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An Implicit Coupling Finite Element and Peridynamic Method for Dynamic Problems of Solid Mechanics with Crack Propagation

International Research Center for Computational Mechanics, State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116023, P. R. China

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Hongwu Zhang

E-mail Address: zhanghw@dlut.edu.cn

Corresponding authors.

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Yonggang Zheng

E-mail Address: zhengyg@dlut.edu.cn

Corresponding authors.

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Hongfei Ye

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, and

Mengkai Lu

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https://doi.org/10.1142/S1758825118500370Cited by:22 (Source: Crossref)

Abstract

An implicit coupling finite element and peridynamic (PD) method is developed in this paper for the dynamic problems of solid mechanics with crack propagation. In this method, an implicit PD formulation is derived from the bond-based pairwise force that is described as a linear function of the displacements by using the first-order Taylor’s expansion technique. The equivalent incremental equations of the PD method and the finite element method are obtained on the basis of the Newmark and the Newton–Raphson schemes. To combine these two methods, the system is partitioned into two subregions and a convenient and efficient coupling strategy is proposed to form the coupling equivalent equation. The coupling domain is achieved by considering that the nodes and material points share the common information. Furthermore, displacement and load control-based incremental-iterative methods are adopted to solve the nonlinear equations. Several representative numerical examples are carried out and the results demonstrate the effectiveness and accuracy of the proposed coupling method.

Keywords:

References

Aksoy, H. G. and Senocak, E. [2011] “ Discontinuous Galerkin method based on peridynamic theory for linear elasticity,” International Journal for Numerical Methods in Engineering 88, 673–692. Web of Science, Google Scholar
Bathe, K. J. [1996] Finite Element Procedures (Prentice-Hall, Englewood Cliffs, NJ). Google Scholar
Belytschko, T. and Black, T. [1999] “ Elastic crack growth in finite elements with minimal remeshing,” International Journal for Numerical Methods in Engineering 45(5), 601–620. Web of Science, Google Scholar
Belytschko, T. and Tabbara, M. [1996] “ Dynamic fracture using element-free Galerkin methods,” International Journal for Numerical Methods in Engineering 39, 923–938. Web of Science, Google Scholar
Bobaru, F. and Duangpanya, M. [2010] “ The peridynamic formulation for transient heat conduction,” International Journal of Heat and Mass Tranfer 53, 4047–4059. Web of Science, Google Scholar
Bobaru, F. and Duangpanya, M. [2012] “ A peridynamic formulation for transient heat conduction in bodies with evolving discontinuities,” Journal of Computational Physics 231, 2764–2785. Web of Science, Google Scholar
Crisfield, M. A. [1991] Non-linear Finite Element Analysis of Solids and Structures (John Wiley & Sons, Chichester, UK). Google Scholar
Dolbow, J., Moës, T. and Belytschko, T. [2000] “ Discontinuous enrichment in finite elements with a partition of unity method,” Finite Elements in Analysis and Design 36(3–4), 235–260. Web of Science, Google Scholar
Foster, C. D., Borja, R. I. and Regueiro, R. A. [2007] “ Embedded strong discontinuity finite elements for fractured geomaterials with variable friction,” International Journal for Numerical Methods in Engineering 72, 549–581. Web of Science, Google Scholar
Galvanetto, U., Mudric, T., Shojaei, A. and Zaccariotto, M. [2016] “ An effective way to couple FEM meshes and Peridynamics grids for the solution of static equilibrium problems,” Mechanics Research Communications 76, 41–47. Web of Science, Google Scholar
Gerstle, W., Sau, N. and Silling, S. A. [2007] “ Peridynamic modeling of concrete structures,” Nuclear Engineering and Design 237, 1250–1258. Web of Science, Google Scholar
Han, F., Lubineau, G., Azdoud, Y. and Askari, A. [2006] “ A morphing approach to couple state-based peridynamics with classical continnum mechanics,” Computer Methods in Applied Mechanics and Engineering 301, 336–358. Web of Science, Google Scholar
Han, F., Lubineau, G. and Azdoud, Y. [2016] “ Adaptive coupling between damage mechanics and peridynamics: A route for objective simulation of material degradation up to complete failure,” Journal of the Mechanics and Physics in Solids 94, 453–472. Web of Science, Google Scholar
Han, F. and Lubineau, G. [2012] “ Coupling of nonlocal and local continnum models by the Arlequin approach,” International Journal for Numerical Methods in Engineering 89(6), 671–685. Web of Science, Google Scholar
Hu, W., Ha, Y. D. and Bobaru, F. [2012] “ Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites,” Computer Methods in Applied Mechanics and Engineering 217–220, 247–261. Web of Science, Google Scholar
Huang, D., Zhang, Q. and Qiao, P. Z. [2011] “ Damage and progressive failure of concrete structures using non-local peridynamic modeling,” Science China Technological Sciences 54(3), 591–596. Web of Science, Google Scholar
Huang, D., Lu, G. D. and Qiao, P. Z. [2015] “ An improved peridynamic approach for quasi-static elastic deformation and brittle fracture analysis,” International Journal of Mechanical Sciences 94–95, 111–122. Web of Science, Google Scholar
Kalthoff, J. F. and Winkler, S. [1987] “ Failure mode transition at high rates of shear loading,” International Conference on Impact Loading and Dynamic Behavior of Materials 1, 185–195. Google Scholar
Katiyar, A., Foster, J. T., Ouchi, H. and Sharma, M. M. [2014] “ A peridynamic formulation of pressure driven convective fluid transport in porous media,” Journal of Computational Physics 261, 209–229. Web of Science, Google Scholar
Lai, X., Ren, B., Fan, H. F., Li, S. F., Wu, C. T., Regueiro, R. A. and Liu, L. S. [2015] “ Peridynamics simulations of geomaterial fragmentation by impulse loads,” International Journal for Numerical and Analytical Methods in Geomechanics 39(12), 1304–1330. Web of Science, Google Scholar
Leon, S. E., Paulino, G. H., Pereira, A., Menezes, I. F. M. and Lages, E. N. [2012] “ A unified library of nonlinear solution schemes,” Applied Mechanics Reviews 64(4) 040803. Web of Science, Google Scholar
Li, H., Zhang, H. W., Zheng, Y. G. and Zhang, L. [2016] “ A peridynamic model for the nonlinear static analysis of truss and tensegrity structures,” Computational Mechanics 57, 843–858. Web of Science, Google Scholar
Linder, C. and Armero, F. [2007] “ Finite elements with embedded strong discontinuities for the modeling of failure in solids,” International Journal for Numerical Methods in Engineering 72, 1391–1433. Web of Science, Google Scholar
Liu, W. Y. and Hong, J. W. [2012] “ A coupling approach of discretized peridynamics with finite element method,” Computer Methods in Applied Mechanics and Engineering 245–246, 163–175. Web of Science, Google Scholar
Lu, M. K., Zhang, H. W., Zheng, Y. G. and Zhang, L. [2016] “ A multiscale finite element method with embedded strong discontinuity model for the simulation of cohesive cracks in solids,” Computer Methods in Applied Mechanics and Engineering 311, 576–598. Web of Science, Google Scholar
Macek, R. W. and Silling, S. A. [2007] “ Peridynamics via finite element analysis,” Finite Elements in Analysis and Design 43, 1169–1178. Web of Science, Google Scholar
Miehe, C., Hofacker, M. and Welschinger, F. [2010a] “ A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits,” Computer Methods in Applied Mechanics and Engineering 199, 2765–2778. Web of Science, Google Scholar
Miehe, C., Welschinger, F. and Hofacker, M. [2010b] “ Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations,” International Journal for Numerical Methods in Engineering 83(10), 1273–1311. Web of Science, Google Scholar
Oterkus, S., Madenci, E. and Agwai, A. [2014] “ Peridynamic thermal diffusion,” Journal of Computational Physics 265, 71–96. Web of Science, Google Scholar
Ouchi, H., Katiyar, A., York, J., Foster, J. T. and Sharma, M. M. [2015] “ A fully coupled porous flow and geomechanics model for fluid driven cracks: a peridynamic approach,” Computational Mechanics 55, 561–576. Web of Science, Google Scholar
Rabczuk, R., Xiao, S. P. and Sauer, A. [2006] “ Coupling of mesh-free methods with finite elements: basic concepts and test results,” International Journal for Numerical Methods in Biomedical Engineering 22(10), 1031–1065. Google Scholar
Seleson, P., Beneddine, S. and Prudhomme, S. [2013] “ A force-based coupling scheme for peridynamics and classical elasticity,” Computational Material Science 66, 34–49. Web of Science, Google Scholar
Shojaei, A., Boroomand, B. and Mossaiby, F. [2015] “ A simple meshless method for challenging engineering problems,” Engineering Computations 32, 1567–1600. Web of Science, Google Scholar
Silling, S. A., Epton, M., Wechner, O., Xu, J. and Askari, E. [2007] “ Peridynamic states and constitutive modeling,” Journal of Elasticity 88, 151–184. Web of Science, Google Scholar
Silling, S. A. and Askari, E. [2005] “ A meshfree method based on the peridynamic model of solid mechanics,” Computers & Structures 83(17–18), 1526–1535. Web of Science, Google Scholar
Silling, S. A. and Lehoucq, R. B. [2010] “ Peridynamic theory of solid mechanics,” Advances in Applied Mechanics 44, 73–168. Web of Science, Google Scholar
Silling, S. A. [2000] “ Reformulation of elasticity theory for discontinuities and long-range forces,” Journal of the Mechanics and Physics of Solids 48, 175–209. Web of Science, Google Scholar
Silling, S. A. [2010] “ Linearized theory of peridynamic states,” Journal of Elasticity 99, 85–111. Web of Science, Google Scholar
Simo, J. C. and Rifai, M. S. [1990] “ A class of mixed assumed strain methods and the method of incompatible modes,” International Journal for Numerical Methods in Engineering 29, 1595–1638. Web of Science, Google Scholar
Strouboulis, T., Copps, K. and Babuska, I. [2001] “ The generalized finite element method,” Computer Methods in Applied Mechanics and Engineering 190, 4081–4193. Web of Science, Google Scholar
Wagner, G. J. and Liu, W. K. [2003] “ Coupling of atomistic and continuum simulations using a bridging scale decomposition,” Journal of Computational Physics 190, 249–274. Web of Science, Google Scholar
Xu, J. F., Askari, A., Weckner, O. and Silling, S. A. [2008] “ Peridynamic analysis of impact damage in composite laminates,” Journal of Aerospace Engineering 21(3), 187–194. Web of Science, Google Scholar