Strain gradient stabilization with dual stress points for the meshfree nodal integration method in inelastic analyses. (English) Zbl 1352.74060
Summary: A nonlinear nodal-integrated meshfree Galerkin formulation based on recently proposed strain gradient stabilization (SGS) method is developed for large deformation analysis of elastoplastic solids. The SGS is derived from a decomposed smoothed displacement field and is introduced to the standard variational formulation through the penalty method for the inelastic analysis. The associated strain gradient matrix is assembled by a B-bar method for the volumetric locking control in elastoplastic materials. Each meshfree node contains two coinciding integration points for the integration of weak form by the direct nodal integration scheme. As a result, a nonlinear stabilized nodal integration method with dual nodal stress points is formulated, which is free from stabilization control parameters and integration cells for meshfree computation. In the context of extreme large deformation analysis, an adaptive anisotropic Lagrangian kernel approach is introduced to the nonlinear SGS formulation. The resultant Lagrangian formulation is constantly updated over a period of time on the new reference configuration to maintain the well-defined displacement gradients as well as strain gradients in the Lagrangian computation. Several numerical benchmarks are studied to demonstrate the effectiveness and accuracy of the proposed method in large deformation inelastic analyses.
MSC:
| 74C05 | Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) |
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
References:
| [1] | ChenJS, YoonS, WuCT. Nonlinear version of stabilized conforming nodal integration for Galerkin meshfree methods. International for Numerical Methods in Engineering2002; 53:2587-2615. · Zbl 1098.74732 |
| [2] | LiS, LiuWK. Meshfree Particle Method. Springer: Berlin, 2004. · Zbl 1073.65002 |
| [3] | PusoMA, ChenJS, ZywiczW, ElmerW. Meshfree and finite element nodal integration methods. International for Numerical Methods in Engineering2008; 74:416-446. · Zbl 1159.74456 |
| [4] | RabczukT, GracieR, SongHJ, BelytschkoT. Immersed particle method for fluid-structure interaction. International for Numerical Methods in Engineering2010; 81:48-71. · Zbl 1183.74367 |
| [5] | WuYC, MagallanesJM, ChoiHJ, CrawfordJE. An evolutionarily coupled finite element‐meshfree formulation for modeling concrete behaviors under blast and impact loadings. Journal of Engineering Mechanics2013; 139:525-536. |
| [6] | WuYC, MagallanesJM, CrawfordJE. Fragmentation and debris evolution modeled by a point‐wise coupled reproducing kernel/finite element formulation. International Journal of Damage Mechanics2014; 23:1005-1034. |
| [7] | BeisselS, BelytschkoT. Nodal integration of the element‐free Galerkin method. Computer Methods in Applied Mechanics and Engineering1996; 139:49-74. · Zbl 0918.73329 |
| [8] | LiuWK, OngJSJ, UrasRA. Finite‐element stabilization matrices – a unification approach. Computer Methods in Applied Mechanics and Engineering1985; 53:13-46. · Zbl 0553.73065 |
| [9] | NagashimaT. Node‐by‐node meshless approach and its applications to structural analyses. International for Numerical Methods in Engineering1999; 46:341-385. · Zbl 0965.74079 |
| [10] | LiuGR, ZhangGY, WangYY, ZhongZH, LiGY, HanX. A nodal integration technique for meshfree radial point interpolation method (NI‐RPIM). International Journal of Solids and Structures2007; 44:3840-3860. · Zbl 1135.74050 |
| [11] | ChenJS, WuCT, YoonS, YouY. A stabilized conforming nodal integration for Galerkin meshfree methods. International for Numerical Methods in Engineering2001; 50:435-466. · Zbl 1011.74081 |
| [12] | DykaCT, RandlesPW, IngelRP. Stress points for tension instability in SPH. International for Numerical Methods in Engineering1997; 40:2325-2341. · Zbl 0890.73077 |
| [13] | ChenJS, WuYC, GuanPC, ThengH, GaidosJ, HofstetterK, AlsalehM. A semi‐Lagrangian reproducing kernel formulation for modeling earth moving operations. Mechanics of Materials2009; 41:670-683. |
| [14] | GuanPC, ChiSW, ChenJS, SlawsonRMJ. Semi‐Lagrangian reproducing kernel particle method for fragment‐impact problems. International Journal of Impact Engineering2011; 38:1033-1047. |
| [15] | ChiSW, LeeCH, ChenJS, GuanPC. A level set enhanced natural kernel contact algorithm for impact and penetration modeling. International for Numerical Methods in Engineering2014; 102:839-866. · Zbl 1352.74139 |
| [16] | HillmanM, ChenJS, ChiSW. Stabilized and variationally consistent nodal integration for meshfree modeling of impact problems. Computational Particle Mechanics2014; 1:245-256. |
| [17] | ChenJS, HillmanM, RüterM. An arbitrary order variationally consistent integration method for Galerkin meshfree methods. International for Numerical Methods in Engineering2013; 95:387-418. · Zbl 1352.65481 |
| [18] | WangD, LiZ. A two‐level strain smoothing regularized meshfree approach with stabilized conforming nodal integration for elastic damage analysis. International Journal of Damage Mechanics2013; 22:440-459. |
| [19] | WuYC, WangDD, WuCT. Three dimensional fragmentation simulation of concrete structures with a nodally regularized meshfree method. Theoretical Applied Fracture Mechanics2014; 27:89-99. |
| [20] | ChenJS, WuCT, BelytschkoT. Regularization of material instabilities by meshfree approximations with intrinsic length scales. International for Numerical Methods in Engineering2000; 47:1303-1322. · Zbl 0987.74079 |
| [21] | WuCT, GuoY, HuW. An introduction to the LS‐DYNA smoothed particle Galerkin method for severe deformation and failure analysis in solids. In 13th International LS‐DYNA Users Conference, Detroit, MI, June 8‐10 2014; 1-20. |
| [22] | WuYC, WangD, WuCT. A direct displacement smoothing meshfree particle formulation for impact failure modeling. International Journal of Impact Engineering2015. DOI: 10.1016/j.ijimpeng.2015.03.013, in early view. |
| [23] | WuCT, KoishiM, HuW. A displacement smoothing induced strain gradient stabilization for the meshfree Galerkin nodal integration method. Computational Mechanics2015; 56:19-37. · Zbl 1329.74292 |
| [24] | HughesTJR. The Finite Element Method. Dover Publications Inc.: New York, 2000. |
| [25] | WuCT, ParkCK, ChenJS. A generalized approximation for the meshfree analysis of solids. International for Numerical Methods in Engineering2011; 85:693-722. · Zbl 1217.74150 |
| [26] | ChenJS, PanC, WuCT, LiuWK. Reproducing kernel particle methods for large deformation analysis of non‐linear structures. Computer Methods in Applied Mechanics and Engineering1996; 139:195-227. · Zbl 0918.73330 |
| [27] | WuCT, HuW, ChenJS. A meshfree‐enriched finite element method for compressible and nearly incompressible elasticity. International for Numerical Methods in Engineering2012; 90:882-914. · Zbl 1242.74174 |
| [28] | WuCT, KoishiM. Three‐dimensional meshfree‐enriched finite element formulation for micromechanical hyperelastic modeling of particulate rubber composites. International for Numerical Methods in Engineering2012; 91:1137-1157. |
| [29] | WuCT, GuoY, AskariE. Numerical modeling of composite solids using an immersed meshfree Galerkin method. Composites Part B2013; 45:1397-1413. |
| [30] | SukumarN. Construction of polygonal interpolants: a maximum entropy approach. International for Numerical Methods in Engineering2004; 61:2159-2181. · Zbl 1073.65505 |
| [31] | CardosoRPR, YoonJW, CracioJJ, BarlatF, Cesar de SaJMA. Development of a one point quadrature shell element for nonlinear applications with contact and anisotropy. Computer Methods in Applied Mechanics and Engineering2002; 191:5177-5206. · Zbl 1083.74583 |
| [32] | RabczukT, BelytschkoT, XiaoSP. Stable particle methods based on Lagrangian kernels. Computer Methods in Applied Mechanics and Engineering2004; 193:1035-1063. · Zbl 1060.74672 |
| [33] | BelytschkoT, LiuWK, MoranB. Nonlinear Finite Elements for Continua and Structures. Wiley: Chichester, 2000. · Zbl 0959.74001 |
| [34] | BelytschkoT, BindemanLP. Assumed strain stabilization of the eight node hexahedral element. Computer Methods in Applied Mechanics and Engineering1993; 105:225-260. · Zbl 0781.73061 |
| [35] | BelytschkoT, GuoY, LiuWK, XiaoSP. A unified stability analysis of meshless particle methods. International for Numerical Methods in Engineering2000; 48:1359-1400. · Zbl 0972.74078 |
| [36] | ParkCK, WuCT, KanCD. On the analysis of dispersion property and stable time step in meshfree method using generalized meshfree approximation. Finite Element in Analysis and Design2011; 47:683-697. |
| [37] | WuCT, RenB. A stabilized non‐ordinary state‐based peridynamics for the nonlocal ductile material failure analysis in metal machining process. Computer Methods in Applied Mechanics and Engineering2015; 291:197-215. · Zbl 1423.74067 |
| [38] | NorrisDM, MoranB, ScudderJK, QuinonesDF. A computer simulation of the tension test. Journal of the Mechanics and Physics of Solids1978; 26:1-19. |
| [39] | LS‐DYNA Users’ Manual. Livermore Software Technology Corporation: Livermore, USA, 2015. |
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.
