The spectral approach is used to examine the wave dispersion in linearized bond-based and state-based peridynamics in one and two dimensions, and comparisons with the classical nonlocal models for damage are made. Similar to the classical nonlocal models, the peridynamic dispersion of elastic waves occurs for high frequencies. It is shown to be stronger in the state-based than in the bond-based version, with multiple wavelengths giving a vanishing phase velocity, one of them longer than the horizon. In the bond-based and state-based, the nonlocality of elastic and inelastic behaviors is coupled, i.e., the dispersion of elastic and inelastic waves cannot be independently controlled. In consequence, the difference between: (1) the nonlocality due to material characteristic length for softening damage, which ensures stability of softening damage and serves as the localization limiter, and (2) the nonlocality due to material heterogeneity cannot be distinguished. This coupling of both kinds of dispersion is unrealistic and similar to the original 1984 nonlocal model for damage which was in 1987 abandoned and improved to be nondispersive or mildly dispersive for elasticity but strongly dispersive for damage. With the same regular grid of nodes, the convergence rates for both the bond-based and state-based versions are found to be slower than for the finite difference methods. It is shown that there exists a limit case of peridynamics, with a micromodulus in the form of a Delta function spiking at the horizon. This limit case is equivalent to the unstabilized imbricate continuum and exhibits zero-energy periodic modes of instability. Finally, it is emphasized that the node-skipping force interactions, a salient feature of peridynamics, are physically unjustified (except on the atomic scale) because in reality the forces get transmitted to the second and farther neighboring particles (or nodes) through the displacements and rotations of the intermediate particles, rather than by some potential permeating particles as on the atomic scale.

References

References
1.
Burt
,
N. J.
, and
Dougill
,
J. W.
,
1977
, “
Progressive Failure in a Model Heterogeneous Medium
,”
J. Eng. Mech. Div.
,
103
(
3
), pp.
365
376
.
2.
Demmie
,
P.
, and
Silling
,
S.
,
2007
, “
An Approach to Modeling Extreme Loading of Structures Using Peridynamics
,”
J. Mech. Mater. Struct.
,
2
(
10
), pp.
1921
1945
.
3.
Foster
,
J. T.
,
Silling
,
S. A.
, and
Chen
,
W. W.
,
2010
, “
Viscoplasticity Using Peridynamics
,”
Int. J. Numer. Methods Eng.
,
81
(
10
), pp.
1242
1258
.
4.
Foster
,
J. T.
,
Silling
,
S. A.
, and
Chen
,
W.
,
2011
, “
An Energy Based Failure Criterion for Use With Peridynamic States
,”
Int. J. Multiscale Comput. Eng.
,
9
(
6
), pp.
675
688
.
5.
Gerstle
,
W.
,
Sau
,
N.
, and
Silling
,
S.
,
2007
, “
Peridynamic Modeling of Concrete Structures
,”
Nucl. Eng. Des.
,
237
(
12
), pp.
1250
1258
.
6.
Gerstle
,
W.
,
2015
,
Introduction to Practical Peridynamics: Computational Solid Mechanics Without Stress and Strain
,
World Scientific
,
Hong Kong
.
7.
Liu
,
W.
, and
Hong
,
J. W.
,
2012
, “
Discretized Peridynamics for Linear Elastic Solids
,”
Comput. Mech.
,
50
(
5
), pp.
579
590
.
8.
Macek
,
R. W.
, and
Silling
,
S. A.
,
2007
, “
Peridynamics Via Finite Element Analysis
,”
Finite Elements Anal. Des.
,
43
(
15
), pp.
1169
1178
.
9.
Madenci
,
E.
, and
Oterkus
,
E.
,
2014
,
Peridynamic Theory and Its Applications
, Vol.
17
,
Springer
,
New York
.
10.
Nishawala
,
V. V.
,
Ostoja-Starzewski
,
M.
,
Leamy
,
M. J.
, and
Demmie
,
P. N.
,
2016
, “
Simulation of Elastic Wave Propagation Using Cellular Automata and Peridynamics, and Comparison With Experiments
,”
Wave Motion
,
60
, pp.
73
83
.
11.
Ostoja-Starzewski
,
M.
,
Demmie
,
P. N.
, and
Zubelewicz
,
A.
,
2013
, “
On Thermodynamic Restrictions in Peridynamics
,”
ASME J. Appl. Mech.
,
80
(
1
), p.
014502
.
12.
Parks
,
M. L.
,
Seleson
,
P.
,
Plimpton
,
S. J.
,
Silling
,
S. A.
, and
Lehoucq
,
R. B.
,
2010
, “
Peridynamics With Lammps: A User Guide v0. 2 Beta
,” Sandia National Laboratories, Albuquerque, NM, pp.
635
662
.
13.
Silling
,
S. A.
,
2000
, “
Reformulation of Elasticity Theory for Discontinuities and Long-Range Forces
,”
J. Mech. Phys. Solids
,
48
(
1
), pp.
175
209
.
14.
Silling
,
S. A.
, and
Askari
,
E.
,
2005
, “
A Meshfree Method Based on the Peridynamic Model of Solid Mechanics
,”
Comput. Struct.
,
83
(
17
), pp.
1526
1535
.
15.
Silling
,
S. A.
,
Epton
,
M.
,
Weckner
,
O.
,
Xu
,
J.
, and
Askari
,
E.
,
2007
, “
Peridynamic States and Constitutive Modeling
,”
J. Elasticity
,
88
(
2
),
151
184
.
16.
Silling
,
S. A.
, and
Lehoucq
,
R. B.
,
2008
, “
Convergence of Peridynamics to Classical Elasticity Theory
,”
J. Elasticity
,
93
(
1
), pp.
13
37
.
17.
Silling
,
S. A.
, and
Lehoucq
,
R. B.
,
2010
, “
Peridynamic Theory of Solid Mechanics
,”
Adv. Appl. Mech.
,
44
, pp.
73
168
.
18.
Bessa
,
M. A.
,
Foster
,
J. T.
,
Belytschko
,
T.
, and
Liu
,
W. K.
,
2014
, “
A Meshfree Unification: Reproducing Kernel Peridynamics
,”
Comput. Mech.
,
53
(
6
), pp.
1251
1264
.
19.
Bažant
,
Z. P.
,
Belytschko
,
T. B.
, and
Chang
,
T.-P.
,
1984
, “
Continuum Theory for Strain-Softening
,”
J. Eng. Mech.
,
110
(
12
), pp.
1666
1692
.
20.
Bažant
,
Z. P.
,
1984
, “
Imbricate Continuum and Its Variational Derivation
,”
J. Eng. Mech.
,
110
(
12
), pp.
1693
1712
.
21.
Pijaudier-Cabot
,
G.
, and
Bažant
,
Z. P.
,
1987
, “
Nonlocal Damage Theory
,”
J. Eng. Mech.
,
113
(
10
), pp.
1512
1533
.
22.
Bažant
,
Z. P.
, and
Pijaudier-Cabot
,
G.
,
1988
, “
Nonlocal Continuum Damage, Localization Instability and Convergence
,”
ASME J. Appl. Mech.
,
55
(
2
), pp.
287
293
.
23.
Bažant
,
Z.
, and
Lin
,
F.-B.
,
1988
, “
Non-Local Yield Limit Degradation
,”
Int. J. Numer. Methods Eng.
,
26
(
8
), pp.
1805
1823
.
24.
Weckner
,
O.
, and
Abeyaratne
,
R.
,
2005
, “
The Effect of Long-Range Forces on the Dynamics of a Bar
,”
J. Mech. Phys. Solids
,
53
(
3
), pp.
705
728
.
25.
Zimmermann
,
M.
,
2005
, “
A Continuum Theory With Long-Range Forces for Solids
,” Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA.
26.
Jirásek
,
M.
,
2004
, “
Nonlocal Theories in Continuum Mechanics
,”
Acta Polytech.
,
44
(
5–6
), pp.
16
34
.
27.
Bažant
,
Z. P.
, and
Cedolin
,
L.
,
1991
,
Stability of Structures: Elastic, Inelastic, Fracture, and Damage Theories
,
Oxford University Press
,
New York
.
28.
Bažant
,
Z. P.
,
1994
, “
Nonlocal Damage Theory Based on Micromechanics of Crack Interactions
,”
J. Eng. Mech.
,
120
(
3
), pp.
593
617
.
29.
Bažant
,
Z. P.
, and
Jirasek
,
M.
,
1994
, “
Nonlocal Model Based on Crack Interactions: A Localization Study
,”
J. Eng. Mater. Technol.
,
116
(
3
), pp.
256
259
.
30.
Jirásek
,
M.
, and
Bažant
,
Z. P.
,
1994
, “
Localization Analysis of Nonlocal Model Based on Crack Interactions
,”
J. Eng. Mech.
,
120
(
7
), pp.
1521
1542
.
31.
Ozbolt
,
J.
, and
Bažant
,
Z. P.
,
1996
, “
Numerical Smeared Fracture Analysis: Nonlocal Microcrack Interaction Approach
,”
Int. J. Numer. Methods Eng.
,
39
(
4
), pp.
635
662
.
32.
Bažant
,
Z. P.
, and
Planas
,
J.
,
1997
,
Fracture and Size Effect in Concrete and Other Quasibrittle Materials
, Vol.
16
,
CRC Press
,
Boca Raton, FL
.
33.
Bažant
,
Z. P.
, and
Jirásek
,
M.
,
2002
, “
Nonlocal Integral Formulations of Plasticity and Damage: Survey of Progress
,”
J. Eng. Mech.
,
128
(
11
), pp.
1119
1149
.
34.
Cusatis
,
G.
,
Pelessone
,
D.
, and
Mencarelli
,
A.
,
2011
, “
Lattice Discrete Particle Model (ldpm) for Failure Behavior of Concrete. I: Theory
,”
Cem. Concr. Compos.
,
33
(
9
), pp.
881
890
.
35.
Demmie
,
P. N.
, and
Ostoja-Starzewski
,
M.
,
2016
, “
Local and Nonlocal Material Models, Spatial Randomness, and Impact Loading
,”
Arch. Appl. Mech.
,
86
(
1–2
), pp.
39
58
.
36.
Cusatis
,
G.
,
Bažant
,
Z. P.
, and
Cedolin
,
L.
,
2003
, “
Confinement-Shear Lattice Model for Concrete Damage in Tension and Compression: I. Theory
,”
J. Eng. Mech.
,
129
(
12
), pp.
1439
1448
.
37.
Zubelewicz
,
A.
, and
Bažant
,
Z. P.
,
1987
, “
Interface Modeling of Fracture in Aggregate Composites
,”
J. Eng. Mech.
,
113
(
11
), pp.
1619
1630
.
38.
Bažant
,
Z. P.
,
1978
, “
Spurious Reflection of Elastic Waves in Nonuniform Finite Element Grids
,”
Comput. Methods Appl. Mech. Eng.
,
16
(
1
), pp.
91
100
.
39.
Bažant
,
Z. P.
, and
Celep
,
Z.
,
1982
, “
Spurious Reflection of Elastic Waves in Nonuniform Meshes of Constant and Linear Strain Unite Elements
,”
Comput. Struct.
,
15
(
4
), pp.
451
459
.
40.
Bažant
,
Z. P.
, and
Chang
,
T.-P.
,
1984
, “
Instability of Nonlocal Continuum and Strain Averaging
,”
J. Eng. Mech.
,
110
(
10
), pp.
1441
1450
.
41.
Liu
,
W. K.
,
Jun
,
S.
, and
Zhang
,
Y. F.
,
1995
, “
Reproducing Kernel Particle Methods
,”
Int. J. Numer. Methods Fluids
,
20
(
8–9
), pp.
1081
1106
.
42.
Achenbach
,
J.
,
Wave Propagation in Elastic Solids
, Vol.
16
,
Elsevier
,
North Holland, Amsterdam, The Netherlands
.
43.
Cemal Eringen
,
A.
,
1972
, “
Linear Theory of Nonlocal Elasticity and Dispersion of Plane Waves
,”
Int. J. Eng. Sci.
,
10
(
5
), pp.
425
435
.
44.
Cemal Eringen
,
A.
,
1983
, “
Theories of Nonlocal Plasticity
,”
Int. J. Eng. Sci.
,
21
(
7
), pp.
741
751
.
45.
Cemal Eringen
,
A.
, and
Edelen
,
D. G. B.
,
1972
, “
On Nonlocal Elasticity
,”
Int. J. Eng. Sci.
,
10
(
3
), pp.
233
248
.
46.
Kröner
,
E.
, “
Elasticity Theory of Materials With Long Range Cohesive Forces
,”
Int. J. Solids Struct.
,
3
(
5
), pp.
731
742
.
47.
Kunin
,
I. A.
,
1966
, “
Theory of Elasticity With Spatial Dispersion- One-Dimensional Complex Structure (Equations for One-Dimensional Model of Macroscopically Homogeneous Linearly Elastic Medium of Complex Structure With Spatial Dispersion)
,”
Prikladnaia Matematika i Mekhanika
,
30
, pp.
866
874
.
48.
David Mindlin
,
R.
,
1964
, “
Micro-Structure in Linear Elasticity
,”
Arch. Ration. Mech. Anal.
,
16
(
1
), pp.
51
78
.
49.
Bažant
,
Z. P.
, and
Chang
,
T.-P.
,
1987
, “
Nonlocal Finite Element Analysis of Strain-Softening Solids
,”
J. Eng. Mech.
,
113
(
1
), pp.
89
105
.
50.
Bažant
,
Z. P.
,
Shang-Ping
,
B.
, and
Ravindra
,
G.
,
1993
, “
Fracture of Rock: Effect of Loading Rate
,”
Eng. Fract. Mech.
,
45
(
3
), pp.
393
398
.
51.
Bažant
,
Z. P.
, and
Oh
,
B. H.
,
1983
, “
Crack Band Theory for Fracture of Concrete
,”
Matériaux et Constr.
,
16
(
3
), pp.
155
177
.
52.
Hirt
,
C. W.
,
1968
, “
Heuristic Stability Theory for Finite-Difference Equations
,”
J. Comput. Phys.
,
2
(
4
), pp.
339
355
.
53.
Wesley Hamming
,
R.
,
1989
,
Digital Filters
,
Oxford University Press
,
New York
.
54.
Silling
,
S. A.
, and
Lehoucq
,
R. B.
,
2010
, “
Peridynamic Theory of Solid Mechanics
,”
Adv. Appl. Mech.
,
44
, pp.
73
168
.
55.
Červenka
,
J.
,
Bažant
,
Z. P.
, and
Wierer
,
M.
,
2005
, “
Equivalent Localization Element for Crack Band Approach to Mesh-Sensitivity in Microplane Model
,”
Int. J. Numer. Methods Eng.
,
62
(
5
), pp.
700
726
.
56.
Bažant
,
Z. P.
,
Le
,
J.-L.
, and
Hoover
,
C. G.
,
2010
, “
Nonlocal Boundary Layer (NBL) Model: Overcoming Boundary Condition Problems in Strength Statistics and Fracture Analysis of Quasibrittle Materials
,”
Fracture Mechanics of Concrete and Concrete Structures—Recent Advances in Fracture Mechanics of Concrete
, B.-H. Oh, Ed.,
Korea Concrete Institute
,
Seoul, Korea
, pp.
135
143
.
57.
Bažant
,
Z. P.
,
1976
, “
Instability, Ductility, and Size Effect in Strain-Softening Concrete
,”
J. Eng. Mech. Div.
,
102
(
2
), pp.
331
344
.
58.
Irwin
,
G. R.
,
1958
,
Fracture in Handbuch der Physik
, Vol. v,
Springer
,
New York
.
59.
Peerlings
,
R. H. J.
,
De Borst
,
R.
,
Brekelmans
,
W. A. M.
, and
Geers
,
M. G. D.
,
2002
, “
Localisation Issues in Local and Nonlocal Continuum Approaches to Fracture
,”
Eur. J. Mech.-A/Solids
,
21
(
2
), pp.
175
189
.
60.
Bažant
,
Z. P.
, and
Beghini
,
A.
,
2005
, “
Which Formulation Allows Using a Constant Shear Modulus for Small-Strain Buckling of Soft-Core Sandwich Structures?
,”
ASME J. Appl. Mech.
,
72
(
5
), pp.
785
787
.
61.
Bažant
,
Z. P.
,
Adley
,
M. D.
,
Carol
,
I.
,
Jirásek
,
M.
,
Akers
,
S. A.
,
Rohani
,
B.
,
Cargile
,
J. D.
, and
Caner
,
F. C.
,
2002
, “
Large-Strain Generalization of Microplane Model for Concrete and Application
,”
J. Eng. Mech.
,
126
(
9
), pp.
971
980
.
62.
Adley
,
M. D.
,
Frank
,
A. O.
, and
Danielson
,
K. T.
,
2012
, “
The High-Rate Brittle Microplane Concrete Model: Part I: Bounding Curves and Quasi-Static Fit to Material Property Data
,”
Comput. Concr.
,
9
(
4
), pp.
293
310
.
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