## 1 Introduction  
Complexity of failure is reflected from sensitivity of strength to small defects and wide scatter of macroscopic behaviors. In engineering practices, spatial information of materials at fine scales, such as atomic defects, dislocations, locations and sizes of micro-particles, micro-voids, micro-cracks, grain boundaries, can only be partially measurable. Random field (RF) models are important to address the uncertainty in spatial distribution. Characteristics of a random field at a fine scale contribute to the pattern of the random field at the next coarse scale, and the contribution is governed by physical laws connecting the two scales. For instance, a failure process initiated from micro-crack propagation is critically determined by the laws of fracture mechanics and the probability distributions for the size and density of micro-cracks. To transform the random field of micro-cracks into failure probability at full structural-scale crossing a number of length scales, the operator representing physics laws need be implemented in a multiscale framework, and to be realized in a stochastic setting to deal with statistical input data. Multiscale stochastic modeling (MSM) of materials is emerging as a new methodology at this research frontier [1], which provides a new multiscale thinking by upscaling fine-scale random fields. Along this line, theoretical and computational MSM methods has been developed on stochastic homogenization [17, 19, 20], variational principles [18], and multiscale stochastic finite element methods (MsSFEM/MSFEM) [1, 21]. Other studies on multiscale stochastic models include [PERSON] and [PERSON] [13] on random media diffusion problems, [PERSON] [14] on characterization of random microstructure, [PERSON] and [PERSON] [15] on fracture of functionally graded materials, [PERSON] et al [16] on strength of carbon nanotube composites, etc.  
A classical example of the multiscale stochastic modeling on materials is Weibull theory that offers a closed-form solution for brittle materials to quantify evolution of failure probability passing large scale separation. Failure probability of most non-brittle materials however is not amenable to asymptotic theoretical/analytical approaches and can only be explored by means of numerical experiments. Nonlinear finite element reliability analysis, using the midpoint or average random field discretization method and first- or second-order reliability method, has been developed since 1990s to evaluate failure behaviors of quasibrittle materials [2, 3]. Attempts have been made to apply the spectral stochastic finite element method to plasticity and failure [4]. To solve multiscale failure problems such as concrete structures, geomaterials, and earth faults with a domain size several orders of magnitude larger than the correlation length, novel multiscale schemes are desired.  
In this study, a preliminary framework of probabilistic upscaling is presented for bottom-up hierarchical modeling of failure propagation across micro-meso-macro scales. In the micro-to-meso process, the strength of stochastic representative volume element (SRVE) is probabilistically assessed by using a lattice model to simulate random micro-cracking. A mixed Weibull-Gaussian distribution is proposed to characterize the statistical strength of SRVE, which can be used as input for the subsequent meso-to-macro upscaling process using smeared crack finite element analysis.