Begell House Inc.
International Journal for Multiscale Computational Engineering
JMC
1543-1649
9
6
2011
PERIDYNAMICS AND MULTISCALE MODELING
vii-ix
Florin
Bobaru
Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0526, USA
A COARSENING METHOD FOR LINEAR PERIDYNAMICS
609-622
Stewart A.
Silling
Multiscale Dynamic Material Modeling Department, Sandia National Laboratories, Albuquerque, New Mexico,87185, USA
A method is obtained for deriving peridynamic material models for a sequence of increasingly coarsened descriptions of a body. The starting point is a known detailed, small scale linearized state-based description. Each successively coarsened model excludes some of the material present in the previous model, and the length scale increases accordingly. This excluded material, while not present explicitly in the coarsened model, is nevertheless taken into account implicitly through its effect on the forces in the coarsened material. Numerical examples demonstrate that the method accurately reproduces the effective elastic properties of a composite as well as the effect of a small defect in a homogeneous medium.
DETERMINATION OF NONLOCAL CONSTITUTIVE EQUATIONS FROM PHONON DISPERSION RELATIONS
623-634
Olaf
Weckner
The Boeing Company, P.O. Box 3707, MC 42-26, Seattle, Washington, 98124-2207, USA
Stewart A.
Silling
Multiscale Dynamic Material Modeling Department, Sandia National Laboratories, Albuquerque, New Mexico,87185, USA
All materials exhibit wave dispersion at "small" wavelengths leading to non-linearities in experimentally determined dispersion curves. Classical local elasticity fails to predict these non-linearities. Nonlocal continuum mechanics allows for the prediction of the elastic behavior over a considerably wider range of lengthscales. Starting from ab initio lattice dynamics calculations we determine the elastic constants and the phonon dispersion relation for silicon. We verify our results using inelastic neutron scattering data. Next we develop the theoretical and numerical framework to construct nonlocal constitutive equations for longitudinal and transverse acoustic modes.
ADAPTIVE REFINEMENT AND MULTISCALE MODELING IN 2D PERIDYNAMICS
635-660
Florin
Bobaru
Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0526, USA
Youn Doh
Ha
Department of Naval Architecture, Kunsan National University, 558 Daehak-ro (San 68, Miryong-dong) Gunsan, Jeonbuk, 573-701, Korea
The original peridynamics formulation uses a constant nonlocal region, the horizon, over the entire domain. We propose here adaptive refinement algorithms for the bond-based peridynamic model for solving statics problems in two dimensions that involve a variable horizon size. Adaptive refinement is an essential ingredient in concurrent multiscale modeling, and in peridynamics changing the horizon is directly related to multiscale modeling. We do not use any special conditions for the "coupling" of the large and small horizon regions, in contrast with other multiscale coupling methods like atomistic-to-continuum coupling, which require special conditions at the interface to eliminate ghost forces in equilibrium problems. We formulate, and implement in two dimensions, the peridynamic theory with a variable horizon size and we show convergence results (to the solutions of problems solved via the classical partial differential equations theories of solid mechanics in the limit of the horizon going to zero) for a number of test cases. Our refinement is triggered by the value of the nonlocal strain energy density. We apply the boundary conditions in a manner similar to the way these conditions are enforced in, for example, the finite-element method, only on the nodes on the boundary. This, in addition to the peridynamic material being effectively "softer" near the boundary (the so-called "skin effect") leads to strain energy concentration zones on the loaded boundaries. Because of this, refinement is also triggered around the loaded boundaries, in contrast to what happens in, for example, adaptive finite-element methods.
CLASSICAL, NONLOCAL, AND FRACTIONAL DIFFUSION EQUATIONS ON BOUNDED DOMAINS
661-674
Nathanial
Burch
Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523, USA
Richard
Lehoucq
Sandia National Labs
The purpose of this paper is to compare the solutions of one-dimensional boundary value problems corresponding to classical, fractional, and nonlocal diffusion on bounded domains. The latter two diffusions are viable alternatives for anomalous diffusion when Fick's first law is an inaccurate model. In the case of nonlocal diffusion, a generalization of Fick's first law in terms of a nonlocal flux is demonstrated to hold. A relationship between nonlocal and fractional diffusion is also reviewed, where the order of the fractional Laplacian can lie in the interval (0, 2]. The contribution of this paper is to present boundary value problems for nonlocal diffusion including a variational formulation that leads to a conforming finite-element method using piecewise discontinuous shape functions. The nonlocal Dirichlet and Neumann boundary conditions used represent generalizations of the classical boundary conditions. Several examples are given where the effect of nonlocality is studied. The relationship between nonlocal and fractional diffusion explains that the numerical solution of boundary value problems, where the order of the fractional Laplacian can lie in the interval (0, 2], is possible.
AN ENERGY BASED FAILURE CRITERION FOR USE WITH PERIDYNAMIC STATES
675-688
John T.
Foster
Mechanical Engineering Department, University of Texas, San Antonio, TX 78249, USA; Terminal Ballistics Technology, Sandia National Laboratories, Albuquerque, New Mexico 87185,USA
Stewart A.
Silling
Multiscale Dynamic Material Modeling Department, Sandia National Laboratories, Albuquerque, New Mexico,87185, USA
Weinong
Chen
Aeronautics and Astronautics and Materials Engineering, Purdue University, West Lafayette, Indiana 47907, USA
Peridynamics is a continuum reformulation of the standard theory of solid mechanics. Unlike the partial differential equations of the standard theory, the basic equations of peridynamics are applicable even when cracks and other singularities appear in the deformation field. Interactions between continuum material points are termed "bonds." In this paper, a method for implementing a rate-dependent plastic material model within a peridynamic numerical code is summarized and a novel failure criterion is then presented by analyzing the energy required to break all bonds across a plane of unit area (energy release rate); with this, one can determine the critical energy density required to irreversibly fail a single bond. By failing individual bonds, this allows cracks to initiate, coalesce, and propagate without a prescribed external crack law. This is demonstrated using experimentally collected fracture toughness measurements to evaluate the energy release rate. Simulations are compared to experimental results.
ON THE ROLE OF THE INFLUENCE FUNCTION IN THE PERIDYNAMIC THEORY
689-706
Pablo
Seleson
Department of Scientific Computing, Florida State University, Tallahassee, Florida 32306-4120, USA
Michael
Parks
Applied Mathematics and Applications, Sandia National Laboratories, Albuquerque, New Mexico 87185-1320, USA
The influence function in the peridynamic theory is used to weight the contribution of all the bonds participating in the computation of volume-dependent properties. In this work, we use influence functions to establish relationships between bond-based and state-based peridynamic models. We also demonstrate how influence functions can be used to modulate nonlocal effects within a peridynamic model independently of the peridynamic horizon. We numerically explore the effects of influence functions by studying wave propagation in simple one-dimensional models and brittle fracture in three-dimensional models.
MODELING DYNAMIC FRACTURE AND DAMAGE IN A FIBER-REINFORCED COMPOSITE LAMINA WITH PERIDYNAMICS
707-726
Wenke
Hu
Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0526, USA
Youn Doh
Ha
Department of Naval Architecture, Kunsan National University, 558 Daehak-ro (San 68, Miryong-dong) Gunsan, Jeonbuk, 573-701, Korea
Florin
Bobaru
Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0526, USA
We propose a peridynamic formulation for a unidirectional fiber-reinforced composite lamina based on homogenization and mapping between elastic and fracture parameters of the micro-scale peridynamic bonds and the macro-scale parameters of the composite. The model is then used to analyze the splitting mode (mode II) fracture in dynamic loading of a 0° lamina. Appropriate scaling factors are used in the model in order to have the elastic strain energy, for a fixed nonlocal interaction distance (the peridynamic horizon), match the classical one. No special criteria for splitting failure are required to capture this fracture mode in the lamina. Convergence studies under uniform grid refinement for a fixed horizon size (m-convergence) and under decreasing the peridynamic horizon (δ-convergence) are performed. The computational results show that the splitting fracture mode obtained with peridynamics compares well with experimental observations. Moreover, in the limit of the horizon going to zero, the maximum crack propagation speed computed with peridynamics approaches the value obtained from an analytical classical formulation for the steady-state dynamic interface debonding found in the literature.