## 2 Overview of Multiscale and Multi-Physical Modelling  
Figures 1 and 2 gives an overview of the multiscale and multi-physical modelling to simulate time-dependent deformation and cracking of concrete structures. The system consists of thermodynamic multi-chemo-physical modelling with DuCOM and nonlinear structural analysis with COM3. The former includes various thermodynamic models, such as multicomponent hydration, micropore structure formation and moisture equilibrium/transport, whilst the latter is a 3D finite-element analysis that implements constitutive laws of uncracked/cracked and hardening/aging/matured concrete. By inputting the basic data, such as mix proportions, structural geometry and boundary conditions in terms of history of environmental exposure of the structure, over time-space domain, the kinematic chemo-physical and mechanical events of representative elementary volume (REV) from different scales would be individually solved in a certain timestep. All other time-dependent properties of concrete such as elastic modulus, temperature, pore pressure, creep, moisture status, total porosity of interlayer, gel and capillary pores are computed internally based on the micromodels of materials inside the DuCOM system [6]. In the multiscale constitutive model, concrete is idealised as a two-phase composite in which cement paste and elastic aggregate co-exist (Figure 2A) [4]. The aggregate is assumed to be rigid and to show elastic deformation, and the shrinkage caused by aggregates is also considered. The hardened cement paste matrix is considered to be an assembly of fictitious clusters, referring to the solidification theory. As cement hydration proceeds, the number of clusters increases. The overall capacity of a cement matrix is obtained by summing the capacities of all clusters, which show time-dependent deformation relevant to the history. Based on the water status in the micro-pores, the time-dependent deformation, which consists of elastic, viscoelastic, viscoplastic and plastic components, is numerically computed. During the drying process, the water in pores is gradually lost. Consequently, in microscale pores, a water meniscus forms, and some shrinkage stress is generated by capillary tension. In contrast, in nanoscale pores, shrinkage stress is generated by disjoining pressure. These shrinkage stresses are quantified in the multiscale model and integrated as the overall shrinkage stresses, as shown in Figure 2B, which are given to compute the volumetric stress of cement paste carried by both skeleton solid and pore pressure, referring to [PERSON]'s theorem of a two-phase continuum [6; 7]. The volumetric change generated by cement hydration and shrinkage is systematically included in the modelling of concrete mechanics, which deals with macroscopic structural responses based on the space-averaged constitutive laws on the fixed four-way cracked concrete model (Figure 2C). It is important to mention that the microfracture cracking criteria are determined based on the local tensile strength depending on the moisture state inside the concrete because drying has both negative and positive effects. Microcracks caused by drying reduce the tensile strength, while negative pore water pressure in a dry state works as pre-stress and enhances local resistance against the tensile load [8]. Thus, in the model, various local tensile strengths were considered according to the moisture status in the micropores and nanopores (Figure 2D).  
Figure 1: Multiscale and multi-physical modelling to simulate time-dependent concrete performance.  
Figure 2: Multiscale and multi-physical modelling to simulate time-dependent deformation and cracking.