Kinematic constraints in the state-based peridynamics with mixed local/nonlocal gradient approximations.

*(English)*Zbl 1311.74019Summary: In contrast to the partial differential equation in the classical continuum mechanics, the equation of motion in standard state-based peridynamics utilizes an integral form and follows an anti-symmetric relationship for the pairwise particle forces. As a consequence, the kinematic constraints such as the boundary displacements and the coupling with other numerical methods in state-based peridynamics cannot be prescribed directly on the geometric boundary for solid mechanics applications. In this paper, an enhanced variant of the state-based peridynamics for the numerical simulation of continuum mechanics problems is presented. The method is first devised based on a convex kernel approximation to localize the influence function on the boundary. A mixed local/nonlocal gradient approximation is introduced to the computation of particle equation of motion and allows a direct imposition of kinematic constraint in the analysis model. The new formulation is shown to retain the conservation nature of state-based peridynamics. Three numerical benchmarks are studied in this paper to demonstrate the effectiveness and accuracy of the proposed method.

##### MSC:

74B05 | Classical linear elasticity |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

35Q74 | PDEs in connection with mechanics of deformable solids |

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##### References:

[1] | Askari, E; Bobaru, F; Lehoucq, RB; Parks, ML; Silling, SA; Weckner, O, Peridynamics for multi-scale materials modelling, J Phys, 125, 012078, (2008) |

[2] | Bazant, Z; Jirasek, M, Nonlocal integration formulations of plasticity and damage: survey and progress, J Eng Mech, 128, 1119-1149, (2002) |

[3] | Bobaru, F; Duangpanya, M, A peridynamic formulation for transient heat conduction in bodies with evolving discontinuities, J Comput Phys, 231, 2764-2785, (2012) · Zbl 1253.80002 |

[4] | Bobaru, F; Hu, W, The meaning, selection, and use of the peridynamic horizon and its relation to crack branching in brittle materials, Int J Fract, 176, 215-222, (2012) |

[5] | Belytschko, T; Lu, YY; Gu, L, Element-free Galerkin methods, Int J Numer Methods Eng, 37, 229-256, (1994) · Zbl 0796.73077 |

[6] | Belytschko T, Liu WK, Moran B, Elkhodary K (2014) Nonlinear finite elements for continua and structures, 2nd edn. Wiley, United Kingdom · Zbl 1279.74002 |

[7] | Bessa MA, Foster JT, Belytschko T, Liu WK (2014) A meshfree unification: reproducing kernel peridynamics. Comp Mech 53:1251-1264. doi:10.1007/s00466-013-0969-x · Zbl 1398.74452 |

[8] | Bonet, J; Lok, TSL, Variational and momentum preservation aspects of smoothed particle hydrodynamic formulations, Comput Methods Appl Mech Eng, 180, 97-115, (1999) · Zbl 0962.76075 |

[9] | Chen, JS; Pan, C; Wu, CT; Liu, WK, Reproducing kernel particle methods for large deformation analysis of nonlinear structures, Comput Methods Appl Mech Eng, 139, 195-227, (1996) · Zbl 0918.73330 |

[10] | Chen, JS; Wu, CT; Belytschko, T, Regularization of material instabilities by meshfree approximations with intrinsic length scales, Int J Numer Methods Eng, 47, 1303-1322, (2000) · Zbl 0987.74079 |

[11] | Chen, X; Gunzburger, M, Continuous and discontinuous finite element methods for a peridynamics model of mechanics, Comput Methods Appl Mech Eng, 200, 1237-1250, (2011) · Zbl 1225.74082 |

[12] | Ha, YD; Bobaru, F, Studies of dynamic crack propagation and crack branching with peridynamic, Int J Fract, 162, 229-244, (2010) · Zbl 1425.74416 |

[13] | Hao, S; Park, HS; Liu, WK, Moving particle finite element method, Int J Numer Methods Eng, 53, 1937-1958, (2002) · Zbl 1169.74606 |

[14] | Hao, S; Liu, WK; Belytschko, T, Moving particle finite element method with global smoothness, Int J Numer Methods Eng, 59, 1007-1020, (2004) · Zbl 1065.74608 |

[15] | Hu, W; Ha, YD; Bobaru, F, Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites, Comput Methods Appl Mech Eng, 217—-220, 247-261, (2012) · Zbl 1253.74008 |

[16] | Foster, JT; Silling, SA; Chen, WW, Viscoelasticity using peridynamics, Int J Numer Methods Eng, 81, 1242-1258, (2010) · Zbl 1183.74035 |

[17] | Kilic, B; Madenci, E, An adaptive dynamic relaxation method for quasi-static simulations using the peridynamic theory, Theor Appl Fract Mech, 53, 194-204, (2010) |

[18] | Li, S; Liu, WK, Synchronized reproducing kernel interpolant via multiple wavelet expansion, Comp Mech, 21, 28-47, (1998) · Zbl 0912.76057 |

[19] | Li, S; Liu, WK, Reproducing kernel hierarchical partition of unity part I—formulation and theory, Int J Numer Methods Eng, 45, 251-288, (1999) · Zbl 0945.74079 |

[20] | Li S, Liu WK (2004) Meshfree particle method. Springer, Berlin |

[21] | Liu, WK; Jun, S; Zhang, YF, Reproducing kernel particle methods, Int J Numer Methods Fluids, 20, 1081-1106, (1995) · Zbl 0881.76072 |

[22] | Liu, W; Hong, JW, A coupling approach of discretized peridynamics with finite element method, Comput Methods Appl Mech Eng, 245—-246, 163-175, (2012) · Zbl 1354.74284 |

[23] | Oterkus, E; Madenci, E; Weckner, O; Silling, SA; Bogert, P; Tessler, A, Combined finite element and peridynamic analyses for predicting failure in a stiffened composite curved panel with a central slot, Compos Struct, 94, 839-850, (2012) |

[24] | Park, ML; Lehoucq, RB; Plimpton, S; Silling, SA, Implementing peridynamics within a molecular dynamics code, Comput Phys Commun, 179, 777-783, (2008) · Zbl 1197.82014 |

[25] | Park, CK; Wu, CT; Kan, CD, On the analysis of dispersion property and stable time step in meshfree method using the generalized meshfree approximation, Finite Elem Anal Des, 47, 683-697, (2011) |

[26] | Pijaudier-Cabot G, Bazant ZP (1987) Nonlocal damage theory. J Eng Mech 113:1512-1533 · Zbl 1220.82074 |

[27] | Seleson, P; Parks, ML; Gunzburger, M; Lehoucq, RB, Peridynamics as an upscaling of molecular dynamics, Multiscale Model Simul, 8, 204-227, (2010) · Zbl 1375.82073 |

[28] | Seleson, P; Beneddine, S; Prudhomme, S, A force-based coupling scheme for peridynamics and class elasticity, Comput Mater Sci, 66, 34-49, (2013) |

[29] | Silling, SA, Reformulation of elasticity theory for discontinuities and long-range forces, J Mech Phys Solids, 48, 175-209, (2000) · Zbl 0970.74030 |

[30] | Silling, SA; Askari, E, A meshfree method based on the peridynamic model of solid mechanics, Comp Struct, 83, 1526-1535, (2005) |

[31] | Silling, SA; Epton, M; Weckner, O; Xu, J; Askari, E, Peridynamic states and constitutive modelling, J Elast, 88, 151-184, (2007) · Zbl 1120.74003 |

[32] | Silling, SA; Lehoucq, R, Peridynamic theory of solid mechanics, Adv Appl Mech, 44, 73-190, (2010) |

[33] | Silling, SA, A coarsening method for linear peridynamics, Int J Multiscale Comput Eng, 9, 609-621, (2011) |

[34] | Tupek, MR; Rimoli, JJ; Radovitzky, R, An approach for incorporating classical continuum damage models in state-based peridynamics, Comput Methods Appl Mech Eng, 263, 20-26, (2013) · Zbl 1286.74022 |

[35] | Wang, HP; Wu, CT; Guo, Y; Botkin, ME, A coupled meshfree/finite element method for automotive crashworthiness simulations, Int J Impact Eng, 36, 1210-1222, (2009) |

[36] | Warren, TL; Siling, SA; Askari, E; Weckner, O; Epton, MA; Xu, J, A non-ordinary state-based peridynamic method to model solid material deformation and fracture, Int J Solids Struct, 46, 1186-1195, (2009) · Zbl 1236.74012 |

[37] | Weckner, O; Mohamed, NAN, Viscoelastic material models in peridynamics, Appl Math Comput, 219, 6039-6043, (2013) · Zbl 1273.74038 |

[38] | Wu, CT; Koish, M, A meshfree procedure for the microscopic analysis of particle-reinforced rubber compounds, Interact Multiscale Mech, 2, 147-169, (2009) |

[39] | Wu, CT; Hu, W, Meshfree-enriched simplex elements with strain smoothing for the finite element analysis of compressible and nearly incompressible solids, Comput Methods Appl Mech Eng, 200, 2991-3010, (2011) · Zbl 1230.74201 |

[40] | Wu, CT; Park, CK; Chen, JS, A generalized approximation for the meshfree analysis of solids, Int J Numer Methods Eng, 85, 693-722, (2011) · Zbl 1217.74150 |

[41] | Wu CT, Koishi M (2012) Three-dimensional meshfree-enriched finite element formulation for micromechanical hyperelastic modelling of particulate rubber composites. Int J Numer Methods Eng 91:1137-1157 |

[42] | Wu, CT; Hu, W; Chen, JS, Meshfree-enriched finite element methods for the compressible and near-incompressible elasticity, Int J Numer Methods Eng, 90, 882-914, (2012) · Zbl 1242.74174 |

[43] | Wu, CT; Guo, Y; Askari, E, Numerical modelling of composite solids using an immersed meshfree Galerkin method, Composit B, 45, 1397-1413, (2013) |

[44] | Wu, CT; Hu, W, Multi-scale finite element analysis of acoustic waves using global residual-free meshfree enrichments, Interact Multiscale Mech, 6, 83-105, (2013) |

[45] | Zhou, K; Du, Q, Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions, SIAM J Numer Anal, 48, 1759-1780, (2011) · Zbl 1220.82074 |

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