Application of the mixed formulation method to eliminate shear-locking phenomenon in the Peridynamic Mindlin plate model
Springer Science and Business Media LLC - Trang 1-16 - 2023
Tóm tắt
Accurately predicting crack propagation in structures remains a key challenge within the finite element method (FEM) framework. Despite developing the local method to mitigate the challenges encountered in the FEM, this method is computationally demanding and inadequate for accurately modeling fracture processes in actual structures. Recently, a powerful nonlocal method, Peridynamics (PD), has been proposed by employing integral equations rather than differential equations to address various discontinuous problems. This study introduces the classical Peridynamic Mindlin plate theory (classical PD) to characterize the kinematics of thick plates. However, the application of the classical PD to extremely thin plate structures results in a shear-locking phenomenon, causing inaccuracies in the solutions. Although the reduced integration with a single-point rule provides an alternative solution to address the shear-locking problem, its sensitivity to the point number in the horizon becomes especially pronounced in the case of discontinuities. To ensure stability and broad applicability, this paper presents a more general mixed formulation method (mixed PD) that can yield accurate results ranging from very thick to thin plate configurations. The efficacy of the mixed PD is demonstrated via several numerical cases.
Tài liệu tham khảo
Onyibo EC, Safaei B (2022) Application of finite element analysis to honeycomb sandwich structures: a review. Rep Mech Eng 3(1):192–209. https://doi.org/10.31181/rme20023032022o
Jafari A, Broumand P, Vahab M, Khalili N (2022) An extended finite element method implementation in comsol multiphysics: solid mechanics. Finite Elem Anal Des 202:103707. https://doi.org/10.1016/j.finel.2021.103707
Wang Y, Gao F, Cui J (2022) A conforming discontinuous Galerkin finite element method for elliptic interface problems. J Comput Appl Math 412:114304. https://doi.org/10.1016/j.cam.2022.114304
Ahn C, Nishizawa Y, Choi W (2020) A finite element method to simulate dislocation stress: a general numerical solution for inclusion problems. AIP Adv 10(1):015111. https://doi.org/10.1063/1.5121149
Borzabadi Farahani E, Sobhani Aragh B, Voges J, Juhre D (2021) On the crack onset and growth in martensitic micro-structures; a phase-field approach. Int J Mech Sci 194:106187. https://doi.org/10.1016/j.ijmecsci.2020.106187
Cockburn B, Karniadakis GE, Shu C-W (2012) Discontinuous Galerkin methods: theory, computation and applications, vol 11. Springer, Berlin. https://doi.org/10.1007/978-3-642-59721-3
BaniHani SM, Al-Oqla FM, Hayajneh M, Mutawe S, Almomani T (2022) A new approach for dynamic crack propagation modeling based on meshless Galerkin method and visibility based criterion. Appl Math Model 107:1–19. https://doi.org/10.1016/j.apm.2022.02.010
Liu GR, Dai KY, Nguyen THOIT (2007) A smoothed finite element method for mechanics problems. Comput Mech 39:859–877. https://doi.org/10.1007/s00466-006-0075-4
Long T, Huang C, Dean H, Liu M (2021) Coupling edge-based smoothed finite element method with smoothed particle hydrodynamics for fluid structure interaction problems. Ocean Eng 225:108772. https://doi.org/10.1016/j.engfracmech.2020.107476
Elices MGGV, Guinea GV, Gomez J, Planas J (2002) The cohesive zone model: advantages, limitations and challenges. Eng Fract Mech 69(2):137–163. https://doi.org/10.1016/s0013-7944(01)00083-2
Agathos K, Bordas SPA, Chatzi E (2019) Improving the conditioning of XFEM/GFEM for fracture mechanics problems through enrichment quasi-orthogonalization. Comput Methods Appl Mech Eng 346:1051–1073. https://doi.org/10.1016/j.cma.2018.08.007
Alder BJ, Wainwright TE (1959) Studies in molecular dynamics. I. General method. J Chem Phys 31(2):459–466. https://doi.org/10.1063/1.1730376
Xiong S, Cao G (2015) Molecular dynamics simulations of mechanical properties of monolayer mos2. Nanotechnology 26(18):185705. https://doi.org/10.1088/0957-4484/26/18/185705
Hospital A, Goi JR, Orozco M, Gelp JL (2015) Molecular dynamics simulations: advances and applications. Adv Appl Bioinform Chem AABC 8:37. https://doi.org/10.2147/AABC.S70333
Wan S, Sinclair RC, Coveney PV (2021) Uncertainty quantification in classical molecular dynamics. Philos Trans R Soc A 379(2197):20200082. https://doi.org/10.5772/intechopen.68507
Hillman M, Pasetto M, Zhou G (2020) Generalized reproducing kernel peridynamics: unification of local and non-local meshfree methods, non-local derivative operations, and an arbitrary-order state-based peridynamic formulation. Comput Part Mech 7:435–469. https://doi.org/10.1007/s40571-019-00266-9
Patnaik S, Sidhardh S, Semperlotti F (2021) Towards a unified approach to nonlocal elasticity via fractional-order mechanics. Int J Mech Sci 189:105992. https://doi.org/10.1016/j.ijmecsci.2020.105992
Patnaik S, Sidhardh S, Semperlotti F (2021) Displacement-driven approach to nonlocal elasticity. arXiv preprint arXiv:2104.05818. https://doi.org/10.1016/j.euromechsol.2021.104434
Voyiadjis GZ (2019) Handbook of nonlocal continuum mechanics for materials and structures. Springer, Berlin. https://doi.org/10.1007/978-3-319-22977-5
Oterkus E, Oterkus S, Madenci E (2021) Peridynamic modeling, numerical techniques, and applications. Elsevier, Berlin. https://doi.org/10.1016/c2019-0-01174-0
Blanc N, Frank X, Radjai F, Mayer-Laigle C, Delenne J-Y (2021) Breakage of flawed particles by peridynamic simulations. Comput Part Mech. https://doi.org/10.1007/s40571-021-00390-5
Bai R, Naceur H, Liang G, Zhao J, Yi J, Li X, Yuan S, Huayan P, Luo J (2023) Alleviation of shear locking in the peridynamic timoshenko beam model using the developed mixed formulation method. Comput Part Mech 10(3):627–643. https://doi.org/10.1007/s40571-022-00517-2
Silling SA, Epton M, Weckner O, Ji X, Askari E (2007) Peridynamic states and constitutive modeling. J Elast 88(2):151–184. https://doi.org/10.1007/s10659-007-9125-1
Silling SA, Lehoucq RB (2010) Peridynamic theory of solid mechanics. Adv Appl Mech 44:73–168. https://doi.org/10.1016/s0065-2156(10)44002-8
Youn Doh Ha and Florin Bobaru (2010) Studies of dynamic crack propagation and crack branching with peridynamics. Int J Fract 162:229–244. https://doi.org/10.1007/s10704-010-9442-4
Giannakeas IN (2020) Peridynamic and finite element coupling strategies for the simulation of brittle fracture. Ph.D. thesis, Brunel University London
Silling SA, Lehoucq RB (2008) Convergence of peridynamics to classical elasticity theory. J Elast 93(1):13. https://doi.org/10.1007/s10659-008-9163-3
Ugur Yolum and Mehmet Ali Güler (2020) On the peridynamic formulation for an orthotropic mindlin plate under bending. Math Mech Solids 25(2):263–287. https://doi.org/10.1177/1081286519873694
Naumenko K, Eremeyev VA (2022) A non-linear direct peridynamics plate theory. Compos Struct 279:114728. https://doi.org/10.1016/j.compstruct.2021.114728
Bazazzadeh S, Zaccariotto M, Galvanetto U (2019) Fatigue degradation strategies to simulate crack propagation using peridynamic based computational methods. Latin Am J Solids Struct. https://doi.org/10.1590/1679-78255022
Giannakeas IN, Papathanasiou TK, Fallah AS, Bahai H (2020) Coupling XFEM and peridynamics for brittle fracture simulation: feasibility and effectiveness. Comput Mech 66(1):103–122. https://doi.org/10.1007/s00466-020-01843-z
Pagani A, Enea M, Carrera E (2022) Quasi-static fracture analysis by coupled three-dimensional peridynamics and high order one-dimensional finite elements based on local elasticity. Int J Numer Methods Eng 123(4):1098–1113. https://doi.org/10.1002/nme.6890
Pagani A, Carrera E (2020) Coupling three-dimensional peridynamics and high-order one-dimensional finite elements based on local elasticity for the linear static analysis of solid beams and thin-walled reinforced structures. Int J Numer Methods Eng 121(22):5066–5081. https://doi.org/10.1002/nme.6510
Diyaroglu C, Oterkus E, Oterkus S, Madenci E (2015) Peridynamics for bending of beams and plates with transverse shear deformation. Int J Solids Struct 69:152–168. https://doi.org/10.1016/j.ijsolstr.2015.04.040
Yang Z, Oterkus E, Nguyen CT, Oterkus S (2019) Implementation of peridynamic beam and plate formulations in finite element framework. Continuum Mech Thermodyn 31:301–315. https://doi.org/10.1007/s00161-018-0684-0
Bai R, Liang G, Naceur H, Zhao J, Yi J, Luo J, Wang L, Huayan P (2023) Locking alleviation technique for the peridynamic Reissner-Mindlin plate model: the developed reduced integration method. Arch Appl Mech 93(3):1167–1188. https://doi.org/10.1007/s00419-022-02320-0
Yang Z, Oterkus E, Oterkus S (2021) A novel peridynamic mindlin plate formulation without limitation on material constants. J Peridyn Nonlocal Model 3:287–306. https://doi.org/10.1007/s42102-021-00050-5
Carpenter N, Belytschko T, Stolarski H (1986) Locking and shear scaling factors in co bending elements. Comput Struct 22(1):39–52. https://doi.org/10.1016/0045-7949(86)90083-0
Hernández E, Vellojin J (2021) A locking-free finite element formulation for a non-uniform linear viscoelastic Timoshenko beam. Comput Math Appl 99:305–322. https://doi.org/10.1016/j.camwa.2021.08.014
Mukherjee S, Prathap G (2001) Analysis of shear locking in Timoshenko beam elements using the function space approach. Commun Numer Methods Eng 17(6):385–393. https://doi.org/10.1002/cnm.413
Ping H, Qingyuan H, Xia Y (2016) Order reduction method for locking free isogeometric analysis of Timoshenko beams. Comput Methods Appl Mech Eng 308:1–22. https://doi.org/10.1016/j.cma.2016.05.010
Bode T, Weißenfels C, Wriggers P (2020) Mixed peridynamic formulations for compressible and incompressible finite deformations. Comput Mech 65:1365–1376. https://doi.org/10.1007/s00466-020-01824-2
Simo JC, Taylor RL, Pister KS (1985) Variational and projection methods for the volume constraint in finite deformation elasto-plasticity. Comput Methods Appl Mech Eng 51(1–3):177–208. https://doi.org/10.1016/0045-7825(85)90033-7
Aldakheel F, Hudobivnik B, Soleimani M, Wessels H, Weissenfels C, Marino M (2022) Current trends and open problems in computational mechanics. Springer, Berlin
Bai R, Naceur H, Liang G, Zhao J, Yi J, Li X, Yuan S, Pu H, Luo J (2022) Alleviation of shear locking in the peridynamic timoshenko beam model using the developed mixed formulation method. Computat Part Mech. https://doi.org/10.1007/s40571-022-00517-2
Oñate E (2013) Structural analysis with the finite element method. Linear statics: volume 2: beams, plates and shells. Springer, Berlin. https://doi.org/10.1007/978-1-4020-8743-1
Bai R, Naceur H, Zhao J, Yi J, Ma J, Pu H, Luo J (2023) Improved numerical integration for locking treatment in the Peridynamic Timoshenko beam model. Eng Comput. https://doi.org/10.1108/EC-07-2022-0442
Ma H, Zhou J, Liang G (2014) Implicit damping iterative algorithm to solve elastoplastic static and dynamic equations. J Appl Math. https://doi.org/10.1155/2014/486171
Zhao M, Li H, Cao S, Xiuli D (2018) An explicit time integration algorithm for linear and non-linear finite element analyses of dynamic and wave problems. Eng Comput. https://doi.org/10.1108/ec-07-2018-0312
Silling SA, Askari E (2005) A meshfree method based on the peridynamic model of solid mechanics. Comput Struct 83(17–18):1526–1535. https://doi.org/10.1016/j.compstruc.2004.11.026
Wang B, Oterkus S, Oterkus E (2020) Determination of horizon size in state-based peridynamics. Continuum Mech Thermodyn. https://doi.org/10.1007/s00161-020-00896-y
Seleson P, Beneddine S, Prudhomme S (2013) A force-based coupling scheme for peridynamics and classical elasticity. Comput Mater Sci 66:34–49. https://doi.org/10.1016/j.commatsci.2012.05.016
Hui W, Qing-Hua Q (2019) Methods of fundamental solutions in solid mechanics. Elsevier, New York. https://doi.org/10.1016/c2018-0-03684-1