1. Damage and Plasticity for Concrete Behavior

European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2004 P. Neittaanmäki, T. Rossi, S. Korotov, E. Oñate, J. Périaux, and D. Knörzer (eds.) Jyväskylä, 24—28 July 2004

DAMAGE AND PLASTICITY FOR CONCRETE BEHAVIOR *,o * † o Ludovic Jason , Gilles Pijaudier-Cabot Antonio Huerta and Shahrokh Ghavamian *

GeM – Institut de Recherche en Génie Civil et Mécanique Ecole Centrale de Nantes – Université de Nantes – CNRS 1, rue de la Noé – BP 92101 – F44300 Nantes, France e-mails : [email protected] [email protected],, [email protected] †

Laboratori de Càlcul Numèric Departament de Matemàtica Aplicada III Universitat Politècnica de Catalunya, Jordi Girona 1-3 E-08034 Barcelona, Spain e-mail : [email protected] o

EDF Recherche et Développement 1, avenue du Général de Gaulle F 92141 Clamart Cedex e-mail : [email protected]

Key words: Damage, plasticity, model, concrete Abstract . Elastic damage models or elastic plastic constitutive laws are not totally sufficient to describe the behavior of concrete. They indeed fail to reproduce the unloading slopes during cyclic loads which define experimentally the value of the damage in the material. When coupled effects are considered, in hydromechanical problems especially, the capability of the numerical model to reproduce the unloading behavior is thus essential, as an accurate value of the damage is needed. A combined plastic – damage formulation is proposed in this contribution. It is applied on simple tension – compression loading to evaluate the ability of the law to simulate elementary situations. Two structural applications are then considered in the form of a composite steel – concrete tube and a representative structural volume of a confinement building for nuclear power plants.

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Ludovic Jason, Gilles Pijaudier-Cabot, Antonio Huerta and Shahrokh Ghavamian

1

INTRODUCTION

Elastic damage models or elastic plastic laws are not totally sufficient to capture the constitutive behavior of concrete correctly. In some cases (using damage mechanics), the calculation of the damage variable (for isotropic cases) or tensor (anisotropic laws) is a key point. It can become essential when coupled effects are considered (coupling between damage and permeability, damage and porosity …). In [1], an experimental law is proposed between the damage distribution in the material and its gas permeability (figure 1). Damage is measured using the unloading slope during cyclic compressive loading. In this case, the capability of the constitutive model to capture the unloading behavior is thus essential if a proper evaluation of the permeability needs to be achieved. An elastic damage model is not appropriate as irreversible strains cannot be captured: a zero stress corresponds to a zero strain and the value of the damage is thus overestimated (figure 2b). An elastic plastic relation is not adapted either (even with softening, see for example [2]) as the unloading curve follows the elastic slope (figure 2c). Another alternative consists in combining these two approaches to propose an elastic plastic damage law. The softening behavior and the decrease of the elastic modulus are reproduced by the damage part while the plasticity effect accounts for the irreversible strains. With this formulation, experimental unloading can be simulated correctly (figure 2a).

K

=

K 0

exp[(11.3 D)1.64 ]

Figure 1: Permeability evolution as a function of the damage value Stress

Stress

Stress

a

b E

E

E

(1-D) E

(1-D) E

c

E

Strain

Strain

Eperimental

Damage model

Strain

Plastic model

Figure 2 : Loading – unloading behavior – Experimental and simulated behaviors

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It is such a model which is presented in this contribution. First, the constitutive law is validated on two elementary tests. A simple tension and a cyclic compression are used to evaluate the capability of the model to simulate simple but relevant applications. Second, two structural tests are considered in the form of a composite steel – concrete tube and a structural representative volume of a containment building for French nuclear power plants. 2 MODEL FORMULATION

Plasticity effects (irreversible strains for example) and damage (softening) are both decribed by the formulation. Nevertheless, they are not entirely coupled. From the total strain tensor ε, an effective stress σ’ is computed from plasticity equations. Then, with the elastic – plastic strain decomposition ( ε = ε e + ε p ), the damage variable D and the real stress σ are calculated. The main advantage of this approach is to fit to numerous constitutive relations. 2.1 Plastic yield surface

In this contribution, the plastic yield surface has been chosen to fulfill three main objectives. First, irreversible effects have to develop during loading (achieved by definition by every plasticity law). Then, the volumetric behavior has to be simulated correctly. Especially, the change from a contractant to a dilatant evolution during simple compression test has to be reproduced. This condition prevents using Von Mises equations, functions of the second stress invariant, that provide an elastic volumetric response. Finally, the brittle – ductile transition has to be reproduced. For high hydrostatic pressures, plastic effects appear experimentally (see [3] for details). It supposes a closed yield surface along the first invariant (plastic threshold in confinement) and eliminates Drücker – Prager equations. The chosen yield surface depends on the three normalized stress invariant ( ρ, ξ ,θ ) and on one hardening internal variable k h ranging from 0 to 1 (definition of a limit surface for k h = 1) [4]

ξ

=

σ 'ii 3r c

ρ

=

s 'ij s ' ji rc

1

3 s 'ij s ' jk s 'ki

3

2 ( s 'ij s ' ji )3/ 2

θ = arcsin(−

)

(1)

with σ ’ij and s’ij the effective and deviatoric stress components respectively. r c is a parameter ˆ (hardening function), ρ (deviatoric of the model. F is defined with three main functions k c invariant) and r (deviatoric shape function): 2

F = ρ (σ σ ') −

kˆ (σ ', k h ) ρ c 2 (σ ' ) r 2 (σ σ ')

(2)

The classical equations of plasticity models are solved using an iterative algorithm based on a Newton Raphson scheme (see [5] for details). Figure (3a) shows the evolution of the yield surface with the hardening parameter in simple compression. Figure (3b) highlights the non symmetry of the plastic law with the Lode angle

3


(in simple compression θ =

π 6

, simple tension θ = −

π 6

and hydrostatic confinement θ = 0 ).

Finally, figure (3c) illustrates the limit surfaces for two values of θ . When the hardening parameter k h reaches its critical level (equal to one), the yield surface becomes a failure one and does not evolve any more.

(a)

(b)

(c) Figure 3: Evolution of the plastic yield surface (hardening, lode angle and f ailure)

2.2.

Damage

The damage model used in this contribution was initially developed in [6]. It describes the constitutive behavior of concrete by introducing a scalar variable D which quantifies the influence of microcracking. To describe the evolution of damage, an equivalent strain εeq is computed from the elastic strain tensor εe e ε ε

= E

−1

σ σ '

(3)

-1

with E the inverse of the elastic stiffness. 3

ε eq

=

 (< ε i > e

+

)2

i =1

e

where <ε i>+ are the positive principal values of the elastic strains.

4

(4)


The damage loading surface g is defined by : g (ε e , D ) = d (ε e ) − D

(5)

 where the damage D takes the maximum value reached by d during the history of loading   D = Max / t (d , 0) . d is computed from an evolution law that distinguishes tensile and

compressive behaviors through two couples of scalars (α ,t Dt ) for tension and (α c , Dc) for compression. d (ε e ) = α t (ε e ) Dt (ε eq ) + α c (ε e )Dc (ε eq )

(6)

The definition of the different parameters can be found in [6]. The damage evolution conditions are finally given by the Kuhn – Tucker expression: 

g

≤ 0,



D ≥ 0,

gD=0

(7)

Once the damage has been computed, the “real” stress is determined using the equation :

σ

= (1 − D )σ '

(8)

3. ELEMENTARY VALIDATION

The elastic plastic damage formulation is now going to be validated on two simple but key applications: tension and cyclic compression. The objective is naturally to compare the numerical results with experiments but also with the elastic damage law for which plastic strains are equal to zero. 3.1. Simple tension 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 concrete is subjected to tension that the first cracks usually appear. That is why the numerical response (elastic plastic damage law) is first compared with such a test [7]. Figure (4a) gives the axial stress – str ain curve. To evaluate the interest of including plasticity in the formulation, a pure damage model is also considered for which the plastic strains are supposed to keep a constant zero value so as the elastic strains equal exactly the total strains (original damage model [6]). Figure (4b) illustrates the simulation with the elastic damage model. As the development of damage is predominant during simple tension tests, the two models are able to reproduce the experiment globally. Especially, the elastic plastic damage constitutive law gives a correct value of the peak position and simulates well the post peak behavior. Choosing the appropriate parameters, the model is thus adapted for simple tension test.

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(a)

(b)

Figure 4 : Simple tension test. Comparison between simulation and experiment for the simple damage and plastic damage laws.

3.2. Cyclic compression simulation

Cyclic compression is the second elementary test. Experimental results are taken from [8]. Figure (5a) illustrates the numerical response without plasticity. With this type of relation, a zero stress corresponds to a zero strain. The unloading curve is elastic with a slope equal to the damage Young’s modulus E d Ed = (1 − D) E 0

(9)

with E0 the virgin Young’s modulus. The numerical response using the elastic plastic damage model is given in figure (6a). This time, damage induces the global softening behavior of concrete while the plastic part reproduces quantitatively the evolution of the irreversible strains. Experimental and numerical unloading slopes are thus similar, contrary to the simple damage formulation response. This difference could seem negligible, it is in fact essential if a correct value of the damage needs to be captured. The elastic damage model overestimates D whereas the full constitutive law provides more acceptable results. Figures (5b) and (6b) illustrate the differences between the two approaches in term of volumetric behaviors. While, with the simple damage law, the volumetric strains are negative, the introduction of plasticity induces a change in the volumetric response, from contractant (negative volumetric strains) to dilatant, a phenomenon which is experimentally observed (see [3] for example) The introduction of plasticity associated with the development of damage plays thus a key role in the numerical simulation of a cyclic compression test. Irreversible strains during loading are quantitatively reproduced, the softening behavior and the unloading slopes are better described. Moreover, the volumetric response that was totally misevaluated by the elastic damage model, is correctly simulated by the full plastic – damage formulation.

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(a)

(b)

Figure 5: Cyclic compression test for the elastic damage law. Axial (a) and volumetric (b) responses

(b)

(a)

Figure 6: Cyclic compression test for the elastic plastic damage model. Axial (a) and volumetric (b) r esponses

4. STRUCTURAL APPLICATIONS 4.1. Circular concrete filled steel tube (passive confinement)

To highlight the interest of the presented constitutive law, the behaviour of a circular concrete filled steel tube (CFT) is going to be simulated. The dimensions and geometry of the sample and the mechanical properties experimentally reported in [9], measured on non wrapped specimens, are listed in table 1. The steel – concrete interface is assumed to be perfect. For the considered compressive strength f’c, Giakoumelis and Lam [10] demonstrate with greased and non greased cylinders, that the steel-concrete interface has little influence on the global behaviour.

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Geometry (mm) D e L 150 4

D

L 450

Steel E s (Pa) 10 21 10

Properties σ e (MPa) 279.9

Concrete E (Pa) 10 2.18 10

Properties ’ f c (MPa) 22

e

Table 1 : Geometry and mechanical properties of the circular CFT as experimentally reported in [8]

A vertical displacement is applied on both steel and concrete plane faces. Two simulations are proposed, the first one using the elastic plastic damage formulation and the second one using the elastic damage constitutive law. The steel tube is modelled with a Von Mises relation (σ e = 279.9 MPa and E t = 2500 MPa for the tangent hardening modulus). One fourth of the cylinder is meshed for a 3D computation. Figures (7a) and (7b) provide a comparison between the simulations with the two approaches and experiments. With the elastic plastic damage relation, numerical and experimental axial forces are in agreement for a given axial strain. The “confinement” effect is highlighted with an increase of the maximal compressive strength (compared to f’c). On the contrary, the damage model underestimates the global behaviour of the column with the apparition of a softening branch which is not observed during measurements. Note that the elastic behaviour does not fit exactly, same as for the simulations performed in [9], it is probably due to experimental differences between the elastic mechanical properties measured on wrapped and non wrapped samples (incomplete hydratation of concrete for example, see [11] for details). Experimentally [8], no confinement effect is noticed at the beginning of the loading. The transversal strain in concrete is lower than in steel due to differences in the Poisson ratio (0.2 and 0.3 respectively). Concrete is thus under lateral tension (figure 8a). As the axial load increases, plasticity is responsible for a change in the Poisson ratio. Lateral expansion in concrete gradually becomes higher than in the steel tube. A radial pressure develops at the interface and a confinement effect appears (passive confinement) (figure 8b). The evolution of the radial stress at the interface as a function of the axial strain is provided in figure (9a) using the full formulation. Figure (9b) illustrates the evolution of the transversal strains εc and εs in concrete and in the steel respectively. Same as what is observed experimentally, concrete is first subjected to tension and then to compression. The change in the sign occurs immediately when εc becomes higher than εs. On the contrary, with the simple elastic damage constitutive law, as εc is always lower than εs, the concrete is only loaded in lateral tension. No passive confinement is observed and that is why the peak in the axial load is so small and inadequate compared with experiment (figures 10a and 10b). The study of the volumetric behaviour yields the same conclusions (figure 11). The change from a contractant to a dilatant evolution obtained with the i ntroduction of plasticity is a direct consequence of the increase of the concrete transversal strains. With the elastic damage relation, the volumetric response is always contractant.

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As a conclusion, the elastic plastic damage constitutive law i s necessary to achieve the change of Poisson ratio that accounts for the passive confinement. For concrete filled steel tubes subjected to axial compressive loading, it is possible to reproduce experimental results correctly and especially the evolution of the axial capacity.

(a) a

b.

(b)

Figure 7 : Simulation of a CFT (a) Evolution of the axial force as a function of the axial strain. Comparison between the elastic plastic damage formulation and the experiment; (b) Comparison between the elastic plastic damage and the elastic damage models

σ r

concrete

σ r steel

εc

< ε s

Figure 8 : Evolution of the radial “pressure” (

εc σr )

> ε s

as a function of the steel ( strains.

εs )

and concrete (

(a)

9

εc )

transversal


(b)

(c)

Figure 9 : Transversal behaviour using the elastic plastic damage model (a) Evolution of the concrete transversal stress as a function of the axial strain; (b) Evolution of the transversal concrete and steel transversal strains as a function of the axial strain; (c) Zoom on the first part of the curve.

(a)

(b)

Figure 10 : Transversal behaviour using the elastic damage model (a) Evolution of the transversal stress as a function of the axial strain, (b) evolution of the concrete and strain transversal strains as function of the axial strain

Figure 11 : Evolution of the vo lumetric strain as a function of the axial force for both approaches.

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4.2. Representative Structural Volume of a containment building

The application presented in this part has been proposed by Electricité 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 12 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 [12] 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. 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 13. Figure 14 provides information about the steel distribution and properties.

Figure 12. Position of the extracted Representative Structural Volume (RSV).

Figure 13. Geometry of the Representative Structural Volume (RSV)

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Type

Horizontal internal reinf. bars Horizontal external reinf.bars Vertical internal reinf. bars Vertical external reinf. bars Horizontal tendons Vertical tendons Hoops*

R

e

D

m

cm

mm

22.60 23.35 22.60 23.35 23.15 22.95 x

20 20 27.297 27.170 40.5 80 x

20 20 25 25 40.5 40.5 3.685

* Hoops are uniformly distributed in the representative volume Figure 14 : Steel geometries and properties

Figure 15: Definition of the F.E. model in dicating the boundary SG, SD, SH and SB

12


PSV

WE

BE

RO

ND

z θ

ND

g

PSH

PSH

IP

z r

VD PSV

Boundary conditions in displacement ND : zero normal displacement on SG and SD NV : zero vertical relation on SB RO : zero rotation on SH

Boundary conditions in stress PSH : horizontal prestress 5.28 MN per tendon PSV : vertical prestress 6.93 MN per tendon WE : weight of the surrounding structure 1.61 MPa : ravit

Loading IP : internal pressure BE : tensile pressure proportional to the internal pressure (bottom effect)

Figure 16. 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 15 and 16). 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 and to faces SB and SH for vertical tendons, then prestressed using internal forces. Then, these elements are restrained to surrounding concrete elements to represent the prestressing technology applied in French PPCVs. The integrity test 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 containing 16,500 Hexa20 elements for concrete and 1200 bar elements for reinforcement and tendons is used in the presented computation. Figure 17 provides the internal pressure applied on the volume as a function of the displacement of a point located at the bottom right of the internal face, using the elastic plastic damage formulation. 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

13


due to the boundary conditions (no normal displacement on the lateral face), it imposes an initial negative radial displacement (see figure 18). Then, upon application of 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 development of damage occurs during state 3. Finally, a partial unloading of the volume occurs (state 4) due to heavy cracking of the structure.

State 3

State 1

State 2

State 4

Figure 17: Displacement – Pressure curve for the representative structural volume Step 2 : Application of the internal pressure Step « zero » Step 1 : Initial state after the application of the prestress

Figure 18: Radial deflection of the RSV through different steps (schematic). View from the top of the volume

Figure 19 describes the distribution of the damage variable in the volume for two different loading steps (just after the apparition of non linearity and after the peak). Damage initially develops along the vertical tendon which is located in the middle of the mesh. Then, it propagates in the depth of the volume and along the vertical axis. It finally forms a localized

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damaged zone in the middle of the structure. Similar results can be found in [13] with the elastic damage model (without plasticity). It tends to prove that for this tension dominant application, the introduction of plasticity does not disturb the development of damage. However, it is the amount of damage at a given loading state which is different according to the damage and plastic damage models. In the second one, damage is lower than in the first one and effects on the material permeability are drastically different (see figure 1). It follows that evaluation of leakage according to these two models are expected to be very different too.

Figure 19 : Damage distribution in the representative structural volume. Initiation of the damage and localized band. The blacker the zone is, the larger the damage is.

5. CONCLUSION

An elastic plastic damage formulation has been proposed and tested on both tension and compression applications. It has been shown that the constitutive law proposes the same advantages as the elastic damage model for tension dominant cases but also improves the constitutive response when compression is considered. For simple tension test, the law is able to simulate both peak and post – peak (especially softening) behaviors. The choice for the plastic yield surface enables to reproduce qualitatively and quantitatively the axial and volumetric responses of a concrete cylinder loaded in compression. Particularly, the development of irreversible strains and the change from a contractant to a dilatant volumetric evolution are correctly simulated. For structural applications, the improvements are also highlighted. Including plasticity is the most appropriate solution to achieve the passive confinement effect of a concrete filled steel tube (increase in the Poisson ratio). Finally, for the representative structural volume, the development of damage is correctly described, following the same path as previously mentioned in former studies. As a conclusion, this constitutive law may represent an appropriate tool to simulate the experimental damage value and may be used for coupled problems (hydro mechanical simulations) for example.

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6. AKNOWLEDGMENTS

Partial financial support from EDF and from the EU through MAECENAS project (FIS52001-00100) is gratefully acknowledged. The authors particularly thank R. Crouch (University of Sheffield) for his help in the design and the numerical development of the plasticity model. The authors would like to thank EDF for scientific support toward the developments in the FE code “Code_Aster”. 7. REFERENCES

[1] V. Picandet, A. Khelidj, G. Bastian, Effect of axial compressive damage on gas permeability of ordinary and high performance concrete, Cement and Concrete Research, 31, 1525-1532, 2001 [2] P. Grassl, K. Lundgren, K. Gylltoft, Concrete in compression: a plasticity theory with a novel hardening law, International Journal of Solids and Structures ,80, 5205-5223, 2002 [3] D. Sfer, I. Carol, R. Gettu, G. Etse, Study of the behavior of concrete under triaxial compression, Journal of Engineering Mechanics, 128, 156-163, 2002 [4] G. Etse, K.J. Willam, Fracture energy formulation for inelastic behaviour of cracking concrete, ASCE Journal of Engineering Mechanics, 106, 1013-1203, 1994 [5] A. Perez-Foguet, A. Rodriguez-Ferran, A. Huerta, Numerical differentiation for non trivial consistent tangent matrices: an application to the MRS – Lade model, International Journal for Numerical Methods in Enginnering , 48, 159-184, 2000 [6] J. Mazars, Application de la mécanique de l’endommagement au comportement non linéaire et à la rupture du béton de structure, PhD Thesis, université Paris VI, 1984. [7] V.S Gopalaratnam, S.P. Shah, Softening response of plain concrete in direct tension, ACI Journal, 310-323, 1985 [8] B.P. sinha, K.H. Gerstle, L.G. Tulin, Stress strain relations for concrete under cyclic loading, Journal of the American Concrete Institute, 195-211, 1964. [9] K.A.S. Susantha, H. Ge, T. Usami, Uniaxial stress strain relationship of concrete confined by various shaped steel tubes, Engineering Structures, 23, 1331-1347, 2001 [10] G. Giakoumelis, D. Lam, Axial capacity of circular concrete filled tube columns, Journal of Constructational Steel Research, in press, 2004 [11] M. Kwon, E. Spacone, E., Three – dimensional finite element analyses of reinforced concrete columns , Computers and Structures 80, 199-212, 2002. [12] Castem, Castem 2000, User’s guide, CEA, DMT, LAMS , 1993 [13] L. Jason, S. Ghavamian, G. Pijaudier-Cabot, A. Huerta, Benchmarks for the validation of a non local damage model, Revue Française de Génie Civil, in press

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1. Damage and Plasticity for Concrete Behavior

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