Representative volume element approximations in elastoplastic spring networks. (English) Zbl 07844040
Summary: We study the large-scale behavior of a small-strain lattice model for a network composed of elastoplastic springs with random material properties. We formulate the model as an evolutionary rate independent system. In an earlier work [two of the authors, Multiscale Model. Simul. 16, No. 2, 857–899 (2018; Zbl 1392.49014)] we derived a homogenized continuum model, which has the form of linearized elastoplasticity, as an evolutionary \(\Gamma\)-limit as the lattice parameter tends to zero. In the present paper we introduce a periodic representative volume element (RVE) approximation for the homogenized system. As a main result we prove convergence of the RVE approximation as the size of the RVE tends to infinity. We also show that the hysteretic stress-strain relation of the effective system can be described with the help of a generalized Prandtl-Ishlinskii operator, and we prove convergence of a periodic RVE approximation for that operator. We combine the RVE approximation with a numerical scheme for rate-independent systems and obtain a computational scheme that we use to numerically investigate the homogenized system in the specific case when the original network is given by a two-dimensional lattice model. We simulate the response of the system to cyclic and uniaxial, monotonic loading, and numerically investigate the convergence rate of the periodic RVE approximation. In particular, our simulations show that the RVE error decays with the same rate as the RVE error in the static case of linear elasticity.
MSC:
| 74C05 | Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) |
| 74Q10 | Homogenization and oscillations in dynamical problems of solid mechanics |
| 74A40 | Random materials and composite materials |
| 74S99 | Numerical and other methods in solid mechanics |
Keywords:
small-strain lattice model; random material; stochastic homogenization; generalized Prandtl-Ishlinskii operator; truncated nonsmooth Newton multigrid methodCitations:
Zbl 1392.49014References:
| [1] | Alicandro, R., Cicalese, M., and Gloria, A., Integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity, Arch. Ration. Mech. Anal., 200 (2011), pp. 881-943. · Zbl 1294.74056 |
| [2] | Bella, P. and Otto, F., Corrector estimates for elliptic systems with random periodic coefficients, Multiscale Model. Simul., 14 (2016), pp. 1434-1462, doi:10.1137/15M1037147. · Zbl 1351.35273 |
| [3] | Bourgeat, A., Mikelić, A., and Wright, S., Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math., 456 (1994), pp. 19-51. · Zbl 0808.60056 |
| [4] | Bourgeat, A. and Piatnitski, A., Approximations of effective coefficients in stochastic homogenization, Ann. Inst. Henri Poincaré Probab. Stat., 40 (2004), pp. 153-165. · Zbl 1058.35023 |
| [5] | Brokate, M. and Sprekels, J., Hysteresis and Phase Transitions, , Springer, New York, 1996. · Zbl 0951.74002 |
| [6] | Drugan, W. J. and Willis, J. R., A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites, J. Mech. Phys. Solids, 44 (1996), pp. 497-524. · Zbl 1054.74704 |
| [7] | Eigel, M., Gittelson, C. J., Schwab, C., and Zander, E., A convergent adaptive stochastic galerkin finite element method with quasi-optimal spatial meshes, ESAIM Math. Model. Numer. Anal., 49 (2015), pp. 1367-1398. · Zbl 1335.65006 |
| [8] | Feyel, F., Multiscale \(FE^2\) elastoviscoplastic analysis of composite structures, Comput. Mater. Sci., 16 (1999), pp. 344-354. |
| [9] | Fischer, J. and Neukamm, S., Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems, Arch. Ration. Mech. Anal., 242 (2021), pp. 343-452. · Zbl 1471.35024 |
| [10] | Gloria, A., Neukamm, S., and Otto, F., Quantification of ergodicity in stochastic homogenization: Optimal bounds via spectral gap on Glauber dynamics, Invent. Math., 199 (2015), pp. 455-515. · Zbl 1314.39020 |
| [11] | Gloria, A., Neukamm, S., and Otto, F., A regularity theory for random elliptic operators, Milan J. Math., 88 (2020), pp. 99-170. · Zbl 1440.35064 |
| [12] | Gloria, A., Neukamm, S., and Otto, F., Quantitative estimates in stochastic homogenization for correlated coefficient fields, Anal. PDE, 14 (2021), pp. 2497-2537, doi:10.2140/apde.2021.14.2497. · Zbl 1485.35156 |
| [13] | Gloria, A. and Otto, F., An optimal variance estimate in stochastic homogenization of discrete elliptic equations, Ann. Probab., 39 (2011), pp. 779-856. · Zbl 1215.35025 |
| [14] | Gloria, A. and Otto, F., An optimal error estimate in stochastic homogenization of discrete elliptic equations, Ann. Appl. Probab., 22 (2012), pp. 1-28. · Zbl 1387.35031 |
| [15] | Gräser, C. and Sander, O., Truncated nonsmooth Newton multigrid methods for block-separable minimization problems, IMA J. Numer. Anal., 39 (2019), pp. 454-481, doi:10.1093/imanum/dry073. · Zbl 1483.65198 |
| [16] | Gräser, C., Kienle, D., and Sander, O., Truncated nonsmooth Newton multigrid for phase-field brittle-fracture problems, with analysis, Comput. Mech., 72 (2023), doi:10.1007/s00466-023-02330-x. · Zbl 1528.74101 |
| [17] | Haberland, S., Analytical and Numerical Investigation of Evolutionary Rate-independent Discrete Systems, Master’s thesis, Technische Universität Dresden, 2021, https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa2-805599. |
| [18] | Hahn, M., Wallmersperger, T., and Kröplin, B. H., Discrete element representation of continua: Proof of concept and determination of the material parameters, Comput. Mater. Sci., 50 (2010), pp. 391-402. |
| [19] | Heida, M., Stochastic homogenization of rate-independent systems and applications, Contin. Mech. Thermodyn., 29 (2017), pp. 853-894. · Zbl 1375.74077 |
| [20] | Heida, M., Neukamm, S., and Varga, M., Stochastic homogenization of \(\lambda \)-convex gradient flows, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), pp. 427-459. · Zbl 1458.49013 |
| [21] | Heida, M., Neukamm, S., and Varga, M., Stochastic two-scale convergence and young measures, Netw. Heterog. Media, 17 (2022), pp. 227-254. · Zbl 1493.74101 |
| [22] | Heida, M. and Schweizer, B., Stochastic homogenization of plasticity equations, ESAIM Control Optim. Calc. Var., 24 (2018), pp. 153-176. · Zbl 1393.74014 |
| [23] | Jagota, A. and Bennison, S. J., Spring-network and finite-element models for elasticity and fracture, in Non-linearity and Breakdown in Soft Condensed Matter, Springer, New York, 1994, pp. 186-201. |
| [24] | Kanit, T., Forest, S., Galliet, I., Mounoury, V., and Jeulin, D., Determination of the size of the representative volume element for random composites: Statistical and numerical approach, Int. J. Solids Struct., 40 (2003), pp. 3647-3679. · Zbl 1038.74605 |
| [25] | Mainik, A. and Mielke, A., Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), pp. 73-99. · Zbl 1161.74387 |
| [26] | Matsui, K., Terada, K., and Yuge, K., Two-scale finite element analysis of heterogeneous solids with periodic microstructures, Comput. Struct., 82 (2004), pp. 593-606, doi:10.1016/j.compstruc.2004.01.004. |
| [27] | Miehe, C., Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation, Internat. J. Numer. Methods Engrg., 55 (2002), pp. 1285-1322. · Zbl 1027.74056 |
| [28] | Mielke, A., Evolution of rate-independent systems, Evol. Equ., 2 (2005), pp. 461-559. · Zbl 1120.47062 |
| [29] | Mielke, A., Generalized Prandtl-Ishlinskii operators arising from homogenization and dimension reduction, Phys. B, 407 (2012), pp. 1330-1335. |
| [30] | Mielke, A., On evolutionary \(\Gamma \)-convergence for gradient systems, in Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, Springer, New York, 2016, pp. 187-249. |
| [31] | Mielke, A. and Roubíček, T., Rate-Independent Systems: Theory and Application, , Springer, New York, 2015. · Zbl 1339.35006 |
| [32] | Mielke, A. and Theil, F., On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004), pp. 151-189. · Zbl 1061.35182 |
| [33] | Mielke, A. and Timofte, A. M., Two-scale homogenization for evolutionary variational inequalities via the energetic formulation, SIAM J. Math. Anal., 39 (2007), pp. 642-668. · Zbl 1185.35282 |
| [34] | Nejad, R. S. and Wieners, C., Parallel inelastic heterogeneous multi-scale simulations, in Multi-scale Simulation of Composite Materials, Springer, New York, 2019, pp. 57-96. |
| [35] | Neukamm, S. and Varga, M., Stochastic unfolding and homogenization of spring network models, Multiscale Model. Simul., 16 (2018), pp. 857-899. · Zbl 1392.49014 |
| [36] | Neukamm, S., Varga, M., and Waurick, M., Two-scale homogenization of abstract linear time-dependent pdes, Asymptot. Anal., 125 (2021), pp. 247-287. · Zbl 1510.35322 |
| [37] | Ostoja-Starzewski, M., Lattice models in micromechanics, Appl. Mech. Rev., 55 (2002), pp. 35-60. |
| [38] | Owhadi, H., Approximation of the effective conductivity of ergodic media by periodization, Probab. Theory Related Fields, 125 (2003), pp. 225-258. · Zbl 1040.60025 |
| [39] | Papanicolaou, G. C. and Varadhan, S. R. S., Boundary value problems with rapidly oscillating random coefficients, in Random Fields, , North-Holland, Amsterdam, 1981, pp. 835-873. · Zbl 0499.60059 |
| [40] | Sander, O. and Jaap, P., Solving primal plasticity increment problems in the time of a single predictor-corrector iteration, Comput. Mech., 65 (2020), pp. 663-685, doi:10.1007/s00466-019-01788-y. · Zbl 1477.74121 |
| [41] | Schneider, M., Josien, M., and Otto, F., Representative volume elements for matrix-inclusion composites - A computational study on the effects of an improper treatment of particles intersecting the boundary and the benefits of periodizing the ensemble, J. Mech. Phys. Solids, 158 (2022), 104652. |
| [42] | Schröder, J. and Hackl, K., Plasticity and Beyond: Microstructures, Crystal-plasticity and Phase Transitions, , Springer, New York, 2013. |
| [43] | Torquato, S., Random Heterogeneous Materials: Microstructure and Macroscopic Properties, , Springer, New York, 2013. |
| [44] | Varga, M., Stochastic Unfolding and Homogenization of Evolutionary Gradient Systems, Ph.D. thesis, Technische Universität Dresden, 2019. |
| [45] | Watanabe, I., Terada, K., de Souza Neto, E. A., and Perić, D., Characterization of macroscopic tensile strength of polycrystalline metals with two-scale finite element analysis, J. Mech. Phys. Solids, 56 (2008), pp. 1105-1125, doi:10.1016/j.jmps.2007.06.001. · Zbl 1151.74011 |
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.
