Heorhiy Sulym, Iaroslav Pasternak, Mykhailo Tomashivskyy
Boundary Integral Equations for an Anisotropic Bimaterial with Thermally Imperfect Interface and Internal Inhomogeneities
DOI 10.1515/ama-2016-0012
BOUNDARY INTEGRAL EQUATIONS FOR AN ANISOTROPIC BIMATERIAL
WITH THERMALLY IMPERFECT INTERFACE AND INTERNAL INHOMOGENEITIES
Heorhiy SULYM*, Iaroslav PASTERNAK**, Mykhailo TOMASHIVSKYY***
*Bialystok
University of Technology, ul. Wiejska 45C, 15-351 Bialystok, Poland
National Technical University, Lvivska Str. 75, 43018 Lutsk, Ukraine
***Ivan Franko National University of Lviv, Universytetska Str. 1, 79000 Lviv, Ukraine
**Lutsk
[email protected], [email protected], [email protected]
received 4 May 2015, revised 25 February 2016, accepted 26 February 2016
Abstract: This paper studies a thermoelastic anisotropic bimaterial with thermally imperfect interface and internal inhomogeneities. Based
on the complex variable calculus and the extended Stroh formalism a new approach is proposed for obtaining the Somigliana type integral
formulae and corresponding boundary integral equations for a thermoelastic bimaterial consisting of two half-spaces with different thermal
and mechanical properties. The half-spaces are bonded together with mechanically perfect and thermally imperfect interface, which model
interfacial adhesive layers present in bimaterial solids. Obtained integral equations are introduced into the modified boundary element
method that allows solving arbitrary 2D thermoelacticity problems for anisotropic bimaterial solids with imperfect thin thermo-resistant interfacial layer, which half-spaces contain cracks and thin inclusions. Presented numerical examples show the effect of thermal resistance
of the bimaterial interface on the stress intensity factors at thin inhomogeneities.
Key words: Bimaterial, Imperfect Interface, Thermoelastic, Anisotropic, Crack, Thin Inclusion
1. INTRODUCTION
Bimaterial parts and structures with different thermal and mechanical properties of their components are widely used in modern engineering design due to their useful anisotropic properties,
which thus operate different way in different directions. As a result
of such bondage of different materials a thin interfacial layer with
finite thickness appears at the interface, which affects the temperature and stress fields in the obtained bimaterial solid. In modeling
of the effect of this layer, due to small thickness of the latter it can
be reduced to consideration of certain boundary conditions
of imperfect interface at the surface of material bondage. However, besides the imperfect interface, another inhomogeneities (such
as cracks, thin inclusions etc.) are usually present in structural
materials. Therefore, in the study of bimaterial solids one should
account for both. Thus, the development of effective methods for
modeling of thermal and mechanical fields’ distribution in bimaterial solids with imperfect interface and internal thin inhomogeneities is an important practical problem.
Development of efficient techniques for analysis of thermoelasticity problems is very important in the study of contact and
friction problems, especially those accounting for thermal emission. Polish scientists vastly develop these studies. Those are
Z. Baczyński, J. Ignaczak, A. Kaczyński, M. Kuciej, S.J. Matysiak,
V. Pauk, E. Wierzbicki, A. Yevtushenko (e.g. see the recent review by Yevtushenko and Kuciej (2012) and monograph
Jewtusheko et al (2014)).
The study of bimaterial solids is widely covered in scientific literature (Benveniste, 2006, Kattis and Mavroyannis, 2006, Pan
and Amadei, 1999, Pasternak et al., 2014, Qin, 2007, Wang and
66
Pan, 2010). In particular, Pan and Amadei (1999) developed
an efficient boundary element approach for defective elastic anisotropic bimaterial solids. Hwu (1992) obtained analytic solutions
for interfacial cracks in thermoelastic anisotropic dissimilar media.
Wang and Pan (2010) derived thermoelastic Green’s functions
for anisotropic thermoelastic bimaterial with Kapitza-type and
spring-type imperfect interface. Pasternak et al. (2014) obtained
truly boundary integral equations for 2D thermoelectroelasticity
of a defective bimaterial solid with a perfect interface.
However, the thermoelasticity of anisotropic bimaterial solids
with imperfect interface containing internal inhomogeneities is less
studied. In general, there are mainly two types of imperfect interfaces in the context of heat conduction (Benveniste, 2006, Chen,
2001, Kattis and Mavroyannis, 2006), namely the weakly conducting interface (or the well known Kapitza thermal contact resistance
model) and the highly conducting interface. At a weakly conducting interface it is assumed that the normal heat flux is continuous
across the interface, and the temperature possesses an interfacial
discontinuity, which is proportional to the normal heat flux (Wang
and Pan, 2010).
Therefore, this paper studies the thermoelastic anisotropic bimaterial with thermally imperfect and mechanically perfect interface. It is assumed the weakly temperature conducting interface,
as in Ref (Wang and Pan, 2010), and tractions and displacements
are assumed to be continuous across the interface (perfect mechanical contact). The general complex variable approach
of Pasternak et al. (2014) is used to derive closed-form boundary
integral formulae and equations for a bimaterial solid with thermally imperfect interface.
DOI 10.1515/ama-2016-0012
acta mechanica et automatica, vol.10 no.1 (2016)
2. GOVERNING EQUATIONS OF PLANE ANISOTROPIC
THERMOELASTICITY AND THE EXTENDED STROH
FORMALISM
is a bondage line, the conditions of imperfect thermal contact are
satisfied:
𝜃 (1) (𝑥1 , 𝑥2 )|𝑥
=
θ(2) (𝑥1 , 𝑥2 )|𝑥
(2)
−𝜌0 (ϑ,1 (𝑥1 , 𝑥2 ))|
2 =0
In a fixed rectangular system of coordinates 𝑂𝑥1 𝑥2 𝑥3 the
equilibrium, heat balance, and constitutive equations of plane
(in a plane 𝑂𝑥1 𝑥2 ) strain and plane heat conduction for a linearly
thermoelastic anisotropic solid can be written as (Hwu, 2010,
Pasternak, 2012, Ting, 1996):
σ𝑖𝑗,𝑗 = 0, ℎ𝑖,𝑖 = 0 (𝑖, 𝑗 = 1,2,3);
(1)
σ𝑖𝑗 = 𝐶𝑖𝑗𝑘𝑚 ε𝑘𝑚 − β𝑖𝑗 θ, ℎ𝑖 = −𝑘𝑖𝑗 θ,𝑗 ,
(2)
where: 𝜀𝑖𝑗 = (𝑢𝑖,𝑗 + 𝑢𝑗,𝑖 )/2 is a strain tensor; 𝜎𝑖𝑗 is a stress
tensor; ℎ𝑖 is a heat flux; 𝑢𝑖 is a displacement vector; 𝜃 is a temperature change with respect to the reference one; 𝐶𝑖𝑗𝑘𝑚
are elastic moduli; 𝑘𝑖𝑗 are heat conduction coefficients; 𝛽𝑖𝑗 =
𝐶𝑖𝑗𝑘𝑚 𝛼𝑘𝑚 (𝑖, 𝑗, 𝑘, 𝑚 = 1, . . ,3) are thermal moduli; αij are
thermal expansion coefficients. Tensors 𝐶𝑖𝑗𝑘𝑚 , 𝑘𝑖𝑗 , 𝛼𝑖𝑗 , and 𝛽𝑖𝑗
are fully symmetric. Here and further, the Einstein summation
convention is used. A comma at subscript denotes differentiation
with respect to a coordinate indexed after the comma, i.e.
𝜕𝑢
𝑢𝑖,𝑗 = 𝑖 .
𝜕𝑥𝑗
According to the extended Stroh formalism (Hwu, 2010, Ting,
1996) the general homogeneous solution of Eqs (1), (2) writes as:
2 =0
ϑ(1) (𝑥1 , 𝑥2 )|𝑥
2 =0
= ϑ(2) (𝑥1 , 𝑥2 )|𝑥
2 =0
𝑥2 =0
,
,
(6)
(7)
and perfect mechanical contact holds
ϕ(1) (𝑥1 , 𝑥2 )|𝑥
2 =0
= ϕ(2) (𝑥1 , 𝑥2 )|𝑥
𝐮(1) (𝑥1 , 𝑥2 )|𝑥
2 =0
= 𝐮(2) (𝑥1 , 𝑥2 )|𝑥
2 =0
2
,
.
=0
(8)
Here and further superscripts 1 or 2 denote corresponding
half-space 𝑆1 or 𝑆2 , respectively, which the field quantity belongs
to. Boundary conditions (6) and (7) from the physical point of view
correspond to a model of a thin layer, for which according to
Fourier law of heat conduction the temperature difference at the
bimaterial interface is proportional to the heat flux through it. Thus,
the parameter ρ0 in Eq (6) is a thermal resistance of the abovementioned layer, and as a consequence, a thermal resistance
of the bimaterial interface. As 𝜌0 → 0 or 𝜌0 → ∞ one obtains
perfect thermal contact or thermally insulated bimaterial interface
(adiabatic contact), respectively.
θ = 2Re{𝑔′(𝑧𝑡 )}, ϑ = 2𝑘𝑡 Im{𝑔′(𝑧𝑡 )},
ℎ1 = −ϑ,2 , ℎ2 = ϑ,1 , 𝑘𝑡 = √𝑘11 𝑘22 − 𝑘12 2 ,
𝐮 = 2Re[𝐀𝐟(𝑧∗ ) + 𝐜𝑔(𝑧𝑡 )],
ϕ = 2Re[𝐁𝐟(𝑧∗ ) + 𝐝𝑔(𝑧𝑡 )],
σ𝑖1 = −φ𝑖,2 , σ𝑖2 = φ𝑖,1 ,
𝑧𝑡 = 𝑥1 + 𝑝𝑡 𝑥2 , 𝑧α = 𝑥1 + 𝑝α 𝑥2 ,
𝐟(𝑧∗ ) = [𝐹1 (𝑧1 ), 𝐹2 (𝑧2 ), 𝐹3 (𝑧3 )]𝐓 ,
(3)
where: 𝜗 is a heat flux function; 𝐹α (𝑧α ) and 𝑔(𝑧𝑡 ) are complex
analytic functions with respect to their arguments; the complex
constant 𝑝𝑡 is a root (with a positive imaginary part) of the characteristic equation for heat conduction 𝑘22 𝑝𝑡2 + 2𝑘12 𝑝𝑡 + 𝑘11 =
0. Constant complex matrices 𝐀, 𝐁, vectors 𝐜, 𝐝, and scalars 𝑝α
(α = 1,2,3) are determined from the extended Stroh eigenvalue
problem (Ting, 1996).
Vector 𝑓(𝑧∗ ) of Stroh complex functions is related to the realvalued stress and displacement functions as (Pasternak, 2012):
𝐟(𝑧∗ ) = 𝐁 T 𝐮 + 𝐀T ϕ − 𝐁 T 𝐮𝑡 − 𝐀T ϕ𝑡 ,
𝐮𝑡 = 2Re{𝐜𝑔(𝑧𝑡 )}, ϕ𝑡 = 2Re{𝐝𝑔(𝑧𝑡 )}.
(4)
According to Eq (3), the Stroh temperature function 𝑔′(𝑧𝑡 )
is related to the heat flux and temperature functions as:
1
ϑ
2
𝑘𝑡
𝑔′(𝑧𝑡 ) = (θ + 𝑖 ).
(5)
3. DERIVATION OF THE INTEGRAL FORMULAE
FOR A BIMATERIAL SOLID
Consider plane strain of a medium consisting of two thermoelastic anisotropic half-spaces 𝑆1 (𝑥2 > 0) and 𝑆2 (𝑥2 < 0) (see
Fig. 1). Along the line 𝑥2 = 0 (actually, the plane 𝑂𝑥2 𝑥3 ), which
Fig. 1. Thermoelastic anisotropic bimaterial medium
It is considered that each of the half-spaces contains systems
of cylindrical holes, which are represented with plane contours
(1)
(2)
𝛤1 =∪𝑖 𝛤𝑖 and 𝛤2 =∪𝑖 𝛤𝑖 , respectively.
For derivation of the integral formulae for the Stroh complex
functions for bonded half-spaces one can use the Cauchy integral
formula (Muskhelishvili, 2008), which relates values of an arbitrary
analytic function 𝜙(𝜏) at the boundary 𝜕𝑆 of the domain 𝑆 with its
value inside this domain:
1
ϕ(𝜏)𝑑𝜏
∫
2𝜋𝑖 ∂𝑆 𝜏−𝑧
ϕ(𝑧)
={
0
∀𝑧 ∈ 𝑆,
∀𝑧 ∉ 𝑆,
(9)
where 𝜏, 𝑧 ∈ ℂ are complex variables, which define the position
of the source and field points, respectively. Herewith, if the domain 𝑆 is infinite it is assumed that the function 𝜙(𝑧) vanishes
at 𝑧 → ∞.
3.1. Heat conduction
Due to the linearity of the problem of heat conduction one can
present its solution as a superposition of the homogeneous solu-
67
Heorhiy Sulym, Iaroslav Pasternak, Mykhailo Tomashivskyy
Boundary Integral Equations for an Anisotropic Bimaterial with Thermally Imperfect Interface and Internal Inhomogeneities
(1)
(𝑖) (𝑗)
𝑞𝑡 (𝑧𝑡 )
(𝑖)
(𝑖)
(𝑖)
𝑔𝑖′ (𝜏𝑡 )𝑑𝜏𝑡
(𝑖)
(𝑗)
𝛤𝑖 𝜏 −𝑧
𝑡
𝑡
=∫
(𝑗)
𝑞̅𝑡 (𝑧𝑡 ) = ∫𝛤
,
(10)
̅̅̅̅̅̅̅̅̅̅̅
(𝑖)
(𝑖)
𝑔𝑖′ (𝜏𝑡 )𝑑𝜏𝑡
(𝑖)
+∞ ϑ(𝑥1 )𝑑𝑥1
(𝑗)
(𝑗)
𝑝𝑡 (𝑧𝑡 )
=
(𝑗)
𝑥1 −𝑧𝑡
+∞ ϑ,1 (𝑥1 )
(𝑗)
𝑥1 −𝑧𝑡
𝑑𝑥1 =
(11)
(𝑗)
𝑥1 −𝑧𝑡
(𝑗)
𝑥1 −𝑧𝑡
|
+
(12)
(𝑗)
(13)
Utilizing Eqs (5), (9)–(11), (13) and accounting for the boundary conditions at the bimaterial interface the Cauchy integral formu(1)
(2)
lae for the functions 𝑔′1 (𝑧𝑡 ) and 𝑔′2 (𝑧𝑡 ) write as
1
(1) (1)
𝑞 (𝑧𝑡 ) +
2𝜋𝑖 𝑡
1
𝜌
(1) (1)
(1)
𝑚 (𝑧𝑡 ) − 0 𝑚𝑡′ (𝑧𝑡 ),
4𝜋𝑘𝑡 𝑡
4𝜋𝑖
(1)
′
𝑔1′ (𝑧𝑡 ) = 𝑔1∞
(𝑧𝑡 ) +
1
(1) (1)
𝑝 (𝑧𝑡 )
4𝜋𝑖 𝑡
+
(1)
(14)
∀Im(𝑧𝑡 ) > 0
(2)
1
1
(2)
4𝜋𝑖
1
(2)
′
𝑔2′ (𝑧𝑡 ) = 𝑔2∞
(𝑧𝑡 ) +
𝑝𝑡 (𝑧𝑡 ) −
(2)
4𝜋𝑘𝑡
(2)
2𝜋𝑖
(2)
(1)
(2) 𝑚𝑡 (𝑧𝑡 )
2𝑘
1
(1)
(1)
(1)
2
(1)
𝑖
(1)
(1)
2𝑘𝑡
(2)
(2)
1
(2)
∀Im(𝑧𝑡 ) < 0: 𝑞̅𝑡 (𝑧𝑡 ) − 𝑝𝑡 (𝑧𝑡 ) +
2
68
(15)
𝑚𝑡 (𝑧𝑡 ) −
𝑚′(𝑧𝑡 ) = 0,
(2)
(1)
[𝑒𝑡 (𝑧𝑡 ) +
2
(2)
𝜌0
(1)
(1)
(1)
[𝑒𝑡 (𝑧𝑡 ) + 𝑒̅𝑡 (𝑧𝑡 )],
(2)
(2)
2𝑖
(1) (2)
(2) [𝑒𝑡 (𝑧𝑡 )
𝜌0 𝑘𝑡
2
(1) (2)
− [𝑒𝑡 (𝑧𝑡 ) +
𝜌
(2)
(2)
+
(18)
0
(𝑗)
(𝑖)
(𝑗)
(𝑖)
(𝑗)
(𝑖)
(𝑖)
(𝑗)
(𝑖)
(19)
(𝑖)
(𝑖)
(𝑗)
(𝑖)
(20)
𝑒𝑡 (𝑧𝑡 ) = ∫Γ 𝑔′𝑖 (𝜏𝑡 )K (𝐵 (𝑗) (𝜏𝑡 − 𝑧𝑡 )) 𝑑𝜏𝑡 ,
𝑖
(𝑗)
𝑒𝑡 (𝑧𝑡 ) = ∫Γ 𝑔′𝑖 (𝜏𝑡 )K (𝐵 (𝑗) (𝜏𝑡 − 𝑧𝑡 )) 𝑑𝜏𝑡 ,
𝑖
where: K(𝑧) = 𝑒 𝑧 𝐸1 (𝑧),
𝐵 (1) =
(1)
(2)
𝑖 𝑘𝑡 +𝑘𝑡
𝜌0 𝑘 (1) ⋅𝑘 (2)
𝑡
𝑡
𝐵 (2) = −
(1)
,
(21)
(2)
𝑖 𝑘𝑡 +𝑘𝑡
𝜌0 𝑘 (1) ⋅𝑘 (2)
𝑡
𝑡
and 𝐸1 (𝑧) is an exponential integral defined as
∞ 𝑒 −𝑡
𝐸1 (𝑧) = ∫𝑧
𝑡
(22)
𝑑𝑡.
Substituting obtained solution into Eq (14) one can derive the
integral formulae for the Stroh complex temperature functions
(1)
(2)
𝑔′1 (𝑧𝑡 ) and 𝑔′2 (𝑧𝑡 ), which do not contain the improper
integrals over the infinite bimaterial interface:
(1)
(1)
(1)
(16)
(1)
(1)
(1)
[𝑞𝑡 (𝑧𝑡 ) − 𝑞̅𝑡 (𝑧𝑡 ) +
(1) (1)
𝑒̅𝑡 (𝑧𝑡 ))]
2𝑖
(2)
(1)
𝜌0 𝑘𝑡
(1)
(𝑒𝑡 (𝑧𝑡 ) +
(23)
,
(2)
(2)
(2)
∀Im(𝑧𝑡 ) > 0: 𝑔′2 (𝑧𝑡 ) = 𝑔′2∞ (𝑧𝑡 ) +
(2)
(2)
(2)
(2)
[𝑞𝑡 (𝑧𝑡 ) −𝑞̅𝑡 (𝑧𝑡 ) −
(2)
2
𝜌0
(2)
2𝜋𝑖
= 0,
1
(2)
(2)
𝜌0 𝑘𝑡
→
(17)
𝑒̅𝑡 (𝑧𝑡 )], 𝑚𝑡 (𝑧𝑡 ) =
1
(1)
𝑡
(1)
2𝑖
(1)
(1)
(1)
2𝜋𝑖
2
𝑞̅𝑡 (𝑧𝑡 ) + 𝑝𝑡 (𝑧𝑡 ) −
(2)
𝑚𝑡 (𝑧𝑡 ) = −
1
(2)
∀Im(𝑧𝑡 ) > 0: 𝑞𝑡 (𝑧𝑡 ) − 𝑝𝑡 (𝑧𝑡 ) −
𝑖
(1)
𝑝𝑡 (𝑧𝑡 ) = 2 ⋅ 𝑞𝑡 (𝑧𝑡 ) +
(1)
𝑚𝑡 (𝑧𝑡 ), ∀Im(𝑧𝑡 ) < 0
(2)
(𝑗)
∀Im(𝑧𝑡 ) > 0: 𝑔′1 (𝑧𝑡 ) = 𝑔′1∞ (𝑧𝑡 ) +
(2)
𝑞𝑡 (𝑧𝑡 ) −
and for determination of the improper integrals (11) through the
Cauchy integrals (10) one obtains the following system of firstorder ordinary differential equations:
(1)
(𝑗)
Satisfying an evident condition that 𝑚𝑡 (𝑧𝑡 ) → 0 as 𝑧𝑡
∞ one obtains the solution of systems (15), (16) as:
(𝑖)
𝑑𝑥1 = 𝑚′ 𝑡 (𝑧𝑡 ).
(1)
𝑚′(𝑧𝑡 ) = 0.
(𝑖)
Accounting for the thermal balance conditions, the function
𝜗(𝑥1 ) tends to zero at the infinity, therefore the first term in Eq
(12) vanishes, and the second term is a derivative of a function
(j)
m′t (zt ). Thus,
+∞ ϑ,1 (𝑥1 )
(2)
𝑚𝑡 (𝑧𝑡 ) −
New functions 𝑒𝑡 (𝑧𝑡 ) and 𝑒̅𝑡 (𝑧𝑡 ) are defined though
the following integral formulae
+∞
−∞
+∞
ϑ(𝑥1 )
(𝑗)
𝑑𝑥
=
𝑚′
(𝑧
∫−∞
2
1
𝑡
𝑡 ).
(𝑗)
(𝑥1 −𝑧𝑡 )
∫−∞
2
(1)
(1)
2𝑘𝑡
(2) (2)
𝑒̅𝑡 (𝑧𝑡 )].
𝑡
ϑ(𝑥1 )
𝜌0
𝑖
(2)
2
(2)
Integration by parts of the improper integral gives:
∫−∞
1
(2)
𝑝𝑡 (𝑧𝑡 ) = 2 ⋅ 𝑞̅𝑡 (𝑧𝑡 ) −
,
+∞ 𝜃 (2) (𝑥1 )𝑑𝑥1
∫−∞
(𝑗) .
𝑥 −𝑧
1
(1)
𝑚𝑡 (𝑧𝑡 ) = 0,
𝑞̅𝑡 (𝑧𝑡 ) + 𝑝𝑡 (𝑧𝑡 ) +
(1)
and the improper integrals over the infinite path −∞ < 𝑥1 <
+∞ as
𝑚𝑡 (𝑧𝑡 ) = ∫−∞
(2)
(2)
2𝑘𝑡
𝑒̅𝑡 (𝑧𝑡 )],
(𝑗)
𝜏̅𝑡 −𝑧𝑡
𝑖
𝑖
(2)
tion given by the functions 𝑔1∞ (𝑧𝑡 ) and 𝑔2∞ (𝑧𝑡 ) (which
should definitely satisfy the boundary conditions (6)), and the
perturbed solution caused by the presence of the contours 𝛤1
and 𝛤2 .
Denote the Cauchy integrals of the complex temperature func(𝑖)
tions 𝑔′𝑖 (𝑧𝑡 ) as:
DOI 10.1515/ama-2016-0012
2𝑖
(2)
𝜌0 𝑘𝑡
(1)
(2)
(𝑒𝑡 (𝑧𝑡 ) +
(24)
(2)
𝑒̅𝑡 (𝑧𝑡 ))].
Using (5) one can derive the integral formulae for the func(1)
(2)
tions 𝑔′1 (𝑧𝑡 ) and 𝑔′2 (𝑧𝑡 ) relating them with the boundary
values of temperature 𝜃 and normal component ℎ𝑛 = ℎ𝑖 𝑛𝑖
of a heat flux vector (𝑛𝑖 is a unit outwards normal vector
to a curve 𝛤𝑖 at its certain point).
According to Eqs (5), (23), and (24), one can obtain the inte-
DOI 10.1515/ama-2016-0012
acta mechanica et automatica, vol.10 no.1 (2016)
gral formulae for temperature and heat flux at the arbitrary point 𝜉
of a bimaterial solid:
(𝐱, ξ) =
Θ𝐼𝑇𝐶∗∗
𝑖
(1)
2Re {𝑔′1 (𝑍𝑡 (ξ))} (∀ξ ∈ 𝑆1 ),
𝜃(ξ) = {
=
(2)
2Re {𝑔′2 (𝑍𝑡 (ξ))} (∀ξ ∈ 𝑆2 ),
−
(25)
∫Γ[ΘITC∗ (𝐱, ξ)ℎ𝑛 (𝐱) − 𝐻ITC∗ (𝐱, ξ)𝜃(𝐱)]𝑑𝑠(𝑥) +
𝜃 ∞ (ξ),
1
(2)
𝜌0 𝜋
(1)
=
(2)
(2)
(2)
2𝑘𝑡 Im {(𝛿2𝑖 − 𝛿1𝑖 𝑝𝑡 )𝑔′′2 (𝑍𝑡 (ξ))} (∀ξ ∈ 𝑆2 ),
(26)
2𝜋
1
∀ξ ∈ 𝑆2 ,
(28)
∀ξ ∈ 𝑆2 .
2𝜋𝑘𝑡
̅𝑡(1,1) )},
(𝑊
1
𝐻𝐼𝑇𝐶∗ (𝐱, ξ) =
(𝐱, ξ)
Θ𝐼𝑇𝐶∗∗
𝑖
2𝜋
=
Im {
2𝑖
(1)
𝜌0 𝑘𝑡
−
1
1
(1)
− Im {𝛿𝑡 ( (1,1)
2𝜋
𝑊
(1,1)
̅𝑡
𝐾(𝐵 (1) 𝑊
𝐻𝑖𝐼𝑇𝐶∗∗ (𝐱, ξ) = −
+
(1)
2𝑖 𝑛̅𝑡
(1)
𝜌0 𝑘𝑡
+
+
(1)
2𝑖 𝑛̅𝑡
(1)
𝜌0 𝑘𝑡
(1,1)
̅𝑡
𝐾(𝐵 (1) 𝑊
1
𝜌0 𝜋
(1)
Re {𝛿𝑡
𝐻
2𝜋
(1)
Re {𝛿𝑡 (
)−
(1)
𝑛𝑡
(1,1) 2
(𝑊𝑡
)
1
̅ (1,1)
𝑊
𝑡
2𝜋𝑘𝑡
1
−
(1)
𝑛̅𝑡
2
̅ (1,1) )
(𝑊
𝑡
))} ;
(2)
(1)
𝜌0 𝜋𝑘𝑡
̅𝑡
𝐾(𝐵 (2) 𝑊
1
))}
(2)
𝑛𝑡
̅ (2,2)
𝑊
𝑡
)},
−
𝑛̅𝑡
2
̅ (2,2) )
(𝑊
𝑡
))},
𝑊𝑡
(𝑖)
(𝑗)
̅𝑡(𝑖,𝑗) = 𝑍̅𝑡(𝑖) (𝐱) − 𝑍𝑡(𝑗) (ξ),
= 𝑍𝑡 (𝐱) − 𝑍𝑡 (ξ), 𝑊
(𝑖)
(𝑖)
(𝑗)
(𝑗)
𝑛𝑡 = 𝑛2 (𝐱) − 𝑝𝑡 𝑛1 (𝐱), 𝛿𝑡 = 𝛿𝑖2 − 𝑝𝑡 𝛿𝑖1 ,
L(𝑖) (𝑧) = K(𝐵 (𝑖) 𝑧) + ln𝑧.
According to Eqs (3) and (4), for derivation of integral formulae for extended displacement and extended stress function one
(𝑖)
should evaluate the anti-derivatives of functions 𝑔′𝑖 (𝑧𝑡 ),
𝑚𝑡 (𝑧), and 𝑝𝑡 (𝑧) as:
(𝑗)
(𝑗)
(30)
𝑔𝑖 (𝑧𝑡 ) = ∫ 𝑔′𝑖 (𝑧𝑡 )𝑑𝑧𝑡 ,
)},
𝑀𝑡 (𝑧) = ∫ 𝑚𝑡 (𝑧)𝑑𝑧 =
+∞
− ∫−∞ ln(𝑥1
(31)
− 𝑧)ϑ(𝑥1 )𝑑𝑥1 ,
(32)
1
(1)
(2)
(2,1)
)−
1
(1) (1)
(1) (1)
[𝑄𝑡 (𝑧𝑡 ) − 𝐾𝑄̅𝑡 (𝑧𝑡 )
(1)
(2)
(1)
(1)
(1)
(2)
(1)
(1)
(1)
(2)
1
(2,1)
𝑊𝑡
(1)
(1)
(1)
+(1 + 𝐾) (𝑒𝑡 (𝑧𝑡 ) + 𝑒̅𝑡 (𝑧𝑡 )) ,
(1)
(1)
(1) (1)
(2) (1)
𝑀𝑡 (𝑧𝑡 ) = −2𝑖𝑘𝑡 (1 − 𝐾)[𝑄̅𝑡 (𝑧𝑡 ) + 𝑄𝑡 (𝑧𝑡 )
(2)
(1)
(1)
(1)
(33)
− (𝑒𝑡 (𝑧𝑡 ) + 𝑒̅𝑡 (𝑧𝑡 ))] ;
)},
Im{𝑖𝛿𝑡 K(𝐵 (1) 𝑊𝑡
(2,1)
2𝜋𝑖
+ (1 − 𝐾)𝑄𝑡 (𝑧𝑡 ) + (𝐾 − 1) (𝑒𝑡 (𝑧𝑡 ) + 𝑒̅𝑡 (𝑧𝑡 ))] ,
)}
(2)
∀Im(𝑧𝑡 ) < 0:
(2)
(2)
1
(1)
𝑔1 (𝑧𝑡 ) = 𝑔1∞ (𝑧𝑡 ) +
(1)
(2,1)
(1)
(2)
𝜌0 𝜋𝑘𝑡
∀Im(𝑧𝑡 ) > 0:
𝑃𝑡 (𝑧𝑡 ) = (1 − 𝐾)𝑄𝑡 (𝑧𝑡 ) − (1 + 𝐾)𝑄̅𝑡 (𝑧𝑡 )
Im{𝑖𝑛𝑡 K(𝐵 (1) 𝑊𝑡
𝑛𝑡 (𝑖𝐵(1) 𝐾(𝐵 (1) 𝑊𝑡
ξ) =
(2,2)
(2)
𝜌0 𝑘𝑡
(2,2) 2
(𝑊𝑡
)
)−
(2,2)
̅𝑡
𝐾(𝐵 (2) 𝑊
(1)
(1)
𝑘𝑡
2𝜋𝑘𝑡
(2)
𝜌0 𝑘𝑡
+∞
(2)
𝑔2 (𝑧𝑡 ) = 𝑔2∞ (𝑧𝑡 ) +
)} ;
𝐱 ∈ 𝑆1 ∧ ξ ∈ 𝑆2 :
1+𝐾
(1,2)
(1,2)
Θ𝐼𝑇𝐶∗ (𝐱, ξ) =
+ 𝐾(𝐵 (2) 𝑊𝑡
)},
(1) Re{ln𝑊𝑡
𝐼𝑇𝐶∗ (𝐱,
(2,2)
̅𝑡
(𝐵 (2) 𝐾(𝐵 (1) 𝑊
2𝑖
2𝑖 𝑛̅𝑡
𝑃𝑡 (𝑧) = ∫ 𝑝𝑡 (𝑧)𝑑𝑧 = − ∫−∞ ln(𝑥1 − 𝑧)𝜃(𝑥1 )𝑑𝑥1 .
̅ (1,1)
𝑊
𝑡
𝐱 ∈ 𝑆2 ∧ ξ ∈ 𝑆1 :
1−𝐾
(2,1)
(2,1)
Θ𝐼𝑇𝐶∗ (𝐱, ξ) =
+ K(𝐵 (1) 𝑊𝑡
)},
(2) Re{ln𝑊𝑡
,𝐻𝑖𝐼𝑇𝐶∗∗ (𝐱, ξ) =
(29)
Utilizing Eqs (23), (24), one obtains:
(1,1)
(𝐱, ξ) = −
Θ𝐼𝑇𝐶∗∗
𝑖
(2)
𝜌0 𝑘𝑡
−
(2)
(2)
1
))} ,
̅𝑡
(𝐵 (1) 𝐾(𝐵 (1) 𝑊
𝐻𝐼𝑇𝐶∗ (𝐱, ξ) =
2𝑖 𝑛̅𝑡
(𝑗)
(1)
𝑛̅𝑡
(1,1)
̅
𝑊
𝑡
𝑡
−
−
̅ (2,2)
𝑊
𝑡
Re {𝛿𝑡 (
2𝜋
(2)
𝑛̅𝑡
̅ (2,2) +
𝑊
𝑡
(1)
−𝑘𝑡
−
1
+
(2,2)
𝑊𝑡
(𝑖,𝑗)
∀ξ ∈ 𝑆1 ,
𝐱 ∈ 𝑆1 ∧ ξ ∈ 𝑆1 :
1
(1,1)
̅𝑡(1,1)
Θ𝐼𝑇𝐶∗ (𝐱, ξ) =
+ ln𝑊
(1) Re{ln𝑊𝑡
(1)
𝑛𝑡
(1,1)
𝑊𝑡
)} ;
where:
According to Eqs (3), (23), (24) the kernels of the heat conduction integral formulae are as follows:
+(𝐾 − 1)𝐿
Im {𝛿𝑡 (
(2)
ℎ𝑖∞ (ξ) =
(1)
(1)
(1)
2𝑘 Im[(𝛿2𝑖 − 𝛿1𝑖 𝑝𝑡 )𝑔′′1∞ (𝑍𝑡 (ξ))]
{ 𝑡(2)
(2)
(2)
2𝑘𝑡 Im[(𝛿2𝑖 − 𝛿1𝑖 𝑝𝑡 )𝑔′′2∞ (𝑍𝑡 (ξ))]
(1)
2𝜋
(2)
𝑛𝑡
(2,2)
𝑊𝑡
1
(2)
𝐻𝑖𝐼𝑇𝐶∗∗ (𝐱, ξ) =
(27)
Im {
(𝐱, ξ) =
Θ𝐼𝑇𝐶∗∗
𝑖
where: the functions 𝜃 ∞ (ξ) and ℎ𝑖∞ (ξ) define the homogeneous
solution for the unnotched bimaterial
∀ξ ∈ 𝑆1 ,
(2)
1
−
(2)
2Re[𝑔′2∞ (𝑍𝑡 (ξ))]
1
(1,2)
𝑊𝑡
̅𝑡(2,2) )},
(𝑊
(𝐱, ξ)ℎ𝑛 (𝐱) − 𝐻𝑖ITC∗∗ (𝐱, ξ)𝜃(𝐱)]𝑑𝑠(𝐱) +
∫Γ[ΘITC∗∗
𝑖
ℎ𝑖∞ (ξ),
𝜃 (ξ) = {
)−
𝐱 ∈ 𝑆2 ∧ ξ ∈ 𝑆2 :
1
(2,2)
̅𝑡(2,2)
Θ𝐼𝑇𝐶∗ (𝐱, ξ) =
+ ln𝑊
(2) Re{ln𝑊𝑡
𝐻𝐼𝑇𝐶∗ (𝐱, ξ) =
(1)
(1)
(1)
2𝑘𝑡 Im {(𝛿2𝑖 − 𝛿1𝑖 𝑝𝑡 )𝑔′′1 (𝑍𝑡 (ξ))} (∀ξ ∈ 𝑆1 ),
(1)
(1,2)
)},
2𝜋𝑘𝑡
ℎ𝑖 (ξ) =
2Re[𝑔′1∞ (𝑍𝑡 (ξ))]
(1,2)
Im{𝑖𝛿𝑡 K(𝐵 (2) 𝑊𝑡
𝑛𝑡 (𝑖𝐵 (2) 𝐾(𝐵 (2) 𝑊𝑡
Re {𝛿𝑡
−(𝐾 + 1)𝐿
∞
(2)
(1)
𝐻𝑖𝐼𝑇𝐶∗∗ (𝐱, ξ) =
(2)
{
1
𝜌0 𝜋𝑘𝑡
(1)
(1,2)
(2)
−
𝑊𝑡
)},
(2) Im{𝑖𝑛𝑡 K(𝐵
𝜌0 𝜋𝑘𝑡
(1)
1
2𝜋𝑖
(2) (2)
(2) (2)
[𝑄𝑡 (𝑧𝑡 ) + 𝐾𝑄̅𝑡 (𝑧𝑡 )
(2)
(1)
(2)
(2)
(2)
+ (1 + 𝐾)𝑄𝑡 (𝑧𝑡 ) − (𝐾 + 1) (𝑒𝑡 (𝑧𝑡 ) + 𝑒̅𝑡 (𝑧𝑡 ))] ,
(2)
(2)
(2)
(1)
(2)
𝑃𝑡 (𝑧𝑡 ) = (1 − 𝐾)𝑄̅𝑡 (𝑧𝑡 ) − (1 + 𝐾)𝑄𝑡 (𝑧𝑡 )
(1)
(2)
(2)
(2)
+(1 + 𝐾) (𝑒𝑡 (𝑧𝑡 ) + 𝑒̅𝑡 (𝑧𝑡 )) ,
69
Heorhiy Sulym, Iaroslav Pasternak, Mykhailo Tomashivskyy
Boundary Integral Equations for an Anisotropic Bimaterial with Thermally Imperfect Interface and Internal Inhomogeneities
+∞ 𝐟 (𝑗) 𝑑𝑥1
(2)
(2)
(2) (2)
(1) (2)
𝑀𝑡 (𝑧𝑡 ) = 2𝑖𝑘𝑡 (1 + 𝐾)[𝑄̅𝑡 (𝑧𝑡 ) + 𝑄𝑡 (𝑧𝑡 )
(1) (2)
− (𝑒𝑡 (𝑧𝑡 )
+
(2) (2)
𝑒̅𝑡 (𝑧𝑡 ))],
(34)
1
𝑖
(𝑖)
∫ 𝑓 ∗ (𝜏̅𝑡 (𝑠) − 𝑧)ℎ𝑛 (𝑠)𝑑𝑠
(𝑖)
(𝑖)
+ ∫Γ (𝑛2 (𝑠) − 𝑝̅𝑡 𝑛1 (𝑠)) ln(𝜏̅𝑡 (𝑠) − 𝑧)𝜃(𝑠)𝑑𝑠,
2
𝑖
(𝑖)
𝑄𝑡 (𝑧)
=
(𝑖)
∫ 𝑞𝑡 (𝑧)𝑑𝑧
=
−𝑖
∗ (𝑖)
(𝑖) ∫Γ 𝑓 (𝜏𝑡 (𝑠)
𝑖
2𝑘
(35)
T
(𝑖)
T
1
(42)
(𝑖)
𝑥1 −𝑧𝛽
+∞ ϕ(𝑥1 )𝑑𝑥1
(𝑗)
𝐦 (𝑧𝛽 ) = ∫−∞
,
(𝑗)
𝑥1 −𝑧𝛽
+∞ 𝐮(𝑥1 )𝑑𝑥1
(𝑗)
𝐩 (𝑧𝛽 ) = ∫−∞
(𝑖)
(𝑖)
+ ∫Γ (𝑛2 (𝑠) − 𝑝𝑡 𝑛1 (𝑠)) ln(𝜏𝑡 (𝑠) − 𝑧) 𝜃(𝑠)𝑑𝑠,
2
,
where
− 𝑧)ℎ𝑛 (𝑠)𝑑𝑠
𝑡
𝑖
(𝑖)
= 𝐀𝑗T 𝐦(𝑧𝛽 ) + 𝐁𝑗T 𝐩(𝑧𝛽 )
+∞ (𝐀𝑗 Re[𝐝𝑗 𝑔𝑗 (𝑥1 )]+𝐁𝑗 Re[𝐜𝑗 𝑔𝑗 (𝑥1 )])𝑑𝑥1
(𝑖) Γ
𝑖
2𝑘𝑡
(𝑖)
𝑥1 −𝑧𝛽
−2 ∫−∞
where according to Eqs (5) and (28), (29),
(𝑖)
(𝑖)
𝑄̅𝑡 (𝑧) = ∫ 𝑞̅𝑡 (𝑧)𝑑𝑧 =
∫−∞
DOI 10.1515/ama-2016-0012
(𝑗)
𝑥1 −𝑧𝛽
(43)
.
𝑓 ∗ (𝑧) = 𝑧(ln(𝑧) − 1), 𝐾 = (𝑘𝑡 − 𝑘𝑡 )⁄(𝑘𝑡 + 𝑘𝑡 ).
Integrating by parts the second term in Eq (42) and accounting for Eq (4), one can obtain that
3.2. Thermoelasticity of an anisotropic bimaterial
∫−∞
(1)
(2)
(1)
(2)
+∞ 𝐟 (1) 𝑑𝑥1
For obtaining the integral formulae of thermoelectroelasticity
one should write the Cauchy integral formula (9) for the Stroh
(1)
(2)
complex vector functions 𝑓 (1) (𝑧∗ ) and 𝑓 (2) (𝑧∗ ), which are
analytic in the domains 𝑆1 and 𝑆2 , respectively. Since the Cauchy
integral formula define the analytic function that vanishes at the
infinity, the complete solution of the problem can be presented as
a sum of the perturbed solution defined by the Cauchy formula
(1) (1)
and a homogeneous solution given by the functions 𝑓∞ (𝑧∗ )
(2) (2)
and 𝑓∞ (𝑧∗ ), which satisfy boundary conditions (8). Consequently, one obtains
(1)
(1)
(1)
𝐟 (1) (𝑧∗ ) = 𝐟∞ (𝑧∗ ) +
+∞
+ ∫−∞ ⟨
(1)
Im𝑧𝛽
𝑑𝑥1
(1)
𝑥1 −𝑧∗
(1)
𝑑𝜏∗
(1)
(1)
1 𝜏∗ −𝑧∗
[∫Γ ⟨
(1)
(Im𝑧∗
+∞
(1)
(1)
(𝜏∗ ) + ∫−∞
(1) ⟩ 𝐟
(1)
1 𝜏∗ −𝑧̅
𝛽
(2)
𝑑𝜏∗
Γ2 𝜏(2) −𝑧 (1)
∗
𝛽
∫ ⟨
(2)
𝐟 (2) (𝑧∗ )
− ∫−∞ ⟨
(36)
> 0),
+∞
(2)
⟩ 𝐟 (2) (𝜏∗ ) − ∫−∞
=
(2) (2)
𝐟∞ (𝑧∗ )
𝑑𝑥1
(2)
𝑥1 −𝑧∗
+
1
2𝜋𝑖
𝑑𝑥1
(1) (𝑥 )
1 = 0,
(1) 𝐟
𝑥1 −𝑧̅𝛽
𝑑𝑥1
(1)
𝑥1 −𝑧𝛽
(37)
𝐟 (2) (𝑥1 ),
(38)
(2)
(𝑖)
(𝑖)
(𝑖)
= 𝐀T2 𝐦(𝑧𝛽 ) + 𝐁2T 𝐩(𝑧𝛽 ) + μ2 𝑀𝑡 (𝑧𝛽 )
where the complex constants 𝜇𝑖 and 𝜆𝑖 are defined as:
μ𝑖 =
1
(𝑖)
𝑘𝑡
(𝐀T𝑖 Im[𝐝𝑖 ] + 𝐁𝑖T Im[𝐜𝑖 ]),
(46)
λ𝑖 = 𝐀T𝑖 Re[𝐝𝑖 ] + 𝐁𝑖T Re[𝐜𝑖 ]
as:
Denoting the Cauchy integrals of the Stroh complex functions
(𝑗)
𝑑𝜏∗
(𝑗)
(𝑖)
𝑗 𝜏∗ −𝑧
𝛽
(𝑖)
̅𝑗 (𝑧𝛽 )
𝐪
(𝑗)
⟩ 𝐟 (𝑗) (𝜏∗ )
(47)
(𝑗)
= ∫Γ ⟨
𝑑𝜏̅∗
(𝑗) (𝜏 (𝑗) ).
(𝑖) ⟩ 𝐟
∗
(𝑗)
𝑗 𝜏
̅ ∗ −𝑧𝛽
(1)
(1)
1
(1)
𝐟 (1) (𝑧∗ ) = 𝐟∞ (𝑧∗ ) +
(39)
2𝜋𝑖
(1)
[𝐪1 (𝑧∗ )
(1)
< 0);
(1)
(1)
(48)
(1)
+⟨𝑀𝑡 (𝑧∗ )⟩μ1 −⟨𝑃𝑡 (𝑧∗ )⟩λ1 + 𝜌0 ⟨𝑚𝑡 (𝑧∗ )⟩λ1 ] ;
(1)
̅T1 𝐦 (𝑧 (1) ) + 𝐁
̅1T 𝐩 (𝑧 (1) ) + 𝑀𝑡 (𝑧 (1) ) μ̅1
̅1 (𝑧𝛽 ) + 𝐀
𝐪
𝛽
𝛽
𝛽
(1)
⟩ 𝐟 (1) (𝜏∗ ) + ∫−∞
(2)
𝑑𝜏∗
(2)
(2)
2 𝜏∗ −𝑧̅
𝛽
⟩ 𝐟 (2) (𝜏∗ ) − ∫−∞
(1)
(2)
+∞
+∞
𝑑𝑥1
(2)
𝑥1 −𝑧𝛽
𝑑𝑥1
(2)
𝑥1 −𝑧̅𝛽
(45)
(𝑖)
−λ2 𝑃𝑡 (𝑧𝛽 )
(1)
−𝑃𝑡 (𝑧𝛽 ) λ̅1 + 𝜌0 𝑚𝑡 (𝑧𝛽 ) λ̅1 = 0;
(1)
𝑑𝜏∗
(1)
(2)
1 𝜏∗ −𝑧
𝛽
∫Γ ⟨
+
(44)
(𝑖)
𝜌0 λ1 𝑚𝑡 (𝑧𝛽 )
(1)
(2) (2)
(𝜏∗ )
(2) ⟩ 𝐟
(2)
2 𝜏∗ −𝑧∗
⟩ 𝐟 (2) (𝑥1 )] (Im𝑧∗
(𝑖)
+ ∑3𝛽=1 𝐈𝛽 (𝐀T1 𝐦 (𝑧𝛽 ) + 𝐁1T 𝐩 (𝑧𝛽 ))
(2)
𝑑𝜏∗
[∫Γ ⟨
(𝑖)
Eqs (36)–(41) can be rewritten as follows:
(2)
𝐟 (1) (𝑥1 ) = 0;
(40)
(1)
(1)
(1)
(1)
𝐟 (2) (𝑥1 ),
(41)
(2)
(2)
1
(2)
𝐟 (2) (𝑧∗ ) = 𝐟∞ (𝑧∗ ) +
2𝜋𝑖
(2)
(2)
(2)
(2)
(50)
[𝐪2 (𝑧∗ )
− ∑3𝛽=1 𝐈𝛽 (𝐀T2 𝐦 (𝑧𝛽 ) + 𝐁2T 𝐩 (𝑧𝛽 ))
(2)
(49)
(1)
𝐪2 (𝑧𝛽 ) − 𝐀T2 𝐦 (𝑧𝛽 ) − 𝐁2T 𝐩 (𝑧𝛽 ) − 𝑀𝑡 (𝑧𝛽 ) μ2
+𝑃𝑡 (𝑧𝛽 ) λ2 = 0;
where: ⟨𝐹(𝑧∗ )⟩ = 𝑑𝑖𝑎𝑔[𝐹1 (𝑧1 ), 𝐹2 (𝑧2 ), 𝐹3 (𝑧3 )],
(𝑖)
(𝑖)
𝑧𝛽 = 𝑥1 + 𝑝𝛽 𝑥2 (𝛽 = 1, . . ,3).
Utilizing Eqs (4) and (8), the improper integrals in Eqs (36)–
(41) can be rewritten as follows:
70
(𝑖)
𝑥1 −𝑧𝛽
(𝑖)
Im𝑧𝛽 < 0:
∫Γ ⟨
+∞ 𝐟 (2) 𝑑𝑥1
∫−∞
> 0:
𝑑𝜏∗
+∞
(𝑖)
−λ1 𝑃𝑡 (𝑧𝛽 )
(1)
⟩ 𝐟 (1) (𝜏∗ )
(𝑖)
= 𝐀T1 𝐦(𝑧𝛽 ) + 𝐁1T 𝐩(𝑧𝛽 ) + μ1 𝑀𝑡 (𝑧𝛽 )
𝐪𝑗 (𝑧𝛽 ) = ∫Γ ⟨
(1)
∫Γ ⟨
⟩ 𝐟 (1) (𝑥1 )]
1
2𝜋𝑖
(𝑖)
𝑥1 −𝑧𝛽
−⟨𝑀𝑡 (𝑧∗ )⟩μ2 + ⟨𝑃𝑡 (𝑧∗ )⟩λ2 ] ;
(51)
DOI 10.1515/ama-2016-0012
(2)
acta mechanica et automatica, vol.10 no.1 (2016)
(2)
(2)
(2)
(2)
−𝑃𝑡 (𝑧𝛽 ) λ1 + 𝜌0 𝑚𝑡 (𝑧𝛽 ) λ1 = 0;
(2)
(2)
ternak et al., 2014), and
(2)
𝐪1 (𝑧𝛽 ) + 𝐀T1 𝐦 (𝑧𝛽 ) + 𝐁1T 𝐩 (𝑧𝛽 ) + 𝑀𝑡 (𝑧𝛽 ) μ1
(2)
(52)
(2)
̅T2 𝐦 (𝑧 ) − 𝐁
̅ 2T 𝐩 (𝑧 ) − 𝑀𝑡 (𝑧 ) μ̅2
̅2 (𝑧𝛽 ) − 𝐀
𝐪
𝛽
𝛽
𝛽
(2)
+𝑃𝑡 (𝑧𝛽 ) λ̅2 = 0,
(53)
where: 𝐈1 = diag[1,0,0], 𝐈2 = diag[0,1,0], 𝐈3 = diag[0,0,1].
Eqs (49), (50), (52), (53) allow to express the improper integrals (43) along the infinite path through the path integrals over
the contours 𝛤𝑗 :
(1)
(1)
− 𝐁2−T 𝐲2 (𝑧𝛽 )),
(1)
(1)
−1 −T ̅−T
̅1 𝐀
̅−1
𝐩(𝑧𝛽 ) = (𝐁
(𝐀1 𝐲1 (𝑧𝛽 ) −
1 − 𝐁2 𝐀2 )
(1)
(1)
(1)
̅1 (𝑧𝛽 ) − 𝑀𝑡 (𝑧𝛽 ) μ̅1 + 𝑃𝑡 (𝑧𝛽 ) λ̅1
𝐲1 (𝑧𝛽 ) = −𝐪
(1)
−𝜌0 𝑚𝑡 (𝑧𝛽 )λ̅1 ,
(1)
𝐲2 (𝑧𝛽 )
(54)
(1)
𝐪2 (𝑧𝛽 )
=
−
(1)
𝑀𝑡 (𝑧𝛽 ) μ2
+
(1)
𝑃𝑡 (𝑧𝛽 ) λ2 ,
(2)
̅2𝐁
̅ 2−1 − 𝐀1 𝐁1−1 )−T (𝐁
̅ 2−T 𝐲3 (𝑧𝛽(2) )
𝐦(𝑧𝛽 ) = (𝐀
=
̅2 𝐀
̅−1
(𝐁
2
−
(2)
𝐲4 (𝑧𝛽 )
(1,1)
𝜋
−
=
=
(2)
−𝐪1 (𝑧𝛽 )
(1)
𝑛̅
̅ (1,1) ⟩ 𝐆1(1) 𝐈𝛽 𝐧
̅𝑡∗(1,1) ⟩δ1(1)
̅(1) − 𝑡 (⟨ln𝑊
+ ∑3𝛽=1 ⟨ln𝑊
𝛽
(2)
𝑀𝑡 (𝑧𝛽 ) μ̅2
−
−
+
(2)
𝑀𝑡 (𝑧𝛽 ) μ1
(2)
𝑃𝑡 (𝑧𝛽 ) λ̅2 ,
+
(55)
̅𝑡∗(1,1) )⟩𝐤 (1) ))
−⟨K(𝐵 (1) 𝑊
−
(2)
𝑃𝑡 (𝑧𝛽 ) λ1
𝐜1
2
(1)
(1,1)
(𝑛𝑡 ln𝑊𝑡
+
𝐜1
̅𝑡(1,1) )
(𝐾 − 1)𝑛̅𝑡(1) L(1) (𝑊
2
(1)
̅𝑡(1,1) )} ,
− 𝑛̅𝑡 ln𝑊
1
(1,1)
(2)
−𝜌0 𝑚𝑡 (𝑧𝛽 ) λ1 .
𝐯 𝐼𝑇𝐶 (𝐱, ξ) = Im {𝐀1 [−⟨𝑓 ∗ (𝑊∗
Substituting Eqs (54), (55) into Eqs (48), (51), accounting
for Eq (35), one can obtain the integral formulae for the Stroh
complex vector-functions of thermoelastic anisotropic bimaterial
with thermally imperfect interface, which do not contain the improper integrals over the infinite path (bimaterial interface):
̅ (1,1) )⟩ 𝐆1(1) 𝐈𝛽 μ̅1 +
− ∑3𝛽=1 ⟨𝑓 ∗ (𝑊
𝛽
(1)
𝜋
(1)
𝐟 (1) (𝑧∗ ) = 𝐟∞ (𝑧∗ ) +
(1)
1
2𝜋𝑖
(1)
(1)
(1)
𝐟
(2)
(2)
(𝑧∗ )
=
(2)
+
(2)
(2)
(2)
(𝑗)
and δ𝑖
)] +
4
1
(1)
Im {𝐁1 𝛿∗ [⟨
𝜋
𝑖
(1)
2𝑘𝑡
𝑖𝐜1
)κ(1) ] −
𝜌0 𝐜1 (𝐾−1)2
+ ∑3𝛽=1 ⟨
1
̅𝑡∗(1,1) )⟩δ1(1)
⟨𝑓 ∗ (𝑊
(1,1)
(1)
2𝑘𝑡
[𝑓 ∗ (𝑊𝑡
(1,1)
̅𝑡
L(1) (𝑊
1
(1,1)
𝑊∗
1
(1)
̅ (1,1)
𝑊
𝛽
)
)} ,
⟩ 𝐧(1)
𝐝1
2
(1)
𝛿𝑡 ([
(1)
̅(1) − 𝑛̅𝑡 (⟨
⟩ 𝐆1 𝐈𝛽 𝐧
2
(1)
−
(1,1)
𝑊𝑡
(1)
𝑛̅𝑡
̅ (1,1)
𝑊
𝑡
]+
2𝑖𝑛̅𝑡
(1)
𝜌0 𝑘𝑡
(1)
̅ (1,1) ⟩ 𝐆1(1) 𝐈𝛽 μ̅1 −
+ ∑3𝛽=1 ⟨ln𝑊
𝛽
(1,1)
are the same as in Ref (Pas-
+
𝑖𝐝1
(1)
2𝑘𝑡
(1)
(1,1)
(1,1)
𝜋
̅𝑡∗
−⟨K(𝐵 (1) 𝑊
(1)
⟩ (δ1 + κ(1) )
̅𝑡
𝐾(𝐵 (1) 𝑊
𝐰𝑗𝐼𝑇𝐶 (𝐱, ξ) = Im {𝐁1 𝛿∗ [⟨ln𝑊∗
(57)
1
̅ (1,1)
𝑊
𝑡∗
)⟩κ(1) )]
(1)
𝑛𝑡
1
(2)
(2)
(𝑗)
(1,1)
̅𝑡
+𝐾𝑓 ∗ (𝑊
−
(2)
[𝐪2 (𝑧∗ )
2𝜋𝑖
+⟨𝑒𝑡 (𝑧∗ ) + 𝑒̅𝑡 (𝑧∗ )⟩κ(2) ] ,
where the constants 𝐆𝑖
(1,1)
̅𝑡∗
L(1) (𝑊
(1,1)
1
(1) (2)
(2)
(2) (2)
(2)
+⟨𝑄𝑡 (𝑧∗ )⟩δ1 + ⟨𝑄̅𝑡 (𝑧∗ )⟩δ2
(2)
4
̅𝑡∗
−𝐵 (1) ⟨𝐾(𝐵 (1) 𝑊
̅2 (𝑧𝛽 ))
− ∑3𝛽=1 𝐈𝛽 (𝐆1 𝐪1 (𝑧𝛽 ) + 𝐆2 𝐪
(1)
(56)
(1) (1)
𝑒̅𝑡 (𝑧∗ )⟩κ(1) ] ,
(2) (2)
𝐟∞ (𝑧∗ )
𝜌0 (1−𝐾)
)⟩μ1
(1)
(1) (1)
(1)
(2) (1)
(1)
+⟨𝑄̅𝑡 (𝑧∗ )⟩δ1 + ⟨𝑄𝑡 (𝑧∗ )⟩δ2
+
−
𝐪𝐼𝑇𝐶
𝑗 (𝐱, ξ) =
[𝐪1 (𝑧∗ )
̅1 (𝑧𝛽 ) + 𝐆2 𝐪2 (𝑧𝛽 ))
+ ∑3𝛽=1 𝐈𝛽 (𝐆1 𝐪
(2) (1)
+⟨𝑒𝑡 (𝑧∗ )
⟩ 𝐧(1)
2
(2)
̅2 (𝑧𝛽 )
𝐪
(1)
(59)
where: the kernels 𝐔 bm (𝐱, ξ), 𝐓 bm (𝐱, ξ), 𝐃𝑗bm (𝐱, ξ), 𝐒𝑗bm (𝐱, ξ)
are the same as those derived by Pasternak et al. (2014)
for a thermoelectroelastic bimaterial with perfect thermal and
mechanical interface, and others kernels are defined by the following equations:
𝐫 𝐼𝑇𝐶 (𝐱, ξ) = − Im {𝐀1 (⟨ln𝑊∗
(2)
𝐀−T
1 𝐲4 (𝑧𝛽 )),
(2)
𝐲3 (𝑧𝛽 )
(58)
Derived equations (56) and (57) for the Stroh complex functions allow writing the integral formulae relating the latter at the
arbitrary point of the bimaterial solid with the boundary values
of temperature, heat flux, displacement and traction at the contours 𝛤𝑖 . Therefore, using Eqs (3), (4), (56), (57) and (Pasternak,
2012, Ting, 1996) one can obtain the following extended
Somigliana integral identity for a thermoelastic bimaterial solid
with imperfect thermal contact at its interface:
1
(2)
−T ̅−T
𝐁1 𝐀−1
(𝐀 2 𝐲3 (𝑧𝛽 )
1 )
(2)
𝐱 ∈ 𝑆1 ∧ ξ ∈ 𝑆1 :
(2)
− 𝐁1−T 𝐲4 (𝑧𝛽 )),
(2)
𝐩(𝑧𝛽 )
(2)
σ𝑗 (ξ) = σ𝑗∞ (ξ) + ∫Γ 𝐃𝑗bm (𝐱, ξ)𝐭(𝐱)𝑑𝑠(𝐱) −
∫Γ 𝐒𝑗bm (𝐱, ξ)𝐮(𝐱)𝑑𝑠(𝐱) + ∫Γ 𝐪ITC
𝑗 (𝐱, ξ)𝜃(𝐱)𝑑𝑠(𝐱) +
ITC
∫Γ 𝐰𝑗 (𝐱, ξ)ℎ𝑛 (𝐱)𝑑𝑠(𝐱),
𝐀−T
2 𝐲2 (𝑧𝛽 )),
(1)
(1)
𝐮(ξ) = 𝐮∞ (ξ) + ∫Γ 𝐔 bm (𝐱, ξ)𝐭(𝐱)𝑑𝑠(𝐱) −
∫Γ 𝐓 bm (𝐱, ξ)𝐮(𝐱)𝑑𝑠(𝐱) + ∫Γ 𝐫 ITC (𝐱, ξ)𝜃(𝐱)𝑑𝑠(𝐱) +
∫Γ 𝐯 ITC (𝐱, ξ)ℎ𝑛 (𝐱)𝑑𝑠(𝐱),
(1)
̅1 𝐁
̅1−1 − 𝐀2 𝐁2−1 )−T (𝐁
̅1−T 𝐲1 (𝑧𝛽 )
𝐦(𝑧𝛽 ) = (𝐀
(1)
(1)
𝜅 (1) = −𝛿2 + 2𝐆2 𝜆2 ; 𝜅 (2) = −𝛿1 − 2𝐆1 𝜆1 .
𝑖
(1)
2𝑘𝑡
))} ,
⟩μ1
̅𝑡∗(1,1) ⟩δ1(1)
(⟨ln𝑊
(60)
)⟩κ(1) )]
(1,1)
𝛿𝑡 (ln𝑊𝑡
̅𝑡(1,1) + (𝐾 − 1)L(1) (𝑊
̅𝑡(1,1) )} ;
+ ln𝑊
71
Heorhiy Sulym, Iaroslav Pasternak, Mykhailo Tomashivskyy
Boundary Integral Equations for an Anisotropic Bimaterial with Thermally Imperfect Interface and Internal Inhomogeneities
1
𝜋
1
(2,1)
𝐫 𝐼𝑇𝐶 (𝐱, ξ) = − Im {𝐀1 [∑3𝛽=1 ⟨ln𝑊𝛽
𝜋
(2)
(2,1)
− 𝑛𝑡 (⟨ln𝑊𝑡∗
+
2
𝐜1
2
(1)
(2,2)
̅
− ∑3𝛽=1 ⟨ln𝑊
𝛽
(1)
⟩ 𝐆2 𝐈𝛽 𝐧(2)
(2,1)
⟩δ2 ) − ⟨K(𝐵 (1) 𝑊𝑡∗
)⟩κ(1) ]
(2,2)
(𝐾 − 1)𝑛𝑡(2) L(1) (𝑊𝑡(2,1) )} ,
−
1
(2,1)
(1)
)⟩ 𝐆2 𝐈𝛽 μ2
𝜋
𝑖
𝜌 (1+𝐾) (1)
(2,1)
(1)
(2,1)
+ (2) ⟨𝑓 ∗ (𝑊𝑡∗ )⟩δ2 − 0
L (𝑊𝑡∗ )κ(1) ]
4
2𝑘
(2,1)
𝑓 ∗ (𝑊𝑡
(2)
2𝑘𝑡
𝐪𝐼𝑇𝐶
𝑗 (𝐱, ξ)
1
(2)
− 𝑛𝑡
2
=
)−
(1)
Im {𝐁1 𝛿∗ [∑3𝛽=1 ⟨
𝜋
1
(1)
(⟨ (2,1)⟩ (δ2
𝑊
+κ
(1)
) −𝐵
(2,1)
𝑊𝑡∗ )⟩κ(1) )]
(2) (1)
(2,1)
− (1) 𝑛𝑡 𝛿𝑡 K(𝐵 (1) 𝑊𝑡
)}
𝜌0 𝑘𝑡
(1)
(2,1)
𝐰𝑗𝐼𝑇𝐶 (𝐱, ξ) = Im {𝐁1 𝛿∗ [∑3𝛽=1 ⟨ln𝑊𝛽
𝜋
+
+
𝑖
(2,1)
(2) (⟨ln𝑊𝑡∗
2𝑘𝑡
𝑖𝐝1
(1)
(2,1)
⟩δ2 − ⟨K(𝐵 (1) 𝑊𝑡∗
(1)
(2,1)
(1)
(𝑊𝑡
(2) 𝛿𝑡 L
(1)
𝑘𝑡 +𝑘𝑡
𝐫
(1)
+ 𝑛𝑡
2
+
𝐜2
2
𝜌0 𝐜2 (𝐾+1)2
4
1
− ⟨K(𝐵
(1)
−
𝐝2
2
(2)
(1,2)
2𝑘𝑡
(2)
)⟩δ1 −
𝜌0 (1−𝐾)
𝑖𝐜2
(1,2)
∗
)−
(1) (1 + 𝐾)𝑓 (𝑊𝑡
2𝑘𝑡
1
(1)
2
4
(2,2)
+
(2)
)⟩ 𝐆1 𝐈𝛽 μ1
(1,2)
L(2) (𝑊𝑡∗
𝑖𝐝2
(2)
2𝑘𝑡
(2)
− Im {𝐁2 𝛿∗ [∑3𝛽=1 ⟨
𝜋
1
(1,2)
𝑖𝐝2
(1,2)
𝑛𝑡 𝛿𝑡 K(𝐵 (2) 𝑊𝑡
1
−
𝑖
(1,2)
(1)
(⟨ln𝑊𝑡∗
𝑖𝐝2
(1)
(2)
(2)
𝑘𝑡 +𝑘𝑡
̅ (𝑖,𝑗)
𝑊
𝛽
𝛽
𝑛̅𝑡
̅ (2,2)
𝑊
𝑡
]−
2𝑖
(2)
𝜌0 𝑘𝑡
(2)
(2)
̅𝑡(2,2) ))} ,
𝑛̅𝑡 K(𝐵 (2) 𝑊
(2,2)
(2)
)} ,
(1,2)
(1,2)
⟩δ1 − ⟨K(𝐵 (2) 𝑊𝑡∗
(1,2)
)} ;
(2)
⟩ 𝐆2 𝐈𝛽 μ̅2 −
𝑖
(2)
2𝑘𝑡
⟩μ2
̅𝑡∗(2,2) ⟩δ(2)
(⟨ln𝑊
2
(63)
)⟩κ(2) )]
(2,2)
̅𝑡(2,2) − (1 + 𝐾)L(2) (𝑊
̅𝑡(2,2) ))}.
− ln𝑊
(𝑗)
(𝑖)
(𝑗)
(𝑖,𝑗)
(𝑖)
(𝑗)
= 𝑍∗ (𝐱) − 𝑍∗ (ξ), 𝑊𝛽
= 𝑍𝛽 (𝐱) − 𝑍∗ (ξ), (64)
(𝑖)
(𝑗)
(𝑖,𝑗)
(𝑖)
(𝑗)
= 𝑍𝛽̅ (𝐱) − 𝑍∗ (ξ), 𝑊𝑡∗ = 𝑍𝑡 (𝐱) − 𝑍∗ (ξ),
̅𝑡∗(𝑖,𝑗) = 𝑍𝑡̅ (𝑖) (𝐱) − 𝑍∗(𝑗) (ξ), 𝛿∗(𝑗) = ⟨𝛿𝑖2 − 𝛿𝑖1 𝑝∗(𝑗) ⟩.
𝑊
(2)
𝛿𝑡 L(2) (𝑊𝑡
𝐱 ∈ 𝑆2 ∧ ξ ∈ 𝑆2 :
72
𝑊∗
(2)
(1)
(1,2) ⟩ 𝐆1 𝐈𝛽 𝐧
𝑊
𝜋
2𝑘𝑡
(2)
⟩ (δ1 + κ(2) )
(𝑖)
(𝑖)
𝐧(𝑖) = λ𝑖 𝑛2 (𝐱) − ρ𝑖 𝑛1 (𝐱), 𝑍∗ (𝐱) = 𝑥1 + 𝑝∗ 𝑥2 ,
1
𝐰𝑗𝐼𝑇𝐶 (𝐱, ξ) = − Im {𝐁2 𝛿∗ [∑3𝛽=1 ⟨ln𝑊𝛽
−
1
̅ (2,2)
𝑊
𝑡∗
)⟩κ(2) )]
(2)
−
𝛿𝑡 (ln𝑊𝑡
(𝑗)
)⟩κ(2) )]
(1) (2)
(2)
𝜌0 𝑘𝑡
(2)
ρ𝑗 = 𝐀𝑗𝑇 Re[𝑝𝑡 𝐝𝑗 ] + 𝐁𝑗𝑇 Re[𝑝𝑡 𝐜𝑗 ],
⟩ (δ1 + κ(2) )
+ 𝐵 (2) ⟨K(𝐵(2) 𝑊𝑡∗
−
1
The following constants and functions are used for contracting
the notations:
)κ(2) ]
(2)
(1,2)
(2)
)κ(2) ]
⟩ 𝐧(2)
2
𝑛𝑡
̅𝑡∗
−⟨K(𝐵 (2) 𝑊
𝜌0 𝐜2 (𝐾 2 −1) (2)
(1,2)
L (𝑊𝑡
)} ,
4
1
𝑊𝑡∗
(2,2)
𝑊∗
(2)
(2,2)
𝑊𝑡
(𝑖,𝑗)
=
(2,2)
̅𝑡∗
L(2) (𝑊
̅(2) − 𝑛̅𝑡 (⟨
⟩ 𝐆2 𝐈𝛽 𝐧
(2)
𝛿𝑡 ([
1
(2)
(2,2)
(1,2)
𝑊𝑡∗ )⟩κ(2) )]
(1,2)
∗
(1) ⟨𝑓 (𝑊𝑡∗
+ 𝑛𝑡 (⟨
1
4
)} ,
𝐪𝐼𝑇𝐶
𝑗 (𝐱, ξ) = Im {𝐁2 𝛿∗ [⟨
̅
− ∑3𝛽=1 ⟨ln𝑊
𝛽
(1)
(1,2)
1)𝑛𝑡 L(2) (𝑊𝑡
)} ,
𝑖
𝜌0 (1+𝐾)
𝜋
(1,2)
(2)
Im {𝐀2 [∑3𝛽=1 ⟨ln𝑊𝛽 ⟩ 𝐆1 𝐈𝛽 𝐧(2)
𝜋
1
𝐪𝐼𝑇𝐶
𝑗 (𝐱, ξ)
(2,2)
̅𝑡
L(2) (𝑊
̅ (2,2)
𝑊
𝛽
)⟩μ2
̅𝑡(2,2) )]
) − 𝐾𝑓 ∗ (𝑊
1
𝜋
−
[𝑓 ∗ (𝑊𝑡
(2)
)⟩δ2 +
(2,2)
(61)
𝐯 𝐼𝑇𝐶 (𝐱, ξ) = Im {𝐀1 [∑3𝛽=1 ⟨𝑓 ∗ (𝑊𝛽
−
(2,2)
(2)
2𝑘𝑡
)
(2)
𝐰𝑗𝐼𝑇𝐶 (𝐱, ξ) = Im {𝐁2 𝛿∗ [⟨ln𝑊∗
(1,2) (2)
(⟨ln𝑊𝑡∗ ⟩δ1
(𝐾 +
𝑖𝐜2
̅𝑡∗
⟨𝑓 ∗ (𝑊
1
ξ) =
1
(2,2)
(2)
̅𝑡∗
+𝐵 (2) ⟨K(𝐵 (2) 𝑊
)⟩κ(1) )]
𝐱 ∈ 𝑆1 ∧ ξ ∈ 𝑆2 :
𝐼𝑇𝐶 (𝐱,
𝑖
2𝑘𝑡
− ∑3𝛽=1 ⟨
(1)
⟩ 𝐆2 𝐈𝛽 μ2
)} ;
(2,2)
)⟩ 𝐆2 𝐈𝛽 μ̅2
𝜋
𝑖𝐝1
1
+
−
𝑡∗
(2)
̅𝑡
𝑛̅𝑡 L(2) (𝑊
(2,2)
̅
+ ∑3𝛽=1 ⟨𝑓 ∗ (𝑊
𝛽
−
𝛽
⟨K(𝐵
2
(2)
̅𝑡(2,2) )} ,
− 𝑛̅𝑡 ln𝑊
(2,2)
(1)
(2)
(2,1) ⟩ 𝐆2 𝐈𝛽 𝐧
𝑊
(1)
(2,2)
(𝑛𝑡 ln𝑊𝑡
2
𝐜2
1
1
(1)
(2)
)⟩κ(2) )) −
𝜋
𝜌0 𝐜1 (𝐾 2 − 1) (1)
(2,1)
L (𝑊𝑡
)},
4
1
1 (2)
(2)
̅𝑡∗(2,2) ⟩δ(1)
̅(2) − 𝑛̅𝑡 (⟨ln𝑊
⟩ 𝐆2 𝐈𝛽 𝐧
2
𝐯 𝐼𝑇𝐶 (𝐱, ξ) = Im {𝐀2 [−⟨𝑓 ∗ (𝑊∗
𝑡
𝑖𝐜1 (1 − 𝐾)
𝐜2
⟩𝐧(2)
2
̅𝑡∗
−⟨K(𝐵 (2) 𝑊
𝐯 𝐼𝑇𝐶 (𝐱, ξ) = − Im {𝐀1 [∑3𝛽=1 ⟨𝑓 ∗ (𝑊𝛽
+
(2,2)
𝐫 𝐼𝑇𝐶 (𝐱, ξ) = − Im {𝐀2 (⟨ln𝑊∗
𝐱 ∈ 𝑆2 ∧ ξ ∈ 𝑆1 :
1
DOI 10.1515/ama-2016-0012
(2)
⟩ 𝐆1 𝐈𝛽 μ1
)⟩κ(2) )]
(62)
The proposed complex variable approach allow not only to derive the boundary integral formulae for anisotropic thermoelasticity, but also to derive singular boundary integral equations for
solving the boundary value problems for thermoelastic bimaterial
solids with thermally imperfect interface. In particular, for derivation of the boundary integral equations it is convenient to apply the
Sokhotskii-Plemelj formula (Muskhelishvili, 2008), which relates
the limit value of the Cauchy integral over a smooth closed contour with its principal value. Thus, according to Eq (59) and (Muskhelishvili, 2008, Pasternak, 2012), for a smooth closed contours
𝛤 in a thermoelastic bimaterial one can obtain the following
boundary integral equations for determination of the unknown
boundary functions:
DOI 10.1515/ama-2016-0012
acta mechanica et automatica, vol.10 no.1 (2016)
1
𝜃(𝐲) = 𝜃 ∞ (𝐲) + RPV ∫Γ Θ𝐼𝑇𝐶∗ (𝐱, 𝐲)ℎ𝑛 (𝐱)𝑑𝑠(𝐱) −
2
CPV ∫Γ 𝐻𝐼𝑇𝐶∗ (𝐱, 𝐲)𝜃(𝐱)𝑑𝑠(𝐱),
1
𝐮(𝐲) = 𝐮∞ (ξ) + RPV ∫Γ 𝐔 bm (𝐱, 𝐲)𝐭(𝐱)𝑑𝑠(𝐱) −
CPV ∫Γ 𝐓 bm (𝐱, 𝐲)𝐮(𝐱)𝑑𝑠(𝐱) +
RPV ∫Γ 𝐫 𝐼𝑇𝐶 (𝐱, 𝐲)𝜃(𝐱)𝑑𝑠(𝐱) +
∫Γ 𝐯 𝐼𝑇𝐶 (𝐱, 𝐲)ℎ𝑛 (𝐱)𝑑𝑠(𝐱),
2
(65)
where RPV stands for a Riemann Principal Value and CPV
for a Cauchy Principal Value of an integral.
The integral equations (65) degenerate, when separate closed
contours 𝛤𝑗 of a line Γ are the faces of mathematical cuts 𝛤𝐶𝑗 (the
simple opened arcs). In this case it is necessary to apply the
theory of dual hypersingular integral equations (Pasternak et al.,
2013a, Pasternak, 2012).
According to (Pasternak et al., 2013a), the stress and displacement discontinuity functions at the tips of thin inhomogeneities, which are not placed at the bimaterial interface, possess
square root singularity. And the stress field in the vicinity of inclusion’s tip is completely defined by the generalized stress intensity
factors. The latter are related to the displacement and stress
discontinuities at thin inhomogeneity with the following equations
𝜋
𝜋𝑠
𝐤 (1) = lim√ 𝐋 ⋅ Δ𝐮(𝑠), 𝐤 (2) = −lim√ Σ𝐭(𝑠),
𝑠→0
8𝑠
𝑠→0
2
are placed. The half-spaces of the bimaterial possess the same
thermal and mechanical properties of glass/epoxy: 𝐸1 = 55 GPa,
𝐸2 = 21 GPa, 𝐺12 = 9.7 GPa, 𝜈12 = 0.25, 𝛼11 = 6.3 ⋅
10−6 K-1,
𝛼22 = 2.0 ⋅ 10−5 K-1,
𝑘11 = 3.46 W/(m∙K),
𝑘22 = 0.35 W/(m∙K). Material symmetry axes coincide with the
reference coordinates. Plane stress is considered.
Fig. 2. Bimaterial with identical properties of half-spaces
(66)
T
(2)
(2)
where: 𝐤 (1) = [𝐾21 , 𝐾11 , 𝐾31 ]T , 𝐤 (2) = [𝐾12 , 𝐾22 , 𝐾32 ] ;
𝐾𝑖𝑗 are generalized stress intensity factors (SIF) (Pasternak et al,
2013b, Sulym, 2007); and 𝑳 = −2√−1𝑩𝑩𝑇 is a real Barnett –
Lothe tensor (Sulym et al., 2014).
Obtained dual boundary integral equations along with the
models of thin thermoelastic inclusions (Pasternak et al., 2013a)
allow solving thermoelastic problems for a bimaterial solid with
thermally imperfect interface, which components contain thin
inhomogeneities.
4. NUMERICAL EXAMPLES
Similarly to (Pasternak et al., 2013a, Pasternak, 2012), obtained integral formulae and equations (25), (26), (59) are introduced entered into the computational algorithm of the modified
boundary element method (Pasternak et al., 2013a, 2014) that
allows numerical solution of a wide range of 2D problems for an
anisotropic thermoelastic bimaterial solid with thermally imperfect
interface containing internal inhomogeneities. Several numerical
examples are considered here for a bimaterial containing thin
thermoelastic inclusions. The boundary element mesh consists of
only 20 discontinuous three-node boundary elements including
two special for convenient determination of stress intensity factors
at inclusion’s tips.
Consider an anisotropic thermoelastic bimaterial (Fig. 2) consisting of two half-space that have identical mechanical and thermal properties. The bimaterial contains a thin rectilinear elastic
isotropic inclusion of length 2𝑎 and thickness 2ℎ = 0.02a
placed at the upper half-space 𝑥2 > 0 parallel to the interface.
The centre of the inclusion is placed at the distance d to the bimaterial interface. In the lower half-space 𝑥2 < 0 at the distance
𝑑/2 to the interface and at the distance a to the central vertical
axis a heat source and a heat drain both of the same magnitude
Fig. 3. Generalized stress intensity factors at the right tip of the inclusion,
which relative rigidity is 102 (a-b) or 10−2 (c-d)
Fig. 3 presents the dependence of the generalized stress intensity factors (SIF) at the right tip of the inclusion on the parameter η, which define the thermal resistance of the interface as 𝜌0 =
𝜂
𝑎⁄
𝑘 ⋅ 10 . The normalization factor is equal to 𝐾0 =
11
√𝜋𝑎 ⋅ 𝐸1 ⋅ 𝛼11 ⁄𝑘11 ⋅ 𝑞. Plots are obtained for different values
of the parameter d and the relative rigidity 𝑘 = 𝐺 𝑖 ⁄𝐺12 of the
inclusion, where Gi is a shear modulus of inclusion’s material. It is
assumed that the inclusion does not possess thermal expansion.
One can see in Fig. 3 that the closer are the heat sources
and the inclusion to the bimaterial interface, the higher are the
values of generalized SIF. Decrease in inclusion’s rigidity considerably increases stress intensity. It should be mentioned that the
73
Heorhiy Sulym, Iaroslav Pasternak, Mykhailo Tomashivskyy
Boundary Integral Equations for an Anisotropic Bimaterial with Thermally Imperfect Interface and Internal Inhomogeneities
most significant growth of all generalized SIF occurs, when the
parameter 𝜂 of normalized thermal resistance of the bimaterial
interface is near zero.
Fig. 4. Generalized stress intensity factors at the right tip of the inclusion,
which relative rigidity is 105 (a-b) or 10−5 (c-d)
The increase in inclusion’s relative rigidity (Fig. 4, a-b) causes
a significant growth in the magnitude of the generalized SIF, and,
in general, their behavior is the same, as well as in case
of a softer inclusion. The same conclusion can be made, when
relative rigidity of inclusion is equal to10−5 .
5. CONCLUSION
The paper presents a general complex variable straightforward approach for obtaining the boundary integral equations and
integral formulae for a defective bimaterial solid with thermally
imperfect interface. The kernels of these equations are obtained
explicitly and in closed-form that allows developing the efficient
boundary element approach for the analysis of thermoelastic
anisotropic bimaterial solids with imperfect interface containing
internal inhomogeneities. The influence of the thermal resistance
of the interface on the field intensity factors at the tips of thin
inhomogeneities is studied. It is shown that thermal resistance of
the interface causes significant influence on the stress intensity at
the tips of cracks even in the bimaterial consisting of two halfspaces with identical thermal and mechanical properties. Also
several examples are considered for bimaterials with thin thermoelastic inclusions, and the intensity of heat flux and stress at
their tips is studied.
.
74
DOI 10.1515/ama-2016-0012
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