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Heorhiy Sulym, Andrii Vasylyshyn, Iaroslav Pasternak DOI 10.2478/ama-2022-0029
Influence of Imperfect Interface of Anisotropic Thermomagnetoelectroelastic Bimaterial Solids on Interaction of Thin Deformable Inclusions
242
INFLUENCE OF IMPERFECT INTERFACE
OF ANISOTROPIC THERMOMAGNETOELECTROELASTIC BIMATERIAL SOLIDS
ON INTERACTION OF THIN DEFORMABLE INCLUSIONS
Heorhiy SULYM* , Andrii VASYLYSHYN** , Iaroslav PASTERNAK***
*Bialystok University of Technology, Wiejska Str 45C, 15-351 Bialystok, Poland
**Ivan Franko National University of Lviv, Universytetska Str. 1, 79000 Lviv, Ukraine
***Lesya Ukrainka Volyn National University, Potapova Str 9, 43025 Lutsk, Ukraine
h.sulym@pb.edu.pl, vasylyshyn.c.h@gmail.com, iaroslav.m.pasternak@gmail.com
received 12 March 2022, revised 30 May 2022, accepted 31 May 2022
Abstract: This work studies the problem of thermomagnetoelectroelastic anisotropic bimaterial with imperfect high-temperature conducting
coherent interface, whose components contain thin inclusions. Using the extended Stroh formalism and complex variable calculus,
the Somigliana-type integral formulae and the corresponding boundary integral equations for the anisotropic thermomagnetoelectroelastic
bimaterial with high-temperature conducting coherent interface are obtained. These integral equations are introduced into the modified
boundary element approach. The numerical analysis of new problems is held and results are presented for single and multiple inclusions.
Key words: anisotropic bimaterial, thermomagnetoelectroelastic, imperfect interface, high-temperature conducting, thin inclusion
1. INTRODUCTION
Pyroelectric, pyromagnetic and multifield material structures
are widely used in modern engineering design, especially in the
developed high-tech manufactures, devices of fine mechanics and
of innovative character. These structures allow combining and
redistributing the energy of four fields of different physical nature
(mechanical, thermal, electrical and magnetic), and therefore have
great potential for use in instrument and sensor systems, preci-
sion positioning devices, energy converters and more.
The development of such bimaterials can be provided by me-
chanical combination of pyroelectric (ferroelectric) and magneto-
strictive (piezomagnetic) materials. As a result, a thin layer ap-
pears at the interface, which affects the temperature and stress
fields in a structurally inhomogeneous solid. When modelling the
effect of this layer, certain boundary conditions of imperfect ther-
mal and magnetoelectromechanical contact of bimaterial compo-
nents are used. Mainly in the scientific literature [1], [2], there are
two types of imperfect thermal conditions of contact of a thin layer
with the environment. These are the high- and low-temperature
conducting. There are also two types of imperfect mechanical
contact conditions, which are the soft and rigid interfaces. In
addition, there can be present some other inhomogeneities in the
structural materials (e.g. cracks, thin inclusions etc.), which can
also be modelled in conditions of imperfect contact, and they
should be taken into account. Thus, the development of effective
methods for modelling and studying the distribution of thermal,
mechanical, electric and magnetic fields in bimaterial deformable
solids with an imperfect material interface and internal thin inho-
mogeneities is an important scientific problem with wide possibili-
ties of practical use.
The study of bimaterial solids with defects is quite widely cov-
ered in the scientific literature. For example, in [13] the authors
studied three models of interfacial cracks (electrically perfectly
permeable, semi-permeable and impermeable) in piezoelectric
materials using the boundary element method. The article [14]
presents an analysis of problems for cracks in homogeneous
piezoelectrics and at the interface of two different piezoelectric
materials; the corresponding explicit analytical solutions are ob-
tained. In [2] author uses specially designed conditions at the
material boundary to model the contact surface between two
anisotropic materials. Pan and Amadei in [11] developed an effec-
tive boundary-element approach to solving problems for elastic
anisotropic bimaterial solids containing cracks and thin inclusions.
Wang and Pan [12] constructed Green's functions for an aniso-
tropic thermoelastic bimaterial with a Kapitza-type interface.
An effective method for solving thermomagnetoelectroelastic
problems for bimaterials is an approach based on the methods of
complex variable calculus and the Stroh formalism. It is widely
used in the analysis of anisotropic [6], [7], piezoelectric [7], [15],
[16] and magnetoelectroelastic [15] solids with through cracks and
inclusions. In [3], [4], boundary integral Somilliana-type equations
for the boundary-element analysis of anisotropic thermomagne-
toelectroelastic bimaterial with holes, cracks and thin inclusions
are obtained.
This paper expands the possibilities of the Stroh formalism-
based approach for a thermomagnetoelectroelastic bimaterial
solid with a high-temperature conducting interface and perfect
magnetoelectromechanical contact of components that may con-
tain thin inclusions sensitive to the influence of fields of different
nature. An appropriate mathematical model has been developed.
Also, there were obtained integral equations of the Somilliana-
type and solved a number of problems for single and interacting
inclusions.
DOI 10.2478/ama-2022-0029 acta mechanica et automatica, vol.16 no.3 (2022)
243
2. GOVERNING EQUATIONS
OF THERMOMAGNETOELECTROELASTICITY
Consider a piecewise homogeneous anisotropic linear ther-
momagnetoelectroelastic medium in the reference coordinate
system . According to [6], [7] and [8], the balance equa-
tions for stress, electric displacement, magnetic induction and
heat flux, as well as constitutive relations can be expressed using
the complex variable calculus. The extended Stroh formalism
allows to write the general solution of these equations through
certain analytical functions and as
, ;
,
;
;
, ; , ; (1)
, ;,
,
where is a stress tensor; is a displacement vector; is
heat flux; is electric displacement; is magnetic induction;
is a density of free charges; is the body force; is the density
of distributed heat; and are electric and magnetic potentials;
is a change of temperature with respect to the reference one;
are thermal conductivity coefficients; is heat flow function;
is a vector of Stroh complex potentials; is a tempera-
ture potential; are certain analytical functions; and is a
complex constant (with a positive imaginary part), which is the
root of the characteristic equation of thermal conductivity
.
Matrices , , constants
and vectors and are determined from the
eigenvalue problem of the Stroh formalism [6] on the basis of
elastic, piezoelectric, dielectric and piezomagnetic constants of
the material.
The Stroh complex potentials, the vector-functions of dis-
placement and stress are related by the following equations [7] :
(2)
Function , temperature and heat flux function are related
as
. (3)
3. FORMULATION OF THE PROBLEM
Consider the problem of thermal conductivity and deformation
for an anisotropic thermomagnetoelectroelastic bimaterial medium
with inclusions. It consists of two thermomagnetoelectroelastic
anisotropic half-spaces, which are separated by a surface
and contains cylindrical holes parallel to the axis on the sur-
face of which arbitrary independent mechanical and thermal
boundary conditions are given (Fig.1). In this case, it suffices to
consider the temperature and thermomagnetoelectroelastic state
in an arbitrary cross-section which is perpendicular to .
Fig. 1. Geometric scheme of a plane problem
for a thermomagnetoelectroelastic anisotropic bimaterial medium
At the interface, the conditions of imperfect thermal contact in
the form of a high-temperature conducting interface
,
, (4)
; (5)
and the conditions of perfect magnetoelectromechanical contact
of components are given
, (6)
. (7)
Here, superscripts 1 and 2 are used to denote the values of
the fields acting in half-spaces and , respectively. A thin
intermediate layer is removed from consideration. Each half-space
contains a system of smooth closed contours
and
. On them, it is possible to set various thermal or
mechanical boundary conditions.
To derive the integral formulas for the Stroh complex poten-
tials, we use the Cauchy integral formula [5]:
(8)
It outlines the relationship between the values of an arbitrary
complex function (analytic in ) at the boundary of the
domain outside and inside it. The function is assumed to
have no poles in . Here are complex variables that
characterise the location of the source points and the field, re-
spectively. Also, in Eq. (8) it is assumed that when the domain
is infinite, then the function should vanish at .
4. DERIVATION OF INTEGRAL REPRESENTATIONS
FOR BIMATERIAL WITH IMPERFECT THERMAL
CONTACT OF COMPONENTS
4.1. Thermal conductivity
The problem of thermal conductivity is linear. Its solution can
be represented as a superposition of homogeneous and perturbed
solutions. Homogeneous solutions and
satisfy Eq. (3). The perturbed solutions are caused by the pres-
ence of contours and and certain boundary conditions set
on them.
Heorhiy Sulym, Andrii Vasylyshyn, Iaroslav Pasternak DOI 10.2478/ama-2022-0029
Influence of Imperfect Interface of Anisotropic Thermomagnetoelectroelastic Bimaterial Solids on Interaction of Thin Deformable Inclusions
244
Let us write the Cauchy formulas for the components of the
bimaterial as follows:
(9.1)
(9.2)
Using the conditions of imperfect thermal contact, Eq. (3) can
be written as
; (9.3)
. (9.4)
Thus, we substitute now Eqs (9.3) and (9.4) into Eqs (9.1) and
(9.2), respectively
;
.
Excluding integrals along the interface of half-spaces we ob-
tain
. (10)
. (11)
Here
,
.
;
;
,
. (12)
;
.
Thus, there are obtained integral representation for the tem-
perature and heat flux at any point bimaterial
; (13)
. (14)
The functions and are homogeneous solutions
for the bimaterial
4.2. Thermomagnetoelectric elasticity
Using Eq. (8), we write the Cauchy integral formula for vectors
and of Stroh complex potentials which are
analytical functions in and , respectively
,
.
Introducing notation
,
, (15)
we rewrite them in the form of
,
(16)
,
(17)
Excluding from Eqs (16) and (17) integrals along the interface
of half-spaces using the Stroh orthogonality conditions we obtain
. (18)
DOI 10.2478/ama-2022-0029 acta mechanica et automatica, vol.16 no.3 (2022)
245
Here
,
.
,
.
,
.
. (19)
and
,
.
,
.
,
.
The obtained (18) and (19) allow to write integral relations
that relate the displacements at an arbitrary point of the thermo-
magnetoelectroelastic bimaterial with temperature, heat flux and
displacement and traction on the contours
i
:
=
, (20)
Also, using Eqs (18) and (19) it is possible to write similar ex-
pressions to determine the stress in an arbitrary point of thermo-
magnetoelectroelastic bimaterial
= (21)
.
According to [10], stress and displacement discontinuities in
the vicinity of tips of thin inhomogeneities are characterised by
generalised stress, electric displacements and magnetic induction
intensity factors. They are determined by the discontinuity func-
tions at the tip of inhomogeneity by formulas
;
,
where
,
– are the vectors of general-
ised stress and electric displacement intensity factors;
– the real tensor Burnett–Lotte [9]. The first two
components
of the vector
differ from generalised
SIFs , , which are introduced for purely elastic problems.
To find , through
we need to use the formula
.
Here – the vector of generalised SIF;
– the second real Burnett–Lotte tensor
[9].
Generalised heat flux intensity factors are defined as
.
Fields of displacements, stresses, temperatures and heat flux
in the vicinity of the inclusion tip are fully characterised by general-
ised stress and electric displacement intensity factors and are
defined by the following relationships:
,
;
,
.
5. NUMERICAL EXAMPLES
The obtained integral equations are introduced into the
scheme of the modified boundary elements method [17]. To solve
them, the curves are approximated using rectilinear
segments (boundary elements) . At each element, three nodal
points are set: one at the centre, and two others at the distance of
of element length at both sides of a central node (discontinuous
three-node boundary element; if the polynomial shape functions
are used it is called the discontinuous quadratic boundary element
[7]). The boundary functions of temperature, heat flux, displace-
ment and stress are approximated at the element using their
nodal values. This allows solving specific two-dimensional prob-
lems of thermomagnetoelectroelasticity for bimaterial solids with
imperfect thermal contact of its components in the presence of
inhomogeneities inside them.
Example 1. To verify the proposed numerical method, con-
sider a test problem for finite square solid with elementary load
given on its faces, which has analytic solution. To proceed with
this, let us cut out a square from the upper half-space of a bimate-
rial solid with high-temperature conducting interface, and consider
Heorhiy Sulym, Andrii Vasylyshyn, Iaroslav Pasternak DOI 10.2478/ama-2022-0029
Influence of Imperfect Interface of Anisotropic Thermomagnetoelectroelastic Bimaterial Solids on Interaction of Thin Deformable Inclusions
246
the prismatic body of a square cross section. To model the latter,
we used only 40 boundary elements. The lower boundary of the
body is at a distance r to the interface (Fig. 2). On the upper
boundary of the body it is given a temperature .
Fig. 2. Cross-sectional scheme of a thermoelastic anisotropic
square body
Let us check the influence of the high-temperature conducting
interface on the temperature distribution in a given finite solid. To
do this, let us fix the coordinate
and
find the temperature value at the points
. The obtained plot shows that the temperature change is
a linear function of coordinate, which is the exact analytic solution
of the problem. Moreover, if one changes the resulting plot
does not change, which also verifies the developed boundary
element approach.
Now let us cut out the same prismatic body, with the same
conditions, from the lower half-space. As in the previous case, we
fix coordinate
and calculate the
temperature value at points . The
obtained schedule of temperature change is identical to the
previous one, which also verifies the obtained kernels of boundary
integral equations.
It should also be noted that the change in thermal conductivity
of the interface does not change the temperature in these bodies.
It is obvious in this case (homogeneous material) that the high
thermal conductivity interface does not affect the temperature
distribution in the considered finite prismatic bodies, which further
verifies the obtained integral formulas and developed computa-
tional programs.
Example 2 Finally, consider the problem where the interface
crosses the square solid (), as shown in Fig. 3. The prop-
erties of the materials are the same as in the previous example.
On the upper boundary of the solid, for
the temperature is
set as follows:
,
.On the bottom
boundary of the cut out square a temperature is given. At
the interface, the conditions of imperfect thermal contact in the
form of a high-temperature conducting interface, Eqs (4) and (5),
are satisfied.
In addition to the boundary elements method, another
approach was used to solve the problem for their mutual
verification.
Fig. 3. Sketch of a thermoelastic anisotropic square solid with HCI
Fig. 4. Temperature change in the cross section of a square body
with HCI
This approach is based on the Stroh formalism. The complex
potentials (3) for a square solid with high-temperature conducting
interface can be taken as the following finite sums of Laurent
series, which are analytic in the selected domain
(22)
Here
N
is the number of terms. Utilising interface conditions
(4) and (5) one can easily find the dependence between and
. After this these coefficients are determined in the following
way. First, one computes the sum of squares of difference
between given boundary conditions and temperature or heat flux
obtained in Eq. (22) in a set of points at the boundary of the
square solid. Then this functional is minimised as in the least
square approach to determine and . Thus, the complex
potentials (22) are obtained explicitly. Using Eqs (1) and (22) it is
then easy to plot temperature change in the cross-section of a
DOI 10.2478/ama-2022-0029 acta mechanica et automatica, vol.16 no.3 (2022)
247
square solid depending on the distance to the interface (for
instance, Fig. 4 depicts plot for HCI with
). Since, on
the boundary
temperature is , then the graph of
temperature change for is parabolic (Fig. 4). The
plot shows that the temperature change is a linear function of
coordinate, since we selected very high-temperature conductivity
of the interface, and thus the temperature of the latter should be
constant.
The same results were obtained using developed modified
boundary elements method, which once again confirms its
correctness.
Example 3. Consider the problem of plane strain for a ther-
momagnetoelectroelastic anisotropic bimaterial consisting of two
half-spaces. It contains a rectilinear elastic isotropic thermally
insulated electro- and magnetically permeable inclusion of finite
length . For its modelling the coupling principle for continua of
different dimension is used [10]. In this example, it is assumed
that the coefficients of linear thermal expansion of the inclusion
material are zero. Its cross section is perpendicular to the bimate-
rial boundary (Fig. 5). One inclusion tip is located.in the half-space
, and the other in the half-space . The singularity
at the point of intersection of the inclusion with the material inter-
face is not accounted for. The centre of inclusion coincides with
the origin. Inclusion thickness is , and its relative
rigidity is
.
Fig. 5. Scheme of the problem for a thermomagnetoelectroelastic
anisotropic bimaterial containing a thin inhomogeneity
The heat source of intensity is located in the half-space
at a distance of to the interface; heat drain of the
same intensity is located in the half-space also, at a
distance of to the interface antisymmetrically.
Two problems are considered:
I) both bimaterial components are made of barium titanate
(BaTiO3);
II) the component is made of barium titanate, and
of cadmium selenide (CdSe).
According to [18], the properties of BaTiO3 are as follows:
elastic moduli: , ,
, , ;
piezoelectric constants: , ,
;
dielectric constants: , ;
heat conduction coefficients: ;
thermal expansion coefficients: ,
; pyroelectric constants: .
The properties of CdSe are as follows [19]:
elastic moduli: , , ,
, , ;
piezoelectric constants: , ,
;
dielectric constants: , ;
heat conduction coefficients: , ;
thermal expansion coefficients: , ;
pyroelectric constants: .
Fig. 6. Dependence of dimensionless generalised SIFs of the inclusion
in an infinite body on the parameter of the interface
The plots in Fig. 6 show the dependence of the dimensionless
stress intensity factors on the thermal conductivity parameter of
the interface . All calculations were performed by the above-
mentioned method of boundary elements [6], [8]. 20 boundary
elements were used to model the inclusion surface. With a further
increase in the number of elements, the results obtained differ by
<0.5%. Generalised SIFs and thermal conductivity parameter are
normalised by
and
, respec-
tively. Here and are coefficients of BaTiO3.
It is noticed that in the first case in Fig. 6 (I), when the compo-
nents of the matrix are made of the same materials with geometric
symmetry of the problem and asymmetry of temperature load, the
values of stress intensity factors at opposite tips of the inclusion
are expected to be the same in magnitude and opposite in sign.
The maximum are dimensionless SIFs
,
for the high-temperature conducting interface at
.
When half-spaces have different properties but the same ge-
ometry of the problem and the same heat load (Fig. 6 (II)), the
symmetry of the solution is obviously not observed. Since the
material of the half-space is the same as in the previous
case, the values of the SIFs at the inclusion tip located in this half-
plane will change very little. However, at the inclusion tip, which is
located in the lower half-space, the change in coefficients is more
noticeable: when
changes from 0.014 to
0.0075, and
from 0.005 to 0.002.
Example 4. Now consider the bimaterial containing two iden-
tical inclusions that are perpendicular to the boundary. Their
properties are the same as in the previous example. They are
placed at a distance to the axis .The heat source and drain of
the same magnitude are located at a distance of to the
material interface and at the distance of to the axis (Fig. 7).
As in the previous problem, we study the dependence of
stress intensity factors on the thermal conductivity parameter of
the interface
.
Heorhiy Sulym, Andrii Vasylyshyn, Iaroslav Pasternak DOI 10.2478/ama-2022-0029
Influence of Imperfect Interface of Anisotropic Thermomagnetoelectroelastic Bimaterial Solids on Interaction of Thin Deformable Inclusions
248
Fig. 7. Scheme of the problem for thermoelastic anisotropic
bimaterial with two inclusions
Fig. 8. Dependence of dimensionless generalised SIFs of two inclusions
in an infinite body on the parameter of the interface , when the
components are made of the same materials
Fig. 9. Dependence of dimensionless generalised SIFs of two inclusions
in an infinite body on the parameter of the interface , when the
components are made of different materials
The plots (Fig. 8) show the values of generalised SIFs for the
(a) first and (b) second inclusions, when the components of the
matrix are made of the same materials.
Due to the fact that the inclusions are identical in their
properties and located symmetrically about the axes and ,
and the materials of the components have the same properties,
the plots for both inclusions have a similar behaviour and differ
only in sign. As in the previous problem, the maximum values of
were obtained for
Somewhat different results are observed in the case when the
component of the bimaterial is made of a material, whose
properties are different from .
Fig. 9(a) shows that in this case the values of SIFs at the
vertex of the first inclusion have undergone significant changes in
comparison with the matrix of Fig. 8(a). The maximum value
increased from 0.011 to 0.014; and
from 0.008
to 0.012. The values
have also increased.
By contrast, at the tip of the second inclusion, the values of
SIFs decreased. Fig. 9(b) shows that the maximum value
decreased from 0.011 to 0.0078;
from 0.008 to 0.0061.
Also, at the upper and lower ends of the second inclusion ap-
proach the value of 0.001 at
.
6. CONCLUSION
A mathematical model of a thermagnetoelectroelastic bimate-
rial solid with a high-temperature conducting coherent interface
and a perfect magnetoelectromechanical contact of components
has been developed, which in turn may contain thin deformable
inclusions. In a closed form, purely boundary integral equations of
the formulated problem are derived. That is, equations in which
there is no need to take into account the integrals along the
interface; thus, the boundary element mesh is required only for
the boundary of the composite body and the midline of the thin
inclusions.
The method can be extended to account for inclusions at the
bimaterial interface; however, the oscillating singularity at its tip
should be accounted for, which is beyond the scope of the present
publication. Nevertheless, the present paper accounts for
imperfect interface, which physically means a thin layer of
different properties on the bimaterial interface, which is very
important in practical applications.
The use of the obtained integral equations in combination with
the boundary element method allows to solve several new
problems for bimaterials consisting of the same and different
anisotropic thermomagnetoelectroelastic materials, as well as
containing thin deformable inclusions. Graphical dependences of
generalised SIFs on the thermal conductivity parameter are
derived. The obtained results show that the high thermal
conductivity interface significantly affects the stress fields at the
vertices of thin inclusions.
All this allows to state that the developed method allows to
solve with high accuracy the problem of thermomagnetoelasticity
for bimaterial solids with a high-temperature conducting interface
with thin ribbon-like deformable inclusions or cracks, which has
not been possible so far using traditional numerical approaches.
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Heorhiy Sulym: https://orcid.org/0000-0003-2223-8645
Andrii Vasylyshyn: https://orcid.org/0000-0002-5703-6894
Iaroslav Pasternak: https://orcid.org/0000-0002-1732-0719
