Peridynamic States and Constitutive Modeling

Silling, Stewart A.; Epton, Michael A.; Weckner, Olaf; Xu, Jiren; Askari, E.

doi:10.1007/s10659-007-9125-1

Cited by 1,844 publications

(1,532 citation statements)

References 18 publications

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“…For instance, the recent paper [38] explains that if the underlying deformation is sufficiently smooth, then peridynamics [36,37] is asymptotically equivalent [38] to classical elasticity as the length-scale decreases, and has a symbiotic relationship with molecular dynamics. As a result, a discretized peridynamic model can be implemented using an off-the-shelf molecular dynamic code as described in [32].…”

Section: Blending Methods For Coupling Atomistic and Continuum Modelsmentioning

confidence: 99%

“…The intrinsic incompatibility of coupling mechanical models with local/nonlocal force interaction suggests that we consider generalizations of classical continuum mechanics as described in [7,13,19,35,36,37], or continuum realizations of molecular dynamics [1,42,12,43,30]. We refer to these classes of methods as generalized continua.…”

Section: Generalized Continuamentioning

confidence: 99%

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Blending Methods for Coupling Atomistic and Continuum Models

Bochev¹,

Lehoucq²,

Parks³

et al. 2009

Multiscale Methods

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In this paper we review recent developments in blending methods for atomisticto-continuum (AtC) coupling in material statics problems. Such methods tie together atomistic and continuum models by using a bridge domain that connects the two models. There are several reasons why AtC coupling methods (of which blending methods are a subset) are important and have been subject to an increased interest in the recent years. Despite tremendous increases in computational power, fully atomistic simulations on an entire model domain remain computationally infeasible for many applications of interest. As a result, attention has focused on hybrid schemes where in all regions with well-behaved solutions, the atomistic (microscopic) model is replaced by a (macroscopic) continuum model enabling a more efficient computational scheme (see [14,29,9] for general information). The main challenge is the synthesis of the two distinct models in a manner that minimizes, or altogether eliminates, undesirable artifacts such as ghost forces, unphysical solutions let alone supporting mathematical analysis. Notable AtC coupling methods include the quasicontinuum method [39], the bridging scale decomposition [41], and [24] where atomistic and continuum models are overlapped (see the latter two references, and those mentioned above for numerous citations to the literature). The numerical analysis of AtC methods has lagged in comparison to the number of methods proposed; see the recent papers [2,3,17,31,26,27] for analyses of the quasicontinuum method.Blending methods couple atomistic/continuum modes via a dedicated blending, or bridge, region inserted between the atomistic and continuum subdomains. The atomistic and continuum models are tied together by using a suitable "continuity" condition (or balance law) for the atomistic and continuum positions or displacements in this region. A complicating factor is that the atomistic and (classical) continuum elastic models rely on nonlocal and local models of force interaction, respectively. In the (classical) elastic context, the "local force" is that exerted on a body by contact forces occurring on the surface (of the body). In contrast, in the atomistic model, forces are summed from atoms separated by a finite distance. The incompatibility arising from coupling local and nonlocal force models is intrinsic. The goal of our paper is threefold. First, we review how blending approaches attempt to ameliorate the various negative effects by relying on an interface between the two models. Second, we discuss the beginnings of a JFISH: "CHAP06" -2009/6/4 -17:33 -PAGE 166 -#2 166Blending methods for coupling atomistic and continuum models numerical analysis by discussing consistency of the resulting numerical schemes. Third, we suggest how the intrinsic limitation associated with coupling nonlocal and local mechanical theories may be avoided by replacing the classical elastic theory with a nonlocal elastic theory.The use of a bridge domain in blending methods bears a resemblance to conventional overlapping...

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Section: Blending Methods For Coupling Atomistic and Continuum Modelsmentioning

confidence: 99%

Section: Generalized Continuamentioning

confidence: 99%

Blending Methods for Coupling Atomistic and Continuum Models

Bochev¹,

Lehoucq²,

Parks³

et al. 2009

Multiscale Methods

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“…We list basic definitions in the SPD theory below; readers can find most from [23] with the exception of the different expressions of strain energy density and force state field. We rewrite those expressions as the functions of a full extension of a bond rather than the deviatoric extension [21].…”

Section: State-based Peridynamic Formulationmentioning

confidence: 99%

“…In this work, we focus on ordinary and elastic materials [23]. Let H δ (x) be a spherical neighborhood centered at an arbitrary point x in R 3 .…”

Section: State-based Peridynamic Formulationmentioning

confidence: 99%

A morphing approach to couple state-based peridynamics with classical continuum mechanics

Han

Lubineau

Azdoud

et al. 2016

Computer Methods in Applied Mechanics and Engineering

137

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Please cite this article as: F. Han, G. Lubineau, Y. Azdoud, A. Askari, A morphing approach to couple state-based peridynamics with classical continuum mechanics, Comput. Methods Appl. Mech. Engrg. (2015), http://dx.doi. org/10.1016/j.cma.2015.12.024 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractA local/nonlocal coupling technique called the morphing method is developed to couple classical continuum mechanics with state-based peridynamics. State-based peridynamics, which enables the description of cracks that appear and propagate spontaneously, is applied to the key domain of a structure, where damage and fracture are considered to have non-negligible effects. In the rest of the structure, classical continuum mechanics is used to reduce computational costs and to simultaneously satisfy solution accuracy and boundary conditions. Both models are glued by the proposed morphing method in the transition region. The morphing method creates a balance between the stiffness tensors of classical continuum mechanics and the weighted coefficients of state-based peridynamics through the equivalent energy density of both models. Linearization of state-based peridynamics is derived by Taylor approximations based on vector operations. The discrete formulation of coupled models is also described. Two-dimensional numerical examples illustrate the validity and accuracy of the proposed technique. It is shown that the morphing method, originally developed for bond-based peridynamics, can be successfully extended to state-based peridynamics through the original developments presented here.

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“…However, the peridynamic theory (Silling 2000;Silling et al 2007) has been proposed as a reformulation of solid mechanics whose basic equations remain applicable regardless of the possible emergence of discontinuities in the deformation. In this paper, the nucleation of discontinuities is studied within the peridynamic theory from the point of view of material stability.…”

Section: Introductionmentioning

confidence: 99%

Crack nucleation in a peridynamic solid

et al. 2010

Self Cite

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A condition for the emergence of a discontinuity in an elastic peridynamic body is proposed, resulting in a material stability condition for crack nucleation. The condition is derived by determining whether a small discontinuity in displacement, superposed on a possibly large deformation, grows over time. Stability is shown to be determined by the sign of the eigenvalues of a tensor field that depends only on the linearized material properties. This condition for nucleation of a discontinuity in displacement can be interpreted in terms of the dynamic stability of plane waves with very short wavelength. A numerical example illustrates that cracks in a peridynamic body form spontaneously as the body is loaded.

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Peridynamic States and Constitutive Modeling

Cited by 1,844 publications

References 18 publications

Blending Methods for Coupling Atomistic and Continuum Models

Blending Methods for Coupling Atomistic and Continuum Models

A morphing approach to couple state-based peridynamics with classical continuum mechanics

Crack nucleation in a peridynamic solid

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