Peridynamic theory and its applications.

*(English)*Zbl 1295.74001
New York, NY: Springer (ISBN 978-1-4614-8464-6/hbk; 978-1-4614-8465-3/ebook). xii, 289 p. (2014).

The book is devoted to a comparatively new computational technique in mechanics of solids – peridynamic (PD) theory which, in particular by using a natural way, introduces the initiation and propagation of fracture without additional failure conditions typical for classical approaches to solid mechanics.

The book is divided into thirteen chapters. The introductory Chapter 1 gives a brief discussion of local (classical) and nonlocal (to which the peridynamics relates) theories discussing their advantages and shortcomings. Chapter 2 presents the main characteristics, ideas, concepts, governing equations of the peridynamic theory with description of vector fields, laws, and initial and constraint conditions of solid mechanics interpreted in the framework of the peridynamic theory. Chapter 3 describes local interactions by using PD based on the equations of motion, Cauchy stresses and strain energy density represented through peridynamic forces. PD for isotropic materials is presented in Chapter 4, introducing the corresponding material parameters and considering 1D, 2D and 3D structures. Moreover, the representation of surface effects by using peridynamics occupies an important place in this chapter. Laminated composites and influence of laminates on PD description are in the center of Chapter 5, where the corresponding governing equations are presented. Peridynamic material parameters are considered for some particular cases of shear strain and stretching of lamina and laminate. The authors also discuss here the surface effects influencing the material parameters of the laminate. Chapter 6 describes the damage prediction in the framework of PD, in which the material damage is introduced via elimination of interactions (micropotentials) among the material points. As a result, failure load and crack path prediction for linear elastic material are introduced in a natural way by using peridynamic simulations.

Numerical (collocation) methods and the corresponding procedures are discussed in Chapter 7. In particular, the authors consider here the issues of spatial discretization, the volume correction procedure required to correct extra volume, time integration, numerical stability and convergence, adaptation procedures and surface effects, boundary conditions, fracture processes and also spatial partitioning and application of parallel computations. Solutions for many benchmark problems and comparison of PD results with those of classical continuum mechanics are given in Chapter 8. As specific examples, the authors consider different quasi-static and dynamic problems for bars, plate lamina and blocks of material in the absence of failure prediction. Chapter 9 presents solutions for various problems considering failure initiation and propagation. The PD predictions are compared with finite element (FE) solutions. With this aim, the examples of isotropic plates with a stretching hole and pre-existing crack are considered together with temperature problems for strips and rectangular plates. Chapter 10 concerns the PD modeling of contact between two bodies in an impact event considering a rigid or deformable indenter and a deformable target body. The PD results are compared with FE predictions and with the results of the Kalthoff-Winkler experiment. The important coupling of the PD theory with finite element modeling is considered in Chapter 11. Discussing the advantages and shortcomings of both methods for different regions of the considered sample, the authors introduce interface elements between FE and PD regions and fulfill a direct coupling. The validity of the direct coupling approach is demonstrated by considering bars and plates with hole.

The basics of PD thermal diffusion are given in Chapter 12. The PD approaches are applied to the equation of thermal diffusion with initial and boundary conditions. The boundary conditions on temperature are imposed in a “fictitious material layer” along the boundary of a non-zero volume. Then the microconductivity is discussed in 1D, 2D and 3D cases. Numerical techniques are employed to solve the PD thermal diffusion equation. Finally, are above-considered surface effects and the developed numerical schemes are applied to different thermal problems. Chapter 13 compares the coupled PD equation based on thermodynamic considerations in the case of local and nonlocal theories. Similar to the classical thermodynamics, the thermodynamics laws, parameters and terms are introduced in the framework of PD, in particular in the non-dimensional form. The strategies for numerical realization of coupled PD equations of the thermoelasticity are discussed, and the validity of the fully coupled PD thermomechanical equations is established by using proper examples of bars, plates and blocks of material.

In a whole, the book is very interesting from the methodical viewpoint, presenting a comparatively new theory of solid mechanics, accompanying the text by many examples, which can be useful to students studying the novel approaches to solid mechanics and related topics, and also to their teachers preparing lectures and practical works.

The book is divided into thirteen chapters. The introductory Chapter 1 gives a brief discussion of local (classical) and nonlocal (to which the peridynamics relates) theories discussing their advantages and shortcomings. Chapter 2 presents the main characteristics, ideas, concepts, governing equations of the peridynamic theory with description of vector fields, laws, and initial and constraint conditions of solid mechanics interpreted in the framework of the peridynamic theory. Chapter 3 describes local interactions by using PD based on the equations of motion, Cauchy stresses and strain energy density represented through peridynamic forces. PD for isotropic materials is presented in Chapter 4, introducing the corresponding material parameters and considering 1D, 2D and 3D structures. Moreover, the representation of surface effects by using peridynamics occupies an important place in this chapter. Laminated composites and influence of laminates on PD description are in the center of Chapter 5, where the corresponding governing equations are presented. Peridynamic material parameters are considered for some particular cases of shear strain and stretching of lamina and laminate. The authors also discuss here the surface effects influencing the material parameters of the laminate. Chapter 6 describes the damage prediction in the framework of PD, in which the material damage is introduced via elimination of interactions (micropotentials) among the material points. As a result, failure load and crack path prediction for linear elastic material are introduced in a natural way by using peridynamic simulations.

Numerical (collocation) methods and the corresponding procedures are discussed in Chapter 7. In particular, the authors consider here the issues of spatial discretization, the volume correction procedure required to correct extra volume, time integration, numerical stability and convergence, adaptation procedures and surface effects, boundary conditions, fracture processes and also spatial partitioning and application of parallel computations. Solutions for many benchmark problems and comparison of PD results with those of classical continuum mechanics are given in Chapter 8. As specific examples, the authors consider different quasi-static and dynamic problems for bars, plate lamina and blocks of material in the absence of failure prediction. Chapter 9 presents solutions for various problems considering failure initiation and propagation. The PD predictions are compared with finite element (FE) solutions. With this aim, the examples of isotropic plates with a stretching hole and pre-existing crack are considered together with temperature problems for strips and rectangular plates. Chapter 10 concerns the PD modeling of contact between two bodies in an impact event considering a rigid or deformable indenter and a deformable target body. The PD results are compared with FE predictions and with the results of the Kalthoff-Winkler experiment. The important coupling of the PD theory with finite element modeling is considered in Chapter 11. Discussing the advantages and shortcomings of both methods for different regions of the considered sample, the authors introduce interface elements between FE and PD regions and fulfill a direct coupling. The validity of the direct coupling approach is demonstrated by considering bars and plates with hole.

The basics of PD thermal diffusion are given in Chapter 12. The PD approaches are applied to the equation of thermal diffusion with initial and boundary conditions. The boundary conditions on temperature are imposed in a “fictitious material layer” along the boundary of a non-zero volume. Then the microconductivity is discussed in 1D, 2D and 3D cases. Numerical techniques are employed to solve the PD thermal diffusion equation. Finally, are above-considered surface effects and the developed numerical schemes are applied to different thermal problems. Chapter 13 compares the coupled PD equation based on thermodynamic considerations in the case of local and nonlocal theories. Similar to the classical thermodynamics, the thermodynamics laws, parameters and terms are introduced in the framework of PD, in particular in the non-dimensional form. The strategies for numerical realization of coupled PD equations of the thermoelasticity are discussed, and the validity of the fully coupled PD thermomechanical equations is established by using proper examples of bars, plates and blocks of material.

In a whole, the book is very interesting from the methodical viewpoint, presenting a comparatively new theory of solid mechanics, accompanying the text by many examples, which can be useful to students studying the novel approaches to solid mechanics and related topics, and also to their teachers preparing lectures and practical works.

Reviewer: I. A. Parinov (Rostov-na-Donu)