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Research Papers

Wave Dispersion and Basic Concepts of Peridynamics Compared to Classical Nonlocal Damage Models

[+] Author and Article Information
Zdeněk P. Bažant

McCormick Institute Professor and W.P. Murphy
Professor of Civil and Mechanical Engineering
and Materials Science,
Northwestern University,
2145 Sheridan Road,
CEE/A135,
Evanston, IL 60208
e-mail: z-bazant@northwestern.edu

Wen Luo, Viet T. Chau, Miguel A. Bessa

Northwestern University,
2145 Sheridan Road,
CEE/A135,
Evanston, IL 60208

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received June 26, 2016; final manuscript received July 23, 2016; published online August 30, 2016. Editor: Yonggang Huang.

J. Appl. Mech 83(11), 111004 (Aug 30, 2016) (16 pages) Paper No: JAM-16-1324; doi: 10.1115/1.4034319 History: Received June 26, 2016; Revised July 23, 2016

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.

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Figures

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Fig. 1

(a) Schematic illustration of how particles in a horizon interact through bond forces. (b) Nodal pair vectors in original and deformed position and displacement vectors of paired nodes.

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Fig. 2

Comparison of phase velocity for wave propagation in bond-based peridynamic (PD) continuum: (1) 1D bond-based peridynamic continuum with uniform micromodulus; and (2) 2D bond-based peridynamic continuum with uniform micromodulus. The dashed line corresponds to local model (classical elasticity).

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Fig. 3

Comparison of phase velocity for wave propagation in state-based peridynamic (PD) continua: (1) 1D state-based peridynamic continuum with uniform micromodulus; and (2) 2D state-based peridynamic continuum with uniform micromodulus. The dashed line corresponds to the local model (classical elasticity).

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Fig. 4

Comparison of phase velocity for 1D wave propagation: (1) bond-based peridynamic continuum, and (2) state-based peridynamic continuum (both with uniform micromodulus). The dashed line corresponds to classical elasticity.

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Fig. 5

Comparison of dispersion diagram for 1D wave propagation: (1) bond-based peridynamic (PD) continuum with uniform micromodulus; and (2) state-based peridynamic (PD) continuum. The dashed line corresponds to classical elasticity.

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Fig. 6

Comparison of phase velocity for 1D wave propagation for bond-based and state-based peridynamic (PD) continuum with uniform micromodulus and original nonlocal model for softening damage considering different nonlocal characteristic length l. The dashed line corresponds to a local model (classical elasticity).

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Fig. 7

Phase velocity for 1D bond-based peridynamic model with a uniform micromodulus, considering a uniform grid with 5 points within the horizon. The thick curve represents the phase velocity envelope of the discretized system, and the points on the curve represent the discrete phase velocities of the system. The dashed line represents the phase velocities of linear elastic model.

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Fig. 8

Dispersion diagram for 1D bond-based peridynamic (PD) model with uniform micromodulus considering a uniform grid with 5 points within the horizon. Note that the y -coordinate stands for the magnitude of normalized frequency, |w/w0|, where w=χv. Note that the region −0.5<χ<0.5 in the plot is the reconstruction zone, and the region outside shows the aliasing region.

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Fig. 9

Phase velocity for 1D state-based peridynamic (PD) model considering a uniform grid with 5 points within the horizon. The dashed line represents the continuum solution, while the dotted line represents the classical elasticity solution. The domain −0.5<χ<0.5 is the reconstruction zone.

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Fig. 10

Dispersion diagram for 1D state-based peridynamic (PD) model considering a uniform grid with 5 points within the horizon. The dashed line represents the continuum solution. Note that the shaded region −0.5<χ<0.5 in the plot is the reconstruction zone.

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Fig. 11

Wave propagation in an elastic bar

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Fig. 12

(a) Direct long-range interparticle interactions in a peridynamic continuum, (b) indirect interactions between particles through intergranular forces, (c) three common cases of micromodulus distribution, (d) micromodulus with a Dirac delta function at |ξ|=δ, (e) approximation of the Dirac delta function at |ξ|=δ, (f) Burt and Dougill's constitutive model for progressive failure of heterogeneous media

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