Continuum Damage Mechanics for hysteresis and fatigue of quasi-brittle materials and structures

Transcription

1 Continuum Damage Mechanics for hysteresis and fatigue of quasi-brittle materials and structures R. Desmorat, F. Ragueneau, H. Pham LMT-Cachan, ENS Cachan/Université Paris 6/CNRS, 61, av. du président Wilson, F CACHAN Cedex, FRANCE SUMMARY For material exhibiting hysteresis such as quasi-brittle materials, it is natural to consider that hysteresis and fatigue are related to each other. One shows in the present work that damage, from the Continuum Damage Mechanics point of view, may be seen as the link between both phenomenon. One attempts hence to set up a unified modeling of hysteresis and damage. Numerical examples are given for concrete and validate the proposed model of internal sliding and friction coupled with damage. The problem of a proper phenomenological modeling of the micro-defects closure effect leading to a dissymetric tension/compression response and to stiffness recovery in compression is also adressed. Cyclic and fatigue applications are in mind but also random fatigue and seismic responses. Copyright c 2006 John Wiley & Sons, Ltd. key words: damage, fatigue, hysteresis, concrete, unilateral conditions INTRODUCTION Models for concrete structures subject to complex loadings, monotonic or not, are represented by constitutive equations often written in a rate form and in the thermodynamics framework [1, 2, 3, 4, 5, 6, 7, 8] when fatigue is usually addressed with specific engineering rules modeling directly the Wöhler curves of materials [9, 10, 11, 12]. A fatigue law for concrete can be a straight line in the maximum applied stress σ Max vs the logarithm of the number of cycles to rupture log N R diagram, generally parametrized by the stress ratio R σ = σ min /σ Max (with σ min the minimum applied stress). In order to reproduce the effect of different mean stresses on Wöhler curves, amplitude laws function of the stress ratio are also often considered in fatigue damage models [13, 14, 15]. The damage increment per cycle δd δn is set as a function of the current damage D, of the stress amplitude σ = σ Max σ min and of R σ. The extension to 3D states of stresses of such modeling is not straightforward. What is then a stress amplitude? How to define a stress ratio? Neither is straightforward the extension to strain controlled tests and to non cyclic loadings encountered in random fatigue or in seismic applications. And the link between a stress and a strain formulation is not so clear for nonlinear materials except if a rate written damage model is defined [16, 17, 18, 19, 20, 21, 22]. The objective of the present work is to model the behavior of quasi-brittle materials with a single set of constitutive equations valid with the same material parameters for monotonic, hysteretic and dynamic loading (at not too high strain rates) but also in fatigue. The damage law established has to allow for damage accumulation under cyclic behavior, a low number of random cycles corresponding in fact to earthquake response and the high number of cycles to fatigue. The damage model proposed next is based on the main dissipative phenomenon activated during unloading-reloading: friction beween cement paste and aggregates, between agregates, between Correspondence to: desmorat@lmt.ens-cachan.fr, tel: , fax:

2 CONTINUUM DAMAGE MECHANICS FOR HYSTERESIS AND FATIGUE 1 microcracks lips, i.e. friction at microscale also referred to as internal friction. Different approaches for internal sliding and friction have been developped: plasticity based on dislocations slips but with no hysteresis without stress sign change, microplane modeling for concrete [23], micromechanics of sliding and friction in long fiber composite materials [24], meso-modeling of microcraking in concrete [5, 25], macroscale representation with phenomenological models for concrete [8] or elastomers [26]. Except the last work on internal sliding and friction of filled elastomers, the models cited do not apply to fatigue. Hysteretic dissipation and damping are closely related, the challenges of numerical computations being to address structural dynamics with no need of a fictitious Finite Element viscous damping matrix, defined at the structure scale. Moreover, Continuum Damage Mechanics is an adequate tool to describe the eigenfrequency decrease of Civil Engineering cracked structures subject to severe loading. In the present work, one will attempt last to gain at the Representative Volume Element (RVE) scale i.e. from the proposed constitutive equations the damage dependency of the global damping parameter for concrete structures. Multifibre beam analysis will be used for application to plain concrete and reinforced concrete structures. 1. MODELING HYSTERESIS OF MATERIALS In order to describe the macroscopic mechanical behavior of quasi-brittle materials like concrete, one has to account for several mechanisms at the heterogeneity level. The crack initiation and growth lead to a decrease of macroscopic Young s modulus. The unilateral behavior of a crack bearing cyclic loading is the source of damage deactivation and stiffness recovery. The roughness of the cracked surfaces as well as the aggregates interlock generates anelastic strains and dilatancy. Under reverse loading, this roughness imposes frictional sliding behavior of the of microcracks lips and of aggregates contacts whose main macroscopic consequence is the experimental observations of hysteresis loops in compression, tension and torsion Micromechanics approaches The behavior of interacting frictional flaws in solid media has been extensively studied in analytical ways [27, 28, 29, 32, 30] or numerically such as by use of the Boundary Element Methods [31]. These different analyses show the importance to account for friction on cracked surfaces, not only to recover the hysteretic loops during cyclic loading but also to accurately describe the mixed mode propagation of a crack in a quasi-brittle material. At the macroscopic level of RVE of continuum mechanics, several models account for these different mechanisms and some of their coupling but only a few handle the frictional sliding behavior coupled with damage. The micromechanics explanations are rather complicated in a 3D framework, preventing a simple and robust expression of constitutive equations at the macroscopic level, necessary feature for numerical analyses of structures. The derivation of the further thermodynamics free energy needs the definition of internal variables. To be relevant, the choice of internal variables has to be motivated by micromechanics evidences and mechanisms [16]. The micromechanics analysis of cracked representative elementary volume allows for the expression of free energy for some particular case studies [32, 24]. The resultant thermodynamics potential is divided into two parts: the elasticity of the cracked matrix and the stored or blocked energy density due to kinematic hardening or friction. Pushed forward in 3D, the homogenization process allows to find the expression of the free energy at the RVE scale accounting for induced anisotropy with open or closed crack conditions [25]. Based on the spectral decomposition of the crack density, induced anisotropy coupled with closed microcracks friction can be recovered [5]. An adequate choice for the invariants of the strain and damage tensors and for their coupling allows for the description of hysteresis loops under load reversal. For cyclic loading, the spectral analysis of the cracked medium leads to the introduction of a fourth order damage tensor in order to ensure continuity of the stress-strain path with the unilateral conditions

3 2 R. DESMORAT, F. RAGUENEAU, H. PHAM of microcracks closure, even if the damage state is initially represented by a second order tensor [34, 5] General form of the macroscopic free energy For efficiency and robustness requirements of the most importance in fatigue implying numerous cycles of loading, the choice of a scalar damage variable is appropriate. To deal with the cracked matrix modeling, Continuum Damage Mechanics proves to be a relevant tool, introducing as phenomenological damage variable the scalar D. For the definition of a thermodynamics potential in adequation with the expressions proposed by Hild and coworkers [24] for composites, Dragon and Halm [5] and Ragueneau et al. [8] for concrete, and by Desmorat and Cantournet [26] for filled elastomers, a general expression is (uniaxial case): ρψ = W 1 (ε, D) + W 2 (ε, ε π, D) + w s (1) where W 1 is a purely elastic part of the free energy, W 2 is an anelastic one and w s is the stored energy density. In 1D the strain ε is the macroscopic total strain and ε π is the macroscopic internal frictional sliding strain. Not based on the classical principle of strain additivity, the thermodynamics variables σ and σ π, associated with the total strain and with the internal frictional sliding strain, are not equal. Note also that ε π is not the plastic or permanent strain. Classically, the elasto-damage coupling is expressed using the following expression for W 1, W 1 = 1 2 E 1(1 D)ε 2 (2) where E 1 is an elasticity modulus eventually equal to the Young s modulus E and where D is the damage variable ranging from 0 for the virgin material to the critical damage D c < 1 for the completely broken one. Following the concept of splitting the free energy into parts, two kinds of partitioning may be adopted for W 2. In the first one, W 2 is directly linked to the level of damage by W 2 = 1 2 DE(ε επ ) 2 [5, 8]. This expression induces a coupling of the energy dissipated trough frictional sliding to the state of cracking. Only one Young s modulus may be introduced by setting E 1 = E. The Cauchy stress is obtained by derivation with respect to the total strain: σ = ρ ψ ε = E(1 D)ε + σπ with σ π = ρ ψ ε = DE(ε ε π ). For a state of cracking (or damage) approaching to D π c 1, this model converges toward a classical plasticity model implying only a frictional sliding behavior σ σ π. Such an approach for the coupling between friction and damage is relevant for instance in the case of bond-slip modelling of reinforced concrete element [33] in which the asymptotic behavior is sliding. For plain concrete in pure tension or compression, the asymptotic behavior should lead to rupture: for D 1, the total stress σ should vanish. The frictional sliding strain can then be expressed as σ π = E 2 (1 D)(ε ε π ) but there is no guide to say that the elasticity loss occurs at the same rate than for the energy part W 1. To conserve the initial stiffness of the virgin material equal to the Young s modulus E, the relationship E = E 1 + E 2 is set. Note that it is in fact quite natural to introduce two Young s modulus in heterogeneous materials as concrete, one for its hard phases (aggregates,...) and one for its soft phases (cement,...). The complete free energy takes the general form: ρψ = 1 2 E 1(1 D)ε g(d)e 2(ε ε π ) 2 + w s (3) where g(d) = 1 D here but where for the general case many other expressions seem possible such as g(d) = a + bd or g(d) = 1/(α + βd) with a, b, α, β as algebraic material dependent parameters. The stresses are obtained as follows: σ = ρ ψ ε = E 1(1 D)ε + σ π σ π = ρ ψ (4) ε π = g(d)e 2(ε ε π ) As one can see for g(d) = 1 D, as D tends toward 1, the stress σ tends toward 0, corresponding to a fully broken material.

4 CONTINUUM DAMAGE MECHANICS FOR HYSTERESIS AND FATIGUE 3 2. GENERALIZED DAMAGE LAW From the thermodynamics of irreversible processes, a damage variable is a state variable representing the fact that microcracks and/or micro-voids are present at microscale. A phenomenological modeling avoids to have to deal with each micro-defect and considers those as averaged on the RVE of continuum mechanics. The damage pattern of an initially isotropic material can be represented in the more general case by a fourth order tensor [34, 16] but for practical reasons a second order tensor D or a scalar variable D are most often used [35, 36, 37, 38, 22]. In any case, the damage variable represents the micro-cracking pattern. Then for a given population of micro-defects the only possibility is to consider one thermodynamics damage variable, whatever the sign of the loading. This is for instance by its coupling with the elasticity and evolution laws that damage acts differently in tension and in compression. One knows from modeling damage and fatigue of metals that this way to proceed is efficient [39, 40] and leads to a single damage evolution law for monotonic, fatigue, creep, creep-fatigue applications [18, 41, 22], ( ) s Y Ḋ = ṗ (5) S where damage is governed by plasticity (through the accumulated plastic strain p) and enhanced by the strain energy (through the strain energy density release rate Y ). Two damage parameters are introduced: the damage strength S and the damage exponent s. The second is related to the slope of the Wöhler (fatigue) curve of the material with usually s 1. The strain energy density release rate Y is the associated variable with D and is derived from the Helmholtz free energy ρψ function of the strains or from the Gibbs free enthalpy ρψ function of the stresses as Y = ρ ψ or Y = ρ ψ (6) D D In the simple uniaxial case where ρψ is [42] ρψ = σ 2 + 2E(1 D) + σ 2 2E(1 hd) with h the microdefects closure parameter (0 < h < 1) and x + (resp. x ) standing for the positive (resp. negative) part of the stress ( x + = x if x > 0, x + = 0 else, x = x + ). The elasticity law reads ε = ρ ψ σ = σ + E(1 D) + σ (8) E(1 hd) or σ = E(1 D)ε in tension and σ = E(1 hd)ε in compression so that there is partial stiffness recovery in compression (quasi-unilateral conditions). The recovery is full if h = 0 (unilateral conditions). The strain energy density release rate is derived as Y = σ 2 + 2E(1 D) 2 + h σ 2 2E(1 hd) 2 (9) so that for the same strain level (in absolute value) it is h times smaller in compression than in tension. Considering the damage evolution law (5) gives a damage rate even smaller in compression as then: Ḋ compression h s Ḋ tension << Ḋtension (10) which leads to a different behavior in tension and in compression and to both a stress dependent damage rate and to the representation of the mean stress effect in fatigue. The damage law (5) has also been used for filled elastomers undergoing large strains (a few hundred of percents). For elastomeric materials plasticity is often meaningless so that the law has been generalized into [43] ( ) s Y Ḋ = π (11) S (7)

5 4 R. DESMORAT, F. RAGUENEAU, H. PHAM corresponding to a damage rate governed by the main dissipative mechanism of internal friction encountered in these materials.the damage evolution law (11) generalizes to non metallic materials the initial law (5). With an adequate definition of the cumulative measure of the internal sliding π, the law (11) will be derived and used next for quasi-brittle materials. We will refer to it as the generalized damage law. The damage exponent s will next be taken equal to the usual default value s = 1 [16]. 3. DAMAGE MODEL WITH INTERNAL SLIDING AND FRICTION As mentioned in section 1, an adequate choice for the state or thermodynamics potential allows for the modeling of hysteresis loops in tension without having to undergo compression. The choice made here is to consider the general form (3) with g(d) = 1 D but extended to 3D introducing the strain ǫ and as internal variables the irreversible strain ǫ π, the damage D and if necessary the consolidation variables r (scalar) and α (tensorial). The state and evolution laws classicaly derive from the thermodynamics and dissipation potentials [44]. Concerning damage, the choices made must finally recover the generalized damage law (11) Model in the thermodynamics framework The state potential is considered quadratic, ρψ = 1 2 (1 D) ǫ : E 1 : ǫ (1 D)(ǫ ǫπ ) : E 2 : (ǫ ǫ π ) Kr2 + 1 Cα : α (12) 2 where E 1 and E 2 are elasticity tensors such as the sum E 1 + E 2 = E is the Hooke tensor of the undamaged material. The last terms of eq. (12) are usually grouped into w s = 1 2 Kr Cα : α, the stored energy density, remaining small in quasi-brittle materials and often neglected. The first two terms of eq. (12) are the elastic energy density ρψ e so that ρψ = ρψ e + w s. The state laws are then derived as: σ = ρ ψ ǫ = E 1 (1 D) : ǫ + E 2 (1 D) : (ǫ ǫ π ) σ π = ρ ψ ǫ π = E 2 (1 D) : (ǫ ǫ π ) R = ρ ψ r = Kr X = ρ ψ α = Cα Y = ρ ψ D = 1 2 ǫ : E 1 : ǫ (ǫ ǫπ ) : E 2 : (ǫ ǫ π ) defining (σ, σ π, R, X, Y ) as the associated variables with (ǫ, ǫ π, r, α, D). And note that an equivalent stress formulation can be proposed in terms of Gibbs free enthalpy ρψ = ρψe + w s, built from the Legendre transformation ρψe of the elastic energy density, Effective stresses σ = laws read ρψ e = (σ σπ ) : E 1 1 : (σ σ π ) 2(1 D) σ 1 D and σ π = σπ 1 D σ = E 1 : ǫ + E 2 : (ǫ ǫ π ) σ π = E 2 : (ǫ ǫ π ) + σπ : E 1 2 : σ π 2(1 D) (13) (14) can then be defined so that the first two state A criterion function f is defined next in order to govern the loading/unloading conditions. A framework similar to plasticity is used with then no viscosity and no loading rate effect: the (15)

6 CONTINUUM DAMAGE MECHANICS FOR HYSTERESIS AND FATIGUE 5 consistency condition f < 0 or f < 0 corresponds to the elastic stage, the internal variables keeping then a constant value, the condition f = 0 and f = 0 corresponds to the irreversible behavior including internal sliding and damage. The choice for the irreversibility function is similar to the yield function in plasticity, using the effective stress concept [45], f = σ π X R σ s (16) but it is written in the σ π -plane. R and X are respectively the isotropic and the kinematic consolidations, σ s is the reversibility limit and. is a norm not specified as long as uniaxial behavior is considered. The evolution laws derive from a dissipation potential F through the normality rule. The damage model is non associated as F is the sum f + F D with F D Lemaitre s damage potential, F D = Y 2 2S(1 D) (17) The normality rule reads, introducing the internal sliding multiplier λ, ǫ π = λ F σ π = ṙ = λ F R = λ α = λ F X Ḋ = λ F Y = λ 1 D σ π X σ π X F π = λ = (1 D) ǫ σ λ ( ) Y = 1 D S ( ) Y π S (18) and leads to the cumulative measure of internal sliding π = t 0 ǫ π dt as: π = ǫ π = ṙ 1 D = λ 1 D (19) As wished, the last equation of the set of evolution laws (18) recovers the generalized damage law (11). Written Ḋ = (Y/S) π, it models a damage governed by the main dissipative mechanism, here internal sliding and friction (through π), and enhanced by the value of the strain energy density (through Y ). Written π = (S/Y )Ḋ, it models the increasing internal sliding due to damage accumulation. As the constitutive equations describing both damage and internal sliding are fully coupled, the interpretation is of course a combination of both points of view: each phenomenon, damage or internal sliding, enhances the other one. Note that using the state law X = Cα allows to write the linear kinematic consolidation law in a more classical form ẊX = C(1 D) ǫ π, similar to Prager linear kinematic hardening law of metals. Last, the positivity of the intrinsic dissipation D is satisfied for any kind of loading, monotonic, cyclic or random, uniaxial or 3D, as D = σ : ǫ ρ ψ = σ π : ǫ π Rṙ X : α + Y Ḋ from Clausius- Duhem inequality and one has ( D = σ π F : σ π + R F R + X : F X + Y F ) λ 0 (20) Y and D 0 when the dissipation potential F(σ π, R, X, Y ; D) is a convex function of its arguments σ π, R, X, Y with F(0, 0, 0, 0; D) = 0 and where the damage D acts as a parameter. The consideration of the evolution laws altogether with the contition f = 0 allows to derive a more practical expression, D = σ s (1 D) ǫ π + Y Ḋ 0 (21)

7 6 R. DESMORAT, F. RAGUENEAU, H. PHAM 3.2. Numerical scheme for non monotonic applications The previous set of constitutive equations is a set of first order differential nonlinear equations. The good thing is that the damage model obtained is written in a rate form so that it applies to any kind of loading, not necessary cyclic. The drawback is that one has to perform the time integration over each time step of the differential equations. This can be too costly for fatigue applications if local Newton or quasi-newton iterations are needed. For seismic and fatigue applications, even non cyclic such as random fatigue, an efficient numerical scheme can be proposed with the realistic assumption of a damage quasi-constant over a time increment, classical assumption for fatigue calculations with Continuum Damage Mechanics [18]. In Finite Element computer codes, the local time integration subroutine has for input at time t n+1 the strain ǫ n+1 = ǫ n + ǫ where the quantity ǫ n denotes the strain at time t n and ǫ the strain increment. These quantities as well as all internal variables are known at time t n. To perform the time integration of the constitutive equations means to determine all quantities at time t n+1 (also the damage D n+1 ). Having also in mind multifibre beam analyses of concrete reinforced structures for which the knowledge of the uniaxial stress-strain response is sufficient, the numerical scheme is presented next in 1D, the tensors E 1, E 2 becoming the scalars E 1, E 2. As classically done for nonlinear constitutive models, an elastic prediction is first made and gives an estimate of the stresses σ and σ π, σ = E 1 (1 D n )ǫ n+1 + E 2 (1 D n )(ǫ n+1 ǫ π n) σ π = σπ = E 2 (ǫ n+1 ǫ π n 1 D ) (22) n If f = σ π X n R n σ s 0 then the loading state is elastic and the internal variables at time t n+1 remain equal to those at time t n : ǫ π n+1 = ǫ π n, r n+1 = r n (and so R n+1 = R n ), α n+1 = α n (and so X n+1 = X n ), D n+1 = D n. If f = σ π X n R n σ s > 0 one needs to correct the value of the stresses and of the internal variables by performing the time integration of the constitutive equations. This is the internal sliding damage correction. If Euler implicit scheme is used, this is done by determining first the cumulative internal sliding increment π from the consistency condition f n+1 = σ π n+1 X n+1 Kr n+1 σ s = 0 rewritten for the uniaxial case as f n+1 = E 2 (ǫ n+1 ǫ π n ) X n (E 2 + C(1 D n )) π sign( σ π X) R n K r σ s = 0 (23) σ π n+1 = E 2(ǫ n+1 ǫ π n+1 ) = E 2(ǫ n+1 ǫ π n ) E 2 ǫ π (24) The internal sliding multiplier λ = r is the positive solution of f n+1 = 0 in which the damage is taken as D D n (consistent assumption in fatigue as the damage does not change much over one cycle, therefore even less over one time increment). The increment π is gained as π = E 2(ǫ n+1 ǫ π n ) X n R n σ s E 2 + (K + C)(1 D n ) Once π is known, all the variables including D may be updated as follows: (25) irreversible strain: ǫ π n+1 = ǫ π n + π sign( σ π X), strain energy density release ( rate: ) Y n+1 = 1 2 E 1 : ǫ 2 n E 2(ǫ n+1 ǫ π n+1 )2, Yn+1 damage: D n+1 = D n + π, S isotropic consolidation: r n+1 = r n + (1 D n+1 ) π, R n+1 = Kr n+1, kinematic consolidation: X n+1 = X n + C(1 D n+1 ) ǫ π, friction stress: σn+1 π = E 2(1 D n+1 )(ǫ n+1 ǫ π n+1 ), stress: σ n+1 = E 1 (1 D n+1 )ǫ n+1 + σn+1 π. To conclude, this is an explicited solution of the implicit discretized equations. The scheme has then the robustness of the implicit schemes but with the efficiency of explicit ones. For cyclic

8 CONTINUUM DAMAGE MECHANICS FOR HYSTERESIS AND FATIGUE 7 loadings a few time increments must be used to describe a whole cycle. Note then that in order to fasten the computations, it may be very efficient to activate a jump-in-cycle procedure [39, 22] which avoids the calculation of all the cycles. 4. HYSTERESIS AND FATIGUE OF QUASI BRITTLE MATERIALS Once the damage model with internal sliding and friction established, it has to be confronted with experiments. The identification of the material parameters E 1, E 2 for elasticity (with as Young s modulus E = E 1 + E 2 ), σ s as irreversibility threshold, S for damage and eventually K and C for the stored energy is not an easy task as there is a full coupling internal sliding/damage evolution. The difficulty is mainly due to the physical feature of hysteresis: the unloadings are not straight lines in the stress-strain diagram making hard the measurement of the internal variables. Note for instance that the irreversible strain ǫ π derived from first equation of (13) rewritten σ = E(1 D)ε E 2 (1 D)ε π = E(1 D)(ε ε an ) (26) is not the permanent strain ǫ an. It is related to ǫ an through the elasticity parameters as ǫ π = Eǫ an /E 2, expression which would have been damage dependent with other choices for the g(d) function. The identification is also not straightforward because one must have in mind the fatigue applications for which the elasticity limit usually defined by damage models becomes the asymptotic fatigue limit σf. From the elasticity law and the definition of the irreversibility criterion one has: σ f E = σ s (27) E 2 which forces σ s to remain small compared to the ultimate or peak stress. The material considered next is concrete. The tensile strength is f t = 4 MPa. In compression, the testing specimen has been loaded under lateral strain control preventing early localization modes. The measured compressive strength is f c = 48.5 MPa and two sets of material parameters are obtained, either with zero values for the consolidation parameters K and C or with non zero K and C (linear consolidations). Set of parameters for tension: E 1 = MPa, E 2 = MPa, σ s = 1 MPa, S = 0.3 MPa, K = 3000 MPa C = 1000 MPa. Set (a) of parameters for compression: E 1 = MPa, E 2 = MPa, σ s = 9 MPa, S = 324 MPa, K = C = 0. Set (b) of parameters for compression: E 1 = MPa, E 2 = MPa, σ s = 6 MPa, S = 476 MPa, K = 130 MPa, C = 110 MPa Hysteresis loops of concrete in compression The model represents the stress-strain response as well as the hysteresis of concrete (Figures 1a and 1b), not perfectly for the hysteresis loops because of the simple modeling of the consolidations R and X (zero or linear) but well enough to envisage the calculations of the fatigue curve of the material: one will have then a unified model for both monotonic and fatigue applications using a single set of material parameters. The hysteresis loops are better represented by use of linear consolidations (case b), but because of a small value for the irrevesibility threshold σ s (better for fatigue), the first loading is not. Details on the model response are given in Figure 2 for the two sets of material parameters. First, for both identifications, one can see the importance of the splitting of the stress into two parts, the elasto-damage stress (σ σ π ) and the friction stress σ π, the second being responsible for the hysteresis and tending to symmetrize. The damage evolves of course more at the high level of compressive stresses (Fig. 3), but evolves also a little during the second (nonlinear) part of each unloading due to intenal sliding reactivation before the minimum stress is reached (friction mechanism). Second, one can see the role played by both the isotropic and kinematic consolidations.

9 8 R. DESMORAT, F. RAGUENEAU, H. PHAM (a) Identification with K=C=0 (b) Identification with linear consolidations Figure 1. Hysteretic response of concrete in compression The splitting between elasto-damage stress and friction stress is different for each two cases, the case (b) leading to cycles of almost constant size more in accordance with fatigue phenomenon. The isotropic consolidation increases the size of the reversibility domain, it is therefore a counterpart to damage which decreases it; it allows for the height of the hysteresis loops to vary slowly. The kinematic consolidation models a stress translation of the reversibility domain. It allows then for a dissymmetry of the friction stress. Figure 2. Evolution of elasto-damage stress (σ σ π ) and friction stress σ π 4.2. Computed fatigue curves Two kinds of fatigue responses can be addressed: the response to a cyclic applied stress between a minimum stress σ min (minimum in absolute value) and a maximum stress σ Max and the response to a cyclic applied strain between ǫ min and ǫ Max. The curves σ Max vs the number of cycles to rupture N R are the Wöhler curves of the material. The curve ǫ Max vs N R is more adapted to softening materials for which experimental testing must be strain controlled. It is simply called the fatigue curve of the material which can eventually be plotted in terms of maximum stress but defined as the stress level obtained at the first loading.

10 CONTINUUM DAMAGE MECHANICS FOR HYSTERESIS AND FATIGUE 9 Figure 3. Damage growth during hysteretic loading The proposed damage model with internal sliding and friction allows for the step by step computation of such curves from the knowledge of the hysteretic response of the material. The time integration is performed by use of the numerical scheme of section 3.2 and the set of parameters are the sets (a) with no consolidations and (b) for linear consolidations. Consider an applied strain which varies cyclically between a minimum strain ǫ min and a maximum strain ǫ Max (maximum in absolute value). Due to the choice of the model (no dissymmetry tension/compression) and of the material parameters the qualitative interpretations of the computations performed address compressive fatigue of concrete. The damage D is calculated at each time increment t n and rupture is assumed to occur when D reaches the critical value D c (two values for D c are considered first, D c = 0.5 and 0.9). The number of cycles to rupture N R is then defined from the Continuum Damage Mechanics point of view by the number of cycles N at which D = D c, N R = N(D = D c ) (28) The computed fatigue curves with the previous sets of material parameters (a) and (b) are compared in Figure 4 for the loadings ǫ min = 0 and ǫ min = ǫ Max. The modeling with linear isotropic consolidation R = Kr, better for the representation of the hysteresis loops, leads to an asymptotic fatigue limit much larger than with the simple modeling R = 0. This is due to the high value of the cumulative internal sliding π in fatigue so that at in the linear consolidation case R = 15.1 MPa for N = 100 cycles (loading ǫ min = 0) to be added to the initial irreversibility limit σ s = 6 MPa and to be compared to the irreversibility limit for the no consolidations case σ s = 9 MPa. A better modeling in fatigue will then generally be obtained with no isotropic consolidation (R = 0). Note that the difference between the two different values D c = 0.5 and D c = 0.9 is within the dispersion always large in fatigue. Also, the computations performed with two time discretizations of the applied cycles, a first one with 10 time increments per cycle, a second one with 50 time increments per cycle, give close results. This convergency feature emphasizes the efficiency of the numerical scheme proposed. The fatigue curve with D c = 0.9 of Fig. 4a is replotted in terms of normalized stress σ Max /f c vs number of cycles to rupture N R (with f c = 48.5 MPa the compressive strength, Fig. 5). The model corresponds to the dot lines, black for the symmetric loading, grey for the ǫ min = 0 loading. The results for ǫ min = 0 give a lower bound of the experimental data for concrete tested in fatigue with a zero stress ratio R σ [46, 47] and seem to be conservative. The computations give the same tendencies than the simple Aas-Jakobsen formula [10] function of the stress ratio R σ = σ min /σ Max

11 10 R. DESMORAT, F. RAGUENEAU, H. PHAM (a) case K=C=0 (b) case of linear consolidations ε min =0 ε min =0 ε min =-ε Max ε min =-ε Max Figure 4. Computed fatigue curves (solid lines: D c = 0.5, dot lines D c = 0.9) and taking then into account the mean stress effect, σ Max f c = 1 β(1 R σ )log N R (29) The material parameter β set to 0.1 allows to recover the fatigue curves for both stress ratios (stress ratio taken in the formula simply as R σ = 1 for the symmteric loading, to zero for the ǫ min = 0 case). As the damage model does not represent the material behavior dissymmetry, the analysis is mainly qualitative. One can nevertheless notice that a mean stress effect is reproduced and that the value obtained from the computations is found of the order of magnitude of the usual value β = for light concrete. Note last that the asymptote is reached for the model around 10 3 cycles so that, as for metals, there is a need of a different approach for High Cycle Fatigue. A two scale damage model for HCF may similarly proves usefull [39, 48, 40, 22]. Model Aas-Jackobsen compressive loading with no load reversal symmetric loading Figure 5. Normalized fatigue curves Comparison between model (D c = 0.9), experiments (marks) and Aas-Jakobsen formula (straight lines) To conclude, a single damage model allows for the representation of both the stress-strain response and the fatigue curves of quasi-brittle materials. The constitutive equations including the generalized damage evolution law being written in a rate form, they apply to any kind of loading as random fatigue or seismic response. Multifibre beam analyses use the uniaxial

12 CONTINUUM DAMAGE MECHANICS FOR HYSTERESIS AND FATIGUE 11 constitutive equations of the material behavior model in order to compute structural response and failure. Computations of reinforced concrete structures submitted to complex loadings can then use straightforwardly the numerical implementation of section 3.2. Examples are given below. The dissymmetry of the stress-strain response of quasi-brittle materials such as concrete has not been taken into account. A way to represent it in the Continuum Damage Mechanics framework is to model at the macroscopic RVE scale the quasi-unilateral conditions of microcracks closure (see section 6). 5. APPLICATION TO STRUCTURES MULTIFIBRE ANALYSIS The following structural case-studies, based on the Finite Element method, make use of the multifibre beam theory. They are built to illustrate the computational possibilities with previous constitutive model of internal sliding and friction coupled with damage. The choice of multifibre Finite Element analysis combines the advantage of using beam-type finite elements with the simplicity of the consideration of an uniaxial material behavior [49, 50]. The kinematics hypothesis assumes no distorsion or warping of a cross section. The general scheme consists in computing the local strains in each fibre from the nodal displacements and rotations through Timoshenko s beams equations. For plane bending and in a cross section of abscissa x, one can compute the strain fields ε xx (x, y), ε xy (x, y) as functions of the longitudinal displacement u 1 (x), of the transversal displacement u 2 (x) and of the rotation θ(x): ε xx (y) = u 1(x) y.θ (x) and 2ε xy (y) = u 2(x) θ(x) (30) The local constitutive equations, at the fibre level, allows for the computation of the Cauchy stress. Different materials in a same cross section, such as steel and concrete, can be accounted for in the multifibre analysis by considering different constitutive equations and material parameters for each fibre. The generalized stresses (moment, normal force) are computed through numerical integration over the cross section and sent back to the global equilibrium solver. The numerical analysis are performed with the set of parameters for tension for the tensile zones (when well identified), with the set (a) of concrete parameters for the compression zones (section 4, no consolidations identification). The dynamics computations use the set of parameters for compression for the whole structure, defining an academic quasi-brittle material. The multifibre facilities of C.E.A. computer code CASTEM is used with Newmark s scheme for the time discretization [51]. No localization limiter is used and the final results are in fact mesh dependent. An adequate choice of the mesh size will allow to conclude on the modeling capability of the proposed approach. Nevertheless, the introduction of a localization limiter is an important point to deal with in further developments Monotonic, cyclic and dynamic response of a beam To emphasize the ability of the model to deal with case-studies of structures subject to cyclic or dynamic loading, a square plain concrete column is studied here for the academic symmetric material response. The effects accounting for damage-hysteresis coupling at the material level are analyzed at the structure level in terms of fatigue resistance and of hysteretic dissipations for vibrations. The structure is L = 10 meters high with a square cross section of 1 m 2. It is anchored at the bottom and is subject to bending. Beam elements of length 50 mm are used. Two kinds of analysis are carried out: i) under quasi-static loading to point out the fatigue predictive ability of the proposed approach for structures and ii) under dynamic loading to assess the physical meaning of hysteresis and damage coupling in the framework of structural dissipation (damping) and eigenfrequencies decrease. The first quasi-static analysis is carried out first for a top monotonic horizontal applied deflection u 2 (L, t) = u(t). Different calculations imposing load-unload sequences at several levels of maximum deflection u Max are then performed and compared to previous monotonic response (Figure 5.1 for which the applied deflection varies cyclically between a zero value and u Max ). Hysteresis is obtained

13 12 R. DESMORAT, F. RAGUENEAU, H. PHAM at the structure scale up to failure. The maximum applied deflection over a cycle normalized by the peak deflection u max /u peak is plotted in Figure 7 as a function of the of cycles to rupture N R : this is the computed fatigue curve of the beam illustrating the ability of the proposed approach to determine the fatigue curve of a structure. load (N) monotonic u max = 0.63 u peak u max = 0.8 u peak u(t) 10 m deflection (m) Figure 6. Column under load reversal loading: low cycle fatigue behavior 1 0,9 u max / u peak 0,8 0,7 0,6 0, number of cycles to rupture Figure 7. Structure fatigue curve Under dynamic loading, the behavior of real structures is generally governed by: the decrease of eigenfrequencies linked to the level of cracking, the increase of damping linked to the level of frictional sliding. The frictional sliding of cracked surfaces is directly related to the average level of damage. To point out these global features, the plain concrete column is monotonically pre-damaged in bending (with

14 CONTINUUM DAMAGE MECHANICS FOR HYSTERESIS AND FATIGUE 13 6 different levels of cracking) and then subject to vibrations analyses. The pre-damaging load F is applied horizontally at the top of the beam in 0.1 second and is suddenly vanished in order to generate free flexural vibrations. The peak load denoted next F peak corresponds to the rupture in monotonic conditions (F peak = F(u peak ) = N, Fig. 5.1). Due to the dynamic effects, the load F = 10 6 N (corresponding to F/F peak = 0.86) leads to direct rupture of the specimen. The 5 other computations performed (F/F peak = 0.17, 0.34, 0.52, 0.69, 0.78) allow to appreciate the evolution of damping and of eigenfrequencies with respect to the level of initial loading and cracking. Figure 8 shows the corresponding transient response of the model for 4 representative levels of pre-damaging loading, results illustrating the damping ratio increase derived next (a) 0.1 (b) displacement (m) displacement (m) time (s) time (s) displacement (m) (c) time (s) displacement (m) (d) time (s) Figure 8. Transient response of a plain concrete column for different pre-damaging loads F: (a) F = 0.17 F peak, (b) F = 0.34 F peak, (c) F = 0.69 F peak, (d) F = 0.86 F peak The evolution of the apparent first eigenmode is plotted in Figure 9 (left) as a function of F Max /F peak. Concerning damping, the logarithmic decrement ξ is evaluated for each computation between m periods of top deflection amplitude decaying from u n to u n+m using ξ = 1 2mπ ln (u n/(u n+m )). The computed evolution of this damping parameter, direct consequence of frictional sliding at the local level, is plotted in Figure 9 (right) as a function of the normalized dynamic load Rupture of a reinforced concrete beam One aims last to compute the response of a reinforced concrete beam of length 1.4 m and of cross-section 0.15 m 0.22 m subject to mechanical loading up to rupture.three points bending

15 14 R. DESMORAT, F. RAGUENEAU, H. PHAM 1st eigengenmode frequency (%) relative damping ratio (%) F / F max peak F / F max peak Figure 9. First eigenfrequency decrease (left) and relative damping ratio increase (right) function of F Max/F peak is studied. Figure 10 shows the geometry of the sample. The two steel reinforcement bars for the upper part of the beam have a diameter φ = 8 mm, the two steel bars for the lower part of the beam have a diameter φ = 14 mm. The reinforcing bars exhibit an elastic threshold of 400 MPa. The behavior used for steel in the analysis is elastic perfectly plastic. Concrete in the tensile parts of the beam is modeled by the set of parameters of section 4 for tension, concrete in compression by the set of parameters (b). loading 2 φ8 2 φ m 1.4 m 0.15 m Figure 10. Three-point bending test The experiment has been performed under load control, the unloads realized at several levels (10, 30, 50 and 70 kn) [8]. The load-deflection diagram is plotted in figure 11 for both experiment and computation. Ten cycles were applied for cyclic top deflections varying between 0 and u i Max at each level i in order to appreciate the hysteretic dissipation. The same global stress-strain response is obtained with 10 or 50 elements in the beam length, but, as expected, the ultimate displacement is smaller for the refined mesh due to strain localization in compression. The model is able to simulate the global behavior of a reinforced concrete structure up to the yielding of the steel reinforcement, describing the three main stages of a reinforced concrete element: elasticity, concrete tension cracking and plasticity of the reinforcement. Even if hysteresis is included at the RVE scale of concrete constitutive equations, it is not transfered from computations to the structure scale. This is due to the presence of the reinforcement, so that this result tends to confirm that most dissipation in reinforced concrete structures is due to bond slip. The computed structure stiffness is too low and no permanent strain are represented. These features are explained again by the fact that the steel-concrete friction mechanism is not taken into account. Due to perfect bonding the steel bars load and damage too much the concrete parts and degrade in a too important manner the global stiffness.

16 CONTINUUM DAMAGE MECHANICS FOR HYSTERESIS AND FATIGUE 15 (a) experiment (b) computation Figure 11. Load-deflection curves 6. QUASI-UNILATERAL CONDITIONS OF MICROCRACKS CLOSURE Many quasi-brittle materials exhibit a stress-strain response different in tension and in compression. Phenomenological constitutive models valid for both tensile and compressive behaviors take into account the micro-defects closure effect either within specific plasticity criterion [52, 53] or within specific damage laws [54, 55, 2, 56, 57, 58]. This effect (mechanically referred to as the quasiunilateral conditions) leads to partial recovery of the elastic properties in compression: most of the cracks responsible for the damage state close. Loading induced damage anisotropy plays also a role on the dissymmetry of concrete behavior [23, 59, 60, 5, 61, 62] but it is not taken into account here. From a theoretical point of vue, the damage state due to the presence of microcracks is represented by the scalar state variable D. A thermodynamics state is independent from both the intensity and the sign of the loading (at constant internal variables) which means that no extra damage variable has to be introduced to model the microcracks closure, D acts differently in tension and in compression. For unidimensional states of stress, a solution has been recalled in section 2 for ductile materials, introducing the microdefects closure parameter h, material dependent. One proposes here to extend it to quasi-brittle materials altogether with the consideration of the damage model with internal sliding and friction of section 3. The stress formulation is used to introduce the microdefects closure effect with in mind a positive stress leading to open microcraks, a negative stress to closed microcracks. The damage D acts then fully on positive stresses here the elasto-damage stress (σ σ π ) and the friction stress σ π and partially as hd on negative stresses, with h the microcracks closure parameter (0 < h < 1). The elastic state potential (14) becomes: ρψe = 1 σ σ π E 1 (1 D) + 1 σ σ π 2 2 E 1 (1 hd) + 1 σ π E 2 (1 D) + 1 σ π 2 2 E 2 (1 hd) (31)

17 16 R. DESMORAT, F. RAGUENEAU, H. PHAM from which the state laws are derived as ǫ = ρ ψ σ = σ σπ + E 1 (1 D) + σ σπ E 1 (1 hd) ǫ π = ρ ψ σ π = σ σπ + E 1 (1 D) + σ σπ E 1 (1 hd) σπ + E 2 (1 D) σ π E 2 (1 hd) R = ρ ψ r = Kr X = ρ ψ α = Cα Y = ρ ψ D = 1 σ σ π E 1 (1 D) 2 + h σ σ π 2 2 E 1 (1 hd) σ π E 2 (1 D) 2 + h σ π 2 2 E 2 (1 hd) 2 so that h = 1 recovers the damage model proposed in section 3. One seek here to quantify the dissymmetry of the tension and compression responses obtained by the single introduction of the microcracks closure parameter h. A first step in modeling is then to keep the criterion function f and the dissipation potential F unchanged compared to the initial damage model with internal sliding and friction. This simple choice has the nice property to also keep the evolution laws (18) unchanged, except for the consideration of h in the variable Y used within the damage evolution law. Figure 12 shows the model response either in monotonic conditions (12a) or in a hysteretic compressive loading following a pre-damaging tension (12b). The material parameters considered are: E 1 = MPa, E 2 = MPa, σ s = 1.5 MPa, S = 16 MPa, h = 0.1, K = 710 MPa, C = 100 MPa. The strain energy density release rate Y is with h << 1 much smaller in compression than in tension. Through this property, the damage rate is quite reduced in compression. With the additional feature of a damage acting partially (as hd) in compression, the model built represents a significantly different behavior in tension and in compression. The dissymmetry is nevertheless not large enough for concrete, point which highlights the important limitations of an initially symmetric elastic domain obtained by use of the criterion function (16). Two tensile pre-damage values D 0 = 0.02 and D 0 = 0.4 are considered in Fig. (12b). The thermodynamics modeling altogether with the definition of a state potential able to be continuously differentiated (as potential (31)) ensure continuous stress-strain responses, even for complex loadings. The elastic stiffness recovery is represented. Due to the presence of irreversible strains the recovery does not occur at zero stress. (32) (a) Monotonic responses (b) Hysteretic responses D 0 =0.02 D 0 =0.4 Figure 12. (a) Monotonic tensile and compressive responses, (b) Hysteretic compressive response after pre-damage D 0 in tension (dot line: D 0 = 0.02, solid line: D 0 = 0.4)

18 CONTINUUM DAMAGE MECHANICS FOR HYSTERESIS AND FATIGUE 17 CONCLUSION The model of internal sliding and friction altogether with the generalized damage law Ḋ = (Y/S) π allow for the representation of damage growth and structural failures in both hysteretic and fatigue loading cases. A few material parameters are introduced: E 1, E 2 for elasticity, σ s as irreversibility threshold, S for damage, eventually h for the microcracks closure effect and K and C for linear consolidations. The hysteresis loops with no load reversal are reproduced. The fatigue curves result from the time integration of the coupled constitutive equations. From the material behavior point of view, the kinematic and isotropic consolidations improve the modeling of the size of the hysteresis loops. The drawback of a linear isotropic consolidation R = Kr is a non realistic increase of the size of the domain at large numbers of cycles leading to a too high asymptotic fatigue limit. For fatigue applications, it will be finally better to set the isotropic consolidation parameter K to zero or even to consider negative values for it (but with positive kinematic consolidation parameter C): due to cyclic frictional sliding, there is asperities erosion of the microcracks surfaces. An efficient numerical scheme for non monotonic applications is derived. It is used within a multifibre computer code for the calculation of monotonic, fatigue and/or dynamic responses of reinforced concrete components. Fatigue curves of structures can be computed and structural damping addressed. Computations of reinforced concrete structures can also be performed efficiently. But recall that the bond-slip mechanism has a main role in such structures. To model this mechanism is in fact essential, for instance for the estimation of structural damping. For complete monotonic, fatigue and or dynamic failures analysis of reinforced structures, one will need to model damage and rupture of the steels. Lemaitre s law of damage governed by plasticity Ḋ = (Y/S) s ṗ and also written in a rate form will be advantageously used [22]. Guidelines to extend the damage model with internal sliding and friction are given in order to take into account the quasi-unilateral condition of microcracks closure and the dissymmetry tension/compression. Recall that even if it is difficult to define an elasticity limit of concrete in compression, the real limit is a few times the elasticity limit measured in tension. For better modeling a dissymmetric irreversibility criterion function should be considered, for example as a Drucker-Prager modified criterion, σπ f = 1 D X + k(d) tr σπ R σ s (33) but inquiries on choices for the k(d) function arise. Such a modeling also has strong consequences on the 3D response of the model, as for dilatancy. It needs a full study by itself, study left to further work. REFERENCES 1. Chen E.S., Buyukozturk O., Constitutive model for concrete in cyclic compression, J. Engrg Mech., Vol No. 6, pp , Laborderie, C., Berthaud, Y., Pijaudier-Cabot, G., Crack closure effect in continuum damage mechanics: numerical implementation, Proc. 2nd Int. Conf. on Computer aided analysis and design of concrete strucutures. Zell am See, Austria, 4-6 april, pp , Yazdani S., Schreyer L., Combined plasticity and damage Mechanics model for plain concrete, J. Engrg Mech., Vol No. 7, pp , Papa E., A damage model for concrete subject to fatigue loading. Eur. J. Mech., A/Solids 12 3, pp , Dragon A., Halm D., An anisotropic model of damage and frictional sliding for brittle materials, European Journal of Mechanics, A/Solids, 17, pp , Meschke G., Lackner R., Mang H.A., An anisotropic elastoplastic-damage model for plain concrete, Int. J. Numer. Meth. Engng., Vol. 42, pp , Al-Gadhib, A.H., Baluch, M.H., Shaalan, A. and Khan, A.R., Damage model for monotonic and fatigue response of high strength concrete. Int. J. Damage Mech. 9(1), pp , Ragueneau F., La Borderie C., Mazars J, Damage Model for Concrete Like Materials Coupling Cracking and Friction, Contribution towards Structural Damping: First Uniaxial Application, Mechanics Cohesive Frictional Materials, Vol. 5, pp , 2000.