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Analytical Solution of 1-Dimensional Peridynamic Equation of Motion

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Abstract

Peridynamics has been introduced to overcome limitations of classical continuum mechanics. Peridynamic equations of motion are in the form of integro-differential equations and analytical solutions of these equations are limited in the literature. In this study, a new analytical solution methodology for 1-dimensional peridynamic equation of motion is presented by utilising inverse Fourier transform. Analytical solutions for both static and dynamic conditions are obtained. Moreover, different boundary conditions including fixed-fixed and fixed-free are considered. Several numerical cases are demonstrated to show the capability of the presented methodology and peridynamic results are compared against results obtained from classical continuum mechanics. A very good agreement between these two different approaches is observed which shows the capability of the current approach.

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Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

References

  1. 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 

  2. Madenci E, Oterkus E (2014) Peridynamic theory and its applications. Springer, New York, NY

    Book  MATH  Google Scholar 

  3. Wu CT, Ren B (2015) 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

    Article  MathSciNet  MATH  Google Scholar 

  4. Sun C, Huang Z (2016) Peridynamic simulation to impacting damage in composite laminate. Compos Struct 138:335–341

    Article  Google Scholar 

  5. Diyaroglu C, Oterkus E, Madenci E, Rabczuk T, Siddiq A (2016) Peridynamic modeling of composite laminates under explosive loading. Compos Struct 144:14–23

    Article  Google Scholar 

  6. Gerstle W, Sau N, Silling S (2007) Peridynamic modeling of concrete structures. Nucl Eng Des 237(12–13):1250–1258

    Article  Google Scholar 

  7. Oterkus E, Guven I, Madenci E (2012) Impact damage assessment by using peridynamic theory. Cent Eur J Eng 2(4):523–531

    Google Scholar 

  8. Cheng Z, Liu Y, Zhao J, Feng H, Wu Y (2018) Numerical simulation of crack propagation and branching in functionally graded materials using peridynamic modeling. Eng Fract Mech 191:13–32

    Article  Google Scholar 

  9. Ozdemir M, Kefal A, Imachi M, Tanaka S, Oterkus E (2020) Dynamic fracture analysis of functionally graded materials using ordinary state-based peridynamics. Compos Struct 244:112296

  10. Silling SA, D’Elia M, Yu Y, You H, Fermen-Coker M (2022) Peridynamic model for single-layer graphene obtained from coarse-grained bond forces. J Peridyn Nonlocal Model 1–22

  11. Liu X, He X, Wang J, Sun L, Oterkus E (2018) An ordinary state-based peridynamic model for the fracture of zigzag graphene sheets. Proc R Soc Math Phys Eng Sci 474(2217):20180019

    MathSciNet  MATH  Google Scholar 

  12. Liu RW, Xue YZ, Lu XK, Cheng WX (2018) Simulation of ship navigation in ice rubble based on peridynamics. Ocean Eng 148:286–298

    Article  Google Scholar 

  13. Vazic B, Oterkus E, Oterkus S (2020) Peridynamic model for a Mindlin plate resting on a Winkler elastic foundation. J Peridyn Nonlocal Model 2(3):229–242

    Article  MathSciNet  MATH  Google Scholar 

  14. Madenci E, Oterkus S (2016) Ordinary state-based peridynamics for plastic deformation according to von Mises yield criteria with isotropic hardening. J Mech Phys Solids 86:192–219

    Article  MathSciNet  Google Scholar 

  15. Huang Y, Oterkus S, Hou H, Oterkus E, Wei Z, Zhang S (2019) Peridynamic model for visco-hyperelastic material deformation in different strain rates. Contin Mech Thermodyn 1–35

  16. Amani J, Oterkus E, Areias P, Zi G, Nguyen-Thoi T, Rabczuk T (2016) A non-ordinary state-based peridynamics formulation for thermoplastic fracture. Int J Impact Eng 87:83–94

    Article  Google Scholar 

  17. Yang Z, Oterkus E, Oterkus S (2021) Peridynamic higher-order beam formulation. J Peridyn Nonlocal Model 3(1):67–83

    Article  MathSciNet  MATH  Google Scholar 

  18. Dorduncu M (2019) Stress analysis of laminated composite beams using refined zigzag theory and peridynamic differential operator. Compos Struct 218:193–203

    Article  Google Scholar 

  19. Yang Z, Vazic B, Diyaroglu C, Oterkus E, Oterkus S (2020) A Kirchhoff plate formulation in a state-based peridynamic framework. Math Mech Solids 25(3):727–738

    Article  MathSciNet  MATH  Google Scholar 

  20. Naumenko K, Eremeyev VA (2022) A non-linear direct peridynamics plate theory. Compos Struct 279:114728

    Article  Google Scholar 

  21. Dorduncu M (2020) Stress analysis of sandwich plates with functionally graded cores using peridynamic differential operator and refined zigzag theory. Thin-Walled Struct 146:106468

  22. Chowdhury SR, Roy P, Roy D, Reddy JN (2016) A peridynamic theory for linear elastic shells. Int J Solids Struct 84:110–132

    Article  Google Scholar 

  23. Jung J, Seok J (2017) Mixed-mode fatigue crack growth analysis using peridynamic approach. Int J Fatigue 103:591–603

    Article  Google Scholar 

  24. Nguyen CT, Oterkus S, Oterkus E (2021) An energy-based peridynamic model for fatigue cracking. Eng Fract Mech 241:107373

  25. Heo J, Yang Z, Xia W, Oterkus S, Oterkus E (2020) Buckling analysis of cracked plates using peridynamics. Ocean Eng 214:107817

  26. Kefal A, Sohouli A, Oterkus E, Yildiz M, Suleman A (2019) Topology optimization of cracked structures using peridynamics. Continuum Mech Thermodyn 31(6):1645–1672

    Article  MathSciNet  Google Scholar 

  27. Imachi M, Tanaka S, Ozdemir M, Bui TQ, Oterkus S, Oterkus E (2020) Dynamic crack arrest analysis by ordinary state-based peridynamics. Int J Fract 221(2):155–169

    Article  Google Scholar 

  28. Vazic B, Wang H, Diyaroglu C, Oterkus S, Oterkus E (2017) Dynamic propagation of a macrocrack interacting with parallel small cracks. AIMS Mater Sci 4(1):118–136

    Article  MATH  Google Scholar 

  29. Diyaroglu C, Oterkus S, Oterkus E, Madenci E, Han S, Hwang Y (2017) Peridynamic wetness approach for moisture concentration analysis in electronic packages. Microelectron Reliab 70:103–111

    Article  Google Scholar 

  30. De Meo D, Russo L, Oterkus E (2017) Modeling of the onset, propagation, and interaction of multiple cracks generated from corrosion pits by using peridynamics. J Eng Mater Technol 139(4):041001

    Article  Google Scholar 

  31. Shi C, Gong Y, Yang ZG, Tong Q (2019) Peridynamic investigation of stress corrosion cracking in carbon steel pipes. Eng Fract Mech 219:106604

  32. Javili A, Morasata R, Oterkus E, Oterkus S (2019) Peridynamics review. Math Mech Solids 24(11):3714–3739

    Article  MathSciNet  MATH  Google Scholar 

  33. Mikata Y (2012) Analytical solutions of peristatic and peridynamic problems for a 1D infinite rod. Int J Solids Struct 49(21):2887–2897

    Article  Google Scholar 

  34. Mikata Y (2021) Peridynamics for fluid mechanics and acoustics. Acta Mech 232(8):3011–3032

    Article  MathSciNet  MATH  Google Scholar 

  35. Silling SA, Zimmermann M, Abeyaratne R (2003) Deformation of a peridynamic bar. J Elast 73(1):173–190

    Article  MathSciNet  MATH  Google Scholar 

  36. Weckner O, Abeyaratne R (2005) The effect of long-range forces on the dynamics of a bar. J Mech Phys Solids 53(3):705–728

    Article  MathSciNet  MATH  Google Scholar 

  37. Weckner O, Brunk G, Epton MA, Silling SA, Askari E (2009) Green’s functions in non-local three-dimensional linear elasticity. Proc R Soc Math Phys Eng Sci 465(2111):3463–3487

    MathSciNet  MATH  Google Scholar 

  38. Aksoylu B, Gazonas GA (2020) On nonlocal problems with inhomogeneous local boundary conditions. J Peridyn Nonlocal Model 2(1):1–25

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan

    Zhenghao Yang & Chien-Ching Ma

  2. PeriDynamics Research Centre, Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, UK

    Erkan Oterkus & Selda Oterkus

  3. Institute of Mechanics, Otto von Guericke University, Magdeburg, Germany

    Konstantin Naumenko

Authors
  1. Zhenghao Yang
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  2. Chien-Ching Ma
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  3. Erkan Oterkus
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  4. Selda Oterkus
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  5. Konstantin Naumenko
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Correspondence to Zhenghao Yang.

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Yang, Z., Ma, CC., Oterkus, E. et al. Analytical Solution of 1-Dimensional Peridynamic Equation of Motion. J Peridyn Nonlocal Model 5, 356–374 (2023). https://doi.org/10.1007/s42102-022-00086-1

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  • DOI: https://doi.org/10.1007/s42102-022-00086-1

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