×

A variational discrete element method for the computation of Cosserat elasticity. (English) Zbl 1479.74137

Summary: The variational discrete element method developed in [the author et al., “A variational discrete element method for quasistatic and dynamic elastoplasticity”, Int. J. Numer. Methods Eng. 121, No. 23, 5295–5319 (2020; doi:10.1002/nme.6460)] for dynamic elasto-plastic computations is adapted to compute the deformation of elastic Cosserat materials. In addition to cellwise displacement degrees of freedom (dofs), cellwise rotational dofs are added. A reconstruction is devised to obtain \(P^1\) non-conforming polynomials in each cell and thus constant strains and stresses in each cell. The method requires only the usual macroscopic parameters of a Cosserat material and no microscopic parameter. Numerical examples show the robustness of the method for both static and dynamic computations in two and three dimensions.

MSC:

74S99 Numerical and other methods in solid mechanics
74A35 Polar materials
74B99 Elastic materials
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Fundamental Algorithms for Scientific Computing in Python, SciPy 1.0, Nat Methods, 17, 261-272 (2020) · doi:10.1038/s41592-019-0686-2
[2] André, D.; Girardot, J.; Hubert, C., A novel DEM approach for modeling brittle elastic media based on distinct lattice spring model, Comput Methods Appl Mech Eng, 350, 100-122 (2019) · Zbl 1441.74193 · doi:10.1016/j.cma.2019.03.013
[3] André, D.; Iordanoff, I.; Charles, J-L; Néauport, J., Discrete element method to simulate continuous material by using the cohesive beam model, Comput Methods Appl Mech Eng, 213, 113-125 (2012) · Zbl 1243.74197 · doi:10.1016/j.cma.2011.12.002
[4] André, D.; Jebahi, M.; Iordanoff, I.; Charles, J-L; Néauport, J., Using the discrete element method to simulate brittle fracture in the indentation of a silica glass with a blunt indenter, Comput Methods Appl Mech Eng, 265, 136-147 (2013) · Zbl 1286.74003 · doi:10.1016/j.cma.2013.06.008
[5] Arnold, D., An interior penalty finite element method with discontinuous elements, SIAM J Numer Anal, 19, 4, 742-760 (1982) · Zbl 0482.65060 · doi:10.1137/0719052
[6] Avci, B.; Wriggers, P., A DEM-FEM coupling approach for the direct numerical simulation of 3D particulate flows, J Appl Mech, 79, 1, 2698 (2012) · doi:10.1115/1.4005093
[7] Balay S, Abhyankar S, Adams M, Brown J, Brune P, Buschelman K, Dalcin L, Dener A, Eijkhout V, Gropp W (2019) Petsc users manual
[8] Belytschko, T.; Hughes, TJR, Computational methods for transient analysis (1983), Amsterdam: North-Holland(Computational Methods in Mechanics, Amsterdam · Zbl 0521.00025
[9] Bleyer J (2018) Numerical tours of computational mechanics with fenics
[10] Boiveau, T.; Burman, E., A penalty-free Nitsche method for the weak imposition of boundary conditions in compressible and incompressible elasticity, IMA J Numer Anal, 36, 2, 770-795 (2016) · Zbl 1433.74101 · doi:10.1093/imanum/drv042
[11] Brocato, M.; Capriz, G., Gyrocontinua, Int J Solids Struct, 38, 6-7, 1089-1103 (2001) · Zbl 1007.74013 · doi:10.1016/S0020-7683(00)00075-5
[12] Burman, E., A penalty-free nonsymmetric Nitsche-type method for the weak imposition of boundary conditions, SIAM J Numer Anal, 50, 4, 1959-1981 (2012) · Zbl 1262.65165 · doi:10.1137/10081784X
[13] Cosserat E, Cosserat F (1909) Théorie des corps déformables. A. Hermann et fils · JFM 40.0862.02
[14] Cundall, P.; Strack, O., A discrete numerical model for granular assemblies, Geotechnique, 29, 1, 47-65 (1979) · doi:10.1680/geot.1979.29.1.47
[15] Dalcin, L.; Paz, R.; Kler, P.; Cosimo, A., Parallel distributed computing using python, Adv Water Resour, 34, 9, 1124-1139 (2011) · doi:10.1016/j.advwatres.2011.04.013
[16] Di Pietro, DA, Cell centered Galerkin methods for diffusive problems, ESAIM Math Modell Numer Anal, 46, 1, 111-144 (2012) · Zbl 1279.65125 · doi:10.1051/m2an/2011016
[17] Ericksen, JL, Liquid crystals and Cosserat surfaces, Q J Mech Appl Math, 27, 2, 213-219 (1974) · Zbl 0284.76010 · doi:10.1093/qjmam/27.2.213
[18] Eymard, R.; Gallouët, T.; Herbin, R., Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces, IMA J Numer Anal, 30, 4, 1009-1043 (2009) · Zbl 1202.65144 · doi:10.1093/imanum/drn084
[19] David, C-LF; Saunders, M., Lsmr: an iterative algorithm for sparse least-squares problems, SIAM J Sci Comput, 33, 5, 2950-2971 (2011) · Zbl 1232.65052 · doi:10.1137/10079687X
[20] Forest, S.; Pradel, F.; Sab, K., Asymptotic analysis of heterogeneous Cosserat media, Int J Solids Struct, 38, 26-27, 4585-4608 (2001) · Zbl 1033.74038 · doi:10.1016/S0020-7683(00)00295-X
[21] Godio, M.; Stefanou, I.; Sab, K.; Sulem, J.; Sakji, S., A limit analysis approach based on Cosserat continuum for the evaluation of the in-plane strength of discrete media: application to masonry, Eur J Mech A/Solids, 66, 168-192 (2017) · Zbl 1406.74029 · doi:10.1016/j.euromechsol.2017.06.011
[22] Hoover, WG; Ashurst, WT; Olness, RJ, Two-dimensional computer studies of crystal stability and fluid viscosity, J Chem Phys, 60, 10, 4043-4047 (1974) · doi:10.1063/1.1680855
[23] Jebahi, M.; André, D.; Terreros, I.; Iordanoff, I., Discrete element method to model 3D continuous materials (2015), New York: Wiley, New York · doi:10.1002/9781119103042
[24] Jeong, J.; Neff, P., Existence, uniqueness and stability in linear Cosserat elasticity for weakest curvature conditions, Math Mech Solids, 15, 1, 78-95 (2010) · Zbl 1197.74009 · doi:10.1177/1081286508093581
[25] Lamb, H., On the propagation of tremors over the surface of an elastic solid, Philos Trans R Soc London Ser A, 203, 359-371, 1-42 (1904) · JFM 34.0859.02
[26] Logg, A.; Mardal, K-A; Wells, G., Automated solution of differential equations by the finite element method: the FEniCS book (2012), Berlin: Springer, Berlin · Zbl 1247.65105 · doi:10.1007/978-3-642-23099-8
[27] Logg, A.; Wells, GN, Dolfin: automated finite element computing, ACM Trans Math Softw, 37, 2, 10298 (2010) · Zbl 1364.65254 · doi:10.1145/1731022.1731030
[28] Marazzato, F.; Ern, A.; Monasse, L., A variational discrete element method for quasistatic and dynamic elastoplasticity, Int J Numer Methods Eng, 121, 23, 5295-5319 (2020) · doi:10.1002/nme.6460
[29] Marazzato F, Ern A, Monasse L (2021) Quasi-static crack propagation with a Griffith criterion using a discrete element method
[30] Mariotti, C., Lamb’s problem with the lattice model Mka3D, Geophys J Int, 171, 2, 857-864 (2007) · doi:10.1111/j.1365-246X.2007.03579.x
[31] Michael, M.; Vogel, F.; Peters, B., DEM-FEM coupling simulations of the interactions between a tire tread and granular terrain, Comput Methods Appl Mech Eng, 289, 227-248 (2015) · Zbl 1423.74673 · doi:10.1016/j.cma.2015.02.014
[32] Monasse, L.; Mariotti, C., An energy-preserving Discrete Element Method for elastodynamics, ESAIM Math Modell Numer Anal, 46, 1527-1553 (2012) · Zbl 1267.74114 · doi:10.1051/m2an/2012015
[33] Neff, P., A finite-strain elastic-plastic Cosserat theory for polycrystals with grain rotations, Int J Eng Sci, 44, 8-9, 574-594 (2006) · Zbl 1213.74032 · doi:10.1016/j.ijengsci.2006.04.002
[34] Neff, P.; Chelminski, K., Well-posedness of dynamic Cosserat plasticity, Appl Math Optim, 56, 1, 19-35 (2007) · Zbl 1120.74006 · doi:10.1007/s00245-007-0878-5
[35] Notsu, H.; Kimura, M., Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity, Netw Heterog Media, 9, 4, 10265 (2014) · Zbl 1302.74019
[36] Potyondy, D.; Cundall, P., A bonded-particle model for rock, Int J Rock Mech Min Sci, 41, 8, 1329-1364 (2004) · doi:10.1016/j.ijrmms.2004.09.011
[37] Providas, E.; Kattis, MA, Finite element method in plane Cosserat elasticity, Comput Struct, 80, 27-30, 2059-2069 (2002) · doi:10.1016/S0045-7949(02)00262-6
[38] Puscas, MA; Monasse, L.; Ern, A.; Tenaud, C.; Mariotti, C., A conservative Embedded Boundary method for an inviscid compressible flow coupled with a fragmenting structure, Int J Numer Methods Eng, 103, 13, 970-995 (2015) · Zbl 1352.76074 · doi:10.1002/nme.4921
[39] Rattez, H.; Stefanou, I.; Sulem, J., The importance of Thermo-Hydro-Mechanical couplings and microstructure to strain localization in 3D continua with application to seismic faults. Part i: Theory and linear stability analysis, J Mech Phys Solids, 115, 54-76 (2018) · doi:10.1016/j.jmps.2018.03.004
[40] Rattez, H.; Stefanou, I.; Sulem, J.; Veveakis, M.; Poulet, T., Numerical analysis of strain localization in rocks with thermo-hydro-mechanical couplings using Cosserat continuum, Rock Mech Rock Eng, 51, 10, 3295-3311 (2018) · doi:10.1007/s00603-018-1529-7
[41] Ries, A.; Wolf, D.; Unger, T., Shear zones in granular media: three-dimensional contact dynamics simulation, Phys Rev E, 76, 5, 051301 (2007) · doi:10.1103/PhysRevE.76.051301
[42] Sautot C, Bordas S, Hale J (2014) Extension of 2D FEniCS implementation of Cosserat non-local elasticity to the 3D case. Technical report, Université du Luxembourg
[43] Shelukhin, VV; Ružička, M., On Cosserat-Bingham fluids, ZAMM-J Appl Math Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 93, 1, 57-72 (2013) · Zbl 1372.76009 · doi:10.1002/zamm.201200037
[44] Stefanou, I.; Sulem, J.; Vardoulakis, I., Three-dimensional Cosserat homogenization of masonry structures: elasticity, Acta Geotech, 3, 1, 71-83 (2008) · doi:10.1007/s11440-007-0051-y
[45] Sulem, J.; Vardoulakis, IG, Bifurcation analysis in geomechanics (1995), London: CRC Press, London · doi:10.1201/9781482269383
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.