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Finite Element Convergence for State-Based Peridynamic Fracture Models


We establish the a priori convergence rate for finite element approximations of a class of nonlocal nonlinear fracture models. We consider state-based peridynamic models where the force at a material point is due to both the strain between two points and the change in volume inside the domain of the nonlocal interaction. The pairwise interactions between points are mediated by a bond potential of multi-well type while multi-point interactions are associated with the volume change mediated by a hydrostatic strain potential. The hydrostatic potential can either be a quadratic function, delivering a linear force–strain relation, or a multi-well type that can be associated with the material degradation and cavitation. We first show the well-posedness of the peridynamic formulation and that peridynamic evolutions exist in the Sobolev space \(H^2\). We show that the finite element approximations converge to the \(H^2\) solutions uniformly as measured in the mean square norm. For linear continuous finite elements, the convergence rate is shown to be \(C_t \Delta t + C_s h^2/\epsilon ^2\), where \(\epsilon \) is the size of the horizon, h is the mesh size, and \(\Delta t\) is the size of the time step. The constants \(C_t\) and \(C_s\) are independent of \(\Delta t\) and h and may depend on \(\epsilon \) through the norm of the exact solution. We demonstrate the stability of the semi-discrete approximation. The stability of the fully discrete approximation is shown for the linearized peridynamic force. We present numerical simulations with the dynamic crack propagation that support the theoretical convergence rate.


In this work, we study the state-based peridynamic theory and obtain an a priori error bound for the finite element approximation. The peridynamic theory is a reformulation of classical continuum mechanics carried out in the work of Silling in [34, 37]. The strain inside the medium is expressed in terms of displacement differences as opposed to the displacement gradients. Acceleration of a point is now due to the sum of the forces acting on the point from nearby points. The new kinematics bypasses the difficulty incurred by juxtaposing displacement gradients and discontinuities as in the case of classical fracture theories. The nonlocal fracture theory has been applied numerically to model the complex fracture phenomenon in materials; see [1, 3, 11, 15, 17, 19, 27, 35, 36, 38, 40]. Every point interacts with its neighbors inside a ball of fixed radius is called the horizon. The size of the horizon sets the length scale of the nonlocal interaction. When the forces between points are linear and the nonlocal length scale tends to zero, it is seen that peridynamic models converge to the classic model of the linear elasticity; see [2, 14, 32, 36]. The work of [39] provides an analytic framework for analyzing FEM for the linear bond and state-based peridynamics. For nonlinear forces associated with double well potentials, the peridynamic evolution converges in the small horizon limit to an evolution with a sharp evolving fracture set and the evolution is governed by the classic linear elastic wave equation away from the fracture set; see [21, 25, 26]. A recent review of the state of the art can be found in [4] and [9].

In this work, we assume small deformation and work with the linearized bond-strain. Let \(D\subset {\mathbb {R}}^d\), for \(d=2,3\), be the material domain. For a displacement field \(\varvec{u}: D \times [0,T] \rightarrow {\mathbb {R}}^d\), the bond-strain between two material points \(\varvec{x}, \varvec{y}\in D\) is given by

$$\begin{aligned} S(\varvec{y}, \varvec{x}, t; \varvec{u}) = \frac{\varvec{u}(\varvec{y}, t) - \varvec{u}(\varvec{x}, t)}{|\varvec{y}- \varvec{x}|} \cdot \frac{\varvec{y}- \varvec{x}}{|\varvec{y}- \varvec{x}|}. \end{aligned}$$

Let \(\epsilon > 0\) be the size of the horizon and \(H_\epsilon (\varvec{x}) = \{ \varvec{y}\in {\mathbb {R}}^d: |\varvec{y}- \varvec{x}| < \epsilon \}\) be the neighborhood of a material point \(\varvec{x}\). For pairwise interaction, we assume the following form of pairwise interaction potential:

$$\begin{aligned} \mathcal {W}^{\,\epsilon} (S(\varvec{y}, \varvec{x}, t;\varvec{u})) = \frac{J^\epsilon (|\varvec{y}- \varvec{x}|)}{\epsilon |\varvec{y}- \varvec{x}|} f(\sqrt{|\varvec{y}- \varvec{x}|} S(\varvec{y}, \varvec{x}, t;\varvec{u})), \end{aligned}$$

where \(J^\epsilon (|\varvec{y}- \varvec{x}|)\) is the influence function. We assume \(J^\epsilon (|\varvec{y}- \varvec{x}|) = J(|\varvec{y}-\varvec{x}|/\epsilon )\) where \(0\leqslant J(r) \leqslant M\) for \(r<1\) and \(J(r) = 0\) for \(r\geqslant 1\). The potential f, see Fig. 1a, is assumed to be convex for small strains and becomes concave for larger strains. In the widely used prototypical micro-elastic brittle (PMB) peridynamic material, the strain vs force profile is linear up to some critical strain \(S_{\text {c}}\) and is zero for any strain above \(S_{\text {c}}\). In contrast, the peridynamic force given by \(\partial _S \mathcal {W}^{\,\epsilon} \) is linear near zero strain and as the strain gets larger and reaches the critical strain, \(S_{\text {c}}^+\) (\(S_{\text {c}}^-\)) for positive (negative) strain, the bond starts to soften, see Fig. 1b. For a given potential function f, the critical strain is given by \(S_{\text {c}}^+ = \frac{r^+}{\sqrt{|\varvec{y}- \varvec{x}|}}\) and \(S_{\text {c}}^- = \frac{r^-}{\sqrt{|\varvec{y}- \varvec{x}|}}\), where \(r^+>0, r^-<0\) are the inflection points of the potential function f as shown in Fig. 1a.

Fig. 1

a Potential function f(r) for tensile force. \(C^+\) and \(C^-\) are two extreme values of f. b Cohesive tensile force

The spherical or hydrostatic strain \(\theta (\varvec{x}, t;\varvec{u})\) at material point is given by

$$\begin{aligned} \theta (\varvec{x},t;\varvec{u})=\frac{1}{\epsilon ^d \omega _d}\int _{H_\epsilon (\varvec{x})} J^\epsilon (|\varvec{y}-\varvec{x}|)S(\varvec{y},\varvec{x},t;\varvec{u}){|\varvec{y}-\varvec{x}|}\,{\text {d}}\varvec{y}, \end{aligned}$$

where \(\omega _d\) is the volume of the unit ball in dimension \(d=2,3\). The potential for hydrostatic interaction is of the form

$$\begin{aligned} {\mathcal {V}}^{\,\epsilon}(\theta (\varvec{x},t;\varvec{u}))=\frac{g(\theta (\varvec{x},t;\varvec{u}))}{\epsilon ^2}, \end{aligned}$$

where g is the potential function associated with the hydrostatic strain. Here g can be of two types: (i) a quadratic function with only one well at zero strain, and (ii) a convex–concave function with a wells at the origin and at \(\pm \infty \); see Fig. 2a. If g is assumed to be quadratic, then the force due to the spherical strain is linear. If g is a multi-well potential, the material softens as the hydrostatic strains exceed the critical value. For the convex–concave type g, the critical values are \(0<\theta _{\text {c}}^+\) and \(\theta ^-_{\text {c}}< 0\) beyond which the force begins to soften is related to the inflection point \(r^+_*\) and \(r_*^-\) of g as follows:

$$\begin{aligned} \theta ^+_{\text {c}}={r^+_*}, \qquad \theta ^-_{\text {c}}={r_*^-}. \end{aligned}$$

The critical compressive hydrostatic strain where the force begins to soften for negative hydrostatic strain is chosen much larger in magnitude than \(\theta _{\text {c}}^+\), i.e., \(\theta ^+_{\text {c}}<< |\theta _{\text {c}}^-|\).

Fig. 2

a Two types of potential function g(r) for hydrostatic force. Dashed line corresponds to the quadratic potential \(g(r) = \beta r^2/2\). Solid line corresponds to the convex–concave type potential g(r). For the convex–concave type potential, there are two special points \(r^-_*\) and \(r^+_*\) at which material points start to soften. \(C^+_*\) and \(C^-_*\) are two extreme values. b Hydrostatic forces

The finite element approximation has been applied to the peridynamic fracture; however, there remains a paucity of literature addressing the rigorous a priori convergence rate of the finite element approximation to peridynamic problems in the presence of material failure. This aspect provides the motivation for the present work. In this paper, we first prove the existence of peridynamic evolutions taking values in \(H^2(D;{\mathbb {R}}^d)\cap H^1_0(D;{\mathbb {R}}^d)\) that are twice differentiable in time; see Theorem 2. We note that as these evolutions will become more fracture like as the region of the nonlocal interaction decreases. These evolutions can be thought of as inner approximations to fracture evolutions. On passing to subsequences it is possible to show that the \(H^2(D;{\mathbb {R}}^d)\cap H^1_0(D;{\mathbb {R}}^d)\) evolutions converge in the limit of vanishing non-locality to a limit solution taking values in the space of special functions of bounded deformation SBD. Here the limit evolution has a well-defined Griffith fracture energy bounded by the initial data; see [26] and [23]. We show here that the higher temporal regularity can be established if the body force changes smoothly in time. Motivated by these considerations, we develop finite element error estimates for solutions that take values in \(H^2(D;{\mathbb {R}}^d)\cap H^1_0(D;{\mathbb {R}}^d)\) and for a bounded time interval.

In this paper, we obtain an a priori \(L^2\) error bound for the finite element approximation of the displacement and velocity using a central in time discretization. Due to the nonlinear nature of the problem, we get a convergence rate using the Lax–Richtmyer stability together with the consistency. Both the stability and consistency are shown to follow from the Lipschitz continuity of the peridynamic force in \(L^2(D;{\mathbb {R}}^d)\); see Sects. 4.2.1 and 4.2.2. The bound on the \(L^2\) error is uniform in time and is given by \(C_t \Delta t + C_s h^2/\epsilon ^2\), where the constants \(C_t\) and \(C_s\) are independent of \(\Delta t\) and the mesh size h; see Theorem 6. A more elaborate discussion of the a priori bound is presented in Sect. 4.2. For the linearized model, we obtain a stability condition on \(\Delta t\), Theorem 9, that is of the same form as those given for linear local and nonlocal wave equations [18, 24]. We demonstrate the stability for the linearized model noting that for small strains the material behaves like a linear elastic material and that the stability of the linearized model is necessary for the stability of nonlinear model. We believe a more constructive CFL stability condition is possible for the linear case and will pursue this in future work.

Previous work [21] treated spatially Lipschitz continuous solutions and addressed the finite difference approximation and obtained bounds on the \(L^2\) error for the displacement and velocity that are uniform in time and of the form \(C_t \Delta t + C_s h/\epsilon ^2\), where the constants \(C_t\) and \(C_s\) are as before. For finite elements, the convergence rate is seen to be slower than for the FEM model introduced here and is of order \(h/\epsilon ^2\) as opposed to \(h^2/\epsilon ^2\). On the other hand, the FEM method increases the computational work due to the inversion of the mass matrix.

We carry out numerical experiments for dynamic crack propagation and obtain convergence rates for Plexiglass that are in line with the theory; see Sect. 5. We also compare the Griffith’s fracture energy with the peridynamic energy of the material softening zone; we show good agreement between the two energies; see Sect. 5.2. Finite difference methods are less expensive than finite element approximations for nonlocal problems; however, the latter offers more control on the accuracy of solution; see [10, 13, 16, 30, 31].

Here the a priori \(L^2\) convergence rates for the FEM given by Theorem 6 include the effects of material degradation through the softening of material properties. The FEM simulations presented in this paper show that the material develops localized softening zones (region where bonds exceed the critical tensile strain) as it deforms. This is in contrast to linear peridynamic models which are incapable of developing softening zones. For nonlinear peridynamic models with material failure, the localization of zones of softening and damage is the hallmark of the peridynamic modeling [15, 19, 34, 37]. One notes that the a priori error involves \(\epsilon \) in the denominator and in many cases \(\epsilon \) is chosen small. However, typical dynamic fracture experiments last only hundreds of microseconds and the a priori error is controlled by the product of simulation time multiplied by \(h^2/\epsilon ^2\). So for material properties characteristic of Plexiglass and \(\epsilon \) of size 4 mm, the a priori estimates predict a relative error of \(\frac{1}{10} \) for simulations lasting around 100 \(\upmu \)s. We point out that the a priori error estimates assume the appearance of nonlinearity anywhere in the computational domain. On the other hand, the numerical simulation and independent theoretical estimates show that the nonlinearity concentrates along “fat” cracks of finite length and width equal to \(\epsilon \); see [25, 26]. Moreover, the remainder of the computational domain is seen to behave linearly and to leading order can be modeled as a linear elastic material up to an error proportional to \(\epsilon \); see [Proposition 6, [22]]. Future work will use these observations to focus on the adaptive implementation and a posteriori estimates. A posteriori convergence for FEM models of peridynamics with material degradation can be seen in the work [7, 31, 33]. For other nonlinear and nonlocal models, the adaptive mesh refinement within FE framework for nonlocal models has been explored in [13] and convergence of the adaptive FE approximation is rigorously shown. A posteriori error analysis of linear nonlocal models is carried out in [12].

The paper is organized as follows. We introduce the equation of motion in Sect. 2 and present the Lipschitz continuity of the force, existence of peridynamic solution, and the higher temporal regularity necessary for the finite element error analysis. In Sect. 3, we consider the finite element discretization. We prove the stability of a semi-discrete approximation in Sect. 3.1. In Sect. 4, we analyze the fully discrete approximation and obtain an a priori bound on errors. The stability of the fully discrete approximation linearized peridynamic force is shown in Sect. 4.3. We present our numerical experiments in Sect. 5. Proofs of the Lipschitz bound on the peridynamic force and higher temporal regularity of solutions is provided in Sect. 6. In Sect. 7, we present our conclusions.

We conclude the introduction by listing the notation used throughout the paper. We denote material domain as D, where \(D\subset {\mathbb {R}}^d\) for \(d=2,3\). Points and vectors in \({\mathbb {R}}^d\) are denoted as bold letters. Some of the key notations are as follows:

[0, T]Time domain
\(\epsilon \)Size of horizon
\(\rho \)Density
\(H_\epsilon (\varvec{x})\)Horizon of \(\varvec{x}\in D\), a ball of radius \(\epsilon \) centered at \(\varvec{x}\)
\(\omega _d\)Volume of unit ball in dimension \(d=2,3\)
\(\omega (\varvec{x}) \in [0,1]\)Boundary function defined on D taking value 1 in the interior and smoothly decaying to 0 as \(\varvec{x}\) approaches \(\partial D\)
\(\varvec{u}\)Displacement field defined over \(D \times [0,T]\). We may also use notation \(\varvec{u}\) to denote field defined over just D
\(\varvec{u}_0, \varvec{v}_0\)Initial condition on displacement
\(\varvec{b}\)Body force defined over \(D\times [0,T]\)
\(\varvec{e}_{\varvec{y}-\varvec{x}}\)The unit vector pointing from a point \(\varvec{y}\) to the point \(\varvec{x}\)
\(S = S(\varvec{y}, \varvec{x}, t; \varvec{u})\)Bond strain \(S=\frac{\varvec{u}(\varvec{y},t)-\varvec{u}(\varvec{x},t)}{|\varvec{y}-\varvec{x}|}\cdot \varvec{e}_{\varvec{y}-\varvec{x}}\). We may also use \(S(\varvec{y},\varvec{x};\varvec{u})\) if \(\varvec{u}\) is a filed defined over just D
\(\theta = \theta (\varvec{x},t;\varvec{u})\)Spherical or hydrostatic strain. We may also use \(\theta (\varvec{x};\varvec{u})\) if \(\varvec{u}\) is a filed defined over just D
\(S^+_{\text{c}}, S^-_{\text{c}}\)Critical bond strain
\(\theta ^+_{\text{c}}, \theta ^-_{\text{c}}\)Critical hydrostatic strain
\(J^\epsilon (r) = J(r/\epsilon )\)Influence function where J is integrable with \(J(r) = 0\) for \(r\geqslant 1\) and \(0\leqslant J(r) \leqslant M\) for \(r < 1\)
\({\bar{J}}_\alpha \)Moment of function J over \(H_1({\mathbf {0}})\) with weight \(1/(\omega _d |{\varvec{\xi }}|^\alpha )\)
fgPotential functions for pairwise and state-based interaction
\(\mathcal {W}^{\,\epsilon} , {\mathcal {V}}^{\,\epsilon}\)Pairwise and state-based potential energy density
\(PD^\epsilon (\varvec{u}(t))\)Total peridynamic potential energy at time t
\({\mathcal {E}}^\epsilon (\varvec{u})(t)\)Total dynamic energy at time t
\({\mathcal {L}}^\epsilon , {\mathcal {L}}^\epsilon _T, {\mathcal {L}}^\epsilon _D\)Total peridynamic force, pairwise peridynamic force, and state-based peridynamic force, respectively
\(a^\epsilon (\varvec{u}, \varvec{v})\)Nonlinear operator where \(\varvec{u}, \varvec{v}\) are vector fields over D
\(a^\epsilon _T, a^\epsilon _D\)Nonlinear pairwise and state-based operator
\(||\cdot ||,||\cdot ||_\infty , ||\cdot ||_n\)\(L^2\) norm over D, \(L^\infty \) norm over D, and Sobolev \(H^n\) norm over D (for \(n=1,2\)), respectively
\(h, \Delta t\)Size of mesh and size of time step
\({\mathcal {T}}_h\)Triangulation of D given by triangular/tetrahedral elements
\({\mathcal {I}}_h\)Continuous piecewise linear interpolation operator on \({\mathcal {T}}_h\)
WSpace of functions in \(H^2(D;{\mathbb {R}}^d)\) such that trace of function is zero on boundary \(\partial D\), i.e., \(W = H^2(D;{\mathbb {R}}^d) \cap H^1_0(D;{\mathbb {R}}^d)\)
\(V_h\)Space of continuous piecewise linear interpolations on \({\mathcal {T}}_h\)
\(\phi _i\)Interpolation function of mesh node i
\(\varvec{r}_h(\varvec{u})\)Finite element projection of \(\varvec{u}\) onto \(V_h\)
\(E^k\)Total error in mean square norm at time step k
\(\varvec{u}^k_h, \varvec{v}^k_h\)Approximate displacement and velocity field at time step k
\(\varvec{u}^k, \varvec{v}^k\)Exact displacement and velocity field at time step k

Equation of Motion, Existence, Uniqueness, and Higher Regularity

We assume D to be an open set with \(C^1\) boundary. To enforce zero displacement boundary conditions at \(\partial D\) and to insure a well-posed evolution, we introduce the boundary function \(\omega (\varvec{x})\). This function is introduced as a factor into the potentials \(\mathcal {W}^{\,\epsilon} \) and \({\mathcal {V}}^{\,\epsilon} \). Here the boundary function takes value 1 in the interior of domain and is zero on the boundary. We assume \(\sup _{\varvec{x}} |\nabla \omega (\varvec{x})| < \infty \) and \(\sup _{\varvec{x}} |\nabla ^2 \omega (\varvec{x})| < \infty \) in our analysis. The hydrostatic strain is modified to include the boundary and is given by

$$\begin{aligned} \theta (\varvec{x},t;\varvec{u})=\frac{1}{\epsilon ^d \omega _d}\int _{H_\epsilon (\varvec{x})} \omega (\varvec{y}) J^\epsilon (|\varvec{y}-\varvec{x}|)S(\varvec{y},\varvec{x},t;\varvec{u}){|\varvec{y}-\varvec{x}|}\,{\text {d}}\varvec{y}. \end{aligned}$$

The peridynamic potentials, Eqs. 2 and 4, are modified to see the boundary as follows:

$$\begin{aligned} \mathcal {W}^{\,\epsilon} (S(\varvec{y}, \varvec{x}, t;\varvec{u}))&= \omega (\varvec{x}) \omega (\varvec{y}) \frac{J^\epsilon (|\varvec{y}- \varvec{x}|)}{\epsilon |\varvec{y}- \varvec{x}|} f(\sqrt{|\varvec{y}- \varvec{x}|} S(\varvec{y}, \varvec{x}, t;\varvec{u})), \end{aligned}$$
$$\begin{aligned} {\mathcal {V}}^\epsilon (\theta (\varvec{x},t;\varvec{u}))&= \omega (\varvec{x}) \frac{g(\theta (\varvec{x},t;\varvec{u}))}{\epsilon ^2} . \end{aligned}$$

We assume that the potential function f is at least four times differentiable and satisfies the following regularity condition:

$$\begin{aligned} C^f_0 := \sup _r |f(r)|< \infty , \qquad C^f_i := \sup _r |f^{(i)}(r)| < \infty , \quad \forall i = 1,2,3,4. \end{aligned}$$

If the potential function g is convex–concave type, then we assume that g satisfies the same regularity condition as f. We denote constants \(C^g_i\), for \(i=0,1,\cdots ,4\), similar to \(C^f_i\) above.

The total potential energy at time t is given by

$$\begin{aligned} \begin{aligned} PD^\epsilon (\varvec{u}(t))&=\frac{1}{\epsilon ^d \omega _d}\int _D \int _{H_\epsilon (\varvec{x})} |\varvec{y}-\varvec{x}|\mathcal {W}^{\,\epsilon} (S(\varvec{y},\varvec{x},t;\varvec{u}))\,{\text {d}}\varvec{y}{\text {d}}\varvec{x}\\&\quad +\int _D {\mathcal {V}}^{\,\epsilon} (\theta (\varvec{x},t;\varvec{u}))\,{\text {d}}\varvec{x}, \end{aligned} \end{aligned}$$

where potential \(\mathcal {W}^{\,\epsilon} \) and \({\mathcal {V}}^\epsilon \) are described above. The material is assumed to be homogeneous and the density is given by \(\rho \). The applied body force is denoted by \(\varvec{b}(\varvec{x},t)\). We define the Lagrangian

$$\begin{aligned} \mathrm{{L}}(\varvec{u},\partial _t \varvec{u},t)=\frac{\rho }{2}||{\dot{\varvec{u}}}||^2 - PD^\epsilon (\varvec{u}(t))+\int _D \varvec{b}(t)\cdot \varvec{u}(t) {\text {d}}\varvec{x}, \end{aligned}$$

here \({\dot{\varvec{u}}}\) is the velocity given by the time derivative of \(\varvec{u}\). Applying the principal of least action together with a straight forward calculation (see, for example, [28] for detailed derivation) gives the nonlocal dynamics

$$\begin{aligned} \begin{aligned} \rho \ddot{\varvec{u}}(\varvec{x},t)={\mathcal {L}}^\epsilon (\varvec{u})(\varvec{x},t)+\varvec{b}(\varvec{x},t)\,\hbox {for}\,\, {\varvec{x}\in D}, \end{aligned} \end{aligned}$$


$$\begin{aligned} {\mathcal {L}}^\epsilon (\varvec{u})(\varvec{x},t) = {\mathcal {L}}^\epsilon _T(\varvec{u})(\varvec{x},t) + {\mathcal {L}}^\epsilon _D(\varvec{u})(\varvec{x},t), \end{aligned}$$

\({\mathcal {L}}^\epsilon _T(\varvec{u})\) is the peridynamic force due to the bond-based interaction and is given by

$$\begin{aligned}&{\mathcal {L}}^\epsilon _T(\varvec{u})(\varvec{x},t) \nonumber \\& =\frac{2}{\epsilon ^d \omega _d}\int _{H_\epsilon (\varvec{x})} \omega (\varvec{x}) \omega (\varvec{y}) \frac{J^\epsilon (|\varvec{y}-\varvec{x}|)}{\epsilon |\varvec{y}-\varvec{x}|}\partial _S f(\sqrt{|\varvec{y}-\varvec{x}|}S(\varvec{y},\varvec{x},t;\varvec{u}))\varvec{e}_{\varvec{y}-\varvec{x}}\,{\text {d}}\varvec{y}, \end{aligned}$$

and \({\mathcal {L}}^\epsilon _D(\varvec{u})\) is the peridynamic force due to the state-based interaction and is given by

$$\begin{aligned}&{\mathcal {L}}^\epsilon _D(\varvec{u})(\varvec{x},t)\nonumber \\& =\frac{1}{\epsilon ^d \omega _d}\int _{H_\epsilon (\varvec{x})} \omega (\varvec{x}) \omega (\varvec{y}) \frac{J^\epsilon (|\varvec{y}-\varvec{x}|)}{\epsilon ^2}\left[ \partial _\theta g(\theta (\varvec{y},t;\varvec{u}))+\partial _\theta g(\theta (\varvec{x},t;\varvec{u}))\right] \varvec{e}_{\varvec{y}-\varvec{x}}\,{\text {d}}\varvec{y}. \end{aligned}$$

The dynamics is complemented with the initial data

$$\begin{aligned} \varvec{u}(\varvec{x},0)=\varvec{u}_0(\varvec{x}), \qquad \partial _t \varvec{u}(\varvec{x},0)=\varvec{v}_0(\varvec{x}). \end{aligned}$$

We prescribe the zero Dirichlet boundary condition on the boundary \(\partial D\)

$$\begin{aligned} \varvec{u}(\varvec{x}) = {\mathbf {0}},\qquad \forall \varvec{x}\in \partial D. \end{aligned}$$

We extend the zero boundary condition outside D to whole \({\mathbb {R}}^d\). In our analysis, we will assume the mass density \(\rho = 1\) without loss of generality.

Existence of Solutions and Higher Regularity in Time

We recall that the space \(H^n_0(D;{\mathbb {R}}^d)\) is the closure in the \(H^n\) norm of the functions that are infinitely differentiable with compact support in D. For suitable initial conditions and body force, we show that solutions exist in

$$\begin{aligned} W=H^2(D;{\mathbb {R}}^d) \cap H^1_0(D;{\mathbb {R}}^d)=\{v\in H^2(D;{\mathbb {R}}^d):\gamma v =0\,\hbox { on }\partial D\}, \end{aligned}$$

where \(\gamma \) is the trace of the function v on the boundary of D. We will assume that \(\varvec{u}\in W\) is extended by zero outside D. We first exhibit the Lipschitz continuity property and boundedness of the peridynamic force for displacements in W. We will then apply [Theorem 3.2, [20]] to conclude the existence of unique solutions.

We note the following Sobolev embedding properties of \(H^2(D;{\mathbb {R}}^d)\) when D is a \(C^1\) domain.

  • From Theorem 2.72 of [8], there exists a constant \(C_{e_1}\) independent of \(\varvec{u}\in H^2(D;{\mathbb {R}}^d)\) such that

    $$\begin{aligned} ||\varvec{u}||_{\infty } \leqslant C_{e_1} ||\varvec{u}||_{2}. \end{aligned}$$
  • Further application of standard embedding theorems (e.g., Theorem 2.72 of [8]) shows there exists a constant \(C_{e_2}\) independent of \(\varvec{u}\) such that

    $$\begin{aligned} ||\nabla \varvec{u}||_{L^q(D;{\mathbb {R}}^{d\times d})} \leqslant C_{e_2} ||\nabla \varvec{u}||_{1} \leqslant C_{e_2} ||\varvec{u}||_{2}, \end{aligned}$$

    for any q such that \(2\leqslant q< \infty \) when \(d=2\) and \(2\leqslant q \leqslant 6\) when \(d=3\).

We have the following result which shows the Lipschitz continuity property of a peridynamic force \({\mathcal {L}}^\epsilon \).

Theorem 1

(Lipschitz continuity of peridynamic force) Letfbe a convex–concave function satisfying\(C^f_i <\infty \)for\(i=0,\cdots ,4\)and letgeither be a quadratic function, orgbe aconvex–concave function with\(C^g_i < \infty \)for\(i=0,\cdots ,4\). Also, let the boundary function\(\omega : D\rightarrow [0,1]\)be such that\(\sup \limits _{\varvec{x}\in D}^{} |\nabla \omega (\varvec{x})| <\infty \) and \(\sup \limits _{\varvec{x}\in D}^{} |\nabla ^2 \omega (\varvec{x})| <\infty \). Then, for any\(\varvec{u},\varvec{v}\in W\), we have

$$\begin{aligned} ||{\mathcal {L}}^\epsilon (\varvec{u}) - {\mathcal {L}}^\epsilon (\varvec{v})||_2&\leqslant \dfrac{{\bar{L}}_1 (1 + ||\varvec{u}||_2 + ||\varvec{v}||_2)^2}{\epsilon ^3} ||\varvec{u}- \varvec{v}||_2, \end{aligned}$$

where constant\({\bar{L}}_1\)does not depend on\(\epsilon \)nor\(\varvec{u},\varvec{v}\). Also, for\(\varvec{u}\in W\), we have

$$\begin{aligned} ||{\mathcal {L}}^\epsilon (\varvec{u})||_2&\leqslant \dfrac{{\bar{L}}_2 (||\varvec{u}||_2 + ||\varvec{u}||_2^2)}{\epsilon ^{5/2}}, \end{aligned}$$

where constant\({\bar{L}}_2\)does not depend on\(\epsilon \)nor\(\varvec{u}\).

Now let \(T>0\) be any positive number, a straight-forward application of [Theorem 3.2, [20]] gives:

Theorem 2

(Existence and uniqueness of solutions over finite time intervals) Letf, g, and\(\omega \)satisfy the hypothesis of Theorem 1. For any initial condition\(\varvec{u}_0,\varvec{v}_0 \in W\), time interval\(I_0=(-T,T)\), and right-hand side\(\varvec{b}(t)\)continuous in time for\(t\in I_0\)such that\(\varvec{b}(t)\)satisfies\(\sup \limits_{t\in I_0} ^{} ||\varvec{b}(t)||_2<\infty \), there is a uniquesolution\(\varvec{u}(t)\in C^2(I_0;W)\)of peridynamic Eq.11. Also, \(\varvec{u}(t)\)and\({\dot{\varvec{u}}}(t)\)are Lipschitz continuous in time for\(t\in I_0\).

We can also show higher regularity in time of evolutions under suitable assumptions on the body force:

Theorem 3

(Higher regularity) Suppose the initial data and righthand side\(\varvec{b}(t)\)satisfy the hypothesis of Theorem 2and suppose further that\({\dot{\varvec{b}}}(t)\)exists and is continuous in time for\(t\in I_0\)and  \(\sup \limits _{t\in I_0} ^{} ||{\dot{\varvec{b}}}(t)||_2 < \infty \). Then, \(\varvec{u}\in C^3(I_0; W)\)and

$$\begin{aligned} || \partial ^3_{ttt} \varvec{u}(\varvec{x},t)||_2&\leqslant \dfrac{C\left(1 + \sup \limits_{s \in I_0} ^{} ||\varvec{u}(s)||_2\right)^2}{\epsilon ^3} \sup _{s\in I_0} ||\partial _t \varvec{u}(s)||_2 + ||{\dot{\varvec{b}}}(\varvec{x},t)||_2, \end{aligned}$$

whereCis a positive constant independent of\(\varvec{u}\).

The proofs of Theorems 1 and 3 are given in Sect. 6. For future reference, we note that for any \(\varvec{u},\varvec{v}\in L^2_0(D;{\mathbb {R}}^d)\), we have

$$\begin{aligned} ||{\mathcal {L}}^\epsilon (\varvec{u}) - {\mathcal {L}}^\epsilon (\varvec{v}) ||&\leqslant \frac{L}{\epsilon ^2} ||\varvec{u}- \varvec{v}||, \end{aligned}$$

where constant L is given by

$$\begin{aligned} L&:= {\left\{ \begin{array}{ll} 4(C^f_2 {\bar{J}}_1 + C^g_2{\bar{J}}_0^2) \qquad \text {if}~ g ~\text {is a convex--concave type}, \\ 4(C^f_2 {\bar{J}}_1 + g''(0){\bar{J}}_0^2) \qquad \text {if}~ g~ \text {is a quadratic function}, \end{array}\right. } \end{aligned}$$

and \({\bar{J}}^\alpha = (\frac{1}{\omega _d})\int _{H_1({\mathbf {0}})} \frac{J(|{\varvec{\xi }}|)}{|{\varvec{\xi }}|^\alpha} {\text {d}}{\varvec{\xi }}\).

Weak Form

We multiply Eq. 11 by a test function \(\tilde{\varvec{u}}\) in \(H^1_0(D;{\mathbb {R}}^d)\) and integrate over D to get

$$\begin{aligned} (\ddot{\varvec{u}}(t),\tilde{\varvec{u}}) = ({\mathcal {L}}^\epsilon (\varvec{u}(t)), \tilde{\varvec{u}}) + (\varvec{b}(t),\tilde{\varvec{u}}). \end{aligned}$$

We have the following integration by parts formula:

Lemma 1

For any\(\varvec{u}, \varvec{v}\in L^2_0(D;{\mathbb {R}}^d)\), we have

$$\begin{aligned} ({\mathcal {L}}^\epsilon (\varvec{u}), \varvec{v}) = - a^\epsilon (\varvec{u}, \varvec{v}), \end{aligned}$$


$$\begin{aligned} a^\epsilon (\varvec{u}, \varvec{v}) = a^\epsilon _T(\varvec{u}, \varvec{v}) + a^\epsilon _D(\varvec{u}, \varvec{v}) \end{aligned}$$


$$\left\{ \begin{aligned} a^\epsilon _T(\varvec{u}, \varvec{v})&= \dfrac{1}{\epsilon ^{d+1}\omega _d} \int _D \int _D \omega (\varvec{x}) \omega (\varvec{y}) J^\epsilon (|\varvec{y}- \varvec{x}|) \nonumber \\& \cdot \partial _S f(\sqrt{|\varvec{y}- \varvec{x}|} S(\varvec{y},\varvec{x};\varvec{u})) S(\varvec{y}, \varvec{x};\varvec{v}) {\text {d}}\varvec{y}{\text {d}}\varvec{x}, \nonumber \\ a^\epsilon _D(\varvec{u}, \varvec{v})&= \dfrac{1}{\epsilon ^2} \int _D \omega (\varvec{x}) g'(\theta (\varvec{x};\varvec{u})) \theta (\varvec{x};\varvec{v}) {\text {d}}\varvec{x}.\end{aligned} \right.$$

The proof of above lemma is identical to the proof of Lemma 4.2 in [28].

Using the above lemma, the weak form of the peridynamic evolution is given by

$$\begin{aligned} (\ddot{\varvec{u}}(t),\tilde{\varvec{u}}) + a^\epsilon (\varvec{u}(t), \tilde{\varvec{u}}) = (\varvec{b}(t),\tilde{\varvec{u}}). \end{aligned}$$

Total dynamic energy We define the total dynamic energy as follows:

$$\begin{aligned} {\mathcal {E}}^\epsilon (\varvec{u})(t) = \frac{1}{2} ||{\dot{\varvec{u}}}(t)||^2_{L^2} + PD^\epsilon (\varvec{u}(t)), \end{aligned}$$

where \(PD^\epsilon \) is defined in Eq. 10. The time derivative of the total energy satisfies

$$\begin{aligned} \frac{\text{d}}{\text{d} t} {\mathcal {E}}^\epsilon (\varvec{u})(t) = (\ddot{\varvec{u}}(t), {\dot{\varvec{u}}}(t)) + a^\epsilon (\varvec{u}(t), {\dot{\varvec{u}}}(t)). \end{aligned}$$

Remark 1

It is readily verified that the peridynamic force and energy are bounded for all functions in \(L^2(D;{\mathbb {R}}^d)\). Here the bound on the force follows from the Lipschitz property of the force in \(L^2(D;{\mathbb {R}}^d)\); see Eq. 23. The peridynamic force is also bounded for functions \(\varvec{u}\) in \(H^1(D;{\mathbb {R}}^d)\). This again follows from the Lipschitz property of the force in \(H^1(D;{\mathbb {R}}^d)\) using arguments established in Sect. 6. The boundedness of the energy \(PD^\epsilon (\varvec{u})\) in both \(L^2(D;{\mathbb {R}}^d)\) and \(H^1(D;{\mathbb {R}}^d)\) follows from the boundedness of the bond potential energy \(\mathcal {W}^{\,\epsilon} (S(\varvec{y}, \varvec{x}, t; \varvec{u}))\) and \({\mathcal {V}}^\epsilon (\theta (\varvec{x}, t; \varvec{u}))\) used in the definition of \(PD^\epsilon (\varvec{u})\); see Eqs. 7 and  8. More generally, this also shows that \(PD^\epsilon (\varvec{u})<\infty \) for \(\varvec{u}\in L^1(D;{\mathbb {R}}^d)\).

We next discuss the spatial and the time discretization of peridynamic equation.

Finite Element Approximation

Let \(V_h\) be given by linear continuous interpolations over tetrahedral or triangular elements \({\mathcal {T}}_h\), where h denotes the size of the finite element mesh. Here we assume the elements are conforming and the finite element mesh is shape regular and \(V_h\subset H^1_0(D;{\mathbb {R}}^d)\).

For a continuous function \(\varvec{u}\) on \({\bar{D}}\), \({\mathcal {I}}_h(\varvec{u})\) is the continuous piecewise linear interpolant on \({\mathcal {T}}_h\). It is given by

$$\begin{aligned} {\mathcal {I}}_h(\varvec{u})\vert _{T} = {\mathcal {I}}_T(\varvec{u}), \qquad \forall T\in {\mathcal {T}}_h, \end{aligned}$$

where \({\mathcal {I}}_T(\varvec{u})\) is the local interpolant defined over the finite element T and is given by

$$\begin{aligned} {\mathcal {I}}_T(\varvec{u}) = \sum _{i=1}^n \varvec{u}(\varvec{x}_i)\phi _i. \end{aligned}$$

Here n is the number of vertices in an element T, \(\varvec{x}_i\) is the position of vertex i, and \(\phi _i\) is the linear interpolant associated to vertex i.

Application of Theorem 4.4.20 and Remark 4.4.27 in [5] gives

$$\begin{aligned}&|| \varvec{u}- {\mathcal {I}}_h(\varvec{u}) || \leqslant c h^2 || \varvec{u}||_2, \hbox { } \qquad \forall \varvec{u}\in W. \end{aligned}$$

Let \(\varvec{r}_h(\varvec{u})\) denote the projection of \(\varvec{u}\in W\) on \(V_h\). For the \(L^2\) norm it is defined as

$$\begin{aligned} ||\varvec{u}- \varvec{r}_h(\varvec{u})||&= \inf _{\tilde{\varvec{u}}\in V_h} ||\varvec{u}- {\tilde{\varvec{u}}}|| \end{aligned}$$

and satisfies

$$\begin{aligned} (\varvec{r}_h(\varvec{u}), {\tilde{\varvec{u}}}) = (\varvec{u}, {\tilde{\varvec{u}}}), \qquad \forall {\tilde{\varvec{u}}} \in V_h. \end{aligned}$$

Since \({\mathcal {I}}_h(\varvec{u}) \in V_h\) and Eq. 34, we see that

$$\begin{aligned}&||\varvec{u}- \varvec{r}_h(\varvec{u})|| \leqslant c h^2 || \varvec{u}||_2, \qquad \forall \varvec{u}\in W. \end{aligned}$$

Semi-discrete Approximation

Let \(\varvec{u}_h(t) \in V_h\) be the approximation of \(\varvec{u}(t)\) satisfying following for all \(t\in [0,T],\)

$$\begin{aligned} (\ddot{\varvec{u}}_h, {\tilde{\varvec{u}}} ) + a^\epsilon (\varvec{u}_h(t), {\tilde{\varvec{u}}})&= ( \varvec{b}(t), {\tilde{\varvec{u}}} ), \qquad \forall {\tilde{\varvec{u}}} \in V_h. \end{aligned}$$

We have the following result:

Theorem 4

(Energy stability of semi-discrete approximation) The semi-discrete scheme is stable and the energy\({\mathcal {E}}^\epsilon (\varvec{u}_h)(t)\), defined in Eq. 30, satisfies the following bound:

$$\begin{aligned} {\mathcal {E}}^\epsilon (\varvec{u}_h)(t)&\leqslant \left[ \sqrt{{\mathcal {E}}^\epsilon (\varvec{u}_h)(0)} + \int _0^t ||\varvec{b}(\tau )|| {\text{d}}\tau \right] ^2. \end{aligned}$$

We note that while proving the stability of semi-discrete scheme corresponding to nonlinear peridynamics, we do not require any assumption on the strain \(S(\varvec{y},\varvec{x}, t;\varvec{u}_h)\). The proof is similar to [Section 6.2, [26]].


Letting \({\tilde{\varvec{u}}} = {\dot{\varvec{u}}}_h(t)\) in Eq. 38 and noting the identity Eq. 31, we get

$$\begin{aligned} \dfrac{\text {d}}{{\text {d}}t} {\mathcal {E}}^\epsilon (\varvec{u}_h)(t) = (\varvec{b}(t), {\dot{\varvec{u}}}_h(t)) \leqslant ||\varvec{b}(t)||\, ||{\dot{\varvec{u}}}_h(t)||. \end{aligned}$$

We also have

$$\begin{aligned} ||{\dot{\varvec{u}}}_h(t)|| \leqslant 2 \sqrt{\frac{1}{2} ||{\dot{\varvec{u}}}_h||^2 + PD^\epsilon (\varvec{u}_h(t))} = 2 \sqrt{{\mathcal {E}}^\epsilon (\varvec{u}_h)(t)}, \end{aligned}$$

where we use the fact that \(PD^\epsilon (\varvec{u})(t)\) is nonnegative. We substitute above inequality in Eq. 39 to get

$$\begin{aligned} \dfrac{\text {d}}{{\text {d}}t} {\mathcal {E}}^\epsilon (\varvec{u}_h)(t)&\leqslant 2 \sqrt{{\mathcal {E}}^\epsilon (\varvec{u}_h)(t)} \, ||\varvec{b}(t)||. \end{aligned}$$

We fix \(\delta > 0\) and define A(t) as \(A(t) = {\mathcal {E}}^\epsilon (\varvec{u}_h)(t) + \delta \). Then, from the above equation, we easily have

$$\begin{aligned} \dfrac{\text {d}}{{\text {d}}t} A(t)&\leqslant 2 \sqrt{A(t)} \, ||b(t)|| \quad \Rightarrow \dfrac{1}{2} \dfrac{\frac{{\text {d}}}{{\text {d}}t} A(t)}{\sqrt{A(t)}} \leqslant ||\varvec{b}(t)||. \end{aligned}$$

Noting that \(\frac{1}{\sqrt{a(t)}}\frac{{\text {d}} a(t)}{{\text {d}}t}= 2\frac{{\text {d}}}{{\text {d}}t} \sqrt{a(t)} \), integrating from \(t=0\) to \(\tau \) and relabeling \(\tau \) as t, we get

$$\begin{aligned} \sqrt{A(t)}&\leqslant \sqrt{A(0)} + \int _0^t ||\varvec{b}(s)|| {\text {d}}s. \end{aligned}$$

Proof is complete once we let \(\delta \rightarrow 0\) and take the square of both sides.

Central Difference Time Discretization

In Sect. 4.2, we calculate the convergence rate for the central difference time discretization of the fully nonlinear problem. We then present a CFL-like condition on the time step \(\Delta t\) for the linearized peridynamic equation in Sect. 4.3.

At time step k, the exact solution is given by \((\varvec{u}^k, \varvec{v}^k)\), where \(\varvec{v}^k = \partial \varvec{u}^k/\partial t\), and their projection onto \(V_h\) is given by \((\varvec{r}_h(\varvec{u}^k), \varvec{r}_h(\varvec{v}^k))\). The solution of fully discrete problem at time step k is given by \((\varvec{u}^k_h, \varvec{v}^k_h)\).

We approximate the initial data on displacement \(\varvec{u}_0\) and the velocity \(\varvec{v}_0\) by their projections \(\varvec{r}_h(\varvec{u}_0)\) and \(\varvec{r}_h(\varvec{v}_0)\). Let \(\varvec{u}^0_h = \varvec{r}_h(\varvec{u}_0)\) and \(\varvec{v}^0_h = \varvec{r}_h(\varvec{v}_0)\). For \(k\geqslant 1\), \((\varvec{u}^k_h, \varvec{v}^k_h)\) satisfies, for all \({\tilde{\varvec{u}}} \in V_h,\)

$$\left\{\begin{aligned} \left( \dfrac{\varvec{u}^{k+1}_h - \varvec{u}^k_h}{\Delta t}, {\tilde{\varvec{u}}} \right)&= (\varvec{v}^{k+1}_h, {\tilde{\varvec{u}}}), \nonumber \\ \left( \dfrac{\varvec{v}^{k+1}_h - \varvec{v}^k_h}{\Delta t}, {\tilde{\varvec{u}}} \right)&= ({\mathcal {L}}^\epsilon (\varvec{u}^k_h), {\tilde{\varvec{u}}} ) + (\varvec{b}^k_h, {\tilde{\varvec{u}}}) , \end{aligned}\right.$$

where we have denoted the projection of \(\varvec{b}(t^k)\), i.e., \(\varvec{r}_h(\varvec{b}(t^k))\), as \(\varvec{b}^k_h\). Combining the two equations delivers central difference equation for \(\varvec{u}^k_h\). We have

$$\begin{aligned} \left( \dfrac{\varvec{u}^{k+1}_h - 2 \varvec{u}^k_h + \varvec{u}^{k-1}_h}{\Delta t^2}, {\tilde{\varvec{u}}} \right)&= ({\mathcal {L}}^\epsilon (\varvec{u}^k_h), {\tilde{\varvec{u}}} ) + (\varvec{b}^k_h, {\tilde{\varvec{u}}}), \qquad \forall {\tilde{\varvec{u}}} \in V_h. \end{aligned}$$

For \(k=0\), we have \(\forall {\tilde{\varvec{u}}} \in V_h,\)

$$\begin{aligned} \left( \dfrac{\varvec{u}^1_h - \varvec{u}^0_h}{\Delta t^2}, {\tilde{\varvec{u}}}\right)&= \dfrac{1}{2}({\mathcal {L}}^\epsilon (\varvec{u}^0_h), {\tilde{\varvec{u}}})+ \dfrac{1}{\Delta t} (\varvec{v}^0_h, {\tilde{\varvec{u}}}) + \dfrac{1}{2}(\varvec{b}^0_h, {\tilde{\varvec{u}}}). \end{aligned}$$

Implementation Details

For completeness, we describe the implementation of the time stepping method using FEM interpolants. Let \(\varvec{N}\) be the shape tensor. Then, \(\varvec{u}^k_h, {\tilde{\varvec{u}}} \in V_h\) are given by

$$\begin{aligned} \varvec{u}^k_h = \varvec{N} \varvec{U}^k, \qquad {\tilde{\varvec{u}}} = \varvec{N} \tilde{\varvec{U}}, \end{aligned}$$

where \(\varvec{U}^k\) and \(\tilde{\varvec{U}}\) are Nd-dimensional vectors, where N is the number of nodal points in the mesh and d is the dimension.

From Eq. 41, for all \(\tilde{\varvec{U}} \in {\mathbb {R}}^{Nd}\) with elements of \(\tilde{\varvec{U}}\) zero on the boundary, then the following holds for \(k\geqslant 1\):

$$\begin{aligned} \left( \varvec{M}\frac{\varvec{U}^{k+1} - 2\varvec{U}^{k} + \varvec{U}^{k-1}}{\Delta t^2} \right) \cdot \tilde{\varvec{U}} = \varvec{F}^k \cdot \tilde{\varvec{U}} . \end{aligned}$$

Here the mass matrix \(\varvec{M}\) and the force vector \(\varvec{F}^k\) are given by

$$\left \{\begin{aligned} \varvec{M}&:= \int _D \varvec{N}^{\text{T}} \varvec{N} {\text {d}}\varvec{x},\nonumber \\ \varvec{F}^k&:= \varvec{F}^k_{pd}+ \int _D \varvec{N}^{\text{T}} \varvec{b}(\varvec{x},t^k) {\text {d}}\varvec{x}, \end{aligned}\right.$$

where \(\varvec{F}^k_{pd}\) is defined by

$$\begin{aligned} \varvec{F}^k_{pd}&:= \int _D \varvec{N}^{\text{T}} ({\mathcal {L}}^\epsilon (\varvec{u}^k_h)(\varvec{x})) {\text {d}}\varvec{x}. \end{aligned}$$

We remark that a similar equation holds for \(k=0\).

At the time step k, we must invert \({\varvec{M}}\) to solve for \({\varvec{U}}^{k+1}\) using

$$\begin{aligned} {\varvec{U}}^{k+1} = \Delta t^2 {\varvec{M}}^{-1} {\varvec{F}}^k + 2 \varvec{U}^k - {\varvec{U}}^{k-1}. \end{aligned}$$

As is well known, this inversion amounts to an increase of computational complexity associated with discrete approximation of the weak formulation of the evolution. Further, the matrix–vector multiplication \(\varvec{M}^{-1} \varvec{F}^k\) needs to be carried out at each time step. On the other hand, the quadrature error in the computation of the force vector \(\varvec{F}^k_{pd}\) is reduced when using the weak form.

We next show the convergence of approximation.

Convergence of Approximation

In this section, we prove the uniform bound on the error and show that the approximate solution converges to the exact solution with rate given by \(C_t \Delta t + C_s h^2/\epsilon ^2\). Here the horizon \(\epsilon > 0\) is assumed to be fixed. We first compare the exact solution with its projection in \(V_h\) and then compare the projection with the approximate solution. We further divide the calculation of error between the projection and the approximate solution in two parts, namely the consistency analysis and error analysis.

The error \(E^k\) is given by

$$\begin{aligned} E^k := ||\varvec{u}^k_h - \varvec{u}(t^k)|| + ||\varvec{v}^k_h - \varvec{v}(t^k)||. \end{aligned}$$

The error is split into two parts as follows:

$$\begin{aligned} E^k&\leqslant \left( ||\varvec{u}^k - \varvec{r}_h(\varvec{u}^k)|| + ||\varvec{v}^k - \varvec{r}_h(\varvec{v}^k)|| \right) + \left( || \varvec{r}_h(\varvec{u}^k) - \varvec{u}^k_h|| + ||\varvec{r}_h(\varvec{v}^k) - \varvec{v}^k_h|| \right) , \end{aligned}$$

where the first term is the error between the exact solution and projection, and the second term is the error between the projection and approximate solution. Let

$$\begin{aligned} \varvec{e}^k_h(\varvec{u}) := \varvec{r}_h(\varvec{u}^k) - \varvec{u}^k_h, \quad \varvec{e}^k_h(\varvec{v}) := \varvec{r}_h(v^k) - \varvec{v}^k_h, \end{aligned}$$


$$\begin{aligned} e^k := ||\varvec{e}^k_h(\varvec{u})|| + ||\varvec{e}^k_h(\varvec{v})||. \end{aligned}$$

Using Eq. 37, we have

$$\begin{aligned} E^k&\leqslant C_p h^2 + e^k, \end{aligned}$$


$$\begin{aligned} C_p := c\left( \sup _t ||\varvec{u}(t)||_2 + \sup _t \left \|\dfrac{\partial \varvec{u}(t)}{\partial t}\right \|_2 \right) . \end{aligned}$$

We have the following a-priori convergence rate given by

Theorem 5

(Convergence of central difference approximation) Let\((\varvec{u},\varvec{v})\)be the exact solution of the peridynamic Eq.11and\((\varvec{u}^k_h, \varvec{v}^k_h)\)be the FE solution of Eq. 40. If\(\varvec{u},\varvec{v}\in C^2([0,T]; W)\), then the scheme is consistent and the error\(E^k\)satisfies the following bound:

$$\begin{aligned}&\sup _{k \leqslant T/\Delta t} E^k \nonumber \\&\quad = C_p h^2 + \exp \left[ T\left(1+{\frac{L}{\epsilon}} ^2\right)\left( \frac{1}{1-\Delta t}\right) \right] \left[ e^0 + \left( \frac{T}{1-\Delta t}\right) \left( C_t \Delta t + C_s \dfrac{h^2}{\epsilon ^2} \right) \right] , \end{aligned}$$

where the constants\(C_p\), \(C_t\), and \(C_s\)are given by Eqs. 51and  58. The constant\(L/\epsilon ^2\)is the Lipschitz constant of the peridynamic force\({\mathcal {L}}^\epsilon (\varvec{u})\)in\(L^2\); see Eq. 23. If the error in initial data is zero, then\(E^k\)is of the order of\(C_t\Delta t + C_s h^2/\epsilon ^2\).

In Theorem 3, we have shown that \(\varvec{u},\varvec{v}\in C^2([0,T]; W)\) for righthand side \(\varvec{b}\in C^1([0,T]; W)\). In Sect. 7, we discuss the behavior of the exponential constant appearing in Theorem 5 for evolution times seen in fracture experiments. Since we are approximating the solution of an ODE on a Banach space, the proof of Theorem 5 will follow from the Lipschitz continuity of the force \({\mathcal {L}}^\epsilon (\varvec{u})\) with respect to the \(L^2\) norm. The proof is given in the following two sections.

Truncation Error Analysis and Consistency

The results in this section follow the same steps as in [20] and, therefore, we will just highlight the major steps. We can write the discrete evolution equation for \((\varvec{e}^k_h(\varvec{u}) = \varvec{r}_h(\varvec{u}^k) - \varvec{u}^k_h, \varvec{e}^k_h(\varvec{v}) = \varvec{r}_h(\varvec{v}^k) - \varvec{v}^k_h)\) as follows:

$$\left \{ \begin{aligned} ( \varvec{e}^{k+1}_h&(\varvec{u}), {\tilde{\varvec{u}}})= (\varvec{e}^{k}_h(\varvec{u}), {\tilde{\varvec{u}}}) + \Delta t (\varvec{e}^{k+1}_h(\varvec{v}), {\tilde{\varvec{u}}} ) + \Delta t ({\varvec{\tau }}^k_h(\varvec{u}),{\tilde{\varvec{u}}}) ,\nonumber \\ (\varvec{e}^{k+1}_h&(\varvec{v}), {\tilde{\varvec{u}}})= (\varvec{e}^k_h(\varvec{v}), {\tilde{\varvec{u}}}) + \Delta t ({\mathcal {L}}^\epsilon (\varvec{u}^k_h) - {\mathcal {L}}^\epsilon (\varvec{r}_h(\varvec{u}^k)), {\tilde{\varvec{u}}}) \nonumber \\ &+ \Delta t ({\varvec{\tau }}^k_h(\varvec{v}), {\tilde{\varvec{u}}})+ \Delta t ({\varvec{\sigma }}^k_{h}(\varvec{u}), {\tilde{\varvec{u}}}), \end{aligned}\right.$$

where consistency error terms \({\varvec{\tau }}^k_h(\varvec{u}), {\varvec{\tau }}^k_h(\varvec{v}), {\varvec{\sigma }}^k_h(\varvec{u})\) are given by

$$\left \{ \begin{aligned} {\varvec{\tau }}^k_h(\varvec{u})&:= \dfrac{\partial \varvec{u}^{k+1}}{\partial t} - \dfrac{\varvec{u}^{k+1} - \varvec{u}^k}{\Delta t}, \nonumber \\ {\varvec{\tau }}^k_h(\varvec{v})&:= \dfrac{\partial \varvec{v}^k}{\partial t} - \dfrac{\varvec{v}^{k+1} - \varvec{v}^k}{\Delta t}, \nonumber \\ {\varvec{\sigma }}^k_{h}(\varvec{u})&:= {\mathcal {L}}^\epsilon (\varvec{r}_h(\varvec{u}^k)) - {\mathcal {L}}^\epsilon (\varvec{u}^k). \end{aligned}\right.$$

When \(\varvec{u},\varvec{v}\) are \(C^2\) in time, we easily see that

$$\begin{aligned} ||{\varvec{\tau }}^k_h(\varvec{u})||&\leqslant \Delta t \sup _{t} \left \|\frac{\partial ^2 \varvec{u}}{\partial t^2}\right \| \qquad \text {and} \qquad ||{\varvec{\tau }}^k_h(\varvec{v})|| \leqslant \Delta t \sup _{t} \left \|\frac{\partial ^2 \varvec{v}}{\partial t^2}\right \|. \end{aligned}$$

To estimate \({\varvec{\sigma }}^k_{h}(\varvec{u})\), we recall the Lipschitz continuity property of the peridynamic force in the \(L^2\) norm; see Eq. 23. This leads us to

$$\begin{aligned} ||{\varvec{\sigma }}^k_{h}(\varvec{u})||&\leqslant \dfrac{L}{\epsilon ^2} ||\varvec{u}^k - \varvec{r}_h(\varvec{u}^k)|| \leqslant \dfrac{L c}{\epsilon ^2} h^2 \sup _{t} ||\varvec{u}(t)||_2, \end{aligned}$$

where the constant L is defined in Eq. 24.

We now state the consistency of this approach.

Lemma 2

(Consistency) Let\(\tau \)be given by

$$\begin{aligned} \tau&:= \sup _{k} \left( ||{\varvec{\tau }}^k_h(\varvec{u})|| + ||{\varvec{\tau }}^k_h(\varvec{v})|| + ||{\varvec{\sigma }}^k_{h}(\varvec{u})|| \right) . \end{aligned}$$

Then, the approach is consistent in that

$$\begin{aligned} \tau&\leqslant C_t \Delta t + C_s \frac{h^2}{\epsilon ^2} , \end{aligned}$$


$$\begin{aligned} C_t&:= \bigg \Vert \frac{\partial ^2 \varvec{u}}{\partial t^2} \bigg \Vert + \bigg \Vert \frac{\partial ^2 \varvec{v}}{\partial t^2}\bigg \Vert \quad {\text {and}} \quad C_s := L c \sup _{t} ||\varvec{u}(t)||_2. \end{aligned}$$

Stability Analysis

In equation for \(\varvec{e}^k_h(\varvec{u})\), we take \({\tilde{\varvec{u}}} = \varvec{e}^{k+1}_h(\varvec{u})\). We have

$$\begin{aligned} ||\varvec{e}^{k+1}_h(\varvec{u})||^2&= (\varvec{e}^{k}_h(\varvec{u}), \varvec{e}^{k+1}_h(\varvec{u})) + \Delta t (\varvec{e}^{k+1}_h(\varvec{v}), \varvec{e}^{k+1}_h(\varvec{u})) + \Delta t ({\varvec{\tau }}^k_h(\varvec{u}), \varvec{e}^{k+1}_h(\varvec{u})), \end{aligned}$$

which implies

$$\begin{aligned} ||\varvec{e}^{k+1}_h(\varvec{u})||&\leqslant ||\varvec{e}^{k}_h(\varvec{u})|| + \Delta t ||\varvec{e}^{k+1}_h(\varvec{v})|| + \Delta t ||{\varvec{\tau }}^{k}_h(\varvec{u})|| . \end{aligned}$$

Similarly, we can show

$$\begin{aligned} ||\varvec{e}^{k+1}_h(\varvec{v})||&\leqslant ||\varvec{e}^{k}_h(\varvec{v})|| + \Delta t ||{\mathcal {L}}^\epsilon (\varvec{u}^k_h) - {\mathcal {L}}^\epsilon (\varvec{r}_h(\varvec{u}^k))|| \nonumber \\&\quad + \Delta t \left( ||{\varvec{\tau }}^{k}_h(\varvec{v})|| + ||{\varvec{\sigma }}^{k}_{per,h}(\varvec{u})|| \right) . \end{aligned}$$

We have from Eq. 23

$$\begin{aligned} ||{\mathcal {L}}^\epsilon (\varvec{u}^k_h) - {\mathcal {L}}^\epsilon (\varvec{r}_h(\varvec{u}^k))||&\leqslant \dfrac{L}{\epsilon ^2} ||\varvec{u}^k_h - \varvec{r}_h(\varvec{u}^k)|| = \dfrac{L}{\epsilon ^2} ||\varvec{e}^k_h(\varvec{u})||. \end{aligned}$$

After adding Eqs. 59 and 60, and substituting Eq. 61, we get

$$\begin{aligned} ||\varvec{e}^{k+1}_h(\varvec{u})|| + ||\varvec{e}^{k+1}_h(\varvec{v})||&\leqslant ||\varvec{e}^k_h(\varvec{u})|| + ||\varvec{e}^k_h(\varvec{v})|| + \Delta t ||\varvec{e}^{k+1}_h(\varvec{v})|| + \dfrac{L}{\epsilon ^2} \Delta t ||\varvec{e}^k_h(\varvec{u})|| + \Delta t \tau , \end{aligned}$$

where \(\tau \) is defined in Eq. 56. Since \(e^k = ||\varvec{e}^k_h(\varvec{u})|| + ||\varvec{e}^k_h(\varvec{v})||\), we can show, assuming \(L/\epsilon ^2 \geqslant 1\),

$$\begin{aligned} e^{k+1}&\leqslant e^k + \Delta t e^{k+1} + \Delta t \dfrac{L}{\epsilon ^2} e^k + \Delta t \tau \\ \Rightarrow e^{k+1}&\leqslant \dfrac{1 + \frac{\Delta t L}{\epsilon ^2}}{1-\Delta t} e^k + \dfrac{\Delta t}{1 - \Delta t} \tau . \end{aligned}$$

Substituting for \(e^k\) recursively in the equation above, we get

$$\begin{aligned} e^{k+1}&\leqslant \left( \dfrac{1 + \frac{\Delta t L}{\epsilon ^2}}{1-\Delta t} \right) ^{k+1} e^0 + \dfrac{\Delta t}{1 - \Delta t} \tau \sum _{j=0}^k \left( \dfrac{1 + \frac{\Delta t L}{\epsilon ^2}}{1-\Delta t} \right) ^{k-j}. \end{aligned}$$

Noting that

$$\begin{aligned} \dfrac{1 + \frac{\Delta t L}{\epsilon ^2}}{1-\Delta t}&= 1 + \frac{1+\frac{L}{\epsilon ^2}}{1-\Delta t} \Delta t \end{aligned}$$

and \((1 +a \Delta t )^k \leqslant \exp (k a\Delta t ) \leqslant \exp (Ta)\) for \(a>0\), we have

$$\begin{aligned} \left( \dfrac{1 + \frac{\Delta t L_1}{\epsilon ^2}}{1-\Delta t} \right) ^k&\leqslant \exp \bigg [\frac{T\left(1+\frac{L_1}{\epsilon ^2}\right)}{1-\Delta t}\bigg ]. \end{aligned}$$

This implies

$$\begin{aligned} e^{k+1}&\leqslant \exp \left[\frac{T\left(1+\frac{L}{\epsilon ^2}\right)}{1-\Delta t}\right] \left( e^0 + \dfrac{\Delta t}{1 - \Delta t} \tau \sum _{j=0}^k 1 \right) \\&\leqslant \exp \left[\frac{T\left(1+\frac{L}{\epsilon ^2}\right)}{1-\Delta t}\right] \left( e^0 + \dfrac{k\Delta t}{1 - \Delta t} \tau \right) . \end{aligned}$$

By substituting above equation in Eq. 50, we get the stability of the scheme.

Lemma 3


$$\begin{aligned} E^k&\leqslant C_p h^2 + \exp \left[ \frac{T\left(1+\frac{L}{\epsilon ^2}\right)}{1-\Delta t}\right] \left( e^0 + \dfrac{k\Delta t}{1 - \Delta t} \tau \right) . \end{aligned}$$

After taking sup over \(k\leqslant T/\Delta t\) and substituting the bound on \(\tau \) from Lemma 2, we get the desired result and proof of Theorem 5 is complete.

We now consider a stronger notion of stability for the linearized peridynamics model.

Linearized Peridynamics and Energy Stability

In this section, we linearize the peridynamics model and obtain a CFL-like stability condition. For problems, where strains are small, the stability condition for the linearized model is expected to apply to the nonlinear model. The slope of peridynamics potential f and g are constant for sufficiently small strain and, therefore, for small strain, the nonlinear model behaves like a linear model.

In Eq. 13, the linearization gives

$$\begin{aligned} {\mathcal {L}}^{\epsilon }_{T,l}(\varvec{u})(\varvec{x}) = \dfrac{2}{\epsilon ^{d+1} \omega _d} \int _{H_{\epsilon }(\varvec{x})} \omega (\varvec{x}) \omega (\varvec{y}) J^\epsilon (|\varvec{y}- \varvec{x}|) f''(0) S(\varvec{y},\varvec{x};\varvec{u}) \varvec{e}_{\varvec{y}- \varvec{x}} {\text {d}}\varvec{y}. \end{aligned}$$

The corresponding bilinear form is denoted as \(a^\epsilon _{T,l}\) and is given by

$$\begin{aligned} a^\epsilon _{T,l}(\varvec{u}, \varvec{v})&= \dfrac{f''(0)}{\epsilon ^{d+1} \omega _d} \int _D \int _{D} \omega (\varvec{x}) \omega (\varvec{y}) J^\epsilon (|\varvec{y}- \varvec{x}|) |\varvec{y}- \varvec{x}| S(\varvec{y},\varvec{x};\varvec{u}) S(\varvec{y},\varvec{x};\varvec{v}) {\text {d}}\varvec{y}{\text {d}}\varvec{x}. \end{aligned}$$

Similarly, the linearization of \({\mathcal {L}}^\epsilon _D\) in Eq. 14 gives

$$\begin{aligned} {\mathcal {L}}^\epsilon _{D,l}(\varvec{u})(\varvec{x})=\frac{g''(0)}{\epsilon ^{d+2} \omega _d}\int _{H_\epsilon (\varvec{x})} \omega (\varvec{x}) \omega (\varvec{y}) J^\epsilon (|\varvec{y}-\varvec{x}|)\left[ \theta (\varvec{y},t;\varvec{u})+\theta (\varvec{x},t;\varvec{u})\right] \varvec{e}_{\varvec{y}-\varvec{x}}\,{\text {d}}\varvec{y}. \end{aligned}$$

The associated bilinear form is given by

$$\begin{aligned} a^\epsilon _{D,l}(\varvec{u},\varvec{v}) = \dfrac{g''(0)}{\epsilon ^2} \int _D \omega (\varvec{x}) \theta (\varvec{x};\varvec{u}) \theta (\varvec{y};\varvec{v}) {\text {d}}\varvec{x}. \end{aligned}$$

The total force after linearization is

$$\begin{aligned} {\mathcal {L}}^\epsilon _l (\varvec{u})(\varvec{x}) = {\mathcal {L}}^\epsilon _{T,l}(\varvec{u})(\varvec{x}) + {\mathcal {L}}^\epsilon _{D,l}(\varvec{u})(\varvec{x}) \end{aligned}$$

and the bilinear operator associated with \({\mathcal {L}}^\epsilon _l\) is given by

$$\begin{aligned} a^\epsilon _l(\varvec{u},\varvec{v}) = a^\epsilon _{T,l}(\varvec{u},\varvec{v}) + a^\epsilon _{D,l}(\varvec{u},\varvec{v}). \end{aligned}$$

We have

$$\begin{aligned} ({\mathcal {L}}^\epsilon _l(\varvec{u}), \varvec{v}) = - a^\epsilon _l(\varvec{u},\varvec{v}). \end{aligned}$$

We now discuss the stability of the FEM approximation to the linearized problem. Let \(\varvec{u}^k_{l,h}\) denote the approximate solution satisfying, for \(k \geqslant 1\),

$$\begin{aligned} \left( \dfrac{\varvec{u}^{k+1}_{l,h} - 2 \varvec{u}^k_{l,h} + \varvec{u}^{k-1}_{l,h}}{\Delta t^2}, {\tilde{\varvec{u}}} \right)&= ({\mathcal {L}}^\epsilon _l(\varvec{u}^k_{l,h}), {\tilde{\varvec{u}}} ) + (\varvec{b}^k_{h}, {\tilde{\varvec{u}}}), \qquad \forall {\tilde{\varvec{u}}} \in V_h \end{aligned}$$

and, for \(k=0\),

$$\begin{aligned} \left( \dfrac{\varvec{u}^1_{l,h} - \varvec{u}^0_{l,h}}{\Delta t^2}, {\tilde{\varvec{u}}}\right)&= \dfrac{1}{2}({\mathcal {L}}^\epsilon (\varvec{u}^0_{l,h}), {\tilde{\varvec{u}}})+ \dfrac{1}{\Delta t} (\varvec{v}^0_{l,h}, {\tilde{\varvec{u}}}) + \dfrac{1}{2}(\varvec{b}^0_h, {\tilde{\varvec{u}}}), \qquad \forall {\tilde{\varvec{u}}} \in V_h. \end{aligned}$$

The following notation will be used to define the discrete energy at each time step k:

$$\left\{\begin{aligned}&{\overline{\varvec{u}}}^{k+1}_h := \frac{\varvec{u}^{k+1}_h + \varvec{u}^k_h}{2}, \, {\overline{\varvec{u}}}^{k}_h := \frac{\varvec{u}^k_h + \varvec{u}^{k-1}_h}{2}, \nonumber \\&{\bar{\partial }}_t \varvec{u}^k_h := \frac{\varvec{u}^{k+1}_h - \varvec{u}^{k-1}_h}{2 \Delta t}, \, {\bar{\partial }}_t^+ \varvec{u}^k_h := \frac{\varvec{u}^{k+1}_h - \varvec{u}^{k}_h}{\Delta t}, \, {\bar{\partial }}_t^- \varvec{u}^k_h := \frac{\varvec{u}^{k}_h - \varvec{u}^{k-1}_h}{\Delta t}. \end{aligned}\right.$$

We also define

$$\begin{aligned} {\bar{\partial }}_{tt} \varvec{u}^k_h&:= \dfrac{\varvec{u}^{k+1}_h - 2\varvec{u}^k_h + \varvec{u}^{k-1}_h}{\Delta t^2} = \dfrac{{\bar{\partial }}^+_t \varvec{u}^k_h - {\bar{\partial }}^-_t \varvec{u}^k_h}{\Delta t}. \end{aligned}$$

We introduce the discrete energy associated with \(\varvec{u}^k_{l,h}\) at time step k as follows:

$$\begin{aligned} {\mathcal {E}}(\varvec{u}^k_{l,h})&:= \frac{1}{2} \left[ ||{\bar{\partial }}^+_t \varvec{u}^k_{l,h}||^2 - \frac{\Delta t^2}{4} a^\epsilon _l({\bar{\partial }}^+_t \varvec{u}^k_{l,h}, {\bar{\partial }}^+_t \varvec{u}^k_{l,h}) + a^\epsilon _l({\overline{\varvec{u}}}^{k+1}_{l,h} , {\overline{\varvec{u}}}^{k+1}_{l,h}) \right]. \end{aligned}$$

Following [Theorem 4.1, [24]], the stability of central difference scheme is given by

Theorem 6

(Energy stability of the central difference approximation of linearized peridynamics) Let\(\varvec{u}^k_{l,h}\)be the approximate solution of Eqs. 70and71. In the absence of body force\(\varvec{b}(t) = 0\)for allt, if\(\Delta t\)satisfies the CFL-like condition

$$\begin{aligned} \frac{\Delta t^2}{4} \sup _{\varvec{u}\in V_h \setminus \{{\mathbf {0}}\}} \dfrac{a^\epsilon _l(\varvec{u},\varvec{u})}{(\varvec{u},\varvec{u})} \leqslant 1, \end{aligned}$$

then the discrete energy is positive and we have the stability

$$\begin{aligned} {\mathcal {E}}(\varvec{u}^k_{l,h})= {\mathcal {E}}(\varvec{u}^{0}_{l,h}) . \end{aligned}$$

We skip the proof of above theorem as it is straightforward extension of Theorem 5.2 in [20].

Numerical Experiments

In this section, we present numerical simulations that are consistent with the theoretical a-priori bound on the convergence rate. We also compare the peridynamic energy of the material softening zone and the classic Griffith’s fracture energy of linear elastic fracture mechanics.

We consider Plexiglass at room temperature and specify the density \(\rho = 1\,200\, {\text {kg/m}}^3\), the bulk modulus \(K = 25\) GPa, the Poisson’s ratio \(\nu = 0.245\), and the critical energy release rate \(G_{\text{c}} = 500 \,{\text {Jm}}^{-2}\). The pairwise interaction and the hydrostatic interaction are characterized by potentials \(f(r) = c (1-\exp (-\beta r^2))\) and \(g(r) = {\bar{C}} r^2/2\), respectively. Here we have used a quadratic hydrostatic interaction potential. The influence function is \(J(r) = 1-r\). Since the pairwise potential f is symmetric for positive and negative strains, the critical strain is given by \(S_{\text{c}}(\varvec{y}, bx) = \frac{\pm {\bar{r}}}{\sqrt{\varvec{y}- \varvec{x}}}\), where \(\pm {\bar{r}}\) is the inflection point of f(r) given by \({\bar{r}} = \frac{1}{\sqrt{\beta }}\). Following Eqs. 94, 95, and 97 of [29], the relation between peridynamic material parameters and Lamé constants \((\lambda , \mu )\) and the critical energy release rate \(G_{\text{c}}\) can be written as (for 2-d)

$$\begin{aligned} c = \frac{\pi G_{\text{c}}}{4M_J}, \qquad \beta = \frac{4\mu }{C M_J}, \qquad {\bar{C}} = \frac{2(\lambda - \mu )}{M_J^2}, \end{aligned}$$

where \(M_J\) is given by

$$\begin{aligned} M_J&= \int _0^1 J(r) r^2 {\text {d}}r = \frac{1}{12}. \end{aligned}$$

By solving Eq. 75, we get \(c = 4\,712.4\), \({\bar{C}}=-1.734\,9\times 10^{11}\), \(\beta = 1.564\,7\times 10^{8}\).

Fig. 3

Material domain \(D = [0,0.1\, {\text {m}}]^2\) with crack of length \(0.02\,{\text {m}}\). The x-component and y-component of displacement are fixed along a collar of thickness equal to the horizon on top. On the bottom the velocity \(\varvec{v}_x = \pm 1\, {\text {m/s}}\) along x-direction is specified on either side of the crack to make the crack propagate upwards

We consider a 2-d domain \(D=[0,0.1\, {\text {m}}]^2\) (with unit thickness in third direction) with the vertical crack of length \(0.02\, {\text {m}}\). The boundary conditions are described in Fig. 3. The simulation time is \(T = 40\, \upmu {\text {s}}\) and the time step is \(\Delta t = 0.004\,\upmu {\text {s}}\). We consider two horizons \(8\, {\text {mm}}\) and \(4\, {\text {mm}}\). We run simulations for mesh sizes \(h = 2, 1, 0.5\, {\text {mm}} \). We consider the central difference time discretization described by Eq. 41 on a uniform mesh consisting of linear triangle elements. The second-order quadrature approximation is used in the simulation for each triangle element. To reduce the load on memory and to avoid the matrix–vector multiplication at each time step, we approximate the mass matrix by the diagonal mass matrix using the lumping (row-sum) technique. Suppose the exact mass matrix is \(\varvec{M} = [m_{ij}]\) where \(m_{ij}\) is the element of \(\varvec{M}\) corresponding to ith row and jth column, then we approximate \(\varvec{M}\) by the diagonal matrix \(\hat{\varvec{M}} = [{\hat{m}}_{ij}]\) where \({\hat{m}}_{ii} = \sum _{j} m_{ij}\) and \({\hat{m}}_{ij} = 0\) if \( j\ne i\).

Convergence Rate

To compute the convergence rate numerically we proceed as follows: consider a fixed horizon \(\epsilon \) and three different mesh sizes \(h_1, h_2, h_3\) such that \(r = h_1/h_2 = h_2/h_3\). Let \(\varvec{u}_1,\varvec{u}_2,\varvec{u}_3\) be approximate solutions corresponding to meshes of size \(h_1,h_2,h_3\), and let \(\varvec{u}\) be the exact solution. We write the error as \(||\varvec{u}_h - \varvec{u}|| =C h^\alpha \) for some constant C and \(\alpha >0\), to get

$$\begin{aligned} \log ( ||\varvec{u}_1 - \varvec{u}_2||)&= C + \alpha \log h_2, \\ \log ( ||\varvec{u}_2 - \varvec{u}_3||)&= C + \alpha \log h_3. \end{aligned}$$

From the above two equations, it is easy to see that the rate of convergence \(\alpha \) is

$$\begin{aligned} \dfrac{\log ( ||\varvec{u}_1 - \varvec{u}_2||) - \log ( ||\varvec{u}_2 - \varvec{u}_3||)}{\log (r)}. \end{aligned}$$

The convergence result for horizons \(\epsilon = 8\,{\text {mm}}\) and \(\epsilon = 4\, {\text {mm}}\) is shown in Fig. 4. In the simulation, we have considered the second-order approximation of integration using quadrature points. The simulations show a rate of convergence that agrees with the a priori estimates given in Theorem 6.

Fig. 4

Convergence rate at different times for two horizons. For both horizons \(\epsilon = 4, 8\, {\text {mm}}\), the three meshes of size \(h = 2, 1, 0.5\, {\text {mm}}\) were considered to compute the convergence rate

Fracture Energy of Crack Zone

The extent of damage at material point \(\varvec{x}\) is given by the function \(Z(\varvec{x})\)

$$\begin{aligned} Z(\varvec{x})&= \max _{\varvec{y}\in H_\epsilon (\varvec{x}) \cap D} \frac{S(\varvec{y},\varvec{x};\varvec{u})}{S^+_{\text{c}}}. \end{aligned}$$

The crack zone is defined as set of material points which have \(Z > 1\). We compute the peridynamic energy of crack zone and compare it with the Griffith’s fracture energy. For a crack of length l, the Griffith’s fracture energy (G.E.) will be \({\text {G.E.}} = G_{\text{c}} \times l\). The peridynamic fracture energy (P.E.) associated with the material softening zone is given by

$$\begin{aligned} {\text {P.E.}} =&\int _{\begin{subarray}{c} \varvec{x}\in D,\\ Z(\varvec{x}) \geqslant 1 \end{subarray}} \left[ \frac{1}{\epsilon ^d \omega _d} \int _{H_\epsilon (\varvec{x})} |\varvec{y}- \varvec{x}| \mathcal {W}^{\,\epsilon} (S(\varvec{y},\varvec{x};\varvec{u}))\,{\text {d}}\varvec{y}\right] {\text {d}}\varvec{x}\\&+\int _{\begin{subarray}{c} \varvec{x}\in D,\\ Z(\varvec{x}) \geqslant 1 \end{subarray}} {\mathcal {V}}^\epsilon (\theta (\varvec{x},t;\varvec{u}))\,{\text {d}}\varvec{x}, \end{aligned}$$

where \(\mathcal {W}^{\,\epsilon} (S(\varvec{y},\varvec{x};\varvec{u}))\) is the bond-based potential; see Eq. 2 and \({\mathcal {V}}^\epsilon (\theta (\varvec{x},t;\varvec{u}))\) is the hydrostatic interaction potential; see Eq. 4.

In Fig. 5, the classical fracture energy and the peridynamic fracture energy are shown at different crack lengths. The error in both energies at different times is shown in Fig. 6. The agreement between two energies is good. The damage profile at time \(30\,\upmu \)s and \(40\,\upmu \)s is shown in Fig. 7. At each node, the damage function Z is computed by treating edges between mesh nodes as bonds. In addition to the damage plots, we show the velocity profile at \(30\, \upmu \)s and \(40\, \upmu \)s in Fig. 8. In Fig. 9, we show the plot of the xx component of symmetric gradient of the displacement. Here the region for which the magnitude of the strain is greater than a multiple of the critical strain is the yellow region. It is seen that the high-strain region surrounds the crack.

As the crack is propagating vertically it is seen that the high-strain region is next to the crack.

Fig. 5

Peridynamic energy and Griffith’s energy as a function of crack length

Fig. 6

Error between Peridynamic energy and Griffith’s energy at different times

Fig. 7

Color plot of damage function Z on deformed material domain at time \(t=30\,\upmu \text{s}\) and \(40\,\upmu \text{s}\). Dark blue represents undamaged material \(Z<1\), \(Z\approx 1\) is yellow at crack tip, red is softening material. Here, the displacements are scaled by 100 and damage function is cut off at 5 to highlight the crack zone

Fig. 8

Velocity profile

Fig. 9

Magnitude of the xx component of strain \(\nabla \varvec{u}+ \nabla \varvec{u}^{\text{T}}\). The region for which the magnitude of the strain is greater than a multiple of the critical strain is the yellow region

Lipschitz Continuity of Peridynamic Force and Higher Temporal Regularity of Solutions

In this section, we prove Theorems 1 and 3. Here \(\varvec{u}\in W\subset H^2(D;{\mathbb {R}}^d)\) and the \(||\varvec{u}||_2\) norm is given by

$$\begin{aligned} ||\varvec{u}||_2 = ||\varvec{u}|| + ||\nabla \varvec{u}|| + ||\nabla ^2 \varvec{u}||. \end{aligned}$$

Proof of Lipschitz Continuity with Respect to the \(\Vert \cdot \Vert _2\) Norm

We assume that the potential function f satisfies \(C^f_i < \infty \) for \(i=0,1,2,3,4\). Recall that \(C^f_0 = \sup _r |f(r)|\) and \(C^f_i = \sup _r |f^{(r)}(r)|\) for \(i=1,\cdots ,4\). \(C^g_i\) is defined similarly for \(i=0,1,\cdots ,4\). If the potential function g is a convex–concave function, then we can assume \(C^g_i < \infty \) for \(i=0,1,2,3,4\). In what follows, we will prove Theorem 1 for convex–concave type g. If g is a purely a quadratic function, the proof follows easily using only a subset of the estimates proved in this section.

Let \(\varvec{u},\varvec{v}\in W\). Using the triangle inequality, we get

$$\begin{aligned} ||{\mathcal {L}}^\epsilon (\varvec{u}) - {\mathcal {L}}^\epsilon (\varvec{v})||_2&\leqslant ||{\mathcal {L}}^\epsilon _T(\varvec{u}) - {\mathcal {L}}^\epsilon _T(\varvec{v})||_2 + ||{\mathcal {L}}^\epsilon _D(\varvec{u}) - {\mathcal {L}}^\epsilon _D(\varvec{v})||_2, \end{aligned}$$

where \({\mathcal {L}}^\epsilon _T\) and \({\mathcal {L}}^\epsilon _D\) is given by Eqs. 13 and 14.

We first write the peridynamic force \({\mathcal {L}}^\epsilon _T(\varvec{u})(\varvec{x})\) as follows:

$$\begin{aligned}&{\mathcal {L}}^\epsilon _T(\varvec{u})(\varvec{x}) \nonumber \\&=\frac{2}{\epsilon ^{d+1} \omega _d}\int _{H_\epsilon (\varvec{x})} \omega (\varvec{x}) \omega (\varvec{y}) \frac{J^\epsilon (|\varvec{y}-\varvec{x}|)}{\sqrt{|\varvec{y}-\varvec{x}|}} f'(\sqrt{|\varvec{y}-\varvec{x}|}S(\varvec{y},\varvec{x};\varvec{u}))\varvec{e}_{\varvec{y}-\varvec{x}}\,{\text {d}}\varvec{y}, \end{aligned}$$

where we substitute \(\partial _S f(\sqrt{|\varvec{y}- \varvec{x}|} S(\varvec{y},\varvec{x};\varvec{u})) = \sqrt{|\varvec{y}- \varvec{x}|}f'(\sqrt{|\varvec{y}- \varvec{x}|} S(\varvec{y},\varvec{x};\varvec{u}))\). The form of the peridynamic force described above is the same as the one given in [Section 6, [20]]. We apply Theorem 3.1 in [20] to show

$$\begin{aligned} ||{\mathcal {L}}^\epsilon _T(\varvec{u}) - {\mathcal {L}}^\epsilon _T(\varvec{v})||_2&\leqslant \dfrac{L_1(1+ (||\varvec{u}||_2 + ||\varvec{v}||_2) + (||\varvec{u}||_2 + ||\varvec{v}||_2)^2)}{\epsilon ^3} ||\varvec{u}- \varvec{v}||_2 \nonumber \\&\leqslant \dfrac{L_1(1+ ||\varvec{u}||_2 + ||\varvec{v}||_2)^2}{\epsilon ^3} ||\varvec{u}- \varvec{v}||_2 \end{aligned}$$


$$\begin{aligned} ||{\mathcal {L}}^\epsilon _T(\varvec{u})||_2&\leqslant \dfrac{L_2(||\varvec{u}||_2 + ||\varvec{u}||_2^2)}{\epsilon ^{5/2}}. \end{aligned}$$

Next we analyze \(||{\mathcal {L}}^\epsilon _D(\varvec{u}) - {\mathcal {L}}^\epsilon _D(\varvec{v})||_2\). We define new terms to simplify the calculations. For \({\varvec{\xi }}\in H_1({\mathbf {0}})\), we set

$$\left\{\begin{aligned} &s_{\varvec{\xi }}= \epsilon |{\varvec{\xi }}|, \, \varvec{e}_{\varvec{\xi }}= \frac{{\varvec{\xi }}}{|{\varvec{\xi }}|}, \nonumber \\ &\omega _{\varvec{\xi }}(\varvec{x})= \omega (\varvec{x}+ \epsilon {\varvec{\xi }}) \omega (\varvec{x}), \nonumber \\ &{\bar{\varvec{u}}}_{\varvec{\xi }}(\varvec{x})= \varvec{u}(\varvec{x}+ \epsilon {\varvec{\xi }}) - \varvec{u}(\varvec{x}), \nonumber \\& (\varvec{u}- \varvec{v})(\varvec{x})= \varvec{u}(\varvec{x}) - \varvec{v}(\varvec{x}). \end{aligned}\right.$$

Similar notations hold if we exchange \(\varvec{x}\), \({\varvec{\xi }}\in H_1({\mathbf {0}})\), and \(\varvec{u}\in W\) by \(\varvec{y}\), \(\varvec{\eta } \in H_1({\mathbf {0}})\), and \(\varvec{v}\in W\), respectively. We will also encounter various moments of the influence function J; therefore, we define the following moments:

$$\begin{aligned} {\bar{J}}_\alpha = \frac{1}{\omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) |{\varvec{\xi }}|^{-\alpha } {\text {d}}{\varvec{\xi }} \qquad \text { for } \alpha \in {\mathbb {R}}. \end{aligned}$$

Recall that \(J(|{\varvec{\xi }}|) = 0\) for \({\varvec{\xi }}\notin H_1({\mathbf {0}})\) and \(0\leqslant J(|{\varvec{\xi }}|) \leqslant M\) for \({\varvec{\xi }}\in H_1({\mathbf {0}})\). The boundary function \(\omega \) is assumed to satisfy

$$\begin{aligned} \sup _{\varvec{x}} |\nabla \omega (\varvec{x})|< \infty ,\qquad \sup _{\varvec{x}} |\nabla ^2 \omega (\varvec{x})| < \infty . \end{aligned}$$

We choose finite constants \(C_{\omega _1}\) and \(C_{\omega _2}\) such that

$$\left\{\begin{aligned} |\nabla \omega _{\varvec{\xi }}(\varvec{x})|&\leqslant C_{\omega _1}, \quad |\nabla \omega (\varvec{x})| \leqslant C_{\omega _1},\nonumber \\ |\nabla ^2 \omega _{\varvec{\xi }}(\varvec{x})|&\leqslant C_{\omega _2}, \quad |\nabla ^2 \omega (\varvec{x})| \leqslant C_{\omega _2}. \end{aligned}\right.$$

We now collect the following estimates which will be used to estimate \(||{\mathcal {L}}^\epsilon _D(\varvec{u}) - {\mathcal {L}}^\epsilon _D(\varvec{v})||_2\).

Lemma 4

Let\(\varvec{u},\varvec{v}\in W\), for any\(\varvec{\eta }\in H_1({\mathbf {0}})\)and\(\delta \leqslant 2\epsilon \), we have

$$\begin{aligned} \sup _{\varvec{x}\in D} |\theta (\varvec{x};\varvec{u})|&\leqslant 2 C_{e_1} {\bar{J}}_0 ||\varvec{u}||_2, \end{aligned}$$
$$\begin{aligned} \int _D |\theta (\varvec{x}+ \delta \varvec{\eta }; \varvec{u})|^2 {\text {d}}\varvec{x}&\leqslant 4 {\bar{J}}^2_0 ||\varvec{u}||_2^2, \end{aligned}$$
$$\begin{aligned} \int _D |\nabla \theta (\varvec{x}+ \delta \varvec{\eta }; \varvec{u})|^2 {\text {d}}\varvec{x}&\leqslant 8 {\bar{J}}^2_0 (1+C_{\omega _1})^2 ||\varvec{u}||_2^2, \end{aligned}$$
$$\begin{aligned} \int _D |\theta (\varvec{x}+ \delta \varvec{\eta }; \varvec{u}- \varvec{v})|^2 \, |\nabla \theta (\varvec{x}+ \delta \varvec{\eta };\varvec{v})|^2 {\text {d}}\varvec{x}&\leqslant 32 {\bar{J}}_0^4 (1+C_{\omega _1})^2 ||\varvec{v}||_2^2 \, ||\varvec{u}- \varvec{v}||_2^2, \end{aligned}$$
$$\begin{aligned} \int _D |\nabla \theta (\varvec{x}+ \delta \varvec{\eta }; \varvec{u})|^4 {\text {d}}\varvec{x}&\leqslant 128 {\bar{J}}^4_0 (C_{e_2}^2 + C_{e_1} C^2_{\omega _1})^2 ||\varvec{u}||_2^4, \end{aligned}$$
$$\begin{aligned} \int _D |\theta (\varvec{x}+ \delta \varvec{\eta }; \varvec{u}- \varvec{v})|^2 \, |\nabla \theta (\varvec{x}+ \delta \varvec{\eta }; \varvec{v})|^4 {\text {d}}\varvec{x}&\leqslant 512 {\bar{J}}_0^6 C_{e_1}^2 (C_{e_2}^2 + C_{e_1}C^2_{\omega _1})^2 ||\varvec{u}- \varvec{v}||_2^2 \, ||\varvec{v}||_2^4, \end{aligned}$$
$$\begin{aligned} \int _D |\nabla ^2 \theta (\varvec{x}+ \delta \varvec{\eta }; \varvec{u})|^2 {\text {d}}\varvec{x}&\leqslant 16 {\bar{J}}^2_0 (1+2C_{\omega _1} + C_{\omega _2})^2 ||\varvec{u}||_2^2 . \end{aligned}$$

Here\(\nabla \)in all the equations above is with respect to\(\varvec{x}\). The constants\(C_{e_1}, C_{e_2}\)are the constants associated with theSobolev embedding property of space\(H^2(D;{\mathbb {R}}^d)\); see Eqs. 18and19.


Using the notation given in Eq. 83, we write \(\theta (\varvec{x};\varvec{u})\) as

$$\begin{aligned} \theta (\varvec{x};\varvec{u})&= \frac{1}{\omega _d} \int _{H_1({\mathbf {0}})} \omega (\varvec{x}+\epsilon {\varvec{\xi }}) J(|{\varvec{\xi }}|) {\bar{\varvec{u}}}_{\varvec{\xi }}(\varvec{x}) \cdot \varvec{e}_{\varvec{\xi }}{\text {d}}{\varvec{\xi }}. \end{aligned}$$

On noting that \(|{\bar{\varvec{u}}}_{\varvec{\xi }}(\varvec{x})| \leqslant 2||\varvec{u}||_{\infty }\) and \(||\varvec{u}||_\infty \leqslant C_{e_1} ||\varvec{u}||_2\), we easily see that

$$\begin{aligned} |\theta (\varvec{x};\varvec{u})|&\leqslant {\bar{J}}_0 2 ||\varvec{u}||_{\infty } \leqslant 2 C_{e_1} {\bar{J}}_0 ||\varvec{u}||_2. \end{aligned}$$

In the rest of the proof, we will let \(\varvec{y}= \varvec{x}+ \delta \varvec{\eta }\), where \(0\leqslant \delta \leqslant 2\epsilon \) and \(\varvec{\eta }\in H_1({\mathbf {0}})\).

To show Eq. 88, we first introduce an important identity which will be used frequently. Let \(p({\varvec{\xi }})\) be some function of \({\varvec{\xi }}\), and \(\alpha , C \in {\mathbb {R}}\). Then,

$$\begin{aligned}&\bigg \vert \frac{C}{\omega _d} \int _{H_1({\mathbf {0}})} \frac{J(|{\varvec{\xi }}|)}{|{\varvec{\xi }}|^\alpha } p({\varvec{\xi }}) {\text {d}}{\varvec{\xi }}\bigg \vert ^2\nonumber \\&= \left( \frac{C}{\omega _d} \right) ^2 \int _{H_1({\mathbf {0}})} \int _{H_1({\mathbf {0}})} \frac{J(|{\varvec{\xi }}|)}{|{\varvec{\xi }}|^\alpha } \frac{J(|\varvec{\eta }|)}{|\varvec{\eta }|^\alpha } p({\varvec{\xi }}) p(\varvec{\eta }) {\text {d}}{\varvec{\xi }}{\text {d}}\varvec{\eta } \nonumber \\&\leqslant \left( \frac{C}{\omega _d} \right) ^2 \int _{H_1({\mathbf {0}})} \int _{H_1({\mathbf {0}})} \frac{J(|{\varvec{\xi }}|)}{|{\varvec{\xi }}|^\alpha } \frac{J(|\varvec{\eta }|)}{|\varvec{\eta }|^\alpha } \frac{p({\varvec{\xi }})^2 + p(\varvec{\eta })^2}{2} {\text {d}}{\varvec{\xi }}{\text {d}}\varvec{\eta }\nonumber \\&= C^2 \frac{{\bar{J}}_\alpha }{\omega _d } \int _{H_1({\mathbf {0}})} \frac{J(|{\varvec{\xi }}|)}{|{\varvec{\xi }}|^\alpha } p({\varvec{\xi }})^2 {\text {d}}{\varvec{\xi }}, \end{aligned}$$

where we used the inequality \(ab \leqslant {\frac{a^2}{2}} + \frac{b^2}{2}\) in the first step, and definition of \({\bar{J}}_\alpha \) and symmetry of terms in the second step.

From the expression of \(\theta (\varvec{y};\varvec{u})\), we can show

$$\begin{aligned} \int _D |\theta (\varvec{y};\varvec{u})|^2 {\text {d}}\varvec{x}&\leqslant 4 {\bar{J}}^2_0 ||\varvec{u}||^2 \leqslant 4 {\bar{J}}^2_0 ||\varvec{u}||^2_2. \end{aligned}$$

We now prove the bound Eq. 89. Taking the gradient of \(\theta (\varvec{y};\varvec{u})\), with respect to \(\varvec{x}\), noting that \(\varvec{y}= \varvec{x}+ \delta \varvec{\eta }\), we get

$$\begin{aligned} \nabla \theta (\varvec{y};\varvec{u})&= \frac{1}{\omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) \omega (\varvec{y}+ \epsilon {\varvec{\xi }}) (\nabla {\bar{\varvec{u}}}_{\varvec{\xi }}(\varvec{y}))^{\text{T}} \varvec{e}_{\varvec{\xi }}{\text {d}}{\varvec{\xi }}\nonumber \\&\quad + \frac{1}{\omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) \nabla \omega (\varvec{y}+ \epsilon {\varvec{\xi }}) {\bar{\varvec{u}}}_{\varvec{\xi }}(\varvec{y}) \cdot \varvec{e}_{\varvec{\xi }}{\text {d}}{\varvec{\xi }}. \end{aligned}$$

We can show using the inequality Eq. 96 and the estimates \(\int _D |\nabla {\bar{\varvec{u}}}_{\varvec{\xi }}(\varvec{y})|^2 {\text {d}}\varvec{x}\leqslant 4 ||\nabla \varvec{u}||^2 \leqslant 4 ||\varvec{u}||_2^2\), \(| \nabla \omega (\varvec{y}+ \epsilon {\varvec{\xi }})| \leqslant C_{\omega _1}\), \(\int _D |{\bar{\varvec{u}}}_{\varvec{\xi }}(\varvec{y})|^2 {\text {d}}\varvec{x}\leqslant 4 ||\varvec{u}||_2^2\), to conclude

$$\begin{aligned} \int _D |\nabla \theta (\varvec{y}; \varvec{u})|^2 {\text {d}}\varvec{x}&\leqslant \dfrac{2{\bar{J}}_0}{\omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) 4 ||\varvec{u}||_2^2 {\text {d}}{\varvec{\xi }}+ \dfrac{2{\bar{J}}_0}{\omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) 4 C_{\omega _1}^2||\varvec{u}||_2^2 {\text {d}}{\varvec{\xi }}\nonumber \\&= 8 {\bar{J}}^2_0 (1+C_{\omega _1}^2) ||\varvec{u}||_2^2 \leqslant 8 {\bar{J}}^2_0 (1+C_{\omega _1})^2 ||\varvec{u}||_2^2. \end{aligned}$$

We now show Eq. 90. We will use Eqs. 87 and 89, and proceed as follows:

$$\begin{aligned}&\int _D |\theta (\varvec{y}; \varvec{u}- \varvec{v})|^2 \, |\nabla \theta (\varvec{y};\varvec{v})|^2 {\text {d}}\varvec{x}\nonumber \\&\leqslant \left( \sup _{\varvec{y}} |\theta (\varvec{y}; \varvec{u}- \varvec{v})| \right) ^2 \int _D |\nabla \theta (\varvec{y};\varvec{v})|^2 {\text {d}}\varvec{x}\nonumber \\&\leqslant 32 {\bar{J}}_0^4 (1+C_{\omega _1})^2 ||\varvec{v}||^2_2 ||\varvec{u}- \varvec{v}||^2_2. \end{aligned}$$

To prove Eq. 91 we note expression of \(\nabla \theta (\varvec{y};\varvec{u})\) in Eq. 98 and inequality \((a+b)^4 \leqslant 8 a^4 + 8 b^4\) and Eq. 96 to get

$$\begin{aligned} |\nabla \theta (\varvec{y};\varvec{u})|^4&\leqslant \dfrac{64 {\bar{J}}^3_0}{\omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) (|\nabla \varvec{u}(\varvec{y}+\epsilon {\varvec{\xi }})|^4 + |\nabla \varvec{u}(\varvec{y})|^4) {\text {d}}{\varvec{\xi }}\nonumber \\&\quad + \dfrac{64 C^4_{\omega _1} {\bar{J}}^3_0}{\omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) (|\varvec{u}(\varvec{y}+\epsilon {\varvec{\xi }})|^4 + |\varvec{u}(\varvec{y})|^4) {\text {d}}{\varvec{\xi }}. \end{aligned}$$

Application of Fubini’s theorem gives

$$\begin{aligned}&\int _D |\nabla \theta (\varvec{y};\varvec{u})|^4 {\text {d}}\varvec{x}\nonumber \\&\leqslant \dfrac{64 {\bar{J}}^3_0}{\omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) \left( \int _D (|\nabla \varvec{u}(\varvec{y}+\epsilon {\varvec{\xi }})|^4 + |\nabla \varvec{u}(\varvec{y})|^4) {\text {d}}\varvec{x}\right) {\text {d}}{\varvec{\xi }}\nonumber \\&\quad + \dfrac{64 C^4_{\omega _1} {\bar{J}}^3_0}{\omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) \left( \int _D (|\varvec{u}(\varvec{y}+\epsilon {\varvec{\xi }})|^4 + |\varvec{u}(\varvec{y})|^4) {\text {d}}\varvec{x}\right) {\text {d}}{\varvec{\xi }}\nonumber \\&\leqslant \dfrac{64 {\bar{J}}^3_0}{\omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) \left( 2 ||\nabla \varvec{u}||^4_{L^4(D;{\mathbb {R}}^{d\times d})} \right) {\text {d}}{\varvec{\xi }}\nonumber \\&\quad + \dfrac{64 C^4_{\omega _1} {\bar{J}}^3_0}{\omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) \left( ||\varvec{u}||^2_\infty \int _D (|\varvec{u}(\varvec{y}+\epsilon {\varvec{\xi }})|^2 + |\varvec{u}(\varvec{y})|^2) {\text {d}}\varvec{x}\right) {\text {d}}{\varvec{\xi }}\nonumber \\&\leqslant 128 {\bar{J}}^4_0 ||\nabla \varvec{u}||^4_{L^4(D;{\mathbb {R}}^{d\times d})} + \dfrac{64 C^4_{\omega _1} {\bar{J}}^3_0}{\omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) \left( ||\varvec{u}||^2_\infty 2 ||\varvec{u}||^2 \right) {\text {d}}{\varvec{\xi }}\nonumber \\&\leqslant 128 {\bar{J}}^4_0 ||\nabla \varvec{u}||^4_{L^4(D;{\mathbb {R}}^{d\times d})} + 128 C^4_{\omega _1} {\bar{J}}^4_0 ||\varvec{u}||^2_\infty \,||\varvec{u}||^2. \end{aligned}$$

Using the Sobolev embedding property, \(||\varvec{u}||_\infty \leqslant C_{e_1}||\varvec{u}||_2\) and \(||\nabla \varvec{u}||_{L^4} \leqslant C_{e_2} ||\varvec{u}||_2\), we obtain

$$\begin{aligned} \int _D |\nabla \theta (\varvec{y};\varvec{u})|^4 {\text {d}}\varvec{x}&\leqslant 128 {\bar{J}}^4_0 (C_{e_2}^4 + C_{e_1}^2 C^4_{\omega _1}) ||\varvec{u}||^4_2 \leqslant 128 {\bar{J}}^4_0 (C_{e_2}^2 + C_{e_1} C^2_{\omega _1})^2 ||\varvec{u}||^4_2 \,. \end{aligned}$$

The estimate Eq. 92 follows by combining estimates Eqs. 87 and 91. It now remains to show Eq. 93. From expression of \(\nabla \theta (\varvec{y};\varvec{u})\) in Eq. 98, we have

$$\begin{aligned} \nabla ^2 \theta (\varvec{y};\varvec{u})&= \frac{1}{\omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) \omega (\varvec{y}+ \epsilon {\varvec{\xi }}) \nabla ^2 ({\bar{\varvec{u}}}_{\varvec{\xi }}(\varvec{y}) \cdot \varvec{e}_{\varvec{\xi }}) {\text {d}}{\varvec{\xi }}\nonumber \\&\quad + \frac{1}{\omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) ((\nabla ({\bar{\varvec{u}}}_{\varvec{\xi }}(\varvec{y}))^{\text{T}} \varvec{e}_{\varvec{\xi }}) \mathbf {\otimes }\nabla \omega (\varvec{y}+ \epsilon {\varvec{\xi }}) {\text {d}}{\varvec{\xi }}\nonumber \\&\quad + \frac{1}{\omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) \nabla \omega (\varvec{y}+ \epsilon {\varvec{\xi }}) \mathbf {\otimes }((\nabla ({\bar{\varvec{u}}}_{\varvec{\xi }}(\varvec{y}))^{\text{T}} \varvec{e}_{\varvec{\xi }}) {\text {d}}{\varvec{\xi }}\nonumber \\&\quad + \frac{1}{\omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) \nabla ^2 \omega (\varvec{y}+ \epsilon {\varvec{\xi }}) {\bar{\varvec{u}}}_{\varvec{\xi }}(\varvec{y}) \cdot \varvec{e}_{\varvec{\xi }}{\text {d}}{\varvec{\xi }}. \end{aligned}$$

Using the equation above, we can show

$$\begin{aligned} \int _D |\nabla ^2 \theta (\varvec{y};\varvec{u})|^2 {\text {d}}\varvec{x}&\leqslant \frac{3 {\bar{J}}_0}{\omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) \left( \int _D |\nabla ^2 {\bar{\varvec{u}}}_{\varvec{\xi }}(\varvec{y})|^2 {\text {d}}\varvec{x}\right) {\text {d}}{\varvec{\xi }}\nonumber \\&\quad + \frac{12 C^2_{\omega _1} {\bar{J}}_0}{\omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) \left( \int _D |\nabla {\bar{\varvec{u}}}_{\varvec{\xi }}(\varvec{y})|^2 {\text {d}}\varvec{x}\right) {\text {d}}{\varvec{\xi }}\nonumber \\&\quad + \frac{3C^2_{\omega _2} {\bar{J}}_0}{\omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) \left( \int _D | {\bar{\varvec{u}}}_{\varvec{\xi }}(\varvec{y})|^2 {\text {d}}\varvec{x}\right) {\text {d}}{\varvec{\xi }}. \end{aligned}$$

The terms \(|{\bar{\varvec{u}}}_{\varvec{\xi }}(\varvec{y})|^2\), \(|\nabla {\bar{\varvec{u}}}_{\varvec{\xi }}(\varvec{y})|^2\), and \(|\nabla ^2 {\bar{\varvec{u}}}_{\varvec{\xi }}(\varvec{y})|^2\) are bounded by \(2 (|\varvec{u}(\varvec{y}+ \epsilon {\varvec{\xi }})|^2 + |\varvec{u}(\varvec{y})|^2)\), \(2 (|\nabla \varvec{u}(\varvec{y}+ \epsilon {\varvec{\xi }})|^2 + |\nabla \varvec{u}(\varvec{y})|^2)\), and \(2 (|\nabla ^2 \varvec{u}(\varvec{y}+ \epsilon {\varvec{\xi }})|^2 + |\nabla ^2 \varvec{u}(\varvec{y})|^2)\), respectively. Therefore, we have

$$\begin{aligned} \int _D |\nabla ^2 \theta (\varvec{y};\varvec{u})|^2 {\text {d}}\varvec{x}&\leqslant \left( 3 {\bar{J}}^2_0 + 12 C^2_{\omega _1} {\bar{J}}^2_0 + 3 C^2_{\omega _2} {\bar{J}}^2_0\right) 4 ||\varvec{u}||^2_2\nonumber \\&\leqslant 16 {\bar{J}}^2_0 (1+C_{\omega _2} + 2C_{\omega _1})^2 ||\varvec{u}||^2_2, \end{aligned}$$

and this completes the proof of lemma.

Estimating\(||{\mathcal {L}}^\epsilon _D(\varvec{u}) - {\mathcal {L}}^\epsilon _D(\varvec{v})||\): We apply the notation described in Eq. 83, and write \({\mathcal {L}}^\epsilon _D(\varvec{u})(\varvec{x})\) as follows:

$$\begin{aligned} {\mathcal {L}}^\epsilon _D (\varvec{u})(\varvec{x})&= \frac{1}{\epsilon ^2 \omega _D} \int _{H_1({\mathbf {0}})} \omega _{\varvec{\xi }}(\varvec{x}) J(|{\varvec{\xi }}|) [g'(\theta (\varvec{x}+ \epsilon {\varvec{\xi }};\varvec{u})) + g'(\theta (\varvec{x};\varvec{u}))] \varvec{e}_{\varvec{\xi }}{\text {d}}{\varvec{\xi }}. \end{aligned}$$

Using the formula above and from the expression for \(\theta \), we can easily show

$$\begin{aligned} ||{\mathcal {L}}^\epsilon _D (\varvec{u}) - {\mathcal {L}}^\epsilon _D (\varvec{v})||&\leqslant \dfrac{L_1}{\epsilon ^2} ||\varvec{u}- \varvec{v}||_2, \end{aligned}$$

where \(L_1 = 4 C^g_2 {\bar{J}}_0^2\).

Estimating\(||\nabla {\mathcal {L}}^\epsilon _D(\varvec{u}) - \nabla {\mathcal {L}}^\epsilon _D(\varvec{v})||\): Taking the gradient of Eq. 107 gives

$$\begin{aligned} \nabla {\mathcal {L}}^\epsilon _D (\varvec{u}) (\varvec{x})&= \dfrac{1}{\epsilon ^2 \omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) \omega _{\varvec{\xi }}(\varvec{x}) \varvec{e}_{\varvec{\xi }}\mathbf {\otimes }[\nabla g'(\theta (\varvec{x}+ \epsilon {\varvec{\xi }}; \varvec{u})) + \nabla g'(\theta (\varvec{x}; \varvec{u}))] {\text {d}}{\varvec{\xi }}\nonumber \\&\quad + \dfrac{1}{\epsilon ^2 \omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) \varvec{e}_{\varvec{\xi }}\mathbf {\otimes }\nabla \omega _{\varvec{\xi }}(\varvec{x}) [g'(\theta (\varvec{x}+ \epsilon {\varvec{\xi }}; \varvec{u})) + g'(\theta (\varvec{x}; \varvec{u}))] {\text {d}}{\varvec{\xi }}\nonumber \\&=: G_1(\varvec{u})(\varvec{x}) + G_2(\varvec{u})(\varvec{x}), \end{aligned}$$

where we have denoted the first and second terms as \(G_1(\varvec{u})(\varvec{x})\) and \(G_2(\varvec{u})(\varvec{x})\) for convenience. On using the triangle inequality, we get

$$\begin{aligned} ||\nabla {\mathcal {L}}^\epsilon _D (\varvec{u}) - \nabla {\mathcal {L}}^\epsilon _D (\varvec{v})||&\leqslant ||G_1(\varvec{u}) - G_1(\varvec{v})|| + ||G_2(\varvec{u}) - G_2(\varvec{u})||. \end{aligned}$$

From the expression of \(G_1(\varvec{u})\), we have

$$\begin{aligned} |G_1(\varvec{u})(\varvec{x}) - G_1(\varvec{v})(\varvec{x}) |&\leqslant \dfrac{1}{\epsilon ^2 \omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) (|\nabla g'(\theta (\varvec{x}+ \epsilon {\varvec{\xi }}; \varvec{u})) - \nabla g'(\theta (\varvec{x}+ \epsilon {\varvec{\xi }}; \varvec{v}))|\\&\quad + |\nabla g'(\theta (\varvec{x}; \varvec{u})) - \nabla g'(\theta (\varvec{x}; \varvec{v}))|) {\text {d}}{\varvec{\xi }}. \end{aligned}$$


$$\begin{aligned} p_1(\varvec{y})&:= |\nabla g'(\theta (\varvec{y}; \varvec{u})) - \nabla g'(\theta (\varvec{y}; \varvec{v}))| \end{aligned}$$

and we get

$$\begin{aligned} ||G_1(\varvec{u}) - G_1(\varvec{v})||^2&\leqslant \left( \dfrac{1}{\epsilon ^2}\right) ^2 \dfrac{2{\bar{J}}_0}{ \omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) \left( \int _D (p_1(\varvec{x}+ \epsilon {\varvec{\xi }})^2 + p_1(\varvec{x})^2) {\text {d}}\varvec{x}\right) {\text {d}}{\varvec{\xi }}. \end{aligned}$$

Note that

$$\begin{aligned} \nabla g'(\theta (\varvec{x}+ \epsilon {\varvec{\xi }}; \varvec{u}))&= g''(\theta (\varvec{x}+ \epsilon {\varvec{\xi }};\varvec{u})) \nabla \theta (\varvec{x}+ \epsilon {\varvec{\xi }};\varvec{u}). \end{aligned}$$

Therefore, from Eq. 110,

$$\begin{aligned} p_1(\varvec{y})&= |g''(\theta (\varvec{y};\varvec{u})) \nabla \theta (\varvec{y};\varvec{u}) - g''(\theta (\varvec{y};\varvec{v})) \nabla \theta (\varvec{y};\varvec{v})|\\&\leqslant C^g_2 |\nabla \theta (\varvec{y};\varvec{u}) - \nabla \theta (\varvec{y};\varvec{v})| + C^g_3 |\theta (\varvec{y};\varvec{u}) - \theta (\varvec{y};\varvec{v})| \, |\nabla \theta (\varvec{y};\varvec{v})|\\&= C^g_2 |\nabla \theta (\varvec{y};\varvec{u}- \varvec{v})| + C^g_3 |\theta (\varvec{y};\varvec{u}-\varvec{v})| \, |\nabla \theta (\varvec{y};\varvec{v})|, \end{aligned}$$

where we have added and subtracted \(g''(\theta (\varvec{y};\varvec{u})) \nabla \theta (\varvec{y};\varvec{v})\) and used the fact that \(g''(r) \leqslant C^g_2\) and \(|g''(r_1) - g''(r_2)| \leqslant C^g_3 |r_1 - r_2|\). We use the estimate on \(p_1\) and proceed as follows:

$$\begin{aligned} \int _D p_1(\varvec{y})^2 {\text {d}}\varvec{x}&\leqslant 2 (C^g_2)^2 \int _D |\nabla \theta (\varvec{y};\varvec{u}- \varvec{v})|^2 {\text {d}}\varvec{x}\\&\quad + 2(C^g_3)^2 \int _D |\theta (\varvec{y};\varvec{u}-\varvec{v})|^2 \, |\nabla \theta (\varvec{y};\varvec{v})|^2 {\text {d}}\varvec{x}, \end{aligned}$$

where we denote \(\varvec{x}+\epsilon {\varvec{\xi }}\) as \(\varvec{y}\). We apply inequality Eqs. 89 and 90 of Lemma 4 to obtain

$$\begin{aligned} \int _D p_1(\varvec{y})^2 {\text {d}}\varvec{x}&\leqslant 2 (C^g_2)^2 8 {\bar{J}}^2_0 (1+C_{\omega _1})^2 ||\varvec{u}- \varvec{v}||_2^2\\&\quad + 2(C^g_3)^2 32 {\bar{J}}_0^4 (1+C_{\omega _1})^2 ||\varvec{v}||_2^2 \, ||\varvec{u}- \varvec{v}||_2^2\\&\leqslant L_2 (1+ ||\varvec{v}||_2)^2 ||\varvec{u}- \varvec{v}||_2^2, \end{aligned}$$

where we have grouped all the constant factors together and denote their product by \(L_2\). Substituting these estimates into Eq. 111 gives

$$\begin{aligned} ||G_1(\varvec{u}) - G_1(\varvec{v})||^2&\leqslant \dfrac{4L_2 {\bar{J}}_0^2 }{\epsilon ^4} (1+ ||\varvec{v}||_2)^2 ||\varvec{u}- \varvec{v}||_2^2\nonumber \\ \Rightarrow ||G_1(\varvec{u}) - G_1(\varvec{v})||&\leqslant \dfrac{L_3(1+ ||\varvec{v}||_2)}{\epsilon ^2} ||\varvec{u}- \varvec{v}||_2, \end{aligned}$$

where we have introduced the new constant \(L_3\).

The formula for \(G_2(\varvec{u})\) is similar to \({\mathcal {L}}^\epsilon _D(\varvec{u})\) and, therefore, we have

$$\begin{aligned} ||G_2(\varvec{u}) - G_2(\varvec{v})||&\leqslant \dfrac{C_{\omega _1} L_1}{\epsilon ^2} ||\varvec{u}- \varvec{v}||_2. \end{aligned}$$

Collecting results, we have shown

$$\begin{aligned} ||\nabla {\mathcal {L}}^\epsilon _D (\varvec{u}) - \nabla {\mathcal {L}}^\epsilon _D (\varvec{v})||&\leqslant \dfrac{L_4(1+||\varvec{v}||_2)}{\epsilon ^2} ||\varvec{u}- \varvec{v}||_2, \end{aligned}$$

where we have introduced new constant \(L_4\).

Estimating\(||\nabla ^2 {\mathcal {L}}^\epsilon _D(\varvec{u}) - \nabla ^2 {\mathcal {L}}^\epsilon _D(\varvec{v})||\): Taking the gradient of Eq. 109, gives

$$\begin{aligned} \nabla ^2 {\mathcal {L}}^\epsilon _D(\varvec{u})(\varvec{x})&= \dfrac{1}{\epsilon ^2 \omega _d} \int _{H_1({\mathbf {0}})} \omega _{\varvec{\xi }}(\varvec{x}) J(|{\varvec{\xi }}|) \varvec{e}_{\varvec{\xi }}\mathbf {\otimes }[\nabla ^2 g'(\theta (\varvec{x}+\epsilon {\varvec{\xi }};\varvec{u})) + \nabla ^2 g'(\theta (\varvec{x};\varvec{u}))] {\text {d}}{\varvec{\xi }}\nonumber \\&\quad + \dfrac{1}{\epsilon ^2 \omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) \varvec{e}_{\varvec{\xi }}\mathbf {\otimes }[\nabla g'(\theta (\varvec{x}+\epsilon {\varvec{\xi }};\varvec{u})) + \nabla g'(\theta (\varvec{x};\varvec{u}))] \mathbf {\otimes }\nabla \omega _{\varvec{\xi }}(\varvec{x}) {\text {d}}{\varvec{\xi }}\nonumber \\&\quad + \dfrac{1}{\epsilon ^2 \omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) \varvec{e}_{\varvec{\xi }}\mathbf {\otimes }\nabla \omega _{\varvec{\xi }}(\varvec{x}) \mathbf {\otimes }[\nabla g'(\theta (\varvec{x}+\epsilon {\varvec{\xi }};\varvec{u})) + \nabla g'(\theta (\varvec{x};\varvec{u}))] {\text {d}}{\varvec{\xi }}\nonumber \\&\quad + \dfrac{1}{\epsilon ^2 \omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) \varvec{e}_{\varvec{\xi }}\mathbf {\otimes }\nabla ^2 \omega _{\varvec{\xi }}(\varvec{x}) [g'(\theta (\varvec{x}+\epsilon {\varvec{\xi }};\varvec{u})) + g'(\theta (\varvec{x};\varvec{u}))] {\text {d}}{\varvec{\xi }}\nonumber \\&=: H_1(\varvec{u})(\varvec{x}) + H_2(\varvec{u})(\varvec{x}) + H_3(\varvec{u})(\varvec{x}) + H_4(\varvec{u})(\varvec{x}). \end{aligned}$$

It is easy to see that estimate on \(||H_2(\varvec{u}) - H_2(\varvec{v})||\) and \(||H_3(\varvec{u}) - H_3(\varvec{v})||\) is similar to the estimate for \(||G_1(\varvec{u}) - G_1(\varvec{v})||\). Thus, from Eq. 112, we have

$$\begin{aligned} ||H_2(\varvec{u}) - H_2(\varvec{v})|| + ||H_3(\varvec{u}) - H_3(\varvec{v})||&\leqslant \dfrac{2C_{\omega _1} L_3 (1+||\varvec{v}||_2) }{\epsilon ^2} ||\varvec{u}- \varvec{v}||_2. \end{aligned}$$

Also the estimate for \(||H_4(\varvec{u}) - H_4(\varvec{v})||\) is similar to the estimate for \(||G_2(\varvec{u}) - G_2(\varvec{v})||\) and we conclude

$$\begin{aligned} ||H_4(\varvec{u}) - H_4(\varvec{v})||&\leqslant \dfrac{C_{\omega _2} L_1 }{\epsilon ^2} ||\varvec{u}- \varvec{v}||_2. \end{aligned}$$

We now work on \(||H_1(\varvec{u}) - H_1(\varvec{v})||\). From expression of \(H_1(\varvec{u})(\varvec{x})\) in Eq. 114, we can easily get the following:

$$\begin{aligned} |H_1(\varvec{u})(\varvec{x}) - H_1(\varvec{v})(\varvec{x})|&\leqslant \dfrac{1}{\epsilon ^2 \omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) (|\nabla ^2 g'(\theta (\varvec{x}+ \epsilon {\varvec{\xi }}; \varvec{u})) - \nabla ^2 g'(\theta (\varvec{x}+ \epsilon {\varvec{\xi }}; \varvec{v}))|\nonumber \\&\quad + |\nabla ^2 g'(\theta (\varvec{x}; \varvec{u})) - \nabla ^2 g'(\theta (\varvec{x}; \varvec{v}))|) {\text {d}}{\varvec{\xi }}. \end{aligned}$$

Let \(p_2(\varvec{y})\), where \(\varvec{y}= \varvec{x}+ \epsilon {\varvec{\xi }}\) and \(\nabla \) is with respect to \(\varvec{x}\), is given by

$$\begin{aligned} p_2(\varvec{y})&:= |\nabla ^2 g'(\theta (\varvec{y}; \varvec{u})) - \nabla ^2 g'(\theta (\varvec{y}; \varvec{v}))|. \end{aligned}$$

We then have

$$\begin{aligned} ||H_1(\varvec{u}) - H_1(\varvec{v})||^2&\leqslant \left( \dfrac{1}{\epsilon ^2}\right) ^2 \dfrac{2{\bar{J}}_0}{ \omega _d} \int _{H_1({\mathbf {0}})} J(|{\varvec{\xi }}|) \left( \int _D (p_2(\varvec{x}+ \epsilon {\varvec{\xi }})^2 + p_2(\varvec{x})^2) {\text {d}}\varvec{x}\right) {\text {d}}{\varvec{\xi }}. \end{aligned}$$

Note that

$$\begin{aligned} \nabla ^2 g'(\theta (\varvec{y}; \varvec{u})) = g'''(\theta (\varvec{y};\varvec{u})) \nabla \theta (\varvec{y};\varvec{u}) \mathbf {\otimes }\nabla \theta (\varvec{y};\varvec{u}) + g''(\theta (\varvec{y};\varvec{u})) \nabla ^2 \theta (\varvec{y};\varvec{u}). \end{aligned}$$

We add and subtract terms to the equation above to get

$$\begin{aligned}&\nabla ^2 g'(\theta (\varvec{y}; \varvec{u}) - \nabla ^2 g'(\theta (\varvec{y}; \varvec{v})\\&= g'''(\theta (\varvec{y};\varvec{u})) [ \nabla \theta (\varvec{y};\varvec{u}) \mathbf {\otimes }\nabla \theta (\varvec{y};\varvec{u}) - \nabla \theta (\varvec{y};\varvec{v}) \mathbf {\otimes }\nabla \theta (\varvec{y};\varvec{v})]\\&\quad + [g'''(\theta (\varvec{y};\varvec{u})) - g'''(\theta (\varvec{y};\varvec{v}))] \nabla \theta (\varvec{y};\varvec{v}) \mathbf {\otimes }\nabla \theta (\varvec{y};\varvec{v})\\&\quad + g''(\theta (\varvec{y};\varvec{u})) [\nabla ^2 \theta (\varvec{y};\varvec{u}) - \nabla ^2 \theta (\varvec{y};\varvec{v})]\\&\quad + [g''(\theta (\varvec{y};\varvec{u})) - g''(\theta (\varvec{y};\varvec{v}))] \nabla ^2 \theta (\varvec{y};\varvec{v}). \end{aligned}$$

Using inequalities \(|g''(r)| \leqslant C^g_2\), \(|g'''(r)|\leqslant C^g_3\), \(|g''(r_1) - g''(r_2)| \leqslant C^g_3 |r_1 - r_2|\), \(|g'''(r_1) - g'''(r_2)| \leqslant C^g_4 |r_1 - r_2|\), and \(|\varvec{a}\mathbf {\otimes }\varvec{a}- \varvec{c}\mathbf {\otimes }\varvec{c}| \leqslant (|\varvec{a}| + |\varvec{c}|) |\varvec{a}- \varvec{c}|\), and the fact that \(\theta (\varvec{y};\varvec{u}) - \theta (\varvec{y};\varvec{v}) = \theta (\varvec{y};\varvec{u}- \varvec{v})\), we have

$$\begin{aligned} p_2(\varvec{y})&\leqslant C^g_3 |\nabla \theta (\varvec{y};\varvec{u})| \, |\nabla \theta (\varvec{y};\varvec{u}- \varvec{v})| + C^g_3 |\nabla \theta (\varvec{y};\varvec{v})|\, |\nabla \theta (\varvec{y};\varvec{u}- \varvec{v})|\\&\quad + C^g_4 | \theta (\varvec{y};\varvec{u}- \varvec{v})| \, |\nabla \theta (\varvec{y};\varvec{v})|^2\\&\quad + C^g_2 |\nabla ^2 \theta (\varvec{y};\varvec{u}-\varvec{v})| + C^g_3 |\theta (\varvec{y};\varvec{u}- \varvec{v})|\, |\nabla ^2 \theta (\varvec{y};\varvec{v})|. \end{aligned}$$

Taking the square of the above equation and using \(\left(\sum \limits_{i=1}^5 a_i\right)^2 \leqslant 5 \sum \limits_{i=1}^5 a^2_i\) gives

$$\begin{aligned} \int _D p_2(\varvec{y})^2 {\text {d}}\varvec{x}&\leqslant 5(C^g_3)^2 \int _D |\nabla \theta (\varvec{y};\varvec{u})|^2 \, |\nabla \theta (\varvec{y};\varvec{u}- \varvec{v})|^2 {\text {d}}\varvec{x}\\&\quad + 5(C^g_3)^2 \int _D |\nabla \theta (\varvec{y};\varvec{v})|^2 \, |\nabla \theta (\varvec{y};\varvec{u}- \varvec{v})|^2 {\text {d}}\varvec{x}\\&\quad + 5 (C^g_4)^2 \int _D | \theta (\varvec{y};\varvec{u}- \varvec{v})|^2 \, |\nabla \theta (\varvec{y};\varvec{v})|^4 {\text {d}}\varvec{x}\\&\quad + 5 (C^g_2)^2 \int _D |\nabla ^2 \theta (\varvec{y};\varvec{u}-\varvec{v})|^2 {\text {d}}\varvec{x}\\&\quad + 5(C^g_3)^2 \int _D |\theta (\varvec{y};\varvec{u}- \varvec{v})|^2 \, |\nabla ^2 \theta (\varvec{y};\varvec{v})|^2 {\text {d}}\varvec{x}\\&=: I_1 + I_2 + I_3 + I_4 + I_5. \end{aligned}$$

We now estimate each term using Lemma 4 as follows. Applying the Hölder inequality and the inequality Eq. 91 of Lemma 4 we get

$$\begin{aligned} I_1&\leqslant 5(C^g_3)^2 \left( \int _D |\nabla \theta (\varvec{y};\varvec{u})|^4 {\text {d}}\varvec{x}\right) ^{1/2} \left( \int _D |\nabla \theta (\varvec{y};\varvec{u}- \varvec{v})|^4 {\text {d}}\varvec{x}\right) ^{1/2} \\&\leqslant 640 (C^g_3)^2 {\bar{J}}_0^4 (C^2_{e_2} + C_{e_1}C^2_{\omega _1})^2 ||\varvec{u}||_2^2\, ||\varvec{u}- \varvec{v}||_2^2. \end{aligned}$$


$$\begin{aligned} I_2&\leqslant 640 (C^g_3)^2 {\bar{J}}_0^4 (C^2_{e_2} + C_{e_1}C^2_{\omega _1})^2 ||\varvec{v}||_2^2\, ||\varvec{u}- \varvec{v}||_2^2. \end{aligned}$$

Using Eq. 92 of Lemma 4, we get

$$\begin{aligned} I_3&\leqslant 2\,560 (C^g_4)^2 {\bar{J}}_0^6 C_{e_1}^2 (C^2_{e_2} + C_{e_1}C^2_{\omega _1})^2 ||\varvec{v}||_2^4\, ||\varvec{u}- \varvec{v}||_2^2. \end{aligned}$$

For \(I_4\), we use the inequality Eq. 93 to get

$$\begin{aligned} I_4&\leqslant 80 (C^g_2)^2 {\bar{J}}_0^2 (1+2C_{\omega _1} + C_{\omega _2})^2 ||\varvec{u}- \varvec{v}||_2^2. \end{aligned}$$

In \(I_5\), we use Eqs. 87 and  93 to get

$$\begin{aligned} I_5&\leqslant 320 {\bar{J}}^4_0 C_{e_1}^2 (1+2C_{\omega _1} + C_{\omega _2})^2 ||\varvec{v}||_2^2\, ||\varvec{u}- \varvec{v}||_2^2. \end{aligned}$$

After collecting results, we can find a constant \(L_5\) such that we have

$$\begin{aligned} \int _D p_2(\varvec{y})^2 {\text {d}}\varvec{x}&\leqslant L_5^2 \,(1+ (||\varvec{u}||_2 + ||\varvec{v}||_2) + (||\varvec{u}||_2 + ||\varvec{v}||_2)^2)^2 \,||\varvec{u}- \varvec{v}||_2^2. \end{aligned}$$

We substitute Eq. 120 into Eq. 119 to show

$$\begin{aligned} ||H_1(\varvec{u}) - H_1(\varvec{v})||^2&\leqslant \frac{4 L^2_5 {\bar{J}}_0^2 (1+ (||\varvec{u}||_2 + ||\varvec{v}||_2) + (||\varvec{u}||_2 + ||\varvec{v}||_2)^2)^2}{\epsilon ^4} ||\varvec{u}- \varvec{v}||_2^2\nonumber \\ \Rightarrow ||H_1(\varvec{u}) - H_1(\varvec{v})||&\leqslant \dfrac{L_6(1+ (||\varvec{u}||_2 + ||\varvec{v}||_2) + (||\varvec{u}||_2 + ||\varvec{v}||_2)^2) }{\epsilon ^2} ||\varvec{u}- \varvec{v}||_2\nonumber \\&\leqslant \dfrac{L_6(1+ ||\varvec{u}||_2 + ||\varvec{v}||_2)^2 }{\epsilon ^2} ||\varvec{u}- \varvec{v}||_2, \end{aligned}$$

where we have introduced the new constant \(L_6\).

We combine the estimates on \(H_1,H_2,H_3,H_4\), introducing a new constant \(L_7\), and get

$$\begin{aligned} ||\nabla ^2 {\mathcal {L}}^\epsilon _D(\varvec{u}) - \nabla ^2 {\mathcal {L}}^\epsilon _D(\varvec{v})||&\leqslant \dfrac{L_7 (1+ ||\varvec{u}||_2 + ||\varvec{v}||_2)^2 }{\epsilon ^2} ||\varvec{u}- \varvec{v}||_2. \end{aligned}$$

On adding the estimates, Eqs. 108,  113,  122, it is evident that the proof of Theorem 1 is complete.

Proof of Higher Temporal Regularity

In this section, we prove that the peridynamic evolutions have higher regularity in time for body forces that that are differentiable in time. To see this we take the time derivative of Eq. 11 to get a second-order differential equation in time for \(\varvec{v}={\dot{\varvec{u}}}\) given by

$$\begin{aligned} \rho \partial ^2_{tt} \varvec{v}(\varvec{x},t)&= Q(\varvec{v}(t); \varvec{u}(t))(\varvec{x}) + {\dot{\varvec{b}}}(\varvec{x},t), \end{aligned}$$

where \(Q(\varvec{v};\varvec{u})\) is an operator that depends on the solution \(\varvec{u}\) of Eq. 11 and acts on \(\varvec{v}\). It is given by

$$\begin{aligned} Q(\varvec{v};\varvec{u})(\varvec{x})&= Q_T(\varvec{v};\varvec{u})(\varvec{x}) + Q_D(\varvec{v};\varvec{u})(\varvec{x}),{ \forall \varvec{x}\in D,} \end{aligned}$$


$$\begin{aligned} Q_T(\varvec{v};\varvec{u})(\varvec{x})&=\frac{2}{\epsilon ^d \omega _d}\int _{H_\epsilon (\varvec{x})} \omega (\varvec{x}) \omega (\varvec{y}) \frac{J^\epsilon (|\varvec{y}-\varvec{x}|)}{\epsilon |\varvec{y}-\varvec{x}|}\nonumber \\&\quad \cdot \partial ^2_{SS} f(\sqrt{|\varvec{y}-\varvec{x}|}S(\varvec{y},\varvec{x},t;\varvec{u})) S(\varvec{y},\varvec{x},t;\varvec{v}) \varvec{e}_{\varvec{y}-\varvec{x}}\,{\text {d}}\varvec{y}, \end{aligned}$$


$$\begin{aligned} Q_D(\varvec{v};\varvec{u})(\varvec{x})&=\frac{1}{\epsilon ^d \omega _d} \int _{H_\epsilon (\varvec{x})} \omega (\varvec{x}) \omega (\varvec{y}) \frac{J^\epsilon (|\varvec{y}-\varvec{x}|)}{\epsilon ^2}\nonumber \\&\quad \; \cdot \left[ \partial ^2_{\theta \theta } g(\theta (\varvec{y},t;\varvec{u})) \theta (\varvec{y},t;\varvec{v}) +\partial ^2_{\theta \theta } g(\theta (\varvec{x},t;\varvec{u})) \theta (\varvec{x},t;\varvec{v}) \right] \varvec{e}_{\varvec{y}-\varvec{x}}\,{\text {d}}\varvec{y}. \end{aligned}$$

Clearly, for \(\varvec{u}\) fixed, the form \(Q(\varvec{v};\varvec{u})\) acts linearly on \(\varvec{v}\) which implies that the equation for \(\varvec{v}\) is a linear nonlocal equation. The linearity of \(Q(\varvec{v};\varvec{u})\) implies the Lipschitz continuity for \(\varvec{v}\in W\) as stated below.

Theorem 7

(Lipschitz continuity of Q) Let\(\varvec{u}\in W\)be any given field. Then, for all\(\varvec{v},\varvec{w}\in W\), we have

$$\begin{aligned} ||Q(\varvec{v};\varvec{u}) - Q(\varvec{w};\varvec{u})||_2&\leqslant \dfrac{L_8(1 + ||\varvec{u}||_2)^2}{\epsilon ^3} ||\varvec{v}-\varvec{w}||_2, \end{aligned}$$

where the constant\(L_8\)does not depend on\(\varvec{u},\varvec{v},\varvec{w}\). This gives for all\(\varvec{v}\in W\)the upper bound,

$$\begin{aligned} ||Q(\varvec{v};\varvec{u})||_2&\leqslant \dfrac{L_8(1 + ||\varvec{u}||_2)^2}{\epsilon ^3} ||\varvec{v}||_2 . \end{aligned}$$

The proof follows the same steps used in the proof of Theorem 1.

If \(\varvec{u}\) is a peridynamic solution such that \(\varvec{u}\in C^2(I_0;W)\), then we have for all \(t \in I_0\), the inequality

$$\begin{aligned} ||Q(\varvec{v};\varvec{u}(t))||_2&\leqslant \dfrac{L_8(1 + \sup \limits_{s \in I_0} ||\varvec{u}(s)||_2)^2}{\epsilon ^3} ||\varvec{v}||_2. \end{aligned}$$

Note that the Lipschitz continuity of \({\dot{\varvec{u}}}(t)\) stated in Theorem 2 implies \(\lim \limits_{t\rightarrow 0^{\pm }}\partial ^2_{tt} \varvec{u}(\varvec{x}, t)=\partial ^2_{tt}\varvec{u}(\varvec{x},0)\). We now demonstrate that \(\varvec{v}(\varvec{x}, t)=\partial _t \varvec{u}(\varvec{x}, t)\) is the unique solution of the following initial boundary value problem.

Theorem 8

(Initial value problem for \(\varvec{v}(\varvec{x},t))\)Suppose the initial data and righthand side\(\varvec{b}(t)\)satisfy the hypothesis of Theorem 2and we suppose further that\({\dot{\varvec{b}}}(t)\)exists and is continuous in time for\(t\in I_0\)and\(\sup \limits_{t\in I_0} ||{\dot{\varvec{b}}}(t)||_2 < \infty \). Then, \(\varvec{v}(\varvec{x}, t)\)is the unique solution to the initial value problem\(\varvec{v}(\varvec{x}, 0)=\varvec{v}_0(\varvec{x})\), \(\partial _t \varvec{v}(\varvec{x},0)=\partial ^2_{tt}\varvec{u}(\varvec{x}, 0)\),

$$\begin{aligned} \rho \partial ^2_{tt} \varvec{v}(\varvec{x},t)&= Q(\varvec{v}(t); \varvec{u}(t))(\varvec{x}) + {\dot{\varvec{b}}}(\varvec{x},t), { t\in I_0},{ \varvec{x}\in D,} \end{aligned}$$

\(\varvec{v}\in C^2(I_0; W)\)and

$$\begin{aligned} || \partial ^2_{tt} \varvec{v}(\varvec{x},t)||_2&\leqslant ||Q(\varvec{v}(t); \varvec{u}(t))(\varvec{x})||_2 + ||{\dot{\varvec{b}}}(\varvec{x},t)||_2. \end{aligned}$$

Theorem 3 now follows immediately from Theorem 8 noting that \(\partial _t \varvec{u}(\varvec{x}, t)=\varvec{v}(\varvec{x},t)\) together with Eqs. 129 and 131. The proof of Theorem 8 follows from the Lipschitz continuity Eq. 127 and the Banach fixed point theorem as in [6].


In this article, we have provided a priori error estimates for finite element approximations to nonlocal state-based peridynamic fracture models. We have shown that the convergence rate applies even over time intervals for which the material is softening over parts of the computational domain. The results are established for two different classes of state-based peridynamic forces. The convergence rate of the approximation is of the form \(C(\Delta t+h^2/\epsilon ^2)\) where the constant C depends on \(\epsilon \) and the \(H^2\) norm of the solution and its time derivatives. For fixed \(\Delta t\) numerical simulations for Plexiglass show that the error decreases at the rate of \(h^2\) at \(40 \, \upmu \)s into the simulation. The simulations were carried out in parallel using 20 threads on a workstation with single Intel Xeon processor and with 32 GB of RAM. We anticipate similar convergence rates for longer times on bigger parallel machines.

We reiterate that the a priori error estimates account for the possible appearance of nonlinearity anywhere in the computational domain. On the other hand, numerical simulation and independent theoretical estimates show that the nonlinearity concentrates along “fat” cracks of finite length and width equal to \(\epsilon \); see [25, 26]. Moreover, the remainder of the computational domain is seen to behave linearly and to leading order can be modeled as a linear elastic material up to an error proportional to \(\epsilon \); see [Proposition 6, [22]]. Future work will use these observations to focus on the adaptive implementation and a-posteriori estimates.


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Correspondence to Robert Lipton.

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This material is based upon work supported by the U. S. Army Research Laboratory and the U. S. Army Research Office under contract/Grant number W911NF1610456.

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Jha, P.K., Lipton, R. Finite Element Convergence for State-Based Peridynamic Fracture Models. Commun. Appl. Math. Comput. 2, 93–128 (2020). https://doi.org/10.1007/s42967-019-00039-4

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  • Nonlocal fracture models
  • Peridynamic
  • State-based peridynamic
  • Numerical analysis
  • Finite element approximation

Mathematics Subject Classification

  • 34A34
  • 34B10
  • 74H55
  • 74S05