Skip to main content
SpringerLink
Log in

Peridynamics: Introduction

  • Living reference work entry
  • First Online:
Handbook of Nonlocal Continuum Mechanics for Materials and Structures

  • 691 Accesses

Abstract

The peridynamic theory is a nonlocal extension of continuum mechanics that is compatible with the physical nature of cracks as discontinuities. It avoids the need to evaluate the partial derivatives of the deformation with respect to the spatial coordinates, instead using an integro-differential equation for the linear momentum balance. This chapter summarizes the peridynamic theory, emphasizing the continuum mechanical and thermodynamic aspects. Formulation of material models is discussed, including details on the statement of models using mathematical objects called peridynamic states that are nonlocal and nonlinear generalizations of second-order tensors. Damage evolution is treated within a nonlocal thermodynamic framework making use of the dependence of free energy on damage. Continuous, stable growth of damage can suddenly become unstable, leading to dynamic fracture. Peridynamics treats fracture and long-range forces on the same mathematical basis as continuous deformation and contact forces, extending the applicability of continuum mechanics to new classes of problems.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

References

  • F. Bobaru, M. Duangpanya, The peridynamic formulation for transient heat conduction. Int. J. Heat Mass Transf. 53, 4047–4059 (2010)

    Article  Google Scholar 

  • F. Bobaru, M. Duangpanya, A peridynamic formulation for transient heat conduction in bodies with evolving discontinuities. J. Comput. Phys. 231, 2764–2785 (2012)

    Article  MathSciNet  Google Scholar 

  • Z. Chen, F. Bobaru, Peridynamic modeling of pitting corrosion damage. J. Mech. Phys. Solids 78, 352–381 (2015)

    Article  MathSciNet  Google Scholar 

  • S.R. Chowdhury, P. Roy, D. Roy, J. Reddy, A peridynamic theory for linear elastic shells. Int. J. Solids Struct. 84, 110–132 (2016)

    Article  Google Scholar 

  • B.D. Coleman, W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Ration. Mech. Anal. 13, 167–178 (1963)

    Article  MathSciNet  Google Scholar 

  • C. Diyaroglu, E. Oterkus, S. Oterkus, E. Madenci, Peridynamics for bending of beams and plates with transverse shear deformation. Int. J. Solids Struct. 69, 152–168 (2015)

    Article  Google Scholar 

  • E. Emmrich, O. Weckner, et al., On the well-posedness of the linear peridynamic model and its convergence towards the navier equation of linear elasticity. Commun. Math. Sci. 5, 851–864 (2007)

    Article  MathSciNet  Google Scholar 

  • J.T. Foster, S.A. Silling, W.W. Chen, Viscoplasticity using peridynamics. Int. J. Numer. Methods Eng. 81, 1242–1258 (2010)

    MATH  Google Scholar 

  • E. Fried, New insights into the classical mechanics of particle systems. Discrete Contin. Dyn. Syst. 28, 1469–1504 (2010)

    Article  MathSciNet  Google Scholar 

  • W. Gerstle, N. Sau, S.A. Silling, Peridynamic modeling of concrete structures. Nucl. Eng. Des. 237, 1250–1258 (2007)

    Article  Google Scholar 

  • W. Gerstle, S. Silling, D. Read, V. Tewary, R. Lehoucq, Peridynamic simulation of electromigration. Comput. Mater. Continua 8, 75–92 (2008)

    Google Scholar 

  • W. Gerstle, N. Sakhavand, S. Chapman, Peridynamic and continuum models of reinforced concrete lap splice compared, in Fracture Mechanics of Concrete and Concrete Structures, Recent Advances in Fracture Mechanics of Concrete, ed. by B.H. Oh, et al. (2010), pp. 306–312

    Google Scholar 

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

    Article  Google Scholar 

  • M.E. Gurtin, W.O. Williams, On the first law of thermodynamics. Arch. Ration. Mech. Anal. 42, 77–92 (1971)

    Article  MathSciNet  Google Scholar 

  • M.E. Gurtin, E. Fried, L. Anand, The mechanics and thermodynamics of continua (Cambridge University Press, Cambridge, 2010), pp. 232–233

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • W. Hu, Y.D. Ha, F. Bobaru, S.A. Silling, The formulation and computation of the nonlocal J-integral in bond-based peridynamics. Int. J. Fract. 176, 195–206 (2012b)

    Article  Google Scholar 

  • W. Hu, Y. Wang, J. Yu, C.-F. Yen, F. Bobaru, Impact damage on a thin glass plate with a thin polycarbonate backing. Int. J. Impact Eng. 62, 152–165 (2013)

    Article  Google Scholar 

  • B. Jeon, R.J. Stewart, I.Z. Ahmed, Peridynamic simulations of brittle structures with thermal residual deformation: strengthening and structural reactivity of glasses under impacts. Proc. R. Soc. A 471, 20150231. (2015)

    Article  Google Scholar 

  • A. Katiyar, J.T. Foster, H. Ouchi, M.M. Sharma, A peridynamic formulation of pressure driven convective fluid transport in porous media. J. Comput. Phys. 261, 209–229 (2014)

    Article  MathSciNet  Google Scholar 

  • B. Kilic, E. Madenci, Prediction of crack paths in a quenched glass plate by using peridynamic theory. Int. J. Fract. 156, 165–177 (2009)

    Article  Google Scholar 

  • R.B. Lehoucq, M.P. Sears, Statistical mechanical foundation of the peridynamic nonlocal continuum theory: energy and momentum conservation laws. Phys. Rev. E 84, 031112 (2011)

    Article  Google Scholar 

  • R.B. Lehoucq, S.A. Silling, Force flux and the peridynamic stress tensor. J. Mech. Phys. Solids 56, 1566–1577 (2008)

    Article  MathSciNet  Google Scholar 

  • R.B. Lehoucq, O.A. von Lilienfeld, Translation of Walter Noll’s derivation of the fundamental equations of continuum thermodynamics from statistical mechanics. J. Elast. 100, 5–24 (2010)

    Article  Google Scholar 

  • E. Lejeune, C. Linder, Modeling tumor growth with peridynamics. Biomech. Model. Mechanobiol., 1–17 (2017a)

    Google Scholar 

  • E. Lejeune, C. Linder, Quantifying the relationship between cell division angle and morphogenesis through computational modeling. J. Theor. Biol. 418, 1–7 (2017b)

    Article  MathSciNet  Google Scholar 

  • R. Lipton, Dynamic brittle fracture as a small horizon limit of peridynamics. J. Elast. 117, 21–50 (2014)

    Article  MathSciNet  Google Scholar 

  • R. Lipton, Cohesive dynamics and brittle fracture. J. Elast. 142, 1–49 (2016)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • J.A. Mitchell, A non-local, ordinary-state-based viscoelasticity model for peridynamics. Technical report SAND2011-8064, Sandia National Laboratories, Albuquerque/Livermore, October 2011a

    Google Scholar 

  • J.A. Mitchell, A nonlocal, ordinary, state-based plasticity model for peridynamics. Technical report SAND2011-3166, Sandia National Laboratories, Albuquerque/Livermore, October 2011b

    Google Scholar 

  • S. Nadimi, State-based peridynamics simulation of hydraulic fracture phenomenon in geological media. Master’s thesis, The University of Utah, 2015

    Google Scholar 

  • W. Noll, Die Herleitung der Grundgleichungen der Thermomechanik der Kontinua aus der statistischen Mechanik. J. Ration. Mech. Anal. 4, 627–646 (1955.) In German, English translation available

    MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  • S. Oterkus, J. Fox, E. Madenci, Simulation of electro-migration through peridynamics, in 2013 IEEE 63rd Electronic Components and Technology Conference (IEEE, 2013), pp. 1488–1493

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • S. Oterkus, E. Madenci, A. Agwai, Peridynamic thermal diffusion. J. Comput. Phys. 265, 71–96 (2014b)

    Article  MathSciNet  Google Scholar 

  • H. Ouchi, A. Katiyar, J. Foster, M.M. Sharma, et al., A peridynamics model for the propagation of hydraulic fractures in heterogeneous, naturally fractured reservoirs. in SPE Hydraulic Fracturing Technology Conference (Society of Petroleum Engineers, 2015)

    Google Scholar 

  • N. Prakash, G.D. Seidel, A coupled electromechanical peridynamics framework for modeling carbon nanotube reinforced polymer composites, in 57th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, p. 0936, (2016)

    Google Scholar 

  • S.A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000)

    Article  MathSciNet  Google Scholar 

  • S.A. Silling, Linearized theory of peridynamic states. J. Elast. 99, 85–111 (2010)

    Article  MathSciNet  Google Scholar 

  • S.A. Silling, Solitary waves in a peridynamic elastic solid. J. Mech. Phys. Solids 96, 121–132 (2016)

    Article  MathSciNet  Google Scholar 

  • S.A. Silling, Stability of peridynamic correspondence material models and their particle discretizations. Comput. Methods Appl. Mech. Eng. 322, 42–57 (2017)

    Article  MathSciNet  Google Scholar 

  • S.A. Silling, E. Askari, A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 83, 1526–1535 (2005)

    Article  Google Scholar 

  • S.A. Silling, R.B. Lehoucq, Convergence of peridynamics to classical elasticity theory. J. Elast. 93, 13–37 (2008)

    Article  MathSciNet  Google Scholar 

  • S.A. Silling, R.B. Lehoucq, The peridynamic theory of solid mechanics. Adv. Appl. Mech. 44, 73–166 (2010)

    Article  Google Scholar 

  • S.A. Silling, M. Epton, O. Weckner, J. Xu, E. Askari, Peridynamic states and constitutive modeling. J. Elast. 88, 151–184 (2007)

    Article  MathSciNet  Google Scholar 

  • S.A. Silling, D. Littlewood, P. Seleson, Variable horizon in a peridynamic medium. J. Mech. Mater. Struct. 10, 591–612 (2015)

    Article  MathSciNet  Google Scholar 

  • S.A. Silling, M.L. Parks, J.R. Kamm, O. Weckner, M. Rassaian, Modeling shockwaves and impact phenomena with Eulerian peridynamics. Int. J. Impact Eng. 107, 47–57 (2017)

    Article  Google Scholar 

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

    Article  Google Scholar 

  • M. Taylor, I. Gözen, S. Patel, A. Jesorka, K. Bertoldi, Peridynamic modeling of ruptures in biomembranes. PLoS One 11, e0165947 (2016)

    Article  Google Scholar 

  • M. Tupek, R. Radovitzky, An extended constitutive correspondence formulation of peridynamics based on nonlinear bond-strain measures. J. Mech. Phys. Solids 65, 82–92 (2014)

    Article  MathSciNet  Google Scholar 

  • C.W. Van Der Merwe, A peridynamic model for sleeved hydraulic fracture. Master’s thesis, Stellenbosch University, Stellenbosch, (2014)

    Google Scholar 

  • T.L. Warren, S.A. Silling, A. Askari, O. Weckner, M.A. Epton, J. Xu, A nonordinary state-based peridynamic method to model solid material deformation and fracture. Int. J. Solids Struct. 46, 1186–1195 (2009)

    Article  Google Scholar 

  • O. Weckner, N.A.N. Mohamed, Viscoelastic material models in peridynamics. Appl. Math. Comput. 219, 6039–6043 (2013)

    MathSciNet  MATH  Google Scholar 

  • R. Wildman, G. Gazonas, A dynamic electro-thermo-mechanical model of dielectric breakdown in solids using peridynamics. J. Mech. Mater. Struct. 10, 613–630 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgment

Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc. for the US Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

Author information

Authors and Affiliations

  1. Sandia National Laboratories, Albuquerque, USA

    S. A. Silling

Authors
  1. S. A. Silling
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. A. Silling .

Editor information

Editors and Affiliations

  1. Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, USA

    George Z. Voyiadjis

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG (outside the USA)

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Silling, S.A. (2018). Peridynamics: Introduction. In: Voyiadjis, G. (eds) Handbook of Nonlocal Continuum Mechanics for Materials and Structures. Springer, Cham. https://doi.org/10.1007/978-3-319-22977-5_29-1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-22977-5_29-1

  • Received:

  • Accepted:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-22977-5

  • Online ISBN: 978-3-319-22977-5

  • eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering

Publish with us

Policies and ethics

Search

Navigation