Abstract
Novel numerical algorithms are presented for the implementation of micro-scale boundary conditions of particle aggregates modelled with the discrete element method. The algorithms are based on a servo-control methodology, using a feedback principle comparable to that of algorithms commonly applied within control theory of dynamic systems. The boundary conditions are defined in accordance with the large deformation theory, and are imposed on a frame of boundary particles surrounding the interior granular micro-structure. Following the formulation presented in Miehe et al. (Int J Numer Methods Eng 83(8–9): 1206–1236, 2010), first three types of classical boundary conditions are considered, in accordance with (1) a homogeneous deformation and zero particle rotation (D), (2) a periodic particle displacement and rotation (P), and (3) a uniform particle force and free particle rotation (T). The algorithms can be straightforwardly combined with commercially available discrete element codes, thereby enabling the determination of the solution of boundary-value problems at the micro-scale only, or at multiple scales via a micro-to-macro coupling with a finite element model. The performance of the algorithms is tested by means of discrete element method simulations on regular monodisperse packings and irregular polydisperse packings composed of frictional particles, which were subjected to various loading paths. The simulations provide responses with the typical stiff and soft bounds for the (D) and (T) boundary conditions, respectively, and illustrate for the (P) boundary condition a relatively fast convergence of the apparent macroscopic properties under an increasing packing size. Finally, a homogenization framework is derived for the implementation of mixed (D)–(P)–(T) boundary conditions that satisfy the Hill–Mandel micro-heterogeneity condition on energy consistency at the micro- and macro-scales of the granular system. The numerical algorithm for the mixed boundary conditions is developed and tested for the case of an infinite layer subjected to a vertical compressive stress and a horizontal shear deformation, whereby the response computed for a layer of cohesive particles is compared against that for a layer of frictional particles.
Highlights
The accurate computation of the non-linear failure and deformation behavior of heterogeneous granular systems commonly requires a resolution of the complex mechanical interactions and deformation mechanisms at the particle scale, which can be adequately accounted for by using the discrete element method (DEM), see e.g., [1,2,3,4,5,6,7,8,9,10] and references therein
The numerical algorithm for the mixed boundary conditions is developed and tested for the case of an infinite layer subjected to a vertical compressive stress and a horizontal shear deformation, whereby the response computed for a layer of cohesive particles is compared against that for a layer of frictional particles
Novel numerical algorithms have been presented for the implementation of three types of classical boundary conditions for a particle aggregate
Summary
The accurate computation of the non-linear failure and deformation behavior of heterogeneous granular systems commonly requires a resolution of the complex mechanical interactions and deformation mechanisms at the particle scale, which can be adequately accounted for by using the discrete element method (DEM), see e.g., [1,2,3,4,5,6,7,8,9,10] and references therein. The computational homogenization framework presented in [16] does include the three aspects mentioned above, and calculates the micro-scale response of a granular packing with a DEM model equipped with a frame of boundary particles at which the finite deformation following from the macro-scale is imposed. Since the present study focuses on two-dimensional particle aggregates, throughout the paper the dimensions related to volume, area, stress and mass density are consistently presented in their reduced form as length, length, force/length and mass/length, respectively
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