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TJMM6 (2014), No. 2, 181-200

A NEW TECHNIQUE TO DERIVE MANY EXPLICIT

THERMOELASTIC GREEN’S FUNCTIONS

VICTOR S, EREMET

Abstract. This study is devoted to a new technique to derive main thermoelastic

Green’s functions (MTGFs), based on their new integral representations via Green’s

functions for Poison’s equation (GFPE). The derived new integral representations forMTGFs permitted to prove a theorem about their constructive formulas expressed

in terms of respective GFPE and some simplest integrals. The influence functions

of thermoelastic displacements are generated by a unitary heat source. This sourceis applied in an arbitrary inner point of a generalized thermoelastic octant at the

different homogeneous mechanical and thermal boundary conditions, prescribed on

its marginal quadrants. According to the proved theorem many MTGFs for a groupof two-and three dimensional BVPs of thermoelasticity for a plane, a half-plane, a

quadrant, a space, a quarter-space and an octant may be obtained by changing therespective well-known GFPE and calculating some simplest integrals. Some concrete

new MTGFs for octant, quarter-space and half-space are presented. The graphical

and numerical computer evaluation of MTGFs for a thermoelastic octant by usingMaple 15 software also is included. All MTGFs are obtained in terms of elementary

functions that are very important for their numerical implementation. The analytical

checking of MTGFs is given for a new BVP for thermoelastic octant. Using theproposed technique it is possible to extend all obtained results for any domain of

Cartesian system of coordinates.

1. Introduction

The most difficult problem of Green’s function method (GFM) is the derivation ofGreen’s functions (GFs). In monographs [1, 2, 3, 4] are presented the methods to deriveGFs for ordinary and partial differential equations. The derivation and application of GFsand Green’s matrices for two dimensional (2D) BVPs for elliptic system in the theory ofelasticity are presented in the monographs [5, 6]. GFs for advanced materials are derivedin the monographs [7, 8]. A large systematic list of GFs for 2D BVPs of Poisson’sequation, derived for canonical Cartesian and polar domains is given in the encyclopedia[9]. Respectively, in the handbook [10] is given GFs and Green’s matrices (GMs) for two–and 3D BVPs in the theory of elasticity, derived for canonical Cartesian domains. Amongtheories of thermoelasticity [11, 12, 13, 14, 15, 16, 17, 18] the best developed theory whichis widely used in practical calculations is the theory of thermal stresses, i.e. the theoryof uncoupled thermoelasticity. This theory presents a synthesis of the theory of heatconduction and of the theory of elasticity. In the theory of uncoupled heat conductionto solve a BVP a Green’s integral formula provides the temperature field resulted from agiven thermal exposure. The analogous Green’s integral formula determines the field ofelastic displacements, produced by the known mechanical actions. But in thermoelasticityaccording to the integral Maysel’s formula [13, 15, 16] the solution of a BVP is not

2010 Mathematics Subject Classification. 74605, 35J08.Key words and phrases. Green’s functions, thermoelasticity, heat conduction, elasticity, thermoelastic

Green’s functions, volume dilatation, integral representations.

181

182 VICTOR S, EREMET

represented directly in terms of the given data, but in terms of a temperature field, whichin the most cases is to be found. This fact introduces certain inconvenience in applicationof Maysel’s formula except the case when the temperature field is known already. Toavoid mentioned above inconvenience the author of this work has proposed the followingnew integral formula in thermoelasticity [10, 19, 20, 21, 22, 23]:

ui (ξ) = a−1∫VF (x)Ui (x, ξ) dV (x)−

∫ΓD

T (y) ∂Ui (y, ξ) /∂nydΓD (y) +

+∫

ΓN∂T (y) /∂nyUi (y, ξ) dΓN (y) +

+a−1∫

ΓM[αT (y) + a∂T (y) /∂ny] Ui (y, ξ) dΓM (y) ; i = 1, 2, 3,

(1)

where ΓD, ΓN and ΓM denote the surfaces on which the boundary conditions of Dirichlet’s,Neumann’s and mixed type are prescribed, respectively: temperature T (y), heat fluxa∂T (y) /∂ny and a heat exchange between exterior medium and surface of the bodyrepresented by αT (y)+a∂T (y) /∂ny law; F is the heat source; a is thermal conductivity;α is the coefficient of convective heat conductivity; γ = αt (2µ+ 3λ) is the thermoelasticconstant; λ, µ are Lame’s constants of elasticity; αt is the coefficient of the linear thermalexpansion. The introduced in Eq. (1) the main thermoelastic Green’s functions (MTGFs)Ui (x, ξ) have the physical sense as thermoelastic displacements at an arbitrary innerpoint ξ ≡ (ξ1, ξ2, ξ3) of observation, generated by a unit heat source applied at anotherarbitrary inner point x ≡ (x1, x2, x3) and described by δ–Dirac’s function. To determinethe MTGFs in the works [10, 19, 20, 21, 22, 23] was proposed the following integralformula [10, 19, 20, 21, 22, 23]:

Ui (x, ξ) = γ

∫V

GT (x, z) Θ(i) (z, ξ) dV (z); x, z, ξ ∈ V, (2)

where GT is the GF for a heat conduction BVP corresponding to an unit internal pointheat source, and Θ(i) are functions of influence of unit concentrated body forces on elasticvolume dilatation.

The main advantage of the formulas (1) and (2) is that the searched thermoelasticdisplacements ui are determined in the integral form directly via the prescribed innerheat source and other thermal data, given on the boundary. However the elegant Maysel’sintegral formula remain still very important because it was generalized to piezoelectricvibrations [24], to piezoelectric bodies [25], to anisotropic thermoelastic bodies [26] andto the dynamic eigenstrain problem [27].

Note, that, using the formulas (1) and (2), the author derived in elementary functionssome new very useful thermoelastic GFs and Green’s type integral formulas for quadrant[28, 29], half–space [30, 31], quarter-space [32, 33], wedge [34, 35] and half-wedge [36]. Forall these BVPs the difficulties associated with the constructing of the additional influencefunctions for elastic volume dilatation and with the computing of the volume integral (2)have been successfully overcome. Furthermore, it was observed that for more complicatedBVPs of thermoelasticity these difficulties are substantially. This is why the author ofthis article starts to find other methods to drive MTGFs. However, the preliminary hisinvestigations have shown that the classical methods [11, 12, 13, 14, 15, 16, 17, 18] such as:method of body-force analogy [11, 16], Goodier’s method [11], method of thermoelasticpotentials [11, 15, 16] and many other methods [12, 13, 14, 15, 16, 17, 18] leads to necessityto solve additional elastic BVPs or to calculate the complicate volume integral (2). So,the mentioned above methods are not effective for constructing MTGFs. But a crucialmoment of author’s next investigations was, when he discovers that thermoelastic Lameequations permit to be presented in a form of three independent Poisson’s type equations.

A NEW TECHNIQUE TO DERIVE MANY EXPLICIT THERMOELASTIC GREEN’S FUNCTIONS183

So, on this base he obtained the following integral representations [37, 38]:

Ui (x, ξ) + λ+µ2µ ξiΘ (x, ξ) =

− γ2(λ+2µ)xiGi (x, ξ) + γξi

2µ GT (x, ξ)−∫

Γ

[Vi (x, y) ∂

∂nΓ− ∂Vi(x,y)

∂nΓ

]Gi (y, ξ) dΓ (y)

Vi (x, y) = Ui (x, y) + yi2µ [(λ+ µ) Θ (x, y)− γGT (x, y)]

(3)

– for MTGFs, and

Θ (x, ξ) =γ

λ+ 2µGΘ (x, ξ) +

∫Γ

[∂Θ (x, y)

∂nΓ−Θ (x, y)

∂

∂nΓ

]GΘ (y, ξ) dΓ (y) (4)

– for thermoelastic volume dilatation Θ (x, ξ) = Uj,j (x, ξ), j = 1, 2, 3.Furthermore, the next author’s preliminary investigations have shown that on the base

of these integral representations it become possible to develop a new very efficient unifiedtechnique of constructing new MTGFs. In turn using this technique it becomes possibleto create a large very useful in applications database of these functions. This is explainedby the fact that, unlike existing classical methods, where each problem may be solvedin isolation (in this case, for each task should be selected his appropriate method), theexpected new technique will solve immediately the whole class of thermoelastic BVPs.

So, the main objective of this paper is developing a new very efficient unified tech-nique of constructing new MTGFs. Also, on the base of this technique we prove atheorem about derivation the constructive formulas for MTGFs to a general BVP foran octant V (0 ≤ x1, x2, x3 <∞) (or quadrant V (0 ≤ x1, x2 <∞)) which is boundedby the quarter-planes Γ10 (y1 = 0, 0 ≤ y2, y3 <∞), Γ20(0 ≤ y1, y3 < ∞, y2 = 0)and Γ30 (0 ≤ y1, y2 <∞, y3 = 0) (or half-planes Γ10 (y1 = 0, 0 ≤ y2 <∞), Γ20(0 ≤ y1 <∞, y2 = 0)) and with different types of homogeneous mechanical and thermal boundaryconditions. The main advantage of obtained constructive formulas is that by changes ofGFPE and by calculating some simplest integrals it is possible easily to write MTGFs inelementary functions for about 12 BVPs of thermoelasticity.

2. Constructive Formulas for MTGFs in Terms of GFPE

Let us consider some canonical semi-infinite domains, whose surfaces represent planesor their parts (straight lines or their parts) of Cartesian system of coordinates. Let alsothese domains have not parallel planes or their parts (parallel straight lines or their parts).

For considered domains, if on the mentioned-above boundaries are given homogeneouslocally-mixed boundary conditions (zero normal stresses and tangential displacements orzero normal stresses and tangential displacements are given in any combinations) are truesome theorems about the constructing MTGFs Ui (x, ξ) and derivation of Green-typeintegral formula, by using integral representations (3) and (4). So, in the work [37] isderived MTGFs and Green-type integral formula for a special BVP of thermoelasticityfor an octant. Also in [38] are derived constructive formulas for MTGFs Ui (x, ξ) for aclass of BVPs of thermoelasticity for an octant, when boundary conditions for thermalGF GT (x, ξ) are linked with mechanical boundary conditions in such a way that derivedMTGFs Ui (x, ξ) are expressed in terms of GFPE only. In the present study boundaryconditions for thermal GF GT (x, ξ) are linked with mechanical boundary conditions inmore complicated form, so that derived MTGFs Ui (x, ξ) are not expressed in terms ofGFPE only, but appear some new additional harmonic functions, so that an obtainedresult differs substantially from those, presented in [37] and [38]. For these cases is truethe following

Theorem 1. Let the field of MTGFs for displacements Ui (x, ξ) and temperature GT (x, ξ)at inner points ξ ≡ (ξ1, ξ2, ξ3) of the generalized BVP for thermoelastic octant V (0 ≤


x1, x2, x3 < ∞) be determined by non-homogeneous Lame thermoelastic equations µ∇2ξ

Ui(x, ξ) + (λ+µ)Θ,ξi(x, ξ)−γGT,ξi (x, ξ) = 0, Poisson equation ∇2xGT (x, ξ) = −δ (x− ξ)

and in the points y ≡ (0, y2, y3), y ≡ (y1, 0, y3) and y ≡ (y1, y2, 0) of boundary quadrantsΓ10 (y1 = 0, 0 ≤ y2, y3 <∞), Γ20(0 ≤ y1 < ∞, y2 = 0, 0 ≤ y3 < ∞) and Γ30(0 ≤ y1, y2 <∞, y3 = 0) the following homogeneous locally-mixed mechanical and thermal conditionsare given:

U1 (x; 0, ξ2, ξ3) = U2 (x; 0, ξ2, ξ3) = U3 (x; 0, ξ2, ξ3) = 0; ∂GT (x; 0, ξ2, ξ3) /∂nξ1 = 0 (5a)

– rigid fixation of the points of boundary quadrant Γ10 (y1 = 0, 0 ≤ y2, y3 <∞);

σ21 (x; ξ1, 0, ξ3) = U2 (x; ξ1, 0, ξ3) = σ23 (x; ξ1, 0, ξ3) = 0;∂GT (x; ξ1, 0, ξ3) /∂nξ2 = 0

(5b)

or

U1 (x; ξ1, 0, ξ3) = σ22 (x; ξ1, 0, ξ3) = U3 (x; ξ1, 0, ξ3) = 0;GT (x; ξ1, 0, ξ3) = 0 (5c)

– on the boundary quadrant Γ20(0 ≤ y1 <∞, y2 = 0, 0 ≤ y3 <∞), and

σ31 (x; ξ1, ξ2, 0) = σ32 (x; ξ1, ξ2, 0) = U3 (x; ξ1, ξ2, 0) = 0;∂GT (x; ξ1, ξ2, 0) /∂nξ3 = 0

(5d)

or

U1 (x; ξ1, ξ2, 0) = U2 (x; ξ1, ξ2, 0) = σ33 (x; ξ1, ξ2, 0) = 0;GT (x; ξ1, ξ2, 0) = 0 (5e)

– on the boundary quadrant Γ30(0 ≤ y1, y2 < ∞, y3 = 0), where σ33 and σ21 σ31 σ23 arethe normal and the tangential stresses which are determined by the well-known Duhamel-Neumann law

σij = µ (Ui,j + Uj,i) + δij (λUk,k − γGT ) ; i, j = 1, 2, 3. (6)

Then the constructive formulae for MTGFs Ui (x, ξ) and thermoelastic volume dilatationΘ (x, ξ) for this class of BVPs of thermoelasticity are the following:

Ui (x, ξ) = γ2(λ+2µ)

{[ξiµ ((λ+ 2µ)GT (x, ξ)− (λ+ µ)G1 (x, ξ))− xiGi (x, ξ)

]−2ξk

λ+2µµ WT (x, ξ)− 2B−1

(x1

∂∂x1− 2λ+2µ

λ+µ

)ξ1

∂∂ξi

∫WT (x, ξ) dξ1

}Θ (x, ξ) = γ

λ+2µ

[G1 (x, ξ) + 2B−1

λ+µ

(µx1

∂∂x1

+ λ+ 2µ)WT (x, ξ)

]i = 1, 2, 3; B = λ+3µ

λ+µ ,

(7)

were GT (x, ξ), Gi (x, ξ) are GFPE and WT (x, ξ) is that regular part of the Green’s func-tions GT (x, ξ) which contains inferior index 1 (that part of the GT (x, ξ) which are re-flected via boundary Γ10). For functions GT (x, ξ), Gi (x, ξ) on the marginal planes ortheir parts (straight lines or their parts) are given homogeneous conditions that are similarto boundary conditions for temperature and thermoelastic displacements Ui (x, ξ) respec-tively. So, as example, under boundary conditions for Gi (x, ξ) it mean that, if Ui = 0,then Gi = 0 and if Ui,n = 0, then Gi,n = 0.

Proof. First, we use the general representations (3) and (4) that in the case of the octantV ≡ (0 ≤ x1, x2, x3 ≤ ∞) can be rewritten in the following form:

Θ (x, ξ) =γ

λ+ 2µGΘ (x, ξ) +

3∑j=1

∫Γj0

[∂Θ (x, y)

∂nyj−Θ (x, y)

∂

∂nyj

]GΘ (y, ξ) dΓj0 (y) (8)


– for thermoelastic volume dilatation Θ (x, ξ), and

Ui (x, ξ) = −λ+µ2µ ξiΘ (x, ξ)− γ

2(λ+2µ)xiGi (x, ξ) + γξi2µ GT (x, ξ)−∑3

j=1

∫Γj0

[Vi (x, y) ∂

∂nyj− ∂Vi(x,y)

∂nyj

]Gi (y, ξ) dΓj0 (y)

(9)

– for MTGFs Ui (x, ξ).Second, we use the following hypotheses presented in works [37] and [38]:

a) Let the surfaces of some domains represent planes or their parts (straight lines or theirparts) of Cartesian system of coordinates. If on the marginal planes or their parts(straight lines or their parts) are given zero normal displacements, zero tangentialstresses and zero normal derivative of Green’s function GT,n = 0 for temperature (seeEqs. (5b) and (5d), then the normal derivative of volume dilatation is equal to zero,Θ,n = 0;

b) Respectively, if on the marginal planes or their parts (straight lines or their parts) aregiven zero normal stresses, zero tangential displacements and zero Green’s functionGT = 0 for temperature (see Eqs. (5c) and (5e), then volume dilatation Θ = 0.

Also here we prove and use in future the following hypothesis:c) If on the marginal planes or their parts (straight lines or their parts) are given zero

normal and tangential displacements and zero Green’s function for temperature (seeEqs. (5a), then volume dilatation on these kind of surfaces (or counters), as exampleboundary quadrant Γ10 (y1 = 0, 0 ≤ y2, y3 <∞), is determined as follows:

Θ |ξ1=0 =2γ

(λ+ 2µ)B−1

[µ

λ+ µx1

∂

∂x1+λ+ 2µ

λ+ µ

]WT |ξ1=0 (10a)

where WT is that regular part of the Green’s function GT (x, ξ) which contains inferiorindex 1 (that part of the GT (x, ξ) which are reflected via boundary Γ10).

To prove Eq. (10a) let us use the relations:

Θ|ξ1=0 = (U1,ξ1 + U2,ξ2 + U3,ξ3) |ξ1=0 =

= U1,ξ1 |ξ1=0 + U2,ξ2 |ξ1=0 + U3,ξ3 |ξ1=0 = U1,ξ1 |ξ1=0(10b)

that follows from boundary conditions (5a): U2 = U3 = 0⇒ U2,ξ2 |ξ1=0 = U3,ξ3 |ξ1=0 = 0.Next, let in the representations (3) and (4) the functions Gi, GΘ and GT are the GFPE

those homogeneous boundary conditions are the similar to the boundary conditions forUi, Θ and GT , respectively. So, it mean that, if on a marginal quadrant are known Uiand Θ, T , then Gi = 0 and GΘ = GT = 0; and if on a marginal quadrant are knownUi,n and Θ,n, T,n, then Gi,n = 0 and GΘ,n = GT,n = 0. In these cases, the hypothesis c)boundary conditions (5a) lead to following equivalent boundary conditions:

U1 = U2 = U3 = 0; ∂GT /∂n10 = 0⇒ G1 = G2 = G3 = GΘ = ∂GT /∂n10 = 0 (11a)

– on the marginal quadrant Γ10 (y1 = 0, 0 ≤ y2, y3 <∞).Also, in the mentioned-above cases, using the hypotheses a) and b) in the works [37,

38] is proved that the boundary conditions (5b), (5c) and (5d), (5e) lead to followingequivalent locally-mixed boundary conditions:

σ21 = U2 = σ23 = 0;GT,2 = 0⇒ U1,2 = 0;U2 = 0;U2,1 = U2,3 = U3,2 = 0⇒ Θ,2 = 0;G1,2 = 0;G2 = 0;G3,2 = 0;GΘ,2 = 0;GT,2 = 0

(11b)

or

U1 = σ22 = U3 = 0;GT = 0;⇒ U1 = U1,1 = U1,3 = U3 == U3,1 = U3,3 = U2,2 = 0⇒ Θ = 0;G1 = G2,2 = G3 = GΘ = GT = 0

(11c)


– on the marginal quadrant Γ20(0 ≤ y1 <∞, y2 = 0, 0 ≤ y3 <∞), and

σ31 = σ32 = U3 = 0; ∂GT /∂nξ3 = 0;⇒ U1,3 = U2,3 = U3 = U3,1 = U3,2

⇒ Θ,3 = 0;G1,3 = 0;G3 = 0;G2,3 = 0;GΘ,3 = 0;GT,3 = 0;(11d)

or

U1 = U2 = σ33 = 0;GT = 0;⇒ U1 = U1,1 = U1,2 = U2 = U2,2 = U2,1 = U3,3

⇒ Θ = G1 = G2 = G3,3 = GΘ = GT = 0(11e)

– on the marginal quadrant Γ30(0 ≤ y1, y2 <∞, y3 = 0).Now substituting the boundary conditions (11c), (11d) and (11e) into Eq. (3) we

obtain the following simplified integral representations for MTGFs:

Ui (x, ξ) = −λ+µ2µ ξiΘ (x, ξ)− γ

2(λ+2µ)xiGi (x, ξ) + γξi2µ GT (x, ξ)−

−∫

Γ10

[Vi (x, y) ∂

∂ny1− ∂Vi(x,y)

∂ny1

]Gi (y, ξ) dΓ10 (y)

(12a)

Taking into account expression (3) for V1 (x, y) and boundary conditions (11a) we canrewrite Eq. (12a) at i = 1 as follows below:

U1 (x, ξ) = −λ+µ2µ ξ1Θ (x, ξ)− γ

2(λ+2µ)x1G1 (x, ξ) + γξ12µ GT (x, ξ)

−∫

Γ10

[U1 (x, y) + y1

2µ [(λ+ µ) Θ (x, y)− γGT (x, y)]]

∂∂ny1

G1 (y, ξ) dΓ10 (y)(12b)

or in the final form:

U1 (x, ξ) = −λ+ µ

2µξ1Θ (x, ξ)− γ

2 (λ+ 2µ)x1G1 (x, ξ) +

γξ12µ

GT (x, ξ) (12c)

because on marginal quadrant Γ10 U1 (x, y) = 0 and y1 = 0.According to (10b) from (12c) we obtain:

U1,ξ1 |ξ1=0 =

[−λ+ µ

2µΘ− γ

2 (λ+ 2µ)x1G1,ξ1 +

γ

2µGT

]|ξ1=0 (12d)

or

Θ|ξ1=0 =γ

(λ+ 2µ)B−1

[λ+ 2µ

λ+ µGT −

µ

λ+ µx1G1,ξ1

]|ξ1=0 (12e)

Next, due equality GT |ξ1=0 = 2WT and due the fact that G1 (x, ξ) has the sameboundary conditions on the marginal quadrants Γ20 and Γ30 as the function GT (x, ξ)except on marginal quadrant Γ10 where the function G1|ξ1=0 = 0 and GT,ξ1 |ξ1=0 = 0,follows that G1,ξ1 |ξ1=0 = −GT,x1 |ξ1=0 = −2WT,x1 |ξ1=0. So, applying the last result in(12e) we obtain volume dilatation on Γ10 given in Eq. (10a).

Substituting the boundary values of the volume dilatation Θ and respective GFPE GΘ

from equations (10a) and (11b), (11c), (11d) and (11e) into representation (4) we can seethat integrals on marginal quadrants Γ20 and Γ30 are zero, so that we obtain:

Θ (x, ξ) = γλ+2µ

[GΘ (x, ξ)− 2B−1

λ+µ

(µx1

∂∂x1

+ λ+ 2µ)

∫Γ10

WT (x; 0, y2, y3) ∂GΘ(0,y2,y3,ξ)∂ny1

dΓ10 (y)] (13a)

As boundary conditions (11a), (11b), (11c), (11d) and (11e) for GΘ (x, ξ) and G1 (x, ξ)on all marginal quadrants are the same, then GΘ (x, ξ) = G1 (x, ξ) and from (13a) follows:

Θ (x, ξ) =γ

λ+ 2µ

[G1 (x, ξ) +

2B−1

λ+ µ

(µx1

∂

∂x1+ λ+ 2µ

)WT (x, ξ)

], (13b)

where WT (x, ξ) is that regular part of the Green’s functions GT (x, ξ) which containsinferior index 1 (that part of the GT (x, ξ) which are reflected via boundary Γ10). So, theformula (13b) coincides with the constructive formula for volume dilatation in the Eq.


(7). Substituting (13b) into (12c) we obtain the final constructive formula for MTGFsU1 (x, ξ):

U1 (x, ξ) = γ2(λ+2µ)

[ξ1µ ((λ+ 2µ)GT (x, ξ)− (λ+ µ)G1 (x, ξ)) −

−x1G1 (x, ξ)− 2B−1ξ1

(x1

∂∂x1

+ λ+2µµ

)WT (x, ξ)

] (14)

that coincides with the result in the Eq. (7) of the theorem obtained at i = 1.Next, applying (13b) into (12a) at i = k = 2, 3 we obtain:

Uk (x, ξ) = γ2(λ+2µ)

[ξkµ ((λ+ 2µ)GT (x, ξ)− (λ+ µ)G1 (x, ξ))−

−xkGk (x, ξ)− 2B−1ξk

(x1

∂∂x1

+ λ+2µµ

)WT (x, ξ)

]+ Ik (x, ξ) ,

(15a)

where taking into account expressions (10a) and (11a) into (12a) the integral Ik can bewritten as follows:

Ik (x, ξ) =

= −γ (λ+ 2µ)−1B−1

(x1

∂∂x1− 2λ+2µ

λ+µ

) ∫Γ10

ykWT (x, y) ∂∂ny1

Gk (y, ξ) dΓ10 (y)

= γ (λ+ 2µ)−1B−1

(x1

∂∂x1− 2λ+2µ

λ+µ

) [ξkWT (x, ξ)− ξ1 ∂

∂ξk

∫WT (x, ξ) dξ1

] (15b)

Finally, introducing integral (15b) into Eq. (15a) we obtain constructive formula

Uk (x, ξ) = γ2(λ+2µ)

{ξkµ [(λ+ 2µ)GT (x, ξ)− (λ+ µ)G1 (x, ξ)]− xkGk (x, ξ)

−2ξkλ+2µµ WT (x, ξ)− 2B−1

(x1

∂∂x1− 2λ+2µ

λ+µ

)ξ1

∂∂ξk

∫WT (x, ξ) dξ1

(15c)

that coincide with the result in Eq. (7) at i = k = 2, 3. �

Note, that one of the most difficult moments of our investigations was the calculationof the integral (15b). But this moment was avoided successfully, when we established thefollowing properties of integral Ik:

• the integral Ik is a harmonic function with respect to coordinates of both points:x ≡ (x1, x2, x3) and ξ ≡ (ξ1, ξ2, ξ3);

• the values of integral Ik on marginal quadrants are determined by the boundaryconditions of his integrands: the integrand Gk (y, ξ) (with respect to coordinatesof the point ξ ≡ (ξ1, ξ2, ξ3)) and of the integrand GT (x, y) (with respect tocoordinates of the point x ≡ (x1, x2, x3). These boundary conditions just aregiven in the Eqs. (11a), (11b), (11c), (11d) and (11e). So, these two propertiesof the left part of integral Ik help us to write his right part as is shown in Eq.(15b).

Note that constructive formulas for MTGFs (7) at i = 1, 2 are applicable also for 2DBVPs of thermoelasticity. Thus the proposed technique permits to derive many MTGFsin thermoelasticity. Indeed, on the base of constructive formula (7) we can easy (bychanging the respective well-known analytical expressions for GFPEs GT (x, ξ), Gi (x, ξ)[10] and calculating some simplest integrals) to write MTGFs Ui (x, ξ) and volume di-latation Θ (x, ξ) in elementary functions for many BVPs of thermoelasticity: eight for 3DBVPs (one for space, one for half-space, two for quarter-space and four for octant) andfour for 2D BVPs (one for plane, one for half-plane and two for quadrant). However, inthis study we give only one example for constructing new MTGFs Ui (x, ξ) and volumedilatation Θ (x, ξ) in elementary functions for a BVP of thermoelasticity for an octant.


3. New Explicit MTGFs Ui (x, ξ) and Green-type Integral Formula for aThermoelastic Octant

Let the field of displacements Ui (x, ξ) and temperature GT (x, ξ) at inner pointsξ ≡ (ξ1, ξ2, ξ3) of the thermoelastic octant V (0 ≤ x1, x2, x3 <∞) be determined by non-homogeneous Lame equations µ∇2

ξUi (x, ξ) + (λ+ µ) Θ,ξi (x, ξ) − γGT,ξi (x, ξ) = 0 and

Poisson equation∇2xGT (x, ξ) = −δ (x− ξ), but in the points y ≡ (0, y2, y3), y ≡ (y1, 0, y3)

and y ≡ (y1y2, 0) of marginal quadrants Γ10 (y1 = 0, 0 ≤ y2, y3 <∞), Γ20(0 ≤ y1 <∞, y2 = 0, 0 ≤ y3 < ∞) and Γ30(0 ≤ y1, y2 < ∞, y3 = 0) the following homogeneousmechanical and thermal conditions are given:

U1 = U2 = U3 = 0; ∂GT /∂n10 = 0 (16a)

– rigid fixation of the points of boundary quadrant Γ10 (y1 = 0, 0 ≤ y2, y3 <∞)

σ21 = U2 = σ23 = 0; ∂GT /∂nξ2 = 0 (16b)

– locally mixed boundary conditions on the marginal quadrant Γ20(0 ≤ y1 < ∞, y2 =0, 0 ≤ y3 <∞) and

σ31 = σ32 = U3 = 0; ∂GT /∂n30 = 0 (16c)

– locally mixed boundary conditions on the boundary quadrant Γ30(0 ≤ y1, y2 <∞, y3 =0).

Then, according to the boundary conditions (11a), (11b), (11d) and (16a), (16b), (16c)the respective boundary conditions for Green’s functions Gi (x, ξ) for Poison’s equationare the following:

G1 = G2 = G3 = ∂GT /∂n10 = GΘ = 0 (17a)

on the boundary quadrant Γ10,

G1,2 = G2 = G3,2 = GT,2 = GΘ,2 = 0 (17b)

on the boundary quadrant Γ20 and

G1,3 = G2,3 = G3 = GT,3 = GΘ,3 = 0 (17c)

on the boundary quadrant Γ30.So, the expressions of GFPEs with boundary conditions (17a), (17b) and (17c) for

octant V are the following [10]:

GΘ (x, ξ) = G1 (x, ξ) =

= (4π)−1 (

R−1 −R−11 +R−1

2 −R−112 +R−1

3 −R−113 +R−1

23 −R−1123

),

(18a)

G2 (x, ξ) = (4π)−1 (

R−1 −R−11 −R

−12 +R−1

12 +R−13 −R

−113 −R

−123 +R−1

123

)(18b)

G3 (x, ξ) = (4π)−1 (

R−1 −R−11 +R−1

2 −R−112 −R

−13 +R−1

13 −R−123 +R−1

123

)(18c)

GT (x, ξ) = (4π)−1 (

R−1 +R−11 +R−1

2 +R−112 +R−1

3 +R−113 +R−1

23 +R−1123

)(18d)

where

R =

√(x1 − ξ1)

2+ (x2 − ξ2)

2+ (x3 − ξ3)

2

R1 =

√(x1 + ξ1)

2+ (x2 − ξ2)

2+ (x3 − ξ3)

2

R2 =

√(x1 − ξ1)

2+ (x2 + ξ2)

2+ (x3 − ξ3)

2

R12 =

√(x1 + ξ1)

2+ (x2 + ξ2)

2+ (x3 − ξ3)

2(18e)

R3 =

√(x1 − ξ1)

2+ (x2 − ξ2)

2+ (x3 + ξ3)

2


R13 =

√(x1 + ξ1)

2+ (x2 − ξ2)

2+ (x3 + ξ3)

2

R23 =

√(x1 − ξ1)

2+ (x2 + ξ2)

2+ (x3 + ξ3)

2

R123 =

√(x1 + ξ1)

2+ (x2 + ξ2)

2+ (x3 + ξ3)

2

On the base of GFPE GT (x, ξ) (18d), (18e) and of the proved theorem, we can rewriteits regular partWT (x, ξ) that contains inferior index 1 (that part of functionGT (x, ξ) thatis reflected via boundary Γ10). So, the function WT (x, ξ) has the following expression:

WT (x, ξ) = (2π)−1 (

R−11 +R−1

12 +R−113 +R−1

123

). (19)

Substituting expressions (18a), (18b), (18c), (18d), (18e) and (19) in the constructiveformula (7) we obtain the final explicit expressions for MTGFs Ui (x, ξ) ; i = 1, 2, 3 andvolume dilatation Θ (x, ξ) as follows:

U1 (x, ξ) = γ8π(λ+2µ)×[

ξ1µ (λ+ 2µ)

(R−1 +R−1

1 +R−12 +R−1

12 +R−13 +R−1

13 +R−123 +R−1

123

)−(

x1 + λ+µµ ξ1

) (R−1 −R−1

1 +R−12 −R

−112 +R−1

3 −R−113 +R−1

23 −R−1123

)−

2B−1ξ1

(x1

∂∂x1

+ λ+2µµ

) (R−1

1 +R−112 +R−1

13 +R−1123

)].

(20a)

U2 (x, ξ) = γ8π(λ+2µ)×{

ξ2µ

[(λ+ 2µ)

(R−1 +R−1

1 +R−12 +R−1

12 +R−13 +R−1

13 +R−123 +R−1

123

)−

(λ+ µ)(R−1 −R−1

1 +R−12 −R

−112 +R−1

3 −R−113 +R−1

23 −R−1123

)]−

x2

(R−1 −R−1

1 −R−12 +R−1

12 +R−13 −R

−113 −R

−123 +R−1

123

)−

2ξ2λ+2µµ

(R−1

1 +R−112 +R−1

13 +R−1123

)− 2B−1

(x1

∂∂x1− 2λ+2µ

λ+µ

)×

ξ1∂∂ξ2

ln (|x1 + ξ1 +R1| · |x1 + ξ1 +R12| · |x1 + ξ1 +R13| · |x1 + ξ1 +R1123|)}

(20b)

U3 (x, ξ) = γ8π(λ+2µ)×{

ξ3µ

[(λ+ 2µ)

(R−1 +R−1

1 +R−12 +R−1

12 +R−13 +R−1

13 +R−123 +R−1

123

)−

(λ+ µ)(R−1 −R−1

1 +R−12 −R

−112 +R−1

3 −R−113 +R−1

23 −R−1123

)]−

x3

(R−1 −R−1

1 +R−12 −R

−112 −R

−13 +R−1

13 −R−123 +R−1

123

)−

2ξ3λ+2µµ

(R−1

1 +R−112 +R−1

13 +R−1123

)− 2B−1

(x1

∂∂x1− 2λ+2µ

λ+µ

)×

ξ1∂∂ξ3

ln (|x1 + ξ1 +R1| · |x1 + ξ1 +R12| · |x1 + ξ1 +R13| · |x1 + ξ1 +R1123|)}

(20c)

– for MTGFs, and

Θ (x, ξ) = γ4π(λ+2µ)

[(R−1 −R−1

1 +R−12 −R

−112 +R−1

3 −R−113 +R−1

23 −R−1123

)+

2B−1

λ+µ

(µx1

∂∂x1

+ λ+ 2µ) (R−1

1 +R−112 +R−1

13 +R−1123

)] (21)

– and for thermoelastic volume dilatation.Also, the final explicit expressions for MTGFs Ui (x, ξ) may be presented in the fol-

lowing compact form:

Ui (x, ξ) = γ8π(λ+2µ)

[∂∂ξi

(R+R1 +R2 +R12 +R3 +R13 +R23 +R123)

+2(∂∂ξi− L(i)

Θ∂∂ξ1

)(ξ1 ln |x1 + ξ1 +R1| |x1 + ξ1 +R12| |x1 + ξ1 +R13|

|x1 + ξ1 +R123| −R1 −R12 −R13 −R123)]

L(i)Θ =

(δ1i −B−1ξ1 (∂/∂ξi)

).

(22)


Indeed, taking the derivatives in (22) it is observed that they coincide with expressions(20a), (20b), (20c).

As example of validation of the obtained MTGFs (20a), (20b), (20c) or (22) in Appen-dix is presented their graphs, constructed using Maple 15 software.

Finally, calculating on the basis of the functions (22) the other influence functions(Ui (0, y2, y3, ξ) on marginal quadrant Γ10, Ui (y, ξ) = Ui (y1, 0, y3; ξ) on marginal quadrantΓ20 and Ui (y, ξ) = Ui (y1, y2, 0; ξ) on marginal quadrant Γ30) and substituting them inthe formula (1) we obtain the following integral solution of above mentioned BVP for thethermoelastic octant in the form:

ui (ξ) = a−1∫∞

0

∫∞0

∫∞0F (z)Ui (z, ξ) dz1dz2dz3−∫ +∞

0

∫ +∞0

∂T (0,y2,y3)∂y1

Ui (0, y2, y3; ξ) dy2dy3−∫ +∞0

∫ +∞0

∂T (y1,0,y3)∂y2

Ui (y1, 0, y3; ξ) dy1dy3

−∫∞

0

∫∞0

∂T (y1,y2,0)∂y3

Ui (y1, y2, 0; ξ) dy1dy2; z ≡ (z1, z2, z3) ; ξ ≡ (ξ1, ξ2, ξ3) ,

(23a)

where Ui (z, ξ) are determined by Eq. (22); the other kernels are determined by thefollowing expressions:

Ui (0, y2, y3; ξ) = γ8π(λ+2µ)

[2 ∂∂ξi

(R+R2 +R3 +R23) + 2(∂∂ξi− L(i)

Θ∂∂ξ1

)× (ξ1 ln |ξ1 +R| |ξ1 +R2| |ξ1 +R3| |ξ1 +R23| −R1 −R12 −R13 −R123)]

(23b)

Ui (y1, 0, y3; ξ) = γ8π(λ+2µ)

[2 ∂∂ξi

(R+R1 +R3 +R13) +

4(∂∂ξi− L(i)

Θ∂∂ξ1

)(ξ1 ln |x1 + ξ1 +R1| |x1 + ξ1 +R13| −R1 −R13)

] (23c)

Ui (x, ξ) = γ8π(λ+2µ)

[2 ∂∂ξi

(R+R1 +R2 +R12) +

4(∂∂ξi− L(i)

Θ∂∂ξ1

)(ξ1 ln |x1 + ξ1 +R1| |x1 + ξ1 +R12| −R1 −R12)

] (23d)

Note, that from formulas (21) and (22) for thermoelastic octant we can obtain respectivenew volume dilatation and MTGFs for half-space V (0 ≤ x1 < +∞,−∞ < x2, x3 < +∞)and also for quarter-space V (0 ≤ x1, x2 < +∞,−∞ < x3 < +∞). To obtain these resultsis enough to omit in the formulas (21) and (22) for thermoelastic octant the terms thatcontain inferior indexes 1, 3 and index 3 respectively.

4. Checking the Derived MTGFs for Thermoelastic Octant

Here we check the obtained MTGFs Ui (x, ξ) for octant by using their proprieties asfunctions of double influence which take in consideration both physical phenomena of solidbody: heat conduction and elasticity [10, 19, 20, 21, 22, 23]. Also in Appendix is presentedMTGFs Ui (x, ξ) and GFPE GT (x, ξ) for octant which are evaluated numerically andgraphically using Maple 15 software.

4.1. Checking the derived MTGFs for thermoelastic octant with respect topoint x ≡ (x1, x2, x3). Here we check the MTGFs for thermoelastic octant derived inthe section 3. So, according to works [10, 19, 20, 21, 22, 23], MTGFs Ui (x, ξ) determinedby (22) must satisfy over the coordinates of the point of observation x ≡ (x1, x2, x3) theequation

∇2xUi (x, ξ) = −γΘ(i) (x, ξ) (24)

and the homogeneous boundary conditions similar to the GFPE GT for octant shown inEqs. (17a), (17b), (17c):

∂Ui (0, y2, y3; ξ) /∂n10 = 0, y ≡ (0, y2, y3) ∈ Γ10 (25a)

∂Ui (y1, 0, y3; ξ) /∂n20 = 0, y ≡ (y1, 0, y3; ξ) ∈ Γ20 (25b)


∂Ui (y1, y2, 0; ξ) /∂n30 = 0, y ≡ (0, y2, y3) ∈ Γ30 (25c)

So, calculating Laplace operator from Ui (x, ξ), determined by equation (22), we obtain:

∇2xUi (x, ξ) = γ [8π (λ+ 2µ)]

−1∇2xUi (x, ξ) = γ [8π (λ+ 2µ)]

−1×[2 ∂∂ξi

(R−1 +R−1

1 +R−12 +R−1

12 +R−13 +R−1

13 +R−123 +R−1

123

)+

+ 4(∂∂ξi− L(i)

Θ∂∂ξ1

) (−R−1

1 −R−112 −R

−113 −R

−1123

)]=

γ [4π (λ+ 2µ)]−1[∂∂ξi

(R−1 −R−1

1+R−1

2−R−1

12+R−1

3 −R−113 +R−1

23 −R−1123

)+2L

(i)Θ

∂∂x1

(R−1

1 +R−112 +R−1

13 +R−1123

)](26a)

But according to handbook [10] (see answer to problem 16.L.9) the obtained in Eq.(26a) last expression may be written as follows:

γ [4π (λ+ 2µ)]−1[∂∂ξi

(R−1 −R−1

1+R−1

2−R−1

12+R−1

3 −R−113 +R−1

23 −R−1123

)+2L

(i)Θ

∂∂ξ1

(R−1

1 +R−112 +R−1

13 +R−1123

)]= −γΘ(i)

(26b)

So, substituting the result (26a) in (26b) in equation (24) we see that it is satisfied.Next taking the derivative in direction to normal n10 on the marginal quadrant Γ10 of

MTGFs Ui (x, ξ), determined by (22), at x → y ≡ (0, y2, y3) we can see that boundarycondition in (25a) is satisfied. Indeed,

∂Ui(x,ξ)∂n10

∣∣x→y≡(0,y2,y3) = −∂Ui(x,ξ)∂x1

∣∣x→y≡(0,y2,y3) =

− γ8π(λ+2µ)

[∂2

∂ξi∂ξ1(−R+R1 −R2 +R12 −R3 +R13 −R23 +R123) +

2(∂∂ξi− L(i)

Θ∂∂ξ1

)(ξ1

∂∂ξ1

ln |x1 + ξ1 +R1| |x1 + ξ1 +R12| |x1 + ξ1 +R13|

|x1 + ξ1 +R123| − ∂∂ξ1

(R1 +R12 +R13 +R123))] ∣∣

x→y≡(0,y2,y3) = 0

(27a)

Taking the derivative in direction to normal n20 on the marginal quadrant Γ20 ofMTGFs Ui (x, ξ), determined by (22), at x → y ≡ (y1, 0, y3) we can see that boundarycondition in (25b) also is satisfied:

∂Ui(x,ξ)∂n20

∣∣x→y≡(y1,0,y3) = −∂Ui(x,ξ)∂x2

∣∣x→y≡(y1,0,y3) = − γ

8π(λ+2µ)×[∂2

∂ξi∂ξ2(−R−R1 +R2 +R12 −R3 −R13 +R23 +R123) + 2

(∂∂ξi− L(i)

Θ∂∂ξ1

)∂∂ξ2

(ξ1 ln |x1+ξ1+R12|·|x1+ξ1+R123|

|x1+ξ1+R1|·|x1+ξ1+R13| +R1 −R12 +R13 −R123

)]x→y≡(y1,0,y3)

= 0

(27b)

Finally, taking the derivative in direction to normal n30 on the marginal quadrant Γ30

of MTGFs Ui (x, ξ), determined by (22), at x→ y ≡ (y1, y2, 0) we can see that boundarycondition in (25c) also is satisfied:

∂Ui(x,ξ)∂n30

∣∣x→y≡(y1,y2,0) = −∂Ui(x,ξ)∂x3

∣∣x→y≡(y1,y2,0) = − γ

8π(λ+2µ)×[∂2

∂ξi∂ξ3(−R−R1 −R2 −R12 +R3 +R13 +R23 +R123) + 2

(∂∂ξi− L(i)

Θ∂∂ξ1

)∂∂ξ3

(ξ1 ln |x1+ξ1+R13||x1+ξ1+R123|

|x1+ξ1+R1||x1+ξ1+R12| +R1 +R12 −R13 −R123

)]x→y≡(y1,y2,0)

= 0.

(27c)

4.2. Checking the derived MTGFs for thermoelastic octant with respect topoint ξ ≡ (ξ1, ξ2, ξ3). According to works [10, 19, 20, 21, 22, 23], MTGFs Ui (x, ξ)determined by (22) must satisfy over the coordinates of the point of application ξ ≡(ξ1, ξ2, ξ3) of the unit point heat source to Lame thermoelastic equations

µ∇2ξUi (x, ξ) + (λ+ µ) Θ, ξi (x, ξ)− γGT, ξi (x, ξ) = 0 (28a)


and boundary conditions in Eqs (16a), (16b), (16c). Indeed, taking Laplace operator fromMTGFs Ui (x, ξ) (22) we obtain:

µ∇2ξUi (x, ξ) = µ γ

4π(λ+2µ)×[∂∂ξi

(R−1 +R−1

1 +R−12 +R−1

12 +R−13 +R−1

13 +R−123 +R−1

123

)+

2B−1 ∂∂ξi

(ln |x1 + ξ1 +R1| |x1 + ξ1 +R12| |x1 + ξ1 +R13| |x1 + ξ1 +R123| −x1

(R−1

1 +R−112 +R−1

13 +R−1123

))] (28b)

Then substituting (28b), (18d), (18e), and (21) into equation (28a) we can see thatLame thermoelastic equations are satisfied.

Also we checked and confirm that the MTGFs Ui (x, ξ) determined by Eq. (22) withrespect to points ξ ≡ (ξ1, ξ2, ξ3) satisfy to boundary conditions (16a), (16b), (16c) or tofollowing identical conditions:

U1 (x; 0, ξ2, ξ3) = U2 (x; 0, ξ2, ξ3) = U3 (x; 0, ξ2, ξ3) = 0 (29a)

σ21 (x; ξ1, 0, ξ3) = U2 (x; ξ1, 0, ξ3) = σ23 (x; ξ1, 0, ξ3) = 0⇒U1,2 (x; ξ1, 0, ξ3) = U2 (x; ξ1, 0, ξ3) = U3,2 (x; ξ1, 0, ξ3) = 0; ξ ≡ (ξ1, 0, ξ3) ∈ Γ20

(29b)

σ31 (x; ξ1, ξ2, 0) = σ32 (x; ξ1, ξ2, 0) = U3 (x; ξ1, ξ2, 0) = 0⇒U1,3 (x; ξ1, ξ2, 0) = U2,3 (x; ξ1, ξ2, 0) = U3 (x; ξ1, ξ2, 0) = 0; ξ ≡ (ξ1, ξ2, 0) ∈ Γ30.

(29c)

Finally we have proved that derived MTGFs for octant in Eq. (22) satisfy to respectiveBVPs of thermoelasticity described by Eqs. (28a) and (16a), (16b).

5. Conclusions

A new technique is proposed for derivation MTGFs Ui (x, ξ) directly from respectiveLame’s equations (28a). This technique is based on new general integral representationsfor MTGFs Ui (x, ξ) which are presented in Eqs. (3) and (4). A theorem about con-structive formulas (7) for MTGFs Ui (x, ξ) in terms of GFPE and other regular harmonicfunctions is proved. According to the obtained constructive formulas the MTGFs for agroup of two-and three dimensional BVPs for a plane, a half-plane, a quadrant, a space,a quarter-space and an octant may be obtained by changing the respective well-knownGFPEs and by calculating simple integrals. New MTGFs for octant, quarter-space andhalf-space are derived. All results are obtained in terms of elementary functions. Thechecking of correctitude of obtained results is given for a new BVP for thermoelasticoctant. The derived MTGFs Ui (x, ξ) and GFPE GT (x, ξ) for octant were evaluatednumerically and graphically by using Maple 15 software. The main advantages of theproposed method in comparison with the GΘ convolution method for MTGFs construct-ing are: a. It is not necessary to derive the functions of influence of a unit concentratedforce onto elastic volume dilatation - Θ(i) and, b. It is not necessary to calculate a com-plicate volume integral of the product of the function Θ(i) and Green’s function GT inthe heat conduction theory. Also the proposed technique may be extended to differentcanonical domains of Cartesian system of coordinates.

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Appendix A. Graphs of MTGFs Ui and GFPE GT within a ThermoelasticOctant

Here we present Figures 1, 2, 3, 4, 5 and 6, showing the MTGFs Ui (x, ξ), determinedby Eqs. (20a), (20b) and (20c) or (22) for 3D BVP of thermoelasticity (28a), (29a), (29b)and (29c) for the octant V (0 ≤ x1, x2, x3 <∞). Also we present Figure 7 showing theGFPE GT (x, ξ) for a 3D BVP of heat conduction theory within octant V that consistfrom Poisson equation ∇2

xGT (x, ξ) = −δ (x− ξ) and boundary conditions (17a), (17b)and (17c) obtained using Maple 15 software. The MTGFs Ui (x, ξ) are generated bythe unitary inner point heat source F = δ (x− ξ). All twelve graphs for the MTGFsUi (x, ξ) were constructed at the following values of the constants: Poisson’s ratio ν = 0.3;elasticity modulus E = 2, 1 · 105Mpa and coefficient of linear thermal expansion αt =1.2 · 10−5 (K)

−1. Graphs of changing MTGFs U1 (x, ξ) in dependence of 0 ≤ ξ1 ≤ 10m;

0 ≤ ξ3 ≤ 10m constructed at ξ1 = 2m; x1 = 2m; x2 = 3m; x3 = 4m - Figure 1a and independence of 0 ≤ x1 ≤ 10m; 0 ≤ x3 ≤ 10m constructed at x1 = 2m; ξ1 = 2m; ξ2 = 3m;ξ3 = 4m - Figure 1b are showed in the Figure 1.

(a) (b)

Figure 1. Graphs of changing MTGFs U1 in dependence of ξ2 ,ξ3 con-structed at ξ1 = 2m; x1 = 2m; x2 = 3m; x3 = 4m – Figure 1a; andin dependence of x2, x3 constructed at x1 = 2m; ξ1 = 2m; ξ2 = 3m;ξ3 = 4m – Figure 1b.

In the Figure 1 it can be observed:

(1) At ξ2 = 0 and ξ3 = 0 the lines tangential to surface U1 in direction ξ2 and ξ3 areparallel to axes 0ξ2 and 0ξ3. This means that boundary condition U1,ξ2(x; ξ1, ξ2 =0, ξ3) = 0 (29b) on the marginal quadrant Γ20 and boundary conditionU1,ξ3(x; ξ1, ξ2, ξ3 = 0) = 0 (29c) on the marginal quadrant Γ30 are satisfied (Figure1a);

(2) At x2 = 0 and x3 = 0 the lines tangential to surface U1 in direction x2 and x3 areparallel to axes 0x2 and 0x3. This means that boundary condition U1,x2

(x, x1, x2 =0, x3; ξ) = 0 (27b) on the marginal quadrant Γ20 and boundary conditionU1,x3

(x1, x2, x3 = 0; ξ) = 0 (27c) on the marginal quadrant Γ30 are satisfied(Figure 1a); This is explaining by fact that boundary conditions for U1 with re-spect to x ≡ (x1, x2, x3) are similar to those for function GT showed in Eqs. (17a),(17b) and (17c);


(3) In the both Figures (Figure 1a and Figure 1b) the MTGF U1 is positive. Thesegraphs are identical. This is explains by Eq. (20a) that at x1 = ξ1 = 2mU1(x, ξ) =U1(ξ, x).

Graphs of changing MTGFs U1 (x, ξ) in dependence of 0 ≤ ξ1 ≤ 10m; 0 ≤ ξ3 ≤ 10m,constructed at x1 = 2m; ξ2 = 3m;x2 = 3m; x3 = 4m – Figure 2a and in dependence of0 ≤ x1 ≤ 10m; 0 ≤ x3 ≤ 10m constructed at x2 = 3m; ξ1 = 2m; ξ2 = 3m; ξ3 = 4m -Figure 2b is showed in the Figure 2.

(a) (b)

Figure 2. Graphs of changing MTGFs U1 in dependence of ξ1, ξ3, con-structed at ξ2 = 3m; x1 = 2m; x2 = 3m; x3 = 4m – Figure 2a; andin dependence of x1, x3, constructed at x2 = 3m; ξ1 = 2m; ξ2 = 3m;ξ3 = 4m – Figure 2b.

In Figure 2 it can be observed:

(1) At ξ1 = 0 the MTGF U1 is zero. This means that boundary conditions andU1(x; ξ1 = 0, ξ2, ξ3) = 0 (29a) on the marginal quadrant Γ10 is satisfied. Atξ3 = 0 the lines tangential to surface U1 in direction ξ3 are parallel to axis 0ξ3.This means that U1,ξ3 (x; ξ1, ξ2, ξ3 = 0) = 0 on the marginal quadrant Γ30, so thatboundary condition (29c) is satisfied (Figure 2a);

(2) At x1 = 0 and x3 = 0 the lines tangential to surface U1 in directions x1 and x3

are parallel to axes 0x1 and 0x3. This means that U1,x1(x1 = 0, x2, x3; ξ) = 0

and U1,x3(x1, x2, x3 = 0; ξ) = 0 on the marginal quadrants Γ10 and Γ30, so that

boundary conditions (27a) and (27c) are satisfied (Figure 2b). This is explainingby fact that boundary conditions for U1 with respect to x ≡ (x1, x2, x3) are similarto those for function GT , shown in equations (17a), (17b) and (17c);

Graphs of changing MTGFs U2 (x, ξ) in dependence of 0 ≤ ξ2 ≤ 10m; 0 ≤ ξ3 ≤ 10m,constructed at ξ1 = 2m; x1 = 2m; x2 = 3m; x3 = 4m – Figure 3a and in dependence of0 ≤ x2 ≤ 10m; 0 ≤ x3 ≤ 10m, constructed at x1 = 2m; ξ1 = 2m; ξ2 = 3m; ξ3 = 4m –Figure 3b are showed in the Figure 3.


(1) At ξ2 = 0 the MTGF U2 is zero. This means that boundary conditionsU2(x; ξ1, ξ2 = 0, ξ3) = 0 (29b) on the marginal quadrants Γ20 are satisfied (Figure3a); At ξ3 = 0 the lines tangential to surface U2 in direction ξ3 are parallel toaxis 0ξ3. This means that boundary conditions U2,ξ3(x; ξ1, ξ2, ξ3 = 0) = 0 (29c)on the marginal quadrants Γ30 are satisfied (Figure 3a);


(a) (b)

Figure 3. Graphs of changing MTGFs U2 in dependence of ξ2, ξ3 con-structed at ξ1 = 2m; x1 = 2m; x2 = 3m; x3 = 4m – Figure 3a; andin dependence of x2, x3 constructed at x1 = 2m; ξ1 = 2m; ξ2 = 3m;ξ3 = 4m – Figure 3b.

(2) At x3 = 0 the lines tangential to surface U2 in direction x3 are parallel to axis0x3. This means that boundary conditions U2,x3

(x1, x2, x3 = 0; ξ) = 0 (27c) onthe marginal quadrants Γ30 are satisfied (Figure 3b); Also, at x2 = 0 the linestangential to surface U2 in direction x2 are parallel to axis 0x2. This means thatboundary condition U2,x2

(x1, x2 = 0, x3; ξ) = 0 (27b) on the marginal quadrantΓ20 are satisfied (Figure 3b). This is explaining by fact that boundary conditionsfor U2 with respect to x ≡ (x1, x2, x3) are similar to those for function GT , shownin equations (17a) and (17c);

(3) In the Figure 3 both graphs (in the Figures 3a and 3b) have jumps: at ξ1 = 2m,ξ3 = 4m (Figure 3a) and at x1 = 2m, x3 = 4m (Figure 3b);

Graphs of changing MTGFs U2 (x, ξ) in dependence of 0 ≤ ξ1 ≤ 10m; 0 ≤ ξ2 ≤ 10m,constructed at ξ3 = 1m; x1 = 2m; x2 = 3m; x3 = 4m – Figure 4a, and in dependence of0 ≤ x1 ≤ 10m; 0 ≤ x2 ≤ 10m, constructed at x3 = 1m; ξ1 = 2m; ξ2 = 3m; ξ3 = 4m –Figure 4b are showed in the Figure 4.


(1) At ξ1 = 0 and at ξ2 = 0 the MTGF U2 is zero. This means that boundaryconditions U2 (x; ξ1 = 0, ξ2, ξ3) = 0 (29a) and U2 (x; ξ1, ξ2 = 0, ξ3) = 0 (29b) onthe marginal quadrants Γ10 and Γ20 are satisfied (Figure 4a);

(2) At x1 = 0 and x2 = 0 (Figures 4a and 4b) the lines tangential to surfaces U2 indirection x1 and x2 are parallel to axes 0x1 and 0x2. This means that boundarycondition U2,x1 (x1 = 0, x2, x3; ξ) = 0 (27a) and U2,x2 (x1, x2 = 0, x3; ξ) = 0 (27b)on the marginal quadrant Γ20 are satisfied (Figure 4b). Note that these boundaryconditions follows from similar boundary conditions for function GT , shown inequations (17a) and (17c);

(3) In the Figure 4 both graphs (in the Figures 4a and 4b) have jumps at ξ1 = 2m,ξ3 = 4m (Figure 4a) and at x1 = 2m, x3 = 4m (Figure 4b);

Graphs of changing MTGFs U3 (x, ξ) in dependence of 0 ≤ ξ2 ≤ 10m; 0 ≤ ξ3 ≤ 10m,constructed at ξ1 = 2m; x1 = 2m; x2 = 3m; x3 = 4m – Figure 5a and in dependence of


(a) (b)

Figure 4. Graphs of changing MTGFs U2 in dependence of ξ1, ξ2 con-structed at ξ3 = 4m; x1 = 2m; x2 = 3m; x3 = 4m – Figure 4a; andin dependence of x1, x2, constructed at x3 = 4m; ξ1 = 2m; ξ2 = 3m;ξ3 = 4m – Figure 4b.

0 ≤ x2 ≤ 10m; 0 ≤ x3 ≤ 10m, constructed at x1 = 2m; ξ1 = 2m; ξ2 = 3m; ξ3 = 4m –Figure 5b are showed in the Figure 5.

(a) (b)



(1) At x2 = 0 and x3 = 0 the lines tangential to surfaces U3 in direction x2 andin direction x3 are parallel to axis 0x2 and 0x3 respectively . This means thatboundary condition U3,x2

(x1, x2 = 0, x3; ξ) = 0 (27b) on the marginal quadrantΓ20 and boundary condition U3,x3

(x1, x2, x3 = 0; ξ) = 0 (27c), on the marginalquadrant Γ30 are satisfied (Figure 5b). This is explaining by fact that boundaryconditions for U3 with respect to x ≡ (x1, x2, x3) are similar to those for functionGT , shown in equations (17a), (17b) and (17c);


(2) At ξ3 = 0 the MTGF U3 is zero. This means that boundary conditionU3(x; ξ1, ξ2, ξ3 = 0) = 0 (29c) on the marginal quadrant Γ30 is satisfied (Fig-ure 5a). At ξ2 = 0 the lines tangential to surfaces U3 in direction ξ2 are parallelto axis 0ξ2. This means that the boundary condition U3,ξ2(x; ξ1, ξ2 = 0, ξ3) = 0(29b) on the marginal quadrant Γ20 is satisfied (Figure 5a);


Graphs of changing MTGF U3 (x, ξ) in dependence of 0 ≤ ξ1 ≤ 10m; 0 ≤ ξ3 ≤ 10m,constructed at ξ2 = 3m; x1 = 2m; x2 = 3m; x3 = 4m – Figure 6a and in dependence of0 ≤ x1 ≤ 10m; 0 ≤ x3 ≤ 10m, constructed at x2 = 3m; ξ1 = 2m; ξ2 = 3m; ξ3 = 4m –Figure 6b are showed in the Figure 6.

(a) (b)



(1) At ξ1 = 0 and ξ3 = 0 the the MTGF U3 is zero. This means that boundaryconditions U3 (x; ξ1 = 0, ξ2, ξ3) = 0 (29a) and U3 (ξ1, ξ2, ξ3 = 0;x) = 0 (29c) onthe marginal quadrants Γ10 and Γ30 are satisfied (Figure 6a);

(2) At x1 = 0 and x3 = 0 the lines tangential to surfaces U3 in directions x1 andx3 are parallel to axes 0x1 and 0x3. This means that boundary conditionsU3,x1 (x1 = 0, x2, x3; ξ) = 0 (27a) and U3,x3 (x1, x2, x3 = 0; ξ) = 0 (27c) on themarginal quadrants Γ10 and Γ30 are satisfied (Figure 6b); This is explaining byfact that boundary conditions for U3 with respect to x ≡ (x1, x2, x3) are similarto those for function GT , showed in Eqs (17a), (17b) and (17c).


Graphs of changing GFPE GT in dependence of 0 ≤ ξ2 ≤ 10m; 0 ≤ ξ3 ≤ 10m,constructed at ξ1 = 2m; x1 = 2m; x2 = 3m; x3 = 4m – Figure 7a and in dependence of0 ≤ ξ1 ≤ 10m; 0 ≤ ξ3 ≤ 10m, constructed at ξ2 = 3m; x1 = 2m; x2 = 3m; x3 = 4m –Figure 7b are showed in the Figure 7.



(a) (b)

Figure 7. Graphs of changing GFPE GT in dependence of ξ2, ξ3, con-structed at ξ1 = 2m; x1 = 2m; x2 = 3m; x3 = 4m – Figure 7a; andin dependence of ξ1, ξ3, constructed at ξ2 = 3m; x1 = 2m; x2 = 3m;x3 = 4m – Figure 7b.

(1) At ξ2 = 0 (Figure 7a) and at ξ3 = 0 (Figure 7a and Figure 7b) the lines tangentialto surface GT in directions ξ2 and ξ3 are parallel to axes 0ξ2 and 0ξ3. Thismeans that boundary condition GT,ξ2 (x; ξ1, ξ2 = 0, ξ3) = 0 (17b) on the marginalquadrant Γ20 (Figure 7a) and boundary condition GT,ξ3 (x; ξ1, ξ2, ξ3 = 0) = 0(17c) on the marginal quadrant Γ30 are satisfied (Figure 7a and Figure 7b);

(2) At ξ1 = 0 the lines tangential to surface GT in directions ξ1 are parallel to axis0ξ1. This means that boundary condition GT,ξ1 (x; ξ1 = 0, ξ2, ξ3) = 0 (17a) onthe marginal quadrant Γ10 is satisfied (Figure 7b);

(3) In the Figures 7a and 7b the GFPE GT ≥ 0. When one of the coordinates of thepoint ξ ≡ (ξ1, ξ2, ξ3) increases the GFPE GT decreases and very soon vanishes.At singular point, when ξ1 = x1 = 2m, ξ2 = x2 = 3m, ξ3 = x3 = 4m the GFPEGT tends to +∞ (Figure 7a and Figure 7b).

Laboratory of Green’s Functions

Agrarian State University of MoldovaMirces,ti Street 44, Chis, inau 2049, Moldova

Laboratory of Mathematical ModelingInstitute of Mathematics and Informatics

Academy of Science of MoldovaE-mail address: [email protected]

mailto:<[email protected]>

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