Dynamic Brittle Fracture as a Small Horizon Limit of Peridynamics
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Abstract
We consider the nonlocal formulation of continuum mechanics described by peridynamics. We provide a link between peridynamic evolution and brittle fracture evolution for a broad class of peridynamic potentials associated with unstable peridynamic constitutive laws. Distinguished limits of peridynamic evolutions are identified that correspond to vanishing peridynamic horizon. The limit evolution has both bounded linear elastic energy and Griffith surface energy. The limit evolution corresponds to the simultaneous evolution of elastic displacement and fracture. For points in spacetime not on the crack set the displacement field evolves according to the linear elastic wave equation. The wave equation provides the dynamic coupling between elastic waves and the evolving fracture path inside the media. The elastic moduli, wave speed and energy release rate for the evolution are explicitly determined by moments of the peridynamic influence function and the peridynamic potential energy.
Keywords
Peridynamics Dynamic fracture Brittle materials Elastic moduli Critical energy release rateMathematics Subject Classification
34A34 74R10 74H551 Introduction
Peridynamics, introduced by Silling in 2000, [30] is a nonlocal formulation of continuum mechanics expressed in terms of regular elastic potentials. The theory is formulated in terms of displacement differences as opposed to spatial derivatives of the displacement field. These features provide the flexibility to simultaneously simulate kinematics involving both smooth deformations and defect evolution. Numerical simulations based on peridynamic modeling exhibit the formation and evolution of sharp interfaces associated with defects and fracture [6, 31, 32, 35], and [15]. These aspects are exploited in the peridynamic scheme for dynamic fracture simulation where the crack path is determined as part of the solution [20, 24]. This type of solution is distinct from the classical setting where the crack path is specified a priori see, [19].
We consider peridynamic formulations with constitutive laws that soften beyond a critical shear strain. Here we discover new quantitative and qualitative information that is extracted from the peridynamic formulation using scaling arguments and by passing to a distinguished small horizon limit. In this limit the dynamics correspond to the simultaneous evolution of elastic displacement and fracture. For points in spacetime not on the crack set the displacement field evolves according to the linear elastic wave equation. The wave equation provides the dynamic coupling between elastic waves and the evolving fracture path inside the media. The limit evolutions have bounded energy expressed in terms of the bulk and surface energies of linear elastic fracture mechanics. They also satisfy an energy inequality expressed in terms of the kinetic energy of the motion together with the bulk elastic energy and a Griffith surface energy. These energies are described by the density ρ, elastic shear modulus μ and energy release rate \(\mathcal{G}_{c}\). The parameters μ and \(\mathcal{G}_{c}\) have explicit formulas given in terms of the moments of the peridynamic influence function and the peridynamic potential energy see, (2.8). These formulas provide a rigorous means to calibrate peridynamic potentials with experimentally measured values of elastic shear modulus and critical energy release rate.
In what follows we examine the family of peridynamic deformations {u ^{ ϵ }(t,x)}_{ ϵ>0} defined for suitable initial data u _{0},v _{0} and investigate the dynamics of the limit u ^{0}(t,x)=lim_{ ϵ→0} u ^{ ϵ }(t,x). To do this we describe peridynamic deformations as trajectories in function space. The nonlocal Dirichlet boundary conditions are incorporated into the function space by defining the class of functions \(L^{2}_{0}(D)\) that are square integrable over D and zero on D _{ α }∖D.^{1} In this context we view peridynamic evolutions as functions of time taking values in the space \(L^{2}_{0}(D)\). It follows from the evolution equation (1.12) that u ^{ ϵ }(t,x) is twice differentiable in time taking values in \(L^{2}_{0}(D)\). This space of functions is denoted by \(C^{2}([0,T];L^{2}_{0}(D))\) see, e.g., [17]. The initial value problem for the peridynamic evolution (1.12) is shown to be well posed on \(C^{2}([0,T];L^{2}_{0}(D))\) see, Sect. 2.2. We apply a scaling analysis to show that the peridynamic evolutions u ^{ ϵ }(t,x) approach a limit evolution u ^{0} in the ϵ→ limit. The limit evolution u ^{0}(t,x) is shown to have bounded linear elastic energy and bounded Griffith surface energy for a wide class of initial conditions. The limit evolution satisfies an energy inequality expressed in terms of the kinetic energy of the motion together with a linear elastic energy in terms of the antiplane shear strain ∇u ^{0} and a Griffith surface energy associated with the evolving jump set \(S_{u^{0}(t)}\) of u ^{0}(t,x) see, Sect. 3.1. The jump set \(S_{u^{0}(t)}\) is the crack set and distinguished limit of the peridynamic model is given by the displacement—crack set pair u _{0}(t,x), \(S_{u^{0}(t)}\). The wave equation provides the dynamic coupling between elastic waves and the evolving fracture path inside the media.
Motivated by the approach given in [34] we investigate the effect of the softening constitutive law (1.3) on the nucleation of fracture inside a peridynamic body. We consider a generic peridynamic neighborhood \(\mathcal{H}_{\epsilon}(x)\) of radius ϵ about the point x. For points x′ inside \(\mathcal{H}_{\epsilon}(x)\) we say that the material between x and x′ (the bond) is critically strained if the magnitude of the shear strain \(\mathcal{S}^{\epsilon}=\eta^{\epsilon}(x)/(\epsilon\xi)>\overline {\eta}/(\epsilon\xi)\), otherwise the shear strain is called subcritical. A linear stability analysis is given that identifies necessary conditions for fracture nucleation inside \(\mathcal{H}_{\epsilon}(x)\). These conditions are directly linked to the appearance of subsets of critically strained bonds inside \(\mathcal{H}_{\epsilon}(x)\) with nonzero area fraction. The fracture nucleation condition given by Proposition 2.1 implies that if the neighborhood contains a nonzero area fraction of critically strained bonds then the neighborhood can be linearly unstable and a displacement jump can be nucleated. These results are presented in Sect. 2.3.
We focus on the peridynamic neighborhoods \(\mathcal{H}_{\epsilon}(x)\) that contain critically strained bonds over an area fraction larger than ϵ ^{ α } with exponent 0<α<1 and 0<ϵ ^{ α }<1. These neighborhoods are referred to as unstable neighborhoods. Under this definition unstable neighborhoods have the potential to nucleate jump discontinuities. We apply this definition to identify a set where unstable neighborhoods for the flows u ^{ ϵ }(t,x) concentrate as the peridynamic horizon approaches zero. To present the idea we fix α and consider the collection of centroids x of all the unstable neighborhoods \(\mathcal{H}_{\epsilon}(x)\) with critically strained bonds over an area fraction greater than ϵ ^{ α } for a family of flows with δ>ϵ>0. This collection is denoted by the set \(C^{\alpha}_{\delta}\). It is shown that the area of \(C^{\alpha}_{\delta}\) vanishes as δ→0 and that the collection of centroids for unstable neighborhoods concentrate onto a set \(C^{\alpha}_{0}\) of zero area as δ→0. This is shown to be true for every choice of 0<α<1 and we take the intersections of these sets denoted here by \(C^{0}_{0}\). The set \(C^{0}_{0}\) is associated with centroids of neighborhoods for which all bonds have become soft. The dynamics associated with each point in the reference configuration belonging to the set \(C_{0}^{0}\) is not affected by the motion of its surroundings. With these ideas in mind the set \(C^{0}_{0}\) presents itself as an alternate description of the crack set in the small horizon limit.
It is shown that the concentration of instability is inevitable for convexconcave peridynamic potentials and is directly linked to the energy budget associated with the peridynamic motion. The analysis shows that for a family of peridynamic flows {u ^{ ϵ }(t,x)}_{ ϵ>0} all driven by the same initial conditions and body forces that the peridydnamic potential energy of each flow is bounded uniformly in time 0≤t≤T independently of the radius of the horizon, see Sect. 4.2. This bound is shown to force the localization see Theorem 2.3. These observations are presented in Sect. 2.4 and established in Sect. 4.5.4.
Within the context of Sect. 3.2 we will apply these observations and adopt the hypothesis that the crack set \(S_{u^{0}}\) for the limit evolution and \(C^{0}_{0}\) are one and the same, see Hypotheses 3.3. We employ a scaling analysis to the peridynamic equation of motion to discover that the limit evolution u ^{0}(t,x) satisfies the wave equation at every point in spacetime where ∇u ^{0} is defined see Theorem 3.4. The wave equation provides the dynamic coupling between elastic waves u ^{0}(t,x) and the evolving fracture path \(S_{u^{0}(t)}\) inside the media. It is important to point out that the limiting dynamic fracture evolution described here follows from scaling arguments and on passing to a distinguished limit in the peridynamic formulation. These results are presented in Sects. 3.1 and 3.2. The mathematical tool set appropriate for extracting the limit behavior from this class of peridynamic models is based on Γconvergence and comes from the literature associated with the analysis of the Mumford Shah functional and free discontinuity problems see, [21, 22], and [23].
We point out here that other related recent work focuses on passing to the small horizon limit for linear peridynamic formulations; establishing a link between linear elasticity and peridynamics see [14, 16, 27, 33].
In closing we note that there is a vast literature on fracture modeling and a complete survey is beyond the scope of this paper. Instead we point out recent proposals for computing crack propagation in dynamic and quasi static settings. Approaches using a phase field for the damage set and a linear elastic field, to represent crack propagation have been proposed and developed in [9, 26], and [7]. Wave equations for fields inside domains with evolving cracks are posed in [11] and variational aspects of sharp interface models are discussed in [25]. For quasi static problems variational phase field methods are developed in the pioneering work of [8, 18]. More recently a two field method using eigendeformations for the fracture field is developed for quasi static problems in [29]. Alternative nonlocal formulations have been developed for quasi static crack propagation in [5, 10].
2 Peridynamic Evolution
We begin this section by introducing a suitable class of initial conditions appropriate for describing the evolution of deformations that can have smooth variation as well as jumps. Here we will choose initial conditions with bounded elastic energy in the sense of fracture mechanics. We show that well posed peridynamic evolutions exist for this class of initial data. These peridynamic evolutions satisfy an energy balance between potential and kinetic energy at each time during the deformation. Next we develop a necessary criterion for fracture initiation inside a peridynamic neighborhood. Here fracture initiation is defined to be the nucleation of a jump in the displacement inside a peridynamic neighborhood. We develop a criterion for the orientation of the nucleated crack based upon the notion of the most unstable direction. The approach taken here is consistent with the analysis of crack nucleation developed in [34]. We conclude with a discussion of the localization of instability in the limit of vanishing peridynamic horizon.
2.1 Initial Conditions and Motivation
Definition 2.1
2.2 Peridynamic Evolutions and Energy Balance
We choose the initial data (u _{0},v _{0}) to be LEFM initial data and the initial crack set at t=0 is prescribed by \(K=S_{u_{0}}\). There is a unique peridynamic evolution for this choice of initial data. This is stated in the following theorem.
Theorem 2.1
(Existence of unique solution for nonlinear peridynamics)
This theorem follows from the Lipschitz continuity of ∇PD ^{ ϵ } and is established in Sect. 4.1.
Theorem 2.2
(Energy balance)
2.3 Instability and Fracture Initiation
Here \(\overline{\eta}=\sqrt{\epsilon\xi}\overline{r}\) where \(\overline{r}\) is the inflection point for the function r:→f(r ^{2}). For \(\mathcal{A}_{\nu}>0\) the jump can grow exponentially. It is evident that this can occur if there are critically strained bonds, \(\mathcal{S}>\overline{\eta}/(\epsilon\xi)\) or equivalently \(\eta>\overline{\eta}\), inside the neighborhood. We summarize these results in the following.
Proposition 2.1
(Facture nucleation condition)
Proposition 2.1 together with (2.19) provides the explicit link between dynamic instability and the critical shear strain where the constitutive law begins to soften.
2.4 Concentration of Fracture Nucleation Sites in the Small Horizon Limit
Here we present results that show that peridynamic neighborhoods likley to nucleate jump sets become concentrated in the small horizon limit. The discussion focuses on the basic unit of peridynamic interaction: the peridynamic neighborhoods \(\mathcal{H}_{\epsilon}(x)\) of diameter ϵ>0 with centroids x∈D. Here we investigate the family of peridynamic evolutions u ^{ ϵ }(t,x) at a fixed time t.
Definition 2.2
Let \(C^{0}_{0,t}={\cap}_{{0<\delta}} C^{0}_{\delta,t}\) denote the concentration set for the set of centroids associated with unstable neighborhoods. In what follows the Lebesgue measure (area) of a set \(\varOmega\subset \mathbb{R}^{2}\) is denoted by \(\mathcal{L}^{2}(\varOmega)\) and we state a theorem on the localization of bond instability as the peridynamic horizon shrinks to zero.
Theorem 2.3
(Localization of bond instability in the small horizon limit)
Theorem 2.3 is established in Sect. 4.5.4. This theorem shows that the nucleation sites concentrate on the centroids associated with the collection of failed neighborhoods \(C^{0}_{0,t}\) for which all bonds have become soft. For points x belonging to \(C^{0}_{0,t}\) the dynamics of the surrounding motion no longer influences the dynamics at x. Intuitively this set of points provides an alternative description of the crack set in the reference configuration as seen in the small horizon limit. The concentration of instability is inevitable for this model and is directly linked to the energy budget associated with the peridynamic motion. It is shown that for a family of peridynamic flows {u ^{ ϵ }(t,x)}_{ ϵ>0} all driven by the same initial conditions and body forces that the peridydnamic potential energy of each flow is bounded uniformly in time 0≤t≤T independently of the radius of the horizon, see Sect. 4.2. This bound forces the localization as shown in Sect. 4.5.4.
3 The Small Horizon, Sharp Interface Limit

Has uniformly bounded linear elastic bulk energy and Griffith surface energy for 0≤t≤T.

Satisfies an energy inequality involving the kinetic energy of the motion together with the bulk elastic and surface energy associated with linear elastic fracture mechanics for 0≤t≤T.

Satisfies the wave equation for points in spacetime not on the crack set.
3.1 Convergence of Peridynamics to Sharp Interface Dynamics Associated with Brittle Fracture
We consider the family of solutions \(u^{\epsilon_{k}}\) to the peridynamic initial value problem with LEFM initial data for a sequence ϵ _{ k }, k=1,2,… . We shall see that we can pass to the limit ϵ _{ k }↘0 to identify a limit evolution u ^{0}(t,x) for 0≤t≤T. The limit flow is found to have an approximate gradient ∇u ^{0}(t,x) almost everywhere in D and the jump set \(S_{u^{0}(t)}\) is the countable union of rectifiable arcs. Moreover the limit evolutions u ^{0}(t,x) have uniformly bounded energy in the sense of linear elastic fracture mechanics over 0≤t≤T. We begin by making the following hypothesis.
Hypothesis 3.1
Theorem 3.1
(Limit evolution with bounded LEFM energy)
Theorem 3.1 is established using Gronwall’s inequality see, Sect. 4.2 and the Γconvergence associated with peridynamic energies see, Sect. 4.3. The proof of Theorem 3.1 is given in Sect. 4.3.
Theorem 3.2
(Energy Inequality)
The proof of Theorem 3.2 given in Sect. 4.4.
Motivated by the energy inequality Theorem 3.2 we conclude this section by showing that the length of the set cracked the by the limiting evolution over the time interval 0≤τ≤t is bounded. Recall the jump set for the deformation u ^{0} at time τ is \(S_{u^{0}(\tau)}\) and its length is given by its one dimensional Hausdorff measure \(\mathcal{H}^{1}(S_{{u^{0}(\tau)}})\). The bound follows from the following theorem.
Theorem 3.3
This shows that the total length of the set cracked by the evolution from t=0 to t=T is bounded. Theorem 3.3 is established in Sect. 4.5.
3.2 Wave Equation for the Displacement
It is shown that the limit evolution u ^{0} solves the wave equation. The following hypothesis on the regularity of the crack set is made.
Hypothesis 3.2
We suppose that the crack set given by \(S_{u^{0}(t)}\) is a closed set for 0≤t≤T.
The next hypotheses applies to the concentration set associated with unstable neighborhoods and its relation to the crack set for the limit flow.
Hypothesis 3.3
Theorem 2.3 shows that the centroids of failed neighborhoods for which all bonds have become soft, see (2.26), concentrate on the lower dimensional set \(C^{0}_{0,t}\). Recall that the dynamics associated with every point in the reference configuration belonging to the set \(C^{0}_{0,t}\) is not affected by the motion of its surroundings. Motivated by this observation we assume \(S_{u^{0}(t)}=C^{0}_{0,t}\) for 0≤t≤T.
The next hypotheses applies to neighborhoods \(\mathcal{H}_{\epsilon_{k}}(x)\) for which the shear strain is subcritical, i.e., \(u^{\epsilon_{k}}(t,y)u^{\epsilon_{k}}(t,x)/yx<\overline{\eta}/yx\), for y in \(\mathcal{H}_{\epsilon_{k}}(x)\). These neighborhoods will be referred to as neutrally stable.
Hypothesis 3.4
We suppose that \(\epsilon_{k}=\frac{1}{2^{k}}<\delta\) and 0≤t≤T. Consider the collection of centroids \(C^{0}_{\delta,t}\). We fatten out \(C^{0}_{\delta,t}\) and consider \(\tilde{C}^{0}_{\delta,t}=\{x\in D: \operatorname{dist}(x,C^{0}_{\delta ,t})<\delta\}\). We suppose that all neighborhoods \(H_{\epsilon_{k}}(x)\) that do not intersect the set \(\tilde{C}^{0}_{\delta,t}\) are neutrally stable.
Passing to subsequences if necessary we apply Theorem 3.1 and take u ^{0} to be the limit evolution of the family of peridynamic evolutions \(\{u^{\epsilon_{k}}\}_{k=1}^{\infty}\) characterized by horizons of radii \(\epsilon_{k}=\frac{1}{2^{k}}\).
Theorem 3.4
(Wave equation)
The proof of Theorem 3.4 is given in Sect. 4.5.
Remark 3.1
The sharp interface limit of the peridynamic model is given by the displacement—crack set pair u _{0}(t,x), \(S_{u^{0}(t)}\). The wave equation provides the dynamic coupling between elastic waves and the evolving fracture path inside the media.
Remark 3.2
We point out that the peridynamic constitutive model addressed in this work does not have an irreversibility constraint and the constitutive law (1.3) applies at all times in the peridynamic evolution. Because of this the crack set at each time is given by \(S_{u^{0}(t)}\). Future work will investigate the effects of irreversibility (damage) in the peridynamic model.
Remark 3.3
We conjecture that Hypotheses 3.2, 3.3 and 3.4 hold true. It is also pointed out that these hypotheses are only used to establish Lemma 4.3 which identifies the directional derivative of u ^{0} at x along the direction e=ξ/ξ with the weak \(L^{2}(D\times\mathcal{H}_{1}(0))\) limit of the shear strain \(\mathcal{S}^{\epsilon_{k}}=\frac{\eta^{\epsilon_{k}}}{\epsilon_{k}\xi }\) restricted to pairs (x,ξ) for which the shear strain is subcritical, i.e., \(\mathcal{S}^{\epsilon_{k}}<\overline{\eta}/(\epsilon _{k}\xi)\).
4 Mathematical Underpinnings and Analysis
From the physical perspective the convexconcave nonlinearity of the peridynamic potential delivers the unstable constitutive law relating force to shear strain. On the other hand from the mathematical viewpoint this class of peridynamic potentials share the same convexconcave structure as the function r:→arctan(r ^{2}) proposed by De Giorgi [21] and analyzed and generalized in the work of Gobbino [21, 22], and Gobbino and Mora [23] for the analysis of the Mumford Shah functional used in image processing [28]. Here we apply the methods developed in these investigations and use them as tools for extracting the limit behavior from the peridynamic model.
In this section we provide the proofs of the theorems stated in sections two and three. The first subsection asserts the Lipschitz continuity of \(\nabla\mathit{PD}^{\epsilon_{k}}(u)\) for u in \(L^{2}_{0}(D)\) and applies the standard theory of ODE to deduce existence of the peridynamic flow see, Sect. 4.1. A Gronwall inequality is used to bound the peridynamic elastic energy and kinetic energy uniformly in time see, Sect. 4.2. We introduce Γconvergence for peridynamic functions in Sect. 4.3 and identify compactness conditions necessary to generate a sequence of peridynamic flows converging to a limit flow. We take limits and apply Γconvergence theory to see that the limit flows have bounded elastic energy in the sense of fracture mechanics. In Sect. 4.4 we pass to the limit in the energy balance equation for peridynamic flows (2.11) to recover an energy inequality for the limit flow. The wave equation satisfied by the limit flow is obtained on identifying the weak L ^{2} limit of the sequence \(\{\nabla \mathit{PD}^{\epsilon_{k}}(u^{\epsilon_{k}})\}_{k=1}^{\infty}\) and passing to the limit in the weak formulation of (1.12) see, Sect. 4.5. We conclude with the proof of Theorem 2.3.
4.1 Existence of Peridynamic Evolution
4.2 Bounds on Kinetic and Potential Energy for Solutions of PD
In this section we apply Gronwall’s inequality to obtain bounds on the kinetic and elastic energy for peridynamic flows. The bounds are used to show that the solutions of the PD initial value problem are Lipschitz continuous in time. The bounds are described in the following theorem.
Theorem 4.1
(Bounds on kinetic and potential energy for peridynamic evolution)
Proof
Theorem 4.1 implies that PD solutions are Lipschitz continuous in time; this is stated explicitly in the following theorem.
Theorem 4.2
(Lipschitz continuity)
Proof
4.3 Compactness and Convergence
We now recall the properties of Γconvergence in order to apply them to the problem considered here. Consider a sequence of functions {F _{ j }} defined on a metric space \(\mathbb{M}\) with values in \(\overline{\mathbb{R}}\) together with a function F also defined on \(\mathbb{M}\) with values in \(\overline {\mathbb{R}}\).
Definition 4.1
 1.for every x in \(\mathbb{M}\) and every sequence {x _{ j }} converging to x, we have that$$\begin{aligned} F(x)\leq\liminf_{j\rightarrow\infty} F_j(x_j), \end{aligned}$$(4.11)
 2.for every x in \(\mathbb{M}\) there exists a recovery sequence {x _{ j }} converging to x, for which$$\begin{aligned} F(x)=\lim_{j\rightarrow\infty} F_j(x_j). \end{aligned}$$(4.12)
We shall see that we can pass to the limit ϵ _{ k }↘0 to find that the limit evolution u ^{0}(t,x) belongs to the class of Generalized SBV functions denoted by GSBV(D). This class of functions has been introduced for the study of free discontinuity problems in [3] and are seen here to naturally arise in the small horizon limit of peridynamics. The space GSBV(D) is composed of all measurable functions u defined on D whose truncations u _{ k }=(u∧k)∨(−k) belong to SBV(B) for every compact subset B of D, see [4, 5]. Every u belonging to GSBV(D) has an approximate gradient ∇u(x) for almost every x in D and the jump set S _{ u } is the countable union of rectifiable arcs up to a set of Hausdorff \(\mathcal{H}^{1}\) measure zero.
4.4 Energy Inequality for the Limit Flow
In this section we prove Theorem 3.2. We begin by showing that the limit evolution u ^{0}(t,x) has a weak derivative \(u_{t}^{0}(t,x)\) belonging to L ^{2}([0,T]×D). This is summarized in the following theorem.
Theorem 4.3
Proof
To establish Theorem 3.2 we require the following inequality.
Lemma 4.1
Proof
Theorem 3.2 now follows immediately on taking the ϵ _{ k }→0 limit in the peridynamic energy balance equation (2.11) of Theorem 2.2 and applying (4.15), (4.17), (4.18), and (4.22) of Lemma 4.1. □
4.5 Stationarity Conditions for the Limit Flow
In this section we prove Theorems 3.3 and 3.4. In the first subsection we give the proof of Theorem 3.3. In the second subsection we provide the proof of Theorem 3.4 using Theorem 4.4. In the last subsection we prove Theorem 4.4.
4.5.1 Proof of Theorem 3.3
4.5.2 Proof of Theorem 3.4
Theorem 4.4
4.5.3 Proof of Theorem 4.4
Lemma 4.2
Lemma 4.3
We have the following string of estimates.
Lemma 4.4
Proof
4.5.4 Proof of Theorem 2.3
5 Summary
The nonlocal continuum model of peridynamic type presented here does not require extra constitutive laws such as a kinetic relation between crack driving force and crack velocity or a crack nucleation condition. Instead this information is intrinsic to the formulation and encoded into the nonlocal constitutive law. Crack nucleation criteria are recovered here by viewing nucleation as a dynamic instability, this is similar in spirit to [34] and the work of [15] for phase transions. The scaling analysis shows that the limit evolution has bounded linear elastic energy and bounded Griffith surface energy and that these are expressed in terms of shear moduli μ and energy release rate \(\mathcal{G}_{c}\). These parameters are determined explicitly by the peridynamic potential f and influence function J. The formulas for μ and \(\mathcal{G}_{c}\) follow directly from the scaling limit without extra hypotheses. With these observations in hand we can turn this correspondence around and use measured values of μ and \(\mathcal{G}_{c}\) to choose peridynamic potentials and influence functions appropriate for a particular material of interest. It is pointed out that the constitutive model (1.3) does not include the effects of damage and bonds strained beyond critical can return to their subcritical constitutive behavior. However with this caveat in mind the theoretical results presented here strongly support the notion that the peridynamic model can be used as a numerical tool for analyzing crack paths associated with brittle fracture by choosing the scaling parameter ϵ sufficiently small in (1.12) and running numerical simulations. In closing we note that future work will attempt to uncover kinetic relations relating crack driving force and velocity for the dynamics in the small horizon limit.
Footnotes
 1.
We denote A∖B=A∼B=A∩B ^{ c }, where B ^{ c } is the complement of B in A.
Notes
Acknowledgements
The author would like to thank Stewart Silling, Richard Lehoucq and Florin Bobaru for stimulating and fruitful discussions. This research is supported by NSF grant DMS1211066, AFOSR grant FA9550050008, and NSF EPSCOR Cooperative Agreement No. EPS1003897 with additional support from the Louisiana Board of Regents.
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