Microsoft Word - 2234


 

A. Spagnoli et alii, Frattura ed Integrità Strutturale, 47 (2019) 394-400; DOI: 10.3221/IGF-ESIS.47.29                                                                
 

394 
 

Focussed on “Crack Paths” 

 
 
 

  
Experimental investigation on the fracture behaviour of natural stone 
exposed to monotonic and cyclic loading 
 
 
Andrea Spagnoli 
Department of Engineering and Architecture, Università di Parma, Parma, Italy 
spagnoli@unipr.it, http://orcid.org/0000-0002-0592-7003 
 

David A. Cendon Franco 
Departamento de Ciencia de los Materiales, Universidad Politécnica de Madrid, Madrid, Spain 
dcendon@mater.upm.es, http://orcid.org/0000-0001-7256-0814 
 

Antonio D’Angelo 
Department of Engineering and Architecture, Università di Parma, Parma, Italy 
dangeloa@pizzarotti.it, https://orcid.org/0000-0003-4704-1048 
 
 
ABSTRACT. The present paper is devoted to an experimental study on the 
fracture behaviour of natural stones, commonly used as elements for building 
cladding, under both monotonic and cyclic loading, with particular emphasis 
to white Carrara marble. The effect of progressive damage produced by in-
service thermal fluctuations can be investigated through the application of 
appropriate cyclic mechanical loads. In the experimental tests conducted, 
some static mechanical properties of marble are characterized by means of 
three-point bending tests on edge-cracked prismatic specimens for the 
determination of Young's modulus, tensile strength and fracture energy. 
Moreover, cyclic three-point bending tests are conducted to determine the 
propagation rate of nominally Mode-I fatigue cracks. Finally, the fatigue 
behaviour of the marble is studied through a cohesive crack model, in which 
the direct tensile strength of the material is determined by a Brazilian test, and 
the behaviour is calibrated by means of a suitable FE model. The effect of 
crack path on the fracture resistance of marble is discussed. 
  
KEYWORDS. Fatigue crack propagation; Marble; Quasi-brittle behaviour, 
Thermal cycles. 
 

 

 
 

Citation: Spagnoli, A., Cendon Franco, D. A., 
D’Angelo, A., Experimental investigation on 
the fracture behaviour of natural stone 
exposed to monotonic and cyclic loading, 
Frattura ed Integrità Strutturale, 47 (2019) 
394-400. 
 
Received: 25.10.2018  
Accepted: 12.11.2018  
Published: 01.01.2019 
 
Copyright: © 2019 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. 

 

 
INTRODUCTION  
 

he Carrara marble is a widely used material in the construction field, both in historical and monumental buildings 
and in recent constructions, for example in the cladding elements of ventilated facades [1]. Installed as coating of 
facades, the marble slabs are exposed to different actions that deteriorate the material, including temperature (daily T 

http://www.gruppofrattura.it/VA/47/2234.mp4


 

                                                              A. Spagnoli et alii, Frattura ed Integrità Strutturale, 47 (2019) 394-400; DOI: 10.3221/IGF-ESIS.47.29 
 

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and seasonal excursions, through-thickness gradient), mechanical loads (wind, self-weight), chemical attacks (acid rain) and 
humidity changes. In particular, temperature may induce stresses due to thermal expansion (restraint effects of the anchorage 
system, non linear temperature fields and non uniform thermal expansion) and thermal fluctuations tend to cause a 
progressive damage of the material, sometimes accompanied by a curvature of the surface of the slabs [2]. Such bowing 
phenomenon can progress up to the collapse of the element, with consequences often critical on safety for the users of the 
buildings where these covering slabs are installed. Bowing is generally accompanied by an overall reduction of strength, 
which increases with increase in degree of bowing, while at the micro structural level of the material bowing is accompanied 
by a decohesion of calcite grains.  
The present paper is devoted to an experimental study on the fracture behaviour of natural stones under both monotonic 
and cyclic loading, with particular emphasis to white Carrara marble. The effect of progressive damage produced by thermal 
fluctuations can be investigated through the application of appropriate cyclic mechanical loads. In the experimental tests 
conducted, some static mechanical properties of marble are characterized by means of three-point bending tests on edge-
cracked prismatic specimens for the determination of Young's modulus, tensile strength and fracture energy [3]. Moreover, 
cyclic three-point bending tests are conducted to determine the propagation rate of nominally Mode-I fatigue cracks [4]. 
Finally, the fatigue behaviour of the marble is studied through a cohesive crack model, in which the direct tensile strength 
of the material is determined by a Brazilian test, and the behaviour is calibrated by means of a suitable FE model [5]. The 
effect of crack path on the fracture resistance of marble is discussed. In particular, it is shown that different level of 
meandering in the intergranular cracking of marble is observed and correlated with the so-called xenoblastic or homoblastic 
texture of calcite grains [4]. 
 
 
EXPERIMENTAL TESTING UNDER MONOTONIC LOADING 
 

 preliminary experimental campaign is carried out to quantify the resistance parameters of a Carrara marble under 
monotonic loading. Four prismatic specimens were tested under three-point bending. The nominal dimensions of 
the specimens are as follows: length L=220 mm; span S=180 mm; height W=60 mm; width B=30 mm (Fig. 1). 

One specimen out of the 4 ones is a smooth specimen. Notched specimens are characterised by a central edge notch 
machined by means of a water jet technique. Notches with different nominal dimensions are machined, so that the notch 
depth is between 6.9 mm and 8.1 mm (the notch width is kept equal to 1.5 mm due to some technical constraints). Tests 
were performed by means of an Instron testing machine. Three LVDT sensors were used to measure the rigid body motion-
free mid-span deflection of the beam and a clip-on gauge to measure Crack Mouth Opening Displacement (CMOD). Tests 
on notched specimens were performed under CMOD control with a of less than 10-3 mm/min. Digital image correlation 
technique was adopted to capture full-field two-dimensional displacement maps on specimen surface in the notch vicinity.  
 

 
Figure 1: Geometrical and testing configuration of notched specimen under three-point bend loading. 

 
The rupture load of the smooth specimen is equal to 5.1 kN (corresponding to a nominal bending strength of 12.8 MPa, 
where the nominal stress  N  is equal to 

26 / (4 )PS BW ). 
For the 3 notched specimens, the mean bending stress at failure is equal to 6.7 MPa (corresponding to a mean rupture load 
of 2.7 kN), with a coefficient of variation equal to 0.13. In Fig. 2, the load against mid-span deflection is shown for two 
notched specimens, along with the corresponding curves of the nominal stress against the crack mouth opening 
displacement. 

A 



 

A. Spagnoli et alii, Frattura ed Integrità Strutturale, 47 (2019) 394-400; DOI: 10.3221/IGF-ESIS.47.29                                                                
 

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From the experimental curves related to notched specimens, fracture toughness of marble is calculated by using two 
different approaches. The first approach is based on the two-parameter model [6]. Note that a modified version of the two-
parameter model has recently been proposed in order to take into account the possible crack deflection during the stable 
crack propagation [7-9]. Accordingly, the initial crack length a0 is assumed to grow steadily before the peak load is attained. 
This nonlinear stable stage terminates when the crack propagates to a critical extent and the SIF KI attains a value KsIC that 
differs from the nominal KIC (computed on the basis of a0). If the geometric and loading conditions are such that the stress 
intensity factor is monotonically increasing with the crack length (being the load constant), as occurs in the case of a 3-point 
bend beam with an edge crack, the critical condition explained before takes place at the peak load. From LEFM formulas 
and from two compliance experimental measurements, the equivalent crack length and the effective toughness KsIC  are 
worked out. The second approach is based on the work-of-fracture method in Ref. [10] recommended by the RILEM 
technical committee. For 3-point bending tests on edge-notched beams, the method is based on the experimental 
determination of the work exerted by the applied mid-span force, which corresponds to the area underneath the complete 
load–displacement curve. Such a work is assumed to be fully spent to produce a mode I crack through the mid-span ligament 
of the beam. 
 

0 0.04 0.08 0.12 0.16
Mid-span deflection,  [mm]

0

1

2

3

0.5

1.5

2.5

L
oa

d,
 P

 [k
N

]

(a)

  
 

Figure 2: Load vs mid-span deflection curves (a) and nominal stress-CMOD curves (b) for notched specimens under three-point 
bending. In the graph experimental results for two specimens (black thin lines) are reported along with numerical results (red thick 
lines). 
 
The mean value of fracture toughness according to the two-parameter model is equal to 1.90 MPam0.5 (coefficient of 
variation equal to 0.19). According to the work-of-fracture method, the mean fracture toughness is 1.91 MPam0.5 (coefficient 
of variation equal to 0.37). 
For the sake of completeness, a splitting test on four prismatic specimens (nominal dimensions D = 60 mm, L = 30 mm, 
B = 60 mm) was performed according to ASTM C496. The resulting indirect tensile strength (  2 / ( )t P DL ) is equal 
on average to 7.1 MPa (coefficient of variation equal to 0.11). 
 
 
EXPERIMENTAL TESTING UNDER CYCLIC LOADING 
 

leven three-point bending tests on notched specimens, with the same geometry as that adopted for monotonic 
loading and with the initial notch length a0 in the range 6.6 to 8.5 mm, under nearly pulsating loading were 
performed. Tests were carried out under load control mode, with blocks of cycles composed by a single square 

cycle with frequency of 0.1 Hz and by 125 sinusoidal cycles with frequency 2.5 Hz. Square cycles are introduced to take 
DIC photos at maximum load. Applied cycles have a loading ratio R = 0.1 and a maximum load expressed as a percentage 
of the mean failure load Pu of three-point bend notched specimens under monotonic loading (Pu=2.7 kN). SN-like data are 

N
o
m

in
a

l s
tr

e
ss

, 
N
 [

M
P

a
]

E 



 

                                                              A. Spagnoli et alii, Frattura ed Integrità Strutturale, 47 (2019) 394-400; DOI: 10.3221/IGF-ESIS.47.29 
 

397 
 

reported in Fig. 3, where the load range level is expressed as the range of the applied SIF (calculated for a crack length equal 
to a0), normalized by the estimated SIF threshold  0.5, = 0.74 MPamI thK  (see the last paragraph of the present Section). 
Run-out specimens (see open circle in Fig. 3) correspond to unbroken specimens after a number of loading cycles larger 
than N0 = 100,000. A log-log linear fitting of the experimental points with 0<fN N  is attempted (negative inverse slope of 
50), bearing in mind that the paucity of experimental data, along with the inherent scatter of the material properties, do not 
allow us to draw any conclusion on the behavioural trend observed. 
For illustrative purposes, Fig. 4a shows the experimental curve of the CMOD at maximum load in the loading cycle against 
the number of cycles to failure for a notched specimen loaded with a ratio   , = 1.16I I thK K . By means of classical LEFM 
formulas, by knowledge of the applied load and the Young modulus of the marble under investigation, the effective crack 
length can be calculated from the measured CMOD. Consequently, crack length against number of cycles to failure curves 
can be obtained (see red curve in Fig. 4a).  
 

10
2

10
3

10
4

10
5

10
6

10
7

Numer of cycles to failure, Nf

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

R
e

la
tiv

e 
S

IF
 r

a
n

g
e,

 
K

I/
K

I,t
h

 
Figure 3: SN plot of fatigue results on notched specimens (open circles indicate run-out data). 

 
The experimental fatigue crack growth results, now expressed in terms of crack length a versus number of loading cycles, 
are analysed. In particular, the material parameters C and m of the Paris law (  / mIda dN C K ) are estimated. Such a law 
has been proved to describe well the experimental trend of the fatigue growth of the so-called long cracks for a variety of 
engineering materials with particular regards to metals. For this purpose, the crack growth rate da/dN versus number of 
cycles is determined according to ASTM E647 standard. The experimental da/dN-N data are analyzed by an incremental 
polynomial method. This method for computing da/dN involves fitting a second-order polynomial (parabola) to sets of 
(2n+1) successive data points a-N, where n is usually 1, 2, 3, or 4 (n = 4 is assumed here). The rate of crack growth at each 
central point ai, in the range ai-n– ai+n, is obtained from the derivative of the above parabola. The SIF range KI is determined 
according to classical LEFM expression for the current crack length a. As an example, in Fig. 4b the da/dN-KI plot is 
reported for the same specimen of Fig. 4a. 
The mean values based on five experimental tests of the estimated parameters of the Paris law are C=0.0002 and m=3.3 
expressing da/dN as m/cycle and KI as MPam0.5, although a large scatter in the results is recorded (coefficient of variation 
of C and m is equal to 0.34 and 0.80, respectively), and a degree of arbitrariness is introduced in the selection of the range 
of Paris law validity for best-fitting of experimental data point. The obtained values are in line with those reported in Ref. 
[4] (C = 0.00029 and m = 3.9). 
Finally, by considering the run-out fatigue tests on notched specimens (considering a conventional number of cycles for the 
fatigue limit N0 = 100,000), an estimation of a fatigue limit for the marble under investigation is attempted. In particular, a 
threshold condition in terms of SIF range (related to the initial notch length a0) is obtained. The resulting mean value of 
Kth is 0.74 MPam0.5 for loading ratio R = 0.1 (    max, , / (1 )I th I thK K R 0.83 MPam0.5). It is instructive to note that the 

ratio max, /I th sICK K  is equal to 0.44, which indicates a behavior slightly in the range of fatigue insensitive materials with flaws 
according to the seminal work of Fleck et al [11]. 



 

A. Spagnoli et alii, Frattura ed Integrità Strutturale, 47 (2019) 394-400; DOI: 10.3221/IGF-ESIS.47.29                                                                
 

398 
 

 

10
1

10
2

10
3

10
4

Numer of loading cycles, N

0

10

20

30

M
a

xi
m

um
 C

M
O

D
 [
m

]

8

12

16

20

24

28

C
ra

ck
 le

ng
th

, 
a 

[m
m

]

CMOD

a

(a)

   
 

        
10-1 100 101

SIF range, KI [MPam
0.5]

10-5

10-4

10-3

10-2

C
ra

ck
 g

ro
w

th
 r

at
e

, d
a/

dN
 [m

/c
yc

le
]

(b)

 
Figure 4: Fatigue crack growth data (a) and da/dN-KI plot (b) for a notched specimen with   , = 1.16I I thK K . 

 
 
NUMERICAL ANALYSIS 
 

n order to analyze the results from a meso-mechanics point of view, a numerical simulation based on the Cohesive 
Zone Model (CZM) was conducted. The CZM was firstly proposed for the numerical simulation of fracture of quasi-
brittle materials, such as concrete and rocks by Hillerborg and coworkers [12]. The evolution of damage in CZM is 

accounted for by means of the so-called softening curve, which is the mathematical function relating the mode I crack opening 
displacement, w, with the stress transferred across the crack lips,  (see Fig. 5). 

 

 
Figure 5: Meso-mechanics basis of the Cohesive Zone Model and parameters of the softening curve. 



w w

=f(w)

fdecoh
Area=Gf

I 



 

                                                              A. Spagnoli et alii, Frattura ed Integrità Strutturale, 47 (2019) 394-400; DOI: 10.3221/IGF-ESIS.47.29 
 

399 
 

The softening curve is defined by three main ingredients: the decohesion strength, fdecoh, which is the value of the softening 
curve for null crack opening; the area comprised between the curve and the abscissa axis, which represents the specific 
fracture energy of the material, GF; and the shape of the softening curve, which depends on the way in which the material 
is damaged as the crack is opened. In the numerical simulations presented in this paper, the CZM was implemented in 
combination with the embedded crack approach. Further details about the implementation can be found in [13, 14]. 
When applying the CZM to quasi-brittle materials, the decohesion strength is usually set equal to the tensile strength of the 
material measured through a splitting tensile test [15], which according to the experimental campaign previously detailed 
was equal to ft=7.1MPa (Young modulus was taken as E = 75 GPa). About the area under the softening curve, it was set 
equal to the specific fracture energy determined in the three-point-bending tests under static conditions, that is to say, GF = 
60 N/m, considering the results of two experimental tests reported in Fig. 2. 
Unfortunately, up to date no general procedure has been set to determine the shape of the softening curve for a given 
material. There are some standard curves, such as the bi-linear curve for plain concrete [16], or the rectangular curve for 
steels [17] but to the authors' knowledge, no standard has been set in the case of marble. For this reason, a trial-and-error 
procedure was made by numerical simulation of the three-point-bending tests conducted for the determination of the 
specific fracture energy, trying different softening curve shapes until achieving a good agreement with the experimental 
results. To facilitate this trial-and-error process, according to [18], the following family of softening curves was used: 

   * 1 1 1n nf e e        , where f* is the non-dimensional stress transferred across the crack lips (stress divided by the 
decohesion strength),  is the non-dimensional crack opening (crack opening divided by the critical crack opening) and n is 
a material constant. With this family of curves it is possible to modify the shape of the softening curve by playing with the 
n parameter (n = 0 leads to a linear softening curve). In the case of the marble analyzed in this paper, it was found that a 
value of n=2 leads to a reasonable agreement between the experimental behaviour and the numerical prediction, as is shown 
in Fig. 2. 
The numerical simulation of the three-point-bending static tests allowed us to provide a meso-mechanics insight on the 
results of cyclic loading tests. In quasi-brittle materials, it has been demonstrated that for samples having dimensions of the 
order of magnitude of the size of the Fracture Process Zone (FPZ), the peak load is reached after such an FPZ has been 
largely developed ahead of the notch tip of the specimen. Thanks to the CZM, it is possible to analyze the initiation and 
evolution of the FPZ, since it is equivalent to the cohesive zone. Fig. 6 shows the length of the cohesive zone (FPZ) as a 
function of the nominal stress intensity factor, KI, applied during the simulation of the three-point bending static test. In 
this plot the KI required to initiate the FPZ is marked, which is considerably lower than the KI corresponding to the peak 
load. In the same plot, the KI corresponding to the fatigue propagation threshold is also highlighted. It can be seen how 
the latter is mostly inside the region of FPZ generation, revealing that a significant level of damage (corresponding to a large 
FPZ) ahead of the notch tip is required for fatigue crack propagation. Actually, according to this analysis, those KI ranges 
below the threshold represented in the figure would not lead to crack propagation in spite of causing appreciable damage 
(FPZ) at the notch tip. It must be reminded that this result is solely based on a numerical simulation, requiring further 
experimental data to confirm the hypothesis. 
 

 
Figure 6: Fracture process zone (FPZ) size in terms of % of the ligament as a function of the stress intensity KI applied. The figure 
also shows the KI corresponding to the fatigue crack propagation threshold. 

0

5

10

15

20

25

30

35

0 0.2 0.4 0.6 0.8 1 1.2

FPZ sizeF
P

Z
 s

iz
e

 (
%

 o
f 

lig
a

m
e

n
t)

K
I 
(MPaꞏm

1/2
)

K
I
 required

for FPZ initiation
K

I
 max in fatigue

propagation threshold

K
I
 min in fatigue

propagation threshold

K
I

corresponding
to peak loadK

I
 for fatigue

propagation threshold



 

A. Spagnoli et alii, Frattura ed Integrità Strutturale, 47 (2019) 394-400; DOI: 10.3221/IGF-ESIS.47.29                                                                
 

400 
 

 
CONCLUSIONS 
 

n experimental campaign on the fracture behavior of Carrara marble under monotonic and cyclic loading has been 
carried out. The results under monotonic loading show a well-known quasi-brittle behavior of the material, which 
requires for instance a two-parameter approach to correctly estimate fracture toughness. The results under cyclic 

loading allow us to estimate the material parameters governing the fatigue crack growth. Also, such results allow an attempt 
to estimate a fatigue limit in terms of SIF threshold. A comparison of experimental monotonic results with those of a 
numerical model based on a cohesive crack approach shows a good correlation. Finally, the numerical results permit to give 
an insight on the correlation between FPZ and fatigue threshold conditions. Further investigation is needed to fully 
understand the experimental observation under cyclic temperature of the different behavior of marble depending on the 
microstructural texture of calcite grains (see xenoblastic and homoblastic types of marble) and in turns on the roughness 
features of intergranular cracking profiles. 
 
 
REFERENCES 
 
[1] Ferrero, A.M., Migliazza, M., Spagnoli, A. Theoretical modelling of bowing in cracked marble slabs under cyclic thermal 

loading (2009) Construction and Building Materials, 23 (6), pp. 2151-2159. 
[2] Spagnoli, A., Ferrero, A.M., Migliazza, M. A micromechanical model to describe thermal fatigue and bowing of marble 

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marble rocks (2016) Fatigue and Fracture of Engineering Materials and Structures, 39 (8), pp. 956-968. 
[4] Migliazza, M., Ferrero, A.M., Spagnoli, A. Experimental investigation on crack propagation in Carrara marble subjected 

to cyclic loads (2011) International Journal of Rock Mechanics and Mining Sciences, 48 (6), pp. 1038-1044. 
[5] Cendon, D. A., Torabi, A. R., Elices, M. Fracture assessment of graphite V-notched and U-notched specimens by using 

the cohesive crack model (2015) Fatigue and Fracure of Engineering Materials and Structures, 38 (5), pp. 563-573. 
[6] Jenq, Y. and Shah, S. P. (1985) Two parameter fracture model for concrete. J. Eng. Mech., 111, 1227–1241.  
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A 

















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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