Peridynamics: A Nonlocal Continuum Theory

Abstract

The peridynamic theory is a nonlocal theory of continuum mechanics based on an integro-differential equation without spatial derivatives, which can be easily applied in the vicinity of cracks, where discontinuities in the displacement field occur. In this paper we give a survey on important analytical and numerical results and applications of the peridynamic theory.

Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the U.S. Department of Energy under contract DE-AC04-94AL85000.

1 Introduction

Peridynamics is a nonlocal theory in continuum mechanics. Linear elastic behaviour of solids is well described by the partial differential equation

$$\begin{array}{rcl} & & \rho (\vec{x}){\partial }_{t}^{2}\vec{u}(\vec{x},t) = (L\vec{u})(\vec{x},t) +\vec{ b}(\vec{x},t)\,,\quad (\vec{x},t) \in \Omega \times (0,T), \\ & & \text{ with }(L\vec{u})(\vec{x},t) := (\lambda + \mu )\mathrm{grad}\mathrm{div}\vec{u}(\vec{x},t) + \mu \mathrm{div}\mathrm{grad}\vec{u}(\vec{x},t),\end{array}$$

(1)

which is derived from Newton’s second law Force = Mass × Acceleration. In (1) ρ describes the density of the body; the right-hand side consists of the external force density $\vec{b}$ as well as inner tensions and macroscopic forces with Lamé parameters $\lambda$ and $\mu$. The variable $\vec{u} :\bar{ \Omega } \times [0,T] \rightarrow {\mathbb{R}}^{d}$ with $\Omega \subset {\mathbb{R}}^{d}$ and d ∈ { 1, 2, 3} is the displacement field. The limitation of the elastic model (1) is the implicit assumption that the deformation is twice continuously differentiable, which manifests in the inability to model spontaneous cracks and fractures. These are discontinuities of the displacement field and thus, (1) is not defined on cracks or fractures.

In 2000, Silling [42] proposed the peridynamic continuum theory that makes minimal regularity assumptions on the deformation. Instead of spatial differential operators, integration over differences of the displacement field is used to describe the existing, possibly nonlinear, forces between particles of the solid body. The resulting derivative-free nonlocal peridynamic equation of motion reads, in the general nonlinear form,

$$\begin{array}{rcl} & & \rho (\vec{x}){\partial }_{t}^{2}\vec{u}(\vec{x},t) \\ & & \quad =\int\limits_{\mathcal{H}(\vec{x})}\vec{f}(\vec{x},\vec{\hat{x}},\vec{u}(\vec{x},t),\vec{u}(\vec{\hat{x}},t),t)d\vec{\hat{x}} +\vec{ b}(\vec{x},t),\;(\vec{x},t) \in \Omega \times (0,T).\end{array}$$

(2)

This theory is named after the greek words peri (near) and dynamis (force), because in (2) for every particle $\vec{x}$ the force exerted by each surrounding particle $\vec{\hat{x}}$ on $\vec{x}$ is included. The integration domain $\mathcal{H}(\vec{x})$ describes the volume of particles interacting and is typically the open ball of radius δ surrounding $\vec{x}$ intersected with $\Omega$. Then δ is called peridynamic horizon. Beyond this horizon we have

$$\vec{f}(\vec{x},\vec{\hat{x}},\vec{u}(\vec{x},t),\vec{u}(\vec{\hat{x}},t),t) = \mathbf{0},\quad \vec{\hat{x}}\notin \mathcal{H}(\vec{x}).$$

(3)

The reader is referred to the recent survey Silling and Lehoucq [48] emphasizing the mechanical aspects of the peridynamic continuum theory. This includes proposing the peridynamic balance of energy and thermodynamic restrictions so that the second law of thermodynamics is not violated. The principles of classical statistical mechanics are used to derive the energy and momentum balance laws in Lehoucq and Sears [34]. The aim of this contribution is to give a survey on important analytical and numerical results and applications of this nonlocal continuum theory.

Nonlocal theories taking into account effects of long-range interaction in elastic materials and their application to problems of solid and fracture mechanics have been studied for a long time, cf. the pioneering work by Kröner [32] and Eringen [28] and the references cited therein, the monographs by Kunin [33] and Rogula [40], and more recently (without being exhaustive) Altan [5, 6], Bažant and Jirásek [9], Chen et al. [15, 16], Lei et al. [35], Pisano and Fuschi [37], Polizzotto [38, 39], and Wang and Dhaliwal [49, 50]. In contrast to the peridynamic theory, all these theories rely on spatial derivatives.

This paper is organized as follows. In Sect. 2 we describe the peridynamic model. First, the bond-based model is demonstrated, where the focus lies on the pairwise force function $\vec{f}$. Then we derive the linearized model by linearizing the pairwise force function. Afterwards, we describe the state-based peridynamic model, which is a generalization of the bond-based model. Finally in Sect. 2, other nonlocal models in elasticity are presented.

In Sect. 3 a survey on the mathematical analysis of the linear bond-based model, the nonlinear bond-based model and state-based model is given. In addition, a nonlocal vector calculus is presented, which is a useful tool for describing the state-based model.

In Sect. 4 peridynamics is studied as a multiscale method. First, results on the limit of vanishing nonlocality are stated, that is the limit of (2) as the peridynamic horizon δ goes to zero. Then the modeling of composite materials including the use of two-scale convergence is considered.

Section 5 deals with the numerical approximation of peridynamics by the quadrature formula method.

Finally in Sect. 6 a few notes on applications and numerical simulations are given.

2 Peridynamic Model

2.1 Bond-Based Model

Since there are no spatial derivatives, boundary conditions are not needed in general for the partial integro-differential equation (2) (although this depends on the singularity behaviour of the integral kernel and the functional analytic setting). Nevertheless, “boundary” conditions can be imposed by prescribing $\vec{u}$ in a strip along the boundary constraining the solution along a nonzero volume. Hence, (2) is complemented with the initial data

$$\vec{u}(\cdot,0) =\vec{ {u}}_{\mathbf{0}}\quad \text{ and}\quad {\partial }_{t}\vec{u}(\cdot,0) =\vec{\dot{ {u}}}_{\mathbf{0}}\,.$$

(4)

The interaction between particles within the given horizon is called bond. Since bonds are defined pairwise, this results in a Poisson ratio of $\nu = 1/4$. Thus, due to the relationship $\lambda = 2\mu \nu /(1 - 2\nu )$, the Lamé parameters $\lambda$ and μ are identical in the linearized bond-based peridynamic theory. Linear elastic, isotropic material can have a Poisson number between − 1 and 1 ∕ 2 though many examples can already be found for a Poisson number close to 1 ∕ 4 such as composite materials, polymers, sand, glass, and iron.

In the following we will use the notation

$$\vec{\xi } =\vec{\hat{ x}} -\vec{ x},\quad \vec{\eta } =\vec{ u}(\vec{\hat{x}},t) -\vec{ u}(\vec{x},t).$$

Note that $\vec{\xi } +\vec{ \eta }$ is the relative position of the deformed configuration (see Fig. 1). Assuming no explicit dependence on time and dependence only on differences in the displacement field, the pairwise force function reads

$$\vec{f}(\vec{x},\vec{\hat{x}},\vec{u}(\vec{x},t),\vec{u}(\vec{\hat{x}},t),t) =\vec{ f}(\vec{x},\vec{\hat{x}},\vec{\eta }).$$

Combining Newton’s third law actio et reactio, which gives

$$\vec{f}(\vec{\hat{x}},\vec{x},-\vec{\eta }) = -\vec{f}(\vec{x},\vec{\hat{x}},\vec{\eta }),$$

(5)

with the balance of angular momentum leads to the conclusion that the pairwise force function is parallel to $\vec{\xi } +\vec{ \eta }$. We call $\vec{f}$ homogenous if

$$\vec{f}(\vec{x},\vec{\hat{x}},\vec{\eta }) =\vec{ f}(\vec{\xi },\vec{\eta })$$

is fulfilled for all $\vec{\xi }$ and $\vec{\eta }$.¹ Furthermore, a material is called microelastic if there exists a so-called pairwise micropotential w such that $\vec{f}(\vec{\xi },\vec{\eta }) = {\nabla }_{\vec{\eta }}w(\vec{\xi },\vec{\eta })$. One of the simplest nonlinear models that has been suggested is the proportional microelastic material model with

$$\vec{f}(\vec{\xi },\vec{\eta }) = {c}_{d,\delta }\,s(\vec{\xi },\vec{\eta })\, \frac{\vec{\xi } +\vec{ \eta }} {\left \vert \vec{\xi } +\vec{ \eta }\right \vert }{\chi }_{(0,\delta )}(\left \vert \vec{\xi }\right \vert ),$$

(6)

where

$$s(\vec{\xi },\vec{\eta }) = \frac{\left \vert \vec{\xi } +\vec{ \eta }\right \vert -\left \vert \vec{\xi }\right \vert } {\left \vert \vec{\xi }\right \vert }$$

denotes the bond stretch² that is the relative change of the length of a bond. By |  ⋅ | we denote the Euclidean norm. In this case the related micropotential is given by

$$w(\vec{\xi },\vec{\eta }) = \frac{{c}_{d,\delta }{s}^{2}(\vec{\xi },\vec{\eta })} {2} \left \vert \vec{\xi }\right \vert {\chi }_{(0,\delta )}(\left \vert \vec{\xi }\right \vert ),$$

where we have chosen the micropotential such that $w(\vec{\xi },\mathbf{0}) = 0$. The constants of proportionality are

$${c}_{1,\delta } = \frac{18K} {5{\delta }^{2}},\;{c}_{2,\delta } = \frac{72K} {5\pi {\delta }^{3}},\;{c}_{3,\delta } = \frac{18K} {\pi {\delta }^{4}}$$

with the bulk modulus $K = \lambda (1 + \nu )/(3\nu ) = 5\lambda /3$.

Fig. 1

Notation for bond-based model

Let the kinetic and elastic energy density be defined as

$$\begin{array}{rcl}{ e}_{kin}(\vec{x},t)& =& \frac{1} {2}\rho (\vec{x}){\left \vert {\partial }_{t}\vec{u}(\vec{x},t)\right \vert }^{2}, \\ {e}_{el}(\vec{x},t)& =& \frac{1} {2}\int\limits_{\mathcal{H}(\vec{x})}w(\vec{\hat{x}} -\vec{ x},\vec{u}(\vec{\hat{x}},t) -\vec{ u}(\vec{x},t))d\vec{\hat{x}}\,, \\ \end{array}$$

and let $l = {e}_{kin} - {e}_{el} - {e}_{ext}$ denote the total energy density with ${e}_{ext}(\vec{x},t) = -\vec{b}(\vec{x},t) \cdot \vec{ u}(\vec{x},t)$ induced by the external force density $\vec{b}$. Then Eq. (2) can be derived from the variational problem of minimizing the functional

$$\vec{u}\mapsto \int\limits_{0}^{T}\int\limits_{\Omega }l(\vec{x},t)d\vec{x}\mathit{dt},$$

see also Emmrich and Weckner [26].

In order to model cracks and to simulate fracture, bonds are allowed to break. This is realized by multiplying the pairwise force function, given, e.g., by (6), with the function

$$\mu (\vec{\xi },\vec{\eta },t) = \left \{\begin{array}{ll} 1&\text{ if}\;s(\vec{\xi },\vec{u}(\vec{\hat{x}},\tau ) -\vec{ u}(\vec{x},\tau )) \leq {s}_{0}\;\forall \tau \leq t\\ 0 &\text{ otherwise} \end{array} \right.$$

for some critical bond stretch s ₀ for breakage (see also Silling and Askari [45] as well as Parks et al. [36]). Note that the resulting pairwise force function now explicitly depends on time t.

2.2 Linearization

A first-order approximation justifies for small relative displacements $\vec{\eta }$ the general linear ansatz

$$\vec{f}(\vec{\xi },\vec{\eta }) =\vec{ {f}}_{\mathbf{0}}(\vec{\xi }) +\vec{ C}(\vec{\xi })\vec{\eta }$$

with the stiffness tensor (or micromodulus function) $\vec{C} =\vec{ C}(\vec{\xi })$ and $\vec{{f}}_{\mathbf{0}}$ denoting forces in the reference configuration. Without loss of generality, we may assume $\vec{{f}}_{\mathbf{0}} \equiv \vec{ 0}$ since otherwise $\vec{{f}}_{\mathbf{0}}$ can be incorporated into the right-hand side $\vec{b}$. In general the stiffness tensor $\vec{C}$ is not definite. However, $\vec{C}$ has to be symmetric with respect to its argument as well as with respect to its tensor structure such that $\vec{C}(\vec{\xi }) =\vec{ C}(-\vec{\xi })$ and $\vec{C}{(\vec{\xi })}^{T} =\vec{ C}(\vec{\xi })$. The corresponding micropotential is then given by $w(\vec{\xi },\vec{\eta }) =\vec{ {\eta }}^{T}\vec{C}(\vec{\xi })\vec{\eta }/2$. In view of (3) we shall require $\vec{C}(\vec{\xi }) =\vec{ 0}$ if $\vert \vec{\xi }\vert \geq \delta$.

The stiffness tensor can be shown to read as

$$\vec{C}(\vec{\xi }) = {\lambda }_{d,\delta }(\left \vert \vec{\xi }\right \vert )\vec{\xi } \otimes \vec{ \xi }$$

for a linear microelastic material (see also Silling [42]), where $\otimes$ denotes the dyadic product. The function ${\lambda }_{d,\delta } : {\mathbb{R}}_{0}^{+} \rightarrow \mathbb{R}$ with ${\lambda }_{d,\delta }(r) = 0$ for $r \geq \delta$ determines the specific material model and depends on the dimension d and the peridynamic horizon δ. The linear peridynamic equation of motion (2) now reads as

$$\begin{array}{rcl} & & \rho (\vec{x}){\partial }_{t}^{2}\vec{u}(\vec{x},t) = ({L}_{ d,\delta }\vec{u})(\vec{x},t) +\vec{ b}(\vec{x},t)\,,\quad (\vec{x},t) \in \Omega \times (0,T)\,, \\ & & \text{ with } \\ & & ({L}_{d,\delta }\vec{u})(\vec{x},t) :=\int\limits_{\mathcal{H}(\vec{x})}\vec{{\lambda }}_{d,\delta }(\left \vert \vec{\hat{x}} -\vec{ x}\right \vert )(\vec{\hat{x}} -\vec{ x}) \otimes (\vec{\hat{x}} -\vec{ x}) \cdot (\vec{u}(\vec{\hat{x}},t) -\vec{ u}(\vec{x},t))d\vec{\hat{x}}.\end{array}$$

(7)

The recent report Du et al. [21] also demonstrates how the one-dimensional equation (7) can be written as two first-order in time nonlocal advection equations. Note that ${\lambda }_{d,\delta }$ can have a singularity at r = 0. The standard example is the linearization of (6) with ${\lambda }_{d,\delta }(r) = {c}_{d,\delta }{r}^{-3}{\chi }_{(0,\delta )}(r)$. Unfortunately, in this model, the interaction jumps to zero if $r = \delta$. This jump discontinuity can be avoided by taking ${\lambda }_{d,\delta }(r) =\tilde{ {c}}_{d,\delta }{r}^{-3}\exp (-{\delta }^{2}/({\delta }^{2} - {r}^{2})){\chi }_{(0,\delta )}(r)$ with a suitable constant of proportionality $\tilde{{c}}_{d,\delta }$, see also Emmrich and Weckner [25]. This is of advantage also for the numerical approximation relying on quadrature.

2.3 State-Based Model

The integral

$$\int\limits_{\mathcal{H}(\vec{x})}\vec{f}\big{(}\vec{\hat{x}} -\vec{ x},\vec{u}(\vec{\hat{x}},t) -\vec{ u}(\vec{x},t)\big{)}\,d\vec{\hat{x}}$$

expresses that the internal force density at $\vec{x}$ is a summation of forces over all vectors $\vec{\hat{x}} -\vec{ x}$. Moreover, the summands are independent from each other. As demonstrated by Silling [42, §11], this effectively limits Poisson’s ratio to a value of one-fourth for homogenous deformations of linear isotropic solids. In the same paper, a generalization is presented that augments the above integral with the term

$$e\big{(}{\vartheta}(\vec{x})\big{)}\,\text{, where }{\vartheta}(\vec{x}) = \int j\big{(}\vert \vec{\hat{x}} -\vec{ x}\vert \big{)}\vert \vec{\hat{x}} -\vec{ x} +\vec{ u}(\vec{\hat{x}},t) -\vec{ u}(\vec{x},t)\vert \,d\vec{\hat{x}}.$$

The quantity ${\vartheta}$ is a weighted average of all the bonds $\vec{\hat{x}} -\vec{ x} +\vec{ u}(\vec{\hat{x}},t) -\vec{ u}(\vec{x},t)$ and provides the volume of a deformed sphere that is centered at $\vec{x}$ in the reference configuration; the quantity e then acts as a volume-dependent strain energy term that incorporates the collective motion of all the $\vec{\hat{x}} -\vec{ x}$ simultaneously. The linearized operator is then shown to circumvent the restriction of Poisson’s ratio to a value of one-fourth to its allowable values (see Silling et al. [46]). The recent report Seleson et al. [41] demonstrates a symbiotic relationship between this generalization and an embedded atom (EAM) molecular dynamic potential.

Silling et al. [46] introduces the peridynamic state theory, a subsequent generalization to the approach in Silling [42, §15]. The integral in (2) is replaced by ${L}_{\text{ PD}}\vec{u}$ where

$$({L}_{\text{ PD}}\vec{u})(\vec{x},t) =\int\limits_{\mathcal{H}(\vec{x})}\Big{(}\underline{\mathbf{T}}[\vec{x},t]\langle \vec{\hat{x}} -\vec{ x}\rangle -\underline{\mathbf{T}}[\vec{\hat{x}},t]\langle \vec{x} -\vec{\hat{ x}}\rangle \Big{)}\;d\vec{\hat{x}}$$

(8)

leading to the equation of motion

$$\rho (\vec{x}){\partial }_{t}^{2}\vec{u}(\vec{x},t) = ({L}_{ \text{ PD}}\vec{u})(\vec{x},t) +\vec{ b}(\vec{x},t),\;(\vec{x},t) \in \Omega \times (0,T).$$

(9)

The term ${L}_{\text{ PD}}\vec{u}$ is a functional of displacement representing the internal force density (per unit volume) that is exerted upon $\vec{x}$ by other points in the body. The force state $\underline{\mathbf{T}}[\vec{x},t]$ at $\vec{x}$ at time t is a mapping from the bond $\vec{\hat{x}} -\vec{ x}$ to a force density (per unit volume) at $\vec{x}$ and is assumed to be zero outside the horizon δ. A material model is therefore provided by a relation between the deformation near $\vec{x}$ and the force state at $\vec{x}$, e.g.,

$$\underline{\mathbf{T}} = \underline{\mathbf{T}}(\underline{\mathbf{Y}})$$

(10)

where $\underline{\mathbf{Y}}$ is the deformation state, a mapping from bonds connected to any $\vec{x}$ to the deformed images of these bonds, e.g.,

$$\begin{array}{rcl} \underline{\mathbf{Y}}[\vec{x},t]\langle \vec{\hat{x}} -\vec{ x}\rangle & =& \vec{\hat{x}} -\vec{ x} +\vec{ u}(\vec{\hat{x}},t) -\vec{ u}(\vec{x},t).\end{array}$$

(11)

Although the deformation or force state maps vectors into vectors, they are more complex than second order tensors because the mapping may be nonlinear and even discontinuous. The constitutive relation (10) explains that the force state maps the collective motion about $\vec{x}$ in contrast to the relation $f(\vec{\hat{x}} -\vec{ x},\vec{u}(\vec{\hat{x}},t) -\vec{ u}(\vec{x},t))$. A bond-based force is recovered via the identification $\underline{\mathbf{T}}[\vec{x},t]\langle \vec{\hat{x}} -\vec{ x}\rangle = \tfrac{1} {2}f\big{(}\vec{\hat{x}} -\vec{ x},\vec{u}(\vec{\hat{x}},t) -\vec{ u}(\vec{x},t)\big{)}$ because then

$$\begin{array}{rcl} & & \underline{\mathbf{T}}[\vec{x},t]\langle \vec{\hat{x}} -\vec{ x}\rangle -\underline{\mathbf{T}}[\vec{\hat{x}},t]\langle \vec{x} -\vec{\hat{ x}}\rangle \\ & =& \frac{1} {2}f\big{(}\vec{\hat{x}} -\vec{ x},\vec{u}(\vec{\hat{x}},t) -\vec{ u}(\vec{x},t)\big{)} -\frac{1} {2}f\big{(}\vec{x} -\vec{\hat{ x}},\vec{u}(\vec{x},t) -\vec{ u}(\vec{\hat{x}},t)\big{)} \\ & =& f\big{(}\vec{\hat{x}} -\vec{ x},\vec{u}(\vec{\hat{x}},t) -\vec{ u}(\vec{x},t)\big{)} \\ \end{array}$$

where we invoked (5) for the last equality. This identification reveals another important distinction between the bond-based and state theories. Namely, that force interaction is carried by the bond in the former theory while the interaction is split between the force state at $\vec{x}$ and $\vec{\hat{x}}$ in the latter theory.

The paper [46] introduces a linear peridynamic state material model generalizing the linear bond-based material model. Silling [44] further generalizes the linear peridynamic state material model. A key concept is that of the modulus state, a second order tensor,

$$\underline{\mathbb{K}} = \nabla \underline{\mathbf{T}}\langle \underline{\mathbf{Y}}\rangle : {\mathbb{R}}^{3} \times {\mathbb{R}}^{3} \rightarrow {\mathbb{R}}^{3\times 3},$$

the Fréchet derivative of the force state with respect to $\underline{\mathbf{Y}}$. The modulus state represents the peridynamic analogue of the fourth order tensor in the classical linear elastic theory and so allows us to linearize a force state about a time-independent deformation state ${\underline{\mathbf{Y}}}^{0}$, e.g.,

$$\begin{array}{rcl} \underline{\mathbf{T}}(\underline{\mathbf{U}})\langle \vec{\hat{x}} -\vec{ x}\rangle = \underline{\mathbf{T}}({\underline{\mathbf{Y}}}^{0})\langle \vec{\hat{x}} -\vec{ x}\rangle +\int\limits_{\mathcal{H}(\vec{x})}\underline{\mathbb{K}}\langle \vec{\hat{x}} -\vec{ x},\vec{\tilde{x}} -\vec{ x}\rangle \underline{\mathbf{U}}\langle \vec{\tilde{x}} -\vec{ x}\rangle \;d\vec{\tilde{x}}& &\end{array}$$

(12)

where $\underline{\mathbf{U}}$ is the displacement state field defined by

$$\begin{array}{rcl} \underline{\mathbf{U}}[\vec{x}]\langle \vec{\hat{x}} -\vec{ x}\rangle & ={ \underline{\mathbf{Y}}}^{0}[\vec{x}]\langle \vec{\hat{x}} -\vec{ x}\rangle - (\vec{\hat{x}} -\vec{ x}).&\end{array}$$

(13)

Assume that the body force b has the decomposition ${\mathbf{b}}^{0} + \hat{\mathbf{b}}$ for some time-independent external force density ${\mathbf{b}}^{0}$ so that

$$\begin{array}{rcl} \int\limits_{\mathcal{H}(\vec{x})}\Big{(}{\underline{\mathbf{T}}}^{0}[\vec{x}]\langle \vec{\hat{x}} -\vec{ x}\rangle -{\underline{\mathbf{T}}}^{0}[\vec{\hat{x}}]\langle \vec{x} -\vec{\hat{ x}}\rangle \Big{)}\;d\vec{\hat{x}} +\vec{ {b}}^{0}(\vec{x}) = \mathbf{0},& &\end{array}$$

(14)

where ${\underline{\mathbf{T}}}^{0} \equiv \underline{\mathbf{T}}({\underline{\mathbf{Y}}}^{0})$. Inserting the linearized force state (12) into the equation of motion (9), and invoking relations (13) and (14) results in an equation of motion with body force $\hat{\vec{b}}$ and force state given by the integral operator of (12).

An important material model is the elastic peridynamic material, defined by

$$\underline{\mathbf{T}}(\underline{\mathbf{Y}}) = \nabla W(\underline{\mathbf{Y}})$$

(15)

where $\nabla W$ is the Fréchet derivative of the scalar valued function W, the strain energy density function. This elastic material model is the peridynamic analogue of the classical elastic model

$$\vec{\sigma } = \frac{\partial W} {\partial \vec{F}},$$

where $\vec{\sigma }$ and $\vec{F}$ are the Piola–Kirchoff stress tensor and deformation gradient, respectively. The modulus state is then $\underline{\mathbb{K}} = \nabla \nabla W(\underline{\mathbf{Y}})$. The reader is referred to Silling [44] for example material models.

2.4 Other Nonlocal Models in Elasticity Theory

In Sect. 1, references to various nonlocal models in elasticity are stated. Two examples are given in the following. In [22], Duruk et al. study the one-dimensional nonlocal nonlinear Cauchy problem

$${\partial }_{t}^{2}u(x,t) = {\partial }_{ x}^{2}\int\limits_{\mathbb{R}}\beta (x -\hat{ x})g(u(\hat{x},t))d\hat{x}\,,\quad (x,t) \in \mathbb{R} \times (0,\infty )\,,$$

(16)

complemented with initial conditions. Here β is an integrable kernel function whose Fourier transform satisfies a certain growth condition, and g is some nonlinear function. Global existence and blow-up conditions are investigated. For a so-called double-exponential kernel β equation (16) turns out to be equivalent to the higher-order Boussinesq equation

$${u}_{tt} - {u}_{xx} - a{u}_{xxtt} + b{u}_{xxxxtt} = {(g(u))}_{xx}$$

with positive constants a, b. This example illustrates that, first, nonlocal models so far still depend on spatial derivatives and, second, nonlocal models may be related with local models described by partial differential equations with higher-order spatial derivatives.

In Eringen [28] and Polizzotto [38] the Eringen model is proposed and developed. Combining the equation of motion

$$\rho (x){\partial }_{t}^{2}\vec{u}(\vec{x},t) = \mathrm{div}\vec{\sigma }(\vec{u}(\vec{x},t)) +\vec{ b}(\vec{x},t)$$

with a nonlocal formulation of the stress tensor

$$\vec{\sigma }(\vec{v}(\vec{x})) =\int\limits_{\Omega }\alpha (\left \vert \vec{\hat{x}} -\vec{ x}\right \vert )\vec{s}(\vec{v}(\vec{\hat{x}}))d\vec{\hat{x}},$$

which incorporates the entire body $\Omega$ with a suitable interaction kernel $\alpha$ and the classical stress $\vec{s}$ from Hooke’s law

$$\vec{s}(\vec{v}) =\vec{ S} : \frac{1} {2}\left (\nabla \vec{v} + {(\nabla \vec{v})}^{T}\right )$$

with stiffness tensor $\vec{S}$, gives a nonlocal equation of motion in linear elasticity.

3 Mathematical Analysis of the Peridynamic Model

3.1 Linear Bond-Based Model in L ²

First results on existence, uniqueness and qualitative behaviour of solutions to the peridynamic equation of motion have been presented in Emmrich and Weckner [24] for the infinite peridynamic bar. Besides well-posedness in ${L}^{\infty }$ also nonlinear dispersion relations as well as jump relations for discontinuous solutions have been studied.

In [23], Emmrich and Weckner have proved results on existence, uniqueness, and continuous dependence of the solution for the linear model in an L ^p-setting for $p > 2$ if $d = 2$ and p > 3 ∕ 2 if d = 3. Moreover, a formal representation of the exact solution and a priori estimates are given. In [25], Emmrich and Weckner have proved well-posedness of the linear model in ${L}^{\infty }{(\Omega )}^{d}$ and in ${L}^{2}{(\Omega )}^{d}$ under the condition

$$\int\limits_{0}^{\delta }\left \vert {\lambda }_{ d,\delta }(r)\right \vert {r}^{d+1}\mathit{dr} < \infty.$$

(17)

Theorem 1 ([25]). 

If (17) is fulfilled, then there exists for every $\vec{{u}}_{\mathbf{0}},\,\vec{\dot{{u}}}_{\mathbf{0}} \in {L}^{2}{(\Omega )}^{d}$, $\vec{b} \in {L}^{1}(0,T;{L}^{2}{(\Omega )}^{d})$ a unique solution $\vec{u} \in {\mathcal{C}}^{1}([0,T],{L}^{2}{(\Omega )}^{d})$ to the initial-value problem (7) and (4) that satisfies the a priori estimate

$${\left \|\vec{u}\right \|}_{{\mathcal{C}}^{1}([0,T],{L}^{2}{(\Omega )}^{d})} \leq {C}_{d,\delta }\left ({\left \|\vec{{u}}_{\mathbf{0}}\right \|}_{{L}^{2}{(\Omega )}^{d}} +{ \left \|\vec{\dot{{u}}}_{\mathbf{0}}\right \|}_{{L}^{2}{(\Omega )}^{d}} +{ \left \|\vec{b}\right \|}_{{L}^{1}(0,T;{L}^{2}{(\Omega )}^{d})}\right ).$$

If $\vec{b} \in \mathcal{C}([0,T],{L}^{2}{(\Omega )}^{d})$ then $\vec{u} \in {\mathcal{C}}^{2}([0,T],{L}^{2}{(\Omega )}^{d})$.

Here, $\Omega$ denotes a bounded Lipschitz domain, and the constant ${C}_{d,\delta }$ depends on the dimension d and the horizon $\delta$. Moreover, other properties of the peridynamic integral operator defined through (7) such as dissipativity and self-adjointness are analyzed in Emmrich and Weckner [25].

In [17, 51], Du and Zhou consider the case $\Omega = {\mathbb{R}}^{d}$. Let ${\mathcal{M}}_{\lambda }{({\mathbb{R}}^{d})}^{d}$ be the space of functions $\vec{u} \in {L}^{2}{({\mathbb{R}}^{d})}^{d}$ with

$$\int\limits_{{\mathbb{R}}^{d}}(\mathcal{F}\vec{u})(\vec{y}) \cdot (\vec{I} +\vec{ {M}}_{\delta }(\vec{y}))(\mathcal{F}\vec{u})d\vec{y} < \infty,$$

depending on ${\lambda }_{d,\delta }$ since

$$\vec{{M}}_{\delta }(\vec{y}) =\int\limits_{0}^{\delta }{\lambda }_{ d,\delta }(\left \vert \vec{\hat{x}}\right \vert )(1 -\cos (\vec{y} \cdot \vec{\hat{ x}}))\vec{\hat{x}} \otimes \vec{\hat{ x}}d\vec{\hat{x}}.$$

Here $\mathcal{F}\vec{u}$ denotes the Fourier transform of $\vec{u}$.

A natural condition coming from the comparison of the deformation energy density of a homogeneous body under isotropic expansion arising from the peridynamic model with the energy density known from the classical linear elasticity theory is

$$\int\limits_{0}^{\delta }{\lambda }_{ d,\delta }(r){r}^{d+3}dr < \infty.$$

(18)

Theorem 2 ([17, 51]). 

Assume ${\lambda }_{d,\delta }(r) > 0$ for $0 < r < \delta$, (18) and $\vec{{u}}_{\mathbf{0}} \in {\mathcal{M}}_{\lambda }{({\mathbb{R}}^{d})}^{d}$, $\vec{\dot{{u}}}_{\mathbf{0}} \in {L}^{2}{({\mathbb{R}}^{d})}^{d}$ and $\vec{b} \in {L}^{2}(0,T;{L}^{2}{({\mathbb{R}}^{d})}^{d})$ . Then the initial-value problem (7) and (4) has a unique solution $\vec{u} \in C([0,T],{\mathcal{M}}_{\lambda }{({\mathbb{R}}^{d})}^{d})$ with ${\partial }_{t}\vec{u} \in {L}^{2}(0,T;{L}^{2}{({\mathbb{R}}^{d})}^{d})$.

If in addition condition (17) is valid, then Du and Zhou show that the space ${\mathcal{M}}_{\lambda }{({\mathbb{R}}^{d})}^{d}$ is equivalent to the space ${L}^{2}{({\mathbb{R}}^{d})}^{d}$.

3.2 Linear Bond-Based Model in H ^σ (σ ∈ (0, 1))

The solution of Theorem 2 can take values in a fractional Sobolev space. Indeed, if

$${c}_{1}{r}^{-2-d-2\sigma } \leq {\lambda }_{ d,\delta }(r) \leq {c}_{2}{r}^{-2-d-2\sigma }\quad \forall \;0 < r \leq \delta $$

(19)

for some exponent $\sigma \in (0,1)$ and positive constants c ₁ and c ₂, then Theorem 2 remains true and the space ${\mathcal{M}}_{\lambda }{({\mathbb{R}}^{d})}^{d}$ is equivalent to the fractional Sobolev space ${H}^{\sigma }{({\mathbb{R}}^{d})}^{d}$, see Du and Zhou [17, 51].

Additionally also the stationary problem is investigated in [17, 51].

3.3 Nonlinear Bond-Based Model

A first result towards the nonlinear model is Erbay et al. [27] analyzing the nonlinear elastic bar. They consider the one-dimensional initial value problem

$$\begin{array}{rcl} & & {\partial }_{t}^{2}u =\int\limits_{\mathbb{R}}\alpha (\hat{x} - x)g(u(\hat{x},t) - u(x,t))d\hat{x},\quad x \in \mathbb{R},\;t > 0, \\ & & u(x,0) = {u}_{0},\quad {\partial }_{t}u(x,0) =\dot{ {u}}_{0},\quad x \in \mathbb{R}. \end{array}$$

(20)

Applying Banach’s fixed point theorem the following theorems are proven.

Theorem 3 ([27]). 

Let $X = {C}_{b}(\mathbb{R})$ or let $X = {L}^{p}(\mathbb{R}) \cap {L}^{\infty }(\mathbb{R})$ with $1 \leq p \leq \infty$ . Assume $\alpha \in {L}^{1}(\mathbb{R})$ and $g \in {\mathcal{C}}^{1}(\mathbb{R})$ with g(0) = 0. Then there exists T > 0 such that the Cauchy problem (20) is locally well-posed with solution in ${\mathcal{C}}^{2}([0,T],X)$ for initial data ${u}_{0},\dot{{u}}_{0} \in X$.

Theorem 4 ([27]). 

Let $X = {\mathcal{C}}_{b}^{1}(\mathbb{R})$ or let $X = {W}^{1,p}(\mathbb{R})$ with $1 \leq p \leq \infty$ . Assume $\alpha \in {L}^{1}(\mathbb{R})$ and $g \in {\mathcal{C}}^{2}(\mathbb{R})$ with g(0) = 0. Then there exists T > 0 such that the Cauchy problem (20) is locally well-posed with solution in ${\mathcal{C}}^{2}([0,T],X)$ for initial data ${u}_{0},\dot{{u}}_{0} \in X$.

The authors of [27] remark that the proofs of the above theorems can be easily adapted to the more general peridynamic equation with a nonlinear pairwise force function $f(\xi,\eta )$, where f is continuously differentiable in η for almost every ξ and fulfils additional assumptions. For a more specific type of nonlinearities, Erbay et al. [27] proved well-posedness in fractional Sobolev spaces.

Theorem 5 ([27]). 

Let $\sigma > 0$ and ${u}_{0},\dot{{u}}_{0} \in {W}^{\sigma,2}(\mathbb{R}) \cap {L}^{\infty }(\mathbb{R})$ . Assume $\alpha \in {L}^{1}(\mathbb{R})$ and $g(\eta ) = {\eta }^{3}$ . Then there exists T > 0 such that the Cauchy problem (20) is locally well-posed with solution in ${\mathcal{C}}^{2}([0,T],{W}^{\sigma,2}(\mathbb{R}) \cap {L}^{\infty }(\mathbb{R}))$.

Furthermore, blow up conditions for these solutions are investigated, which we shall not present here.

3.4 State-Based Model and Nonlocal Vector Calculus

The recent reports Du et al. [18, 19] consider a nonlocal vector calculus building upon the ideas of Gunzburger and Lehoucq [29]. The nonlocal vector calculus is applied in Du et al. [20] to establish the well-posedness of the linear peridynamic state equilibrium equation. We briefly summarize the vector calculus and well-posedness for the case of a linear isotropic material.

Let $\Omega$ denote an open subset of ${\mathbb{R}}^{d}$ and let $\vec{\alpha } =\vec{ \alpha }(\vec{x},\vec{y})$ denote an anti-symmetric mapping, i.e., $\vec{\alpha }(\vec{x},\vec{y}) = -\vec{\alpha }(\vec{y},\vec{x})$. Define the nonlocal divergence operator of a tensor $\vec{\Psi }$ and its adjoint operator,

$$\begin{array}{rcl} {\mathcal{D}}_{t}(\vec{\Psi })(\vec{x})& =& \int\limits_{{\mathbb{R}}^{d}}\big{(}\vec{\Psi }(\vec{x},\vec{y}) +\vec{ \Psi }(\vec{y},\vec{x})\big{)} \cdot \vec{ \alpha }(\vec{x},\vec{y})\,d\vec{y}\quad \mbox{ for}\ \vec{x} \in \Omega, \\ {\mathcal{D}}_{t}^{{_\ast}}(\vec{v})(\vec{x},\vec{y})& =& -\big{(}\vec{v}(\vec{y}) -\vec{ v}(\vec{x})\big{)} \otimes \vec{ \alpha }(\vec{x},\vec{y})\quad \mbox{ for}\ \vec{x},\vec{y} \in \Omega, \\ \end{array}$$

established via $\big{(}{\mathcal{D}}_{t}(\vec{\Psi }),\vec{v}{\big{)}}_{\Omega } = \big{(}\vec{\Psi },{\mathcal{D}}_{t}^{{_\ast}}(\vec{v}){\big{)}}_{\Omega \times \Omega }$, inner products on ${L}^{2}(\Omega )$ and ${L}^{2}(\Omega \times \Omega )$, respectively. The operators ${\mathcal{D}}_{t}(\vec{\Psi })$ and ${\mathcal{D}}_{t}^{{_\ast}}(\vec{v})$ return a vector and tensor respectively. Let $\omega = \omega (\vec{x},\vec{y})$ and $\mathbf{U} = \mathbf{U}(\vec{x})$ denote a nonnegative scalar function and tensor, respectively. The choice $\vec{\Psi }(\vec{x},\vec{y}) = \omega (\vec{x},\vec{y})\mathbf{U}(\vec{x})$ results in the weighted nonlocal divergence operator ${\mathcal{D}}_{t,\omega }$ for tensors and its adjoint operator,

$$\begin{array}{rcl} {\mathcal{D}}_{t,\omega }\big{(}\mathbf{U}\big{)}(\vec{x})& =& {\mathcal{D}}_{t}\big{(}\omega (\vec{x},\vec{y})\mathbf{U}(\vec{x})\big{)}(\vec{x})\quad \mbox{ for}\ \vec{x} \in \Omega, \\ {\mathcal{D}}_{t,\omega }^{{_\ast}}\big{(}\vec{u}\big{)}(\vec{x})& =& \int\limits_{{\mathbb{R}}^{d}}{\mathcal{D}}_{t}^{{_\ast}}(\vec{u})(\vec{x},\vec{y})\,\omega (\vec{x},\vec{y})\,d\vec{y}\quad \mbox{ for}\ \vec{x} \in \Omega.\end{array}$$

The distinction between ${\mathcal{D}}_{t}^{{_\ast}}(\vec{v})$ and ${\mathcal{D}}_{t,\omega }^{{_\ast}}\big{(}\vec{u}\big{)}$ is that the former is a tensor on $\Omega \times \Omega$ while the latter is a tensor on Ω.

In the case of a linear peridynamic solid, define the operator

$$\mathcal{L}(\vec{u}) = {\mathcal{D}}_{t}\big{(}\eta \big{(}{\mathcal{D}}_{t}^{{_\ast}}(\vec{u}){\big{)}}^{T}\big{)} + {\mathcal{D}}_{ t,w}\big{(}\gamma \mathrm{Tr}\big{(}{\mathcal{D}}_{t,\omega }^{{_\ast}}(\vec{u})\big{)}\mathrm{I}\big{)}$$

(21)

where η and γ are materials constants. Mild conditions may also be supplied so that

$$\mathcal{L}\rightarrow L \equiv (\lambda + \mu )\mathrm{grad}\mathrm{div} + \mu \mathrm{div}\mathrm{grad}$$

in ${H}^{-1}({\mathbb{R}}^{d})$ as the horizon decreases. Here, μ and λ are the Lamé parameters that are given as expressions in terms of η and γ. The first term on the right-hand side of (21) corresponds to the bond-based material model and is the peridynamic analogue of $\mu \mathrm{div}\mathrm{grad} + \mu \mathrm{grad}\mathrm{div}$. The second term on the right-hand side of (21) accounts for volumetric terms associated with Poisson ratio not equal to one fourth and is the peridynamic analogue of $\lambda \mathrm{grad}\mathrm{div}$.

In words, the peridynamic operator $\mathcal{L}$ converges to the Navier operator as the horizon tends to zero where the horizon is given by the support of $\vec{\alpha } \cdot \vec{ \alpha }$ and $\omega$ in the two operators in the right-hand side of (21). In contrast to the Navier operator, the peridynamic operator $\mathcal{L}$ only depends upon spatial differences in the displacement field and avoids an assumption of the weak differentiability. Hence discontinuities in the displacement field do not lead to an ill-posed system.

A minor simplification of the main result of Du et al. [20] is given by the following result associated with a linear isotropic peridynamic material.

Theorem 6 ([20]). 

Let the nonzero volume ${\Omega }_{c} \subset \Omega$ with ${\Omega }_{s} \cup {\Omega }_{c} = \Omega$ and ${\Omega }_{s} \cap {\Omega }_{c} = \varnothing$ be given. The volume-constrained problem

$$\left \{\begin{array}{ll} \mathcal{L}(\vec{u})(\vec{x})\,& =\,\vec{ b}(\vec{x}),\quad \vec{x} \in {\Omega }_{s}, \\ \vec{u}(\vec{x}) & ={ \vec{h}}_{d}(\vec{x}),\quad \vec{x} \in {\Omega }_{c}, \end{array} \right.$$

(22)

is well-posed on ${L}^{2}{({\Omega }_{s})}^{d}$ when $\vec{b} \in {L}^{2}{({\Omega }_{s})}^{d}$ , given the volume-constraint ${\vec{h}}_{d}$ and mild conditions on $\eta$ and $\gamma$.

The volume-constraint is the nonlocal analogue of a boundary condition, necessary for the well-posedness of a boundary value problem. The above theorem is established by demonstrating that (22) is the optimality system associated with a quadratic functional with a unique minimizer. Nonlocal analogues of Korn’s inequality and Green’s identities are needed.

4 Peridynamics as a Multiscale Method

4.1 Limit of Vanishing Nonlocality

A fundamental question of the peridynamic theory was if it generalizes the conventional linear elastic theory. More precisely, does the nonlocal linear peridynamic equation of motion converge towards the Navier equation of linear elasticity? Indeed in [25], Emmrich and Weckner proved convergence in an interior subdomain under smoothness assumptions of the solution. Therefore, let $\Lambda \subset {\mathbb{R}}^{+}$ be a null sequence bounded from above by some ${\delta }_{0} > 0$, ${\Omega }_{0}$ be the interior subdomain defined as all $\vec{x} \in \Omega$ such that $\mathrm{dist}(x,\partial \Omega ) > {\delta }_{0}$, ${L}_{d,\delta }$ the linear operator defined through (7) and L the operator corresponding to the Navier equation of linear elasticity defined through (1) (with $\lambda = \mu$ since $\nu = 1/4$).

Theorem 7 ([25]). 

Let (17) be valid for all $\delta \in \Lambda$ and ${\lambda }_{d,\delta }$ be nonnegative. If $\vec{v} \in {\mathcal{C}}^{2}{(\Omega )}^{d}$ then

$${\left \|{L}_{d,\delta }\vec{v} - L\vec{v}\right \|}_{{L}^{\infty }{({\Omega }_{0})}^{d}} \rightarrow 0\quad \text{ as}\quad \delta \rightarrow 0\quad (\delta \in \Lambda ).$$

In addition, an expansion of ${L}_{d,\delta }\vec{v}$ in terms of a series of differential operators of even order 2n ($n = 1,2,\ldots$) applied to $\vec{v}$ can be shown for smooth $\vec{v}$, where the second-order differential operator is the Navier operator and where the coefficients of the differential operators behave like ${\delta }^{2(n-1)}$. This resembles, in the one-dimensional case, the expansion in Arndt and Griebel [7] for the evolution equation of an atomistic system as well as the higher-order Boussinesq equation appearing in the work of Duruk et al. [22]. A similar observation has also been made in Emmrich and Weckner [24] when discussing the relation between the peridynamic elastic bar and nonlinear dispersion relations: different dispersion relations correspond to different micromodulus functions, which then correspond (for smooth solutions) to different higher-order differential equations.

Furthermore in [17], Du and Zhou have also investigated the limit of vanishing nonlocality in the case of the full space $\Omega \,=\,{\mathbb{R}}^{d}$ being then able to show convergence of the sequence of solutions.

Theorem 8 ([17]). 

Let (18) be valid and ${\lambda }_{d,\delta }(r) > 0$ for $0 < r < \delta$ . If $\vec{{u}}_{\mathbf{0}} \in {H}^{1}{({\mathbb{R}}^{d})}^{d}$, $\vec{\dot{{u}}}_{\mathbf{0}} \in {L}^{2}{({\mathbb{R}}^{d})}^{d}$ and $\vec{b} \in {L}^{2}(0,T;{L}^{2}{({\mathbb{R}}^{d})}^{d})$ , then the solution of the initial-value problem (7) and (4) converges to the solution of the initial-value problem (1) and (4) as $\delta \rightarrow 0$ in the conventional norms of ${L}^{2}(0,T;{\mathcal{M}}_{\lambda }{({\mathbb{R}}^{d})}^{d}) \cap {H}^{1}(0,T;{L}^{2}{({\mathbb{R}}^{d})}^{d})$ if

$$\int\limits_{{B}_{\delta }(\mathbf{0})}{\lambda }_{d,\delta }(\left \vert \vec{x}\right \vert ){\left \vert \vec{x}\right \vert }^{4}d\vec{x} \rightarrow 2d(d + 2)\mu \quad \text{ as}\quad \delta \rightarrow 0.$$

Note that here no extra regularity of the solution is assumed. Recall that if (17) is also fulfilled, then ${\mathcal{M}}_{\lambda }{({\mathbb{R}}^{d})}^{d} = {L}^{2}{({\mathbb{R}}^{d})}^{d}$ and (18) is always fulfilled if the deformation energy density for isotropic expansion arising from the peridynamic model coincides with the energy density known from the classical linear elasticity theory.

Results for the nonlinear bond-based model are not known so far. The limit of vanishing nonlocality of the state-based model is investigated in Lehoucq and Silling [47], see also Sect. 3.4.

4.2 Composite Material and Two-Scale Convergence

In [4], Alali and Lipton focus on the modeling and analysis of heterogeneous media using the peridynamic formulation. Considering both macroscopic and microscopic dynamics in one model, the so-called two-scale convergence is essential to approximate a solution.

In detail, a peridynamic long-range force is perturbed by an oscillating short-range force representing the heterogeneities. For example, one can think of a fibre-reinforced composite. The deformations are assumed to be small. This justifies that both the long-scale macroscopic force function f _long and the short-scale microscopic force function f _short are linearizations of the bond-strech model (see Sect. 2.2). The connection between those two scales is described by a parameter $\epsilon$. The displacement inside the composite is denoted by $\vec{{u}}_{\epsilon } =\vec{ {u}}_{\epsilon }(\vec{x},t)$. The peridynamic equation of motion for heterogeneous material reads

$$\begin{array}{rcl}{ \rho }_{\epsilon }(\vec{x}){\partial }_{t}^{2}\vec{{u}}_{ \epsilon }(\vec{x},t)& =& \int\limits_{{\mathcal{H}}_{long}(\vec{x})}\vec{{f}}_{long}(\vec{\hat{x}} -\vec{ x},\vec{{u}}_{\epsilon }(\vec{\hat{x}},t) -\vec{ {u}}_{\epsilon }(\vec{x},t))d\vec{\hat{x}} \\ & +& \int\limits_{{\mathcal{H}}_{short}(\vec{x})}\vec{{f}}_{short}(\vec{x},\vec{\hat{x}} -\vec{ x},\vec{{u}}_{\epsilon }(\vec{\hat{x}},t) -\vec{ {u}}_{\epsilon }(\vec{x},t))d\vec{\hat{x}} \\ & +& \vec{{b}}_{\epsilon }(\vec{x},t),\quad (\vec{x},t) \in \Omega \times (0,T). \end{array}$$

(23)

Note that the microscopic force function now explicitly depends on $\vec{x}$ since the bond strength is influenced by the distance of the fast parameters of $\vec{x}/\epsilon$ and $\vec{\hat{x}}/\epsilon$. The macroscopic integration volume ${\mathcal{H}}_{long}(\vec{x})$ is the open ball with centre $\vec{x}$ and radius ${\delta }_{1} > 0$ intersected with $\Omega$, the microscopic integration volume ${\mathcal{H}}_{short}(\vec{x})$ is the open ball with centre $\vec{x}$ and radius $\epsilon {\delta }_{2} > 0$ intersected with $\Omega$. Alali and Lipton [4] show well-posedness of (23) complemented with initial conditions. Furthermore they show that the displacement $\vec{{u}}_{\epsilon } =\vec{ {u}}_{\epsilon }(\vec{x},t)$ can be approximated by a function $\vec{u} =\vec{ u}(\vec{x},\vec{x}/\epsilon,t)$. Indeed, setting $\vec{y} =\vec{ x}/\epsilon$, $\vec{u} =\vec{ u}(\vec{x},\vec{y},t)$ is the two-scale limit of $\vec{{u}}_{\epsilon } =\vec{ {u}}_{\epsilon }(\vec{x},t)$, that is

$$\int\limits_{\Omega \times (0,T)}\vec{{u}}_{\epsilon }(\vec{x},t)\Psi \left (\vec{x},\vec{x}/\epsilon,t\right )d\vec{x}dt \rightarrow \int\limits_{\Omega \times Y \times (0,T)}\vec{u}(\vec{x},\vec{y},t)\Psi \left (\vec{x},\vec{y},t\right )d\vec{x}d\vec{y}dt$$

for all $\Psi \in {C}_{c}^{\infty }({\mathbb{R}}^{3} \times Y \times {\mathbb{R}}^{+})$ that are Y -periodic in y, as $\epsilon$ tends to zero. Here $Y \subset {\mathbb{R}}^{3}$ is the unit period cube with the origin at the center. The function $\vec{u} =\vec{ u}(\vec{x},\vec{y},t)$ is obtained from a partial integro-differential initial value problem, which is similar to (23) but independent of $\epsilon$.

5 Numerical Approximation

In what follows, we only describe the quadrature formula method. For a finite element approximation, we refer to Chen and Gunzburger [14].

Let $\Omega \subset {\mathbb{R}}^{d}$ ($d \in \{ 2,3\}$) be (for simplicity) a polyhedral domain and let $\mathbb{T} =\{ {\mathcal{T}{}_{\mathcal{l}}\}}_{\mathcal{l}\in \mathbb{N}}$ be a sequence of partitions of $\Omega$ such that

• 1. $\mathcal{T} =\{ {T{}_{i}\}}_{i=1}^{M}$ for any $\mathcal{T} \in \mathbb{T}$ with T _i ($i = 1,\ldots,M$, $M \in \mathbb{N}$) being a closed triangle (tetrahedron) or parallelogram (parallelepiped) with nonempty interior $\mathring{T}_{i}$ such that

  $$\bigcup\limits_{i=1}^{M}{T}_{ i} = \overline{\Omega }\,,\;\mathring{T}_{i} \cap \mathring{T}_{j} = \varnothing \text{ for }i\neq j\;(i,j = 1,\ldots,M)\,;$$

• 2. There exist constants c ₁ > 0 and c ₂ > 0 such that for all $\mathcal{T} \in \mathbb{T}$ and all $T \in \mathcal{T}$

  $${h}_{\mathrm{max}} :{=\max }_{T\in \mathcal{T}}{h}_{T} \leq {c}_{1}{h}_{T} \leq {c}_{2}{\rho }_{\mathrm{min}} :{=\min }_{T\in \mathcal{T}}{\rho }_{T}$$

  with ${h}_{T} :=\mathrm{ diam}\,T$ being the diameter of T and ρ _T being the radius of the largest inscribed ball of T (quasi-uniformity);

• 3. ${h}_{\mathrm{max}}({\mathcal{T}}_{\mathcal{l}}) :{=\max }_{T\in {\mathcal{T}}_{\mathcal{l}}}{h}_{T} \rightarrow 0$ as $\mathcal{l} \rightarrow \infty$.

We now describe the quadrature formula method for the spatial approximation of (7). Let $\phi \in \mathcal{C}(\overline{\Omega })$. On each $T \in \mathcal{T} \in \mathbb{T}$, we consider the quadrature formula

$${Q}_{T}[\phi ] :=\sum\limits_{\mu =1}^{{m}_{T} }{\omega }_{T,\mu }\phi (\vec{{x}}_{T,\mu }) \approx \int\limits_{T}\phi (\vec{\hat{x}})d\vec{\hat{x}} =: {I}_{T}[\phi ]$$

with the quadrature points $\vec{{x}}_{T,\mu } \in T$ (remember that T is closed and quadrature points can also lie on the boundary of T) and weights ${\omega }_{T,\mu } > 0$ ($\mu = 1,\ldots,{m}_{T} \in \ \mathbb{N}$). We then get the composed quadrature formula

$${Q}_{\Omega }[\phi ] :=\sum\limits_{T\in \mathcal{T}}{Q}_{T}[\phi ] \approx {I}_{\Omega }[\phi ]\,.$$

By introducing a global numbering of the quadrature points $\vec{{x}}_{T,\mu }$ (each quadrature point is only counted once), we find

$${Q}_{\Omega }[\phi ] =\sum\limits_{j=1}^{N}{\omega }_{ j}\phi (\vec{{x}}_{j})$$

with globally numbered weights ω _j . Note that ${\omega }_{j}$ is the sum of all the local weights corresponding to the quadrature point $\vec{{x}}_{j}$. For having a consistent quadrature, certain additional standard assumptions are required, which we are not going to specify here. Using the quadrature points as collocation points for the approximation of (7), we arrive at

$$ \begin{array}{rcl} & & \rho (\vec{{x}}_{i}){\partial }_{t}^{2}\vec{{u}}_{ i}(t) = ({L}_{d,\delta }^{(h)}\vec{{u}}^{(h)})(\vec{{x}}_{ i},t) +\vec{ b}(\vec{{x}}_{i},t)\,,\quad i = 1,\ldots,N\,, \\ & & \text{ with }({L}_{d,\delta }^{(h)}\vec{{u}}^{(h)})(\vec{{x}}_{ i},t) \\ & & :=\sum\limits_{j\in J(i)}{\omega }_{j}{\lambda }_{d,\delta }(\vert \vec{{x}}_{j} -\vec{ {x}}_{i}\vert )(\vec{{x}}_{j} -\vec{ {x}}_{i}) \otimes (\vec{{x}}_{j} -\vec{ {x}}_{i})(\vec{{u}}_{j}(t) -\vec{ {u}}_{i}(t)) \\ \end{array} $$

and with $\vec{{u}}_{i}(t)$ being an approximation of $\vec{u}(\vec{{x}}_{i},t)$ and $\vec{{v}}^{(h)}(\vec{{x}}_{i}) :=\vec{ {v}}_{i}$ for a grid function $\vec{{v}}^{(h)}$. Since the integration only takes place in the ball of radius δ intersected with Ω (see also Fig. 2), we take

$$ J(i) :=\{ j \in \{ 1,2,\ldots,N\} \setminus \{ i\} : \left \vert \vec{{x}}_{j} -\vec{ {x}}_{i}\right \vert < \delta \}\,. $$

As is seen, we take the global weight corresponding to a quadrature point even if not all the elements to which this quadrature point belongs are fully inside the peridynamic horizon. Moreover, if a quadrature point lies on the boundary of the peridynamic horizon than it is not taken into account. One may object that it should be better not to work with the global but with the local weight here. However, this cannot influence the order of approximation.

Fig. 2

Discretization of the integration volume

Let us assume that the displacement field is smooth and that the local quadrature rules are exact at least for polynomials of degree r − 1. Moreover, let us consider the case of the linearization of the bond-based peridynamic model with a suitable regularization in order to avoid the jump discontinuity along the boundary of the peridynamic horizon such as ${\lambda }_{d,\delta }(r) =\tilde{ {c}}_{d,\delta }{r}^{-3}\exp (-{\delta }^{2}/({\delta }^{2} - {r}^{2})){\chi }_{(0,\delta )}(r)$ (see Emmrich and Weckner [25]).

Based upon an estimate of the local error of the quadrature rule, taking into account the growth behaviour of the weak singularity of the integral kernel, and employing stability of the solution with respect to perturbations of the integral kernel, an error estimate of order $\mathcal{O}({h}_{\text{ max}}^{\min (r,d)})$ and $\mathcal{O}({h}_{\text{ max}}^{d}\log 1/h)$ if r = d, respectively, has been shown in Büsing [13], where d denotes the dimension of the domain. The restriction of the order to the dimension d could not be overcome due to the singularity behaviour of the integral kernel.

6 Applications and Numerical Simulations

6.1 Simulation of Nanofibres

Investigating the behaviour of nanofibre networks and thin membranes of nanofibres is of great economical and scientific interest. Bobaru [10] and Bobaru et al. [12] have analyzed the behaviour of these fabrics under dynamic deformation including the influence of van der Waals forces. Assuming linear-elastic behaviour on each fibre, we have a micropotential as given in Sect. 2.2. In order to avoid modeling atoms, the van der Waals forces acting between fibres are modeled by adding the so-called Lennard–Jones type potential of the form

$${w}^{LJ}(\left \vert \vec{\xi } +\vec{ \eta }\right \vert ) = \alpha {\left ( \frac{a} {\left \vert \vec{\xi } +\vec{ \eta }\right \vert }\right )}^{12} - \beta {\left ( \frac{a} {\left \vert \vec{\xi } +\vec{ \eta }\right \vert }\right )}^{6}$$

with parametres α, β and a to the microelastic potential. Since van der Waals forces are long-range forces, the linearized peridynamic equation of motion (7) is complemented with an integral of the Lennard–Jones force density over a horizon, which is different to δ.

6.2 Simulation of Cracks

Accurate crack prediction is of topical interest and many authors have worked on it over the last years, see, e.g., Aidun and Silling [3], Agwai et al. [1, 2], Askari et al. [8], Bobaru and Ha [11], Huang et al. [30], and Kilic and Madenci [31]. According to [3], the so-called Kalthoff–Winkler experiment is one of 12 characteristic phenomena of dynamic fracture, which a good numerical simulation should be able to reproduce. In this experiment, a cylindrical steel impactor strikes on the edge of a plate, which has been primed with two parallel notches. If the strike is fast enough, the crack propagates in a characteristic angle (see Fig. 3 for a simulation in Matlab with the quadrature formula method as described above and a standard Runge–Kutta solver for the time integration). Silling [43] shows that the Kalthoff–Winkler experiment is correctly simulated with the peridynamic theory. In [3] the simulation of further characteristic phenomena of dynamic fracture with the peridynamic theory is investigated.

Fig. 3

Kalthoff–Winkler experiment