Numerical Simulation of the Damage Behavior of a Concrete Beam with an Anisotropic Damage Model
An anisotropic damage model was proposed to describe the nonlinear behavior of concrete beams under monotonic and cyclic loading. The hysteresis effect of concrete is approximately modeled by employing nonlinear loading/linear reloading stress paths in the model, which was implemented into ABAQUS. L...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2018
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| Цитувати: | Numerical Simulation of the Damage Behavior of a Concrete Beam with an Anisotropic Damage Model / Y.C. Long, C.T. Yu // Проблемы прочности. — 2018. — № 5. — С. 57-65. — Бібліогр.: 22 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860032435420397568 |
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| author | Long, Y.C. Yu, C.T. |
| author_facet | Long, Y.C. Yu, C.T. |
| citation_txt | Numerical Simulation of the Damage Behavior of a Concrete Beam with an Anisotropic Damage Model / Y.C. Long, C.T. Yu // Проблемы прочности. — 2018. — № 5. — С. 57-65. — Бібліогр.: 22 назв. — англ. |
| collection | DSpace DC |
| container_title | Проблемы прочности |
| description | An anisotropic damage model was proposed to describe the nonlinear behavior of concrete beams under monotonic and cyclic loading. The hysteresis effect of concrete is approximately modeled by employing nonlinear loading/linear reloading stress paths in the model, which was implemented into ABAQUS. Linear, bilinear, exponential and Reinhardt strain softening functions are introduced to investigate their influence on accuracy of calculations.
Предложена модель анизотропного повреждения для описания нелинейного поведения бетонных балок в условиях монотонного и циклического нагружения. Поведение бетона при гистерезисе ориентировочно моделируют с использованием траектории напряжения при нелинейной нагрузке/линейной перегрузке с последующей реализацией модели в программе ABAQUS. Введены линейная, билинейная, экспоненциальная функции и функция Рейнхардта деформации разуплотнения для изучения их влияния на точность расчетов.
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| first_indexed | 2025-12-07T16:52:34Z |
| format | Article |
| fulltext |
UDC 539.4
Numerical Simulation of the Damage Behavior of a Concrete Beam with an
Anisotropic Damage Model
Y. C. Long
a,b,1
and C. T. Yu
b
a Key Laboratory of New Technology for Construction of Cities in Mountain Area, Chongqing
University, Chongqing, China
b School of Civil Engineering, Chongqing University, Chongqing, China
1 longyc02@cqu.edu.cn
An anisotropic damage model was proposed to describe the nonlinear behavior of concrete beams
under monotonic and cyclic loading. The hysteresis effect of concrete is approximately modeled by
employing nonlinear loading/linear reloading stress paths in the model, which was implemented into
ABAQUS. Linear, bilinear, exponential and Reinhardt strain softening functions are introduced to
investigate their influence on accuracy of calculations. The load–deflection responses obtained by the
damage model reflect the damage-induced nonlinear behavior of concrete beams, results are
comparable with the test data. The strain softening functions significantly affect simulation accuracy,
and the responses obtained by the Reinhardt function are in the best agreement with experimental
ressults. The numerical data under cyclic loading are consistent with those obtained in the
experiment, characterizing the degradation of stiffness and hysteresis effect of concrete beams. It is
concluded that the anisotropic damage model can be used to simulate the nonlinear behavior of
concrete structures.
Keywords: concrete structure, anisotropic damage model, hysteresis effect, numerical
simulation, cyclic loading.
Introduction. Numerical simulation is an important method to investigate the damage
and fracture mechanisms of the concrete structure. One of the critical issue for numerical
simulation is to develop a constitutive model of concrete which represents its nonlinear
behavior. The continuum damage mechanics theory (CDMT) was by far the most popular
theory for developing a constitutive model of concrete.
Based on the assumption of isotropic damage, Mazars et al. [1, 2] and Papa [3]
developed damage models of concrete by utilizing scalar damage variable to represent the
degradation of stiffness, which established the fundamental framework of continuum
damage mechanics. Cervera et al. [4] and Richard et al. [5] introduced weighted damage
evolution rules into the scalar damage model to model the stiffness recovery effect resulting
from the closing of microcracks upon load reversals. Cervera et al. [4] and Haussler-Combe
and Kuhn [6] proposed rate-dependent damage models within the framework of CDMT,
which evolve from the fundamental rate-independent scalar damage models via an
additional viscoelastic contribution, to take into account the strain-rate sensitivity of
concrete. Desprez et al. [7] formulated an isotropic damage model for FRP-confined
concrete and analyzed the nonlinear response of FRP-confined concrete column under axial
and flexural loads. Although those scalar damage models didn’t account for the anisotropic
features of concrete, they were widely used in the nonlinear analyses of concrete structures
because of their high efficiency.
Second- or fourth-order damage tensor was employed by Papa and Taliercio [8],
Alliche [9], Lu et al. [10], Badel et al. [11], and Francois [12], within the fundamental
framework of continuum damage mechanics, to model the anisotropic properties of
concrete damage and develop anisotropic damage models. The above models were applied
to the analyses of damage and failure prediction of concrete structures subjected to cyclic
© Y. C. LONG, C. T. YU, 2018
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5 57
and fatigue loads. Wu and Xu [13] introduced thermodynamically consistent projection
operators into the damage models employing the spectral decomposition of the stress and
strain tensor, which guarantees the fulfillment of energy conservation under arbitrary load
history. Based on the concept of thermodynamics, Ngo et al. [14] proposed an anisotropic
damage model for thermo-mechanical coupling and modeled the nonlinear response of
concrete members under thermal and mechanical loading. Hariri-Ardebili and Mirzabozorg
[15] employed anisotropic damage mechanics approach to model the cracking behavior in
mass concrete and then conducted a numerically seismic assessment of the arch–reservoir–
foundation system. Compared to the isotropic damage models, the governing equations of
the anisotropic ones were much more complicated. Most of these models, therefore,
employed linear paths to represent unloading/reloading behavior to obtain high numerical
efficiency. They might get a closer result, when the hysteretic behavior could be modeled,
to the realistic response of the concrete structure.
Long and He [16] developed an anisotropic damage model in which nonlinear
unloading/linear reloading stress paths were employed to model the hysteretic behavior.
This model was implemented into the finite element analysis (FEA) program ABAQUS
[17] by programming its constitutive relation in user subroutine UMAT [18]. Although this
model could reflect the stiffness degradation and hysteretic response of concrete sample
under uniaxial tension or compression, its applicability to the concrete structure should be
demonstrated.
1. Fundamental Governing Equations. There are three granted assumptions in
Long’s model [16]. First, the strain can be, for any arbitrary material point, decomposed
into elastic and inelastic parts. Then, the material axes are mutually orthogonal and parallel
to the principal directions of the stress tensor. Last, the damage relations are defined
regarding stress and inelastic strain corresponding to the rotating material axes with the
assumption that the damage is decoupled. According to these assumptions, the governing
equations of the model are as follows:
� �� D , (1)
where � and � denote the stress and strain tensor, respectively, and D is the constitutive
matrix representing the stress–strain relation and given as
D E T GT� �
� �( ) ,0
1 1
r t
T
(2)
in which E0 is the linear elastic stiffness matrix, and Tr is the transformation matrix of
strain from global coordinates to the local one defined by the rotating material axes. G stands
for the damage matrix corresponding to flexibility in local coordinates and is defined as
G�
g e
g e
g e
in
in
in
1 1
2 2
3 3
0 0
0 0
0 0
( )
( )
( )
, (3)
where g ei i
in( ) and ei
in denote the secant flexibility and inelastic strain on damage axis
i�1, 2,3. Because the above-mentioned governing equations are nonlinear, the linearization
method should be utilized to obtain the constitutive relation in the incremental form that
can be referred to work [16].
As shown in Fig. 1, stress paths subjected to cyclic loading comprises the loading
path, unloading path, reloading path, partial unloading path and partial reloading one. The
Y. C. Long and C. T. Yu
58 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5
loading path is defined by the yielding function, given in Eq. (4), considering the effect of
stress state on the material strength:
F q p
f f
f f
c c
in
bc c
bc c
�
�
� � � �
�
�
�
1
1
3 0
2
�
� � � � �
�
( ) ( ) ,
( )
( )
max
,
( )
( )
( ) ( ),
,
�
� �
� �
� �
�
� � � �
�� �
c c
in
t t
in
ii ij ijp q s s
1 1
1
3
3
2
, ,s pij ij� ��
(4)
in which p denotes the hydrostatic pressure, q stands for the Mises stress, � and � are
dimensionless parameters, fbc and fc represent the maximum compressive stresses under
equibiaxial and uniaxial compression, respectively, and � �c c
in( ) and � �t t
in( ) denote the
equivalent stresses corresponding to uniaxial compression and tension. The symbol is
the McCauley bracket which means x x x� �
1
2
( | | ).
The hysteretic behavior is approximately modeled by employing the nonlinear
unloading path and linear reloading one. Figure 1 shows the unloading/reloading paths
defined regarding stress and inelastic strain. The nonlinear unloading path is given in Eq. (5):
| | | | | | | | ,s B e e C e e
B
i i m i m
in
i
in
i m
in
i
in
� � � � �
�
�
�
�
�
� �
[ | | | | | | ],i m i m
in
i
p
i
t
i m
in
i
p
e e H e e
C
� � �
�
�1
� �
�
�
[ | | | | ],
| |
H e e
e e
i
t
i m
in
i
p
i m
i m
in
i
p
� �
�
�
�
1
1
��
,
(5)
where i denotes the ith damage axis and
� ( , )t c represents tension or compression, si
and ei
in are the present principal stress and inelastic strain, �
i m and ei m
in
denote the
stress and inelastic strain at the start of unloading, ei
p
is the plastic strain, Hi
t
denotes
Numerical Simulation of the Damage Behavior ...
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5 59
Fig. 1. The definition of unloading and reloading paths subjected to cyclic loading.
the tangent modulus of unloading path corresponding to zero stress, and �
� ( , )0 1 is a
constant.
Besides, the reloading path is defined by Eq. (6) as a linear path
| | | | | | ,
, [ , ],
s H e e
H
d
d
E d
i i a i i
in
i a
in
i
i
i
i
� � �
�
�
�
�
1
0 1
(6)
in which �
i a and ei a
in
are the stress and inelastic strain at the start of the reloading path,
and Hi
is the modulus determined by the damage factor di . Herein, di is defined
concerning tensile and compressive damage variables, which can be referred to reference
[16]. The definitions of partial unloading and partial reloading are similar to those of
unloading and reloading and also can be referred to [16].
2. A Numerical Example of a Notched Beam Test. This model is used to simulate
the damaging process of plain concrete beam tests conducted by Hordijk [19]. As shown in
Fig. 2, square-cross-section beams, whose dimensions were 500 100 50� � mm, had an
initial notch depth
� 10, 30, and 50 mm, respectively, and were subjected to four-point
loading. The material properties used herein are as follows: elastic modulus E0
43 8 10� .
MPa, Poisson’s ratio �� 0.2, tensile strength ft � 3.0 MPa, tensile fracture energy density
Gt
f
� 125 N/m. The details of Hordijk’s experiments can be referred to study [19].
Numerical analyses are carried out by the finite element analysis program ABAQUS since
this model has been embedded into ABAQUS by its user-defined subroutine UMAT.
Four-point plane stress element with dimensions 5 5� mm is utilized to model those
concrete beams in the numerical simulations.
2.1. Monotonic Loading Tests. The stress–inelastic strain curve is an important effect
on numerical simulation. Four types of the strain–softening curve are used in the monotonic
loading analyses, as illustrated in Fig. 3. The definitions of these curves are as follows:
60 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5
Y. C. Long and C. T. Yu
Fig. 2. Testing plan and finite element mesh of notched beam test.
(1) The linear softening curve is given by Eq. (7):
� �
�ft
in
f
� �1 , � f
t
f
t
G
f h
�
2
, (7)
where � and �
in are the stress and inelastic strain, and Gt
f
, ft , and h denote the tensile
fracture energy density, tensile strength, and characteristic size of the damaged element,
respectively.
(2) The bilinear softening curve is defined by Eq. (8) is proposed in [20]:
�
�
�
� �
� �
� �
� � �
ft
in
f
in
f
in
f
in
f
�
� � �
�
�
� �
1 0 85 0
0 15
1
1
1
. , ,
. , ,
. , ,
�
�
�
�
�
�
�
� � �� � �
�
1 2 0 15
G
f h
G
f h
t
f
t
f f
F t
f
t
(8)
where it is assumed that the maximum diameter of aggregate is 16 mm and �F � 7 is
given.
(3) The exponent one proposed by Karihaloo [21] is given as follows:
� �
�ft
in
� �
�
�
�
�
�
�
�
�
exp . ,0 405
0
�0 6 405� . .
G
f h
t
f
t
(9)
(4) The Reinhardt’s one [22] is defined as
� �
�
�
�f
c c
t
in
f
in
f
� �
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�1 1
3
2exp e c
c
in
f
�
�
2 1 1
3( ) ,
�
�
(10)
in which c1 3� and c2 � 6.93 are used and, therefore, � f t
f
tG f h� 5 136. .
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5 61
Numerical Simulation of the Damage Behavior ...
Fig. 3. Tensile strain softening curves of concrete.
Figure 4 shows the responses between load and deflection, i.e., the P fl�
curves,
compared to the experimental ones. The peak value of load in each case is given in Table 1,
where !� �( )max max maxP P Pe e denotes the error between the numerical result and the
experimental one. As illustrated in Fig. 4, softening curves have a significant influence on
the structural responses of concrete beams although the calculated P fl�
curves are
comparable to the experimental ones. In the case with the linear softening curve, numerical
responses are significantly different from experimental ones, as they overestimate the
values of peak load from 12 to 29% and their post-peak curves are much steeper than the
62 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5
Y. C. Long and C. T. Yu
T a b l e 1
The Comparison of the Calculated Peak Value of Load Pmax
and Experimental One Pe max [19]
Notch
depth
, mm
Experimental Linear Bilinear Exponential Reinhardt’s
Pe max ,
kN
Pmax ,
kN
!,
%
Pmax ,
kN
!,
%
Pmax ,
kN
!,
%
Pmax ,
kN
!,
%
10 3.62 4.67 29.0 4.14 14.4 4.21 16.3 3.98 9.9
30 2.53 2.93 15.8 2.61 3.2 2.65 4.7 2.51 �0.8
50 1.42 1.59 12.0 1.43 0.7 1.44 1.4 1.37 �3.5
Fig. 4. The P fl�
curves of concrete beams subjected to monotonic loading (compared with
experimental results [19]).
experimental ones. When utilizing bilinear and exponential softening curves, the numerical
P fl�
curves are much closer to the experimental results than those obtained by the linear
softening curve. Furthermore, the numerical responses in the case with the Reinhardt’s
curve are in the closest agreement with experimental ones, in which the errors of peak load
are from�3.5 to 9.9%. Consequently, the Reinhardt’s curve best reflects the influence of
strain-softening effect on the structural response of the notched concrete beams.
2.2. Cyclic Loading Tests. Cyclic tests of the notched beams are also modeled by the
anisotropic damage model. The loading path is determined by the Reinhardt curve, and the
unloading and reloading ones are defined by Eqs. (5) and (6). Equation (11) gives the
plastic strain eit
p
and tangent modulus Hit
t at a zero stress:
e c k eit
p
t t i
in
� � , H E A B
e
it
t
t t
i
in
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
0
0
exp ,
�
(11)
where ct � 0, kt � 0.4, At � 0.5, Bt � 1.0, and �0 0� f Et . The exponential constant
�
defining the unloading curve is given as 0.5.
Figure 5 shows the P fl�
curves of notched beams obtained by numerical
simulation, subjected to monotonic and cyclic loading, compared to the experimental
results. The P fl�
curves under monotonic loading are identical to the envelopes of
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5 63
Numerical Simulation of the Damage Behavior ...
Fig. 5. The P fl�
curves of concrete beams subjected to cyclic loading [19].
those under cyclic loading because they are determined by the loading path that gives the
boundary of structural responses of concrete beams. The calculated results are highly
consistent with experimental ones, which indicates that this anisotropic damage model
reasonably reflects the stiffness degradation and hysteretic behavior of concrete. Slight
difference between numerical and experimental results, such as the peak load, the post-peak
envelope, and the unloading/reloading paths, results from the fact that the envelopes of the
numerical responses are different from those of experiments. The hysteretic behavior in the
material level is approximately modeled by employing the nonlinear unloading and linear
reloading paths, which obtains a reasonably hysteretic response in the structural level.
Therefore, this method applies to modeling the effect of hysteretic behavior on the
structural response of concrete structures.
Conclusions. The fracture behavior of concrete beams was investigated by an
anisotropic damage model. The results draw the following conclusions:
1. The strain softening curves significantly affect the numerical response of concrete
beams. The linear softening curve overestimates the peak value of the load and causes
steeper post-peak responses than experimental ones while the Reinhardt’s one obtains the
closest responses compared with experimental ones.
2. The method of nonlinear unloading/linear reloading applies to represent the
hysteretic behavior of concrete. Using this method, the calculated load-deflection responses
agree well with experimental ones. The numerical results make it possible to conclude that
the stiffness degradation and hysteretic behavior can be well represented by the present
anisotropic damage model.
Acknowledgments. The present work has been supported by the National Natural
Science Foundation of China, with Grant No. 51578088. The authors are grateful for this
support.
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64 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5
Y. C. Long and C. T. Yu
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Redeived 05. 03. 2018
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5 65
Numerical Simulation of the Damage Behavior ...
|
| id | nasplib_isofts_kiev_ua-123456789-173992 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0556-171X |
| language | English |
| last_indexed | 2025-12-07T16:52:34Z |
| publishDate | 2018 |
| publisher | Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| record_format | dspace |
| spelling | Long, Y.C. Yu, C.T. 2020-12-28T18:42:59Z 2020-12-28T18:42:59Z 2018 Numerical Simulation of the Damage Behavior of a Concrete Beam with an Anisotropic Damage Model / Y.C. Long, C.T. Yu // Проблемы прочности. — 2018. — № 5. — С. 57-65. — Бібліогр.: 22 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/173992 539.4 An anisotropic damage model was proposed to describe the nonlinear behavior of concrete beams under monotonic and cyclic loading. The hysteresis effect of concrete is approximately modeled by employing nonlinear loading/linear reloading stress paths in the model, which was implemented into ABAQUS. Linear, bilinear, exponential and Reinhardt strain softening functions are introduced to investigate their influence on accuracy of calculations. Предложена модель анизотропного повреждения для описания нелинейного поведения бетонных балок в условиях монотонного и циклического нагружения. Поведение бетона при гистерезисе ориентировочно моделируют с использованием траектории напряжения при нелинейной нагрузке/линейной перегрузке с последующей реализацией модели в программе ABAQUS. Введены линейная, билинейная, экспоненциальная функции и функция Рейнхардта деформации разуплотнения для изучения их влияния на точность расчетов. The present work has been supported by the National Natural Science Foundation of China, with Grant No. 51578088. The authors are grateful for this support. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Numerical Simulation of the Damage Behavior of a Concrete Beam with an Anisotropic Damage Model Численное моделирование поведения бетонной балки с помощью модели анизотропного повреждения Article published earlier |
| spellingShingle | Numerical Simulation of the Damage Behavior of a Concrete Beam with an Anisotropic Damage Model Long, Y.C. Yu, C.T. Научно-технический раздел |
| title | Numerical Simulation of the Damage Behavior of a Concrete Beam with an Anisotropic Damage Model |
| title_alt | Численное моделирование поведения бетонной балки с помощью модели анизотропного повреждения |
| title_full | Numerical Simulation of the Damage Behavior of a Concrete Beam with an Anisotropic Damage Model |
| title_fullStr | Numerical Simulation of the Damage Behavior of a Concrete Beam with an Anisotropic Damage Model |
| title_full_unstemmed | Numerical Simulation of the Damage Behavior of a Concrete Beam with an Anisotropic Damage Model |
| title_short | Numerical Simulation of the Damage Behavior of a Concrete Beam with an Anisotropic Damage Model |
| title_sort | numerical simulation of the damage behavior of a concrete beam with an anisotropic damage model |
| topic | Научно-технический раздел |
| topic_facet | Научно-технический раздел |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/173992 |
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