A general framework for modeling long-term behavior of earth and concrete dams

Abstract

Many problems are linked with the long-term behavior of both earthdams and concrete dams. They range from hydraulic fracturing to alkali-silica reaction (ASR) and to repair work in concrete dams, from seismic behavior to secondary consolidation in earthdams. A common framework for the simulation of such systems is shown, based on the mechanics of multiphase porous media. The general model is particularized to specific situations and several examples are shown.

Keywords

earth dams / concrete dams / multiphase porous materials / coupled problems / hydraulic fracture / concrete hydration / alkali-silica reaction (ASR) / finite element method

Introduction

Dams are very challenging structures, depending on their geometrical complexity and the variety of stress/strain histories they can undergo. Other problems are related to the material they are composed of: different and complex phenomena can take place ranging from creep, shrinkage, thermal and hygral effects, chemical reactions in concrete dams to dynamical behavior, stability and liquefaction of earth dams. Traditional structural models account for some of them in a purely phenomenological way, i.e., by assuming simple constitutive relationships based on experience, as, for instance, in the case of shrinkage. In other cases these simple relationships are not available and a quantitative solution of the problem at hand is not possible, for instance in the case of alkali-silica reaction ASR effects. In the present discussion the material of the dam body (soil or concrete) is assumed to be a multiphase porous medium composed of a deformable solid skeleton and one or several fluids saturating the interstitial voids. This is a more general frame, which allows for a deeper analysis of the physics of the phenomena and the solution of a wider set of mechanical problems.

In particular, the mathematical model is formulated by using two scales starting from micro-level, i.e., from a local form of the governing equations at the pore scale. The final form of the mathematical model is subsequently obtained by applying some averaging operators to the equations at the micro-level, while the constitutive laws are defined directly at the upper scale, according to the so-called hybrid mixture theory. This approach allows for taking into account both bulk phases and interfaces of the multiphase system, assures that the second law of thermodynamics is satisfied at the macro-level, that no unwanted dissipations are generated and that the definition of the relevant quantities involved is thermodynamically correct. In particular, if the thermodynamically constrained averaging theory (TCAT) is used, the satisfaction of the second law of thermodynamics for all constituents at micro-level is also guaranteed. Within this last approach some stress measures are obtained and their form is described. The chosen procedure does not exclude however the use of a numerical multiscale approach in the formulation of the material properties. The numerical solution is obtained directly at the macro-level by discretizing the governing equations in their final form.

Governing and constitutive equations

The general mathematical model for thermo-hydro-mechanical problems taking place in a medium composed of a deformable solid skeleton and one or several fluids saturating the interstitial voids encompasses the equations for the conservation of the usual thermodynamic properties of every component. Each one of these equations contains, in its general form, terms representing the interaction effects of all the present fields (coupling terms). The mathematical model is represented by [1,2]:

1) One mass balance equation for each constituent (solid, liquid and gaseous phase). For the sake of simplicity, mass balance equation for solid skeleton is usually summed with the equation of each fluid phase. The ensuing equation comprises a rate capacity term related to changes of that constituent contained in a representative volume element dependent on deformation of all components, flux terms, dependent on transport law of the fluid phase (usually of Darcy’s type and/or Fick’s type), source/sink terms and exchange terms of the quantity with other phases;

2) The linear momentum balance (equilibrium) equation for the whole multiphase system. This is based on Terzaghi’s effective stress generalized to partially saturated conditions as described in detail in Section 3. When dealing with soil dynamic problems, the acceleration frequencies are supposed to be low as in the case of earthquake motion, which allows the relative acceleration of the pore fluid to be disregarded [3]. Hence, only an inertia term has to be added to the equilibrium equation;

3) The energy balance equation for the mixture solid plus fluid (s), assuming that all phases of the material are locally in thermodynamic equilibrium (hence their temperatures are the same). When mechanical terms are neglected, internal energy depends on temperature only; in particular it is related to rate of heat capacity of the mixture at constant volume. Volume heat sources are frequently retained, owing that they can represent very different phenomena, e.g., heat production due to hydration of concrete or to plastic work. Hence, they represent coupling effects between chemo/mechanical and thermal fields. Source terms may also arise along the boundary and represent frictional effects. Energy terms related to phase changes, convective and flux contributions complete the balance equation. The last ones are generally expressed by means of Fourier’s law.

More simple models can be easily obtained from the previous one by properly omitting some terms or equations. For instance, when isothermal problems are analyzed, the energy balance can be disregarded or when gaseous phase is always at constant pressure (for instance the atmospheric one [3]) the corresponding mass balance can be dropped.

A proper choice of state variables for hygro-thermo-chemo-mechanical modeling of concrete, in particular at early ages for repair work, as well as for soil is of great importance. From a practical point of view, the physical quantities used, should be easily measurable during experimental tests, and from a theoretical point of view, they must uniquely describe the thermodynamic state of the medium. They should assure a good numerical performance of the computer code, based on the resulting mathematical model, as well. The number of the state variables may be reduced if the existence of local thermodynamic equilibrium at each point of the medium is assumed. In such a case all constituents of the porous medium have the same temperature and several additional equations valid only at thermodynamic equilibrium may be used to reduce the number of variables. Use of solid skeleton displacement vector, gas pressure and temperature (the same for all constituents of the porous medium) is rather obvious. However, a choice of variables describing hygrometric state or advancement of cement hydration is not so obvious. Capillary pressure is the moisture state variable used by the authors for analysis of concrete and it is clearly related to stresses through the generalized Bishop stress tensor. For soil, water/capillary and gas pressures are used [4] and in case of static air phase assumption the gas balance equation can be dropped.

In real applications there is an obvious necessity to reduce the model complexity by introducing simplifying assumptions, and partial models are often used in practice. Of course the admissibility and the correctness of such simplifications have to be carefully assessed to obtain meaningful numerical models and reliable forecasts.

The mathematical model is completed by the constitutive equations. In the simplest formulation, linear relationships are generally used; however, several nonlinearities are also incorporated for the solution of specific problems. For the mechanical behavior of the overall system traditional elasto-plastic models are introduced; cohesive models are used in fracture mechanics applications and damage models for diffused degradation of the material strength. Fluid permeability may depend on porosity or capillary pressure; other nonlinearities of material properties due to temperature, gas pressure, moisture content and material degradation are also taken into account. In advanced formulations, special constitutive models can accommodate, even though in a phenomenological description, very complicated problems such as ASR, thermo-chemical strains, creep and shrinkage and other variation of material parameters due to aging.

For the spatial discretization of governing equations we have chosen the finite element method, following in particular the continuous Galerkin approach [5]. Special techniques are sometimes necessary to handle situations in which geometric singularities play a fundamental role, as, for instance, in the presence of discrete fracture problems. In similar cases continuous changes in the spatial mesh are automatically made (a posteriori mesh refinement) based on the evolution of the crack path and information on the solution are transferred from the old to the new mesh. Once the spatial approximation is obtained, the equations are discretized in the time domain. Usually this step is accomplished by means of the finite difference method (generalised trapezoidal and Newmark method) [5], but finite element approximation in time (discontinuous Galerkin method) has also been used. In the latter technique computational difficulties depending on the increment of the unknowns (usually doubled) are balanced by the availability of time discretization error measures, which allows for time adaptivity and optimal distribution of time steps increments with an overall gain of the solution process [6].

Effective stress principle and stress measures

In the last century many researchers have attempted to formulate an equation capable of describing the mechanical behavior of a porous medium taking into account its multiphase nature. The main adopted approaches were to define an effective stress equation and/or a certain number of independent stress variables.

The first stress measures in partially saturated porous media mechanics have arisen in the soil mechanics and reservoir engineering community. They are the Bishop [7] stress, the Skempton [8] stress, the net stress [9], the generalized Bishop stress [1], the generalized Skempton stress [10] and the stresses proposed by other authors (e.g., Coussy [11], Khalili et al. [12], etc.). All these measures contain the total stress and several of them include an effective stress. Up to now only the generalized Bishop stress [13], the Coussy stress [11], and a Bishop-like stress [14] have been proven to be thermodynamically consistent. On the other hand the mechanics community uses within mixture and hybrid mixture theories, the solid stress tensor and the fluid stress tensors, the sum of which, weighted by the respective volume fractions, gives the total stress (see for example Refs. [13] and [15]). Based on a survey of the literature, the most used stress tensors today are the generalized Bishop stress tensor and the Skempton stress tensor. A Skempton-like stress tensor has been shown to be thermodynamically consistent only very recently by means of TCAT by introducing a new form of the solid stress tensor and a mathematical expression of the Biot coefficient [16]. A proof that the new solid stress tensor also satisfies the Hill-Mandel condition can be found in Ref. [17].

In the following we use the results obtained in Refs. [16] and [17] to identify some forms of the effective stress equation and the links between the various solid pressure definitions. In view of this theory, analyzing the stress state and the deformation of the material it is necessary to consider not only the action of an external load, but also the pressure exerted on the skeleton by fluids present in its voids. Hence, the total stress tensor BoldItalic^total acting in a point of the porous medium may be split into the effective stress {\eta }^{\mathrm{s}}{\mathit{ϵ}}^{\mathrm{s}}, which accounts for stress effects due to changes in porosity, spatial variation of porosity and the deformations of the solid matrix, and a part accounting for the solid phase pressure exerted by the pore fluids [16-18]:

(1){\mathit{t}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}={\eta }^{\mathrm{s}}{\mathit{ϵ}}^{\mathrm{s}}-\alpha P^{\mathrm{s}}\mathit{I}\mathit{,}

where BoldItalic is the second order unit tensor, α is the Biot coefficient and P^s is some measure of solid pressure acting in the system, also simply called solid pressure. Many different forms of P^s have been proposed in the past decades in Geomechanics, but for porous media with a fine microstructure, like for example concrete in which the interactions between molecules of water and the solid skeleton on micro structural level are rather complex, the formulation of Gray et al. [16] has been adopted here. This formulation takes into account the degree of contact of each fluid phase with the solid one. Including the interface in the analysis allows the interpretation of the Biot coefficient as the ratio of the hydrostatic part of the total stress tensor (p^total) to the normal force exerted on the solid surface by the surrounding fluids, i.e., -{\langle {\mathit{n}}_{\mathrm{s}}{\mathit{t}}_{\mathrm{s}}{\mathit{n}}_{\mathrm{s}}\rangle }^{\mathrm{s}\mathrm{s}}:

(2)\alpha =-{\displaystyle \frac{p^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}}{{\langle {\mathit{n}}_{\mathrm{s}}{\mathit{t}}_{\mathrm{s}}{\mathit{n}}_{\mathrm{s}}\rangle }^{\mathrm{s}\mathrm{s}}}}=\mathrm{1}-{\displaystyle \frac{{\tilde{K}}_{\mathrm{T}}}{{\tilde{K}}_{\mathrm{S}}}}.

This relationship accounts for different values of bulk modulus for solid phase (grain) and the skeleton, {\tilde{K}}_{\mathrm{S}} and {\tilde{K}}_{\mathrm{T}}, respectively. Here BoldItalic_s is the stress tensor of the solid phase at microscopic level, BoldItalic_s is the unit vector normal to the solid phase in each point, while the Macaulay brackets {\langle \rangle }^{\mathrm{s}\mathrm{s}} indicate an averaging over the solid surface. With these results, P^s is selected to be the average normal force exerted on the solid surface by the fluids in the pore space:

(3)P^{\mathrm{s}}=-{\langle {\mathit{n}}_{\mathrm{s}}{\mathit{t}}_{\mathrm{s}}{\mathit{n}}_{\mathrm{s}}\rangle }^{\mathrm{s}\mathrm{s}}.

By considering the interfaces and by formulating the model from the micro-level, the following form of the so-called “standard solid pressure”, p^s is obtained:

(4)p^{\mathrm{s}}=x_{\mathrm{s}}^{\mathrm{w}\mathrm{s}}{p}^{\mathrm{w}}+x_{\mathrm{s}}^{\mathrm{g}\mathrm{s}}{p}^{\mathrm{g}}+x_{\mathrm{s}}^{\mathrm{w}\mathrm{s}}{\gamma }^{\mathrm{w}\mathrm{s}}{J}_{\mathrm{w}\mathrm{s}}^{\mathrm{s}}+x_{\mathrm{s}}^{\mathrm{g}\mathrm{s}}{\gamma }^{\mathrm{g}\mathrm{s}}{J}_{\mathrm{g}\mathrm{s}}^{\mathrm{s}},

where x_{\mathrm{s}}^{\mathrm{w}\mathrm{s}} and x_{\mathrm{s}}^{\mathrm{g}\mathrm{s}} are the fractions of skeleton area in contact with water and gas, respectively, while J_{\mathrm{w}\mathrm{s}}^{\mathrm{s}} and J_{\mathrm{g}\mathrm{s}}^{\mathrm{s}} are the curvature of the water/solid and gas/solid interfaces in that order. γ^ws and γ^gs are surface tension-like terms. The two forms of the solid pressure are related to each other by means of

(5)p^s=\alpha P^s.

For further details see Refs. [16-18].

By using the following simplified version of the capillary pressure [18], valid at thermodynamic equilibrium and neglecting the direct contribution of the fluid-solid interfaces

(6)p^{\mathrm{c}}\equiv p^{\mathrm{g}}-p^{\mathrm{w}}={\mathit{\Pi }}^{\mathrm{f}}-{\gamma }^{\mathrm{w}\mathrm{g}}{J}_{\mathrm{w}\mathrm{g}}^{\mathrm{w}}\mathrm{.}

Eq. (4) can be transformed in

(7)p^{\mathrm{s}}=p^{\mathrm{g}}+x_{\mathrm{s}}^{\mathrm{w}\mathrm{s}}{\gamma }^{\mathrm{w}\mathrm{g}}{J}_{\mathrm{w}\mathrm{g}}^{\mathrm{w}}-x_{\mathrm{s}}^{\mathrm{w}\mathrm{s}}{\mathit{\Pi }}^{\mathrm{f}}+x_{\mathrm{s}}^{\mathrm{w}\mathrm{s}}{\gamma }^{\mathrm{w}\mathrm{s}}{J}_{\mathrm{w}\mathrm{s}}^{\mathrm{s}}+(\mathrm{1}-x_{\mathrm{s}}^{\mathrm{w}\mathrm{s}}){\gamma }^{\mathrm{g}\mathrm{s}}{J}_{\mathrm{g}\mathrm{s}}^{\mathrm{s}}.

Equation (6) considers the disjoining pressure Π^f and can be applied in the hygroscopic region (i.e., when the saturation level is lower than the solid saturation point and the water is present only as a thin film on the skeleton surface) as well as in the non-hygroscopic region (i.e., for higher levels of moisture content, for which saturation values exceed the solid saturation point).

In the case of concrete one can recognize in Eq. (7) terms corresponding to the main physical phenomena leading to shrinkage: the first term on the RHs describes an effect of gas pressure, the second one of capillary tension, the third one of disjoining pressure, and the last two terms, resulting from action of surface tension of solids on the interfaces with the pore fluids, are negligible. Taking into account such simplifications and relation (5), the so-called “effective stress principle”, i.e., Eq. (1), can be rewritten in the following manner:

(8){\eta }^{\mathrm{s}}{\mathit{ϵ}}^{\mathrm{s}}={\mathit{t}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}+(p^{\mathrm{g}}-x_{\mathrm{s}}^{\mathrm{w}\mathrm{s}}{p}^{\mathrm{c}})\mathit{I}\mathrm{.}

The formulations of the stress measures described above allow for a correct interpretation of the thermo-hygral-mechanical behavior of a porous medium, as shown in Fig. 1 for the case of a drying process of an ordinary concrete sample [18]. In the range of RH<30% the model proposed fits the experimental results better than more classical formulations; see Ref. [18] for the details about material properties and boundary conditions.

Applications

Three different applications are presented in the following to show the versatility and the efficiency of the multiphase approach previously summarized.

Hydraulic fracture propagation

The propagation of a discrete fracture caused by water pressure in a cohesive medium (concrete or rock) is first analyzed. Water pressure could be caused artificially (as in extraction engineering) or could be caused by the filled reservoir water. In this application, failure conditions as a consequence of overtopping wave acting on a concrete gravity dam are investigated. Fracture propagates in mode II conditions, which are accounted for as in Ref. [19], usually enucleating from a geometric singularity. The problem is similar to ICOLD benchmark [20], but the dam foundation is also considered, which has been assumed homogeneous with the dam body (see Fig. 2). In such a situation, the crack path is unknown.

Applied loads are the dam self-weight and the hydrostatic pressure due to water in the reservoir, growing from zero to the overtopping level h (which is higher than the dam). The geometry of the dam is shown in Fig. 2 together with an intermediate discretization. For more details on material parameters and boundary/initial conditions, we refer to Ref. [21]. A continuous updating of the geometry and discretization are made according to crack movements. This also allows for the proper representation of the process zone and the cohesive forces. This is shown in Fig. 3, which represents the process zone when the fracture length is 3.5 m and corresponds to an intermediate step of the analysis when the water level is 78 m. Finally, the formation of the fluid lag is studied. This is due to the different velocities of propagation of the crack tip and the one of seepage inside the fracture. Depending on the characteristics of the fluid lag, different water pressure distributions inside the process zone take place resulting in different stress field in the tip area. Figure 4 depicts cohesive stresses and water pressure distribution, putting in evidence the fluid lag zone.

In similar cases, crack path cannot be forecast; hence, the traditional use of special/interface elements to simulate fracture propagation in large structures is prevented. The alternative to the successive remeshing within this approach to fracture is the use of cumbersome discretizations of the fracture areas, but also this strategy is not viable in analyzing the cohesive behavior of dams. Furthermore, the used technique for the analysis of the nucleation of the fracture does not require the presence of an initial notch and requires a very limited amount of information to be initially defined.

Dynamic behavior of an earth dam

Seismic behavior of an earth dam is shown as the second example. In this case a partial saturated zone also appears which needs an extension of the above model to partial saturation [1,2]. In the following the simulation was performed with the passive air phase assumption, i.e., with air pressure in the unsaturated region equal to the atmospheric pressure [3]. This dam is located near Perugia in Italy and was stressed by the Umbria-Marche earthquake of September 1997 while the water level in the lake was only 5 m [22].

Some damage was observed in the dam and displacements were measured after the earthquake.

Numerical simulations of the seismic behavior of the dam are performed by using the Swandyne code [3,23,24] and a comparison with the in-situ damage is carried out. The dam and the surrounding soil are discretized with triangular isoparametric finite elements as can be seen from Fig. 7. Small strains, small displacements and plane strain conditions are assumed. The most important material parameters used in the computation are obtained from the data of the in-situ geotechnical analysis. The others are estimated by direct comparison between the given materials and similar well-known soils. In particular, the solid skeleton is considered to be elasto-plastic and the Mohr-Coulomb and Pastor-Zienkiewicz [3,25] laws are used for cohesive and granular materials, respectively. The Pastor-Zienkiewicz law was originally developed also to model the liquefaction phenomenon under undrained shearing of very loose sands, and was used successfully to simulate the collapse of the Lower San Fernando dam near Los Angeles during the 1971 earthquake [3]. It was further extended for partially saturated materials by Refs. [26,27]. The partial saturation conditions are described using Safai-Pinder’s law [28].

First, an initial elastic static analysis is performed assuming the gravity and external water pressure to be applied without dynamic effects. The phreatic line is hence obtained, revealing the existence of a zone above it where capillary pressures develop. The cohesion resulting from these capillary pressures is taken into account in the model. Then, a full nonlinear dynamic computation is performed, using vertical and horizontal accelerations of the Nocera Umbra registration as seismic input excitation (see Fig. 5). As far as the results are concerned, the calculated displacements of the dam are close to those measured (see Fig. 6). The numerical analysis reveals, also, the damage accumulated by the concrete barrier in the core of the foundation of the dam as indicated in Fig. 7 where the equivalent plastic strain contour after 200 s is shown. As a consequence, the barrier should become more pervious. Water pressure distribution after 200 s is shown in Fig. 8. From Figs. 6(b) and (c) it appears that the displacements are increasing for a considerable period after the end of the earthquake. This was also observed by Zienkiewicz et al. [3] and is aided by the redistribution of pore pressures. The good agreement obtained between in situ and simulated damage reveals the validity of the proposed approach.

Mechanical behavior of concrete at early stages

In this section the general model presented briefly in the first part of this work will be particularized to the case of cementitious materials at early ages like for example concrete during the hydration process and to the long-term behavior assessment of concrete structures. This version of the model can be fruitfully applied for the cases of repairing works of damaged concrete structures exposed to aggressive environments, e.g., dams subject to ASR.

In addition to what was described in the previous sections it is necessary to formulate an evolution equation accounting for the chemical reaction the hydration/hardening process is based on. To meet this requirement a non-dimensional measure related to the chemical reaction extent, known as hydration degree, can be defined as follows:

(9){\mathit{\Gamma }}_{\mathrm{h}\mathrm{y}\mathrm{d}\mathrm{r}}=\chi /{\chi }_{\infty }=m_{\mathrm{h}\mathrm{y}\mathrm{d}\mathrm{r}}/m_{\mathrm{h}\mathrm{y}\mathrm{d}\mathrm{r}\infty },

where m_hydr means mass of hydrated water (chemically combined), χ is the hydration extent and χ_∞, m_hydr∞ are the final values of hydration extent and mass of hydrated water, respectively. From the macroscopic point of view, hydration of cement is a complex interactive system of competing chemical reactions of various kinetics and amplitudes. For this reason, a thermodynamically based approach is used herein, see Refs. [29,30]. In this approach the hydration extent χ is the advancement of the hydration reaction and its rate is related to the affinity of the chemical reaction through an Arrhenius-type relationship, as usual for thermally activated chemical reactions. This formulation can be written in terms of hydration degree, defined as in Eq. (9), and relative humidity by means of a function β_ϕ(ϕ) (ϕ is the relative humidity) [29,30]:

(10){\displaystyle \frac{\mathrm{d}{\mathit{\Gamma }}_{\mathrm{h}\mathrm{y}\mathrm{d}\mathrm{r}}}{\mathrm{d}t}}={\tilde{A}}_{\mathit{\Gamma }}({\mathit{\Gamma }}_{\mathrm{h}\mathrm{y}\mathrm{d}\mathrm{r}}){\beta }_{\phi }(\phi )\exp \left(-{\displaystyle \frac{E_{\mathrm{a}}}{RT}}\right),

where {\tilde{A}}_{\mathit{\Gamma }} is the normalized affinity (it accounts both for chemical non-equilibrium and for the nonlinear diffusion process), E_a is the hydration activation energy, and R is the universal gas constant. An analytical formula for the description of the normalized affinity can be found in Ref. [31].

Taking into account that the material properties change with time, i.e., they are a function of hydration degree, the constitutive relationship describing the stress-strain behavior of concrete can be written in the following form:

(11){\eta }^{\mathrm{s}}{\dot{\mathit{ϵ}}}^{\mathrm{s}}=\mathit{D}\left({\dot{\mathit{ϵ}}}_{\mathrm{t}\mathrm{o}\mathrm{t}}-{\dot{\mathit{ϵ}}}_{\mathrm{c}}-{\dot{\mathit{ϵ}}}_{\mathrm{c}\mathrm{h}}-{\dot{\mathit{ϵ}}}_{\mathrm{t}}\right)+\dot{\mathit{D}}\left({\mathit{ϵ}}_{\mathrm{t}\mathrm{o}\mathrm{t}}-{\mathit{ϵ}}_{\mathrm{c}}-{\mathit{ϵ}}_{\mathrm{c}\mathrm{h}}-{\mathit{ϵ}}_{\mathrm{t}}\right),

where BoldItalic is the tangent matrix of the material, BoldItalic_ch is the chemical strain, BoldItalic_c is the creep strain, BoldItalic_t is the thermal strain (infinitesimal deformations) and, finally, {\eta }^{\mathrm{s}}{\dot{\mathit{ϵ}}}^{\mathrm{s}} is the so-called “effective stress tensor”. The chemical strains account for thermo-chemical processes that take place in concrete at early ages, for further details see Refs. [29,30].

Creep is modeled here by means of the solidification theory [32], for the description of the basic creep, and microprestress theory [33], for the description of the long-term creep and the stress induced creep (part of the so-called drying creep). In the following, a brief description of the necessary constitutive relationships will be presented. According to the solidification theory proposed by Bažant and Prasannan [32], aging is considered as a result of a solidification process involving basic constituents, which do not have aging properties. In view of this theory the total creep strain increment, dBoldItalic_c, is decomposed into two components, the visco-elastic, and the viscous flow strain increments, dBoldItalic_v and dBoldItalic_f:

(12)\mathrm{d}{\mathit{ϵ}}_{\mathrm{c}}=\mathrm{d}{\mathit{ϵ}}_{\mathrm{v}}+\mathrm{d}{\mathit{ϵ}}_{\mathrm{f}}={\displaystyle \frac{\mathrm{d}\gamma }{{\mathit{\Gamma }}_{\mathrm{h}\mathrm{y}\mathrm{d}\mathrm{r}}}}+{\displaystyle \frac{\mathrm{1}}{\eta }}\mathit{G}{\eta }^{\mathrm{s}}{\mathit{ϵ}}^{\mathrm{s}}\mathrm{d}t,

where γ is the visco-elastic microstrain and η is the apparent microscopic viscosity. The matrix BoldItalic is defined in such a way that \mathit{ϵ}=E^{-\mathrm{1}}\mathit{G}\mathit{t}{\displaystyle \square } , where BoldItalic and BoldItalic are stress and strain vectors, respectively [33]. In the current model concrete is treated as a multiphase porous medium, so the stresses in Eq. (12) should be interpreted as the effective ones, {\eta }^{\mathrm{s}}{\dot{\mathit{ϵ}}}^{\mathrm{s}}, [29] and not the total ones, BoldItalic^total, as in the original theory by Bažant et al. [32]. In such a way it is possible to couple the free shrinkage with the creep, obtaining creep strains even if the concrete structure is externally unloaded. The capillary forces represent, in fact, an “internal” load for the microstructure of the material skeleton. It is now possible to define the creep compliance function [18]

(13)J(t,t^{\prime })=q_{\mathrm{1}}^{\prime }+\mathit{\Phi }(t,t^{\prime })+\mathrm{1}/\eta .

The apparent macroscopic viscosity, η, is not constant in time and is defined by means of microprestress theory, while the micro-compliance function Φ(t,t’) can be expanded in Dirichlet series. For further details see Ref. [29].

By applying this version of the general model, the assessment of the hydration process and autogenous shrinkage evolution for a high performance cement paste is presented and discussed. The results of the numerical computations are compared with the ones of the experimental tests performed by Lura et al. [34].

The autogenous shrinkage is a phenomenon the physical nature of which is still not well understood. In recent years various approaches, taking into account possible factors influencing autogenous shrinkage such as surface tension in the cement gel, disjoining pressure and tension in capillary water, have been proposed to explain this phenomenon.

The main parameters (for the fully matured material) used in our simulations are listed in Ref. [29]. Initially the HCP was at temperature T_o = 293.15 K and relative humidity ϕ_o = 99.9% RH, while hydration degree was equal to Γ_hydr0 = 0.3 (i.e., material had already some rigidity). The boundary conditions corresponded to adiabatic heat exchange and sealed conditions for mass exchange. Figure 9(a) shows the comparison of the hydration degree calculated from the experimental data for heat of hydration [34] and the evolution resulting from FE calculation. A good agreement of the results can be noticed. The relative humidity development in time clearly shows the self-desiccation process of the material, Fig. 9(b). The RH values measured during the test have been scaled to 100%RH because we refer to the RH due to menisci formation, neglecting the effects of salts dissolved in the pore fluid. Figure 10 shows the measured deformation and the shrinkage strains calculated in Ref. [34] in comparison with the present simulations. The agreement is good especially as far as the shrinkage strains are concerned. Some significant differences between the values of strains measured during the test and those calculated using the capillary tension model can be ascribed to the creep of the cement paste which becomes more evident at lower RH, e.g., for longer time span. This effect is considered due to the coupling between the creep and the shrinkage via the effective stress tensor [35].

Patch structural reparation—replacement of spalled area because of ASR

In this subsection an application of a simplified version of the full model described above (and in Refs. [29, 30]) is used for analyzing the effects of a patch structural reparation of a concrete element damaged because of ASR.

The following simplifications and changes have been introduced:

1) Elimination of the unknown p^g (gas pressure), set it equal to the atmospheric pressure (passive air phase assumption);

2) Phase changes are not considered (optional);

3) Diffusive transport of water vapour is considered as negligible but vapour is present (optional);

4) Total stress is used;

5) The model used for the assessment of aging and hydration is essentially the same.

The resulting model has been implemented in this new form in the FE code in CAST3M, by using a partitioned approach.

The analysis deals with the concrete element shown in Fig. 11 in which the patch (5 cm×20 cm) is highlighted in red. The initial conditions used in the simulations are described in Table 1 for the three zones: original concrete zone, interfacial zone and repaired zone. For the first 12 hours the concrete element is sealed, then the reparation is considered in contact with a surrounding environment characterized by a relative humidity equal to 70%.

Figure 12 shows the distribution of relative humidity at different time instants (0, 12 h, 24 h, 7 d, 28 d, 100 d). One can observe that due to the size of the concrete element under consideration, the thermo-hygral equilibrium can be reached only after a long time, approximately 100 d (see Fig. 12). Similarly, the hydration process takes several days to be completed, see Fig. 13 for the distribution of the hydration degree at 48 h: after this period the patch is still hardening.

Figure 14 shows the thermal field at different time instants along a time span equal to 24 h. This is the period during which a large part of the reaction energy is released. Indeed, the concrete element is not in adiabatic conditions, i.e., it can exchange heat with the surrounding environment.

The calculation has been split into two parts: thermo-hygral and mechanical one. The latter was carried out by including or not a visco-elastic model for the analysis of the mechanical behavior of concrete. Moreover, a special constitutive law has been formulated for assessing the effects of the use of some expansive agents in the mix of concrete. We remind that in general the reaction of the expansive products depends on several factors:

1) expansive agent type;

2) porosity and fineness of the additive;

3) temperature;

4) relative humidity.

The expansive phenomenon can be properly modeled by following an approach similar to the one used for the hydration process (see Eqs. (9) and (10)). First, it is necessary to define a sort of expansion degree:

(14){\mathit{\Gamma }}_{\mathrm{e}\mathrm{s}\mathrm{p}}=m_{\mathrm{e}\mathrm{s}\mathrm{p}-\mathrm{h}\mathrm{y}\mathrm{d}\mathrm{r}}/m_{\mathrm{e}\mathrm{s}\mathrm{p}},

where m_esp-hydr is the mass of hydrated additive and m_esp is the total mass of agent in the concrete admixture.

The evolution equation can be formulated in the following form, i.e., an Arrhenius-type relationship as usual in thermal-activated processes:

(15){\displaystyle \frac{\mathrm{d}{\mathit{\Gamma }}_{\mathrm{e}\mathrm{s}\mathrm{p}}}{\mathrm{d}t}}=A_{\mathrm{e}\mathrm{s}\mathrm{p}}({\mathit{\Gamma }}_{\mathrm{e}\mathrm{s}\mathrm{p}}){\beta }_{\mathrm{e}\mathrm{s}\mathrm{p}-\phi }(\phi )\exp \left(-{\displaystyle \frac{E_{\mathrm{a}-\mathrm{e}\mathrm{s}\mathrm{p}}}{RT}}\right),

in which A_esp is the chemical affinity of the expansion reaction, β_esp-ϕ is a function of the relative humidity and finally E_a-esp is the activation energy of the process.

The functions used in Eq. (15) are similar to those used in the simulation of the hydration of concrete, and in their formulation two basic assumptions were made: the water consumption due to the additive is negligible and the heat generation produced in the hydration of the chemical agent is not relevant.

From a mechanical point of view, it is necessary to add a new strain component to Eq. (11) to take into account the expansion owing to the additive. This contribution in terms of the increment of strain can be considered related to the variation of the expansion degree and then given by

(16)\mathrm{d}{ϵ}_{\mathrm{e}\mathrm{s}\mathrm{p}}=\xi \mathrm{d}{\mathit{\Gamma }}_{\mathrm{e}\mathrm{s}\mathrm{p}},

where \xi is a parameter depending on the quantity of additive used in the admixture.

It is worth mentioning that the chemical reaction corresponding to the hydration of the cement and the one related to the reaction of the expansive agent, take place at different velocities, so that these chemical processes can be considered as independent of each other.

Figure 15 shows the distribution of the σ_xx stress at 100 d in four different cases:

1) Concrete admixture without expansive agent; concrete is treated as pure elastic material (see Fig 15(a));

2) Concrete admixture without expansive agent, concrete is treated as visco-elastic material (see Fig 15(b));

3) Concrete admixture with expansive agent; concrete is treated as pure elastic material (see Fig 15(c));

4) Concrete admixture with expansive agent; concrete is treated as visco-elastic material (see Fig 15(d)).

The effects of the expansive additive are positive contributing to a decrease of the tensile stress especially in the interfacial zone between the patch and the original concrete element (compare Fig. 15(a) with Fig. 15(c) and Fig. 15(b) with Fig. 15(d)). Also, the viscosity plays an important role, by leading to a relaxation of the stress as expected (compare Figs. 15(b) and (d) with Figs. 15(a) and (c) respectively).

Conclusions

A multiphase approach for the dam body material has been presented as a general framework which allows the analysis of a wide variety of problems. It is based on the hybrid mixture theories used in conjunction with TCAT. The resulting model can be fruitfully applied to relevant engineering problems like, for instance, the analysis of the behavior of earth dams, the analysis of hydraulic fracture in concrete dams or the assessment of the effects of repairing operations to concrete elements damaged by an aggressive environment (for example ASR).

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