continuum damage mechanics for hysteresis and fatigue of...

Continuum Damage Mechanics for hysteresis and fatigueof quasi-brittle materials and structures

R. Desmorat∗, F. Ragueneau, H. Pham

LMT-Cachan, ENS Cachan/Universite Paris 6/CNRS,61, av. du president Wilson, F-94235 CACHAN Cedex, FRANCE

SUMMARY

For material exhibiting hysteresis such as quasi-brittle materials, it is natural to consider that hysteresisand fatigue are related to each other. One shows in the present work that damage, from the ContinuumDamage Mechanics point of view, may be seen as the link between both phenomenon. One attempts henceto set up a unified modeling of hysteresis and damage. Numerical examples are given for concrete andvalidate the proposed model of internal sliding and friction coupled with damage. The problem of a properphenomenological modeling of the micro-defects closure effect leading to a dissymetric tension/compressionresponse and to stiffness recovery in compression is also adressed. Cyclic and fatigue applications are inmind 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 representedby 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 modelingdirectly the Wohler curves of materials [9, 10, 11, 12]. A fatigue law for concrete can be a straightline in the maximum applied stress σMax vs the logarithm of the number of cycles to rupture logNRdiagram, generally parametrized by the stress ratio Rσ = σmin/σMax (with σmin the minimumapplied stress).

In order to reproduce the effect of different mean stresses on Wohler curves, amplitude lawsfunction of the stress ratio are also often considered in fatigue damage models [13, 14, 15].The damage increment per cycle δD

δNis set as a function of the current damage D, of the

stress amplitude ∆σ = σMax − σmin and of Rσ. The extension to 3D states of stresses of suchmodeling 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 loadingsencountered in random fatigue or in seismic applications. And the link between a stress and astrain formulation is not so clear for nonlinear materials except if a rate written damage model isdefined [16, 17, 18, 19, 20, 21, 22].

The objective of the present work is to model the behavior of quasi-brittle materials with a singleset of constitutive equations valid with the same material parameters for monotonic, hysteretic anddynamic loading (at not too high strain rates) but also in fatigue. The damage law establishedhas to allow for damage accumulation under cyclic behavior, a low number of random cyclescorresponding 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 duringunloading-reloading: friction beween cement paste and aggregates, between agregates, between

∗Correspondence to: [email protected], tel: 33 1 47 40 74 60, fax: 33 1 47 40 74 65

CONTINUUM DAMAGE MECHANICS FOR HYSTERESIS AND FATIGUE 1

microcracks lips, i.e. friction at microscale also referred to as internal friction. Different approachesfor internal sliding and friction have been developped: plasticity based on dislocations slips but withno hysteresis without stress sign change, microplane modeling for concrete [23], micromechanicsof sliding and friction in long fiber composite materials [24], meso-modeling of microcrakingin concrete [5, 25], macroscale representation with phenomenological models for concrete [8] orelastomers [26]. Except the last work on internal sliding and friction of filled elastomers, the modelscited do not apply to fatigue.

Hysteretic dissipation and damping are closely related, the challenges of numerical computationsbeing to address structural dynamics with no need of a fictitious Finite Element viscous dampingmatrix, defined at the structure scale. Moreover, Continuum Damage Mechanics is an adequatetool to describe the eigenfrequency decrease of Civil Engineering cracked structures subject tosevere loading. In the present work, one will attempt last to gain at the Representative VolumeElement (RVE) scale – i.e. from the proposed constitutive equations – the damage dependency ofthe global damping parameter for concrete structures.

Multifibre beam analysis will be used for application to plain concrete and reinforced concretestructures.

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 andgrowth lead to a decrease of macroscopic Young’s modulus. The unilateral behavior of a crackbearing cyclic loading is the source of damage deactivation and stiffness recovery. The roughnessof 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 lipsand of aggregates contacts whose main macroscopic consequence is the experimental observationsof hysteresis loops in compression, tension and torsion.

1.1. Micromechanics approaches

The behavior of interacting frictional flaws in solid media has been extensively studied in analyticalways [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 onlyto recover the hysteretic loops during cyclic loading but also to accurately describe the mixed modepropagation of a crack in a quasi-brittle material. At the macroscopic level of RVE of continuummechanics, several models account for these different mechanisms and some of their couplingbut only a few handle the frictional sliding behavior coupled with damage. The micromechanicsexplanations are rather complicated in a 3D framework, preventing a simple and robust expressionof constitutive equations at the macroscopic level, necessary feature for numerical analyses ofstructures.

The derivation of the further thermodynamics free energy needs the definition of internalvariables. To be relevant, the choice of internal variables has to be motivated by micromechanicsevidences and mechanisms [16]. The micromechanics analysis of cracked representative elementaryvolume allows for the expression of free energy for some particular case studies [32, 24]. Theresultant thermodynamics potential is divided into two parts: the elasticity of the cracked matrixand the stored or blocked energy density due to kinematic hardening or friction. Pushed forwardin 3D, the homogenization process allows to find the expression of the free energy at the RVEscale accounting for induced anisotropy with open or closed crack conditions [25]. Based on thespectral decomposition of the crack density, induced anisotropy coupled with closed microcracksfriction can be recovered [5]. An adequate choice for the invariants of the strain and damage tensorsand for their coupling allows for the description of hysteresis loops under load reversal. For cyclicloading, the spectral analysis of the cracked medium leads to the introduction of a fourth orderdamage tensor in order to ensure continuity of the stress-strain path with the unilateral conditions

Copyright c© 2006 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 00:0–0Prepared using nagauth.cls

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].

1.2. General form of the macroscopic free energy

For efficiency and robustness requirements of the most importance in fatigue implying numerouscycles of loading, the choice of a scalar damage variable is appropriate. To deal with the crackedmatrix modeling, Continuum Damage Mechanics proves to be a relevant tool, introducing asphenomenological damage variable the scalar D. For the definition of a thermodynamics potentialin adequation with the expressions proposed by Hild and coworkers [24] for composites, Dragonand Halm [5] and Ragueneau et al. [8] for concrete, and by Desmorat and Cantournet [26] for filledelastomers, a general expression is (uniaxial case):

ρψ = W1(ε,D) +W2(ε, επ, D) + ws (1)

where W1 is a purely elastic part of the free energy, W2 is an anelastic one and ws is the storedenergy density. In 1D the strain ε is the macroscopic total strain and επ is the macroscopic internalfrictional sliding strain. Not based on the classical principle of strain additivity, the thermodynamicsvariables σ and σπ, associated with the total strain and with the internal frictional sliding strain,are not equal. Note also that επ is not the plastic or permanent strain.

Classically, the elasto-damage coupling is expressed using the following expression for W1,

W1 =1

2E1(1 −D)ε2 (2)

where E1 is an elasticity modulus eventually equal to the Young’s modulus E and where D isthe damage variable ranging from 0 for the virgin material to the critical damage Dc < 1 for thecompletely broken one.

Following the concept of splitting the free energy into parts, two kinds of partitioning may beadopted for W2. In the first one, W2 is directly linked to the level of damage by W2 = 1

2DE(ε−επ)2

[5, 8]. This expression induces a coupling of the energy dissipated trough frictional sliding to thestate of cracking. Only one Young’s modulus may be introduced by setting E1 = E. The Cauchystress is obtained by derivation with respect to the total strain: σ = ρ∂ψ

∂ε= E(1 −D)ε+ σπ with

σπ = −ρ ∂ψ∂επ

= DE(ε−επ). For a state of cracking (or damage) approaching to Dc ≈ 1, this modelconverges toward a classical plasticity model implying only a frictional sliding behavior σ ≈ σπ.Such an approach for the coupling between friction and damage is relevant for instance in thecase of bond-slip modelling of reinforced concrete element [33] in which the asymptotic behavioris sliding. For plain concrete in pure tension or compression, the asymptotic behavior should leadto rupture: for D ≈ 1, the total stress σ should vanish. The frictional sliding strain can then beexpressed as σπ = E2(1 −D)(ε− επ) but there is no guide to say that the elasticity loss occurs atthe same rate than for the energy part W1. To conserve the initial stiffness of the virgin materialequal to the Young’s modulus E, the relationship E = E1 +E2 is set. Note that it is in fact quitenatural to introduce two Young’s modulus in heterogeneous materials as concrete, one for its hardphases (aggregates,...) and one for its soft phases (cement,...).

The complete free energy takes the general form:

ρψ =1

2E1(1 −D)ε2 +

1

2g(D)E2(ε− επ)2 + ws (3)

where g(D) = 1 − D here but where for the general case many other expressions seem possiblesuch as g(D) = a + bD or g(D) = 1/(α + βD) with a, b, α, β as algebraic material dependentparameters. The stresses are obtained as follows:

σ = ρ∂ψ

∂ε= E1(1 −D)ε+ σπ

σπ = −ρ∂ψ

∂επ= g(D)E2(ε− επ)

(4)

As one can see for g(D) = 1−D, as D tends toward 1, the stress σ tends toward 0, correspondingto a fully broken material.



2. GENERALIZED DAMAGE LAW

From the thermodynamics of irreversible processes, a damage variable is a state variablerepresenting the fact that microcracks and/or micro-voids are present at microscale. Aphenomenological modeling avoids to have to deal with each micro-defect and considers those asaveraged on the RVE of continuum mechanics. The damage pattern of an initially isotropic materialcan be represented in the more general case by a fourth order tensor [34, 16] but for practical reasonsa second order tensor DDD or a scalar variable D are most often used [35, 36, 37, 38, 22]. In anycase, the damage variable represents the micro-cracking pattern. Then for a given population ofmicro-defects the only possibility is to consider one thermodynamics damage variable, whateverthe sign of the loading. This is for instance by its coupling with the elasticity and evolution lawsthat 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-fatigueapplications [18, 41, 22],

D =

(

Y

S

)s

p (5)

where damage is governed by plasticity (through the accumulated plastic strain p) and enhancedby the strain energy (through the strain energy density release rate Y ). Two damage parametersare introduced: the damage strength S and the damage exponent s. The second is related to theslope of the Wohler (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 fromthe Helmholtz free energy ρψ function of the strains or from the Gibbs free enthalpy ρψ⋆ functionof the stresses as

Y = −ρ∂ψ

∂Dor Y = ρ

∂ψ⋆

∂D(6)

In the simple uniaxial case where ρψ⋆ is [42]

ρψ⋆ =〈σ〉2+

2E(1 −D)+

〈σ〉2−

2E(1 − hD)(7)

with h the microdefects closure parameter (0 < h < 1) and 〈x〉+ (resp. 〈x〉−) standing for thepositive (resp. negative) part of the stress (〈x〉+ = x if x > 0, 〈x〉+ = 0 else, 〈x〉− = −〈−x〉+). Theelasticity law reads

ε = ρ∂ψ⋆

∂σ=

〈σ〉+E(1 −D)

+〈σ〉−

E(1 − hD)(8)

or σ = E(1 −D)ε in tension and σ = E(1 − hD)ε in compression so that there is partial stiffnessrecovery in compression (quasi-unilateral conditions). The recovery is full if h = 0 (unilateralconditions).

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 intension. Considering the damage evolution law (5) gives a damage rate even smaller in compressionas then:

Dcompression ≈ hsDtension << Dtension (10)

which leads to a different behavior in tension and in compression and to both a stress dependentdamage 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 fewhundred of percents). For elastomeric materials plasticity is often meaningless so that the law hasbeen generalized into [43]

D =

(

Y

S

)s

π (11)



corresponding to a damage rate governed by the main dissipative mechanism of internal frictionencountered in these materials.The damage evolution law (11) generalizes to non metallic materialsthe initial law (5). With an adequate definition of the cumulative measure of the internal slidingπ, the law (11) will be derived and used next for quasi-brittle materials. We will refer to it asthe generalized damage law. The damage exponent s will next be taken equal to the usual defaultvalue 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 allowsfor the modeling of hysteresis loops in tension without having to undergo compression. The choicemade here is to consider the general form (3) with g(D) = 1 −D but extended to 3D introducingthe strain ǫǫǫ and as internal variables the irreversible strain ǫǫǫπ, the damage D and if necessary theconsolidation variables r (scalar) and ααα (tensorial). The state and evolution laws classicaly derivefrom the thermodynamics and dissipation potentials [44]. Concerning damage, the choices mademust finally recover the generalized damage law (11).

3.1. Model in the thermodynamics framework

The state potential is considered quadratic,

ρψ =1

2(1 −D) ǫǫǫ : EEE1 : ǫǫǫ+

1

2(1 −D)(ǫǫǫ− ǫǫǫπ) : EEE2 : (ǫǫǫ− ǫǫǫπ) +

1

2Kr2 +

1

2Cααα : ααα (12)

where EEE1 and EEE2 are elasticity tensors such as the sum EEE1 +EEE2 = EEE is the Hooke tensor of theundamaged material. The last terms of eq. (12) are usually grouped into ws = 1

2Kr2 + 1

2Cααα : ααα,

the stored energy density, remaining small in quasi-brittle materials and often neglected. The firsttwo terms of eq. (12) are the elastic energy density ρψe so that ρψ = ρψe + ws.

The state laws are then derived as:

σσσ = ρ∂ψ

∂ǫǫǫ= EEE1(1 −D) : ǫǫǫ+EEE2(1 −D) : (ǫǫǫ− ǫǫǫπ)

σσσπ = −ρ∂ψ

∂ǫǫǫπ= EEE2(1 −D) : (ǫǫǫ− ǫǫǫπ)

R = ρ∂ψ

∂r= Kr

XXX = ρ∂ψ

∂ααα= Cααα

Y = −ρ∂ψ

∂D=

1

2ǫǫǫ : EEE1 : ǫǫǫ+

1

2(ǫǫǫ− ǫǫǫπ) : EEE2 : (ǫǫǫ− ǫǫǫπ)

(13)

defining (σσσ, −σσσπ, R, XXX, −Y ) as the associated variables with (ǫǫǫ, ǫǫǫπ, r, ααα, D). And note that anequivalent stress formulation can be proposed in terms of Gibbs free enthalpy ρψ⋆ = ρψ⋆e + ws,built from the Legendre transformation ρψ⋆e of the elastic energy density,

ρψ⋆e =(σσσ − σσσπ) : EEE−1

1 : (σσσ − σσσπ)

2(1 −D)+σσσπ : EEE−1

2 : σσσπ

2(1 −D)(14)

Effective stresses σσσ =σσσ

1 −Dand σσσπ =

σσσπ

1 −Dcan then be defined so that the first two state

laws read

σσσ = EEE1 : ǫǫǫ+EEE2 : (ǫǫǫ− ǫǫǫπ)

σσσπ = EEE2 : (ǫǫǫ− ǫǫǫπ)(15)

A criterion function f is defined next in order to govern the loading/unloading conditions. Aframework similar to plasticity is used with then no viscosity and no loading rate effect: the



consistency condition f < 0 or f < 0 corresponds to the elastic stage, the internal variableskeeping then a constant value, the condition f = 0 and f = 0 corresponds to the irreversiblebehavior including internal sliding and damage.

The choice for the irreversibility function is similar to the yield function in plasticity, using theeffective stress concept [45],

f = ‖σσσπ −XXX‖ −R− σs (16)

but it is written in the σσσπ-plane. R and XXX are respectively the isotropic and the kinematicconsolidations, σs is the reversibility limit and ‖.‖ is a norm not specified as long as uniaxialbehavior is considered.

The evolution laws derive from a dissipation potential F through the normality rule. The damagemodel is non associated as F is the sum f + FD with FD Lemaitre’s damage potential,

FD =Y 2

2S(1 −D)(17)

The normality rule reads, introducing the internal sliding multiplier λ,

ǫǫǫπ = λ∂F

∂σσσπ=

λ

1 −D

σσσπ −XXX

‖σσσπ −XXX‖

r = −λ∂F

∂R= λ

ααα = −λ∂F

∂XXX= λ

∂F

∂σσσ= (1 −D) ǫǫǫπ

D = λ∂F

∂Y=

λ

1 −D

(

Y

S

)

=

(

Y

S

)

π

(18)

and leads to the cumulative measure of internal sliding π =∫ t

0‖ǫǫǫπ‖dt as:

π = ‖ǫǫǫπ‖ =r

1 −D=

λ

1 −D(19)

As wished, the last equation of the set of evolution laws (18) recovers the generalized damagelaw (11). Written D = (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 energydensity (through Y ). Written π = (S/Y )D, it models the increasing internal sliding due to damageaccumulation. As the constitutive equations describing both damage and internal sliding are fullycoupled, 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 XXX = Cααα allows to write the linear kinematic consolidation law ina more classical form XXX = C(1−D)ǫǫǫπ, 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 = σσσ : ǫǫǫ− ρψ = σσσπ : ǫǫǫπ −Rr −XXX : ααα + Y D from Clausius-Duhem inequality and one has

D =

(

σσσπ :∂F

∂σσσπ+R

∂F

∂R+XXX :

∂F

∂XXX+ Y

∂F

∂Y

)

λ ≥ 0 (20)

and D ≥ 0 when the dissipation potential F (σσσπ, R,XXX, Y ;D) is a convex function of its argumentsσσσπ, R, XXX , Y with F (000, 0,000, 0;D) = 0 and where the damage D acts as a parameter. Theconsideration of the evolution laws altogether with the contition f = 0 allows to derive a morepractical expression,

D = σs(1 −D)‖ǫǫǫπ‖ + Y D ≥ 0 (21)



3.2. Numerical scheme for non monotonic applications

The previous set of constitutive equations is a set of first order differential nonlinear equations. Thegood thing is that the damage model obtained is written in a rate form so that it applies to anykind of loading, not necessary cyclic. The drawback is that one has to perform the time integrationover each time step of the differential equations. This can be too costly for fatigue applicationsif local Newton or quasi-Newton iterations are needed. For seismic and fatigue applications, evennon cyclic such as random fatigue, an efficient numerical scheme can be proposed with the realisticassumption of a damage quasi-constant over a time increment, classical assumption for fatiguecalculations with Continuum Damage Mechanics [18].

In Finite Element computer codes, the local time integration subroutine has for input at timetn+1 the strain ǫǫǫn+1 = ǫǫǫn+∆ǫǫǫ where the quantity ǫǫǫn denotes the strain at time tn and ∆ǫǫǫ the strainincrement. These quantities as well as all internal variables are known at time tn. To perform thetime integration of the constitutive equations means to determine all quantities at time tn+1 (alsothe damage Dn+1). Having also in mind multifibre beam analyses of concrete reinforced structuresfor which the knowledge of the uniaxial stress-strain response is sufficient, the numerical scheme ispresented next in 1D, the tensors EEE1, EEE2 becoming the scalars E1, E2.

As classically done for nonlinear constitutive models, an elastic prediction is first made and givesan estimate of the stresses σ and σπ ,

σ = E1(1 −Dn)ǫn+1 + E2(1 −Dn)(ǫn+1 − ǫπn)

σπ =σπ

1 −Dn

= E2(ǫn+1 − ǫπn)(22)

If f = |σπ −Xn| −Rn − σs ≤ 0 then the loading state is elastic and the internal variables at timetn+1 remain equal to those at time tn: ǫ

πn+1 = ǫπn, rn+1 = rn (and so Rn+1 = Rn), αn+1 = αn (and

so Xn+1 = Xn), Dn+1 = Dn.If f = |σπ −Xn| − Rn − σs > 0 one needs to correct the value of the stresses and of

the internal variables by performing the time integration of the constitutive equations. Thisis the internal sliding–damage correction. If Euler implicit scheme is used, this is done bydetermining first the cumulative internal sliding increment ∆π from the consistency conditionfn+1 = ‖σσσπn+1 −XXXn+1‖ −Krn+1 − σs = 0 rewritten for the uniaxial case

fn+1 = |E2(ǫn+1 − ǫπn) −Xn − (E2 + C(1 −Dn))∆π · sign(σπ −X)| −Rn −K∆r − σs = 0 (23)

asσπn+1 = E2(ǫn+1 − ǫπn+1) = E2(ǫn+1 − ǫπn) − E2∆ǫ

π (24)

The internal sliding multiplier ∆λ = ∆r is the positive solution of fn+1 = 0 in which the damageis taken as D ≈ Dn (consistent assumption in fatigue as the damage does not change much overone cycle, therefore even less over one time increment). The increment ∆π is gained as

∆π =|E2(ǫn+1 − ǫπn) −Xn| −Rn − σs

E2 + (K + C)(1 −Dn)(25)

Once ∆π is known, all the variables including D may be updated as follows:

• irreversible strain: ǫπn+1 = ǫπn + ∆π · sign(σπ −X),• strain energy density release rate: Yn+1 = 1

2E1 : ǫ2n+1 + 1

2E2(ǫn+1 − ǫπn+1)

2,

• damage: Dn+1 = Dn +

(

Yn+1

S

)

∆π,

• isotropic consolidation: rn+1 = rn + (1 −Dn+1)∆π, Rn+1 = Krn+1,• kinematic consolidation: Xn+1 = Xn + C(1 −Dn+1)∆ǫ

π,• friction stress: σπn+1 = E2(1 −Dn+1)(ǫn+1 − ǫπn+1),• stress: σn+1 = E1(1 −Dn+1)ǫn+1 + σπn+1.

To conclude, this is an explicited solution of the implicit discretized equations. The scheme hasthen the robustness of the implicit schemes but with the efficiency of explicit ones. For cyclic



loadings a few time increments must be used to describe a whole cycle. Note then that in orderto 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 withexperiments. The identification of the material parameters E1, E2 for elasticity (with as Young’smodulus E = E1 + E2), σs as irreversibility threshold, S for damage and eventually K and C forthe 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 straightlines in the stress-strain diagram making hard the measurement of the internal variables. Note forinstance that the irreversible strain ǫπ derived from first equation of (13) rewritten

σ = E(1 −D)ε− E2(1 −D)επ = E(1 −D)(ε− εan) (26)

is not the permanent strain ǫan. It is related to ǫan through the elasticity parameters asǫπ = Eǫan/E2, expression which would have been damage dependent with other choices forthe g(D) function. The identification is also not straightforward because one must have in mindthe fatigue applications for which the elasticity limit usually defined by damage models becomesthe asymptotic fatigue limit σ∞

f . From the elasticity law and the definition of the irreversibilitycriterion one has:

∣

∣σ∞

f

∣

∣ =E

E2

σs (27)

which forces σs to remain small compared to the ultimate or peak stress.The material considered next is concrete. The tensile strength is ft = 4 MPa. In compression, the

testing specimen has been loaded under lateral strain control preventing early localization modes.The measured compressive strength is fc = 48.5 MPa and two sets of material parameters areobtained, either with zero values for the consolidation parameters K and C or with non zero K andC (linear consolidations).

• Set of parameters for tension: E1 = 10000 MPa, E2 = 25000 MPa, σs = 1 MPa, S = 0.3MPa, K = 3000 MPa C = 1000 MPa.

• Set (a) of parameters for compression: E1 = 20000 MPa, E2 = 15000 MPa, σs = 9 MPa,S = 324 MPa, K = C = 0.

• Set (b) of parameters for compression: E1 = 16000 MPa, E2 = 19000 MPa, σs = 6 MPa,S = 476 MPa, K = 130 MPa, C = 110 MPa.

4.1. Hysteresis loops of concrete in compression

The model represents the stress-strain response as well as the hysteresis of concrete (Figures 1aand 1b), not perfectly for the hysteresis loops because of the simple modeling of the consolidationsR and X (zero or linear) but well enough to envisage the calculations of the fatigue curve of thematerial: one will have then a unified model for both monotonic and fatigue applications usinga single set of material parameters. The hysteresis loops are better represented by use of linearconsolidations (case b), but because of a small value for the irrevesibility threshold σs (better forfatigue), 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 twoparts, the ”elasto-damage” stress (σ− σπ) and the friction stress σπ, the second being responsiblefor the hysteresis and tending to symmetrize. The damage evolves of course more at the high levelof compressive stresses (Fig. 3), but evolves also a little during the second (nonlinear) part ofeach unloading due to intenal sliding reactivation before the minimum stress is reached (frictionmechanism). Second, one can see the role played by both the isotropic and kinematic consolidations.



(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, thecase (b) leading to cycles of almost constant size more in accordance with fatigue phenomenon. Theisotropic consolidation increases the size of the reversibility domain, it is therefore a counterpartto damage which decreases it; it allows for the height of the hysteresis loops to vary slowly. Thekinematic consolidation models a stress translation of the reversibility domain. It allows then fora dissymmetry of the friction stress.

Figure 2. Evolution of elasto-damage stress (σ − σπ) and friction stress σπ

4.2. Computed fatigue curves

Two kinds of fatigue responses can be addressed: the response to a cyclic applied stress between aminimum stress σmin (minimum in absolute value) and a maximum stress σMax and the responseto a cyclic applied strain between ǫmin and ǫMax. The curves σMax vs the number of cycles torupture NR are the Wohler curves of the material. The curve ǫMax vs NR is more adapted tosoftening materials for which experimental testing must be strain controlled. It is simply calledthe fatigue curve of the material which can eventually be plotted in terms of maximum stress butdefined as the stress level obtained at the first loading.



Figure 3. Damage growth during hysteretic loading

The proposed damage model with internal sliding and friction allows for the step by stepcomputation of such curves from the knowledge of the hysteretic response of the material. The timeintegration is performed by use of the numerical scheme of section 3.2 and the set of parametersare the sets (a) with no consolidations and (b) for linear consolidations.

Consider an applied strain which varies cyclically between a minimum strain ǫmin and amaximum strain ǫMax (maximum in absolute value). Due to the choice of the model (nodissymmetry tension/compression) and of the material parameters the qualitative interpretationsof the computations performed address compressive fatigue of concrete. The damageD is calculatedat each time increment tn and rupture is assumed to occur when D reaches the critical value Dc

(two values for Dc are considered first, Dc = 0.5 and 0.9). The number of cycles to rupture NR isthen defined from the Continuum Damage Mechanics point of view by the number of cycles N atwhich D = Dc,

NR = N(D = Dc) (28)

The computed fatigue curves with the previous sets of material parameters (a) and (b) arecompared in Figure 4 for the loadings ǫmin = 0 and ǫmin = −ǫMax. The modeling with linearisotropic consolidation R = Kr, better for the representation of the hysteresis loops, leads to anasymptotic fatigue limit much larger than with the simple modeling R = 0. This is due to thehigh value of the cumulative internal sliding π in fatigue so that at in the linear consolidation caseR = 15.1 MPa for N = 100 cycles (loading ǫmin = 0) to be added to the initial irreversibility limitσs = 6 MPa and to be compared to the irreversibility limit for the no consolidations case σs = 9MPa. 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 Dc = 0.5 and Dc = 0.9 iswithin the dispersion always large in fatigue. Also, the computations performed with two timediscretizations of the applied cycles, a first one with 10 time increments per cycle, a second onewith 50 time increments per cycle, give close results. This convergency feature emphasizes theefficiency of the numerical scheme proposed.

The fatigue curve with Dc = 0.9 of Fig. 4a is replotted in terms of normalized stress σMax/fcvs number of cycles to rupture NR (with fc = 48.5 MPa the compressive strength, Fig. 5). Themodel corresponds to the dot lines, black for the symmetric loading, grey for the ǫmin = 0 loading.The results for ǫmin = 0 give a lower bound of the experimental data for concrete tested in fatiguewith a zero stress ratio Rσ [46, 47] and seem to be conservative. The computations give the sametendencies than the simple Aas-Jakobsen formula [10] function of the stress ratio Rσ = σmin/σMax



εmin=0

εmin=-εMax

εmin=0

εmin=-εMax

(a) case K=C=0 (b) case of linear consolidations

Figure 4. Computed fatigue curves (solid lines: Dc = 0.5, dot lines Dc = 0.9)

and taking then into account the mean stress effect,

σMax

fc= 1 − β(1 −Rσ) logNR (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 Rσ = −1 for the symmteric loading, to zero for theǫmin = 0 case). As the damage model does not represent the material behavior dissymmetry, theanalysis is mainly qualitative. One can nevertheless notice that a mean stress effect is reproducedand that the value obtained from the computations is found of the order of magnitude of the usualvalue β = 0.0685 for light concrete. Note last that the asymptote is reached for the model around103 cycles so that, as for metals, there is a need of a different approach for High Cycle Fatigue. Atwo scale damage model for HCF may similarly proves usefull [39, 48, 40, 22].

Model

Aas-Jackobsen

compressive loadingwith no load reversal

symmetric loading

Figure 5. Normalized fatigue curves – Comparison between model (Dc = 0.9), experiments (marks)and Aas-Jakobsen formula (straight lines)

To conclude, a single damage model allows for the representation of both the stress-strainresponse and the fatigue curves of quasi-brittle materials. The constitutive equations includingthe generalized damage evolution law being written in a rate form, they apply to any kindof loading as random fatigue or seismic response. Multifibre beam analyses use the uniaxial



constitutive equations of the material behavior model in order to compute structural responseand failure. Computations of reinforced concrete structures submitted to complex loadings canthen 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 notbeen taken into account. A way to represent it in the Continuum Damage Mechanics framework isto model at the macroscopic RVE scale the quasi-unilateral conditions of microcracks closure (seesection 6).

5. APPLICATION TO STRUCTURES – MULTIFIBRE ANALYSIS

The following structural case-studies, based on the Finite Element method, make use of themultifibre beam theory. They are built to illustrate the computational possibilities with previousconstitutive model of internal sliding and friction coupled with damage.

The choice of multifibre Finite Element analysis combines the advantage of using beam-typefinite 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 schemeconsists in computing the local strains in each fibre from the nodal displacements and rotationsthrough Timoshenko’s beams equations. For plane bending and in a cross section of abscissa x,one can compute the strain fields εxx(x, y), εxy(x, y) as functions of the longitudinal displacementu1(x), of the transversal displacement u2(x) and of the rotation θ(x):

εxx(y) = u′1(x) − y.θ′(x) and 2εxy(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 themultifibre analysis by considering different constitutive equations and material parameters for eachfibre. The generalized stresses (moment, normal force) are computed through numerical integrationover 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 tensilezones (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 parametersfor compression for the whole structure, defining an academic quasi-brittle material. The multifibrefacilities of C.E.A. computer code CASTEM is used with Newmark’s scheme for the timediscretization [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 theproposed approach. Nevertheless, the introduction of a localization limiter is an important pointto deal with in further developments.

5.1. Monotonic, cyclic and dynamic response of a beam

To emphasize the ability of the model to deal with case-studies of structures subject to cyclicor dynamic loading, a square plain concrete column is studied here for the academic symmetricmaterial response. The effects accounting for damage-hysteresis coupling at the material level areanalyzed at the structure level in terms of fatigue resistance and of hysteretic dissipations forvibrations. The structure is L = 10 meters high with a square cross section of 1 m2. It is anchoredat the bottom and is subject to bending. Beam elements of length 50 mm are used. Two kindsof analysis are carried out: i) under quasi-static loading to point out the fatigue predictive abilityof the proposed approach for structures and ii) under dynamic loading to assess the physicalmeaning 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 deflectionu2(L, t) = u(t). Different calculations imposing load-unload sequences at several levels of maximumdeflection uMax are then performed and compared to previous monotonic response (Figure 5.1 forwhich the applied deflection varies cyclically between a zero value and uMax). Hysteresis is obtained



at the structure scale up to failure. The maximum applied deflection over a cycle normalized bythe peak deflection umax/upeak is plotted in Figure 7 as a function of the of cycles to rupture NR:this is the computed fatigue curve of the beam illustrating the ability of the proposed approach todetermine the fatigue curve of a structure.

0

2 105

4 105

6 105

8 105

1 106

1.2 106

0 0.05 0.1 0.15 0.2

umax

= 0.63 upeak

monotonic

umax

= 0.8 upeak

load

(N

)

deflection (m)

10 m

u(t)

Figure 6. Column under load reversal loading: low cycle fatigue behavior

0,5

0,6

0,7

0,8

0,9

1

0 5 10 15 20

um

ax /

up

eak

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 pointout these global features, the plain concrete column is monotonically pre-damaged in bending (with



6 different levels of cracking) and then subject to vibrations analyses. The pre-damaging load Fis applied horizontally at the top of the beam in 0.1 second and is suddenly vanished in order togenerate free flexural vibrations. The peak load denoted next Fpeak corresponds to the rupture inmonotonic conditions (Fpeak = F (upeak) = 1.16 106 N, Fig. 5.1).

Due to the dynamic effects, the load F = 106 N (corresponding to F/Fpeak = 0.86)leads to direct rupture of the specimen. The 5 other computations performed (F/Fpeak =0.17, 0.34, 0.52, 0.69, 0.78) allow to appreciate the evolution of damping and of eigenfrequencieswith respect to the level of initial loading and cracking. Figure 8 shows the corresponding transientresponse of the model for 4 representative levels of pre-damaging loading, results illustrating thedamping ratio increase derived next.

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

dis

pla

cem

ent

(m)

time (s)

-0.04

-0.02

0

0.02

0.04

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

dis

pla

cem

ent

(m)

time (s)

-0.1

-0.05

0

0.05

0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

dis

pla

cem

ent

(m)

time (s)

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

dis

pla

cem

ent

(m)

time (s)

(a) (b)

(c) (d)

Figure 8. Transient response of a plain concrete column for different pre-damaging loads F :(a) F = 0.17 Fpeak, (b) F = 0.34 Fpeak, (c) F = 0.69 Fpeak, (d) F = 0.86 Fpeak

The evolution of the apparent first eigenmode is plotted in Figure 9 (left) as a functionof FMax/Fpeak. Concerning damping, the logarithmic decrement ξ is evaluated for eachcomputation between m periods of top deflection amplitude decaying from un to un+m usingξ = 1

2mπln (un/(un+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 normalizeddynamic load.

5.2. Rupture of a reinforced concrete beam

One aims last to compute the response of a reinforced concrete beam of length 1.4 m and ofcross-section 0.15 m×0.22 m subject to mechanical loading up to rupture.Three points bending



80

85

90

95

100

0 0.2 0.4 0.6 0.8 1

1st

eig

eng

enm

ode

freq

uen

cy

(%)

Fmax

/ Fpeak

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1

rela

tiv

e d

amp

ing

rati

o

(%)

Fmax

/ Fpeak

Figure 9. First eigenfrequency decrease (left) and relative damping ratio increase (right) function ofFMax/Fpeak

is studied. Figure 10 shows the geometry of the sample. The two steel reinforcement bars for theupper part of the beam have a diameter φ = 8 mm, the two steel bars for the lower part of thebeam 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 partsof the beam is modeled by the set of parameters of section 4 for tension, concrete in compressionby the set of parameters (b).

1.4 m 0.15 m

0.22 m2 φ8

2 φ14

loading

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 experimentand computation. Ten cycles were applied for cyclic top deflections varying between 0 and uiMax ateach level i in order to appreciate the hysteretic dissipation. The same global stress-strain responseis obtained with 10 or 50 elements in the beam length, but, as expected, the ultimate displacementis 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 theyielding 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 includedat the RVE scale of concrete constitutive equations, it is not transfered from computations to thestructure scale. This is due to the presence of the reinforcement, so that this result tends to confirmthat most dissipation in reinforced concrete structures is due to bond slip. The computed structurestiffness is too low and no permanent strain are represented. These features are explained againby the fact that the steel-concrete friction mechanism is not taken into account. Due to perfectbonding the steel bars load and damage too much the concrete parts and degrade in a too importantmanner the global stiffness.



(a) experiment (b) computation

Figure 11. Load-deflection curves

6. QUASI-UNILATERAL CONDITIONS OF MICROCRACKS CLOSUREMany 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 intoaccount the micro-defects closure effect either within specific plasticity criterion [52, 53] or withinspecific damage laws [54, 55, 2, 56, 57, 58]. This effect (mechanically referred to as the quasi-unilateral conditions) leads to partial recovery of the elastic properties in compression: most of thecracks responsible for the damage state close. Loading induced damage anisotropy plays also a roleon the dissymmetry of concrete behavior [23, 59, 60, 5, 61, 62] but it is not taken into accounthere.

From a theoretical point of vue, the damage state due to the presence of microcracks isrepresented by the scalar state variable D. A thermodynamics state is independent from boththe 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 extendit to quasi-brittle materials altogether with the consideration of the damage model with internalsliding and friction of section 3.

The stress formulation is used to introduce the microdefects closure effect with in mind a positivestress leading to open microcraks, a negative stress to closed microcracks. The damage D acts thenfully on positive stresses – here the elasto-damage stress (σ − σπ) and the friction stress σπ – andpartially as hD on negative stresses, with h the microcracks closure parameter (0 < h < 1). Theelastic state potential (14) becomes:

ρψ⋆e =1

2

〈σ − σπ〉2

+

E1(1 −D)+

1

2

〈σ − σπ〉2

−

E1(1 − hD)+

1

2

〈σπ〉2

+

E2(1 −D)+

1

2

〈σπ〉2

−

E2(1 − hD)(31)



from which the state laws are derived as

ǫ = ρ∂ψ⋆

∂σ=

〈σ − σπ〉+

E1(1 −D)+

〈σ − σπ〉−

E1(1 − hD)

ǫπ = −ρ∂ψ

∂σπ=

〈σ − σπ〉+

E1(1 −D)+

〈σ − σπ〉−

E1(1 − hD)−

〈σπ〉+

E2(1 −D)−

〈σπ〉−

E2(1 − hD)

R = ρ∂ψ⋆

∂r= Kr

X = ρ∂ψ⋆

∂α= Cα

Y = ρ∂ψ⋆

∂D=

1

2

〈σ − σπ〉2

+

E1(1 −D)2+h

2

〈σ − σπ〉2

−

E1(1 − hD)2+

1

2

〈σπ〉2

+

E2(1 −D)2+h

2

〈σπ〉2

−

E2(1 − hD)2

(32)

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 thento keep the criterion function f and the dissipation potential F unchanged compared to the initialdamage model with internal sliding and friction. This simple choice has the nice property to alsokeep the evolution laws (18) unchanged, except for the consideration of h in the variable Y usedwithin the damage evolution law.

Figure 12 shows the model response either in monotonic conditions (12a) or in a hystereticcompressive loading following a pre-damaging tension (12b). The material parameters consideredare: E1 = 14000 MPa, E2 = 21000 MPa, σs = 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 compressionthan in tension. Through this property, the damage rate is quite reduced in compression. With theadditional feature of a damage acting partially (as hD) in compression, the model built representsa significantly different behavior in tension and in compression. The dissymmetry is neverthelessnot large enough for concrete, point which highlights the important limitations of an initiallysymmetric elastic domain obtained by use of the criterion function (16).

Two tensile pre-damage values D0 = 0.02 and D0 = 0.4 are considered in Fig. (12b). Thethermodynamics modeling altogether with the definition of a state potential able to be continuouslydifferentiated (as potential (31)) ensure continuous stress-strain responses, even for complexloadings. The elastic stiffness recovery is represented. Due to the presence of irreversible strainsthe recovery does not occur at zero stress.

(a) Monotonic responses (b) Hysteretic responsesD0=0.02

D0=0.4

Figure 12. (a) Monotonic tensile and compressive responses, (b) Hysteretic compressive response afterpre-damage D0 in tension (dot line: D0 = 0.02, solid line: D0 = 0.4)



CONCLUSION

The model of internal sliding and friction altogether with the generalized damage law D = (Y/S)πallow for the representation of damage growth and structural failures in both hysteretic and fatigueloading cases. A few material parameters are introduced: E1, E2 for elasticity, σs as irreversibilitythreshold, S for damage, eventually h for the microcracks closure effect and K and C for linearconsolidations.

The hysteresis loops with no load reversal are reproduced. The fatigue curves result from thetime 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 ofthe domain at large numbers of cycles leading to a too high asymptotic fatigue limit. For fatigueapplications, it will be finally better to set the isotropic consolidation parameter K to zero or evento consider negative values for it (but with positive kinematic consolidation parameter C): due tocyclic frictional sliding, there is asperities erosion of the microcracks surfaces.

An efficient numerical scheme for non monotonic applications is derived. It is used within amultifibre computer code for the calculation of monotonic, fatigue and/or dynamic responses ofreinforced concrete components. Fatigue curves of structures can be computed and structuraldamping addressed. Computations of reinforced concrete structures can also be performedefficiently. But recall that the bond-slip mechanism has a main role in such structures. To modelthis mechanism is in fact essential, for instance for the estimation of structural damping. Forcomplete monotonic, fatigue and or dynamic failures analysis of reinforced structures, one willneed to model damage and rupture of the steels. Lemaitre’s law of damage governed by plasticityD = (Y/S)sp 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 orderto take into account the quasi-unilateral condition of microcracks closure and the dissymmetrytension/compression. Recall that even if it is difficult to define an elasticity limit of concrete incompression, the real limit is a few times the elasticity limit measured in tension. For bettermodeling a dissymmetric irreversibility criterion function should be considered, for example as aDrucker-Prager modified criterion,

f = ‖σσσπ

1 −D−XXX‖ + k(D) tr σσσπ −R− σs (33)

but inquiries on choices for the k(D) function arise. Such a modeling also has strong consequenceson the 3D response of the model, as for dilatancy. It needs a full study by itself, study left tofurther work.

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Aussois, France, 14-16 October 2004.


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