Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

56, 4, pp. 927-938, Warsaw 2018
DOI: 10.15632/jtam-pl.56.4.927

XFEM SIMULATION OF THE INFLUENCE OF CRACKING

INTRODUCED BY PRE-LOADING ON THE STRENGTHENING OF

A CEMENT TREATED MIXTURE

Karol Brzeziński
Warsaw University of Technology, Faculty of Civil Engineering, Institute of Roads and Bridges, Warsaw, Poland

e-mail: k.brzezinski@il.pw.edu.pl

The paper attempts to explain the phenomenon of a static strength increase due to the
previous cyclic loading. Finite Element Method simulation of the potential strengthening
mechanism is presented. The crack growth is controlled by the energy criterion used in the
linear fracture theory. A wedge splitting test has been performed in order to determine the
critical energy release rate of the tested mixtureGIC. It was observed that the appearance
of additional cracks in the material may lead to an increase of its strength. Analysis of the
tensile stress distribution allows for a qualitative explanation of the observed phenomenon.

Keywords: fatigue life, cement-treated base layers, strengthening effect, load history, XFEM

1. Introduction

The fatigue strength is an important aspect of estimating durability ofmany types of structures
subjected to cyclic loadings. Particularly, road pavement structures, which are often subject to
manymillions of load cycles, are designedwith regard to their fatigue life. Themodern approach
to fatigue was initiated by Wöhler (1970). He measured fatigue strength as a feature of the
material, so the number of load cycles that the structure can carry depends on the amplitude
of the load. This dependency, presented later by his successor in the form of a graph, is known
as the “Wöhler curve”. By using such formulas, the number of constant amplitude load cycles
causing destruction of the material can be estimated.
Taking into account variable amplitude loads requires additional assumptions about fati-

gue accumulation. The simplest theory, the so-called linear damage accumulation model, is the
Palmgren-Miner hypothesis (Miner, 1945). Based on the assumptions of this theory, the durabi-
lity of the structure does not depend on the order of the applied load with a variable amplitude.
As shown in the literature, the impact of load history can be so significant that we overestimate
or underestimate the fatigue life by up to 4 times (Bijak-Żochowski et al., 1999). Accurate asses-
sment of the load amplitude order is the subject of many hypotheses (Brzeziński and Zbiciak,
2014; Fatemi andYang, 1998;Marco andStarkey, 1954; Xi andSonglin, 2009; Zhao et al., 2014).
In particular, the results of the research conducted by Koba (2000) show that samples of

cement-treated soil subjected to a pre-load exhibit amuch higher fatigue strength. Furthermore,
an increase in the static strengthwas also observed.Based on the above observations, the fatigue
hypothesis has been developed. The hypothesis takes into account the impact of the history on
the fatigue of cement-treated soils (Brzeziński and Zbiciak, 2014). In contrast to the previously
mentioned criteria, the proposed hypothesis assumes that the static strength also varies due
to the cyclic loading. This approach allows for the modeling of strengthening as well as the
weakening of fatigue strength, depending on the applied preload. This hypothesis led to the
onset of the research concerning changes in static strength of cement-treated mixtures used in
road pavement structures (Brzeziński et al., 2017).



928 K. Brzeziński

This paper attempts to explain the phenomenon of static strength increasing by the previous
cyclic loading. The change of the mechanical properties of the loaded sample is undoubtedly
related to changes in the structure andmicrostructure of cementitiousmaterials (e.g. cracks and
micro-cracks). An effective method for detecting and locating the cracks within the sample is
acoustic detection (Landis andShah, 1995;Moczko, 1996, Ohtsu, 1996). Formation andmerging
ofmicro-cracks usually results indegradation of thematerial or evendestructionof the structure.
However, in some cases the increase in density of dislocation may be responsible for an increase
in the fracture toughness (Ritchie, 1999). The knowledge of changes in the internal structure
of the material allows for a better understanding of the investigated phenomena. However, a
quantitative approach requires an appropriate research. In this work, Finite Element Analysis
(FEM) analysis is used in order to explain how cracks formed during cyclic loading can cause
strengthening of the sample.

2. Microstructure changes under cyclic loads

The change in static strength of a material as a result of the cyclic loading can be explained by
transformation of its internal structure. This phenomenon is easier to observe and investigate
in more homogeneous materials (e.g. metal alloys). In the case of composite materials, this is a
much more difficult task. The internal structure is much more heterogeneous, and mechanical
properties depend on characteristics of individual components and the ratio between them.
The polycrystalline grain structure is significantly different from crystal in its ideal form,

which is due to numerous defectswhich significantly affectmechanical properties of thematerial.
The analysis of the influence of dislocations on the mechanical properties of the material is
promising. Especially, since dislocations may be introduced into the material by cyclic pre-
loading. Due to their presence, the stress required to produce plastic deformation is 103-104

times smaller than that of the ideal crystal (Wawszczuk, 2012). The energy supplied during the
cyclic loading may turn into potential energy of misalignment. Furthermore, during long-term
operation, the amount of dislocationmay increase. Initially, it facilitates slipping.However,when
the number of dislocations increases, they block each other, so that their movement (and hence
the slip phenomenon) is impeded. In such a situation, strengthening is observed, in the case
of metal alloys represented as the increase of the yield point (Moćko and Kowalewski, 2014).
In brittle materials, such as cement-treated mixtures, different types of imperfections usually
cause strength reduction. However, in some cases, the strengthening effect may occur. The
strengtheningmechanism at the dislocation level can be explained by the fact that they impede
propagation and joining of micro-cracks by cutting the path of their potential development. In
this way, they can eliminate critical cracks that would otherwise damage the structure.
Themicro-cracks can have a similar influenceon thematerial in theprocess of static strength

changing. The accumulation ofmicro-cracks in the structure is usually causingweakening of the
material because over time it merges into larger cracks and eventually results in destruction of
the structure. On the other hand, differently orientedmicro-cracks canmake propagation of the
dominant crack difficult, thus increasing the static or fatigue strength of the structure.
As mentioned above, observing such a phenomenon in the case of cement-bond materials is

muchmore difficult due to heterogeneity of the structure. In addition, the critical element is the
cement matrix which is a small part of the material volume. Therefore, the effects observed are
much less marked. In spite of this, detection of dislocation motion in cement matrix materials
is possible. Studies conducted byKyriazopoulos et al. (2011) showed the possibility of detecting
dislocation by recording the flow of small electrical charges generated in the material. Another
method of observation of changes occurring inside the materials during load application is the
method of acoustic detection (Landis and Shah, 1995; Moczko, 1996, Ohtsu, 1996). The energy



XFEM simulation of the influence of cracking introduced by pre-loading... 929

released during micro-cracking or dislocation motion causes a sound wave propagating in the
sample. The analysis of the signal recorded from several piezoelectric transducers located on
the sample not only allows one to observe the moment of the micro-crack but also allows its
localization as well as estimation of the size and direction of the slip plane.
The cracks caused by the cyclic loading may be involved in the mechanisms responsible for

improving toughness of the material (Ritchie, 1999). Development of dominant cracks can be
prevented if the crack tip encounters several interconnected micro-cracks. Then several crack
tips are formed, which reduces the intensity factor of stresses. According to the linear fracture
mechanics, based onGriffith’s theory (Griffith, 1920; Prokopski, 2008), crack enlargement requ-
ires the work of forming a free surface of the crack within the body. Therefore, development of
several cracks at the same time can be difficult because it requires more energy.

3. Laboratory tests

3.1. Scope of the research

Themain part of the studies consists of the comparison of static strength of not pre-loaded
samples and samples subjected to cyclic pre-loading. In these studies, a static change has been
observed (even strengthening of the material). The preliminary results are presented in (Brze-
ziński et al., 2017). However, in this paper, only supplementary laboratory tests related to FEM
analysis will be presented. They have been conducted in order to determine the fracture energy
(critical energy release rateGIC) of the tested material.

3.2. Materials and preparation of samples

Quartz sand of the standard grain size (PN-EN 196-1:2006) has been used to prepare the
mix. The use of this type of aggregate was intended to ensure repeatability of tests and to
increase accuracy of the analysis.
The cement used for the preparation was CEM I 42.5 RPortland cement. Themass content

of the cement relative to sand was 10%. The cement content in cement-bond soils should be
between 4% and 10% (Rafalski, 2007), so the amount of cement used could be considered as
typical. In addition, tap water was used in the amount needed to achieve optimum moisture
(determined by the Proctor test).

Fig. 1. Sample prepared for theWedge Splitting Test

Theprimary objective of theWedge SplittingTest (WST) is to determine the critical energy
release rate GIC (Brühwiler and Wittmann, 1990; Löfgren et al., 2004). This parameter deter-



930 K. Brzeziński

mines the material susceptibility to cracking and corresponds to the energy dissipated during
cracking of thematerial. TheWST is commonlyused for brittlematerials such as concrete (Sitek
et al., 2014). The test stand is shown in Fig. 1.
A sample of 20×20cm×10cm rectangular shape was prepared with a proper cut allowing

installation of a force application mechanism and a notch indicating the origin of the crack.
Samples were made of the samemixture as the beams subjected to cyclic tests.
The wedge with an angle of 2ξ = 10◦ was pushed with a constant velocity of 2mm/s. The

load was passed to the specimen via rollers located on the sides of thewedge. Every sample was
subjected to a single load until it was destroyed. The vertical force F and the Crack Opening
Displacement (COD) were measured during the test.

3.3. Laboratory test results

The test result is usually illustrated as a graph of the dependence of the horizontal force on
theCOD.The area under the graph defines its work to open the cracksWF . The critical energy
release rate is given by

GIC =
WF

A
(3.1)

whereWF is thework of the force forming the crack [J],A – the area of the created crack surface
[m2].
The test was performed on 3 samples and then the results were averaged. In the FEM

simulation, the average value of critical energy release rateGIC =50.12J/m2 was used.

4. Potential strengthening effect simulation by FEM

The analysis of microstructural mechanisms capable of explaining the changes in strength, in
particular the strengthening effect, allows for amore comprehensive analysis and understanding
of the phenomena studied. It can improve the effectiveness of the conducted research and also
help one to draw additional conclusions useful in determining the future research goals.
The FEM was used in the analysis. The influence of additional cracks in the sample on

the propagation delay of the dominant crack was investigated, which might be interpreted as
an increase of the static strength. It should be emphasized that the analysis performed can be
considered both as simulation of the strengthening or weakening mechanism of the material. In
this case, the interpretation of what is strengthening and what is weakening depends on what
cracks arrangement is adopted as the reference point.Nevertheless, in order tomake the analysis
more readable, the reference point is the simplest cracks arrangement (single crack in themiddle
of the sample), which is also most unfavorable considering the strength of the sample.

4.1. Theoretical assumptions

The FEM simulation has been performed with the Abaqus R○ program. Advanced modeling
techniques such as eXtended Finite Element Method (XFEM) or Virtual Crack Closure Tech-
nique (VCCT) were used for this purpose. Therefore, before describing the model, the basic
theoretical assumptions of the mentioned techniques and their implementation in the program
will be presented.
The first concept of the XFEM application, a method for simulating cracks development

independently from the model grid, was presented by Belytschko and Black (1999). In the
following years, thismethodwas developed and used in numerous applications (Remmers et al.,
2008; Song et al., 2006) and also implemented in commercial software (ABAQUS, 2013). The



XFEM simulation of the influence of cracking introduced by pre-loading... 931

idea of the method consists of enriching typical finite elements with additional shape functions
that allow fordiscontinuitymodelingandavoiding the singularity problemat the tip of the crack.
The analysis is based on the linear fracture mechanics theory, so development of the crack is
modeled in such a way that it propagates immediately throughout the finite element. For this
reason, the above mentioned problem of singularity does not occur. The detailed description of
this technique is given in (Dolbow and Belytschko, 1999).
Description of cohesive properties of the crack is the same as in the classical approach.

It is based on the linear fracture theory proposed by Griffith (1920). In the VCCT criterion,
deformation energy released during crack propagation by δa is calculated by assuming that it is
the same as the work required to close the crack.
The crack propagation occurs when the energy released in the VCCT procedure is greater

than the critical value, which can be written as follows

f =
GI

GIC
> 1.0 (4.1)

where f is the condition of crack propagation, GIC – critical energy release rate, GI – energy
release rate (the ratio of the released energy to the additional surface resulting from crack
propagation).
When the XFEM elements are used, the crack propagation direction is independent of the

mesh. Therefore, the criterion determining the crack propagation direction must be assumed.
In this analysis, it is assumed that the crack will propagate perpendicular to the direction of
the maximum tensile stress. These criteria require knowledge of the critical value of the energy
release rateGIC. It has been determined experimentally in the wedge splitting test.
The XFEM modeling technique is often referred to be as mesh independent. It should be

understood that the course of the crack is not determinedby the shapeof themesh.However, the
solution is sensitive to size of the elements. First of all, this is obviously related to the accuracy
of reflecting the state of stress in the structure. Less obvious is the impact resulting from the
linear fracturemechanics. In theVCCT technique, which is used in theXFEM, the propagation
of the crack occurs abruptly (each time in length determined by the mesh density). Thus, the
size of the element corresponds with the characteristic crack length a, which is a component
of the crack propagation criterion (Griffith, 1920). Therefore, the specified mesh density is not
versatile. However, in the author’s opinion, a mesh with a given characteristic density can be
used to compare similar models with the samematerial.

4.2. Geometry and parameters of the FEM model

After conducting tests necessary to obtain characteristic material parameters of the mixtu-
re, a proper FEM model has been prepared. A 3-point bending strength test was performed.
Geometry of the sample and the static scheme corresponding to the actual test were used. The
load was applied in a kinematic manner. In the first simulation, the crack was applied at the
bottom of the sample, at themost strained cross section. The load velocity was adjusted so that
the result of the strength test simulation was similar to the results obtained in real test. Then,
the effect of additional cracks occurrence on the resulting strength was analyzed.
In order to speed up the calculation, a plain stress condition was assumed. The model

consisted of a small beam that was freely based on rounded supports, loaded from the top
through a rounded loading nose. Geometrical invariability of the static system was ensured by
boundary conditions and by defining contact between the elements (Fig. 2).
The loading nose moved in the vertical direction at a speed of 0.2mm/s. The value of the

loading force was calculated on the basis of vertical stresses in the supports. At each stage of
simulation, at least one pre-fracture crack was placed at the bottom of the sample, in the most
strained section. The enriched elements were used in a limited area because of practical reasons.



932 K. Brzeziński

Fig. 2. The static scheme of the FEMmodel

Various crack configurations were adopted on the basis of the crack development presented in
the paper (Landis and Shah, 1995). However, in themodel crack configurations were simplified.
The linear elastic material model was applied in order to define constitutive properties of all

elements. The applied values of material parameters are summarized in Table 1.

Table 1.Applied parameters of the materials

Element Young’s modulus [GPa] Poisson’s const. [–] GIC [J/m2]

Supports and nose 210.00 0.30 –
Beam 3.50 0.25 50.12

4.3. Simulation results

The procedure of the analysis has been as follows. It was assumed that not pre-loaded
elements were not cracked, so in the simulation only the initial crack was placed in thematerial.
Therefore, in the strength test, the development of the dominant crack is not disturbed.On the
other hand, in the case of pre-loaded samples, cracks and micro-cracks appeared, which might
affect the development of the dominant crack. In many cases, the appearance of additional
cracks might reduce the observed strength of the sample because destruction progressed easier
bymerging existing discontinuities. Sometimes, however, the crack propagation of the dominant
crackmight be difficult due to proximity to other cracks. Accordingly, the simulation of a single-
-cracked specimen strength testwas treated as a benchmark. In the subsequent stages of analysis
another crack was placed at the bottom of the sample at a distance of 4.0, 5.0 or 6.0mm. In
addition, the results of triple fracture (two additional cracks symmetrically located at a distance
of 6.0mm from the base fracture) were also compared. Figure 3 shows exemplary results of
the 3-point bending tests and the simulation result of this test in the basic variant (single
crack).
From Fig. 3 it can be concluded that the first step of the simulation - the strength test

modeling, was performed correctly. Both the slope of the pre- and post-destruction graphs,
as well as the maximum value of the stress, are very similar. Further simulations were made
by adding one or two cracks. The results are shown in Fig. 4. One can see that the strength
increased in all cases. The maximum increase in strength was approximately 20% (in the case
of an additional crack located at a distance of 4.0mm).



XFEM simulation of the influence of cracking introduced by pre-loading... 933

Fig. 3. Comparison of exemplary results of laboratory strength tests and FEM simulation of this test
(3-point bending)

Fig. 4. Comparison of FEM simulation results of a strength test obtained by applying single
pre-fracture and additional cracks at a distance of 4.0, 5.0 and 6.0mm

Amore detailed analysis of the stress state in subsequent load phases (illustrated in Figs. 5
and 6) may allow for a better understanding of the observed phenomenon. In the case of the
single crack, shown in Fig. 5, general conclusions regarding the effect of crack evolution on the
stress distribution in the sample can be drawn.The appearance of the initial crack at a low level
of the stress (about 50%) results in reduction of the effective area of the cross-section. Certain
stresses are carried between the crack edges until the critical crackmouth opening. It reflects the
cohesive zone of interaction at the crack tip (Elfgren and Shah, 1991; Hillerborg, 1989). As the
load increases, the gap widens until it stops carrying the stresses at all. It should be noted that
the recorded force does not decrease immediately, at the time of initiation of cracks propagation.
The crack is 6mm long at the moment of the maximum stress.
The next figure (Fig. 6) shows a simulation with additional cracks located symmetrically

at a distance of 6.0mm. The introduced cracks lead to redistribution of the internal forces in
the loaded sample. The additional crack forms a peculiar “cover” of the dominant crack. This
phenomenon reduces tensile stresses near the dominant crack and slows its propagation. Finally,
this led to an increase in themeasured strengthbyabout 14%.Thedevelopment of thedominant
crack is stopped and destruction is determined by propagation of one of the adjacent cracks. In
all analyzed cases, the beneficial effect of the occurrence of additional cracks is observed. This is
due to the fact that the reference is the single crack located in themost strained cross section. It



934 K. Brzeziński

Fig. 5. Development of cracks in subsequent load phases – single cracked sample
(displacement scaled 3 times)

should be noted that the maximum strengthening obtained in the simulation is approximately
2 times greater than that presented in (Brzeziński et al., 2017). In fact, some individual samples
tested in the laboratory showed evenmore than 15% increase in strength. The results presented
in (Brzeziński et al., 2017) have beenalready averaged tominimize the impact of randomfactors.
In order to compare stress rearrangement in different crack configurations, it is summarized

in Fig. 7.
At the level of analysis it should be emphasized that the observed change of strength is a

characteristic of the structure rather than the material. Nevertheless, in the phenomenological
point of view, it manifests as strengthening of thematerial. In the case of both approaches, it is
reasonable to assume that the observed increase in static strength can result into longer fatigue
life.



XFEM simulation of the influence of cracking introduced by pre-loading... 935

Fig. 6. Development of cracks in subsequent load phases – two additional cracks at a distance of 6.0mm
(displacement scaled 3 times)

5. Conclusions

In this paper, FEM simulation of the potential strengthening mechanism was presented. The
analysis is strongly related to the theoretical considerations concerning structural changes of
the material. It has been based on themodeling of the 3-point bending test performed in order
to determine the strength of the material. The discontinuity modeling XFEM technique was
used. The crack growthwas controlled by the energy criterion used in the linear fracture theory.
The implementation of the criterion in the Abaqus R○ program was based on the Virtual Crack
Closure Technic (VCCT). In order to apply this criterion, it was necessary to know the critical
energy release rate of the tested mixture GIC. Therefore, a wedge splitting test (WST) was
performed in order to determine the desired property of the material.



936 K. Brzeziński

Fig. 7. Stress rearrangement near the crack tip at 100% level of effort in different crack configurations
(displacement scaled 3 times)

During the simulation, it was observed that the appearance of additional cracks in the
material before the start of the strength test (for example as a result of pre-loading) may lead
to an increase in the maximum force required to destroy the sample. Analysis of the tensile
stress distribution in the horizontal direction (approximately perpendicular to the crack) allows
for a qualitative explanation of the observed strength increase. The additional crack, located
outside of themost strained cross section, forms a “cover” of the dominant crack. The reduction
of the tensile stress near the dominant crack slows its propagation. It should be emphasized
that this is a characteristic of the structure rather than of the material itself. Nevertheless,
in phenomenological terms, the result of the simulation, interpreted as a result of the 3-point
bending test, indicates strengthening of the material.

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Manuscript received November 27, 2017; accepted for print February 1, 2018