586 Pages 230 B/W Illustrations
    by Chapman & Hall

    586 Pages 230 B/W Illustrations
    by Chapman & Hall

    Continue Shopping

    This handbook covers the peridynamic modeling of failure and damage. Peridynamics is a reformulation of continuum mechanics based on integration of interactions rather than spatial differentiation of displacements. The book extends the classical theory of continuum mechanics to allow unguided modeling of crack propagation/fracture in brittle, quasi-brittle, and ductile materials; autonomous transition from continuous damage/fragmentation to fracture; modeling of long-range forces within a continuous body; and multiscale coupling in a consistent mathematical framework.


    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


    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


    Local limits and asymptotically compatible discretizations


    Local PDE limits of linear peridynamic models

    Discretization schemes and discrete local limits

    Asymptotically compatible schemes for peridynamics


    Roadmap for Software Implementation


    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


    III Material Models and Links to Atomistic Models

    Constitutive Modeling in Peridynamics


    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


    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


    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


    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


    Analytical Verification

    IV Modeling Material Failure and Damage

    Dynamic brittle fracture as an upscaling of unstable mesoscopic dynamic


    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


    Crack Branching in Dynamic Brittle Fracture


    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 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


    Relations Between Peridynamic and Classical Cohesive Models 


    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


    Peridynamic modeling of fiber-reinforced composites


    Peridynamic analysis of a lamina

    Peridynamic analysis of a laminate 

    Numerical results


    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


    Convergence studies and damage models that influence the damage


    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


    V Multiphysics and Multiscale Modeling

    Coupling Local and Nonlocal Models


    Energy-based blending schemes

    The Arlequin method

    Description of the coupling model

    A numerical example

    The morphing method


    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


    A Peridynamic model for corrosion damage



    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 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



    Convergence study for 1D diffusion-controlled corrosion

    Convergence study for 2D activation-controlled corrosion

    with Concentration-Dependent Damage model

    Peridynamics for Coupled Field Equations


    Diffusion Equation

    Thermal diffusion

    Moisture diffusion

    Electrical conduction

    Coupled Field Equations


    Thermal diffusion with a structural coupling term

    Equation of motion with a thermal coupling term


    Mechanical deformation due to fluid pressure

    Fluid flow in porous medium



    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


    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


    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