Fracture Behavior and Energy Analysis of 3D Concrete Mesostructure under Uniaxial Compression

Abstract

In order to investigate the fracture behavior of concrete mesostructure and reveal the inner failure mechanisms which are hard to obtain from experiments, we develop a 3D numerical model based on the Voronoi tessellation and cohesive elements. Specifically, the Voronoi tessellation is used to generate the aggregates, and the cohesive elements are applied to the interface transition zone (ITZ) and the potential fracture surfaces in the cement matrix. Meanwhile, the mechanical behavior of the fracture surfaces is described by a modified constitutive which considers the slips and friction between fracture surfaces. Through comparing with the experiments, the simulated results show that our model can accurately characterize the fracture pattern, fracture propagation path, and mechanical behaviors of concrete. In addition, we found that the friction on the loading surfaces has a significant effect on the fracture pattern and the strength of concrete. The specimens with low-friction loading surfaces are crushed into separate fragments whereas those with high-friction loading surfaces still remain relatively complete. Also, the strength of concrete decreases with the increase of the specimen height in the high friction-loading surfaces condition. Further, the energy analysis was applied to estimate the restraint impact of loading surfaces restraint on the compressive strength of concrete. It shows that the proportion of the friction work increases with the increase of the restraint degree of loading surfaces, which finally causes a higher compressive strength. Generally, based on the proposed model, we can characterize the complicated fracture behavior of concrete mesostructure, and estimate the inner fracture mode through extracting and analyzing the energies inside the cohesive elements.

Full text

Materials 2019, 12, 1929; doi:10.3390/ma12121929 www.mdpi.com/journal/materials
Article
Fracture Behavior and Energy Analysis of 3D
Concrete Mesostructure under Uniaxial Compression
Yiqun Huang
1
, Shaowei Hu
2,
*, Zi Gu
2
and Yueyang Sun
3
1
College of Mechanics and Materials, Hohai University, Nanjing 210098, China;
yiqunhuang@hhu.edu.cn
2
School of civil Engineering, Chongqing University, Chongqing400044, China;
albert_gu@foxmail.com
3
College of Water Conservancy and Hydropower Engineering, Hohai University,
Nanjing 210098, China; syyang0118@126.com
* Correspondence: hushaowei@cqu.edu.cn
Received: 15 May 2019; Accepted: 13 June 2019; Published: 14 June 2019
Abstract: In order to investigate the fracture behavior of concrete mesostructure and reveal the inner
failure mechanisms which are hard to obtain from experiments, we develop a 3D numerical model
based on the Voronoi tessellation and cohesive elements. Specifically, the Voronoi tessellation is
used to generate the aggregates, and the cohesive elements are applied to the interface transition
zone (ITZ) and the potential fracture surfaces in the cement matrix. Meanwhile, the mechanical
behavior of the fracture surfaces is described by a modified constitutive which considers the slips
and friction between fracture surfaces. Through comparing with the experiments, the simulated
results show that our model can accurately characterize the fracture pattern, fracture propagation
path, and mechanical behaviors of concrete. In addition, we found that the friction on the loading
surfaces has a significant effect on the fracture pattern and the strength of concrete. The specimens
with low-friction loading surfaces are crushed into separate fragments whereas those with high-
friction loading surfaces still remain relatively complete. Also, the strength of concrete decreases
with the increase of the specimen height in the high friction-loading surfaces condition. Further, the
energy analysis was applied to estimate the restraint impact of loading surfaces restraint on the
compressive strength of concrete. It shows that the proportion of the friction work increases with
the increase of the restraint degree of loading surfaces, which finally causes a higher compressive
strength. Generally, based on the proposed model, we can characterize the complicated fracture
behavior of concrete mesostructure, and estimate the inner fracture mode through extracting and
analyzing the energies inside the cohesive elements.
Keywords: meso; concrete; voronoi tessellation; cohesive element; compression; fracture
1. Introduction
Concrete is a composite material consisting of aggregates, cement matrix, interface transition
zone (ITZ) and some other content such as pores. These mesoscopic components determine the macro
mechanical behaviour of concrete. In order to research the effect of these mesoscopic components on
the mechanical behaviors, researchers have presented many modelling and simulation methods.
In terms of the modelling, there are two typical methods to rebuild the concrete mesostructure.
One is to generate the aggregates and pores in a random way such as randomly throwing
circles/spheres [1–4] or polygon/polyhedrons [5–8], and the other is to directly restructure the internal
components of concrete based on the tomography which are obtained from computed tomography
(CT) scanning [9-11]. In terms of the simulation methods, previous works have proposed many

Materials 2019, 12, 1929 2 of 24
effective numerical methods to research the rebuilt concrete including the traditional finite element
method (FEM) with the continuing solid elements [12–16], the lattice model with the lattice elements
[17–21], the homogeneous model with the homogenized elements for the heterogeneity of concrete
[22–26], and the improved FEM method with cohesive elements [27–35]. Besides the aforementioned
FEM methods, the discrete element method (DEM) with separated elements such as the particle flow
method [36–38] and rigid-body-spring method (RBSM) [39–41] is also applied in simulation of
concrete.
In recent years, more researchers have adopted the model with cohesive elements to simulate
the fracture process of concrete [31–35]. As a transitional element inserted into the edges of solid
elements [42], the cohesive element can set its geometric thickness as any non-negative value
including zero, and it can only transfer the normal stress and shear stress. Due to the aforementioned
characteristics, the cohesive element is very suitable to characterize the mechanical behavior of the
interfaces with extremely small thickness such as ITZ, and it is also applicable for the representation
of the potential/existing fracture surfaces. In addition, previous works indicated that the model with
cohesive elements is size independent in a reasonable range of mesh size [33–35], which means the
extremely fine mesh is unnecessary. Hence, the cohesive element can save significant calculation time
and space. However, existing research about the cohesive elements mainly focused on the tensile
fracture behavior of concrete [30–35], and only a few studies were about the fracture behavior under
compression [28]. In these studies, the friction effect is rarely considered. Some only considered the
tensile and shear damage without the consideration of friction in Refs. [33–35], and the other only
considered the friction as an enhancement of shear stress in Refs. [28–30]. However, the fracture
behavior of concrete under compression is pretty complicated. It contains almost all fracture modes
which may occur inside the concrete including the tensile-shear fracture, the compressive-shear
fracture, the separation and close of cracks, and the friction and slips between fracture surfaces. For
this reason, a more appropriate constitutive of cohesive elements should be established to
characterize the complicated fracture behavior of concrete accurately.
The fracture behaviour of concrete under uniaxial compression has been studied by several
methods, which includes the traditional FEM method such as the homogeneous model [22] and the
lattice model [20,21], and the DEM method such as rigid-body-spring methods [40,41] and particle
flow [37,38]. These researches have provided some valuable results and conclusions such as the effect
of aggregates content and loading condition on the mechanical behavior of concrete. Based on these
researches and the cohesive element model mentioned above, we can establish a more effective
numerical model to investigate and reveal the inner failure mechanisms of concrete under various
loading condition including compression.
In this paper, we present a 3D numerical model based on 3D Voronoi tessellation and zero
thickness cohesive elements. Based on the modified constitutive that takes into account the effect of
the norm-tangential coupling and the friction between cracks, the proposed model can characterize
the complicated fracture behavior of concrete mesostructure. Moreover, this model can also be used
to estimate the inner fracture mode through extracting energies inside the cohesive elements. The
fracture process of concrete specimens under compression is simulated through the ABAQUS [42]
with subroutine VUMAT that the modified constitutive is implemented in. Then, we investigate the
effect of the loading surfaces friction and the specimen height on the fracture behavior of concrete
under uniaxial compression. Finally, the restraint influence of loading surfaces is estimated by the
energy analysis.
2. Numerical Model
2.1. Generation of Aggregates Based on 3D Voronoi Tessellation.
Based on Voronoi polyhedron tessellation [43–45], a 3D aggregates generation method is
presented in this section. The Voronoi tessellation can generate seamlessly connected cells according
to the seeds that were placed randomly in advance within a given region. Normally, one seed can
generate one corresponding polyhedron. Thus, we can generate aggregates by reasonably throwing

Materials 2019, 12, 1929 3 of 24
seeds and reducing the size of generated polyhedrons. In this paper, we obtain seeds by randomly
generating the spheres without overlapping. Also, it is noted that the generated polyhedrons
normally have too many vertices and surfaces, which could cause great difficulties in modeling. For
this reason, simplifying the vertices’ number of polyhedrons is necessary. The method to reduce the
vertices number of polyhedrons is established on finding the intersection points between the pre-set
vectors and the surfaces of polyhedrons, and the new surfaces of polyhedrons can be obtained by
determining the convex hull of the reduced vertices, as shown in Figure 1. The pre-set vectors starting
from the center of polyhedron are generated by rotating the basic vectors randomly. In this paper,
the basic vectors are {[1,0,0], [0,1,0], [1,1,0], [−1,1,0], [0,0,1], [1,1,1], [−1,1,1], [−1−1,1], [1,−1,1], [0,1,1],
[1,0,1]}, which means 22 intersection points can be obtained by these 11 vectors.
(a)
(b)
(c)
Figure 1. The simplified process of the polyhedron: (a) determine the pre-set vectors
starting from the center of the polyhedron, (b) find the intersection points between the
vectors and surfaces, (c) obtain the new surfaces of the polyhedron by determining the
convex hull of intersection points.
On the basis of the simplified polyhedrons, the whole process (Figure 2) to generate the
aggregates can be written as follows:
Step1: randomly throw the spheres in a specified region (the spheres should have no overlap
and satisfy the size distribution of aggregates), and then take the centers of the spheres as the seeds
to generate Voronoi polyhedrons.
Step2: generate the Voronoi polyhedrons based on the generated seeds.
Step3: simplify the polyhedrons by reducing the vertices and determine the convex hull of the
reduced vertices.
Step4: zoom the simplified polyhedrons through the scaling ratio of the sphere volume to the
polyhedron one which shares the same seed. The scaling ratio is normally less than 1 (in rare cases
the ratio may greater than 1, and the ratio would be set to 1). The ratio can be calculated by:
33
3
if 1
,
1 if 1
ss
dd
s
d
VV
c
VV
ratio
V
V
⋅≤


=
>


(1)
where
s
V is the volume of the sphere,
d
V is the volume of the simplified polyhedron, and
c
is an
empirical correction factor whose range is 1 to 1.1.
(a)
(b) (c) (d)

Materials 2019, 12, 1929 4 of 24
Figure 2. The generating process of the aggregates in concrete: (a) throw the spheres
randomly in a specified region and extract the spheres centers as the seeds, (b) generate the
Voronoi polyhedrons based on extracted seeds, (c) simplify the generated polyhedrons, (d)
obtain the aggregates by zooming simplified polyhedrons.
2.2. Application of Cohesive Elements
After the generation of aggregates in concrete, a two-phase mesostructure of concrete can be
generated easily by merging the cement matrix and aggregates as shown in Figure 3. Moreover, ITZ
is an important component of the concrete mesostructure, it should be also considered in the model.
There are two problems for traditional continuing solid elements. First, the solid element is hard to
represent the ITZ whose thickness is far less than the size of aggregates. Secondly, the solid element
is hard to characterize the behavior of the fracture surfaces in concrete. In order to solve these
problems, the cohesive elements are inserted into the interfaces of the solid elements when we build
the model of concrete mesostructure. The unique advantages of the cohesive element are that it only
transfers the normal stress and shear stress and its geometry thickness is independent of its
calculation thickness. These characteristics make the cohesive element can be used for the
characterization of ITZ and potential fracture surfaces in concrete. In our model, the geometry
thickness of the cohesive elements is set to 0 and the calculation one is set to 1, which means the
behavior of ITZ and potential fracture surfaces in concrete are characterized by the stress–
displacement relationship.
(a) (b) (c)
Figure 3. Generation of the two-phase mesostructure of concrete: (a) aggregates, (b) cement
matrix, (c) concrete merged by aggregates and cement matrix.
Based on the method in Ref. [35], the zero thickness cohesive elements are inserted into the
interfaces of the solid elements in the initial mesh of the two-phase structure, and the process of
inserting the cohesive elements is shown in Figure 4. Three types of zero cohesive elements exist in
the model: (1) CE_ITZ, the cohesive element inserted into the interface of the aggregate and cement
matrix; (2) CE_AGG, the cohesive element inserted into the interface inside the aggregate; (3)
CE_CEM, the cohesive element inserted into the interface inside the cement matrix. Specially, because
the aggregate strength is much higher than the mortar and ITZ, we assume that the aggregate will
not be damaged. Thus, CE_AGG is not considered in this paper.
(a)
(b)
(c)
Figure 4. Insertion of the cohesive elements into the interfaces of the solid elements: (a)
initial mesh of the two-phase structure, (b) the mesh with zero thickness cohesive elements,
(c) element types in the mesh.

Materials 2019, 12, 1929 5 of 24
2.3. Modified Constitutive Applied in Cohesive Elements
The none-linear mechanical behavior is defined in the cohesive elements according to the
previous work [27–35]. Thus, a reasonable and effective constitutive relationship to the cohesive
elements is the basis of obtaining accurate numerical results. For the tensile condition, the constitutive
can be obtained easily by determining the relation between the normal stress and normal
displacement without the consideration of the shear stresses and friction stress. However, the shear
stresses and friction stress are important in compression condition. For this reason, a reasonable
constitutive which can characterize the normal stress, shear stresses and friction stress is needed for
the simulation of the concrete under compression condition.
A modified constitutive with the consideration of the friction stress is established based on the
bilinear damage model. The constitutive is given by the relation of stresses and displacements. In the
case of 3D model, one normal stress ( n
t) and two orthogonal shear stresses (
s
t,t
t) should be
determined in the constitutive.
In the stresses and displacements relation, two phases exist: (1) elastic phase, (2) damage phase.
In elastic phase, the relation between the stresses and displacements can be expressed as:
___
, , ,
nnn sss t st
tk tk tk
δδδ
=⋅ =⋅ =⋅ (2)
where
_
n
k is the initial stiffness of the normal direction,
_
s
k is the initial stiffness of the tangential
direction, n
δ
is the normal displacement,
s
δ
is the tangential displacement in first direction, and
t
δ
is the tangential displacement in second direction.
The element initiates the damage phase (the potential fracture surface begins to develop) once:
222
000
+ 1 ,
nst
nss
ttt
ttt


+≥



 (3)
where 0n
t is the tensile strength, 0
s
t is the shear strength, and
x
is the Macaulay bracket, which
can be expressed as:
if 0
0 if 0
xx
xx
>

=≤
 (4)
The stress-displacement relationship in a single direction is shown in Figure 5. The area under
the curve represents the normal fracture energy n
G (tangential fracture energy
s
G). The fracture
energies can be calculated by:
00
,
22
nf n sf s
ns
tt
GG
δδ
==, (5)
where nf
δ
is the failure displacement in normal direction, and
s
f
δ
is the failure displacement in
tangential direction.

Materials 2019, 12, 1929 6 of 24
Figure 5. Stress–displacement relationship in a single direction.
The normal (tangential) initial damage displacement
0n
δ
(
0s
δ
) in Figure 5 can be expressed
by:
00
00
,
ns
ns
ns
tt
kk
δδ
==
(6)
The initial damage and failure surfaces controlled by normal and tangential displacements are
established (Figure 6), which can be calculated by:
222
000
222
+ 1 (initial damage surface) ,
1 (failure surface)
nst
nss
nst
nf sf sf
δδδ
δδδ
δδδ
δδδ


+=




 
++=
 
 
 
(7)
Figure 6. Initial damage and failure surfaces, and the relative displacement points: point B
as the historic maximum displacement point, point A as the intersection point of vector OB
and the initial damage surface, point C as the intersection point of vector OB and the failure
surface.

Materials 2019, 12, 1929 7 of 24
As shown in Figure 6, the point B ( ,nmax
δ
, ,
s
max
δ
, t,max
δ
) is the historic maximum displacement
point, and 222
,,
++
n max s max t,max
δδδ
is the historic maximum displacement during the loading
process. ,nmax
δ
,,
s
max
δ
, t,max
δ
are the normal and two orthogonal tangential components of the
historic maximum displacement respectively. The point A ( ,0n
δ
, ,0
s
δ
, 0t,
δ
) is the relative initial
damage displacement point, which can be obtained by finding the intersection point of vector OB
and the initial damage surface. The point C ( ,nf
δ
,
s
,f
δ
, t, f
δ
) is the relative failure displacement
point, and it is the intersection point of vector OB and the failure surface. The coordinate value of the
point A can be calculated through the geometry relation in Figure 6:
00
,0 2
22 2 2
00
00
,0 2
22 2 2
00
00
2
22 2 2
00
,
(+)
,
(+)
(+)
n,max n s
n
n s,max t,m ax n,max s
s,max n s
s
n s,max t,max n,max s
t,max n s
t,0
n s,max t,max n,m ax s
δδδ
δ
δδ δ δ δ
δδδ
δ
δδ δ δ δ
δδδ
δ
δδ δ δ δ
=
+
=
+
=
+
(8)
The coordinate value of the point C can also be similarly calculated by:
,2
22 2 2
2
22 2 2
2
22 2 2
,
(+)
,
(+)
(+)
n,max nf sf
nf
nf s,max t,max n,max sf
s,max nf sf
s, f
nf s,max t,max n,max sf
t,max nf sf
t, f
nf s,max t,max n,max sf
δδδ
δ
δδ δ δ δ
δδδ
δ
δδ δ δ δ
δδδ
δ
δδ δ δ δ
=
+
=
+
=
+
(9)
So the relative displacements can be expressed as:
222
,,
222
0,0,00
222
=+ ,
= +
=+
max n max s max t,max
nst,
fn,fs,ft,f
δδ δ δ
δδ δδ
δδ δδ
+
+
+
,
,
(10)
where max
δ
is the relative historic maximum displacement, 0
δ
is the relative initial damage
displacement, and
f
δ
is the relative failure displacement.
According to the bilinear damage model in Figure 5, the damage factor
D
can be rewritten as
follows:
0
0
()
()
max f
f
max
D
δδδ
δδδ
−
=
− (11)

Materials 2019, 12, 1929 8 of 24
During the damage phase in the loading process, the friction stresses would occur when the
cohesive element (fracture surface) is under compressive conditions. In the damaged area, the friction
stress
f
T can be calculated by:
if
if
def
max max s def
def
f
sdef max s def
k
T
kk
δ
ττδ
δ
δτδ
⋅≤⋅


=
⋅>⋅


，
(12)
where max
τ
is the maximum static friction stress which controls the calculation of
f
T. When max
τ
cannot prevent the relative slip of crack,
f
T should be calculated by the first formula in Equation
12. When the relative slip of the crack is prevented,
f
T can be calculated by the second formula.
According to the friction law, max
τ
can be calculated by:
,
max n n
fk
τδ
=⋅ ⋅− (13)
where
f
is the internal friction coefficient.
The tangential deformation displacement def
δ
in Equation 12 can be calculated through two
orthogonal tangential displacements (
s
δ
,t
δ
) and two orthogonal slip displacements ( ,
s
slip
δ
, t,slip
δ
):
22
,,
()+()
def s s slip t t slip
δδδ δδ
=− − (14)
After the calculation of the friction stress, the orthogonal slip displacements should be updated
when the slip of the crack occurs:
,
,max
22
,
,max
22
,
( ) ,
()+()
( ) ,
()+()
old
ssslip
new max
s slip s sdef
old old
ss s slip t t,slip
old
tt,slip
new max
tslip t sdef
old old
ss s slip t t,slip
k
k
k
k
δδ
τ
δδ τ δ
δδ δδ
δδ
τ
δδ τ δ
δδ δδ
−
=− ≤
−−
−
=− ≤
−−
(15)
where ,
new
s
slip
δ
, ,
new
tslip
δ
is the updated slip displacements, and old
s
,slip
δ
, ,
old
t slip
δ
is the slip
displacements before the update.
According to the direction of the tangential deformation, the frictional stress components ( ,
f
s
T
, ,
f
t
T) in two orthogonal direction can be rewritten as follows:
,
,,
22 22
,,
= , =
()+() ()+()
old old
s s slip t t,slip
fs f ft f
old old old old
s s slip t t,slip s s slip t t,slip
TT TT
δδ δδ
δδ δδ δδ δδ
−−
−− −−
(16)
Finally, the stresses in the damage phase can be calculated by:

Materials 2019, 12, 1929 9 of 24
,
,
(1 ) if >0 ,
if 0
(1 ) ,
(1 )
nn n
n
nn n
sssfs
tstft
Dk
t
k
tDkDT
tDkDT
δδ
δδ
δ
δ
−

=≤


=− +⋅
=− +⋅
(17)
2.4. Calculation of Three Different Types of Energy
To analyze the fracture characteristic of the concrete, we extract the energies inside the cohesive
elements during the loading process. Because three different kinds of stress (normal stress, tangential
stress, and friction stress) exist in the cohesive element, three corresponding energies should be
extracted respectively. These three energies are: (1) the normal stress work n
Ed caused by the
normal stress, (2) the shear stress work
s
Ed caused by the pure shear stress, (3) the friction stress
work
f
Ed caused by the friction stress. These energies can be calculated by:
0
,,
00
,,
00
( ) ,
(2 ( - ) ) + (2 ( - ) ) ,
(2 ) + (2 ) ,
n
st
st
nn
A
ssfs tft
AA
ffs ft
AA
Ed t d dA
Ed t D T d dA t D T d dA
Ed D T d dA D T d dA
δ
δδ
δδ
δ
δδ
δδ
=
=⋅ ⋅
=⋅ ⋅
 
   
   
(18)
where
A
is the area of the cohesive element.
The energies of the cohesive elements in whole concrete structure can be calculated through:
= ,
= ,
= ,
nn
cohesive elements
ss
cohesive elements
ff
cohesive elements
E
Ed
E
Ed
E
Ed



(19)
where n
E (
s
E,
f
E) is the normal (shear, friction) stress work of the cohesive elements in whole
concrete structure.
3. Numerical Examples and Results
3.1. Input Data of the Finite Element Model
Square concrete specimens of 10 cm side are chosen for simulation. The density of the aggregates
is assumed as 2800 kg/m3, and the gradation of aggregate size distribution based on Ref. [46] is listed
in Table 1. To save the computing resources, only coarse aggregates larger than 2 mm are considered
in this study. The loading scheme of the concrete specimen is shown in Figure 7. The specimen is
placed between two rigid plates with the lower plate being constrained and the other plate being
applied by a vertical displacement loading. The interaction between the concrete and the rigid plate
includes normal contact and tangential friction. According to the friction coefficient of the interface
between the concrete and the rigid plate, the loading condition can be divided into low-friction (LF)
condition with the friction coefficient = 0.001 and high-friction (HF) condition with the friction
coefficient = 0.2.

Materials 2019, 12, 1929 10 of 24
Table 1. The size distribution of aggregates in experiment [Error! Reference source not
found.].
Grain Size (mm) Unit Content (kg/m3) Volume Content (%)
8~4 540 19.29
4~2 363 12.91
2~1 272 9.71
1~0.5 272 9.71
0.5~0.25 234 8.35
Figure 7. Loading scheme of the concrete specimen.
Without the consideration of interfaces inside the aggregates, the mesh of the concrete specimen
is shown in Figure 8. The previous works [33–35] have proved that the model with zero thickness
cohesive elements is size-independent in a reasonable range. Thus, the model in this paper is meshed
in a reasonable size (average element size = 5 mm) based on the repeat trial. One typical model has
about 200,000 nodes, 60,000 solid tetrahedron elements and 150,000 zero thickness cohesive elements.
It takes about 10 h to solve one model by parallel computation using 6 Intel I7 8700k CPUs @3.7Hz
(School of Civil Engineering, Chongqing University, Chongqing, China).
(a)
(b)
Figure 8. Mesh of the concrete specimen: (a) solid elements, (b) zero thickness cohesive
elements.
According to the previous works about cohesive element model [28–30], repeated trial
calculations, and the experimental results [46], the material parameters value applied in the solid
elements are E = 70 GPa, v = 0.2, ρ = 2800 kg/m
3
for aggregate; E = 25 GPa, v = 0.2, ρ = 2400 kg/m3 for
cement matrix. The material parameters value applied in the cohesive elements of concrete are
_
n
k
=

Materials 2019, 12, 1929 11 of 24
_
s
k
= 10
6
GPa, ρ = 2400 kg/m3,
0n
t
= 3.0 MPa,
0s
t = 10.5 MPa,
n
G = 40 N∙m,
s
G = 400 N∙m, f = 0.45
for cement matrix;
_
n
k
=
_
s
k
= 10
6
GPa, ρ = 2400 kg/m3,
0n
t
= 1.5 MPa,
0s
t
= 5.25 MPa,
n
G
= 20 N∙m,
s
G
= 200 N∙m, f = 0.45 for ITZ.
The simulations all end when the displacement d = 1 mm, corresponding to the macro strain ε =
0.01. The models are solved in ABAQUS/EXPLICIT solver, with the user subroutine VUMAT [Error!
Reference source not found.] in which the modified constitutive is implemented. According to the
repeat trial calculation, the loading time is set to 1s which can ensure the quasi-static loading
conditions. The cracks are represented by deleting the zero thickness cohesive elements whose
damage coefficient is equal to 1. All specimens with random aggregates all show a similar result. For
this reason, one typical specimen is chosen to be discussed in the next section.
3.2. Effect of Loading Surface Friction Coefficient under Compression
3.2.1. Low-Friction (LF) Condition
The numerical experiments of concrete under uniaxial compression in the LF condition are
carried out. The average stress–strain curve of concrete is shown in Figure 9. The numerical result is
similar to the experimental one [46] in terms of peak stress and the curve shape. To analyze the
fracture patterns of concrete during the loading process in the LF condition, four typical states are
chosen and marked in the stress–strain curve (Figure 9).
Figure 9. Average stress–strain curve of concrete under uniaxial compression in low-
friction (LF) condition.
Figure 10 shows the fracture pattern (deformation is magnified 5 times) and fracture surfaces of
concrete in four chosen states. At the initial state of failure, the cracks start developing by ripping
through the loading surfaces and ITZ. With the increase of displacement loading, the cracks expand
continually, and the opening degree of cracks increase either. It can be seen that the fracture surfaces
always remain substantially parallel to the loading direction during the loading process. Finally, the
specimen is penetrated by cracks and divided into several parts. The final fracture pattern of the
numerical result shows an agreement to the experimental one.

Materials 2019, 12, 1929 12 of 24
Fracture pattern
Fracture surfaces
(a)
Fracture pattern
Fracture surfaces
(b)
Fracture pattern
Fracture surfaces
(c)
Fracture pattern
Fracture surfaces
(d)

Materials 2019, 12, 1929 13 of 24
(e)
Figure 10. Fracture pattern (aggregates in blue and cement matrix in gray) and fracture
surfaces (interface transition zone (ITZ) in blue and cement matrix interfaces in gray) in
different states during the loading process in Figure 9: (a) state A, (b) state B, (c) state C, (d)
state D and (e) the final fracture pattern in experiments [46] (License number:
4607510939744).
More details of the final fracture pattern can be obtained from the loading-direction
displacement nephogram (Figure 11a) of the concrete inner section at the final fracture status. The
concrete can be divided into two parts according to the displacement value. One, namely the moving
part, has a larger displacement value (the absolute value is greater than 1e
−3
) and is shown in light
blue (Figure 11a), and the other, namely the unmoving part, has a small displacement value (the
absolute value is smaller than 1.2e
−4
) and is shown in red. These two parts are separated by cracks
with highly discontinuous. Thus, we can obtain the shape of these two parts by extracting the
elements with the displacement value higher than 1e
−3
and smaller than 1.2e
−4
in the loading direction
(Figure 11b).
(a)
(b)
Figure 11. Two main parts of concrete divided through the distribution of the loading-
direction displacement: (a) the loading-direction displacement nephogram of the concrete
inner section, (b) the moving part and unmoving part.
By amplifying the deform shape of moving part (unmoving part) 20 times as shown in Figure
12a, it can be found that the concrete specimen has been crushed into several separate fragments
(Figure 12b). These fragments are all awl-shaped with one side wide and the other side sharp.
Through the shape of the crushed fragments, it can be inferred that at the initial stage of failure, the
awl-shaped fragments are shaped by the development of cracks and not totally separated yet. With
the proceeding of loading process, these unseparated awl-shaped fragments of moving part wedge
the ones of unmoving part, which finally causes the concrete being totally crushed into separate
fragments.

Materials 2019, 12, 1929 14 of 24
(a)
(b)
Figure 12. Fragments of the failure concrete specimen: (a) the deform shape of the
unmoving part which is magnified 20 times, (b) the separated fragments of specimen.
3.2.2. High-Friction (HF) Condition
The average stress-strain curve in the HF condition is shown in Figure 13. The numerical result
is similar to the experimental one [46] in terms of peak stress and the curve shape. The strength of
concrete is much higher than the one in the LF condition. To analyze the fracture patterns of concrete
during the loading process, four typical states are chosen and marked in the stress-strain curve
(Figure 13).
Figure 13. Stress-strain curve of concrete under uniaxial compression in high-friction (HF)
condition.
The fracture pattern (deformation is magnified 5 times) and the fracture surfaces of concrete in
four different states are shown in Figure 14. At the initial stage of the fracture process, the cracks
mainly occur near the four free surfaces, while the surfaces contacted to the rigid plate remain
complete. With the increase of loading, the cracks gradually developed into the interior of the
concrete specimen, and the concrete fragments near the free surfaces were gradually split from the
concrete specimen. The surfaces contacted to the rigid plate always remained relatively complete
compared to those in the LF condition. The final fracture pattern shows an agreement with the
experimental one.

Materials 2019, 12, 1929 15 of 24
Fracture pattern
Fracture surfaces
(a)
Fracture pattern
Fracture surfaces
(b)
Fracture pattern
Fracture surfaces
(c)
Fracture pattern
Fracture surfaces
(d)

Materials 2019, 12, 1929 16 of 24
(e)
Figure 14. Fracture pattern (aggregate in blue and cement matrix in gray) and fracture
surfaces (ITZ in blue and cement matrix interfaces in gray) in different states during the
loading process in Figure 13: (a) state A, (b) state B, (c) state C, (d) state D and (e) the final
fracture pattern in experiments [46].
By observing the nephogram of displacement whose direction is vertical to the loading one
(Figure 15), the concrete can be divided into two parts according to the displacement value. One
namely residual part has a small displacement, and the other namely separation part has a larger
displacement (the displacement value of the separation part is 10 times larger than the residual part).
These two parts are separated by cracks with high discontinuity, like the situation in the LF condition.
Thus, we can obtain the residual part during the fracture process by deleting the elements with higher
displacement values (Figure 16a–c). It can be seen that as the fracture process proceeded, the residual
part gradually shrank with the splitting of the separation part. However, due to the influence of
surface friction, the residual part still remained dumbbell shape with a thin middle part and two thick
ends.
Figure 15. The displacement nephogram whose direction is vertical to the loading one.
(a)
(b)
(c)
Figure 16. The residual part of the concrete specimen at different loading states in Figure
13: (a) state B, (b) state C, (c) state D.

Materials 2019, 12, 1929 17 of 24
3.2.3. Energy Analysis
The fracture modes of concrete under compression are investigated through the energy analysis.
The energy evolution of concrete specimens during the loading process are shown in Figure 17. It can
be seen that the shear stress work and the friction stress work dominate the energy consumption
during the whole loading process, and the normal stress work always keeps a low value. At the final
stage of failure, the friction stress work is higher than the shear stress one in both loading condition.
As the friction can only occur when the cracks are closed and under compression, it can be inferred
that in the compression condition, the compression-shear fracture mode is the main fracture mode
inside the concrete, and the tensile-shear fracture mode is not so important.
(a)
(b)
Figure 17. Energy evolution of the concrete specimens under compression in different
loading conditions: (a) LH condition, (b) HF condition.
For comparison purposes, we have carried out a group of numerical experiments of concrete
specimens under uniaxial tension. The final fracture pattern and the energy evolution of concrete
specimens are shown in Figure 18. Unlike the compression condition, the normal stress work
dominates the energy consumption during the whole loading process, and the shear stress work only
occupies a small proportion of energy consumption in the post-peak stage. Also, the friction work
has no contribution during the whole loading process. That is also why the previous work [33–35]
can obtain reasonable results without considering the friction effect. Thus, according to the energy
consumption proportion during the loading process, we can identify the main fracture mode of
concrete structures (e.g., concrete beam or concrete column) in the future studies using the proposed
numerical method. When the shear and friction stress work account for a large proportion of energy
consumption, the main fracture mode is compression fracture. When the normal stress work
dominates the energy consumption, the main fracture m ode is tension fracture. It is worth noting that
there are still many complicated fracture modes of concrete needing analysis, and we will analyze
them in future studies.
(a)
(b)
Figure 18. Fracture pattern and energy evolution of the concrete specimen under uniaxial
tension: (a) fracture pattern and main fracture surfaces, (b) energy evolution.
0246810
0.0
0.1
0.2
0.3
0.4
0.5
Energy (N•m)
ε (10-4)
En
Es
Ef
0
1
2
3
Stress
σ
(MPa)

Materials 2019, 12, 1929 18 of 24
To investigate the effect of loading condition (LF, HF) on the compressive strength of concrete
in terms of energy, the energy increments at the pre-peak stage (Figure 17) are extracted and listed in
Table 2. Also, the proportion of different kinds of energy increment are shown in Figure 19. It can be
seen that the proportion of the shear stress work increment in the HF condition is significantly lower
than that in the LF condition, while the proportion of the friction stress work increment in the HF
condition is higher than that in the LF condition. The reason why these differences exist is the restrain
effect of the loading surface friction. Due to the restraint of the loading surface, the concrete specimen
in the HF condition cannot separate freely as that in the LF condition. For this reason, more cracks
inside the concrete are closed and subjected to compression in the HF condition, which causes a
higher proportion of the friction stress work and a lower proportion of the shear stress work.
Table 2. Energy increments and proportion of the concrete specimen in different loading
condition at the pre-peak stage.
Loading Condition Energy Type Energy Value
(N∙m)
Energy Proportion
(%)
LF condition
n
EΔ
3.31 8.88
s
EΔ
21.85 58.57
f
EΔ
12.14 32.55
HF condition
n
EΔ
3.12 5.12
s
EΔ
30.47 49.98
f
EΔ
27.37 44.90
Figure 19. Energy increments proportion of the concrete specimen in different loading
conditions at the pre-peak stage.
3.3. Effect of Specimen Height
To investigate the effect of the specimen height on the fracture pattern and the mechanical
behavior of concrete, three kinds of the concrete specimen with different height (50 mm, 100 mm, 200
mm) were chosen for the numerical experiments in the HF condition as shown in Figure 20. All
experiments were ended when the displacement d = 1 mm.

Materials 2019, 12, 1929 19 of 24
(a)
(b)
(c)
Figure 20. Concrete specimens with the different heights: (a) 50 mm, (b) 100 mm, (c) 200
mm.
3.3.1. Fracture Pattern and Mechanical Behavior
Figure 21 shows the average stress-strain curves of the concrete specimens with different
heights. The strength of the concrete specimen decreases with the increase of the specimen height,
but the shape of the stress–strain curves is basically unchanged. The numerical stress-strain curves
fit well with the experimental one, especially in the pre-peak stage. To investigate the fracture pattern
of concrete specimens during the loading process, three typical states were chosen from the concrete
specimens with different height (height = 50 mm, 200 mm).
Figure 21. Stress–strain curve of concrete specimens with different heights.
The evolution of concrete fracture pattern (deformation is magnified 5 times) with the specimen
height = 200 mm is shown in Figure 22. The final fracture pattern is similar to that obtained from
experiments [46]. Due to the restraint effect of the loading surfaces, the cracks mainly occur at the
middle part along the height direction, and the loading surfaces always remain complete during the
loading process.

Materials 2019, 12, 1929 20 of 24
(a)
(b)
(c)
(d)
Figure 22. Fracture pattern (aggregate in blue and cement matrix in gray) of the concrete
specimen with 200 mm height at different states in Figure 20: (a) state A, (b) state B, (c) state
C, and (d) the final fracture pattern in experiments [46].
The evolution of concrete fracture pattern (deformation is magnified 5 times) with the specimen
height = 50 mm is shown in Figure 23. Due to the small specimen height, the fracture region of the
concrete specimen is strongly affected by the restraint of the loading surfaces. Thus, the deformation
of this specimen is smaller than the other two kinds with different height, and the cracks mainly
occurs near the free surfaces.
(a)
(b)
(c)
(d)
Figure 23. Fracture pattern (aggregate in blue and cement matrix in gray) of the concrete
specimen with 50 mm height at different states in Figure 20: (a) state D, (b) state E, (c) state
F, and (d) the final fracture pattern in experiments [46].
3.3.2. Energy Analysis
The fracture modes of concrete with different height are investigated through the energy
analysis. The energy evolution of the concrete specimens with different heights (200 mm, 50 mm) are
shown in Figure 24. The energy evolution regulation of these two specimens are similar to that with
the height = 100 mm (Figure 17b). The normal stress work still has little effect on the energy
consumption, while the shear stress work and the friction stress work always play a dominated role
during the fracture process. However, the proportion of friction stress work increases with the
decrease of the specimen height.

Materials 2019, 12, 1929 21 of 24
(a) (b)
Figure 24. Energy evolution of the concrete specimen under compression with different
specimen heights: (a) 200 mm, (b) 50 mm.
The energies increment at the pre-peak stage (Figure 24) were extracted and are listed in Table
3, and the proportion of different kinds of energy increments are shown in Figure 25. As shown in
results, the proportion of friction stress work decreases with the increase of the specimen height,
while the proportion of the normal stress work and the shear stress work increase. This change means
the restraint effect of loading surfaces has a limited effect range. when the fracture region is inside
the effect range of the restraint of the loading surfaces, the free separation of cracks would be
restrained, which causes a higher proportion of friction stress work; but when the fracture region is
far enough to get rid of the restraint of loading surfaces, the cracks can separate freely, and the
fracture process at the pre-peak stage would be similar to that in the LF condition at the pre-peak
stage. For example, the strength and the energy increments proportion of the specimens with 200 mm
height in the HF condition (Figure 24a and Figure 25) are basically same as the one in the LF condition
(Figure 17a and Figure 19). However, with the development of the cracks, by which the fracture
region would be affected by the restraint effect of loading surfaces, the fracture pattern and the
mechanical behavior would be different from the one in the LF condition at the post-peak stage.
Table 3. Energy increments and proportion of the concrete specimens with different
specimen heights at the pre-peak stage.
Height of Specimen Energy Type Energy Value
(N∙m)
Energy Proportion
(%)
200mm
n
EΔ
4.31 8.49
s
EΔ
29.72 58.51
f
EΔ
16.77 33.00
50mm
n
EΔ
2.26 3.96
s
EΔ
24.05 42.14
f
EΔ
30.77 53.90
Figure 25. Energy increments proportion of the concrete specimen with different specimen
heights at the pre-peak stage.

Materials 2019, 12, 1929 22 of 24
4. Conclusions
A 3D numerical model of concrete mesostructure is proposed based on Voronoi tessellation and
cohesive elements. A modified constitutive is established to characterize the complex behavior of the
potential fracture surfaces. Based on the proposed model, the effect of the loading condition and the
height of the concrete specimens under uniaxial compression were investigated by analyzing the
fracture patterns and the energy increments proportion. Several conclusions can be obtained as
follows:
(1) The mechanical behavior and the fracture patterns of concrete under compression obtained
from the numerical results shows an agreement with that obtained from experiments, which means
the proposed model in this paper is suitable for the mesoscopic study of concrete.
(2) The energy consumption proportion can be used to identify the fracture mode of concrete
during the fracture process. The compressive fracture of concrete is always accompanied by high
shear and friction stress work, and the tensile fracture is accompanied by high normal stress work.
(3) The effect of loading surfaces restraint on the compressive strength can be estimated by the
proportion of the energies increment at the pre-peak stage: a higher proportion of the friction stress
work and a lower proportion of the shear stress always cause a higher compressive strength.
In this study, we have established a 3D numerical model and used it to analyze the fracture
behavior of concrete under compression. However, there are still many things worth studying based
on this research. First, the aggregate generated in this paper is based on pure theory, and we can try
to replace it with more realistic aggregates reconstructed by CT scanning in further studies. Secondly,
there are still many factors that can affect the mechanical behavior of concrete such as the loading
form, the loading rate, the aggregate/cement content, and the strength range of concrete. These factors
are meaningful to the design of concrete and worth investigating in further studies.
Author Contributions: Methodology, Y.H. and Z.G.; writing-original draft preparation, Y.H.; Supervision, S.H.;
writing-review and editing, Y.S.
Funding: This research received no external funding
Acknowledgments: This study was funded by the State Key Program of National Natural Science of China
(Grant no. 51739008), the National Major Scientific Instruments Development Project of China (Grant
no.51527811), and the Postgraduate Research and Practice Innovation Program of Jiangsu Province (Grant
no.2018B683X14). We thank Mr. Jiang zhaofei for his contribution to the research.
Conflicts of Interest: We have no conflicts of interest to declare.
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