A viscoelastic poromechanical model for shrinkage and creep of concrete

Abstract

Long-term delayed strain of concrete impacts the lifetime of civil engineering structures such as dams, nuclear power plants, nuclear waste storage tunnels or large bridges. In design practice, the long-term delayed strain of concrete is decomposed into four components: autogenous shrinkage, basic creep, drying shrinkage and drying creep. The four components are first computed separately and then summed up to obtain the total delayed strain of concrete, without wondering about any potential correlation between them. In this work, we aim at modeling the total delayed strain in a unified manner, without assuming this decomposition a priori. Such model is derived in the framework of viscoelastic poromechanics. The influence of relative humidity on the creep properties is taken into account. We assume that drying creep is due to the fact that the mechanical consequences of capillary effects are larger in loaded drying specimens than in non-loaded drying specimens. The model is validated by comparing the prediction with experimental results of delayed strain from the literature and discussed with respect to existing models.

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.