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The discrete element modelling (DEM) of triaxial tests plays a critical role in 2 unveiling fundamental properties of particulate materials, but the numerical 3 implementation of a flexible membrane boundary for the testing still imposes 4 problems. In this study, a robust algorithm was proposed to reproduce a flexible 5 membrane boundary in triaxial testing. The equivalence of strain energy enables the 6 particle-scale parameters representing the flexible membrane to be directly 7 determined from the real geometric and material parameters of the membrane. Then 8 the proposed flexible membrane boundary was implemented in the context of discrete 9 element simulation of triaxial testing and was validated with laboratory experiments. 10 Furthermore, comparisons of triaxial tests with flexible and rigid boundaries were 11 performed from macro-scale to meso-scale. The results show that the boundary 12 condition has limited influences on the stress-strain behaviour but a relatively large 13 impact on the volumetric change, the failure mode, the distribution of contact forces, 14 and the fabric evolution of particles in the specimen during triaxial testing. 15

Efficient selections of particle-scale contact parameters in discrete element modelling remain an open question. The aim of this study is to provide a hybrid calibration framework to estimate linear contact stiffnesses (normal and tangential) for both two-dimensional and three-dimensional simulations. Analytical formulas linking macroscopic parameters (Young's modulus, Poisson's ratio) to mesoscopic particle parameters for granular systems are derived based on statistically isotropic packings under small-strain isotropic stress conditions. By taking the derived analytical solutions as initial approximations, the gradient descent algorithm automatically obtains a reliable numerical estimation. The proposed framework is validated with several numerical cases including randomly distributed monodisperse and polydisperse packings. The results show that this hybrid method practically reduces the time for artificial trials and errors to obtain reasonable stiffness parameters. The proposed framework can be extended to other parameter calibration problems in DEM.