Benchmarks for the validation of a non local damage model Ludovic Jason*'**- Shahrokh Ghavamian**-Gilles Pijaudier-Cabot*-Antonio Huerta*** * R&DO, Jnstitut de Recherches en Genie Civil et canique, Ecole Centrale de Nantes, 1 rue de la Noe, BP 92101 F-44321Nantes Cedex 3 Ludovic.Jason@ec-nantes.fr ** EDF Recherche et Developpement, 1 avenue du General de Gaulle, 92141 Clamart cedex *** Laboratori de Calcul Numeric, Departament de Matematica Aplicada l Universitat Politecnica de Catalunya, Jordi Girona 1-3 E-08034 Barcelona, Spain ABSTRACT: The aim of this contribution is to present a series of organised benchmar that helps al validating the robustness of the FE implementation of a constitutive relation, its pertinence with respect to experiments, and quantitative and qualitative comparisons of structural elements. It is applied to an isotropic damage model used for concrete, coupled with a non local gradient formulation to avoid a spurious description of strain localisation. Aſter the elementary (uniial monotonic, cyclic or triial loadin and structural (three point bending tests) simulations, an industrial application is presented in the form of a representative structural volume of a containment building for French nuclear power plants. KEYWOS :concrete, nonlocal, isotropic, damage, structure. Jason, L., Ghavamian, S., Pijaudier-Cabot, G. and Huerta, A., Benchmarks for the validation of a non local damage model, Revue française de Génie civil, Vol. 8, Issue 2 -3, pp. 303-328, 2004 1
Transcript
Benchmarks for the validation of a non local damage model
Ludovic Jason*'**- Shahrokh Ghavamian**-Gilles
Pijaudier-Cabot*-Antonio Huerta***
* R&DO, Jnstitut de Recherches en Genie Civil et Mecanique,Ecole Centrale de Nantes, 1 rue de la Noe, BP 92101 F-44321Nantes Cedex [email protected]
** EDF. Recherche et Developpement, 1 avenue du General de Gaulle, 92141Clamart cedex*** Laboratori de Calcul Numeric, Departament de Matematica Aplicada IllUniversitat Politecnica de Catalunya, Jordi Girona 1-3 E-08034 Barcelona, Spain
ABSTRACT: The aim of this contribution is to present a series of organised benchmarks that helps al validating the robustness of the FE implementation of a constitutive relation, its pertinence with respect to experiments, and quantitative and qualitative comparisons of
structural elements. It is applied to an isotropic damage model used for concrete, coupled
with a non local gradient formulation to avoid a spurious description of strain localisation.
After the elementary (uniaxial monotonic, cyclic or triaxial loading) and structural (three point bending tests) simulations, an industrial application is presented in the form of a
representative structural volume of a containment building for French nuclear power plants.
Jason, L., Ghavamian, S., Pijaudier-Cabot, G. and Huerta, A., Benchmarks for the validation of a non local damage model, Revue française de Génie civil, Vol. 8, Issue 2 -3, pp. 303-328, 2004
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where criJ and &k/ are respectively the stress and strain components, CiJkl is the fourth order elastic tensor and D is the damage variable.
For the description of the damage growth, an equivalent strain is introduced from the local strain tensor:
.I
e,� == })<e, >+)2
[2] 1=1
where <e;>+ are the positive principal strains.
The loading surface g is defined by:
g(e, D) == d(e)-D [3]
where the damage variable D is also the history variable which takes the maximum value reached by d during the history of loading, D = Maxrt( d, 0).
d is defined by an evolution law which distinguishes the mechanical responses of the material in tension and in compression by introducing two scalars D, and De.
D = 1- Ko(l-A,)6,q
A, [4]
exp[B,(e,q -K0)]
In the definition of d, De and Cle have analogous formulations as D, and a, . Ko is a parameter of the model and represents the initial threshold from which damage grows. D, and De are the tensile and compressive part of the damage. A,,c , B,.c are four parameters of the model. The weights a, and ac are computed from the strain tensor &. They are defined as functions of the principal values of the strains e/ and e/ due to positive and negative stresses. In uniaxial tension, a, = l and ae = 0. In uniaxial compression, Cle = I, a, = 0. The exponent /3 reduces the effect of damage on the response of the material under shear compared to tension.
The evolution of damage is determined by the Kuhn - Tucker conditions
0 0
g $ 0, d '?. 0, g d = 0 [5)
3
•
Different techniques exist to limit strain localisation {Pijaudier-Cabot et al.,
1987, Peerlings et al., 1996 b, Lorentz et al., 2003 ). In every case, the principle is to include a spatial information in the form of a characteristic length. Here a gradient approach is chosen. It is based on the calculation of a non local strain tensor that solves, for each component of the strain tensor, the equation:
[6]
where 'v2 denotes the second order gradient operator. The constant c is proportional to the square of a length and characterises the non local interaction. The following boundary conditions are adopted for the non local strains (Peerlings et al., 1996 b) :
'vs.n=O [7]
The physical interpretation of these additional boundary conditions remains an unresolved issue. Nevertheless, with these assumptions, the model yields exactly the same response as the local formulation if the deformation is kept homogeneous. The global numerical problem is finally governed by the equilibrium equation and by the definition of the non local strains.
\7.a+ f = 0
&,1 -c\72&,1 = &if
[8]
where/ represents an external load density.
This gradient approach is applied to the isotropic damage model described in section 2.1 simply by replacing the local equivalent strain [2] with its non local counterpart [9] in equation [ 4].
3 -
li,q = �)< li; >.)2
i=l
[9]
Figure 2 (b) illustrates the non local response of the bar considered in figure 1. Damage is no longer localised in a single element. The mesh dependency problem is solved {Peerlings et al., 1996 b).
Compared to the classical non local methods which apply a non local equation directly on the equivalent strain of Mazars' model (Pijaudier-Cabot et al., 2001) either in an integral format or in a gradient one, this method fits any constitutive law since it is based on the computation of a non local strain tensor from which any invariant can be derived in order to control plasticity or damage. The obvious
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counterpart of the versatility of this model is the full computation of the non local strain tensor instead of the calculation of a non local scalar invariant.
3. Elementary tests
The model was implemented in the open source FE code "Code Aster", using a
full Newton-Raphson scheme to solve the problem (Godard, 2003)
The first step of the validation process consists in single element simulations in
order to evaluate the ability of the model to reproduce an experimental behaviour.
3.1. Monotonic tensile test
For concrete, tension is the most relevant loading that a model has to predict as
far as cracking is concerned. It is indeed when the concrete is subjected to tension
that the first cracks usually appear. That is why the numerical constitutive response
is first compared with an experimental tensile test (Gopalaratnam et al., 1985).
Figure 3 gives the axial stress strain curve.
0 200 400 600 soo
Axial strain ( 10·9
)
Figure 3. Axial stress strain curve.
The parameters chosen for this simulation are given in table 1. The Young's
modulus is measured from the initial experimental slope. s00 is used to calibrate the peak position, B t for the post peak behaviour and At for the final asymptote. From
the formulation of the model, Ac and Be play no role in the tensile response of the material.
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E (GPa) At Bt Eoo
31.25 0.88 8000 6 10 5
Table 1. Parameters for the simulation of the tensile test
The peak is captured correctly which validates the choice for the initial threshold
i::00. With the appropriate parameters At and Bt, the post peak behaviour, even not exactly fitted, is well reproduced by the simulation. From the comparison between experiment and simulation, one can thus consider that the isotropic damage model is adapted for the description of monotonic tensile tests.
3.2. Alternated uniaxial loading
To go further in the validation, an alternated tension compression application is considered. The test is still performed on a single element. Figure 4 illustrates the loading path. The element is twice loaded - unloaded in tension and twice in compression. Figure 5 shows the numerical response. The parameters, initially chosen for the benchmark (Ghavamian, 1999), are given in table 2. Since the concrete used for tension (3 .1) and compression tests are different, the parameters chosen for the two simulations are also different.
E (GPa) v At Bi Ac Be 1:oo
32 0.2 1 10000 1.15 1391 9.375 10-5
Table 2. Parameters for the simulation of the alternated tension - compression test
This result yields some remarks. First, for total unloading, zero stress corresponds to zero strain. The isotropic damage model described in this contribution is a pure damage model. "Plasticity" is not taken into account that is why irreversible strains do not appear. The unloading is elastic with a slope equal to the damaged Young's modulus Ed
[1 O]
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3.3. Triaxial test with confinement pressure
A triaxial test is modelled using the scalar constitutive law. A vertical displacement is applied on the plane face of a cylindrical sample after a confinement pressure is applied. The numerical results are compared with the experiment (Sfer et
al., 2002) for different levels of confinement. Figure 6 shows the axial stress strain curve from the application of the vertical displacement (the confinement phase is not represented). The parameters used for this simulation (table 3) are calibrated from the uniaxial compression curve. Once again, the concrete mix used for this test is not the same as the previous ones.
E (GPa) v Ai Bi Ac Be
27.3 0.2 0.8 10000 1.5 1700
Table 3. Parameters for the simulation of the triaxial with confinement test
From this result, one can consider that the isotropic damage model gives acceptable results for low confinement pressures only. Using the appropriate parameters, the uniaxial compression test is correctly modelled. For a confinement pressure of 4.5 MPa the simulation and the experiment are still in agreement. Out, as soon as the pressure takes important values (30 MPa or 60 MPa) the model gives an overestimated prevision of the real response. In fact, lhis comes from the defnition of the equivalent strain (9] that characterises the material extension during loading. When the hydrostatic pressure is applied, the sample is not subjected to
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tension, the equivalent strain keeps a zero value and the material is elastic if the principal strains remain negative during loading. For 30 or 60 MPa for example, damage has already initiated when the application of the vertical displacement begins. This cannot be described by the present constitutive relation. The isotropic damage model is thus adapted for the simulation of triaxial tests with low confinement pressures. Note that this conclusion concerns the axial behavior only. Indeed, due to the lack of irreversible strains in the model, the volumetric strains are not correctly reproduced. Especially, the change from a contractant to a dilatant behavior cannot be captured (Sfer et al., 2002).
4. Structural tests
Elementary tests are not sufficient to validate the isotropic constitutive Jaw described in section 2. The non local part of the model plays indeed no role if homogeneous tests are considered only. After elementary applications, the following step is thus structural tests. They are usually three dimensional applications. In the next part, two bending tests will be considered to validate the non local formulation and to compare some numerical results with experiment.
4.1. Numerical validation on a notched bending beam
Let us consider a notched bending beam whose geometry and boundary conditions are depicted in figure 7. A major problem of the local formulation comes from the fact that the numerical response depends on the size and the orientation of the mesh.
To validate the numerical results, a comparison with (Rodriguez-Ferran et al., 2002) is proposed. In this paper, another well-known non local theory (PijaudierCabot et al., 200 I) is used. The local equivalent strain [2] is replaced by the non local quantity:
c,/x)= 1 fv(s)c,/s+x)ds with V,(x)= f'I'(s)ru [II] V,(x) n n
where fl is the volume of the structure, V,(x) is the representative volume at point x and v(s) is the weight function, for instance:
lf(s) = exp(- 411:112
) c
[12]
le is the internal length of the non local continuum.
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geometry and the load system are presented in figure 11. Figure 12 depicts the steel distribution. The aim of this test is to evaluate the three dimensional performance of the model. Figure 13 provides the numerical load - deflection curve and a compari on with experiment. The parameters chosen for the simulatioD are Ac
= 1.'276 Bc= l 768 A 1=0.9, B1
=8000, £oo= l 10·4, E=20 GPa, v=0.2 c=3.35 I 5 cm2•
The steel bars are modelled with a Von Mises plasticity law (linear hardening) using the parameters E=200 GPa, v=O, cr0=400 MPa (yield stress), ET
=3245 MPa (plastic tangent stiffness) The steel - concrete interface is assumed to be perfect. The study was stopped when a deflection of 4 mm was reached. Only one fourth of the beam is meshed as depicted in figure 14 (3600 hexa8 elements).
The results show qualitative and quantitative similarities: the elastic part is correctly modelled and the inelastic behaviour agrees with the experiments. The role of the steel in the global response is also underlined by the lack of softening branch on the load deflection curve. Moreover, a major localisation band appears in the damage distribution (figure 14) in the middle of the beam, followed by some secondary bands that characterise the presence of steel in the concrete. This "discrete" damage distribution illustrates well the formation of cracks in a R.C beam. Thus, for bending tests, the isotropic gradient model seems sufficient to capture the structural response correctly
This test gives also the opportunity to highlight the interest of the non local formulation. Figure 15 shows the damage distribution for local and non local simulations, as a function of the number of elements used to discretize the longitudinal section of the half beam (20, 40 or 100 elements). For numerical reasons, the non local parameter c has been increased to allow the use of coarser meshes ( c = 87 cm2). That is why a continuous zone of damage is observed instead of the discrete localisation bands.
In the local case, damage geometrically localises, as already mentioned in section 2. The distribution is dependent on the size and the orientation of the mesh (figures l Sa, I Sc). This dependency gradually yields a critical state where the mesh is so fine that damage and strains localise in infinitely small bands (see figure I Sc ). It so results in a fracture without energy dissipation. In this case, the local model simulates a numerical response which is physically unacceptable.
On the contrary, the non local approach gives a continuous zone of localisation which is no longer influenced by the mesh (Figure 15b, d). Figure 16 illustrates the differences in term of load deflection curves.
y
l1111111111H111111 mt=--��� ���� ������J�
Figure 11. Three point bending beam
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The simulations with 20 or 40 longitudinal element meshes give exactly the same result when the non local regularised approach is used. On the contrary, if a local computation is considered, one can observe a mesh dependency and the apparition of characterislic numerical snap back which correspond to unrealistic partial unloading of the beam. In this case, including a regularised approach is still necessary to avoid a spurious strain localisation despite an expected regularisation effect due to the reinforcement. (a) local simulation (20 element mesh) (b) non local simulation (20 element mesh)
----------------======== ----------... -!!!!!!!!
----=========-·--------------
(c) local simulation (100 element mesh) (d) non local simulation (40 element mesh)
I I ---·==
·------�:�======== -··-=============;======
-===========:ea::eeseasaBE::=
Figure 15. Damage for local and non local simulations (half beam). Clear zones
correspond to damaged ones: O<DOJ for grey zones and D = 0 for black zones.
8 104
7104
s 10•
l 5 104
4104
_g
3 104
2 104
1 104
0 0 0.001 0.002 0.003 0.004 0.005
Deflection (m)
Figure 16. Local and non local simulations
5. Pre industrial application
The previous elementary and structural tests have so far provided interesting information about the validity of the constitutive damage law. The model reproduces well the behaviour of concrete in simple tension and for three point bending tests. It is also adapted for triaxial tests with low confinement pressures (axial behavior). With these results, the final step is the simulation of a pre i.ndustrial test in the fom1 of a representative part of a containment building of a nuclear power plant.
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. ..... . ~-- I• __ ~;.-
. e,. - • • I ~·•~ · v - . - -
5.1. Presentation of the simulation
The application presented in this part has been recently proposed by Electricite
de France. The test, named PACE 1300, is a Representative Structural Volume
(RSV) of a prestressed pressure containment vessel (PPCV) of a French 1300 MWe
nuclear power plant. Figure 17 illustrates the location of the RSV within the entire
PPCV structure. The model incorporates almost all components of the real structure:
concrete, vertical and horizontal reinforcement bars, transversal reinforcements, and
prestressed tendons in both horizontal and vertical directions. The size of the RSV is chosen to respect 3 conditions: large enough to include a sufficient number of
components (specially prestress tendons) and to offer a significant observation area in the centre, far enough from boundary conditions, while remaining as small as
possible to ease computations. The model was prepared using Gibiane [Castem,
1993] mesh generating scripts which create models with different mesh refinements.
This was an important aspect of this application, where the mesh size effect was of
great concern on various nonlinear calculations.
With these conditions, the RSV includes 11 horizontal and 10 vertical
reinforcement bars (on both internal and external faces), 5 horizontal and 3 vertical
prestressed tendons, and 24 reinforcement hoops uniformly distributed in the
volume. The geometry of the problem is given in figure 18. Figure 19 provides
information about the steel distribution and properties .
.-.......
. :.--·-·
·-·-·-.-·-
Repreullf ive
Stmct11ra
Vo/11111e
Figure 17. Position of the extracted Representative Structural Volume (RSV) .
..... -···t::·(;� 0.10458 rad
R 22.95 m ••• · :
.········· .........
.
h 2m
_____ ye= 0.9 m
Figure 18. Geometry of the Representative Structural Volume (RSV)
Boundary conditions in stress PSH : horizontal prestress
5.28 MN per tendon PSV : vertical prestress
6.93 MN per tendon WE : structure weight
1.61 MPa g: gravity
Figure 21. Boundary conditions and loading for the Representative Volume
The behaviour of the RSV needs to be as close as possible to the in situ situation. The following boundary conditions have been chosen : face SB blocked along OZ,
on face SH all nodes are restrained to follow the same displacement along OZ and
no rotations are allowed for faces SG and SD (see Figures 20 and 21). A more adequate boundary condition would have probably been a periodic one on SG and
SD.
In order to model the effect of prestressed tendons, bar elements were anchored
to faces SG and SD for horizontal cables, SB and SH for vertical tendons, and then
prestressed using internal forces. Then, these elements are restrained to surrounding
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concrete elements to represent the prestressing technology applied in French PPCVs. The integrity tests loading is represented by a radial pressure on the internal face SI and the bottom effect applied on face SH (tensile pressure proportional to the internal pressure to simulate the effect of the neighbouring structure). The body weight of RSV and that of the surrounding upper-structure are also taken into account. With these conditions, a mesh has been defined containing 16,500 Hexa20 elements for concrete and 1200 bars elements for reinforcement and tendons.
Figure 22 provides the evolution of the stress distribution (radial and orthoradial stresses respectively) for a simple elastic linear simulation with an increasing internal pressure. One can especially notice the distribution of stresses around the prestressed tendons that highlights their role in the global response.
Using the non local model presented in the paper for this type of application would require a mesh small enough to take into account non local interaction. It corresponds to a computation with up to 1 million degrees of freedom which is still to be performed. That is why the simulation is performed with the local model to get first ideas about the response of the structure. The parameters chosen for the concrete are A0
=1.15, A1=0.8, B0=1,391, 8 1
= 10,000, e00=9.37 10-5, E=31 GPa and
v=0.22, for the reinforcement bars (Von Mises model): E=200 GPa, v=0.3, av=500 MPa (yield stress), E,=20000 MPa (tangent modulus), and for the prestressed tendons : E= l90 GPa, v=O, av=l550 MPa, E1
=0 MPa, f=0.16 rad- 1 (curved friction coefficient) cp=0.0015 m- 1 (decrease in tension). A perfect steel - concrete interface is considered.
With these geometry, parameters and components, the test aims at reproducing qualitatively the global behavior of a RSV of a French containment building. Nevertheless, the material characteristics and the loading conditions are not exactly those of a real French structure. That is why the mechanisms and especially the failure pressures presented below are not those expected in situ quantitatively.
Figure 23 provides the internal pressure applied on the volume as a function of the displacement of a point located at the top right of the internal face. This curve can be divided in four parts. The initial state corresponds to the application of the prestress on the tendons. This yields a compaction of the volume and due to the boundary conditions (no normal displacement on the lateral face), it imposes an initial negative radial displacement (see figure 24). Then, by applying the internal pressure, there is a zone of linear behaviour where the compaction is reduced and the structure returns towards its initial rest position before undergoing tension for higher values of internal pressure. Damage does not evolve during state 2. The developments of damage occurs during state 3. Finally, a partial unloading of the volume occurs (state 4) due to heavy cracking of the structure.
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.. -- I (
Initial state (state l) Development of damage (state 3) Partial unloading
Figure 23. Pressure - Displacement plot, using local simulation
Step 2: Application of the .... ------�---- internal pressure 0
'1al'tllililfl•fi(----- Step« zero» C:!11•···�
:.__ ____ Step 1 : Initial state after the
I I
I I
I I
I I
I I
I I
I I \ I
I I
I I
I I
\ I
I I
I I
I I
I I \I
v
application of the prestress
Figure 24. Radial deflection of the RSV through different steps (schematic). View
from the top of the volume.
21
A first damage band appears in the middle of the volume along the vertical
prestressed tendons. With an increasing internal pressure, one can notice the
development of four additional localised zones. At the end of the computation,
almost all the RSV is damaged.
Figure 26 illustrates the damage state at the first part of the computation with a
view from the top. As already mentioned, damage initially develops along the
vertical tendon which is located in the middle of the volume. Then, a propagation of
the damage can be noticed along the horizontal prestressed tendons. Subsequently,
the second damage band appears along another neighbouring vertical tendon.
These different representations give further information about the behaviour of
the structure. They particularly locate heavily damaged zones that appear at different
levels of internal pressures. Nevertheless, it also raises some questions :
The prestressed tendons seem to play a relevant role in the global response
of the structure : the vertical tendon is the starting point for the development of
damage and the horizontal bars help its propagation. Moreover, one can notice, just
after the application of the prestress, a small growth of damage along the horizontal
prestress tendons (see figure 27), probably due to the boundary conditions and the
steel - concrete "interface" which is assumed to be perfect.
Concerning the differences between local and non local simulations, studies
have still to be performed. Nevertheless, we face a computational limit due to the
size of the calculation. Using a non local approach with appropriate parameters and
specially an appropriate internal length would require a mesh small enough to
reflect non local interactions. This would correspond to a computation with up to 1
million degrees of freedom, which remains to be performed.
In performing further investigations we have noticed that modelling
reinforcements and prestressed tendons using bar elements is not adequate if
concrete is represented by isoparametric volume elements. Stress singularities arise
which are amplified with finer meshes, thus creating non linearities just by changing
the size of concrete elements for the same load. This was even observed for linear
elastic calculation of the RSV PACE 1300.
6. Conclusions
A set of benchmarks was presented in this paper to propose a gradual validation
of a regularised non local isotropic damage model.
Elementary tests defned the advantages and limits of the constitutive relation. It
is able to reproduce in a good way a simple tension test and triaxial compression with low confinement pressures. Due to the lack of plasticity and crack closure
effect, it fails in simulating a cyclic loading entirely.
Two structural bending applications illustrated the necessity to include a non
local approach in the isotropic model to avoid a spurious strain and damage
localisation. They succeeded in showing how the non local approach was able to
propose a mesh objective numerical response.
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Finally, a pre industrial test was presented in the form of a representative volume of a containment building of a French nuclear power plant. It was shown how the
isotropic damage model was able to provide first information about the global (pressure - displacement curve) and local (damage) responses. Nevertheless, the necessity to carry out further studies was underlined, especially in order to understand the role of the prestressed tendons on damage initiation and to evaluate the benefits of the non local approach. These studies are still in progress with the developments of two tests with a smaller geometry.
Acknowledgements
Partial financial support from the EU through MAECENAS project (FISS-2001-00100) is gratefully acknowledged. The authors would like to thank EDF for scientific support toward the developments in the FE code "Code_Aster". The first author also thanks LaCan laboratory from UPC (Barcelona) for their welcome and their help for part of the computations presented in this contribution.
