Convergence of Peridynamics to Classical Elasticity Theory

Silling, Stewart A.; Lehoucq, Richard B.

doi:10.1007/s10659-008-9163-3

Cited by 411 publications

(247 citation statements)

References 31 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%

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%

Blending Methods for Coupling Atomistic and Continuum Models

Bochev¹,

Lehoucq²,

Parks³

et al. 2009

Multiscale Methods

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“…In addition, a force state similar to the stress tensor in the classical continuum mechanics is introduced. Based on the study of convergence by Silling and Lehoucq [131], it is shown that the peridynamics stress tensor convergences to a Piola-Kirchhoff stress tensor as the length scale goes to zero. Using the state-based peridynamic method, Warren et al [132] presented elastic deformation and fracture of a bar, and Foster et al [133,134] studied viscoplasticity.…”

Section: Peridynamicsmentioning

confidence: 99%

Multiscale modeling and simulation of rolling contact fatigue

Gharehbagh

Ali

2018

International Journal of Fatigue

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iii ABSTRACT In this thesis, a hierarchical multiscale method was developed to predict rolling contact fatigue lives of mechanical systems. In the proposed multiscale method, the molecular modeling and simulation of lubricant was conducted to investigate the friction between rolling contact surfaces. The calculated friction coefficient was passed to the continuum model of rolling contact components to predict fatigue lives.Molecular dynamics modeling and simulation of thin film lubrication and lubricated contact surfaces were carried out to investigate mechanisms of hydrodynamic lubrication at nano-scale first. Although various lubricant alkane chains were considered in the molecular model, the chain length of eight united molecules were mainly employed in this thesis. In addition, the effects of temperature and nano-particles (debris) on the friction forces were discussed. It was found that the existing of nano-particles (debris) could increase the friction force between contact surfaces with hydrodynamic lubrication.In the continuum model of the developed multiscale method, finite element analysis was employed to predict rolling contact fatigue life of rolling contact components, including bearing and gear-tooth. Specifically, the fatigue crack initiation of bearing was studied, and then the fatigue crack initiation and propagation in gear-tooth. In addition, the enhancement of gear-tooth fatigue life by using composite patches was discussed as well. It should be noted that the friction coefficient used in the continuum model was calculated in the molecular model. It is one-way message passing in iv the developed multiscale method.Another continuum method was studied and developed in this thesis to provide alternate methods for the continuum model in the proposed multiscale framework.Peridynamics method has advantages in modeling and simulation of discontinuities, including cracks, over the conventional finite element methods. The applications of Peridynamics in predicting fatigue crack initiation and propagation lives were discussed in this thesis. v PUBLIC ABSTRACTThis research aims to develop a hierarchical multiscale method to predict rolling contact fatigue lives of mechanical systems. In the proposed multiscale method, the molecular modeling and simulation of lubricant was conducted to investigate the friction between rolling contact surfaces. The calculated friction coefficient was passed to the continuum model of rolling contact components to predict fatigue lives.Molecular dynamics modeling and simulation of thin film lubrication and lubricated contact surfaces were carried out to investigate mechanisms of hydrodynamic lubrication at nano-scale first. In addition, the effects of temperature and nano-particles (debris) on the friction forces were discussed. It was found that the existing of nanoparticles (debris) could increase the friction force between contact surfaces with hydrodynamic lubrication.In the continuum model of the developed multiscale method, finite element analysis was employed to pred...

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“…Раз-ница заключается в том, что в перидинамической модели силы не локальны. Предельный переход в континуум Коши происходит при r→0 [Silling, Lehoucq, 2008].…”

Section: способы прямой континуализации дискретной средыunclassified

On Discrete Structure of Geologic Medium and Continual Approach to Modeling Its Movements

Мухамедиев

2016

Geodin. tektonofiz.

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This paper discusses the structure of a geologic medium represented by accessible lithified rocks and provides an overview of methods used to describe its movements. Two basic opinions are considered in the framework of the discussion: (1) an initially homogeneous and continuous geologic medium acquires the structure composed of blocks in the process of the geologic medium's deformation/destruction/degradation, and (2) a geologic medium is composed of blocks (and often has hierarchic, active, energy-saturated features), and the continuity model is thus not valid for describing the geologic medium's deformation. Proponents of the first point of view actively apply the standard or modified continuum model of a solid deformed body (SDB) in estimations of the stress-strain state, but the input parameters of this model do not contain any information on discreteness in principle. Authors who support the second opinion, either explicitly or implicitly assume that the block structure of the geologic medium, which is detectable by geological methods, makes a direct and unambiguous impact on all other mechanical properties of the geologic medium and, above all, on the nature of its movements.Based on results obtained by interpreting the data collected in our long-term field studies of rock fracturing, mathematical processing of GPS-measurements, and theoretical models, we agree with the concept of the geologic medium's block structure, but argue that the geologic block-structure property is not acquired but congenital. Regarding sedimentary rocks, it means that the discrete structure has been already embodied in the rock before sediment lithification, regardless of the intensity of macroscopic deformations. A discrete structure is the form of the geologic medium existence and a cause of the congenital anisotropy of the geologic medium's strength characteristics. Due to subsequent deformation of the geologic medium, some elements of the structure can be manifested more clearly, and the structure itself can become more complex due to secondary effects. At the same time, the structure of geologic blocks is not directly manifested in the spatial and temporal features of the recent movements in the geologic medium. However, it is not an obstacle to developing continuum models of such movements in the same way as, for instance, the adequacy of the continuum general theory of relativity is not denied in view of the discrete-hierarchical structure of the Universe.The key requirements to a model include its applicability, testability, confirmability/deniability of its predictions in the investigated space-time scale, and compliance with conservation laws. This paper briefly discusses the most important aspects of the continuum approach based on the concept of an effective continuous medium and, above all, the Cauchy continuum model, envisaging that the dynamic response of the medium in spatial descriptions is given only by the Cauchy symmetric stress tensor (T). In more general continuum models of the medium (such as mom...

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Convergence of Peridynamics to Classical Elasticity Theory

Cited by 411 publications

References 31 publications

Blending Methods for Coupling Atomistic and Continuum Models

Blending Methods for Coupling Atomistic and Continuum Models

Multiscale modeling and simulation of rolling contact fatigue

On Discrete Structure of Geologic Medium and Continual Approach to Modeling Its Movements

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