Concentrated Couple in a Plane and in a Half-Plane in the Framework of Fractional Nonlocal Elasticity

Abstract

In nonlocal elasticity, the constitutive equation for the stress tensor is written in an integral form, with the weight function, referred to as the nonlocality kernel, often being the Green’s function for the partial differential equation. In this paper, we obtain solutions to elasticity problems for a concentrated couple in a plane and on the boundary of a half-plane within framework of a new theory of nonlocal elasticity, where the nonlocal kernel is the Green’s function of the Cauchy problem for the fractional diffusion equation. The obtained solutions are free from nonphysical singularities that appear in the classical local elasticity solutions.

1. Introduction

The main weakness of solutions derived within the framework of classical elasticity theory is the appearance of nonphysical singularities at the points of application of concentrated loads. This has resulted in approaches that improve classical elastic solutions and eliminate nonphysical singularities; in particular, there have been great advances using nonlocal elasticity.The foundations of nonlocal theories, based on different schemes, were laid down by Kröner [1], Podstrigach [2], Krumhansl [3], Eringen [4,5], and Kunin [6,7]. In nonlocal elasticity, the constitutive equation for the stress tensor is written in the integral form, with the weight function referred to as the nonlocality kernel. Starting from Eringen’s investigations on nonlocal elasticity [8,9,10], the nonlocal kernels were often chosen as Green’s functions for partial differential operators (see also [11,12,13], where additional references can be found).

The fractional calculus has numerous applications in different branches of science (see, for example, refs. [14,15,16,17,18,19,20,21,22] among many others) as well as recent books on this subject [23,24,25,26]. Fractional calculus was also used in various approaches to nonlocal theories [27,28,29,30,31,32,33,34,35]. Fractional nonlocal elasticity theory with the kernel being the Green’s function of the Cauchy problem for the fractional diffusion equation was elaborated in [36]. In the present paper, this theory is used to obtain solutions for a concentrated couple in a plane and on the boundary of a half-plane. Concentrated couples as fundamental solutions for different elasticity theories were considered in [37,38,39].

A system of the basic equations for a linear isotropic nonlocal elastic solid [4,5,8,9,10] consists of the equilibrium equation

$${\displaystyle \nabla }·\mathbf{t}\left(\mathbf{x},\tau \right)=0,$$

(1)

the constitutive equation for the nonlocal stress tensor

$$\mathbf{t}\left(\mathbf{x},\tau \right)={\int }_{V}K\left(|\mathbf{x}-{\mathbf{x}}^{\prime }|,\tau \right)\mathbf{\sigma }\left({\mathbf{x}}^{\prime }\right)dv\left({\mathbf{x}}^{\prime }\right),$$

(2)

the constitutive equation for the local stress tensor

$$\mathbf{\sigma }\left({\mathbf{x}}^{\prime }\right)=2\mu \mathbf{e}\left({\mathbf{x}}^{\prime }\right)+\lambda \mathrm{tr}\mathbf{e}\left({\mathbf{x}}^{\prime }\right)\mathbf{I},$$

(3)

and geometrical relations

$$\mathbf{e}\left({\mathbf{x}}^{\prime }\right)={\displaystyle \frac{1}{2}}\left[{\nabla }^{\prime }\mathbf{u}\left({\mathbf{x}}^{\prime }\right)+\mathbf{u}\left({\mathbf{x}}^{\prime }\right){\nabla }^{\prime }\right].$$

(4)

Here $\mathbf{\sigma }\left({\mathbf{x}}^{\prime }\right)$ and $\mathbf{t}\left(\mathbf{x},\tau \right)$ are the classical and nonlocal stress tensors, ${\mathbf{x}}^{\prime }$ and $\mathbf{x}$ are the running and reference points, $\mathbf{u}$ is the displacement vector, $\mathbf{e}$ is the linear strain tensor, $\lambda$ and $\mu$ denote the Lamé constants, ∇ and ${\nabla }^{\prime }$ stand for the gradient operators with respect to $\mathbf{x}$ and ${\mathbf{x}}^{\prime }$, $\mathbf{I}$ is the unit tensor.

The nonlocal kernel $K\left(|\mathbf{x}-{\mathbf{x}}^{\prime }|,\tau \right)$ is a function of the distance $|\mathbf{x}-{\mathbf{x}}^{\prime }|$ between the reference and running points, and includes the parameter $\tau$ associated with a characteristic length ratio $l_0/l$, where $l_0$ is an internal characteristic length and l is an external characteristic length [9,10]. The nonlocal kernel $K\left(|\mathbf{x}-{\mathbf{x}}^{\prime }|,\tau \right)$ is a delta sequence and in the classical elasticity limit, $\tau \to 0$ tends to the Dirac delta function $\delta \left(\right|\mathbf{x}-{\mathbf{x}}^{\prime }\left|\right)$.

Eringen considered the nonlocal modulus $K\left(\mathbf{x},\tau \right)$ as the Green function of the Cauchy problem for the diffusion operator [8,10]:

$${\displaystyle \frac{\partial K\left(\mathbf{x},\tau \right)}{\partial \tau }}=a\mathrm{\Delta }K\left(\mathbf{x},\tau \right),$$

(5)

$$\tau =0:K\left(\mathbf{x},\tau \right)=\delta \left(\mathbf{x}\right),$$

(6)

with the solution

$$K\left(x,\tau \right)={\displaystyle \frac{1}{2\sqrt{\pi a\tau }}}exp\left(-{\displaystyle \frac{x^2}{4a\tau }}\right)$$

(7)

for one spatial variable in Cartesian coordinates;

$$K\left(r,\tau \right)={\displaystyle \frac{1}{4\pi a\tau }}exp\left(-{\displaystyle \frac{r^2}{4a\tau }}\right)$$

(8)

in polar coordinates in the axisymmetric case;

$$K\left(r,\tau \right)={\displaystyle \frac{1}{8{\left(\pi a\tau \right)}^{3/2}}}exp\left(-{\displaystyle \frac{r^2}{4a\tau }}\right)$$

(9)

in spherical coordinates in the central symmetric case.

In this instance, the corresponding nonlocal stress tensor $\mathbf{t}\left(\mathbf{x},\tau \right)$ is a solution to the associated Cauchy problem

$${\displaystyle \frac{\partial \mathbf{t}\left(\mathbf{x},\tau \right)}{\partial \tau }}=a\mathrm{\Delta }\mathbf{t}\left(\mathbf{x},\tau \right),$$

(10)

$$\tau =0:\mathbf{t}\left(\mathbf{x},\tau \right)=\mathbf{\sigma }\left(\mathbf{x}\right).$$

(11)

It should be emphasized that though the parameter $\tau$ in Equations (5) and (10) looks like time in the standard diffusion equation, in fact, it has quite different physical meaning: this is the parameter describing the spatial nonlocality. The parameter $\tau$ is proportional to the characteristic length ratio $l_0/l$ [9,10] and is nondimensional. The coefficient a in Equations (5) and (10) has the physical dimension $\left[a\right]={\mathrm{m}}^2$. This is also consistent with the nondimensional quantities introduced by Equation (39). The limit $\tau \to 0$ does not mean the transition to the initial time value, but describes the transition from the nonlocal to local stress tensor.

In the fractional nonlocal elasticity theory proposed in the paper [36], the nonlocal kernel $K\left(\mathbf{x},\tau \right)$ is introduced as the Green’s function of the Cauchy problem for the fractional diffusion operator with the fractional derivative ${\partial }^{\alpha }/\partial {\tau }^{\alpha }$ with respect to the nonlocality parameter $\tau$

$${\displaystyle \frac{{\partial }^{\alpha }K\left(\mathbf{x},\tau \right)}{\partial {\tau }^{\alpha }}}=a\mathrm{\Delta }K\left(\mathbf{x},\tau \right),$$

(12)

$$\tau =0:K\left(\mathbf{x},\tau \right)=\delta \left(\mathbf{x}\right),$$

(13)

under the assumption $0<\alpha \le 1$.

Here, $d^{\alpha }f\left(\tau \right)/d{\tau }^{\alpha }$ is the Caputo fractional derivative [40,41]

$${\displaystyle \frac{d^{\alpha }f\left(\tau \right)}{d{\tau }^{\alpha }}}={\displaystyle \frac{1}{\mathrm{\Gamma }(m-\alpha )}}{\int }_0^{\tau }{\left(\tau -u\right)}^{m-\alpha -1}{\displaystyle \frac{d^{m}f\left(u\right)}{d{u}^m}}du,m-1<\alpha <m,$$

(14)

for which the following Laplace transform rule is valid:

$$\mathcal{L}\left\{{\displaystyle \frac{{\mathrm{d}}^{\alpha }f\left(\tau \right)}{\mathrm{d}{\tau }^{\alpha }}}\right\}=s^{\alpha }{f}^{*}\left(s\right)-\sum _{k=0}^{m-1}{f}^{\left(k\right)}\left(0^{+}\right)s^{\alpha -1-k},m-1<\alpha <m,$$

(15)

where $\mathrm{\Gamma }\left(\alpha \right)$ is the gamma function, the asterisk marks the Laplace transform, s is the transform variable.

In the framework of fractional nonlocal elasticity, the nonlocality kernel $K\left(\mathbf{x},\tau \right)$ is expressed as [36,42]

$$K\left(x,\tau \right)={\displaystyle \frac{1}{2\sqrt{a}{\tau }^{\alpha /2}}}M\left({\displaystyle \frac{\alpha }{2}};{\displaystyle \frac{\left|x\right|}{\sqrt{a}{\tau }^{\alpha /2}}}\right),0\le \alpha \le 1,$$

(16)

for one spatial variable in Cartesian coordinates;

$$K\left(r,\tau \right)={\displaystyle \frac{1}{2\pi }}{\int }_0^{\infty }{E}_{\alpha }\left(-a{\xi }^2{\tau }^{\alpha }\right)J_0\left(r\xi \right)\xi \mathrm{d}\xi ,0\le \alpha \le 1,$$

(17)

in polar coordinates in the axisymmetric case;

$$K\left(r,\tau \right)={\displaystyle \frac{1}{4\pi a{\tau }^{\alpha }r}}W\left(-{\displaystyle \frac{\alpha }{2}},1-\alpha ;-{\displaystyle \frac{r}{\sqrt{a}{\tau }^{\alpha /2}}}\right),0\le \alpha \le 1,$$

(18)

in spherical coordinates in the central symmetric case.

Here

$$E_{\alpha }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\mathrm{\Gamma }(\alpha k+1)}},\alpha >0,z\in C,$$

(19)

is the Mittag-Leffler function (ML function) in one parameter $\alpha$ [40,41];

$$M\left(\alpha ;z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{{(-1)}^k{z}^k}{k!\mathrm{\Gamma }\left[-\alpha k+(1-\alpha )\right]}},0\le \alpha \le 1,z\in C,$$

(20)

is the Mainardi function [16,40];

$$W\left(\alpha ,\beta ;z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{k!\mathrm{\Gamma }\left(\alpha k+\beta \right)}},\alpha >-1,\beta \in C,z\in C,$$

(21)

is the Wright function [43].

Equations (7)–(9) are particular cases of Equations (16)–(18), respectively, when $\alpha \to 1$. For $\alpha \to 0$, we have

$$K\left(x\right)={\displaystyle \frac{1}{2\sqrt{a}}}exp\left(-{\displaystyle \frac{\left|x\right|}{\sqrt{a}}}\right)$$

(22)

for one spatial variable in Cartesian coordinates;

$$K\left(r\right)={\displaystyle \frac{1}{2\pi a}}{K}_0\left({\displaystyle \frac{r}{\sqrt{a}}}\right)$$

(23)

in polar coordinates in the axisymmetric case, where $K_0\left(r\right)$ is the modified Bessel function;

$$K\left(r\right)={\displaystyle \frac{1}{4\pi ar}}exp\left(-{\displaystyle \frac{r}{\sqrt{a}}}\right)$$

(24)

in spherical coordinates in the central symmetric case.

The Cauchy problem for the nonlocal stress tensor $\mathbf{t}\left(\mathbf{x},\tau \right)$ associated with Equations (12) and (13) takes the form:

$${\displaystyle \frac{{\partial }^{\alpha }\mathbf{t}\left(\mathbf{x},\tau \right)}{\partial {\tau }^{\alpha }}}=a\mathrm{\Delta }\mathbf{t}\left(\mathbf{x},\tau \right),$$

(25)

$$\tau =0:\mathbf{t}\left(\mathbf{x},\tau \right)=\mathbf{\sigma }\left(\mathbf{x}\right).$$

(26)

In Equation (5) and (10), Eringen [8,10] put $a=1$; we have introduced the coefficient a in Equations (12) and (25) to obtain the accurate nondimensional quantities in the limiting cases of these equations.

2. Concentrated Couple in a Plane

We consider a concentrated clockwise couple M applied at the origin of coordinates in a plane. In polar coordinates $(r,\theta )$, the components of classical (local) stress tensor $\mathbf{\sigma }$ are [44]

$${\sigma }_{rr}=0,{\sigma }_{\theta \theta }=0,{\sigma }_{r\theta }={\displaystyle \frac{A}{r^2}},$$

(27)

where

$$A={\displaystyle \frac{M}{2\pi }}.$$

(28)

According to (27) and Equations (A1) and (A2) from Appendix A, we obtain that components of the nonlocal stress tensor

$$t_{rr}=0,t_{\theta \theta }=0.$$

(29)

From key equations of the theory (25) and (26), taking into account Equation (29) and Equations (A3) and (A4) from Appendix A in the axisymmetric case with $\partial /\partial \theta =0$, it follows that the component $t_{r\theta }$ is determined from the Cauchy problem

$${\displaystyle \frac{{\partial }^{\alpha }{t}_{r\theta }}{\partial {\tau }^{\alpha }}}=a\left({\displaystyle \frac{{\partial }^2{t}_{r\theta }}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial t_{r\theta }}{\partial r}}-{\displaystyle \frac{4}{r^2}}{t}_{r\theta }\right),$$

(30)

$$\tau =0:t_{r\theta }={\displaystyle \frac{A}{r^2}}.$$

(31)

The Laplace transform with respect to the nonlocality parameter $\tau$ and the Hankel transform of the second order with respect to the radial coordinate r result in the solution in the transform domain

$${\widehat{t}}_{r\theta }^{*}(\xi ,s)={\displaystyle \frac{A}{2}}{\displaystyle \frac{s^{\alpha -1}}{s^{\alpha }+a{\xi }^2}},$$

(32)

where the hat marks the Hankel transform, $\xi$ is the transform variable, and Equation (A5) from Appendix B has been used.

Using the inverse Laplace transform formula [40,41]

$${\mathcal{L}}^{-1}\left\{{\displaystyle \frac{s^{\alpha -1}}{s^{\alpha }+b}}\right\}=E_{\alpha }\left(-b{\tau }^{\alpha }\right),$$

(33)

where $E_{\alpha }\left(z\right)$ is the ML function in one parameter $\alpha$ (19), we get

$$t_{r\theta }\left(r,\tau \right)={\displaystyle \frac{A}{2}}{\int }_0^{\infty }{E}_{\alpha }\left(-a{\xi }^2{\tau }^{\alpha }\right)J_2\left(r\xi \right)\xi \mathrm{d}\xi .$$

(34)

For $\alpha =1$,

$$E_1\left(-z^2\right)={\mathrm{e}}^{-z^2},$$

(35)

and taking into account Equation (A9) from Appendix B, we have

$$t_{r\theta }\left(r,\tau \right)={\displaystyle \frac{A}{r^2}}\left[1-\left(1+{\displaystyle \frac{r^2}{4a\tau }}\right)exp\left(-{\displaystyle \frac{r^2}{4a\tau }}\right)\right].$$

(36)

The classical elasticity solution ${\sigma }_{r\theta }=A/r^2$ (see Equation (27)) follows from Equation (36) when $\tau \to 0$. As the ML function $E_{\alpha }\left(0\right)=1$, in the general case described by the solution (34), the classical elasticity solution is obtained using the integral (A7) from Appendix B.

Another particular case of the solution (34) is obtained for $\alpha \to 0$ when

$$E_0\left(-z^2\right)={\displaystyle \frac{1}{1+z^2}}.$$

(37)

Evaluating the corresponding integral (see Equation (A11) from Appendix B), we arrive at

$$t_{r\theta }\left(r\right)={\displaystyle \frac{A}{2a}}\left[{\displaystyle \frac{2a}{r^2}}-K_2\left({\displaystyle \frac{r}{\sqrt{a}}}\right)\right].$$

(38)

Figure 1 shows the dependence of the nondimensional stress component ${\overline{t}}_{r\theta }$ on the nondimensional distance $\overline{r}$ for different values of the order of fractional derivative $\alpha$. The following nondimensional quantities are introduced:

$${\overline{t}}_{r\theta }={\displaystyle \frac{a{\tau }^{\alpha }}{A}}{t}_{r\theta },\overline{r}={\displaystyle \frac{r}{\sqrt{a}{\tau }^{\alpha /2}}}.$$

(39)

3. Concentrated Couple on the Boundary of a Half-Plane

Next, we consider a half-plane loaded by a counterclockwise couple M applied at the origin of coordinates. In the polar coordinate system $(r,\theta )$, the components of the classical (local) stress tensor $\mathbf{\sigma }$ are [44]

$${\sigma }_{rr}=-{\displaystyle \frac{B}{r^2}}sin2\theta ,{\sigma }_{\theta \theta }=0,{\sigma }_{r\theta }={\displaystyle \frac{B}{r^2}}{cos}^2\theta ={\displaystyle \frac{B}{2r^2}}+{\displaystyle \frac{B}{2r^2}}cos2\theta ,$$

(40)

where

$$B={\displaystyle \frac{2M}{\pi }}.$$

(41)

We assume that

$$t_{rr}(r,\theta ,\tau )=T_{rr}(r,\tau )sin2\theta ,$$

(42)

$$t_{\theta \theta }(r,\theta ,\tau )=T_{\theta \theta }(r,\tau )sin2\theta ,$$

(43)

$$t_{r\theta }(r,\theta ,\tau )=T_{r\theta }^{\left(1\right)}(r,\tau )+T_{r\theta }^{\left(2\right)}(r,\tau )cos2\theta .$$

(44)

According to Equations (A1)–(A4) from Appendix A, for the part of the stress tensor component $T_{r\theta }^{\left(1\right)}(r,\tau )$, we obtain the Cauchy problem

$${\displaystyle \frac{{\partial }^{\alpha }{T}_{r\theta }^{\left(1\right)}}{\partial {\tau }^{\alpha }}}a\left({\displaystyle \frac{{\partial }^2{T}_{r\theta }^{\left(1\right)}}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial T_{r\theta }^{\left(1\right)}}{\partial r}}-{\displaystyle \frac{4}{r^2}}{T}_{r\theta }^{\left(1\right)}\right),$$

(45)

$$\tau =0:T_{r\theta }^{\left(1\right)}={\displaystyle \frac{B}{2r^2}}.$$

(46)

The Cauchy problem of such a type was considered in Section 2. The solution reads

$$T_{r\theta }^{\left(1\right)}(r,\tau )={\displaystyle \frac{B}{4}}{\int }_0^{\infty }{E}_{\alpha }\left(-a{\xi }^2{\tau }^{\alpha }\right)J_2\left(r\xi \right)\xi \mathrm{d}\xi .$$

(47)

In compliance with the assumption (42)–(44) and Equations (A1)–(A4) from Appendix A, the coefficients $T_{rr}(r,\tau )$, $T_{\theta \theta }(r,\tau )$ and $T_{r\theta }^{\left(2\right)}(r,\tau )$ satisfy the equations

$${\displaystyle \frac{{\partial }^{\alpha }{T}_{rr}}{\partial {\tau }^{\alpha }}}=a\left[{\displaystyle \frac{{\partial }^2{T}_{rr}}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial T_{rr}}{\partial r}}-{\displaystyle \frac{4}{r^2}}{T}_{rr}+{\displaystyle \frac{8}{r^2}}{T}_{r\theta }^{\left(2\right)}-{\displaystyle \frac{2}{r^2}}\left(T_{rr}-T_{\theta \theta }\right)\right],$$

(48)

$$\tau =0:T_{rr}=-{\displaystyle \frac{B}{r^2}},$$

(49)

$${\displaystyle \frac{{\partial }^{\alpha }{T}_{\theta \theta }}{\partial {\tau }^{\alpha }}}=a\left[{\displaystyle \frac{{\partial }^2{T}_{\theta \theta }}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial T_{\theta \theta }}{\partial r}}-{\displaystyle \frac{4}{r^2}}{T}_{\theta \theta }-{\displaystyle \frac{8}{r^2}}{T}_{r\theta }^{\left(2\right)}+{\displaystyle \frac{2}{r^2}}\left(T_{rr}-T_{\theta \theta }\right)\right],$$

(50)

$$\tau =0:T_{\theta \theta }=0,$$

(51)

$${\displaystyle \frac{{\partial }^{\alpha }{T}_{r\theta }^{\left(2\right)}}{\partial {\tau }^{\alpha }}}=a\left[{\displaystyle \frac{{\partial }^2{T}_{r\theta }^{\left(2\right)}}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial T_{r\theta }^{\left(2\right)}}{\partial r}}-{\displaystyle \frac{8}{r^2}}{T}_{r\theta }^{\left(2\right)}+{\displaystyle \frac{4}{r^2}}\left(T_{rr}-T_{\theta \theta }\right)\right],$$

(52)

$$\tau =0:T_{r\theta }^{\left(2\right)}={\displaystyle \frac{B}{2r^2}}.$$

(53)

The system of coupled Equations (48)–(53) can be splitted into independent equations by introducing the following new sought-for functions:

$$f(r,\tau )=T_{rr}(r,\tau )+T_{\theta \theta }(r,\tau ),$$

(54)

$$g(r,\tau )=T_{rr}(r,\tau )-T_{\theta \theta }(r,\tau )+2T_{r\theta }^{\left(2\right)}(r,\tau ),$$

(55)

$$h(r,\tau )=T_{rr}(r,\tau )-T_{\theta \theta }(r,\tau )-2T_{r\theta }^{\left(2\right)}(r,\tau ).$$

(56)

Hence,

$$T_{rr}(r,\tau )={\displaystyle \frac{1}{2}}f(r,\tau )+{\displaystyle \frac{1}{4}}\left[g(r,\tau )+h(r,\tau )\right],$$

(57)

$$T_{\theta \theta }(r,\tau )={\displaystyle \frac{1}{2}}f(r,\tau )-{\displaystyle \frac{1}{4}}\left[g(r,\tau )+h(r,\tau )\right],$$

(58)

$$T_{r\theta }^{\left(2\right)}(r,\tau )={\displaystyle \frac{1}{4}}\left[g(r,\tau )-h(r,\tau )\right].$$

(59)

For the auxiliary functions $f(r,\tau )$, $g(r,\tau )$ and $h(r,\tau )$, we obtain a splitted system of the Cauchy problems

$${\displaystyle \frac{{\partial }^{\alpha }f}{\partial {\tau }^{\alpha }}}=a\left({\displaystyle \frac{{\partial }^{2}f}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial f}{\partial r}}-{\displaystyle \frac{4}{r^2}}f\right),$$

(60)

$$\tau =0:f(r,\tau )=-{\displaystyle \frac{B}{r^2}},$$

(61)

$${\displaystyle \frac{{\partial }^{\alpha }g}{\partial {\tau }^{\alpha }}}=a\left({\displaystyle \frac{{\partial }^{2}g}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial g}{\partial r}}\right),$$

(62)

$$\tau =0:g(r,\tau )=0,$$

(63)

$${\displaystyle \frac{{\partial }^{\alpha }h}{\partial {\tau }^{\alpha }}}=a\left({\displaystyle \frac{{\partial }^{2}h}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial h}{\partial r}}-{\displaystyle \frac{16}{r^2}}h\right),$$

(64)

$$\tau =0:h(r,\tau )=-{\displaystyle \frac{2B}{r^2}}.$$

(65)

It follows from Equations (62) and (63) that

$$g(r,\tau )=0.$$

(66)

The Laplace transform with respect to the nonlocality parameter $\tau$ and the Hankel transform (of the second order for the function $f(r,\tau )$ and of the fourth order for the function $h(r,\tau )$) with respect to the radial coordinate r with taking into account Equations (A5) and (A6) from Appendix B give the expressions in the transform domain

$${\widehat{f}}^{*}(\xi ,s)=-{\displaystyle \frac{B}{2}}{\displaystyle \frac{s^{\alpha -1}}{s^{\alpha }+a{\xi }^2}},$$

(67)

$${\widehat{h}}^{*}(\xi ,s)=-{\displaystyle \frac{B}{2}}{\displaystyle \frac{s^{\alpha -1}}{s^{\alpha }+a{\xi }^2}},$$

(68)

and after inverting the integral transforms, we get

$$f(r,\tau )=-{\displaystyle \frac{B}{2}}{\int }_0^{\infty }{E}_{\alpha }\left(-a{\xi }^2{\tau }^{\alpha }\right)J_2\left(r\xi \right)\xi \mathrm{d}\xi ,$$

(69)

$$h(r,\tau )=-{\displaystyle \frac{B}{2}}{\int }_0^{\infty }{E}_{\alpha }\left(-a{\xi }^2{\tau }^{\alpha }\right)J_4\left(r\xi \right)\xi \mathrm{d}\xi .$$

(70)

The components of the nonlocal stress tensor take the form:

$$t_{rr}(r,\theta ,\tau )=-{\displaystyle \frac{B}{8}}{\int }_0^{\infty }{E}_{\alpha }\left(-a{\xi }^2{\tau }^{\alpha }\right)\left[2J_2\left(r\xi \right)+J_4\left(r\xi \right)\right]\xi \mathrm{d}\xi sin2\theta ,$$

(71)

$$t_{\theta \theta }(r,\theta ,\tau )=-{\displaystyle \frac{B}{8}}{\int }_0^{\infty }{E}_{\alpha }\left(-a{\xi }^2{\tau }^{\alpha }\right)\left[2J_2\left(r\xi \right)-J_4\left(r\xi \right)\right]\xi \mathrm{d}\xi sin2\theta ,$$

(72)

$$\begin{array}{ccc}\hfill t_{r\theta }(r,\theta ,\tau )& =& {\displaystyle \frac{B}{4}}{\int }_0^{\infty }{E}_{\alpha }\left(-a{\xi }^2{\tau }^{\alpha }\right)J_2\left(r\xi \right)\xi \mathrm{d}\xi \hfill \\ & +& {\displaystyle \frac{B}{8}}{\int }_0^{\infty }{E}_{\alpha }\left(-a{\xi }^2{\tau }^{\alpha }\right)J_4\left(r\xi \right)\xi \mathrm{d}\xi cos2\theta .\hfill \end{array}$$

(73)

The particular case of the solution (71)–(73) corresponding to $\alpha =1$ is obtained using the integrals (A9) and (A10) from Appendix B and reads:

$$t_{rr}(r,\theta ,\tau )=-{\displaystyle \frac{B}{r^2}}\left[1-{\displaystyle \frac{6a\tau }{r^2}}+\left({\displaystyle \frac{1}{2}}+{\displaystyle \frac{6a\tau }{r^2}}-{\displaystyle \frac{r^2}{16a\tau }}\right)exp\left(-{\displaystyle \frac{r^2}{4a\tau }}\right)\right]sin2\theta ,$$

(74)

$$t_{\theta \theta }(r,\theta ,\tau )=-{\displaystyle \frac{B}{r^2}}\left[{\displaystyle \frac{6a\tau }{r^2}}-{\displaystyle \frac{3}{2}}\left(1+{\displaystyle \frac{4a\tau }{r^2}}+{\displaystyle \frac{r^2}{8a\tau }}\right)exp\left(-{\displaystyle \frac{r^2}{4a\tau }}\right)\right]sin2\theta ,$$

(75)

$$\begin{array}{ccc}\hfill t_{r\theta }(r,\theta ,\tau )& =& {\displaystyle \frac{B}{2r^2}}\left[1-\left(1+{\displaystyle \frac{r^2}{4a\tau }}\right)exp\left(-{\displaystyle \frac{r^2}{4a\tau }}\right)\right]\hfill \\ & +& {\displaystyle \frac{B}{2r^2}}\left[1-{\displaystyle \frac{12a\tau }{r^2}}+\left(2+{\displaystyle \frac{r^2}{8a\tau }}+{\displaystyle \frac{12a\tau }{r^2}}\right)exp\left(-{\displaystyle \frac{r^2}{4a\tau }}\right)\right]cos2\theta .\hfill \end{array}$$

(76)

Figure 2 presents the dependence of the nondimensional stress coefficient ${\overline{T}}_{rr}$ on the nondimensional distance $\overline{r}$ for different values of the order of fractional derivative $\alpha$. The nondimensional distance $\overline{r}$ is the same as in Equation (39), the nondimensional ${\overline{T}}_{rr}$ is introduced as

$${\overline{T}}_{rr}={\displaystyle \frac{a{\tau }^{\alpha }}{B}}{T}_{rr}.$$

(77)

4. Concluding Remarks

We have investigated the nonlocal stress fields caused by a concentrated couple in a plane and by a concentrated couple acting on the boundary of a half-plane in the framework of the new theory of fractional nonlocal elasticity in which the nonlocal kernel in the stress constitutive equation is the Green’s function of the Cauchy problem for the fractional diffusion operator.

The obtained solutions are free from nonphysical singularities appearing in the classical local elasticity solutions. The nonlocal stress equals zero at the point of application of the couple, achieves maximum at some distance from this point and approaches the classical local stress at a large distance from the point of application. The nonlocal solutions for concentrated forces and concentrated couples are important for stress analysis in nanostructures and composite materials (see, for example, [45]). The fractional nonlocal elasticity used in the present paper allows better matching of results at macro- and micro- (nano-) structural levels with different scale lengths. The impotance of such matching was accented in [9,10,46].

It should be emphasized that Figure 1 presents the nonlocal stress component $t_{r\theta }$, whereas Figure 2 shows only the coefficient $T_{rr}$ (see Equation (42)). As the ML function $E_{\alpha }\left(0\right)=1$, from Equations (71)–(73) with taking into account integrals (A7) and (A8) from Appendix B it is evident that the solution (71)–(73) satisfies the conditions (40) for $\tau \to 0.$ It should be noted that though in the case of a concentrated couple on the boundary of a half-plane the local stress component ${\sigma }_{\theta \theta }=0$, the nonlocal stress component $t_{\theta \theta }\ne 0$ and tends to zero when the nonlocality parameter $\tau \to 0$.