A strain-morphed nonlocal meshfree method for the regularized particle simulation of elastic-damage induced strain localization problems.

*(English)*Zbl 1336.74007Summary: In this work, a strain-morphed nonlocal meshfree method is developed to overcome the computational challenges for the simulation of elastic-damage induced strain localization problem when the spatial domain integration is performed based on the background cells and Gaussian quadrature rule. The new method is established by introducing the decomposed strain fields from a meshfree strain smoothing to the penalized variational formulation. While the stabilization strain field circumvents the onerous zero-energy modes inherent in the direct nodal integration scheme, the regularization strain field aims to avoid the pathological localization of deformation in Galerkin meshfree solution using the weak-discontinuity approach. A strain morphing algorithm is introduced to couple the locality and non-locality of the decomposed strain approximations such that the continuity condition in the coupled strain field is met under the Galerkin meshfree framework using the direct nodal integration scheme. Three
numerical benchmarks are examined to demonstrate the effectiveness and accuracy of the proposed method for the regularization of elastic-damage induced strain localization problems.

##### MSC:

74A45 | Theories of fracture and damage |

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\textit{C. T. Wu} et al., Comput. Mech. 56, No. 6, 1039--1054 (2015; Zbl 1336.74007)

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[1] | Bažant, ZP; Belytschko, T; Chang, TP, Continuum theory for strain soften, J Eng Mech, 110, 1666-1692, (1984) |

[2] | Bažant ZP, Planas J (1998) Fracture and size effect in concrete and other quasibrittle materials. CRC Press, Boca Raton |

[3] | Beissel, S; Belytschko, T, Nodal integration of the element-free Galerkin method, Comput Methods Appl Mech Eng, 139, 49-74, (1996) · Zbl 0918.73329 |

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

[5] | Belytschko, T; Guo, Y; Liu, WK; Xiao, SP, A unified stability analysis of meshless particle methods, Int J Numer Methods Eng, 48, 1359-1400, (2000) · Zbl 0972.74078 |

[6] | Cardoso, RPR; Yoon, JW; CrJcio, JJ; Barlat, F, J.M.A. C.M.A. atrd, development of a one point quadrature shell element for nonlinear applications with contact and anisotropy, Comput Methods Appl Mech Eng, 191, 5177-5206, (2002) · Zbl 1083.74583 |

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

[8] | 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 |

[9] | Chen, JS; Wu, CT; Yoon, S; You, Y, A stabilized conforming nodal integration for Galerkin meshfree methods, Int J Numer Methods Eng, 50, 435-466, (2001) · Zbl 1011.74081 |

[10] | Chen, JS; Zhang, X; Belytschko, T, An implicit gradient model by a reproducing kernel strain regularization in strain localization problems, Comput Methods Appl Mech Eng, 193, 2827-2844, (2004) · Zbl 1067.74564 |

[11] | Chen, JS; Hillman, M; Rüter, M, An arbitrary order variationally consistent integration for Galerkin meshfree methods, Int J Numer Methods Eng, 95, 361-450, (2013) · Zbl 1352.65481 |

[12] | Borst, R; Muhlhaus, HB, Gradient-dependent plasticity: formulation and algorithm aspects, Int J Numer Methods Eng, 35, 521-539, (1992) · Zbl 0768.73019 |

[13] | Dyka, CT; Randles, PW; Ingel, RP, Stress points for tension instability in SPH, Int J Numer Methods Eng, 40, 2325-2341, (1997) · Zbl 0890.73077 |

[14] | Hillman, M; Chen, JS; Chi, SW, Stabilized and variationally consistent nodal integration for meshfree modeling of impact problems, Comput Part Mech, 1, 245-256, (2014) |

[15] | Hill, R, Acceleration waves in solids, J Mech Phys Solids, 10, 1-16, (1962) · Zbl 0111.37701 |

[16] | Hill, R, General theory of uniqueness and stability in elastic-plastic solids, J Mech Phys Solids, 6, 236-249, (1958) · Zbl 0091.40301 |

[17] | Lasry, D; Belytschko, T, Localization limiters in transient problems, Int J Solids Struct, 23, 581-597, (1988) · Zbl 0636.73021 |

[18] | Li, S; Liu, WK, Numerical simulations of strain localization in inelastic solids using mesh-free methods, Int J Numer Methods Eng, 48, 1285-1309, (2000) · Zbl 1052.74618 |

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

[20] | Lian, YP; Yang, PF; Zhang, X; Zhang, F; Liu, Y; Huang, P, A mesh-grading material point method and its parallelization for problems with localized extreme deformation, Comput Methods Appl Mech Eng, 289, 291-315, (2015) |

[21] | Lian, YP; Zhang, X; Zhang, F; Cui, XX, Tied interface grid material point method for problems with localized extreme deformation, Int J Impact Eng, 70, 50-61, (2014) |

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

[23] | Peerlings, RHJ; Borst, R; Brekelmans, WAM; Geers, MGD, Localization issues in local and nonlocal continuum approaches to fracture, Eur J Mech A, 21, 175-189, (2002) · Zbl 1041.74006 |

[24] | Rabczuk, T; Belytschko, T; Xiao, SP, Stable particle methods based on Lagrangian kernels, Comput Methods Appl Mech Eng, 193, 1035-1063, (2004) · Zbl 1060.74672 |

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

[26] | Simo, JC; Hughes, TJR, On the variational foundation of assumed strain methods, ASME J Appl Mech, 53, 51-54, (1986) · Zbl 0592.73019 |

[27] | Sukumar, N, Construction of polygonal interpolants: a maximum entropy approach, Int J Numer Methods Eng, 61, 2159-2181, (2004) · Zbl 1073.65505 |

[28] | Wang, DD; Peng, H, A Hermite reproducing kernel Galerkin meshfree approach for buckling analysis on thin plates, Comput Mech, 51, 1013-1029, (2013) · Zbl 1366.74023 |

[29] | Wang, DD; Li, Z, A two-level strain smoothing regularized meshfree approach with stabilized conforming nodal integration for elastic damage analysis, Int J Damage Mech, 22, 440-459, (2013) |

[30] | Wang, DD; Li, L; Li, Z, A regularized Lagrangian meshfree method for rainfall infiltration triggered slope failure analysis, Eng Anal Bound Elem, 42, 51-59, (2014) · Zbl 1297.76172 |

[31] | Wang, DD; Chen, P, Quasi-convex reproducing kernel meshfree method, Comput Mech, 54, 689-709, (2014) · Zbl 1311.65152 |

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

[33] | 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 |

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

[35] | Wu CT, Guo Y, Hu W (2014) 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. 8-10 June, pp 1-20 · Zbl 0091.40301 |

[36] | Wu, CT; Ren, B, A stabilized non-ordinary state-based peridynamics for the nonlocal ductile material failure analysis in metal machining process, Comput Methods Appl Mech Eng, 291, 197-215, (2015) |

[37] | Wu, CT; Koishi, M; Hu, W, A displacement smoothing induced strain gradient stabilization for the meshfree Galerkin nodal integration method, Comput Mech, 56, 19-37, (2015) · Zbl 1329.74292 |

[38] | Wu, YC; Wang, DD; Wu, CT, Three dimensional fragmentation simulation of concrete structures with a nodally regularized meshfree method, Theor Appl Fract Mech, 27, 89-99, (2014) |

[39] | Wu YC, Wang DD, Wu CT (2015) A direct displacement smoothing meshfree particle formulation for impact failure modeling. Int J Impact Eng. doi:10.1016/j.ijimpeng.2015.03.013 |

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