Handbook of Peridynamic Modeling

"I The Need for Nonlocal Modeling and Introduction to Peridynamics Why Peridynamics? The mixed blessing of locality Origins of nonlocality in a model Long-range forces Coarsening a fine-scale material system Smoothing of a heterogeneous material system Nonlocality at the macroscale The mixed blessing of nonlocality Introduction to Peridynamics Equilibrium in terms of integral equations Material modeling Bond based materials Relation between bond densities and flux Peridynamic 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 models Damage in correspondence material models Nucleation strain Treatment of boundaries and interfaces Bond based materials State based materials Emu numerical method 2.7 Conclusions II Mathematics, Numerics, and Software Tools of Peridynamics Nonlocal Calculus of Variations and Well-posedness of Peridynamics Introduction . A brief review of well-posedness results Nonlocal balance laws and nonlocal vector calculus Nonlocal calculus of variations - an illustration Nonlocal 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 Summary Roadmap for Software Implementation Introduction Evaluating the internal force density Bond damage and failure The tangent stiffness matrix Modeling contact Meshfree discretizations for peridynamics Proximity search for identification of pairwise interactions Time integration Explicit 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 test Summary III Material Models and Links to Atomistic Models Constitutive Modeling in Peridynamics Introduction Kinematics, momentum conservation, and terminology Linear peridynamic isotropic solid Plane elasticity Plane stress Plane strain ""Bond-based” theories as a special case On the role of the influence function Finite Deformations Invariants of peridynamic scalar-states Correspondence models Non-ordinary correspondence models for solid mechanics Ordinary correspondence models for solid mechanics Plasticity Yield surface and flow rule Loading/unloading and consistency Non-ordinary models A non-ordinary beam model A non-ordinary plate/shell model Other non-ordinary models Final Comments Links between Peridynamic and Atomistic Models Introduction Molecular dynamics Meshfree discretization of peridynamic models Upscaling molecular dynamics to peridynamics A one-dimensional nonlocal linear springs model A three-dimensional embedded-atom model Computational speedup through upscaling Concluding 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 Application Numerical Examples Verification of Cone and Center Crack Problems Dimensional Analysis of Hertzian Cone Crack Development in Brittle Elastic Solids State-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 dynamic Introduction The macroscopic evolution of brittle fracture as a small horizon limit of mesoscopic dynamics Dynamic instability and fracture initiation Localization of dynamic instability in the small horizon-macroscopic limit Free crack propagation in the small horizon-macroscopic limit Summary Crack Branching in Dynamic Brittle Fracture Introduction A brief review of literature on crack branching Theoretical models and experimental results on dynamic brittle 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 models Brief Review of the bond-based Peridynamic model An accurate and efficient quadrature scheme Peridynamic 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 boundaries Load case 2: stress on pre-crack surfaces Load case 3: velocity boundary conditions Influence of sample geometry 10.5.3.1 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 Introduction Analytical PD-based normal cohesive law Case 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 law Case 1 – No bonds have reached critical stretch Case 2 – Bonds have exceeded the critical stretch PD-based mixed-mode cohesive law Conclusion Peridynamic modeling of fiber-reinforced composites Introduction 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 direction Uniaxial 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 damage behavior Damage-dependent critical bond strain Critical bond strain dependence on compressive strains along other directions Surface 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 elastic response of a multilayered system to impact 13.4 Computational results for damage progression in the seven-layer glass system Damage evolution for the cross-section Damage evolution in the first layer Damage evolution in the second layer Damage evolution in the fourth layer Damage evolution in the seventh layer Conclusions V Multiphysics and Multiscale Modeling Coupling Local and Nonlocal Models Introduction Energy-based blending schemes The Arlequin method Description of the coupling model A numerical example The morphing method Overview 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 Summary A Peridynamic model for corrosion damage Abstract Introduction Electrochemical Kinetics Problem formulation of 1D pitting corrosion The peridynamic formulation for 1D pitting corrosion Results and discussion of 1D pitting corrosion Pit corrosion depth proportional to square root t Activation-controlled, diffusion-controlled, and IR-controlled corrosion 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 for pitting 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 corrosion with Concentration-Dependent Damage model Peridynamics for Coupled Field Equations Introduction 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 Hygrothermomechanics Numerical solution to peridynamic field equations Correction of PD material parameters Boundary conditions Essential boundary conditions Natural boundary conditions Example 1 Example 2 Example 3 Applications Coupled nonuniform heating and deformation Coupled nonuniform moisture and deformation in a square plate Coupled fluid pore pressure and deformation Coupled electrical, temperature, deformation, and vacancy diffusion Remarks"