Engineering Structures 27 (2005) 239–250 www.elsevier.com/locate/engstruct

Interaction between drying, shrinkage, creep and cracking phenomena in concrete F. Benboudjemaa, F. Meftahb,∗, J.M. Torrentic a Laboratoire de Mécanique et Technologie, Ecole Normale Supérieure de Cachan, 61 Avenue du président Wilson, 94235 Cachan, France b Laboratoire de Mécanique, Université Marne La Vallée, 5 Boulevard Descartes, 77420 Champs sur Marne, France c IRSN, BP 6, 92265 Fontenay-Aux-Roses Cedex, France

Received 7 June 2004; received in revised form 27 September 2004; accepted 29 September 2004 Available online 10 December 2004

Abstract In this paper, a hydro-mechanical model, accounting for the full coupling of drying, shrinkage, creep and cracking is presented. In this model, a new basic creep constitutive law, based on microscopic considerations of the role of water, is elaborated. Further, the existing model for drying creep of Bažant and Chern is improved by the introduction of a second material parameter which accounts for the interaction between the solid skeleton and liquid water layers in a drying and creeping material. The model is, thereafter, used for investigating the effect of cracking on the delayed behavior of concrete and displaying the frontier between the intrinsic behavior of the material and the structural effects. © 2004 Elsevier Ltd. All rights reserved. Keywords: Drying; Shrinkage; Creep; Cracking; Coupling; Finite element analysis

1. Introduction Drying of concrete occurs in a non-homogeneous manner leading to a strong structural effect; self-equilibrated stresses do arise within the material that induce cracking [1, 2]. Therefore, cracking process interacts with the creep mechanisms of drying concrete. On one hand, creep strains tend to relax drying generated self-equilibrated stresses and then to attenuate crack propagation. On the other hand, crack occurrence leads to redistribution of the overall stress-state (self-equilibrated stresses and stresses due to an applied mechanical load) which affects creep mechanisms [3]. These considerations mean that structural effect affects, simultaneously, measurements of shrinkage and creep deformations in any experiment. Unfortunately, no experimental procedure allows to separate easily intrinsic behaviors from structural effects when shrinkage and creep are concerned. The identification of constitutive laws from ∗ Corresponding author. Tel.: +33 1 6095 7789; fax: +33 1 6095 7799.

E-mail address: [email protected] (F. Meftah). 0141-0296/$ - see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2004.09.012

experimental measurements cannot be performed in a straightforward way. It needs the use of a robust mechanical model for cracking in order to quantify correctly, by means of computations, the structural part [4]. The intrinsic behavior of the drying and creeping concrete can therefore be deduced in a sort of inverse analysis by focusing on the conventional components: drying shrinkage, basic creep and drying creep. Note that drying also affects, in an intrinsic way, the creep process. At the micro-scale level, water migration during a sustained load amplifies creep process [1]. This leads, at the macro-scale level, to the socalled drying creep strain. In this work, a constitutive model for cracking, based on an orthotropic elasto-plastic damage approach, is used. It incorporates, in a full-coupled manner, drying and creep strains and their interaction. Adopting such an approach enables to quantify the effect of both crack onset and crack closure on measured delayed strains, and therefore, to identify the effective structural contribution all over the drying process. It also enables to reproduce experimentally observed results [5], in which drying in specific directions

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Nomenclature a, b, D0 diffusivity-relative humidity relationship parameters A, k, Vm desorption isotherm parameters ax , bx nominal strength parameters At c , Bt c , Ct c second order creep stress tensors cx damage-cumulative plastic strain relationship parameter C water content D effective diffusion coefficient D, Dc , Dt second order damage tensor, damage scalar in compression and second order damage tensor in tension E, Et c second order stiffness tensor and stiffness tensor corrected by creep effect f x0 initial yield stress F, G yield surface and plastic potential function h relative humidity I second order unit tensor 1 unit vector first invariant I1 second invariant J2 Jbc , Jdc , Jt c second order basic creep, intrinsic drying creep and total creep compliance tensors stiffness of the load-bearing absorbed water kdc layers hydrous compressibility factor kds sph sph apparent stiffness of the porous material and kr , k i of the skeleton p, s0 parameters governing the crack closure effect α f , β Drucker–Prager criterion parameters αg parameter governing the dilatancy µ material parameter for the stress-induced shrinkage model plastic multiplier, cumulative plastic strain λx , κ θ unit conversion factor sph εdev bc , εbc deviatoric basic creep strain vector and spherical basic creep strain dev εdev bc_i , ε bc_r irreversible and reversible deviatoric creep strain vectors sph sph εbc_i , εbc_r irreversible and reversible spherical creep strains εt c , εbc , εdc total, basic and intrinsic drying creep strain vectors εe , ε p , εds elastic, plastic and drying shrinkage strain vectors apparent viscosity of the micro-diffusing ηdc water apparent viscosity related to the unabsorbed ηidev water in the nano-porosity apparent viscosity related to the strongly ηrdev absorbed water in the nano-porosity sph sph ηi , ηr apparent viscosity of the water at the microscopic and macroscopic level

τ τdc

nominal strength characteristic pseudo-time for intrinsic drying creep σ , σ dev , σ sph total, deviatoric and spherical stress vectors T second order standard transformation tensor Subscripts 0 initial value ∼ effective quantity ˆ principal component c compression t tension n value at time tn

leads to an orthotropic behavior of concrete. Furthermore, a linear relationship between drying shrinkage and concrete water content variation is adopted [6–8]. It is shown that this simple linear relationship enables to describe a major part of experimental results obtained by Granger [7]. Therefore, observed non-linearity, in the experimental drying shrinkage versus water content variation diagram is imputed mainly to cracking, elucidating thus the frontier between the structural and intrinsic behavior of drying shrinkage. Note that, drying process is considered to be governed by a single diffusion type equation, which gives the evolution of water content in the material. Moreover, the basic creep model explicitly takes into account the role of water micro-diffusion and hydrates micro-sliding in creep process. Finally, the intrinsic effect of drying on creep mechanism is incorporated by modifying the model previously proposed by Bažant and Chern [1], in order to take into account the interaction between the absorbed water and the skeleton. The numerical simulations based on these models highlight that drying creep strains and drying process kinetics have to be dissociated in order to retrieve accurately creep experimental results of Granger [7]. 2. Hydro-mechanical modeling 2.1. Drying model The drying of concrete is modeled here by a diffusiontype equation which takes into account the migration of both liquid and vapor phases in concrete: C˙ = ∇ · (D(C)∇C)

(1)

where C is the water content and D is the diffusivity, which depends on the water content. The dot represents the derivative with respect to time. The diffusivity is given by the relationship derived by Xi et al. [9]:
  b(h−1) (2) D(h) = D0 1 + a 1 − 2−10

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where h is the relative humidity, D0 , a and b are constant material parameters, depending upon the concrete mix. Note that a desorption isotherm, which relates the relative humidity to the water content in concrete, is needed for Eq. (2) to make the model completely formulated in terms of the water content variable. Therefore, the BSB model [9], referred also to as the three-parameter BET model, is used such that: AkVm h (3) C= (1 − kh)[1 + (A − 1)kh] where A, k and Vm are material parameters which can be determined from the desorption isotherm curve. Furthermore, the diffusivity (Eq. (2)) is not made dependent on the damaged state of the material, since it is considered, here, that the drying-induced cracking does not influence significantly the drying process. Indeed, experimental results show that a non-loaded specimen and a loaded one (by a compressive load) dry in the same manner [10,11], even if the compressive loading prevents from pronounced micro-cracking. Further, it has been shown [12,13] that the opening of drying induced cracks ranges from 25 to 50 µm, while Bažant et al. [14] consider that cracks have a negligible effect on drying when their opening is less than 100 µm. Moreover, in the studied specimens, crack depth is restricted to a small part of the radius.

2.3. Basic creep model Mechanisms of basic creep of concrete are not well understood yet. Many theories have been proposed in the literature to retrieve the collected experimental evidences [23]. However, no theory has been universally accepted, although it is well admitted that water plays a fundamental role [24]. In this paper, a multiaxial model developed by the authors is used [4,25], where the role of water is integrated in an original manner. It is assumed that the creep process can be split into a spherical and a deviatoric part. Indeed, existing multiaxial experimental results show that spherical and deviatoric creep strains are proportional to the spherical and deviatoric part of the stress tensor, respectively [25]. In this model, each part of the creep strain is therefore associated with a physical mechanism, which occurs at different observation scales of the material. The spherical part is driven by the migration of adsorbed water in the micro and nano-porosity while the deviatoric part is caused by the sliding of the C–S–H layers [3]. These mechanisms lead to analytical expressions of sph spherical creep strain εbc and deviatoric basic creep strain εdev bc . The basic creep strain decomposition is written as: sph

Drying of concrete occurs in a non-homogeneous manner and leads in general to a structural effect (cracking of the specimen skin). Thus, experimental measurements of drying shrinkage are very sensitive to the dimension, the shape of specimen and the boundary conditions [15]. This makes it difficult to interpret obtained results in order to derive a constitutive relationship for drying shrinkage. To overcome this shortcoming, a very thin specimen can be used to minimize the risk of cracking. Experimental results highlight then that drying shrinkage strains are proportional to relative humidity variation in the range 50%–100% [16, 17]. Thereby, models for drying shrinkage are often based on this finding [1,2,18,19]. In this paper, however, the drying shrinkage is taken as proportional to the water content variation [6–8]: ˙ ε˙ ds = kds C1

monitoring or gammadensimetry). Furthermore, the material parameter kds can be here identified easily from the linear part of the drying shrinkage versus weight loss curve [7].

εbc = εbc 1 + ε dev bc

2.2. Drying shrinkage model

(4)

where ε˙ ds is the drying shrinkage strain (free drying shrinkage strain) rate vector, kds is the hydrous compressibility factor and 1 is the unit vector. Note that both approaches, use of water content instead of relative humidity, are equivalent since the desorption isotherm curve is almost linear in the range 50%–100% of relative humidity [17,21,22]. The major reason for dealing with water content rather than relative humidity is that water content can be more easily measured (by weight loss

241

(5)

where εbc is the basic creep strain vector. It should be emphasized that such a decomposition of the creep strain (in a spherical and a deviatoric part) has been previously proposed for concrete in the case of multiaxial loadings [26,27]. The spherical basic creep strains are written as [4]: 
 sph sph sph sph sph sph sph  ηr ε˙ bc_r = hσ − kr εbc_r − 2ηr ε˙ bc_i   

+
  sph sph sph sph sph sph ηi ε˙ bc_i = 2kr εbc_r − ki εbc_i − hσ sph (6)  +  with x = (x + |x|)/2     sph sph sph εbc = εbc_r + εbc_i sph

sph

where εbc_r and εbc_i are the reversible and the irreversible spherical creep strains respectively; h is the relative sph sph humidity; ηr and ηi are the apparent viscosities of the water at two different scales of the material (microscopic and nanoscopic level, respectively). These apparent quantities depend upon the water viscosity and the connected porosity sph sph geometry. Further, kr and ki are the apparent stiffness associated to the precedent viscosities and related to the stiffness of the porous material and the skeleton; σ sph is the spherical stress. It is recalled that the spherical part of creep is due to the migration of adsorbed water under hydrostatic stresses. Water migration in both nano and micro-porosities is therefore modeled by the Poiseuille’s equation, assuming a cylindrical connected porosity. The use of the water

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mass conservation law for a representative volume leads to Eq. (6). The constitutive relations for deviatoric creep strains read [4]: dev dev εdev bc = ε bc_r + ε bc_i ; dev dev dev ηrdev ε˙ dev ; bc_r + kr ε bc_r = hσ

(7)

dev ηidev ε˙ dev bc_i = hσ dev where εdev bc_r and ε bc_i are the reversible and the irreversible deviatoric creep strain respectively; ηrdev and ηidev are the apparent viscosities related to strongly absorbed and unabsorbed water in the C–S–H porosity and krdev is the stiffness of the strongly absorbed water, associated to its bearing capacity. σ dev is the deviatoric stress vector. In Eqs. (6) and (7), the spherical and deviatoric basic creep strains are proportional to the internal relative humidity. This feature has been experimentally observed on pre-dried and sealed specimens [28]. Moreover, the model takes into account the fact that creep strains are partially reversible, according to experimental results [29]. Eqs. (6) and (7) can be solved analytically for constant stresses and a constant relative humidity. The basic creep strains vector εbc can be expressed as:

εbc (t) = hJbc (t) · σ

(8)

where σ is the stress vector and Jbc is the second order basic creep compliance tensor, depending upon the material parameters (see Appendix). It is well known that aging affects considerably basic creep of concrete [15,20]. It is emphasized that aging has not been considered in this study. Nevertheless, the present basic creep model is only intended to be used for one specific age at loading. If a different age of loading is considered, the parameters of the model should be identified again. 2.4. Drying creep model The drying creep corresponds to the additional strain observed when concrete is stressed together with a change in the internal moisture state. This behavior, called the Picket effect [30], is quite paradoxical. Indeed, a previously dried specimen creeps less than a saturated one, although the more the relative humidity decreases during a creep test, the more concrete creeps [31]. The same behavior has been reported later on wood specimens [32], while the phenomenon is rather called mechano-sorption than drying creep. It is well accepted nowadays that one mechanism at least explains one part of the drying creep of concrete: the microcracking effect [2,20]. When a specimen is submitted to both drying and compressive mechanical loads, the dryinginduced micro-cracking is less pronounced compared to a non-loaded specimen. Therefore the measured delayed strain is greater than the sum of the elementary components: drying shrinkage strain and basic creep strain. The additional strain corresponds to the micro-cracking effect and can be

modeled by using an adequate mechanical model, which describes correctly both strain softening and irreversible strain behaviors [3]. Experimental results highlight that even thin cement paste specimens exhibit drying creep strains [16], although they are not submitted to a prominent drying-induced cracking. Moreover, numerical simulations show that the micro-cracking effect fails to retrieve alone the whole part of drying creep [33]. Therefore, many authors proposed an intrinsic mechanism to explain the additional part that micro-cracking can not reproduce. Among them, one can find the seepage theory [34], the stress-induced shrinkage [1, 2], the drying-induced creep [35], the pore stress effect [36], or the micro-prestress relaxation [20]. None of these theories have been universally accepted in the scientific community yet. The most used theory is probably the stress-induced shrinkage one, proposed by Bažant and Chern [1]. These authors suggested that simultaneous drying and loading cause micro-diffusion of water molecules between micropores and macro-pores. This enhances bond breakage in cement gel, which results in intrinsic drying creep strain. This assumption leads to a constitutive relationship for the stress-induced shrinkage strain vector εdc [1]: ˙ ε˙ dc = µ|h|σ

(9)

where µ is a material parameter. This expression is quite similar to a simple rheological model: a single dashpot ˙ −1 . where the viscosity is η = (µ|h|) This model gives a purely viscous response for a constant applied stress. In this case, the drying creep kinetic is completely controlled by the drying process kinetic. However, numerical simulations, performed using this model [3], show that no suitable value of the material parameter µ can be identified in order to retrieve experimental drying creep results [7]. This can be explained by the fact that the model does not take into account the interaction between the material skeleton and bond breakage mechanism due to water micro-diffusion during the drying creep process. In this work, the constitutive law, proposed for drying creep, corresponds to a Kelvin–Voigt model with ˙ and a stiffness kdc . Accordingly, the a viscosity ηdc /θ |h| intrinsic drying creep strain reads: ˙ dc εdc = θ |h|σ ˙ ηdc ε˙ dc + θ |h|k

(10)

where ηdc is the viscosity of the micro-diffusing water; kdc is the stiffness of the load-bearing absorbed water layers; θ = 1s is a unit conversion factor. In this model, it is assumed that the material skeleton bounds the micro-diffusion process in adsorbed water layers. A similar relation has been used previously to model mechano-sorption (drying creep) of ˙ dc ) can be wood [37]. A characteristic time τdc = ηdc /(|h|k then defined. Thus, this model enables to control both the kinetic (by adjusting the ratio ηdc /kdc ) and the amplitude (by adjusting the parameter kdc ) of the intrinsic drying creep strain. Moreover, the characteristic time depends

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upon the relative humidity rate. This is consistent with the experimental findings of Day et al. [16]. For a constant rate of relative humidity and constant stresses, the intrinsic drying creep strains vector can be written as: εdc (t) = Jdc (t) · σ

(11)

where Jdc is the intrinsic drying creep compliance tensor given by: Jdc (t) = [1 − exp(−t/τdc )]/kdc I

(12)

where I is the second order unit tensor. 2.5. Cracking model The mechanical behavior of concrete is modeled by a damage model coupled with softening plasticity. The use of the plasticity theory enables to describe inelastic strains observed experimentally. The damage variable is associated to the mechanical degradation process of concrete induced by the development of micro-cracks. In order to describe properly the differences in the damage process of concrete in compression and in tension, the damage variable is separated into a compressive one and a tensile one. The damage process is assumed here to be isotropic in compression and orthotropic in tension, where orthotropy is induced by cracking in preferential directions, which prevails during drying. Therefore, a scalar damage variable Dc is used in compression, while a tensorial (of second order) one Dt is considered in tension. The damage tensor D is then given by: I − D = (1 − Dc )(I − Dt ).

(13)

The constitutive behavior of the material can, therefore, be established by relating effective stresses σ˜ to nominal ones σ , on the basis that the undamaged part of the material remains elastic. The stress–strain relationship reads: σ = (I − D) · σ˜ = (I − D) ·E0 · εe

(14)

where E0 is the initial elastic stiffness tensor, εe is the elastic strains vector obtained by the classical decomposition of the total strain rate vector ε˙ into an irreversible strain rate vector, given by the plastic component ε˙ p , and the previously defined delayed strain components, as: ε˙ e = ε˙ − ε˙ p − ε˙ ds − ε˙ bc − ε˙ dc .

(15)

In order to take into account the crack closure effect, each principal component Dtii (i ∈ [1, 2, 3] and repeated indices do not imply summation) of the damage tensor in tension Dt has been pre-multiplied by a weight function p [38]: 1 − Dtii = (1 − pi (σ˜ˆ )Dtii )

(16)

which reflects the elastic stiffness recovery during the unloading process from tension to compression in each i th concerned direction: pi (σˆ ) = s0 + (1 − s0 )(σ˜ˆ i + /|σ˜ˆ i |)

(17)

243

where s0 is a material parameter governing the crack closure effect and giving the proportion of recovered stiffness. The principal effective stress vector σˆ˜ is obtained from the effective stress vector σ (related to the global coordinate system) by the standard transformation tensor T (3 × 6 matrix): σ˜ˆ = T · σ˜ .

(18)

Once micro-cracks are initiated, local stresses are redistributed to undamaged material micro-bonds over the effective area. Accordingly, it appears reasonable to state that the plastic flow occurs in the undamaged material micro-bounds by means of effective quantities [39]. In this approach, softening plasticity is considered to govern crack occurrence and growth in all the material. Therefore, damage traduces the accompanying stiffness degradation due to crack opening. Thus, the evolution of the damage is considered to be related to the relative crack opening given by cumulated plastic strains κc and κˆ ti (used as hardening/softening parameters). An exponential form is adopted here, which reads in the principal effective stresses system [40]: Dc = 1 − exp(−cc κc ) Dtii

=

1 − exp(−ct κˆ ti )

and (19)

where ct and cc are material parameters identified from experimental stress–strain curves. This model is a rotating crack based one. In tension, only the cumulated plastic strain rate κˆ˙ t is aligned with the current direction of the principal effective stresses σˆ˜ . Therefore, the cumulated plastic strain rate vector κˆ˙ t in the principal effective stresses system is obtained from κ˙ t , expressed in the global coordinate system, by the same standard transformation tensor T used for the effective stresses (Eq. (18)): κˆ˙ t = T · κ˙ t .

(20)

This requires that applied stresses do not rotate significantly which is the case in the performed numerical investigations. The yield surfaces F are given as functions of the effective state of stresses. In order to reproduce suitable behavior in compression and in tension, a Drucker–Prager criterion in compression and three Rankine criteria in tension [41] are used (Fig. 1). The Drucker–Prager criterion is written as:  (21) Fc (σ˜ , κc ) = 3 J2 (σ˜ ) + α f I1 (σ˜ ) − β τ˜c (κc ) where J2 (σ˜ ) is the second invariant of the effective stress vector σ˜ ; I1 (σ˜ ) is the first invariant of the effective stresses vector σ˜ ; τ˜c is the effective strength in compression related to the compressive cumulated plastic strain κc ; α f and β are two material parameters, which depend upon the ratio of the uniaxial to biaxial compressive strength.

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read in the principal effective stress space: τc = fc0 [(1 + ac ) exp(−bc κc ) − ac exp(−2bc κc )] τti = ft [(1 + at ) exp(−bt κˆ ti ) − at exp(−2bt κˆ ti )]

(27)

where ac and bc are material parameters identified from a uniaxial compression test; f c0 is the elastic strength in compression; at and bt are material parameters identified from a uniaxial tension test and f t is the strength in tension. In order to avoid mesh dependency, a characteristic length lc is introduced. This length is related to the mesh size [42] in order to dissipate the same amount of energy after mesh refinement, when strains localize in one row of finite elements. 2.6. Full coupling between creep and cracking The creep process is assumed here to occur only in the undamaged part of the material, driven by effective stresses. Total creep strains εt c read therefore: Fig. 1. Yield surfaces in the two-dimensional space of the effective principal stresses (σ˜ I , σ˜ I I ).

For each principal tensile stress direction, a Rankine criterion is considered: Fti (σ˜ˆ , κˆti ) = σ˜ˆ i − τ˜ti (κˆ ti )

(22)

where τ˜ti the corresponding effective strength related to the tensile cumulated plastic strain vector κˆ ti . Moreover, a non-associative plastic flow theory is adopted in compression in order to predict the correct amount of dilatancy of concrete since damage is isotropic in compression. The plastic potential reads:  (23) G c = 3 J2 (σ˜ ) + αg I1 (σ˜ ) − β τ˜c (κc ) where αg is a material parameter which controls the dilatancy of concrete. Further, the plastic strain rate vector is then obtained by Koiter assumption: ε˙ p =

i

∂F ∂ Gc λ˙ it t + λ˙ c ∂ σ˜ ∂ σ˜ i

(24)

where λit and λc are the tensile and compressive plastic multipliers, respectively, which gives the cumulated plastic strain rates as: κ˙ c = (1 + 2αg2 )1/2 λ˙ c

and

κˆ˙ it = λ˙ it .

(25)

For both tension and compression, the effective strength of the material is written as: τ˜c (κc ) = τc (κc )/(1 − Dc ) τ˜ti (κˆ ti )

=

τti (κˆ ti )/(1 −

Dtii )

and (26)

where τc and τti are the nominal strengths of the material obtained from a uniaxial tension and a compression test, respectively. They are defined by generic functions [40] and

ε t c (t) = εbc (t) + εdc (t) = hJbc (t) · σ˜ + Jdc (t) · σ˜ = Jt c (t) · σ˜

(28)

where Jt c is the total creep compliance tensor. In order to take into account the load history, the stresses and the relative humidity are approximated by linear functions during a time-step: (t − tn ) and tn (t − tn ) σ˜ (t) = σ˜ n + σ˜ n (29) tn with t ∈ [tn , tn+1 ], tn = tn+1 − tn , h n = h n+1 − h n , σ˜ n = σ˜ n+1 − σ˜ n and where tn is the time at time-step number n; h n is the relative humidity at time-step number n; σ˜ n is the effective stress vector at time step number n. The approximation of stresses and relative humidity (Eq. (29)) enables to solve analytically the differential equations (6), (7) and (10), for both basic and drying creep components. The total creep strains can be now expressed as [4]: h(t) = h n + h n

n ˜ n + Ct c · σ˜ n+1 ε n+1 t c = At c · ε t c + B t c · σ

(30)

where εntc is the total creep strains vector at time-step number n; At c , Bt c and Ct c are second order tensors which depend only upon material parameters, relative humidity (h n and h n+1 ), tn and tn [4]. Therefore, we just need to save the stresses and the total creep strains at time step number n to know the total creep strains at time-step number n+1. The effective stresses at the end of the time step number n are updated by the following relationship: σ˜ n+1 = E0 · εn+1 e n+1 = E0 · (εn+1 − εn+1 − ε n+1 p ds − ε t c )

(31)

n+1 and ε n+1 are the elastic, plastic and total where εn+1 e , εp strains vectors at time step number n + 1, respectively.

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245

Finally, if one makes use of Eq. (30), the effective stresses vector σ˜ n+1 at the end of the time step reads: tr σ˜ n+1 = σ˜ n+1 − Et c · εn+1 p tr n σ˜ n+1 = Et c · (εn+1 − εnp − εn+1 ds − At c · ε t c

− Bt c · σ˜ n )

(32)

Et c = (I + E0 · Ct c )−1 · E0 where Et c is the stiffness tensor, corrected by creep effect, tr is the trial stress vector, corrected by creep effect. and σ˜ n+1 They can be calculated at the beginning of the time step, since all the involved quantities are known at this time. Eq. (32) shows that the creep effect can be taken into account without any noticeable changes in existing return mapping algorithms for softening/hardening plasticity [41, 43]. The computed stress state is therefore simultaneously affected by creep and cracking. The governing equations of the softening plastic model are non-linear. Hence, a local iterative procedure is used. During a time step, a Euler backward integration scheme is adopted and the non-linear equations are solved by the Newton–Raphson method. The calculus of the trial stress (Eq. (32)) allows to identify the active surfaces. The plastic multipliers must be positive (or equal to zero) during the plastic process. If a plastic multiplier is found to be negative during this process, the associated surface is deactivated, and the iterative procedure restarts. 3. Numerical simulations It is recalled that the attempt here is to separate and thus to identify, by means of numerical simulations and “backanalysis”, the contribution of the structural effect and of the intrinsic behavior in drying shrinkage and drying creep tests, which present inhomogeneous drying conditions. Therefore, the components of the delayed strains are considered in a progressive way: intrinsic and structural drying shrinkage, basic creep, intrinsic and structural drying creep, such that the contribution of each component can be assessed in situations where creep, drying and cracking are concomitant. Numerical simulations are carried out with CAST3M finite element code, developed by the French Atomic Energy Commission. The numerical results are illustrated through experimental ones obtained by Granger [7] on a concrete specimen. The experiments were carried out on a 28 days old specimen, which was previously sealed. The drying test was performed in a room at a constant temperature (20 ◦ C±1 ◦ C) and a constant relative humidity (50% ± 5%). The weight loss was measured on 16 cm × 15 cm (diameter × height) cylindrical specimens. Shrinkage and creep tests were performed on 16 cm × 100 cm (diameter × height) cylindrical specimens. Displacements were measured on a 50 cm base, located in the center of the specimen. This avoids boundary effects. For the basic and total creep tests,

Fig. 2. Finite element discretization and boundary conditions of the simulations of drying shrinkage, basic creep and total creep tests in axisymmetric configuration.

a compressive stress equal to 12 MPa is applied. The mechanical properties measured by Granger [7] are the Young’s modulus E = 33.7 GPa, the compressive stress f c = 49.6 MPa and the tensile stress f t = 3.7 MPa. Numerical simulations are performed in an axisymmetric configuration using 44 × 50 (height × width) four-nodes rectangular finite elements. The used discretization and the boundary conditions are presented in Fig. 2. Note that a fine mesh is considered at the drying face of the sample. In the numerical reported results, the strains correspond to the average values obtained, as for the experimental tests, from the longitudinal displacements computed at the half height of the specimen on the external surface of the cylindrical specimen. 3.1. Drying It has been shown above that the delayed strains are closely related to the water content at each point of the material. Therefore, drying simulations are performed in order to get the most representative water content profiles within the specimen and their time-evolution. This is considered to be achieved if the global (of the entire sample) weight loss evolution with time is retrieved by the simulation (Fig. 3). For this purpose, the parameters of the drying model have been identified by fitting the experimental data of Granger [7]. Thereafter, the water content profiles, obtained with this set of parameters, listed in Table 1, can be retained for the calculation of the strains. 3.2. Drying shrinkage component In Eq. (4) the intrinsic drying shrinkage behavior is given by a linear relationship of the water content variation, that is, of the weight loss up to a multiplicative constant. However, the experimentally measured strain versus weight

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Fig. 3. Identification of drying model parameters–Experimental and numerically fitted evolutions of weight loss with time.

Table 1 Drying model parameter values a

b

D0 (m2 /s)

A

k

Vm (l/m3 )

225

4

1.4 × 10−12

1.3

0.61

74.1

Fig. 4. Experimental and simulated drying shrinkage versus weight loss.

Table 2 Material parameters for drying shrinkage, basic creep and drying creep models kds (m3 /l) 1.22 × 10−5 sph

loss shows a non-linearity at the beginning and at the end of the drying process (Fig. 4). This non-linearity is, therefore, explained by the occurrence of cracking: cracks opening at the beginning and crack closure at the end of the drying. Thus, a coupling of the linear (intrinsic) drying shrinkage model and the cracking model (the orthotropic elasto–plastic damage (OEPD) model) should enable to retrieve, by simulation, both of the linear and the non-linear parts of Fig. 4, while the parameter kds is identified from the linear part only. Its value is given in Table 2. For this purpose, two numerical simulations are performed by using, first, an elastic model and then the OEPD model. The comparison between the results obtained with the two models unveils the influence of the microcracking on the apparent drying shrinkage strain, since for the elastic model no structural effect is taken into account. For the two cases, the evolutions of drying shrinkage strain versus the weight loss are displayed in Fig. 4, together with experimental results. As expected, the computations show that the best agreement with the experimental results is reached with the OEPD model, especially at the beginning of the drying process (low values of the weight loss), when the crack opening at the specimen skin affects significantly the evolution of the apparent drying shrinkage. Thereafter, cracking has less and less influence and the proportionality between drying shrinkage and weight loss is retrieved as an intrinsic behavior. The difference between the two numerical curves (obtained with the elastic and OEPD models) corresponds to a structural strain, induced by the micro-cracking. The amplitude of the obtained structural strain reaches a maximum value of about 150 µm/m (25% of the total drying shrinkage strain), which is non-negligible. Therefore,

ηr (MPa s) 2.08 × 1010

sph

kr (MPa) 4.29 × 104 sph

ηi (MPa s) 1.04 × 1012

sph

ki (MPa) 2.22 × 104

krdev (MPa) 3.53 × 104

kdc (MPa) 12

ηrdev (MPa s) ηidev (MPa s) ηdc (MPa) 6.63 × 1010 2.46 × 1012 1.6 × 105

µ1 (Pa−1 ) µ2 (Pa−1 ) 3.03 × 10−10 9.03 × 10−10

a suitable mechanical model should be used in order to separate accurately the contributions of each strain component (especially intrinsic and structural effects). 3.3. Basic creep component Basic creep parameters can be separately identified from basic creep experimental results. The identification procedure requires that the longitudinal and transversal basic creep strains be measured. The spherical and deviatoric components can, therefore, be separated and serve for the sph sph sph sph identification of the parameters (kr , ηr , ki , ηi ) and (krdev , ηrdev , ηidev ), respectively. The stiffness parameters are identified from the asymptotic values of strains while the viscosity ones are computed by fitting the kinetics, at loading for the reversible parameters and unloading for the irreversible ones [4]. Fig. 5 shows the simulated longitudinal basic creep strain obtained by fitting the experimental curve, with the set of parameters reported in Table 2. It can be observed that the developed basic creep model is able to reproduce the evolution of the strain at both short and long-term with the same parameter values. These latter will, therefore be used for the study of the drying creep component. 3.4. Drying creep component Drying shrinkage, structural effect due to micro-cracking and basic creep contribution have been previously identified.

F. Benboudjema et al. / Engineering Structures 27 (2005) 239–250

Fig. 5. Experimental and numerically fitted evolutions with time of longitudinal basic creep strain.

This will allow to study the intrinsic drying creep behavior, which is done by simulating a total creep test, using both Bažant’s model (Eq. (8)) and the proposed one (Eq. (9)). Therefore, the issue is to verify whether the drying process and the drying creep have the same kinetics or not. Note that in the proposed drying creep model, the parameter kdc traduces the interaction between the solid skeleton and the liquid phase which bounds the micro-diffusion process in adsorbed water layers and then separates the two kinetics. In Fig. 6 the simulated delayed strains, for both of the models, are reported with the experimental ones used for the identification of the parameters, which values are reported in Table 2. It can be observed that the proposed model is able to reproduce the experimental results during all the test, without changing the values of the drying creep parameters. With Bažant’s model, however, simulations were performed by using two values of the parameter µ. The first value µ1 is obtained such that we retrieve the final amplitude of the delayed strain, while the second value µ2 enables to reproduce the evolution of the delayed strains at the beginning of the drying process only. It can be observed that the delayed strains are overestimated by about 45% after 900 days when the value µ2 is used, while this value gives a good agreement until about 200 days. This issue highlights the fact that the drying process and the intrinsic drying creep mechanisms occur with different kinetics, which is considered in the proposed model. Further, one should pay attention to the test duration, since the prediction of shrinkage or creep is often based on short-time tests (duration of 1 to 3 months) [44]. Another aspect discussed here is the separation between the micro-cracking effect and the intrinsic drying creep components. The micro-cracking effect εmc corresponds to the extra strain, which results from the closure of microcracks (due to drying) by the applied compressive load. This component can therefore be obtained by removing basic t c_test t c_test creep εbc , drying creep ε dc (obtained from a total _test (obtained creep test) and drying shrinkage strains εds ds from a drying shrinkage test), from the total delayed strains

247

Fig. 6. Experimental and numerically fitted evolutions with time of longitudinal total delayed strain.

Fig. 7. Contributions of each component of delayed strains obtained with the proposed model in the total creep test.

t c_test εtds : t c_test (t) εmc (t) = εtds t c_test (t) + εt c_test (t) + εsh _test (t)]. −[ε bc

dc

ds

(33)

The contributions of each delayed strain component are displayed in Figs. 7 and 8, for both of the proposed model and Bažant’s one (with the parameter’s value µ1 ), respectively. It has been previously reported that the micro-cracking effect lasts only a few days [2]. In these simulations, we find that its contribution remains important at least during several hundred days. Moreover, its evolution depends here strongly on the used model for intrinsic drying creep. This is due to the different kinetics of the two models. In the case of the proposed Kelvin chain based model, the intrinsic drying creep strain reaches rapidly its asymptotic value. The Bažant model, however, gives a strain which increases slowly according to the drying kinetic: for this model drying creep and drying process have the same kinetic.

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Acknowledgment The authors gratefully acknowledge EDF R&D for the financial support.

Appendix

Fig. 8. Contributions of each component of delayed strains obtained with the Bažant’s model (case with µ1 value) in the total creep test.

4. Conclusions 1. An orthotropic elasto-plastic damage model, where orthotropy is induced by cracking in tension, has been presented. This model has been coupled with a drying shrinkage, a basic creep and a drying creep model, based on chemo-physical mechanisms. The drying process is modeled by Fick’s second law, where the diffusivity depends strongly upon the water content. The adopted framework enables, therefore, to study each strain component and then to separate the intrinsic behavior of the material from the structural one. 2. A simple linear relationship between drying shrinkage and water content is sufficient to retrieve experimental drying shrinkage results obtained by Granger [7] until about 1 year (weight loss of 2%). The drying process causes a strong structural effect: micro-cracking arises in the concrete skin and attenuates the free drying shrinkage strain. The amplitude of the structural strain is quite large. It highlights the need of an adequate cracking model to separate accurately the intrinsic behavior of the material from the structural induced effect. Nevertheless, after one year, the numerical curve deviates from the experimental one. This behavior seems to be intrinsic, since the adopted mechanical model takes into account most of the characteristics of cracked concrete. 3. The comparison between numerical and experimental results of total delayed strains (including drying shrinkage, basic and drying creep) underlines the need to dissociate the drying creep strains kinetic from the drying process one. Indeed, a better agreement with experimental results has been achieved with a model separating both kinetics. Moreover, identification of intrinsic drying creep parameters based on short-time measurements (about 200 days) has been found to be misleading, for predicting long-term deformation. Indeed the drying process of the studied specimen (16 cm diameter) lasts about ten years.

The detailed basic creep constitutive equations are presented here. The expressions of the spherical creep strains (reversible and irreversible parts) can be derived analytically from Eq. (6) in the eigenvector base and then in the original base after a base transformation. The expressions of the deviatoric creep strains (reversible and irreversible parts) can be obtained, in a straightforward way, from Eq. (7). They are given in the case of constant stresses and relative humidity. The expressions of the reversible and irreversible creep strains are [25]: sph

sph

εbc_r (t, h) = h Jr (t)σ sph sph εbc_i (t, h)

=

sph

and

sph h Ji (t)σ sph

(34)

sph

where Jr and Ji are the reversible and irreversible spherical basic creep compliance functions, respectively. sph sph sph sph If the term [2kr εbc_r − ki εbc_i ] − hσ sph + is equal to zero (Eq. (6)), the compliance functions read: sph

sph

sph

sph

Jr (t) = [1 − exp(−(kr /ηr )t)]/kr sph Ji (t)

= 0.

and (35)

In this case, the strain is entirely reversible. sph sph sph sph If the term [2kr εbc_r −ki εbc_i ]−hσ sph+ is not equal to zero (Eq. (6)), the compliance functions are written as: 1 1 sph Jr (t) = sph + (u 1 u 2 − 1) kr

1 × (−u 1 u 2 exp(ω1 (t − t p )) + exp(ω2 (t − t p ))) 2kr  u1 + sph (exp(ω1 (t − t p )) − exp(ω2 (t − t p ))) (36a) ki 1 1 sph Ji (t) = sph + (u 1 u 2 − 1) ki

u2 (− exp(ω1 (t − t p )) + exp(ω2 (t − t p ))) × 2kr  1 + sph (exp(ω1 (t − t p )) − u 1 u 2 exp(ω2 (t − t p ))) (36b) kr where

F. Benboudjema et al. / Engineering Structures 27 (2005) 239–250

u 1 = (ω1 + u ii )/(2u ri )

and

u 2 = 2u ri /(ω2 + u ii )

√ ω1 = [−(u rr + 4u ri + u ii ) − ]/2 √ ω2 = [−(u rr + 4u ri + u ii ) + ]/2

(37a) and (37b) sph

sph

with  = (u rr + 4u ri + u ii )2 − 4u rr , u ii , u rr = kr /ηr u ii sph sph sph sph = ki /ηi and u ri = kr /ηi . The time t p corresponds to the time when the term sph sph sph sph [2kr εbc_r − ki εbc_i ] − hσ sph + is equal to zero (transition between the reversible and the irreversible state). Since the irreversible strain is initially equal to zero, Eq. (6) leads to: sph

tp =

ηr

sph

ln(2)

(38)

kr

The deviatoric basic creep strain can be expressed analytically by [25]: dev dev εdev bc_r (t, h) = h Jr (t)σ

εdev bc_i (t, h)

=

and

h Jidev (t)σ dev

(39)

where Jrdev and Jidev are the reversible and irreversible deviatoric basic creep compliance functions, respectively. They read: Jrdev (t) = [1 − exp(−t/τrdev )]/krdev Jidev (t)

=

and

t/ηidev .

(40)

The spherical J sph and the deviatoric J dev basic creep compliance functions read therefore: sph

sph

J sph (t) = Jr (t) + Ji J

dev

(t) =

Jrdev (t)

+

(t)

and

Jidev (t).

(41)

Thus, the basic creep compliance tensor (second order) Jbc is written as: J sph (t) − J dev (t) J sph (t) + 2 J dev (t) I+ 3 3 × (1 ⊗ 1 − I).

Jbc (t) =

(42)

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