## 2 Micro-to-Meso Upscaling
### Probabilistic characterization of SRVE strength  
By virtue of the weakest-link or series model, strength of brittle materials can be well described by Weibull distribution. On the other end of spectrum, by using the parallel model, strength of ductile materials (plasticity) exhibit Gaussian distribution. Our lattice model shows that the failure pattern is more ductile when the size of the lattice is reduced and becomes brittle for large size, which suggests that the distribution of the SRVE strength lies between Weibull and Gaussian as noted by [PERSON] and [PERSON] [PERSON] and [PERSON] (2009), whereas a grafted Weibull-Gaussian distribution was proposed. Hereby we further note that physical mechanisms related to the lower tail of extreme statistics are associated with high sensitivity of local defect or imperfection, which can be modeled as a weakest link system; in other words, the lower tail of strength distribution follows the Weibull distribution. Physical mechanisms for the upper tail of extreme statistics can be interpreted as flaw tolerant, i.e. a parallel system whereas the potential of the material strength is fully exploited; in other words, the upper tail follows the Gaussian distribution. Based on this understanding, we propose a mixed Weibull-Gaussian cdf with Weibull lower tail and Gaussian upper tail as  
\[F(s)=e^{-(s/\rho)^{s}}F_{w}\left(s\right)+\left\|-e^{-(s/\rho)^{s}}\right\|F_{ G}\left(s\right) \tag{11}\]  
\[F_{w}\left(s\right)=1-\exp\left[-\left(\frac{s}{b}\right)^{m}\right] \tag{12}\]  
\[F_{G}\left(s\right)=\frac{1}{2}\left[1+erf\left(\frac{s-\mu_{{}_{G}}}{\sigma_{ G}\sqrt{2}}\right)\right] \tag{13}\]  
where \(F_{w}\left(s\right)\) is the Weibull cumulative distribution function, and \(F_{G}\left(s\right)\) is the Gaussian cumulative distribution function, and erf( ) is the error function. The weight \(e^{-(s/\rho)^{s}}\) is in the range [0,1]. When the weight is small approaching zero, the distribution is close to Gaussian; conversely when the weight increases toward the value 1, the distribution becomes more Weibull. The six parameters \(\alpha,\ \beta,\ m,\ b,\ \mu_{{}_{G}}\) and \(\sigma_{{}_{G}}\) can be determined by using the nonlinear least-squares regression algorithm, which returns the optimized values for the parameters that minimize the sum of the squared differences between the observed responses and their fitted values.  
As an illustrative example, the proposed mixed Weibull-Gaussian distribution is applied to fit the strength distribution of flawed inert silicon nitride (SNW-1000) [PERSON] and [PERSON] (2009). The testing data in [PERSON] and [PERSON] (2009) and many other publications show that strength distribution of some brittle and quasi-brittle materials deviates from the two-parameter Weibull distribution, and modified versions of Weibull distribution have been proposed. As shown in Fig 6, the sample data deviate from the linearity in the Weibull plot, and the mixed Weibull-Gaussian distribution provides a nonlinear fitting with a coefficient of correlation \(R^{2}=0.9494\). While the example in Fig 6 is more of illustrative purpose, we emphasize the most advantage of the mixed Weibull-Gaussian distribution lies on the asymptotic matching of the lower and upper tails, and the flexibility of smooth transition between them. Further investigation isneeded with respect to experimental verification, and detailed discussion of the mixed Weibull-Gaussian distribution will be provided in a separate report.  
By performing Monte Carlo simulation for an ensemble of SRVE, the statistics of SRVE strength can be input into the regression model to obtain the optimized parameters of the mixed Weibull-Gaussian distribution. By using 10,000 SRVE samples, an example of the fitted mixed Gaussian-Weibull distribution is shown in Fig. 7, which demonstrates good agreement with the numerical data obtained from Monte Carlo simulation.