The hydration model and averaged stoichiometry of the reaction

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Cement chemists use abbreviated symbols which describes each oxide by one letter: CaO = C; SiO2 = S; Al2O3=A; and Fe2O3=F. Similarly H2O in hydrated cement is denoted by H, and SO3 by S. The compounds listed in the previous table react with water forming the solid hydrated cement paste. The two calcium silicates are the main constituents of cements and so the physical behaviour of cement during hydration is similar to that of these two compounds alone. The mechanics of hydration is not yet perfectly known but probably hydration proceeds by a gradual reduction of the size of the anhydrous grain of cement. For instance, Giertz-Hedstrom (1938) found that after 28 days in contact withwater only a depth of 4 μm of cement grains is hydrated and 8 μm after a year. Furthermore Powers (1949) calculated that complete hydration is possible only for cement particles smaller than 50 μm since in greater particles water cannot reach the core. The main hydrates are the calcium silicate hydrates C3S2H3 (the so called C-S-H) and the tricalcium aluminate hydrate C3AH6. C3S2H3 consists of fibrous particles with a very irregular shape (see Figure 2.1).

Microstructure of the cement paste

The microstructure of the cement paste consists of the hydration products (essentially CS- H gel and Ca(OH)2), anhydrous cement grains and capillary pores which are partially saturated by water (see Figure 2.4). Actually also hydrates are porous but their pores are very small compared to the capillary ones (from one to two orders of magnitude smaller), and for the relative humidities higher than the 50% are completely water-filled.
The porosity of hydrates is approximately equal to 0.28 and the order of magnitude of the pores size is about 2 nm. The mass of non-evaporable water (chemically combined) has been estimated as 23% of the mass of the anhydrous cement. The volume of the solid part of hydrates is smaller than the sum of the volumes of the anhydrous cement and the chemically bound water by about 0.254 of the volume of the latter. These averaged relationships have been obtained experimentally by Power (1947 and 1960) and are the basis for the equations proposed by Jensen and Hansen (1996, 2001 and 2002) where silica fume is also considered (eqs (1.32) of Chapter 1). These equations are valid for an isolated system (i.e. without mass exchanges with the external environment) and give the evolution with hydration of the volume fractions of chemical shrinkage, capillary water, gel water, gel solid, anhydrous cement and silica fume.

Heat of hydration of cement

The hydration reaction is exothermic, energy of up to 500 J per gram of cement being liberated. Consequently in massive structures hydration can result in a large rise in temperature and also in large thermal gradients which may induce diffuse or localized cracking. This behavior modified by creep and autogenous shrinkage can be very difficult to predict. Moreover, especially in the summer months (depending evidently on the geographic region) also solar radiation on the surfaces must be taken into account since it has a not negligible effect (Sciumè et al. 2012a). The knowledge of the heat production of cement is then critical especially in mass concrete.


After setting, with the progress of hydration the volume changes of the different phases present in the cement paste have as consequence the development of a volume of gas which leads to the self-desiccation of the cement paste (Jensen, 1993). The selfdesiccation is very important in high performance concretes and cement pastes with a low water/binder ratio. In Figure 2.17 the internal relative humidity measured in an ordinary concrete (OC) and in a high-performance concrete (HPC) is plotted over time (the specimen are sealed). A strong self-desiccation for the high-performance concrete is shown. Being the self-desiccation related to the hydration process the decrease of relative humidity is very important during the first 3 weeks after the casting.

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In the previous paragraph the self-desiccation of concrete and its consequences have been discussed, and numerical examples have been presented. When the material is hygrally isolated from the environment, if not important thermal gradients are established during the hydration process, the hydration degree increases homogenously and so also the selfdesiccation is homogenous. This means that in sealed conditions hygral gradients are negligible and then no mass transport of water occurs; nonetheless it is important to remember that for mass concrete this statement is not valid because in that case serious thermal gradients leads to non homogenous hydration advancement, and so also to a non homogenous self-desiccation of the material which induces a weak hygral transport from the colder border to the hydrated core of the structure; an example is presented in the following pages.
After the exposure of concrete to the environment, as happens in civil engineering structures when the formworks are removed, if the environmental relative humidity is lower than that of the material, a movement of the internal water from the concrete structure to the environment occurs. Taking into account this phenomenon in concrete structure design is of critical significance because drying is the cause of shrinkage and has effect on creep strain. Moreover hygral gradient induces gradient of strain which can produce cracks due to the self-restrained shrinkage.

Table of contents :

General and TCAT symbols
Symbols in the concrete sections (Chapters 1, 2 and 3)
Symbols in the tumor section (Chapter 4)
1.1 Introduction
1.2 Brief overview of TCAT
1.2.1 Microscale
1.2.2 Macroscale and concept of representative elementary volume REV
1.2.3 Closure techniques
1.2.4 The TCAT procedure
1.2.5 Advantages of the TCAT approach
1.3 The multiphase system
1.4 General governing equations
1.4.1 Mass.
1.4.2 Momentum
1.4.3 Energy
1.5 Constitutive equations
1.5.1 The hydration model and averaged stoichiometry of the reaction
1.5.2 Fluid phases velocities
1.5.3 Water vapour diffusion
1.5.4 A hydration-dependent desorption isotherm
1.5.5 The effective stress principle
1.5.6 Effective thermal conductivity and thermal capacity
1.5.7 Mechanical constitutive model
1.5.8 Creep rheological model
1.5.9 Thermal and hygral strains
1.5.10 Damage model
1.6 Final system of equations
1.7 Numerical solution and computational procedure
References of Chapter 1
2.1 Introduction
2.2 Hydration process and related phenomena
2.2.1 Microstructure of the cement paste
2.2.2 Heat of hydration of cement
2.3 Mechanical properties
2.4 Self-desiccation and autogenous shrinkage
2.5 Hygral transport and drying shrinkage
2.6 Basic and drying creep
References of Chapter 2
3.1 Introduction
3.2 The ConCrack benchmark
3.2.1 Identification of the model parameters
3.2.2 Finite element mesh of the structure and boundary conditions
3.2.3 Thermo-hygro-chemical results
3.2.4 Mechanical results and four point bending test
3.3 Application to repairs of concrete structures
3.3.1 Identification of the model parameters
3.3.2 Modeling of the two repaired beam and of the reference one
3.4 Conclusions
References of Chapter 3
4.1 Introduction
4.2 TCAT procedure for biological system
4.3 Context and bibliographic review of tumor growth models
4.4 The multiphase system
4.5 General governing equations
4.6 Constitutive equations
4.6.1 Tumor cell growth.
4.6.2 Tumor cell death.
4.6.3 The rate of nutrient consumption.
4.6.4 The diffusion of nutrients through the ECM.
4.6.5 The interaction force among the phases.
4.6.6 The mechanical behaviour of the ECM.
4.6.7 The differential pressure between the three fluid phases
4.7 Final system of equations
4.8 Spatio-temporal discretization and computational procedure
4.9 Three applications of biological interest
4.9.1 Growth of a multicellular tumor spheroid (MTS) in vitro
4.9.2 Multicellular tumor spheroid (MTS) in vivo
4.9.3 Tumor growth along microvessels (tumor cord model).
4.10 Conclusions and perspectives
References of Chapter 4


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