A viscoelastic poromechanical model for shrinkage and creep of concrete
Introduction
Long-term time-dependent strains of concrete are of great importance when it comes to analyze the security of major civil engineering concrete structures such as dams, nuclear power plants, nuclear waste storage tunnels or large bridges, as they are normally designed for a service lifetime of several decades. Conventionally, the delayed behavior of concrete is decomposed into autogenous shrinkage, basic creep, drying shrinkage and drying creep.
Autogenous shrinkage is the time-dependent strain of a non-loaded specimen exchanging no water with its surroundings. Many consider that autogenous shrinkage is caused by the capillary depression due to self-desiccation [[1], [2], [3], [4], [5], [6], [7], [8], [9], [10]] while, recently, autogenous shrinkage was also attributed to eigenstresses that prevail in the solid skeleton as a result of hydration [[11], [12], [13]]. For what concerns the modeling of autogenous shrinkage as a response to capillary forces induced by self-desiccation, several authors [3,5,6,9] considered autogenous shrinkage as an elastic response to those forces, while others [1,4,7,8,10,[14], [15], [16], [17], [18]] considered it as a viscoelastic response to those same forces.
Basic creep is the difference between the strain of a loaded specimen exchanging no water with its surroundings and the autogenous shrinkage. Several theories exist in the literature to explain the physical origin of basic creep. For example, Bažant et al. [19] explained it with the microprestress theory, in which they postulated that, for load levels below 40% of the strength, the origin of the creep is the shear slip at overstressed creep sites. As a result of a progressive relaxation of microprestress at the creep sites and consecutive increase of their apparent viscosity, the creep rate observed under a constant applied stress declines over time. Recently, Vandamme [20] proposed, by joint analysis of results of macroscopic creep tests and microindentation creep tests, that basic creep originates from local microscopic relaxations.
Drying shrinkage is the difference between the strain of a non-loaded specimen exchanging water with its surroundings and the autogenous shrinkage. For cases where the relative humidity is larger than 40%, drying shrinkage is often considered to be caused by capillary effects. Again, some researchers [7,15,18,21,22] model drying shrinkage as the viscoelastic response of the material to those capillary effects while others (e.g., Refs.[4,[23], [24], [25], [26], [27]]) consider it as an elastic response to those same forces.
Drying creep is the additional time-dependent strain of a loaded specimen exchanging water with its surroundings, with respect to the sum of autogenous shrinkage, basic creep and drying shrinkage. Drying creep is also known under the name of Pickett effect as it was observed first by Pickett [28]. The drying creep measured on a specimen can be divided into two parts: an intrinsic part and a part due to a structural effect. In a specimen in contact with dry air, drying takes place faster at the surface of the specimen than in its center, which hence creates a gradient of relative humidity through the specimen. In case of drying in absence of any mechanical load, the surface may crack, whereas cracking is limited by the load in case of drying under load. Hence, shrinkage in a cracking specimen should be less than in a non-cracking specimen (if it is possible to prevent a drying specimen from cracking) [29,30]. When the specimen is loaded in compression, cracking is more limited and a supplementary part of the shrinkage is mobilized. This latter part of the strain is called drying creep related to the structural effect. The structural effect is not sufficient to explain the totality of drying creep [[30], [31], [32]]. The fraction of drying creep that cannot be explained by the structural effect is called the intrinsic drying creep, as it relates solely to the material. In the following, whenever drying creep is mentioned, we refer to its intrinsic part. The origin of (intrinsic) drying creep is still not known. Bažant et al. [33] consider that microdiffusion of water due to drying promotes shear slip in the microprestress theory [19], so that the kinetics of creep is faster in presence of drying than in absence of drying. This explanation is also supported by Vlahinic et al. [34] who propose that the migration of water molecules plays a lubricant role that amplifies delayed strain. Sinko et al. [35] propose that creep deformations are accelerated by the movement of water. In contrast, to explain drying creep, Sellier et al. [15] consider that the mechanical consequence of capillary forces applied to the solid skeleton is greater in presence of mechanical load than in absence of it.
Most design codes (e.g., Eurocode 2 [36], ACI [37] and code model [38]) and academic models in the literature [[39], [40], [41], [42], [43], [44]] follow such classical decomposition of delayed strains. In these models, each of the four components of delayed strain is computed individually with a different kinetic law. Then, by summing them up, the total delayed strain is obtained. However, with such approach, any potential correlation between the four components of delayed strain is not considered. For instance, drying creep and drying shrinkage are observed to be proportional to each other [45,46]. In this work, we aim at proposing a predictive model that does not assume this classical decomposition of delayed strains a priori.
From the origin of the four different components of delayed strains mentioned above, we can see that each of the four components can be considered to be a viscoelastic response to an applied external mechanical load or to internal capillary forces. Thus, in the spirit of the approach proposed by Sellier et al. [15], we could envision to model the various components of delayed strain in a unified manner, which will be attempted with poromechanics in the framework of non-aging isotropic linear viscoelasticity. By doing so, we aim at modeling the delayed strain of concrete without having to assume the classical decomposition a priori.
In this study, first, we list the main experimental tendencies that are to be taken into account in the model. The influence of relative humidity on both creep compliance and effective stress is discussed. Then, we present non-aging linear viscoelasticity with environment-dependent properties. Next, we estimate the elastic and long-term asymptotic values of the viscoelastic Biot coefficient, based on elastic and viscoelastic homogenization schemes. Then, the model is derived and calibrated against experimental results of delayed strain from the literature. In the end, the derived model is compared to other models from the literature and we list the advantages and weak points of the model and draw several conclusions.
Section snippets
Main lines of the model
Since, in this present work, we are mostly interested in the long-term delayed behavior of concrete (e.g., for containment buildings in nuclear power plants, where presetressing is applied when concrete is more than one-year-old), the model is dedicated to mature cementitious materials, for which we assume that the hydration slowed down significantly, that the microstructure does not change significantly anymore and that aging effect due to polymerization can be neglected [[47], [48], [49]].
Influence of relative humidity on creep modulus
In this section, we aim at modeling the dependence of the bulk creep compliance JK(t) on the relative humidity hr (see Eq. (6)). There is very few experimental data on the creep of predried specimens under autogenous conditions: Refs.[64,65] by macroscopic testing and Refs.[54,61] by microindentation testing. We use the microindentation results to infer the relation between bulk creep modulus CK and relative humidity hr.
The experimental microindentation results in Refs.[54,61] provide the
Influence of relative humidity on effective stress
This section is dedicated to model the influence of relative humidity on the effective stress (see Eq. (3)).
The capillary stress σh in Eq. (2) depends on relative humidity. On one hand, the saturation degree Sl is directly related to the relative humidity hr via the desorption isotherm Sl(hr(t)). On the other hand, the capillary pressure can be related to relative humidity through Kelvin's equation: where R = 8.314 J ⋅ K−1 ⋅ mol−1, T and Vw = 1.8 × 10−5 m3/mol are the ideal
Non-aging linear viscoelasticity with environmental dependency
In this section, we aim at extending the constitutive Eq. (1) to the case where viscoelastic properties (i.e., compliances) depend on environmental parameters that are evolving with time. For the sake of simplicity, in this particular section we only consider a unidimensional case.
From the definition of compliance, the viscoelastic strain ε(t,t0) at time t under the application of a constant stress σ0 that is applied since time t0 is expressed as: where J(t − t0) is the
Biot coefficient
While some properties of concrete such as creep compliance can be measured experimentally, the viscoelastic Biot coefficient is difficult to measure. Hence, in this section, we aim at estimating the viscoelastic Biot coefficient bc(t) of concrete through homogenization.
Concrete is a heterogeneous material made of various solid phases whose characteristic lengths vary from nanometers to centimeters. Based on the characteristic size of the various phases in microstructure of concrete, researchers
Constitutive equations
In this section, we list the procedure to model the time-dependent shrinkage and creep strains. We recall that the concrete/cement paste is considered as a non-aging viscoelastic porous material. We assume that the viscoelastic Poisson's ratio ν(t) is constant over time and equal to 0.2.
The model is a semi-coupled hydro-mechanical model: the relative humidity hr(t) affects the mechanical response but we neglect the impact of mechanical load on the history hr(t) of relative humidity (as
Application of model
In this section, we calibrate the model against experimental results from the literature. Since the model is intended for long-term delayed strain of mature concrete, we choose to consider the tests of Granger [41] and of the VeRCoRs project [92]. In both examples, we suppose that stresses and hydric state are homogeneous within each specimen.
Discussion
In this section, we discuss the following three points: (1) alternative way of taking into account drying creep, (2) general comparison of the model here derived with existing models from the literature, (3) aspects to be considered in more details for improving the model.
Conclusions
In this work, we proposed a non-aging linear viscoelastic poromechanical model to predict the delayed strain of cement-based materials, without decomposing the delayed strain classically into autogenous/drying shrinkage and basic/drying creep. The model is dedicated to delayed strains of mature materials, i.e., materials whose microstructure does not change significantly with time anymore. Consequently, the model cannot simulate shrinkage and creep behavior related to or caused by hydration at
Declaration of competing interest
The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: The authors acknowledge financial support from EDF.
Acknowledgments
The authors acknowledge financial support from EDF and thank EDF for this support.
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Now at Graduate School of Environmental Studies, Naogoya University, Japan.