# Complete general solutions for equilibrium equations of isotropic strain
gradient elasticity

Y.O. Solyaev  Institute of Applied Mechanics of Russian Academy of Sciences,
Moscow, Russia  Moscow Aviation Institute, Moscow, Russia

###### Abstract

In this paper, we consider isotropic Mindlin-Toupin strain gradient elasticity
theory in which the equilibrium equations contain two additional length-scale
parameters and have the fourth order. For this theory we developed an extended
form of Boussinesq-Galerkin (BG) and Papkovich-Neuber (PN) general solutions.
Obtained form of BG solution allows to define the displacement field through
the single vector function that obeys the eight-order bi-harmonic/bi-Helmholtz
equation. The developed PN form of the solution provides an additive
decomposition of the displacement field into the classical and gradient parts
that are defined through the standard Papkovich stress functions and modified
Helmholtz decomposition, respectively. Relations between different stress
functions and completeness theorem for the derived general solutions are
established. As an example, it is shown that a previously known fundamental
solution within the strain gradient elasticity can be derived by using the
developed PN general solution.

##  1 Introduction

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

In the present study, we consider the strain gradient elasticity theory
(SGET), which general formulation for isotropic materials have 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 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]. Boundary-value problem of SGET
consists of the fourth-order equilibrium equations and extended number of
boundary conditions, which form can be obtained based on the variational
approach [16, 17]. In SGET there arise an extended definition of surface
traction 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 remain the same to
classical elasticity since the number of primary field variables (components
of the displacement vector) does not change.

Notably, that Mindlin-Toupin SGET contains as the special cases a number of
famous simplified gradient theories [20, 21, 22] and also several kinds of
incomplete gradient theories like the couple stress theory, the dilatation
gradient elasticity, etc. [23, 18, 24, 25, 26]. Nowadays, these theories
attracts an increasing attention in applications to fracture mechanics and
dislocations problems [27, 28, 29, 30, 31], in the studies of small-scale and
high-frequency processes [32, 33], in the description of mechanical behaviour
of composites and metamaterails[34, 35, 36, 37, 38].

First variant of general solution for equilibrium equations of SGET have been
presented in the initial work by Mindlin [16], though the particular variant
of this solution within the couple stress theory have been established earlier
by Mindlin and Tiersten [23]. The form of Mindlin solution [16] can be treated
as the generalized variant of the classical Papkovich-Neuber (PN) solution. It
defines the displacement field in rather complicated form through the vector
and scalar functions that obey the fourth order governing equations. Later,
the simpler variants of the PN solutions with stress functions that obey the
second-order equations have been established within SGET [39, 35, 40, 41, 42,
43], though the completeness of these solutions have not been proven or it was
implied on the basis of heuristic reasoning [44].

Lurie et al. [39, 35] 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. Gradient part of
the solution was defined by using two additional vector functions that
satisfied the modified Helmholtz equations. Solyaev et al. [42] used similar
form of PN solution within SGET and reduced the representation for the
gradient part of solution to the 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
[44]. Similar result have been also established within the simplified strain
gradient elasticity theory by Charalambopoulos et al. [41], though the
possibility of the use of solenoidal vector stress function for the gradient
part of the displacement field have not been considered. Representation of the
displacement field in terms of Lame’s potentials have been used by Placidi and
Dhaba within the Saint-Venant’s problems [43].

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 [45], Gurtin
[46] 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 have 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
obtaine the extended variant of Boussinesq-Galerkin (BG) solution within SGET.
Generalizing the approach of classical elasticity, we introduce then the
relations between the Galerkin vector and 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 a
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 [44].

Combining both results for BG and PN forms of solution we prove the theorem of
their completeness within SGET in a sense of similar proofs developed by
Mindlin [5] and Gurtin [46] 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” [47]. 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
relation between Galerkin stress function and the displacement field and also
involving established PN representation of solution within SGET. Then, based
on the relations between the stress functions of PN and BG solutions we prove
completeness of the former. Thus, in the presented proof it is essential to
have both representation of solutions in BG and PN forms.

The key point of the presented results is the proposed modified definition of
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
[40, 48, 44]. Incorporation of complete general solutions into the numerical
schemes (like in a Trefftz method) can be also an important issue for the
development of stable and flexible numerical solvers within SGET [39, 49].

##  2 Preliminaries

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

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

where $\boldsymbol{u}(\boldsymbol{r})$ is the vector of mechanical
displacements at a point $\boldsymbol{r}=\\{x_{1},x_{2},x_{3}\\}$;
$\boldsymbol{b}(\boldsymbol{r})$ 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 the
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 corresponds to the formulation of
SGET in terms of 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 and in the modified couple stress theory $l_{1}=0$ [23, 50], and in the
dilatation gradient elasticity $l_{2}=0$ [24]. Assuming that both length scale
parameters equal to zero $l_{1}=l_{2}=0$ equation (1) reduces to classical
elasticity equilibrium equation. Therefore, all considerations presented below
for equation (1) will be valid for any kind of mentioned gradient theories and
all presented solutions must contain the corresponding classical solutions as
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 $\mathcal{F}$ is the scalar (or the vector) field that should be found
in the bounded or in the unbounded region of three-dimensional euclidean space
$D$; $\mathcal{B}$ 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
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 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 Laplace equation and
$\mathcal{F}^{*}_{c}$ is some particular solution of Eq. (6).

Then we represent the general solution of inhomogeneous Helmholtz equation (5)
in the following 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 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 can identify $\mathcal{F}_{g}$ as the general solution of
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:

$F_{c}=(1-l^{2}\nabla^{2})\mathcal{F},\qquad\mathcal{F}_{g}=l^{2}\nabla^{2}\mathcal{F}$
(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 given arbitrary field
$\mathcal{F}$ that satisfies (3). For the further analysis we will also need
the explicit form of particular solutions of equations (4), that can be
represented as follows [51]:

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

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

Notations.

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 ($\bar{\psi}$, $\bar{\boldsymbol{\Psi}}$). Particular
solutions will be denoted with star superscripts ($\psi^{*}$,
$\boldsymbol{\Psi}^{*}$). Potentials (stress functions) that satisfy the
Laplace or the Poisson equations will be denoted with subscript ”c” –
”classic” ($\psi_{c}$, $\boldsymbol{B}_{c}$). Potentials that satisfy the
modified Helmholtz equation will be denoted with subscript ”g” – ”gradient”
($\psi_{g}$, $\boldsymbol{B}_{g}$).

##  3 Boussinesq-Galerkin solution

According to Helmholtz theorem every suitably regular vector field
$\boldsymbol{u}(\boldsymbol{r})$ admits the representation:

$\boldsymbol{u}=\nabla\phi+\nabla\times\boldsymbol{S}$ (15)

where $\phi$ is the scalar potential and $\boldsymbol{S}$ is the vector
potential for which we can assume that $\nabla\cdot\boldsymbol{S}=0$ 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 $\boldsymbol{u}$ that are used in
classical elasticity [6], though the validity of Helmholtz theorem for the
fields with sub-linear growth have been also established [52, 53].

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

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

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

$\displaystyle\phi$
$\displaystyle=\frac{1}{\alpha}(1-l_{2}^{2}\nabla^{2})\nabla\cdot\boldsymbol{W}$
(17) $\displaystyle\boldsymbol{S}$
$\displaystyle=-(1-l_{1}^{2}\nabla^{2})\nabla\times\boldsymbol{W}$

Combining (16), (17) we obtain:

$\displaystyle(1-l_{1}^{2}\nabla^{2})(1-l_{2}^{2}\nabla^{2})\nabla^{2}\nabla^{2}\boldsymbol{W}=-\frac{\boldsymbol{b}}{\mu}$
(18)

Using (15), (17) we found:

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

or alternatively:

$\boldsymbol{u}=(1-l_{1}^{2}\nabla^{2})\nabla^{2}\boldsymbol{W}-\kappa(1-l_{3}^{2}\nabla^{2})\nabla\nabla\cdot\boldsymbol{W}$
(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 fuction $\boldsymbol{W}$ is Galerkin
stress function that obey 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).

Notably, that bi-Helmholtz/bi-Laplace equation have 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 similar eight-order equation defines the
fundamental solution for the modified Airy stress function within the edge
dislocations problems [54, 55].

##  4 Papkovich-Neuber solution

Let us introduce the following vector function:

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

From equilibrium equations written in terms of Galerkin stress function (18),
we found that $\boldsymbol{B}$ has to satisfy

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

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

$\boldsymbol{B}=\boldsymbol{B}_{c}+\boldsymbol{B}_{g}$ (23)

where

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

and according to Corollary 1 we also have

$\boldsymbol{B}_{c}=(1-l_{2}^{2}\nabla^{2})\boldsymbol{B},\qquad\boldsymbol{B}_{g}=l_{2}^{2}\nabla^{2}\boldsymbol{B}$
(25)

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

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

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

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

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

$\displaystyle\boldsymbol{u}$
$\displaystyle=(1-l_{1}^{2}\nabla^{2})(\nabla^{2}\bar{\boldsymbol{W}}+\nabla^{2}\boldsymbol{W}^{*})-\kappa(1-l_{3}^{2}\nabla^{2})\nabla(\nabla\cdot\bar{\boldsymbol{W}}+\nabla\cdot\boldsymbol{W}^{*})$
(28) $\displaystyle=\boldsymbol{B}-\kappa\nabla(\bar{\varphi}+\varphi^{*})$

where we introduce new scalar potentials:

$\displaystyle\bar{\varphi}=(1-l_{3}^{2}\nabla^{2})(\nabla\cdot\bar{\boldsymbol{W}}),\qquad\varphi^{*}=(1-l_{3}^{2}\nabla^{2})(\nabla\cdot\boldsymbol{W}^{*})$
(29)

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

$\displaystyle(1-l_{1}^{2}\nabla^{2})\nabla^{2}\bar{\varphi}=0$ (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, (29)2:

$\displaystyle(1-l_{1}^{2}\nabla^{2})\nabla^{2}\varphi^{*}$
$\displaystyle=(1-l_{1}^{2}\nabla^{2})(1-l_{3}^{2}\nabla^{2})\nabla^{2}(\nabla\cdot\boldsymbol{W}^{*})$
(31) $\displaystyle=(1-l_{3}^{2}\nabla^{2})(\nabla\cdot\boldsymbol{B})$

From this relation it is seen that potential $\varphi^{*}$ should be treated
as the particular solution of equation (31) since the general solution of
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
relation between $l_{3}$ and $l_{1}$, $l_{2}$ into (31) we can find then

$\displaystyle(1-l_{1}^{2}\nabla^{2})\nabla^{2}\varphi^{*}=\frac{1}{\kappa}(1-l_{1}^{2}\nabla^{2})(\nabla\cdot\boldsymbol{B})-\frac{1}{\kappa\alpha}(1-l_{2}^{2}\nabla^{2})(\nabla\cdot\boldsymbol{B})$
(32)

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

$\displaystyle(1-l_{1}^{2}\nabla^{2})\nabla^{2}\varphi^{*}$
$\displaystyle=\frac{1}{\kappa}(1-l_{1}^{2}\nabla^{2})(\nabla\cdot\boldsymbol{B})-\frac{1}{\kappa\alpha}\nabla\cdot\left(\boldsymbol{B}_{c}+l_{2}^{2}\frac{\boldsymbol{b}}{\mu}\right)+\frac{l_{2}^{2}}{\kappa\alpha}\frac{\nabla\cdot\boldsymbol{b}}{\mu}$
(33) $\displaystyle\implies(1-l_{1}^{2}\nabla^{2})\nabla^{2}\varphi^{*}$
$\displaystyle=\frac{1}{\kappa}(1-l_{1}^{2}\nabla^{2})(\nabla\cdot\boldsymbol{B})-\frac{1}{\kappa\alpha}\nabla\cdot\boldsymbol{B}_{c}$

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

$(1-l_{1}^{2}\nabla^{2})\boldsymbol{B}_{c}-l_{1}^{2}\frac{\boldsymbol{b}}{\mu}$

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

$\displaystyle(1-l_{1}^{2}\nabla^{2})\nabla^{2}\varphi^{*}$
$\displaystyle=\frac{l_{1}^{2}}{\kappa\alpha\mu}\nabla\cdot\boldsymbol{b}+(1-l_{1}^{2}\nabla^{2})\nabla\cdot\left(\boldsymbol{B}_{c}+\frac{1}{\kappa}\boldsymbol{B}_{g}\right)$
(34)

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

Particular solution for the obtained equation (34) can be decomposed into
three parts:

$\displaystyle\varphi^{*}=\varphi^{*}_{b}+\varphi_{c}^{*}+\varphi_{g}^{*}$
(35)

so that

$(1-l_{1}^{2}\nabla^{2})\nabla^{2}\varphi^{*}_{b}=\frac{l_{1}^{2}}{\kappa\alpha\mu}\nabla\cdot\boldsymbol{b}$
(36)
$(1-l_{1}^{2}\nabla^{2})\nabla^{2}\varphi_{c}^{*}=(1-l_{1}^{2}\nabla^{2})\nabla\cdot\boldsymbol{B}_{c}$
(37)
$(1-l_{1}^{2}\nabla^{2})\nabla^{2}\varphi_{g}^{*}=\frac{1}{\kappa}(1-l_{1}^{2}\nabla^{2})\nabla\cdot\boldsymbol{B}_{g}$
(38)

Definition of particular solution $\varphi_{c}^{*}$ (37) can be reduced to the
classical problem, in which similar particular solution was found for equation
$\nabla^{2}\varphi_{c}^{*}=\nabla\cdot\boldsymbol{B}_{c}$ in the following
form [3, 46]:

$\varphi_{c}^{*}=\frac{1}{2}(\boldsymbol{r}\cdot\boldsymbol{B}_{c}+\beta_{c}),$
(39)

where

$\nabla^{2}\boldsymbol{B}_{c}=-\frac{\boldsymbol{b}}{\mu},\qquad\nabla^{2}\beta_{c}=\frac{\boldsymbol{r}\cdot\boldsymbol{b}}{\mu}$
(40)

Particular solution $\varphi_{g}^{*}$ (38) can be defined by using Helmhlotz
decomposition for the vector potential $\boldsymbol{B}_{g}$, which can be used
without any additional restrictions since $\boldsymbol{B}_{g}$ satisfies the
vector Helmholtz equation (24)2 [56]. Thus, we define:

$\displaystyle\boldsymbol{B}_{g}=\nabla\Psi+\nabla\times\boldsymbol{\Psi},\qquad\nabla\cdot\boldsymbol{B}_{g}=\nabla^{2}\Psi,\qquad\nabla\times\boldsymbol{B}_{g}=-\nabla^{2}\boldsymbol{\Psi}$
(41)

where function $\Psi$ and solenoidal field $\boldsymbol{\Psi}$ are the scalar
and vector potentials, respectively, that have to satisfy the following
equations:

$\displaystyle(1-l_{2}^{2}\nabla^{2})\nabla^{2}\Psi=-l_{2}^{2}\frac{\nabla\cdot\boldsymbol{b}}{\mu},\qquad(1-l_{2}^{2}\nabla^{2})\nabla^{2}\boldsymbol{\Psi}=l_{2}^{2}\frac{\nabla\times\boldsymbol{b}}{\mu},$
(42)

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

Based on comparison of equation for the scalar potential $\Psi$ (41)2 and
equation for the particular solution $\varphi_{g}^{*}$ (38) 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:

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

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

$\boldsymbol{u}=\boldsymbol{B}_{c}-\frac{\kappa}{2}\nabla(\boldsymbol{r}\cdot\boldsymbol{B}_{c}+\beta_{c})+\nabla\times\boldsymbol{\Psi}-\kappa\nabla\varphi$
(45)

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

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

Obtained representation (45) can be additionally simplified. Namely, according
to Theorem 1 we can define:

$\displaystyle\varphi$
$\displaystyle=\tilde{\varphi}_{c}+\varphi_{g},\qquad\nabla^{2}\tilde{\varphi}_{c}=\frac{l_{1}^{2}}{\kappa\alpha\mu}\nabla\cdot\boldsymbol{b},\qquad(1-l_{1}^{2}\nabla^{2})\varphi_{g}=\frac{l_{1}^{4}}{\kappa\alpha\mu}\nabla\cdot\boldsymbol{b}$
(47)

so that the classical harmonic part of this solution $\tilde{\varphi}_{c}$ can
be combined with corresponding standard Papkovich stress function $\beta_{c}$
in (45). It is convenient to introduce then the following 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 defining by
corresponding linear combination of the right hand sides of equations (47)2
and (40)2.

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

$\displaystyle\boldsymbol{u}$
$\displaystyle=\boldsymbol{u}_{c}+\boldsymbol{u}_{g},$ (49)
$\displaystyle\boldsymbol{u}_{c}$
$\displaystyle=\boldsymbol{B}_{c}-\frac{1}{4(1-\nu)}\nabla(\boldsymbol{r}\cdot\boldsymbol{B}_{c}+\varphi_{c})$
$\displaystyle\boldsymbol{u}_{g}$
$\displaystyle=\boldsymbol{B}_{g}+l_{1}^{2}\nabla\varphi_{g}$

in which the stress functions $\boldsymbol{B}_{c}$, $\boldsymbol{B}_{g}$,
$\varphi_{c}$, $\varphi_{g}$ have to satisfy:

$\displaystyle\nabla^{2}\boldsymbol{B}_{c}$
$\displaystyle=-\frac{\boldsymbol{b}}{\mu},$ (50)
$\displaystyle\nabla^{2}\varphi_{c}$
$\displaystyle=\frac{\boldsymbol{r}\cdot\boldsymbol{b}}{\mu}+2l_{1}^{2}(1-2\nu)\frac{\nabla\cdot\boldsymbol{b}}{\mu}$
$\displaystyle(1-l_{2}^{2}\nabla^{2})\boldsymbol{B}_{g}$
$\displaystyle=-l_{2}^{2}\frac{\boldsymbol{b}}{\mu},\qquad\nabla\cdot\boldsymbol{B}_{g}=0,$
$\displaystyle(1-l_{1}^{2}\nabla^{2})\varphi_{g}$
$\displaystyle=-l_{1}^{2}\,\frac{1-2\nu}{2(1-\nu)}\,\frac{\nabla\cdot\boldsymbol{b}}{\mu}$

Obtained representation (49) validates the possibility of additive
decomposition of general solution for the displacement field within Mindlin-
Toupin SGET into the so-called classical $\boldsymbol{u}_{c}$ and gradient
$\boldsymbol{u}_{g}$ parts. Such representation have been heuristically
assumed in Refs. [42, 44] and and it was explicitly established previously
only within the simplified gradient theories [20, 39, 41]. Similar
decomposition have been also obtained within the analysis of fundamental
solutions of different gradient theories [57, 58, 59].

In the derived form of PN solution (49) the classical part of the displacement
field $\boldsymbol{u}_{c}$ 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
$\boldsymbol{u}_{g}$ is represented via the modified Helmholtz decomposition
that is the linear combination of the potential part $\nabla\varphi_{g}$ and
solenoidal part $\boldsymbol{B}_{g}$ 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 $\boldsymbol{u}_{g}$, 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
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), (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 [46] we state:

######  Theorem 2.

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

###### Proof.

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

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

This relation can be rewritten in the following equivalent form:

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

where $\hat{\alpha}=1/\alpha$, $\hat{\boldsymbol{b}}=-\mu\boldsymbol{u}$,
$\hat{l}_{1}=l_{2}$, $\hat{l}_{2}=l_{1}$.

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

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

in which the stress functions can be defined based on (50) and (13), (14):

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

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 $\hat{\boldsymbol{B}}_{g}=\nabla\times\hat{\boldsymbol{\Psi}}$
in which the vector field $\hat{\boldsymbol{\Psi}}$ is defined based on
equation (42)2 and Theorem 1.

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

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

$\displaystyle\boldsymbol{B}_{c}$
$\displaystyle=(1-l_{1}^{2}\nabla^{2})(1-l_{2}^{2}\nabla^{2})\nabla^{2}\boldsymbol{W},$
(55) $\displaystyle\boldsymbol{B}_{g}$
$\displaystyle=-l_{2}^{2}\,(1-l_{1}^{2}\nabla^{2})\nabla^{2}(\nabla\times\nabla\times\boldsymbol{W})$

Scalar potentials $\varphi_{c}$ and $\varphi_{g}$ in PN solution (49) can be
defined based on relations (46)-(48) and Corollary 1 as follows:

$\displaystyle\varphi_{c}$ $\displaystyle=\beta_{c}+2\tilde{\varphi}_{c}$ (56)
$\displaystyle\varphi_{g}$ $\displaystyle=l_{1}^{2}\nabla^{2}\varphi$

where $\beta_{c}$ is indeterminate harmonic function and $\tilde{\varphi}_{c}$
has to satisfy

$\displaystyle\tilde{\varphi}_{c}=(1-l_{1}^{2}\nabla^{2})\varphi$ (57)

Scalar potential $\varphi$ was defined in Section 4 based on the following
relation:

$\displaystyle\varphi=\bar{\varphi}+\varphi^{*}_{b}$ (58)

in which according to (27)1:

$\displaystyle\bar{\varphi}=(1-l_{3}^{2}\nabla^{2})(\nabla\cdot\bar{\boldsymbol{W}})$
(59)

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

$\displaystyle\varphi^{*}_{b}=\varphi^{*}-\psi_{c}^{*}-\psi_{g}^{*}=(1-l_{3}^{2}\nabla^{2})(\nabla\cdot\boldsymbol{W}^{*})-\frac{1}{2}(\boldsymbol{r}\cdot\boldsymbol{B}_{c}+\beta_{c})-\frac{1}{\kappa}\Psi$
(60)

By its definition, $\boldsymbol{B}_{g}$ (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:

$\displaystyle\varphi=(1-l_{3}^{2}\nabla^{2})(\nabla\cdot\boldsymbol{W})-\frac{1}{2}(\boldsymbol{r}\cdot\boldsymbol{B}_{c}+\beta_{c})$
(61)

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

$\displaystyle\tilde{\varphi}_{c}=(1-l_{1}^{2}\nabla^{2})(1-l_{3}^{2}\nabla^{2})(\nabla\cdot\boldsymbol{W})-\frac{1}{2}(\boldsymbol{r}\cdot\boldsymbol{B}_{c}+\beta_{c})+l_{1}^{2}\nabla\cdot\boldsymbol{B}$
(62)

Finally, classical scalar PN stress functions $\varphi_{c}$ can be defined by
using (LABEL:fcfg1)1, (62) as follows:

$\displaystyle\varphi_{c}=2(1-l_{1}^{2}\nabla^{2})(1-l_{3}^{2}\nabla^{2})(\nabla\cdot\boldsymbol{W})-\boldsymbol{r}\cdot\boldsymbol{B}_{c}+2l_{1}^{2}\nabla\cdot\boldsymbol{B}_{c}$
(63)

Gradient scalar PN stress function $\varphi_{g}$ can be defined based on
relations (LABEL:fcfg1)2, (61):

$\displaystyle\varphi_{g}=l_{1}^{2}\nabla^{2}(1-l_{3}^{2}\nabla^{2})(\nabla\cdot\boldsymbol{W})-l_{1}^{2}\nabla\cdot\boldsymbol{B}_{c}$
(64)

Derived relations (LABEL:tbgbc), (63), (64) allow us to define PN stress
functions $\boldsymbol{B}_{c}$, $\boldsymbol{B}_{g}$, $\varphi_{c}$,
$\varphi_{g}$ by using given Galerkin stress vector $\boldsymbol{W}$, which
representation for a given arbitrary displacement field $\boldsymbol{u}$ have
been already found (54). Then, according to derivations presented in previous
section, equation (18) implies (50), and (20) implies (49). Therefore, PN
solution (49), (50) is also complete within SGET. ∎

##  6 Fundamental solution

Derivation of the fundamental solution based on PN general solution within the
classical elasticity can be found, e.g. in [9, 46]. Within SGET similar result
can be obtained by using developed form of PN solution (49), (50). Thus, let
us consider the concentrated body force $\boldsymbol{Q}$ applied at the origin
of coordinate system:

$\boldsymbol{b}=\boldsymbol{Q}\delta(\boldsymbol{r})$ (65)

where $\delta(\boldsymbol{r})$ is Dirac delta function.

Based on (49) the fundamental (Kelvin) solution of SGET that corresponds to
the body force (65) can be presented as follows:

$\displaystyle\boldsymbol{u}=\boldsymbol{B}^{*}_{c}-\frac{1}{4(1-\nu)}\nabla(\boldsymbol{r}\cdot\boldsymbol{B}^{*}_{c}+\varphi^{*}_{c})+\boldsymbol{B}^{*}_{g}+l_{1}^{2}\nabla\varphi^{*}_{g}$
(66)

where the stress functions $\boldsymbol{B}^{*}_{c}$, $\boldsymbol{B}^{*}_{g}$,
$\varphi^{*}_{c}$, $\varphi^{*}_{g}$ are the particular solutions of the
following equations

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

The remaining task is to found the particular solutions of equations
(67)-(70). Solution for vector stress function $\boldsymbol{B}^{*}_{c}$(67) is
similar to classical elasticity and according to potential theory it is given
by [46]:

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

where $r=\lvert\boldsymbol{r}\rvert$.

Scalar stress function $\varphi^{*}_{c}$ in classical elasticity vanishes
since it is valid that:

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

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

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

Particular solution to this equation can be presented in the following form:

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

which follows from formula for the particular solutions of Poisson equation
(14) and the identity that can be proven based on divergence theorem:

$\int\limits_{D}f(\boldsymbol{r}-\boldsymbol{\xi})\nabla_{\boldsymbol{\xi}}\cdot(\boldsymbol{Q}\delta(\boldsymbol{\xi}))dv_{\boldsymbol{\xi}}=\boldsymbol{Q}\cdot\nabla_{\boldsymbol{r}}\,f(\boldsymbol{r})$
(75)

where $f(\boldsymbol{r}-\boldsymbol{\xi})$ is an arbitrary function;
$\boldsymbol{\xi}$ is integration variable; $\boldsymbol{r}$ is radial
distance.

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

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

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

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

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

$\displaystyle\nabla\cdot\tilde{\boldsymbol{B}}^{*}_{g}=\nabla^{2}\Psi^{*}$
(78)

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

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

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

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

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

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

Substitution of relations (71), (74), (80), (81) into the representation for
the displacement field (66) provides us

$\displaystyle\boldsymbol{u}$ $\displaystyle=\frac{\boldsymbol{Q}}{4\pi\mu
r}-\frac{1}{16\pi\mu(1-\nu)}\nabla\left(\frac{\boldsymbol{r}\cdot\boldsymbol{Q}}{r}\right)$
(82)
$\displaystyle+l_{1}^{2}\frac{1-2\nu}{8\pi(1-\nu)\mu}\,\boldsymbol{Q}\cdot\nabla\nabla\left(\frac{1}{r}\right)$
$\displaystyle-\frac{\boldsymbol{Q}}{4\pi\mu}\frac{e^{-r/l_{2}}}{r}-\frac{l_{2}^{2}}{4\pi\mu}\boldsymbol{Q}\cdot\nabla\nabla\left(\frac{1-e^{-r/l_{2}}}{r}\right)$
$\displaystyle-
l_{1}^{2}\frac{1-2\nu}{8\pi(1-\nu)\mu}\,\boldsymbol{Q}\cdot\nabla\nabla\left(\frac{e^{-r/l_{1}}}{r}\right)$

Finally, after some standard simplifications from (82) we obtain the form of
fundamental solution of SGET that exactly coincides to those one presented
previously, e.g. in [59]:

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

##  7 Conclusion

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

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