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The generation of non-ordinary state-based peridynamics by the weak form of the peridynamic method. (English) Zbl 07272700
Summary: A weak form of the peridynamic (PD) method derived from the classical Galerkin framework by substituting the traditional derivatives into the PD differential operators is proposed. The attractive features of the proposed weak form of PD method include the following: (1) a higher-order approximation than the non-ordinary state-based peridynamic (NOSB-PD) in the strain construction; (2) the NOSB-PD is demonstrated as a special case of the weak form of the PD method; (3) as an extension of the NOSB-PD, the zero-energy mode oscillations in the weak form of the PD can be significantly reduced by introducing higher-order PD derivatives. In addition, a series of numerical tests are conducted. The results show the following: (1) the three proposed stabilization items containing higher-order PD derivatives have a better accuracy and stability than the traditional items of the NOSB-PD. In particular, the stress point stabilization item is preferred since it has the highest accuracy and efficiency and does not introduce any additional parameters; (2) the weak form of PD method is very suitable in dealing with the crack propagation and bifurcation problems.
MSC:
74 Mechanics of deformable solids
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[1] 1. Zienkiewicz, OC , et al. The finite element method: solid mechanics. Vol. 2. London: Butterworth-heinemann, 2000. · Zbl 0991.74003
[2] 2. Moës, N, Gravouil, A, Belytschko, T. Non-planar 3D crack growth by the extended finite element and level sets—Part I: mechanical model. Int J Numer Meth Eng 2002; 53: 2549-2568. · Zbl 1169.74621
[3] 3. Shi, G-H. Manifold method of material analysis. Army Research Office Research Triangle Park NC, 1992.
[4] 4. Yang, Y, Zheng, H. A three-node triangular element fitted to numerical manifold method with continuous nodal stress for crack analysis. Eng Fract Mech 2016; 162: 51-75.
[5] 5. Zheng, H, Yang, YT, Shi, GH. Reformulation of dynamic crack propagation using the numerical manifold method. Eng Anal Bound Elem 2019; 105: 279-295. · Zbl 07063077
[6] 6. Ren, HL , et al. An explicit phase field method for brittle dynamic fracture. Comput Struct 2019; 217: 45-56.
[7] 7. Silling, SA. Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solid 2000; 48: 175-209. · Zbl 0970.74030
[8] 8. Gerstle, WH. Introduction to practical peridynamics: computational solid mechanics without stress and strain. Singapore: World Scientific, 2015.
[9] 9. Silling, SA , et al. Peridynamic states and constitutive modeling. J Elast 2007; 88: 151-184. · Zbl 1120.74003
[10] 10. Tupek, M, Rimoli, J, Radovitzky, R. An approach for incorporating classical continuum damage models in state-based peridynamics. Comput Meth Appl Mech Eng 2013; 263: 20-26. · Zbl 1286.74022
[11] 11. Wang, Y, Zhou, X, Xu, X. Numerical simulation of propagation and coalescence of flaws in rock materials under compressive loads using the extended non-ordinary state-based peridynamics. Eng Fract Mech 2016; 163: 248-273.
[12] 12. Rahaman, MM , et al. A peridynamic model for plasticity: micro-inertia based flow rule, entropy equivalence and localization residuals. Comput Meth Appl Mech Eng 2017; 327: 369-391. · Zbl 1439.74075
[13] 13. Wu, CT, Ren, B. A stabilized non-ordinary state-based peridynamics for the nonlocal ductile material failure analysis in metal machining process. Comput Meth Appl Mech Eng 2015; 291: 197-215. · Zbl 1423.74067
[14] 14. Gu, X, Zhang, Q, Madenci, E. Non-ordinary state-based peridynamic simulation of elastoplastic deformation and dynamic cracking of polycrystal. Eng Fract Mech 2019; 218: 106568.
[15] 15. Chorzepa, MG, Yaghoobi, A. Innovative meshless computational method for the analysis of fiber-reinforced concrete (FRC) structures. In: Geotechnical and Structural Engineering Congress 2016, 2016.
[16] 16. Shou, YD, Zhou, XP, Berto, F. 3D numerical simulation of initiation, propagation and coalescence of cracks using the extended non-ordinary state-based peridynamics. Theor Appl Fract Mech 2019; 101: 254-268.
[17] 17. Ren, HL , et al. Dual-horizon peridynamics. Int J Numer Meth Eng 2016; 108: 1451-1476.
[18] 18. Javili, A , et al. Peridynamics review. Math Mech Solid 2019; 24: 3714-3739.
[19] 19. Silling, SA. Stability of peridynamic correspondence material models and their particle discretizations. Comput Meth Appl Mech Eng 2017; 322: 42-57. · Zbl 1439.74017
[20] 20. Littlewood, DJ . Simulation of dynamic fracture using peridynamics, finite element modeling, and contact. In: ASME 2010 International Mechanical Engineering Congress and Exposition, 2010.
[21] 21. Breitenfeld, MS , et al. Non-ordinary state-based peridynamic analysis of stationary crack problems. Comput Meth Appl Mech Eng 2014; 272: 233-250. · Zbl 1296.74099
[22] 22. Li, P, Hao, ZM, Zhen, WQ. A stabilized non-ordinary state-based peridynamic model. Comput Meth Appl Mech Eng 2018; 339: 262-280. · Zbl 1440.74028
[23] 23. Wan, J , et al. Improved method for zero-energy mode suppression in peridynamic correspondence model. Acta Mechanica Sinica 2019; 35: 1021-1032.
[24] 24. Yaghoobi, A, Chorzepa, MG. Higher-order approximation to suppress the zero-energy mode in non-ordinary state-based peridynamics. Comput Struct 2017; 188: 63-79.
[25] 25. Javili, A, McBride, AT, Steinmann, P. Continuum-kinematics-inspired peridynamics. Mechanical problems. J Mech Phys Solid 2019; 131: 125-146.
[26] 26. Luo, J, Sundararaghavan, V. Stress-point method for stabilizing zero-energy modes in non-ordinary state-based peridynamics. Int J Solid Struct 2018; 150: 197-207.
[27] 27. Bessa, MA , et al. A meshfree unification: reproducing kernel peridynamics. Computat Mech 2014; 53: 1251-1264. · Zbl 1398.74452
[28] 28. Ganzenmüller, GC, Hiermaier, S, May, M. On the similarity of meshless discretizations of peridynamics and smooth-particle hydrodynamics. Comput Struct 2015; 150: 71-78.
[29] 29. Gu, X , et al. Possible causes of numerical oscillations in non-ordinary state-based peridynamics and a bond-associated higher-order stabilized model. Comput Meth Appl Mech Eng 2019; 357: 112592. · Zbl 1442.74031
[30] 30. Dyka, CT, Ingel, RP. An approach for tension instability in smoothed particle hydrodynamics (SPH). Comput Struct 1995; 57: 573-580. · Zbl 0900.73945
[31] 31. Randles, PW, Libersky, LD. Normalized SPH with stress points. Int J Numer Meth Eng 2000; 48: 1445-1462. · Zbl 0963.74079
[32] 32. Rabczuk, T, Belytschko, T, Xiao, SP. Stable particle methods based on Lagrangian kernels. Comput Meth Appl Mech Eng 2004; 193: 1035-1063. · Zbl 1060.74672
[33] 33. Belytschko, SBT. Nodal integration of the element-free Galerkin method. Comput Meth Appl Mech Eng 1996; 139: 49-74. · Zbl 0918.73329
[34] 34. Belytschko, T, Yong, G, Liu, WK. A unified stability analysis of meshless particle methods. Int J Numer Meth Eng 2000; 48: 1359-1400. · Zbl 0972.74078
[35] 35. Nagashima, T. Node-by-node meshless approach and its applications to structural analyses. Int J Numer Meth Eng 1999; 46: 341-385. · Zbl 0965.74079
[36] 36. Liu, GR , et al. A nodal integration technique for meshfree radial point interpolation method (NI-RPIM). Int J Solid Struct 2007; 44: 3840-3860. · Zbl 1135.74050
[37] 37. Madenci, E, Barut, A, Futch, M. Peridynamic differential operator and its applications. Comput Meth Appl Mech Eng 2016; 304: 408-451. · Zbl 1425.74043
[38] 38. Madenci, E , et al. Numerical solution of linear and nonlinear partial differential equations using the peridynamic differential operator. Numer Meth Partial Diff Eq 2017; 33: 1726-1753. · Zbl 1375.65124
[39] 39. Madenci, E , et al. Weak form of peridynamics for nonlocal essential and natural boundary conditions. Comput Meth Appl Mech Eng 2018; 337: 598-631. · Zbl 1440.74030
[40] 40. Madenci, E, Barut, A, Dorduncu, M. Peridynamic differential operator for numerical analysis. Berlin: Springer, 2019. · Zbl 1375.65124
[41] 41. Wang, H, Oterkus, E, Oterkus, S. Predicting fracture evolution during lithiation process using peridynamics. Eng Fract Mech 2018; 192: 176-191.
[42] 42. Dorduncu, M. Stress analysis of laminated composite beams using refined zigzag theory and peridynamic differential operator. Compos Struct 2019; 218: 193-203.
[43] 43. Rabczuk, T, Ren, HL, Zhuang, XY. A nonlocal operator method for partial differential equations with application to electromagnetic waveguide problem. Comput Mater Continua 2019; 59: 31-55.
[44] 44. Gu, X, Madenci, E, Zhang, Q. Revisit of non-ordinary state-based peridynamics. Eng Fract Mech 2018; 190: 31-52.
[45] 45. Sun, G , et al. The virtual element method strength reduction technique for the stability analysis of stony soil slopes. Comput Geotechn 2020; 119: 103349.
[46] 46. Ferziger, JH, Perić, M. Computational methods for fluid dynamics. Vol. 3. Berlin: Springer, 2002. · Zbl 0998.76001
[47] 47. Timoshenko, SP, Goodier, JN. Theory of elasticity. New York: McGraw-Hill, 1970.
[48] 48. Zhang, ZN, Chen, YQ. Numerical simulation for fracture propagation of multi-cracked rock materials using virtual multidimensional internal bonds. Chin J Geotechn Eng 2008; 319: 516-516.
[49] 49. Ni, T , et al. Peridynamic simulation of fracture in quasi brittle solids using irregular finite element mesh. Eng Fract Mech 2018; 188: 320-343.
[50] 50. Zhang, HH , et al. Numerical analysis of 2-D crack propagation problems using the numerical manifold method. Eng Anal Bound Elem 2010; 34: 41-50. · Zbl 1244.74238
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