## 2 Micro-to-Meso Upscaling  
### Micro-cracking in a random field  
Deterministic multiscale models, hierarchical or concurrent, use representative volume element (RVE) as a vehicle to transport local effects into global solutions. In stochastic problems, the so-called stochastic RVE plays a similar role but rather than transporting deterministic or mean values, as implied in deterministic models, it transports uncertainty information, e.g. probability distribution. In the micro-to-meso upscaling process, the RFs of SRVE strength and fracture energy can be used to account for the profusion of microscopic flaws and cracking. Local cohesive laws, including fracture surface energy and strength, at a certain fine-scale resolution are assumed to be intrinsic material properties independent of damage evolution. Let the coordinates \(y\) at micro-scale be defined as \(x/\varepsilon\), where \(\varepsilon\) is a resolution parameter representing the ratio between macro-scale coordinates \(x\) and micro-scale coordinates \(y\) (Fig.1). With a sample point \(\omega\in\Omega\) the probability space, \(r(y,\omega)\) and \(s(y,\omega)\) denote a fracture energy RF and a strength RF, respectively. Following the generalized [PERSON] energy-balance concept [9], the criterion for micro-cracking or local damage initiation is to have a local positive crack driving force  
\[\mathbf{g}(y,\omega)=g(y,\omega)-r(y,\omega)>0 \tag{1}\]  
where \(g(y,\omega)\) is the strain energy release rate that varies spatially dependent on local material properties. The micro-cracking in a two-dimensional RF will choose favorite initiation and propagation orientation such that the decrease of total system free energy is maximized  
\[U(y_{0},\theta_{0})=\underset{(y,\theta)}{Max}\int_{0}^{\alpha a}\mathbf{g}(y, \omega)da \tag{2}\]  
where the path \(da\) follows the orientation \(\theta\), and \(\Delta a\) is the crack propagation incremental length chosen in a particular simulation. In a discrete model, e.g., the lattice system in Fig.1, the path \(da\) becomes discrete, i.e. \(\Delta a\), and the latter corresponds to the lattice spacing between a broken bond and its nearest neighbors. It is noted that the idea of energy minimization have appeared in a number of publications, e.g. the variational model in [12]. The attempts to generalize [PERSON] fracture theory remain largely at mathematical formulations, and little work has been done to provide practical tools for numerical implementation. The initiation criterion (1) and the propagation criterion (2), explicitly given in our model, are directly applicable for crack simulation in continuum media [9] and discrete media as given in this study. A number of studies have been conducted on stochastic microcracks [5, 22, 23]. To simulate the micro-cracking in a RF, one of the most effective numerical approaches is to use lattice models that allow disorder to be introduced naturally. Various types of discrete lattice models such as central force model, electrical fuse model, bond-bending model, and beam-type model have been used to study progressive damage of disordered quasi-brittle materials [5]. In this study a random spring model is used and the network is constructed from a lattice where all the bonds between nearest-neighbor sites are Hooke's law springs with unit cross-section area.