1 Introduction

The peridynamic continuum theory was introduced by Silling in [17] to allow for discontinuous deformation. The internal force density given by the divergence of the stress tensor was replaced by an integral operator and leads to a nonlocal model of force interaction. An important generalization of the peridynamic theory was given by Silling et al. in [19]. In particular, a sophisticated deformation operator—the deformation state about an equilibrium—was introduced generalizing the approach discussed in [17, §15] to allow for deformation dependent upon collective motion. This generalization, the “state-based” peridynamic theory, allowed the force interaction at a point to be dependent upon the deformation state, thus generalizing the “bond-based” theory introduced in [17, §1–§14] where the force interaction at a point was a sum over all pairs of points. The immediate significance is that for homogeneous deformations of a linear isotropic material, the Poisson ratio is no longer limited to be equal to one-fourth as it was for the bond-based theory. The state-based peridynamic theory for linear isotropic materials introduced in [19, §15] was also considered within the context of a sophisticated linearization procedure in [18] also applicable to nonlinear material models.

The goal of our paper is to establish the well-posedness of the volume-constrained peridynamic equation of motion for a linear anisotropic heterogeneous material by exploiting the nonlocal vector calculus developed in [9]. In particular, the well-posedness of the peridynamic analogue of the boundary value problem for the Navier equilibrium equation of linear elasticity on the space of square integrable functions is established. We introduce volume-constraints that represent the nonlocal analogue of the boundary conditions. The constraints are explicitly enforced on volumes of non-zero measure. Volume constraints are necessary because a trace operator, or informally, the restriction of a function to a lower-dimensional manifold on the space of square integrable functions is not defined. Said another way, because functions with jump discontinuities are square integrable, a meaningful notion of restricting a function to a lower-dimensional manifold is not possible within a Hilbert space setting.

We also demonstrate that the peridynamic analogue of the Navier operator for a linear isotropic homogeneous material converges to the classical Navier operator as the extent of the nonlocality vanishes. This is shown to be a simple application of the ideas in [9] demonstrating that the nonlocal vector operators converge to their classical analogues. An important conclusion is that the peridynamic Navier operator is a generalization of the classical operator over spaces containing discontinuous functions, i.e., discontinuous deformations are allowed.

Previous work has focused on the well-posedness of the bond-based model introduced in [17]; see [4, 11, 12, 15] when the deformation is vector-valued and [2, 3, 5, 6, 10, 14, 16] when the deformation is scalar-valued or the related case of nonlocal diffusion. Thus, the present study fills a significant gap in the analyses of peridynamic models and also demonstrates the efficacy of the nonlocal vector calculus framework introduced in [9]. Our analysis depends crucially upon a few key ingredients; for instance, that only rigid displacement fields are in the null space of the nonlocal peridynamic Navier operator, which is established under lower regularity assumptions than those associated with the classical Navier operator. We also establish a nonlocal Korn’s inequality that provides the coercivity of the nonlocal Navier operator on the space of square integrable functions.

The paper is organized as follows. The strain energy density for a linear anisotropic heterogeneous peridynamic material is introduced in Sect. 2. In Sect. 3, we recast the strain energy density in terms of the nonlocal calculus introduced in [9]. The variational problem is given in Sect. 4 and the well-posedness results are given in Sects. 5 and 6. Appendix A establishes the equivalence of the peridynamic energy space with the space of square integrable functions.

2 An Elastic Linear Peridynamic Material

The strain energy density function for an elastic constitutively linear anisotropic peridynamic heterogeneous solid is written as

$$ W(\underline{\mathbf{Y}}) = \frac{\kappa}{2} \vartheta^2 + \frac {\eta }{2} \int_{\mathbb{R}^3} \varpi(\mathbf{x},\boldsymbol{ \xi}) \biggl( \underline{e}\langle\boldsymbol{\xi} \rangle- \frac{\vartheta|\boldsymbol{\xi}|}{3} \biggr)^2 d \boldsymbol{\xi}, $$
(1)

where the kinematic quantities \(\underline{\mathbf{Y}}\) and \(\underline {e}\) are the deformation and extension states, respectively, ϑ is the nonlocal dilatation, ϖ is an influence function, κ=κ(x) is the bulk modulus function, and η=η(x) is a material function proportional to the shear modulus at x. The above energy density is a generalization of the density of [18, Example 3, p. 103] for an elastic constitutively linear isotropic peridynamic homogeneous solid which is recovered by supposing that κ,η are constants and that ϖ=ϖ(|ξ|).

The kinematic quantity \(\underline{\mathbf{Y}}\), the deformation state, is a mapping from bonds ξ=x′−x at x to the deformed images of these bonds, e.g.,

$$ \underline{\mathbf{Y}}[\mathbf{x}]\bigl\langle\mathbf{x}^\prime- \mathbf{x}\bigr\rangle= \underline {\mathbf{Y}}\bigl\langle\mathbf{x}^\prime- \mathbf{x}\bigr\rangle =\mathbf{y}\bigl(\mathbf{x}^\prime\bigr)- \mathbf{y}(\mathbf{x}), $$
(2)

where y(x) denotes the position of x after deformation. The extension state

$$\underline{e}\langle\boldsymbol{\xi}\rangle= \bigl|\underline{\mathbf{Y}}\langle\boldsymbol{\xi} \rangle\bigr|- |\boldsymbol{\xi}| $$

represents the change in the length of any bond ξ due to deformation. The nonlocal dilatation is given by

$$ \vartheta(\mathbf{x}) = \frac{3}{m(\mathbf{x})} \int_{\mathbb {R}^3} |\boldsymbol{\xi}| \varpi(\mathbf{x} ,\boldsymbol{\xi}) \underline{e}\langle\boldsymbol{\xi}\rangle \, d \boldsymbol{\xi}, $$
(3)

where

$$ m(\mathbf{x}) = \int_{\mathbb{R}^3} |\boldsymbol{\xi}|^2 \, \varpi (\mathbf{x},\boldsymbol{\xi}) \, d \boldsymbol{\xi}. $$
(4)

The influence function ϖ is typically of compact support, say over a sphere. The associated radius ε of the ball B ε (0), referred to as the peridynamic horizon, can then be used to define the neighborhood over which the collective deformation associated with a point x occurs.

When κ,η are constants and ϖ=ϖ(|ξ|), the strain energy density (1) is the peridynamic analogue of the classical isotropic energy density

$$ \mathbf{T} \colon\mathbf{E} = \kappa(\operatorname{Tr} \mathbf{E})^2 + 2 \mu \operatorname{dev} \mathbf{E} \colon\operatorname{dev} \mathbf{E}, $$
(5)

where

$$\operatorname{dev} \mathbf{E} = \mathbf{E} - \frac{1}{3} ( \operatorname{Tr} \mathbf {E} ) \mathbf{I}, \quad\mathbf{E} = \frac{1}{2} \bigl( \nabla \mathbf{u} + (\nabla\mathbf{u})^T \bigr), $$

and κ and μ are the bulk and shear moduli; “Tr” denotes the trace operator and u denotes the displacement, i.e., u(x)=y(x)−x. The material parameter η in (1) is proportional to the shear modulus via the relationship

$$ \eta= \frac{15 \mu}{m}, $$
(6)

where the constant m is given by (4); see [18, Example 3, p. 103].

In stark contrast to the classical energy density (5), the peridynamic density (1) depends only upon the collective deformation, i.e., differences in the deformation, and not its derivatives. Therefore W remains well-defined at points of discontinuity.

A small deformation, possibly discontinuous, is defined by Silling [18, p. 92] and leads to a linearized version of the peridynamic theory. This linearization procedure is applied to the Fréchet derivative of \(W(\underline{\mathbf{Y}})\) with respect to the deformation state \(\underline{\mathbf{Y}}\). The resulting derivative defines the force state and leads to the peridynamic Navier operator; see [18] for details.

2.1 Bond-Based Peridynamic Mechanics

The nonlocal dilatation (3) and (4) allows us to rewrite the energy density (1) as

$$ W(\underline{\mathbf{Y}}) = \frac{1}{2} \biggl( \kappa- \frac{\eta m}{9} \biggr)\vartheta^2 + \frac{\eta}{2} \int_{\mathbb{R}^3} \varpi (\mathbf{x},\boldsymbol{\xi}) \underline{e}^2\langle \boldsymbol{\xi}\rangle\,d \boldsymbol{\xi}. $$

Because bond-based peridynamic mechanics implicitly assumes a Poisson’s ratio ν=1/4, then κ−ηm/9=0 by (6). Therefore the energy density (1) leads to the bond-based density

$$ W_{\mathrm{bond}}(\underline{e}) = \frac{\eta}{2} \int_{\mathbb {R}^3} \varpi(\mathbf{x} ,\boldsymbol{\xi}) \underline{e}^2\langle \boldsymbol{\xi}\rangle\, d \boldsymbol{\xi}, $$

where the collective deformation given by \(\underline{\mathbf{Y}}\) is replaced by the extension state \(\underline{e}\). The bond-based density is a summation of independent pairs of interactions.

3 A Reformulation in Terms of the Nonlocal Vector Calculus

Because our goal is to establish the well-posedness of the peridynamic Navier equation, we linearize the dilatation ϑ and extension state \(\underline{e}\langle\boldsymbol{\xi}\rangle\). This leads to a second-order approximation \(\tilde{W}\) of the energy density W given by (1). The approximation \(\tilde{W}\) is conveniently phrased in terms of the nonlocal divergence and its adjoint introduced in [9]. An immediate benefit is that the peridynamic Navier operator converges to the classical Navier operator as the horizon vanishes; see Sect. 4.1. We first introduce the necessary vector calculus followed by the approximation \(\tilde{W}\). The nonlocal vector calculus developed in [9] generalized the nonlocal operators defined for scalar functions as discussed in, for instance, [13, 14], to those defined for vector and tensor fields and allows us to provide a systematic and concise formulation of the state-based peridynamic model.

Let α=α(x,y) denote an anti-symmetric mapping from ℝ3×ℝ3 to ℝ3, i.e., α(x,y)=−α(y,x). Given the tensor function Ψ:ℝ3×ℝ3→ℝ3×3 and the function v:ℝ3→ℝ3, the nonlocal divergence operator \(\mathcal{D}\boldsymbol{\varPsi}:\mathbb {R}^{3}\to\mathbb{R}^{3}\) for tensors and its adjoint operator \(\mathcal{D}^{\ast}\mathbf{v}:\mathbb{R}^{3}\times \mathbb{R}^{3}\to\mathbb{R}^{3\times3}\) are defined as

(7a)
(7b)

Let ω:Ω×Ω→ℝ denote a non-negative scalar function. Given the tensor function U:ℝ3→ℝ3×3 and vector functions u:ℝ3→ℝ3, the special choice Ψ=ω U results in the weighted nonlocal divergence operator \(\mathcal{D}_{\omega}\mathbf{U}:\mathbb {R}^{3}\to\mathbb{R}^{3}\) for tensors and its adjoint operator defined by

(8a)
(8b)

By direct calculation, the relations \((\mathcal{D}\boldsymbol {\varPsi},\mathbf{v} )_{\mathbb{R}^{3}} = (\boldsymbol{\varPsi},\mathcal{D}^{\ast}\mathbf{v} )_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\) and \((\mathcal{D}_{\omega}\mathbf{U}, \mathbf{u} )_{\mathbb{R}^{3}} = (\mathbf{U}, \mathcal{D}_{\omega}^{\ast}\mathbf{u} )_{\mathbb{R}^{3}}\) imply that the operators \(\mathcal{D}^{\ast}\) and \(\mathcal{D}_{\omega}^{\ast}\) are adjoint to \(\mathcal{D}\) and \(\mathcal{D}_{\omega}\), respectively, as claimed, where \((\cdot, \cdot )_{\mathbb{R}^{3}}\) denotes the standard L 2 inner product on ℝ3 and \((\cdot,\cdot )_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\) denotes the Frobenius inner product on ℝ3×ℝ3. The distinction between \(\mathcal{D}^{\ast}\mathbf{v}\) and \(\mathcal{D}_{\omega}^{\ast}\mathbf{u}\) is that the former is a tensor on ℝ3×ℝ3 whereas the latter is a tensor on ℝ3.

Let u denote the displacement of the position of x associated with an equilibrium state. A second-order approximation to the peridynamic strain energy density (1) is then given by

$$ \tilde{W}(\mathbf{u}) = \frac{\kappa}{2} \bigl( \operatorname{Tr}\bigl(\mathcal {D}_\omega^{*}\mathbf{u}\bigr) \bigr)^2 + \frac{\eta}{2}\int_{\mathbb{R}^3} \varpi( \mathbf{x},\mathbf {y}) \biggl(\operatorname{Tr}\bigl(\mathcal{D}^{*} \mathbf{u}\bigr) - \frac{\operatorname{Tr}(\mathcal{D}_\omega^{*}\mathbf {u})|\mathbf{y}-\mathbf{x}|}{3} \biggr)^2\,d\mathbf{y}. $$
(9)

The second-order approximation is immediate because the extension state \(\underline{e}\) is replaced by its linearization, with respect to u(y)−u(x), e.g.,

where

$$ \boldsymbol{\alpha}(\mathbf{x},\mathbf{y}) = \frac{\mathbf{x}-\mathbf{y}}{|\mathbf{y}-\mathbf{x}|}, \qquad \omega(\mathbf{x},\mathbf{y}) = \frac{|\mathbf{y}-\mathbf {x}|\varpi(\mathbf{x},\mathbf{y})}{\phi(\mathbf{x})},\qquad\phi (\mathbf{x}) = \frac{1}{3}m(\mathbf{x}), $$
(10)

and m is given by (4). In a similar fashion, the nonlocal dilatation ϑ is replaced by \(\operatorname{Tr}(\mathcal{D}_{\omega}^{*}\mathbf{u})\).

4 Variational Principle

Let Ω denote the solution domain of the model equation and \(\varOmega_{\mathcal{I}} \) denote the interaction domain over which volume-constraints are to be applied. The interaction domain contains points y not in Ω interacting with points x in Ω. When the support of ϖ is finite, then \(\varOmega_{\mathcal{I}}\) is of finite volume when Ω is also of finite volume. A variety of choices for the region \(\varOmega _{\mathcal{I}}\) having positive volume are possible; see Fig. 1 for illustrations of some of the possibilities. Most of our results can also be applied to cases where \({\varOmega\cup\varOmega_{\mathcal{I}}}\) is a union of the domains displayed. As explained in the introduction, the interaction domain provides a mechanism for demonstrating the well-posedness of the peridynamic Navier equation on the space of square integrable functions where a trace operator is not in general available. A mechanical justification is given by Silling [17, §13]. In particular, the depiction in the lower-lefthand in Fig. 1 suggests how the boundary of Ω can be approximated by the interaction domain.

Fig. 1
figure 1

Four of the possible configurations for Ω and \(\varOmega_{\mathcal{I}}\)

Full size image

We further assume that \(\varOmega_{\mathcal{I}}= \varOmega _{n} \cup\varOmega_{d}\), where Ω n ∩Ω d =∅. Here, Ω n and Ω d are volumes over which “Neumann” and “Dirichlet” volume constraints are to be applied.

The potential energy of a linear isotropic peridynamic material under an external force density b(x) and an external “traction” g(x) is then given by

$$ E(\mathbf{u};\mathbf{b},\mathbf{g}) = \int_{\varOmega\cup \varOmega_{\mathcal{I}}}\tilde{W}( \mathbf{u})\,d\mathbf{x}- \int_\varOmega\mathbf{u} \cdot\mathbf{b} \,d\mathbf{x} - \int_{\varOmega_n}\mathbf{u}\cdot\mathbf{g}\,d \mathbf{x}, $$

where the strain energy density \(\tilde{W}\) is given by (9).

The displacement u can be characterized as the solution of the constrained optimization problem

$$ \min_{\mathbf{u}\in U({\varOmega\cup\varOmega_{\mathcal{I}}})} E(\mathbf{u};\mathbf{b},\mathbf{g}) \quad \mbox{subject to}\quad \mathbf{u}= \mathbf{h}\quad\mbox{for}\ \mathbf{x}\in\varOmega_d, $$
(11)

where \(U({\varOmega\cup\varOmega_{\mathcal{I}}})\) denotes a function space defined on \({\varOmega\cup\varOmega_{\mathcal{I}}}\) such that the potential energy is finite. The space of test functions for the minimization problem is then given by

$$U_0({\varOmega\cup\varOmega_{\mathcal{I}}}) = \{\mathbf{v}\in U({\varOmega\cup\varOmega_{\mathcal{I}}})\colon\mathbf{v}= 0\mbox { on }\varOmega_d\}. $$

The following result provides the Euler–Lagrange equations for the optimization problem (11). The somewhat tedious proof is given in Appendix A.1.

Lemma 1

Let \(\mathbf{v}\in U_{0}({\varOmega\cup\varOmega_{\mathcal{I}}})\). Then

We define the operator

$$ -\mathcal{L}:= \mathcal{D} \bigl(\eta\varpi \bigl( \mathcal{D}^\ast \bigr)^T \bigr) + \mathcal{D}_{\omega} \bigl(\sigma\operatorname{Tr} \bigl(\mathcal{D}_\omega^\ast \bigr)\mathbf {I} \bigr), $$
(12)

where

(13a)
(13b)

and recall that the functions κ and η represent the bulk modulus and a material function proportional to the shear modulus, respectively, and ϕ is given by (10). Lemma 1 then implies that the constrained energy minimization problem (11) results in the equivalent nonlocal volume-constrained problem

$$ \left\{ \begin{array}{@{}l@{\quad}l} -\mathcal{L}\mathbf{u}=\mathbf{b} & \mbox{on}\ \varOmega,\\ \mathbf{u}=\mathbf{h} & \mbox{on}\ \varOmega_d,\\ -\mathcal{L}\mathbf{u}=\mathbf{g} & \mbox{on}\ \varOmega_n. \end{array} \right. $$
(14)

The optimization problem (11) and the equivalent volume constrained problem (14) can be specialized to two specific problems of interest. For brevity, we only consider these specializations in the rest of the paper, although all results can be easily extended to hold for the more general problems (11) and (14). When \(\varOmega_{\mathcal{I}}=\varOmega_{d}\), the “Dirichlet” problem is determined by solving

$$ \min_{\mathbf{u}\in U({\varOmega\cup\varOmega_{\mathcal{I}}})} E(\mathbf{u};\mathbf{b},{\bf0}) \quad \mbox{subject to}\quad \mathbf{u}= \mathbf{h}\quad\mbox{for}\ \mathbf {x}\in \varOmega_{\mathcal{I}}, $$
(15a)

whereas when \(\varOmega_{\mathcal{I}}=\varOmega_{n}\), the “Neumann” problem is given by

$$ \min_{\mathbf{u}\in\widehat{U}({\varOmega\cup\varOmega_{\mathcal {I}}})} E(\mathbf{u};\mathbf{b},\mathbf{g}) , $$
(15b)

where \(\widehat{U}({\varOmega\cup\varOmega_{\mathcal{I}}})\) denotes the quotient space of \(U({\varOmega\cup\varOmega_{\mathcal{I}}})\) with respect to the space of rigid displacements; see Sect. 5.1. The specialized volume-constrained problems corresponding to the minimization problems (15a) and (15b) are given, respectively, by the nonlocal “Dirichlet” volume-constrained problem

$$ \left\{ \begin{array}{@{}l} -\mathcal{L}\mathbf{u}=\mathbf{b}\quad\mbox{on}\ \varOmega,\\ \mathbf{u}=\mathbf{h}\quad\mbox{on}\ \varOmega_{\mathcal{I}} \end{array} \right. $$
(16a)

and the nonlocal “Neumann” volume-constrained problem

$$ \left\{ \begin{array}{@{}l@{\quad}l} -\mathcal{L}\mathbf{u}=\mathbf{b}\quad\mbox{on}\ \varOmega,\\ -\mathcal{L}\mathbf{u}=\mathbf{g}\quad\mbox{on}\ \varOmega _{\mathcal{I}}. \end{array} \right. $$
(16b)

Relabeling g as b over the interaction domain \(\varOmega_{\mathcal{I}}\), the problem (16b) has the equivalent form

$$ -\mathcal{L}\mathbf{u}=\mathbf{b}\quad\mathrm{on}\ \varOmega \cup\varOmega_{\mathcal{I}}. $$
(17)

The operator in the problem (17) (or equivalently, in problem (16b)) has a non-trivial null space. Thus, to ensure the existence of solutions of the problem (16b), compatibility conditions must be imposed on the data functions b and g; these are given in (24). To ensure the uniqueness of solutions, additional constraints should be imposed in the displacement u, such as those in (22). We postpone presenting these until after we have discussed the nature of that null space; see Lemma 2.

The problems (16a) and (16b) are, respectively, the nonlocal analogs of the pure displacement and pure traction boundary-value problems for the Navier equations of linear elasticity. As such, we could refer to (16a) and (16b), respectively, as the (nonlocal) pure displacement and pure traction volume-constrained problems for the peridynamic Navier equations of linear elasticity.

4.1 Classical and Peridynamic Navier Equations

For an isotropic homogeneous linear elastic material, κ and η are constants and ϖ=ϖ(|y−x|) is a radial function so that σ and τ of (13a), (13b) are also constants. In the case of Ω≡ℝ3, the following result demonstrates that the peridynamic Navier equation converges to the classical Navier equation as the horizon, the radius of the support of ϖ, vanishes. The result depends upon the properties established in [9] demonstrating how the nonlocal vector operators converge to their classical analogues.

Theorem 1

Let \(\mathcal{L}\) be given by (12), let u∈[H 1(ℝ3)]3, and let η, κ, and μ be positive constants (where the latter two represent the bulk and shear moduli). If

$$\frac{\eta}{15}\int_{B_\varepsilon(0)} |\mathbf{x}|^2\varpi\bigl(| \mathbf{x}|\bigr)\,d\mathbf{x}\rightarrow\mu\quad \mathit{as}\ \varepsilon\to0, $$

where ϖ(|y−x|) is positive on B ε (0), the sphere of radius ε centered at the origin, then

$$\mathcal{L}\mathbf{u}\to\mu\nabla\cdot\nabla\mathbf{u}+ (\mu+ \lambda)\nabla \nabla \cdot\mathbf{u}\quad\mathit{in}\ H^{-1}\bigl(\mathbb{R}^3 \bigr)\ \mathit{as}\ \varepsilon\to0 $$

with Lamé first parameter and Poisson’s ratio, respectively, given by

$$\lambda=\sigma+ \mu\quad\mathit{and}\quad \nu= \frac{\lambda}{2(\lambda+\mu)}, $$

where the constant σ is given by (13a).

Proof

The hypothesis on the influence function ϖ and (4) imply that m is a constant, and therefore, by (10) and (13a), both ϕ and σ, respectively, are also constants. Let ω be given by (10). Then, [9, Corollary 5.4] implies that \(\mathcal{D}_{\omega}( \operatorname{Tr}(\mathcal{D} _{\omega}^{\ast}\mathbf{u})\mathbf{I})\rightarrow-\nabla (\nabla\cdot\mathbf{u})\), and an argument similar to the proof of [11, Theorem 2.19] grants that \(\mathcal{D}(\eta\varpi(\mathcal{D}^{\ast}\mathbf{u})^{T})\rightarrow -\mu\nabla\cdot(\nabla\mathbf{u})-2\mu\nabla(\nabla\cdot \mathbf{u})\) in H −1(ℝ3). The conclusion of the theorem now follows by (6) and the well-known relationship 3κ=2μ+3λ. □

The theorem explains that the peridynamic Navier operator is a generalization of the classical operator over spaces containing discontinuous functions.

5 Preliminaries

We first make the following assumptions about the domain \({\varOmega \cup\varOmega_{\mathcal{I}}}\), the material functions κ and η, and the influence function ϖ.

Assumption 1

\({\varOmega\cup\varOmega_{\mathcal{I}}}\) is an open and connected domain in ℝ3.

Assumption 2

\({\varOmega\cup\varOmega_{\mathcal{I}}}\) is bounded and satisfies the interior cone condition with parameters r 0>0 and θ 0>0, as defined by the property that (see [1] for details) for any point \(\mathbf{x}\in{\varOmega\cup\varOmega_{\mathcal {I}}}\), the intersection between the ball centered at x with radius r 0 and the domain \({\varOmega \cup\varOmega_{\mathcal{I}}}\) contains a cone with an angle no smaller than θ 0.

Assumption 3

The material functions κ and η satisfy

$$ \kappa_0 \leq\kappa(\mathbf{x}) \leq\kappa_1 \quad\mathit{and} \quad \eta_0 \leq\eta(\mathbf{x}) \leq\eta_1 $$

for some positive constants κ 0, κ 1, η 0, and η 1.

Assumption 4

The influence function ϖ is nonnegative and non-degenerate on \({\varOmega\cup\varOmega_{\mathcal{I}}}\times{\varOmega\cup \varOmega_{\mathcal{I}}}\). A non-degenerate influence function is defined by the existence of two positive constants ϖ 0 and ϖ 1 such that ϖ(x,y)≥ϖ 1 for all \(\mathbf{x},\mathbf{y}\in{\varOmega\cup\varOmega_{\mathcal{I}}}\) satisfying ∥x−y∥≤ϖ 0. The influence function ϖ is symmetric, i.e., ϖ(x,y)=ϖ(y,x), and, for some constant M>0, we have \(\int_{\varOmega\cup \varOmega_{\mathcal{I}}}\varpi^{2}(\mathbf{x},\mathbf{y})\, d\mathbf{y} \leq M\) for all \(\mathbf{x}\in{\varOmega\cup\varOmega_{\mathcal{I}}}\).

The non-degeneracy condition in Assumption 4 is automatically satisfied if the function under consideration is bounded below uniformly by a positive constant over the domain \({\varOmega\cup\varOmega_{\mathcal{I}}} \). When the influence function is radial, e.g., ϖ=ϖ(|y−x|), ϖ is non-degenerate if the support of ϖ contains a small ball centered at the origin which is typically the case for a peridynamic model; for instance, see [17–19]. The constant ϖ 0 is strictly smaller than the peridynamic horizon.

5.1 Bilinear Form

Define the bilinear form

(18)

Setting v=u, we obtain

$$B(\mathbf{u}, \mathbf{u}) = 2 E(\mathbf{u};\mathbf{0},\mathbf{0}) $$

from which it follows that B(u,u)≥0 for any \(\mathbf{u}\in U({\varOmega\cup\varOmega_{\mathcal{I}}} )\). Hence, we define a formal peridynamic inner product and its associated formal peridynamic norm by

respectively. The peridynamic energy space \(U({\varOmega\cup \varOmega_{\mathcal{I}}})\) can therefore be defined as

We also define the space

In order to demonstrate that ((⋅,⋅)) and ⦀⋅⦀ define an inner product and norm, respectively, on some subspace of \(U({\varOmega\cup\varOmega_{\mathcal{I}}})\), we need the following technical result.

Lemma 2

Assume the domain \({\varOmega\cup\varOmega_{\mathcal{I}}}\) satisfies Assumption 1 and \(\mathbf{u}\in L^{2}({\varOmega\cup\varOmega_{\mathcal{I}}})\). If for almost all \(\mathbf{x}\in{\varOmega\cup\varOmega_{\mathcal{I}}}\),

$$ \bigl(\mathbf{u}(\mathbf{y})-\mathbf{u}(\mathbf{x}) \bigr)\cdot ( \mathbf{y}-\mathbf{x}) = 0\quad\mathit{for\ a.e.}\ \mathbf{y}\in({\varOmega\cup \varOmega_{\mathcal{I}}}) \cap B_{\delta_0}(\mathbf{x}) $$
(19)

for a given δ 0>0, then there exists a constant-valued skew-symmetric matrix A=−A T and a constant-valued vector \(\bold c\) such that

$$ \mathbf{u}(\mathbf{x}) = \mathbf{A}\mathbf{x}+ \mathbf{c}\quad \mathit{a.e.}\ \mathbf{x}\in{\varOmega\cup\varOmega_{\mathcal{I}}}. $$
(20)

Proof

For any \(\mathbf{x}_{0}\in{\varOmega\cup\varOmega_{\mathcal{I}}}\), we may choose δ(x 0)∈(0,δ 0/2) such that

$$\bigcup_{\mathbf{y}\in B_{\delta(\mathbf{x}_0)}(\mathbf{x}_0)} B_{\delta(\mathbf{x}_0)}(\mathbf{y}) \subset ({\varOmega\cup \varOmega_{\mathcal{I}}})\cap B_{\delta_0}(\mathbf{x}_0). $$

Let \(\{\tilde{\mathbf{e}}_{i}\}_{i=1}^{3}\) denote an orthonormal basis in ℝ3. By the assumption, the relation (19) holds for \(\mathbf{y}\in({\varOmega\cup\varOmega_{\mathcal {I}}})\cap B_{\delta_{0}}(\mathbf{x}_{0})\) for almost all \(\mathbf{x} _{0}\in{\varOmega\cup\varOmega_{\mathcal{I}}}\). Let \(B_{\delta (\mathbf{x}_{0})/N}(\mathbf{x}_{0}+\delta(\mathbf{x}_{0})\tilde {\mathbf{e}}_{i})\) denote three small balls centered at \(\mathbf {x}_{0}+\delta(\mathbf{x} _{0})\tilde{\mathbf{e}}_{i}\) with radii δ(x 0)/N, where the positive constant N is chosen such that the balls are small enough and do not intersect. We may also choose points \(\{\mathbf{x}_{0i}\}_{i=1}^{3}\) such that, \(\mathbf{x}_{0i}\in B_{\delta(\mathbf{x}_{0})/N}(\mathbf {x}_{0}+\delta(\mathbf{x}_{0})\tilde {\mathbf{e}}_{i})\) for 1≤i≤3. By the definition of \(B_{\delta(\mathbf{x}_{0})/N}(\mathbf{x}_{0}+\delta (\mathbf{x}_{0})\tilde {\mathbf{e}}_{i})\), the set \(\{\mathbf{x}_{0i}-\mathbf{x}_{0}\}_{i=1}^{3}\) form a basis. For notational simplicity, we set e i =(x 0i −x 0)/∥x 0i −x 0∥.

Then, we have for x 0 and some set \(\mathfrak{N}_{\mathbf {x}_{0}}\) of measure zero,

$$\bigl(\mathbf{u}(\mathbf{y})-\mathbf{u}(\mathbf{x}_0) \bigr)\cdot ( \mathbf{y}-\mathbf{x}_0) = 0\quad\forall \mathbf{y}\in ({\varOmega \cup\varOmega_{\mathcal{I}}})\cap B_{\delta_0(\mathbf {x}_0)}(\mathbf{x}_0) \setminus\mathfrak{N}_{\mathbf{x}_0}. $$

Thus, for any \(\mathbf{x}\in B_{\delta(\mathbf{x}_{0})}(\mathbf {x}_{0})\cap(({\varOmega\cup\varOmega_{\mathcal{I}}})\cap B_{\delta_{0}(\mathbf{x}_{0})}(\mathbf{x}_{0}) \setminus\mathfrak {N}_{\mathbf{x}_{0}})\), we have

$$\bigl(\mathbf{u}(\mathbf{x}) - \mathbf{u}(\mathbf{x}_0) \bigr)\cdot (\mathbf{x}-\mathbf{x}_0) = 0. $$

Moreover,

$$\bigl(\mathbf{u}(\mathbf{x}) - \mathbf{u}\bigl(\mathbf{x}_0 +\delta (\mathbf{x}_0) \mathbf{e}_i\bigr) \bigr)\cdot \bigl( \mathbf{x}-\mathbf{x}_0- \delta(\mathbf{x}_0) \mathbf{e}_i \bigr) = 0. $$

This implies that u(x)⋅e i is linear in x for any i, which gives the linearity of u in \(B_{\delta _{0}}(\mathbf{x}_{0})\). Then, (19) implies that u has to be of the form (20) in \(B_{\delta_{0}}(\mathbf{x}_{0})\).

Because \({\varOmega\cup\varOmega_{\mathcal{I}}}\) is a bounded connected open set, for any two points x 0 and x 1 in \({\varOmega\cup\varOmega _{\mathcal{I}}}\), there exists a finite number of balls \(\{B_{\delta(\mathbf{y}_{k}) }(\mathbf{y}_{k})\}_{k=1}^{K}\) with sufficiently small radii δ(y k )>0 with the properties that their union is completely contained in \({\varOmega\cup\varOmega_{\mathcal{I}}}\), that covers a connected path between x 0 and x 1, and \(B_{\delta(\mathbf{y}_{k}) }(\mathbf {y}_{k})\cap B_{\delta(\mathbf{y}_{k+1}) }(\mathbf{y}_{k+1})\) has a non-empty interior. Therefore u is given by (20) in each ball that must be the same in neighboring balls so that u is a rigid displacement over the domain \({\varOmega\cup\varOmega_{\mathcal{I}}}\). □

The proof of the above lemma uses an argument that is substantially different from that given in, for example [21, Prop. 1.2], for continuous vector fields satisfying (19) everywhere. We also note that similar results can be established for more general domains, in particular domains with multiple connected components that possibly allow different pairs {A,c} to be associated with the different connected components.

Lemma 2 implies that the null spaces for the adjoints of the nonlocal point divergence and weighted operators, respectively, both defined in Sect. 3, are of the form (20), i.e., they consists of only rigid displacement fields. In addition, Lemma 2 also implies that the peridynamic Navier operator \(\mathcal{L}\) has the same null space as that of the classical Navier operator.

Corollary 1

Under Assumptions 2–4, if \(-\mathcal {L}\mathbf{u} =\mathbf{0} \) on \({\varOmega\cup\varOmega_{\mathcal{I}}}\), then there exists a constant-valued skew-symmetric matrix A and a constant-valued vector \(\bold c\) such that (20) is satisfied. Moreover, \(Z({\varOmega\cup\varOmega_{\mathcal{I}}})\) is the space of rigid displacements.

Proof

By the calculation given in Sect. 4, we can see that

$$0=-\int_{{\varOmega\cup\varOmega_{\mathcal{I}}}} \mathbf{u} \cdot\mathcal{L}\mathbf{u}\, d \mathbf{x}= B(\mathbf{u},\mathbf {u}) . $$

Thus, (7b), (10)1 and Assumption 4 imply the existence of a constant δ 0<ϖ 0 such that (19) is satisfied. Lemma 2 then shows that (20) holds. □

We then have the following result.

Proposition 1

Under Assumptions 2–4, ⦀u⦀ and ((u,v)) define a norm and an inner product, respectively, on both \(U_{0}({\varOmega\cup \varOmega_{\mathcal{I}}})\), provided \(\varOmega_{\mathcal{I}}\) has a non-empty interior, and on \(U({\varOmega\cup\varOmega_{\mathcal{I}}})/Z({\varOmega\cup \varOmega_{\mathcal{I}}})\).

Proof

We note that ⦀u⦀ defines a semi-norm on \(U_{0}({\varOmega \cup\varOmega_{\mathcal{I}}})\). Thus, it suffices to prove that B(u,u)=0 implies u=0. Assumption 3 implies that

$$ \mathcal{D} \bigl(\eta\varpi \bigl(\mathcal{D}^\ast \mathbf{u} \bigr)^T \bigr) (\mathbf{x}) = 0= \mathcal{D}_{\omega} \bigl(\sigma\operatorname{Tr} \bigl(\mathcal{D}_\omega^\ast \mathbf {u} \bigr)\mathbf{I} \bigr) (\mathbf{x}). $$
(21)

The first equality implies that \(\operatorname{Tr}(\mathcal{D}_{\omega}^{\ast}\mathbf{u})= 0\) and thus, the second equality grants that \(\mathrm{Tr}(\mathcal{D}^{\ast}\mathbf{u})= (\mathbf{u}(\mathbf{y})-\mathbf{u}(\mathbf{x}))\cdot(\mathbf {y}-\mathbf{x} )=0\) on the support of ϖ. Given the non-degeneracy condition given on ϖ, from Assumption 1 and Lemma 2 we can deduce that the kernel of the operator \(\operatorname{Tr}(\mathcal{D} ^{\ast})\) is the set of rigid body motions u(x)=Ax+c for a constant vector c and constant skew-symmetric matrix A. So, in either \(U_{0}({\varOmega\cup\varOmega_{\mathcal {I}}})\) with \(\varOmega_{\mathcal{I}}\) having a non-empty interior or in \(U(\varOmega)/Z({\varOmega\cup\varOmega_{\mathcal {I}}})\), the only u satisfying (21) is \(\mathbf{u}\equiv\bold0\). Thus we conclude that ⦀⋅⦀ defines a norm and ((⋅,⋅)) defines an inner product on \(U_{0}({\varOmega\cup\varOmega_{\mathcal{I}}})\) and \(U(\varOmega )/Z({\varOmega\cup\varOmega_{\mathcal{I}}})\). □

Then, for all \(\mathbf{u}\in Z({\varOmega\cup\varOmega_{\mathcal {I}}})\) and \(\mathbf{v}\in U_{0}({\varOmega\cup\varOmega_{\mathcal {I}}})\) we have B(u,v)=0. Thus, we conclude that:

Corollary 2

$$ U({\varOmega\cup\varOmega_{\mathcal{I}}}) = U_0({\varOmega\cup \varOmega_{\mathcal{I}}}) \oplus Z({\varOmega\cup\varOmega_{\mathcal{I}}}), $$

and that the quotient space \(\widehat{U}({\varOmega\cup\varOmega _{\mathcal{I}}}) = U({\varOmega\cup\varOmega_{\mathcal{I}}}) / Z({\varOmega\cup\varOmega_{\mathcal{I}}})\).

In words, any function in \(U({\varOmega\cup\varOmega_{\mathcal {I}}})\) can be written as a direct sum of two functions that are orthogonal with respect to the inner product ((⋅,⋅)), the first a function that vanishes on \(\varOmega_{\mathcal{I}}\) whereas the second is a rigid displacement. An element of the quotient space can be determined by imposing the conditions

$$ \int_{{\varOmega\cup\varOmega_{\mathcal{I}}}} \mathbf{u}(\mathbf {x})\,d\mathbf{x}= \mathbf{0} \quad\mbox{and}\quad\int_{{\varOmega\cup\varOmega_{\mathcal{I}}}}\mathbf{x}\times\mathbf {u}(\mathbf{x}) \,d\mathbf{x}= \mathbf{0}. $$
(22)

5.2 Nonlocal Dual Spaces and Nonlocal Trace Spaces

If \(\varOmega_{\mathcal{I}}\) is not empty, we can define the dual space of \(U_{0}({\varOmega\cup\varOmega_{\mathcal{I}}})\) by

where, for a function b:Ω→ℝ3, the dual norm is defined by

Similarly, whether or not \(\varOmega_{\mathcal{I}}\) is empty, we can define the dual space of \(U({\varOmega\cup\varOmega_{\mathcal{I}}})\) by

where, for functions b:Ω→ℝ3 and \(\mathbf{g}\colon {\varOmega_{\mathcal{I}}}\rightarrow\mathbb{R}^{3}\), the dual norm ⦀{b,g}⦀∗ is defined by

Let \(\mathbf{h}\colon{\varOmega_{\mathcal{I}}}\rightarrow\mathbb {R}^{3}\) denote a mapping; then define the space

where, for functions \(\mathbf{h}\colon{\varOmega_{\mathcal {I}}}\rightarrow\mathbb{R}^{3}\), the norm is defined by

The space \(U_{\mathcal{I}}\) and norm are the nonlocal analogs of a trace space and a trace norm, respectively.

The nonlocal dual and trace spaces and their associated norms may also be defined in terms of subsets of \(\varOmega_{\mathcal{I}}\). For example, the nonlocal trace norm can be defined in terms of \(\varOmega_{d}\subseteq\varOmega_{\mathcal{I}}\) and the dual norm ⦀{b,g}⦀∗ can be defined in terms of \(\varOmega_{n}\subseteq\varOmega_{\mathcal{I}}\).

5.3 The Variational Problems

Recall that for simplicity, we consider only the cases \(\varOmega _{d}=\varOmega_{\mathcal{I}}\) which leads to Dirichlet volume constrained problems and \(\varOmega _{n}=\varOmega_{\mathcal{I}} \) which leads to Neumann volume constrained problems. In the first case, we obtain the “Dirichlet” volume-constrained variational problem

$$ \left\{ \begin{array}{@{}l} \mathrm{given}\ \mathbf{b}\in{U}_0^\ast({\varOmega\cup\varOmega _{\mathcal{I}}})\ \mathrm{and}\ \mathbf{h}\in U_{\mathcal {I}},\ \mathrm{seek}\ \mathbf{u}\in U({\varOmega\cup\varOmega_{\mathcal {I}}})\\[1mm] \mathrm{such\ that}\ \mathbf{u}=\mathbf{h}\ \mathrm{on}\ \varOmega _{\mathcal{I}}\ \mathrm{and}\ B(\mathbf{u},\mathbf{v}) = F_d(\mathbf{v})\quad\forall \mathbf {v}\in U_0({\varOmega\cup\varOmega_{\mathcal{I}}}), \end{array} \right. $$
(23)

where the linear functional F d is defined by

$$F_{d}(\mathbf{v}) = \int_{\varOmega} \mathbf{b}\cdot \mathbf{v}\, d\mathbf{x} \quad\forall\mathbf{v}\in U_0({\varOmega\cup \varOmega_{\mathcal{I}}}). $$

In the second case, we obtain the “Neumann” volume-constrained variational problem

$$ \left\{ \begin{array}{@{}l} \mathrm{given}\ \{\mathbf{b},\mathbf{g}\}\in{U^\ast({\varOmega \cup\varOmega_{\mathcal{I}}})}\ \mathrm{satisfying\ the\ compatibility\ conditions}\\[2mm] \quad\int_\varOmega\mathbf{b}\,d\mathbf{x}+ \int_{\varOmega _{\mathcal{I}}} \mathbf{g}\,d\mathbf{x}= \mathbf{0}\quad\mathrm{and}\quad \int_\varOmega\mathbf{x}\times\mathbf{b}\,d\mathbf{x}+ \int _{\varOmega_{\mathcal{I}}} \mathbf{x}\times\mathbf{g}\,d\mathbf {x}= \mathbf{0},\\[2mm] \mathrm{seek}\ \mathbf{u}\in U({\varOmega\cup\varOmega_{\mathcal {I}}})/Z({\varOmega\cup\varOmega_{\mathcal{I}}})\ \mathrm{such\ that}\\[2mm] \quad B(\mathbf{u},\mathbf{v}) = F_n(\mathbf{v})\quad\forall \mathbf {v}\in U({\varOmega\cup\varOmega_{\mathcal{I}}})/Z({\varOmega\cup \varOmega_{\mathcal{I}}}), \end{array} \right. $$
(24)

where the linear functional F n is defined by

$$F_n(\mathbf{v})= \int_{\varOmega} \mathbf{b}\cdot \mathbf{v}\, d\mathbf{x}+ \int_{\varOmega_{\mathcal{I}}} \mathbf{g}\cdot \mathbf{v} \,d\mathbf{x} \quad\forall \mathbf{v}\in U({\varOmega\cup\varOmega_{\mathcal {I}}})/Z({ \varOmega\cup\varOmega_{\mathcal{I}}}). $$

The quotient space \(U({\varOmega\cup\varOmega_{\mathcal {I}}})/Z({\varOmega\cup\varOmega_{\mathcal{I}}})\) is used to guarantee the uniqueness of the solution. An element of the quotient space can be selected by imposing the constraints in (22). The compatibility conditions in (24) guarantee the existence of solutions and follow from the Fredholm alternative [7, 20] and the fact that B(u,v)=0 for all rigid motion test functions v.

Because the bilinear form B(⋅,⋅) induces an inner product on both \(U_{0}({\varOmega\cup\varOmega_{\mathcal{I}}})\) and \(U({\varOmega\cup\varOmega_{\mathcal{I}}})/Z({\varOmega\cup \varOmega_{\mathcal{I}}})\), it is continuous and coercive on these spaces. If we assume that \(U({\varOmega\cup \varOmega_{\mathcal{I}}})\) is a complete space and the data are such that the functionals F d (⋅) and F n (⋅) are continuous, the Lax-Milgram theorem can be applied to show that both (23) and (24) have unique solutions and, moreover, those solutions respectively satisfy, for some positive constant C,

6 Well-posedness of the Peridynamic Navier Equation

A technique for establishing that \(U({\varOmega\cup\varOmega _{\mathcal{I}}})\) is a Hilbert space is to, under appropriate conditions on the influence function ϖ, demonstrate that \(U({\varOmega\cup\varOmega_{\mathcal{I}}})\) is equivalent to a well-known Sobolev space. In the classical linear elastic theory, a crucial result is Korn’s inequality whose proof generally relies on a compactness argument. In [22], the equivalence for \(U({\varOmega\cup \varOmega_{\mathcal{I}}})\) is established for special volume constraints associated with a linear bond-based peridynamic model that allows for a precise characterization of the energy spaces in terms of a set of explicitly constructed basis functions. In Appendix A.2, we demonstrate the equivalence of \(U({\varOmega \cup\varOmega_{\mathcal{I}}})\) and \(L^{2}({\varOmega\cup\varOmega_{\mathcal{I}}})\) via a nonlocal Korn inequality combined with compactness results associated with Hilbert–Schmidt operators. For the sake of brevity, only the case corresponding to homogeneous Dirichlet volume-constrained problems is considered, but similar conclusions can be made for the inhomogeneous Dirichlet and Neumann constrained-value problems and even for nonlocal problems of the type (14) having mixed constraints.

6.1 Equilibrium Equation

We now establish the following well-posedness results on the space of square integrable functions for the peridynamic Navier equation subject to Dirichlet or Neumann volume constraints.

Theorem 2

The Dirichlet volume-constrained peridynamic Navier problem (23) is well-posed in \(U_{0}({\varOmega\cup\varOmega_{\mathcal {I}}})\) provided that Assumptions 1–4 are satisfied. Moreover, in this case, the peridynamic energy space is equivalent to \(L^{2}({\varOmega\cup \varOmega_{\mathcal{I}}})\).

Theorem 3

The Neumann volume-constrained peridynamic Navier problem (24) is well-posed in \(U({\varOmega\cup\varOmega_{\mathcal {I}}})/Z({\varOmega\cup\varOmega_{\mathcal{I}}})\) provided that Assumptions 1–4 are satisfied. Moreover, in this case, the peridynamic energy space is equivalent to \(L^{2}({\varOmega\cup\varOmega_{\mathcal{I}}})\).

These theorems represent the first available well-posedness results for the volume-constrained peridynamic Navier equation. The proofs of these theorems are immediate from the discussions at the end of Sect. 5.3 and the results of Appendix A.2, where we demonstrate that the energy space is indeed a Hilbert space and further, given the assumptions on the influence function ϖ, the energy space is \(L^{2}({\varOmega \cup\varOmega_{\mathcal{I}}})\).

In many practical applications of peridynamic models, ϖ=ϖ(x,y) has compact support on a sphere centered at x with radius equal to the peridynamic horizon, which is a special case of Theorems 2 and 3. In this case, the integrability condition of Assumption 4 becomes \(\int_{B_{\delta}(\mathbf {x})}\varpi ^{2}(\mathbf{x},\mathbf{y})\,d\mathbf{y}< M\). This is only an example of a sufficient condition on the influence function assuring well-posedness. Also possible is the weaker condition requiring only the L 1 integrability of ϖ for well-posedness of solutions in \(L^{2}({\varOmega\cup\varOmega _{\mathcal{I}}})\), as shown in [22] for a system of bond-based models with special constraints. The stronger condition used in Theorems 2 and 3 enables the immediate application of standard Hilbert-Schmidt theory. In addition, as studied in [16, 22] for a bond-based peridynamic systems and for scalar nonlocal equations in [10, 16], more general conditions on ϖ(x,y) may be defined so that space equivalence between \(U({\varOmega\cup\varOmega _{\mathcal{I}}})\) and a fractional Sobolev space may be derived.

6.2 Equation of Motion

Given results on the variational problems discussed above associated with the peridynamic Navier operator, we may readily obtain well-posedness of associated time dependent problems as well. For example, we consider the nonlocal time dependent peridynamic Navier equation with homogeneous Dirichlet type constrained values and initial conditions:

$$ \left\{ \begin{array}{@{}l@{\quad}l} \mathbf{u}_{tt}(t,x ) - \mathcal{L}\mathbf{u}(t,x) = \mathbf {b}&\forall (t, x)\in(0, T)\times\varOmega,\\ \mathbf{u}(0, x)=\mathbf{u}_{0},\mathbf{u}_{t}(0, x) = \mathbf{v}_{0} & \forall x\in\varOmega,\\ \mathbf{u}(t,x)=\mathbf{0} &\forall (t, x)\in(0, T)\times \varOmega_{\mathcal{I}}. \end{array} \right. $$
(25)

Let us define some relevant function spaces. Given a Banach space X, in particular \(X=U_{0}({\varOmega\cup \varOmega_{\mathcal{I}}})=\{ \mathbf{u}\in L^{2}({\varOmega\cup\varOmega_{\mathcal{I}}}),\ \mathbf{u}=0\ \mathrm{in}\ \varOmega_{\mathcal{I}}\}\), the space of functions L 2((0,T);X) is defined as

Similar definitions apply for the spaces C((0,T);X), L ∞((0,T);X) and H s((0,T);X) for s≥1. Then, for u 0∈X, v 0∈L 2(Ω), and b∈L 2((0,T)×Ω), a weak formulation of (25) is given by: find u∈H 1((0,T);X)∩H 2((0,T);L 2(Ω)) such that u(0,⋅)=u 0 and u t (0,⋅)=v 0 in Ω and

$$\frac{d^2}{dt^2}(\mathbf{u}, \mathbf{v})_{\varOmega} + B(\mathbf {u}, \mathbf{v}) =(\mathbf{b},\mathbf{v})_{\varOmega}, \quad\forall \mathbf{v}\in X=U_0({\varOmega\cup\varOmega_{\mathcal{I}}}) $$

with (⋅,⋅) Ω being the standard inner product in L 2(Ω).

Under Assumptions 2–4, by Proposition 1, B(u,v) defines an inner product on X, which in this case is equivalent to the conventional inner product on L 2(Ω). Using standard variational techniques (see for example [8]), the following result ensues.

Theorem 4

Suppose b∈L 2((0,T)×Ω), u 0∈X, and v 0∈L 2(Ω). Then (25) has a unique solution u such that u∈L 2((0,T);X) and u t ∈L 2((0,T)×Ω) and such that there exists a positive constant C=C(T) such that

Results similar to those in Theorem 4 have been established for the peridynamic bond-based model; see for instance, [4, 11, 12]. Similar results concerning the Neumann-type constrained-value problems can also be established.

7 Concluding Remarks

We applied the nonlocal vector calculus of [9] to analyze the peridynamic Navier equation. Well-posedness of the associated variational and time-dependent problems are established for some nonlocal problems with volume-constraints. Whereas detail proofs are presented mostly for homogeneous Dirichlet-type volume constraints, similar to the discussion about scalar nonlocal diffusion models considered in [9, 10], the analysis can be extended to nonlocal Neumann problems and to nonlocal mixed Dirichlet-Neumann problems.