Statistics-assisted fracture modelling of small un-notched and large notched sandstone specimens with specimen-size/grain-size ratio from 30 to 900

•A linear closed-form model is proposed for quasi-brittle fracture of sandstone.•95 sandstone specimens with volume ratio close to 1000 were tested and analyzed.•One normal distribution considering scatters/reliability applies to all results.•The closed-form solution is a predictive model more suitable for design analysis.•Statistical analysis of discrete cracking removes the need of curve-fitting. Quasi-brittle fracture properties of a medium-grain sandstone with an average grain size G around 0.3–0.4 mm were investigated under three-point-bending (3-p-b) conditions. In total, 95 specimens were tested with the beam width W varying from 10 to 300 mm, or the specimen-size/grain-size ratio from 30 to 900. 45 medium-sized specimens (W = 30, 60, 100 mm) were tested first to determine the tensile strength ft, which was then used as a reference for tests of 8 large notched specimens (W = 300 mm) and 42 small un-notched specimens (W = 10 mm). Statistical fracture modelling, based on normal distributions and the characteristic microstructure measurement (the average grain size G in this study), was used to quantify the quasi-stable fictitious crack growth Δafic at the peak load Pmax and the characteristic crack length ach* defined by the bulk toughness and strength properties. The statistics-assisted modelling has changed the previous curve-fitting boundary effect model (BEM) to a predictive closed-form solution, providing a useful option when large scatters in experimental data and reliability in design need to be focused. The well-known size effect law (SEL) proposed for geometrically similar specimens was also used to fit the sandstone results, and compared with the closed-form BEM with built-in statistical functions. Applications of SEL and BEM and their key differences were explained.