## Abstract

In this work, we develop the Green’s function method for the solution of the peridynamic non-local diffusion model in which the spatial gradient of the generalized potential in the classical theory is replaced by an integral of a generalized response function in a horizon. We first show that the general solutions of the peridynamic non-local diffusion model can be expressed as functionals of the corresponding Green’s functions for point sources, along with volume constraints for non-local diffusion. Then, we obtain the Green’s functions by the Fourier transform method for unsteady and steady diffusions in infinite domains. We also demonstrate that the peridynamic non-local solutions converge to the classical differential solutions when the non-local length approaches zero. Finally, the peridynamic analytical solutions are applied to an infinite plate heated by a Gauss source, and the predicted variations of temperature are compared with the classical local solutions. The peridynamic non-local diffusion model predicts a lower rate of variation of the field quantities than that of the classical theory, which is consistent with experimental observations. The developed method is applicable to general diffusion-type problems.

## 1. Introduction

Non-locality is inherent in the transfer of thermal energy, for energy carriers, such as phonons and electrons, collide with each other and bring energy from one point to the other. Nevertheless, macroscopic models adopting a local formulation may successfully represent heat conduction in many cases in continua in that heat conduction at the macroscopic scale necessitates a sufficiently large domain and a sufficiently long time to ensure a meaningful statistical ensemble. However, in the last decades, invalidation of local formulations has been found in more and more situations. When the size of electronic devices decreases to less than the mean free path of the phonon of the material, violation of the local theory has been revealed experimentally in these nanostructures [1,2]. Large temperature gradients may appear in emergency cooling [3] and rapid transient heat transfer [4]. In such situations, the heat flux is found to be smaller than that predicted by the local theory [5]. Besides heat conduction, non-locality also exists in other diffusion problems where the particle description of carriers is valid, such as phase transition [6] and electron transport [7]. Non-locality may also arise in modelling the overall properties of heterogeneous media [8].

In fact, several non-local theories of heat conduction have been proposed in the past [9–11]. These theories are developed particularly considering collisions of electrons or involve high-order gradients of temperature. On the other hand, equations for heat conduction in the peridynmaic framework have been developed by some researchers, following the peridynamic theory [12,13] which reformulates the governing equations of continuum mechanics by replacing the spatial differentiation with a spatial integration. This reformulation also has advantages in dealing with damages, discontinuities, and friction and wear, at macro- and nano-scales [14–19].

Gerstle *et al.* [20] firstly developed a one-dimensional peridynamic model to simulate electro-migration that is affected by coupled heat conduction, deformation and diffusion of electrons in a thin film of copper. Bobaru & Duangpanya [21] developed a peridynamic transient heat transfer model for a bar, where a mesh-free numerical method based on the one-point Gauss quadrature was used to compute the temperature field, and it was demonstrated that the numerical results converge to the classical solutions of the Fourier’s law in the limit when the characteristic length scale tends to zero for several one-dimensional examples with Dirichlet and Neumann boundary conditions. Subsequently, Bobaru & Duangpanya [22] extended their formulation to heat conduction in two dimensions, and investigated transient heat conduction in heterogeneous bodies containing growing and intersecting cracks, and fibres [22]. Recently, Oterkus *et al.* [23] proposed an ordinary state-based peridynamic heat conduction theory based on the Lagrangian formalism. It can degenerate into the bond-based formulation; thus, Oterkus *et al.* [23] demonstrated the validity of the bond-based peridynamic formulation for heat conduction in multi-dimensional domains with a series of problems of different boundary conditions by comparison with the classical solutions. To be consistent with the integral formulation of peridynamics and for the sake of well-posedness, volume constraints are proposed by Gunzburger & Lehoucq [24] and Du *et al.* [25]. Gunzburger & Lehoucq [24] gave the fundamental solution for steady-state non-local diffusion problems which is expressed in terms of the Green’s functions and the volume constraints. Besides the mesh-free method used in the above numerical simulations, Du *et al.* [26] proposed a convergent adaptive finite-element algorithm to deal with non-local diffusion problems with the volume constraints. Recently, Chen & Bobaru [27] numerically studied the convergent properties of different kernels in the integral formulation of peridynamics with the one-point Gauss quadrature method. It is noted that Chen & Bobaru [28] have applied the peridynamic non-local diffusion formulation to study corrosion of metals along with numerical computations. Therefore, until now, all solutions of diffusion within the formalism of peridynamics are numerical computations.

Among analytical solutions of boundary-value problems, the Green’s functions as basic solutions play a special role in many aspects. Weckner *et al.* [29] have presented the three-dimensional static and dynamic Green’s functions for a point force within the peridynamic formalism. In this paper, we first show that the solution of the peridynamic non-local diffusion model can be obtained from the solution of a point source. Thus, we drive the transient and steady-state peridynamic Green’s functions for a point source in infinite domains. The method is applied to a two-dimensional infinite plate heated by a Gauss source. To facilitate expressions, we use the terminology of thermal diffusion in the derivations, but, because of the mathematical analogy [30], the model and the method can be applied to general diffusion problems such as electrical conduction, electrostatics, magnetostatics and mass diffusion. The non-local diffusion model is also closely associated with the master equation that describes the jump process (see [24,25,31] for extended references, e.g. [32].)

## 2. General solution of peridynamic thermal diffusion

In this section, we will develop the Green’s function method for non-local peridynamic formulation of thermal diffusion, and propose general solutions of boundary-value problems of unsteady and steady heat conduction in one- to three-dimensional isotropic media.

In a three-dimensional isotropic medium, the non-local governing equation of heat conduction expressed in the peridynamic formulation is
*T* at a material point ** x** at time

*t*;

*ρ*is the mass density;

*c*

_{v}is the specific heat capacity;

*l*with centre

**;**

*x**V*is the volume of

*q*

_{b}is the rate of heat generation per unit volume.

*f*

_{T}is the peridynamic dual heat flow density or referred to as thermal response function. For bond-based peridynamic thermal diffusion,

*f*

_{T}can be represented as a linear function

*ξ*=∥

**′−**

*x***∥;**

*x**K*(

*ξ*) is the peridynamic microconductivity function and can be determined from the thermal conductivity of the classical theory as given in appendix A.

Associated with the linear thermal response function *f*_{T}, equation (2.1) is a linear equation, the solution of which needs boundary conditions. In the classical theory, commonly used linear boundary conditions can be expressed as
*κ*_{i} and *h*_{i} are the thermal conductivity and heat transfer coefficient between the boundary surface and its surrounding, respectively, and *S*_{i} represents the surface of boundary *i*. *s* is the total number of the boundaries. *g*_{i}(** x**,

*t*) is a given function. If

*κ*

_{i}=0, equation (2.3) represents the first-type (or Dirichlet) boundary condition; if

*h*

_{i}=0, equation (2.3) represents the second-type (or Neumann) boundary condition. Otherwise, it is the third-type (or Robin) boundary condition.

It should be noted that boundary conditions in the classical form (2.2) are not necessary for the solution of equation (2.1) which does not contain any spatial derivatives; instead, according to the work of Gunzburger & Lehoucq [24], and Du *et al.* [25], equation (2.1) is augmented with the following volume constraints:
*Γ* denotes a non-zero volume which is disjoint from

Equations (2.1), (2.4) and (2.6) constitute the formulation of a general three-dimensional non-homogeneous heat conduction problem. Now we consider the following auxiliary problem of a point heat source, where the governing equation, the volume constraints (V.C.) and the initial condition (I.C.) are
*a*
*b*
*c*in which the temperature is denoted by *G*, and the thermal response function *f*_{G} has the same form as *f*_{T}. *δ* is the Dirac function. The problem described by (2.7) has homogeneous boundary conditions which are the homogeneous parts of the non-homogeneous conditions in equations (2.4) and (2.6). The solution of the problem in equation (2.7) is denoted as the Green’s function *G*(** x**,

*t*|

***,**

*x**t**), which describes the temperature field in the region

*t*>

*t**. This relation implies that the temperature response only depends on the time interval and spatial distance, not related to the detail moment and position.

With this reciprocal relation, equation (2.7a) can be rewritten as the form represented by *G*(** x***,−

*t** |

**,−**

*x**t*), that is,

*t*and

**by**

*x**t** and

***, respectively, leads to**

*x**T*from equation (2.10) after multiplied by

*G*, we get

*** and**

*x**t**. Integrating equation (2.11) over the entire domain of variable

***, and simultaneously from 0 to**

*x**t*+

*ε*with respect to

*t**, where

*ε*is an arbitrary but positive small quantity, we get

In addition, *G*(** x**,

*t*|

***,**

*x**t*+

*ε*)=0, because the response time is before the heat source is imposed. Then, the first integrand of equation (2.12) can be rewritten as

*G*|

_{t*=0}=

*G*(

**,**

*x**t*|

***,0).**

*x*Substituting equations (2.13) and (2.14) into equation (2.12) and letting *G*=*G*(** x**,

*t*|

***,**

*x**t**). From the volume constraints in equation (2.4) and equation (2.7b), we obtain

*G*|

_{Γi}represents the value of the Green’s function in the volume

*Γ*

_{i}. Thus, the general solution can be represented by the Green’s function as

By eliminating the terms related to time *t*, the general solutions for steady heat conduction can be obtained, wherein the governing equation and the volume constraints are
*a*
*b*The corresponding auxiliary problem is defined by
*a*
*b*The general solution is

The peridynamc general solutions expressed in terms of the Green’s functions have the same form as those of the classical theory, except for the boundary conditions and Green’s functions. Thus, we shall solve the peridynamic Green’s functions in the next section.

## 3. Green’s functions in infinite domains

In this section, we derive the Green’s functions in one-, two- and three-dimensional infinite domains, in which case, *s*=0 in (2.17) and (2.20), namely, no boundary conditions are involved.

### (a) Unsteady heat conduction

For unsteady heat conduction in a three-dimensional infinite medium, the auxiliary problem equation (2.7) can be rewritten as
*a*
*b*in which *f*_{G}=*K*(*ξ*)[*G*(** x**′,

*t*)−

*G*(

**,**

*x**t*)] and Δ(

**,**

*x**t*)=

*δ*(

**−**

*x****)**

*x**δ*(

*t*−

*t**). By the three-dimensional Fourier transform with respect to

**, equation (3.1) becomes**

*x**a*

*b*where

**=**

*ξ***′−**

*x***and**

*x**l*with its centre at the origin. Owing to the symmetry of

*K*with respect to

**′ and**

*x***, equation (3.3) can be rewritten as**

*x**k*=∥

**∥.**

*k*Now equation (3.2a) is a first-order partial differential equation with respect to *t*. The solution of equation (3.2) can be expressed as
*t*>*t**, due to Δ(** x**,

*t*)=

*δ*(

**−**

*x****)**

*x**δ*(

*t*−

*t**), equation (3.6) becomes

*r*=∥

**−**

*x****∥. While**

*x**t*<

*t**,

*G*(

**,**

*x**t*|

***,**

*x**t**)=0.

Similarly, for the two-dimensional problem, the Green’s function is
*B*(*kξ*)=1−*J*_{0}(*kξ*). *J*_{0}(*kξ*) is the zero-order Bessel function.

For one dimension, the Green’s function is

### (b) Steady heat conduction

For steady heat conduction, two methods will be introduced to obtain the Green’s functions, namely, a direct method and an indirect method.

We firstly introduce the direct method. For steady heat conduction in a three-dimensional infinite medium, the auxiliary problem equation (2.19) can be expressed as
** x**)=

*δ*(

**−**

*x****). By the three-dimensional Fourier transform with respect to**

*x***, equation (3.12) becomes**

*x*On the other hand, the Green’s function of the steady heat conduction could be obtained from that of the unsteady heat conduction by

For the auxiliary problem (2.7) of unsteady heat conduction, applying the integral operator *a*
*b*
*c*Denoting *a*
*b*
*c*If *h*_{i} is not zero, the problem (3.17) can be decomposed into the superposition of a steady heat conduction problem and an unsteady heat conduction problem, that is,
*G*_{st}(** x**) satisfies the conditions of steady heat conduction

*a*

*b*and

*G*

_{ust}(

**,**

*x**t*) satisfies the conditions of unsteady heat conduction

*a*

*b*

*c*The governing equation and the volume constraints in equation (3.19) are simply those of the auxiliary problem described in equation (2.19). For the problem described in equation (3.20), the temperature field will decay to zero as time

It is noted that the integral obtained from equation (3.14) or equation (3.15) is divergent. Some special treatments are required. For three dimensions, equation (3.14) can be rewritten as the sum of a Dirac function and a convergent integral

For two dimensions, the divergent integral obtained by any of the above two methods could be expressed as a conventional solution plus a Dirac function and a convergent non-local integral

For one dimension, the Green’s function can be similarly expressed as

## 4. Relation between peridynamic solutions and classical solutions

In this section, we investigate the relation between the peridynamic solutions and the classical solutions from two aspects. Firstly, as the non-local length *l* and the microconductivity function *K*(*ξ*).

In fact, the relation between the peridynamic and classical solutions can be determined by examining the values of *M*(** k**). This is because the peridynamic and classical solutions have the same form except for the Green’s functions. At the same time, the classical Green’s functions have the same form as those of peridynamics. For example, the classical Green’s functions in infinite domains can be obtained by replacing

*M*(

**) with**

*k**κk*

^{2}. For simplicity,

*M*(

**) in the peridynamic solutions are generically denoted by**

*k**M*

_{peri}, and those of the classical theory are denoted by

*M*

_{cl}. Next, we examine the values of

*M*

_{peri}.

### (a) Convergence of peridynamic solutions to classical solutions

We first investigate the limit of *M*(** k**) in infinite domains. As

*ξ*will become an infinitely small quantity. For example, in the three-dimensional domain, using the Taylor expansion,

*B*(

*kξ*) in equation (3.4) can be expressed as

*M*(

**) are equal to those of the classical theory.**

*k*### (b) The effect of non-local factors

From the forms of *M*_{peri}, we could readily find that they are determined by two factors: the microconductivity function *K*(*ξ*) and the non-local length *l*. Several forms of the microconductivity function *K*(*ξ*) have been seen in the literature (e.g. [20–23]). They can be generally expressed as
*n* is an integer normally selected to be 0 or 1 or 2. In the work of Chen & Bobaru [27], *χ*(*ξ*) is defined as a ‘shape factor’, and two kinds of shape factor are selected: a constant one and a linear one, which can be expressed as
*a*
*b*
where *K*(*ξ*) is determined, the corresponding *M*_{peri} can be obtained. For instance, substituting *K*(*ξ*) into equation (3.4) yields *M*_{peri} for a three-dimensional domain as
*a*
*b*in which superscripts C and L represent the values for the constant and linear shape factors, respectively. Similarly, for a two-dimensional domain, there are
*a*
*b*For one dimension, there are
*a*
*b*In the classical theory, *M*_{cl} is always equal to *κk*^{2} in which, *k*=∥** k**∥ for infinite domains. In order to reflect the effect of the non-local factors more clearly, we compare the values of

*M*for the constant and linear shape factors are defined as

*M*

^{C}and

*M*

^{L}, respectively, as shown in figure 1.

Figure 1 shows that the peridynamic results are always smaller than, but approach, those of the classical theory as *n* increases. This feature has two implications. First, the non-local effect weakens with the increase of *n*. Second, for unsteady heat conduction, the variation rate of the temperature predicted by the peridynamic non-local model will be always smaller than that predicted by the classical Fourier’s Law of the differential type, which is consistent with the experimental observation and the prediction of a non-local heat-conduction model based on the Boltzmann equation [10,33]. Moreover, for the same dimension and *n*, the values of *M*_{peri} for the linear shape factor are larger than those for the constant shape factor, and the former are closer to the classical solutions. Finally, for the same *n* and shape factor, the values of *M*_{peri} are related to the dimension, whereas the classical values remain the same for all dimensions.

We choose equation (4.9b) with *n*=2 as an example to investigate the effect of the non-local length *l*. Introducing an external characteristic length *L*_{0}, we plot the non-dimensionalized values of *M* in figure 2. As *l* decreases, the peridynamic solutions approach the classical one.

As a matter of fact, transitions of non-local integral diffusion formulations to corresponding partial differential equations in the limit of the characteristic length have been proved under suitable conditions on the kernel functions [24,25,32]; therefore, the results in this section are in line with the established theoretical analyses.

## 5. An example

In this section, an infinite plate heated by a Gauss heat source is used to demonstrate the results of the peridynamic analytical solutions. The initial condition is *T*(** x**,0)=

*T*

_{c}. The Gauss heat source is described as follows:

*Q*is the power in a unit length and

*R*represents the effective heated radius.

The Green’s function for this problem can be expressed as equation (3.8) in which the microconductivity function is taken as

The peridynamic solutions for the temperature are plotted in figure 3*a*, where *a*, we just show the variations of the temperature at different times at *b*. We can clearly see that the peridynamic analytical solutions approach the classical solutions as the non-local length decreases, which is consistent with the results shown in figure 2. Moreover, the peridynamic analytical solutions are larger than classical solutions in the central region, which is due to the slower change of the non-local solutions as analysed in §4b.

## 6. Conclusion

In this paper, we present the Green’s function method to solve diffusion problems described by the peridynamic non-local formulation. We derive the general solutions expressed by the Green’s functions along with the volume constraints. Then, the Green’s functions for unsteady and steady heat conduction are obtained for infinite domains. The relations between the peridynamic solutions and classical solutions are investigated. We demonstrate the convergence of the peridynamic solutions to the classical solutions. The effect of the non-local shape factor and non-local length are quantitatively analysed. The peridynamic analytical solutions are applied to prediction of the temperature field in an infinite plate heated by a Gauss source. The method proposed in this paper can not only be used to obtain solutions of a variety of complicated problems, but can also be applied to other diffusion-type models, such as neutronic diffusion, electrical conduction, electrostatics, magnetostatics and mass diffusion.

## Authors' contributions

All authors participated in the development of the study and in writing the manuscript. All authors gave approval for publication.

## Competing interests

We have no competing interests.

## Funding

This research is supported by the National Natural Science Foundation of China under grant no. 11521202.

## Appendix A. Peridynamic material parameters

Peridynamic material parameters can be determined by equating the thermal potential of the peridynamic theory to that of the classical continuum theory [23]. By this approach [23], the peridynamic parameters for one-, two- and three-dimensional domains are derived as follows.

The peridynamic thermal potential at point ** x** is defined as

For a one-dimensional medium, the peridynamic material parameters can be determined by considering a linear temperature field in the form *T*(*x*)=*x*. Substituting this temperature field into equation (A 1) and equation (A 3) leads to the peridynamic thermal potential

For a two-dimensional medium, the peridynamic material parameters can be determined by considering a linear temperature field in the form *T*(*x*,*y*)=*x*+*y*. Substituting this temperature field into equation (A 1) and equation (A 3) leads to the peridynamic thermal potential

For a three-dimensional medium, the peridynamic material parameters can be determined by considering a linear temperature field in the form *T*(*x*,*y*,*z*)=*x*+*y*+*z*. Substituting this temperature field into equations (A 1) and (A 3) leads to the peridynamic thermal potential

- Received March 14, 2016.
- Accepted September 1, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.