Memoirs on Differential Equations and Mathematical Physics Volume 58, 2013, 25–64

Memoirs on Differential Equations and Mathematical Physics Volume 58, 2013, 25–64

L. Giorgashvili, D. Natroshvili, and Sh. Zazashvili

TRANSMISSION AND INTERFACE CRACK PROBLEMS OF THERMOELASTICITY FOR HEMITROPIC SOLIDS

Abstract. The purpose of this paper is to investigate basic transmission and interface crack problems for the differential equations of the theory of elasticity of hemitropic materials with regard to thermal effects. We con- sider the so called pseudo-oscillation equations corresponding to the time harmonic dependent case. Applying the potential method and the theory of pseudodifferential equations first we prove uniqueness and existence theo- rems of solutions to the Dirichlet and Neumann type transmission-boundary value problems for piecewise homogeneous hemitropic composite bodies. Af- terwards we investigate the interface crack problems and study regularity properties of solution.

2010 Mathematics Subject Classification. 31B10, 35J57, 47G30, 47G40, 74A60, 74G30, 74G40, 74M15.

Key words and phrases. Elasticity theory, elastic hemitropic materi- als, integral Equations, pseudodifferential equations, transmission problems, interface crack problems, potential theory.

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1. Introduction

Technological and industrial developments, and also recent important progress in biological and medical sciences require the use of more general and refined models for elastic bodies. In a generalized solid continuum, the usual displacement field has to be supplemented by a microrotation field.

Such materials are called micropolar or Cosserat solids. They model con- tinua with a complex inner structure whose material particles have 6 degree of freedom (3 displacement components and 3 microrotation components).

Recall that the classical elasticity theory allows only 3 degrees of freedom (3 displacement components).

Experiments have shown that micropolar materials possess quite different properties in comparison with the classical elastic materials (see, e.g., [3], [4], [7], [15], [23], [25], [26], and the references therein). For example, in non- centrosymmetric micropolar materials the propagation of left-handed and right-handed elastic waves is observed. Moreover, the twisting behaviour under an axial stress is a purely hemitropic (chiral) phenomenon and has no counterpart in classical elasticity. Such solids are calledhemitropic non- centrosymmetric, acentric, orchiral. Throughout the paper we use the term hemitropic.

Hemitropic solids are not isotropic with respect to inversion, i.e., they are isotropic with respect to all proper orthogonal transformations but not with respect to mirror reflections.

Materials may exhibit chirality on the atomic scale, as in quartz and in biological molecules - DNA, as well as on a large scale, as in composites with helical or screw–shaped inclusions, certain types of nanotubes, fabricated structures such as foams, chiral sculptured thin films and twisted fibers. For more details see the references [3], [4], [14], [15], [20], [23], [24], [26]–[30], [34], [35], [46]–[50], [53], [56], [57].

Mathematical models describing the chiral properties of elastic hemitropic materials have been proposed by A´ero and Kuvshinski [3], [4] (for historical notes see also [14], [15], [46], and the references therein).

In the present paper we deal with the model of micropolar elasticity for hemitropic solids when the thermal effects are taken into consideration.

In the mathematical theory of hemitropic thermoelasticity there are in- troduced the asymmetric force stress tensor and couple stress tensor, which are kinematically related with the asymmetric strain tensor, torsion (cur- vature) tensor and the temperature function via the constitutive equations.

All these quantities along with the heat flux vector are expressed in terms of the components of the displacement and microrotation vectors, and the tem- perature function. In turn, the displacement and microrotation vectors, and the temperature satisfy a coupled complex system of second order partial differential equations of dynamics. When the mechanical and thermal char- acteristics (displacements, microrotations, temperature, body force, body couple vectors, and heat source) do not depend on the time variable t we

have the differential equations of statics. If time dependence is harmonic (i.e., the pertinent fields are represented as the product of the time depen- dent exponential function exp{−iσt} and a function of the spatial variable x∈R³) then we have the steady state oscillation equations. Hereσis a real frequency parameter. Note that if σ= 0, then we obtain the equations of statics. Ifσ=σ1+iσ2is a complex parameter, then we have the so called pseudo-oscillation equations(which are related to the dynamical equations via the Laplace transform). All the above equations generate a strongly elliptic, formally non-self-adjoint 7×7 matrix differential operator.

The Dirichlet, Neumann and mixed type boundary value problems (BVP) corresponding to this model are well investigated for homogeneous bodies of arbitrary shape and the uniqueness and existence theorems are proved, and regularity results for solutions are established by the potential method, as well as by variational methods (see [39]–[43] and the references therein).

The main goal of our investigation is to study the Dirichlet and Neumann type transmission and interface crack problems of the theory of elasticity with regard to thermal effects for piecewise homogeneous hemitropic com- posite bodies of arbitrary geometrical shape. We develop the boundary integral equations method to obtain the existence and uniqueness results in various H¨older (C^k,α), Sobolev–Slobodetski (W_p^s) and Besov (B^s_p,q) func- tional spaces. We study regularity properties of solutions at the crack edges and characterize the corresponding stress singularity exponents.

2. Field Equations

2.1. Constitutive relations and basic differential equations. Denote byR³the three-dimensional Euclidean space and let Ω⁺⊂R³be a bounded domain with a boundary S := ∂Ω⁺, Ω⁺ = Ω⁺∪S. Further, let Ω⁻ = R³\Ω⁺. We assume that Ω ∈ {Ω⁺,Ω⁻} is filled with an elastic material possessing the hemitropic properties.

Denote by u = (u1, u2, u3)^> and ω = (ω1, ω2, ω3)^> the displacement vector and the microrotation vector, respectively. Byϑwe denote the tem- perature increment – temperature distribution function. Here and in what follows the symbol (·)^> denotes transposition. Note that the microrotation vector in the hemitropic elasticity theory is kinematically distinct from the macrorotation vector ¹₂ curl u.

Throughout the paper the central dot denotes the real scalar product, i.e.,a·b:= P^N

k=1

akbk for complex-valuedN-dimensional vectorsa, b∈C^N. The force stress {τpq} and the couple stress {µpq} tensors in the lin- ear theory of hemitropic thermoelasticity read as follows (the constitutive equations)

τpq=τpq(U) := (µ+α)∂puq+ (µ−α)∂qup+λδpqdivu+δδpqdivω+

+(κ+ν)∂pωq+ (κ−ν)∂qωp−2α X3

k=1

εpqkωk−δpqηϑ, (2.1)

µpq=µpq(U) :=δδpqdivu+ (κ+ν) h

∂puq− X3

k=1

εpqkωk

i

+βδpqdivω+

+(κ−ν)h

∂qup− X3

k=1

εqpkωk

i

+(γ+ε)∂pωq+(γ−ε)∂qωp−δpqζϑ, (2.2) where U = (u, ω, ϑ)^>, δpq is the Kronecker delta, εpqk is the permutation (Levi–Civit´a) symbol, and α, β, γ, δ, λ, µ, ν, κ, and ε are the material constants, while η >0 and ζ >0 are constants describing the coupling of mechanical and thermal fields (see [3], [14]), ∂ = (∂1, ∂2, ∂3),∂j =∂/∂xj, j= 1,2,3.

The linear equations of dynamics of the thermoelasticity theory of hemi- tropic materials have the form (see, e.g., [14])

X3

p=1

∂pτpq(x, t) +%Fq(x, t) =%∂_tt²uq(x, t), q= 1,2,3, X3

p=1

∂pµpq(x, t)+

X3

l,r=1

εqlrτlr(x, t)+%Gq(x, t) =I∂_tt²ωq(x, t), q= 1,2,3, κ⁰∆ϑ(x, t)−η∂tdivu(x, t)−ζ∂tdivω(x, t)−κ⁰⁰∂tϑ(x, t) +Q(x, t) = 0, where t is the time variable, ∂t = ∂/∂t, F = (F1, F2, F3)^> and G = (G1, G2, G3)^> are the body force and body couple vectors per unit volume, Qis the heat source density, % is the mass density of the elastic material, and I is a constant characterizing the so called spin torque corresponding to the microrotations (i.e., the moment of inertia per unit volume); here κ⁰ = ^λ_T⁰

0 and κ⁰⁰ = _T^c0

0, where λ0 > 0 is the heat conduction coefficient, T0>0 is an initial natural state temperature andc0>0 is the specific heat coefficient.

Using the relations (2.1)–(2.2) we can rewrite the above dynamic equa- tions as

(µ+α)∆u(x, t) + (λ+µ−α) grad divu(x, t) + (κ+ν)∆ω(x, t)+

+(δ+κ−ν) grad divω(x, t) + 2αcurlω(x, t)−

−ηgradϑ(x, t) +%F(x, t) =%∂_tt²u(x, t),

(κ+ν)∆u(x, t) + (δ+κ−ν) grad divu(x, t) + 2αcurlu(x, t)+

+(γ+ε)∆ω(x, t) + (β+γ−ε) grad divω(x, t) + 4νcurlω(x, t)−

−4αω(x, t)−ζgradϑ(x, t) +%G(x, t) =I∂_tt²ω(x, t),

κ⁰∆ϑ(x, t)−η∂tdivu(x, t)−ζ∂tdivω(x, t)−κ⁰⁰∂tϑ(x, t) +Q(x, t) = 0, where ∆ is the Laplace operator.

If all the quantities involved in these equations are harmonic time de- pendent, i.e., u(x, t) =u(x)e^−itσ,ω(x, t) =ω(x)e^−itσ,ϑ(x, t) =ϑ(x)e^−itσ, F(x, t) = F(x)e^−itσ, G(x, t) = G(x)e^−itσ and Q(x, t) = Q(x)e^−itσ with σ∈Randi=√

−1, we obtain thesteady state oscillation equationsof the hemitropic theory of thermoelasticity:

(µ+α)∆u(x) + (λ+µ−α) grad divu(x) +%σ²u(x)+

+(κ+ν)∆ω(x) + (δ+κ−ν) grad divω(x) + 2αcurlω(x)−

−ηgradϑ(x) =−%F(x),

(κ+ν)∆u(x) + (δ+κ−ν) grad divu(x) + 2αcurlu(x)+

+(γ+ε)∆ω(x) + (β+γ−ε) grad divω(x) + 4νcurlω(x)−

−ζgradϑ(x) + (Iσ²−4α)ω(x) =−%G(x), (κ⁰∆ +iσκ⁰⁰)ϑ(x) +iησdivu(x) +iζσdivω(x) =−Q(x),

(2.3)

hereu,ω,F, andGare complex-valued vector functions, whileϑandQare complex-valued scalar functions, andσis a frequency parameter.

If σ = σ1+iσ2 is a complex parameter with σ2 6= 0, then the above equations are called thepseudo–oscillation equations, while forσ= 0 they represent theequilibrium equations of statics.

Let us introduce the block wise 7×7 matrix differential operator corre- sponding to the system (2.3):

L(∂, σ) :=



L⁽¹⁾(∂, σ) L⁽²⁾(∂, σ) L⁽⁵⁾(∂, σ) L⁽³⁾(∂, σ) L⁽⁴⁾(∂, σ) L⁽⁶⁾(∂, σ) L⁽⁷⁾(∂, σ) L⁽⁸⁾(∂, σ) L⁽⁹⁾(∂, σ)





7×7

, (2.4)

where

L⁽¹⁾(∂, σ) :=£

(µ+α)∆ +%σ²¤

I3+ (λ+µ−α)Q(∂), L⁽²⁾(∂, σ) =L⁽³⁾(∂, σ) := (κ+ν)∆I3+ (δ+κ−ν)Q(∂) + 2αR(∂), L⁽⁴⁾(∂, σ) := [(γ+ε)∆ + (Iσ²−4α)]I3+ (β+γ−ε)Q(∂) + 4νR(∂),

L⁽⁵⁾(∂, σ) :=−η∇^>, L⁽⁶⁾(∂, σ) :=−ζ∇^>, L⁽⁷⁾(∂, σ) :=iησ∇, L⁽⁸⁾(∂, σ) :=iζσ∇, L⁽⁹⁾(∂, σ) :=κ⁰∆ +iσκ⁰⁰.

Here and in the sequelIk stands for thek×kunit matrix and

R(∂) := [−εkjl∂l]3×3, Q(∂) := [∂k∂j]3×3, ∇:= [∂1, ∂2, ∂3]. (2.5) Throughout the paper summation over repeated indexes is meant from one to three if not otherwise stated. It is easy to see that forv= (v₁, v₂, v₃)^>

R(∂)v= curlv, Q(∂)v= grad divv, (2.6) R(−∂) =−R(∂) = [R(∂)]^>, Q(∂)R(∂) =R(∂)Q(∂) = 0,

Q(∂) = [Q(∂)]^>, [R(∂)]²=Q(∂)−∆I3, [Q(∂)]²=Q(∂)∆.

Due to the above notation, the system (2.3) can be rewritten in matrix form as

L(∂, σ)U(x) = Φ(x), U = (u, ω, ϑ)^>, Φ = (−%F,−%G,−Q)^>. Note that L(∂, σ) is not formally self-adjoint. Further, let us remark that the differential operator

L(∂) :=L(∂,0) (2.7)

corresponds to the static equilibrium case, while the formally self-adjoint differential operator

L0(∂) :=





L⁽¹⁾₀ (∂) L⁽²⁾₀ (∂) [0]3×1

L⁽³⁾₀ (∂) L⁽⁴⁾₀ (∂) [0]3×1

[0]_1×3 [0]_1×3 κ⁰∆





7×7

(2.8) with

L⁽¹⁾₀ (∂) := (µ+α)∆I3+ (λ+µ−α)Q(∂), L⁽²⁾₀ (∂) =L⁽³⁾₀ (∂) := (κ+ν)∆I3+ (δ+κ−ν)Q(∂), L⁽⁴⁾₀ (∂) := (γ+ε)∆I3+ (β+γ−ε)Q(∂),

represents the principal homogeneous part of the operators (2.4) and (2.7).

Denote

L(∂, σ) :=e

·L⁽¹⁾(∂, σ) L⁽²⁾(∂, σ) L⁽³⁾(∂, σ) L⁽⁴⁾(∂, σ)

¸

6×6

, Le0(∂) :=

"

L⁽¹⁾₀ (∂) L⁽²⁾₀ (∂) L⁽³⁾₀ (∂) L⁽⁴⁾₀ (∂)

#

6×6

.

(2.9)

The operators (2.9) correspond to the equations of hemitropic elasticity when thermal effects are not taken into consideration ([40]). It is clear that the operatorL0(∂),L(∂, σ) ande Le0(∂) are formally self-adjoint.

2.2. Generalized stress operators. The components of the force stress vector τ⁽ⁿ⁾ and the couple stress vector µ⁽ⁿ⁾, acting on a surface element with a unite normal vectorn= (n1, n2, n3), read as

τ⁽ⁿ⁾=¡

τ₁⁽ⁿ⁾, τ₂⁽ⁿ⁾, τ₃⁽ⁿ⁾¢_>

, µ⁽ⁿ⁾=¡

µ⁽ⁿ⁾₁ , µ⁽ⁿ⁾₂ , µ⁽ⁿ⁾₃ ¢_>

, where

τ_q⁽ⁿ⁾= X3

p=1

τpqnp, µ⁽ⁿ⁾_q = X3

p=1

µpqnp, q= 1,2,3.

It is also well known that the normal component of the heat flux vector across a surface element with a normal vectorn= (n1, n2, n3) is expressed with the help of the normal derivative of the temperature function

κ⁰n· ∇ϑ=κ⁰ X3

p=1

np∂pϑ=κ⁰∂nϑ,

where∂n=∂/∂ndenotes the usual normal derivative.

Throughout the paper we will refer the six vector (τ⁽ⁿ⁾, µ⁽ⁿ⁾)^> as the mechanical thermo-stress vector, while the seven vector (τ⁽ⁿ⁾, µ⁽ⁿ⁾, κ⁰∂nϑ)^>

asthe generalized thermo-stress vector.

Let us introduce the generalized thermo-stress operators T(∂, n) =

·T⁽¹⁾(∂, n) T⁽²⁾(∂, n) −ηn^>

T⁽³⁾(∂, n) T⁽⁴⁾(∂, n) −ζn^>

¸

6×7

, (2.10)

P(∂, n) =



T⁽¹⁾(∂, n) T⁽²⁾(∂, n) −ηn^>

T⁽³⁾(∂, n) T⁽⁴⁾(∂, n) −ζn^>

[0]1×3 [0]1×3 κ⁰∂n





7×7

, (2.11)

where

T^(j)= [T_pq^(j)]3×3, j= 1,4, n^>= (n1, n2, n3)^>, T_pq⁽¹⁾(∂, n) = (µ+α)δpq∂n+ (µ−α)nq∂p+λnp∂q, T_pq⁽²⁾(∂, n) = (κ+ν)δpq∂n+ (κ−ν)nq∂p+δnp∂q−2α

X3

k=1

εpqknk, T_pq⁽³⁾(∂, n) = (κ+ν)δpq∂n+ (κ−ν)nq∂p+δnp∂q,

T_pq⁽⁴⁾(∂, n) = (γ+ε)δpq∂n+ (γ−ε)nq∂p+βnp∂q−2ν X3

k=1

εpqknk.

One can easily check that for an arbitrary vectorU = (u, ω, ϑ)^>, T(∂, n)U =¡

τ⁽ⁿ⁾, µ⁽ⁿ⁾¢_>

, P(∂, n)U =¡

τ⁽ⁿ⁾, µ⁽ⁿ⁾, κ⁰∂nϑ¢_>

, i.e., the six vector T(∂, n)U corresponds to the mechanical thermo-stress vector and the seven vectorP(∂, n)Ucorresponds to the generalized thermo- stress vector.

Further, let us introduce the boundary differential operators which occur in Green’s formulas and are associated with the adjoint differential operator L^∗(∂, σ) :=L^>(−∂, σ):

T^∗(∂, n) =

·T⁽¹⁾(∂, n) T⁽²⁾(∂, n) −iσηn^>

T⁽³⁾(∂, n) T⁽⁴⁾(∂, n) −iσζn^>

¸

6×7

,

P^∗(∂, n) =



T⁽¹⁾(∂, n) T⁽²⁾(∂, n) −iσηn^>

T⁽³⁾(∂, n) T⁽⁴⁾(∂, n) −iσζn^>

[0]1×3 [0]1×3 κ⁰∂n





7×7

.

(2.12)

It is easy to see that the principal homogeneous parts of the operators T(∂, n) and T^∗(∂, n) are the same, as well as the principal homogeneous parts of the operatorsP(∂, n) andP^∗(∂, n).

Note that when the thermal effects are not taken into consideration the hemitropic stress operator reads as [40]

T(∂, n) =

·T⁽¹⁾(∂, n) T⁽²⁾(∂, n) T⁽³⁾(∂, n) T⁽⁴⁾(∂, n)

¸

6×6

. (2.13)

Evidently, forU = (u, ω,0)^>andUe = (u, ω)^>we haveT(∂, n)U =T(∂, n)Ue in view of (2.10) and (2.13).

2.3. Green’s identities. For vector functions

Ue = (u, ω)^>,Ue⁰ = (u⁰, ω⁰)^> ∈[C²(Ω⁺)]⁶, we have the following Green formula [40]

Z

Ω⁺

hUe⁰·L(∂,e 0)Ue+E(Ue⁰,Ue)i dx=

Z

∂Ω⁺

{Ue⁰}⁺· {T(∂, n)Ue}⁺dS, (2.14)

where the operatorsL(∂,e 0) andT(∂, n) are given by (2.9) and (2.13) respec- tively, ∂Ω⁺ is a piecewise smooth manifold, n is the outward unit normal vector to∂Ω⁺, the symbols{ · }^± denote the limiting values on∂Ω^± from Ω^± respectively,E(·,·) is the so calledenergy bilinear form,

E(Ue⁰,Ue) =E(U ,e Ue⁰) = X3

p,q=1

n

(µ+α)u⁰_pqupq+ (µ−α)u⁰_pquqp+ + (κ+ν)(u⁰_pqωpq+ω_pq⁰ upq)+(κ−ν)(u⁰_pqωqp+ω_pq⁰ uqp)+(γ+ε)ω_pq⁰ ωpq+

+ (γ−ε)ω⁰_pqωqp+δ(u⁰_ppωqq+ω⁰_qqupp) +λu⁰_ppuqq+βω_pp⁰ ωqq

o

(2.15) with

u_pq=∂_pu_q− X3

k=1

ε_pqkω_k, ω_pq=∂_pω_q, p, q= 1,2,3. (2.16) In what follows the over bar denotes complex conjugation. The necessary and sufficient conditions for the quadratic formE(U ,e Ue) to be positive def- inite with respect to the variablesupq andωpq, read as (see [4], [14], [18])

µ >0, α >0, γ >0, ε >0, λ+ 2µ >0, µγ−κ²>0, αε−ν²>0, (λ+µ)(β+γ)−(δ+κ)²>0, (3λ+ 2µ)(3β+ 2γ)−(3δ+ 2κ)²>0,

(µ+α)(γ+ε)−(κ+ν)²>0, (λ+ 2µ)(β+ 2γ)−(δ+ 2κ)²>0, µ£

(λ+µ)(β+γ)−(δ+κ)²¤

+ (λ+µ)(µγ−κ²)>0, µ£

(3λ+ 2µ)(3β+ 2γ)−(3δ+ 2κ)²¤

+ (3λ+ 2µ)(µγ−κ²)>0.

Let us note that, if the condition 3λ+ 2µ > 0 is fulfilled, which is very natural in the classical elasticity, then the above conditions are equivalent

to the following simultaneous inequalities

µ >0, α >0, γ >0, ε >0, 3λ+ 2µ >0, µγ−κ²>0, αε−ν²>0, (µ+α)(γ+ε)−(κ+ν)²>0,

(3λ+ 2µ)(3β+ 2γ)−(3δ+ 2κ)²>0.

(2.17)

For simplicity in what follows we assume that 3λ+ 2µ >0 and therefore the conditions (2.17) imply positive definiteness of the energy quadratic form E(U ,e Ue) defined by (2.15). From (2.17) it follows that

γ >0, ε >0, λ+µ >0, β+γ >0, d1:= (µ+α)(γ+ε)−(κ+ν)²>0, d2:= (λ+ 2µ)(β+ 2γ)−(δ+ 2κ)²>0.

Formula (2.15) can be rewritten as E(U ,e Ue⁰) =3λ+2µ

3

³

divu+3δ+2κ 3λ+2µ divω

´³

divu⁰+3δ+2κ 3λ+2µ divω⁰

´ + +1

3

³

3β+ 2γ−(3δ+2κ)² 3λ+2µ

´

(divω)(divω⁰)+

+

³ ε−ν²

α

´

curlω·curlω⁰+ +µ

2 X3

k,j=1, k6=j

·∂uk

∂xj + ∂uj

∂xk +κ µ

³∂ωk

∂xj +∂ωj

∂xk

´¸

×

×

·∂u⁰_k

∂xj

+∂u⁰_j

∂xk

+κ µ

³∂ω⁰_k

∂xj

+∂ω_j⁰

∂xk

´¸

+ +µ

3 X3

k,j=1

·∂uk

∂xk −∂uj

∂xj +κ µ

³∂ωk

∂xk −∂ωj

∂xj

´¸

×

×

·∂u⁰_k

∂xk

−∂u⁰_j

∂xj

+κ µ

³∂ω⁰_k

∂xk

−∂ω_j⁰

∂xj

´¸

+ +

³ γ−κ²

µ

´ X³

k,j=1, k6=j

·1 2

³∂ωk

∂xj +∂ωj

∂xk

´³∂ω_k⁰

∂xj +∂ω⁰_j

∂xk

´ +

+1 3

³∂ωk

∂xk

−∂ωj

∂xj

´³∂ω⁰_k

∂xk

−∂ω_j⁰

∂xj

´¸

+ +α

³

curlu+ν

αcurlω−2ω

´

·

³

curlu⁰+ν

αcurlω⁰−2ω⁰

´ . In particular,

E(U ,e Ue) =3λ+ 2µ 3

¯¯

¯divu+3δ+ 2κ 3λ+ 2µ divω

¯¯

¯²+ +1

3

³

3β+ 2γ−(3δ+ 2κ)² 3λ+ 2µ

´

|divω|²+

+µ 2

X3

k,j=1, k6=j

¯¯

¯¯∂uk

∂xj +∂uj

∂xk +κ µ

³∂ωk

∂xj +∂ωj

∂xk

´¯¯¯

¯

2

+

+µ 3

X3

k,j=1

¯¯

¯¯∂uk

∂xk −∂uj

∂xj +κ µ

³∂ωk

∂xk −∂ωj

∂xj

´¯¯¯

¯

2

+

+³ γ−κ²

µ

´ X³

k,j=1, k6=j

·1 2

¯¯

¯∂ωk

∂xj

+∂ωj

∂xk

¯¯

¯²+1 3

¯¯

¯∂ωk

∂xk

−∂ωj

∂xj

¯¯

¯²

¸ +

+³ ε−ν²

α

´

|curlω|²+α

¯¯

¯curlu+ν

αcurlω−2ω

¯¯

¯². We formulate here the following technical lemma.

Lemma 2.1. Let Ue = (u, ω)^> ∈ [C¹(Ω⁺)]⁶ and E(U ,e Ue) = 0 in Ω⁺. Then

u(x) = [a×x] +b, ω(x) =a, x∈Ω⁺, (2.18) whereaandb are arbitrary three-dimensional constant complex vectors.

Moreover,

(i) for an arbitrary vector Ue = (u, ω)^> defined by formulas (2.18)and an arbitrary unit vectorn= (n1, n2, n3)the generalized hemitropic stress vector T(∂, n)Ue vanishes identically, i.e., T(∂, n)Ue(x) = 0 for allx∈Ω⁺.

(ii) for an arbitrary vector U := (U ,e 0)^> = (u, ω,0)^>, where u and ω are given by formulas(2.18), and for an arbitrary unit vectorn= (n1, n2, n3)the generalized hemitropic thermo-stress vectorP(∂, n)U vanishes identically, i.e.,P(∂, n)U(x) = 0 for allx∈Ω⁺.

Proof. The first part of the lemma is shown in [40]. The second part easily follows from the first part and from the formulas (2.10), (2.11), (2.13). ¤ Throughout the paper Lp,W_p^s, H_p^s, andB_p,q^s (withs∈R, 1 < p <∞, 1≤q≤ ∞) denote the well–known Lebesgue, Sobolev–Slobodetski, Bessel potential, and Besov spaces, respectively (see, e.g., [54], [55], [31]). We recall thatH₂^s=W₂^s=B_2,2^s ,W_p^t=B_p,p^t , andH_p^k =W_p^k, for any s∈R, for any positive and non-integert, and for any non-negative integerk.

Further, letM0be a Lipschitz surface without boundary. For a Lipschitz sub-manifold M ⊂ M0 we denote byHe_p^s(M) and Be_p,q^s (M) the subspaces ofH_p^s(M0) andB^s_p,q(M0), respectively,

He_p^s(M) = n

g: g∈H_p^s(M0), suppg⊂ M o

, Be_p,q^s (M) =

n

g: g∈B^s_p,q(M0), suppg⊂ M o

,

whileH_p^s(M) andB_p,q^s (M) denote the spaces of restrictions onMof func- tions fromH_p^s(M0) andB^s_p,q(M0), respectively,

H_p^s(M) =©

r_Mf : f ∈H_p^s(M0)ª

, B_p,q^s (M) =©

r_Mf : f ∈B^s_p,q(M0)ª .

Herer_M is the restriction operator.

IfUe =Ue⁽¹⁾+iUe⁽²⁾is a complex–valued vector, whereUe^(j)= (u^(j), ω^(j))^>

(j= 1,2) are real–valued vectors, then

E(U ,e Ue) =E(Ue⁽¹⁾,Ue⁽¹⁾) +E(Ue⁽²⁾,Ue⁽²⁾),

and, due to the positive definiteness of the energy form for real–valued vector functions, we have

E(U ,e U)e ≥c^∗ X3

p,q=1

h

(u⁽¹⁾_pq)²+ (u⁽²⁾_pq)²+ (ω⁽¹⁾_pq)²+ (ω_pq⁽²⁾)²i ,

where c^∗ is a positive constant depending only on the material constants, and u^(j)pq and ω^(j)pq are defined by formulae (2.16) with u^(j) and ω^(j) for u andω.

From the positive definiteness of the energy formE(·,·) with respect to the variables (2.16) it follows that there exist positive constants c1 and c2

such that for an arbitrary real–valued vectorUe ∈[C¹(Ω⁺)]⁶ B(e U ,e Ue) :=

Z

Ω⁺

E(U ,e Ue)dx≥

≥c1

Z

Ω⁺

½ X3

p,q=1

£(∂puq)²+ (∂pωq)²¤ +

X3

p=1

[u²_p+ω_p²]o dx−

−c2

Z

Ω⁺

X3

p=1

[u²_p+ω_p²]dx,

i.e., the following Korn’s type inequality holds (cf. [17, Part I, §12], [32, Ch. 10])

B(eU ,e Ue)≥c1kUek²_[H1

2(Ω⁺)]⁶−c2kUek²_[H0

2(Ω⁺)]⁶, (2.19) where k · k_[H^s

2(Ω⁺)]⁶ denotes the norm in the Sobolev space [H₂^s(Ω⁺)]⁶. Clearly, the counterpart of (2.19) holds for an arbitrary complex-valued vectorUe ∈[H₂¹(Ω⁺)]⁶ as well,

B(eU ,e Ue)≥c₁kU|e ²_[H1

2(Ω⁺)]⁶−c₂kUke ²_[H0

2(Ω⁺)]⁶. (2.20) These results imply that the differential operatorsL(∂, σ) ande Le0(∂) are strongly ellipticand the following inequality (the accretivity condition) holds (cf., e.g., [17, Part I,§5], [32, Ch. 4, Lemma 4.5])

c⁰₂|ξ|²|η|²≥Le0(ξ)η·η= X6

k,j=1

Le0kj(ξ)ηjηk ≥c⁰₁|ξ|²|η|² (2.21) with some constants c⁰_k > 0, k = 1,2, for arbitrary ξ ∈ R³ and arbitrary complex vectorη ∈C⁶.

Consequently, in view of (2.8) and (2.21) the differential operatorL(∂, σ) is strongly elliptic as well, since

C₂⁰|ξ|²|η|²≥L₀(ξ)η·η= X6

k,j=1

L_0kj(ξ)η_jη_k ≥C₁⁰|ξ|²|η|²

with some constantsC_k⁰ >0,k= 1,2,for arbitraryξ∈R³and for arbitrary complex vectorη ∈C⁷.

Now let U = (U , ϑ)e ^> = (u, ω, ϑ)^> andU⁰ = (Ue⁰, ϑ⁰)^> = (u⁰, ω⁰, ϑ⁰)^> be vector functions of the class [C²(Ω⁺)]⁷. With the help of relation (2.14) and standard manipulations we can show that the following Green’s formulas

hold Z

Ω⁺

U⁰·L(∂, σ)U dx= Z

∂Ω⁺

{U⁰}⁺·©

P(∂, n)Uª₊ dS−

− Z

Ω⁺

h

E(Ue⁰,U)e −%σ²u⁰·u− Iσ²ω⁰·ω−ηϑdivu⁰−ζϑdivω⁰−

−iησϑ⁰divu−iζσϑ⁰divω−iσκ⁰⁰ϑϑ⁰+κ⁰gradϑ⁰·gradϑ i

dx, (2.22) Z

Ω⁺

h

U⁰·L(∂, σ)U−L^∗(∂, σ)U⁰·U i

dx=

= Z

∂Ω⁺

h

{U⁰}⁺·©

P(∂, n)Uª₊

−©

P^∗(∂, n)U⁰ª₊

· {U}⁺ i

dS, (2.23) where L^∗(∂, σ) = L^>(−∂, σ) is the operator formally adjoint to L(∂, σ), the differential operatorsL(∂, σ),P(∂, n) andP^∗(∂, n) are defined by (2.4), (2.11) and (2.12) respectively. The proof of (2.22) and (2.23) easily follows from (2.14) in view of the identity

U⁰·L(∂, σ)U−Ue⁰·L(∂,e 0)Ue =%σ²u⁰·u−ηgradϑ·u⁰+Iσ²ω⁰·ω−

−ζgradϑ·ω⁰+κ⁰ϑ⁰∆ϑ+iησϑ⁰divu+iσζϑ⁰divω+iσκ⁰⁰ϑϑ⁰. By the standard limiting approach, Green’s formula (2.22) can be extended to Lipschitz domains (see, e.g., [45], [32]) and to the case of complex–valued vector functionsU ∈[W_p¹(Ω⁺)]⁷ andU⁰∈[W_p¹0(Ω⁺)]⁷ with 1/p+ 1/p⁰ = 1, 1< p <∞, andL(∂, σ)U ∈[Lp(Ω⁺)]⁷ (cf. [31], [10], [32])

D

{U⁰}⁺,©

P(∂, n)Uª+E

∂Ω⁺= Z

Ω⁺

U⁰·L(∂, σ)U dx+

+ Z

Ω⁺

h

E(Ue⁰,Ue)−%σ²u⁰·u− Iσ²ω⁰·ω−ηϑdivu⁰−ζϑdivω⁰−

−iησϑ⁰divu−iζσϑ⁰divω−iσκ⁰⁰ϑϑ⁰+κ⁰gradϑ⁰·gradϑ i

dx, (2.24)

where h ·,· i∂Ω⁺ denotes the duality between the spaces [Bp,p^1p (∂Ω⁺)]⁷ and [B_p⁻0,p^p10(∂Ω⁺)]⁷, which extends the usual real L2-scalar product, i.e., for f, g∈[L2(S)]⁷

hf, giS = X7

k=1

Z

S

fkgkdS= (f, g)[L2(S)]⁷.

Clearly, the generalized trace functional {P(∂, n)U}⁺ ∈ [Bp,p^−1p(∂Ω⁺)]⁷ is well defined by the relation (2.24).

Let us introduce the sesquilinear form related to the operatorL(∂, σ) B(U, U⁰) :=

Z

Ω⁺

h

E(U ,e Ue⁰)−%σ²u·u⁰−Iσ²ω·ω⁰−ηϑdivu⁰−ζϑdivω⁰−

−iησϑ⁰divu−iζσϑ⁰divω−iσκ⁰⁰ϑϑ⁰+κ⁰gradϑ·gradϑ⁰ i

dx. (2.25) With the help of (2.20) and (2.25) we derive the inequality

ReB(U, U)≥C1kUk²_[H1

2(Ω⁺)]⁷−C2kUk²_[H0

2(Ω⁺)]⁷, (2.26) with some positive constantsC1andC2. This inequality plays a crucial role in the study of boundary value problems of the micropolar elasticity theory for hemitropic continua by means of the variational methods based on the well known Lax–Milgram theorem.

3. Formulation of Transmission Problems and Uniqueness Theorems

Let Ω be a bounded region in R³ with the smooth connected boundary

∂Ω =S0. Let Ω1⊂Ω be a sub-domain of Ω with a smooth simply connected boundary ∂Ω1 = S1 ⊂ Ω. Put Ω0 := Ω\Ω1. In what follows, by n(z), z∈S0∪S1, we denote the outward unit normal vector with respect to the domains Ω₁ and Ω, at the pointz. We assume thatS_`∈C^2,γ0, 0< γ⁰ ≤1,

`= 0,1,if not otherwise stated. Let the domains Ω` be filled up by elastic continua heaving different hemitropic material constants, α^{(`)</sup>, β<sup>(`)}, γ^{(`)</sup>, δ<sup>(`)}, λ^{(`)</sup>, µ<sup>(`)}, ν^{(`)</sup>, κ<sup>(`)} and ε<sup>(`)</sup>, `= 0,1; η^{(`)</sup> >0 andζ<sup>(`)}>0,`= 0,1, are constants describing the coupling of mechanical and thermal fields in Ω`

(see [3], [14]),∂= (∂1, ∂2, ∂3),∂j=∂/∂xj,j= 1,2,3.

Analogously, for the mechanical characteristics, e.g., the displacement and microrotation vectors, the force stress and couple stress vectors, and also for the differential operators, fundamental matrices and potentials re- lated to the hemitropic material occupying the domain Ω`, ` = 0,1, we also employ the superscript (`). In particular, u<sup>(`)</sup> = (u^{(`)</sup><sub>1</sub> , u<sup>(`)}₂ , u^{(`)</sup><sub>3</sub> )<sup>T</sup>, ω<sup>(`)}= (ω₁^{(`)</sup>, ω<sup>(`)}₂ , ω₃^{(`)</sup>)<sup>T</sup> andϑ<sup>(`)}denote the displacement and microrotation vectors and temperature function in the domain Ω`;E<sup>(`)</sup>(U^{(`)</sup>, U<sup>(`)}) desig- nates the appropriate potential energy density,L^{(`)</sup>(∂, σ),L<sup>(`)}(∂),L^(`)₀ (∂),

P^{(`)</sup>(∂, n) andP<sub>0</sub><sup>(`)}(∂, n) are the corresponding differential operators given by the formulae (2.4), (2.7), (2.8), (2.5) and (2.6).

In what follows we treat transmission problems for the differential equa- tions of pseudo-oscillations, i.e., we assume that

σ=σ1+iσ2 with σ2>0. (3.1) It is clear that the nonhomogeneous differential equationL^{(`)</sup>(∂, σ)U<sup>(`)} = Ψ^{(`)</sup> in Ω` we can reduce to the homogeneous one,L^{(`)</sup>(∂, σ)V<sup>(`)}= 0, with the help of the volume Newtonian potential NΩ`(Ψ<sup>(`)}) (see Appendix A).

Therefore, without loss of generality we can assume that the body force and body couple vectors absent.

We will study the following boundary-transmission problems:

Find regular complex-valued vector-functionsU<sup>(`)</sup>∈[C<sup>1</sup>(Ω`)]⁷∩[C²(Ω`)]⁷,

`= 0,1,satisfying the differential equations

L^{(`)</sup>(∂, σ)U<sup>(`)}(x) = 0 in Ω`, `= 0,1, (3.2) the transmission conditions onS1

{U⁽¹⁾(z)}⁺− {U⁽⁰⁾(z)}⁻ =f(z) on S1, (3.3)

©P⁽¹⁾(∂, n)U⁽¹⁾(z)ª₊

−©

P⁽⁰⁾(∂, n)U⁽⁰⁾(z)ª₋

=F(z) o S1, (3.4) and either the Dirichlet boundary condition onS0

{U⁽⁰⁾(z)}⁺=f^(D)(z) n S₀, (3.5) or the Neumann boundary condition onS0

©P⁽⁰⁾(∂, n)U⁽⁰⁾(z)ª₊

=F^(N)(z) on S0. (3.6) We assume that the given transmission and boundary data are complex- valued vectors and

f ∈[C^1,β0(S0)]⁷, F ∈[C^0,β0(S0)]⁷, f^(D)∈[C^1,β0(S1)]⁷, F^(N)∈[C^0,β0(S1)]⁷,

with 0< β⁰ < γ⁰ ≤1.We refer to the boundary-transmission problem (3.2)–

(3.5) as Problem (TD) and the boundary-transmission problem (3.2)–(3.4) and (3.6) as Problem (TN).

The above problem setting is aclassicalone in the space of continuously differentiable vector-functions.

In the case of a weak setting of the problems we look for a solution pair (U⁽⁰⁾, U⁽¹⁾) in the Sobolev spaces, U^{(`)</sup> ∈ [W<sub>p</sub><sup>1</sup>(Ω`)]⁷, ` = 0,1, with L<sup>(`)}(∂, σ)U<sup>(`)</sup>∈[Lp(Ω`)]⁷. Therefore, equations (3.2) are understood in the distributional sense. However, we remark that solutions to these homoge- neous equations actually are analytical vector-functions of the real spatial variablexin the open domains Ω0 and Ω1, since the differential operators L^(`)(∂, σ) are strongly elliptic.

The Dirichlet type boundary and transmission conditions are understood in the usual trace sense, while the Neumann type conditions are understood in the generalized trace sense defined by Green’s identity (2.24) (for details see [37], [42]).

We start with the study of uniqueness of solutions to these problems.

Theorem 3.1. Problems(TD)and(TN)may have at most one solution in the space of regular vector-functions.

Proof. Due to linearity of the problems under consideration, it suffices to show that the corresponding homogeneous problems have only the trivial solutions. Let a pair of regular vectors

(U⁽⁰⁾, U⁽¹⁾)∈¡

[C¹(Ω₀)]⁷∩[C²(Ω₀)]⁷¢

ס

[C¹(Ω₁)]⁷∩[C²(Ω₁)]⁷¢ be a solution of either the homogeneous Problem (TD) or Problem (TN).

Using Green’s formulae for the vector-functions U⁽⁰⁾ and U⁽¹⁾ and taking into account the chosen direction of the normal vector on the boundaries S0 andS1, we get

Z

Ω1

h

−E⁽¹⁾¡Ue⁽¹⁾,Ue⁽¹⁾¢

+%1σ²|u⁽¹⁾|²+I1σ²|ω⁽¹⁾|²−C0κ⁰₁|∇ϑ⁽¹⁾|²−κ⁰⁰₁|ϑ⁽¹⁾|²i dx+

+ Z

S1

n

T⁽¹⁾(∂, n)U⁽¹⁾·Ue⁽¹⁾+C0κ⁰₁ϑ⁽¹⁾∂nϑ⁽¹⁾ o₊

dS= 0, (3.7) Z

Ω0

h

−E⁽⁰⁾¡Ue⁽⁰⁾,Ue⁽⁰⁾¢

+%0σ²|u⁽⁰⁾|²+I0σ²|ω⁽⁰⁾|²−C0κ⁰₀|∇ϑ⁽⁰⁾|²−κ⁰⁰₀|ϑ⁽⁰⁾|² i

dx+

+ Z

S0

n

T⁽⁰⁾(∂, n)U⁽⁰⁾·Ue⁽⁰⁾+C0κ⁰₀ϑ⁽⁰⁾∂nϑ⁽⁰⁾ o₊

dS−

− Z

S1

n

T⁽⁰⁾(∂, n)U⁽⁰⁾·Ue⁽⁰⁾+C0κ⁰₀ϑ⁽⁰⁾∂nϑ⁽⁰⁾ o₋

dS= 0, (3.8)

where C0=−i

σ, κ⁰<sub>`</sub>= λ<sup>(`)</sup>₀

T₀<sup>(`)</sup>, κ<sup>00</sup><sub>`</sub> = c^(`)₀

T₀^{(`)</sup>, Ue<sup>(`)}= (u^{(`)</sup>, ω<sup>(`)})^>, `= 0,1.

The homogeneous boundary and transmission conditions, f^{(`)</sup> =F<sup>(`)} = 0, yield

X1

`=0

Z

Ω`

h

E^{(`)</sup>¡Ue<sup>(`)},Ue^(`)¢

−%`σ<sup>2</sup>|u<sup>(`)</sup>|²− I`σ<sup>2</sup>|ω<sup>(`)</sup>|²+ +C0κ⁰<sub>`</sub>|∇ϑ<sup>(`)</sup>|²+κ⁰⁰<sub>`</sub>|ϑ<sup>(`)</sup>|²

i

dx= 0. (3.9)