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Citation: Piskozub, Y.; Piskozub, L.;
Sulym, H. Effect of the Transverse
Functional Gradient of the Thin
Interfacial Inclusion Material on the
Stress Distribution of the Bimaterial
under Longitudinal Shear. Materials
2022,15, 8591. https://doi.org/
10.3390/ma15238591
Academic Editor: Francisco J.
G. Silva
Received: 3 November 2022
Accepted: 28 November 2022
Published: 2 December 2022
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materials
Article
Effect of the Transverse Functional Gradient of the Thin
Interfacial Inclusion Material on the Stress Distribution of the
Bimaterial under Longitudinal Shear
Yosyf Piskozub 1,* , Liubov Piskozub 2and Heorhiy Sulym 3
1Department of Applied Mathematics, Faculty of Computer Science and Telecommunications,
Cracow University of Technology, Warszawska Str. 24, 31-155 Cracow, Poland
2Department of Applied Mathematics and Physics, Ukrainian Academy of Printing, Pidgolosko Str. 19,
79020 Lviv, Ukraine
3Department of Mechanics and Applied Computer Science, Faculty of Mechanical Engineering,
Bialystok University of Technology, Wiejska 45C, 15-351 Bialystok, Poland
*Correspondence: yosyf.piskozub@pk.edu.pl; Tel.: +48-574-560-665 or +38-073-222-4246
Abstract:
The effect of a functional gradient in the cross-section material (FGM) of a thin ribbon-like
interfacial deformable inclusion on the stress–strain state of a piecewise homogeneous linear–elastic
matrix under longitudinal shear conditions is considered. Based on the equations of elasticity theory,
a mathematical model of such an FGM inclusion is constructed. An analytic–numerical analysis of the
stress fields for some typical cases of the continuous functional gradient dependence of the mechanical
properties of the inclusion material is performed. It is proposed to apply the constructed solutions to
select the functional gradient properties of the inclusion material to optimize the stress–strain state in
its vicinity under the given stresses. The derived equations are suitable with minor modifications for
the description of micro-, meso- and nanoscale inclusions. Moreover, the conclusions and calculation
results are easily transferable to similar problems of thermal conductivity and thermoelasticity with
possible frictional heat dissipation.
Keywords:
functionally graded material; thin inclusion; composites; nonperfect contact; frictional
heating; crack; stress intensity factor
1. Introduction
Many structural materials contain numerous thin inhomogeneities in the form of
inclusions of different origins [
1
–
7
]. Quite often, these inclusions are used as elements to
reinforce the structural parts of machines and structures or as fillers in composite materials
or coatings [
8
–
15
]. The use of nanocomposites with specific properties in engineering and
technology has significantly shifted the interest from the study of objects at the macro level
(100–10−1m) and micro level (10−3–10−6m) to the nano level (10−9m) [16–20].
One of the typical examples of composite materials is the structure with thin ribbon
inclusions. Structural elements made with the use of FGM have proven to be rather effective
in practice [
21
–
23
]. In this way, it is possible to achieve a significant improvement in their
mechanical, rheological, thermal, or other properties or the formation of protective thin
layers [18,24–27].
The mathematical modeling of nanostructures requires the construction of more
complex constitutional laws in comparison with the macro level [
28
,
29
]. Therefore, it is
important to construct methods for studying the stress–strain state of such structures. To
model a thin inclusion, there are mainly two basic approaches using analytical methods.
The first one is based on the use of Eshelby’s analytical solution [
21
,
30
,
31
] for an ellipsoidal
inclusion, in which a limiting transition with a decrease in one of the characteristic dimen-
sions of the inclusion is performed. However, its application to thin interphase inclusions
Materials 2022,15, 8591. https://doi.org/10.3390/ma15238591 https://www.mdpi.com/journal/materials
Materials 2022,15, 8591 2 of 12
is impossible. The second one is based on the principle of the conjugation of continua of
different measurability [
32
–
37
] and the method of jump functions [
28
,
29
,
38
–
41
]. According
to this method, the inclusions are replaced by a certain surface (in the two-dimensional
case, a line) of the discontinuity of the physical–mechanical fields, which describes the
perturbing effect of a thin inclusion. Successful attempts were made in [
17
,
33
,
42
–
45
] to
apply it to consider the influence of various physical and contact nonlinearities in the
antiplane problem of elasticity theory for two compressed half-spaces with interfacial
defects. The frictional slip with possible heat generation for contacting bodies [
34
,
44
,
46
–
49
]
and the boundary element approach [
50
–
52
] were also considered here. Inhomogeneity of
the mechanical properties of structural materials can be both designed for a specific pur-
pose (FGM) and a consequence of technological processes of obtaining new materials and
their processing (FSW, ball-burnishing process, etc.) [53,54]. Such factors cause additional
complexity in the constitutive relations for the mathematical modeling of the behavior of
such materials. However, the case of a thin inclusion of an inhomogeneous material has
not been practically studied.
The process of improving the mathematical models of FGM [
11
,
25
,
26
,
55
–
58
] is com-
plicated by the complex geometry of structural elements and the consideration of imper-
fections in the contact of their components. This is especially important to ensure their
qualitative design, both in terms of mechanical strength [
11
,
59
–
63
] and in terms of the con-
sideration of thermal, magnetic, and piezoelectric load factors [13,21,24,44,53,54,58,64,65].
The works [
17
,
31
,
35
,
39
] have been devoted to the consideration of surface energy and
stresses in nanocomposites. Consideration of the heterogeneity of inclusions’ properties
at the micro and nano scale is particularly important because the heterogeneity density of
matter (discrepancy variation in mechanical and other properties) of matter bodies with
a decrease in their scale usually increases, and the impact of this heterogeneity increases
even further.
This publication aims to develop the method of jump functions and to construct a
convenient structured and highly versatile approach to study the longitudinal displacement
and thermal heating of bodies with thin inclusions of arbitrary physical nature, including
those made from FGM.
2. Formulation of the Problem
Consider an unbounded isotropic structure consisting of two half-spaces with elastic
moduli
Gk(k=
1, 2
)
, which is subjected to an external longitudinal shear load determined
by uniformly distributed infinity stresses
σ∞
yz
and
σ∞
yz
, concentrated forces of intensity
Qk
and screw dislocations with vector Burgers component
bk
at points
ς∗k∈Sk(k=1, 2)
. The
stresses must satisfy the conditions
σ∞
xz2G1=σ∞
xz1G2
at infinity to ensure the straightness of
the interface.
Let us investigate the stress–strain state (SSS) of the body section with a plane
xOy
perpendicular to the direction
Oz
of its longitudinal displacement (external problem). The
plane sections of half-spaces perpendicular to this axis form two half-planes
Sk(k=
1, 2
)
and the abscissa axis corresponds to the interface
L∼x
between them (Figure 1). At the
interface of half-spaces (plane xOz), there is a tunnel section
L0= [−a
;
a]
in the direction of
the shear axis z, in which a certain object of general thickness 2h(ha)is inserted.
According to the paradigm of the method of jump functions [
36
], the presence of a thin
inclusion in the bulk is modeled by jumps in the components of stress and displacement
vectors in [38,40,41]:
σyzh∼
=σ−
yz −σ+
yz =f3(x),x∈L0
h∂w
∂xih∼
=∂w−
∂x−∂w+
∂x=σxz
Gh≡σ−
xz
G1−σ+
xz
G2=f6(x);(1)
f3(x) = f6(x) = 0, if x/∈L0. (2)
Materials 2022,15, 8591 3 of 12
Materials2022,15,xFORPEERREVIEW3of13
Figure1.Theloadingandgeometricschemeoftheproblem.
Accordingtotheparadigmofthemethodofjumpfunctions[36],thepresenceofa
thininclusioninthebulkismodeledbyjumpsinthecomponentsofstressanddisplace‐
mentvectorsin[38,40,41]:
3
6
12
,
;
yz yz yz
h
xz xz xz
hh
fx xL
www fx
xxxGGG
(1)
36
0fx fx
,if
xL
.(2)
Itishereinaftermarked
,,
h
x
hxh
,
,,
h
x
hxh
;super‐
scripts“+”and“−”correspondtotheboundaryvaluesofthefunctionsontheupperand
lowerbanksoftheline
L
.
Themathematicalmodelofathininclusionisgivenascomplicatedconditionsof
imperfectcontactbetweenoppositematrixsurfacesadjacenttotheinclusion(internal
problem)[28,38–40,65].Thegeneralmodelofathin,physicallynonlinearinclusionispre‐
sentedin[28,29,38],wherethemethodsofmodelingthinobjectsinvolvetheintegration
ofthedefiningrelationsdescribingthephysicalandmechanicalpropertiesofthematerial
oftheinclusion,withthesubsequentconsiderationofthesmallnessofoneofthelinear
dimensionsoftheinclusion.
Letusconsiderasimilarmodelforathininclusion,assumingthatthemechanical
propertiesoftheinclusionmaterialarecoordinate‐dependent.Thiswillallowustomodel
theinclusionsofafunctionallygradedmaterial:
1
() () 2 ( ) () 0,
() () () 0.
x
in
in in in
xxzyz
h
a
h
in in in
yyz
hh
w
Gx x a d
xh
Gxw x h x
(3)
Here,
()
in
x
Gx
,()
in
y
Gx
arethevariableshearmodulioftheinclusion’smaterial.As
aspecialcase,consideringtheirvaluestobeconstant,weobtainHooke’slaw.Theupper
index“in”denotesthetermsdescribingtheinclusionmaterial’sSSScomponents.
Contactbetweenmatrixcomponentsandtheinclusionat
L
andbetweenthebima‐
terialstructurecomponentsalongaline
\LL
issupposedtobemechanicallyperfect,
(,) (,), (,) (,), ,
in in
yz yzk
wxh w xh xh xh xL
21
(, 0) (, 0),
( , 0) ( , 0), \ ,
yz yz
wx wx
xxxLL
(4)
orfrictionalinsomeareas
f
xL L
,aswasconsideredinworks[41,46–48],
Figure 1. The loading and geometric scheme of the problem.
It is hereinafter marked
[•]h=•(x,−h)− •(x,+h)
,
h•ih=•(x,−h)+•(x,+h)
;
superscripts “+” and “
−
” correspond to the boundary values of the functions on the upper
and lower banks of the line L.
The mathematical model of a thin inclusion is given as complicated conditions of
imperfect contact between opposite matrix surfaces adjacent to the inclusion (internal
problem) [
28
,
38
–
40
,
65
]. The general model of a thin, physically nonlinear inclusion is
presented in [
28
,
29
,
38
], where the methods of modeling thin objects involve the integration
of the defining relations describing the physical and mechanical properties of the material
of the inclusion, with the subsequent consideration of the smallness of one of the linear
dimensions of the inclusion.
Let us consider a similar model for a thin inclusion, assuming that the mechanical
properties of the inclusion material are coordinate-dependent. This will allow us to model
the inclusions of a functionally graded material:
Gin
x(x)D∂win
∂xEh(x)−2σin
xz (−a)−1
h
x
R
−ahσin
yzih(ξ)dξ=0,
Gin
y(x)win h(x) + hDσin
yzEh(x) = 0.
(3)
Here,
Gin
x(x)
,
Gin
y(x)
are the variable shear moduli of the inclusion’s material. As a
special case, considering their values to be constant, we obtain Hooke’s law. The upper
index “in” denotes the terms describing the inclusion material’s SSS components.
Contact between matrix components and the inclusion at
L0
and between the bimaterial
structure components along a line L\L0is supposed to be mechanically perfect,
w(x,±h) = win (x,±h),σin
yz(x,±h) = σyzk(x,±h),x∈L0,
w(x,+0) = w(x,−0),
σyz2(x,+0) = σyz1(x,−0),x∈L\L0,
(4)
or frictional in some areas x∈Lf⊂L0, as was considered in works [41,46–48],
σin
yz(x,±h) = σyzk(x,±h) = −sgnhwin ihτmax
yz . (5)
Here,
τmax
yzK
is the limit value of shear stresses, at which the slippage begins. In this
case, however, additional iterative methods should be applied to determine the area of the
slip zones depending on the specific types of external loading of the composite [41].
Materials 2022,15, 8591 4 of 12
3. Materials and Methods
Expressions for the components of the stress tensor and the derivatives of displace-
ments on the line
L
of the infinite plane
S=S1∪S2
, as well as inside the latter, can be
obtained by applying the results of [37] to the solution of the external problem
σyz(z) + iσxz (z) = σ0
yz(z) + iσ0
xz (z)+
+ipkg3(z)−Cg6(z)(z∈Sk;r=3, 6; k=1, 2);(6)
σ±
yzk(x) = ∓pkf3(x)−Cg6(x) + σ0±
yz (x),
σ±
xzk (x) = ∓C f6(x) + pkg3(x) + σ0±
xz (x),(7)
where the notation [28,41] is introduced:
gr(z)≡1
πR
L0
fr(x)dx
x−z,sr(x)≡
x
R
−a
fr(x)dx,
p=1
G1+G2,pk=Gkp,C=G1G2p.
(8)
The superscript “+” corresponds to k = 2 and “–” corresponds to k = 1. Values marked
with superscript “0” correspond to values in a continuous medium without inclusions
under the same external load (homogeneous solution) [28,41].
Using (7), (8) and boundary conditions (4), it is easy to obtain from model (3) a system
of singular integral equations:
(p2−p1)f6(x) + 2pg3(x)−s3(x)
hGin
x(x)=F3(x),
(p2−p1)f3(x) + 2Cg6(x)−Gi n
y(x)s6(x)
h=F6(x),
F3(x) = 2
Gin
xσin
xz (−a)−σ0
xz2(x)/G2+σ0
xz1(x)/G1,
F6(x) = Dσ0
yzE(x)−Gin
yσ0
yzk(x)
Gk−Gin
y
h0
w(−a).
(9)
Balance conditions on the power balance and unambiguity of displacements while
moving around the thin defect must be added to the solution of the external problem:
a
R
−a
f3(ξ)dξ=2hσin
xz (a)−σin
xz (−a),
a
R
−a
f6(ξ)dξ= [w](a)−[w](−a).
(10)
Solving (9) and (10) using the methods in [
29
,
38
,
41
], it is easy to obtain a system of
linear algebraic equations with unknown coefficients of the decomposition of the jump
functions fr(x)into a series by orthogonal Jacobi or Chebyshev polynomials.
An important aspect of the study of the strength of such structures is the improvement
of their strength criteria. In fracture mechanics, it is acceptable to use the stress intensity
factor to describe the behavior of the SSS in the vicinity of the crack tip [
42
,
45
,
61
–
63
,
66
].
This is not sufficient for the case of a thin deformable inclusion. In [
45
], the authors obtained
the two-term asymptotical expressions for the distribution of SSS in the vicinity of the thin
inclusion tips using the introduced generalized stress intensity factors (GSIF):
K31 +iK32 =lim
r→0(θ=0,π)
√2πrσyz +iσxz. (11)
Here,
(r
,
θ)
is a system of polar coordinates with the origin near the right or the left tip
of the inclusion z=±rexp(iθ)±a.
Materials 2022,15, 8591 5 of 12
Considering the well-known mathematical analogy [
67
], the obtained solutions to the
antiplane problem can be regarded as solutions to the accordant heat conduction problem,
if we take into account the correspondence of the values
w∼T,∂w
∂x∼∂T
∂x,∂w
∂y∼∂T
∂y,qx∼σxz,qy∼σyz,
Gx∼λx,Gy∼λy,K31 ∼kqy,K32 ∼kqx.
The terms are as follows:
T
—temperature,
qx
,
qy
—heat flows,
λx
,
λy
—thermal con-
ductivity coefficients, kqy,kq x—heat flow intensity factors [40].
4. Numerical Results and Discussion
Since the main focus of this article is to investigate the effect of the functional gradient
on the mechanical properties of the inclusion material, we will limit ourselves to one of
the most representative variants of the structure loading: homogeneous longitudinal shear
σ∞
yz =τ
and
σ∞
xzk =τk(k=1, 2)
at infinity. However, the calculations for loading by
concentrated force factors or dislocations do not make any fundamental difference except
for the necessity to consider the locality of their application [29,41].
The dependence
Gx(x)
,
Gy(x)
on coordinate
x
for mathematical modeling can be
defined as an arbitrary function (linear, exponential, power, periodic [
58
], etc.), which
adequately reflects the desired practical properties of the material. To illustrate the method,
let us consider one of the illustrative variants of the functional gradient of the inclusion
material—the piecewise linear one:
Gx(x) = Gy(x) = (G01 −G0)x
a+G01,x∈[−a, 0];
(G02 −G01)x
a+G01,x∈[0, a],(12)
where G0,G01,G02—some given constants.
To significantly reduce the number of calculations without loss of generality, it is
convenient to use the following dimensionless quantities, marked by symbol “~” (tilde)
on top:
e
x=x/a,e
h=h/a,e
y=y/a,
e
Gin
x(e
x) = Gin
x(x)/Ggav ,e
Gin
y(e
x) = Gin
y(x)/Ggav ,
e
τk=τk/Ggav,e
τ=τ/Ggav ,Ggav =√G1G2,
e
G0=G0/Ggav,e
G01 =G01/Ggav,e
G02 =G02/Ggav,
e
σxz(e
x) = σxz(x)/Ggav,e
σyz(e
x) = σyz(x)/Ggav.
e
K31 =K+
31
2e
CGga v√πa,e
K32 =K+
32
2p2Ggav √πa,
where K+
31 K+
32 are the GSIFs near the tip x= +aof the inclusion.
Figures 2–11 illustrate the dependence of the stress–strain behavior of the matrix
in the inclusion vicinity on the variation in the parameters
e
G0
,
e
G01
,
e
G02
, the values of
which were chosen to reveal a qualitative picture of the FGM effect on the stress–strain
parameters. It can be immediately concluded from Figures 2and 3that under the load
e
τ
,
the dimensionless
e
K+
31
are expected to decrease with the increasing shear moduli of any
part of the inclusion, while at e
K+
32 they appear to increase with increasing load e
τk.
The effect of changes in the moduli
e
Gx(x)
,
e
Gy(x)
on the stresses
e
σyz
,
e
σxz
on the inclu-
sion surface is more obvious if we choose a linear growth law for them along the inclusion
axis (Figures 4–6). The magnitude of the surface stresses increases significantly in the stiffer
part of the inclusion. The larger the stiffness gradient, the more significant the increase.
The choice of the piecewise linear law of moduli
e
Gx(x)
,
e
Gy(x)
change in the
Formulae (14)
as
e
G01 =e
G02
(variant 1) or
e
G0=e
G01
(variant 2) has a more contrasting effect on the
surface stresses
e
σyz
,
e
σxz
, especially in the vicinity of the gradient breaking point
x=0
(
Figures 7and 8
). Moreover, variant 2 of the functional dependence of the inclusion mate-
rial moduli leads to partial unloading in the softer part of the inclusion near the breaking
point x=0 (Figure 8).
Materials 2022,15, 8591 6 of 12
Materials2022,15,xFORPEERREVIEW6of13
Figures2–11illustratethedependenceofthestress
–
strainbehaviorofthematrixin
theinclusionvicinityonthevariationintheparameters
00102
,,GG G
,thevaluesofwhich
werechosentorevealaqualitativepictureoftheFGMeffectonthestress
–
strainparame‐
ters.ItcanbeimmediatelyconcludedfromFigures2and3thatundertheload
,the
dimensionless
31
K
areexpectedtodecreasewiththeincreasingshearmoduliofanypart
oftheinclusion,whileat
32
K
theyappeartoincreasewithincreasingloadk
.
Figure2.Influenceoftheparameters
00102
,,GG G
ontheGSIF
31
K
undertheload,uniformly
distributedoninfinitystress
yz
.
Figure3.Influenceoftheparameters
00102
,,GG G
ontheGSIF
32
K
undertheload,uniformly
distributedoninfinitystress
(1,2)
xzk k k
.
Figure 2.
Influence of the parameters
e
G0
,
e
G01
,
e
G02
on the GSIF
e
K+
31
under the load, uniformly
distributed on infinity stress σ∞
yz =τ.
Materials2022,15,xFORPEERREVIEW6of13
Figures2–11illustratethedependenceofthestress
–
strainbehaviorofthematrixin
theinclusionvicinityonthevariationintheparameters
00102
,,GG G
,thevaluesofwhich
werechosentorevealaqualitativepictureoftheFGMeffectonthestress
–
strainparame‐
ters.ItcanbeimmediatelyconcludedfromFigures2and3thatundertheload
,the
dimensionless
31
K
areexpectedtodecreasewiththeincreasingshearmoduliofanypart
oftheinclusion,whileat
32
K
theyappeartoincreasewithincreasingloadk
.
Figure2.Influenceoftheparameters
00102
,,GG G
ontheGSIF
31
K
undertheload,uniformly
distributedoninfinitystress
yz
.
Figure3.Influenceoftheparameters
00102
,,GG G
ontheGSIF
32
K
undertheload,uniformly
distributedoninfinitystress
(1,2)
xzk k k
.
Figure 3.
Influence of the parameters
e
G0
,
e
G01
,
e
G02
on the GSIF
e
K+
32
under the load, uniformly
distributed on infinity stress σ∞
xzk =τk(k=1, 2).
Materials2022,15,xFORPEERREVIEW7of13
Figure4.Stressdistributionalongwiththeupperinterface(inclusion–matrixhalf‐space
2
S
)with
alineardistributionofmaterialstiffness.
Figure5.Stressdistributionalongwiththeupperinterface(inclusion–matrixhalf‐space
2
S
)with
alineardistributionofmaterialstiffness.
Figure6.Stressdistributionalongwiththeupperinterface(inclusion–matrixhalf‐space
2
S
)with
alineardistributionofmaterialstiffness.
Figure 4.
Stress distribution along with the upper interface (inclusion–matrix half-space
S2
) with a
linear distribution of material stiffness.
Materials 2022,15, 8591 7 of 12
Materials2022,15,xFORPEERREVIEW7of13
Figure4.Stressdistributionalongwiththeupperinterface(inclusion–matrixhalf‐space
2
S
)with
alineardistributionofmaterialstiffness.
Figure5.Stressdistributionalongwiththeupperinterface(inclusion–matrixhalf‐space
2
S
)with
alineardistributionofmaterialstiffness.
Figure6.Stressdistributionalongwiththeupperinterface(inclusion–matrixhalf‐space
2
S
)with
alineardistributionofmaterialstiffness.
Figure 5.
Stress distribution along with the upper interface (inclusion–matrix half-space
S2
) with a
linear distribution of material stiffness.
Materials2022,15,xFORPEERREVIEW7of13
Figure4.Stressdistributionalongwiththeupperinterface(inclusion–matrixhalf‐space
2
S
)with
alineardistributionofmaterialstiffness.
Figure5.Stressdistributionalongwiththeupperinterface(inclusion–matrixhalf‐space
2
S
)with
alineardistributionofmaterialstiffness.
Figure6.Stressdistributionalongwiththeupperinterface(inclusion–matrixhalf‐space
2
S
)with
alineardistributionofmaterialstiffness.
Figure 6.
Stress distribution along with the upper interface (inclusion–matrix half-space
S2
) with a
linear distribution of material stiffness.
Materials2022,15,xFORPEERREVIEW8of13
Figure7.Stressdistributionalongwiththeupperinterface(inclusion—matrixhalf‐space
2
S
)
withapiecewiselineardistributionofmaterialstiffness.
Figure8.Stressdistributionalongwiththeupperinterface(inclusion—matrixhalf‐space
2
S
)
withapiecewiselineardistributionofmaterialstiffness.
Figure9.Stressdistributioninthematrixatthevicinityoftheinclusionwithalineardistribution
ofmaterialstiffness.
Figure 7.
Stress distribution along with the upper interface (inclusion—matrix half-space
S2
) with a
piecewise linear distribution of material stiffness.
Materials 2022,15, 8591 8 of 12
Materials2022,15,xFORPEERREVIEW8of13
Figure7.Stressdistributionalongwiththeupperinterface(inclusion—matrixhalf‐space
2
S
)
withapiecewiselineardistributionofmaterialstiffness.
Figure8.Stressdistributionalongwiththeupperinterface(inclusion—matrixhalf‐space
2
S
)
withapiecewiselineardistributionofmaterialstiffness.
Figure9.Stressdistributioninthematrixatthevicinityoftheinclusionwithalineardistribution
ofmaterialstiffness.
Figure 8.
Stress distribution along with the upper interface (inclusion—matrix half-space
S2
) with a
piecewise linear distribution of material stiffness.
Materials2022,15,xFORPEERREVIEW8of13
Figure7.Stressdistributionalongwiththeupperinterface(inclusion—matrixhalf‐space
2
S
)
withapiecewiselineardistributionofmaterialstiffness.
Figure8.Stressdistributionalongwiththeupperinterface(inclusion—matrixhalf‐space
2
S
)
withapiecewiselineardistributionofmaterialstiffness.
Figure9.Stressdistributioninthematrixatthevicinityoftheinclusionwithalineardistribution
ofmaterialstiffness.
Figure 9.
Stress distribution in the matrix at the vicinity of the inclusion with a linear distribution of
material stiffness.
Materials2022,15,xFORPEERREVIEW9of13
Figure10.Stressdistributioninthematrixatthevicinityoftheinclusionwiththepiecewiselinear
distributionofmaterialstiffness.
Figure11.Stressdistributioninthematrixatthevicinityoftheinclusionwiththepiecewiselinear
distributionofmaterialstiffness.
Theeffectofchangesinthemoduli(), ()
xy
GxGx
onthestresses
yz
,
xz
onthe
inclusionsurfaceismoreobviousifwechoosealineargrowthlawforthemalongthe
inclusionaxis(Figures4–6).Themagnitudeofthesurfacestressesincreasessignificantly
inthestifferpartoftheinclusion.Thelargerthestiffnessgradient,themoresignificant
theincrease.
Thechoiceofthepiecewiselinearlawofmoduli(), ()
xy
GxGx
changeintheformulae
(14)as
01 02
GG
(variant1)or
001
GG
(variant2)hasamorecontrastingeffectonthe
surfacestresses
yz
,
xz
,especiallyinthevicinityofthegradientbreakingpoint
0x
(Figures7and8).Moreover,variant2ofthefunctionaldependenceoftheinclusionma‐
terialmodulileadstopartialunloadinginthesofterpartoftheinclusionnearthebreaking
point
0x
(Figure8).
Figures9–11illustratethechangesinthestressfieldinthematrixintheinclusion
vicinityunderdifferentvariantsofthelawoffunctionalchangeoftheinclusionmaterial
moduli.Thetrendstowardsadecreaseinthestressmagnitudesinthevicinityofthestiffer
partsoftheinclusionarevisible.
Figure 10.
Stress distribution in the matrix at the vicinity of the inclusion with the piecewise linear
distribution of material stiffness.
Materials 2022,15, 8591 9 of 12
Materials2022,15,xFORPEERREVIEW9of13
Figure10.Stressdistributioninthematrixatthevicinityoftheinclusionwiththepiecewiselinear
distributionofmaterialstiffness.
Figure11.Stressdistributioninthematrixatthevicinityoftheinclusionwiththepiecewiselinear
distributionofmaterialstiffness.
Theeffectofchangesinthemoduli(), ()
xy
GxGx
onthestresses
yz
,
xz
onthe
inclusionsurfaceismoreobviousifwechoosealineargrowthlawforthemalongthe
inclusionaxis(Figures4–6).Themagnitudeofthesurfacestressesincreasessignificantly
inthestifferpartoftheinclusion.Thelargerthestiffnessgradient,themoresignificant
theincrease.
Thechoiceofthepiecewiselinearlawofmoduli(), ()
xy
GxGx
changeintheformulae
(14)as
01 02
GG
(variant1)or
001
GG
(variant2)hasamorecontrastingeffectonthe
surfacestresses
yz
,
xz
,especiallyinthevicinityofthegradientbreakingpoint
0x
(Figures7and8).Moreover,variant2ofthefunctionaldependenceoftheinclusionma‐
terialmodulileadstopartialunloadinginthesofterpartoftheinclusionnearthebreaking
point
0x
(Figure8).
Figures9–11illustratethechangesinthestressfieldinthematrixintheinclusion
vicinityunderdifferentvariantsofthelawoffunctionalchangeoftheinclusionmaterial
moduli.Thetrendstowardsadecreaseinthestressmagnitudesinthevicinityofthestiffer
partsoftheinclusionarevisible.
Figure 11.
Stress distribution in the matrix at the vicinity of the inclusion with the piecewise linear
distribution of material stiffness.
Figures 9–11 illustrate the changes in the stress field in the matrix in the inclusion
vicinity under different variants of the law of functional change of the inclusion material
moduli. The trends towards a decrease in the stress magnitudes in the vicinity of the stiffer
parts of the inclusion are visible.
5. Conclusions
The proposed sufficiently simple and mathematically correct methodology made it
possible for us to construct, for the first time, a mathematical model of a deformable thin
linear interfacial inclusion with essentially inhomogeneous linear mechanical properties.
Such a model can be used to simulate a thin inclusion from a functionally graded material
and to solve the corresponding problems of defining the stress–strain field of the corre-
sponding micro- or nanostructures by efficient analytical–numerical methods (the jump
function method and its modifications), without the need to involve purely numerical
approaches (in particular, FEM).
The calculations of the stress–strain field components for simple test cases of the
functional dependence of the shear moduli of inclusion material have demonstrated the
expected qualitative picture of their effect on the variation in the FGM parameters. In
particular, (1) the stress magnitude increases significantly in the vicinity of the inclusion
regions with increased stiffness; (2) the combination of the inclusion materials from parts
with piecewise linear mechanical characteristics may lead to partial unloading of the
inclusion and matrix in their softer part in the vicinity of the breaking point of the gradient
dependence of the inclusion material parameters; (3) the contrast of the stress field changes
of the inclusion and matrix is proportional to the increase in the gradient dependence.
The conclusions and calculation results are easily transferable to analogous problems
of thermal conductivity and thermoelasticity with possible frictional heat generation and
can be used for recommendations on the optimal operating parameters of structures.
The discussed conclusions can be useful in designing the functionally gradient me-
chanical properties of the material of inclusions and in the optimization of engineering
structures to increase their strength and service life. The proposed method is effective for
solving a wide class of problems of deformation of solids with thin deformable inclusions
of finite length and can be used for SSS calculation for different FGM inclusions.
Author Contributions:
Conceptualization, Y.P. and H.S.; methodology, Y.P. and H.S.; software, Y.P.
and L.P.; validation, Y.P. and H.S.; formal analysis, Y.P.; investigation, Y.P. and H.S.; resources,
H.S.; data curation, Y.P.; writing—original draft preparation, Y.P.; writing—review and editing, Y.P.;
Materials 2022,15, 8591 10 of 12
visualization, Y.P. and L.P.; supervision, H.S.; project administration, H.S.; funding acquisition, H.S.
All authors have read and agreed to the published version of the manuscript.
Funding:
This investigation was performed within the framework of research project No. 2017/27/B/
ST8/01249, funded by the National Science Centre, Poland, and with project financing through the
program of the Minister of Education and Science of Poland named “Regional Initiative of Excellence”
in 2019–2022, project No. 011/RID/2018/19; amount of financing: 12,000,000 PLN.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement:
The data presented in this study are openly available at https://ua1lib.org
/book/665574/5c937e (accessed on 9 January 2022), reference number [
37
]; https://doi.org/10.3390/ma
15041435 (accessed on 9 January 2022), reference number [
38
]; and https://doi.org/10.3390/ma14174928
(accessed on 9 January 2022), reference number [41].
Conflicts of Interest: The authors declare no conflict of interest.
Nomenclature
FGM functionally gradient material;
GSIF generalized stress intensity factor;
SSS stress–strain state
x,y,zCartesian coordinates;
frjump functions;
Ek,νk,Gk,GinK
x,GinK
yelastic properties of the materials;
Skhalf-planes (sections of the solid);
w,σxz,σyz,σxx,σyy displacement, stresses (components of SSS);
L0= [−a;a]line, modeling the presence of thin inclusion;
Qk,bkmagnitudes of concentrated forces and screw dislocations;
σ∞
yz,σ∞
xzk uniformly distributed in infinity shear stresses.
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