# Microstructure-Based Prediction of the Elastic Behaviour of Hydrating Cement Pastes
## 1 Introduction  
Most existing modelling practices for analysis of cement-based materials are based on continuum descriptions of their responses to external driving forces, namely stresses due to deformation gradients, heat fluxes due to temperature differences, diffusion due to concentration gradients, etc. These continuum approaches basically relate responses to the driving forces, considering the governing differential equations and the appropriate boundary conditions, which are usually solved with numerical methods such as the Finite Element Method (FEM). A fundamental aspect of these continuum approaches is the definition of constitutive laws that link the driving forces to the phenomenological responses which are generally based on accrued knowledge from macroscopic experiments. However, the problem complexity rises considerably when the approach aims to account for discontinuities, such as due to the pores network and micro-cracks. One practical limitation of using continuum approaches is the derivation of multi-variable phenomenological constitutive laws,which usually demands unaffordable experimental programmes, particularly regarding coupled mechanisms, as it is the case of mechanical-transport modelling in porous media. The basic idea of the present work is to derive the macroscopic constitutive laws of cement-based materials by modelling the physical processes at a small length scale, so that the constitutive laws may be looked as micromechanics-based, rather than being phenomenologically approximated. This aids scientists and engineers to bridge the gap between knowledge of microscopic mechanisms and the macroscopic responses, while they are still using continuum approaches with an acceptable accuracy for modelling complex phenomena. This modelling strategy is the genesis of continuum micromechanics approaches [1; 2], in which a cement-based material is looked as a macro-homogeneous body, yet micro-heterogeneous, filling a Representative Elementary Volume (REV) with a prescribed characteristic length. Such REV includes homogeneous subdomains with the identification of known physical quantities that are called material phases [3]. Once their mechanical behaviour, shapes, and interactions are known at a small length scale, the overall homogenised mechanical behaviour of the material can be estimated. One of the promising strategies for deriving the macroscopic constitutive law of a cement-based material is to model the cement paste microstructure using discretisation methods rather than continuum approaches, an example of the former being the lattice approach. A wide set of microstructural behaviour representations inside the cement paste can be constructed using lattices of space-filling polyhedra within the REV. Each polyhedron becomes a material phase in the microstructure, computationally represented as a lattice 'node' at the polyhedron centre, which is linked to the neighbouring 'nodes' using 'linear lattice beam elements'. The material mechanical behaviour at the macroscale is obtained by defining appropriate properties to the lattice elements and taking into account a homogenisation technique for upscaling. A fundamental requirement from the lattice model, to be highlighted for the present study, is that it has to reproduce the macroscopic elastic properties of the cement paste. In addition, the cement paste microstructure can be appropriately reproduced only if the hydration of cement is well understood, and adequately accounted for in the numerical model. The total process of cement hydration is a complex chemical and physical set of phenomena, discussed worldwide by many researchers [4; 5; 6]. Several theoretical models have been developed to simulate the microstructure of the hardened cement paste, by considering the most dominant reactions that occur during hydration. Some of those well-known numerical hydration models are: (i) the Jennings and Johnson Model [7], (ii) the CEMHYD3D [8], (iii) the HYMOSTRUC3D [9; 10] and (iv) the \(\upmu\)ic [11].