Handbook of Peridynamic Modeling  book cover
1st Edition

Handbook of Peridynamic Modeling

ISBN 9781482230437
Published October 25, 2016 by Chapman and Hall/CRC
586 Pages 230 B/W Illustrations

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Book Description

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.

Table of Contents


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


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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