## 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 [PERSON] [1], [PERSON] [2], [PERSON] [3], [PERSON] [4, 5], and [PERSON] [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 [PERSON]'s investigations on nonlocal elasticity [8, 9, 10], the nonlocal kernels were often chosen as [PERSON]'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 [PERSON]'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  
\[\mathbf{\
abla}\cdot\mathbf{t}(\mathbf{x},\tau)=0, \tag{1}\]  
the constitutive equation for the nonlocal stress tensor  
\[\mathbf{t}(\mathbf{x},\tau)=\int_{V}K\big{(}|\mathbf{x}-\mathbf{x}^{\prime}|, \tau\big{)}\mathbf{\sigma}\big{(}\mathbf{x}^{\prime}\big{)}dv(\mathbf{x}^{\prime}), \tag{2}\]  
the constitutive equation for the local stress tensor  
\[\mathbf{\sigma}\big{(}\mathbf{x}^{\prime}\big{)}=2\mu\,\mathbf{e}\big{(}\mathbf{x }^{\prime}\big{)}+\lambda\,\mathbf{tr}\,\mathbf{e}\big{(}\mathbf{x}^{\prime} \big{)}\mathbf{I}, \tag{3}\]  
and geometrical relations  
\[\mathbf{e}\big{(}\mathbf{x}^{\prime}\big{)}=\frac{1}{2}\big{[}\mathbf{\
abla}^{ \prime}\mathbf{u}\big{(}\mathbf{x}^{\prime}\big{)}+\mathbf{u}\big{(}\mathbf{ x}^{\prime}\big{)}\mathbf{\
abla}^{\prime}\big{]}. \tag{4}\]  
Here \(\mathbf{\sigma}(\mathbf{x}^{\prime})\) and \(\mathbf{t}(\mathbf{x},\tau)\) 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 Lame constants, \(\mathbf{\
abla}\) and \(\mathbf{\
abla}^{\prime}\) stand for the gradient operators with respect to \(\mathbf{x}\) and \(\mathbf{x}^{\prime}\), \(\mathbf{I}\) is the unit tensor. The nonlocal kernel \(K(|\mathbf{x}-\mathbf{x}^{\prime}|,\tau)\) 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 \(K(|\mathbf{x}-\mathbf{x}^{\prime}|,\tau)\) is a delta sequence and in the classical elasticity limit, \(\tau\to 0\) tends to the Dirac delta function \(\delta(|\mathbf{x}-\mathbf{x}^{\prime}|)\).