Wave Dispersion and Basic Concepts of Peridynamics Compared To
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Wave Dispersion and Basic Concepts of Peridynamics Zdenek P. Bazant1 McCormick Institute Professor and W.P. Murphy Professor of Civil and Mechanical Engineering Compared to Classical Nonlocal and Materials Science, Northwestern University, Damage Models 2145 Sheridan Road, CEE/A135, The spectral approach is used to examine the wave dispersion in linearized bond-based Evanston, IL 60208 and state-based peridynamics in one and two dimensions, and comparisons with the clas- e-mail: [email protected] sical 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 Wen Luo stronger in the state-based than in the bond-based version, with multiple wavelengths Northwestern University, giving a vanishing phase velocity, one of them longer than the horizon. In the bond-based 2145 Sheridan Road, and state-based, the nonlocality of elastic and inelastic behaviors is coupled, i.e., the dis- CEE/A135, persion of elastic and inelastic waves cannot be independently controlled. In conse- Evanston, IL 60208 quence, the difference between: (1) the nonlocality due to material characteristic length for softening damage, which ensures stability of softening damage and serves as the Viet T. Chau localization limiter, and (2) the nonlocality due to material heterogeneity cannot be dis- Northwestern University, tinguished. This coupling of both kinds of dispersion is unrealistic and similar to the orig- 2145 Sheridan Road, inal 1984 nonlocal model for damage which was in 1987 abandoned and improved to be CEE/A135, nondispersive or mildly dispersive for elasticity but strongly dispersive for damage. With Evanston, IL 60208 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 Miguel A. Bessa that there exists a limit case of peridynamics, with a micromodulus in the form of a Delta Northwestern University, function spiking at the horizon. This limit case is equivalent to the unstabilized imbricate 2145 Sheridan Road, continuum and exhibits zero-energy periodic modes of instability. Finally, it is empha- CEE/A135, sized that the node-skipping force interactions, a salient feature of peridynamics, are Evanston, IL 60208 physically unjustified (except on the atomic scale) because in reality the forces get trans- mitted to the second and farther neighboring particles (or nodes) through the displace- ments and rotations of the intermediate particles, rather than by some potential permeating particles as on the atomic scale. [DOI: 10.1115/1.4034319] 1 Introduction types of micromoduli. Zimmermann [25] compared the dispersion behavior of discrete bond-based peridynamic networks based on Peridynamics is a recently promulgated radically new nonlocal intermolecular forces with classical gradient and integral type theory of solid mechanics with an intriguing new name.2 It nonlocal continua. For integral-type nonlocal elasticity and for eschews Cauchy’s concept of stress and strain and is based on various gradient formulations, the dispersion relation was studied, central force interactions between material points at various finite e.g., by Jirasek [26]. However, a comprehensive investigation of distances3 [e.g., Refs. 2–18]. The novel idea of peridynamics [13] the various versions of peridynamics is still lacking. is to perform the nonlocal spatial (weighted) averaging over dis- This work analyzes the wave propagation behavior obtained placement differences, whereas in the classical nonlocal damage from the continuous and discrete descriptions for both bond- and model [19–23], the averaging is performed over inelastic strains state-based peridynamics. The effects originating from the discre- or over a scalar softening parameter that depends on the displace- tization of continuum peridynamics are isolated from the effects ment derivatives. The introduction of nonlocality into the contin- of the governing laws of the different peridynamic formulations. uum model is essential for preventing spurious localization of A comparison with the classical nonlocal continua is also pro- softening damage [19]. On the other hand, an artificial or exces- vided. The main objectives of this study are fourfold: sive nonlocality introduced into elasticity may raise challenging questions when compared to the nonlocal damage model, (1) Analytical examination of the wave dispersion properties especially with respect to wave dispersion, material property char- of linear peridynamics; acterization, and numerical discretization. (2) Comparison of peridynamic continuum to the earlier con- This study is focused on the wave dispersion properties. Silling tinuous nonlocal models for quasibrittle materials [13], and Weckner and Abeyaratne [24], obtained a general [19–23,27–33] (although comparisons to the extensive expression for the dispersion relation of the bond-based peridy- work on lattice particle models for quasibrittle materials namic continuum and evaluated the expression by using different [e.g., Refs. 34–37] are beyond the present scope); (3) Comparison of important metrics such as critical time step, 1Corresponding author. numerical error as well as rate of convergence of the dis- 2In Greek, “peri” means “about” and “dyna” means “force”. Although crete peridynamic formulations with classical numerical “dynamics” is normally understood as the opposite of “statics”, “peridynamics” is methods, such as finite difference schemes and the finite applied to statics, too. element method. 3In this regard, it bears partial similarity to the 1977 “network model” of Burt and Dougill [1] for concrete, which was, however, used only to simulate the strain- (4) Discussion of some problematic or improvable physical softening constitutive relation aspects of the current forms of peridynamics. Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received June 26, 2016; final manuscript The wave dispersion analysis will pursue an approach similar received July 23, 2016; published online August 30, 2016. Editor: Yonggang Huang. to that used earlier for spurious wave reflection in nonuniform Journal of Applied Mechanics Copyright VC 2016 by ASME NOVEMBER 2016, Vol. 83 / 111004-1 Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 08/30/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use finite element meshes [38,39] and for the original nonlocal models where uS ¼ uXðÞS; t and u ¼ uXðÞ; t ; of distributed damage [40] (see also p. 926 in Ref. [27]). uS þ XS À u À X nuðÞS À u; XS À X; t ¼ (4) 2 Overview of Peridynamic Continuum Models jjuS þ XS À u À Xjj 2.1 Bond-Based Peridynamics. The continuum limit of a jjuS þ XS À u À Xjj discrete (bond-based) peridynamic network [13] is a special kind sðÞuS À u; XS À X ¼ À 1 (5) jjX À Xjj of a nonlocal continuum defined at every point X by the following S equation of motion [9,14]: Here CðÞXS À X is a specified material property called micromo- dulus which has the same dimension as f (although these dimen- qðÞX uX€ðÞ; t ¼ FXðÞ; t þ bXðÞ; t (1) sions have been perfectly justified mathematically, it is strange ð from the physical viewpoint that the material stiffness needs to be where FXðÞ; t ¼ fuX½ðÞS; t À uXðÞ; t ; XS À X; t dV (2) fundamentally characterized by quantities of typical dimensions 6 V such as N/m in 3D). where X; X ¼ Lagrangian coordinate vectors of paired continuum S 2.2 State-Based Peridynamics. The state-based version of points, u; u ¼ displacement vectors of these points, t ¼ time, S peridynamics, developed by Silling et al. [15], brings about a sig- q ¼ mass density, V ¼ nonlocal averaging domain bounded by a nificant improvement in the material modeling capability. In this horizon radius, d (or briefly “horizon”), b ¼ distributed body version, an effective strain tensor is constructed from dyadic prod- force, FXðÞ; t ¼ elastic resisting force, and f ¼ force density trans- ucts of all the pairwise force vectors emanating from a node. The mitted by the nodal pair to point of coordinate vector X in the use of tensors circumvents the Poisson ratio restrictions of bond- undeformed state, called the “pairwise response function,” whose based peridynamics (1/4 in 3D, 1/3 in 2D) and allows using the dimension is N/m6, N/m5, or N/m4, respectively, for three- standard tensorial constitutive relations (as well as the microplane dimensional (3D), two-dimensional (2D), and one-dimensional models). The average of the displacement differences over all the (1D) continuum. Force f is calculated as [9,14] (Fig. 1) pairs is used to approximate the displacement derivative, which was shown in Ref. [18] to coincide, for uniform grids, with the so- called Lanczos derivative (a particular case of the reproducing fuðÞÀ u; X À X; t ¼ CðÞX À X sðÞu À u; X À X; t S S S S S kernel approximation [41]). Connection to other mesh-free meth- nuðÞS À u; XS À X; t (3) ods was elucidated by Bessa et al. [18]. In state-based peridynamics, with correspondence to classical constitutive laws, the conservation of linear momentum equation is formulated as [15] ð  À1 qðÞX uX€ðÞ; t ¼ aðÞXS À X rðÞXS; t Á K ðÞXS V à À1 þrðÞX; t Á K ðÞX Á XS dV þ bXðÞ; t (6) where aðÞXS À X is the weight function, typically used in the peri- dynamics literature as the window (or separator) function, and KXðÞis the so-called shape tensor ð KXðÞ¼ aðÞXS À X ðÞXS À X ðÞXS À X dV (7) V and where the stress at each point of the peridynamic continuum is obtained from the constitutive law for that material, rðÞX; t rðÞFdefðÞX; t , applied to the nonlocal deformation gradi- ent tensor, which is ð FdefðÞX; t ¼ aðÞXS À X ðÞXS À X ðÞXS À X dV V Á KÀ1ðÞX dV (8) This tensor is substituted into the aforementioned constitutive law giving r, which is then substituted into Eq.