we analyze the effect of the initial length and location of the crack on the fracture path and the evolution of the horizontal 
crest displacement. In this line, we have conducted four simulations for each Gc-value: 

- Case I. Dam with an initial horizontal fracture on the upstream face at height 66.51 m. 
- Case II. Dam with an initial horizontal fracture on the upstream face at height 44.34 m. 
- Case III. Dam with an initial horizontal fracture on the upstream face at height 22.17 m  
- Case IV: Dam with three initial horizontal fractures on the upstream face at the heights 22.17, 44.34, and 66.51 m 

respectively. 
We simulate the fracture propagation with a 2-dimensional model under plane strain conditions. We discretize the domain 
around the fracture path with structured quadrilateral elements of 8 cm size and triangular elements in the remaining parts 
of the dam. We include the fracture paths for Case I and Gc = 100 N/m reported in previous works in Fig. 4(b), and the 
evolution of the overflow against the horizontal displacement of the crest in Fig. 4(c). 
 
Simulation for a rate Gc=100 N/m 
The evolution of the overflow against the horizontal crest displacement for Case I is depicted in Fig. 5(a). We depict the 
contour plots of the phase-field at four-time steps, denoted from A to C in the plot of the panel (a). The initial fracture 
begins to propagate when the overflow is 6.11 m (point A). The crest displacement is linear with the overflow, but once the 
water level is 6.11 m above the crest, a sudden growth of the fracture occurs. The propagation of the fracture makes the 
crest displacement increase suddenly while the water level remains constant (transition from time step A to B). The fracture 
path at time step B is shown in Fig. 5(b). 
After the initial sudden fracture growth, the crack propagates slowly. The crest displacement is no longer linear with 
overflow, indeed the displacement grows with a lower increase of the water level than before. This behavior changes at time 
step C, where a sudden propagation of the fracture occurs again. Between the time steps B and C the fracture branches, a 
new sub-vertical fracture appears that heads towards the change of slope in the downstream face. After time step D, both 
fractures continue the propagation. The crest displacement is then higher for lower rises of the water level. 
The results of Case II are included in Fig. 6. The evolution of the crest displacement against the overflow is plotted in panel 
(a), and the fracture pattern at the last time step of the simulation is shown in panel (b). As in the previous case, initially, the 
crest displacement is linear with the overflow. The fracture begins to propagate when the water level is about 7 m. Afterward, 
the fracture grows slowly, but the crest displacement is no longer linear with the water level, and the slope of the curve 
water level-crest displacement changes. We stop the simulation when the overflow is 11 m.  
 



 

                                                              H. Mazighi et alii, Frattura ed Integrità Strutturale, 61 (2022) 154-175; DOI: 10.3221/IGF-ESIS.61.11 
 

163 
 

 
(a) 
 

 
(b) (c)

 

Figure 4: (a) Geometry of the Koyna dam. (b) Crack paths reported in previous works. c) Overflow against crest displacement reported 
in previous works. 
 
The results of Case III are depicted in Fig. 7. The crest displacement is linear with the overflow up to the water level is 7.72 
m above the crest of the dam (time step A). At that moment, the initial fracture suddenly propagates, and the crest 
displacements increase with a slight rise of the water level until the time step B. Afterward, the fracture propagates slowly, 
but the crest displacement increases with lower a lower rise of the water level. We stop the simulation when the overflow is 
11 m. 
The results of the last configuration, Case IV, are included in Fig. 8. This configuration includes the three initial fractures. 
Initially, the dam displacement is linear with the water level, until the level is 6.11 m above the crest at time step A. Between 



 

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the time steps A and B, the initial fracture located at the higher level –denoted crack I– experiences a sudden increase which 
makes the crest displacement also increase suddenly without apparent water level rise. Afterward, and between the time 
steps B and C, the previous fracture goes on propagating and makes the crest displacement no longer linear with the water 
level. Later, at time step C the initial fracture located at the lowest level –crack III– starts to propagate. The propagation 
continues until the fracture reaches the foundation at time step E. At this moment, the crack I branches, which makes the 
crest displacement increase suddenly until the time step F. Afterward, both branches of the fracture continue the 
propagation. But at time step G, the initial fracture at the middle level –crack II– begins to propagate. The simulation is 
stopped when the overflow is 11 m. 
The crest displacement at the end of the simulation is almost 90 mm. This configuration, Case IV, provided higher crest 
displacements than the other three due to the presence of three initial fractures. The three initial fractures propagate at 
different time steps. The initial crack I --the one located at the highest level-- is the first fracture to grow and the propagation 
starts for the same overflow as for Case I, i.e., the configuration with only crack I.  

 
(a) 

 
Step A        Step B          Step C             Step D 

(b) 
Figure 5: (a) Overflow against crest displacement for Case I with Gc=100 N/m. (b) Contour plots of the phase-field at four-time steps. 
 



 

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(a) 

 
(b) 

Figure 6: (a) Overflow against crest displacement for Case II with Gc=100 N/m. (b) Contour plots of the phase-field at the end of the 
simulation. 
 
 
 
 
 
 



 

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(a) 

  
Step A        Step B 

(b) 
Figure 7: (a) Overflow against crest displacement for Case III with Gc=100 N/m. (b) Contour plots of the phase-field at two-time 
steps. 
 
 
 



 

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(a) 

 
Step A        Step B          Step C             Step D 

 

 
Step E        Step F          Step G             Step H 

Figure 8: (a) Overflow against crest displacement for Case IV with Gc=100 N/m. (b) Contour plots of the phase-field at eight-time 
steps. 



 

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Simulation for a rate Gc=200 N/m 
We describe and analyze the results of the simulation of our four adopted cases in this section with Griffith’s critical energy 
release rate is now Gc = 200 N/m. The result for the model with crack I –the initial fracture at the highest level, Case I– 
are included in Fig. 9. Both simulations with Gc = 100 N/m and Gc = 200 N/m present the same pattern of crest 
displacement evolution. Both simulations present two plateaus, the first one due to a sudden propagation of the initial 
fracture, and the second due to the branching of the fracture. However, the model with the highest toughness requires a 
higher overflow to initiate the fracture propagation. Nevertheless, for the same final overflow, 11 m, the final crest 
displacement is 80 mm in both simulations, i.e., the final crest displacement is almost independent of the toughness of 
concrete, as well as the final fracture pattern. 
 

 
(a) 

 
Step A        Step B          Step C             Step D 

(b) 
Figure 9: (a) Overflow against crest displacement for Case I with Gc=200 N/m. (b) Contour plots of the phase-field at four-time 
steps. 
 



 

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The evolution of the crest displacement and final fracture pattern of Case II with toughness Gc = 200 N/m, Fig. 10, differs 
from the case with toughness Gc = 100 N/m. The case with the highest toughness requires a higher overflow to initiate the 
propagation of the fracture. Once the crack starts to grow, a sudden but short propagation occurs which makes the crest 
displace without almost water rise. Afterward, the crest goes on increasing with the rise in water level, although the fracture 
almost does not grow. The final overflow is 11 m, which produces a final crest displacement of 30.5 mm, slightly lower than 
the model with toughness Gc = 100 N/m where it is 32 mm. The plateau in the overflow-crest displacement curve is not 
given in the model with toughness Gc = 100 N/m, as well as the final fracture is much longer. 
 

 
(a) 

 
Step A        Step B 

(b) 
Figure 10: (a) Overflow against crest displacement for Case II with Gc=200 N/m. (b) Contour plots of the phase-field at two-time 
steps. 
 
Case III –initial fracture at the lowest level– with toughness Gc = 200 N/m, Fig. 11, has a similar pattern to the case with 
toughness Gc = 100 N/m. The fracture begins to propagate for a higher overflow in the case with Gc=200 N/m, and 



 

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although both simulations have a plateau in the overflow-crest displacement, it becomes more pronounced for the case with 
the highest toughness. The final fracture pattern, as well as crest displacement, are almost identical for both cases, and 
consequently, independent of the concrete toughness. 
Case IV with Gc = 200 N/m, Fig. 12, includes the three initial fractures. The evolution of the crest displacement is quite 
similar to the case with Gc = 100 N/m, although two main differences arise: one of the plateaus are not present in this case, 
and the first fracture begins to propagate for a higher overflow. The plateau is not presented because the initial fracture at 
the middle level, crack II, does not propagate, as shown in panel (b). The final patterns of the fractures are quite similar to 
the model with Gc = 100 N/m, except for the absence of crack II. The value of the final crest displacement is also equal to 
the model with Gc = 100 N/m. 

 
(a) 

 
Step A        Step B 

(b) 
Figure 11: (a) Overflow against crest displacement for Case III with Gc = 200 N/m. (b) Contour plots of the phase-field at two-time 
steps. 



 

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(a) 

 
Step A        Step B          Step C 

 
Step D        Step E          Step F 

Figure 12: (a) Overflow against crest displacement for Case IV with Gc = 200 N/m. (b) Contour plots of the phase-field at six-time 
steps. 



 

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CONCLUSIONS 
 

n this work, we present a hybrid phase-field model to simulate the propagation of fractures under mode-I in brittle 
materials. We studied the behavior of a life-size structure, a concrete gravity dam, taking into account multi basic 
parameters that can considerably influence the structure stability. 

We validated our model with a benchmark: the propagation of a fracture in a notched beam. Experimental and numerical 
results are available in the literature. Our numerical simulations provided similar results to those reported in the literature. 
In this line, our evolution of the applied load against the crack mouth opening displacement (CMOD) is quite similar to the 
results of other authors. 
We applied our model to the study of a life-size structure, the Koyna dam. The structure is a 103 m height gravity dam 
located in India. It has been widely used as a benchmark model for static analysis of fracture propagation with 2-dimensional 
mathematical models. We analyzed four cases, three of them with only one initial small fracture in the upstream face at three 
levels, and the fourth case includes the three previous initial fractures, and we considered two values for the toughness of 
the concrete: Griffith critical energy release, Gc, equal to 100 N/m and 200 N/m. We simulated the propagation of the 
fractures under a flood episode.  
The most adverse level of the initial fracture is the highest one. This case provided the highest crest displacements at the 
end of the simulation when the overflow was 11 m. This case was the most unfavorable because of fracture propagates 
along the thinnest part of the dam. The simulations with the two considered values of the Griffith energy release rate 
provided the same fracture patterns for the same initial fracture configurations. The overflow required to initiate the fracture 
propagation increased as Gc does. Nevertheless, once the fracture started to grow, the crest displacement for the final 
overflow, 11 m in all the simulations, was independent of the Gc-value. Consequently, concrete with enhanced toughness 
requires higher values of the overflow, but once fracture starts to propagate, the final fracture length and crest displacement 
are almost independent of concrete toughness. 
Our simulations showed that phase-field models are suitable for life-size problems in civil engineering. They can be useful 
for design engineers as well as stakeholders and infrastructure operators in their safety assessment tasks. Our recommended 
phase-field approach turns out to be very convenient from the point of view of computations and ease of implementation 
for quasi-brittle structures. 
 
 
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    /NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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    /ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing.  Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
  >>
  /Namespace [
    (Adobe)
    (Common)
    (1.0)
  ]
  /OtherNamespaces [
    <<
      /AsReaderSpreads false
      /CropImagesToFrames true
      /ErrorControl /WarnAndContinue
      /FlattenerIgnoreSpreadOverrides false
      /IncludeGuidesGrids false
      /IncludeNonPrinting false
      /IncludeSlug false
      /Namespace [
        (Adobe)
        (InDesign)
        (4.0)
      ]
      /OmitPlacedBitmaps false
      /OmitPlacedEPS false
      /OmitPlacedPDF false
      /SimulateOverprint /Legacy
    >>
    <<
      /AddBleedMarks false
      /AddColorBars false
      /AddCropMarks false
      /AddPageInfo false
      /AddRegMarks false
      /ConvertColors /ConvertToCMYK
      /DestinationProfileName ()
      /DestinationProfileSelector /DocumentCMYK
      /Downsample16BitImages true
      /FlattenerPreset <<
        /PresetSelector /MediumResolution
      >>
      /FormElements false
      /GenerateStructure false
      /IncludeBookmarks false
      /IncludeHyperlinks false
      /IncludeInteractive false
      /IncludeLayers false
      /IncludeProfiles false
      /MultimediaHandling /UseObjectSettings
      /Namespace [
        (Adobe)
        (CreativeSuite)
        (2.0)
      ]
      /PDFXOutputIntentProfileSelector /DocumentCMYK
      /PreserveEditing true
      /UntaggedCMYKHandling /LeaveUntagged
      /UntaggedRGBHandling /UseDocumentProfile
      /UseDocumentBleed false
    >>
  ]
>> setdistillerparams
<<
  /HWResolution [2400 2400]
  /PageSize [612.000 792.000]
>> setpagedevice