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 ". 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.