Peridynamics for fluid mechanics and acoustics

Abstract

Peridynamic governing equations of fluid mechanics for barotropic flow are developed. As a special case of such a flow, linearized acoustics is also considered. The peridynamic governing equation of acoustics is also developed and discussed in detail. In order to obtain these new peridynamic governing equations, integral operators called “peridynamic D operators” are developed systematically, which are obtained by directly requiring the peridynamic D operators to converge to corresponding classical differential operators as the generalized material horizon approaches 0. Even though peridynamic D operators are applied only to fluid mechanics, acoustics, and heat conduction in an anisotropic inhomogeneous material in the present paper, it is clear that these peridynamic D operators can be used in any other field of mathematical physics to obtain a peridynamic (nonlocal) version of the governing equations. As an application of the newly obtained peridynamic governing equations, a time-dependent 3D (three-dimensional) peridynamic acoustics equation with a sound source is analytically solved for two different initial pressure disturbance profiles, and the results are discussed. These are believed to be the first exact analytical solutions for peridynamic acoustics.

Appendices

Appendix 1: Convergence proof of peridynamic barotropic fluid mechanics to classical barotropic fluid mechanics

Let us first note that the Navier–Stokes equation (1) can be written as

$$ \rho \frac{{\partial {\varvec{v}}}}{\partial t} = - \nabla p^{\prime} + \mu \nabla^{2} \user2{v + }(\mu + \lambda )\nabla (\nabla \bullet {\varvec{v}}) - \rho ({\varvec{v}} \bullet \nabla ){\varvec{v}} + \rho {\varvec{b}}, $$

(70)

We would like to prove that the peridynamic Navier–Stokes equation (7) and the peridynamic continuity equation (8) approach the classical Navier–Stokes equation (70) and the classical continuity equation (2), respectively, in the limit of the material horizon $\delta$ going to zero ($\delta \to 0$). First, we start with (7) and (70). Since the left-hand side (L.H.S) of both (7) and (70) is the same, we only need to show that the right-hand side (R.H.S) of (7) approaches the R.H.S of (70) in the limit of the material horizon $\delta$ going to zero ($\delta \to 0$). In other words, we need to show that

$$ \int_{ - \infty }^{\infty } {\tilde{\user2{C}}({\varvec{\xi}})({\varvec{v}}({\varvec{x}} + {\varvec{\xi}},t) - {\varvec{v}}({\varvec{x}},t))d{\varvec{\xi}}} \to \mu \nabla^{2} \user2{v + }(\mu + \lambda )\nabla (\nabla \bullet {\varvec{v}}), $$

(71)

$$ {\begin{aligned} & \int_{ - \infty }^{\infty } {{\varvec{C}}_{G} ({\varvec{\xi}})(p^{\prime}({\varvec{x}} + {\varvec{\xi}},t) - p^{\prime}({\varvec{x}},t))d{\varvec{\xi}}} \to \nabla p^{\prime}({\varvec{x}},t), \\ & \int_{ - \infty }^{\infty } {C_{CV} ({\varvec{\xi}},{\varvec{v}}({\varvec{x}},t))({\varvec{v}}({\varvec{x}} + {\varvec{\xi}},t) - {\varvec{v}}({\varvec{x}},t))d{\varvec{\xi}}} \to ({\varvec{v}} \bullet \nabla ){\varvec{v}} \\ \end{aligned}} $$

(72)

in the limit of $\delta \to 0$. It is noted here that the Navier–Stokes equation (70) is very similar to Navier’s equation for isotropic elastodynamics,

$$ \rho \frac{{\partial {\varvec{u}}}}{\partial t} = \mu \nabla^{2} \user2{u + }(\mu + \lambda )\nabla (\nabla \bullet {\varvec{u}}) + \rho {\varvec{b}}, $$

(73)

where ${\varvec{u}}$ is a displacement vector. It has been rigorously shown (Mikata [26]) that the peridynamic Navier’s equation is given by

$$ \rho \frac{{\partial {\varvec{u}}}}{\partial t} = \int_{ - \infty }^{\infty } {\tilde{\user2{C}}({\varvec{\xi}})({\varvec{u}}({\varvec{x}} + {\varvec{\xi}},t) - {\varvec{u}}({\varvec{x}},t)){\text{d}}{\varvec{\xi}}} + \rho {\varvec{b}} $$

(74)

where $\tilde{\user2{C}}({\varvec{\xi}})$ is the same $\tilde{\user2{C}}({\varvec{\xi}})$ defined in (10). In other words, it has been proved that

$$ \int_{ - \infty }^{\infty } {\tilde{\user2{C}}({\varvec{\xi}})({\varvec{u}}({\varvec{x}} + {\varvec{\xi}},t) - {\varvec{u}}({\varvec{x}},t)){\text{d}}{\varvec{\xi}}} \to \mu \nabla^{2} \user2{u + }(\mu + \lambda )\nabla (\nabla \bullet {\varvec{u}}) $$

(75)

in the limit of $\delta \to 0$. Obviously, (75) is mathematically equivalent to (71), which proves (71). The proof of (72) is given in Appendix 3.

In order to show that the peridynamic continuity equation (8) approaches the classical continuity equation (2) in the limit of $\delta \to 0$, we need to show that

$$ \int_{ - \infty }^{\infty } {{\varvec{C}}_{D} ({\varvec{\xi}}) \bullet (\rho ({\varvec{x}} + {\varvec{\xi}},t){\varvec{v}}({\varvec{x}} + {\varvec{\xi}},t) - \rho ({\varvec{x}},t){\varvec{v}}({\varvec{x}},t)){\text{d}}{\varvec{\xi}}} \to \nabla \bullet (\rho {\varvec{v}}) $$

(76)

in the limit of $\delta \to 0$. The proof of (76) is also given in Appendix 3.

Appendix 2: Convergence proof of peridynamic incompressible fluid mechanics to classical incompressible fluid mechanics and peridynamic acoustics to classical acoustics

We would like to prove that the incompressible peridynamic Navier–Stokes equation (13), the peridynamic incompressibility condition (14), and the peridynamic acoustics governing equation (16) approach the corresponding classical equations (4), (5), and (6), respectively, in the limit of the material horizon $\delta$ going to zero ($\delta \to 0$). In other words, we need to show

$$ {\begin{aligned} & \int_{ - \infty }^{\infty } {{\varvec{C}}_{G} ({\varvec{\xi}})(p^{\prime}({\varvec{x}} + {\varvec{\xi}},t) - p^{\prime}({\varvec{x}},t)){\text{d}}{\varvec{\xi}}} \to \nabla p^{\prime}({\varvec{x}},t)\quad {\text{as}}\;\delta \to 0, \\ & \int_{ - \infty }^{\infty } {C_{L} ({\varvec{\xi}})({\varvec{v}}({\varvec{x}} + {\varvec{\xi}},t) - {\varvec{v}}({\varvec{x}},t)){\text{d}}{\varvec{\xi}}} \to \nabla^{2} {\varvec{v}}({\varvec{x}},t)\quad {\text{as}}\;\delta \to 0, \\ & \int_{ - \infty }^{\infty } {C_{CV} ({\varvec{\xi}},{\varvec{v}}({\varvec{x}},t))({\varvec{v}}({\varvec{x}} + {\varvec{\xi}},t) - {\varvec{v}}({\varvec{x}},t)){\text{d}}{\varvec{\xi}}} \to ({\varvec{v}} \bullet \nabla ){\varvec{v}}\quad {\text{as}}\;\delta \to 0, \\ & \int_{ - \infty }^{\infty } {C_{D} ({\varvec{\xi}})\bullet({\varvec{v}}({\varvec{x}} + {\varvec{\xi}},t) - {\varvec{v}}({\varvec{x}},t)){\text{d}}{\varvec{\xi}}} \to \nabla \bullet {\varvec{v}}\quad {\text{as}}\;\delta \to 0, \\ & \int_{ - \infty }^{\infty } {C_{L} ({\varvec{\xi}})(p^{\prime}({\varvec{x}} + {\varvec{\xi}},t) - p^{\prime}({\varvec{x}},t)){\text{d}}{\varvec{\xi}}} \to \nabla^{2} p^{\prime}({\varvec{x}},t)\quad {\text{as}}\;\delta \to 0. \\ \end{aligned}} $$

(77)

The last limit of (77) is mathematically equivalent to the second limit of (77), since both of them deal with a peridynamic Laplacian of a function, a vector function for the second limit, and a scalar function for the last limit. Thus, we need to prove the first four limits of (77) as $\delta \to 0$. These all involve the del (nabla) operator, $\nabla$, and its operation on a function such as gradient, divergence, and Laplacian. All of these are discussed and proved in Appendix 3.

Appendix 3: Peridynamic del operator and peridynamic Laplacian

As mentioned in Sect. 3, it seems clear that the peridynamic (nonlocal) version of many governing equations in mathematical physics can be obtained by following the scheme developed in Mikata [27] and the present paper. In order to facilitate the process, the peridynamic del operator and some other peridynamic D operators are discussed in this Appendix.

First, the following correspondence is proved between some of the classical differential operators including the del operator and the corresponding peridynamic D operators in 3D (three dimensions):

$$ {\begin{aligned} \nabla f \leftrightarrow \nabla_{p}f \equiv {\varvec{J}}_{G} (f) & = \int_{ - \infty }^{\infty } {{\varvec{K}}_{G} ({\varvec{\xi}})(f({\varvec{x}} + {\varvec{\xi}},t) - f({\varvec{x}},t)){\text{d}}{\varvec{\xi}}}, \\ \nabla \bullet {\varvec{f}} \leftrightarrow \nabla_{p}\bullet{\varvec{f}}\equiv J_{D} ({\varvec{f}}) & = \int_{ - \infty }^{\infty } {{\varvec{K}}_{D} ({\varvec{\xi}})\bullet({\varvec{f}}({\varvec{x}} + {\varvec{\xi}},t) - {\varvec{f}}({\varvec{x}},t)){\text{d}}{\varvec{\xi}}}, \\ \nabla \times {\varvec{f}} \leftrightarrow \nabla_{p}\times {\varvec{f}}\equiv {\varvec{J}}_{C} ({\varvec{f}}) & = \int_{ - \infty }^{\infty } {{\varvec{K}}_{C} ({\varvec{\xi}}) \times ({\varvec{f}}({\varvec{x}} + {\varvec{\xi}},t) - {\varvec{f}}({\varvec{x}},t)){\text{d}}{\varvec{\xi}}}, \\ \nabla^{2} {\varvec{f}} \leftrightarrow \nabla_{p}^{2}{\varvec{f}}\equiv {\varvec{J}}_{L} ({\varvec{f}}) & = \int_{ - \infty }^{\infty } {K_{L} ({\varvec{\xi}})({\varvec{f}}({\varvec{x}} + {\varvec{\xi}},t) - {\varvec{f}}({\varvec{x}},t)){\text{d}}{\varvec{\xi}}}, \\ ({\varvec{v}} \bullet \nabla ){\varvec{f}} \leftrightarrow ({\varvec{v}}\bullet\nabla_{p}){\varvec{f}}\equiv {\varvec{J}}_{CV} ({\varvec{f}}) & = \int_{ - \infty }^{\infty } {K_{CV} ({\varvec{\xi}},{\varvec{v}}({\varvec{x}},t))({\varvec{f}}({\varvec{x}} + {\varvec{\xi}},t) - {\varvec{f}}({\varvec{x}},t)){\text{d}}{\varvec{\xi}}} \\ \end{aligned}} $$

(78)

where $\nabla_p$ is the peridynamic del operator in 3D, and

$$ {\varvec{K}}_{G} ({\varvec{\xi}}) = {\varvec{K}}_{D} ({\varvec{\xi}}) = {\varvec{K}}_{C} ({\varvec{\xi}}) = {\varvec{K}}_{\nabla } ({\varvec{\xi}}) = \frac{3}{4\pi }(\xi_{1} ,\xi_{2} ,\xi_{3} )g_{\delta } (\xi ), $$

(79)

$$ {\begin{aligned} K_{L} ({\varvec{\xi}}) & = \frac{3}{2\pi }g_{\delta } (\xi ), \\ K_{CV} ({\varvec{\xi}},{\varvec{v}}({\varvec{x}},t)) & = {\varvec{v}}\bullet{\varvec{K}}_{\nabla }({\varvec{\xi}}) = \frac{3}{4\pi }(v_{1} ({\varvec{x}},t)\xi_{1} + v_{2} ({\varvec{x}},t)\xi_{2} + v_{3} ({\varvec{x}},t)\xi_{3} )g_{\delta } (\xi ). \\ \end{aligned}} $$

(80)

Here, $g_{\delta } (\xi )$ is a normalized nonlocality function discussed in Mikata [27], and the suffixes G, D, C, L, and CV stand for “gradient”, “divergence”, “curl”, “Laplacian”, and “convection”, respectively. It should be noted that in the present paper only the spherically symmetric nonlocality functions are considered. The only requirement for the normalized nonlocality function $g_{\delta } (\xi )$ in 3D is given by (see also Mikata [27])

$$ \mathop {\lim }\limits_{\delta \to 0} \int_{0}^{\infty } {\xi^{2m + 2} g_{\delta } (\xi )d\xi } = \left\{ {\begin{array}{*{20}c} {1} & {(m = 1)} \\ 0 & {(m \ge 2)}. \\ \end{array} } \right. $$

(81)

It should also be mentioned that the last 2 equations of (78) and (80) can be equivalently expressed using a matrix kernel as

$$ {\begin{aligned} \nabla^{2} {\varvec{f}} \leftrightarrow \nabla_{p}^{2}{\varvec{f}}\equiv {\varvec{J}}_{L} ({\varvec{f}}) & = \int_{ - \infty }^{\infty } {{\varvec{K}}_{L} ({\varvec{\xi}})({\varvec{f}}({\varvec{x}} + {\varvec{\xi}},t) - {\varvec{f}}({\varvec{x}},t)){\text{d}}{\varvec{\xi}}}, \\ ({\varvec{v}} \bullet \nabla ){\varvec{f}} \leftrightarrow ({\varvec{v}}\bullet\nabla_{p}){\varvec{f}}\equiv{\varvec{J}}_{CV} ({\varvec{f}}) & = \int_{ - \infty }^{\infty } {{\varvec{K}}_{CV} ({\varvec{\xi}},{\varvec{v}}({\varvec{x}},t))({\varvec{f}}({\varvec{x}} + {\varvec{\xi}},t) - {\varvec{f}}({\varvec{x}},t)){\text{d}}{\varvec{\xi}}} \\ \end{aligned}} $$

(82)

where

$$ {\begin{aligned} {\varvec{K}}_{L} ({\varvec{\xi}}) & = \frac{{3\delta_{ij} }}{2\pi }g_{\delta } (\xi ), \\ {\varvec{K}}_{CV} ({\varvec{\xi}},{\varvec{v}}({\varvec{x}},t)) & = \frac{{3\delta_{ij} }}{4\pi }(v_{1} ({\varvec{x}},t)\xi_{1} + v_{2} ({\varvec{x}},t)\xi_{2} + v_{3} ({\varvec{x}},t)\xi_{3} )g_{\delta } (\xi ). \\ \end{aligned}} $$

(83)

The focus here is to prove (78). In order to do that, we need to show

$$ \begin{aligned} \mathop {\lim }\limits_{\delta \to 0} \nabla_{p} f & = \nabla f, \\ \mathop {\lim }\limits_{\delta \to 0} \nabla_{p}\bullet {\varvec{f}} & = \nabla \bullet {\varvec{f}}, \\ \mathop {\lim }\limits_{\delta \to 0} \nabla_{p}\times {\varvec{f}} & = \nabla \times {\varvec{f}}, \\ \mathop {\lim }\limits_{\delta \to 0} \nabla_{p}^{2}{\varvec{f}} & = \nabla^{2} {\varvec{f}}, \\ \mathop {\lim }\limits_{\delta \to 0} ({\varvec{v}}\bullet\nabla_{p}){\varvec{f}} & = ({\varvec{v}} \bullet \nabla ){\varvec{f}}. \\ \end{aligned} $$

(84)

We have

$$ {\begin{aligned} \nabla_{p}{f}\equiv {\varvec{J}}_{G} (f)& = \sum\limits_{n = 1}^{\infty } {\frac{1}{n!}} {\varvec{J}}_{G}^{(n)} (f), \\ \nabla_{p}\bullet{\varvec{f}}\equiv J_{D} ({\varvec{f}}) & = \sum\limits_{n = 1}^{\infty } {\frac{1}{n!}} J_{D}^{(n)} ({\varvec{f}}), \\ \nabla_{p}\times {\varvec{f}}\equiv{\varvec{J}}_{C} ({\varvec{f}}) & = \sum\limits_{n = 1}^{\infty } {\frac{1}{n!}} {\varvec{J}}_{C}^{(n)} ({\varvec{f}}), \\ \nabla_{p}^{2}{\varvec{f}}\equiv {\varvec{J}}_{L} ({\varvec{f}}) & = \sum\limits_{n = 1}^{\infty } {\frac{1}{n!}} {\varvec{J}}_{L}^{(n)} ({\varvec{f}}), \\ ({\varvec{v}} \bullet\nabla_{p}){\varvec{f}}\equiv{\varvec{J}}_{CV} ({\varvec{f}}) & = \sum\limits_{n = 1}^{\infty } {\frac{1}{n!}} {\varvec{J}}_{CV}^{(n)} ({\varvec{f}}) \\ \end{aligned}} $$

(85)

where

$$ {\begin{aligned} {\varvec{J}}_{G}^{(n)} (f) & = \int_{ - \infty }^{\infty } {{\varvec{K}}_{\nabla} ({\varvec{\xi}})} \left( {\xi_{1} \frac{\partial }{{\partial x_{1} }} + \xi_{2} \frac{\partial }{{\partial x_{2} }} + \xi_{3} \frac{\partial }{{\partial x_{3} }}} \right)^{n} f({\varvec{x}},t){\text{d}}{\varvec{\xi}}, \\ J_{D}^{(n)} ({\varvec{f}}) & = \int_{ - \infty }^{\infty } {{\varvec{K}}_{\nabla} ({\varvec{\xi}})}\bullet\left( {\xi_{1} \frac{\partial }{{\partial x_{1} }} + \xi_{2} \frac{\partial }{{\partial x_{2} }} + \xi_{3} \frac{\partial }{{\partial x_{3} }}} \right)^{n} {\varvec{f}}({\varvec{x}},t){\text{d}}{\varvec{\xi}}, \\ {\varvec{J}}_{C}^{(n)} ({\varvec{f}}) & = \int_{ - \infty }^{\infty } {{\varvec{K}}_{\nabla} ({\varvec{\xi}})} \times \left( {\xi_{1} \frac{\partial }{{\partial x_{1} }} + \xi_{2} \frac{\partial }{{\partial x_{2} }} + \xi_{3} \frac{\partial }{{\partial x_{3} }}} \right)^{n} {\varvec{f}}({\varvec{x}},t){\text{d}}{\varvec{\xi}}, \\ {\varvec{J}}_{L}^{(n)} ({\varvec{f}}) & = \int_{ - \infty }^{\infty } {{{K}}_{L} ({\varvec{\xi}})} \left( {\xi_{1} \frac{\partial }{{\partial x_{1} }} + \xi_{2} \frac{\partial }{{\partial x_{2} }} + \xi_{3} \frac{\partial }{{\partial x_{3} }}} \right)^{n} {\varvec{f}}({\varvec{x}},t){\text{d}}{\varvec{\xi}}, \\ {\varvec{J}}_{CV}^{(n)} ({\varvec{f}}) & = \int_{ - \infty }^{\infty } {\varvec{v}}({\varvec{x}}, t)\bullet{{\varvec{K}}_{\nabla} ({\varvec{\xi}})} \left( {\xi_{1} \frac{\partial }{{\partial x_{1} }} + \xi_{2} \frac{\partial }{{\partial x_{2} }} + \xi_{3} \frac{\partial }{{\partial x_{3} }}} \right)^{n} {\varvec{f}}({\varvec{x}},t){\text{d}}{\varvec{\xi}}. \\ \end{aligned}} $$

(86)

Using (85), and noting that most of the terms in each infinite series in the limit of δ → 0 are zero due to the property of the normalized nonlocality function $g_{\delta } (\xi )$ (i.e., (81)), we obtain

$$ {\begin{aligned} \mathop {\lim }\limits_{\delta \to 0} {\nabla}_{p} f & = \mathop {\lim }\limits_{\delta \to 0} {\varvec{J}}_{G}^{(1)} (f), \\ \mathop {\lim }\limits_{\delta \to 0} \nabla_{p}\bullet {\varvec{f}} & = \mathop {\lim }\limits_{\delta \to 0} J_{D}^{(1)} ({\varvec{f}}), \\ \mathop {\lim }\limits_{\delta \to 0} {\nabla}_{p}\times{\varvec{f}} & = \mathop {\lim }\limits_{\delta \to 0} {\varvec{J}}_{C}^{(1)} ({\varvec{f}}), \\ \mathop {\lim }\limits_{\delta \to 0} {\nabla}_{p}^{2} {\varvec{f}} & = \frac{1}{2}\mathop {\lim }\limits_{\delta \to 0} {\varvec{J}}_{L}^{(2)} ({\varvec{f}}), \\ \mathop {\lim }\limits_{\delta \to 0} ({\varvec{v}}\bullet{\nabla}_{p}) {\varvec{f}} & = \mathop {\lim }\limits_{\delta \to 0} {\varvec{J}}_{CV}^{(1)} ({\varvec{f}}). \\ \end{aligned}} $$

(87)

After working out the necessary algebras using (86), we finally obtain (84), which proves (78). It should be noted that all of the equations needed to be proved in Appendices 1 and 2 (i.e., (72), (76), and (77)) are included in the correspondence (78).

We have correspondences similar to (78) between classical differential operators and peridynamic D operators in 2D (two dimensions) and in 1D (one dimension). We list the final results without proof below, which can be proved in a similar manner as in 3D.

For 2D:

$$ {\begin{aligned} \nabla f \leftrightarrow \nabla_{p}f \equiv {\varvec{J}}_{G} (f) & = \int_{ - \infty }^{\infty } {{\varvec{K}}_{G} ({\varvec{\xi}})(f({\varvec{x}} + {\varvec{\xi}},t) - f({\varvec{x}},t)){\text{d}}{\varvec{\xi}}}, \\ \nabla \bullet {\varvec{f}} \leftrightarrow \nabla_{p} \bullet {\varvec{J}} \equiv J_{D} ({\varvec{f}}) & = \int_{ - \infty }^{\infty } {{\varvec{K}}_{D} ({\varvec{\xi}}) \bullet ({\varvec{f}}({\varvec{x}} + {\varvec{\xi}},t) - {\varvec{f}}({\varvec{x}},t)){\text{d}}{\varvec{\xi}}}, \\ \nabla^{2} {\varvec{f}} \leftrightarrow \nabla_{p}^{2} {\varvec{f}} \equiv {\varvec{J}}_{L} ({\varvec{f}}) & = \int_{ - \infty }^{\infty } {K_{L} ({\varvec{\xi}})({\varvec{f}}({\varvec{x}} + {\varvec{\xi}},t) - {\varvec{f}}({\varvec{x}},t)){\text{d}}{\varvec{\xi}}}, \\ ({\varvec{v}} \bullet \nabla ){\varvec{f}} \leftrightarrow ({\varvec{v}}\bullet\nabla_{p}){\varvec{f}}\equiv{\varvec{J}}_{CV} ({\varvec{f}}) & = \int_{ - \infty }^{\infty } {K_{CV} ({\varvec{\xi}},{\varvec{v}}({\varvec{x}},t))({\varvec{f}}({\varvec{x}} + {\varvec{\xi}},t) - {\varvec{f}}({\varvec{x}},t)){\text{d}}{\varvec{\xi}}} \\ \end{aligned}} $$

(88)

where $\nabla_{p}$ is the peridynamic del operator in 2D, and

$$ {\varvec{K}}_{G} ({\varvec{\xi}}) = {\varvec{K}}_{D} ({\varvec{\xi}}) = {\varvec{K}}_{\nabla } ({\varvec{\xi}}) = \frac{1}{\pi }(\xi_{1} ,\xi_{2} )g_{\delta } (\xi ), $$

(89)

$$ K_{L} ({\varvec{\xi}}) = \frac{2}{\pi }g_{\delta } (\xi ), $$

(90)

$$ K_{CV} ({\varvec{\xi}},{\varvec{v}}({\varvec{x}},t)) = \frac{1}{\pi }(v_{1} ({\varvec{x}},t)\xi_{1} + v_{2} ({\varvec{x}},t)\xi_{2} )g_{\delta } (\xi ). $$

(91)

Here, the requirement for the normalized nonlocality function $g_{\delta } (\xi )$ in 2D is given by

$$ \mathop {\lim }\limits_{\delta \to 0} \int_{0}^{\infty } {\xi^{2m + 1} g_{\delta } (\xi )d\xi } = \left\{ {\begin{array}{*{20}c} {1} & {(m = 1)} \\ 0 & {(m \ge 2)}. \\ \end{array} } \right. $$

(92)

For 1D:

$$ {\begin{aligned} \frac{{{\text{d}}f}}{{{\text{d}}x}} \leftrightarrow D_{x} f \equiv J_{1} (f) & = \int_{ - \infty }^{\infty } {K_{1} (\xi )(f(x + \xi ,t) - f(x,t)){\text{d}}\xi }, \\ \frac{{{\text{d}}^{2} f}}{{{\text{d}}x^{2} }} \leftrightarrow D_{x}^{2} f \equiv J_{2} (f) & = \int_{ - \infty }^{\infty } {K_{2} (\xi )(f(x + \xi ,t) - f(x,t)){\text{d}}\xi }, \\ \frac{{{\text{d}}^{n} f}}{{{\text{d}}x^{n} }} \leftrightarrow D_{x}^{n} f \equiv J_{n} (f) & = \int_{ - \infty }^{\infty } {K_{n} (\xi )(f(x + \xi ,t) - f(x,t)){\text{d}}\xi } \\ \end{aligned}} $$

(93)

where $D_{x}$ is a peridynamic D operator in 1D, and

$$ {\begin{aligned} K_{1} (\xi ) & = \frac{\xi }{2}g_{\delta } (\xi ), \\ K_{2} (\xi ) & = g_{\delta } (\xi ), \\ K_{n} (\xi ) & = \frac{n!}{{2\xi^{n - 2} }}g_{\delta } (\xi ). \\ \end{aligned}} $$

(94)

Here, the requirement for the normalized nonlocality function $g_{\delta } (\xi )$ in 1D is given by

$$ \mathop {\lim }\limits_{\delta \to 0} \int_{0}^{\infty } {\xi^{2m} g_{\delta } (\xi )d\xi } = \left\{ {\begin{array}{*{20}c} {1} & {(m = 1)} \\ 0 & {(m \ge 2)}. \\ \end{array} } \right. $$

(95)

By using (93) and (94), for example, we have the following correspondence:

$$ E\frac{{\partial^{2} u}}{{\partial x^{2} }} + b(x,t) = \rho \frac{{\partial^{2} u}}{{\partial t^{2} }} \leftrightarrow \int_{ - \infty }^{\infty } {C(\xi )} (u(x - \xi ,t) - u(x,t))d\xi + b(x,t) = \rho \ddot{u}(x,t) $$

(96)

where the right side of the above correspondence is 1D peridynamics studied in Silling et al. [3] and Mikata [18] in detail, and

$$ C(\xi ) = Eg_{\delta } (\xi ). $$

(97)

As was discussed in Silling et. al [3] and Mikata [18], there is a relation between $C(\xi )$ and Young’s modulus E,

$$ E = \int_{0}^{\infty } {\xi^{2} } C(\xi ){\text{d}}\xi. $$

(98)

The above relation (98) was “derived” by a physical argument in Silling et al. [3]. It is important to mention here that this relation (98) can be derived immediately from (97) and (95), where (97) (i.e., (96)) is a special case of the correspondence between the classical differential operators and the peridynamic D operators systematically developed in this Appendix.