Introduction to Practical Peridynamics

The following sections are included:

• Dedication
• Preface
• Contents

Engineering is the application of mathematics, science, and human ingenuity to devise the trappings of civilization. Today we call the early engineers and scientists of the Age of Reason “natural philosophers”, and most of them were practical people who invented abstract engineering models at the same time as they were designing and constructing philosophical frameworks and technologies to serve humanity…

Before we attempt to revise the theory of deformable solids, it is well to understand the history of its development. Following Newton's 1687 masterpiece entitled “Philosophiæ Naturalis Principia Mathematica”, (Newton, 1999), in which the concepts of force, mass, and acceleration were established, and the calculus simultaneously invented, natural philosophers, such as Euler and the Bernoulli brothers, successfully modeled elastic axial members such as beams and columns – without the concepts of strain and stress…

The reason for including this chapter is to help in understanding the similarities and differences between continuum mechanics, the state based peridynamic model, and the state based peridynamic lattice model (SPLM)…

The entire discipline of fracture mechanics is necessary because of limitations in the scope of applicability of continuum mechanics. Fracture is a mode of deformation not easily represented by continuum mechanics…

Silling (Silling, 2000) wrote the seminal journal paper describing the first theory of peridynamics. In a generalization of Navier's approach, this paper assumes that the force between any two particles in a material body is a function only of the states of the two particles – and the inter-particle force is insensitive to the states of other nearby particles. This original theory of peridynamics was later named the “bond-based peridynamic theory”, to distinguish it from the more general “state-based peridynamic theory” published in (Silling, S.A., Epton, M., Weckner, O., Xu, J., and Askari, E., 2007). The bond-based theory seems reasonable, but was found by Silling to be insufficiently general to capture some known behaviors of real materials. We present the bond-based theory in this chapter as a way to motivate an understanding of the more general state-based theory of peridynamics, which we defer to Chapter 8…

In Chapter 5, following (Silling, 2000), we presented the bond-based continuum peridynamic model. From Eq. 5.5, we see that this model is represented by an integro-differential equation that must be satisfied at each of an infinite number of points X. In addition, the spatial integral involves an infinite number of terms in the summation. Discretization of the system of equations is necessary for computational solution…

We have introduced the continuum bond-based peridynamic model in Chapter 5. To implement the model on a computer, however, we must first discretize the model. In Chapter 6, we developed a lattice-based model for solids. The advantage of the lattice-based model is that it provides an already-discretized solid model that can then serve as the domain supporting deformed solids, including those with cracks…

In the last chapter, we saw that the bond-based peridynamic model is unable to simulate three-dimensional elastic bodies with Poisson's ratio different from one-quarter. In this chapter, we provide background and motivation for the more general state-based peridynamic theory…

In this chapter, we develop an elastic state-based peridynamic lattice model (SPLM). We defined the lattice Λ^N_R in Definition 6.3, where N_R is the dimensionality of the space spanned by the lattice. In the reference configuration, for one-dimensional problems we assume a straight row of equally spaced particles Λ¹, a close-packed hexagonal lattice Λ² for two-dimensional problems, and a face-centered cubic close-packed lattice Λ³ for three-dimensional problems. We show that the SPLM is capable of simulating a linear elastic classical continuum, and furthermore that it is capable of simulating geometrically nonlinear large-deformation behavior. The SPLM presented in this chapter is “regular” as opposed to micropolar, in that we do not consider particle rotations as degrees of freedom; only particle translations are included as degrees of freedom. Furthermore, we assume that the material is simple, ordinary, and mobile, as defined by Definitions 8.18, 8.20 and 8.21, respectively…

The theory of plasticity was developed in recognition of the fact that many materials do not “spring back” after the load has been removed. At sufficiently high deviatoric stress levels, solid materials flow almost like fluids. Plasticity models of solids, used in conjunction with elasticity models, replicate the fluid-like behavior seen at sufficiently high levels of deviatoric stress. Fig. 10.1 shows a clay body plastically deforming…

The subject of this chapter is damage mechanics, which goes under various names: the “smeared crack model”, “continuum damage mechanics”, and “nonlocal continuum damage mechanics”. We question under what conditions damage can be, or should be, thought of as a continuum phenomenon. The bulk of the literature has treated damage mechanics as a branch of continuum mechanics, and as such, engineers have modelled damage primarily using the finite element method…

The purpose of this chapter is to present methods for solving the peridynamic equations of particle motion, Eqs. 5.5 and 8.30. These second-order differential equations represent the application of Newton's second law, F = ma, to each particle in each lattice body ℒ_ℛ…

The purpose of this chapter is to describe how one can implement the state-based peridynamic lattice model (SPLM) either on a single processor (a laptop or desktop computer) or on many processors running in parallel using the MPI (Message Passing Interface) protocol. The molecular dynamics research community has led the way in developing methods for simulating particle dynamics on massively parallel computers…

The desire to model reinforced concrete structures has played a major role in driving advances in computational solid modeling. Reinforced concrete is a nonhomogeneous composite involving a concrete medium interacting with embedded steel reinforcing bars. Characterizing bond-slip behavior between concrete and rebar has been less than successful using traditional ideas of classical continuum mechanics (Gerstle, W. H. and Ingraffea, A.R., 1991). Complicating the situation even more, plain concrete is a composite material at the mesoscale, consisting of gravel, sand, cement and water. Featuring nonlinear damage and fracture behavior in the tensile regime and quasi-continuous plastic-like behavior in the compressive regime, concrete has been a challenge to model using finite element methods. With so many size scales, it has been difficult for modelers to settle upon the size scale of interest in modeling concrete structures. The SPLM developed in this book provides a natural framework for specifying the minimum size scale of the model…

The following sections are included:

• Bibliography
• Index