1 Introduction

The development and the proof of completeness of general solutions in classical elasticity were the subject of research for more than a hundred years. The most famous general solutions are known after Boussinesq [1] and Galerkin [2] and Papkovich [3] and Neuber [4]. The interrelations between these solutions and their completeness have been discussed by Mindlin [5], Gurtin [6], Noll [7], Sternberg [8] and Eubanks [9], Sokolnikoff [10], Slobodyansky [11], Wang and Wang [12] and others. The universal constructive scheme for the development of general solutions and evaluation of their completeness and non-uniqueness within classical elasticity have been established based on the matrix methods of the theory of differential operators by Lurie [13], Wang [14] and others [15].

In the present study, we consider the strain gradient elasticity theory (SGET), which general formulation for isotropic materials has been developed by Mindlin [16] and Toupin [17]. The main feature of SGET is the assumption that the potential-energy density of the media depends on the gradient of strain in addition to strain. In the constitutive equations of SGET, there arise five additional material constants in addition to two classical Lame parameters for isotropic materials, though only two additional parameters arise in the equilibrium equations [18]. The boundary-value problem of SGET consists of the fourth-order equilibrium equations and an extended number of boundary conditions, which form can be obtained based on the variational approach [16, 17]. In SGET, there arise an extended definitions of surface tractions accounting for the normal and for the curvature of the Cauchy cut, as well as additional definitions for double tractions and edge tractions [16, 17, 19]. The number of equilibrium equations in SGET remains the same as in classical elasticity since the number of primary field variables (components of the displacement vector) does not change.

Note that Mindlin–Toupin SGET contains as special cases a number of famous simplified gradient theories [20–22] and also several kinds of incomplete gradient theories like the couple stress theory, the dilatation gradient elasticity, etc. [18, 23–26]. Nowadays, these theories attract an increasing attention in applications to fracture mechanics and dislocation problems [27–31], in contact mechanics and indentation problems [32–34], in the studies of small-scale and high-frequency processes [35, 36], and in the description of the mechanical behavior of composites and metamaterials [37–41].

The first variant of the general solution for equilibrium equations of SGET has been presented in the initial work by Mindlin [16], though the particular variant of this solution within the couple stress theory has been established earlier by Mindlin and Tiersten [23]. The form of the Mindlin solution [16] can be treated as a generalized variant of the classical Papkovich–Neuber (PN) solution. It defines the displacement field in a rather complicated form through the vector and scalar functions that obey the fourth-order governing equations. Later, the simpler variants of PN solutions with stress functions that obey the second-order equations were established within SGET [38, 42–46], though the completeness of these solutions was not proven or was implied on the basis of heuristic reasoning [47].

Lurie et al. [38, 42] introduced PN general solution within the simplified variant of gradient theory and represented the displacement field through the additive decomposition into the sum of classical and gradient parts. The classical part of the solution was defined similarly to the standard PN solutions through the harmonic vector and scalar functions. The gradient part of the solution was defined by using two additional vector functions that satisfied the modified Helmholtz equations. Solyaev et al. [45] used a similar form of PN solution within SGET and reduced the representation for the gradient part of the solution to six scalar functions for the arbitrary curvilinear coordinates, which allows the separation of variables in the Helmholtz equation. Recently, this representation was additionally simplified such that the gradient part of PN solution was defined through the modified variant of Helmholtz decomposition, in which the scalar and the vector potentials obey the modified Helmholtz equations with different coefficients [47]. A similar result has also been established within the simplified strain gradient elasticity theory by Charalambopoulos et al. [44], though the possibility of the use of a solenoidal vector stress function for the gradient part of the displacement field has not been considered. Representation of the displacement field in terms of Lame’s potentials has been used by Placidi and Dhaba within the Saint-Venant’s problems [46].

In the present paper, we derive the general solutions for equilibrium equations of Mindlin–Toupin SGET and prove its completeness in a deductive manner that was suggested by Papkovich [3] and used by Gurtin and Sternberg [48], Gurtin [49] within the classical elasticity. We start from the Helmholtz decomposition for the displacement field, which is valid for the arbitrary smooth vector fields in the bounded domains and which validity in the infinite domains has been proven for weakly decaying fields in the classical work by Gurtin [6]. Based on Helmholtz decomposition and analysis of equilibrium equations of SGET, we introduce the definition of the modified Galerkin vector that appears to obey the eight-order governing equation and that allows to define the displacement field in the presence of arbitrary bulk force, i.e., we obtain the extended variant of Boussinesq–Galerkin (BG) solution within SGET. Generalizing the approach of classical elasticity, we introduce then relations between the Galerkin vector and the Papkovich–Neuber stress functions. The resulting modified form of PN solution becomes very attractive for applications because it provides a very simple form of the general solution of Mindlin–Toupin SGET with additive decomposition into the classical and gradient parts of the displacement field. The classical part is defined in standard PN form through the harmonic vector and harmonic function, while the gradient part is simply represented through the modified Helmholtz decomposition. In such a way, we prove the correctness of the previously supposed simplified form of PN solution within SGET [47].

Combining both results for BG and PN forms of solution, we prove the theorem of their completeness within SGET in the sense of similar proofs developed by Mindlin [5] and Gurtin [49] within classical elasticity. Considered kind of a proof can be formulated, according to Truesdell, as follows: “corresponding to any stress field satisfying the given equations there exists at least one suitable choice of the stress functions” [50]. Hence, to prove the completeness of BG solution, we show an explicit representation for the modified Galerkin stress function through a given displacement field that satisfies the equilibrium equation of SGET. Similarly to classical elasticity, this is done by using the analogy between SGET equilibrium equation and the relation between the Galerkin stress function and the displacement field and also involving established PN representation of the solution within SGET. Then, based on the relations between the stress functions of PN and BG solutions, we prove the completeness of the former. Thus, in the presented proof, it is essential to have both representations of solutions in BG and PN forms.

The key point of the presented results is the proposed modified definition of the Galerkin stress function and its specific relations to the Papkovich stress functions within SGET that have not been established previously for the best of author’s knowledge. Applications of the obtained results can be related to the wide class of boundary value problems that can be solved analytically in a simple manner by using PN representation, for which we show the completeness [43, 47, 51]. Incorporation of complete general solutions into the numerical schemes (like in a Trefftz method) can also be an important issue for the development of stable and flexible numerical solvers within SGET [42, 52].

2 Preliminaries

Equilibrium equations of SGET can be represented in the following form [16]:

$$ \alpha (1-l_{1}^{2} \nabla ^{2}) \nabla \nabla \cdot \pmb u - (1-l_{2}^{2} \nabla ^{2}) \nabla \times \nabla \times \pmb u = -\frac{\pmb b}{\mu}, $$
(1)

where: \(\pmb u(\pmb{r})\) is the vector of mechanical displacements at a point \(\pmb r = \{x_{1}, x_{2}, x_{3}\}\); \(\pmb b(\pmb{r})\) is the body-force density vector; \(\alpha = (\lambda +2\mu )/\mu = 2(1-\nu )/(1-2\nu )\) is classical non-dimensional parameter; \(\lambda \), \(\mu \) are the classical Lame constants; \(\nu \) is Poisson’s ratio; \(l_{1}\) and \(l_{2}\) are the length scale parameters of isotropic elastic material that arise in the equilibrium equations of Mindlin–Toupin strain gradient elasticity.

Note that the length scale parameters \(l_{1}\) and \(l_{2}\) are differently defined through the additional gradient material constants within the so-called Mindlin Forms I, II and III that correspond to the formulation of SGET in terms of the second gradient of displacement, strain gradients, and symmetric/anti-symmetric parts of strain gradients, respectively [16]. Nevertheless, the form of equilibrium equations (1) will be the same in all of these variants of SGET [16]. Equilibrium equations of different simplified gradient theories can be obtained from Eq. (1) as the particular cases. For example, in the Aifantis theory, it is valid \(l_{1}=l_{2}\) [20], in the couple stress theory and in the modified couple stress theory \(l_{1}=0\) [23, 53], and in the dilatation gradient elasticity \(l_{2}=0\) [24]. Assuming that both length scale parameters are equal to zero \(l_{1}=l_{2}=0\) equation (1) reduces to the classical elasticity equilibrium equation. Therefore, all considerations presented below for equation (1) will be valid for any kind of gradient theories mentioned above, and all presented solutions must contain the corresponding classical solutions in the particular cases.

For the purpose of the following derivations, we also need to define the general solution for the fourth-order scalar (or vector) equation of the following form:

$$ (1-l^{2} \nabla ^{2}) \nabla ^{2} \mathcal {F} = \mathcal {B}, $$
(2)

where ℱ is the scalar (or vector) field that should be found in the bounded or in the unbounded region of three-dimensional euclidean space \(D\); ℬ is the prescribed continuous scalar (or vector) field of class \(C^{n}\) (\(n\geq 1\)) defined in \(D\); and we assume that \(l\in \mathbb{R}\) so that operator \((1-l^{2} \nabla ^{2})\) corresponds to the modified Helmholtz equation, also known as the screened Poisson equation.

Theorem 1

General solution of equation (2) is given by

$$ \mathcal {F} = \mathcal {F}_{c} + \mathcal {F}_{g}, $$
(3)

where \(\mathcal {F}_{c}\) and \(\mathcal {F}_{g}\) are the general solutions of the Poisson equation and inhomogeneous modified Helmholtz equations, respectively

$$ \nabla ^{2}\mathcal {F}_{c} =\mathcal {B}, \qquad (1-l^{2} \nabla ^{2}) \mathcal {F}_{g} = l^{2}\mathcal {B}, $$
(4)

Proof

Let us define \(\mathcal {F}_{c}\) as

$$ \mathcal {F}_{c} = (1-l^{2} \nabla ^{2})\mathcal {F}, $$
(5)

Then from (2), (5) it follows that \(\mathcal {F}_{c}\) is the general solution of the Poisson’s equation

$$ \nabla ^{2}\mathcal {F}_{c} = \mathcal {B} $$
(6)

and that we can define:

$$ \mathcal {F}_{c} = \bar{\mathcal {F}}_{c} + \mathcal {F}_{c}^{*}, \quad \nabla ^{2} \bar {\mathcal {F}}_{c} = 0, \quad \nabla ^{2} \mathcal {F}_{c}^{*} = \mathcal {B}, $$
(7)

where \(\bar{\mathcal {F}}_{c}\) is the general solution of the Laplace equation and \(\mathcal {F}^{*}_{c}\) is some particular solution of (6).

Then we represent the general solution of the inhomogeneous Helmholtz equation (5) in the form

$$ \mathcal {F} = \bar {\mathcal {F}} + \mathcal {F}^{*}, \quad (1-l^{2} \nabla ^{2})\bar {\mathcal {F}} = 0, \quad (1-l^{2} \nabla ^{2}) \mathcal {F}^{*} = \mathcal {F}_{c}, $$
(8)

where \(\bar {\mathcal {F}}\) is the general solution of homogeneous Helmholtz equations and the particular solution \(\mathcal {F}^{*}\) can be defined as

$$ \mathcal {F}^{*} = \mathcal {F}_{c} + \mathcal {F}_{g}^{*}, $$
(9)

where \(\mathcal {F}_{g}^{*}\) is some unknown function.

Substituting (9) into (8)3 we find that \(\mathcal {F}_{g}^{*}\) should be the particular solution of the following inhomogeneous Helmholtz equation

$$ (1-l^{2} \nabla ^{2}) \mathcal {F}_{g}^{*} = l^{2}\mathcal {B}, $$
(10)

where we take into account (7).

Then, introducing \(\mathcal {F}_{g} = \bar {\mathcal {F}} + \mathcal {F}_{g}^{*}\) and using (8), (10) we find that \(\mathcal {F}_{g}\) is the general solution of the inhomogeneous Helmholtz equation

$$ (1-l^{2} \nabla ^{2}) \mathcal {F}_{g} = l^{2}\mathcal {B} $$
(11)

and according to given definitions (8), (9) we obtain \(\mathcal {F} = \mathcal {F}_{c} + \mathcal {F}_{g}\). □

Corollary 1

General solutions for equations (4) can be presented in the following form:

$$ \mathcal {F}_{c} =(1-l^{2} \nabla ^{2})\mathcal {F}, \qquad \mathcal {F}_{g} = l^{2} \nabla ^{2}\mathcal {F}. $$
(12)

Proof

The proof follows from definitions (3) and (5). □

Corollary 1 can be treated as the completeness theorem for representation of general solution (3) to equation (2) since it shows how to define the parts of solution \(\mathcal {F}_{c}\) and \(\mathcal {F}_{g}\) for a given arbitrary field ℱ that satisfies (3). Theorem 1 is also known as Bifield ansatz [54].

For the further analysis, we will also need the explicit form of particular solutions of equations (4) that can be represented as [55]

$$ \mathcal {F}_{c}^{*}(\pmb r) = (\mathcal {N} \ast \mathcal {B}) (\pmb r) = - \frac{1}{4\pi} \int _{D} \frac{\mathcal {B}(\pmb \xi )}{\lvert \pmb r - \pmb \xi \rvert} dv_{\pmb\xi}, $$
(13)
$$ \mathcal {F}_{g}^{*}(\pmb r) = (\mathcal {H} \ast \mathcal {B}) (\pmb r)= \frac{1}{4\pi}\int _{D} \frac{ e^{-\lvert \pmb r - \pmb \xi \rvert /l} \,\mathcal {B}(\pmb \xi )}{\lvert \pmb r - \pmb \xi \rvert} dv_{\pmb\xi}, $$
(14)

where \(\mathcal {N}(\pmb r,\pmb\xi )\) is the Newtonian potential and \(\mathcal {H}(\pmb r,\pmb\xi )\) is the Green’s function of inhomogeneous modified Helmholtz equation (4)2.

Notations and definitions.

In the following derivations we will use bold symbols for vectors and italic symbols for scalars. General solutions of homogeneous equations will be denoted with bar symbols (\(\bar{\psi}\), \(\bar{\pmb\Psi}\)). Particular solutions will be denoted with star superscripts (\(\psi ^{*}\), \(\pmb\Psi ^{*}\)). Potentials (stress functions) that satisfy the Laplace or the Poisson equations will be denoted with subscript “c” – “classic” (\(\psi _{c}\), \(\pmb B_{c}\)). Potentials that satisfy the modified Helmholtz equation will be denoted with subscript “g” – “gradient” (\(\psi _{g}\), \(\pmb B_{g}\)). By the term “general solution”, we will denote the representation of the solution for a given partial differential equation (in terms of potentials or stress functions) that includes all possible solutions for this equation [15]. By the term “fundamental solution” (considered in Sect. 6), we denote the solution for the equilibrium state of the linearly elastic infinite body subjected to a concentrated force.

3 Boussinesq–Galerkin Solution

According to the Helmholtz theorem every suitably regular vector field \(\pmb u(\pmb r)\) admits the representation:

$$ \pmb u = \nabla \phi + \nabla \times \pmb S, $$
(15)

where \(\phi \) is the scalar potential and \(\pmb S\) is the vector potential for which we can assume that \(\nabla \cdot \pmb S=0\) without loss of generality.

For the further analysis, it will be enough to assume that in the infinite domains, representation (15) can be introduced under assumptions of weak decay conditions for the displacement field \(\pmb u\) that are used in classical elasticity [6], though the validity of the Helmholtz theorem for the fields with sub-linear growth has also been established [56, 57].

Substituting (15) into (1) and using standard vector calculus identities, we obtain equilibrium equations in terms of potentials:

$$ \nabla ^{2} \left ( \alpha (1-l_{1}^{2} \nabla ^{2}) \nabla \phi + (1-l_{2}^{2} \nabla ^{2}) \nabla \times \pmb S \right ) = -\frac{\pmb b}{\mu}. $$
(16)

Equation (16) can be reduced to the high-order equation with respect to the single vector function \(\pmb W(\pmb r)\) by using the following definitions of potentials \(\phi \) and \(\pmb S\):

$$ \begin{aligned} \phi &= \frac{1}{\alpha}(1-l_{2}^{2} \nabla ^{2})\nabla \cdot \pmb W, \\ \pmb S &= -(1-l_{1}^{2} \nabla ^{2})\nabla \times \pmb W. \end{aligned} $$
(17)

Combining (16)–(17) we obtain

$$ \begin{aligned} (1-l_{1}^{2} \nabla ^{2})(1-l_{2}^{2} \nabla ^{2})\nabla ^{2} \nabla ^{2} \pmb W = -\frac{\pmb b}{\mu}. \end{aligned} $$
(18)

Using (15) and (17) we find that

$$ \pmb u = \frac{1}{\alpha}(1-l_{2}^{2} \nabla ^{2})\nabla \nabla \cdot \pmb W -(1-l_{1}^{2} \nabla ^{2})\nabla \times \nabla \times \pmb W $$
(19)

or alternatively

$$ \pmb u = (1-l_{1}^{2} \nabla ^{2})\nabla ^{2}\pmb W - \kappa (1-l_{3}^{2} \nabla ^{2})\nabla \nabla \cdot \pmb W, $$
(20)

where \(\kappa = \frac{\alpha -1}{\alpha} = \frac{1}{2(1-\nu )}\) and \(l^{2}_{3} = \frac{\alpha}{1-\alpha}\left (\frac{1}{\alpha} l_{2}^{2} - l_{1}^{2}\right )\) (it can be shown that \(l_{3}\in \mathbb{R}\) that is the consequence of definitions of \(l_{1}\), \(l_{2}\) and requirements for the positive definition of strain energy density in isotropic SGET, see [18]).

Representation (20) (or (19)) should be treated as Boussinesq–Galerkin (BG) solution generalized for SGET. Vector function \(\pmb W\) is Galerkin stress function that obeys the eight-order bi-harmonic/bi-Helmholtz governing equation (18). In absence of gradient effects (\(l_{1}=l_{2}=0\)) this representation reduces to classical BG solution. The key idea in the presented form of BG solution is the appropriate choice of definitions for the Helmholtz potentials (17).

Note that bi-Helmholtz/bi-Laplace equation has been considered previously within the second strain gradient elasticity theory (accounting for the dependence of strain energy on strain, gradient of strain, and second gradient of strain), where it was shown that a similar eight-order equation defines the fundamental solution for the modified Airy stress function within the edge dislocations problems [58, 59]. The Green’s function of the bi-Helmholtz/bi-Laplace equation was given in Refs. [30, 58].

4 Papkovich–Neuber Solution

Let us introduce the following vector function:

$$ \pmb B = (1-l_{1}^{2} \nabla ^{2})\nabla ^{2}\pmb W. $$
(21)

From equilibrium equations written in terms of the Galerkin stress function (18), we found that \(\pmb B\) has to satisfy

$$ (1-l_{2}^{2} \nabla ^{2})\nabla ^{2}\pmb B = -\frac{\pmb b}{\mu}. $$
(22)

By making use of Theorem 1, the general solution of equation (22) can be constructed as

$$ \pmb B = \pmb B_{c} + \pmb B_{g}, $$
(23)

where

$$ \nabla ^{2}\pmb B_{c} = -\frac{\pmb b}{\mu}, \qquad (1-l_{2}^{2} \nabla ^{2})\pmb B_{g} = -l_{2}^{2}\frac{\pmb b}{\mu}, $$
(24)

and according to Corollary 1 we also have

$$ \pmb B_{c} = (1-l_{2}^{2}\nabla ^{2})\pmb B, \qquad \pmb B_{g} = l_{2}^{2} \nabla ^{2}\pmb B. $$
(25)

Then, let us define the general solution of (21), namely

$$ \pmb W = \bar {\pmb W} + \pmb W^{*}, $$
(26)

where \(\bar {\pmb W}\) is the general solution of corresponding homogeneous equations and \(\pmb W^{*}\) is appropriate particular solution, i.e.:

$$ (1-l_{1}^{2} \nabla ^{2})\nabla ^{2} \bar {\pmb W} = 0, \qquad (1-l_{1}^{2} \nabla ^{2})\nabla ^{2} \pmb W^{*} = \pmb B. $$
(27)

Substituting (26) into BG solution for the displacement field (20) and taking into account (27) we obtain

$$ \begin{aligned} \pmb u &= (1-l_{1}^{2} \nabla ^{2})(\nabla ^{2}\bar {\pmb W} + \nabla ^{2}\pmb W^{*}) - \kappa (1-l_{3}^{2} \nabla ^{2})\nabla ( \nabla \cdot \bar {\pmb W} + \nabla \cdot \pmb W^{*}) \\ &= \pmb B - \kappa \nabla (\bar{\varphi}+ \varphi ^{*}), \end{aligned} $$
(28)

where we introduce new scalar potentials:

$$ \begin{aligned} \bar{\varphi}= (1-l_{3}^{2} \nabla ^{2})(\nabla \cdot \bar {\pmb W}), \qquad \varphi ^{*} = (1-l_{3}^{2} \nabla ^{2})(\nabla \cdot \pmb W^{*}). \end{aligned} $$
(29)

Using (27)1, (29)1 we immediately find that potential \(\bar{\varphi}\) obeys the homogeneous equation

$$ \begin{aligned} (1-l_{1}^{2} \nabla ^{2})\nabla ^{2}\bar{\varphi}= 0, \end{aligned} $$
(30)

which general solution can be found based on Theorem 1.

In order to derive the representation for potential \(\varphi ^{*}\) let us consider the relation that follows from (27)2 and (29)2:

$$ \begin{aligned} (1-l_{1}^{2} \nabla ^{2})\nabla ^{2}\varphi ^{*} &= (1-l_{1}^{2} \nabla ^{2})(1-l_{3}^{2} \nabla ^{2})\nabla ^{2}(\nabla \cdot \pmb W^{*}) \\ &= (1-l_{3}^{2} \nabla ^{2})(\nabla \cdot \pmb B). \end{aligned} $$
(31)

From this relation, it is seen that potential \(\varphi ^{*}\) should be treated as the particular solution of (31) since the general solution of the corresponding homogeneous equation is given by \(\bar{\varphi}\) (30) and it is already included into the displacement representation (28) (similar particular solution in classical PN solution obeys Poisson equation). Substituting the relation between \(l_{3}\) and \(l_{1}\), \(l_{2}\) into (31) we then find that

$$ \begin{aligned} (1-l_{1}^{2} \nabla ^{2})\nabla ^{2}\varphi ^{*} = \frac{1}{\kappa}(1-l_{1}^{2} \nabla ^{2})(\nabla \cdot \pmb B) -\frac{1}{\kappa \alpha}(1-l_{2}^{2} \nabla ^{2})(\nabla \cdot \pmb B). \end{aligned} $$
(32)

Using representation (23), (24) for the second term in the right hand side of equation (32) we obtain:

$$ \begin{aligned} (1-l_{1}^{2} \nabla ^{2})\nabla ^{2}\varphi ^{*} &= \frac{1}{\kappa}(1-l_{1}^{2} \nabla ^{2})(\nabla \cdot \pmb B) -\frac{1}{\kappa \alpha} \nabla \cdot \left (\pmb B_{c} + l_{2}^{2} \frac{\pmb b}{\mu} \right ) + \frac{l_{2}^{2}}{\kappa \alpha} \frac{\nabla \cdot \pmb b}{\mu}, \\ \implies (1-l_{1}^{2} \nabla ^{2})\nabla ^{2}\varphi ^{*} &= \frac{1}{\kappa}(1-l_{1}^{2} \nabla ^{2})(\nabla \cdot \pmb B) - \frac{1}{\kappa \alpha}\nabla \cdot \pmb B_{c}. \end{aligned} $$
(33)

Then, we take into account that \(\pmb B_{c}\) is the general solution for the Poisson equation (24)2. Therefore, without loss of generality, vector \(\pmb B_{c}\) can be replaced by

$$ (1 -l_{1}^{2} \nabla ^{2}) \pmb B_{c} - l_{1}^{2} \frac{\pmb b}{\mu}. $$

In such a way, from (33) we obtain the final form of the governing equation for potential \(\varphi ^{*}\), namely

$$ \begin{aligned} (1-l_{1}^{2} \nabla ^{2})\nabla ^{2}\varphi ^{*} &= \frac{l_{1}^{2}}{\kappa \alpha \mu} \nabla \cdot \pmb b + (1-l_{1}^{2} \nabla ^{2}) \nabla \cdot \left ( \pmb B_{c} +\frac{1}{\kappa}\pmb B_{g} \right ), \end{aligned} $$
(34)

where we also take into account the decomposition (23) and definitions for \(\alpha \) and \(\kappa \).

Particular solution for the obtained equation (34) can be decomposed into a sum,

$$ \begin{aligned} \varphi ^{*} = \varphi ^{*}_{b} + \varphi _{c}^{*} + \varphi _{g}^{*}, \end{aligned} $$
(35)

of three parts, so that

$$ (1-l_{1}^{2} \nabla ^{2})\nabla ^{2}\varphi ^{*}_{b} = \frac{l_{1}^{2}}{\kappa \alpha \mu} \nabla \cdot \pmb b, $$
(36)
$$ (1-l_{1}^{2} \nabla ^{2})\nabla ^{2}\varphi _{c}^{*} = (1-l_{1}^{2} \nabla ^{2}) \nabla \cdot \pmb B_{c}, $$
(37)
$$ (1-l_{1}^{2} \nabla ^{2})\nabla ^{2}\varphi _{g}^{*} = \frac{1}{\kappa}(1-l_{1}^{2} \nabla ^{2}) \nabla \cdot \pmb B_{g}. $$
(38)

Definition of particular solution \(\varphi _{c}^{*}\) (37) can be reduced to the classical problem, in which similar particular solution was found for equation \(\nabla ^{2}\varphi _{c}^{*} = \nabla \cdot \pmb B_{c} \) in the form [3, 49]

$$ \varphi _{c}^{*} = \frac{1}{2}(\pmb r\cdot \pmb B_{c} + \beta _{c}), $$
(39)

where

$$ \nabla ^{2}\pmb B_{c} = -\frac{\pmb b}{\mu}, \qquad \nabla ^{2}\beta _{c} = \frac{\pmb r\cdot \pmb b}{\mu}. $$
(40)

Particular solution \(\varphi _{g}^{*}\) (38) can be defined by using Helmhlotz decomposition for the vector potential \(\pmb B_{g}\), which can be used without any additional restrictions since \(\pmb B_{g}\) satisfies the vector Helmholtz equation (24)2 [60]. Thus, we define

$$ \begin{aligned} \pmb B_{g} = \nabla \Psi + \nabla \times \pmb\Psi ,\qquad \nabla \cdot \pmb B_{g} = \nabla ^{2} \Psi , \qquad \nabla \times \pmb B_{g} = -\nabla ^{2} \pmb\Psi, \end{aligned} $$
(41)

where function \(\Psi \) and solenoidal field \(\pmb\Psi \) are the scalar and vector potentials, respectively, that have to satisfy the equations

$$ \begin{aligned} (1-l_{2}^{2}\nabla ^{2})\nabla ^{2}\Psi = -l_{2}^{2} \frac{\nabla \cdot \pmb b}{\mu}, \qquad (1-l_{2}^{2}\nabla ^{2}) \nabla ^{2}\pmb\Psi = l_{2}^{2}\frac{\nabla \times \pmb b}{\mu}, \end{aligned} $$
(42)

which are obtained based on equation (24)2 and definitions (41).

Based on the comparison of equation (41)2 for the scalar potential \(\Psi \) and equation (38) for the particular solution \(\varphi _{g}^{*}\), we can define the last one as

$$ \varphi _{g}^{*} = \frac{1}{\kappa}\Psi. $$
(43)

Then, substituting (23), (35) into (28) and taking into account (39), (43) we obtain

$$ \pmb u = \pmb B_{c} + \pmb B_{g} - \kappa \nabla \left ( \bar{\varphi}+ \varphi ^{*}_{b} + \frac{1}{2}(\pmb r\cdot \pmb B_{c} + \beta _{c}) + \frac{1}{\kappa}\Psi \right ). $$
(44)

In this relation, we observe that the sum \(\bar{\varphi}+ \varphi ^{*}_{b}\) is the general solution of the inhomogeneous equation (36) that will be denoted hereafter as \(\varphi = \bar{\varphi}+ \varphi ^{*}_{b}\). Substituting the Helmholtz decomposition for \(\pmb B_{g}\) (41) into (44), we also find that the potential part of this field \(\nabla \Psi \) is cancelled and the definition of the displacement field becomes

$$ \pmb u = \pmb B_{c} - \frac{\kappa}{2}\nabla (\pmb r\cdot \pmb B_{c} + \beta _{c}) + \nabla \times \pmb\Psi - \kappa \nabla \varphi, $$
(45)

in which: \(\pmb B_{c}\) and \(\beta _{c}\) should be treated as standard harmonic scalar and vector stress functions of PN solution defined by (40); \(\pmb\Psi \) and \(\varphi \) are the additional stress functions of gradient theory; \(\pmb\Psi \) is the general solution of (42)2 and \(\varphi \) is the general solution of

$$ (1-l_{1}^{2} \nabla ^{2})\nabla ^{2}\varphi = \frac{l_{1}^{2}}{\kappa \alpha \mu} \nabla \cdot \pmb b. $$
(46)

The obtained representation (45) can be simplified. Namely, according to Theorem 1 we can define

$$ \begin{aligned} \varphi &= \tilde{\varphi}_{c} + \varphi _{g}, \qquad \nabla ^{2} \tilde{\varphi}_{c} = \frac{l_{1}^{2}}{\kappa \alpha \mu} \nabla \cdot \pmb b, \qquad (1-l_{1}^{2}\nabla ^{2})\varphi _{g} = \frac{l_{1}^{4}}{\kappa \alpha \mu} \nabla \cdot \pmb b, \end{aligned} $$
(47)

so that the classical harmonic part of this solution \(\tilde{\varphi}_{c}\) can be combined with the corresponding standard Papkovich stress function \(\beta _{c}\) in (45). It is convenient to then introduce a single scalar function in relation (45)

$$ \varphi _{c} = \beta _{c} + 2\tilde{\varphi}_{c} $$
(48)

that should obey the Poisson equation with the right-hand side defined by the corresponding linear combination of the right-hand sides of equations (47)2 and (40)2.

In representation (45), we can also replace term \(\nabla \times \pmb\Psi \) by the initial vector potential \(\pmb B_{g}\) (see (41)) with additional requirement that this vector field should be solenoidal. As a result, after some simplifications and renormalization for the potential \(\varphi _{g}\), one can obtain the final form of the Papkovich–Neuber general solution within SGET, namely

$$ \begin{aligned} \pmb u &= \pmb u_{c} + \pmb u_{g}, \\ \pmb u_{c} &= \pmb B_{c} - \frac{1}{4(1-\nu )}\nabla (\pmb r\cdot \pmb B_{c} + \varphi _{c}), \\ \pmb u_{g} &= \pmb B_{g} + l_{1}^{2}\nabla \varphi _{g}. \end{aligned} $$
(49)

in which the stress functions \(\pmb B_{c}\), \(\pmb B_{g}\), \(\varphi _{c}\), \(\varphi _{g}\) have to satisfy:

$$ \begin{aligned} \nabla ^{2}\pmb B_{c} &= -\frac{\pmb b}{\mu}, \\ \nabla ^{2}\varphi _{c} &= \frac{\pmb r\cdot \pmb b}{\mu} + 2l_{1}^{2}(1-2 \nu )\frac{\nabla \cdot \pmb b}{\mu} \\ (1-l_{2}^{2}\nabla ^{2})\pmb B_{g} &= -l_{2}^{2}\frac{\pmb b}{\mu}, \qquad \nabla \cdot \pmb B_{g} = 0, \\ (1-l_{1}^{2}\nabla ^{2})\varphi _{g} &= -l_{1}^{2}\, \frac{1-2\nu}{2(1-\nu )}\, \frac{\nabla \cdot \pmb b}{\mu} \end{aligned} $$
(50)

The obtained representation (49) validates the possibility of additive decomposition of the general solution for the displacement field within Mindlin–Toupin SGET into the so-called classical \(\pmb u_{c}\) and gradient \(\pmb u_{g}\) parts. Such representation has been heuristically assumed in Refs. [45, 47] and it was explicitly established previously within the simplified gradient theories in Refs. [20, 42, 44]. Based on the operator-split technique, such decomposition has been obtained within the general Mindlin–Toupin theory in Ref. [61]. A similar decomposition has also been obtained within the analysis of fundamental solutions of different gradient theories [62–64].

In the derived form of PN solution (49) the classical part of the displacement field \(\pmb u_{c}\) is defined through the standard Papkovich stress functions (with the only modification of the body force in the Poisson equation for the scalar function). The gradient part of the displacement field \(\pmb u_{g}\) is represented via the modified Helmholtz decomposition that is the linear combination of the potential part \(\nabla \varphi _{g}\) and solenoidal part \(\pmb B_{g}\) defined as the solutions of modified Helmoltz equations with different length scale parameters (see (50)). In this modified decomposition, the length scale parameter \(l_{1}\) defines the potential part of \(\pmb u_{g}\), while \(l_{2}\) defines its rotational part that is in agreement with the initial structure of equilibrium equations of SGET (1).

General solutions for different simplified gradient theories can be obtained by assuming corresponding values for the length scale parameters. Namely, assuming \(l_{1}=0\) we obtain the general solution for the couple stress theory [16]. For the case \(l_{2}=0\) we obtain the general solution for the dilatation gradient elasticity theory [24]. For the simplified Aifantis theory, it should be used \(l_{1}=l_{2}\). Classical PN solution follows from (49) and (50) if \(l_{1}=l_{2}=0\).

5 Completeness Theorem

The main next step is to prove the completeness of the developed solutions. Reformulating the similar classical theorem given by Gurtin [49] we state:

Theorem 2

Let \(\pmb u\) be a displacement field that satisfies equilibrium equation (1) and corresponds to the body force \(\pmb b\). Then there exists a field \(\pmb W\) that satisfies (18), (20); and fields \(\pmb B_{c}\), \(\pmb B_{g}\), \(\varphi _{c}\), \(\varphi _{g}\) that satisfy (49) and (50).

Proof

Consider the definition for the displacement field through the modified Galerkin stress function within SGET (19):

$$ \pmb u = \frac{1}{\alpha}(1-l_{2}^{2} \nabla ^{2})\nabla \nabla \cdot \pmb W -(1-l_{1}^{2} \nabla ^{2})\nabla \times \nabla \times \pmb W. $$
(51)

This relation can be rewritten in the equivalent form

$$ \begin{aligned} \hat{\alpha}(1-\hat{l}_{1}^{\,2} \nabla ^{2})\nabla \nabla \cdot \pmb W -(1- \hat{l}_{2}^{\,2} \nabla ^{2})\nabla \times \nabla \times \pmb W = - \frac{\hat {\pmb b}}{\mu}, \end{aligned} $$
(52)

where \(\hat{\alpha}= 1/\alpha \), \(\hat{\pmb b} = -\mu \pmb u\), \(\hat{l}_{1} = l_{2}\), and \(\hat{l}_{2} = l_{1}\).

The form of equation (52) with respect to \(\pmb W\) is exactly the same to the equilibrium equation (1) of SGET that is defined with respect to \(\pmb u\). Therefore, the task of finding a field \(\pmb W\) that satisfies (51) is reduced to finding a particular solution to the equilibrium equation of SGET corresponding to prescribed body forces \(\hat {\pmb b}\). This can be done by using the derived form of the Papkovich–Neuber solution of SGET (49) and (50), namely

$$ \begin{aligned} \pmb W = \hat {\pmb B}_{c} - \frac{1}{4(1-\hat{\nu})}\nabla (\pmb r \cdot \hat {\pmb B}_{c} + \hat{\varphi}_{c}) + \hat {\pmb B}_{g} + \hat{l}_{1}^{\,2}\nabla \hat{\varphi}_{g} \end{aligned} $$
(53)

in which the stress functions can be defined based on (50) and (13) and (14). This gives

$$ \begin{aligned} \hat {\pmb B}_{c} &= \mathcal {N} \ast \pmb u, \qquad \hat{\varphi}_{c} = -\mathcal {N} \ast (\pmb r\cdot \pmb u + 2\hat{l}_{1}^{\,2}(1-2\hat{\nu}) \nabla \cdot \pmb u), \\ \hat {\pmb B}_{g} &= \nabla \times (\hat{\pmb\Psi}_{c} + \hat{\pmb\Psi}_{g}), \quad \hat{\pmb\Psi}_{c} = -\hat{l}^{\,2}_{2}\, \mathcal {N} \ast (\nabla \times \pmb u), \quad \hat{\pmb\Psi}_{g} = - \hat{l}^{\,2}_{2}\, \mathcal {H}_{2} \ast (\nabla \times \pmb u), \\ \hat{\varphi}_{g} &= \hat{l}_{1}^{\,2}\, \frac{1-2\hat{\nu}}{2(1-\hat{\nu})} \mathcal {H}_{1} \ast \left (\nabla \cdot \pmb u \right ), \end{aligned} $$
(54)

where \(\mathcal {H}_{1}\) and \(\mathcal {H}_{2}\) are the Green’s functions of Helmholtz equations (14) defined with the length scale parameters \(l_{1}\) and \(l_{2}\), respectively; and for the solenoidal gradient potential we use its representation \(\hat{\pmb B}_{g} =\nabla \times \hat{\pmb\Psi}\), in which the vector field \(\hat{\pmb\Psi}\) is defined based on equation (42)2 and Theorem 1.

Thus, we prove that there exists a vector field \(\pmb W\) that corresponds to a given displacement field \(\pmb u\). The governing equation for \(\pmb W\) (18) follows from equilibrium equation (1) by its definition, and consequently, the completeness of BG solution (18) and (20) within SGET is proven.

Consider then the relations between the Papkovich–Neuber and the Galerkin stress functions. By using relations (21) and (25) and taking into account that \(\nabla \cdot \pmb B_{g}=0\), we obtain the definitions for the vector stress functions of PN solution (49):

$$ \begin{aligned} \pmb B_{c} &= (1-l_{1}^{2} \nabla ^{2})(1-l_{2}^{2}\nabla ^{2}) \nabla ^{2}\pmb W, \\ \pmb B_{g} &= -l_{2}^{2}\, (1-l_{1}^{2} \nabla ^{2})\nabla ^{2}( \nabla \times \nabla \times \pmb W). \end{aligned} $$
(55)

The scalar potentials \(\varphi _{c}\) and \(\varphi _{g}\) in PN solution (49) can be defined based on relations (46)–(48) and Corollary 1 as

$$ \begin{aligned} \varphi _{c} &= \beta _{c} + 2\tilde{\varphi}_{c}, \\ \varphi _{g} &= l_{1}^{2}\nabla ^{2}\varphi, \end{aligned} $$
(56)

where \(\beta _{c}\) is indeterminate harmonic function and \(\tilde{\varphi}_{c}\) has to satisfy

$$ \begin{aligned} \tilde{\varphi}_{c} = (1-l_{1}^{2}\nabla ^{2})\varphi. \end{aligned} $$
(57)

Scalar potential \(\varphi \) was defined in Sect. 4 based on the relation

$$ \begin{aligned} \varphi = \bar{\varphi}+ \varphi ^{*}_{b}, \end{aligned} $$
(58)

in which, according to (27)1:

$$ \begin{aligned} \bar{\varphi}= (1-l_{3}^{2} \nabla ^{2})(\nabla \cdot \bar {\pmb W}) \end{aligned} $$
(59)

and, according to (35), (39), and (43)

$$ \begin{aligned} \varphi ^{*}_{b} = \varphi ^{*} - \psi _{c}^{*} - \psi _{g}^{*} = (1-l_{3}^{2} \nabla ^{2})(\nabla \cdot \pmb W^{*}) - \frac{1}{2}(\pmb r\cdot \pmb B_{c} + \beta _{c}) - \frac{1}{\kappa}\Psi. \end{aligned} $$
(60)

By its definition, \(\pmb B_{g}\) (50) does not have the potential part, so that in (60) we can set \(\Psi =0\) (see (41)). Then, substituting (59), (60), into (58) and taking into account (26) we obtain

$$ \begin{aligned} \varphi = (1-l_{3}^{2} \nabla ^{2})(\nabla \cdot \pmb W) - \frac{1}{2}(\pmb r\cdot \pmb B_{c} + \beta _{c}). \end{aligned} $$
(61)

Using the definition for \(\tilde{\varphi}_{c}\) (57), obtained relation (61) and the definition of particular solution \(\varphi _{c}^{*}\) (37), (39), we find that

$$ \begin{aligned} \tilde{\varphi}_{c} = (1-l_{1}^{2}\nabla ^{2})(1-l_{3}^{2} \nabla ^{2})( \nabla \cdot \pmb W) - \frac{1}{2}(\pmb r\cdot \pmb B_{c} + \beta _{c}) + l_{1}^{2}\nabla \cdot \pmb B. \end{aligned} $$
(62)

Finally, classical scalar PN stress functions \(\varphi _{c}\) can be defined by using (56)1 and (62) as follows:

$$ \begin{aligned} \varphi _{c} = 2(1-l_{1}^{2}\nabla ^{2})(1-l_{3}^{2} \nabla ^{2})( \nabla \cdot \pmb W) - \pmb r\cdot \pmb B_{c} + 2l_{1}^{2}\nabla \cdot \pmb B_{c}. \end{aligned} $$
(63)

Gradient scalar PN stress function \(\varphi _{g}\) can be defined based on relations (56)2 and (61):

$$ \begin{aligned} \varphi _{g} = l_{1}^{2}\nabla ^{2}(1-l_{3}^{2} \nabla ^{2})(\nabla \cdot \pmb W) - l_{1}^{2}\nabla \cdot \pmb B_{c}. \end{aligned} $$
(64)

Derived relations (55), (63), and (64) allow us to define PN stress functions \(\pmb B_{c}\), \(\pmb B_{g}\), \(\varphi _{c}\), \(\varphi _{g}\) by using given Galerkin stress vector \(\pmb W\), which representation for a given arbitrary displacement field \(\pmb u\) has already been found (54). Then, according to derivations presented in the previous section, equation (18) implies (50), and (20) implies (49). Therefore, PN solution (49)–(50) is also complete within SGET. □

6 Fundamental Solution

The derivation of the fundamental solution based on PN general solution within the classical elasticity can be found, e.g., in [9, 49]. Within SGET a similar result can be obtained by using the developed form of PN solution (49)–(50). Thus, let us consider the concentrated body force \(\pmb Q\) applied at the origin of the coordinate system, viz.,

$$ \pmb b = \pmb Q \delta (\pmb r) $$
(65)

where \(\delta (\pmb r)\) is Dirac delta function.

Based on (49), the SGET solution that corresponds to the body force (65) can be presented as

$$ \begin{aligned} \pmb u = \pmb B^{*}_{c} - \frac{1}{4(1-\nu )}\nabla (\pmb r\cdot \pmb B^{*}_{c} + \varphi ^{*}_{c}) + \pmb B^{*}_{g} + l_{1}^{2} \nabla \varphi ^{*}_{g}, \end{aligned} $$
(66)

where the stress functions \(\pmb B^{*}_{c}\), \(\pmb B^{*}_{g}\), \(\varphi ^{*}_{c}\), \(\varphi ^{*}_{g}\) are the particular solutions of the equations

$$ \nabla ^{2}\pmb B^{*}_{c} = -\frac{\pmb Q}{\mu}\delta (\pmb r), $$
(67)
$$ \nabla ^{2}\varphi ^{*}_{c} = \frac{\pmb r\cdot \pmb Q}{\mu}\delta ( \pmb r) + 2l_{1}^{2}(1-2\nu ) \frac{\nabla \cdot (\pmb Q \delta (\pmb r))}{\mu}, $$
(68)
$$ (1-l_{2}^{2}\nabla ^{2})\pmb B^{*}_{g} = -l_{2}^{2}\frac{\pmb Q}{\mu} \delta (\pmb r), \qquad \nabla \cdot \pmb B^{*}_{g} = 0, $$
(69)
$$ (1-l_{1}^{2}\nabla ^{2})\varphi ^{*}_{g} = - l_{1}^{2}\, \frac{1-2\nu}{2(1-\nu )}\, \frac{\nabla \cdot (\pmb Q \delta (\pmb r))}{\mu}. $$
(70)

The remaining task is to find the particular solutions of equations (67)–(70). The solution for vector stress function \(\pmb B^{*}_{c}\) (67) is similar to classical elasticity, and according to potential theory, it is given by [49]

$$ \pmb B^{*}_{c} = \frac{\pmb Q}{4\pi \mu r}, $$
(71)

where \(r = \lvert \pmb r\rvert \).

The scalar stress function \(\varphi ^{*}_{c}\) in classical elasticity vanishes since it is valid that

$$ \nabla ^{2}\varphi ^{*}_{c} = \frac{\pmb r\cdot \pmb Q}{\mu}\delta ( \pmb r) \quad \implies \quad \varphi ^{*}_{c}\equiv 0; $$
(72)

therefore, equation (68) can be expressed in a simpler form as

$$ \nabla ^{2}\varphi ^{*}_{c} = 2l_{1}^{2}(1-2\nu ) \frac{\nabla \cdot (\pmb Q \delta (\pmb r))}{\mu}, $$
(73)

The particular solution to this equation can be presented in the form

$$ \varphi ^{*}_{c} = -l_{1}^{2}\frac{1-2\nu}{2\pi \mu}\,\pmb Q\cdot \nabla \left (\frac{1}{r}\right ), $$
(74)

which follows from the formula for the particular solutions of the Poisson equation (14) and the identity

$$ \int _{D} f(\pmb r-\pmb \xi )\nabla _{\pmb\xi}\cdot (\pmb Q \delta (\pmb \xi ))dv_{\pmb\xi} = \pmb Q\cdot \nabla _{\pmb r} \,f( \pmb r), $$
(75)

which an be proven based on the divergence theorem, where \(f(\pmb r-\pmb \xi )\) is an arbitrary function; \(\pmb \xi \) is integration variable; and \(\pmb r\) is radial coordinate.

The solution for the gradient vector stress function \(\pmb B^{*}_{g}\) (69) can be found in several steps. At first, we neglect the requirement that \(\pmb B^{*}_{g}\) is the divergence-free field and find the particular solution \(\tilde{\pmb B}^{*}_{g}\) to the equation:

$$ (1-l_{2}^{2}\nabla ^{2})\tilde{\pmb B}^{*}_{g} = -l_{2}^{2} \frac{\pmb Q}{\mu}\delta (\pmb r) \quad \implies \quad \tilde{\pmb B}^{*}_{g} = -\frac{\pmb Q}{4\pi \mu} \frac{e^{-r/l_{2}}}{r}, $$
(76)

Then we should subtract the potential part of the obtained solution \(\tilde{\pmb B}^{*}_{g}\) to find the field \(\pmb B^{*}_{g}\), which will satisfy \(\nabla \cdot \pmb B^{*}_{g}=0\). The potential part of \(\tilde{\pmb B}^{*}_{g}\) can be found by using Helmholtz decomposition and a representation similar to (41). As a result, we obtain

$$ \pmb B^{*}_{g} = \tilde{\pmb B}^{*}_{g} - \nabla \Psi ^{*}, $$
(77)

in which the potential \(\Psi ^{*}\) is defined according to relation (see (41))

$$ \begin{aligned} \nabla \cdot \tilde{\pmb B}^{*}_{g} = \nabla ^{2} \Psi ^{*}. \end{aligned} $$
(78)

Then, combining the equation for \(\tilde{\pmb B}^{*}_{g}\) (76) and the relation (78) and using (14) we obtain

$$ \begin{aligned} &\nabla \cdot \left (l_{2}^{2}\nabla ^{2}\tilde{\pmb B}^{*}_{g} -l_{2}^{2} \frac{\pmb Q}{\mu}\delta (\pmb r)\right ) = \nabla ^{2} \Psi ^{*} \\ &\implies \Psi ^{*} = \frac{l_{2}^{2}}{4\pi \mu}\pmb Q\cdot \nabla \left (\frac{1-e^{-r/l_{2}}}{r}\right ). \end{aligned} $$
(79)

Substituting (76), (79) into (77), we find the desirable solenoidal field \(\pmb B^{*}_{g}\) in the form

$$ \begin{aligned} \pmb B^{*}_{g} = -\frac{\pmb Q}{4\pi \mu} \frac{e^{-r/l_{2}}}{r} - \frac{l_{2}^{2}}{4\pi \mu} \pmb Q\cdot \nabla \nabla \left ( \frac{1 - e^{-r/l_{2}}}{r}\right ). \end{aligned} $$
(80)

The solution for the gradient scalar stress function \(\varphi ^{*}_{g}\) (70) is given by

$$ \begin{aligned} &(1-l_{1}^{2}\nabla ^{2})\varphi ^{*}_{g} = - l_{1}^{2}\, \frac{1-2\nu}{2(1-\nu )}\, \frac{\nabla \cdot (\pmb Q \delta (\pmb r))}{\mu} \\ &\implies \quad \varphi ^{*}_{g} = - \frac{1-2\nu}{8\pi (1-\nu )\mu} \, \pmb Q\cdot \nabla \left (\frac{e^{-r/l_{1}}}{r}\right ). \end{aligned} $$
(81)

The substitution of relations (71), (74), (80) and (81) into the representation for the displacement field (66) yields

$$ \begin{aligned} \pmb u &= \frac{\pmb Q}{4\pi \mu r} - \frac{1}{16\pi \mu (1-\nu )} \nabla \left (\frac{\pmb r\cdot \pmb Q}{r}\right ) \\ &+ l_{1}^{2}\frac{1-2\nu}{8\pi (1-\nu )\mu}\,\pmb Q\cdot \nabla \nabla \left (\frac{1}{r}\right ) \\ &- \frac{\pmb Q}{4\pi \mu} \frac{e^{-r/l_{2}}}{r} - \frac{l_{2}^{2}}{4\pi \mu} \pmb Q\cdot \nabla \nabla \left ( \frac{1 - e^{-r/l_{2}}}{r}\right ) \\ &- l_{1}^{2} \frac{1-2\nu}{8\pi (1-\nu )\mu}\, \pmb Q\cdot \nabla \nabla \left (\frac{e^{-r/l_{1}}}{r}\right ). \end{aligned} $$
(82)

Finally, after some standard simplifications, from (82) we can obtain the resulting form of the solution for the considered problem, namely

$$ \begin{aligned} \pmb u &= \frac{1}{4\pi \mu}\left ( \frac{\pmb Q}{r} - \frac{1}{2(1-\nu )}\pmb Q\cdot \nabla \nabla r \right ) \\ &+ l_{1}^{2}\frac{1-2\nu}{8\pi (1-\nu )\mu}\,\pmb Q\cdot \nabla \nabla \left (\frac{1-e^{-r/l_{1}}}{r}\right ) \\ &- \frac{\pmb Q}{4\pi \mu} \frac{e^{-r/l_{2}}}{r} - \frac{l_{2}^{2}}{4\pi \mu} \pmb Q\cdot \nabla \nabla \left ( \frac{1 - e^{-r/l_{2}}}{r}\right ), \end{aligned} $$
(83)

which can be used to obtain the form of SGET fundamental solution assuming an arbitrary shift (\(\pmb r'\)) of the loaded point with respect to the origin of the coordinate system, i.e., by using \(r = \lvert \pmb r - \pmb r'\rvert \) in (83). This form of fundamental solution will exactly coincide with those one presented previously, e.g. in Refs. [61, 64, 65].

7 Conclusion

We derived an extended form of Boussinesq–Galerkin and Papkovich–Neuber general solutions within Mindlin–Toupin strain gradient elasticity. We prove the theorem of completeness for these solutions, i.e., we establish their generality [50]. In the presented proof, it is essential to have both forms of general solution (BG and PN) and various established relations between different stress functions. Further work should be related to the analysis of the nonuniqueness and the degree of nonuniqueness of the derived general solutions within SGET.