Handbook of Peridynamic Modeling

Bobaru, Florin; Foster, John T.; Geubelle, Philippe H; Silling, Stewart A.

Editors Bobaru, Foster, Geubelle, and Silling present readers with a collection of academic and research perspectives toward a comprehensive guide to contemporary peridynamic modeling in a variety of applications. The editors have organized the sixteen selections that make up the main body of the text in five parts devoted to the need for nonlocal modeling and introduction toperidynamics; mathematics, numeric’s, and software tools of peridynamics; material models and links to atomsistic models; and other related subjects. Florin Bobaru is a faculty member of the University of Nebraska-Lincoln. John T. Foster is a faculty member of the University of Texas at Austin. Philippe H. Geubelle is a faculty member of the University of Illinois. Stewart A. Silling is with Sandia National Laboratories in New Mexico~ProtoView, 2017

• I The Need for Nonlocal Modeling and Introduction to PeridynamicsWhy Peridynamics? The mixed blessing of locality Origins of nonlocality in a modelLong-range forces Coarsening a fine-scale material system Smoothing of a heterogeneous material system Nonlocality at the macroscale The mixed blessing of nonlocalityIntroduction to Peridynamics Equilibrium in terms of integral equations Material modeling Bond based materialsRelation between bond densities and fluxPeridynamic states Ordinary state based materials Correspondence materials Discrete particles as peridynamic bodies Setting the horizon Linearized peridynamics Plasticity Bond based microplastic material LPS material with plasticity Damage and fracture Damage in bond based models Damage in ordinary state based material modelsDamage in correspondence material models Nucleation strain Treatment of boundaries and interfaces Bond based materialsState based materials Emu numerical method 2.7 Conclusions II Mathematics, Numerics, and Software Tools of PeridynamicsNonlocal Calculus of Variations and Well-posedness of Peridynamics Introduction .A brief review of well-posedness resultsNonlocal balance laws and nonlocal vector calculus Nonlocal calculus of variations - an illustrationNonlocal calculus of variations - further discussions Summary Local limits and asymptotically compatible discretizations Introduction Local PDE limits of linear peridynamic models Discretization schemes and discrete local limits Asymptotically compatible schemes for peridynamics SummaryRoadmap for Software Implementation Introduction Evaluating the internal force density Bond damage and failureThe tangent stiffness matrix Modeling contact Meshfree discretizations for peridynamics Proximity search for identification of pairwise interactions Time integrationExplicit time integration for transient dynamics Estimating the maximum stable time step Implicit time integration for quasi-statics Example simulations Fragmentation of a brittle disk resulting from impact Quasi-static simulation of a tensile testSummary III Material Models and Links to Atomistic ModelsConstitutive Modeling in Peridynamics Introduction Kinematics, momentum conservation, and terminologyLinear peridynamic isotropic solid Plane elasticityPlane stress Plane strain "Bond-based” theories as a special caseOn the role of the influence function Finite Deformations Invariants of peridynamic scalar-statesCorrespondence modelsNon-ordinary correspondence models for solid mechanics Ordinary correspondence models for solid mechanics Plasticity Yield surface and flow rule Loading/unloading and consistencyNon-ordinary models A non-ordinary beam modelA non-ordinary plate/shell model Other non-ordinary modelsFinal Comments Links between Peridynamic and Atomistic Models Introduction Molecular dynamicsMeshfree discretization of peridynamic modelsUpscaling molecular dynamics to peridynamics A one-dimensional nonlocal linear springs model A three-dimensional embedded-atom model Computational speedup through upscalingConcluding remarks Absorbing Boundary Conditions with Verification Introduction A PML for State-based Peridynamics Two-dimensional (2D), State-based Peridynamics Review Auxiliary Field Formulation and PML ApplicationNumerical ExamplesVerification of Cone and Center Crack ProblemsDimensional Analysis of Hertzian Cone Crack Developmentin Brittle Elastic SolidsState-based Verification of a Cone Crack Bond-based Verification of a Center Crack Verification of an Axisymmetric Indentation Problem Formulation Analytical Verification IV Modeling Material Failure and Damage Dynamic brittle fracture as an upscaling of unstable mesoscopic dynamicIntroduction The macroscopic evolution of brittle fracture as a small horizon limitof mesoscopic dynamics Dynamic instability and fracture initiation Localization of dynamic instability in the small horizon-macroscopic limitFree crack propagation in the small horizon-macroscopic limit SummaryCrack Branching in Dynamic Brittle Fracture Introduction A brief review of literature on crack branching Theoretical models and experimental results on dynamicbrittle fracture and crack branching Computations of dynamic brittle fracture based on FEM Dynamic brittle fracture results based on atomistic modeling Dynamic brittle fracture based on particle and lattice-based methods Phase-field models in dynamic fracture Results on dynamic brittle fracture from peridynamic modelsBrief Review of the bond-based Peridynamic modelAn accurate and efficient quadrature schemePeridynamic results for dynamic fracture and crack branching Crack branching in soda-lime glass Load case 1: stress on boundaries Load case 2: stress on pre-crack surfaces Load case 3: velocity boundary conditions Crack branching in Homalite Load case 1: stress on boundariesLoad case 2: stress on pre-crack surfaces Load case 3: velocity boundary conditions Influence of sample geometry10.5.3.1 Load case 1: stress on boundaries Load case 2: stress on pre-crack surfaces Load case 3: velocity boundary conditions Discussion of crack branching results Why do cracks branch? The importance of nonlocal modeling in crack branching Conclusions Relations Between Peridynamic and Classical Cohesive Models IntroductionAnalytical PD-based normal cohesive lawCase 1 – No bonds have reached critical stretch Case 2 – Bonds have exceeded the critical stretch Numerical approximation of PD-based cohesive law PD-based tangential cohesive lawCase 1 – No bonds have reached critical stretchCase 2 – Bonds have exceeded the critical stretch PD-based mixed-mode cohesive lawConclusionPeridynamic modeling of fiber-reinforced compositesIntroduction Peridynamic analysis of a lamina Peridynamic analysis of a laminate Numerical results Conclusions Appendix A: PD material constants of a lamina Simple shear Uniaxial stretch in the fiber directionUniaxial stretch in the transverse direction Biaxial stretch Appendix B: Surface correction factors for a composite lamina Appendix C: PD interlayer and shear bond constants of a laminate Peridynamic Modeling of Impact and Fragmentation Introduction Convergence studies and damage models that influence the damagebehavior Damage-dependent critical bond strainCritical bond strain dependence on compressive strains alongother directionsSurface effect in impact problems Convergence study for impact on a glass plate Impact on a multilayered glass system Model description A comparison between FEM and peridynamics for the elasticresponse of a multilayered system to impact13.4 Computational results for damage progression in the seven-layerglass system Damage evolution for the cross-sectionDamage evolution in the first layerDamage evolution in the second layerDamage evolution in the fourth layerDamage evolution in the seventh layer Conclusions V Multiphysics and Multiscale Modeling Coupling Local and Nonlocal ModelsIntroductionEnergy-based blending schemesThe Arlequin method Description of the coupling modelA numerical exampleThe morphing methodOverview Description of the morphing method One-dimensional analysis of ghost forces Numerical examples Force-based blending schemes Convergence of peridynamic models to classical models Derivation of force-based blending schemes A numerical example SummaryA Peridynamic model for corrosion damageAbstract Introduction Electrochemical Kinetics Problem formulation of 1D pitting corrosion The peridynamic formulation for 1D pitting corrosionResults and discussion of 1D pitting corrosion Pit corrosion depth proportional to square root tActivation-controlled, diffusion-controlled, and IR-controlledcorrosion Corrosion damage and the Concentration-Dependent Damage(CDD) model Damage evolution Saturated concentration Formulation and results of 2D and 3D pitting corrosion PD formulation of 2D and 3D pitting corrosion The Concentration-Dependent Damage (CDD) model forpitting corrosion: example in 2D A coupled corrosion/damage model for pitting corrosion: 2D example Diffusivity affects the corrosion rate Pitting corrosion with the CDD+DDC model in 3D Pitting corrosion in heterogeneous materials: examples in 2D Pitting corrosion in layer structures Pitting corrosion in a material with inclusions: a 2D example Conclusions Appendix Convergence study for 1D diffusion-controlled corrosion Convergence study for 2D activation-controlled corrosionwith Concentration-Dependent Damage model Peridynamics for Coupled Field EquationsIntroduction Diffusion Equation Thermal diffusion Moisture diffusion Electrical conduction Coupled Field Equations Thermomechanics Thermal diffusion with a structural coupling term Equation of motion with a thermal coupling term Porelasticity Mechanical deformation due to fluid pressure Fluid flow in porous medium Electromigration HygrothermomechanicsNumerical solution to peridynamic field equations Correction of PD material parameters Boundary conditions Essential boundary conditions Natural boundary conditionsExample 1Example 2 Example 3 Applications Coupled nonuniform heating and deformation Coupled nonuniform moisture and deformation in a square plate Coupled fluid pore pressure and deformationCoupled electrical, temperature, deformation, and vacancy diffusion Remarks