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## Aspects of the discretized peridynamic theory and the finite element method for concurrent multiscale simulation : Aspectos da teoria peridinâmica discretizada e do método dos elementos finitos para simulação em múltiplas escalas concorrentes; Aspectos da teoria peridinâmica discretizada e do método dos elementos finitos para simulação em múltiplas escalas concorrentes

Fabiano Fernandes Bargos
Fonte: Biblioteca Digital da Unicamp Publicador: Biblioteca Digital da Unicamp
Tipo: Tese de Doutorado Formato: application/pdf
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Nesse trabalho, considera-se a simulação em múltiplas escalas concorrentes, usando a teoria peridinâmica e a elasticidade clássica, para a simulação de problemas de engenharia. Primeiramente a teoria peridinâmica em uma dimensão é estudada em detalhes com o foco na aplicação de condições de contorno de Dirichlet. Problemas de estado plano de tensão em chapas com e sem furo são considerados. É proposto um método de pós-processamento dos resultados de peridinâmica para o cálculo das tensões no material. Em seguida, a peridinâmica discretizada é acoplada ao método dos elementos finitos por meio de dois diferentes programas de computador, um especializado em peridinâmica e o outro em elementos finitos. A modelagem acoplada é usada para prever a formação e a propagação de uma trinca em uma chapa com furo. O fenômeno macroscópico de formação e propagação de trincas é resultado de processos físicos com origem na escala atomística. No entanto, as simulações existentes deste tipo problema são normalmente feitas com abordagens baseadas na teoria do contínuo, como a mecânica da fratura e o dano contínuo, que não consideram aspectos atomísticos do problema. A teoria peridinâmica é uma formulação da mecânica do contínuo em termos de equações integrais...

## Dinâmica molecular e peridynamics aplicadas a nanotecnologia : um estudo sobre filmes finos e nanofios metálicos; Molecular dynamics and peridynamics applied to nanotechnology : a study of thin films and metallic nanowires

Zenner Silva Pereira
Fonte: Biblioteca Digital da Unicamp Publicador: Biblioteca Digital da Unicamp
Tipo: Tese de Doutorado Formato: application/pdf
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Nas últimas décadas uma geração de nanodispositivos foi desenvolvida. Estes dispositivos nanoeletrônicos são fabricados por novas técnicas fundamentadas em física, química e engenharia. Muitos desses nanomateriais têm suas propriedades físicas alteradas pelo efeito de tamanho, por causa desses novos efeitos é importante entender como estes dispositivos trabalham propriamente a fim de encontrarmos formas de obter novas aplicações baseadas nestes novos efeitos. Nanofios metálicos estão sendo largamente estudados tanto teoricamente como experimentalmente. Recentemente uma nova possibilidade de soldagem foi mostrada experimentalmente entre nanofios de ouro em temperatura ambiente, sem necessidade de aplicação de calor adicional e com baixa pressão, chamada de solda fria (cold welding). Usando Dinâmica Molecular (MD) com potenciais efetivos, nós simulamos o processo de soldagem fria em nanofios de ouro, prata e ouro-prata com diâmetros de 4.3nm em 300 K. Nós mostramos que a soldagem fria é um processo possível até mesmo quando os nanofios sofrem fortes deformações e defeitos antes do processo de soldagem. Durante o processo de soldagem os nanofios resultaram com poucos defeitos. Pequenas pressões foram necessárias para que a soldagem fosse atingida. Nós também realizamos cálculos de Dinâmica Molecular com embedded-atom-method para modelar o crescimento de filmes-finos de paládio depositados em um substrato de ouro para um sistema de aproximadamente 100 mil átomos. Nós mostramos que o filmes-finos de paládio cresceu sob stress sobre o substrato de ouro. Após a deposição de 9 monocamadas o stress armazenado no filmes de paládio relaxou formando defeitos na estrutura do cristal. Defeitos do tipo falhas de empilhamento surgiram nos filmes de paládio formando um padrão de deformação no mesmo. Para quantificar o stress nós também calculamos a evolução do tensor de stress durante o crescimento. Existem fenômenos físicos como fraturas em materiais que são caracterizados pela quebra das ligações atômicas que levam a efeitos macroscópicos. Para estudarmos este tipo de problema...

## Propagation via a Peridynamics Formulation: A StochasticDeterministic Perspective

Evangelatos, Georgios
ENG
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Novel numerical methods for treating fractional differential and integrodifferential equations arising in non local mechanics formulations are proposed. For fractional differential equations arising in modeling oscillatory systems incorporating viscoelastic elements governed by fractional derivatives, the devised scheme is based on the Grunwald-Letnikov fractional derivative representation, dual time meshing technique and Taylor expansion. The proposed algorithm transforms the governing fractional differential equation into a second order differential equation with appropriate effective coefficients. The enhanced efficiency of the scheme hinges upon circumventing the calculation of the non local fractional derivative operator. Several examples of application are provided. Further, the concept of non locality, specifically viscoelasticity, governed by fractional derivatives is utilized to accurately model polyester materials. Specifically, the linear standard solid (Zener model) is extended to capture non linear viscoelastic behavior. Then, experimental data of polyester ropes are utilized using the Gauss Newton and Levenberg-Marquart minimization algorithm to determine the model parameters. Next, for integrodifferential equations arising in peridynamics theory of mechanics...

## Peridynamics and Material Interfaces

Alali, Bacim; Gunzburger, Max
Tipo: Artigo de Revista Científica
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The convergence of a peridynamic model for solid mechanics inside heterogeneous media in the limit of vanishing nonlocality is analyzed. It is shown that the operator of linear peridynamics for an isotropic heterogeneous medium converges to the corresponding operator of linear elasticity when the material properties are sufficiently regular. On the other hand, when the material properties are discontinuous, i.e., when material interfaces are present, it is shown that the operator of linear peridynamics diverges, in the limit of vanishing nonlocality, at material interfaces. Nonlocal interface conditions, whose local limit implies the classical interface conditions of elasticity, are then developed and discussed. A peridynamics material interface model is introduced which generalizes the classical interface model of elasticity. The model consists of a new peridynamics operator along with nonlocal interface conditions. The new peridynamics interface model converges to the classical interface model of linear elasticity.

## A nonlocal biharmonic operator and its connection with the classical analogue

Radu, Petronela; Toundykov, Daniel; Trageser, Jeremy
Tipo: Artigo de Revista Científica
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We introduce here a nonlocal operator as a natural generalization to the biharmonic operator that appears in plate theory. This operator is built in the nonlocal calculus framework defined by Du et al. and its connected with the recent theory of peridynamics. For the steady state equation coupled with different boundary conditions we show existence and uniqueness of solutions, as well as regularity of solutions. The boundary conditions considered are nonlocal counterparts of the classical clamped and hinged boundary conditions. For each system we show convergence of the nonlocal solutions to their local equivalents using compactness arguments developed by Bourgain, Brezis and Mironescu.

## On the similarity of meshless discretizations of Peridynamics and Smooth-Particle Hydrodynamics

Ganzenmüller, Georg C.; Hiermaier, Stefan; May, Michael
Tipo: Artigo de Revista Científica
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This paper discusses the similarity of meshless discretizations of Peridynamics and Smooth-Particle-Hydrodynamics (SPH), if Peridynamics is applied to classical material models based on the deformation gradient. We show that the discretized equations of both methods coincide if nodal integration is used. This equivalence implies that Peridynamics reduces to an old meshless method and all instability problems of collocation-type particle methods apply. These instabilities arise as a consequence of the nodal integration scheme, which causes rank-deficiency and leads to spurious zero-energy modes. As a result of the demonstrated equivalence to SPH, enhanced implementations of Peridynamics should employ more accurate integration schemes.

## A Generalized Nonlocal Calculus with Application to the Peridynamics Model for Solid Mechanics

Alali, Bacim; Liu, Kuo; Gunzburger, Max
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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A nonlocal vector calculus was introduced in [2] that has proved useful for the analysis of the peridynamics model of nonlocal mechanics and nonlocal diffusion models. A generalization is developed that provides a more general setting for the nonlocal vector calculus that is independent of particular nonlocal models. It is shown that general nonlocal calculus operators are integral operators with specific integral kernels. General nonlocal calculus properties are developed, including nonlocal integration by parts formula and Green's identities. The nonlocal vector calculus introduced in [2] is shown to be recoverable from the general formulation as a special example. This special nonlocal vector calculus is used to reformulate the peridynamics equation of motion in terms of the nonlocal gradient operator and its adjoint. A new example of nonlocal vector calculus operators is introduced, which shows the potential use of the general formulation for general nonlocal models.

## On a Class of Nonlocal Wave Equations from Applications

Beyer, Horst Reinhard; Aksoylu, Burak; Celiker, Fatih
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We study equations from the area of peridynamics, which is an extension of elasticity. The governing equations form a system of nonlocal wave equations. Its governing operator is found to be a bounded, linear and self-adjoint operator on a Hilbert space. We study the well-posedness and stability of the associated initial value problem. We solve the initial value problem by applying the functional calculus of the governing operator. In addition, we give a series representation of the solution in terms of spherical Bessel functions. For the case of scalar valued functions, the governing operator turns out as functions of the Laplace operator. This result enables the comparison of peridynamic solutions to those of classical elasticity as well as the introduction of local boundary conditions into the nonlocal theory. The latter is studied in a companion paper.; Comment: 36 pages, 8 figures

## Dual-horizon Peridynamics

Ren, Huilong; Zhuang, Xiaoying; Cai, Yongchang; Rabczuk, Timon
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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In this paper we develop a new Peridynamic approach that naturally includes varying horizon sizes and completely solves the "ghost force" issue. Therefore, the concept of dual-horizon is introduced to consider the unbalanced interactions between the particles with different horizon sizes. The present formulation is proved to fulfill both the balances of linear momentum and angular momentum. Neither the "partial stress tensor" nor the "`slice" technique are needed to ameliorate the ghost force issue in \cite{Silling2014}. The consistency of reaction forces is naturally fulfilled by a unified simple formulation. The method can be easily implemented to any existing peridynamics code with minimal changes. A simple adaptive refinement procedure is proposed minimizing the computational cost. The method is applied here to the three Peridynamic formulations, namely bond based, ordinary state based and non-ordinary state based Peridynamics. Both two- and three- dimensional examples including the Kalthof-Winkler experiment and plate with branching cracks are tested to demonstrate the capability of the method in solving wave propagation, fracture and adaptive analysis .

## Incorporating local boundary conditions into nonlocal theories

Aksoylu, Burak; Beyer, Horst Reinhard; Celiker, Fatih
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We study nonlocal equations from the area of peridynamics on bounded domains. In our companion paper, we discover that, on $\mathbb{R}^n$, the governing operator in peridynamics, which involves a convolution, is a bounded function of the classical (local) governing operator. Building on this, we define an abstract convolution operator on bounded domains. The abstract convolution operator is a function of the classical operator, defined by a Hilbert basis available due to the purely discrete spectrum of the latter. As governing operator of the nonlocal equation we use a function of the classical operator, this allows us to incorporate local boundary conditions into nonlocal theories. For the homogeneous wave equation with the considered boundary conditions, we prove that continuity is preserved by time evolution. We give explicit solution expressions for the initial value problems with prominent boundary conditions such as periodic, antiperiodic, Neumann, and Dirichlet. In order to connect to the standard convolution, we give an integral representation of the abstract convolution operator. We present additional "simple" convolutionsbased on periodic and antiperiodic boundary conditions that lead Neumann and Dirichlet boundary conditions. We present a numerical study of the solutions of the wave equation. For discretization...

## Almost Finite Speed of Propagation for Linear Peridynamics

Stalker, John
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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The peridynamic analogue of the wave equation does not have finite speed propogation. We show, for one dimensional linear peridynamics, that solutions do nonetheless satisfy estimates analogous to those satisfied by solutions of the wave equations. More precisely, the solution and all derivatives become small as we go away from the domain of dependence of the initial data.

## Improvements to the Prototype Micro-Brittle Linear Elasticity Model of Peridynamics

Ganzenmüller, Georg C.; Hiermaier, Stefan; May, Michael
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.59%
This paper assesses the accuracy and convergence of the linear-elastic, bond-based Peridynamic model with brittle failure, known as the prototype micro-brittle (PMB) model. We investigate the discrete equations of this model, suitable for numerical implementation. It is shown that the widely used discretization approach incurs rather large errors. Motivated by this observation, a correction is proposed, which significantly increases the accuracy by cancelling errors associated with the discretization. As an additional result, we derive equations to treat the interactions between differently sized particles, i.e., a non-homogeneous discretization spacing. This presents an important step forward for the applicability of the PMB model to complex geometries, where it is desired to model interesting parts with a fine resolution (small particle spacings) and other parts with a coarse resolution in order to gain numerical efficiency. Validation of the corrected Peridynamic model is performed by comparing longitudinal sound wave propagation velocities with exact theoretical results. We find that the corrected approach correctly reproduces the sound wave velocity, while the original approach severely overestimates this quantity. Additionally...

## A Micropolar Peridynamic Theory in Linear Elasticity

Chowdhury, S. Roy; Rahaman, Md Masiur; Roy, Debasish; Sundaram, Narayan
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.88%
A state-based micropolar peridynamic theory for linear elastic solids is proposed. The main motivation is to introduce additional micro-rotational degrees of freedom to each material point and thus naturally bring in the physically relevant material length scale parameters into peridynamics. Non-ordinary type modeling via constitutive correspondence is adopted here to define the micropolar peridynamic material. Along with a general three dimensional model, homogenized one dimensional Timoshenko type beam models for both the proposed micropolar and the standard non-polar peridynamic variants are derived. The efficacy of the proposed models in analyzing continua with length scale effects is established via numerical simulations of a few beam and plane-stress problems.

Lipton, Robert