Skip to main content
SpringerLink
Log in

Introduction

  • Chapter
  • First Online:
Peridynamic Theory and Its Applications

Abstract

One of the underlying assumptions in the classical theory is its locality. The classical continuum theory assumes that a material point only interacts with its immediate neighbors; hence, it is a local theory. The interaction of material points is governed by the various balance laws. Therefore, in a local model a material point only exchanges mass, momentum, and energy with its closest neighbors. As a result, in classical mechanics the stress state at a point depends on the deformation at that point only. The validity of this assumption becomes questionable across different length scales. In general, at the macroscale this assumption is acceptable. However, the existence of long-range forces is evident from the atomic theory and as such the supposition of local interactions breaks down as the geometric length scale becomes smaller and approaches the atomic scale. Even at the macroscale there are situations when the validity of locality is questionable, for instance when small features and microstructures influence the entire macrostructure.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Agwai A (2011) A peridynamic approach for coupled fields. Dissertation, University of Arizona

    Google Scholar 

  • Agwai A, Guven I, Madenci E (2011) Predicting crack propagation with peridynamics: a comparative study. Int J Fracture 171:65–78

    Article  Google Scholar 

  • Agwai A, Guven I, Madenci E (2012) Drop-shock failure prediction in electronic packages by using peridynamic theory. IEEE Trans Adv Pack 2(3):439–447

    Google Scholar 

  • Alali B, Lipton R (2012) Multiscale dynamics of heterogeneous media in the peridynamic formulation. J Elast 106:71–103

    Article  MathSciNet  MATH  Google Scholar 

  • Ari N, Eringen AC (1983) Nonlocal stress field at Griffith crack. Cryst Latt Defect Amorph Mater 10:33–38

    Google Scholar 

  • Askari E, Xu J, Silling SA (2006) Peridynamic analysis of damage and failure in composites. Paper 2006–88 presented at the 44th AIAA aerospace sciences meeting and exhibit. Grand Sierra Resort Hotel, Reno, 9–12 Jan 2006

    Google Scholar 

  • Askari A, Nelson K, Weckner O, Xu J, Silling S (2011) Hail impact characteristics of a hybrid material by advanced analysis techniques and testing. J Aerosp Eng 24:210–217

    Article  Google Scholar 

  • Barenblatt GI (1962) The mathematical theory of equilibrium cracks in brittle fracture. Adv Appl Mech 7:56–125

    Google Scholar 

  • Bazant ZP (1991) Why continuum damage is nonlocal—micromechanics arguments. J Eng Mech 117:1070–1087

    Article  Google Scholar 

  • Bazant ZP, Jirasek M (2002) Nonlocal integral formulations of plasticity and damage: survey of progress. J Eng Mech 128(11):1119–1149

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Bobaru F (2007) Influence of Van Der Waals forces on increasing the strength and toughness in dynamic fracture of nanofiber networks: a peridynamic approach. Model Simul Mater Sci Eng 15:397–417

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Bobaru F, Ha YD (2011) Adaptive refinement and multiscale modeling in 2D peridynamics. Int J Multiscale Comput Eng 9(6):635–660

    Article  Google Scholar 

  • Bobaru F, Hu W (2012) The meaning, selection and use of the peridynamic horizon and its relation to crack branching in brittle materials. Int J Fract 176(2):215–222

    Article  Google Scholar 

  • Bobaru F, Yang M, Alves LF, Silling SA, Askari E, Xu J (2009) Convergence, adaptive refinement, and scaling in 1D peridynamics. Int J Numer Meth Eng 77:852–877

    Article  MATH  Google Scholar 

  • Celik E, Guven I, Madenci E (2011) Simulations of nanowire bend tests for extracting mechanical properties. Theor Appl Fract Mech 55:185–191

    Article  Google Scholar 

  • Colavito KW, Kilic B, Celik E, Madenci E, Askari E, Silling S (2007a) Effect of void content on stiffness and strength of composites by a peridynamic analysis and static indentation test. Paper 2007–2251 presented at the 48th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, Waikiki, 23–26 Apr 2007

    Google Scholar 

  • Colavito KW, Kilic B, Celik E, Madenci E, Askari E, Silling S (2007b) Effect of nano particles on stiffness and impact strength of composites. Paper 2007–2001 presented at the 48th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, Waikiki, 23–26 Apr 2007

    Google Scholar 

  • Cox BN, Gao H, Gross D, Rittel D (2005) Modern topics and challenges in dynamic fracture. J Mech Phys Solid 53:565–596

    Article  MathSciNet  MATH  Google Scholar 

  • Dayal K, Bhattacharya K (2006) Kinetics of phase transformations in the peridynamic formulation of continuum mechanics. J Mech Phys Solids 54:1811–1842

    Article  MathSciNet  MATH  Google Scholar 

  • Demmie PN, Silling SA (2007) An approach to modeling extreme loading of structures using peridynamics. J Mech Mater Struct 2(10):1921–1945

    Article  Google Scholar 

  • Dugdale DS (1960) Yielding of steel sheets containing slits. J Mech Phys Solids 8(2):100–104

    Article  Google Scholar 

  • Elliott HA (1947) An analysis of the conditions for rupture due to Griffith cracks. Proc Phys Soc 59:208–223

    Article  Google Scholar 

  • Emmrich E, Weckner O (2007) The peridynamic equation and its spatial discretization. J Math Model Anal 12(1):17–27

    Article  MathSciNet  MATH  Google Scholar 

  • Eringen AC (1972a) Nonlocal polar elastic continua. Int J Eng Sci 10:1–16

    Article  MathSciNet  MATH  Google Scholar 

  • Eringen AC (1972b) Linear theory of nonlocal elasticity and dispersion of plane waves. Int J Eng Sci 10:425–435

    Article  MATH  Google Scholar 

  • Eringen AC, Edelen DGB (1972) On nonlocal elasticity. Int J Eng Sci 10:233–248

    Article  MathSciNet  MATH  Google Scholar 

  • Eringen AC, Kim BS (1974a) Stress concentration at the tip of crack. Mech Res Commun 1:233–237

    Article  Google Scholar 

  • Eringen AC, Kim BS (1974b) On the problem of crack tip in nonlocal elasticity. In: Thoft-Christensen P (ed) Continuum mechanics aspects of geodynamics and rock fracture mechanics. Proceedings of the NATO advanced study institute held in Reykjavik, 11–20 Aug 1974. D. Reidel, Dordrecht, pp 107–113

    Google Scholar 

  • Eringen AC, Speziale CG, Kim BS (1977) Crack-tip problem in non-local elasticity. J Mech Phys Solids 25:339–355

    Article  MathSciNet  MATH  Google Scholar 

  • Foster JT, Silling SA, Chen WW (2010) Viscoplasticity using peridynamics. Int J Numer Meth Eng 81:1242–1258

    MATH  Google Scholar 

  • Foster JT, Silling SA, Chen W (2011) An energy based failure criterion for use with peridynamic states. Int J Multiscale Comput Eng 9(6):675–688

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Gerstle W, Silling S, Read D, Tewary V, Lehoucq R (2008) Peridynamic simulation of electromigration. Comput Mater Continua 8(2):75–92

    Google Scholar 

  • Griffith AA (1921) The phenomena of rupture and flow in solids. Philos Trans R Soc Lond A 221:163–198

    Article  Google Scholar 

  • Ha YD, Bobaru F (2011) Characteristics of dynamic brittle fracture captured with peridynamics. Eng Fract Mech 78:1156–1168

    Article  Google Scholar 

  • Hillerborg A, Modeer M, Petersson PE (1976) Analysis of crack formation and crack growth by means of fracture mechanics and finite elements. Cem Concr Res 6(6):773–781

    Article  Google Scholar 

  • Hu W, Ha YD, Bobaru F (2012a) Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites. Comput Meth Appl Mech Eng 217–220:247–261

    Article  MathSciNet  Google Scholar 

  • Hu W, Ha YD, Bobaru F, Silling SA (2012b) The formulation and computation of the non-local J-integral in bond-based peridynamics. Int J Fract 176:195–206

    Article  Google Scholar 

  • Kadau K, Germann TC, Lomdahl PS (2006) Molecular dynamics comes of age: 320 billion atom simulation on BlueGene/L. Int J Mod Phys C 17:1755–1761

    Article  Google Scholar 

  • Kilic B (2008) Peridynamic theory for progressive failure prediction in homogeneous and heterogeneous materials. Dissertation, University of Arizona

    Google Scholar 

  • Kilic B, Madenci E (2009a) Structural stability and failure analysis using peridynamic theory. Int J Nonlinear Mech 44:845–854

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Kilic B, Madenci E (2010a) An adaptive dynamic relaxation method for quasi-static simulations using the peridynamic theory. Theor Appl Fract Mech 53:194–201

    Article  Google Scholar 

  • Kilic B, Madenci E (2010b) Coupling of peridynamic theory and finite element method. J Mech Mater Struct 5:707–733

    Article  Google Scholar 

  • Kilic B, Madenci E (2010c) Peridynamic theory for thermomechanical analysis. IEEE Trans Adv Packag 33:97–105

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Klein PA, Foulk JW, Chen EP, Wimmer SA, Gao H (2001) Physics-based modeling of brittle fracture: cohesive formulations and the application of meshfree methods. Theor Appl Fract Mech 37:99–166

    Article  Google Scholar 

  • Kroner E (1967) Elasticity theory of materials with long range cohesive forces. Int J Solids Struct 3:731–742

    Article  Google Scholar 

  • Kunin IA (1982) Elastic media with microstructure I: one dimensional models. Springer, Berlin

    Book  MATH  Google Scholar 

  • Kunin IA (1983) Elastic media with microstructure II: three-dimensional models. Springer, Berlin

    Book  MATH  Google Scholar 

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

    Article  Google Scholar 

  • Lehoucq RB, Silling SA (2008) Force flux and the peridynamic stress tensor. J Mech Phys Solids 56:1566–1577

    Article  MathSciNet  MATH  Google Scholar 

  • Liu W, Hong J (2012a) Discretized peridynamics for brittle and ductile solids. Int J Numer Meth Eng 89(8):1028–1046

    Article  MathSciNet  MATH  Google Scholar 

  • Liu W, Hong J (2012b) A coupling approach of discretized peridynamics with finite element method. Comput Meth Appl Mech Eng 245–246:163–175

    Article  MathSciNet  Google Scholar 

  • Lubineau G, Azdoud Y, Han F, Rey C, Askari A (2012) A morphing strategy to couple non-local to local continuum mechanics. J Mech Phys Solids 60:1088–1102

    Article  MathSciNet  Google Scholar 

  • Macek RW, Silling SA (2007) Peridynamics via finite element analysis. Finite Elem Anal Des 43(15):1169–1178

    Article  MathSciNet  Google Scholar 

  • Melenk JM, Babuska I (1996) The partition of unity finite element method: basic theory and applications. Comput Meth Appl Mech Eng 139:289–314

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Mitchell JA (2011a) A nonlocal, ordinary, state-based plasticity model for peridynamics. SAND2011-3166. Sandia National Laboratories, Albuquerque

    Google Scholar 

  • Mitchell JA (2011b) A non-local, ordinary-state-based viscoelasticity model for peridynamics. SAND2011-8064. Sandia National Laboratories, Albuquerque

    Google Scholar 

  • Moes N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Meth Eng 46:131–150

    Article  MATH  Google Scholar 

  • Ostoja-Starzewski M (2002) Lattice models in micromechanics. Appl Mech Rev 55:35–60

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Oterkus E, Barut A, Madenci E (2010) Damage growth prediction from loaded composite fastener holes by using peridynamic theory. In: Proceedings of the 51st AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, April 2010. AIAA, Reston, Paper 2010–3026

    Google Scholar 

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

    Article  Google Scholar 

  • Oterkus E, Madenci E, Weckner O, Silling S, Bogert P, Tessler A (2012b) Combined finite element and peridynamic analyses for predicting failure in a stiffened composite curved panel with a central slot. Compos Struct 94:839–850

    Article  Google Scholar 

  • Ozbolt J, Bazant ZP (1996) Numerical smeared fracture analysis: nonlocal microcrack interaction approach. Int J Numer Meth Eng 39:635–661

    Article  MATH  Google Scholar 

  • Polleschi M (2010) Stability and applications of the peridynamic method. Thesis, Polytechnic University of Turin

    Google Scholar 

  • Rogula D (1982) Nonlocal theory of material media. Springer, Berlin, pp 137–243

    MATH  Google Scholar 

  • Schlangen E, van Mier JGM (1992) Simple lattice model for numerical simulation of fracture of concrete materials and structures. Mater Struct 25:534–542

    Article  Google Scholar 

  • Seleson P, Parks ML (2011) On the role of influence function in the peridynamic theory. Int J Multiscale Comput Eng 9(6):689–706

    Article  Google Scholar 

  • Seleson P, Parks ML, Gunzburger M, Lehocq RB (2009) Peridynamics as an upscaling of molecular dynamics. Multiscale Model Simul 8(1):204–227

    Article  MathSciNet  MATH  Google Scholar 

  • Seleson P, Beneddine S, Prudhomme S (2013) A force-based coupling scheme for peridynamics and classical elasticity. Comput Mater Sci 66:34–49

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Silling SA (2003) Dynamic fracture modeling with a meshfree peridynamic code. In: Bathe KJ (ed) Computational fluid and solid mechanics. Elsevier, Amsterdam, pp 641–644

    Google Scholar 

  • Silling SA (2010) Linearized theory of peridynamic states. J Elast 99:85–111

    Article  MathSciNet  MATH  Google Scholar 

  • Silling SA (2011) A coarsening method for linear peridynamics. Int J Multiscale Comput Eng 9(6):609–622

    Article  Google Scholar 

  • Silling SA, Askari A (2004) Peridynamic modeling of impact damage. In: Moody FJ (ed) PVP-vol. 489. American Society of Mechanical Engineers, New York, pp 197–205

    Google Scholar 

  • Silling SA, Askari A (2005) A meshfree method based on the peridynamic model of solid mechanics. Comput Struct 83(17–18):1526–1535

    Article  Google Scholar 

  • Silling SA, Bobaru F (2005) Peridynamic modeling of membranes and fibers. Int J Nonlinear Mech 40:395–409

    Article  MATH  Google Scholar 

  • Silling SA, Lehoucq RB (2008) Convergence of peridynamics to classical elasticity theory. J Elast 93:13–37

    Article  MathSciNet  MATH  Google Scholar 

  • Silling SA, Lehoucq RB (2010) Peridynamic theory of solid mechanics. Adv Appl Mech 44:73–168

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Silling SA, Weckner O, Askari A, Bobaru F (2010) Crack nucleation in a peridynamic solid. Int J Fract 162:219–227

    Article  MATH  Google Scholar 

  • Taylor MJ (2008) Numerical simulation of thermo-elasticity, inelasticity and rupture in membrane theory. Dissertation, University of California, Berkeley

    Google Scholar 

  • Wang H, Tian H (2012) A fast Galerkin method with efficient matrix assembly and storage for a peridynamic model. J Comput Phys 231:7730–7738

    Article  MathSciNet  MATH  Google Scholar 

  • Warren TL, Silling SA, Askari A, Weckner O, Epton MA, Xu J (2009) A non-ordinary state-based peridynamic method to model solid material deformation and fracture. Int J Solids Struct 46:1186–1195

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Xu XP, Needleman A (1994) Numerical simulations of fast crack growth in brittle solids. J Mech Phys Solids 42:1397–1434

    Article  MATH  Google Scholar 

  • Xu J, Askari A, Weckner O, Razi H, Silling S (2007) Damage and failure analysis of composite laminates under biaxial loads. In: Proceedings of the 48th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, April 2007. AIAA, Reston. doi: 10.2514/6.2007-2315

    Google Scholar 

  • Xu J, Askari A, Weckner O, Silling SA (2008) Peridynamic analysis of impact damage in composite laminates. J Aerosp Eng 21(3):187–194

    Article  Google Scholar 

  • Yu K, Xin XJ, Lease KB (2011) A new adaptive integration method for the peridynamic theory. Model Simul Mater Sci Eng 19:45003

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ, USA

    Erdogan Madenci

  2. Naval Architecture and Marine Engineering, University of Strathclyde, Glasgow, UK

    Erkan Oterkus

Authors
  1. Erdogan Madenci
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Erkan Oterkus
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media New York

About this chapter

Cite this chapter

Madenci, E., Oterkus, E. (2014). Introduction. In: Peridynamic Theory and Its Applications. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8465-3_1

Download citation

  • DOI: https://doi.org/10.1007/978-1-4614-8465-3_1

  • Published:

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4614-8464-6

  • Online ISBN: 978-1-4614-8465-3

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation