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Talkbcam20120604/05Mgunzburger NONLOCAL DIFFUSION, NONLOCAL MECHANICS, AND A NONLOCAL VECTOR CALCULUS Max Gunzburger Department of Scientific Computing, Florida State University Basque Center for Applied Mathematics, June 2012 • In this course, \we" includes the people I collaborate on in studying nonlocal models for diffusion and mechanics Qiang Du at Penn State Rich Lehoucq at the Sandia National Laboratories Xi Chen (student) at Florida State Marta D'Elia (postdoc) at Florida State Mike Parks at the Sandia National Laboratories Qingguang Guan (student) at Florida State Pablo Seleson at the University of Texas at Austin Miro Stoyanov at the Oak Ridge National Laboratory Feifei Xu (student) at Florida State Guannan Zhang (student) at Florida State Kun Zhou (student) at Penn State MOTIVATION { SOLID MECHANICS Multiscale material mechanics modeling • We define two requirements of any model, including a material mechanics model { a valid model is one that provides a faithful description of the physics { a tractable model is one for which useful information can be extracted, e.g., through discretization, at a manageable cost • These two requirements allows us to define a multiscale model as one that is valid and tractable over a wide range of scales { multiscale modeling usually (but not always) implies that model behavior changes depending on the scale at which the model is applied • A common approach towards developing a multiscale material mechanics model couples two or more well-known models, e.g., molecular dynamics (MD) and classical elasticity (CE) each of which is useful at a different scale ) thus creating a multiscale multi-model or composite model { although there have been impressive accomplishments in multi-model materials modeling and simulations, challenges remain • Alternately, one can look for a single model for materials that remains valid and useful over a wide range of scales ) thus acting as a multiscale mono-model { quasicontinuum methods (QC) are sometimes presented as being multiscale mono-models - although it is unlikely one would want to use QC in regions where CE is valid • \A wide range of scales" is measured, e.g., relative to the range of scales over which MD is tractable and CE is valid • Many of the shortcomings of composite methods involving, e.g., MD and CE, stem from the coupling of a nonlocal model to a local model • Furthermore, except perhaps for problems posed at the nanoscale, there is a gap between scales for which MD is tractable and CE is valid { perhaps QC can be used to bridge or, most probably, to just narrow that gap - there would remain the problems attendant to the coupling of a nonlocal model to a local model • We are motivated to sidestep the difficulties encountered when coupling local and nonlocal models by looking for either { a model that can be used as a bridge between a local continuum model and a nonlocal atomistic model ) a model that exhibits local behavior on scales for which local CE models are valid and nonlocal behavior on scales over which nonlocal MD models are tractable or, if we want to reach for Utopia, { a single nonlocal multiscale material mono-model that has the potential to accurately capture desired physics over a wide range of scales • We study the use of the peridynamics (PD) model for solid mechanics developed by Stewart Silling of Sandia for both of these purposes { although a relatively recent development (2000), the effectiveness of PD has already been demonstrated in several sophisticated applications, including - fracture and failure of composites - crack instability - fracture of polycrystals - nanofiber networks { the successful implementation and application of PD within multiscale en- gineering analyses include - the EMU package (http://www.sandia.gov/emu/emu.htm) a three-dimensional meshless Lagrangian code - PD has also been implemented within the massively parallel MD code LAMMPS (http://lammps.sandia.gov/) { Mike Parks is leading a team at Sandia that has nearly finished developing (release planned sometime in 2012) a new open source, massively parallel peridynamics software package (PERIDIGM) for elastic, elastic-plastic, and viscoelastic models https://software.sandia.gov/trac/peridigm (b) (a) (b) (a) (c) (d) PD EMU simulation of cracking in a PD EMU simulation of notched graphite/epoxy composite laminate oscillatory crack path in for two layups; the panel on the right has a thin membrane being a greater percentage of fibers in the ±45 dragged past a rigid degree directions than the panel on the left cylinder (c) (d) (b) (a) (c) (b) (d) (a) PD EMU simulation of dynamic PD EMU simulation of stretching fracture in a polycrystal, showing and failure of initially square nanofiber interface cracks at grain boundaries pulled from the vertical edges (c) (d) PD LAMMPS simulation of the PD LAMMPS simulation of the fracture patterns in a cylinder fracture patterns in a disc after impact with a hard ball after impact with a hard ball which, for clarity, is not displayed which, for clarity, is not displayed Undeformed cylinder PERIDIGM simulations of a fragmenting brittle cylinder PERIDIGM simulation of a fragmenting metal ring with a new peridynamics plasticity model; note regions of necking and failure Dr. Paul Fussel, a Boeing Senior Manager, says Our understanding of composites, even though they have been in service for many years, is less than our understanding of metallic systems. Peridynamic multi-scale approaches enable us to model and understand the micro, meso, and macro performance of matrix, reinforcement, and ensemble materials under varying load conditions that is unparalleled by the more traditional FEM approaches. (Emphases are mine) Why peridynamics? • At \small" scales, PD behaves like the nonlocal MD model ) simple discretizations of the PD model have been shown to reduce to MD ) hence its successful implementation into a production MD code • At \large" scales, PD behaves like the local CE model ) in certain limits, PD has been shown to reduce to CE • In between, PD can \smoothly" effect the transition between nonlocal and local behaviors • Some examples (linear and nonlinear) comparing PD, CE, MD MD CE { atomistically-spaced grid m PD { atomistically-spaced grid PD { coarser grid MD & PD CE MD & PD CE Leonard-Jones simulation MD PD Peridynamics • The peridynamics (PD) model for mechanics introduced by Stewart Silling has the following features: { it is a continuum model { it is a nonlocal model { it is free of spatial derivatives which result in the following desirable features of solutions: { mathematically speaking, spatial solutions operator are less smoothing than classical elliptic operators - indeed, they need not be smoothing at all { solutions with jump discontinuities are admissible { the nucleation and propagation of cracks and other defects can automatically be accounted for • The PD local balance of linear momentum is given by Z 0 0 0 3 ρ(x)u¨(x; t) = k (u − u; x ; x) dx + b(x; t) 8 x 2 B ⊂ R ; t 2 (0;T ) B x = material point in the reference configuration B = the body in the reference configuration u(x; t) = displacement of the material point x η(x; t) = x + u(x; t) = position of the material point x in the deformed configuration u0 = u(x0; t) η0 = η(x0; t) ρ(x) = mass density b(x; t) = external body force density • PD is a continuum model { all material points in the body B are involved • The first term on the right-hand side Z ρ(x)u¨(x; t) = k (u0 − u; x0; x) dx0 + b(x; t) B is an integral operator representing the force density of the body due to deformation { this operator is nonlocal because material points x0 =6 x are involved - classical elasticity ) material points only interact through contact - peridynamics ) material points a finite distance apart interact • The kernel k(·; ·; ·) Z ρ(x)u¨(x; t) = k (u0 − u; x0; x) dx0 + b(x; t) B of the nonlocal operator does not involve any spatial derivatives • The kernel k(·; ·; ·) is assumed to be anti-symmetric k (u0 − u; x0; x) = −k (u − u0; x; x0) { this is a necessary and sufficient condition for the conservation of linear momentum • Linearized PD model Z 0 0 0 ρ(x)u¨(x; t) = C(x ; x)(η − η) dx + b(x; t) B C(x0; x) = micromodulus function • Bond-based PD model ) k(u0 − u; x0; x) is collinear with η0 − η { for linear materials ) C(x0; x) = ζ(x0 − x)(x0 − x) ⊗ (x0 − x) { for isotropic, linear, microelastic materials, this results in a Poisson ratio = 1=4 • State-based PD model ) k(u0 − u; x0; x) may depend on the collective behavior at x and x0 { PD states are the continuum equivalent of the multibody potentials of classical particle mechanics that go beyond central force interactions 0 0 0 { for linear materials ) Cs(x ; x) = Cb(x ; x) + C(x ; x) 0 Cb(x ; x) = contribution from bond-based potential 0 Cs(x ; x) = contribution from influence of the bonds around the bond x0 − x { for isotropic, linear, microelastic materials, any Poisson ratio is possible • A constitutive assumption in PD is that the extent of nonlocal interactions is finite { there exists a horizon δ such that k (u0 − u; x0; x) = 0 if jx0 − xj > δ • In CE, the locality of force interactions implies that there is no length scale independent of specific material behavior e.g., outside of material inhomogeneities • On the other hand, in PD, the nonlocal force density operator does contain a length scale, the PD horizon δ { this length scale enables PD to be a multiscale model which - behaves like MD at small scales - behaves like CD at large scales The multiscale nature of the PD model • We partition the body B into a collection of non-overlapping covering
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