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A general anisotropic peridynamic plane model based on micro-beam bond

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Abstract

Peridynamic (PD) theory can overcome the shortcomings of classical continuum mechanics (CCM) in simulating crack initiation and propagation. In this paper, a general anisotropic PD plane model is proposed based on the micro-beam bond. First, the general anisotropic expression of the micromoduli in different directions can be expressed through a certain transformation relation, which can make the bond exhibit general anisotropy. Then, with the use of the Allman interpolation method, the deformations of the bond can be obtained, and the PD strain energy density of the general anisotropic plane model can be expressed. Finally, the general anisotropic PD parameters in the micromoduli expression can be obtained by equating the strain energy densities of PD and CCM models. Numerical examples of static uniaxial tension, static shear, dynamic fracture of the compact tension test and dynamic fracture of the three-hole plate test prove the effectiveness of the proposed model in handling static in-plane problems and dynamic in-plane fracture problems.

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References

  1. Shi D, Xiao X (2017) A new shell-beam element modeling method and its use in crash simulation of triaxial braided composites. Compos Struct 160:792–803

    Article  Google Scholar 

  2. Murotani K, Yagawa G, Choi JB (2013) Adaptive finite elements using hierarchical mesh and its application to crack propagation analysis. Comput Methods Appl Mech Eng 253:1–14

    Article  MathSciNet  MATH  Google Scholar 

  3. Schöllmann M, Fulland M, Richard H (2003) Development of a new software for adaptive crack growth simulations in 3D structures. Eng Fract Mech 70(2):249–268

    Article  Google Scholar 

  4. El Khaoulani R, Bouchard P (2012) An anisotropic mesh adaptation strategy for damage and failure in ductile materials. Finite Elem Anal Des 59:1–10

    Article  MathSciNet  Google Scholar 

  5. Barenblatt GI (1959) The formation of equilibrium cracks during brittle fracture. General ideas and hypotheses. Axially-symmetric cracks. J Appl Math Mech 23(3):622–636

    Article  MathSciNet  MATH  Google Scholar 

  6. Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45(5):601–620

    Article  MATH  Google Scholar 

  7. Klein P, Foulk J, Chen E et al (2001) Physics-based modeling of brittle fracture: cohesive formulations and the application of meshfree methods. Theoret Appl Fract Mech 37(1–3):99–166

    Article  Google Scholar 

  8. Zi G, Rabczuk T, Wall W (2007) Extended meshfree methods without branch enrichment for cohesive cracks. Comput Mech 40(2):367–382

    Article  MATH  Google Scholar 

  9. Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48(1):175–209

    Article  MathSciNet  MATH  Google Scholar 

  10. Mikata Y (2019) Linear peridynamics for isotropic and anisotropic materials. Int J Solids Struct 158:116–127

    Article  Google Scholar 

  11. Sun S, Sundararaghavan V (2014) A peridynamic implementation of crystal plasticity. Int J Solids Struct 51(19–20):3350–3360

    Article  Google Scholar 

  12. Pashazad H, Kharazi M (2019) A peridynamic plastic model based on von mises criteria with isotropic, kinematic and mixed hardenings under cyclic loading. Int J Mech Sci 156:182–204

    Article  Google Scholar 

  13. Diana V, Ballarini R (2020) Crack kinking in isotropic and orthotropic micropolar peridynamic solids. Int J Solids Struct 196:76–98

    Article  Google Scholar 

  14. D’Antuono P, Morandini M (2017) Thermal shock response via weakly coupled peridynamic thermo-mechanics. Int J Solids Struct 129:74–89

    Article  Google Scholar 

  15. Oterkus S, Madenci E, Agwai A (2014) Fully coupled peridynamic thermomechanics. J Mech Phys Solids 64:1–23

    Article  MathSciNet  MATH  Google Scholar 

  16. O’Grady J, Foster J (2014) Peridynamic plates and flat shells: a non-ordinary, state-based model. Int J Solids Struct 51(25–26):4572–4579

    Article  Google Scholar 

  17. O’Grady J, Foster J (2014) Peridynamic beams: a non-ordinary, state-based model. Int J Solids Struct 51(18):3177–3183

    Article  Google Scholar 

  18. Zhang Q, Li S, Zhang A-M et al (2022) A nonlocal nonlinear stiffened shell theory with stiffeners modeled as geometrically-exact beams. Comput Methods Appl Mech Eng 397:115150

    Article  MathSciNet  MATH  Google Scholar 

  19. Kilic B, Agwai A, Madenci E (2009) Peridynamic theory for progressive damage prediction in center-cracked composite laminates. Compos Struct 90(2):141–151

    Article  Google Scholar 

  20. Hu W, Ha YD, Bobaru F (2012) Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites. Comput Methods Appl Mech Eng 217:247–261

    Article  MathSciNet  MATH  Google Scholar 

  21. Oterkus E, Madenci E (2012) Peridynamic analysis of fiber-reinforced composite materials. J Mech Mater Struct 7(1):45–84

    Article  Google Scholar 

  22. Ghajari M, Iannucci L, Curtis P (2014) A peridynamic material model for the analysis of dynamic crack propagation in orthotropic media. Comput Methods Appl Mech Eng 276:431–452

    Article  MathSciNet  MATH  Google Scholar 

  23. Hu Y, Madenci E (2016) Bond-based peridynamic modeling of composite laminates with arbitrary fiber orientation and stacking sequence. Compos Struct 153:139–175

    Article  Google Scholar 

  24. Zhou W, Liu D, Liu N (2017) Analyzing dynamic fracture process in fiber-reinforced composite materials with a peridynamic model. Eng Fract Mech 178:60–76

    Article  Google Scholar 

  25. Hu Y, Yu Y, Wang H (2014) Peridynamic analytical method for progressive damage in notched composite laminates. Compos Struct 108:801–810

    Article  Google Scholar 

  26. Azdoud Y, Han F, Lubineau G (2013) A morphing framework to couple non-local and local anisotropic continua. Int J Solids Struct 50(9):1332–1341

    Article  Google Scholar 

  27. Trageser J, Seleson P (2019) Anisotropic two-dimensional, plane strain, and plane stress models in classical linear elasticity and bond-based peridynamics. arXiv: Classical Physics

  28. Prakash N (2020) A novel numerical method for modeling anisotropy in discretized bond-based peridynamics. CoRR 2011.08013

  29. Diana V, Casolo S (2019) A full orthotropic micropolar peridynamic formulation for linearly elastic solids. Int J Mech Sci 160:140–155

    Article  Google Scholar 

  30. Zhang H, Qiao P (2019) A state-based peridynamic model for quantitative elastic and fracture analysis of orthotropic materials. Eng Fract Mech 206:147–171

    Article  Google Scholar 

  31. Hattori G, Trevelyan J, Coombs WM (2018) A non-ordinary state-based peridynamics framework for anisotropic materials. Comput Methods Appl Mech Eng 339:416–442

    Article  MathSciNet  MATH  Google Scholar 

  32. Liu S, Fang G, Liang J et al (2020) A new type of peridynamics: element-based peridynamics. Comput Methods Appl Mech Eng 366:113098

    Article  MathSciNet  MATH  Google Scholar 

  33. Tian DL, Zhou XP (2021) A continuum-kinematics-inspired peridynamic model of anisotropic continua: elasticity, damage, and fracture. Int J Mech Sci 199:106413

  34. Shen G, Xia Y, Hu P et al (2021) Construction of peridynamic beam and shell models on the basis of the micro-beam bond obtained via interpolation method. Eur J Mech A Solids 86:104174

  35. Nguyen-Van H, Mai-Duy N, Tran-Cong T (2009) An improved quadrilateral flat element with drilling degrees of freedom for shell structural analysis. CMES Comput Model Eng Sci 49(2):81–110

    Google Scholar 

  36. Zheng G, Yan Z, Xia Y (2023) Peridynamic shell model based on micro-beam bond. CMES Comput Model Eng Sci 3:1975-1995

  37. Hughes TJ, Brezzi F (1989) On drilling degrees of freedom. Comput Methods Appl Mech Eng 72(1):105–121

    Article  MathSciNet  MATH  Google Scholar 

  38. Lubineau G, Azdoud Y, Han F et al (2012) A morphing strategy to couple non-local to local continuum mechanics. J Mech Phys Solids 60(6):1088–1102

    Article  MathSciNet  Google Scholar 

  39. Zheng G, Li L, Han F et al (2022) Coupled peridynamic model for geometrically nonlinear deformation and fracture analysis of slender beam structures. Int J Numer Methods Eng 123(16):3658–3680

    Article  MathSciNet  Google Scholar 

  40. Xia Y, Wang H, Zheng G et al (2022) Discontinuous Galerkin isogeometric analysis with peridynamic model for crack simulation of shell structure. Comput Methods Appl Mech Eng 398:115193

    Article  MathSciNet  MATH  Google Scholar 

  41. Diana V, Carvelli V (2021) A continuum-molecular model for anisotropic electrically conductive materials. Int J Mech Sci 211:106759

    Article  Google Scholar 

  42. Yu H, Li S (2020) On energy release rates in peridynamics. J Mech Phys Solids 142:104024

    Article  MathSciNet  Google Scholar 

  43. Van Buskirk W, Cowin S, Ward RN (1981) Ultrasonic measurement of orthotropic elastic constants of bovine femoral bone. J Biomech Eng 103:67–72

    Article  Google Scholar 

  44. Li J, Li S, Lai X et al (2022) Peridynamic stress is the static first Piola–Kirchhoff Virial stress. Int J Solids Struct 241:111478

    Article  Google Scholar 

  45. Behiri J, Bonfield W (1989) Orientation dependence of the fracture mechanics of cortical bone. J Biomech 22(8–9):863–872

    Article  Google Scholar 

  46. Afshar A, Daneshyar A, Mohammadi S (2015) XFEM analysis of fiber bridging in mixed-mode crack propagation in composites. Compos Struct 125:314–327

    Article  Google Scholar 

Download references

Acknowledgements

The authors wish to express their appreciation to the reviewers for their helpful suggestions which greatly improved the presentation of this paper.

Funding

This work was supported by Project of the National Natural Science Foundation of China (Grant No. 12072065) and Applied Basic Research Program of Liaoning Province (Grant Nos. 2022JH2/101300224). These supports are gratefully acknowledged.

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Authors and Affiliations

  1. School of Automotive Engineering, Dalian University of Technology, Dalian, 116024, China

    Guozhe Shen, Bo Xu, Yang Xia, Weidong Li & Guojun Zheng

  2. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, 116024, China

    Guozhe Shen, Yang Xia, Weidong Li & Guojun Zheng

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  1. Guozhe Shen
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  2. Bo Xu
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  3. Yang Xia
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Correspondence to Weidong Li.

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Shen, G., Xu, B., Xia, Y. et al. A general anisotropic peridynamic plane model based on micro-beam bond. Comput Mech 71, 1065–1079 (2023). https://doi.org/10.1007/s00466-023-02274-2

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  • DOI: https://doi.org/10.1007/s00466-023-02274-2

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