The Pennsylvania State University The Graduate School
THE ANALYSIS OF THE PERIDYNAMIC THEORY OF SOLID
MECHANICS
A Dissertation in Mathematics by Kun Zhou
c 2012 Kun Zhou
Submitted in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
May, 2012 The thesis of Kun Zhou was reviewed and approved∗ by the following:
Qiang Du Verne M. Willaman Professor of Mathematics Dissertation Advisor, Chair of Committee
Ludmil Zikatanov Professor of Mathematics
Wen Shen Professor of Mathematics
Xiantao Li Professor of Mathematics
Suzanne Shontz Professor of Computer Science and Engineering
Svetlana Katok Professor of Mathematics Graduate Program Chair
∗Signatures are on file in the Graduate School. Abstract
The peridynamic model proposed by Silling [1] is an integral-type nonlocal contin- uum theory. It depends crucially upon the non-locality of the force interactions and does not explicitly involve the notion of deformation gradients. It provides a more general framework than the classical theory for problems involving discontinuities or other singularities in the deformation. In this dissertation, we focus on the re- cent developed peridynamic models including the ordinary bond-based, state-based models. The linear ordinary bond-based peridynamic model is analyzed under a rigorous analytical framework. Meanwhile the relation between the peridynamic energy space and fractional sobolev spaces is established for various micromodulus functions. And for better assisting the nonlocal mechanical modeling and nonlocal mathematical analysis, a vector calculus for the nonlocal operators is developed. Nonlocal analogs of several theorem and identities of the vector calculus for differ- ential operators are also presented. Relationships between the nonlocal operators and their differential counterparts are established. We further apply the nonlocal vector calculus to express the constitutive relation for the ordinary state-based peridynamic elastic material. The linear peridynamic models and associated non- local volume-constraints problems are defined and analyzed within the nonlocal vector calculus framework. Especially, the well-posedness of the ordinary state- based peridynamic model for a linear homogeneous and anisotropic material is demonstrated. Moreover, we establish relation between the classical elasticity and nonlocal peridynamic theory as the nonlocal horizon converges to zero. And un- der the mathematical framework introduced, we conduct the numerical analysis of the finite-dimensional approximations to the bond-based peridynamic models. A posterior error estimator for the peridynamic model is also proposed and studied. To recover the full elasticity theory in its local limit, we also developed a peri- dynamic double-bonds model. Finally an open issue involved in the peridynamic crack nucleation theory is discussed.
iii Table of Contents
List of Figures viii
Acknowledgments ix
Chapter 1 Overview 1 1.1 Motivation to the peridynamic theory ...... 1 1.2 The current linear PD models ...... 2 1.2.1 The ordinary bond-based PD model ...... 3 1.2.2 The ordinary state-based PD model ...... 4 1.3 Content of the dissertation work ...... 6
Chapter 2 Mathematical Analysis of Bond-based Linear PD models 10 2.1 Introduction ...... 10 2.2 Bond-based PD model on Rd ...... 12 2.2.1 Mathematics analysis of the PD model ...... 12 2.2.2 Space equivalence for special influence functions ...... 19 2.2.3 Properties of stationary PD Model ...... 24 2.2.4 The time-dependent PD model ...... 26 2.3 The bond-based PD model on a finite bar ...... 29 2.3.1 The PD operators and related function spaces ...... 30 2.3.2 Space equivalence for special influence functions ...... 32 2.3.3 Properties of the stationary PD Model ...... 35 2.3.4 The time-dependent PD model on a finite bar ...... 37 2.4 Conclusion ...... 38
iv Chapter 3 A Nonlocal Vector Calculus, Nonlocal Balance Laws and Non- local Volume-Constrained Problems 39 3.1 Introduction ...... 39 3.1.1 Notation ...... 41 3.2 Nonlocal fluxes and nonlocal action-reaction principles ...... 43 3.2.1 Local fluxes ...... 43 3.2.2 Nonlocal fluxes ...... 44 3.3 Nonlocal operators ...... 45 3.3.1 Nonlocal point divergence, gradient, and curl operators . . . 45 3.3.2 Nonlocal adjoint operators ...... 46 3.3.3 Further observations and results about nonlocal operators . 48 3.3.3.1 A nonlocal divergence operator for tensor functions and a nonlocal gradient operator for vector functions 48 3.3.3.2 Nonlocal vector identities ...... 49 3.3.3.3 Nonlocal curl operators in two and higher dimensions 51 3.4 A nonlocal vector calculus ...... 52 3.4.1 Nonlocal interaction operators ...... 53 3.4.2 Nonlocal integral theorems ...... 54 3.4.3 Nonlocal Green’s identities ...... 57 3.4.4 Special cases of the vector calculus ...... 59 3.4.4.1 The free space vector calculus ...... 59 3.4.4.2 The vector calculus for interactions of infinite extent 60 3.4.4.3 The vector calculus for localized kernels ...... 60 3.5 Relations between nonlocal and differential operators ...... 61 3.5.1 Identification of nonlocal operators with differential opera- tors in a distributional sense ...... 62 3.5.2 Relations between weighted nonlocal operators and weak representations of differential operators ...... 65 3.5.2.1 Nonlocal weighted operators ...... 66 3.5.2.2 Relationships between weighted operators and dif- ferential operators ...... 68 3.5.3 A connection between the nonlocal and local Gauss theorems 74 3.6 The nonlocal volume-constrained problems ...... 75 3.7 Local and nonlocal balance laws ...... 78 3.7.1 Abstract balance laws ...... 79 3.7.1.1 Abstract local balance laws ...... 82 3.7.1.2 Abstract nonlocal balance laws ...... 83 3.7.2 The peridynamics nonlocal theory of continuum mechanics . 84
v Chapter 4 Application of the Nonlocal Vector Calculus to the Analysis of state-based Linear PD Materials 88 4.1 Introduction ...... 88 4.2 Useful identities in the nonlocal vector calculus ...... 89 4.3 Constitutive relations in peridynamic modeling ...... 90 4.4 Variational principles for linear peridynamic models ...... 92 4.4.1 Variation of the potential energy ...... 92 4.5 Well-posedness of the state-based PD solid models ...... 97 4.5.1 Decomposition of the solution space ...... 101 4.5.2 Nonlocal dual spaces and nonlocal trace spaces ...... 102 4.5.3 Well-posedness of variational problems ...... 103 4.5.4 An example of the peridynamic energy space ...... 104 4.6 Concluding remarks ...... 112
Chapter 5 Convergence of the PD models to the classical elasticity theory 114 5.1 Introduction ...... 114 5.2 Model reformulation ...... 115 5.3 Convergence of the bond-based PD model to the classical elasticity 116 5.4 The convergence of the ordinary state-based PD equation to the classical elasticity ...... 121 5.5 Conclusion ...... 126
Chapter 6 The Numerical Analysis of the bond-based PD model 127 6.1 Introduction ...... 127 6.2 Numerical Analysis of the Bond-based PD model on 1-D bar . . . . 128 6.2.1 Finite-dimensional approximations ...... 128 6.3 A posterior error estimate of bond-based PD equations ...... 136 6.3.1 Finite element discretization and a posterior error estimate . 138 6.3.2 Relation with local case ...... 141 6.4 Conclusion ...... 149
Chapter 7 The PD double-bonds model 151 7.1 Introduction ...... 151 7.2 The full nonlocal Peridynamic double-bonds model ...... 151 7.3 Relation between PD double-bonds and PD bond-based models . . 158 7.4 Conclusion ...... 159
vi Chapter 8 An open issue – The nonlinear simulation 160 8.1 Introduction ...... 160 8.2 A one dimensional experiment ...... 162 8.2.1 Simulation method ...... 163 8.2.2 Simulation results ...... 164 8.3 Open questions ...... 168
Bibliography 169
vii List of Figures
1.1 Point q interacts indirectly with x even though they are outside each other’s horizon, since they are both within the horizon of in- termediate points such as p ...... 4
3.1 Four of the possible configurations for Ωs and Ωc...... 53 3.2 For a localized kernel, the domain Ωs and the interaction regions Ωc and Ω whose thicknesses are given by the horizon ε...... 61 − 4.1 Four of the possible configurations for Ω = (Ω Ω ) (Ω Ω )0 . 93 s ∪ c ∪ s ∩ c
6.1 Ωs and Ωc...... 142 6.2 K = Kin Kout, an illustration in the two-dimensional space. . . . . 143 6.3 The δ-neighborhood∪ of an edge e...... 146 6.4 A projection of x onto e...... 146 6.5 An illustration of the integral domain SC(δ, δ + ξ) that denotes the spherical cap of a sphere with radius δ and height δ + ξ...... 147
8.1 Displacement field with δ = 0.1, h = 0.005, t = 0.00001 and t = 0.2 164 8.2 Compare the displacement under the same4 condition but different grid sizes ...... 165 8.3 Compare the displacement under the same condition but different time step ...... 166 8.4 P(x) for the particle in the middle of the bar, i.e. P (0), with mag- nitude of external force 85 ...... 166 8.5 Displacements at t = 0.15, t = 0.19303 and t = 0.24 ...... 167 8.6 Displacements at t = 0.15, t = 0.19303 and t = 0.24 ...... 167 8.7 ∂f/∂η at t = 0.15, t = 0.19303 and t = 0.24 ...... 168
viii Acknowledgments
It is a pleasure to thank those who made this thesis possible. First and foremost, I would like to show my deepest gratitude to my thesis advisor Professor Qiang Du. During my PhD studies, I received extraordinary support from him; his brilliant and patient guidance enabled me to develop a deep understanding of mathematics and a set of skills tackling quantitative challenges; his generous support exempted me from heavy teaching load and provided me plenty of time to think and con- duct my research project. Besides, his great personality and rigorous scholarship influence me a lot. I would also like to thank Professor Max Gunzburger and Dr. Richard B. Lehoucq for their invaluable educations on mathematics, writing, logicality and collaboration. Every communication with them through email, during the confer- ences is exciting and fruitful. They have made available their support in many ways. Moreover I would like to show my gratitude to Dr. Michael L. Parks for his insights and enthusiasm in the conferences in Philadelphia and Lincoln and during my visit to Sandia National Labs. Likewise, I am grateful to Professor Ludmil Zikatanov, Professor Xiantao Li and Professor Wen Shen from math department and Professor Suzanne Shontz from department of computer science and engineering for their time to serve on my committee and for their nice classes that I attended during my PhD studies. Further, I would like to give my appreciation to my friends in the group, Lei Zhang, Tianjiang Li, Manlin Li, Yanxiang Zhao, Yanping Ma and Li Tian as well as Professor Lili Ju, from University of South Carolina, for their help and support. We had a great time together. Last but not least, I am indebted to my parents, Yongqiang Zhou and Xin Huang, who have always gave me their trust and love, and to my wife, Jingyan Zhang, who has given me the strength and support needed during my academic life and being always there on my side.
ix Chapter 1
Overview
1.1 Motivation to the peridynamic theory
The peridynamic(PD) theory of mechanics attempts to unite the mathematical modeling of continuous media, fractures and particles within a single framework, see [1, 2, 3, 4]. It achieves this goal by replacing the partial differential equations of the classical elasticity theory with integral type equations. The model equations are based on the model of internal forces within a body in which material points interact with each other directly over finite distance. The classical elasticity theory is based on the assumption of a continuous distri- bution of mass within a body. And it assumes that the internal forces are contact forces that act across zero distance. The mathematical description of classical elas- ticity that follows from these assumptions relies on PDEs that additionally assume sufficient smoothness of the deformation for the well-posedness of the equation. The classical theory has been demonstrated to provide a good approximation to the response of real materials down to small length scales. Moreover, the PDEs of the classical elasticity do not apply directly on a crack or dislocation because of deformation is discontinuous on these features. Consequently, the techniques of fracture mechanics introduce relations that are extraneous to the basic field equations of the classical theory. For example, linear elastic fracture mechanics considers a crack to evolve according to a separate constitutive model that pre- dicts, on the basis of nearby conditions, how fast a crack grows, in what direction, whether it should arrest, branch, and so on. Although the methods of fracture 2 mechanics provide important and reliable tools in my applications, it is uncertain to what extent this approach can meet the future needs of fracture modeling in complex media under general conditions. Aside from requiring these supplemental constitutive equations for the growth of defects within linear elastic fracture mechanics, the classical theory predicts some well-known nonphysical features in the vicinity of these singularities. The unbounded stresses and energy densities predicted by the classical PDEs are con- ventionally treated in idealized cases by assuming that their effect is confined to a small process zone near the crack tip. However, the reasoning behind neglecting the singularities in this way becomes more troublesome as conditions and geometries become more complex. Molecular dynamics(MD) provides an approach to understanding the mechan- ics of materials at the smallest length scales that has met with important successes in recent years. However, even with the fastest computers, it is widely recognized that MD can not model systems of sufficient size to make it a viable replacement for continuum modeling. These considerations motivate the development of the PD theory which at- tempts to treat the evolution of discontinuities according to the same field equa- tions as for continuous deformation. The PD theory also has the goal of treating discrete particles according to the same field equations as for continuous media. The ability to treat both the nanoscale and the macroscale within the same mathe- matical system may make the method an attractive framework in which to develop multiscale and atomistic-to-continuum methods.
1.2 The current linear PD models
The peridynamic model [1, 2, 3] is a reformulation of solid mechanics in which a point in a continuum interacts directly with other points separated from it by a finite distance. The maximum interaction distance provides a length scale for a material model, called material horizon δ, although the model may additionally contain smaller length scales. The central assumption in the peridynamic model is that the strain energy density W (x) at a point x depends collectively on the deformation of all the points 3 in a neighborhood x with radius, δ > 0. Let x and y be the positions of two particles in the reference configuration, u(t, x) and u(t, y) denote the displacement of the two particles with respect to the reference configuration. And we denote f u(t, y) u(t, x), y x the interaction force density between x and y. Then the − − peridynamic equation of motion is
Z ρ(x)u¨(t, x) = f(u(t, y) u(t, x), y x) dy + b(t, x) (1.1) Ω − − where ρ is the mass density, b the external force density. The material horizon δ is built in the interaction force density f. In this dissertation, we mainly consider the linear peridynamic equation,
Z ρ(x)u¨(t, x) = C(x, y) u(t, y) u(t, x) dy + b(t, x) (1.2) Ω − where C(x, y) is a matrix-valued function obtained by taking the derivative of f u(t, y) u(t, x), y x with respect to the relative displacement u(t, y) u(t, x). − − − And the strain energy density is Z T W (t, u)(x) = u(t, y) u(t, x) C(x, y) u(t, y) u(t, x) dy. (1.3) Ω − −
The function C(x, y) is a 2nd order tensor, called micromodulus function which contains all the information including the material horizon δ that uniquely deter- mines the material properties. Depending on the form of C(x, y), the peridynamic model equations are divided into two groups, the ordinary bond-based PD model, the ordinary state-based PD model.
1.2.1 The ordinary bond-based PD model
The ordinary bond-based model interaction between two particles x and y only depends on the two particles themselves, independent of any others surrounding them. And the direction of the force is parallel to the bond x y. Then the − micromodulus can be expressed as
ω(x y) C(x, y) = − (y x) (y x) (1.4) y x 2 − ⊗ − | − | 4
where the function ω(x y) is called influence function defined in [2, Definition − 3.2], which measures the strength of the direct interaction between two particles, is normally defined as ( ω(x y) x y < δ ω(x y) = − | − | (1.5) − 0 otherwise. i.e. two particles lose interaction if their distance is larger than δ. Thus the bond-based PD strain energy density function is given by
1 Z ω(x y) 2 W (u) = − (u(t, y) u(t, x)) (y x) dy (1.6) b 2 y x 2 − · − Ω | − | 1.2.2 The ordinary state-based PD model
The state-based PD model assume that two particles can interact through an intermediate particle, see figure 1.2.2.
Figure 1.1. Point q interacts indirectly with x even though they are outside each other’s horizon, since they are both within the horizon of intermediate points such as p 5
In the figure, the point q interacts indirectly with x even though they are outside each other’s neighborhood with horizon δ since they are both within the horizon of the intermediate point such as p. It’s still a model for the ordinary material, so the direction of the force is also parallel to the bond x y. Based on − the above discussion the micromodulus function is given by
ω(x y) C(x, y) = − (y x) (y x) + C (x, y) (1.7) y x 2 − ⊗ − s | − | where Cs(x, y) describes the indirect interaction force between x and y whose ex- act expression will be derived in chapter 4.
To introduce the energy density function, we first review the peridynamic con- stitutive relation defined in [2, section 13]. Let y = u(t, y) + y u(t, x) x and | − − | x = y x for any given pair x and y. The fundamental deformation quantities | − | used in the peridynamic modeling can be expressed as follows:
extension scalar state e = y x, (1.8a) − weighted volume m = (ωx) x, (1.8b) • d dilatation θb = ωx e, (1.8c) m • θxb isotropic part of e ei = , (1.8d) d deviatoric part of e ed = e ei, (1.8e) − where d denotes the space dimension. The operation between different states • may be viewed as an inner product in a Hilbert space whose precise form is not required in the current dissertation; interested readers can refer to [2, Definition 2.6] for detailed explanations. Then the energy density function is defined as the following.
k(x)θb2 η(x) W (u) = + (ωed) ed. (1.9) s 2 2 •
The energy density is composed by the dilatation part and the shear deformation 6 part, i.e. deviatoric part in this language.
1.3 Content of the dissertation work
In this dissertation, we focus on various mathematical aspects of the PD models and nonlocal modeling.
In chapter 2, we study a linear PD model for a spring network systems in Rd, one dimensional PD bar with a periodic boundary condition, as some illustrative examples. The detailed analysis on the related PD operators and the associated functional spaces are conducted. And we prove the well-posedness of weak solu- tions to the PD equation, together with studies on the solution regularity. We point out, in particular, that for some special cases of singular micromodulus func- tions, the solution operators still share certain smoothing properties in fractional Sobolev spaces. These mathematical results can become useful in analyzing the output of numerical simulations based on the PD models and in assessing the qual- ity of the numerical solutions. Some of the materials can be found in [25] and [28], coauthored with my dissertation advisor Professor Qiang Du.
In chapter 3, we introduce a vector calculus for nonlocal operators that mimics the classical vector calculus for differential operators. We define nonlocal analogs of the divergence, gradient, and curl operators and deduce the corresponding nonlocal adjoint operators. Nonlocal analogs of the Gauss theorem and the Green’s identi- ties of the vector calculus for differential operators are also derived. We establish relationships between the nonlocal operators and their differential counterparts. The nonlocal vector calculus is also used to define nonlocal volume-constrained problems that are analogous to boundary-value problems for partial differential operators and to express the nonlocal balance laws1 in the sense that subregions not in direct contact may have a non-vanishing interaction. Our nonlocal vector calculus, then, provides an alternative to standard approaches for circumventing
1A balance law postulates that the rate of change of an extensive quantity over any subregion of a body is given by the rate at which that quantity is produced in the subregion minus the flux out of the subregion through its boundary. 7 the technicalities associated with the lack of sufficient regularity in local balance laws. Some of the materials can be found in [44], and coauthored with Professor Qiang Du from Penn State University, Professor Max Gunzburger from Florida State University and Dr. Rich. B. Lehoucq from Sandia National Laboratories.
In chapter 4, our major contribution here is the well-posedeness of the state- based linear peridynamic equilibrium equation, a first of a kind result. We rep- resent the peridynamic deformation fields such as the extension scalar, dilatation, etc., in terms of operators from the nonlocal vector calculus. Based upon the rep- resentation of the deformation state, we then formulate variational principles and rewrite both the linear bond-based and the linear state-based peridynamic elastic model via the nonlocal divergence operators and their corresponding adjoints. Our variational formulation coincides with the linearization of the original peridynamic equation derived in [1, 2]. Moreover, we prove the well-posedness of peridynamic “boundary”-value problem, or more appropriately, volume-constrained problem, that is, the linear peridynamic elastic model subject to suitable constraints that are either explicitly enforced on volumes of non-zero measure or naturally implied by the variational principle. Some of the materials can be found in [50], and coauthored with Professor Qiang Du from Penn State University, Professor Max Gunzburger from Florida State University and Dr. Rich. B. Lehoucq from Sandia National Laboratories.
In chapter 5, we demonstrate that the anisotropic PD operator including the ordinary bond-based and state-based operators, in the local limit, tends to the classical elasticity operator CE(u) = (C:( u + uT ). The clear relation − −∇ · ∇ ∇ between the nonlocal parameters, such as the influence function ω(x, y), and the 4th order stiffness tensor in the classical elasticity is established for the three kinds of models. We also showed that how the parameters relate to the coupling of elon- gations and shears in the limiting sense. As special cases, it is illustrated that the isotropic bond-based model converges the classical Navier equation with Poisson 1/4, and the isotropic ordinary state-based model, as a general model, converges to the Navier equation with arbitrary Possion ratio. Some of the materials can be found in [25, 28, 50], Professor Qiang Du from Penn State University, Professor 8
Max Gunzburger from Florida State University and Dr. Rich. B. Lehoucq from Sandia National Laboratories.
In chapter 6, based on the analytical framework established in chapter 2, we consider various finite dimensional approximations to the nonlocal BVPs. A num- ber of theorems are given on the convergence and error estimates of these approx- imations which appear to be the first of their kind in the literature. Estimates of the condition numbers of the resulting linear systems of equations are also provided here. Our findings are established in very general settings, and they are consistent with the results derived or observed for various specialized cases in earlier studies [5, 6, 7, 8]. In order to have a better understanding of the PD models, their nu- merical approximations, and their various applications, detailed analytical studies are becoming increasingly important. The results given in this chapter can be considered as another step forward in this direction. Some of the materials can be found in [28] and in [19]. The work is coauthored with Professor Qiang Du from Penn State University, Professor Lili Ju from University of South Carolina and Dr. Li Tian from Penn State University.
In chapter 7, we develop the PD Double-bonds model in order to recover the full elasticity in its local limit. In chapter 5, it is shown that the current PD models can not recover the 21 independent coefficients in the stiffness matrix in the classi- cal material. The PD double-bonds model effectively incorporate the interactions among elongations and shear deformations in different directions in a single-integral operator. We use Fourier analysis to show that the PD Double-bonds model con- verges to the classical elasticity model with 21 independent coefficients. And it can be reduced to the PD bond-based model by specifying the influence function ω.
In chapter 8, we list some open issues in the crack nucleation theory. We first review the analytical and numerical result in [63] for the crack nucleation condition. We conduct an similar one dimensional experiment in order to investigate more precise relation between the different physical quantities in the PD model. Some preliminary results are provided. It illustrates some different scenarios from the 9 discussion in [63] which is listed to be an open question at this moment. Chapter 2
Mathematical Analysis of Bond-based Linear PD models
2.1 Introduction
The effectiveness of PD model has already been demonstrated in several sophis- ticated applications, including the fracture and failure of composites, crack in- stability, fracture of polycrystals, and nanofiber networks. Yet, from a rigorous mathematical point of view, many important and fundamental issues remain to be studied. In this chapter, we intend to formulate a rigorous functional analyt- ical framework of the PD models so as to provide a better understanding of the PD model and to guide us in the development and analysis of the numerical al- gorithms. This in turn will help us utilize the PD theory for multiscale materials modeling. Indeed, PD can be effectively used in the multiscale modeling of mate- rials in different ways: it can serve as a bridge between molecular dynamics (MD) and continuum elasticity (CE) to help mitigate the difficulties encountered when one attempts to couple MD and CE directly [9, 10, 11, 12, 13] and, in some situa- tions, PD can be used as a stand-alone model to capture the behavior of materials over a wide range of spatial and temporal scales. For example, to study problems involving defects, one can use the same equations of motion over the entire body and no special treatment is needed near or at defects. Such properties make PD a powerful tool for modeling problems involving cracks, interfaces or defects, we 11 refer to [14] for a review of the recent applications of the PD framework. Parallel to the modeling and application to practical problems, there also have been efforts to establish a sound theoretical foundation for the PD model. For instance, an abstract variational formulation is presented in [15]. Some results on the existence and uniqueness of L2 solutions of the PD models associated with bounded integral PD operators have been given in [16]. Though much of the focus of [17] is on developing homogenization theory for the PD model, some existence and uniqueness results are also provided, again for bounded integral PD operators. In [18], a nonlocal vector calculus was developed which also provided a rigorous framework for studying the boundary value problems of the nonlocal peridynamic models. In this chapter, we study a linear peridynamic model for a spring network sys- tems in Rd and boundary value problem on a one-dimensional bar. The bond-based model in Rd is discussed in section 2.2 The detailed analysis on the related PD operators and the associated functional spaces are given in section 2.2.1 and 2.2.2. We consider the stationary and time-dependent PD models in section 2.2.3 and Section 2.2.4 respectively. Following the same line, we discuss the bond-based PD model in section 2.3. The properties of the models and their solutions depend crucially on the particular influence functions used to specify the spring network systems. Our results are valid for more general influence functions than consid- ered in the earlier literature. Indeed, the only essential assumption on the influence functions is that appropriate elastic modulus can be defined for the material model. For these more general cases, we prove the well-posedness of weak solutions to the peridynamic equation, together with studies on the solution regularity. We point out, in particular, that for some special cases of singular influence functions, the solution operators still share certain smoothing properties in fractional Sobolev spaces. These mathematical results can become useful in analyzing the output of numerical simulations based on the PD models and in assessing the quality of the numerical solutions. 12
2.2 Bond-based PD model on Rd
In this section, we analyze the bond-based PD model (1.6) on the whole space Rd. We set the micromodulus function
ω( x y ) C(x, y) = | − | (y x) (y x) y x 2 − ⊗ − | − | i.e. the influence function is set to be a radial, ω(x y) = ω( y x ), so the model − | − | under consideration is isotropic and homogenous.
Then, the bond-based PD equation of motion is give by d utt(t, x) = Lδu(t, x) + b(t, x) , (t, x) (0,T ) R ∀ ∈ × u(0, x) = g(x) , x Rd (2.1) ∀ ∈ d ut(0, x) = h(x) , x R ∀ ∈ where
Z ω( y x ) L u(x) = | − | (y x) (y x) u(y) u(x) dy. (2.2) δ y x 2 − ⊗ − − Bδ(x) | − | 2.2.1 Mathematics analysis of the PD model
To set up a suitable functional setting to discuss the well posedness and properties of peridynamic model equations, we first make some assumptions on the influence function ω: Z 2 ω( x ) > ρ( x ) 0, x Bδ(0) , and τδ := x ω( x ) dx < , (2.3) | | | | ≥ ∀ ∈ Bδ(0) | | | | ∞ where ρ( x ) is non-negative function belonging to L1 B (0) which is strictly pos- | | δ itive in at least a measure nonzero subset of B (0). The condition that ω( x ) > δ | | ρ( x ) is easily satisfied by all practical choices of the influence functions. | |
We note that, as pointed out in the literature (see for instance, equation (2.10) in [16], and equation (93) in [1]), the assumption on τδ being finite is needed in or- der to have a suitable definition of the elastic moduli for the corresponding material 13 under consideration. This assumption, in fact, allows us to study the bond-based PD models with much more general influence functions, and thus more general micromodulus functions, than those considered in the existing mathematical anal- ysis. As a notational convention, we use uˆ = uˆ(ξ) to denote the Fourier transform of u = u(x). Moreover, u¯ denotes the complex conjugate of u, and uT denotes the transpose of u. By performing the Fourier transform, we can introduce an equivalent definition of our multidimensional peridynamic operator,
Z 1 ix ξ ξ ξ · ξ Lδu(x) = d/2 Mδ( )uˆ( )e d (2.4) − (2π) Rd where M (ξ), a real-valued and symmetric positive semi-definite d d matrix, is δ × the Fourier symbol of the pseudo-differential operator, see [20], L : − δ Z ω( y x ) M (ξ) = | − | 1 cos(ξ y) y y dy (2.5) δ y x 2 − · ⊗ Bδ(0) | − | for any ξ Rd and for any PD horizon parameter δ > 0. Moreover referring to ∈ [16], the PD operator vanishes identically when applied to rigid body motions that read as u(x) = Rx + r for some skew-symmetric R and some vector r. Since here PD operator is defined on L2(Rd), its kernel contains only u(x) = 0.
By the equivalent form of peridynamic operator, we can define the following functional space, equipped with an associated norm:
d Definition 1. The space ω(R ), which depends on the influence function ω, M 2 d d consists of all the functions u L (R ) for which the ω(R ) norm ∈ M
1 Z 2
u ω = uˆ(ξ) (I + Mδ(ξ))uˆ(ξ)dξ , (2.6) M k k Rd · is finite. We also define the corresponding inner product associated with the Mω norm: Z
(u, v) ω = vˆ(ξ) (I + Mδ(ξ))uˆ(ξ)dξ , (2.7) M Rd · d 1 d for any u, v ω(R ). In addition, we use ω− (R ) to denote the dual space of ∈ M M d ω(R ). M 14
Remark: The norm is well-defined since I + Mδ(ξ) is real-valued symmetric pos- itive definite matrix and it is uniformly bounded below by I.
Meanwhile, we can have the following properties:
d Lemma 1. The space ω(R ) is a Hilbert space corresponding to the inner product M ( , ) ω . · · M d Proof: Let un be a Cauchy sequence in ω(R ). By definition, it is equivalent { } M to say n 1 o (I + Mδ) 2 uˆn is a Cauchy sequence in L2(Rd). So by the completeness of L2(Rd), there exists an element v L2(Rd), such that ∈
1 (I + M (ξ)) 2 uˆ (ξ) v(ξ) 2 0 k δ n − kL → as n . Then we set → ∞
1 1 u(x) = F − [(I + Mδ(ξ))− 2 v(ξ)],
1 where F − denotes the inverse Fourier transform. Then one can see that
1 2 2 un u ω = (I + Mδ(ξ)) (uˆn(ξ) uˆ(ξ)) L 0. k − kM k − k →
d So the space ω(R ) is complete, and it is thus a Hilbert space. M
d Lemma 2. The dual space of ω(R ) is the space of distributions: M Z 1 d 1 ω− (R ) = u : uˆ(ξ) (I + Mδ(ξ))− uˆ(ξ)dξ < , M Rd · ∞ equipped with the norm
1 Z 2 1 1 ξ ξ − ξ ξ u ω− = uˆ( ) (I + Mδ( )) uˆ( )d . k kM Rd · 15
d Proof: Let l = l(u) be a bounded linear functional on ω(R ), then by the M d Riesz Representation Theorem we know that there exists a unique w ω(R ) ∈ M d d such that l(u) = (u, w) ω(R ) for any u ω(R ). Using the inner product given M ∈ M in (2.7), we have Z l(u) = wˆ (ξ) (I + Mδ(ξ))uˆ(ξ)dξ . Rd · 1 2 d Let vˆ(ξ) = (I + Mδ(ξ))wˆ (ξ). We have (I + Mδ(ξ)) 2 wˆ (ξ) L (R ) since w ∈ ∈ d ω(R ). Thus, M
1 1 2 d (I + Mδ(ξ))− 2 vˆ(ξ) = (I + Mδ(ξ)) 2 wˆ (ξ) L (R ) . ∈
1 d So, v ω− (R ) and ∈ M Z l(u) = uˆ(ξ) vˆ(ξ)dξ . Rd · Again by the Riesz Representation Theorem, we have
2 2 l = w ω k k Zk kM = wˆ (ξ) (I + Mδ(ξ))wˆ (ξ)dξ d · ZR 1 = vˆ(ξ) (I + Mδ(ξ))− vˆ(ξ)dξ Rd · 1 = v ω− k kM
1 d d Meanwhile, if v ω− (R ), for any u ω(R ), ∈ M ∈ M Z (u, v)L2 = uˆ(ξ) vˆ(ξ) dξ | | | Rd { · } | Z n 1 1 o = uˆ(ξ) (I + Mδ(ξ)) 2 (I + Mδ(ξ))− 2 vˆ(ξ) dξ | Rd · d 1 6 u ω(R ) v −ω . k kM k kM
1 d d So, an element v ω− (R ) corresponds a bounded linear functional on ω(R ). ∈ M M
d Lemma 3. The peridynamic operator Lδ is self-adjoint on ω(R ). The oper- − M 16
d 1 d ator Lδ + I is also an isometry from ω(R ) to ω− (R ), and the norm and − M M d inner product in ω(R ) can also be formulated as M
1 1 2 2 u ω = (u, u) ω = [(u, u) + ( Lδu, u)] k kM M − 1 Z Z 2 2 ω( y x ) 2 = u L2 + | − 2| (u(y) u(x)) (y x) dydx k k d y x − · − R Bδ(x) | − | (2.8)
d for any u ω(R ). ∈ M Remark 1. The result of this Lemma is analogous to the classical result corre- sponding to the differential operator , that is, + I is self-adjoint and it is −4 −4 1 d 1 d an isometry from H (R ) to H− (R ).
Proof: By the equivalent definition of the PD operator in (2.4) as a pseudo- differential operator with a real, nonnegative symbol, we immediately see that d Lδ is self-adjoint in ω(R ). The fact that Lδ + I defines an isometry follows − M − d 1 d directly from the definitions of the norms in ω(R ) and ω− (R ). In addition, M M using the Parseval formula, we have Z ( Lδu, u) + (u, u) = ( Ldδu, uˆ) + (uˆ, uˆ) = uˆ(ξ) (I + Mδ(ξ)) uˆ(ξ)dξ , − − Rd · which implies the equation (2.8).
For the purpose of discussing the regularity of the weak solutions, we also need to define the following space Z 2 d 2 ω(R ) = u : uˆ(ξ) (I + Mδ(ξ)) uˆ(ξ)dξ < , M Rd · ∞ with the dual space Z 2 d 2 ω− (R ) = u : uˆ(ξ) (I + Mδ(ξ))− uˆ(ξ)dξ < M Rd · ∞ which share the similar properties as the ones in Lemma 1, Lemma 2 and Lemma 3. 17
Given any y B (0), let us define a difference operator D by ∈ δ y
D v(x) = v(x + y) v(x) y − for function v defined in a suitable function space. By the representation of Lδ given in (7.8), we have
Z Z ω( y ) 1 Dy + D y L = | | y yD dy = ω( y )(y y) − dy (2.9) δ y 2 ⊗ y 2 | | ⊗ y 2 Bδ(0) | | Bδ(0) | | which gives an interesting formulation of Lδ as a linear combination difference operators (first or second order). The matrix-valued weight ω( y )(y y) is in fact | | ⊗ in L1(Rd) under the assumption (2.3). Then,
Lemma 4. Let P be an scalar operator which commutes with difference operator D for all y B (0), then P also commutes with L . y ∈ δ δ
Proof: For u = u(x) with Lδu suitably defined, we have
Z ω( y ) Z ω( y ) | | y yP D u(x)dy = | | y yD P u(x)dy , y 2 ⊗ y y 2 ⊗ y Bδ(0) | | Bδ(0) | |
d That is, P (Lδu)(x) = Lδ(P u)(x), for any x R , so the lemma follows. ∈
We note that in particular, all scalar linear differential operators with constant coefficients, and their inverses (when they exist), commute with the difference operator Dy for any y, so we have,
Corollary 1. Let P be a scalar linear differential operator with constant coeffi- cients, then PLδ = LδP .
Let us remark that the results of Lemma 4 and Corollary 1 depend crucially on the facts that in the representation of the PD operator given by (7.8), the horizon parameter δ is a uniform constant in space and that the influence function ω = ω( y x ) is only a function of y x . | − | | − | The above corollary in particular allows us to have the well-posedness of the generalized (in distribution sense) solutions to the peridynamic equation with rough data. For instance, even when the external force b = b(t, x) is not in 18
L2, we may still get a unique solution to the equation (2.29) in the appropriate 1 1 weak sense by lifting b to P − b for some P , if P − b can be properly defined, to 1 1 get a generalized solution of (2.29) in the form P ( L + I)− P − b. − δ
d Now we can discuss the equivalence between the defined space ω(R ) and M standard Sobolev spaces. Similar results for the special case of scalar valued func- tions can be found in [21]. To treat the vector valued case, we adopt the convention that for two symmetric matrices A and B, A < B means B A is positive definite, − and A B means B A is positive semi-definite. Let , denote the continuous ≤ − → embedding of function spaces.
The following general embedding results can be shown:
Lemma 5. Let the influence function ω = ω( y ) satisfy the condition (2.3). Then | |
τδ 2 d 0 < Mδ(ξ) ξ I , ξ R (2.10) 6 2 | | ∀ ∈
Consequently, we have
1 d d 2 d 2 d H (R ) , ω(R ) ,H (R ) , ω(R ). → M → M
Proof: Using the inequality
(ξ y)2 0 1 cos(ξ y) · ξ 2 y 2/2, 6 − · 6 2 6 | | | | we have
Z 1 cos(ξ y) M (ξ) = ω( y ) − · y y dy δ | | y 2 ⊗ Bδ(0) | | Z 1 cos(ξ y) ω( y ) − · y 2dy I 6 | | y 2 | | Bδ(0) | | Z 1 2 2 6 ω( y ) ξ y dy I 2 Bδ(0) | | | | | | τ = δ ξ 2I 2 | | 19 which gives (2.10). The rest of the conclusions follow from the above inequality and the definitions of the relevant function spaces.
2.2.2 Space equivalence for special influence functions
We now focus on some influence functions with special properties to establish d relations between ω(R ) and the more conventional Sobolev spaces. To examine M more singular influence functions, we let
s d 2 d d s 2 d d H (R ) = u (L (R )) : ξ uˆ (L (R )) ∈ | | ∈ denote the fractional Sobolev space on Rd for s (0, 1). ∈
For convenience, we define the following functions for χ > 0 and α 0: ≥ Z 1 cos(z1) C (α, χ) = − min(z2)dz , (2.11) min z 2+d+2α Bχ(0) | | Z 1 cos(z1) C (α, χ) = − max(z2)dz . (2.12) max z 2+d+2α Bχ(0) | | Here, we have min(z2) = min z2 with z d being the components of z, and { i } { i}i=1 similarly, max(z2) = max z2 . { i } For influence functions ω such that ω( x ) L1(B (0)), there are already some | | ∈ δ 2 d results in [17, 16] showing that Lδ is a bounded linear operator from L (R ) to − L2(Rd). In fact, we have:
Lemma 6. Let ω = ω( y ) satisfy the additional condition that | | Z ω( y ) dy < , (2.13) Bδ(0) | | ∞ we then have 2 d d 2 d ω(R ) = ω(R ) = L (R ) , M M 20 and Z 1 2 d 2 d 2 2 u L 6 u ω(R ) 6 u L (1 + ω( y ) dy) , u L (R ) , (2.14) M k k k k k k Bδ(0) | | ∀ ∈ Z 2 d 2 2 d 2 u L 6 u ω(R ) 6 u L (1 + ω( y ) dy) , u L (R ) . (2.15) M k k k k k k Bδ(0) | | ∀ ∈ 1 Moreover, the operators L and ( L + I)− are bounded linear operators from − δ − δ L2(Rd) to L2(Rd).
Proof: Under the condition of ω, we can see
Z 0 < Mδ(ξ) 6 ω( y ) dy I . Bδ(0) | |
By Parseval identity, we have (2.14) and (2.15) which in turn implies that L − δ 1 2 d 2 2 1 d and ( Lδ + I)− are bounded operators from L (R ) to L (R ), and ω− (R ) = − M d 2 d ω(R ) = L (R ). M
For more general influence functions, i.e. ω satisfying (2.3), the PD operator 2 d Lδ may become unbounded in L (R ) when the function ω( x ) is no longer − | | 1 in L (Bδ(0)). Yet, as we demonstrate below, the basic existence and uniqueness results remain valid but with the discussion taking place in other function spaces d such as ω(R ), as defined earlier. This is due to the fact that Lδ becomes a M − d 1 d bounded operator from ω(R ) to ω− (R ). To see how such spaces are related M M to the conventional Sobolev spaces, we first consider the space equivalence for some special influence functions.
Lemma 7. Let the influence function ω = ω( y ) satisfy the assumption (2.3) and | | the condition
d 2β ω( y ) γ1 y − − , y δ (2.16) | | > | | ∀| | 6 for some exponent β (0, 1) and positive constant γ1, then we have ∈
d β d 2 d 2β d ω(R ) , H (R ) , ω(R ) , H (R ). M → M → 21
β d 1 d 2β d 2 d H− (R ) , ω− (R ) ,H− (R ) , ω− (R ). → M → M Moreover, we have
d β d d C1 u H (R ) 6 u ω(R ) , u ω(R ) , (2.17) k k k kM ∀ ∈ M
β d C u 1 d u β d , u H− ( ) , (2.18) 1 ω− (R ) 6 H− (R ) R k kM k k ∀ ∈ and 2 d ˜ 2β d 2 d C1 u H (R ) 6 u ω(R ) , u ω(R ) , (2.19) k k k kM ∀ ∈ M 2β d C˜ u 2 d u 2β d , u H− ( ) , (2.20) 1 ω− (R ) 6 H− (R ) R k kM k k ∀ ∈ 1 with constants C1 = min(1, (γ1Cmin(β, δ)) 2 ) and C˜1 = min(1, γ1Cmin(β, δ)).
Proof: Under the condition on σ, when ξ 1, we do a change of variable | | >
z = ξ Ry | | where R is an orthogonal matrix from Rd to Rd with the first row being ξ/ ξ . | | So we have
Z 1 cos(ξ y) M (ξ) = ω( y ) − · y y dy δ | | y 2 ⊗ Bδ(0) | | Z 1 cos(ξ y) γ1 − · y y dy > y 2+d+2β ⊗ Bδ(0) | | Z 2+2β 1 cos(z1) 1 T 1 T = γ1 ξ − R z R zdz 2+d+2β ξ ξ | | B ξ δ(0) z ⊗ | | | | | | | | Z 2β 1 cos(z1) T ξ 0 = γ1 −2+d+2β R z z Rdz | | B ξ δ(0) z ⊗ | | | | Z 1 cos(z ) ξ 2β 1 T 2 = γ1 −2+d+2β R diag(zi )Rdz | | B ξ δ(0) z | | | | Z 2β 1 cos(z1) 2 γ1 ξ − min(z )dz I > | | z 2+d+2β Bδ(0) | | 2β = γ1C (β, δ) ξ I . min | | 22 where the equality in the 5th line holds due to the symmetry of the integration domain. Thus the space embedding results follow. Moreover (2.17) and (2.19) are satisfied. The inequalities (2.18) and (2.20) follow by duality estimates. Meanwhile, we also have
Lemma 8. Let the influence function ω = ω( y ) satisfy the condition | |
d 2α ω( y ) γ2 y − − , y δ (2.21) | | 6 | | ∀| | 6 for some exponent α (0, 1) and positive constant γ2, then we have ∈
α d d 2α d 2 d H (R ) , ω(R ) ,H (R ) , ω(R ) , → M → M and 1 d α d 2 d 2α d ω− (R ) , H− (R ) , ω− (R ) , H− (R ). M → M → Moreover, we have
α d d α d u ω(R ) 6 C2 u H (R ), u H (R ) , (2.22) k kM k k ∀ ∈
1 d u α d C u 1 d , u − ( ) , (2.23) H− (R ) 6 2 −ω (R ) ω R k k k kM ∀ ∈ M and 2α d 2 d ˜ 2α d u ω(R ) 6 C2 u H (R ), u H (R ) , (2.24) k kM k k ∀ ∈ 2 d u 2α d C˜ u 2 d , u − ( ) , (2.25) H− (R ) 6 2 −ω (R ) ω R k k k kM ∀ ∈ M 1 with constants C2 = max(1, (γ2C (α, )) 2 ) and C˜2 = max(1, γ2C (α, )). max ∞ max ∞ Proof: Similar to the proof of the previous lemma, under the condition on σ and for ξ > 0, we make a change of variable z = ξ Ry where R is an orthogonal | | | | matrix from Rd to Rd with the first row ξ/ ξ . Then we have | | Z 1 cos(ξ y) M (ξ) = ω( y ) − · y y dy δ | | y 2 ⊗ Bδ(0) | | Z 1 cos(ξ y) γ2 − · y y dy 6 y 2+d+2α ⊗ Bδ(0) | | 23
Z 2+2α 1 cos(z1) 1 T 1 T = γ2 ξ − R z R zdz 2+d+2α ξ ξ | | B ξ δ(0) z ⊗ | | | | | | | | Z 2α 1 cos(z1) T ξ 0 = γ2 −2+d+2α R z z Rdz | | B ξ δ(0) z ⊗ | | | | Z 1 cos(z ) ξ 2α 1 T 2 = γ2 −2+d+2α R diag(zi )Rdz | | B ξ δ(0) z | | | | Z 1 cos(z ) ξ 2α 1 2 6 γ2 −2+d+2α max(z )dz I | | d z R | | 2α = γ2C (α, ) ξ I . max ∞ | | where the equality in the 5th line holds due to the symmetry of the integration domain. Thus the space embedding results follow from the respective definitions of the function spaces. Moreover, (2.22) and (2.24) are satisfied. The inequalities (2.23) and (2.25) then follow from duality estimates. Consequently, we see that under certain conditions on the influence function, d the space ω(R ) is equivalent to some standard fractional Sobolev spaces. M Theorem 1. Let the influence function ω = ω( y ) satisfy the condition | |
d 2α d 2α γ1 y − − ω( y ) γ2 y − − , y δ (2.26) | | 6 | | 6 | | ∀| | 6 for some exponent α (0, 1) and positive constants γ1 and γ2, then we have ∈
d α d 2 d 2α d ω(R ) = H (R ) , ω(R ) = H (R ). M M
d Moreover, for any u ω(R ), ∈ M
α d d α d C1 u H (R ) 6 u ω(R ) 6 C2 u H (R ) , (2.27) k k k kM k k
2 d and for any u ω(R ), ∈ M
˜ 2α d 2 d ˜ 2α d C1 u H (R ) 6 u ω(R ) 6 C2 u H (R ) , (2.28) k k k kM k k with the positive constants C1, C2, C˜1 and C˜2 defined in Lemmas 7 and 8. 24
We see from the above discussions that, under additional assumptions on the form of the influence function ω, we have the equivalence or continuous embedding d theories between ω(R ) and certain fractional Sobolev spaces. M
2.2.3 Properties of stationary PD Model
In this section, we give some results on the existence and uniqueness of weak so- lutions to the stationary (equilibrium) PD model with general influence functions, 2 d i.e. Lδ may be unbounded in L (R ):
L u + u = b (2.29) − δ and some convergence properties of the solution of the stationary PD model. The term u is added for two purposes, one is to eliminate the need to imposing far field conditions at infinity and the other is to eliminate the nonuniqueness of solution when no boundary condition is imposed.
First, we may also establish the corresponding variational theory and some regularity properties for the stationary PD model. Then, using the properties of the PD operator provided earlier, we have
1 d Lemma 9. Let ω = ω( y ) satisfy the condition (2.3), for any b ω− (R ), the | | ∈ M d problem (2.29) has a unique solution u ω(R ) which is the minimizer of the ∈ M functional:
1 2 E(u) = u d (u, b) 2 d ω(R ) L (R ) 2k kM − 1 Z = uˆ(ξ) (I + Mδ(ξ))uˆ(ξ)dξ (u, b)L2(Rd) (2.30) 2 Rd · −
d in ω(R ). M Proof: The conclusion follows directly from the fact that E = E(u) is a convex 1 d quadratical functional with Lδu + u b ω− (R ) being its variational deriva- − − ∈ M d tive at u ω(R ). ∈ M 25
We note that more general variational descriptions of the PD models can be found in [15, 18]. As for regularity, we have for some special influence functions that:
Lemma 10. Let ω = ω( y ) satisfy the condition (2.26), the problem (2.29) has a | | unique solution u Hm+2α(Rd), whenever b Hm(Rd) for any m 2α. ∈ ∈ > − Proof: Taking the Fourier transform of the equation (2.29), we get
(Mδ(ξ) + I)uˆ(ξ) = bˆ(ξ) . (2.31)
Then we have
2 m ˆ 2 2 m 1 d ((Mδ(ξ) + I)uˆ(ξ)) ((Mδ(ξ) + I)) uˆ(ξ)( ξ + 1) = b(ξ) ( ξ + 1) L (R ) . · | | | | | | ∈
By Theorem 1, we have
2 2 + 2 1 ( ξ + 1) α m uˆ(ξ) L (Rd) . | | | | ∈
So the result follows.
We now go back to the general influence functions to consider the regularity of solution of the equilibrium equations (2.29).
Lemma 11. Let the influence function ω = ω( y ) satisfy (2.3), the problem (2.29) | | d 1 d 2 d has a unique solution u ω(R ) for b ω− (R ). Moreover, if b L (R ), ∈ M ∈ M ∈ the solution of the equilibrium equation (2.29) satisfies
2 d u ω(R ). ∈ M
Proof: The first part follows from the isometry property given in the Lemma 3. The proof of regularity is similar to that of the Lemma 10.
By Lemma 4 and Corollary 1, we also have the following regularity
Lemma 12. Let P be a linear scalar operator with constant coefficients, if u is a solution of the equation (2.29) with a given function b, then P u is the solutions 26 of (2.29) with the right hand side being P b.
Proof: From Lemma 4 and Corollary 1, we know L P = PL . So − δ − δ
L (P u) = P ( L u) = P b. − δ − δ
This leads to the result of the lemma.
2.2.4 The time-dependent PD model
With the suitable function spaces for the PD operator and the stationary peridy- namic model given earlier, we now proceed to discuss the existence and uniqueness of the solutions of the time-dependent PD model (2.1) in these spaces, again for more general influence functions ω = ω( y ). | | Using the Fourier transform, we first rewrite the PD equation (2.1) as
ˆ uˆtt(t, ξ) + Mδ(ξ)uˆ(t, ξ) = b(t, ξ) , uˆ(0, ξ) = gˆ(ξ) , (2.32) uˆt(0, ξ) = hˆ(ξ) .
By Duhamel’s principle, we formally have p p sin ( Mδ(ξ)t) ˆ uˆ(t, ξ) = cos ( Mδ(ξ)t)gˆ(ξ) + p h(ξ) Mδ(ξ) Z t 1 p ˆ + p sin ( Mδ(ξ)s)b(t s, ξ)ds . (2.33) Mδ(ξ) 0 −
Then by taking the inverse Fourier transform, we can get
Z d Z u(t, x) = G(t, y)g(x y) dy + G(t, y)h(x y) dy Rd dt − Rd − Z t Z + G(t, y)b(t s, x y) dyds , (2.34) 0 Rd − − p 1 sin ( Mδ(ξ)t) where G(t, y) = F − ( p ), see also [22]. Mδ(ξ) 27
From the equation (2.33), we can see
Theorem 2. If the influence function ω = ω( y ) satisfies (2.3), and | |
d 2 d 2 2 d g ω(R ) , h L (R ) , b L (0,T ; L (R )), (2.35) ∈ M ∈ ∈ for some T > 0, then the PD equation (2.1) has a unique solution u = u(t, x) given by (2.34). Moreover,
d 2 2 d u C([0,T ]; ω(R )) , ut L (0,T ; L (R )). (2.36) ∈ M ∈
Proof: From (2.33) and (2.34), we can see the solution u can be expressed by the given quantities b, g and h, i.e. the solution of the PD equation (2.1) uniquely exists, so it is suffice to give the proper space that the solution belongs to.
First, we note that since Mδ(ξ) is real symmetric and positive definite, it can T be diagonalized by an orthogonal matrix Q = Q(ξ), i.e. Mδ(ξ) = Q diag(λδ,i)Q.
And we denote the three terms on the right hand side of (2.33) as uˆ1(t, ξ), uˆ2(t, ξ) and uˆ3(t, ξ) respectively so that
uˆ(t, ξ) = uˆ1(t, ξ) + uˆ2(t, ξ) + uˆ3(t, ξ) .
By the condition (2.35), we readily have Z 2 p p u1(t, x) ω = gˆ(ξ) cos( Mδ(ξ)t)(I + Mδ(ξ)) cos( Mδ(ξ)t)gˆ(ξ) dξ k kM Rd · Z T p T p = gˆ(ξ) Q diag(cos( λδ,it)Q(I + Mδ(ξ))Q diag(cos( λδ,it)Qgˆ(ξ) dξ Rd · Z T p p = gˆ(ξ) Q diag(cos( λδ,it)diag(1 + λδ,i)diag(cos( λδ,it)Qgˆ(ξ) dξ Rd · Z T 2 p = gˆ(ξ) Q diag((1 + λδ,i) cos ( λδ,it))Qgˆ(ξ) dξ Rd · Z T 6 gˆ(ξ) Q diag(1 + λδ,i)Qgˆ(ξ) dξ Rd · Z 2 = gˆ(ξ) (I + Mδ(ξ))gˆ(ξ) dξ = g ω . Rd · k kM 28
d So we can see that u1 = u1(t, x) is uniformly bounded in C([0,T ]; ω(R )). M Similarly, we can also deduce that
2 2 2 2 3 2 u2(t, x) ω (1 + T ) h L2( d) , and u3(t, x) ω (T + T ) b L2(0,T ;L2( d)) k kM ≤ k k R k kM ≤ k k R uniformly in [0,T ]. Therefore, it implies that u = u(t, x) is bounded uniformly in d d ω(R ) for any t [0,T ], i.e. u C([0,T ]; ω(R )). M ∈ ∈ M
Differentiating (2.33) with respect to t, we get
p p p uˆ (t, ξ) = M (ξ) sin ( M (ξ)t)gˆ(ξ) + cos ( M (ξ)t)hˆ(ξ) t − δ δ δ Z t p + cos ( Mδ(ξ)(t s))bˆ(s, ξ)ds . (2.37) 0 −
Then through a similar calculation as in the above, we can get that ut is uniformly bounded in L2(0,T ; L2(Rd)). These a priori estimates, together with standard PDE theory [23, 24], lead d to the existence and uniqueness of the solution u of (2.1) in C([0,T ]; ω(R )) M ∩ 1 2 H (0,T ; L (Rd)). Note that for the linear time-dependent equation, we can easily get the following lemma.
Lemma 13. Let P be a time-independent linear operator which commutes with
Lδ, then it commutes with the solution operator of the system (2.1).
Similar to the stationary case, it again allows us to establish the well-posedness of even more generalized solutions to (2.1).
Theorem 3. Let the influence function ω = ω( y ) satisfy (2.3), and P be an | | time-independent linear operator which commutes with Lδ. Then for the initial conditions and the forcing term satisfying
d 2 d 2 2 d P g ω(R ) ,P h L (R ) ,P b L (0,T ; L (R )), ∈ M ∈ ∈ the PD equation (2.1) has a unique solution u = u(t, x) with
d 1 2 d P u C([0,T ]; ω(R )) H (0,T ; L (R )). ∈ M ∩ 29
1/2 In particular, we can take P = ( L + I)− , then we get the existence and − δ uniqueness of weak solution u = u(t, x) to (2.1) with
2 d 1 1 d u C([0,T ]; L (R )) H (0,T ; ω− (R )) ∈ ∩ M
2 d 1 d 2 1 d for g L (R ), h ω− (R ) and b L (0,T ; ω− (R )). ∈ ∈ M ∈ M The proof is straightforward by verifying that P u is also the solution of the PD equation with the transformed data. Note that the theorem also implies the well-posedness of the Cauchy problem for the time-dependent PD equation even when the initial displacement is given as a distribution.
2.3 The bond-based PD model on a finite bar
In this section, we investigate the properties of the bond-based PD model defined on a finite bar, represented by the interval (0, π),
Z x+δ utt(t, x) = ω( y x ) u(t, y) u(t, x) dy, for x (0, π) (2.38) x δ | − | − ∈ − with the solution u = u(x) satisfying either
u is odd in ( δ, δ) and (π δ, π + δ) , (2.39) − − or u is even in ( δ, δ) and (π δ, π + δ) . (2.40) − − The condition (2.39) resembles (and thus is called) a Dirichlet-like condition, while (2.40) resembles a Neumann-like condition. Indeed, for smooth enough func- tions, (2.39) implies u(0) = u(π) = 0, while (2.40) leads to ux(0) = ux(π) = 0. In practice, one often also studies other displacement loading or force loading conditions. We note that the odd and even extensions given in (2.39) and (2.40) allow us to more easily formulate the spectrum of the corresponding PD operator than, for example, the case of force loading condition. We thus focus on the former cases in this work and leave discussions on other general BVPs to future works. 30
2.3.1 The PD operators and related function spaces
Similar to the section 2.2 and [25], we begin with a definition of the PD operators on (0, π).
Definition 2. For ω satisfying (2.3), the PD operator o is defined by −Lδ Z x+δ o δu(x) = ω( y x )(u(y) u(x)) dy x (0, π) (2.41) −L x δ | − | − ∀ ∈ − if u is an integrable function satisfying the Dirichlet-like displacement loading condition given in (2.39). Similarly, e is defined for functions satisfying the −Lδ Neumann-like displacement loading condition (2.40). More precisely, with the Fourier sine and cosine series expansions,
X o X e u(x) = uk sin(kx) and v(x) = vk cos(kx) k k with the coefficients uo and ve given by { k} { k} Z π Z π o 2 e 2 uk = u(x) sin(kx)dx , vk = v(x) cos(kx)dx , k 1 . π 0 π 0 ∀ ≥
We have the following representations of the PD operators o and e: −Lδ −Lδ X X ou(x) = η (k)uo sin(kx) , ev(x) = η (k)ve cos(kx) , (2.42) −Lδ δ k −Lδ δ k k k where ηδ is the Fourier symbol of the bond-based PD operator on the 1-D bar, defined as
Z δ ηδ(k) = ω( y )(1 cos(ky)) dy k 1 . (2.43) δ | | − ∀ ≥ − X Note that we have used to denote the sum over all positive integers. In k the case of the Neumann boundary condition with the Fourier cosine expansion, we have the constant term removed from the expansion so that the corresponding function always has zero mean. We now define the associated functional spaces and explicit spectra and eigenfunctions for the PD operators, similar to the study 31 for the PD equation in Rd in section 2.2 and in [25].
o Definition 3. The space Mω, which depends on the influence function ω, consists o of all functions u for which the Mω-norm
( )1/2 2 o 1/2 X o2 u o = [ ( u, u)] = η (k)u (2.44) k kMω −π Lδ δ k k
o is finite. The corresponding inner product in Mω is given by
X o o o (u, v) o := η (k)u v u, v M . (2.45) Mω δ k k ∀ ∈ ω k
so In addition, given an exponent s, we let Mω be generalized energy spaces with ( ) so 2 X s o2 M = u : u so = η (k)u < . ω k kMω δ k ∞ k
se Similarly, we have the definitions of Mω and their norms and inner products.
so We can see that Mω is a Hilbert space, with its dual space with respect to the 2 so o standard L duality pairing given by M − . Moreover, the PD operator is a ω −Lδ o o 1o self-adjoint operator, and it is also an isometry from Mω to Mω− = Mσ− . Let Hs denote the fractional Sobolev space on (0, π) for s [0, 1), which is o ∈ s the closure of Co∞ in H (0, π). Here Co∞ is the space of functions which are the s restrictions on (0, π) of functions in C∞(R) satisfying (2.39), and H (0, π) is the 0 2 0o standard fractional order Sobolev space [26]. Note that Ho = Lo = Mσ , and one s can characterize Ho and its norm in the Fourier space as ( ) X X Hs = v = v sin(kx) v 2 = v 2k2s < . o k | k ks | k| ∞ k k
Similarly, we have the same results for spaces M se related to e and Hs. The for- ω Lδ e mulations given in (2.42) are motivated by the explicit eigenfunctions and spectra of the PD-operators o and e. For o, it has a complete set of eigenfunc- −Lδ −Lδ −Lδ tions sin(kx) with the corresponding spectra η (k) . Analogously, e has a { } { δ } −Lδ complete set of eigenfunctions cos(kx) with spectra η (k) . { } { δ } 32
2.3.2 Space equivalence for special influence functions
o We now establish relations between Mω and the Sobolev spaces. Similar results e hold for the space Mω. Our discussion here is similar to that in section 2.1 for func- tions in Rd. We also refer to [21] for discussions of nonlocal spaces in more general non-Hilbert space settings and its application to nonlinear variational problems. Under the assumption (2.3) and with the symbol , being the conventional → notation for the continuous embedding between spaces, we first have what follows.
Lemma 14. Let ω = ω( y ) satisfy (2.3). We have for k 1, | | ≥ Z δ ηδ(k) inf (1 cos(ky))ρ( y )dy > 0 , (2.46) ≥ k 1 δ − | | ≥ − k4δ2τ 0 τ k2 2η (k) δ . (2.47) ≤ δ − δ ≤ 12
Moreover, H1 , M o , L2, H2 , M 2o , L2. Similar results hold for spaces o → ω → o o → ω → o associated with (2.40).
Proof. By the assumption on ω = ω( x ), we have for any k 1 that | | ≥ Z δ ηδ(k) (1 cos(ky))ρ( y )dy > 0 . ≥ δ − | | − Since ρ( x ) L1( δ, δ), by the Riemann lemma, we have | | ∈ − Z δ lim ρ( y ) cos(ky)dy = 0, k δ | | →∞ − which gives Z δ Z δ lim (1 cos(ky))ρ( y )dy = ρ( y )dy > 0 . k δ − | | δ | | →∞ − − Thus, we have the infimum attainable in (2.46), which remains strictly positive, thus implying that M o , L2. ω → o Next, using the inequality 1 cos(x) x 2/2, we have | − | ≤ | | Z δ 2 2 2 0 < 2ηδ(k) ω( y )k y dy τδk k 1. (2.48) ≤ δ | | | | ≤ ∀ ≥ − 33
We get the first inequality in (2.47). Using Taylor expansions of the cosine function, we can also get the second inequality in (2.47). The continuous space embedding results then follow from the definitions of the corresponding spaces. We omit the details here (for a proof of similar results in Rd, we refer to [25]).
The above lemma implies in particular that the solution spaces of the nonlocal BVPs of the PD model are subspaces of L2. In fact, for an influence function ω such that ω( x ) L1(0, δ), L2 is the natural solution space. As shown in [17, 25, 16] | | ∈ for problems defined in Rd, the linear PD operator is a bounded linear operator from L2 to itself. For the finite bar case, we also have what follows.
Lemma 15. Let ω = ω( y ) satisfy the additional condition that | |
ω( y ) L1(0, δ) ; (2.49) | | ∈ then M 2o = M o = L2 and o is a bounded linear operator from L2 to L2, and ω ω o −Lδ o o Z δ 0 < inf ηδ(k) ηδ(k) 4 ω( y )dy k 1. (2.50) k 1 ≥ ≤ ≤ 0 | | ∀ ≥
Similar results hold for spaces associated with (2.40).
Using the definition of ηδ(k) given in (2.43), it is obvious that the rightmost inequality of (2.50) holds, while the lower bound can be derived in the same way as (2.46) with ρ( y ) being replaced by ω( y ). | | | | We note that while the above lemma implies the equivalence of norms between M 2o, M o, and L2 and the boundedness of o from L2 to L2, such equivalence ω ω o −Lδ o o relations and the operator bound of o are not uniform with respect to the −Lδ horizon parameter δ. In fact, the upper bound in (2.50) tends to grow to infinity as δ 0. This is not surprising since, based on the analysis given later in the chapter → 5, the PD operator actually converges to a second order differential operator and 2 2 is thus unbounded from Lo to Lo in such a limit. For more general micromodulus functions, the PD operator may become −Lδ unbounded in L2, yet the basic existence and uniqueness results remain valid as we demonstrate below, with the discussion taking place in other function spaces 34 defined earlier. For convenience, we define the following function:
Z χ 1 cos(z) ωδ(α, χ) = − 1+2α dz . (2.51) χ z − | |
Lemma 16. Let ω = ω( y ) satisfy (2.3),and for some positive constant γ1, | |
1 2α ω( y ) γ1 y − − y δ (2.52) | | ≤ | | ∀| | ≤ for some exponent α (0, 1). Then we have ∈
0 η (k) Cδ(α)2k2α k 1 , (2.53) ≤ δ ≤ 1 ∀ ≥
δ 1 α o 2α 2o for C (α) = (γ1ω (α, )) 2 , and H , M and H , M . Moreover, 1 δ ∞ o → ω o → ω
δ α u o C (α) u u H , (2.54) k kMω ≤ 1 k kα ∀ ∈ o δ 2 2α u 2o C (α) u 2 u H . (2.55) k kMω ≤ 1 k k α ∀ ∈ o
Similar results hold for spaces associated with (2.40).
Proof. The lemma follows straightforwardly from the observation that for any k 1, the coefficient η (k) as defined in (2.43) satisfies ≥ δ Z δ 1 δ 2 2α 0 ηδ(k) γ1 (1 cos(ky)) 1+2α dy C1 (α) k , ≤ ≤ δ − y ≤ − | | which in turn leads to the continuous space embeddings using the definitions of the corresponding spaces.
Similarly, we also have the following lemma.
Lemma 17. Let ω = ω( y ) satisfy (2.3) and for some positive constant γ2, | |
1 2β ω( y ) γ2 y − − y δ (2.56) | | ≥ | | ∀| | ≤ for some exponent β < 1. Then we have
η (k) Cδ(β)2k2β k 1 , (2.57) δ ≥ 2 ∀ ≥ 35
δ 1 o β 2o 2β for C (β) = (γ2ω (β, δ)) 2 , and M , H and M , H . Moreover, 2 δ ω → o ω → o
δ o C (β) u u o u M , (2.58) 2 k kβ ≤ k kMω ∀ ∈ ω δ 2 2o C (β) u 2 u 2o u M . (2.59) 2 k k β ≤ k kMω ∀ ∈ ω
Similar results hold for spaces associated with (2.40).
The proof again follows from similar estimates on η (k) for any k 1: δ ≥ Z δ 1 δ 2 2β ηδ(k) γ2 (1 cos(ky)) 1+2β dy C2 (β) k . ≥ δ − y ≥ − | | We note that for β < 0, Lemma 15 can be applied, which gives stronger results. Based on the above discussion, we see that under suitable conditions on the influence function, for example, if (2.52) and (2.56) are satisfied with β = α ∈ o 2o (0, 1), then the spaces Mω and Mω are equivalent to standard fractional Sobolev α 2α spaces Ho and Ho , respectively. We caution again that the equivalence relations are not uniform as δ 0. → Let us also note that for many influence functions used in the existing studies of the linear PD models, conditions (2.49), (2.52), and (2.56) are often very relevant. 1 2β In fact, influence functions of the form ω( y ) = γ2 y − − with β < 1, especially | | | | 1 the case with β = 0 and ω = γ2 y − , have been frequently used in the literature | | [5, 7, 27].
2.3.3 Properties of the stationary PD Model
We now discuss the existence and uniqueness of weak solutions to the stationary o (equilibrium) PD model. We again focus on the solution in Mω of the equation
ou = f . (2.60) −Lδ
For completeness, parallel conclusions are also stated (without proof) on the solu- e tion in Mω of the equation eu = f . (2.61) −Lδ 36
Similar to elliptic equations, we may establish the variational theory and regularity properties for the stationary PD model. First, we obviously have what follows.
o Lemma 18. Let ω = ω( y ) satisfy (2.3); then for any f M − , the problem | | ∈ ω (2.60) has a unique solution u M o which is the minimizer of the functional ∈ ω
1 2 1X o2 u o (u, f)L2 = ηδ(k)u (u, f)L2 2k kMω − 2 k − k
o in Mω. A similar result holds for (2.61). As for the regularity, we have for some special micromodulus functions what follows.
Lemma 19. Let ω = ω( y ) satisfy (2.3) and (2.56). For β [0, 1), the problem | | ∈ (2.60) has a unique solution u = u(x) Hm+2β whenever f Hm for any m ∈ o ∈ o ≥ 2β. Moreover, − δ 2 u +2 C (β)− f . k km β ≤ 2 k km Similar results hold for (2.61).
Proof. Considering (2.60), with f o being the coefficients of the Fourier sine { k } expansion of f, obviously we have a unique solution u = u(x) with uo = f o/η (k) { k k δ } being the coefficients of its Fourier sine expansion. Moreover,
X X X Cδ(β)4k2m+4βuo2 η2(k)k2muo2 = k2mf o2 < . 2 k ≤ δ k k ∞ k k k
So, we have the solution u = u(x) Hm+2β with the desired bound. ∈ o The above lemma implies a lifting in regularity on the order of 2β (0, 2) for ∈ influence functions satisfying (2.52) and (2.56). We now go back to consider the regularity of solutions of (2.60) and (2.61) for more general cases.
Lemma 20. Let ω = ω( y ) satisfy (2.3); the problem (2.60) has a unique solution | | o o 2 u M for f M − , and u o = f o . Moreover, if f L , we have ∈ ω ∈ ω k kMω k kMω− ∈ o 2o u M and u 2o = f 0. Similar results hold for (2.61). ∈ ω k kMω k k Using the linearity of the model, we have the following corollary, similar as in [25]. We now let D be a difference operator given by D v(x) = v(x + y) v(x) for y y − 37 any function v defined in a suitable space. Then let P be a linear operator which 1 commutes with Dy for any y with P − its inverse (defined in a suitable space). Moreover, for any measurable function v that is odd in ( δ, δ) and (π δ, π + δ), − − we assume that P v, in the sense of distributions, remains odd in ( δ, δ) and − (π δ, π + δ). Examples of such operators include linear differential operators of − even orders with constant coefficients.
Corollary 2. Under the above assumptions on P , we have P o = oP and Lδ Lδ 1 o o 1 P − = P − . This in turn implies that if u is a solution of (2.60) with a Lδ Lδ 1 given function f, then P u and P − u are the solutions of (2.60) with the right 1 hand side being P f and P − f, respectively. A similar conclusion holds for (2.61).
Although the above corollary is trivial to derive, we note that problems with singular data are indeed of practical interests; see [6] for examples where Dirac- delta functions are used for f as concentrated forces. While o is a nonlocal integral operator, it shares some properties of an −Lδ elliptic differential operator such as those associated with the elliptic regularity, as shown earlier. As another illustration, we know that if u satisfies (2.60), with ω satisfying (2.49) and f being nonnegative, then
Z δ Z δ ω( y ) dy u(x) u(x + y)ω( y ) dy x (0, π) . δ | | ≥ δ | | ∀ ∈ − − Thus, we can readily get the following maximum principle.
Lemma 21. Let ω = ω( y ) satisfy (2.49), let f be nonnegative almost everywhere, | | and let u be a solution of (2.60). Then u is nonnegative in (0, π), and its maximum is attained in the interior or it is identically zero. A similar conclusion holds for (2.61).
2.3.4 The time-dependent PD model on a finite bar
Similar results can be established for the time-dependent PD models by adapting the approaches given in [25] to the setting of nonlocal BVPs as the case for the equilibrium models. We thus simply state a result without detailed derivation. 38
Theorem 4. Let ω = ω( y ) satisfy (2.3), and let P be a time-independent operator | | o that satisfies the same assumptions as those made in the Corollary 2. If P g0 M , ∈ ω 2 2 2 P g1 L , and P b L (0,T ; L ), then (2.38) with the nonlocal boundary condition ∈ o ∈ o (2.39), the initial conditions u(x, 0) = g0(x), and ut(x, 0) = g1(x) and the forcing term b = b(x, t) has a unique solution u = u(x, t) with P u C([0,T ],M o) ∈ ω ∩ H1(0,T ; L2). In particular, for P = I, we get u C([0,T ],M o) H1(0,T ; L2) o ∈ ω ∩ o o 2 2 2 2 with g0 M , g1 L , and b L (0,T ; L ). We also have u C([0,T ],L ) ∈ ω ∈ o ∈ o ∈ o ∩ 1 o 2 o 2 o H (0,T ; M − ) with g0 L , g1 M − , and b L (0,T ; M − ). Similar results ω ∈ o ∈ ω ∈ ω hold for problems associated with the nonlocal boundary condition (2.40).
2.4 Conclusion
In this chapter, a general functional analytic framework is provided for the mathe- matical and numerical analysis of the linear peridynamic models. For illustration, we focus on the case of the linear constitutive relations corresponding to the spring system in multi-dimensional space and one-dimensional boundary value problem. Various analytical issues are established here under the unified framework, extend- ing some of the results given in the literature. The techniques developed here can be extended to study more general nonlocal peridynamic state models [3]. We note that the analytic frameworks and the stud- ies of the solution regularity properties associated with the PD models can also be useful in establishing basic convergence and error estimates of their numerical ap- proximations such as the Galerkin finite element approximation [28] and chapter 6. While the Fourier based techniques similar to that developed here can still be used in the analysis of certain special nonlocal boundary value problems for the linear bond based PD models defined on box-like domains [28], other techniques need to be further developed in the future to treat more generic boundary conditions, arbitrary geometry and nonlinear models. Chapter 3
A Nonlocal Vector Calculus, Nonlocal Balance Laws and Nonlocal Volume-Constrained Problems
3.1 Introduction
In this chapter, we introduce a vector calculus for nonlocal operators that mimics the classical vector calculus for differential operators. We define nonlocal analogs of the divergence, gradient, and curl operators and deduce the corresponding nonlocal adjoint operators. Nonlocal analogs of the Gauss theorem and the Green’s identi- ties of the vector calculus for differential operators are also derived. We establish relationships between the nonlocal operators and their differential counterparts. The nonlocal vector calculus can be used to define nonlocal volume-constrained problems that are analogous to boundary-value problems for partial differential operators. In addition, The nonlocal vector calculus we develop has an impor- tant application to balance laws1 that are nonlocal in the sense that subregions not in direct contact may have a non-vanishing interaction. This is accomplished by defining a nonlocal flux in terms of interactions between disjoint regions of positive measure possibly sharing no common boundary. Our nonlocal vector cal-
1A balance law postulates that the rate of change of an extensive quantity over any subregion of a body is given by the rate at which that quantity is produced in the subregion minus the flux out of the subregion through its boundary. 40 culus, then, provides an alternative to standard approaches for circumventing the technicalities associated with the lack of sufficient regularity in local balance laws. Preliminary attempts at a nonlocal calculus are found in [29] and [18] which included applications to image processing and steady-state diffusion, respectively. In particular, the authors of [29] and [30] where a discrete nonlocal divergence and gradient are introduced within the context of machine learning; see also of of [31], [32], and [33] where a discrete calculus is also discussed. However, the discussion in these papers is limited to scalar problems. In contrast, this paper extends the ideas in [29] and [18] to vector and tensor fields and beyond the consideration of image processing and steady-state diffusion. For example, the ideas presented here enable an abstract formulation of the balance laws of momentum and energy and for the peridynamic theory for solid mechanics2 that parallels the classical vector calculus formulation of the balance laws of elasticity. The nonlocal vector calculus presented in this chapter, however, is sufficiently general that we envisage application to balance laws beyond those of elasticity, e.g., to the laws of fluid mechanics and electromagnetics. This chapter is organized as follows. The remainder of this section is devoted to establishing notation. In Section 3.2, the notions of local and nonlocal fluxes into or out off a region are briefly compared and contrasted. In Section 3.3, the nonlocal divergence, gradient, and curl operators are introduced as are the corre- sponding adjoint operators, several vector identities, and other results about the operators. The nonlocal vector calculus is developed in Section 3.4; in particular, nonlocal integral theorems and nonlocal Green’s identities are derived. In Sec- tion 3.5, connections between the nonlocal operators and distributional and weak representations of the associated classical differential operators are made. The connections made in Section 3.5 justify the use of the terminology “nonlocal di- vergence, gradient, and curl” to refer to the nonlocal operators defined in Section 3.3. Sections 3.6 and 3.7 deal with applications of the nonlocal vector calculus. In Section 3.6, examples are given of nonlocal volume-constrained problems formu- lated in terms of the nonlocal operators. Then, in Section 3.7, a brief review of the conventional notion of a balance law is provided after which abstract nonlocal
2Peridynamics was introduced in [1] and [2]; [4] reviews the peridynamic balance laws of mo- mentum and energy and provides many citations for the peridynamic theory and its applications. See Section 3.7.2 for a brief discussion. 41 balance laws are discussed. The notion of nonlocal fluxes discussed in Section 3.2 is used in developing nonlocal balance laws and the vector calculus developed in Section 3.4 plays a crucial role in transforming balance laws into field equations. Also, in Section 3.7, a brief discussion is given of the application of our nonlocal vector calculus to the peridynamic theory for continuum mechanics. Throughout, wherever it is illuminating, we associate definitions and results of the nonlocal vector calculus with the analogous definitions and results of the classical differential vector calculus.
3.1.1 Notation
We have need of two types of functions and two types of nonlocal operators. Point functions refer to functions defined at points whereas two-point functions refer to functions defined for pairs of points. Point operators map two-point functions to point functions whereas two-point operators map point functions to two-point func- tions so that the nomenclature for operators refer to their ranges. Point and two- point operators are both nonlocal. Point operators involve integrals of two-point functions whereas two-point operators explicitly involve point functions evaluated at two different points. We now make more precise the definitions given above. Points in Rn are denoted by the vectors x, y, or z and the natural Cartesian basis is denoted by e1,..., en. n m k Let m, k, and n denote positive integers. For Ω R , functions from Ω into R × ⊆ or Rm or R are referred to as point functions or point mappings and are denoted by Roman letters, upper-case bold for tensors, lower-case bold for vectors, and plain face for scalars, respectively, e.g., U(x), u(x), and u(x), respectively. Functions m k m from Ω Ω into R × or R or R are referred to as two-point functions or two- × point mappings and are denoted by Greek letters, upper-case bold for tensors, lower-case bold for vectors, and plain face for scalars, respectively, e.g., Ψ(x, y), ψ(x, y), and ψ(x, y), respectively. Symmetric and antisymmetric scalar two-point functions ψ(x, y) satisfy ψ(x, y) = ψ(y, x) and ψ(x, y) = ψ(y, x), respectively, − and similarly for vector and tensor two-point functions. A useful observation is 42 that Z Z if ψ(x, y) is antisymmetric, then ψ(x, y) dydx = 0 Ω Rn (3.1) Ω Ω ∀ ⊆ and similarly for antisymmetric vector and tensor two-point functions. For the sake of notational simplicity, in much of the rest of the chapter, we introduce the following notation:
α := α(x, y) α0 := α(y, x) ψ := ψ(x, y) ψ0 := ψ(y, x)
u := u(x) u0 := u(y) u := u(x) u0 := u(y) and similarly for other functions. The dot (or inner) product of two vectors u, v Rm is denoted by u v R; ∈ · ∈ m k k the dyad (or outer) product is denoted by u w R × whenever w R ; given ⊗ ∈ ∈ k m a second-order tensor (matrix) U R × , the tensor-vector (or matrix-vector) ∈ product is denoted by U v and is given by the vector whose components are the · dot products of the corresponding rows of U with v.3 For n = 3, the cross product of two vectors u and v is denoted by u v R3. The Frobenius product of two × ∈ m k m k second-order tensors A R × and B R × , denoted by A: B, is given by ∈ ∈ m m the sum of the element-wise product of the two tensors. The trace of B R × , ∈ denoted by tr B, is given by the sum of the diagonal elements of B. Inner products in L2(Ω) and L2(Ω Ω) are defined in the usual manner. For × example, for vector functions, we have Z (u, v)Ω = u v dx for u(x), v(x) Ω Ω · ∈ Z Z (µ, ν)Ω Ω = µ ν dydx for µ(x, y), ν(x, y) Ω × Ω Ω · ∈ with analogous expressions involving the Frobenius product and the ordinary prod- uct for tensor and scalar functions, respectively.
3In matrix notation, the inner, outer, matrix-vector products are given by x y = xT y, x y = xyT , and U v = Uv. · ⊗ · 43
3.2 Nonlocal fluxes and nonlocal action-reaction principles
A key concept in the development of a vector calculus is the notion of a flux which accounts for the interaction of points in a domain with points outside the domain. As a result, the notion of a flux is also fundamental to the understanding of balance laws in mechanics, heat transfer, and many other settings; see Section 3.7. In the classical setting of local interactions, that interaction occurs at the boundary of the domain, whereas in the nonlocal case, the interaction must occur over volumes external to the domain. In order to contrast the notion of a nonlocal flux with the classical local flux, we begin by briefly reviewing the latter notion.
3.2.1 Local fluxes
n n Let Ω1 R and Ω2 R denote two disjoint open regions. If Ω1 and Ω2 have a ⊂ ⊂ nonempty common boundary ∂Ω12 := Ω1 Ω2, then, for a vector-valued function ∩ q(x), the expression Z q ~n1 dA (3.2) ∂Ω12 · represents the classical local flux out of Ω1 into Ω2, where ~n1 denotes the unit normal on ∂Ω12 pointing outward from Ω1 and dA denotes a surface measure in n R ; q ~n1 is referred as the flux density along ∂Ω12. The vector q is often expressed · in terms of an intensive variable through a constitutive relation.4 The flux, then, conveys a notion of direction out of and into a region and is a proxy for the interaction between Ω1 and Ω2. It is important to note that the flux from Ω1 into
Ω2 occurs across their common boundary and that if the two disjoint regions have no common boundary, then the flux from one to the other is zero. The classical flux
(3.2) is then deemed to be local because there is no interaction between Ω1 and Ω2 when separated by a finite distance. The classical flux satisfies the action-reaction
4 For example, if q ~n1 denotes the heat flux density, then q is related to the temperature via Fourier’s heat law; see· Section 3.7.1.1. 44 principle5 Z Z q ~n1 dA + q ~n2 dA = 0, (3.3) ∂Ω12 · ∂Ω21 · where, of course, ∂Ω12 = ∂Ω21 and ~n2 = ~n1 denotes the unit normal on ∂Ω21 − R pointing outward from Ω2. In words, the flux q ~n1 dA from Ω1 into Ω2 across ∂Ω12 · R their common boundary ∂Ω12 is equal and opposite to the flux q ~n2 dA from ∂Ω21 · Ω2 into Ω1 across that same surface.
3.2.2 Nonlocal fluxes
We identify Z Z ψ(x, y) dydx (3.4) Ω1 Ω2 as a scalar interaction, or nonlocal flux, from Ω1 into Ω2, where ψ : (Ω1 Ω2) R ∪ × (Ω1 Ω2) denotes an antisymmetric function. We have that ψ(x, y) dy is R Ω2 ∪ → R the flux density into Ω2 from the point x Ω1 and likewise, ψ(x, y) dx is the ∈ Ω1 flux density into Ω1 from the point y Ω2. As is the case for the local flux density R ∈ q ~n1, the nonlocal flux density ψ(x, y) dy is related to an intensive variable · Ω2 through a constitutive relation; see Section 3.7.1.2. From (3.1), it is easily seen that the antisymmetry of ψ(x, y) is equivalent to the nonlocal action-reaction principle Z Z Z Z n ψ(x, y) dy dx + ψ(x, y) dy dx = 0 Ω1, Ω2 R ; (3.5) Ω1 Ω2 Ω2 Ω1 ∀ ⊂
(3.5) is the nonlocal analogue of (3.3). In words, (3.5) states that the flux (or interaction) from Ω1 into Ω2 is equal and opposite to the flux (or interaction) from
Ω2 into Ω1. The flux is nonlocal because, by (3.5), the interaction may be nonzero even when the closures of Ω1 and Ω2 have an empty intersection. This is in stark contrast to classical local interactions for which we have seen that the interaction between Ω1 and Ω2 vanishes if their closures have empty intersection, i.e., have no common boundary.
5An example is in mechanics for which Newton’s third law, i.e., the force exerted upon on object is equal and opposite to the force exerted by the object, is an action-reaction archtype. 45
3.3 Nonlocal operators
The nonlocal vector calculus developed in Section 3.4 involves nonlocal operators that mimic the classical local differential divergence, gradient, and curl operators. An important distinction between local and nonlocal operators is that the adjoint operators for the former involve the same operators, i.e., the adjoint of is , ∇· −∇ of is , and of is , whereas the adjoint of nonlocal operators involve ∇ −∇· ∇× ∇× differently defined nonlocal gradient, divergence, and curl operators. At this point, the association of the nonlocal operators given in Definition 4 and Theorem 5 with the operators of the classical differential vector calculus is purely conjectural. Justifications for making these associations are provided in Section 3.5.
3.3.1 Nonlocal point divergence, gradient, and curl opera- tors
The nonlocal point divergence, gradient, and curl operators map two-point func- tions to point functions and are defined in terms of their action on two-point functions as follows. These operators along with their adjoints are the building blocks of our nonlocal calculus.
Definition 4. [Nonlocal operators] Given the vector two-point function ν : Rn × Rn Rk and the antisymmetric vector two-point function α: Rn Rn Rk, the → × → action of the nonlocal point divergence operator on ν is defined as D Z n ν (x) := ν + ν0 α dy for x R , (3.6a) D Rn · ∈ where ν : Rn R. Given the scalar two-point function η : Rn Rn R and D → × → the antisymmetric vector two-point function β : Rn Rn Rk, the action of the × → nonlocal point gradient operator on η is defined as G Z n η (x) := η + η0 β dy for x R , (3.6b) G Rn ∈
3 where η : Rn Rk. Given the vector two-point function µ: Rn Rn R and G → × → the antisymmetric vector two-point function γ : Rn Rn R3, the action of the × → 46 nonlocal point curl operator on µ is defined as C Z n µ (x) := γ µ + µ0 dy for x R , (3.6c) C Rn × ∈
3 where µ : Rn R . C → The nonlocal point operators , , and map vectors to scalars, scalars to D G C vectors, and vectors to vectors, respectively, as is the case for the divergence, gradient, and curl differential operators. Relationships between the nonlocal point operators and differential operators are made in Section 3.5 where we demonstrate circumstances under which the nonlocal point operators are identified with the corresponding differential operators in the sense of distributions and also as weak representations. Because the integrands in (3.6a)–(3.6c) are antisymmetric, (3.1) immediately implies that Z Z Z (ν) dx = 0, (η) dx = 0, and (µ) dx = 0. (3.7) Rn D Rn G Rn C
These relations may be viewed as free-space nonlocal integral theorems.6
3.3.2 Nonlocal adjoint operators
The adjoint operators corresponding to the nonlocal point operators are two-point operators that are defined as follows.
Definition 5. Given a point operator that maps two-point functions F to point Q n functions defined over R , the adjoint operator ∗ is a two-point operator that Q maps point functions G to two-point functions defined over Rn Rn that satisfies × G, (F ) ∗(G),F = 0, (3.8) Rn Rn Rn Q − Q × where, for = , , or , we have that F and G denote pairs of vector-scalar, Q D G C scalar-vector, or vector-vector functions, respectively.
6 R For example, n (ν) dx = 0 can be viewed as a nonlocal analog of the free space classical R RD local Gauss theorem n u dx = 0. R ∇ · 47
Definition 3.8 can be used to determine the nonlocal adjoint two-point operators corresponding to the nonlocal point operators introduced in Definition 4.
Theorem 5. [Nonlocal adjoint operators] Given the point function u: Rn → R, the adjoint of is the two-point operator whose action on u is given by D
n ∗ u (x, y) = (u0 u)α for x, y R , (3.9a) D − − ∈
n n k n k where ∗ u : R R R . Given the point function v: R R , the adjoint D × → → of is the two-point operator whose action on v is given by G
n ∗ v (x, y) = (v0 v) β for x, y R , (3.9b) G − − · ∈
n n n 3 where ∗ v : R R R. Given the point function w: R R , the adjoint of G × → → is the two-point operator whose action on w is given by C
n ∗ w (x, y) = γ (w0 w) for x, y R , (3.9c) C × − ∈
n n 3 where ∗ w : R R R . C × → Proof. Let ξ = uν. Then,
(ξ + ξ0) α = (uν + u0ν0) α = u(ν + ν0) α + (u0 u)ν0 α (3.10) · · · − · so that, from (3.6a) and the first equation in (3.7), we have
Z Z Z 0 = (ξ) dx = u(ν + ν0) α + (u0 u)ν0 α dydx n D n n · − · R ZR RZ Z Z = u (ν + ν0) α dydx + (u0 u)ν0 α dydx n n · n n − · ZR R Z Z R R = u (ν) dx + (u0 u)ν α dydx, Rn D Rn Rn − · (3.11) where the last equality follows because we have, due to the antisymmetry of α, that (u0 u)(ν ν0) α is an antisymmetric two-point function so that, by (3.1), − − · Z Z (u0 u)(ν ν0) α dydx = 0. (3.12) Rn Rn − − · 48
Associating F with ν, G with u, and with , we see that (3.11) is exactly of Q D the form (3.8) with ∗ = ∗, where ∗ is given by (3.9a). Q D D In a similar manner, (3.9b) can be derived from (3.6b) and the second equation in (3.7) and (3.9c) can be derived from (3.6c) and the third equation in (3.7).
3.3.3 Further observations and results about nonlocal op- erators
In this subsection, we collect several observations and results that can be deduced from the definitions and results of Sections 3.3.1 and 3.3.2.
3.3.3.1 A nonlocal divergence operator for tensor functions and a non- local gradient operator for vector functions
A nonlocal divergence operator for tensor functions is defined by applying (3.6a) to each row of the tensor.
n n m k Definition 6. Given the tensor two-point function Ψ: R R R × and × → the antisymmetric vector two-point function α: Rn Rn Rk, the action of the × → nonlocal point divergence operator for tensors on Ψ is defined as Dt Z n t(Ψ)(x) := Ψ + Ψ0 α dy for x R , (3.13a) D Rn · ∈
n m where t(Ψ)(x): R R . D → Definition 5 implies that, given the point function v(x): Rn Rm, the action → of the adjoint of on v is given by Dt
n t∗(v)(x, y) = (v0 v) α for x, y R , (3.13b) D − − ⊗ ∈
n n m k where t∗(v): R R R × , i.e., t∗ maps a vector point function to a second- D × → D order tensor two-point function.7
7 We could, in a slight abuse of notation, abbreviate t(Ψ) by (Ψ). In an analogous fashion, ∗ ∗ D D t (v) may be abbreviated by (v) when the argument is a vector point function. Note that Dthis notational abuse is customaryD for the differential divergence and gradient operators for which is used to denote the divergence operator for both vectors and tensors and is used to denote the∇· gradient operator on both scalars and vectors. ∇ 49
One can also extend Definition (3.6b) of the nonlocal gradient operator of two-point scalar functions to define the nonlocal gradient operator acting on a Gv two-point vector function ψ(x, y) as the point tensor function Z v(ψ)(x) = (ψ0 + ψ) β dy . G Rn ⊗
The corresponding nonlocal adjoint operator ∗ acting on a point tensor function Gv U(x) is then given by the two-point vector function
∗(U)(x, y) = (U0 U) β . Gv − − ·
3.3.3.2 Nonlocal vector identities
Compositions of the point operators defined in Sections 3.3.1 and 3.3.3.1 with the corresponding adjoint two-point operators derived in Sections 3.3.2 and 3.3.3.1 lead to the following nonlocal vector identities. In this proposition, we set α = β = γ and, for the first, second, and fourth results, i.e., those involving nonlocal curl operators, we set m = k = 3; we also set m = k for the third result.
Proposition 1. The nonlocal divergence, gradient, and curl operators and the corresponding adjoint operators satisfy8
n 3 ∗(u) = 0 for u: R R (3.14a) D C → n ∗(u) = 0 for u: R R (3.14b) C D → 8The four identities in (3.14) are analogous to the vector identities associated with the differ- ential divergence, gradient and curl operator:
( u) = 0, ( u) = 0, u = tr( u), ∇ · ∇ × ∇ × ∇ ∇ · ∇ ( u) + ( u) = ( u), − ∇ · ∇ ∇ ∇ · ∇ × ∇ × respectively. Because ( )∗ = , ∗ = , and ( )∗ = ( ), these identities can be written in the form ∇· −∇ ∇ −∇· ∇× ∇× ( )∗u) = 0, ( )∗u) = 0, ∗u = tr ( )∗u, ∇ · ∇× ∇ × ∇· ∇ ∇· ( )∗u ( )∗u = ( )∗u ∇ · ∇· − ∇ ∇ ∇ × ∇× that more directly correspond to (3.14a)–(3.14d), respectively. In particular, the identities (3.14a), (3.14b), and (3.14d) suggest that ∗, ∗, and ∗ can also be viewed as nonlocal analogs of the differential gradient, divergence,−D and−G curl operators,C respectively, that, when op- erating on point functions, result in two-point functions. 50
n k ∗(u) = tr t∗(u) for u: R R (3.14c) G D → n 3 t t∗(u) ∗(u) = ∗(u) for u: R R . (3.14d) D D − G G C C →
Proof. We prove (3.14d); the proofs of (3.14a)–(3.14c) are immediate after direct substitution of the operators involved. Let u: Rn R3. Then, by (3.6c), (3.9c), (3.13a), and (3.13b) and recalling → that α is an antisymmetric function, we have Z t t∗(u) ∗(u) = (u0 u) α + (u u0) α0 α dy D D − G G − Rn − ⊗ − ⊗ · Z + (u0 u) α + (u u0) α0 α dy n − · − · Z R = 2 (u0 u)(α α) (u0 u) α α dy − n − · − − · ZR = 2 α (u0 u) α dy, − Rn × − × where, for the last equality, we have used the vector identity a (b c) = b(c × × · a) c(a b). A simple computation shows that the last expression is equal to − · ∗(u) so that (3.14d) is proved. C C
Functions of the form ∗(u) do not entirely comprise the null space of the oper- C ator . In fact, it is obvious that for any antisymmetric two-point function ν(x, y), D we have ν = 0. However, functions of the form ∗(u) are the only symmetric D C two-point functions belonging to the null space of . Analogous statements can D be made for the null space of the operator and two-point functions of the form C ∗(u). Note that, because of the nonlocality of the operators, ∗ µ = 0 and D G C 6 ∗ η = 0. C G 6 Another set of results for the nonlocal operators that mimic obvious properties of the corresponding differential operators are given in the following proposition whose proof is a straightforward consequence of the definitions given in Theorem 5.
Proposition 2. Let b and b denote a constant scalar and vector, respectively. Then, the adjoints of the nonlocal divergence, gradient, and curl operators satisfy
∗(b) = 0, ∗(b) = 0, and ∗(b) = 0. D G C 51
These results do not hold for the point divergence, gradient, and curl operators, i.e., (b), (b), and (b) do not necessarily vanish for constants b and constant D G C vectors b.
3.3.3.3 Nonlocal curl operators in two and higher dimensions
The nonlocal point and two-point curl operators defined in (3.6c) and (3.9c), re- spectively, were defined in three dimensions. These definitions can be generalized to arbitrary dimensions by replacing the vector cross product with the wedge prod- n n r uct so that, e.g., instead of (3.9c) we would have ∗(w): R R R given by C × →
∗(w)(x, y) = γ (w0 w), C ∧ − where γ(x, y): Rn Rn Rr with r a positive integer. This suggests a possible × → exterior calculus-based formalism. However, such a formalism is beyond the scope of this chapter so that only the special cases of r = 3 and r = 2 are considered and the vector cross product is retained. We define nonlocal point and two-point curl operators in two space dimensions, i.e., for R2; in fact, we have two types of each kind of curl operator.9 First, we assume, without loss of generality, that µ e3 = 0, w e3 = 0, and γ e3 = 0 in · · · (3.6c) and (3.9c). Then, for a vector two-point function µ(x, y): Rn Rn R2, × → we can view µ as the nonlocal scalar point function defined by C Z n µ (x) = (µ2 + µ20 )γ1 (µ1 + µ10 )γ2 dy for x R C Rn − ∈
n 2 and, for a vector point function w: R R , we can view ∗(w) as the nonlocal → C scalar two-point function defined by
n ∗(w)(x, y) = (w20 w2)γ1 (w10 w1)γ2 for x, y R . C − − − ∈ 9This is analogous to the two types of differential curl operators in two dimensions, one operating on vectors, the other on scalars, respetively given by
∂u1 ∂u2 ∂u ∂u curl u = and curl u = e1 + e2. ∂x2 − ∂x1 −∂x2 ∂x1 52
Next, assume, again without loss of generality, that µ = µe3, w = we3, and γ e3 = · 0 in (3.6c) and (3.9c). Then, for a scalar two-point function µ(x, y): Rn Rn R, × → we can view (µ) as the nonlocal vector point function defined by C Z n (µ)(x) = (µ + µ0)(γ2e1 γ1e2) dy for x R C Rn − ∈
n and, for a scalar point function w(x): R R, we can view ∗(w) as the nonlocal → C vector two-point function defined by
n ∗(w)(x, y) = (w0 w)(γ2e1 γ1e2) for x, y R . C − − ∈
3.4 A nonlocal vector calculus
We develop a nonlocal vector calculus that mimics the classical vector calculus for differential operators. In the classical calculus, interactions are local which is why, e.g., in the Gauss theorem, the contribution coming from interactions with points outside a domain Ω appears in the form of a flux through the boundary ∂Ω. If interactions are nonlocal, then points outside of Ω interact with points in Ω so that that interaction cannot be accounted for merely by an integral along the boundary ∂Ω. In fact, one must account for interactions with points in the complement domain Rn Ω. \ n To treat the most general case, we define, for a given open subset Ωs R , the ⊂ corresponding interaction domain
n Ωc := y R Ωs such that α(x, y) = 0 for some x Ωs (3.15) { ∈ \ 6 ∈ } so that Ωc consists of those points outside of Ωs that interact with points in Ωs. For simplicity, we assume that the corresponding interactions domains for β and γ are also given by (3.15). No assumption is made about the geometric relation between Ωs and Ωc so that, e.g., the four configurations of Figure 3.1 are possible. n n Note that the situation Ωc = R Ωs is allowable as is Ωs = R . \ n We set Ωsc = Ωs Ωc. From (3.15) we see that points in R Ωsc do not interact ∪ \ 53
Ω c Ω c Ω Ω s s
Ω c Ω s Ω Ω s c
Figure 3.1. Four of the possible configurations for Ωs and Ωc.
with points in Ωs, i.e.,
n α(x, y) = 0 for x Ωs and y R Ωsc ∈ ∈ \ and, by antisymmetry,
n α(x, y) = 0 for y Ωs and x R Ωsc. ∈ ∈ \
Note that we do not assume that points in Ω interact only with points in Ω Ω , c s ∪ c n i.e., in general, points in Ωc may interact with points in R Ωsc as well. \
3.4.1 Nonlocal interaction operators
The nonlocal operators of Definition 4 describe the interaction of points in Ωs with points not only in Ωs but also with those in Ωc. Naturally, we need the same information for points in Ωc.
n Definition 7. [Nonlocal interaction operators] Given a domain Ωs R , let ⊂ the interaction domain Ωc be defined by (3.15). Then, corresponding to the point divergence operator ν : Rn R defined in (3.6a), we define the action of the D → point interaction operator (ν):Ωs R on ν by N → Z ν (x) := (ν + ν0) α dy for x Ωs. (3.16a) N Ωc · ∈ 54
Corresponding to the point gradient operator η : Rn Rk defined in (3.6b), we G → k define the action of the point interaction operator (η):Ωs R on η by S → Z η (x) := (η + η0)β dy for x Ωs. (3.16b) S Ωc ∈
3 Corresponding to the point curl operator µ : Rn R defined in (3.6c), we C → 3 define the action of the point interaction operator (µ):Ωs R on µ by T → Z µ (x) := γ (µ + µ0) dy for x Ωs. (3.16c) T Ωc × ∈
Letting ψ(x, y) = (ν + ν0) α and letting Ω1 = Ω and Ω2 = Ω , we see from · c s the discussion following (3.4) that (ν) represents a nonlocal flux density into Ω N c from points in Ω , and similarly for (η) and (µ). s S T
3.4.2 Nonlocal integral theorems
Definitions 4 and 7 lead to integral theorems for the nonlocal vector calculus that mimic the basic integral theorems of the vector calculus for differential operators.
Theorem 6. [Nonlocal integral theorems] Assuming the notations and defi- nitions found in Definitions 4 and 7, we have10 Z Z ν dx = ν dx (3.17a) Ωs D Ωs N Z Z η dx = η dx (3.17b) Ωs G Ωs S Z Z µ dx = µ dx. (3.17c) Ωs C Ωs T 10The nonlocal integral theorems (3.17) are analogous to the classical differential integral the- orems given by Z Z Z Z Z Z v dx = v ~ndx, v dx = v~ndx, and v dx = ~n v dx Ω ∇ · ∂Ω · Ω ∇ ∂Ω Ω ∇ × ∂Ω × for functions v and v defined on Rn, with n = 3 for the third one, for which the integrals are well defined. 55
Proof. We have that Z Z Z Z Z ν dx = ν + ν0 α dy dx = ν + ν0 α dy dx n n Ωs D Ωs R · Ωs R /Ωs · Z Z Z = ν + ν0 α dy dx = ν dx, Ωs Ωc · Ωs N where the four equalities follow in succession from (3.6a), the antisymmetry11 of α and (3.1), (3.15), and (3.16a). Thus, (3.17a) follows; (3.17b)and (3.17c) follow in exactly the same way starting with (3.6b) and (3.6c), respectively. Alternately, they can be derived from (3.17a); one simply chooses ν = ηb and ν = b µ, × respectively, in that equation, where b is a constant vector; one also has to associate β and γ with α.
In words, (3.17a) states that the integral of the nonlocal divergence of ν over 12 Ωs is equal to the total flux out of Ωs into Ωc. Similar interpretations of (3.17b) and (3.17c) hold. The nonlocal integral theorems (3.17a)–(3.17c) are action-reaction principles. For example, from (3.17a) and the assumptions made, we have that Z Z 0 = ν dx ν dx Ωs N − Ωs D Z Z Z Z = (ν + ν0) α dydx (ν + ν0) α dydx n Ωs Ωc · − Ωs R · Z Z Z Z (3.18) = (ν + ν0) α dydx (ν + ν0) α dydx Ωs Ωc · − Ωs Ωc · Z Z Z Z = (ν + ν0) α dydx + (ν + ν0) α dydx Ωs Ωc · Ωc Ωs · which is exactly of the form (3.5) with Ω1 = Ω ,Ω2 = Ω , and ψ = (ν + ν0) α; s c · (3.18) simply states that the flux out of Ωs into Ωc is equal and opposite to the
flux from Ωc into Ωs. The nonlocal integral theorems (3.17a)–(3.17c) have the simple consequences
11Because α is antisymmetric, the integrand in the double integral is antisymmetric, i.e., we have that (ν + ν0) α = (ν0 + ν) α0. 12This observation· is analogous− to· the observation for the classical Gauss theorem R u = R Ω ∇ · ∂Ω u ~ndA that, by (3.2), the integral of the local divergence of u over Ω is equal to the total flux out· of Ω. 56 listed in the following corollary that can be viewed as providing nonlocal integration by parts formulas.
Corollary 3. [Nonlocal integration by parts formulas] Adopt the hypotheses of Theorem 6 and let the nonlocal adjoint operators be given as in Theorem 5. Then, given the point functions u(x): Rn R, v(x): Rn Rk, and w(x): Rn R3, → → → we have13,14 Z Z Z Z u ν dx ∗ u ν dydx = u (ν) dx (3.19a) Ωs D − Ωs Ωs D · Ωs N Z Z Z Z v η dx ∗ v η dydx = v (η) dx (3.19b) Ωs ·G − Ωs Ωs G Ωs ·S Z Z Z Z w µ dx ∗ w µ dydx = w (µ) dx. (3.19c) Ωs ·C − Ωs Ωs C · Ωs ·T
Proof. Let ξ = uν; then, from (3.6a) and (3.10), we have
Z (ξ) = u(ν + ν0) α + (u0 u)ν0 α dy D n · − · RZ Z = u (ν + ν0) α dy + (u0 u)ν0 α dy n · n − · R Z R (3.20) = u (ν) + (u0 u)ν0 α dy D n − · ZR = u (ν) + (u0 u)ν0 α dy x Ωs, D Ωsc − · ∀ ∈ where the last equality follows from (3.15). Also, from (3.16a), we have
Z (ξ) = u(ν + ν0) α + (u0 u)ν0 α dy N Ωc · − · Z (3.21) = u (ν) + (u0 u)ν0 α dy x Ωs. N Ωc − · ∀ ∈ 13If ν and α are scalar-valued functions and for free space, a version of the integration by parts formula (3.19a) appears in Lemma 2.1 of [34]. 14The classical analog of (3.19a) is given by, for a scalar function u(x) and vector function v(x), Z Z Z u v dx + v u dx = uv ~ndA Ω ∇ · Ω · ∇ ∂Ω · and similarly for (3.19b) and (3.19c). 57
Then, Z Z 0 = (ξ) dx ξ dx Ωs D − Ωs N Z Z = u (ν) dx u ν dx Ωs D − Ωs N Z Z Z Z + (u0 u)ν0 α dydx (u0 u)ν0 α dydx Ωs Ωsc − · − Ωs Ωc − · Z Z Z Z = u (ν) dx u ν dx + (u0 u)ν0 α dydx, Ωs D − Ωc N Ωs Ωs − · where the first equality follows from (3.17a), the second from (3.20) and (3.21), and the third from the linearity of the integration operator. Then, (3.19a) follows from the antisymmetry of α and (3.12). In a similar manner, (3.19b) and (3.19c) can be derived from (3.17a) along with (3.6b) and (3.6c), respectively. Alternately, (3.19b) and (3.19c) easily follow by setting, for an arbitrary constant vector b, ν = ηb and ν = b µ, respectively, × in (3.19a) and also associating β and γ with α.
3.4.3 Nonlocal Green’s identities
Nonlocal (generalized) Green’s identities are simple consequences of Corollary 3. For the following two corollaries, we carry over the notations, definitions, and results obtained above.
Corollary 4. [Nonlocal (generalized) Green’s first identities] Given the scalar point functions u(x): Rn R and v(x): Rn R and the two-point second- → → 58
n n k k 15 order tensor function Θ(x, y): R R R × , then × → Z Z Z u Θ ∗(v) dx ∗(u) Θ ∗(v)ν dydx Ωs D ·D − Ωs Ωs D · ·D Z (3.22a) = u Θ ∗(v) dx. Ωs N ·D
Given the vector point functions u(x): Rn Rk and v(x): Rn Rk and the → → two-point scalar function θ(x, y): Rn Rn R, then × → Z Z Z v θ ∗(u) dx θ ∗(v) ∗(u) dydx Ωs ·G G − Ωs Ωs G G Z (3.22b) = v θ ∗(u) dx. Ωs ·S G
Given the vector point functions u(x): Rn R3 and w(x): Rn R3 and the → → n n 3 3 two-point second-order tensor function Θ(x, y): R R R × , then × → Z Z Z w Θ ∗(u) dx ∗(w) Θ ∗(u) dydx Ωs ·C ·C − Ωs Ωs C · ·C Z (3.22c) = w Θ ∗(u) dx. Ωs ·T ·C
Proof. The results (3.22a)–(3.22c) follow by setting ν = Θ ∗(u) in (3.19a), ·D η = θ ∗(u) in (3.19b), and µ = Θ ∗(u) in (3.19c), respectively. G ·C Corollary 5. [Nonlocal (generalized) Green’s second identities] We as- sume the same notation as in Corollary 4 and we also assume that the tensors Θ
15We have that (3.22a) and (3.23a) are the nonlocal analogs of the local classical (generalized) first Green’s identity Z Z Z u (Θ v) dx + u (Θ v) dx = u~n (Θ v) dA Ω ∇ · · ∇ Ω ∇ · ∇ ∂Ω · · ∇ and of the local classical (generalized) second Green’s identity Z Z Z Z u (Θ v) dx v (Θ u) dx = u~n (Θ u) dA v~n (Θ u) dA, Ω ∇ · · ∇ − Ω ∇ · · ∇ ∂Ω · · ∇ − ∂Ω · · ∇ respectively, and similarly for (3.22b), (3.23b), (3.22c) and (3.23c). These are generalizations of the classical local Green’s identities for which θ = 1 and Θ is the identity tensor. 59 appearing in (3.22a) and (3.22c) are symmetric. Then, Z Z u Θ ∗(v) dx v Θ ∗(u) dx Ωs D ·D − Ωs D ·D Z Z (3.23a) = u Θ ∗(v) dx v Θ ∗(u) dx Ωs N ·D − Ωs N ·D Z Z v θ ∗(u) dx u θ ∗(v) dx Ωs ·G G − Ωs ·G G Z Z (3.23b) = v θ ∗(u) dx u θ ∗(v) dx Ωs ·S G − Ωs ·S G Z Z w Θ : ∗(u) dx u Θ : ∗(w) dx Ωs ·C C − Ωs ·C C Z Z (3.23c) = w Θ : ∗(u) dx u Θ : ∗(v) dx. Ωs ·T C − Ωs ·T C Proof. The result (3.23a) is obtained by reversing the roles of u and v in (3.22a) and then subtracting the result from (3.22a). The results (3.23b) and (3.23c) are obtained from (3.22b) and (3.22c), respectively, in a similar manner.
3.4.4 Special cases of the vector calculus
We consider some special cases of the vector calculus we developed in Sections 3.4.1–3.4.3.
3.4.4.1 The free space vector calculus
n As was mentioned previously, the vector calculus allows for the choice Ωs = R in which case we also simply set Ω = in all definitions and results. The resulting c ∅ nonlocal integral theorems have already been stated; see (3.7). Corresponding to the nonlocal divergence operator , we have the nonlocal generalized Green’s first D identity Z Z Z u Θ ∗(v) dx ∗(u) Θ ∗(v)ν dydx = 0 Rn D ·D − Rn Rn D · ·D 60
and the generalized nonlocal Green’s second identity Z Z u Θ ∗(v) dx v Θ ∗(u) dx = 0 Rn D ·D − Rn D ·D corresponding to (3.22a) and (3.23a), respectively, and similarly for the other op- erators.
3.4.4.2 The vector calculus for interactions of infinite extent
n n In case points in Ωs interact with all points in R , we have that Ωc = R Ωs. In \ this case, all definitions and results remain unchanged.
3.4.4.3 The vector calculus for localized kernels
An important special case is that of localized kernels for which we have that
α(x, y) = 0 if y x ε (3.24) | − | ≥ and similarly for β and γ; here, ε > 0 denotes a cut-off or horizon parameter which is not necessarily small and which defines the extent of interactions. We then have that n Ωc = y R Ωs : y x < ε for x Ωs . { ∈ \ | − | ∈ } We also define the domain
n Ω := y Ωs : y x < ε for x R Ωs = y Ωs : y x < ε for x Ωc , − { ∈ | − | ∈ \ } { ∈ | − | ∈ } where the last equality follows from (3.24). See Figure 3.2 for an illustration. From (3.16a) for the first and third equalities and (3.24) for the second, we have that Z Z Z (ν) dx = (ν + ν0) α dydx Ωs N Ωs Ωc · Z Z Z = (ν + ν0) α dydx = (ν) dx. Ω Ωc · Ω N − −
The first term is the nonlocal flux from Ωs into Ωc; according to the last term, 61
Figure 3.2. For a localized kernel, the domain Ωs and the interaction regions Ωc and Ω whose thicknesses are given by the horizon ε. − only points in Ω Ωs contribute to that flux. Similar observations can be made − ⊂ for the operators and . S T Of course, all the definitions and results of Sections 3.3.1–3.4.3 hold for the case of localized kernels.
3.5 Relations between nonlocal and differential operators
In this section, we choose particular kernel functions α, β, and γ that lead to iden- tifications of the nonlocal operators with their differential counterparts. In Section 3.5.1, the identification is made in a distributional sense whereas, in Section 3.5.2, we demonstrate circumstances under which the nonlocal point operators are weak representations of the divergence, gradient, and curl differential operators. For the sake of brevity, we mostly consider the nonlocal point divergence operator D and its adjoint operator ∗. However, analogous results also hold for the non- D local gradient and curl operators and , respectively, and their adjoints. We G C then consider, in Section 3.5.3, a connection between the nonlocal and local Gauss theorems made with the help of two remarkable results given in [35]. Throughout this section, we set k = n in Definition 5, Theorem 4, and all subsequent results. 62
3.5.1 Identification of nonlocal operators with differential operators in a distributional sense
We begin with the identification of the nonlocal point divergence operator with D the divergence differential operator . This identification is subject to the un- ∇· derstanding that the former operates on two-point functions whereas the latter operates on point functions. Where there might be ambiguity in applying differ- ential operators to two-point function, we identify explicitly the variable of dif- ferentiation, e.g., ν(x, y) denotes the divergence of ν(x, y) with respect to ∇y · y. We consider special cases of localized kernels (see Section (3.4.4.3)) for which ε 0. As such, the “proofs” provided in this subsection are purely formal; → they could be made rigorous through limiting processes involving sequences of ε-dependent kernels with ε 0. → We first relate, in a distributional sense, the nonlocal divergence with the local differential divergence.
n n Proposition 3. Let ν 0∞(R R ), i.e., ν belongs to the space of compactly ∈ C × supported infinitely differentiable functions. Select the (antisymmetric) distribution
α(y, x) = δ(y x), (3.25) −∇y − where and δ(y x) denote the differential gradient with respect to y and the ∇y − Dirac delta distribution, respectively. Then,
n ν (x) = ν(x, x) x R , (3.26) D ∇ · ∀ ∈ where denotes the differential divergence operator. ∇· Proof. We have, by the chain rule and for α given by (3.25),
ν(x, x) = ν(x, y) = + ν(x, y) = ∇ · ∇x · |y x ∇y · |y x = ν(y, x) = + ν(x, y) = ∇y · |y x ∇y · |y x = y ν(y, x) + ν(x, y) y=x Z∇ · | = y ν(y, x) + ν(x, y) δ(y x) dy Rn ∇ · − 63
Z = ν(y, x) + ν(x, y) yδ(y x) dy − n · ∇ − Z R = ν(y, x) + ν(x, y) α(x, y) dy = ν(x), Rn · D where the last equality follows from (3.6a).
The following proposition shows that the average of the action of ∗ on a point D function can be related, in a distributional sense, to the action of on that −∇ function.
Proposition 4. Let u ∞(Ω) and select α as in (3.25). Then, ∈ C0 Z n ∗(u) dy = u x R . Rn D −∇ ∀ ∈
Proof. Starting with (3.9a) and (3.25), we have that Z Z ∗(u) dy = u(y) u(x) yδ(y x) dy n D n − ∇ − R RZ = δ(y x) y u(y) u(x) dy − n − ∇ − ZR = δ(y x) yu(y) dy = u(x) = ( )∗u(x). − Rn − ∇ −∇ ∇·
We next relate, in a distributional sense, the interaction operator with the N normal component along the boundary.
Proposition 5. Assume the hypotheses of Proposition 3. Then, for all smooth n functions u(x):Ωs R , → Z Z u (ν) dx = u(x)ν(x, x) ~ndA, (3.27) Ωs N ∂Ωs · where ~n denotes the unit outward-pointing normal vector along ∂Ω . Thus, (ν) s N is a delta measure concentrated on ∂Ω with coefficient (weight) ν(x, x) ~n. s · Proof. With α chosen as in (3.25) and using the definition (3.9) for the operator 64
∗, we have that D Z Z Z Z ∗(u) ν dydx = u(y) u(x) ν(x, y) α(x, y) dydx − Ωs Ωs D · Ωs Ωs − · Z Z = u(y) u(x) ν(x, y) yδ(y x) dydx − Ωs Ωs − · ∇ − Z Z = y u(y) u(x) ν(x, y)δ(y x) dydx Ωs Ωs ∇ − · − Z Z + u(y) u(x) δ(y x) y ν(x, y) dydx Ωs Ωs − − ∇ · Z Z u(y) u(x)δ(y x)ν(x, y) ~ndydx − Ωs ∂Ωs − − · Z Z = yu(y) ν(x, y)δ(y x) dydx Ωs Ωs ∇ · − Z = u(x) ν(x, x) dx, Ωs ∇ · where the third equality follows because u(y) u(x) = 0 whenever y = x and the − fourth equality follows because u(x) = 0. Also, with α chosen as in (3.25) and ∇y using (3.26), we have that Z Z u (ν) dx = u(x) ν(x, x) dx Ωs D Ωs ∇ · Z Z = u(x)ν(x, x) ~ndA u(x) ν(x, x) dx ∂Ωs · − Ωs ∇ ·
Substituting the last two results into (3.19) results in Z Z Z Z Z u (ν) dx = u (ν) dx ∗(u) ν dydx = u(x)ν(x, x) ~ndA. Ωs N Ωs D − Ωs Ωs D · ∂Ωs ·
Now consider the composition of the nonlocal divergence operator and its ad- joint, i.e., ∗ , which, according to the next proposition, can be identified, in D D the sense of distributions, with ∆ = ( ), where ∆ denotes the Laplace − ∇ · −∇ operator . 65
n 2 1 Proposition 6. Let u 0∞(R ) and select α(x, y) = ∆yδ(y x). Then, ∈ C | | 2 −
n ∗u(x) = ∆u(x) x R . D D − ∀ ∈
Proof. From (3.6a) and (3.9a) we have Z 2 ∗u (x) = 2 u(y) u(x) α(x) dy D D − n − | | Z R = u(y) u(x) ∆yδ(y x) dy − n − − ZR = δ(y x)∆y u(y) u(x) dy − n − − ZR = δ(y x)∆yu(y) dy − Rn − = ∆u(x) = u(x) = ( )∗u(x), − −∇ · ∇ ∇ · ∇· where the differential Green’s second identity is used for the third equality.
3.5.2 Relations between weighted nonlocal operators and weak representations of differential operators
The classical differential calculus only involves operators mapping point functions to point functions; on the other hand, the nonlocal operators defined earlier map two-point functions to point functions or, in case of the nonlocal adjoint operators, point functions to two-point functions. To further demonstrate that the nonlocal operators correspond to nonlocal versions of the classical divergence, gradient, and curl differential operators, we use the nonlocal operators , , and to define, in D G C Section 3.5.2.1, corresponding nonlocal weighted operators that map point func- tions to point functions. We also show that the adjoint operators corresponding to the weighted operators are weighted integrals of the nonlocal adjoint operators
∗, ∗, and ∗. Then, in Section 3.5.2.2, the weighted operators are rigorously D G C shown to be nonlocal versions of the corresponding differential operators. 66
3.5.2.1 Nonlocal weighted operators
The nonlocal point operators defined in (3.6) induce new operators, referred to as weighted operators. Definition 8 and 9 and Proposition 7 hold for general k and m although, when we apply them in Section 3.5.2.2, we set k = m = n.
Definition 8. Let ω(x, y): Rn Rn R denote a non-negative scalar-valued two- × → point function. Let the operators , , and be defined as in (3.6). Then, given D G C the point function u(x): Rn Rk, the weighted nonlocal divergence operator → n ω(u): R R is defined by its action on u by D →
n ω(u)(x) := ω(x, y)u(x) for x R . (3.28a) D D ∈
Given the point function u(x): Rn R, the weighted nonlocal gradient operator → n k ω(u): R R is defined by its action on u by G →
n ω(u)(x) := ω(x, y)u(x) for x R . (3.28b) G G ∈
Given the point function w(x): Rn R3, the weighted nonlocal curl operator → n 3 ω(u): R R is defined by its action on w by C →
n ω(w)(x) := ω(x, y)w(x) for x R . (3.28c) C C ∈
The following result shows that the adjoint operators corresponding to the weighted nonlocal operators are determined as weighted integrals of the corre- sponding nonlocal two-point adjoint operators (3.9).
Proposition 7. Let ω(x, y): Rn Rn R denote a non-negative scalar two-point × → function and let ∗, ∗, and ∗ denote the nonlocal adjoint operators given in (3.9). D G C n k The action on u of the adjoint operator ω∗ (u)(x): R R corresponding to the D → weighted nonlocal divergence operator is given by Dω Z n ω∗ (u)(x) = ∗(u)(x, y) ω(x, y) dy for x R (3.29a) D Rn D ∈ for scalar point functions u(x): Rn R. The action on u of the adjoint operator → n ω∗ (v)(x): R R corresponding to the weighted nonlocal gradient operator ω is G → G 67 given by Z n ω∗ (v)(x) = ∗(v)(x, y) ω(x, y) dy for x R . (3.29b) G Rn G ∈ for vector point functions v(x): Rn Rk. The action on w of the adjoint operator → n 3 ω∗ (w)(x): R R corresponding to the weighted nonlocal curl operator ω is C → C given by Z n ω∗ (w)(x) = ∗(w)(x, y) ω(x, y) dy for x R (3.29c) C Rn C ∈ for vector point functions w(x): Rn R3. → Proof. We have that Z ω(u), u n = u(x) ω(x, y)u(x) dx D R n D ZR Z = ∗(u)(x, y) ω(x, y)u(x) dydx Rn Rn D · Z Z = ω(x, y) ∗(u)(x, y) dy u(x) dx, Rn Rn D · where (3.28a) is used for the first equality and (3.8) and (3.9a) for the second.
But, by definition, the adjoint operator ∗ ( ) corresponding to the operator ( ) Dω · Dω · satisfies u, ω(u) n = ∗ (u), u n . D R Dω R Comparing the last two results yields (3.28a). The conclusions (3.28b) and (3.28c) are derived in a similar fashion.
The definition given in (3.28a) and the result (3.29a) can be extended to tensors and vectors, respectively.
Definition 9. Let ω(x, y): Rn Rn R denote a non-negative scalar two- × → point function and let be given as in (3.13b). Given the tensor point function Dt n ` k n ` U(x): R R × , the weighted nonlocal divergence operator t,ω(U): R R → D → for tensors is defined by
n t,ω U (x) := t ω(x, y)U(x) (x) for x R . (3.30) D D ∈ 68
n ` k Corollary 6. The adjoint operator t,ω∗ (u)(x): R R × corresponding to the D → operator is given by Dt,ω Z n t,ω∗ u (x) = t∗(u)(x, y) ω(x, y) dy for x R (3.31) D Rn D ∈ for vector point functions u: Rn R`. → Similar results hold for the operator . Gv
3.5.2.2 Relationships between weighted operators and differential op- erators
We have seen that the nonlocal operators satisfy many properties that mimic their differential counterparts. In particular, we established, in Section 3.5.1, that the nonlocal point operators can be identified with the corresponding differential operators in a distributional sense. Here, we establish rigorous connections between the weighted operators and their differential counterparts for the case of special localized kernels (see Section 3.24), i.e., we introduce the horizon parameter ε > 0 and analyze what occurs as ε 0, that is, in the local limit. To this end, we define →
n Bε(x) := y R : y x < ε { ∈ | − | } and let φ denote a positive radial function satisfying a normalization condition Z y x 2φ( y x ) dy = n, (3.32) Bε(x) | − | | − | where n denotes the space dimension. We then choose
y x α(x, y) = − for x = y y x 6 | − | y x φ( y x ) y B (x) (3.33) ε ω(x, y) = | − | | − | ∈ 0 otherwise.
Note that α is an antisymmetric function whereas ω is a symmetric function. We then have, for a scalar function u, that the components of u and ∗ u given Gω Dω 69 by (3.28b) and (3.29a), respectively, are given by, for j = 1, . . . , n, Z dju(x) := u(x + z) + u(x) zj φ( z ) dz (3.34a) Bε(0) | | Z dj∗u(x) := u(x + z) u(x) zj φ( z ) dz, (3.34b) − Bε(0) − | | where zj denotes the j-th component of z. The following result ensues.
Lemma 22. Assume that ω and α are defined as in (3.33). Let u: Rn R and → let d u(x) and d∗u(x) denote the j-th components of (u) and ∗ (u), respectively. j j Gω Dω Then,
d u = d∗u. (3.35) j − j Proof. From (3.34) we have Z dju(x) + dj∗u(x) = 2u(x) yjφ( y )dy = 0 − Bε(0) | | from which the result directly follows.
Based on this lemma and the definition of the weighted operators, we have the following results. Corollary 7. Under the same conditions as in Lemma 22, we have
= ∗ , = ∗ , and = ∗ . (3.36) Dω −Gω Gω −Dω Cω Cω
Thus, under the hypotheses of this corollary, the identities in (3.36) mimic their counterparts in the vector calculus for differential operators. Identities such as those in Proposition 1 do not generally hold for the weighted operators due to their nonlocal properties.16 Synergistic with the results of Section 3.5.1, we have the following. Corollary 8. Select the singular distribution
z φ( z ) = ∂ δ(z), j | | − j 16The identities in Lemma 22 and Corollary 7 also hold for more general choices of α and ω, namely, those such that α is antisymmetric and ω is a radial function with support in Bε(x). 70
where ∂ju denotes the weak derivative of u with respect to xj. Then,
d = ∂ and d∗ = ∂ . j j j − j
The following proposition demonstrates that the components of the weighted gradient operator or, by (3.36), ∗ , converge, as ε 0, to the corresponding Gω Dω → spatial derivatives.
Proposition 8. Let ω be defined as in (3.33) with φ satisfying (3.32). Then, for j = 1, . . . , n, the weighted operators dj and dj∗ defined by (3.34) are bounded linear operators from H1(Rn) to L2(Rn). Moreover, if u H1(Rn), then as ε 0 ∈ →
d u ∂ u 2 n 0 (3.37a) k j − j kL (R ) → d∗u + ∂ u 2 n 0, (3.37b) k j j kL (R ) → where ∂ju denotes the weak partial derivative of u with respect to xj. Moreover, if Z z 1+sφ( z ) dz < for some 0 s 1, (3.37c) Bε(0) | | | | ∞ ≤ ≤ then, for j = 1, . . . , n, the weighted operators dj and dj∗ are bounded linear operators t n t s n from H (R ) to H − (R ) for any t 0. ≥
Proof. The conclusion (3.37a) is equivalent to dcju ∂cju L2( n) 0, where dcju k − k R → and ∂cju denote the Fourier transforms of dju and ∂ju, respectively. Under the condition (3.37c) for s [0, 1], if u Ht(Rn) for t 0, then ∈ ∈ ≥ Z iy ξ dcju = (e · + 1) ub(ξ) yj φ( y ) dy Bε(0) | | Z = i sin(y ξ) ub(ξ) yj φ( y ) dy Bε(0) · | | Z X = i sin(yjξj) cos( ykξk) Bε(0) k=j 6 X + cos(y ξ ) sin( y ξ ) u(ξ) y φ( y ) dy j j k k b j | | k=j 6 71
Z X = iu(ξ) sin(y ξ ) cos( y ξ ) y φ( y ) dy, b j j i k j | | Bε(0) k=j 6 where the second and fourth equalities hold because of the symmetry of the domain of integration and the antisymmetry of the integrand with respect to the integration P variable, respectively. Because for any 0 s 1, sin(yjξj) cos( k=j yiξk) ≤ ≤ | 6 | ≤ y ξ s, we have | j j| Z s 1+s dcju ξ ub(ξ) yj φ( y ) dy | | ≤ | | Bε(0) | | | |
t s n so that dcju H − (R ). In particular, for t = s = 1, we obtain that, under the ∈ condition (3.32), the weighted operators dj, j = 1, . . . , n, defined in (3.34a) are bounded linear operators from H1(Rn) to L2(Rn). If u H1(Rn), we have that ∈
dcju ∂cju dcju + ∂cju | − | ≤Z | | | | X sin(yjξj) cos( ykξk) u(ξ) yjφ( y ) dy + ξju ≤ b | | | b| Bε(0) k=j Z 6 2 ξjub yj φ( y ) dy + ξjub ≤ | | Bε(0) | | | | 2 n 2 ξju L (R ), ≤ | b| ∈ where the third inequality holds because sin(x) x and cos(x) 1. By | | ≤ | | | | ≤ Taylor’s theorem, we have Z 3 3 X 2 dcju = i (yjξj + y ξ cos(θ1)/6)(1 + ( ykξk) cos(θ2)/2) u yjφ( y ) dy j j b | | Bε(0) k=j Z 6 2 = iξjub yj φ( y ) dy + Iε(ξ) = iξjub + Iε(ξ), Bε(0) | | where, for some θ1 and θ2, Z i 4 3 Iε(ξ) : = yj cos(θ1)φ( y ) dy ξj ub 6 Bε(0) | | Z i 2 X 2 + y ( y ξ ) cos(θ2)φ( y ) dy ξ u 2 j i i | | j b Bε(0) k=j 6 72
Z i 4 X 2 3 + y ( y ξ ) cos(θ1) cos(θ2)φ( y ) dy ξ u. 12 j k k | | j b Bε(0) k=j 6
Because ub is bounded a.e., we see that, for any ξ, X X I (ξ) ε2 ξ3u + ε2 ( ξ )2ξ u + ε4 ( ξ )2ξ3u 0 as ε 0. | ε | ≤ | j b| | k j b| | k j b| → → k=j k=j 6 6 Hence, we have n dcju ∂cju a.e. for ξ R as ε 0. → ∈ → Then, by the dominated convergence theorem, we obtain
d u ∂ u 2 n 0 as ε 0. k j − j kL (R ) → →
Lemma 22 implies the same results hold for the operator dj∗.
The above lemma implies that if ω satisfies (3.37c) for s = 0, then the weighted operators are bounded operators from L2(Rn) to L2(Rn). More generally, for φ satisfying (3.37c) with positive s, the operators dj and dj∗ actually map a subspace of L2(Rn), for instance the fractional Sobolev space Hs(Rn), to L2(Rn), or even 2 n s n map L (R ) to H− (R ). We refer to [25] for related work. A direct consequence of Lemma 8 is the following result.
Corollary 9. Under the condition of Lemma 8, the weighted operators , , and Dω Gω and their adjoint operators ∗ , ∗ , and ∗ are bounded linear operators from Cω Dω Gω Cω t n t s n H (R ) to H − (R ) for 0 s 1, where n = 3 for the weighted curl operators. ≤ ≤ Moreover, if u H1(Rn) and u [H1(Rn)]n, then ∈ ∈
(u) u ∗ (u) u Dω → ∇ · Dω → −∇
(u) u ∗ (u) u (3.38) Gω → ∇ Gω → −∇ ·
(u) u ∗ (u) u, Cω → ∇ × Cω → ∇ × where the convergence as ε 0 is with respect to L2(Rn). → Proof. We prove the convergence of the operator (u) to the divergence differ- Dω 73 ential operators. First, note that
n n X X (u) = d u and u = ∂ u Dω i i ∇ · i i i=1 i=1 so that
n X (u) u 2 n d u ∂ u 2 n 0 as ε 0. kDω − ∇ · kL (R ) ≤ k i i − i ikL (R ) → → i=1
The remaining results can be proved in a similar fashion.
For the purpose of discussing nonlocal equations, we also need to consider combinations of the weighted operators such as (C1 ∗ (u)), where C1(x) is a Dω Dω “constitutive” tensor point function that, for example, describes a point property of a material. Note that the two-point property is involved in the definition of the weighted operators. In the next corollary, we illustrate that whenever the horizon ε goes to zero, the combinations of the weighted operators converge to their local counterparts.
1 n 1 n n n n n n Corollary 10. Let u H (R ), u [H (R )] , C1 : R R R in L∞(R ∈ ∈ → × × n n n R ), and c2 : R R in L∞(R ). Then, → C1 ∗ (u) (C1 u) Dω ·Dω → −∇ · · ∇ c2 ∗ (u) (c2 u) (3.39) Gω Gω → −∇ ∇ · C1 ∗ (u) C1 ( u) , Cω ·Cω → ∇ × · ∇ ×
1 n where the convergence as ε 0 is with respect to H− (R ). → Proof. Using Proposition 9 and a similar method of proof as for Lemma 8, the results are obtained.
We also have the following result.
Lemma 23. Let denote a linear operator that commutes with the differential and L nonlocal operators. Then, if u [H1(Rn)]n, u H1(Rn), and u [H1(R3)]3 L ∈ L ∈ L ∈ 74 as need be, we have
( u) u ∗ ( u) u Dω L → ∇ · L Dω L → −∇L ( u) u ∗ ( u) u (3.40) Gω L → ∇L Gω L → −∇ · L ( u) u ∗ ( u) u, Cω L → ∇ × L Cω L → ∇ × L where the convergence as ε 0 is with respect to L2(Rn). → If is selected as a differential operator with constant coefficients, or its formal L inverse, we can observe convergence in either stronger or weaker norms of the weighted operators.
3.5.3 A connection between the nonlocal and local Gauss theorems
Two interesting lemmas found in [35]17 allow us to make another connection be- tween nonlocal and local operators or, more precisely, between the nonlocal Gauss theorem (3.17a) and the classical Gauss theorem. Given a two-point function ν : Rn Rn Rk, we define the vector-valued point × → function q(x): Rn Rk by → Z q(x) := (y x)ψ(x, y x) dy, (3.41) − Rn − −
n n where, with p(x, y) = (ν + ν0) α and z = y x, the function ψ : R R R · − × → is given by Z 1 ψ(x, z) = p x + λz, x (1 λ)z dλ. 0 − − Then, Lemmas I and II in [35] state that Z n q(x) = (ν + ν0) α dy x R (3.42) ∇ · Rn · ∀ ∈ and Z Z Z q(x) ~ndA = (ν + ν0) α dydx, (3.43) n ∂Ωs · Ωs R Ωs · \ 17See [36] for an English translation of [35]. 75
respectively, where ~n denotes the outward pointing unit normal vector along ∂Ωs. From (3.6a) and (3.42) we then have that
q = (ν) (3.44) ∇ · D and from (3.16a) and (3.43), using the by now familiar sequence of steps, we have that Z Z Z q ~ndA = (ν + ν0) α dydx n ∂Ωs · Ωs R Ωs · Z Z \ Z (3.45) = (ν + ν0) α dydx = (ν) dx. Ωs Ωc · Ωs N Then, the nonlocal Gauss theorem (3.17a) for ν, (3.44), and (3.45) imply that Z Z Z Z 0 = (ν) dx (ν) dx = q dx q ~ndA, Ωs D − Ωs N Ωs ∇ · − ∂Ωs · i.e., the classical Gauss’s theorem for the vector-valued function q. Thus, we have shown that the nonlocal Gauss’s theorem (3.17a) for the nonlocal vector two-point function ν(x, y) formally implies the classical Gauss theorem for the local vector point function q(x) derived from ν through (3.41). Evidently, the Gauss theorem can be given a meaning without the notions of the divergence operator, unit normal vector, or surface.
3.6 The nonlocal volume-constrained problems
The nonlocal point operators, the corresponding nonlocal adjoint operators, and the corresponding nonlocal interaction operators given in (3.6), (3.9), and (3.16), respectively, can be used to define nonlocal “boundary-value” problems that are analogous to classical boundary-value problems for partial differential equations. Here, we merely state problems involving scalar and vector “second-order” oper- ators so that we are in the setting of “elliptic” problems. Specifically, we define nonlocal problems that are are analogous to the second-order differential boundary- 76 value problems C2 u = f in Ω −∇ · · ∇ u = g on ∂Ωd (3.46a) C2 u n = h on ∂Ω · ∇ · n C4 : u = f in Ω −∇ · ∇ u = g on ∂Ωd (3.46b) C4 : u n = h on ∂Ω ∇ · n C2 u k u = f in Ω ∇ × · ∇ × − ∇ ∇ · u = g on ∂Ωd (3.46c) o n C2 u = h1 × · ∇ × on ∂Ωn, k u = h2 ∇ · respectively, where ∂Ω = ∂Ω ∂Ω denotes the boundary of Ω with ∂Ω ∂Ω = ; d∪ n d∩ n ∅ ∂Ωd and ∂Ωn are the parts of the boundary ∂Ω on which Dirichlet and Neumann boundary conditions are applied, respectively. In (3.46a)–(3.46c), C4, C2, and c denote fourth-order tensor, second-order tensor, and scalar point functions, respec- tively. Each of the problems (3.46a)–(3.46c) for an unknown function, e.g., u in (3.46a), consists of a partial differential equation and boundary conditions, where the latter may be viewed as constraints placed on possible solutions of the partial differential equation. Thus, in the case of local operators, constraints are applied along the boundary ∂Ω of the domain Ω on which the partial differential equation is applied. A consequence of the nonlocality of operators is that constraints analogous to the boundary conditions in (3.46a)–(3.46c) are applied instead over sets with positive measure in Rn. Thus, in the case of nonlocal operators, constraints are applied on the interaction domain Ωc that has positive volume and that corresponds to the do- main Ωs on which the nonlocal operator equation is applied. As a result, we refer to nonlocal problems with constraints imposed on volumes having positive vol- ume as volume-constrained problems in contrast to local problems with constraints imposed on boundary surfaces that are universally referred to as boundary-value problems. We now describe the nonlocal problems analogous to (3.46a)–(3.46c). In the 77
differential equation setting we had to divide the boundary ∂Ω in the two parts ∂Ωd and ∂Ωn over which we applied Dirichlet and Neumann conditions, respectively.
Similarly, in the nonlocal case we divide the interaction domain Ωc into two sub- domains that we use to apply Dirichlet-like and Neumann-like volume constraints. Let Ω = Ω Ω , where Ω Ω = although either Ω or Ω may be empty. c cd ∪ cn cd ∩ cn ∅ cd cn Over Ωcd we specify function values, e.g., for a given function g, we set
u(x) = g(x) for x Ω (3.47) ∈ cd which is, of course, a straightforward generalization of the Dirichlet boundary condition u(x) = g(x) for x ∂Ω for the partial differential equation case; see, ∈ d e.g., (3.46a). The Neumann boundary condition in (3.46a) specifies q ~n = (C2 · · u) n for x ∂Ω , i.e., the flux density our of Ω through x ∂Ω is specified. ∇ · ∈ n ∈ n For the nonlocal case we similarly specify the nonlocal flux density out of Ωs into
Ωcn, i.e., for a given function h(x) and tensor function Θ2(x, y), we set Z cn(ν)(x) = (ν + ν0) α dA = h(x) for x Ωs with ν = Θ2 ∗(u). N Ωcn · ∈ ·D (3.48) To define volume-constrained problems analogous to the boundary-value prob- lems (3.46a)–(3.46c), we let Θ4, Θ2, and θ denote fourth-order tensor, second-order tensor, and scalar two-point functions, respectively, where the tensors are symm- teric in the function and matrix senses and are positive definite. Using (3.47) and (3.48), the nonlocal volume-constrained problems corresponding to the boundary- value problems (3.46a)–(3.46c) are then given by
Θ2 ∗(u) = f in Ωs D ·D u = g in Ωcd (3.49a) Ω Θ2 ∗(u) = h in Ω N cn ·D s t Θ4 : t∗(u) = f in Ω D D u = g in Ωcd (3.49b) Ω Θ4 : ∗(u) = h in Ω N cn,t Dt s 78
Θ2 ∗(u) + θ ∗(u) = f in Ωs C ·C G G u = g in Ωcd (3.49c) Ω Θ2 ∗(u) + Ω θ ∗(u) = h in Ω , T cn ·C S cn G s respectively. The nonlocal Green’s first identities (4) are useful for defining variation formu- lations of the nonlocal volume-constrained problems. For example, from (3.22a) and (3.49a), we have that u(x) satisfies u = g on Ωcd, and for suitable test functions v(x) that vanish on Ωcd, Z Z Z Z ∗(u) Θ2 ∗(v) dydx = vf dx vh dx. (3.50) Ωs Ωs D · ·D Ωs − Ωs
The kernel function α dictates the choice of function spaces for the trial and test functions u and v, respectively, so that the problem (3.50) is well posed. A special form of the nonlocal volume-constrained problem (3.49a) correspond- ing to a scalar-valued solution is studied in [18] by appealing to a variational formu- lation. Well-posedness results are provided in [18] for the case in which the natural energy space (used to define the variational problem) is equivalent to L2(Ω), the space of square integrable functions. Although the notion of a nonlocal vector calculus is not introduced, the recent book [37] describes, using semigroup anal- ysis, substantial recent work on nonlocal diffusion and its relationship to (3.46a). The report [38] extends these previous results to the case when the natural energy space is equivalent a fractional Sobolev space, a proper subspace of L2(Ω). In this case, the variational problem possess smoothing properties akin to those for elliptic partial differential equations but with possibly reduced order.
3.7 Local and nonlocal balance laws
Along with kinematics and constitutive relations, balance laws are a cornerstone of continuum mechanics. A balance law postulates that the rate of change of the amount of a quantity in any subregion of a body is given by the rate at which that quantity is produced within the subregion minus the rate at which the quantity exits the subregion; the latter is referred to as the flux out of the subregion. In the 79 classical differential equation setting of continuum mechanics, the flux is local, i.e., the quantity exits or enters the subregion through its boundary. Difficulties arise, however, in the classical setting due to, e.g., shock waves, corner singularities, and material failure, all of which are troublesome when defining an appropriate notion for the “flux through the boundary of a subregion.” The nonlocal vector calculus we develop has an important application to non- local balance laws for which subregions not in direct contact may have a non-zero interaction, i.e., there is a nonzero flux between the subregions. This is accom- plished by defining the flux in terms of interactions between disjoint open regions of positive measure that are possibly a finite distance apart. An important fea- ture of nonlocal balance laws is that the significant technical details associated with determining normal and tangential traces along boundaries of suitable re- gions is obviated when fluxes are induced through interactions between volumes. Our nonlocal calculus, then, provides an alternative to standard approaches for cir- cumventing the technicalities associated with lack of sufficient regularity in local balance laws such as measure-theoretic generalizations of the Gauss-Green theo- rem (see, e.g., [39] and [40]) or the use of the fractional calculus (see, e.g., [41] and [42]).
3.7.1 Abstract balance laws
n We start with an abstract balance law for an open subset Ωs R given by ⊆
n (Ω; q) = (Ω) (Ω, R Ω; q) Ω Ωs (3.51) A P − I \ ∀ ⊂ which postulates that (Ω; q) (the rate of change of the amount of the intensive A quantity q in any open subregion Ω Ω ) is equal to (Ω) (the rate at which ⊂ s P the intensive quantity is produced in the subregion by external sources) minus (Ω, Rn Ω; q) (the rate at which the intensive quantity exits the subregion). In I \ (3.51), q(x, t) may be a scalar or vector time-dependent point function and we assume that (Ω), the rate of production of q in any subregion Ω is given, e.g., P through the specification of an external source function. Here, we take the usual 80 choices
∂ Z Z (Ω; q) = q(x, t) dx and (Ω) = b(x, t) dx (3.52) A ∂t Ω P Ω for the rate of change and production terms, respectively, where b(x, t) denotes a given function. Note that if we assume the general case for which points in Ωs only n interact with points in Ωc R Ωs, then (3.51) reduces to ⊆ \
(Ω; q) = (Ω) (Ω, Ω ; q) Ω Ω (3.53) A P − I c ∀ ⊂ s
We assume that the interaction or flux operator ( , ; q) in (3.51) or (3.53) is I · · alternating, i.e., (Ω, Ω; q) = 0 Ω Rn (3.54) I ∀ ⊆ and that (Ω1 Ω2, Ω3; q) I ∪ = (Ω1, Ω3; q) + (Ω2, Ω3; q) (Ω1 Ω2, Ω3; q) n I I − I ∩ Ω1, Ω2, Ω3 R (Ω , Ω Ω ; q) ∀ ⊆ 3 1 2 I ∪ = (Ω3, Ω1; q) + (Ω3, Ω2; q) (Ω3, Ω1 Ω2; q) I I − I ∩ (3.55) The assumed properties (3.54) and (3.55) of ( , ; q) immediately imply a third I · · property.
Proposition 9. Assume the interaction operator ( , ) satisfies (3.54) and (3.58). I · · Then, that operator is antisymmetric, i.e., we have
n (Ω1, Ω2; q) + (Ω2, Ω1; q) = 0 Ω1, Ω2 R . (3.56) I I ∀ ⊆ 81
Proof. Through repeated applications of (3.54) and (3.55), we obtain
0 = (Ω1 Ω2, Ω1 Ω2; q) I ∪ ∪ = (Ω1, Ω1 Ω2; q) + (Ω2, Ω1 Ω2; q) (Ω1 Ω2, Ω1 Ω2; q) I ∪ I ∪ − I ∩ ∪ = (Ω1, Ω1; q) + (Ω1, Ω2; q) (Ω1, Ω1 Ω2; q) I I − I ∩ + (Ω2, Ω1; q) + (Ω2, Ω2; q) (Ω2, Ω1 Ω2; q) I I − I ∩ (Ω1 Ω2, Ω1; q) (Ω1 Ω2, Ω2; q) + (Ω1 Ω2, Ω1 Ω2; q) − I ∩ − I ∩ I ∩ ∩ = (Ω1, Ω2; q) + (Ω2, Ω1; q) (Ω1, Ω1 Ω2; q) (Ω1 Ω2, Ω1; q) I I − I ∩ − I ∩ (Ω2, Ω1 Ω2; q) (Ω1 Ω2, Ω2; q) − I ∩ − I ∩ = (Ω1, Ω2; q) + (Ω2, Ω1; q) (3.57) I I (Ω1 (Ω1 Ω2), Ω1 Ω2; q) (Ω1 Ω2, Ω1 Ω2; q) − I \ ∩ ∩ − I ∩ ∩ (Ω1 Ω2, Ω1 (Ω1 Ω2); q) (Ω1 Ω2, Ω1 Ω2; q) − I ∩ \ ∩ − I ∩ ∩ (Ω2 (Ω1 Ω2), Ω1 Ω2; q) (Ω1 Ω2, Ω1 Ω2; q) − I \ ∩ ∩ − I ∩ ∩ (Ω1 Ω2, Ω2 (Ω1 Ω2); q) (Ω1 Ω2, Ω1 Ω2; q) − I ∩ \ ∩ − I ∩ ∩ = (Ω1, Ω2; q) + (Ω2, Ω1; q) I I (Ω1 (Ω1 Ω2), Ω1 Ω2; q) (Ω1 Ω2, Ω1 (Ω1 Ω2); q) − I \ ∩ ∩ − I ∩ \ ∩ (Ω2 (Ω1 Ω2), Ω1 Ω2; q) (Ω1 Ω2, Ω2 (Ω1 Ω2); q). − I \ ∩ ∩ − I ∩ \ ∩
Now, assume that the regions Ω1 and Ω2 are disjoint, i.e., Ω1 Ω2 = . Then, ∩ ∅ (3.56) immediately follows from (3.57). Thus, we have proven (3.56) for disjoint regions.
If Ω1 Ω2 = , we have that Ω1 (Ω1 Ω2) and Ω1 Ω2 are disjoint regions as ∩ 6 ∅ \ ∩ ∩ are Ω2 (Ω1 Ω2) and Ω1 Ω2. Then, from (3.56) for disjoint regions, which we \ ∩ ∩ just proved, and (3.57), we obtain (3.56) for the case Ω1 Ω2 = . ∩ 6 ∅ The following result, which holds only for disjoint subsets, immediately follows from (3.55).
Proposition 10. Assume the interaction operator ( , ) satisfies relations (3.54) I · · 82 and (3.55). Then, ( , ; q) is bilinear for disjoint subsets, i.e., I · · n (Ω1 Ω2, Ω3; q) = (Ω1, Ω3; q) + (Ω2, Ω3; q) Ω1, Ω2, Ω3 R such I ∪ I I ∀ ⊆ (Ω3, Ω1 Ω2; q) = (Ω3, Ω1; q) + (Ω3, Ω2; q) that Ω1 Ω2 = . I ∪ I I ∩ ∅ (3.58)
The assumption (3.54) postulates that there are no self-interactions. The result
(3.56) is an abstract action-reaction principle, i.e., the flux from Ω1 into Ω2 is equal and opposite to the flux from Ω2 into Ω1. The result (3.58) states that the total interaction or flux from two disjoint regions Ω1 and Ω2 into a region Ω3 is simply the sum of the individual fluxes from Ω1 into Ω3 and from Ω2 into Ω3; (3.58) also states that the total flux from a region Ω3 into two disjoint regions Ω1 and Ω2 is the sum of the individual fluxes from Ω3 into the two subregions.
3.7.1.1 Abstract local balance laws
n For disjoint open regions Ω1, Ω2 R , we assume that the interaction operator ⊂ (Ω1, Ω2; q) is given by (3.2) so that I Z loc(Ω1, Ω2; q) = q ~ndA for Ω1 Ω2 = with ∂Ω12 = Ω1 Ω2 (3.59) I ∂Ω12 · ∩ ∅ ∩ for some vector q, where ∂Ω12 denotes the common boundary of Ω1 and Ω2 and ~n the unit normal along ∂Ω12 pointing out of Ω1. Note that
if ∂Ω12 = , then (Ω1, Ω2; q) = 0. ∅ Iloc
The interaction operator given in (3.59) satisfies (3.58) and, by (3.3), is antisym- metric. From (3.51), (3.52), (3.59), and the Gauss theorem, we have that Z (qt + q b) dx = 0 Ω Ωs Ω ∇ · − ∀ ⊂ from which, because Ω is arbitrary in Ωs, we obtain the local field equation
q + q = b x Ω (3.60) t ∇ · ∀ ∈ s 83 corresponding to the balance law (3.51) and the interaction operator (3.59). To obtain a field equation in terms of the intensive variable q, a constitutive equation must be postulated relating the flux vector q to the intensive variable q. For example, consider the case of heat conduction for which the variable q denotes the temperature. Then, a constitutive equation relating the heat flux vector q to the temperature q is given by the Fourier heat law q = κ q, where κ denotes − ∇ the thermal diffusivity. Then, from (3.60), we have the field equation
qt = κ∆q + b (3.61) for the temperature q, i.e., the heat equation.
3.7.1.2 Abstract nonlocal balance laws
From (3.4), in the nonlocal case, we define the interaction operator18 Z
nonloc(Ω1, Ω2; q) := Ω2 (ν) dx I Ω1 N Z Z (3.62) n = (ν + ν0) α dydx Ω1, Ω2 R Ω1 Ω2 · ∀ ⊂
that gives the nonlocal flux from Ω1 into Ω2; in (3.62), Ω2 (ν):Ω1 R denotes an N → interaction density.19 The interaction operator given in (3.62) satisies (3.58), i.e., is bilinear for disjoint regions, and, because α(x, y) is antisymmetric, is alternating and therefore antisymmetric. In stark contrast to that for the interaction operator ( , ; q) given in (3.59), Iloc · · the flux induced by ( , ; q) dispenses with the need for determining a unit Inonloc · · normal to an orientable surface and instead considers volume interactions among regions. Such a relation and (3.62) can also be interpreted more broadly. Fur- thermore, the local interaction operator ( , ; q) vanishes whenever Ω1 Ω2 = Iloc · · ∩ ∅ whereas, in general, the nonlocal interaction operator ( , ; q) does not. Inonloc · · We derive a nonlocal field equation analogous to the classical heat equation.
18 Comparing loc(Ω1, Ω2; q) and nonloc(Ω1, Ω2; q), we see that the role of the flux density q ~n for the local balanceI is now assumedI by the nonlocal flux density (ν). Just as was the case for· q, ν should be related to the intensive variable q through a constitutiveN law. 19See also page 85 in [4] and page 29 in [43]. 84
Due to (3.54), we have that Z Z Z nonloc(Ω, Ωc) = (ν) dx = (ν + ν0) α dydx Ω Ωs I Ω N Ω Ωc · ∀ ⊂ that we substitute, along with (3.52), into (3.51) to obtain Z (qt + (ν) b) dx Ω Ωs. Ω N − ∀ ⊂
Then, using the nonlocal Gauss theorem (3.17a), we obtain Z (qt + (ν) b) dx Ω Ωs Ω D − ∀ ⊂ from which it follows that, because Ω is arbitrary in Ωs,
q + (ν) = b x Ω . t D ∀ ∈ s
The interaction vector ν is related to the extensive quantity q through the consti- tutive relation ν = κ ∗(q), where the adjoint operator ∗ is defined in (3.9a). We D D then obtain the field equation for q given by
q + κ ∗(q) = b x Ω t DD ∀ ∈ s which may be viewed as a nonlocal heat equation analogous to the classical local heat equation (3.61). Note that that the operator κ ∗( ) is exactly that that DD · appears in the steady-state volume constrained problem (3.49a) with Θ2 = κI, where κ is constant and I denotes the identity tensor.
3.7.2 The peridynamics nonlocal theory of continuum me- chanics
We now demonstrate that the nonlocal calculus developed in the earlier sections is synergistic with the peridynamic nonlocal theory of continuum mechanics. The abstract balance law (3.51) and the discussion that ensues generalize, in straightforward manner, to vector valued intensive quantities. In particular, we 85 obtain the field equation
∂m + (Ψ) = b x Ω , (3.63) ∂t Dt ∀ ∈ s where the operator is defined in (3.13a) and the tensor Ψ is related to m through Dt a constitutive relation. In particular, consider the case of m denoting the momen- tum density for a continuum solid material so that m = ρ(x) ∂ u(x, t), where ρ( ) ∂t · denotes the material density and u(x, t) the displacement. A constitutive equation 1 T is given by Ψ(x, y) = ∗(u) , where the adjoint operator ∗( ) is defined in 2 Dt Dt · (3.13b). We then have, from (3.63), that
2 ∂ u 1 T ρ + ( ∗u = b x Ω . (3.64) ∂t2 2Dt Dt ∀ ∈ s
Substituting the definitions of and ∗, we obtain, invoking (3.15), Dt Dt Z T t ( t∗u = 2 (α α) (u0 u) dx (3.65) D D − Ωsc ⊗ · − so that, from (3.64), we have that
∂2u Z ρ 2 = (α α) (u0 u) dx + b x Ωs. (3.66) ∂t Ωsc ⊗ · − ∀ ∈
T A mechanical perspective indicates that ∗u describes the deformation of u Dt and that the constitutive relation maps the deformation to the force density given by the integral operator. Because the integrand of (3.65) is antisymmetric with respect to the arguments x and y, the operator (3.65) induces an interaction, i.e., that of forces between subregions.20 Equation (3.66) is a generalization of the conservation of momentum equation in the linearized bond-based peridynamic theory introduced in [1]. In fact, if we require that the null space of the integral operator in (3.64) contains rigid motions,
20In the local case, (3.59) is an abstraction of Cauchy’s postulate in mechanics. There, the intensive variable q is the vector momentum density and q is the stress tensor with q n then being the stress force density at a point of a surface. Then, Cauchy’s postulate states· that the interaction between two abutting regions occurs at the common interface between the two regions and is given by the integral of the stress force along that interface. This is exactly a word description of (3.59). 86 i.e., that
T ( ∗(Ax + c)) = 0 Dt Dt for all constant skew-symmetric tensors A and all constant vectors c, then a sufficient condition is that α(x, y) = ω( y x )(y x)/ y x , where | − | − | − | ω : R+ R. Then, from (3.66), we have that → ∂2u Z y x y x ρ = ω( y x ) − ⊗ − (u0 u) dx + b x Ω , (3.67) ∂t2 | − | y x 2 · − ∀ ∈ s Ωsc | − | which is the linearized peridynamic bond-based model of [1]. In [25], analytical conditions on ω describing the amount of smoothing associated with the integral operator are given and the well posedness of the balance of linear momentum is discussed. That chapter also demonstrates that, for the case of localized kernels, as ε 0, → 1 T ( ∗u) µ ( u) 2µ ( u) 2Dt Dt → − ∇ · ∇ − ∇ ∇ · which is the Navier operator of linear elasticity for Poisson ratio one-quarter; see also [16] and chapter 5. The subsequent chapter [28] considers volume-constrained problems on bounded domains in R and squares in R2. The state-based peridynamic theory [2] requires the consideration of both vol- umetric and shear deformations. Similar to the above discussion for bond-based peridynamics models, we may also formulate the state-based peridynamic model in terms of the nonlocal operators. Let α and ω be given by (3.33). Then, the linear state-based peridynamic integral operator [3] is given by
T η ∗u + (λTr ∗ u I , (3.68) Dt Dt Dt,ω Dt,ω where η and λ are materials constants and t, t∗, t,w, and t,w∗ are given by D D D D (3.13a), (3.13b), (3.31), and (3.30), respectively. The scalar Tr t,ω∗ u measures D the volumetric change, or dilatation, in the material so that Tr ∗ u I is a diago- Dt,ω nal tensor representing volumetric stress. This allows us to readily apply the nonlo- cal calculus to study the well-posedness of both free-space and volume-constrained linear peridynamic state-based balance laws, and also suggests why, in the limit as 87
ε 0, the operator given by (3.68) leads to the linear Navier operator of elasticity → for linear isotropic materials with general Poisson ratios, e.g., not relegated to a value of one-quarter. This is discussed in chapter 5. Chapter 4
Application of the Nonlocal Vector Calculus to the Analysis of state-based Linear PD Materials
4.1 Introduction
The significance of the generalized peridynamic theory, the PD state model, intro- duced in [2], is that for homogeneous deformations of a linear isotropic material, Poisson’s ratio is no longer limited to be equal to one-fourth. The bond- and state- based peridynamic theory refer to the case when Poisson’s ratio is one-fourth or not, respectively. The goal of our chapter is to determine the well-posedeness of the linear peridy- namic equilibrium equation by exploiting the nonlocal vector calculus developed in [44] given volume constraints. These constraints represent the nonlocal ana- logue [1, section 13] of boundary conditions necessary for the stability of the linear peridynamic equilibrium operator. The framework developed in [44] generalized the nonlocal operators defined for scalar functions as discussed in, for instance, [29, 18, 45] to those defined for vector fields which allows us to provide a system- atic formulation of the state-based peridynamic model in terms of the nonlocal operators. Our major contribution here is the well-posedeness of the state-based linear peridynamic equilibrium equation, a first of a kind result. Previous work 89 has focused on the well-posedness of the bond-based model; see [17, 25, 16] when the deformation is vector-valued and [46, 47, 37, 21, 18] when the deformation is scalar-valued or the related case of nonlocal diffusion. Thus, the present study fills a significant gap in the analysis of peridynamic models and also demonstrates the efficacy of the nonlocal vector calculus framework introduced by [44] for the analysis of nonlocal problems. We represent the peridynamic deformation fields such as the extension scalar, dilatation, etc., in terms of operators from the nonlocal vector calculus. Based upon the representation of the deformation state, we then formulate variational principles and rewrite both the linear bond-based and the linear state-based peridy- namic elastic model via the nonlocal divergence operators and their corresponding adjoints. Our variational formulation coincides with the linearization of the origi- nal peridynamic equation derived in [1, 2]. Moreover, we prove the well-posedness of peridynamic “boundary”-value problem, or volume-constrained problem, that is, the linear peridynamic elastic model subject to suitable constraints that are either explicitly enforced on volumes of non-zero measure or naturally implied by the variational principle. This chapter is organized as follows. In Section 4.2, in additional to the opera- tors and identities introduced in chapter 3, we present two other nonlocal Green’s identities that are used for nonlocal state-based peridynamic modeling. In Sec- tions 4.3 and 4.4, we rewrite the peridynamic constitutive relation and the lin- earized peridynamic models for elastic materials as variational problems involving the nonlocal calculus operators by minimizing the total potential energy. Finally, in Sections 4.5, we establish the well-posedness of the peridynamic models.
4.2 Useful identities in the nonlocal vector cal- culus
Our formulation of the linear peridynamic constitutive relation employs the nonlo- cal divergence operator for tensor functions , the weighted nonlocal divergence Dt operators for tensor functions and their adjoint operators ∗ and ∗ , re- Dt,ω Dt Dt,ω spectively. These operators were introduced within the nonlocal vector calculus in 90 chapter 3 and [44]. Then from the definition of (3.13b) and (3.31), we have Z Tr t,ω∗ (u) (x) = Tr t∗(u) (x, y) ω(x, y) dy for x Ω. (4.1a) D Ω D ∈
By the relations (a b) c = (a c) b = (b c) a that holds for any d-dimensional ⊗ ⊗ · vectors a, b, c and that Tr ∗(u) = ∗(u) and Tr ∗ (u) = ∗ (u) in which ∗ Dt G Dt,ω Gω G and ∗ are defined in chapter 3 and [44], we obtain the nonlocal Green’s identities Gω for the trace of tensor divergences: Z Z Z T Tr( t∗u)Tr( t∗v) dy dx = t t∗(u) v dx, (4.1b) Ω Ω D D Ω D D · Z Z Tr( t,ω∗ u)Tr( t,ω∗ v) dy dx = t,ω Tr( t,ω∗ u)I v dx. (4.1c) Ω D D Ω D D ·
4.3 Constitutive relations in peridynamic mod- eling
In this section, we rewrite the constitutive relation introduced in (1.8a) - (1.8e) in section 1.2.2, for a peridynamic ordinary elastic material in terms of the nonlocal vector calculus operators. We now specialize the definitions of the nonlocal calculus to the peridynamic theory. Let α(x, y) = (x y)/ y x , − | − | 1 Z n(x) = y x 2ω(x, y) dy, (4.2) d Ω | − |
ω(x, y) = y x ω(x, y)/n(x). | − | From this definition, we have the relation
n(x) = m/d. where m is the weighted volume defined in (1.8b). Then, after linearizing the 91 extension scalar state with respect to u(y) u(x), we obtain − e = u(y) + y u(x) x y x | − − | − | − | = y x u(y) u(x)/ y x (4.3) − · − | − | = Tr ∗u . Dt By the definition of the scalar product of two states, we have the following descrip- tion of a linear peridynamic elastic material:
θb = Tr ∗ u , (4.4a) Dt,ω i e = Tr ∗ u y x /d, (4.4b) Dt,ω | − | d e = Tr ∗u Tr ∗ u y x /d, (4.4c) Dt − Dt,ω | − | where ∗u = ∗(u), ∗ u = ∗ (u) and “Tr” denotes the trace operator. We Dt Dt Dt,ω Dt,ω see that, when small deformations are considered, see [3, Definition 4.1] for more explanations, the peridynamic deformation quantities can be represented in terms of the nonlocal adjoint tensor divergence operator ∗ and the weighted adjoint Dt tensor divergence operator ∗ . Dt,ω Let δ > 0 denote the horizon, which is treated as a material property. For any two points x , y Ω, if x y < δ, the vector ξ = y x is called a bond. ∈ | − | −
In the sequel, we focus on the linearized theory of a peridynamic solid whose microelastic energy is given by [2, page 23],
k η W (θ, ed) = θb2 + (ωed) ed , (4.5) 2 2 • where k and η are material parameters. We can rewrite the state-based peridy- namic strain energy in terms of nonlocal vector calculus as
2 Z d k(x) Tr( t,ω∗ u) η(x) Tr( t,ω∗ u) y x 2 W (θ, e ) = D + ω(x, y) Tr( t∗u) D | − | dy. 2 2 Ω D − d 92
4.4 Variational principles for linear peridynamic models
In this section, we use an energy minimization principle to derive equations for state-based linear peridynamic materials. By (4.5), the total potential energy of a state-based peridynamic material under an external force b(x) is given by
Z 2 Z Z Z k(x) θb η(x) d 2 Es(u) = dx + ω(x, y)(e ) dy dx u b dx Ω 2 Ω Ω 2 − Ω · 2 Z k(x) Tr( ∗ u) Z = Dt,ω dx u b dx (4.6) Ω 2 − Ω · Z Z η(x) Tr( t,ω∗ u) y x 2 + ω(x, y) Tr( t∗u) D | − | dy dx. Ω Ω 2 D − d
4.4.1 Variation of the potential energy
The model problems for peridynamic materials we consider in this chapter are described by the minimization of the potential energy so as to find the equilibrium state of the material. Thus, to determine a minimizer of the potential energy, we seek a displacement field u such that δE(u)/δu = 0 with certain constraints. To define the contraints, we first introduce the following notation:
Ωs : the solution domain of the model equation
Ωc : the constraint domain where the displacement field u is specified. Ω = (Ω Ω ) (Ω Ω )0 s ∪ c ∪ s ∩ c U(Ω): a function space defined on Ω such that the total potential energy is finite.
Note that the constraint domain Ωc is either empty or has positive volume, that is, it cannot be a lower-dimensional manifold.
A variety of choices for the region Ωc having positive volume can be allowed as part of the the material domain. See Figure 4.1 for illustrations of some of the possibilities. We note that most of our results can also be applied to cases where 93
Ω is a union of the domains shown in the figure.
Figure 4.1. Four of the possible configurations for Ω = (Ω Ω ) (Ω Ω )0 s ∪ c ∪ s ∩ c We consider the following constrained energy minimization problem that cor- responds to conventional boundary-value problems with essential boundary condi- tions:
min Es(u) subject to u = hd for x Ωc. (4.7) u U(Ω) ∈ ∈
We define the test-function space U0 as
U0 = v U(Ω): v = 0 on Ω . { ∈ c}
Let v denote an arbitrary function in U0 and f() = Es(u + v). We compute f 0() =0 to obtain | Z Z f 0(0) = k(x)Tr( t,ω∗ u)Tr( t,ω∗ v) dx b v dx Ω D D − Ω · Z Z y x + η(x)ω(x, y) Tr( t∗u) Tr( t,ω∗ u)| − | Ω Ω D − D d y x Tr( ∗v) Tr( ∗ v)| − | dy dx Dt − Dt,ω d Z Z = k(x)Tr( t,ω∗ u)Tr( t,ω∗ v) dx b v dx Ω D D − Ω · Z Z + η(x)ω(x, y)Tr( t∗u)Tr( t∗v) dx Ω Ω D D Z Z 1 2 + η(x)Tr( t,ω∗ u)Tr( t,ω∗ v) 2 ω(x, y) y x dy dx Ω D D d Ω | − | Z 1 Z η(x)Tr( t,ω∗ v) ω(x, y)Tr( t∗u) y x dy dx − Ω D d Ω D | − | 94
Z 1 Z η(x)Tr( t,ω∗ u) ω(x, y)Tr( t∗v) y x dy dx . − Ω D d Ω D | − |
From (4.2) and (4.1a) and the Green’s identities (4.1b) and (4.1c), we obtain Z Z f 0(0) = k(x)Tr( t,ω∗ u)Tr( t,ω∗ v) dx b v dx Ω D D − Ω · Z Z + η(x)ω(x, y)Tr( t∗u)Tr( t∗v) dx Ω Ω D D Z η(x)n(x) + Tr( t,ω∗ u)Tr( t,ω∗ v) dx Ω d D D Z η(x)n(x) Tr( t,ω∗ u)Tr( t,ω∗ v) dx − Ω d D D Z η(x)n(x) Tr( t,ω∗ u)Tr( t,ω∗ v) dx − Ω d D D Z 1 = t,ω k(x) η(x)n(x) Tr( t,ω∗ u)I v dx Ω D − d D · Z T + t η(x)ω(x, y)( t∗u) v dx b v dx . D D · − Ω ·
Thus, the constrained energy minimization problem (4.7) results in the equivalent nonlocal system (u)(x) = b(x), x Ωs, −L ∈ (4.8) u(x) = h (x), x Ω , d ∈ c where the peridynamic operator ( ) is given by L · T ( ) = η(x)ω(x, y) ∗( ) + k(x) n(x)η(x)/d Tr ∗ ( ) I . (4.9) −L · Dt Dt · Dt,w − Dt,ω ·
By direct calculation, one sees that the nonlocal system (4.8) derived using a variational principle is exactly the same as the one obtained in [2] based upon the balance of linear momentum. Setting c(x) = (k(x) η(x)n(x)/d)/n2(x), (4.10) − we have the following equivalent form of the peridynamic operator.
Lemma 24. The peridynamic operator can be alternatively expressed in the fol- 95 lowing form. For a given function u U(Ω), ∈ Z (u)(x) = C(x, y) u(y) u(x) dy (4.11) L Ω − where C(x, y) = K1(x, y) + C0(x, y) with
ω(x, y) K1(x, y) = 2η(x) (y x) (y x), (4.12a) x y 2 − ⊗ − | − | Z C0(x, y) = c(z)ω(x, z)ω(z, y)(x z) (z y) Ω − ⊗ − c(y)ω(x, y)ω(y, z)(x y) (y z) (4.12b) − − ⊗ − + c(x)ω(x, z)ω(x, y)(x z) (x y) dz. − ⊗ −
T Proof. We first calculate η(x)ω(x, y) ∗(u) : Dt Dt T t η(x)ω(x, y) t∗(u) D Z D = 2η(x) ω(x, y)α(x, y) α(x, y) u(y) u(x) dy − Ω ⊗ − Z ω(x, y) = 2η(x) 2 (y x) (y x) u(y) u(x) dy − Ω x y − ⊗ − − Z | − | = K1(x, y) u(y) u(x) dy. − Ω − Next, we consider k(x) n(x)η(x)/d Tr ∗ (u) I : Dt,w − Dt,ω k(x) n(x)η(x)/d Tr ∗ (u) I Dt,w − Dt,ω Z Z = c(y) u(z) u(y) α(y, z)ω(y, z) y z dz Ω Ω − · | − | Z + c(x) u(z) u(x) α(x, z)ω(x, z) x z dz Ω − · | − | α(x, y)ω(x, y) y x dy · | − | Z Z = c(y)ω(x, y)ω(y, z) (y x) (z y) u(z) u(y) dz dy Ω Ω − ⊗ − − 96
Z Z + c(x)ω(x, y)ω(x, z) (y x) (z x) u(z) u(x) dz dy Ω Ω − ⊗ − − Z Z = c(y)ω(x, y)ω(y, z) (y x) (z y)u(z) dz dy Ω Ω − ⊗ − Z Z c(y)ω(x, y)ω(y, z) (y x) (z y)u(y) dz dy − Ω Ω − ⊗ − Z Z + c(x)ω(x, y)ω(x, z) (y x) (z x)u(z) dz dy Ω Ω − ⊗ − Z Z c(x)ω(x, y)ω(x, z) (y x) (z x)u(x) dz dy. − Ω Ω − ⊗ −
We also note that, by changing the order of integration, Z Z c(y)ω(x, y)ω(y, z) (y x) (z y)u(z) dz dy Ω Ω − ⊗ − Z Z = c(z)ω(x, z)ω(z, y) (z x) (y z)u(y) dz dy Ω Ω − ⊗ − and Z Z c(x)ω(x, y)ω(x, z) (y x) (z x)u(z) dz dy Ω Ω − ⊗ − Z Z = c(x)ω(x, z)ω(x, y) (z x) (y x)u(y) dz dy. Ω Ω − ⊗ −
Thus, we have Z t,w k(x) n(x)η(x)/d Tr t,ω∗ (u) I = C0(x, y) u(x) u(y) dy . D − D Ω −
The conclusion of the lemma now follows.
By (4.12a) and (4.12b), the double state K[x] z x, y x defined in [3, eqns h − − i 23, 24] for the peridynamic solid is obtained as
K[x] z x, y x = K1(x, y)∆(z x, y x) h − − i − − + c(x)ω(x, z)ω(x, y)(z x) (y x), − ⊗ − in which ∆( , ) denotes the delta-function, and the force state T(x, y) can be found · · 97 by [3, eqn 23]. Both of these states coincide with the ones derive from the balance law. We can also refer to Table 1 in [3, page 27] for the mechanical explanations for the nonlocal quantities.
4.5 Well-posedness of the state-based PD solid models
In the previous section, the peridynamic models are derived from energy mini- mization principles. The derivations are implicitly based on corresponding weak formulations of the nonlocal models, that is,
Z Z η(x)ω(x, y)Tr( t∗u)Tr( t∗v)) dy dx Ω Ω D D Z Z + k(x) n(x)η(x)/d Tr( ∗ u)Tr( ∗ v) dx = b v dx v U0, − Dt,ω Dt,ω · ∀ ∈ Ω Ω u(x) = h (x), x Ω d ∈ c (4.13) for the linearized state-based peridynamic solid. We then define the bilinear form Z Z Z B(u, v) = ψ1(x, y)Tr( t∗u)Tr( t∗v) dy dx+ ψ2(x)Tr( t,ω∗ u)Tr( t,ω∗ v) dx, Ω Ω D D Ω D D (4.14) where
ψ1(x, y) = η(x)ω(x, y) (4.15a)
ψ2(x) = k(x) n(x)η(x)/d (4.15b) − In the following, we focus on the well-posedness of the linearized state-based peridynamic model with a given volume-constraint. To establish our findings, we need to make some assumptions which are listed separately below.
Assumption 1. Ω is an open and connected domain in Rd.
Assumption 2. Ω is bounded and satisfies the interior cone condition with pa- rameters r0 > 0 and θ0 > 0, as defined by the property that (see [48] for details) if 98
for any point x Ω, the intersection between the ball centered at x with radius r0 ∈ and the domain Ω contains a cone with an angle no smaller than θ0.
Assumption 3. η(x) η0 > 0 and k(x) k0 > 0 for any x Ω, and ω(x, y) is ≥ ≥ ∈ non-degenerate and is nonnegative in Ω Ω, where the non-degeneracy condition × is specified by the property that any function β(x, y) is said to be non-degenerate if there exist two positive constants δ0 > 0 and β0 > 0 such that
β(x, y) β0 > 0 x, y Ω satisfying x y δ0. ≥ ∀ ∈ k − k ≤
Assumption 4. k(x) k1 < , η(x) η1 < and ω = ω(x, y) is symmetric, ≤ ∞ ≤ ∞ i.e. ω(x, y) = ω(y, x), and for some constant M > 0, Z ω2(x, y) dy M, x Ω . Ω ≤ ∀ ∈
Remark 2. This non-degeneracy condition in Assumption 3 is automatically sat- isfied if the function under consideration is bounded below uniformly by a positive constant over the domain Ω. In the special case for which β(x, y) = βe(x y), − we also have that β is non-degenerate if the support of βe contains a small ball centered at the origin which is usually the case for peridynamic models [1, 3, 2]. In such a case, δ0 is a parameter that is strictly smaller than the peridynamic horizon parameter.
Remark 3. In the following discussions, the above assumptions are used in dif- ferent places, but they all serve to establish the well-posedness of the PD models. For instance, we use the Assumption 1 to help us clarify, in Lemma 25, that the kernel of the operator from u to u(y) u(x) (y x) is the rigid body motion. In − · − addition, Assumptions 1 and 3 are used to prove the general well-poseness of the nonlocal variational problems associated with the PD-state models (as documented in Lemma 26 and Theorems 7 and 8), provided that the related energy space is a well-defined Hilbert space. The latter result on the energy space is established in the section 4.5.4 for the PD-state energy under the additional Assumptions 2 and 4, see Lemmas 27, 28, 29, 30 and Theorems 9, 10.
Remark 4. We also note that it is possible to relax the non-negativity assumptions 99 on η(x), k(x), ω(x, y) to consider more general cases that allow for sign changes. Such generalizations are left to future studies.
Setting v = u in (4.14), we obtain Z Z B(u, u) = η(x)ω(x, y)Tr( t∗u)Tr( t∗u) dy dx Ω Ω D D Z + k(x) n(x)η(x)/d Tr( t,ω∗ u)Tr( t,ω∗ u) dx Ω − D D
Z k(x) θb2 Z Z η(x) = dx + ω(x, y)(ed)2 dy dx, (4.16) Ω 2 Ω Ω 2 where we have used the definition (4.4c) and the derivation given in (4.6). Note that B(u, u) 0 for any u U(Ω), we thus formally define an inner ≥ ∈ product and its associated norm by
((u, v)) = B(u, v) and u = (B(u, u))1/2, ||| ||| respectively. One readily sees that the function space U(Ω) can be expressed as
U(Ω) = u: u < . { ||| ||| ∞}
We define the space as Z = u: u = 0 . Z { ||| ||| } To show that (( , )) and actually define an inner product and norm, · · ||| · ||| respectively, in a suitable subspace, we need the following lemma that is analogous to a similar result stated for Rd in, for instance, in [49, Prop. 1.2]. The proof requires an argument for the nonlocal case which is substantially different from that given in [49] for the local counterpart.
Lemma 25. Assume the domain Ω satisfies Assumption 1 and u L2(Ω). If for ∈ a.e. x Ω, ∈
u(y) u(x) (y x) = 0 for a.e. y Ω B (x), (4.17) − · − ∈ ∩ δ0 then u is given by a rigid body motion in Ω, that is, there exists a constant-valued 100 skew-symmetric matrix A and a constant-valued vector c such that
u(x) = Ax + c a.e. x Ω. (4.18) ∈
Proof. For any x0 Ω, we may choose δ(x0) (0, δ0/2) such that ∈ ∈
y B (x0)Bδ(x0)(y) Ω Bδ0 (x0). ∪ ∈ δ(x0) ⊂ ∩
d Let e˜i i=1, ,d be an orthonormal basis in R . By the assumption of the lemma, { } ··· we may find for some x0 Ω at which (4.17) holds for y Ω B (x0) except a ∈ ∈ ∩ δ0 measure zero set. Let Bδ(x0)/N (x0 +δ(x0)˜ei) be small balls centering at x0 +δ(x0)˜ei with radii δ(x0)/N where the positive constant N is chosen such that the balls are small enough and do not intersect. d Since (4.17) holds for x Ω almost everywhere, we may choose d points x0 ∈ { i}i=1 such that, x0 B ( ) (x0 + δ(x0)˜e ) for 1 i d. By the definitions of i ∈ δ x0 /N i ≤ ≤ d B ( ) (x0 +δ(x0)˜e ), one readily sees that x0 x0 form a basis. For notation δ x0 /N i { i − }i=1 simplicity, we set e = (x0 x0)/ x0 x0 . i i − k i − k Then we have for x0 and some zero measure set Nx0 ,
u(y) u(x0) (y x0) = 0 y Ω − · − ∀ ∈ x0 where Ω = Ω B (x0) N . x0 ∩ δ0 \ x0 Thus, for any x B ( )(x0) Ω , we have ∈ δ x0 ∩ x0 u(x) u(x0) (x x0) = 0. − · −
Moreover, u(x) u(x0 + δ(x0)e ) x x0 δ(x0)e = 0. − i · − − i This implies that u(x) e is linear in x for any i, which gives the linearity of u in · i Bδ0 (x0). Then, (4.17) implies that u has to take on the form given by (4.18).
Because Ω is a bounded connected open set, for any two points x0 and x1 in K Ω, there exists a finite number of balls B ( )(y ) with sufficiently small radii { δ yk k }k=1 δ(yk) > 0 with the properties that their union is completely contained in Ω, that covers a connected path between x0 and x1, and B ( )(y ) B ( )(y +1) has δ yk k ∩ δ yk+1 k 101 a non-empty interior. Then, we see that in each ball, u is given by a rigid body motion and their form must be the same in neighboring balls and thus it is a global rigid body in the whole domain Ω.
We then have the following result.
Lemma 26. If the Assumptions 1 and 3 hold, u and ((u, v)) define a norm ||| ||| and an inner product, respectively, on both U0(Ω), provided Ωc has a non-empty interior, and U(Ω) . \Z Proof. We establish the result for the state-based models first. We note that u ||| ||| defines a semi-norm on U0(Ω). Thus, it suffices to prove that B(u, u) = 0 implies u = 0. Because the Assumption 3 holds, from (4.16), we have
Z k(x) θb2 Z Z η(x) dx = 0 and ω(x, y)(ed)2 dy dx = 0. (4.19) Ω 2 Ω Ω 2
The first equation in (4.19) implies that Tr( t,ω∗ u) = 0 by the first equation in D (4.19) and thus, by the second equation in (4.19), Tr( ∗u) = u(y) u(x) (y Dt − · − x) = 0 in the support of ω(x, y). Given the non-degeneracy condition given on ω, from Assumption 1 and Lemma 25 we can deduce that the kernel of the operator
Tr( ∗) is the set of rigid body motions u(x) = Ax + c for a constant vector c Dt and constant skew-symmetric matrix A. So, in either U0(Ω) with Ωc having a non-empty interior or in U(Ω) , the only u satisfying (4.19) is u 0. Thus we \Z ≡ conclude that defines a norm and (( , )) defines an inner product on U0(Ω) ||| · ||| · · and U(Ω) . \Z
4.5.1 Decomposition of the solution space
Let the space S(Ω) consist of functions u U(Ω) that satisfy ∈ T ψ1 ∗(u) + ψ2Tr ∗ (u) I = 0 x Ω . (4.20) Dt Dt Dt,w Dt,ω ∀ ∈ s
Then, for all u S(Ω) and v U0(Ω) we have ∈ ∈
B(u, v) = 0. 102
Thus, we conclude that
U(Ω) = U0(Ω) S(Ω), (4.21) ⊕ that is, any function in U(Ω) can be written as a direct sum of two functions that are orthogonal with respect to the inner product (( , )), the first a function that · · vanishes on Ωc whereas the second a function satisfying (4.20).
4.5.2 Nonlocal dual spaces and nonlocal trace spaces
Define the dual norm by Z v b dx Ωs · b o∗ = sup ||| ||| v U0(Ω), v=0 v ∈ 6 ||| ||| and define the dual space of U0(Ω) as
U ∗(Ω) = b : b ∗ < . 0 { ||| |||o ∞}
Similarly, we can define the dual space of U(Ω) by
U ∗(Ω) = b : b ∗ < , { ||| ||| ∞} where Z v b dx Ω · b ∗ = sup . ||| ||| v U(Ω), v=0 v ∈ 6 ||| ||| d As for the nonlocal trace space, let hd :Ωc R denote a mapping; then the → trace space is defined as
U = h : h < , d { d ||| d|||d ∞} where
hd d = inf v . ||| ||| v Ωc =hd ||| ||| | 103
4.5.3 Well-posedness of variational problems
The variational problems discussed above can take on quite general forms with respect to the different constraints imposed. For example, a “Dirichlet” volume- constraint problem is formulated as given b U and h U , seek u U (Ω) such that 0∗ d d 0 ∈ ∈ ∈ B(u, v) = Fd(v) v U0(Ω) (4.22) ∀ ∈ and u(x) = hd(x) whereas the form of a “Neumann” volume-constraint problem is given by given b U ∗, seek u U(Ω) such that ∈ ∈ \Z (4.23) B(u, v) = F (v) v U(Ω) , n ∀ ∈ \Z where the linear functionals F ( ) and F ( ) are defined by d · n · Z Fd(v) = v b dx v U0(Ω) Ω · ∀ ∈ and Z Fn(v) = v b dx v U(Ω) , Ω · ∀ ∈ \Z respectively. We note that in the “Neumann” volume-constrained problem, the quotient space U(Ω) is considered which is to guarantee the uniqueness of the solution \Z to the problem. Equivalently, we can instead pose a condition on the constraint domain Ω such as c Z u(x) dx = 0. Ωc
Because B( , ) defines an inner product on both U0(Ω) and U(Ω) , it is a · · \Z continuous and coercive bilinear form on those spaces. If we assume that U(Ω) is a complete space and the data are such that the functionals F ( ) and F ( ) are d · n · continuous, the Lax-Milgram theorem can be applied to show that both (4.22) and 104
(4.23) have unique solutions and, moreover, those solutions satisfy
u b ∗ + C h and u b ∗, ||| ||| ≤ ||| |||o ||| d|||d ||| ||| ≤ ||| ||| respectively. Thus, for the “Dirichlet” and “Neumann” volume-constraint prob- lems, we have the following pair of results where the completeness of the space U(Ω) is demonstrated in the next section.
Theorem 7. If Ω satisfies Assumption 1 and U0(Ω) is a Hilbert space, b U ∗(Ω), ∈ 0 and h U (Ω), the Dirichlet volume-constraint problem (4.22) for the linear state- d ∈ d based peridynamic material (see (4.8) and (4.9)) has a unique solution u U0(Ω) ∈ provided that the Assumption 3 is satisfied.
Theorem 8. If Ω satisfies Assumption 1 and U(Ω)/ is a Hilbert space and Z b U ∗(Ω), the Neumann volume-constraint problem (4.23) for the linear state- ∈ based peridynamic material (see (4.8) and (4.9)) has a unique solution u U(Ω)/ ∈ Z provided that the Assumption 3 is satisfied.
Remark 5. The above theorems represent the first available results on the rigorous well-posedness of the linear state-based peridynamic elasticity models with volume- constraints, though it remains to verify the energy space is indeed a Hilbert space. In the next subsection, we provide an illustrative example as a demonstration.
4.5.4 An example of the peridynamic energy space
A technique for proving that U(Ω) is a Hilbert space is to establish, under appro- priate conditions on the influence function ω(x, y), that U(Ω) is in fact equivalent to a well-known Sobolev space. In the classical (local) elasticity theory, this is established using some useful facts such as the Korn inequality that generally re- lies on a compactness argument. In [28], the equivalence is established for special volume-constraints that makes the peridynamic operators commute with the dif- ferential operators and allows for a precise characterization of the energy spaces in terms of the eigenspaces of the associated differential operators. While that tech- nique is special, the results/conclusions are expected to remain valid in general. Here, we present another illustrative example showing the equivalence of spaces 105 via a nonlocal Korn inequality combined with compactness results associated with Hilbert-Schmidt operators. In this section, for much of the discussions here, we consider in details a state- based peridynamic material which is possibly inhomogeneous and anisotropic. In addition to the Assumptions 1 and 3, we need to further use the Assumptions 2 and 4. We now define the tensor field Z P (x) = K1(x, y) + C0(x, y) dy (4.24) Ω where K1 and C0 are defined in (4.12a) and (4.12b). Under the interior cone condition given in the Assumption 2 on the domain, for any x Ω,¯ we may assume ∈ that a cone Br0,θ0 (x), centered at x with radius r0 and angle θ0 is contained in the domain Ω. Let us define
1 Z π(x) = y x 2ω(x, y) dy . d | − | Br0,θ0 (x)
We now show that, under the Assumptions 1-4, the energy space U(Ω) is in fact equivalent to L2(Ω). First, we note that n(x) and c(x), defined in (4.2) and (4.10) respectively, have the following upper and lower bounds.
Lemma 27. If the domain and coefficients in the peridynamic model (4.8) satisfy Assumptions 1-4, we have that
2 1/2 1/2 0 < π0(r0, θ0) n(x) π1(Ω) = diam(Ω) Ω M /d , (4.25a) ≤ ≤ | | | |
c(x) (k1 + η1π1(Ω)/d)/π0(r0, θ0) , (4.25b) | | ≤ where π0(r0, θ0) = minx Ω π(x). ∈ Proof. We first prove the inequality for n(x).
1 Z n(x) = y x 2ω(x, y) dy d Ω | − | Z Z 1 1/2 1/2 y x 4 dy ω2(x, y) dy ≤ d Ω | − | Ω 106
1 diam(Ω) 2 Ω 1/2M 1/2 ≤ d| | | |
= π1(Ω).
By the interior cone condition of the domain Ω, we have n(x) π(x). On the ≥ other hand, since y x 2ω(x, y) > 0 on any nonzero measure set in B (x) | − | r0,θ0 by the non-degeneracy condition, we have π(x) > 0 for any x in Ω.¯ Given the integrability condition of ω(x, y), we have that π(x) is continuous in Ω,¯ thus we have
n(x) min π(x) = π0(r0, θ0) > 0. ≥ x Ω ∈ Then one readily sees that
2 2 c(x) k(x) η(x)n(x)/d /n (x) < (k1 + η1π1(Ω)/d)/π (r0, θ0). | | ≤ | − | 0
This proves the lemma.
We next show that the tensor field P (x) is uniformly positive definite.
Lemma 28. If the domain and coefficients in the peridynamic model (4.8) satisfy
Assumptions 1-4, we have P (x) is uniformly positive definite, i.e. P (x) P0 > 0, ≥ where Z ω(x, y) P0 = η0 min 2 (y x) (y x) dy, (4.26) x Ω B (x) y x − ⊗ − ∈ r0,θ0 | − | and Br0,θ0 (x) represents the intersection between a ball with the radius r0 centered at x and a cone with vertex x and angle θ0.
Proof. First of all, for any unit vector u Rd, we may use a similar argument as ∈ in the proof of the Lemma 27 to get that
"Z # Z ω(x, y) ω(x, y) 2 u 2 (y x) (y x) dy u = 2 [u(y x)] dy · B (x) y x − ⊗ − B (x) y x − r0,θ0 | − | r0,θ0 | − | is uniformly bounded below by a positive constant, independent of x Ω. Taking ∈ d the minimum over all u in the unit sphere in R , we see that P0 is a positive definite matrix. 107
From the definitions (4.12b) and (4.10), we get
Z Z Z C0(x, y) dy = c(z)ω(x, z)ω(z, y)(x z) (z y) Ω Ω Ω − ⊗ − c(y)ω(x, y)ω(y, z)(x y) (y z) − − ⊗ − + c(x)ω(x, z)ω(x, y)(x z) (x y) dz dy. − ⊗ −
By changing the order of integrals, we obtain Z Z Z C0(x, y) dy = c(z)ω(x, z)ω(z, y)(x z) (z y) Ω Ω Ω − ⊗ − c(z)ω(x, z)ω(z, y)(x z) (z y) − − ⊗ − + c(x)ω(x, z)ω(x, y)(x z) (x y) dz dy − ⊗ − Z Z = c(x)ω(x, z)ω(x, y)(x z) (x y) dz dy Ω Ω − ⊗ − Z Z = c(x) ω(x, y)(x y) dy ω(x, y)(x y) dy Ω − ⊗ Ω − k(x) Z Z = 2 ω(x, y)(x y) dy ω(x, y)(x y) dy n (x) Ω − ⊗ Ω − η(x) Z Z ω(x, y)(x y) dy ω(x, y)(x y) dy − dn(x) Ω − ⊗ Ω −
= A1 A2. −
Since k(x) > k0 > 0 and n(x) π0(r0, θ0) > 0, we conclude that A1 0. By the ≥ ≥ definition of K1 in (4.12a), we get
Z Z ω(x, y) K1(x, y) = 2η(x) (y x) (y x) dy y x 2 − ⊗ − Ω Ω | − | = 2A3
First, because of the interior cone condition of the domain Ω, we have
Z ω(x, y) A3 = η(x) (y x) (y x) dy y x 2 − ⊗ − Ω | − | 108
P0 . ≥
We can rewrite P (x) as P (x) = 2A3 + A1 A2 = A3 + A1 + (A3 A2). Since − − A3 P0 > 0 and A1 0, we only need to show that A3 A2 is semi-positive ≥ ≥ − definite, i.e. A3 A2 0, which is equivalent to show that d n(x)(A3 A2) 0. − ≥ − ≥ Let u is an arbitrary nonzero vector. By the definition of n(x), we have
u d n(x)(A3 A2)u · − Z Z ω(x, y) = u η(x) ω(x, y) x y 2 dy (y x) (y x) dy · | − | x y 2 − ⊗ − Ω Ω | − | Z Z η(x) ω(x, y)(x y) dy ω(x, y)(x y) dy u − Ω − ⊗ Ω − Z Z ω(x, y) 2 = η(x) ω(x, y) x y 2 dy u(y x) dy | − | x y 2 − Ω Ω | − | Z 2 ω(x, y) u(y x) dy − Ω − where d is space dimension.
By Cauchy-Schwarz inequality, we have
Z 2 ω(x, y) u(y x) dy Ω − Z ω1/2(x, y) 2 = ω1/2(x, y) x y u(y x) dy | − | x y − Ω | − | Z Z ω(x, y) 2 ω(x, y) x y 2 dy u(y x) dy ≤ | − | x y 2 − Ω Ω | − | which implies u d n(x)(A3 A2)u 0 for any nonzero u, i.e. d n(x)(A3 A2) 0. · − ≥ − ≥ The result of the Lemma then follows.
We then have the following lemma that is a nonlocal Korn inequality for the state-based model.
Lemma 29. Let the domain and coefficients in the bilinear form (4.16) satisfy the 109
Assumptions 1-4. Then, there exists a constant c1 > 0 such that for u in U0(Ω), provided that Ωc has a non-empty interior, we have
u c1 u 2 . (4.27) ||| ||| ≥ k kL (Ω)
An inequality similar to the above also holds for u U(Ω) . ∈ \Z
Proof. We show K1(x, y) and C0(x, y) are square integrable which in turn implies
K1(x, y) + C0(x, y) is square integrable. First, Z Z Z Z 2 2 2 2 K1(x, y) dy dx = 4η (x) ω (x, y) dy dx 4η1 Ω M < . (4.28) Ω Ω | | Ω Ω ≤ | | ∞
Similarly, Z Z 2 C0(x, y) dx dy Ω Ω| | Z Z Z
Ω c(z)ω(x, z)ω(z, y) x z z y ≤ | | Ω Ω Ω | − || − | + c(y)ω(x, y)ω(y, z) x y y z | − || − | 2 + c(x)ω(x, z)ω(x, y) x z x y dz dy dx Z Z Z | − || − | 2 3 Ω c(z)ω(x, z)ω(z, y) x z z y dz dy dx ≤ | | Ω Ω Ω | − || − | Z Z Z 2 + c(y)ω(x, y)ω(y, z) x y y z dz dy dx Ω Ω Ω | − || − | Z Z Z 2 + c(x)ω(x, z)ω(x, y) x z x y dz dy dx . Ω Ω Ω | − || − |
By a change of variables, we have Z Z 2 C0(x, y) dx dy Ω Ω| | Z Z Z 2 9 Ω c(x)ω(x, z)ω(x, y) x z x y dz dy dx ≤ | | Ω Ω Ω | − || − | Z Z 2 = 9 Ω c(x) 2 ω2(x, z) x z 2 dz dx | | Ω | | Ω | − | Z 9 diam(Ω) 4M 2 Ω c(x) 2 dx ≤ | | | | Ω | | 110 where diam(Ω) denotes the diameter of the domain Ω. The final inequality holds | | by the integrability condition on ω(y, x). By the inequality (4.25b), we can see that Z Z 2 C0(x, y) dx dy < . Ω Ω | | ∞
For simplicity, we separate the PD operator , as the following L Z Z u = C(x, y)u(y) dy + C(x, y) dyu(x) = Au + Bu. −L − Ω Ω
We can then invoke the Hilbert-Schmidt theory to conclude that the operator A has a sequence of real eigenvalues λ such that λ is monotonically non-increasing i | i| and λ 0 as i N. Then 0 is the only accumulation point of the spectrum of i → → A. As we have verified that the PD operator = A + B is non-negative, and −L under the Assumptions 1-4, , the kernel of only contains the zero element in ||| · ||| U0(Ω) or U(Ω) , we have = A + B > 0. Since operator B P0 > 0 and 0 is \Z −L ≥ the only accumulation point of A, we conclude that the smallest eigenvalue of the state-based peridynamic operator under study, which becomes a self-adjoint and compact operator in L2(Ω), is strictly positive. This gives (4.27).
We can also prove the upper bound of the energy norm. Lemma 30. Under the condition of Lemma 29, we have
u c2 u 2 , (4.29) ||| ||| ≤ k kL (Ω) where c2 is a constant. Proof. We prove the two relations Z Z 2 η(x)ω(x, y)Tr( t∗u)Tr( t∗u) dy dx c u L2 (4.30a) Ω Ω D D ≤ k k and
Z 2 k(x) n(x)η(x)/d Tr( t,ω∗ u)Tr( t,ω∗ u) dx c u L2 , (4.30b) Ω − D D ≤ k k 111 wheret c is a generic constant. First, Z Z η(x)ω(x, y)Tr( t∗u)Tr( t∗u) dy dx Ω Ω D D Z Z 2 2 η1 ω(x, y) u(y) u(x) α(x, y) dx dy ≤ Ω Ω | − | | | Z Z 2 2 2η1 ω(x, y) u(y) + u(x) dx dy ≤ Ω Ω | | | | Z Z Z Z 2 2 = 2η1 ω(x, y) u(y) dx dy + 2η1 ω(x, y) u(x) dy dx Ω Ω | | Ω Ω | | Z Z Z Z 2 2 = 2η1 u(y) ω(x, y) dx dy + 2η1 u(x) ω(x, y) dy dx Ω | | Ω Ω | | Ω 1/2 1/2 2 4η1 Ω M u 2 . ≤ | | k kL (Ω)
The final inequality holds because Z Z 1/2 ω(x, y) dy < Ω 1/2 ω2(x, y) dy < Ω 1/2M 1/2. Ω | | Ω | |
By the fact that k(x) η(x)n(x)/d k1 + η1π1(Ω)/d, we then have | − | ≤ Z
k(x) n(x)η(x)/d Tr( t,ω∗ u)Tr( t,ω∗ u) dx Ω − D D Z Z 2 = k(x) n(x)η(x)/d (u(y) u(x)) α(x, y)ω(x, y) y x dy dx Ω − Ω − · | − | Z Z Z 2 2 2 k1 + η1π1(Ω)/d u(y) u(x) dy ω (x, y) y x dy dx ≤ Ω Ω | − | Ω | − | Z Z Z 2 2 2 2 2 k1 + η1π1(Ω)/d ( u(x) + u(y) ) dy ω (x, y) y x dy dx ≤ Ω Ω | | | | Ω | − | Z Z 2 2 2 2 diam(Ω) M(k1 + η1π1(Ω)/d) u(x) + u(y) dy dx ≤ | | Ω Ω | | | | 2 2 4 Ω diam(Ω) M k1 + η1d1(Ω)/d u 2 . ≤ | || | k kL (Ω)
Combining the two inequalities, the result of the lemma follows.
Thus, the energy space for the homogeneous and anisotropic state-based peri- dynamic model is a Hilbert Space, in fact, is L2(Ω). This in turn implies that the model equation for the state-based peridynamic material is well-posed. As a conclusion, we have the following theorem. 112
Theorem 9. The Dirichlet volume-constrained problem for the state-based peridy- namic model (4.22) is well-posed provided that the assumptions 1-4 are satisfied. Moreover, the energy space, in this case, is equivalent to L2(Ω).
Similar results hold for the “Neumann” volume-constraint problem of the state- based peridynamic model (4.23). We state the results and omit the derivations.
Theorem 10. If Assumptions 1-4 are satisfied, the Neumann volume-constrained problem for the state-based peridynamic model (4.23) is well-posed in U(Ω) and \Z the energy space is equivalent to L2(Ω).
Remark 6. In many practical applications of peridynamic models, ω = ω(x, y) has compact support for x y B (0) which is a special case of the above general − ∈ δ discussion. In this case, the integrability condition in the Assumption 4 becomes Z ω2(x, y) dy < M. Bδ(x)
This is only one of the sufficient conditions to assure the well-posedness. It is commonly believed that a weaker condition requiring only the L1 integrability of ω would suffice for the well-posedness of solutions in L2(Ω), as shown in [28] for a system of bond-based models with special constraints. Yet, the stronger conditions used in the theorems here allow the immediate application of the standard Hilbert- Schmidt theory. In addition, as studied in [28], for a special bond-based peridynamic system and more recently for scalar nonlocal equations in [38], if we change the integrability condition on ω(x, y), one may also derive a more regular energy space U(Ω), say, that equivalent to a fractional Sobolev space.
4.6 Concluding remarks
In this study, we apply the nonlocal vector calculus developed in [44] to peridy- namic models. We express the constitutive relations of the peridynamic models in terms of nonlocal calculus operators. We rewrite the peridynamic equation in terms of a variational principle using the nonlocal calculus operators and prove that the state-based peridynamic models is well-posed in their intrinsic energy 113 norm. The whole theory can be directly extended to the bond-based peridynamic models, see [50] for detail. Chapter 5
Convergence of the PD models to the classical elasticity theory
5.1 Introduction
In this chapter, we establish the relation between the PD models and the classical elasticity theory as the PD horizon parameter goes to zero. It is explained in [51] how the general state-based PD material model converges to the continuum elasticity model as the ratio of the PD horizon to effective length scale decreases, assuming that the underlying deformation is sufficiently smooth. And the isotropic homogeneous PD model was proved to converge to the classical Navier equation with Poisson ratio one-quarter in [16]. Here, we examine the relation between the three kinds of PD models, i.e. the bond-based, ordinary state-based PD models and the classical elasticity theory in detail. The correspondence between the 4th order stiffness tensor in the classical elasticity and the nonlocal parameters is drawn. It is shown that the bond-based anisotropic PD material converge to the classical material model with 15 independent parameters; the ordinary anisotropic state- based model is a little more. This chapter is organized as following. We first reformulate the PD models by the nonlocal operators in another way which mimics the expressions of the classical strain and stress in section 5.2. In section 5.3, we discuss the convergence of the bond-based PD model including the anisotropic and isotropic equations. 115
The similar issue for the ordinary state-based PD models is discussed in section 5.4.
5.2 Model reformulation
Using the nonlocal tensor divergence and nonlocal tensor weighted divergence Dt as well as their adjoint operators defined in chapter 3, we express the PD Dt,ω strain energy density in another which mimics the form of the classical elasticity model. Z 1 T T W (u, x) = t∗(u) + t∗(u) : Tb : t∗(u) + t∗(u) 8 Ω D D D D 1 T T + t,ω∗ (u) + t,ω∗ (u) : Ts : t,ω∗ (u) + t,ω∗ (u) (5.1) 8 D D D D
It can be reduced to the existing PD models, such as the ordinary bond-based and state-based model, by specifying α(x, y) and ω(x, y) which are contained in the nonlocal operators and fourth order tensors Tb and Ts in the energy density function. In the following discussion, we let ω(x, y) = ω(x y) and let the material − domain Ω = R3 for simplicity. Moreover, we need the following assumption.
Assumption 5. The influence function ω(y) is a even function, i.e. ω(y) = Z 2 ω( y) and τδ = x ω(x) dx converges to a nonzero constant as δ 0. − Bδ | | → Remark 7. This assumption guarantees the validness of the local limit of the anisotropic PD operators. Since by Fourier transformation, the bond-based PD operator is written as
Z ω(y) iξ y Lcu = 1 e · y y dyu y 2 − ⊗ Bδ(0) | | Z ω(y) = 1 cos(ξ y) i sin(ξ y)y y dyu y 2 − · − · ⊗ Bδ(0) | | If the ω(y) is an even function, so as ω(y)/ y 2, the term with the odd function | | i sin(ξ y) can be ruled out. And if τ does not converge to zero,when we take · δ δ 0, only the second order derivatives exist. → 116
5.3 Convergence of the bond-based PD model to the classical elasticity
In this section, we let (y x)/ y x if y x < δ, α(x, y) = − | − | | − | (5.2) 0 otherwise, where δ denotes the material horizon, notably, the maximum distance at which two particles can interact.
Furthermore, we set Ts to be a zero tensor, and ω(x, y) if i = j and k = l, Tb,ijkl = 0 otherwise, where ω(x, y) is the kernel function of the PD model that carries the material properties. Then we can derive the bond-based PD model,see [1], from (5.1), Z 1 T T Wb(u, x) = t∗(u) + t∗(u) : Tb : t∗(u) + t∗(u) 8 Ω D D D D Z 1 2 = ω(x, y) Tr( t∗u) dy 2 Ω D 1 Z ω(x, y) 2 = (u(y) u(x)) (y x) dy 2 y x 2 − · − Ω | − | whose operator in the strong form is
1 T Lbu = t Tb : t∗(u) + t∗(u) − D 2 D D
T = ω(x, y)( ∗u) (5.3) Dt Dt Z ω(x, y) = (y x) (y x) u(y) u(x) dy. y x 2 − ⊗ − − Ω | − | Comparing (5.3) with classical elasticity operator,
u + ( u)T Eu = C: ∇ ∇ (5.4) − ∇ · 2 117
1 T we can treat =: ∗(u)+ ∗(u) as the bond-based nonlocal strain tensor, N 2 Dt Dt 1 T and σN =: Tb : N = Tb : t∗(u) + t∗(u) as the bond-based nonlocal stress 2 D D tensor. The nonlocal divergence of the nonlocal stress tensor (σ ) gives the Dt N total internal force acting on a point x.
We apply Fourier transform to find the limit operator of the anisotropic bond- based PD model when δ 0. → Theorem 11. Assume the displacement field u is smooth enough and the influence function of the PD operator ω(y x) satisfies assumption 5. As δ 0, the ordinary − → bond-based PD operator (5.3) converges to the classical elasticity operator (5.4) with the Voigt expression of the 4th order stiffness tensor given by
A11 A12 A13 B13 B11 B12 A12 A22 A23 B23 B21 B22 A A A B B B 1 13 23 33 33 31 32 [C] = (5.5) 2 B13 B23 B33 A12 B22 B11 B11 B21 B31 B22 A23 B33 B12 B22 B32 B11 B33 A13
Proof. From the discussion in Remark 7, the Fourier symbol of the ordinary bond- based PD operator L under consideration is
Z ω(y) M (ξ) = 1 cos(ξ y)y y dy δ y 2 − · ⊗ Bδ(0) | | 1 Z ω(y) = (ξ y)2 y y dy + (δ) 2 y 2 · ⊗ Bδ(0) | | 2 Z y1 y1y2 y1y3 3 1 ω(y) 2 X 2 2 X = 2 y2y1 y2 y2y3 ξi yi + ξiξjyiyj + (δ) 2 (0) y Bδ | | 2 i=1 i=j y3y1 y3y2 y3 6
where 1 Z (ξ y)4 cos(θ) (δ) = ω(y) · y ydy for some θ, 24 y 2 ⊗ Bδ(0) | | 118 which converges to 0 when δ goes to 0. And for notation simplicity, we define the following parameters. Z Z ω(y) 4 ω(y) 4 A11 = lim y1 dy A22 = lim y2 dy δ 0 y 2 δ 0 y 2 → Bδ(0) | | → Bδ(0) | | Z Z ω(y) 4 ω(y) 2 2 A33 = lim y3 dy A12 = lim y1y2 dy (5.6) δ 0 y 2 δ 0 y 2 → Bδ(0) | | → Bδ(0) | | Z Z ω(y) 2 2 ω(y) 2 2 A13 = lim y1y3 dy A23 = lim y2y3 dy δ 0 y 2 δ 0 y 2 → Bδ(0) | | → Bδ(0) | | and Z Z ω(y) 3 ω(y) 3 B13 = lim y1y2 dy B23 = lim y1y2 dy δ 0 y 2 δ 0 y 2 → Bδ(0) | | → Bδ(0) | | Z Z ω(y) 3 ω(y) 3 B12 = lim y1y3 dy B32 = lim y1y3 dy δ 0 y 2 δ 0 y 2 → Bδ(0) | | → Bδ(0) | | Z Z ω(y) 3 ω(y) 3 B21 = lim y2y3 dy B31 = lim y2y3 dy (5.7) δ 0 y 2 δ 0 y 2 → Bδ(0) | | → Bδ(0) | | Z Z ω(y) 2 ω(y) 2 B11 = lim y1y2y3 dy B22 = lim y1y2y3 dy δ 0 y 2 δ 0 y 2 → Bδ(0) | | → Bδ(0) | | Z ω(y) 2 B33 = lim y1y2y3 dy δ 0 y 2 → Bδ(0) | |
Then we let δ 0, the Fourier symbol of lim Mδ(ξ) is expressed as the following. δ 0 → →
The first column of lim Mδ(ξ) is δ 0 →
2 2 2 A11ξ1 + A12ξ2 + A13ξ3 + 2B13ξ1ξ2 + 2B12ξ1ξ3 + 2B11ξ2ξ3 1 2 2 2 B13ξ + B23ξ + B33ξ + 2A12ξ1ξ2 + 2B11ξ1ξ3 + 2B22ξ2ξ3 . 2 1 2 3 2 2 2 B12ξ1 + B22ξ2 + B32ξ3 + 2B11ξ1ξ2 + 2A13ξ1ξ3 + 2B33ξ2ξ3 119
The second column of lim Mδ(ξ) is δ 0 →
2 2 2 B13ξ1 + B23ξ2 + B33ξ3 + 2A12ξ1ξ2 + 2B11ξ1ξ3 + 2B22ξ2ξ3 1 2 2 2 A12ξ + A22ξ + A23ξ + 2B23ξ1ξ2 + 2B22ξ1ξ3 + 2B21ξ2ξ3 . 2 1 2 3 2 2 2 B11ξ1 + B21ξ2 + B31ξ3 + 2B22ξ1ξ2 + 2B33ξ1ξ3 + 2A23ξ2ξ3
The third column of lim Mδ(ξ) is δ 0 →
2 2 2 B12ξ1 + B22ξ2 + B32ξ3 + 2B11ξ1ξ2 + 2A13ξ1ξ3 + 2B33ξ2ξ3 1 2 2 2 B11ξ + B21ξ + B31ξ + 2B22ξ1ξ2 + 2B33ξ1ξ3 + 2A23ξ2ξ3 . 2 1 2 3 2 2 2 A13ξ1 + A23ξ2 + A33ξ3 + 2B33ξ1ξ2 + 2B32ξ1ξ3 + 2B31ξ2ξ3
By directly calculation, we can also find the Fourier symbol, CE(ξ), of the classical elasticity operator, (5.4) too.
The first column of CE(ξ) is
2 2 2 2C1111ξ1 + 2C1212ξ2 + 2C1313ξ3 + 2(C1112 + C1211)ξ1ξ2 + 2(C1113 + C1311)ξ1ξ3 + 2(C1213 + C1312)ξ2ξ3 1 2 2 3 2C1211ξ + 2C2212ξ + 2C2313ξ + 2(C1212 + C2211)ξ1ξ2 + 2(C1213 + C2311)ξ1ξ3 + 2(C2213 + C2312)ξ2ξ3 . 2 1 2 3 2 2 2 2C1311ξ1 + 2C2312ξ2 + 2C3313ξ3 + 2(C1312 + C2311)ξ1ξ2 + 2(C1313 + C3311)ξ1ξ3 + 2(C2313 + C3312)ξ2ξ3
The second column of CE(ξ) is
2 2 2 2C1112ξ1 + 2C1222ξ2 + 2C1323ξ3 + 2(C1122 + C1212)ξ1ξ2 + 2(C1123 + C1312)ξ1ξ3 + 2(C1223 + C1322)ξ2ξ3 1 2 2 2 2C1212ξ + 2C2222ξ + 2C2323ξ + 2(C1222 + C2212)ξ1ξ2 + 2(C1223 + C2312)ξ1ξ3 + 2(C2223 + C2322)ξ2ξ3 . 2 1 2 3 2 2 2 2C1312ξ1 + 2C2322ξ2 + 2C3323ξ3 + 2(C1322 + C2312)ξ1ξ2 + 2(C1323 + C3312)ξ1ξ3 + 2(C2323 + C3322)ξ2ξ3
The third column of CE(ξ) is
2 2 2 2C1113ξ1 + 2C1223ξ2 + 2C1333ξ3 + 2(C1123 + C1213)ξ1ξ2 + 2(C1133 + C1313)ξ1ξ3 + 2(C1233 + C1323)ξ2ξ3 1 2 2 3 2C1213ξ + 2C2223ξ + 2C2333ξ + 2(C1223 + C2213)ξ1ξ2 + 2(C1233 + C2313)ξ1ξ3 + 2(C2233 + C2323)ξ2ξ3 . 2 1 2 3 2 2 2 2C1313ξ1 + 2C2323ξ2 + 2C3333ξ3 + 2(C1323 + C2313)ξ1ξ2 + 2(C1333 + C3313)ξ1ξ3 + 2(C2333 + C3323)ξ2ξ3
Then one can see that the Fourier symbol of the ordinary bond-based PD model converges to the Fourier symbol of the classical elasticity equation with 4th order 120 stiffness tensor A11 B13 B12 B13 A12 B11 B12 B11 A13 B13 A12 B11 A12 B23 B22 B11 B22 B33 B12 B11 A13 B11 B22 B33 A13 B33 B32 B13 A12 B11 A12 B23 B22 B11 B22 B33 1 C = A12 B23 B22 B23 A22 B21 B22 B21 A23 2 B11 B22 B33 B22 B21 A23 B33 A23 B31 B12 B11 A13 B11 B22 B33 A13 B33 B32 B11 B22 B33 B22 B21 A23 B33 A23 B31 A13 B33 B32 B33 A23 B31 B32 B31 A33 whose expression under Voigt notation can be found as in (5.5)
From the theorem, one can see that the anisotropic bond-based PD material corresponds to the classical material with 15 independent parameters. The param- eter Aij is the coefficient of the coupling of elongations in the i- and j-directions.
Bij is the coefficient of the coupling of the elongation in the i-direction and the shear in the plane normal to j direction. Together from the Voigt expression of the stiffness tensor, thus, the 6 missing degrees of freedom come from the coupling of shears. However, the coupling of the shear deformations are, indeed, considered in the bond-based PD model, but they share the parameters with the coupling of elongations, say Aij, and coupling of elongation and shear, say Bij. To make it clear, A12, A23 and A13, at the same time, are the coefficients of shears in the planes normal to 3-direction, 1-direction and 2-direction respectively. B22 is the coefficient of the coupling of shears in the plane normal to 3-direction and in the plane normal to 1-direction. B33 is the coefficient of the coupling of the shears in the plane normal to 1-direction and in the plane normal to 2-direction. And B11 is the one of the coupling of the shears in the plane normal to 2-direction and in the plane normal to 3-direction.
If we set ω(y x) = ω( y x ), B = 0 for i, j = 1, 2, 3 and A11 = A22 = − | − | ij A33 = 3A12 = 3A13 = 3A23. Let A12 = λ. Then, the homogeneous and isotropic bond-based PD model converges to the classical elasticity model with the Voigt 121 expression of the 4th order stiffness tensor
3λ λ λ 0 0 0 λ 3λ λ 0 0 0 λ λ 3λ 0 0 0 1 [C] = (5.8) 2 0 0 0 λ 0 0 0 0 0 0 λ 0 0 0 0 0 0 λ i.e. the classical homogeneous, isotropic elasticity model with Poisson 1/4.
5.4 The convergence of the ordinary state-based PD equation to the classical elasticity
In this section, we discuss the convergence of the state-based PD model as the horizon δ goes to 0. To obtain the ordinary state-based PD model, defined in section 1.2.2, we let
α(x, y) be defined by (5.2) and make the following settings of Tb and Ts. η ω(x, y) if i = j and k = l Tb,ijkl = (5.9a) 0 otherwise.
k η n(x)/3 if i = j and k = l Ts,ijkl = − (5.9b) 0 otherwise. where k and η are positive constants and n(x) is the normalization function defined by 1 Z n(x) = y x 2ω(x y) dy. 3 Bδ(x) | − | − And function ω(x, y) that is contained in the equation (3.30) and (3.31) is defined as the following ω(x y) = y x ω(x y)/n(x). − | − | − Plugging the definitions above into the strain energy density (5.1), we obtain the 122 following Z 1 T T Ws(u, x) = t∗(u) + t∗(u) : Tb : t∗(u) + t∗(u) 8 Ω D D D D 1 T T + t,ω∗ (u) + t,ω∗ (u) : Ts : t,ω∗ (u) + t,ω∗ (u) 8 D D D D Z 1 2 1 2 = k η n(x)/3 Tr( t,ω∗ u) + ηω(x y) Tr( t∗u) dy 2 − D 2 Ω − D Z k 2 η 2 = Tr( t,ω∗ u) + ω(x y) Tr( t∗u) Tr( t,ω∗ u) y x /d dy. 2 D 2 Ω − D − D | − |
By the definition of the nonlocal tensor divergence operators and the constitutive relation discussed in section 4.3 and section 1.2.2, we can readily see that
k 2 η d d Ws(u, x) = θb + (ωe ) e , 2 2 • which is strain energy density of the ordinary state-based PD equation defined in section 1.2.2.
We apply the variational principle and green’s identities to obtain the strong form operator of the ordinary state-based PD model
1 T 1 T Lsu = t Tb : t∗(u) + t∗(u) + t,ω Ts : t,ω∗ (u) + t,ω∗ (u) , − D 2 D D D 2 D D or
T L u = ηω(x y) ∗(u) + k n(x)η/3 Tr ∗ (u) I . (5.10) − s Dt − Dt Dt,w − Dt,ω
To discuss the convergence of the state-based PD operator, we also set Ω = R3. As a consequence, the normalization function n(x) becomes a constant. Without loss of generality, in the following context, we set n(x) = 1. Then we have the following theorem.
Theorem 12. Assume the displacement field u is smooth enough and the influence function ω(x y) satisfies the assumption 5. As δ 0, the ordinary state-based − → PD operator (5.10) converges to the classical elasticity operator (5.4) with the 4th 123 order stiffness tensor under Voigt notation
A11 A12 A13 B13 B11 B12 A12 A22 A23 B23 B21 B22 A A A B B B η 13 23 33 33 31 32 [C] = 2 B13 B23 B33 A12 B22 B11 B11 B21 B31 B22 A23 B33 B12 B22 B32 B11 B33 A13
2 P1 P1P2/2 P1P3/2 0 0 0 2 P1P2/2 P2 P2P3/2 0 0 0 P P /2 P P /2 P 2 0 0 0 1 3 2 3 3 + (k η/3) (5.11) − 0 0 0 P1P2/2 0 0 0 0 0 0 P2P3/2 0 0 0 0 0 0 P1P3/2 where P1 + P2 + P3 = 3n(x) = 3.
T Proof. The limit of the first part, ηω(x y) ∗(u) , is the same as the one Dt − Dt in theorem 11. We only focus on the second part of the state-based PD operator, Tr ∗ (u) I . Dt,w Dt,ω
Because ω(x y) satisfies the assumption 5, the discussion in section 3.5.2.2 is − still valid. By equation (5.10a), (5.10b), Lemma 5.1 and proof in Proposition 5.6 of [44], we have
Z \ dj∗u = u(x + z) u(x) zj ω(z) dz − Bδ(0) − Z \ = u(x + z) zj ω(z) dz − Bδ(0) i P ξ u (5.12) → − j j b Z \ dcju = u(x + z) + u(x) zj ω(z) dz Bδ(0) Z = i sin(y ξ)ub(ξ) yiω(y) dy Bδ(0) · 124
i P ξ u (5.13) → j j b where i is the imaginary unit, and Z 2 Pj = lim yj ω(y) dy, for j = 1, 2, 3 and P1 + P2 + P3 = 3n(x) = 3 δ 0 → Bδ(0) By definition of Tr ∗ (u) , we have Dt,ω Z 3 X Tr ∗ (u) = u (x + z) u (x) z ω(z) dz Dt,ω i − i i Bδ(0) i=1
Thus, by (5.12),
∂u1 ∂u2 ∂u3 Tr t,ω∗ (u) P1 P2 P3 . D → − ∂x1 − ∂x2 − ∂x3
And by definition of (U), Dt,ω Z t,ω(U) = ω(y x)U(x) + ω(x y)U(y) (y x) dy D Bδ(x) − − − because of the order property of ω(y x), we have − Z Z ω(y x)U(x)(y x) dy = U(x) ω(z)z dz = 0, Bδ(x) − − Bδ(0) and Z Z t,ω(U) = ω(x y)U(y)(y x) dy = ω(z)U(x z)( z) dz D Bδ(x) − − Bδ(0) − −
Using Fourier transformation, Z iy ξ \t,ω(U) = e− · Ub (ξ)( y) ω(y) dy D Bδ(0) − Z = i sin(y ξ) Ub (ξ) y ω(y) dy Bδ(0) · 125
By equation (5.13), we conclude that
∂ ∂ ∂ T t,ω(U) (P1 ,P2 ,P3 )U (x) as δ 0. D → ∂x1 ∂x2 ∂x3 → So the limit of the Fourier symbol of Tr ∗ (u) I is −Dt,w Dt,ω
2 2 P1 ξ1 P1P2ξ1ξ2 P1P3ξ1ξ3 2 2 P1P2ξ1ξ2 P ξ P2P3ξ2ξ3 2 2 2 2 P1P3ξ1ξ3 P2P3ξ2ξ3 P3 ξ3
Then combining with theorem 11, we can get the 4th order stiffness tensor of the limiting operator of the ordinary state-based PD operator.
A11 B13 B12 B13 A12 B11 B12 B11 A13 B13 A12 B11 A12 B23 B22 B11 B22 B33 B12 B11 A13 B11 B22 B33 A13 B33 B32 B13 A12 B11 A12 B23 B22 B11 B22 B33 η C = A12 B23 B22 B23 A22 B21 B22 B21 A23 2 B11 B22 B33 B22 B21 A23 B33 A23 B31 B B A B B B A B B 12 11 13 11 22 33 13 33 32 B11 B22 B33 B22 B21 A23 B33 A23 B31 A13 B33 B32 B33 A23 B31 B32 B31 A33 2 P1 0 0 0 P1P2/2 0 0 0 P1P3/2 0 P1P2/2 0 P1P2/2 0 0 0 0 0 0 0 P1P3/2 0 0 0 P1P3/2 0 0 0 P1P2/2 0 P1P2/2 0 0 0 0 0 2 + (k η/3) P1P2/2 0 0 0 P 0 0 0 P2P3/2 − 2 0 0 0 0 0 P2P3/2 0 P2P3/2 0 0 0 P P /2 0 0 0 P P /2 0 0 1 3 1 3 0 0 0 0 0 P2P3/2 0 P2P3/2 0 2 P1P3/2 0 0 0 P2P3/2 0 0 0 P3 whose expression under Voigt notation is given by (5.11).
Thus the ordinary state-based PD model adds some more freedoms to the bond-based equation, but still does not represent the full elasticity. From the Voigt notation of the stiffness tensor, we can see that the freedom being added is contributing to the coupling of elongations, and so as the coupling of shears in certain directions of the material body because they share the same coefficients as discussion in last section.
If the isotropic body is considered, i.e. ω(y x) = ω( y x ), P1 = P2 = P3 = − | − | 126
n(x) = 1 and A11 = A22 = A33 = 3A12 = 3A13 = 3A23. Then the model converges to an isotropic elasticity material model with 4th order stiffness tensor under Voigt notation λ + 2µ λ λ 0 0 0 λ 3λ + 2µ λ 0 0 0 λ λ 3λ + 2µ 0 0 0 [C] = (5.14) 0 0 0 λ 0 0 0 0 0 0 λ 0 0 0 0 0 0 λ where λ = (ηA12 + k η/3)/2 and µ = (ηA12 + (k η/3)/2)/2. So the model − − represents the material with Poisson ratio
λ 3ηA12 + 3k η ν = = − , 2(λ + µ) 12ηA12 + 9k 3η − which could be any reasonable value by adjusting k and η.
5.5 Conclusion
In this chapter, we show the local limits of the ordinary bond-based, ordinary state- based PD models. And the clear relation between the nonlocal quantities and the entries in the 4th order stiffness tensor in the classical elasticity is established, which provide the guidance for the modeling of the PD material by the classical elasticity theory. Chapter 6
The Numerical Analysis of the bond-based PD model
6.1 Introduction
In this chapter, we consider various finite dimensional approximations and the a posterior error estimate of the finite element approximation the nonlocal bond- based PD models. A number of theorems are given on the convergence and error estimates of these approximations (see section 6.2.1 for detail), which appear to be the first of their kind in the literature. Estimates of the condition numbers of the resulting linear systems of equations are also provided here. Our findings are established in very general settings, and they are consistent with the results derived or observed for various specialized cases in earlier studies [5, 6, 7, 8]. Moreover, motivated by the need of adaptive computation for large scale simula- tions based on the nonlocal models with possible singular solutions, we also develop the a posteriori error estimators for the nonlocal model equations to provide more theoretical guidance on the design of automatic error control and adaptive numer- ical schemes, see section 6.3. There have been some works on adaptive methods and a posteriori analysis for integral equations, see [52, 53, 54]. While adaptive re- finement and coarsening method for PD model have been studied in [55, 6, 56, 57], there has not been any serious attempt to derive rigorous a posteriori error esti- mators. On the other hand, although PD models are integral equations, given the 128 variational formulations presented in [44] based on the framework of nonlocal cal- culus and nonlocal balance laws, there is a close resemblance between the nonlocal PD models and the classical differential equations, which can be utilized to de- velop a posteriori error estimates for PD models. The latter is the main objectives of the current work. We provide a rigorous a posteriori error estimator which is shown to be both reliable and efficient. Moreover, we pay close attention to not only the connections but also the key differences between the a posterior analysis of local and nonlocal models. Dependence of the error estimators on the model parameters is also considered. Some preliminary numerical experiments provide further substantiation of our theoretical analysis.
6.2 Numerical Analysis of the Bond-based PD model on 1-D bar
The model we consider in this section is the bond-based stationary PD model
ou(x) = b(x) x (0, π) (6.1) −Lδ ∈ with the periodic boundary condition,
u is odd in ( δ, δ) and (π δ, π + δ) . (6.2) − − where Z x+δ o δu = ω( x y ) u(y) u(x) dy −L − x δ | − | − − .
6.2.1 Finite-dimensional approximations
In practice, finite difference, finite element, quadrature, and particle-based meth- ods have all been used to solve PD models [5, 6, 7, 58, 59, 27, 8]. Given the nonlocal nature, in all of these approximations, the discrete schemes are also nonlocal. The functional setting presented in section 2.3.2 can be readily used to analyze the con- vergence of finite-dimensional approximations and numerical solutions. We now 129 present a few illustrative examples. First, we consider the problem (6.1). Let V be finite-dimensional subspaces { n} of M o, and assume that as n , V is dense in M o; that is, for any u M o, ω → ∞ { n} ω ∈ ω there exists a sequence of u V such that n ∈ n
u u o 0 as n . (6.3) k − nkMω → → ∞
We seek approximations to the PD model in Vn, which fall into the so-called internal or conforming approximations. Examples include the space formed by the first n Fourier sine modes or the space formed by continuous piecewise finite element polynomials, satisfying (6.2), on a mesh having n grid points and the mesh parameter h 0 as n . In cases where the influence function satisfies the n → → ∞ s o suitable conditions so that Ho = Mω for some s < 1/2, we may also take the space of odd piecewise polynomials which are not required to be continuous across grid points.
We let un be Galerkin–Ritz approximation to the problem (6.1); that is,
1 2 min E(v ) = min v o (v , f) 2 . (6.4) n n Mω n L vn Vn vn Vn ∈ ∈ 2k k −
It follows that un is in fact the best approximation of u in the subspace Vn measured o by the norm of Mω. Thus, we easily get the following theorem.
o Theorem 13. If ω = ω( y ) satisfies (2.3), then for any f M − , we have | | ∈ ω
o o u un Mω min u vn Mω 0 as n . (6.5) vn Vn k − k ≤ ∈ k − k → → ∞ Given appropriate regularity of the solution u, one could also use the best ap- proximation property to derive error estimates for various different approximation methods. For example, for the nonlocal BVPs considered here, Fourier spectral methods are quite natural and can be easily implemented. In this regard, we have what follows.
Theorem 14. Let ω = ω( y ) satisfy (2.3) and (2.56) with β < 1. With f Hm | | ∈ o for some m min 0, 2β and V is the subspace spanned by the first n Fourier ≥ { − } n 130 sine modes, then
δ 2 m 2β+γ u u C (β)− n− − f as n (6.6) k − nkγ ≤ 2 k km → ∞ for any γ min m + 2β, m . A similar result holds for (2.61). ≤ { } Proof. Under the assumptions on ω, we have from (2.57) that
η (k) Cδ(β)2k2β k 1 . δ ≥ 2 ∀ ≥
We also have the identity
( o )1/2 ( o )1/2 X k2γ X 1 k4β u u = f 2 = (kmf )2 . k − nkα η2(k) k k2m+4β 2γ η2(k) k k=n+1 δ k=n+1 − δ
The error estimates then follow by applying the regularity on f to get
( o )1/2 1 m+γ 2β X m 2 δ 2 m+γ 2β u u n− − (k f ) C (β)− n− − f . k − nkγ ≤ Cδ(β)2 k ≤ 2 k km 2 k=n+1
The condition m min 0, 2β is stated in the above theorem only for assur- ≥ { − } ing that the solution to the PD model has enough regularity. We note in addition that if ω = ω( y ) satisfy (2.52) and (2.56) with α = β [0, 1), then by the | | ∈ definitions of ωδ(β, δ), τδ, we have
1 Z δ Z δ − ωδ(β, δ) 1 cos(y) 1 β γ1 − lim lim γ1 − 1+β dy y dy = , (6.7) δ 0 τδ ≥ δ 0 δ y δ | | 2 → → − | | − δ 2 so that C (β) = ω (β, δ)/γ2 (cγ1)/(2γ2) for δ small, if τ c as δ 0. Thus, 2 δ ≥ δ → → δ 2 the constant C (β)− is uniformly bounded as δ 0, provided that τ c. Thus, 2 → δ → the estimate (6.6) gives dependence on both the number of Fourier modes and the horizon parameter. Similarly, for finite element methods, we have what follows. Theorem 15. Let ω = ω( y ) satisfy (2.3) and additional conditions (2.52) and | | (2.56) with 0 β α (0, 1). Given a regular and quasi-uniform mesh having ≤ ≤ ∈ 131 the mesh parameter h 0, let V be made up either by continuous piecewise finite → n element polynomials that are of degree m 1 and odd in ( δ, δ) and (π δ, π + δ), ≥ − − or, for α < 1/2, Vn can also be made by piecewise polynomials that are of degree m 0, not necessarily continuous across grid points, and odd in ( δ, δ) and ≥ − m 2α (π δ, π + δ). Then for f H 0− with 0 m0 m + 1 and s [0, β], we have, − ∈ o ≤ ≤ ∈ as h 0, →
δ 1+s0 δ 3 s0 m0 α+(2β α)s0 u un s csC1 (α) C2 (β)− − h − − f m 2β (6.8) k − k ≤ k k 0− for s0 = 1 s/β and some generic constant c independent of h, δ, and f. − s Proof. For s = β, we first use the best approximation property and the norm equivalence to get that
δ 1 δ 1 o o u un β C2 (β)− u un Mω C2 (β)− min u vn Mω vn Vn k − k ≤ k − k ≤ ∈ k − k δ δ 1 C1 (α)C2 (β)− min u vn α . vn Vn ≤ ∈ k − k By the standard finite element approximation theory in the fractional Sobolev space norms, which are consequences of the approximation theory in the integer order Sobolev space norm based on space interpolation techniques (see a direct derivation in [60]), we have, for h small,
δ δ 1 u un β C1 (α)C2 (β)− min u vn α vn Vn k − k ≤ ∈ k − k δ δ 1 m α C (α)C (β)− ch 0− u ≤ 1 2 k km0 for some constant c, independent of h, δ, and u. Now using the regularity result in Lemma 10, we have that for m0 2β, ≥
δ 2 u m C2 (β)− f m 2β , k k 0 ≤ k k 0− which thus implies
δ δ 3 m0 α u un β cC1 (α)C2 (β)− h − f m 2β (6.9) k − k ≤ k k 0− for a generic constant c, independent of h, δ, and f. This gives (6.8) for s = β. 132
By the choice of m, we can always take m0 = 2β m+1 in the above derivation ≤ to get δ δ 2 2β α u u o cC (α)C (β)− h − f 0 . (6.10) k − nkMω ≤ 1 2 k k Next, we may invoke the duality argument to prove (6.8) for s = 0. For β = 0, we already have the estimate. So we consider only β > 0. Let w be the solution of
ow = u u , −Lδ − n and let wn be the finite element approximation of w in Vn; from (6.10), we have
δ δ 2 2β α w w o cC (α)C (β)− h − u u 0 . k − nkMω ≤ 1 2 k − nk
On the other hand,
δ δ 2 m0 α u un M o cC1 (α)C2 (β)− h − f m 2β . k − k ω ≤ k k 0−
So, we get that
2 δ 2 δ 4 m0+2β 2α u un 0 = (w wn, u un)M o cC1 (α) C2 (β)− h − u un 0 f m 2β k − k − − ω ≤ k − k k k 0− for a constant c, independent of h, δ, and f. This leads to (6.8) for s = 0. The general conclusion (6.8) follows from these two special cases of s = 0 or s = β based on the standard space interpolation techniques.
Note that for α (0, 0.5), all piecewise polynomial spaces, discontinuous or ∈ not, automatically become conforming elements for the internal discretization of the PD nonlocal BVPs considered here. In addition, for the case β = 0, we simply get from (6.9) that, for any 0 m0 m + 1, ≤ ≤
δ δ 3 m0 α u u 0 cC (α)C (0)− h − f , (6.11) k − nk ≤ 1 2 k km0 which gives sharper dependence on δ. Since α can be arbitrarily small, the estimate shows the almost optimal order error estimates, with respect to the mesh size h, 2 in L even for m = 0, that is, Vn being the space of piecewise constant functions. Due to the use of the various bounds on the different norms in the derivation, 133 the estimates in (6.8) provide only some conservative bounds that may only be tight in the worst-case scenerio. In fact, if the bound on the best approximation
u v o can be directly estimated instead of using the known results on the k − nkMω δ Sobolev norms, then we may avoid the explicit dependence on the parameter C1 (α). Yet, the above analysis does provide some upper bounds on how the error might be dependent on the mesh size h, horizon parameter δ, and the regularity of the 2 1 solution and data in the model. For example, with ω( y ) = cδ− y − , we can take | | | | 2 2α 2 β = 0, γ2 = cδ− , and for any α (0.5, 1), γ1 = δ − . Moreover, from (6.7), we ∈ δ have that C2 (0) is uniformly bounded from below. Direct calculation shows that Z δ 2 2α 2 ∞ 1 cos(z) 2α 2 C (α) = 2δ − − dz = O(δ − ) , 1 z 1+2α 0 | | δ α 1 so that C (α) is uniformly bounded from the above by cδ − as δ 0. Then the 1 → m α 1+α error bound for u u 0 becomes O(h 0− δ− ). In turn, this implies that one k − nk can expect the convergence of the finite element approximation when both h and δ m α 1+α approach to zero, as long as one keeps h 0− δ− 0. Naturally, in this case, the → numerical solutions actually converge to the solutions of the classical differential equations. We refer to [6, 7] for numerical results. Concerning the finite-dimensional approximations, an issue of interests is the condition number estimation for the resulting stiffness matrices associated with the finite element approximations; see [5, 6] for related discussions. The analytic framework provided in chapter 2 can again be readily applied. We give the follow- ing as an illustration.
Theorem 16. Let ω = ω( y ) satisfy (2.3) and additional conditions (2.52) and | | (2.56) for some 0 β α (0, 1). Given a regular and quasi-uniform mesh ≤ ≤ ∈ having the mesh parameter h 0, let V be as defined in the theorem 15 with a → n n o o set of nodal basis functions φ . Let A = (A ) = ((φ , φ ) o ) be the n n { j}j=1 ij i j Mω × stiffness matrix corresponding to the finite element approximation of (2.60) with
λ1 and λn being the smallest and largest eigenvalues. Then, for h small, we have
δ 2 δ 2 1 2α 0 < c˜1C (β) h λ1 λ c˜2C (α) h − (6.12) 2 ≤ ≤ n ≤ 1 134 and o δ δ 2 2α cond(A ) c(C (α)/C (β)) h− (6.13) ≤ 1 2 for some generic constant c, c˜1, and c˜2, independent of h and δ. A similar result holds for (2.61).
Proof. The result is a simple consequence of some space and norm equivalences and the inverse inequality. We first have, for the given finite element nodal basis, that there are some generic constants c1 c2 > 0 such that [61, 60] ≥ n 2 2 2 X c2h z u c1h z u = z φ V , | | ≤ k nk0 ≤ | | ∀ n j j ∈ n j=1
n where z R is the vector with components zj . Then, for α > 0, ∈ { }
2 2 2 c2h z u u | | ≤ k nk0 ≤ k nkβ δ 2 2 δ 2 T o C (β)− un o = C (β)− z A z ≤ 2 k kMω 2 δ δ 2 2 δ δ 2 2α 2 (C (α)/C (β)) u c(C (α)/C (β)) h− u ≤ 1 2 k nkα ≤ 1 2 k nk0 δ δ 2 1 2α 2 cc1(C (α)/C (β)) h − z ≤ 1 2 | | for a generic constant c, independent of h and δ. The eigenvalue and condition number estimates then follow immediately.
We pay special attention to the dependence of the condition number bound on 3 the value of δ. Take ω(y) = 3δ− /2 as an illustration (a case studied in [5]) which implies τδ = 1; (2.48) and a direct calculation then show that
3 Z δ 3 sin(kδ) ηδ(k) = 3 (1 cos(kx))dx = 2 1 , 2δ 0 − 2δ − kδ which satisfies, for small δ < 1,
3 2 2 c˜ 3δ− (δ sin(δ))/2 η (k) min 6δ− , τ k /2 k 1, (6.14) ≤ − ≤ δ ≤ { δ } ∀ ≥ wherec ˜ is a positive constant independent of δ. We thus can take, for this special ω = ω( y ), β = 0. Then, using (6.14) and the inverse inequality for finite element | | 135 functions, we get
2 2 2 2 2 2 2 2 c˜ un 2 un o min 3δ− un 2 , τδ un 1 /2 min 3δ− , ch− un 2 , k kL ≤ k kMω ≤ { k kL k kH } ≤ { }k kL so that o 2 2 cond(A ) c min δ− , h− , ≤ { } 2 which, in the limit h 0, is consistent with the bound cδ− established both → rigorously and computationally in [5] for ω( y ) = c with a finite δ, in which c | | δ δ is a constant only depending on δ. This property can also be easily interpreted from the fact that this particular influence function satisfies (2.49) so that the PD operator is bounded from L2 to L2 for any finite δ. Meanwhile, for δ 0, the → 2 above estimate also recovers the standard bound of ch− for the finite element stiffness matrices associated with second order elliptic equations. Theorem 16 gives a more general estimates which also covers other interesting 1 β cases. For example, if we take instead ω( y ) = y − − , for some β [0, 1), | | | | ∈ δ δ 2 by Theorem 16, we notice that the crucial quantity is (C1 (α)/C2 (β)) . Let us consider the case β = 0 in more detail which, as mentioned in the discussion of error estimate, corresponds to a choice of common interests. Following the δ estimates on C1 (α) given earlier, we have that, for any arbitrarily small α > 0,
o α 2α 2 2 cond(A ) c min h− δ − , h− . ≤ { }
This shows a very mild dependence on the mesh size for any finite δ. Furthermore, we note that this line of discussion may also have other implications. For example, for time-dependent PD models solved via explicit marching schemes, the time step constraint for the numerical stability may also be tied to the condition number estimate on Ao. We then expect that similar conclusions concerning the mild dependence of the time-step stability conditions on the spatial mesh size hold for the influence function with ω( y ) = c . This is indeed the case as illustrated in | | δ [27] where more detailed discussions can be found. 136
6.3 A posterior error estimate of bond-based PD equations
In this section, we develop and analyze the a posterior error estimator of the bond- based PD equation. And the relation between the a posterior error estimator and its local counterpart is established.
Let Ω be a bounded, open, connected spatial domain in Rd and
Ω = (Ω Ω ) (Ω¯ Ω¯ )o, (6.15) s ∪ c ∪ s ∩ c as seen in figure 4.1, where Ωs denotes the solution domain,Ωc denotes the con- straint domain such that Ω Ω = and (Ω¯ Ω¯ )o denotes the interior of the s ∩ c ∅ s ∩ c region Ω¯ Ω¯ . We assume that Ω has convex polygonal boundaries with Ω being s ∩ c s c a domain surrounding Ωs. The model we consider in this section is the bond-based PD model defined on the domain Ω, T t ω ( t∗u) (x) = f(x) for x Ωs D D ∈ (6.16) u(x) = 0 for x Ω . ∈ c where the influence function ω = ω( y x ) is defined as | − | ω( y x ) when y x < δ, ω(y, x) = | − | | − | 0 otherwise.
Then the bilinear form of the equation is given by Z Z B(u, v) = ω( y x )Tr( t∗u)Tr( t∗v) dxdy Ω Ω | − | D D and weak formulation of the nonlocal model is
2 B(u, v) = f, v) v Ln0(Ω), ∈ (6.17) u(x) = 0 x Ω , ∈ c 137 where L2 (Ω) := v L2(Ω ): v(x) = 0 for x Ω . n0 { ∈ s ∈ c}
We also define α(x, y) involved in the definition of the operators and ∗ as Dt Dt
α(x, y) = (y x)/ y x − | − |
and we define the kernel function for PD model with z = y x, − z z Λ(z) = (α αβ)(z) = ⊗ ω( z ). (6.18) ⊗ z 2 | | | | In this section, we assume that Z ω( z ) 0 , 0 < c(δ) ω( z ) dz M(δ) < , (6.19) | | ≥ ≤ Bδ(0) | | ≤ ∞ where c(δ) and M(δ) are two constants that depend on δ.
In [50], the well-posedness of linear state-based and bond-based PD models with solutions in L2(Ω) = (L2(Ω))d has been established for general symmetric square integrable kernel functions by showing the corresponding nonlocal bilinear 2 d form being continuous and coercive in the volume constrained space (Ln0(Ω)) . The coercivity is a particular consequence of spectral properties of the Hilbert- Schmidt operator and the precise characterization of the energy space. These results generalized similar properties proved under more special nonlocal boundary conditions in [28] with isotropic and homogeneous L1 kernel functions. For the constrained volume condition considered here, similar to [46, 21], it is also possible to establish similar properties for the bilinear form with other kernels, say those homogeneous, isotropic and L1 integrable kernels. In this paper, we focus on the finite element approximation by assuming the continuity and coercivity in the 2 d constrained energy space (Ln0(Ω)) of the bilinear form corresponding to the linear bond-based PD model. That is, for any u, v L2 (Ω), we assume that there exist ∈ n0 non-negative constants C1 and C2(δ) such that
2 B(u, v) C1 M(δ) u 2 v 2 and B(u, v) C2(δ) u 2 . (6.20) | | ≤ k kL (Ω)k kL (Ω) ≥ k kL (Ω) 138
And we review the following lemma about nonlocal green’s identity,
Lemma 31. Assume the kernel function Λ be defined as in (6.18), with ω satisfying (6.19). Then for any u, v L2 (Ω), ∈ n0 Z Z Z T Tr( t∗u)Tr( t∗v) dy dx = t t∗(u) v dx. (6.21) Ω Ω D D Ω D D ·
We note that such results have been established in [44] with the implicit as- sumption that the integrands are well defined. By the properties of the bilinear form and thus the PD operator, the identity can be stated in the suitable spaces. As δ 0, the problem (6.16) over Ω converges to Navier-Lam´e equation over Ω → s with homogeneous Dirichlet constrained volume condition [50, 28]: T µ ( u(x) + u(x)) µ ( u(x)) = f(x) for x Ωs − ∇ · ∇ ∇ − ∇ ∇ · ∈ (6.22) u(x) = 0 for x ∂Ω , ∈ s with the coefficient µ, under the above assumption for Λ, can be evaluated as Z β 2 2 µ = lim z1z2 dz, (6.23) δ 0 z 2 → Bδ(0) | |
T where z = (z1, ..., zd) . The weak form of problem (6.22) can be formulated as: find u H1(Ω) (= ∈ 0 L2(Ω) (H1(Ω))d) such that for any v H1(Ω) 0 ∩ ∈ 0
µ( u + T u : v) + µ( u, v) = (f, v), (6.24) ∇ ∇ ∇ ∇ · ∇ · where the symbol ‘:’ denotes the Frobenius product.
6.3.1 Finite element discretization and a posterior error estimate
To study the numerical solution of the linear nonlocal bond-based PD model (6.17) h 2 h by finite element methods, we let S be a finite dimensional subspace of Ln0, and S consists of piecewise smooth function over K , where K denotes the mesh of Ω { } { } s 139
(Ω = S K). Moreover, we denote uh (Sh )d the finite element approximation s ∈ local of u by solving (6.24). Applying the framework in [62], we can get a residual-type identity that states, for any v H1(Ω), ∈ 0
µ(( + T )(u uh): v) + µ( (u uh), v) ∇ ∇ − ∇ ∇ · − ∇ · X Z = (f + µ ( uh + T uh) + µ ( uh)) v dx ∇ · ∇ ∇ ∇ ∇ · · K K Z 1 h T h h µ u + u + ( u )I r v dν , (6.25) −2 ∂K J∇ ∇ ∇ · K ·
h T h h where u + u + ( u )I r denotes the normal flux on an edge (face) r J∇ ∇ ∇ · K defined as follows: suppose r is the common edge(face) of elements K and K0, then
h T h h h T h h u + u + ( u )I r = ( uK + uK + ( uK )I) ~nK J∇ ∇ ∇ · K ∇ ∇ ∇ · · h T h h + ( u + u + ( u ) I) ~nK . (6.26) ∇ K0 ∇ K0 ∇ · K0 · · 0
We now can define the interior residual, similarly as for the scalar case discussed earlier, rh = f + µ ( uh + T uh) + µ ( uh) in K, (6.27) ∗ ∇ · ∇ ∇ ∇ ∇ · and the boundary residual
h h T h h R = µ u + u + ( u )I r on any interior edge (face) r, (6.28) ∗ J∇ ∇ ∇ · K and then obtain the local a posteriori error estimator. Let uh Sh be the Galerkin approximation of u in Sh obtained using the weak ∈ form (6.17) and u∗ denote the weak solution of the model, then we have that
B(uh, vh) = (f , vh), vh Sh. ∀ ∈
h h By setting e = u∗ u and using the Green’s identity, Lemma 31, we get, for any − 140 v L2 (Ω), ∈ n0
h h B(e , v) = B(u∗ , v) B(u , v) Z − h T = v(x) f t ω ( t∗u ) dx Ωs − D D = (Rh , v) where h h T R (x) = f(x) ω ( ∗u ) , x K, (6.29) − Dt Dt ∀ ∈ denotes the residual of the peridynmaic problem (6.16). For v L2 , we define the energy norm as ∈ n0 p v = B(v , v) (6.30) k kE
Since (6.20) implies the equivalence between v and v 2 , so we can k kE k kL (Ω) define the dual energy norm as
(v , w) v E∗ = sup (6.31) k k w L2 (Ω) w E ∈ n0 k k and we have
Lemma 32. For u , uh L2 (Ω), the energy norm of error eh equals the dual ∗ ∈ n0 energy norm of the corresponding residual Rh, i.e.
eh = Rh . k kE k kE∗
Proof. First, by Cauchy-Schwartz inequality, we have that for any v L2 (Ω), ∈ n0
(Rh , v) = B(eh , v) eh v , ≤ k kE k kE so h h (R , v) h R E∗ = sup e E. k k v v ≤ k k k kE 141
Second, since eh L2 (Ω), ∈ n0 h h h h (R , v) B(e , e ) h R E∗ = sup h = e E, k k v v ≥ e k k k kE k kE then the conclusion follows.
The reliability and efficiency also can be proved in the following theorem.
2 h h d Theorem 17. Suppose that u∗ L (Ω) is the weak solution of (6.17), u (S ) ∈ n0 ∈ h h h is the finite element approximation of u, R is defined as in (6.29), e = u∗ u is − the exact error. Then there exists non-negative constants C3 and C4(δ) such that
h h h C3η e C4(δ)η , (6.32) δ ≤ k kE ≤ δ
h h p where ηδ = R L2(Ω)/ M(δ) denotes the a posteriori error estimator, E = p k k k · k B( , ) denotes the energy norm. · · Proof. By (6.20), we have for any v L2 (Ω), ∈ n0 p p C2(δ) v 2 v C1M(δ) v 2 , k kL (Ω) ≤ k kE ≤ k kL (Ω) using the definition of and Cauchy-Schwartz inequality, we obtain k · kE∗ 1 1 v 2 v v 2 , p L (Ω) E∗ p L (Ω) C1M(δ)k k ≤ k k ≤ C2(δ)k k let v = Rh in the above inequality, applying Lemma 32, we reach the conclusion.
Next, we will find the connection between (6.29) and (6.25).
6.3.2 Relation with local case
For simplicity, we assume Ωc is a layered strip, with thickness δ, surrounding Ωs, see Figure 6.1. Since, from chapter 5, the PD problem (6.16) converges to the classical Navier problem (6.22), it is thus natural to study if the a posteriori error 142
Figure 6.1. Ωs and Ωc. estimator derived earlier for a nonlocal problem would converge to the correspond- ing estimator of its local limit in some suitable sense. To study the limiting behavior as δ 0 of the a posterior error estimator on a → given mesh, we assume that δ is sufficiently small such δ << hK , in the remaining discussion provided in this section. For the finite element analysis carried out here, we focus on the case of having globally continuous and piecewise smooth basis functions so that a conforming Galerkin approximation of the PDE model is given in the limit with the corresponding a posteriori error estimation briefly reviewed in the previous subsection. More general analysis involving those based on discontinuous basis functions will be given in the future. And we further assume that
0 1 d u, v (C (Ω) H (Ω)) , u Ωc = 0, v Ωc = 0, ∈ ∩ | | (6.33) u , v (Cτ (K))d for any K, where τ 2. K K ∈ ≥ For any K Ω, Γ is defined by ⊂ K
Γ := x K Dist(x,K) δ , (6.34) K { 6∈ | ≤ } then it follows that Z Z Tr( t∗u)ωTr( t∗v) dy dx − Ω Ω D D Z T = v(x) t ω( t∗u) dy dx − Ωs ·D D Z X T = v(x) ω( ∗u) dy dx − ·Dt Dt K K 143
X Z Z = v(x) 2Λ (u(y) u(x)) dy dx · · − K K K X Z Z + v(x) 2Λ (u(y) u(x)) dy dx. (6.35) · · − K K ΓK
ΓK
Kout
Kin
Figure 6.2. K = K K , an illustration in the two-dimensional space. in ∪ out Denote K = K K , where K = K B (∂K) (the δ neighborhood of in ∪ out out ∩ δ − ∂K), while Kin = K/Kout (see Figure 6.2 for an illustration of the two-dimensional case). When y, x K, since u is smooth within an element K, by Taylor expan- ∈ sion, we get
1 u(y) = u(x) + u(x) (y x) + (y x)T H (x)(y x) + o(δ2), ∇ · − 2 − u − where Hu denotes the Hessian of u. Then for any K, consider the first term in (6.35), Z Z v(x) 2Λ (u(y) u(x)) dy dx K · K · − Z Z = v(x) 2Λ u(x) (y x) dy dx K · K · ∇ · − Z Z 1 T + v(x) 2Λ (y x) Hu(x) (y x) dy dx K · K · 2 − · − Z Z + v(x) 2Λ o(δ2) dy dx K · K ·
= T1 + T2 + T3, (6.36) where H ( ) is the Hessian tensor of u. u · 144
Denote z = y x, and evaluate T2 as − Z Z 1 T v(x) 2Λ (y x) Hu(x)(y x) dy dx K · K · 2 − − Z Z ω = v(x) z z zT H (x)z dz dx + o(1), (6.37) · z 2 ⊗ · u Kin Bδ(0) | |
T T Let z = (z1, z2, ..., z ) , u = (u1, u2, ..., u ) , then we can compute the i th com- d d − ponent of (z z zT H (x)z) as ⊗ · u X z z zT H (x)z = z z z z u (x), (6.38) ⊗ · u i i j k l l,jk j,k,l where ul,jk denotes the second-order mixed derivative of ul along xj and xk direc- tions.
Using the symmetry of integral over Bδ(0), we have that the only non-zero terms from (6.38) after integration, are
! Z ω X z4u (x) + z2z2(u (x) + u (x) + u (x)) dz z 2 i i,ii i j i,jj j,ij j,ji Bδ(0) j=i | | 6 Z ω X = z2z2 dz (3 u (x) + (u (x) + u (x) + u (x))) z 2 1 2 i,ii i,jj j,ij j,ji Bδ(0) j=i | | 6 Z ω X = z2z2 dz (u (x) + u (x) + u (x)) z 2 1 2 i,jj j,ij j,ji Bδ(0) | | j Z ω = z2z2 dz ( u + T u) + ( u) , (6.39) z 2 1 2 ∇ · ∇ ∇ ∇ ∇ · i Bδ(0) | | since we have the relationship
Z z4 Z z4 Z z2 z2 Z z2z2 ω l dz = ω s dz = 3 ω m n dz = 3 ω t p dz, z 2 z 2 z 2 z 2 Bδ(0) | | Bδ(0) | | Bδ(0) | | Bδ(0) | | for any l, m, n (m = n), s, t, p (t = p) = 1, 2, ..., d. 6 6 Therefore, as δ 0, → Z T T2 v(x) µ( ( u + u) + ( u)) dx. (6.40) → Kin · ∇ · ∇ ∇ ∇ ∇ · 145
Since u is piecewise smooth for x K and y Γ respectively, we define s as ∈ ∈ K the intersection of ∂K and the vector (y x) (as shown in Figure 6.4), one then − has the following Taylor expansion
uΓ (y) = u(x) + u (x) (s x) + uΓ (s) (y s) + o(δ). K ∇ K · − ∇ K · −
Then the second term in (6.35) can be computed as following. Z Z v(x) 2Λ (u(y) u(x)) dy dx K · ΓK · − Z Z = v(x) 2Λ uK (x) (y x) dy dx K · ΓK · ∇ · − Z Z
+ v(x) 2Λ [ uΓK (s) uK (x)] (y s) dy dx K · ΓK · ∇ − ∇ · − Z Z + v(x) 2Λ o(δ) dy dx K · ΓK ·
= S1 + S2 + S3. (6.41)
Adding up T1 and S1 gives us Z Z T1 + S1 = v(x) 2Λ uK (x) (y x) dy dx K · K ΓK · ∇ · − Z Z ∪ = v(x) 2Λ u(x) z dz dx, K · Bδ(0) · ∇ · = 0. (6.42)
By the symmetry of Bδ(0), it is also not hard to show ( considering the scaling factor in ω) that
T3 + S3 0 as δ 0. (6.43) → →
Next, we will evaluate the term S2. Assume that the elements K and K0 have a common edge e and, without loss of generality, the outward normal of K on e is T ~ne = (1, 0, 0, ..., 0) . Here we use the coordinate system ξ η, where ξ R and × ∈ d 1 η R − , as shown in Figure 6.3, i.e., edge (or face) e is on the (d 1) dimensional ∈ − − plane ξ = 0. 146
ξ = δ ξ ΓK (Kout ) δ e η ξ =0 δ Kout (ΓK ) ξ = δ − Figure 6.3. The δ-neighborhood of an edge e.
One can easily observe that
(e [ δ, 0]) K B (e) Γ B (e), (6.44) × − ≈ out ∩ δ ≈ K0 ∩ δ the difference between any two of the above sets is of magnitude O(δd), which means the difference can be ignored in limit sense. Similarly, we can also have
(e [0, δ]) K0 B (e) Γ B (e), (6.45) × ≈ out ∩ δ ≈ K ∩ δ where the difference between any two sets again can be ignored in limit sense.
ξ ΓK (Kout ) x
xe (0,ηx) e η s
x (ξx,ηx) Kout (ΓK )
Figure 6.4. A projection of x onto e.
Since v( ) is continuous across elements, we then have · Z Z
S2 = v(x) 2Λ ( uΓK (s) uK (x)) (y s) dy dx K · ΓK · ∇ − ∇ · − X Z Z = v(x) 2Λ ( ) · Γ ( ) · r edge(K)/ e Kout Bδ r K Bδ x ∈ { } ∩ ∩ ( uΓ (s) u (x)) (y s) dy dx ∇ K − ∇ K · − Z Z 0 Z + v(xe) 2Λ e · δ Bδ(x) ΓK · − ∩ ( uΓ (x ) u (x )) (y s) dy dξ d~η + o(1), (6.46) ∇ K e − ∇ K e · − 147
where xe = (ξ, 0). For the K0 part, we change the order of integration and obtain Z Z
S20 = v(x) 2Λ ( uΓK (s) uK0 (x)) (y s) dy dx K · Γ · ∇ 0 − ∇ · − 0 K0 X Z Z = v(x) 2Λ K0 Bδ(r) · Γ Bδ(x) · r edge(K )/ e out K0 ∈ 0 { } ∩ ∩
( uΓ (s) uK (x)) (y s) dy dx ∇ K0 − ∇ 0 · − Z Z 0 Z + v(xe) 2Λ e · δ Bδ(x) ΓK · − ∩ ( uΓ (x ) u (x )) (s x) dy dξ d~η + o(1). (6.47) ∇ K e − ∇ K e · −
Adding up (6.46) and (6.47), the resulting integral on edge (face) e can be presented as
Z Z 0 Z 2ω v(xe) 2 z z u˜(xe)z dz dξ d~η + o(1), (6.48) e · δ SC(δ,δ+ξ) z ⊗ ∇ − | | where z = y x, u˜( ) = uΓ ( ) u ( ). Notice the contributions from S2 − ∇ · ∇ K · − ∇ K · and S0 to (6.48) are the same as δ 0. 2 →
ΓK (Kout )
e δ
x (z = 0) Kout (ΓK )
Figure 6.5. An illustration of the integral domain SC(δ, δ+ξ) that denotes the spherical cap of a sphere with radius δ and height δ + ξ.
The i th component of z z u˜(x )z can be computed as − ⊗ ∇ e
d X (z z u˜(x )z) = z z z u˜ (x ), (6.49) ⊗ ∇ e i i j k j,k e j,k=1 whereu ˜ denotes the derivative of u Γ u along x direction. Since the j,k j| K − j|K k outward normal of K on e is (1, 0, 0, ..., 0)T , by the symmetry of the integral, when 148 i = 1 we have
d Z 0 Z 2ω X 2 ( zizjzku˜j,k(xe)) dz dη δ Bδ(x) ΓK z − ∩ | | j,k=1 Z 0 Z d 2ω X 2 = 2 ( z1zj u˜j,j(xe)) dz dη δ Bδ(x) ΓK z − ∩ | | j=1 Z Z 0 d 2ω X 2 = 2 ( z1zj u˜j,j(xe)) dη dz Bδ(xe) ΓK z1 z ∩ − | | j=1 d Z ω X = ( z2z2u˜ (x )) dz z 2 1 j j,j e Bδ(0) | | j=1 d Z ω X = z2z2 dz (3˜u (x ) + u˜ (x )). (6.50) z 2 1 2 1,1 e j,j e Bδ(0) | | j=2
Similarly, when i > 1 we get
d Z 0 Z 2ω X 2 ( zizjzku˜j,k(xe)) dz dη δ Bδ(x) ΓK z − ∩ | | j,k=1 Z 0 Z 2ω 2 = 2 (z1zi (˜u1,i(xe) +u ˜i,1(xe)) dz dη δ Bδ(x) ΓK z − ∩ | | Z ω = (z2z2(˜u (x ) +u ˜ (x )) dz z 2 1 i 1,i e i,1 e Bδ(0) | | Z ω = z2z2 dz (˜u (x ) +u ˜ (x )). (6.51) z 2 1 2 1,i e i,1 e Bδ(0) | | Combining (6.50) and (6.51), we get that the expression in (6.48) becomes Z T v(xe) µδ( u˜(xe) + u˜ (xe) + ( u˜(xe))I) ~ne dξ + o(1), e · ∇ ∇ ∇ · · where Z ω µ = z2z2 dz. δ z 2 1 2 Bδ(0) | | So Z 1 T S2 v(ν) µ u(ν) + u(ν) + ( u(ν))I r dν, (6.52) → −2 ∂K · J∇ ∇ ∇ · K 149
where µ = lim µδ. δ 0 →
Finally, combining (6.35), (6.36), (6.40), (6.42) and (6.52), we have
Theorem 18. Under the assumption (6.19), with u and v satisfying (6.33). For any element K, as δ 0, → Z T v(x) t ω( t∗u) dx − K ·D D Z v(x) µ( ( u + T u) + ( u)) dx → K · ∇ · ∇ ∇ ∇ ∇ · Z 1 T v(ν) µ u(ν) + u(ν) + ( u(ν))I r dν. (6.53) − 2 ∂K · J∇ ∇ ∇ · K
We therefore have the corollary
Corollary 11. Under assumption (6.19), with uh and v satisfy (6.33). For any element K, as δ 0, → Z Z 1 Z v(x) Rh dx v(x) rh dx v(ν) Rh dν. (6.54) K · → K · ∗ − 2 ∂K · ∗
Thus we can see that the a posteriori error estimation for the PD bond-based model is connected to its local counterpart as well. Summing up (6.54) for all the K’s implies the local limit of PD operator being the Navier operator (with Poisson ratio 1/4), as stated in [44].
6.4 Conclusion
Utilizing the analytic framework established in chapter 2.3, we are also able to show the convergence of finite-dimensional approximations to the nonlocal BVPs and to derive typical error estimates for both Fourier spectral and finite element methods. Important issues such as the dependence on the model parameter (PD horizon, for example) and the mesh parameter of the condition numbers of the stiffness matrices can be discussed within the framework. Moreover, a rigorous mathematical framework is established for the a posteriori error analysis of the finite element solutions PD equations. The estimators are carefully derived and 150 the reliability and efficiency are proven. Meanwhile, the weak convergence of the nonlocal estimator to its local analogue is also demonstrated for PD model problem, drawing interesting connections to the limit case for which a lot of studies can be found in the literature. The nonlocal nature of the problem also offers some sharp contrasts with the local counterpart as explored in our analysis. While the analysis is done for problems with an operator-induced norm being equivalent to L2 norm, due to our choice of influence function, it is possible to generalize such a framework of a posteriori error analysis for problems involving more general influence functions. This will be pursued in our future works, along with the development of the adaptive convergence analysis and practical implementation for nonlocal models. Chapter 7
The PD double-bonds model
7.1 Introduction
As discussed in the previous chapters, the current existing PD bond-based and state-based models can not recover the 21-independent parameters full elasticity model. In this chapter, we proposed an alternative PD model, the PD double-bonds model, using two bonds simultaneously determining the internal force acting on a certain point. And the relation between the traditional bond-based model and the PD double-bonds model are discussed. The generality in the internal force calculation provides more freedom to the nonlocal material modeling.
7.2 The full nonlocal Peridynamic double-bonds model
In this model, to incorporate both the elongation and shear deformation in the system. We define the following two extension scalars.
e y x (u) = u(y + z x) u(x) + y x y x h − i − − − − −
e z x (u) = u(y + z x) u(x) + z x z x h − i − − − − −
Then by linearizing the the extensions with respect to u(y + z x) u(x), we − − 152 get their linearized versions.
e y x (u) = y x u(y + z x) u(x)/ y x (7.1) h − i − · − − | − | e z x (u) = z x u(y + z x) u(x)/ z x (7.2) h − i − · − − | − | where (7.1) measures the projection of the extension of the bond (y x + z x + − − x) x on the direction y x and (7.2) shares the similar meaning. − −
Using the definition of the domains in section 4.4.1, the potential energy the PD double-bonds model can be defined.
Z Z Z 1 Ep(u) = ω(y x, z x)χ y+z x Ω e y x (u) y x { − ∈ } 4 Ω Bδ(x) Bδ(x) − − h − i | − | e z x (u) z x dy dz dx h − i | − | Z Z Z 1 T = ω(y x, z x)χ y+z x Ω u(y + z x) u(x) { − ∈ } 8 Ω Bδ(x) Bδ(x) − − − − (y x) (z x) + (z x) (y x) u(y + z x) u(x) dy dz dx · − ⊗ − − ⊗ − − − (7.3) where ω(y x, z x) is symmetric, i.e. ω(y x, z x) = ω(z x, y x) and − − − − − − even, i.e. ω(y x, z x) = ω(x y, z x) and ω(y x, z x) = ω(y x, x z). − − − − − − − −
Then we can study the following energy minimization problem Z min Ep(u) u b dx (7.4) u − Ω ·
u(x) = h(x), x Ω ∈ c
By variational principle, we can readily get the equality that Z Z Z 1 ω(y x, z x)χ y+z x Ω (y x) (z x) + (z x) (y x) 4 Ω Bδ(x) Bδ(x) − − { − ∈ } − ⊗ − − ⊗ − Z u(y + z x) u(x) v(y + z x) v(x) dy dz dx = v b dx − − · − − Ω · 153
Next we show that Z Z Z 1 ω(y x, z x)χ y+z x Ω (y x) (z x) 4 Ω Bδ(x) Bδ(x) − − { − ∈ } − ⊗ − + (z x) (y x) u(y + z x) u(x) v(y + z x) dy dz dx − ⊗ − − − · − Z Z Z 1 = ω(y x, z x)χ y+z x Ω (y x) (z x) −4 Ω Bδ(x) Bδ(x) − − { − ∈ } − ⊗ − + (z x) (y x) u(y + z x) u(x) v(x) dy dz dx (7.5) − ⊗ − − − ·
Let w = y + z x, we have y w = z x < δ and z w = y x < δ. − | − | | − | | − | | − | Moreover from the characteristic function χ y+z x Ω , we have w Ω. Thus the { − ∈ } ∈ following equalities hold. Z Z Z 1 ω(y x, z x)χ y+z x Ω (y x) (z x) { − ∈ } 4 Ω Bδ(x) Bδ(x) − − − ⊗ − + (z x) (y x) u(y + z x) u(x) v(y + z x) dy dz dx − ⊗ − − − · − Z Z Z 1 = ω(w z, w y)χ y+z w Ω (w z) (w y) { − ∈ } 4 Ω Bδ(w) Bδ(w) − − − ⊗ − + (w y) (w z) u(w) u(y + z w) v(w) dy dz dw − ⊗ − − − ·
We replace the dummy variable w by x and use the evenness and symmetry of ω(y x, z x) to get (7.5). − −
Through the derivation above, one can easily get the strong form of the PD double-bonds model. pu(x) = b(x) , x Ωs −L ∈ (7.6) u(x) = h(x) x Ω ∈ c where denotes the PD double-bonds operator and is defined as −Lp Z Z 1 pu = ω(y x, z x) (y x) (z x) + (z x) (y x) −L 2 Bδ(x) Bδ(x) − − − ⊗ − − ⊗ − 154
u(y + z x) u(x) χ y+z x Ω dy dz (7.7) − − { − ∈ }
Compared with the existing bond-based PD models,
1 Z ω(y x) u = − (y x) (y x) u(y) u(x) dy (7.8) −Lb 2 y x 2 − ⊗ − − Bδ(x) | − | we can see that instead of integrating the pairwise-force density on the bond y x with respect to the particles y B (x), the PD double-bonds model com- − ∈ δ pute the internal force acting on x by cumulating the force density on the bond (y + z x) x = (y x) + (z x) + x x with respect to two points y, z B (x). − − − − − ∈ δ
In the following analysis, we assume the model is defined in the whole space, Rd. By Fourier transform, the PD double-bonds operator can be expressed in the following way. Z Z 1 (y+z) ξ dpu = ω(y, z) 1 e · y z + z y dy dz ub(ξ) L 2 Bδ(0) Bδ(0) − ⊗ ⊗ 1 Z Z = ω(y, z) 1 cos((y + z) ξ) i sin((y + z) ξ) 2 Bδ(0) Bδ(0) − · − · y z + z y dy dz u(ξ) ⊗ ⊗ b
Because of the evenness of ω(y, z), Z Z 1 dpu = ω(y, z) cos(y ξ + z ξ) y z + z y dy dz ub(ξ) L −2 Bδ(0) Bδ(0) · · ⊗ ⊗
= Mδ(ξ)u (7.9)
Theorem 19. Assume the displacement field u is smooth enough and the influence function of the PD double-bonds operator ω(y, z) is even and symmetric. As δ 0, → the PD double-bonds operator (7.7) converges to the classical elasticity operator
T CE(u) = C( u + u ) − −∇ · ∇ ∇ 155 with the Voigt expression of the 4th order stiffness tensor given by
A11 A12 A13 B11 B12 B13 A12 A22 A23 B21 B22 B23 A A A B B B 13 23 33 31 32 33 [C] = (7.10) B11 B12 B13 C11 C12 C13 B21 B22 B23 C12 C22 C23 B31 B32 B33 C13 C23 C33
Proof. By equation (7.9), Z Z 1 Mδ(ξ) = ω(y, z) cos(y ξ) cos(z ξ) sin(y ξ) sin(z ξ) −2 Bδ(0) Bδ(0) · · − · · y z + z y dy dz ⊗ ⊗ 1 Z Z = ω(y, z)(y ξ)(z ξ) y z + z y dy dz + (δ) 2 Bδ(0) Bδ(0) · · ⊗ ⊗ Z Z 2y1z1 y1z2 + y2z1 y1z3 + y3z1 1 = ω(y, z) y1z2 + y2z1 2y2z2 y2z3 + y3z2 2 Bδ(0) Bδ(0) y1z3 + y3z1 y2z3 + y3z2 2y3z3 3 X ( yizjξiξj) dy dz + (δ) i,j=1 where (δ) converges to 0 when δ goes to 0. And for notation simplicity, we define the following parameters. Z Z Z Z 2 2 2 2 A11 = lim ω(y, z) y1z1 dy dz A22 = lim ω(y, z) y2z2 dy dz δ→0 δ→0 Bδ (0) Bδ (0) Bδ (0) Bδ (0) Z Z Z Z 2 2 2 2 A33 = lim ω(y, z) y3z3 dy dz A12 = lim ω(y, z) y1z2 dy dz (7.11) δ→0 δ→0 Bδ (0) Bδ (0) Bδ (0) Bδ (0) Z Z Z Z 2 2 2 2 A13 = lim ω(y z) y1z3 dy dz A23 = lim ω(y, z) y2z3 dy dz δ→0 δ→0 Bδ (0) Bδ (0) Bδ (0) Bδ (0) and Z Z Z Z 2 2 B13 = lim ω(y, z)y1z1z2 dy dz B23 = lim ω(y, z) y2z1z2 dy dz δ→0 δ→0 Bδ (0) Bδ (0) Bδ (0) Bδ (0) Z Z Z Z 2 2 B12 = lim ω(y, z) y1z1z3 dy dz B32 = lim ω(y, z) y3z1z3 dy dz δ→0 δ→0 Bδ (0) Bδ (0) Bδ (0) Bδ (0) 156
Z Z Z Z 2 2 B21 = lim ω(y, z) y2z2z3 dy dz B31 = lim ω(y, z) y3z2z3 dy dz δ→0 δ→0 Bδ (0) Bδ (0) Bδ (0) Bδ (0) (7.12) Z Z Z Z 2 2 B11 = lim ω(y, z) y1z2z3 dy dz B22 = lim ω(y, z) y2z1z3 dy dz δ→0 δ→0 Bδ (0) Bδ (0) Bδ (0) Bδ (0) Z Z 2 B33 = lim ω(y, z) y3z1z2 dy dz δ→0 Bδ (0) Bδ (0) moreover, Z Z Z Z C11 = lim ω(y, z) y2y3z2z3 dy dz C22 = lim ω(y, z) y1y3z1z3 dy dz δ→0 δ→0 Bδ (0) Bδ (0) Bδ (0) Bδ (0) Z Z Z Z C33 = lim ω(y, z) y1y2z1z2 dy dz C12 = lim ω(y, z) y2y3z1z3 dy dz δ→0 δ→0 Bδ (0) Bδ (0) Bδ (0) Bδ (0) (7.13) Z Z Z Z C13 = lim ω(y, z) y2y3z1z2 dy dz C23 = lim ω(y, z) y1y3z1z2 dy dz δ→0 δ→0 Bδ (0) Bδ (0) Bδ (0) Bδ (0)
Then we let δ 0, by the symmetry of ω(y, z), the Fourier symbol of lim Mδ(ξ) is expressed → δ→0 as the following.
The first column of lim Mδ(ξ) is δ→0
2 2 2 A11ξ1 + C33ξ2 + C22ξ3 + (B13 + B13)ξ1ξ2 + (B12 + B12)ξ1ξ3 + (C23 + C23)ξ2ξ3 2 2 2 B13ξ + B23ξ + C12ξ + (A12 + C33)ξ1ξ2 + (B11 + C23)ξ1ξ3 + (B22 + C13)ξ2ξ3 . 1 2 3 2 2 2 B12ξ1 + C13ξ2 + B32ξ3 + (B11 + C23)ξ1ξ2 + (A13 + C22)ξ1ξ3 + (B33 + C12)ξ2ξ3
The second column of lim Mδ(ξ) is δ→0
2 2 2 B13ξ1 + B23ξ2 + C12ξ3 + (A12 + C33)ξ1ξ2 + (B11 + C23)ξ1ξ3 + (B22 + C13)ξ2ξ3 2 2 2 C33ξ + A22ξ + C11ξ + (B23 + B23)ξ1ξ2 + (C13 + C13)ξ1ξ3 + (B21 + B21)ξ2ξ3 . 1 2 3 2 2 2 C23ξ1 + B21ξ2 + B31ξ3 + (B22 + C13)ξ1ξ2 + (B33 + C12)ξ1ξ3 + (A23 + C11)ξ2ξ3
The third column of lim Mδ(ξ) is δ→0
2 2 2 B12ξ1 + C13ξ2 + B32ξ3 + (B11 + C23)ξ1ξ2 + (A13 + C22)ξ1ξ3 + (B33 + C12)ξ2ξ3 2 2 2 C23ξ + B21ξ + B31ξ + (B22 + C13)ξ1ξ2 + (B33 + C12)ξ1ξ3 + (A23 + C11)ξ2ξ3 . 1 2 3 2 2 2 C22ξ1 + C11ξ2 + A33ξ3 + (C12 + C21)ξ1ξ2 + (B32 + B32)ξ1ξ3 + (B31 + B31)ξ2ξ3
By directly calculation, we can also find the Fourier symbol, CE(ξ), of the classical elasticity 157 operator too.
The first column of CE(ξ) is
2 2 2 C1111ξ1 + C1212ξ2 + C1313ξ3 + (C1112 + C1211)ξ1ξ2 + (C1113 + C1311)ξ1ξ3 + (C1213 + C1312)ξ2ξ3 2 2 3 C1211ξ + C2212ξ + C2313ξ + (C1212 + C2211)ξ1ξ2 + (C1213 + C2311)ξ1ξ3 + (C2213 + C2312)ξ2ξ3 . 1 2 3 2 2 2 C1311ξ1 + C2312ξ2 + C3313ξ3 + (C1312 + C2311)ξ1ξ2 + (C1313 + C3311)ξ1ξ3 + (C2313 + C3312)ξ2ξ3
The second column of CE(ξ) is
2 2 2 C1112ξ1 + C1222ξ2 + C1323ξ3 + (C1122 + C1212)ξ1ξ2 + (C1123 + C1312)ξ1ξ3 + (C1223 + C1322)ξ2ξ3 2 2 2 C1212ξ + C2222ξ + C2323ξ + (C1222 + C2212)ξ1ξ2 + (C1223 + C2312)ξ1ξ3 + (C2223 + C2322)ξ2ξ3 . 1 2 3 2 2 2 C1312ξ1 + C2322ξ2 + C3323ξ3 + (C1322 + C2312)ξ1ξ2 + (C1323 + C3312)ξ1ξ3 + (C2323 + C3322)ξ2ξ3
The third column of CE(ξ) is
2 2 2 C1113ξ1 + C1223ξ2 + C1333ξ3 + (C1123 + C1213)ξ1ξ2 + (C1133 + C1313)ξ1ξ3 + (C1233 + C1323)ξ2ξ3 2 2 3 C1213ξ + C2223ξ + C2333ξ + (C1223 + C2213)ξ1ξ2 + (C1233 + C2313)ξ1ξ3 + (C2233 + C2323)ξ2ξ3 . 1 2 3 2 2 2 C1313ξ1 + C2323ξ2 + C3333ξ3 + (C1323 + C2313)ξ1ξ2 + (C1333 + C3313)ξ1ξ3 + (C2333 + C3323)ξ2ξ3
Then one can see that the Fourier symbol of the ordinary bond-based PD model converges to the Fourier symbol of the classical elasticity equation with 4th order stiffness tensor A11 B13 B12 B13 C33 C23 B12 C23 C22 B13 A12 B11 C33 B23 C13 C23 B22 C12 B12 B11 A13 C23 C13 B33 C22 C12 B32 B13 C33 C23 A12 B23 B22 B11 C13 C12 C = C33 B23 C13 B23 A22 B21 C13 B21 C11 C23 C13 B33 B22 B21 A23 C12 C11 B31 B12 C23 C22 B11 C13 C12 A13 B33 B32 C23 B22 C12 C13 B21 C11 B33 A23 B31 C22 C12 B32 C12 C11 B31 B32 B31 A33 whose expression under Voigt notation can be found as in (7.10). 158
7.3 Relation between PD double-bonds and PD bond-based models
As a special case of the PD double-bonds model, the PD bond-based model could be derived by restricting the the influence function ω(y x, z x) on some special − − form. In the expression (7.7), we let
ω(y x, z x) = ω 2(y x)∆(y x, z x). (7.14) − − − − −
The PD double-bonds operator reduces to the PD bond-based equation operator. Z Z 1 pu = ω(y x, z x) (y x) (z x) + (z x) (y x) −L 2 Bδ(x) Bδ(x) − − − ⊗ − − ⊗ − u(y + z x) u(x) χ y+z x Ω dy dz − − { − ∈ } 1 Z Z = ω 2(y x)∆(y x, z x) (y x) (z x) 2 Bδ(x) Bδ(x) − − − − ⊗ − + (z x) (y x) u(y + z x) u(x) χ y+z x Ω dy dz − ⊗ − − − { − ∈ } Z = ω 2(y x) (y x) (y x) u(2(y x) + x) u(x) Bδ(x) − − ⊗ − − −
χ 2(y x)+x Ω dy { − ∈ } Z 1 = ω(t) t t u(t + x) u(x) χ t+x Ω dy 4 B2δ(0) ⊗ − { ∈ } 1 Z = ω(y x)(y x) (y x) u(y) u(x) dy 4 B2δ(x) Ω − − ⊗ − − ∩ If we let ω 2(y x) = ω 2 y x in (7.14), the PD double-bonds model becomes − | − | an isotropic model corresponding to the material with Poisson ratio one quarter.
Remark 8. It remains to calculate the specific forms of the influence function ω(y x, z x) such that the PD double-bonds equation converges to the local − − elastic model for a certain material. 159
7.4 Conclusion
The proposed PD double-bonds model recovers the 21 independent parameters in the full classical elasticity by choosing appropriate influence function ω(y x, z x), − − from which, the bond-based PD model can be derived as a special case. Chapter 8
An open issue – The nonlinear simulation
8.1 Introduction
In the dissertation, we analyze the linearized PD theories in which the micromodu- lus function C(y, x) uniquely determines the material properties. Since the micro- modulus function in the linear case is independent of the current displacement field u(x), it is not convenient to use it solely to discuss the dynamic problems, such as crack nucleation and propagation. Thus, the nonlinear PD model is a useful tool for the main issue - the crack nucleation problem. In this chapter, we will review the existing work on the crack nucleation theory and discuss some open issues to be resolved. In [63], Silling et al. discussed the crack nucleation condition of the PD ma- terials in the context of the linearized PD theory. In this work, assuming the micromodulus C(y, x) is continuous, the linearized PD model was written as Z ρ(x)u¨(x, t) = C(y, x)u(y, t)dy P(x)u(x, t) + b(x, t) Bδ(x) − where the tensor P(x) = R C(y, x) dy. Then the jumps in displacement and Bδ(x) acceleration at some point x in the material domain are defined as
+ + [[u]] = u(x ) u(x−) , [[u¨]] = u¨(x ) u¨(x−). − − 161
If we applying the jump-operator [[ ]] to both sides of the PD equation and assume · ρ(x) and b(x) are continuous, it is easy to get Z ρ(x)[[u¨]] = [[ C(y, x)u(y, t)dy]] [[P(x)u]]. Bδ(x) −
Since C, therefore P are continuous and the discontinuity we considered is a measure-zero set, it can be shown that
ρ(x)[[u¨]] = P(x)[[u]]. −
Intuitively, a small discontinuity at point x is considered unstable if it gets larger with time. Thus, the material becomes unstable if
[[u¨]] [[u]] < 0 (8.1) · thus (P(x)[[u]]) [[u]] < 0. Then the condition for crack nucleation in this linear · setting is that the minimal eigenvalue of P(x) is negative. In paper [63], an intuitive understanding of the negativity of the minimal eigenvalue of the tensor P was given with the aid of dispersion relation, Z ρω2(k, n)a = P C(ξ) cos(kξ n)dξa − Bδ(0) · where a is a unit eigenvector of P. If let k and dot a on both sides → ∞
ρω2( , n) = a Pa. ∞ ·
Then the negativity of minimal eigenvalue of P implies that the frequency of plane waves in the limit of zero wavelength is imaginary. Note that the dispersion relation is based on the assumption of homogeneous deformation, the tensor P is constant in x. While in the real case, the deformation can hardly be homogeneous, P(x) becomes a function of position x. The authors also use a computational example to illustrate the crack nucleation process. In the example, the body is a nonlinearly elastic 2-D plate containing a hole. Forces are applied on the two ends to stretch to plate. This example uses 162 the force-loading boundary condition around the hole and at the ends of the plate. The constitutive model the paper discussed is the multidimensional version of the equation (8.2) in the next section. But neither detailed parameter setting nor the way the force is applied to the body is given. The simulation result showed that given the external force is large enough, the crack nucleates from the middle of the plate and before the crack nucleates, the minimum eigenvalue of the tensor P(x) becomes negative.
The paper use one specific example to illustrate that the condition (8.1) is a necessary condition of the crack nucleation. However, to get more understanding of this nucleation process of PD model and a precise necessary and sufficient condition for the crack nucleation, we conduct a one dimensional experiment for a simple nonlinear example.
8.2 A one dimensional experiment
In this one dimensional example, we let the domain be an interval [ L, L]. And − let ξ = y x and η = u(y, t) u(x, t) + y x y x , we define the nonlocal | − | | − − | − | − | strain s = η/ξ. In this study, we focus on the nonlinear peridynamic equation with strain-softening, notably,
δ L Z ∧ u(y, t) + y u(x, t) x u¨(x, t) = f(u(x, t), u(y, t), x, y) − − dy + b(x, t) (δ L) u(y, t) + y u(x, t) x − ∧ | − − | u(x, 0) = u0(x, 0) u˙(x, 0) = u1(x, 0) (8.2) where c(ξ)(1 1/(1 + s)) s 0 − ≤ c(ξ)s 0 < s s1 f(u(x, t), u(y, t), x, y) = ≤ c(ξ)s1 c0(ξ)(s s1) s1 < s s0 − − ≤ 0 s0 < s 163
p and c0(ξ) = c(ξ)s1/(s0 s1). In this study, we focus on the form c(ξ) = cξ . We − use force loading ”boundary” condition in this example.
To compute the “tensor” P (x), we linearize the equation with respect to u(y) − u(x), see [3] for more detail. Then we have
1 ∂f K < y x, z x > [x] = z x,y x − − 2 ∂η 4 − − and
Z Z 1 Z ∂f P (x) = K < y x, z x > [x]dzdy = (y, x)dy. (8.3) B B − − 2 B ∂η where 2 c(ξ)ξ/(ξ + η) s 0 ≤ ∂f(η, ξ) c(ξ)/ξ 0 < s s1 = ≤ , (8.4) ∂η c0(ξ)/ξ s1 < s s0 − ≤ 0 s0 < s where c0(ξ) is defined as before.
Remark 9. This constitutive relation represent the three phases: when the nonlocal strain s is in between 0 and s1, the material is in the loading phase; if s becomes larger than s1 but still less than s0, it falls into the strain-softening phase; while s is larger than s0 the bond y x breaks. The constitutive relation for the case s < 0 − is just for preventing the exchange of particles’ positions.
8.2.1 Simulation method
The PD bar [ L, L] is discretized into n grids with equal length. We take the mid- − dle point of each grid as our mesh point. The total internal force, i.e. the integral operator is computed by the mid-point quadrature rule. Thus the computational equation was turned into a second order ODE system,
n X u(xj, t) + xj u(xi, t) xi u¨(x , t) = f(u(x , t), u(x , t), x , x ) − − + b(x , t), i i j i j u(x , t) + x u(x , t) x i j=0 | j j − i − i| 164
for i = 1 n. Then we use velocity verlet method in time to compute the dis- ··· placement fields.
8.2.2 Simulation results
In the simulation, we set the initial displacement and velocity to be 0 and set the interval to be [ 1, 1], c = 10000, s1 = 0.5 and s0 = 1.0. And let p = 0.1. The − external force b(x) exerts on the the δ-wide boundary layers, i.e.
( F x [ L, L + δ] b(x) = − ∈ − − F x [L δ, L + δ] ∈ −
From the experiment, two force boundaries can be seen, F1 and F2. If F < F1, the material remains continuous when the two waves propagating from the two ends meet at the middle. When F > F1 and F < F2, the material will crack at the middle when the two waves meet. While if F > F2, the material will crack immediately at the points x = L + δ and x = L δ which are the discontinuities − − of the external forces, see Figure 8.1.
Figure 8.1. Displacement field with δ = 0.1, h = 0.005, t = 0.00001 and t = 0.2 4
The green line represents the displacement field when the the external force has magnitude 150 at time 0.2. In this case the one dimensional bar starts to break in the middle because the strain between the particles on the left half and right half of the bar becomes larger than s0, which is 1 in this simulation. When the external 165
force F is up to 200, the bar breaks immediately at the boundary layers, around x = 0.9 and x = 0.9, because of the discontinuity of the external force. However, − if the external force is very small, say 80 in this example, the displacement remains continuous. So clearly, in this particular example, F1 is between 80 and 150 and
F2 lies between 150 and 200. Moreover, this simulation is not resolution sensitive, i.e., the result does not depend on the grid size h and time step t. To illustrate this point, we take the 4 breaking bar as an example, first plot the displacements of the equation with the same magnitude of external forces 150 but different grid sizes, h1 = 0.005 and h2 = 0.0025, see Figure 8.2. We can see that the displacements calculated under the two different grid sizes are almost equal.
Figure 8.2. Compare the displacement under the same condition but different grid sizes
With grid size h = 0.005, we also compare the simulation results of the same equation with different time step t = 0.00005 and t = 0.00001, see Figure 8.3. 4 4 We can see it is almost not distinguishable.
Remark 10. In the future, we can also calculate the exact force boundaries F1 and
F2 under different grid sizes and under different deltas so as to find the relation between the critical forces and material parameters.
Now we consider the value of P (x) defined in (8.3) at the middle point of the PD bar, i.e. x = 0. In Figure 8.4, we can see that although the bar does not break, because the magnitude of the external force (F = 85) is large enough, P (x) falls below 0 for some time before it reverses back to positive. This evidence shows that at some point x, the material, in the existence of nonlinearity, can have negative 166
Figure 8.3. Compare the displacement under the same condition but different time step
P value with displacement remains continuous. Moreover, this phenomenon is mesh-independent.
Figure 8.4. P(x) for the particle in the middle of the bar, i.e. P (0), with magnitude of external force 85
The smallest P (0) value is at t = 0.19303. From the following figure 8.5, the time t = 0.19303 seems the meeting time of waves from the ends of the bar. At this time, most of the strain around the middle point, x = 0, falls into the strain- softening phase, i.e. s > s1. By the definition of P (x), it becomes negative at this moment. Because the force is not large enough to break the bar, after the wave reflects back, the strains in the middle relieve which leads P (x) back to 0. Figure 8.6 shows the evolution of the displacement field along time. In the 3-D picture, we plot the displacement of the bar from x = 0 to x = 0.12 and from t = 0.18 to t = 0.22. We can see that the displacement field of this bar peaks around t = 0.19 167
and goes down after that. The big displacement around t = 0.19 makes the strain to fall in the strain-softening phase and leads to a negative P (0). In figure 8.7, we plot the integrand of P (0), ∂f/∂η as defined in (8.3) at t = 0.18, t = 0.19303 and t = 0.22. Together with figures 8.5 and 8.6, one can see that small change in displacement field can result big change in ∂f/∂η, thus big change in P (0).
Figure 8.5. Displacements at t = 0.15, t = 0.19303 and t = 0.24
Figure 8.6. Displacements at t = 0.15, t = 0.19303 and t = 0.24 168
Figure 8.7. ∂f/∂η at t = 0.15, t = 0.19303 and t = 0.24
Remark 11. The picture above shows that the condition (8.1) is a necessary con- dition for the crack nucleation. For searching for the sufficient one, one can try to think in terms of wave propagation, i.e. find a wave number such that if the wave with the certain wave number ceases to propagate the material has to break, or consider it in the energy point of view.
8.3 Open questions
Based on the experiments above, some further interesting experiments can be done.
First, the exact critical forces F1 and F2 can be calculated under different mesh size so as to find the critical values when grid size h goes to 0. More interestingly, a sufficient condition for the crack nucleation still remains open. Bibliography
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Kun Zhou was born in Tianjin, China, P. R. in 1983. He finished the high school education from Yaohua High School in 2002. Zhou majored in applied mathematics at University of Science and Technology of China and earned a bachelor’s degree in 2007. In August 2007, he enrolled in the Ph.D. program in Mathematics at Pennsylvania State University to study Applied and Computational Mathematics under the supervision of Professor Qiang Du. His current research interest is in the fields of nonlocal modeling and peridynamics.