1. Introduction

The behavior of concrete, a fragile material in extension, heterogeneous and porous, is governed by numerous and complex physicochemical phenomena. Prestress losses induced by the delayed behavior of concrete (shrinkage and creep) reduce the load range that the structure can support over time. These delayed deformations that appear in concrete during the life of this one are composed by shrinkage at a young age (endogenous shrinkage specific to hydration and thermal shrinkage), by desiccation shrinkage with drying modeling, and as soon as it undergoes stresses, by clean creep and desiccation creep.

In design rules, delayed deformations of concrete are generally based on empirical rules based on a large number of results from the literature, taking into account the main parameters, such as temperature, humidity, aggregate content, aggregate content, water/cement ratio. The kinetics of phenomena uses equivalent times calculated using an Arrhenius law to take into account aging and temperature.

A detailed analysis of the physicochemical phenomena that are at the origin of the various delayed deformations of concrete makes it possible to propose a modeling based on an equivalent continuous medium model [bib2], which was introduced in Code_Aster (clean creep and desiccation creep are not treated here).

1.1. Phenomenological aspects of the behavior of concrete at a young age: thermohydration

Young age is defined as the first 100 days of concrete’s life. Endogenous withdrawal or hydration withdrawal, and thermal withdrawal occur from the first moments of absorption (at a young age), for a period ranging from a few hours to a few days, for thermal withdrawal, and from a few months to a year, for the withdrawal of hydration, and from a few months to a year, for the withdrawal of hydration, generally completed during prestress. Phenomena of prevented withdrawals or differential withdrawals, under formwork, can be the cause of stresses or cracks that must be evaluated. In the liquid phase, concrete is a viscous fluid in which solid grains are suspended in the hydraulic binder containing solid particles (cements, etc.). Following the formation of the first hydrates, the concrete sets in, about ten hours after its manufacture, which corresponds to the establishment of related hydrate bridges between the cement grains in the totality of the material. At the very beginning, the grains are relatively dispersed in the mixing water. Over time, the hydration of cement grains is accompanied by the consumption of this mixing water. Experimentally, we note that the volume balance of the reaction is negative; it is the Le Chatelier contraction. Simply put, the total volume of hydrates is almost 10% less than the total volume of these components. Mechanically, at the scale of cement grains, the phenomenon stops when the hydrate bridges formed between the grains are sufficiently rigid to prevent a possible relative approximation of the grains. The macroscopic consequences on the structures are practically non-existent since throughout this phase, the concrete is still deformable, and any contraction is compensated by a granular readjustment of the material against the walls of the formwork. Although of relatively small magnitude, and of insufficient mechanical effect to cause real concrete cracking, the stresses generated at the interface of two consecutive lifts can affect the tensile strength margin of the material by 50%.

The setting of concrete accompanied by the hydration of the cement leads to an exothermic reaction. In massive structures the temperature can then rise to more than \(50°C\). Hydration is a thermo-activated reaction, i.e. the rate of hydration increases with temperature. As the hydration rate decreases, the temperature decreases, causing thermal shrinkage. In addition, the mechanical properties of concrete vary according to its degree of hydration, and finally, the consumption of water that occurs during hydration leads to capillary shrinkage. The various withdrawals can cause stresses much greater than the (low) tensile strength of concrete and lead to cracking of the material.

The calculation of the temperature and hydration degree fields is available with the THER_NON_LINE command (cf. [U4.30.02]). The calculation of the mechanical fields taking into account endogenous shrinkage is performed with the command STAT_NON_LINE.

1.2. Drying and desiccation removal

Modeling drying is important because the physicochemical and mechanical properties of the material are highly dependent on the humidity inside the material. The objective is to propose a macroscopic modeling of concrete drying based on a limited number of parameters, easily measurable experimentally, based on a law of non-linear transient diffusion of humidity, chained to temperature, by avoiding complex mechanical, physical and chemical couplings, at the material scale.

Upon dismantling, the concrete is immersed in an external environment which generally has a humidity level of the order of 60 to 80% RH (relative humidity = ratio of vapor pressure to saturated vapor pressure for a given temperature). It then undergoes a real water shock (by analogy to a thermal shock). The concrete is then in thermodynamic imbalance with the atmosphere. Drying will allow it to regain a water balance with the external environment.

Physically, drying involves complex phenomena that are closely linked to each other, depending on the heterogeneous and granular structure of concrete. At the macroscopic scale, it is possible [bib2] to model drying as a phenomenon of non-linear diffusion, with diffusion in liquid phase of the Darcy type, as long as there is continuity of the liquid phase, and with diffusion in the gaseous phase of the Fick type, for water vapor.

Desiccation shrinkage is the primary macroscopic consequence of concrete drying. It is the direct extension of the phenomena of capillary tension that are at the origin of endogenous shrinkage. By its intensity, the deformations being of the order of 400.10—6 to 800.10—6 for 50% humidity and for common concrete, it is one to three times greater than the elastic deformation for a loading close to \(10\mathit{MPa}\).

First, we present the modeling of thermohydration in the non-linear thermal operator of Code_Aster, then the modeling of drying, and finally, the introduction of endogenous shrinkage and desiccation shrinkage in the nonlinear mechanics operator.