Numerical simulation of brittle damage in concrete specimens
A framework for damage mechanics of concrete is applied to simulate the non-linear elastic deformation behavior of concrete using finite element method (FEM). A rather simple isotropic damage model containing essentially no adjustable parameters is shown to produce results in remarkably good a...
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| Date: | 2005 |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2005
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| Cite this: | Numerical simulation of brittle damage in concrete specimens / Y. Labadi, N.E. Hannachi // Проблемы прочности. — 2005. — № 3. — С. 57-74. — Бібліогр.: 34 назв. — англ. |
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| author | Labadi, Y. Hannachi, N.E. |
| author_facet | Labadi, Y. Hannachi, N.E. |
| citation_txt | Numerical simulation of brittle damage in concrete specimens / Y. Labadi, N.E. Hannachi // Проблемы прочности. — 2005. — № 3. — С. 57-74. — Бібліогр.: 34 назв. — англ. |
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| description | A framework for damage mechanics of concrete
is applied to simulate the non-linear elastic
deformation behavior of concrete using
finite element method (FEM). A rather simple
isotropic damage model containing essentially
no adjustable parameters is shown to produce
results in remarkably good agreement with sample
experimental data: the damage law requires
only the fracture energy to be defined completely.
The model is achieved by introducing a
damage surface that is similar to the yield function
in the conventional theory of plasticity. A
special form of damage surfaces is constructed
to illustrate the application of the model. A new
damage criterion, defined as an equivalent
strain norm, is proposed, in order to take into
consideration the asymmetric behavior of concrete.
For verifying the FEM program including
the model, deformations predicted by this
model are compared with both the experimental
ones for the concrete structural model and
the ones calculated without application of the
continuum damage mechanics.
Запропоновано критерій руйнування конструкцій з бетону, що отримав назву
норми еквівалентних деформацій. Критерій дозволяє враховувати асиметричну
механічну поведінку бетону при розтязі та стиску. Розроблено ізотропну
модель, що базується на використанні поверхні пошкодження, яка аналогічна
функції плинності теорії пластичності. За допомогою запропонованої
моделі виконано скінченноелементний розрахунок нелінійно-пружної деформації
зразків із бетону. Проведено порівняння отриманих результатів з експериментальними.
Установлено, що точність розрахунків більш висока, аніж
прогнози без використання підходів механіки пошкодження твердого тіла.
Предложен критерий повреждения конструкций из бетона, получивший название нормы
эквивалентных деформаций, который позволяет учитывать асимметричное механическое
поведение бетона при растяжении и сжатии. Разработана изотропная модель, базирующаяся
на использовании поверхности повреждения, аналогичной функции течения теории
пластичности. С помощью предложенной модели выполнен конечноэлементный расчет
нелинейно-упругой деформации образцов из бетона. Проведено сравнение полученных результатов
с экспериментальными. Показана хорошая корреляция между расчетными и экспериментальными
результатами. Установлено, что точность расчетов более высокая по
сравнению с прогнозами, выполненными без применения подходов механики повреждения
твердого тела.
|
| first_indexed | 2025-12-07T16:02:43Z |
| format | Article |
| fulltext |
UDC 539.4
Numerical Simulation of Brittle Damage in Concrete Specimens
Y. Labadia and N. E. Hannachib
a Université de Technologie de Troyes, Troyes, France
b Faculté de Génie de la Construction, Université de Tizi-Ouzou, Tizi-Ouzou, Algeria
УДК 539.4
Численное моделирование хрупкого разрушения бетонных
образцов
Ю. Лабадиа, Н. Э. Ханнаши6
а Технологический университет г. Труа, Франция
6 Университет г. Тизи-Узу, Алжир
Предложен критерий повреждения конструкций из бетона, получивший название нормы
эквивалентных деформаций, который позволяет учитывать асимметричное механическое
поведение бетона при растяжении и сжатии. Разработана изотропная модель, базиру
ющаяся на использовании поверхности повреждения, аналогичной функции течения теории
пластичности. С помощью предложенной модели выполнен конечноэлементный расчет
нелинейно-упругой деформации образцов из бетона. Проведено сравнение полученных резуль
татов с экспериментальными. Показана хорошая корреляция между расчетными и экспери
ментальными результатами. Установлено, что точность расчетов более высокая по
сравнению с прогнозами, выполненными без применения подходов механики повреждения
твердого тела.
Ключевые слова : упругость, повреждение, нелинейное поведение, бетон,
конечные элементы, численные методы.
No t a t i o ns
D - scalar value representing the local damage parameter (ranging from 0 for
the virgin material to 1, which represents the failure or zero stress)
p о - material specific mass
£ j - strain tensorУ
о И - effective stress tensorУ
ф 0 - denotes the initial (undamaged) elastic stored energy
^°jkl - fourth order tensor of the elastic stiffness of the undamaged material
^ijkl - fourth order tensor of the elastic stiffness of the damaged material
a - parameter taking into consideration the shear resistance
f t - material yield stress in tension
f c - material yield stress in compression
© Y. LABADI, N. E. HANNACHI, 2005
ISSN 0556-171X. Проблемы прочности, 2005, № 3 57
Y. Labadi, N. E. Hannachi
k - coefficient accounting for the large difference in tensile strength and
comressive strengths, k = f j f t
G f - fracture energy by unit surface (J/m2)
g f - specific fracture energy (J/m3)
E - Young modulus of the material
G - shear modulus
lch - characteristic length of the element of volume representing the material’s
average behavior
Introduction. It is well known that the deformation of most engineering materials is often accompanied by irreversible changes in their internal structure. The nucleation and the growth of distributed microscopic cavities and cracks will not only induce the occurrence of macro-cracks, but also lead to the deterioration of material properties due to internal micro structural changes. Therefore, proper understanding and knowledge of the damaging process and its effects on the macroscopic practical problems are required. An approach to these problems can be provided by the recently developed theory of continuum damage mechanics (CDM), which involves irreversible micro structural changes implicitly. Several types of CDM models have been proposed for brittle materials. For example, the Chow-Yang model [1] is based on the hypothesis of damage surface that is similar to the yield function in plasticity theory and uses one scalar valued internal variable to represent the damage state. The Chaboche model [2] basically assumes anisotropic damage and introduces tensorial damage variables. Therefore, it can be considered that there exists a relationship between small cracks (damage) and strain direction. However, a detailed method to apply this model to anisotropic damage in any direction is still under development. Moreover, CDM theories, which incorporate vectors [3], scalars [4], second-order tensors [5, 6], and fourth-order tensors [7], have been proposed. However, these theories show either a discontinuous stress-strain response when the unilateral condition takes place or an unacceptable non-symmetric elastic behavior for some loading conditions [8]. In this paper, we try to describe the non-linear elastic constitutive equation using CDM and evaluate the structural integrity of brittle material components by the damage parameter introduced in this type of theory. In concrete, the inelasticity is generally associated with irreversible thermodynamics process involving elastic degradation. Strength and stiffness degradation can be effectively modeled in the CDM framework: isotropic damage models with a single damage variable [9] or a tension damage variable and a compression damage variable [10, 11], anisotropic damage models [12, 13], to cite a few.In this paper, a rather simple isotropic damage model for concrete in the framework for damage mechanics is presented and used to simulate the non-linear elastic deformation behavior of concrete, using the finite element method (FEM). This model introduces a damage surface that is similar to the yield function in the conventional theory of plasticity. A special form of damage surfaces is constructed to illustrate the application of the model. This model contains essentially no adjustable parameters; indeed the damage law requires only the
58 ISSN 0556-171X. npo6n.eubi npounocmu, 2005, № 3
Numerical Simulation o f Brittle Damage
fracture energy to be completely defined. To verify the FEM program including
the present model, the deformations predicted by this model are compared with
both the experimental ones in the concrete structural model and the calculated
ones without using the CDM; they employed the concept of internal state
variables and a phenomenological method to describe the state of a material
element. For simplicity, we only consider the rate-independent behavior of the
element in an isothermal process. Consequently, time and temperature will not
appear in the formulation. The final objective is being able to simulate the
behavior of concrete structures until fracture with the relatively simple models of
use and implementation. This paper is organized as follows. In Section 1, using
the phenomenological approach of continuum damage mechanics; the damage
model is formulated into a general framework of the thermodynamics of
irreversible processes. The numerical aspects related to the use of this model in
finite element analysis are discussed in Section 2. A modified Newton-Raphson
algorithm, that uses the initial stiffness matrix as iteration matrix, has been used
for the solution of the non-linear system. Finally, numerical results obtained with
the presented model are compared with experimental and numerical results
concerning the damage development in tension test and on in singleuedge notched
concrete beam under bending load.
1. The Damage Model.
1.1. Energetic Considerations. The model is formulated in the strain-space,
and it is based on the equivalent strain concept. Since the observable variable is
the strain and in order to simplify the numerical implementation of this model in
finite element code, we choose the Helmholtz free energy as the state potential
where a is a material parameter varying from 0 (without shear retention) to 1
(damage ineffective for shear stress). It is introduced in order to take into
consideration the friction between the two surfaces of crack and it accounts the
shear resistance.
As the damage increases towards the unit, the free energy is not equal zero
because the shear resistance of cracks. For the energetic consistence of the model,
the inequality of Clausius-Duhem that states the no-negative character of the rate
of mechanical energy dissipation for any arbitrary infinitesimal variation £ j
from an equilibrium condition has to be ensured
(1)
(2)
Equation (2) can be written as
(3)
ISSN 0556-171X. npo6n.eubi npounocmu, 2005, N 3 59
Y. Labadi, N. E. Hannachi
Since the variation £ j is completely arbitrary, this inequality can be fulfilled
only if the term in parentheses is identically zero
1 dp— o a - — = 0.
P 0 3 d£ a (4)
From the Eq. (4) we obtain
o ij = P 0 = [1 - D + (1 - 6 a6 kl )aD ̂ i j k l £ kl = ^ ijk l£ kl. (5)
For the normal stresses (i.e., i = j and k = l) the term ( 1 - 6 3 6 kl )aD
vanishes in Eq. (5), while for the shear stresses it introduces a reduction of the
damage action in order to assure a minimum strength even for the completely
damaged material.
Equation (3) can be simplified like follows:
dp . 0 •
<P d =-dD D D = p 0 D ^ 0. (6)
This last equation implies the irreversibility of the damage. It is satisfied if
D is increasing (that is to say, being always defined positive).
1.2. Damage Criterion. Within the brittle damage approach, the concept of
yield surface as the familiar plasticity law is used. There exists a reversible
(elastic) domain such that damage does not develop from any interior state but
may evolve from a state on the boundary of the domain. So, the damage growth
will be governed by a loading surface of equation f (£ y,D ) = 0 (convex in D).
This threshold function depends on the equivalent strain and the damage. It is
defined in the strain space [6, 9, 12, 14-16]. In this study, a new form for the
equivalent strain is proposed. The domain or loading surface is defined as
f ( £ j , D ) = £ - K ( £, D ) < 0, (7)
where £ is an equivalent strain or a norm of the strain tensor £ ^ , K (~ , D ) is the
softening parameter and takes the largest value of £ ever reached at the
considered point in the material. Initially K (~ ,D ) = £d o, in which £d o is the
threshold of damage. Value of £ d o may be regarded as the tensile strain at which
damage is initiated. We consider that the threshold is reached when the stress is
maximal in uniaxial tension £ d o = f t j E
K ( £, D ) = max £ D 0; max £
T<t (8)
In concrete, there is a large difference in tensile strength and compressive
strength. Thus, the following expression is proposed to define the damage
60 ISSN 0556-171X. npo6n.eubi npounocmu, 2005, № 3
Numerical Simulation o f Brittle Damage
threshold of concrete, taking into consideration the asymmetric behavior in
tension and compression:
£ =
[1+ r(k — 1)]
k (9)
where J is a fourth-order symmetrical tensor. For initially isotropic material, J
may be written as
1 —v —v
E
J = —
- E
E E
0 0 0
1
E
—v
E
0 0 0
1
E
0 0 0
1
G
0 0
S
1
G
0
1
G J
(10)
with E = max (E ,G ), and
0 3if î=1
r = ■3É (£ i) 3i=13 •f ^1̂ «il
i=1 i=1
(11)
where £ i denotes the tensor of principal strains.
1.3. Damage Evolution. While considering a non-standard material,
evolution of damage is defined as
. dF(~)
D = — £ ).
d£
(12)
The evolution of damage is governed by an equivalent strain. So that the
evolution law writes:
D =
0
dF ( £ ) /~
if f (£ ij ,D ) < 0 (unloading),
Ч if f (£ ij ,D ) = 0 and f (£ j ,D ) = 0 (loading),
(13)
d£ П У y" i]
ISSN 0556-171X. Проблемы прочности, 2005, N2 3 61
Y. Labadi, N. E. Hannachi
where dF ( ? ) /d? is a positive definite function ?. The shape of the function
F ( e ) is determined from experimental observations available in the literature [9].
According to [17], the strain-softening curve of concrete must be concave. Then
we write d F (? ) /d? under exponential form as
dF ( e )
de
' D 0
e 2 /
(1+ b e )exp [-b ( e - e D o)], (14)
while proceeding to the integration of Eq. (14) we obtain
D = 1- ( e D 0 ̂exp[-b( e - eD0)]A ~ )
(15)
D is defined in the strain interval [eD0 ,» ], so that for ? = e D0 it gives D = 0
and for ? it gives D = 1. The total energy dissipated during the deformation
process has an upper bound value that is equal to the specific fracture energy of
the material g f , can be obtained by the following integration along the whole
strain path:
g f = f o( e )de= f 0 dD. (16)
If we consider a uniaxial test, then e = e n and =
obtains the following form:
0 (e°1)2
2 P 0 E, and Eq. (16)
g f = f
eD 0
(e°1)2̂
2p 0 dD
de 0 de 11.
11
(17)
A ^er some rearrangements, with introducing (15) into (17), the following
results are obtained:
g f ^ e[- + 1
P 0 V2 b
(18)
Finally b is given like following:
I
b=
g f P 0E 1
f / 2
-1
(19)
Here b is a dimensionless constant. The relation (19) shows that the parameter b
in Eq. (15) depends on the specific fracture energy dissipated during the whole
damaging process. The fracture energy per unit surface G f is defined by
G f = g f l ch . Hence we obtain
62 ISSN 0556-171X. npo6n.eubi npounocmu, 2005, N2 3
Numerical Simulation o f Brittle Damage
b _ 1 ) 1 £ 0 ,
L , f 2 2
(20)
The value of the fracture energy G f can be measured during a bending test on notched concrete specimens. Value of G f is a known material property and
lch is a characteristic length of the element volume representing the material’s average behavior. The length lch characterizes the size of the smallest region over which damage can localize. As is well known, concrete exhibits strain softening leading to a complete loss of strength. In these materials, the secant modulus decreases with increasing strain. At the beginning, a linear relationship between stress and strain exists for up to 60 percent of the maximum stress. Then, micro cracks develop within the specimen, which is indicated by the non-linearity in the curve up to the tensile strength. In the post-peak regime, more micro cracks are developed in the weakest cross section of the specimen called the fracture process zone (FPZ) and they cause a continual decrease of its tensile strength from a peak stress f t to zero, together with an increase in deformation. More and more micro cracks are formed until finally they coalesce into macro cracks. In the post-peak regime, all fracture energy is consumed in FPZ. This behavior is called strain softening. The characteristic length is a geometrical constant, which is introduced as a measure of the length of the FPZ in a specimen.Similarly, a uniaxial compression test can be considered and the symbol g c is adopted to identify the global energy dissipation. We write then
This parameter plays the role of a localization margin that limits the size of the localization zone.Note that expression (15) is similar to the isotropic damage models given by [9,13, 18] and also to the anisotropic damage models given by [12, 19].2. Finite Element Formulation. The non-linear solution scheme selected in this study uses the secant stiffness matrix at the beginning of the step with combined with a constant stiffness matrix during the subsequent iterations, that is, the incremental-iterative method. A program was written in a FORTRAN code, and is expressed in a modular form consisting of various subroutines called from the main program and from within themselves. It has been written along the same format of elasticity program CALSEF as developed by the International Centre for Numerical Methods in Engineering (CIMNE Barcelona). This software is an adaptation of the PLAST2 program, as developed by Owen and Hinton in their work on finite elements modeling [20]. Also, this program has been adapted for the structural damage analysis by [21]. The same idea has been used in [22] for the 3D damage structures analysis. The preparation of data, as well as the visualization of results, is achieved with the help of the GID (CIMNE) pre-post-processor. The algorithm of calculation is an incremental type, based on the secant method, this type of algorithm is widely used, e.g., [11, 13, 23].
-1
^ g c = k l G f . (21)
ISSN 0556-171X. npo6n.eubi npounocmu, 2005, № 3 63
Y. Labadi, N. E. Hannachi
Let’s recall some theoretical aspects below. The local equilibrium condition
of the continuum can be written as:
on Q (i = 1 ,3 and j = 1, 3)
on T c
d° i j, j + f vi = 0;
^ ijnj = f si ,
(22)
(23)
where Q and r s are the body volume and surface, f vi is the vector of the mass
loads, f si is the vector of surface loads, and n j is the unit vector normal to the
surface.
We rewrite the Eq. (22) under a global weak form through the principle of*
virtual work. We choose an arbitrary virtual displacement {u } as a test function:
-J
Q
dQ = 0. (24)
We proceed then by integrating by parts to get the weak form:
J ° ijd Q ~ J u* f s id r - J u* fv id Q = 0.
Q aXj r Q
Under matrix form we write
J {£ * }r {<7}dQ - J {u * }r { f s} d r - J {u * }r { fv }dQ = 0.
Q r Q
While using the stress-strain relation, Eq. (23) becomes
J {£ *}r [As ]{£}dQ-J {u y { f s} d r - J {u *}T { fv }dQ = 0.
(25)
(26)
Q Q
(27)
We proceed then with the approximation of the variational form. The domain
Q is divided in subdomains Q e (finite elements). The displacements in an
element may be approximated as
n
u l = X N i(X,y )uiei = i=1
n
u 2 = 2 N i(X,y )u2i
i=1
n
u 3
(28)
= 2 N i(x, y)u 3i
i =1
64 ISSN 0556-171X. npo6n.eubi npounocmu, 2005, № 3
Numerical Simulation o f Brittle Damage
where N t are the global shape functions at node i of coordinates (x t , y t ) in the
global reference mark (x , y ) and of coordinates (£ i , ̂ i ) in the local reference
(£ ,^) and n is the number of nodes in the element.
According to the Galerkin method, we use the same approximations for the
virtual displacement, so real strains and virtual strains can be written as
\{u( x , y )}= [ N ]{un},
l{e(x , y )}= [B ]{un },
I{u (x, y )}= [ N ]{un},l{£ *( x, y )}= [B ]{un }, (29)
where
[Bj ] =
dNj
dx
0dN;
0
N
dydN;
dy dx
for the plane problems
Then we obtain
f [B ]T [As ][B ]dQ e{uen }-f [N ]T f }dT e - f [N ]T { f ve }dQ e = 0. (30)
That conducted to the following elementary shape:
[k (e)]{un } = { f (e)}, (31)
where [k(e) ] = f [B]T [AS(D) ][B]dQe is the element stiffness matrix and { f (e)} =
Qe
= f [N ]T {f V }dQe + f [N ]T {f se }dT e is the element nodal force,
Qe r e
Assembly of matrixes and elementary vectors provides the global system:
f [B]T [AS(d ) ][B]dQ{^n} = f [N ]T { f v}dQ - f [N ]T { f s}dT.
Q
Under condensed form:
Q
[Ks (D )]{U } = {F },
(32)
(33)
where [K (D )] is the global secant rigidity matrix and {F } are the global vector
forces.
The system of equations (33) is non-linear due to its dependence to the
damage parameter D (that is to the stiffness matrix K ) on the strain tensor £
(that is on the displacement vector u). Therefore, a step-by-step procedure must
ISSN 0556-171X. npoÖÄeubi npounocmu, 2005, № 3 65
Y. Labadi, N. E. Hannachi
be adopted, where the external loads on the right-hand side of Eq. (33) are applied
incrementally. Within each step, the adoption of a regular Newton-Raphson
algorithm for the solution of the non-linear system, with the proper tangent
stiffness, would provide quadratic convergence to the solution. However, with a
full Newton-Raphson scheme the tangent stiffness matrix has to be formed and
refactoried for each iteration. Moreover, the tangent stiffness matrix becomes
non-symmetric and the computational time drastically increases. In our numerical
test, we have found that the adoption of a modified Newton-Raphson scheme
requires a computational time, to reach the same tolerance, which is smaller than
that required for the full Newton-Raphson scheme. In fact, the greater number of
iterations necessary at each time step due to the crudeness of the predictor, is
more than counterbalanced by the inversion of the tangent stiffness matrix.
Moreover, the modified Newton-Raphson scheme is always convergent to the
solution, even in the presence of flex points in the load-displacement curve. With
the modified Newton-Raphson algorithm, the iteration scheme for the solution of
system (33) becomes:
[K s ]{A U r }= {W (U r )}, (34)
{U r+1} = {U r } + {A U r }. (35)
Solution of the nonlinear problem will be achieved when the residual force
{W(U)} ({W(U)}= [K s(D )]{U} — {F }^ 0) is sufficiently small. At convergence,
{W(U)} = ^ is a tolerance chosen by the operator (for example, ^ = 0.001).
The accuracy of satisfying the global equilibrium equations is controlled by
the magnitude of the unbalanced nodal forces. In this study, the convergence
criterion employed is the following Euclidean norm:
N
2 ( { w r })2
------------ ^ , (36)
| 2 ( { f }i ) 2
where N is the total number of degrees of freedom in the case, W is the residual
force value, f is the external force acting on the member, r denotes the iteration
number, and ^ is the specified tolerance.
3. Num erical Applications and Results.
3.1. SE N B Specimen (Single-Edge Notched Beam test). We carry out a
finite element analysis, until the rupture, on plain concrete Iosipescu single-edge
specimen, under four-point bending conditions. This type of experiment has been
simulated extensively in the literature using experimental [24, 25], analytical and
numerical methods [18, 26-30]. Besides, this example is also treated in the
examples manual of the commercial finite element software ABAQUS/Explicit
[31].
66 ISSN 0556-171X. npo6n.eubi npounocmu, 2005, N 3
Numerical Simulation o f Brittle DamageThe material properties used in the simulations are chosen as shown in Table 1. The geometry, loads and supports are shown in Fig. 1. The finite element discretization of the specimen is presented in Fig. 2. In the analysis 673 constant strain triangular elements are employed.
T a b l e 1
Mechanical Properties of Concrete for SENB Specimen and Tension Test
Characterictic SENB Specimen Tension Test
Young modulus (N/mm2) 24,800 36,000
Poisson’s ratio 0.18 0.15
Uniaxial tensile strength (N/mm2) 2.8 3.4
Fracture energy (N/mm) 0.05 0.07
0.13F
I V
1
1
1
;
i
r
/ \
— U —
203 | 397 |61[61| 397------------ H----' — ----------- J c 203 ,
224
',82
Fig. 1. Single-edge noteched beam: geometry and loading conditions.
Fig. 2. Mesh used in the analysis for SENB specimen.
In Fig. 3, we present crack pattern of the structure at different instants of this analysis. In Fig. 4, we present a calculation done with a non-local damage model using an adaptive mesh strategy [29]. For a qualitative comparison we refer to the experimental result given in Fig. 5 [25].With regard to the obtained results, we can see that the damage is localized in a strip, the crack profile corresponds well to the experimental results, and the crack propagates in the predicted direction. The load-CMSD (Crack Mouth Sliding Displacement) curve product by the model (Fig. 6) is comparable with the
ISSN 0556-171X. npoôëeubi npounocmu, 2005, N 3 67
Y. Labadi, N. E. Hannachione from the experiment [25], and to the one given by [17]. In this test, the behavior is a combination of two modes of fracture. In Fig. 7, we show that the crack opens up in Mode I and in Mode II.
1
a
Damage
0,49128■ 0.43669
0.38211■ 0.32752■ 0.27293■ 0.21S34
■ 0.1637B
0.10917
■ 0.054583
■ 0
L
b
Damage
_■ 0.59366
1 0.5277
0.46173
] 0,395?'?
!i| 0.32381
Id- 0.26305
Ifj 0.19788
1 0.13192
' 0.065957
■ 0
I ) n m a g c
0.93028
0.32692
0.72355
0.62019
0.51682
m» 0.41345
H 0.31009
■ 0.20672
■ 010336
■ 0
Fig. 3. Damage evolution in a SENB at various instants of the analysis.
c
a b
Fig. 4. Crack profile in non local damage analysis (Rodriguez-Ferran et al. 2001): (a) adaptive
mesh; (b) damage pattern.
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Numerical Simulation o f Brittle Damage
Fig. 5. Experimental crack pattern in SENB (Shlangen 1993).
CMSD in mm
Fig. 6. Load-CMSD curve.
Fig. 7. Final deformed mesh and damage distribution of SENB.
Fig. 8. Tension test: geometrical data (dimensions in mm).
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Y. Labadi, N. E. Hannachi
3.2. Tensile Test. This example concerns the simulation of the experimental
tension test performed by Hassanzadeh [32] on four edge notched concrete
specimen. Di Prisco [33] considered this test as a benchmark. O f this fact, some
authors [11, 34] also considered this test. The geometry is shown in Fig. 8. The
specimen has been discretised by constant strain triangular elements as shown in
Fig. 9. The material properties used in the simulations are chosen as shown in
Table 1.
In Fig.10 we present crack pattern of the structure at different instants of the
analysis and for a qualitative comparison we consider the numerical results done
by Comi [11] as shown in Fig. 11. Finally, the computed total reactions versus the
imposed displacement of the top of the specimen are plotted in Fig. 12.
Fig. 9. Tension test: FE mesh and boundary conditions.
Damage
b
Fig. 10. Predicted damage pattern evolution during the analysis: u = 0.004 (a) and 0.05 mm (b).
70 ISSN 0556-171X. Проблемы прочности, 2005, N 3
Numerical Simulation o f Brittle Damage
a b
Fig. 11. Damage pattern evolution given by Comi: u = 0.004 (a) and 0.05 mm (b).
Displacemen in mm
Fig. 12. Load versus displacement under the load point (tension test).
Conclusions. An isotropic damage model for the macroscopic description of concrete behavior, together with its finite element implementation was presented. The damage is directly related to the effective strain through a suitable exponential equation in which a limited number of physically defined material parameters, identifiable by means of standard laboratory tests. Indeed, only one material parameter, that is, the fracture energy, needs to be given together with the classical material mechanical characteristics. The value of the fracture energy can be easily measured by means of a bending test on notched concrete specimens. The damage model and its evolution law in the present work are of particular convenience because of their easy implementation in finite element program and, because the small computational effort that is required in addition to the usual elastic analysis, if a suitable solution scheme is adopted. The form given for the evolution law allows determination of the local damage in direct way and thus allows one to avoid the step-by-step integration of the damage evolution. Due to its simplicity, it is of clear and immediate applicability in existing finite element code. Despite its intrinsic simplicity, it makes possible an effective description of stiffness degradation and strength reduction of the material response. The application of the model to structural analysis predicts satisfactory results and is in good agreement with the experimental data and others numerical results. This model is suitable for the study of quasi-static problems where monotonically increasing loads are applied because the presence of single damage parameter the effect of unilateral effect is not taken into consideration.
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Y. Labadi, N. E. Hannachi
Резюме
Запропоновано критерій руйнування конструкцій з бетону, що отримав назву
норми еквівалентних деформацій. Критерій дозволяє враховувати асимет
ричну механічну поведінку бетону при розтязі та стиску. Розроблено ізотроп
ну модель, що базується на використанні поверхні пошкодження, яка анало
гічна функції плинності теорії пластичності. За допомогою запропонованої
моделі виконано скінченноелементний розрахунок нелінійно-пружної дефор
мації зразків із бетону. Проведено порівняння отриманих результатів з експе
риментальними. Установлено, що точність розрахунків більш висока, аніж
прогнози без використання підходів механіки пошкодження твердого тіла.
1. C. L. Chow and F. Yang, “On one-parameter description of damage state for
brittle material,” Eng. Fract. Mech., 40, No. 2, 335-343 (1991).
2. J. L. Chaboche, P. M. Lesne, and J. F. Maire, “Continuum damage
mechanics, anisotropy and damage deactivation for brittle materials like
concrete and ceramic composites,” Int. J. Damage M ech, 4, No. 1, 5-22
(1995).
3. D. Krajcinovic and G. U. Fonseka, “The continuous damage theory of brittle
materials, parts II and I,” J. Appl. Mech. ASME, 48, 809-824 (1981).
4. P. Ladeveze and J. Lemaitre, “Damage effective stress in quasi-unilateral
material conditions,” IUTAM Congress, Lyngby, Denmark (1984).
5. C. Cordebois and F. Sidoroff, “Endommagement anisotrope en élasticité et
plasticité,” Journal de Mécanique appliquée, 25, 45-60 (1979).
6. S. Ramtani, “Contribution à la modélisation du comportement multiaxial du
béton endommagé avec description de l ’effet unilateral,” Thèse de doctorat
de l’université Paris VI, France (1990).
7. L. W. Ju, “On energy-based coupled elastoplastic damage theories: constitutive
modeling and computational aspects,” Int. J. Solids Struct., 25, No. 7,
803-833 (1989).
8. J. L. Chaboche, “Damage induced anisotropy: on the difficulties associated
with the active/passive unilateral condition,” Int. J. Damage Mech., 1,
148-171 (1992).
9. J. Lemaitre and J. Mazars, “Application de la théorie de l’endommagement
au comportement non linéaire et à la rupture du béton de structure,” Annales
de l'ITBTP, 401, 114-138 (1982).
10. J. Mazars and G. Pijaudier-Cabot, “Continuum damage theory - application
to concrete,” ASCE J. Eng. Mech., 115, No. 2, 345-365 (1989).
11. C. Comi, “A non local model with tension and compression damage
mechanisms,” Euro. J. Mech.: A/Solids, 20, 1-22 (2001).
12. F. Ghrib and R. Tinawi, “Non linear behavior of concrete dams using
damage mechanics,” J. Eng. M ech, 121, No. 4, 513-527 (1995).
13. F. Ragueneau, J. Mazars, and C. La Borderie, “Damage model with frictional
sliding. Contributions towards structural damping for concrete structures,”
Rev. Euro. Elém. Finis, 10, 1-15 (2001).
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14. J. W. Dougill, “On stable progressively fracturing solids,” Z. Angew. Math.
P hys, 27, No. 4, 423-437 (1976).
15. Z. P. Bazant and S. S. Kim, “Plastic fracturing theory for concrete,” ASCE J.
M ech, 105, 407-428 (1979).
16. J. C. Simo, J. W. Ju, R. L. Taylor, and K. S. Pister, “On the strain-based
continuum damage models: Form ulation and computational aspects.
Constitutive laws for engineering materials: Theory and applications,” C. S.
Desai et al. (Eds.), Elsevier Science Publishing (1987).
17. J. G. Rots, “Smeared and discrete representations of localized fracture,” Int.
J. Fract., 51, 45-59 (1991).
18. P. H. Feenstra, “Computational aspects of biaxial in plain and reinforced
concrete,” Ph.D. Thesis, Delft University of Technology, The Netherlands
(1993).
19. S. Fichant, G. Pijaudier-Cabot, and C. Laborderie, “Continuum damage
modeling: Approximation of crack induced anisotropy,” Mech. Res. Commun.,
24, No. 2, 109-114 (1997).
20. D. R. J. Owen and E. Hinton, Finite Element in Plasticity. Theory and
Practice, Pineridge Press Limited, Swansea, United Kingdom (1980).
21. Y. Labadi, K. Saanouni, and N. E. Hannachi, “Anisotropie induite du
dommage avec effet de désactivation/activation,” Annales Maghrébines de
l ’Ingénieur, 12, 671-675 (1998).
22. A. Al-Ghadib, K. Asad-ur-Raman, and M. Baluch, “CDM based finite
element code for concrete in 3-D,” Comput. Struct., 67, 451-462 (1998).
23. L. Davenne, C. Saouridis, and J. M. Piau, “Un code de calcul pour la
prévision du comportement des structures endommageables en béton, en
béton armé, ou en béton de fibres,” Annales de l ’ITBTP, 478, 139-155
(1989).
24. M. Arrea and A. R. Ingraffea, Mixed-Mode Crack Propagation in Mortar
and Concrete, Department of Structure Engineering, Cornell University,
Report 81-13 (1981).
25. E. Shlangen, “Experimental and numerical analysis of fracture process in
concrete,” Ph.D. Thesis, Delft University of Technology, The Netherlands
(1993).
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University of Technology, The Netherlands (1986).
27. J. G. Rots, “Removal of finite elements in strain-softening analysis of tensile
fracture,” in: Proc. of the Third Conference on Computational Plasticity-
Fundamentals and Applications, Pineridge Press (1992), Part I, pp. 669-680.
28. J. Lubliner, J. Oller, S. Oliver, and S. Onate, “A plastic-damage model for
concrete,” Int. J. Solids Struct, 25, No. 3, 299-326 (1989).
29. A. Rodriguez-Ferran, I. Arbos, and A. Huerta, “Adaptive analysis based on
error estimation for nonlocal damage models,” Rev. Euro. Elém. Finis, 10,
193-207 (2001).
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Y. Labadi, N. E. Hannachi
30. M. G. D. Geers, R. De Borst, and R. H. J. Peerlings, “Damage and crack
modeling in single-edge and double-edge notched concrete beam,” Eng.
Fract. M ech, 65, 247-261 (2000).
31. ABAQUS, User’s Manual, Version 4.8. Hibbit, Karlson and Sorensen, Inc.,
Palo Ato, California (1990).
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by simultaneously applied normal and shear displacements,” Ph.D. Thesis,
Delft University of Technology, The Netherlands (1991).
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reinforced concrete: some results on benchmark tests,” Int. J. Fracture, 103,
127-148 (2000).
34. L. Ferrara, “Numerical simulation of mixed-mode fracture in concrete via a
non local damage model,” in: Proc. of the second Int. PhD Symposium in
Civil Engineering (Budapest 26-28 August) (1998), pp. 1-8.
Received 10. 09. 2004
74 ISSN 0556-171X. npoôëeubi npounocmu, 2005, N 3
|
| id | nasplib_isofts_kiev_ua-123456789-47683 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0556-171X |
| language | English |
| last_indexed | 2025-12-07T16:02:43Z |
| publishDate | 2005 |
| publisher | Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| record_format | dspace |
| spelling | Labadi, Y. Hannachi, N.E. 2013-07-25T12:35:14Z 2013-07-25T12:35:14Z 2005 Numerical simulation of brittle damage in concrete specimens / Y. Labadi, N.E. Hannachi // Проблемы прочности. — 2005. — № 3. — С. 57-74. — Бібліогр.: 34 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/47683 539.4 A framework for damage mechanics of concrete is applied to simulate the non-linear elastic deformation behavior of concrete using finite element method (FEM). A rather simple isotropic damage model containing essentially no adjustable parameters is shown to produce results in remarkably good agreement with sample experimental data: the damage law requires only the fracture energy to be defined completely. The model is achieved by introducing a damage surface that is similar to the yield function in the conventional theory of plasticity. A special form of damage surfaces is constructed to illustrate the application of the model. A new damage criterion, defined as an equivalent strain norm, is proposed, in order to take into consideration the asymmetric behavior of concrete. For verifying the FEM program including the model, deformations predicted by this model are compared with both the experimental ones for the concrete structural model and the ones calculated without application of the continuum damage mechanics. Запропоновано критерій руйнування конструкцій з бетону, що отримав назву норми еквівалентних деформацій. Критерій дозволяє враховувати асиметричну механічну поведінку бетону при розтязі та стиску. Розроблено ізотропну модель, що базується на використанні поверхні пошкодження, яка аналогічна функції плинності теорії пластичності. За допомогою запропонованої моделі виконано скінченноелементний розрахунок нелінійно-пружної деформації зразків із бетону. Проведено порівняння отриманих результатів з експериментальними. Установлено, що точність розрахунків більш висока, аніж прогнози без використання підходів механіки пошкодження твердого тіла. Предложен критерий повреждения конструкций из бетона, получивший название нормы эквивалентных деформаций, который позволяет учитывать асимметричное механическое поведение бетона при растяжении и сжатии. Разработана изотропная модель, базирующаяся на использовании поверхности повреждения, аналогичной функции течения теории пластичности. С помощью предложенной модели выполнен конечноэлементный расчет нелинейно-упругой деформации образцов из бетона. Проведено сравнение полученных результатов с экспериментальными. Показана хорошая корреляция между расчетными и экспериментальными результатами. Установлено, что точность расчетов более высокая по сравнению с прогнозами, выполненными без применения подходов механики повреждения твердого тела. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Numerical simulation of brittle damage in concrete specimens Численное моделирование хрупкого разрушения бетонных образцов Article published earlier |
| spellingShingle | Numerical simulation of brittle damage in concrete specimens Labadi, Y. Hannachi, N.E. Научно-технический раздел |
| title | Numerical simulation of brittle damage in concrete specimens |
| title_alt | Численное моделирование хрупкого разрушения бетонных образцов |
| title_full | Numerical simulation of brittle damage in concrete specimens |
| title_fullStr | Numerical simulation of brittle damage in concrete specimens |
| title_full_unstemmed | Numerical simulation of brittle damage in concrete specimens |
| title_short | Numerical simulation of brittle damage in concrete specimens |
| title_sort | numerical simulation of brittle damage in concrete specimens |
| topic | Научно-технический раздел |
| topic_facet | Научно-технический раздел |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/47683 |
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