Multiphasic model of early stage hydration in concrete using the theory of porous media

The aim of this paper is to model the behavior of concrete during the initial stage of maturing. Most models consider only thermo‐mechanical phenomena assuming the hygral phenomena of less importance due to high liquid saturation. In order to grasp those effects, the proposed model is an extension of Gawin's model (D. Gawin, F. Pesavento, and B. A. Schrefler, Int J Numer Methods Eng 67(3), 299–331, 2006). The fresh concrete, modeled as porous material, is described within the well‐founded framework of the Theory of Porous Media. The multiphase material consists of a solid phase representing, for example, cement, gravel, etc., and a fluid and gas phase representing the pores filled with water and dry air. Compared to Gawin's model, here, beside the effect of hydration‐dehydration, the effect of evaporation‐condensation will be taken into account. A new formulation for the evaporation process in the fresh concrete is proposed and validated by an experimental study. The presented model is implemented in the research code PANDAS. The proposed model investigates and proves the assumption of Gawin's model to have a linear connection between porosity and hydration degree.


INTRODUCTION
Predicting the behavior of concrete is of great practical importance and, as such, it has been researched throughout the years.During the initial period of concrete's maturing, the material is undergoing non-uniform deformations due to highly exothermic and thermally activated reactions of cement particles.This exothermic process, caused from chemical reactions, leads to heat production inside of the material, which can be up to 50 • C in case of massive structures.Due to temperature evolution in the material, the kinetics of the hydration process is triggered and with the temperature increase the time of reaction increases as well [1][2][3].
The proposed model of concrete at the early stage of hydration includes not only thermo-mechanical phenomena as, for example, in the literature [3][4][5], but also takes into account the hygral phenomena in fresh concrete.The latter is of particular interest in case of considerable changes of relative humidity and water saturation conditions.The presented model assumes the fresh concrete as a porous material and uses a macroscopic continuum-mechanical approach within the framework of the Theory of Porous Media (TPM) [6].A porous material is a material with internal structure made of F I G U R E 1 REV of the microstructure of a partially saturated porous material, (left), and the macro model of a multiphase material based on TPM after a homogenisation process (right) [9].REV, representative elementary volume; TPM, Theory of Porous Media.a solid phase, representing the solid skeleton of fresh concrete, and pores, which can be filled with multiple fluid phases, where liquid and gas phase represent the water and dry air [7].
The model description at the microlevel is showing a change of the mechanical properties of fresh concrete as a change in concentrations of its aggregates [3].The intent is to consider full coupling and interactions between these various phenomena.The properties of the materials are highly dependent on the degree of hydration [1].The balance equations of the porous material will consider the effects hydration-dehydration and evaporation-condensation, which play main roles in material's phase changes.

THEORY OF POROUS MEDIA
The model is based on a continuum-mechanical approach to obtain the governing balance equations by the averaging theory of Hassanizadeh and Gray [8], also known as Hybrid Mixture Theory.The Theory of Porous Media (TPM) uses as a basis the Hybrid Mixture Theory and extends it by the concept of volume fractions.TPM provides an excellent framework to describe at the macroscopic level the complicated microstucture of a material without knowing the exact structure.
The representative elementary volume (REV) is defined locally and its aggregates are assumed to be in a state of ideal disarrangement (for a schematic sketch see Figure 1).By using a real or a virtual averaging process, the statistically smeared out microstructure is achieved, meaning all geometrical and physical quantities are seen as a statistical mean value of the occurring actual quantities [9].Constitutive laws will be introduced directly at the macrolevel of the material.
Partially saturated porous materials are commonly described by a triphasic material model, where the single aggregates are the materially incompressible solid skeleton   , and the two pore fluids, that is, the incompressible pore water   and the compressible dry air   .The volume fractions   of each aggregate    ∈ {, , } are defined as the local averages of the respective partial volume element d  with respect to the representative volume element d of the overall body [10] In Equation (1),  is the position vector at the time .In this multiphase body the saturation condition (1) for the sum of all volume fractions   must be fulfilled at any time.The partial density of each constituent   =     is related to the real density   = d  ∕ d  via the volume fraction   and the mass element is d  .

KINEMATIC RELATIONS
In the framework of the TPM, the body  consists of material of all aggregates   .At any given time , every spatial point  is occupied by the material of all constituents   in the current configuration.Moreover, in reference configuration for the time  0 , the material point of each aggregate is defined with the reference position vector   .The motion, which leads to the individual motion of each aggregate, is described by the actual position vector  at time  in the Lagrangean setting The assumption of individual motion functions for all phases  allows to define different velocity fields  ′  using the material time derivative () ′ .The requirement of the relation between positions of material points in current (actual) and reference configuration is based on the non-singular Jacobian determinant and the inverse motion function The restriction of   = det   > 0 needs to be fulfilled during the deformation of the multiphasic body, where   represents the deformation gradient of each aggregate.

BALANCE EQUATIONS
The deformation and thermal processes of a continuum mechanical body  are described by the usual balance relations -Balance of mass, linear momentum, moment of momentum, energy (1 law of thermodynamics) and entropy (2 law of thermodynamics).They are formulated for each constituent   and are based on Truesdell's metaphysical principles within the mixture theory.Each balance equation describes a single aggregate of the multiphasic body taking into consideration so-called production terms, which represent the interaction mechanism between all constituents of the body .
For each constituent the equations [7,9] (  ) ′ +   div  ′  = ρ , 0 = div   +    + p ,   = (  )  ,   (  ) ′ =   ∶   − div   +     + ε ,   (  ) ′ = div have to be fulfilled, where   represents Cauchy's stress tensor,  the gravity,   the velocity gradient tensor,   the flux, and   the heat supply.Θ is the temperature, which is assumed to be the same for all constituents.In Equations ( 4), nearly all balances contain an additional production term.For the mass balance equation the production term ρ represents the mass exchange between the phases.The total momentum production ŝ = p + ρ  ′  contains the direct momentum exchange p and an additional term resulting from the mass exchange from the mass balance ρ  ′  , which is neglected due to its low impact.For the energy and entropy balance the production terms also consist of a total production and additional terms.Those will be neglected and, hence, the production terms reduce to the direct production parts ε and ξ .Hence, the requirement for the sum of all production terms needs to be fulfilled.

CONSTITUTIVE MODELLING
To close the multiphasic model, constitutive assumptions have to be formulated.The solid part is, essentially, modelled with a Neo-Hookean material and the fluid or gas as inviscid fluid.

General setting
The Helmholtz free energy function   of the solid constituent is with the initial volume fraction   0 and density of solid   0 , the first invariant  1 = tr   of the right Cauchy-Green tensor   =      and a dimensionless parameter  which accounts for the change of porosity.Further, the macroscopic Lamé constants (  ,   ) are used.An evaluation of the whole aggregate's entropy inequality defines the stresses of both, the solid and fluid constituents, as well as the linear momentum production terms [6,11].For the pore pressure, the relation  =   =     +     holds with the saturations   and   .Neglecting the frictional fluid forces and with it the fluid extra stresses [12,13] results in the set of stress/pressure definitions and the definition of the production term where with  ∈ (, ) representing the liquid and gas phase.Hence, using the well-known concept of effective stress and the saturation condition, the Cauchy stress is Using the above definition of the fluid stresses and the balance of momentum for the seepage velocity   =  ′  −  ′  , Darcy's law is obtained for each phase in the fluid The permeability tensor in Equation ( 9) is reduced to  =  because it is assumed that concrete is an isotropic material.  and   denote the dynamic viscosity and relative permeability of the gas and liquid [1,14].

Mass exchange
The mass exchange plays a significant role in this multiphasic model.There are two different processes happening simultaneously inside of the porous material.As the cement comes in contact with water, chemical reactions are triggered and with it the first process of hydration starts.It is the interaction solely between the liquid and solid phase, where the volume fraction of solid increases in time.The mass exchange ρ is modelled with [1] where dΓ ℎ d describes the rate of hydration degree inside of the material.It is dependent on the temperature Θ, the gas constant , the activation energy   , the final stage of material hydration ρ ∞ and the normalized affinity  Γ , determined from adiabatic experiments in the literature [3,15].The coefficient  Φ is shown as F I G U R E 3 Boundary conditions and geometry given by Gawin et al. [1].
in relation to relative humidity Φ [1].Using the difference between the pressures of gas and liquid phase   =   −   under conditions of thermodynamic equilibrium the relative humidity is formulated in dependence of   using the Coleman-Noll method, as demonstrated in the literature [1,16,17].In order to investigate the evaporation process during the initial stage of concrete's hardening, a new formulation for ρ is postulated as function of the norm of the seepage velocity of the liquid phase ‖  ‖.The parameter  represents the maximum of mass exchange and  1 is a material parameter describing the relation between the seepage velocity and mass exchange rate, see Figure 2.

NUMERICAL EXAMPLE AND EXPERIMENTAL VALIDATION OF THE EVAPORATION PROCESS
The numerical treatment of this triphasic material model of fresh concrete is based on the weak formulation of the governing balance relations of all its constituents.This follows the usual steps and the set of primary variables are the displacement of the solid phase   , capillary pressure   , gas pressure   , the volume fraction of the solid   , and the temperature of the whole aggregate Θ.The latter is assumed to be the same for each constituent Θ  = Θ.Different to the weak form in the literature [1], the mass balance of the solid is included.The formulation is implemented in the Finite Element code PANDAS developed at the University of Stuttgart.Taylor Hood elements are used for the spatial discretization, where we select quadratic shape function for the solid displacements   and linear shape functions for all other variables.
In the literature [1], a linear connection between the porosity and hydration degree of the material is assumed instead of the proposed balance equation.In a first example, this assumption is confirmed by the numerical results using the proposed extended formulation.The problem geometry (see Figure 3), as well as its boundary and initial conditions, are taken from the literature [1].Initially, the material has a temperature Θ 0 = 293.15K,relative humidity Φ 0 = 0.999 and initial hydration degree Γ 0 = 0.1.Figure 4 shows the temperature evolution in the extended model over the period of 48  hours of fresh concrete's hardening in comparison to the results of Gawin et al. [1].The red line represents the result of the proposed extended model, the blue spots Gawin's results and the gray spots the experimental results also taken from literature [1].A good agreement can be observed.The assumption in Gawin's model of a linear connection between the porosity   and the hydration degree Γ ℎ is proven in the Figure 5.
But here, it is the result of the extended model.Next, the calibration of the new evolution equation ( 10) will be shown.The experiment uses the Dynamic Vapor Sorption (DVS) method (see Figure 6) to measure over time the change in mass of fresh concrete poured in small rectangular samples.The samples are kept in an environment at constant temperature Θ = 293K, relative humidity Φ = 0.94, and atmospheric pressure   =   = 101.325kPa.As the top surface of the sample is open, it allows the gas phase to enter and exit.This setup is simulated with the proposed model.The geometry of this two-dimensional problem is presented in Figure 7, where the boundary conditions are adjusted to the experimental setup.The top surface is drained and the other boundaries are undrained surfaces.The bottom and side edges are also fixed in the and -directions, respectively.The set of material parameters is shown in Table 1 and taken from literature [1,18] with the exception of  and  1 .
The simulation represents the first 3 days of concrete's hardening.After that period the evaporation process becomes stationary.At the time  = 0, the processes of hydration-dehydration and evaporation-condensation start happening inside the porous material.These two processes represent the mass exchange between all three phases.During the hydration, the liquid phase (water) comes into contact with cement particles and the hardening process starts resulting in an increase of solid volume fraction in the material.The evaporation process, which is the focus of this numerical example, causes liquid water to evaporate into the dry air and exit at the top of the domain.In Equation (12), the evaporation rate is modelled dependent on the seepage velocity of the water particles.As the velocity increases, it causes the water evaporation to increase rapidly.The relation is governed by two variables  and  1 .The values given in Table 1 are the outcome of the calibration process in comparison with the experimental results.In Figure 8, the blue line represents the experimental results and the red spots indicate the numerical solution of the extended model.There are obvious differences, however the trend is good.One of the future trails will be making the material parameter  dependent of the hydration degree Γ ℎ of the material.

CONCLUSIONS
The presented model describes concrete at the early stage of hydration as a triphasic material model based on the Theory of Porous Media.In order to develop a thermodynamically consistent fully coupled model with individual phases, it is assumed that the solid skeleton and liquid phase (water) are materially incompressible contrary to the gas phase (dry air).Furthermore, interaction between all three phases is triggered by two processes inside of the material, hydrationdehydration and evaporation-condensation.The numerical model shows that the assumption in the literature of a linear connection between porosity and hydration degree was suitable.Further, the evaporation process has been computed comparing to an experimental study.

F I G U R E 2
Mass exchange rate due to evaporation ρ (  ) for  = 1 kg∕sm 3 and  1 = 5.0 s 2 ∕m 2 .

F I G U R E 4
Temperature evolution of fresh concrete during the initial 48 hours of hydration.F I G U R E 5Ratio between porosity and hydration degree of fresh concrete.

F I G U R E 6
Experimental setup via DVS method.DVS, Dynamic Vapor Sorption.

F I G U R E 7
Boundary conditions of fresh concrete.TA B L E 1Material parameters and initial conditions for fresh concrete.

F I G U R E 8
Comparison of numerical and experimental results of fresh concrete for evaporation rate ρ in [kg∕m3 s].