Peridynamics (PD) is a novel continuum mechanics theory established by Stewart Silling in 2000. The roots of PD can be traced back to the early works of Gabrio Piola according to dell’Isola et al. PD has been attractive to researchers as it is a non-local formulation in an integral form, unlike the local differential form of classical continuum mechanics. Although the method is still in its infancy, the literature on PD is fairly rich and extensive. The prolific growth in PD applications has led to a tremendous number of contributions in various disciplines. This manuscript aims to provide a concise description of the PD theory together with a review of its major applications and related studies in different fields to date. Moreover, we succinctly highlight some lines of research that are yet to be investigated.
BarenblattGI.The mathematical theory of equilibrium cracks in brittle fracture. Adv Appl Mech1962; 7: 55–129.
6.
NeedlemanA.A continuum model for void nucleation by inclusion debonding. J Appl Mech1987; 54: 525–531.
7.
AlfanoGCrisfieldMA.Finite element interface models for the delamination analysis of laminated composites: Mechanical and computational issues. Int J Numer Meth Eng2001; 50(7): 1701–1736.
8.
GasserTCHolzapfelGA.Geometrically non-linear and consistently linearized embedded strong discontinuity models for 3D problems with an application to the dissection analysis of soft biological tissues. Comput Meth Appl Mech Eng2003; 192(47–48): 5059–5098.
9.
ParkKPaulinoGH.Cohesive zone models: A critical review of traction-separation relationships across fracture surfaces. Appl Mech Rev2013; 64(6): 060802.
10.
EringenAC.Nonlocal Continuum Field Theories. New York: Springer, 2002.
11.
SillingSA.Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids2000; 48(1): 175–209.
12.
dell’IsolaFCorteADEspositoRRussoL.Some cases of unrecognized transmission of scientific knowledge: From antiquity to Gabrio Piola’s peridynamics and generalized continuum theories. Adv Struct Mater2016; 42: 77–128.
13.
FriedE.New insights into the classical mechanics of particle systems. Discrete Continuous Dynam Syst2010; 28(4): 1469–1504.
14.
FosdickR.A causality approach to particle dynamics for systems. Arch Rat Mech Anal2013; 207: 247–302.
15.
FaillaGSantiniAZingalesM.Solution strategies for 1D elastic continuum with long-range interactions: Smooth and fractional decay. Mech Res Commun2010; 37(1): 13–21.
16.
ReddyJNSrinivasaAArbindAKhodabakhshiP.On gradient elasticity and discrete peridynamics with applications to beams and plates. Adv Mater Res2013; 745: 145–154.
17.
ReddyJNSrinivasaAR.Non-linear theories of beams and plates accounting for moderate rotations and material length scales. Int J Non-Lin Mech2014; 66: 43–53.
18.
LubineauGAzdoudYHanFReyCAskariA.A morphing strategy to couple non-local to local continuum mechanics. J Mech Phys Solids2012; 60(6): 1088–1102.
19.
AzdoudYHanFLubineauG.A morphing framework to couple non-local and local anisotropic continua. Int J Solids Struct2013; 50(9): 1332–1341.
20.
KraczekBMillerSTHaberRBJohnsonDD.Adaptive spacetime method using Riemann jump conditions for coupled atomistic-continuum dynamics. J Computat Phys2010; 229(6): 2061–2092.
21.
MurdochA.Physical Foundations of Continuum Mechanics. Cambridge: Cambridge University Press, 2012.
22.
DrapacaCSSivaloganathanS.A fractional model of continuum mechanics. J Elasticity2012; 107(2): 105–123.
23.
PoissonS.Mémoire sur l’équilibre et le mouvement des corps élastiques. Mém Acad Sci1829; 8: 357.
24.
MottPHRolandCM.Limits to Poisson’s ratio in isotropic materials—general result for arbitrary deformation. Physica Scripta2013; 87(5): 055404.
25.
MadenciEOterkusE.Peridynamic Theory and Its Applications. New York: Springer, 2014.
26.
SillingSAEptonMWecknerOXuJAskariE.Peridynamic states and constitutive modeling. J Elasticity2007; 88(2): 151–184.
27.
Podio-GuidugliP.Primer in elasticity. J Elasticity2000; 58(1): 1–104.
28.
TupekMRRadovitzkyR.An extended constitutive correspondence formulation of peridynamics based on nonlinear bond-strain measures. J Mech Phys Solids2014; 65(1): 82–92.
SillingSA.Stability of peridynamic correspondence material models and their particle discretizations. Comput Meth Appl Mech Eng2017; 322: 42–57.
31.
SillingSALehoucqRB.Peridynamic theory of solid mechanics. Adv Appl Mech2010; 44: 73–168.
32.
Ostoja-StarzewskiMDemmiePNZubelewiczA.On thermodynamic restrictions in peridynamics. J Appl Mech2012; 80(1): 014502.
33.
BobaruFFosterJTGeubellePHSillingSA.Handbook of Peridynamic Modeling. Boca Raton, FL: CRC Press, 2017.
34.
AguiarARFosdickR.A constitutive model for a linearly elastic peridynamic body. Math Mech Solids2014; 19(5): 502–523.
35.
MitchellJSillingSLittlewoodD.A position-aware linear solid constitutive model for peridynamics. J Mech Mater Struct2015; 10(5): 539–557.
36.
RenHZhuangXRabczukT.Dual-horizon peridynamics: A stable solution to varying horizons. Comput Meth Appl Mech Eng2017; 318: 762–782.
37.
RenHZhuangXCaiYRabczukT.Dual-horizon peridynamics. Int J Numer Meth Eng2016; 108: 1541–1476.
38.
SillingSLittlewoodDSelesonP.Variable horizon in a peridynamic medium. J Mech Mater Struct2015; 10(5): 591–612.
39.
QueirugaAFMoridisG.Numerical experiments on the convergence properties of state-based peridynamic laws and influence functions in two-dimensional problems. Comput Meth Appl Mech Eng2017; 322: 97–122.
40.
LeQVChanWKSchwartzJ.A two-dimensional ordinary, state-based peridynamic model for linearly elastic solids. Int J Numer Meth Eng2014; 98(8): 547–561.
41.
OuchiHKatiyarAYorkJFosterJTSharmaMM.A fully coupled porous flow and geomechanics model for fluid driven cracks: a peridynamics approach. Computat Mech2015; 55(3): 561–576.
42.
GanzenmüllerGCHiermaierSMayM. Improvements to the Prototype Micro-brittle Linear Elasticity Model of Peridynamics ( Lecture Notes in Computational Science and Engineering, Vol. 100). New York: Springer, 2013.
43.
AhadiAHanssonPMelinS.Simulating nanoindentation of thin Cu films using molecular dynamics and peridynamics. Solid State Phenomena2017; 258: 25–28.
44.
OterkusEDiyarogluCZhuNOterkusSMadenciE. Utilization of peridynamic theory for modeling at the nano-scale. In: Nanopackaging: From Nanomaterials to the Atomic Scale. Berlin: Springer, 2013, pp. 1–16.
45.
EbrahimiSSteigmannDJKomvopoulosK.Peridynamics analysis of the nanoscale friction and wear properties of amorphous carbon thin films. J Mech Mater Struct2015; 10(5): 559–572.
46.
Roy ChowdhurySMasiur RahamanMRoyDSundaramN. A micropolar peridynamic theory in linear elasticity. Int J Solids Struct2015; 59: 171–182.
47.
LehoucqRBSillingSA.Force flux and the peridynamic stress tensor. J Mech Phys Solids2008; 56(4): 1566–1577.
48.
SillingSA.A coarsening method for linear peridynamics. Int J Multiscale Computat Eng2011; 9(6): 609–622.
49.
BobaruFYangMAlvesFSillingSAAskariEXuJ.Convergence, adaptive refinement, and scaling in 1D peridynamics. Int J Numer Meth Eng2009; 77(6): 852–877.
50.
MadenciEBarutAFutchM.Peridynamic differential operator and its applications. Comput Meth Appl Mech Eng2016; 304: 408–451.
51.
BellidoJCMora-CorralCPedregalP.Hyperelasticity as a -limit of peridynamics when the horizon goes to zero. Calculus Variations Partial Differential Equations2015; 54(2): 1643–1670.
52.
SelesonPDuQParksML.On the consistency between nearest-neighbor peridynamic discretizations and discretized classical elasticity models. Comput Meth Appl Mech Eng2016; 311: 698–722.
53.
RahmanRFosterJT.Peridynamic theory of solids from the perspective of classical statistical mechanics. Physica A2015; 437: 162–183.
54.
EmmrichEPuhstD.Survey of existence results in nonlinear peridynamics in comparison with local elastodynamics. Computat Meth Appl Math2015; 15(4): 483–496.
55.
MengeshaTDuQ.Characterization of function spaces of vector fields and an application in nonlinear peridynamics. Nonlin Anal Theory Meth Applications2016; 140: 82–111.
56.
LiuWHongJW.Discretized peridynamics for linear elastic solids. Computat Mech2012; 50(5); 579–590.
OterkusSMadenciE.Peridynamics for antiplane shear and torsional deformations. J Mech Mater Struct2015; 10: 167–193.
59.
MadenciEOterkusS.Ordinary state-based peridynamics for plastic deformation according to von Mises yield criteria with isotropic hardening. J Mech Phys Solids2016; 86: 192–219.
60.
WildmanRAGazonasGA.A perfectly matched layer for peridynamics in two dimensions. J Mech Mater Struct2012; 7(8–9): 765–781.
61.
DayalKBhattacharyaK.Kinetics of phase transformations in the peridynamic formulation of continuum mechanics. J Mech Phys Solids2006; 54(9): 1811–1842.
62.
HuangDLuGQiaoP.An improved peridynamic approach for quasi-static elastic deformation and brittle fracture analysis. Int J Mech Sci2015; 94–95: 111–122.
63.
MikataY.Analytical solutions of peristatic and peridynamic problems for a 1D infinite rod. Int J Solids Struct2012; 49(21): 2887–2897.
64.
ZaccariottoMLuongoFSaregoGGalvanettoU.Examples of applications of the peridynamic theory to the solution of static equilibrium problems. Aeronaut J2015; 119(1216): 677–700.
65.
BuryachenkoVA.Effective thermoelastic properties of heterogeneous thermoperistatic bar of random structure. Int J Multiscale Computat Eng2015; 13(1): 55–71.
66.
GerstleWSillingSReadDTewaryVLehoucqR.Peridynamic simulation of electromigration. Comput Mater Continua2008; 8(2): 75–92.
67.
BobaruFDuangpanyaM.The peridynamic formulation for transient heat conduction. Int J Heat Mass Transfer2010; 53(19–20): 4047–4059.
68.
BobaruFDuangpanyaM.A peridynamic formulation for transient heat conduction in bodies with evolving discontinuities. J Computat Phys2012; 231(7): 2764–2785.
ChenZBobaruF.Selecting the kernel in a peridynamic formulation: A study for transient heat diffusion. Comput Phys Commun2015; 197: 51–60.
71.
DiyarogluCOterkusSOterkusEMadenciEHanSHwangY.Peridynamic wetness approach for moisture concentration analysis in electronic packages. Microelectron Reliab2017; 70: 103–111.
72.
TianXDuQ.Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations. SIAM J Numer Anal2013; 51(6): 3458–3482.
73.
AgwaiAGuvenIMadenciE.Crack propagation in multilayer thin-film structures of electronic packages using the peridynamic theory. Microelectron Reliab2011; 51(12): 2298–2305.
74.
KilicBMadenciE.Peridynamic theory for thermomechanical analysis. IEEE Trans Adv Packaging2010; 33(1): 97–105.
75.
RenBFanHBergelGLRegueiroRALaiXLiS.A peridynamicsâ”SPH coupling approach to simulate soil fragmentation induced by shock waves. Computat Mech2014; 55(2): 287–302.
76.
FanHLiS.A peridynamics-SPH modeling and simulation of blast fragmentation of soil under buried explosive loads. Comput Meth Appl Mech Eng2017; 318: 349–381.
77.
WildmanRAGazonasGA.A dynamic electro-thermo-mechanical model of dielectric breakdown in solids using peridynamics. J Mech Mater Struct2015; 10(5): 613–630.
78.
PrakashNSeidelGD.Electromechanical peridynamics modeling of piezoresistive response of carbon nanotube nanocomposites. Computat Mater Sci2016; 113: 154–170.
79.
KilicBMadenciE.Coupling of peridynamic theory and the finite element method. J Mech Mater Struct2010; 5(5): 707–733.
80.
OterkusEMadenciEWecknerOSillingSBogertPTesslerA.Combined finite element and peridynamic analyses for predicting failure in a stiffened composite curved panel with a central slot. Composite Struct2012; 94(3): 839–850.
81.
GalvanettoUMudricTShojaeiAZaccariottoM.An effective way to couple FEM meshes and peridynamics grids for the solution of static equilibrium problems. Mech Res Commun2016; 76: 41–47.
82.
LeeJOhSEHongJ-W.Parallel programming of a peridynamics code coupled with finite element method. Int J Fracture2017; 203(1–2): 99–114.
83.
SelesonPBeneddineSPrudhommeS.A force-based coupling scheme for peridynamics and classical elasticity. Computat Mater Sci2013; 66: 34–49.
84.
OterkusSMadenciEOterkusE.Fully coupled poroelastic peridynamic formulation for fluid-filled fractures. Eng Geol2017; 225: 19–28.
85.
TongQLiS.Multiscale coupling of molecular dynamics and peridynamics. J Mech Phys Solids2016; 95: 169–187.
86.
RahmanRFosterJTHaqueA.A multiscale modeling scheme based on peridynamic theory. J Multiscale Computat Eng2014; 12(3): 223–248.
87.
ShojaeiAMudricTZaccariottoMGalvanettoU.A coupled meshless finite point/Peridynamic method for 2D dynamic fracture analysis. Int J Mech Sci2016; 119: 419–431.
88.
HanFLubineauGAzdoudY.Adaptive coupling between damage mechanics and peridynamics: A route for objective simulation of material degradation up to complete failure. J Mech Phys Solids2016; 94: 453–472.
89.
KatiyarAFosterJTOuchiHSharmaMM.A peridynamic formulation of pressure driven convective fluid transport in porous media. J Computat Phys2014; 261: 209–229.
90.
JabakhanjiRMohtarRH.A peridynamic model of flow in porous media. Adv Water Resources2015; 78: 22–35.
91.
SillingSAParksMLKammJRWecknerORassaianM.Modeling shockwaves and impact phenomena with Eulerian peridynamics. Int J Impact Eng2017; 107: 47–57.
BobaruFHaYD.Adaptive refinement and multiscale modeling in 2D peridynamics. Int J Multiscale Computat Eng2011; 9(6): 635–660.
94.
BobaruF.Peridynamics and multiscale modeling. J Multiscale Computat Eng2011; 9(6): 7–9.
95.
AlaliBLiptonR.Multiscale dynamics of heterogeneous media in the peridynamic formulation. J Elasticity2012; 106(1): 71–103.
96.
RahmanRFosterJT.Bridging the length scales through nonlocal hierarchical multiscale modeling scheme. Computat Mater Sci2014; 92: 401–415.
97.
TalebiHSilaniMBordasSPAKerfridenPRabczukT.A computational library for multiscale modeling of material failure. Computat Mech2014; 53(5): 1047–1071.
98.
Sandia National Laboratories. LAMMPS Users Manual, 2003.
99.
DuQLiptonRMengeshaT.Multiscale analysis of linear evolution equations with applications to nonlocal models for heterogeneous media. ESAIM Math Modell Numer Anal2016; 50(5): 1425–1455.
100.
XuFGunzburgerMBurkardtJ.A multiscale method for nonlocal mechanics and diffusion and for the approximation of discontinuous functions. Comput Meth Appl Mech Eng2016; 307: 117–143.
101.
XuFGunzburgerMBurkardtJDuQ.A multiscale implementation based on adaptive mesh refinement for the nonlocal peridynamics model in one dimension. Multiscale Model Sim2016; 14(1): 398–429.
102.
YaghoobiAChorzepaMKimSDurhamSA.Mesoscale fracture analysis of multiphase cementitious composites using peridynamics. Materials2017; 10(2): 162.
103.
ShelkeABanerjeeSKunduTAmjadUGrillW.Multi-scale damage state estimation in composites using nonlocal elastic kernel: An experimental validation. Int J Solids Struct2011; 48(7–8): 1219–1228.
104.
JivkovAPYatesJR.Elastic behaviour of a regular lattice for meso-scale modelling of solids. Int J Solids Struct2012; 49(22): 3089–3099.
105.
SillingSABobaruF.Peridynamic modeling of membranes and fibers. Int J Non-Lin Mech2005; 40(2–3): 395–409.
106.
DiyarogluCOterkusEOterkusSMadenciE.Peridynamics for bending of beams and plates with transverse shear deformation. Int J Solids Struct2015; 69–70: 152–168.
107.
O’GradyJFosterJ.Peridynamic beams: A non-ordinary, state-based model. Int J Solids Struct2014; 51(18): 3177–3183.
108.
DiyarogluCOterkusEOterkusS.An Euler–Bernoulli beam formulation in an ordinary state-based peridynamic framework. Math Mech Solids2017. doi: 10.1177/1081286517728424.
109.
TaylorMSteigmannDJ.A two-dimensional peridynamic model for thin plates. Math Mech Solids2015; 20(8): 998–1010.
110.
IsakssonP.An implicit stress gradient plasticity model for describing mechanical behavior of planar fiber networks on a macroscopic scale. Eng Fract Mech2010; 77(8): 1240–1252.
111.
IsakssonPHagglundR.Acoustic emission assisted fracture zone analysis of cellulose fibre materials. J Composite Mater2012; 47(22): 2865–2874.
112.
IsakssonPDumontPJJRollandDuRoscoatS.Crack growth in planar elastic fiber materials. Int J Solids Struct2012; 49(13): 1900–1907.
113.
IsakssonPHägglundR.Strain energy distribution in a crack-tip region in random fiber networks. Int J Fracture2009; 156(1): 1–9.
114.
ChowdhurySRRoyPRoyDReddyJN.A peridynamic theory for linear elastic shells. Int J Solids Struct2016; 84: 110–132.
115.
CelikEGuvenIMadenciE.Simulations of nanowire bend tests for extracting mechanical properties. Theoret Appl Fractu Mech2011; 55(3): 185–191.
116.
LiHZhangHZhengYZhangL.A peridynamic model for the nonlinear static analysis of truss and tensegrity structures. Computat Mech2016; 57(5): 843–858.
117.
LeQVChanWKSchwartzJ.Two-dimensional peridynamic simulation of the effect of defects on the mechanical behavior of Bi2 Sr2CaCu2Ox round wires. Superconductor Sci Technol2014; 27(11): 115007.
118.
SilhavyM.Higher gradient expansion for linear isotropic peridynamic materials. Math Mech Solids2017; 22: 1483–1493.
119.
FosterJTSillingSAChenWW.Viscoplasticity using peridynamics. Int J Numer Meth Eng2010; 81(10): 1242–1258.
120.
SunSSundararaghavanV.A peridynamic implementation of crystal plasticity. Int J Solids Struct2014; 51(19): 3350–3360.
121.
WecknerONik MohamedNA.Viscoelastic material models in peridynamics. Appl Math Computat2013; 219(11): 6039–6043.
122.
MadenciEOterkusS.Ordinary state-based peridynamics for thermoviscoelastic deformation. Eng Fract Mech2017; 175: 31–45.
123.
GriebelMSchweitzerMA (eds.). Meshfree Methods for Partial Differential Equations VIII. 8th ed.New York: Springer, 2017.
124.
ChenJSHillmanMChiSW.Meshfree methods: progress made after 20 years. J Eng Mech2017; 143(4): 04017001.
125.
SillingSAAskariE.A meshfree method based on the peridynamic model of solid mechanics. Comput Struct2005; 83(17–18): 1526–1535.
GanzenmüllerGCHiermaierSMayM. On the similarity of meshless discretizations of peridynamics and smooth-particle hydrodynamics. Comput Struct2015; 150: 71–78.
O’GradyJFosterJ.A meshfree method for bending and failure in non-ordinary peridynamic shells. Computat Mech2016; 57(6): 921–929.
130.
SelesonPLittlewoodDJ.Convergence studies in meshfree peridynamic simulations. Comput Math Applicat2016; 71(11): 2432–2448.
131.
MacekRWSillingSA.Peridynamics via finite element analysis. Finite Elements Anal Design2007; 43(15): 1169–1178.
132.
ChenXGunzburgerM.Continuous and discontinuous finite element methods for a peridynamics model of mechanics. Comput Meth Appl Mech Eng2011; 200(9–12): 1237–1250.
133.
WangHTianH.A fast Galerkin method with efficient matrix assembly and storage for a peridynamic model. J Computat Phys2012; 231(23): 7730–7738.
134.
LiuZChengAWangH.An HP-Galerkin method with fast solution for linear peridynamic models in one dimension. Comput Math Applications2017; 73(7): 1546–1565.
135.
DuQJuLTianLZhouK.A posteriori error analysis of finite element method for linear nonlocal diffusion and peridynamic models. Math Computat2013; 82(284): 1889–1922.
136.
DuQTianLZhaoX.A convergent adaptive finite element algorithm for nonlocal diffusion and peridynamic models. SIAM J Numer Anal2013; 51(2): 1211–1234.
137.
TianXDuQ.Nonconforming discontinuous Galerkin methods for nonlocal variational problems. SIAM J Numer Anal2015; 53(2): 762–781.
138.
BeckmannRMellaRWenmanMR.Mesh and timestep sensitivity of fracture from thermal strains using peridynamics implemented in Abaqus. Comput Meth Appl Mech Eng2013; 263: 71–80.
139.
SchweitzerAWuS.Numerical integration of on-the-fly-computed enrichment functions in the PUM. In: GriebelMSchweitzerMA (eds), Meshfree Methods for Partial Differential Equations VII, Vol. 100. Cham: Springer, 2015, pp. 247–267.
140.
ReddyJSrinivasaA.On the force–displacement characteristics of finite elements for elasticity and related problems. Finite Elements Anal Design2015; 104: 35–40.
141.
KhodabakhshiPReddyJNSrinivasaA.GraFEA: a graph-based finite element approach for the study of damage and fracture in brittle materials. Meccanica2016; 51(12): 3129–3147.
142.
RenBWuCTAskariE.A 3D discontinuous Galerkin finite element method with the bond-based peridynamics model for dynamic brittle failure analysis. Int J Impact Eng2017; 99: 14–25.
143.
De MeoDOterkusE. Finite element implementation of a peridynamic pitting corrosion damage model. Ocean Eng2017; 135: 76–83.
144.
GerstleWSauNSillingS.Peridynamic modeling of concrete structures. Nucl Eng Design2007; 237(12–13): 1250–1258.
145.
GerstleWSauNSakhavandN.On peridynamic computational simulation of concrete structures. In: American Concrete Structure. ACI Special Publication, 2009, pp. 245–264.
146.
DiyarogluCOterkusSOterkusEMadenciE.Peridynamic modeling of diffusion by using finite-element analysis. IEEE Trans Components Packaging Manufacturing Technol2017; 7(11): 1823–1831.
147.
SadeghiradAChoppDLRenXFangELuaJ.A novel hybrid approach for level set characterization and tracking of non-planar 3D cracks in the extended finite element method. Eng Fract Mech2016; 160: 1–14.
148.
NorouzzadehAAnsariR.Finite element analysis of nano-scale Timoshenko beams using the integral model of nonlocal elasticity. Physica E2017; 88: 194–200.
149.
WangHYangDZhuS.A Petrov–Galerkin finite element method for variable-coefficient fractional diffusion equations. Comput Meth Appl Mech Eng2015; 290: 45–56.
150.
RichartNMolinariJF.Implementation of a parallel finite-element library: Test case on a non-local continuum damage model. Finite Elements Anal Design2015; 100: 41–46.
151.
LiuWYangQDMohammadizadehSSuXY.An efficient augmented finite element method for arbitrary cracking and crack interaction in solids. Int J Numer Meth Eng2014; 99(6): 438–468.
152.
LiuWYangQDMohammadizadehSSuXYLingDS.An accurate and efficient augmented finite element method for arbitrary crack interactions. J Appl Mech2013; 80(4): 041033.
153.
ZingalesMDi PaolaMInzerilloG.The finite element method for the mechanically basedmodel of non-local continuum. Int J Numer Meth Eng2011; 86(13): 1558–1576.
154.
PisanoAASofiAFuschiP.Finite element solutions for nonhomogeneous nonlocal elastic problems. Mech Res Commun2009; 36(7): 755–761.
155.
NguyenVPNguyenGDNguyenCT, et al. Modelling complex cracks with finite elements: a kinematically enriched constitutive model. Int J Fract2017; 203(1–2): 21–39.
156.
NaderiMIyyerN.3D modeling of arbitrary cracking in solids using augmented finite element method. Composite Struct2017; 160: 220–231.
157.
MoësNDolbowJBelytschkoT.A finite element method for crack growth without remeshing. Int J Numer Meth Eng1999; 46(1): 131–150.
158.
FriesTPBelytschkoT.The extended/generalized finite element method: An overview of the method and its applications. Int J Numer Meth Eng2010; 84: 253–304.
159.
AgwaiAGuvenIMadenciE.Predicting crack propagation with peridynamics: A comparative study. Int J Fract2011; 171(1): 65–78.
160.
FosterJSillingSAChenW.An energy based failure criterion for use with peridynamic states. Int J Multiscale Computat Eng2011; 9(6): 675–688.
161.
KilicBMadenciE.Structural stability and failure analysis using peridynamic theory. Int J Non-Lin Mech2009; 44(8): 845–854.
162.
IbrahimRA.Recent advances of structural life assessment and related problems. Springer Proc Physics2015; 168: 1–27.
163.
AskariAAzdoudYHanFLubineauGSillingS.Peridynamics for analysis of failure in advanced composite materials. In: CamanhoPHallettS (eds.), Numerical Modelling of Failure in Advanced Composite Materials. Woodhead Publishing, 2015, pp. 331–350.
164.
Yi-leHYinYWangH.Peridynamic analytical method for progressive damage in notched composite laminates. Composite Struct2014; 108(1): 801–810.
165.
JungJSeokJ.Fatigue crack growth analysis in layered heterogeneous material systems using peridynamic approach. Composite Struct2016; 152: 403–407.
166.
EmmrichEPuhstD.A short note on modeling damage in peridynamics, J Elasticity2016; 123(2): 245–252.
167.
De MeoDDiyarogluCZhuNOterkusESiddiqMA.Modelling of stress-corrosion cracking by using peridynamics. Int J Hydrogen Energy2016; 41(15): 6593–6609.
ChenZZhangGBobaruF.The influence of passive film damage on pitting corrosion. J Electrochem Soc2016; 163(2): C19–C24.
170.
JiangHHeLFanLZhanG.Numerical analysis method of cemented carbide turning tool’s micro breakage based on peridynamic theory. Int J Adv Manufacturing Technol2017; 88(5–8): 1619–1628.
171.
KilicBMadenciE.Prediction of crack paths in a quenched glass plate by using peridynamic theory. Int J Fracture2009; 156(2): 165–177.
172.
HaYDBobaruF.Characteristics of dynamic brittle fracture captured with peridynamics. Eng Fract Mech2011; 78(6): 1156–1168.
173.
ShiY.Creating atomic models of brittle glasses for in silico mechanical tests. Int J Appl Glass Sci2016; 7(4): 464–473.
174.
LiuWHongJ-W.Discretized peridynamics for brittle and ductile solids. Int J Numer Meth Eng2012; 89(8): 1028–1046.
175.
SillingSA.Dynamic fracture modeling with a meshfree peridynamic code. Computat Fluid Solid Mech2003; 1: 641–644.
176.
SillingSAWecknerOAskariEBobaruF.Crack nucleation in a peridynamic solid. Int J Fract2010; 162(1–2): 219–227.
177.
HaYDBobaruF.Studies of dynamic crack propagation and crack branching with peridynamics. Int J Fract2010; 162(1–2): 229–244.
178.
BobaruFHuW.The meaning, selection, and use of the peridynamic horizon and its relation to crack branching in brittle materials. Int J Fract2012; 176(2): 215–222.
179.
DipasqualeDZaccariottoMGalvanettoU.Crack propagation with adaptive grid refinement in 2D peridynamics. Int J Fract2014; 190(1–2): 1–22.
180.
LiptonR.Cohesive dynamics and brittle fracture. J Elasticity2016; 124(2): 143–191.
181.
PanchadharaRGordonPA.Application of peridynamic stress intensity factors to dynamic fracture initiation and propagation. Int J Fract2016; 201(1): 81–96.
182.
ZhouXWangYXuX.Numerical simulation of initiation, propagation and coalescence of cracks using the non-ordinary state-based peridynamics. Int J Fract2016; 201(2): 213–234.
183.
OterkusSMadenciE.Peridynamic modeling of fuel pellet cracking. Eng Fract Mech2017; 176: 23–37.
184.
MadenciEColavitoKPhanN.Peridynamics for unguided crack growth prediction under mixed-mode loading. Eng Fract Mech2016; 167: 34–44.
185.
LeeJHongJW.Dynamic crack branching and curving in brittle polymers. Int J Solids Struct2016; 100–101: 332–340.
186.
BobaruF.Influence of van der Waals forces on increasing the strength and toughness in dynamic fracture of nanofibre networks: a peridynamic approach. Modell Sim Mater Sci Eng2007; 15(5): 397–417.
187.
HuangDLuGLiuY.Nonlocal peridynamic modeling and simulation on crack propagation in concrete structures. Math Problems Eng2015: 1–11.
188.
HuWHaYDBobaruF.Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites. Comput Meth Appl Mech Eng2012; 217–220: 247–261.
189.
GhajariMIannucciLCurtisP.A peridynamic material model for the analysis of dynamic crack propagation in orthotropic media. Comput Meth Appl Mech Eng2014; 276: 431–452.
190.
ZhangGLeQLoghinASubramaniyanABobaruF.Validation of a peridynamic model for fatigue cracking. Eng Fract Mech2016; 162: 76–94.
191.
ChengZZhangGWangYBobaruF.A peridynamic model for dynamic fracture in functionally graded materials. Composite Struct2015; 133: 529–546.
192.
De MeoDZhuNOterkusE. Peridynamic modeling of granular fracture in polycrystalline materials. J Eng Mater Technol2016; 138(4): 041008.
193.
WangYZhouXXuX.Numerical simulation of propagation and coalescence of flaws in rock materials under compressive loads using the extended non-ordinary state-based peridynamics. Eng Fract Mech2016; 163: 248–273.
194.
ZhouXPShouYDNumerical simulation of failure of rock-like material subjected to compressive loads using improved peridynamic method. Int J Geomech2017; 17(3): 04016086.
195.
HaYDLeeJHongJW.Fracturing patterns of rock-like materials in compression captured with peridynamics. Eng Fract Mech2015; 144: 176–193.
196.
LeeJHaYDHongJW.Crack coalescence morphology in rock-like material under compression. Int J Fract2017; 203(1–2): 211–236.
197.
EvangelatosGISpanosPD.A collocation approach for spatial discretization of stochastic peridynamic modeling of fracture. J Mech Mater Struct2011; 6(7–8): 1171–1195.
198.
HuYMadenciE.Peridynamics for fatigue life and residual strength prediction of composite laminates. Composite Struct2017; 160: 169–184.
199.
SillingSAAskariA.Peridynamic model for fatigue cracks. Technical Report 18590, Sandia National Laboratories, 2014.
200.
SillingSAAskariE. Peridynamic modeling of impact damage. In: ASME Pressure Vessels and Piping Conference, Problems Involving Thermal Hydraulics, Liquid Sloshing, and Extreme Loads on Structures, Vol. 489, 2004, pp. 197–205.
201.
DemmiePNSillingSA.An approach to modeling extreme loading of structures using peridynamics. J Mech Mater Struct2007; 2(10): 1921–1945.
202.
OterkusEGuvenIMadenciE.Impact damage assessment by using peridynamic theory. Central Eur J Eng2012; 2(4): 523–531.
203.
SunCHuangZ.Peridynamic simulation to impacting damage in composite laminate. Composite Struct2016; 138: 335–341.
204.
BobaruFHaYDHuW.Damage progression from impact in layered glass modeled with peridynamics. Central Eur J Eng2012; 2(4): 551–561.
205.
HuWWangYYuJYenCFBobaruF.Impact damage on a thin glass plate with a thin polycarbonate backing. Int J Impact Eng2013; 62: 152–165.
206.
DiehlPSchweitzerA.Simulation of wave propagation and impact damage in brittle materials using peridynamics. In: MehlMBischoffMSchäferM (eds), Recent Trends in Computational Engineering - CE2014: Optimization, Uncertainty, Parallel Algorithms, Coupled and Complex Problems. Berlin: Springer, 2015, pp. 251–265.
207.
LiuNLiuDZhouW.Peridynamic modelling of impact damage in three-point bending beam with offset notch. Appl Math Mech2017; 38(1): 99–110.
208.
LeeJLiuWHongJW.Impact fracture analysis enhanced by contact of peridynamic and finite element formulations. Int J Impact Eng2016; 87: 108–119.
209.
KosteskiLEIturriozICisilinoAP, et al. A lattice discrete element method to model the falling-weight impact test of PMMA specimens. Int J Impact Eng2016; 87: 120–131.
210.
DemmiePNOstoja-StarzewskiM.Local and nonlocal material models, spatial randomness, and impact loading. Arch Appl Mech2016; 86(1–2): 39–58.
OmidvarMIskanderMBlessS.Response of granular media to rapid penetration. Int J Impact Eng2014; 66: 60–82.
213.
WarrenTLHanchakSJPoormonKL.Penetration of limestone targets by ogive-nosed VAR 4340 steel projectiles at oblique angles: Experiments and simulations. Int J Impact Eng2004; 30(10): 1307–1331.
214.
BorgJPVoglerTJ.Mesoscale calculations of the dynamic behavior of a granular ceramic. Int J Solids Struct2008; 45(6): 1676–1696.
215.
SillingSA.Origin and effect of nonlocality in a composite. J Mech Mater Struct2014; 9(2): 245–258.
216.
BuryachenkoVA.Some general representations in thermoperistatics of random structure composites. Int J Multiscale Computat Eng2014; 12(4): 331–350.
217.
KilicBAgwaiAMadenciE.Peridynamic theory for progressive damage prediction in center-cracked composite laminates. Composite Struct2009; 90(2): 141–151.
218.
HuWHaYDBobaruF.Modeling dynamic fracture and damage in a fiber-reinforced composite lamina with peridynamics. Int J Multiscale Computat Eng2011; 9(6): 707–726.
219.
OterkusEMadenciE.Peridynamic theory for damage initiation and growth in composite laminate. Key Eng Mater2012; 488–489(1): 355–358.
220.
OterkusEMadenciE.Peridynamic analysis of fiber-reinforced composite materials. J Mech Mater Struct2012; 7(1): 47–84.
221.
HuYLDe CarvalhoNVMadenciE. Peridynamic modeling of delamination growth in composite laminates. Composite Struct132(2015) 610–620.
222.
DiyarogluCOterkusEMadenciERabczukTSiddiqA.Peridynamic modeling of composite laminates under explosive loading. Composite Structures2016; 144: 14–23.
223.
SadowskiTPankowskiB.Peridynamical modelling of nanoindentation in ceramic composites. Solid State Phenomena2016; 254: 55–59.
224.
ShenFZhangQHuangD.Damage and failure process of concrete structure under uniaxial compression based on peridynamics modeling. Math Problems Eng2013: 1–5.
225.
YaghoobiAChorzepaMG.Fracture analysis of fiber reinforced concrete structures in the micropolar peridynamic analysis framework. Eng Fract Mech2017; 169: 238–250.
226.
DuzziMZaccariottoMGalvanettoU.Application of peridynamic theory to nanocomposite materials. Adv Mater Res2014; 1016: 44–48.
227.
HanFAzdoudYLubineauG.Computational modeling of elastic properties of carbon nanotube/polymer composites with interphase regions. Part II: Mechanical modeling. Computat Mater Sci2014; 81: 652–661.
228.
DeckleverJSpanosP.Nanocomposite material properties estimation and fracture analysis via peridynamics and Monte Carlo simulation. Probabil Eng Mech2015; 44: 77–88.
229.
DengQChenYLeeJ.An investigation of the microscopic mechanism of fracture and healing processes in cortical bone. Int J Damage Mech2009; 18(5): 491–502.
230.
TaylorMGözenIPatelSJesorkaABertoldiK.Peridynamic modeling of ruptures in biomembranes. PLoS ONE2016; 11(11): e0165947.
231.
PerréPAlmeidaGAyouzMFrankX.New modelling approaches to predict wood properties from its cellular structure: image-based representation and meshless methods. Ann Forest Sci2016; 73(1): 147–162.
232.
LejeuneELinderC.Modeling tumor growth with peridynamics. Biomech Model Mechanobiol2017; 16(4): 1141–1157.
233.
LejeuneELinderC.Quantifying the relationship between cell division angle and morphogenesis through computational modeling. J Theoret Biol2017; 418: 1–7.
234.
LejeuneELinderC.Modeling mechanical inhomogeneities in small populations of proliferating monolayers and spheroids. Biomech Model Mechanobiol2018; in press.
235.
LejeuneELinderC.Understanding the relationship between cell death and tissue shrinkage via an agent-based model. J Biomech2018; 7: 9–17.
236.
JaviliADortdivanliogluBKuhlELinderC.Computational aspects of growth-induced instabilities through eigenvalue analysis. Computat Mech2015; 56: 405–420.
237.
KuhlE.Unfolding the brain. Nat Phys2016; 12(6): 533–534.
238.
HattoriGTrevelyanJAugardeCECoombsWMAplinAC.Numerical simulation of fracking in shale rocks: current state and future approaches. Arch Computat Meth Eng2017; 24(2): 281–317.
239.
TurnerDZ.A non-local model for fluid-structure interaction with applications in hydraulic fracturing. Int J Computat Meth Eng Sci Mech2013; 14(5): 391–400.
240.
NadimiSMiscovicIMcLennanJ.A 3D peridynamic simulation of hydraulic fracture process in a heterogeneous medium. J Petrol Sci Eng2016; 145: 444–452.
241.
WuFLiSDuanQLiX.Application of the method of peridynamics to the simulation of hydraulic fracturing process. Springer Proc Phys2017; 188: 561–569.
242.
PanchadharaRGordonPAParksML.Modeling propellant-based stimulation of a borehole with peridynamics. Int J Rock Mech Mining Sci2017; 93: 330–343.
243.
LaiXRenBFanH, et al. Peridynamics simulations of geomaterial fragmentation by impulse loads. Int J Numer Analyt Meth Geomech2015; 39(12): 1304–1330.
244.
FengYChenXXuXF.Current status and potentials of enhanced geothermal system in China: A review. Renew Sustain Energy Rev2014; 33: 214–223.
245.
ZhouXPGuXBWangYT.Numerical simulations of propagation, bifurcation and coalescence of cracks in rocks. Int J Rock Mech Mining Sci2015; 80: 241–254.
246.
ZingalesM.Wave propagation in 1D elastic solids in presence of long-range central interactions. J Sound Vibr2011; 330(16): 3973–3989.
247.
ButtSNTimothyJJMeschkeG.Wave dispersion and propagation in state-based peridynamics. Computat Mech2017; 60: 725–738.
248.
BeyerHRAksoyluBCelikerF.On a class of nonlocal wave equations from applications. J Math Phys2014; 57(6): 062902.
249.
BazantZPLuoWChauVTBessaMA.Wave dispersion and basic concepts of peridynamics compared to classical nonlocal damage models. J Appl Mech2016; 83(11): 111004.
250.
RahmanRFosterJT.Onto resolving spurious wave reflection problem with changing nonlocality among various length scales. Commun Nonlin Sci Numer Sim2016; 34: 86–122.
251.
NishawalaVVOstoja-StarzewskiMLeamyMJDemmiePN.Simulation of elastic wave propagation using cellular automata and peridynamics, and comparison with experiments. Wave Motion2016; 60: 73–83.
252.
SillingSA.Solitary waves in a peridynamic elastic solid. J Mech Phys Solids2016; 96: 121–132.
253.
VoglerTJBorgJPGradyDE.On the scaling of steady structured waves in heterogeneous materials. J Appl Phys2012; 112(12): 123507.
254.
WildmanRAGazonasGA.A finite difference-augmented peridynamics method for reducing wave dispersion. Int J Fract2014; 190(1–2): 39–52.
255.
MartowiczAStaszewskiWJRuzzeneMUhlT. Peridynamics as an analysis tool for wave propagation in graphene nanoribbons. In: Proc SPIE2015; 9435: .
256.
KavianiFMirdamadiHR.Wave propagation analysis of carbon nano-tube conveying fluid including slip boundary condition and strain/inertial gradient theory. Comput Struct2013; 116: 75–87.
257.
ParksMLLehoucqRBPlimptonSJSillingSAImplementing peridynamics within a molecular dynamics code. Comput Phys Commun2008; 179(11): 777–783.
258.
DaiZBessaMALiSLiuWK.Particle method modeling of nonlocal multiresolution continua. In: GriebelMSchweitzerMA (eds), Meshfree Methods for Partial Differential Equations VII, Vol. 100. Cham: Springer, 2015, pp. 43–60.
259.
Podio-GuidugliP.On the modeling of transport phenomena in continuum and statistical mechanics. Discrete Continuous Dynam Syst Ser S2017; 10(6): 1393–1411.
260.
MargolinLG.Finite scale theory: The role of the observer in classical fluid flow. Mech Res Commun2014; 57: 10–17.
HenkeSFShanbhagS.Mesh sensitivity in peridynamic simulations. Comput Phys Commun2014; 185(1): 181–193.
263.
FreimanisAPaeglitisA.Mesh sensitivity in peridynamic quasi-static simulations. Procedia Eng2017; 172: 284–291.
264.
ParksMLLittlewoodDJMitchellJASillingSA.Peridigm Users’ Guide. Sandia National Laboratories, September2012.
265.
BrothersMDFosterJTMillwaterHR.A comparison of different methods for calculating tangent-stiffness matrices in a massively parallel computational peridynamics code. Comput Meth Appl Mech Eng2014; 279: 247–267.
266.
KilicBMadenciE.An adaptive dynamic relaxation method for quasi-static simulations using the peridynamic theory. Theoret Appl Fract Mech2010; 53(3): 194–204.
267.
YuKXinXJLeaseKB.A new adaptive integration method for the peridynamic theory. Modell Sim Mater Sci Eng2011; 19(4): 045003.
268.
SelesonP.Improved one-point quadrature algorithms for two-dimensional peridynamic models based on analytical calculations. Comput Meth Appl Mech Eng2014; 282: 184–217.
269.
GuXMHuangTZZhaoXLXuWRLiHBLiL.Circulant preconditioned iterative methods for peridynamic model simulation. Appl Math Computat2014; 248: 470–479.
