Hydration Model Description

Introduction

Hydration is defined as the chemical absorption of water into a substance, a process by which heat is generated (hydration heat). The setting of concrete (which can be considered as a transition from liquid to solid phase) is the most relevant example for the hydration process in the engineering world.

The effects of the hydration process can be separated into different physical parts, where the thermal and mechanical parts are the most relevant. The implementation of hydration models in FLAC3D follows this separation, as the hydration heat generation and heat transfer are dealt with in thermal models, material hardening and strength development are implemented as constitutive models of mechanical behavior. The hydration model is based on a procedure that considers empirical rules, theoretical considerations and practical experiences (Onken and Rostásy 1995).

The coupling parameter between the different parts of the constitutive model are the hydration grade \(\alpha\) and the concrete age \(t_e\) [s]. The hydration grade is defined as the ratio between the accumulated hydration heat up to the current time \(Q\) [J/m³] and the ultimate hydration heat generated until total completion \(Q_{max}\) [J/m³].

(1)\[\alpha (t_e) = {{Q (t_e)} \over {Q_{\rm max}}}\]

The time-dependent evolution of the hydration grade depends on the temperature of the concrete. Thus, the heat flow above the boundaries and the heat conductivity within the hydrating material are of specific importance for this evolution as well as the initial temperature. Lower temperatures lead to a longer process with lower hydration heat generation, whereas higher temperatures lead to a shorter process with higher hydration heat generation. An equivalent age \(t_e\) is introduced as a measure of time that is independent of these real-time effects.

(2)\[t_e = \int_0^t e^{{E_A \over R} \left({1 \over T_{{\rm Re} f}} - {1 \over T} \right)} d \tau\]

where \(R\) is the universal gas constant (8.314 J/K/mol), and \(E_{A}\) is the activation energy [J/mol]. The description of the thermal behavior is related to the reference temperature \(T_{Ref}\) [K], which is set to 20°C in Onken and Rostásy (1995); \(T\) is the material’s temperature [K].

A thermal hydration model, hydration, is implemented in FLAC3D. For simulating a hydration process, a mechanical constitutive model that can adjust the mechanical properties corresponding to the hydration grade (or equivalent concrete age) is required. The modified Drucker-Prager model for hydration — hydration-drucker-prager model, is provided to handle the mechanical aspects. These models are described in the following sections.

Thermal Model for Hydration

It is primarily the hydration of the binding material (cement, in the case of concrete), that leads to the hydration process.

(3)\[Q(t_e) = C Q_{Ce} (t_e) \ \ \ \ \ \ Q_{\rm max} = C Q_{Ce, {\rm max}}\]

where \(C\) is the content of binding material [kg/m³], \(Q_{Ce}\) is the amount of hydration heat from binding material produced up to the current time [J/kg], and \(Q_{Ce, {\rm max}}\) is the ultimate hydration heat generated until total completion [J/kg].

With respect to Equation (1) and (3), the current heat release is

(4)\[q(t_e) = \dot Q(t_e) = \dot \alpha C Q_{Ce, {\rm max}}\]

Without any restriction, the hydration process would run over the full range of the hydration grade, which is \(\alpha = [0,1]\). The heat release is always active, and at the end, a heat in the amount of \(Q_{max}\) (see Equation (1)) has been generated. However, this process can be restricted by physical or numerical reasons. Therefore, two independent limitations are available, one depending on temperature and the other on hydration grade:

(5)\[\begin{split} q(t_e) = \begin{cases} q(t_e) & \forall T \leq T_{{\rm max}, q} \\ 0 & \forall T \geq T_{{\rm max}, q} \\ \end{cases}\end{split}\]

or

(6)\[\begin{split} q(t_e) = \begin{cases} q(t_e) & \forall T \leq \alpha_{{\rm max}, q} \\ 0 & \forall T \geq \alpha_{{\rm max}, q} \\ \end{cases}\end{split}\]

The behavior of hydration grade, depending on equivalent age, Equation (2), is an exponential description with the two Jonasson material parameters, \(b\) [-] and \(t_1\) [1/s]:

(7)\[\alpha(t_e) = e^{-\left[1{\rm n}\left(1 + {t_e \over t_1}\right)\right]^b}\]

The equivalent age is given in Equation (2). As shown in Equation (2), the only material parameter of equivalent age is the activation energy.

Two versions of activation energy are currently available: a generalized model and a German model.

Generalized Concrete Model

The generalized version for the activation energy as a function of the temperature, \(E_A\) is implemented as

(8)\[E_A (T) = E_{A, 2} + {{E_{A, 1} - E_{A, 2}} \over {1 + e^{{T - T_{0, EA}} \over {d T_{EA}}}}}\]

This defines the activation energy between the two limits, \(E_{A, 1}\) [J/mol] and \(E_{A, 2}\) [J/mol]. \(T_{0, EA}\) [K] gives the temperature where the difference between the limits is divided in half, and \(dT_{EA}\) [K] characterizes the S-shape of the function.

German Concrete Model

For the German concrete model (Onken and Rostásy 1995), values for the activation energy are given in two ranges of temperature:

(9)\[\begin{split} E_A (T) = \begin{cases} E_{A, 1} + d E_{A, T} \cdot (T_{0, EA} - T) & \forall T \leq T_{0, EA} \\ E_{A, 1} & \forall T > T_{0, EA} \\ \end{cases}\end{split}\]

Typical parameters are \(E_{A, 1}\) = 30 J/mol, \(d_{EA, T}\) = 1.47 J/mol/K, and \(T_{0, EA}\) = 293 K.

Both versions of the activation energy are implemented and can be selected by the parameter flag-law (0: German concrete; 1: general model).

The heat transfer is assumed to be isotropic, with the following functions of specific heat, \(c_p\), and thermal conductivity, \(\lambda\):

(10)\[c_p = c_{p, 1} \cdot (1 + dc_{p, \alpha} \cdot \alpha) \cdot (1 + dc_{p, T} \cdot T)\]

(11)\[\lambda = \lambda_1 \cdot (1 + d \lambda_{\alpha} \cdot \alpha) \cdot (1 + d \lambda_T \cdot T)\]

and constant coefficient of thermal expansion.

An alphabetical list of properties of the hydration thermal model is given here.

Mechanical Model for Hydration

• Hydration-Drucker-Prager Model

The mechanical aspects of hydration in FLAC3D are handled by a modified Drucker-Prager model (see model description).

Reference

Onken, P., and F. Rostásy. Wirksame Betonzugfestigkeit im Bauwerk bei früh einsetzendem Temperaturzwang, DAfStb Heft 449. Berlin: Beuth-Verlag (1995).