Memoirs on Diﬀerential Equations and Mathematical Physics Volume 39, 2006, 69–103

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Volume 39, 2006, 69–103

R. Gachechiladze, J. Gwinner, and D. Natroshvili

A BOUNDARY VARIATIONAL INEQUALITY APPROACH TO UNILATERAL CONTACT WITH HEMITROPIC MATERIALS

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Abstract. We study unilateral frictionless contact problems with hemitropic materials in the theory of linear elasticity. We model these problems as unilateral (Signorini type) boundary value problems, give their variational formulation as spatial variational inequalities, and transform them to boundary variational inequalities with the help of the potential method for hemitropic materials. Using the self-adjointness of the Steklov–Poincar´e operator, we obtain the equivalence of the boundary variational inequality formulation and the corresponding minimization problem. Based on our variational inequality approach we derive existence and uniqueness theorems. Our investigation includes the special particular case of only traction- contact boundary conditions without prescribing the displacement and microrotation vectors along some part of the boundary of the hemitropic elastic body.

2000 Mathematics Subject Classification: 35J85, 74A35, 49J40.

Key words and phrases: Elasticity theory, hemitropic material, boundary variational inequalities, potential method, unilateral problems.

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1. Introduction

In recent years continuum mechanical theories in which the deformation is described not only by the usual displacement vector field, but by other scalar, vector or tensor fields as well, have been the object of intensive re- search. Classical elasticity associates only the three translational degrees of freedom to material points of the body and all the mechanical characteristics are expressed by the corresponding displacement vector. On the contrary, micropolar theory, by including intrinsic rotations of the particles, provides a rather complex model of an elastic body that can support body forces and body couple vectors as well as force stress vectors and couple stress vectors at the surface. Consequently, in this case all the mechanical quantities are written in terms of the displacement and microrotation vectors.

The origin of the rational theories of polar continua goes back to brothers E. and F. Cosserat [6] who gave a development of the mechanics of contin- uous media in which each material point has the six degrees of freedom of a rigid body (for the history of the problem see [28], [33], [22], [9], and the references therein).

A micropolar solid which is not isotropic with respect to inversion is called hemitropic, noncentrosymmetric, orchiral. Materials may exhibit chirality on the atomic scale, as in quartz and in biological molecules, as well as on a large scale, as in composites with helical or screw-shaped inclusions.

Mathematical models describing the hemitropic properties of elastic materials have been proposed by Aero and Kuvshinski [1], [2]. We note that the governing equations in this model become very involved and generate 6×6 matrix partial differential operator of second order. Evidently, the corresponding 6×6 matrix boundary differential operators describing the force stress and couple stress vectors have also an involved structure in comparison with the classical case.

In [29], [30], [31], the fundamental matrices of the associated systems of partial differential equations of statics and steady state oscillations have been constructed explicitly in terms of elementary functions and the basic boundary value problems of hemitropic elasticity have been studied by the potential method for smooth and non-smooth Lipschitz domains.

Particular problems of the elasticity theory of hemitropic continuum have been considered in [10], [23], [24], [25], [26], [33], [34], [35], [39] (see also [3], [4] and the references therein for electromagnetic scattering by a homogeneous chiral obstacle).

The main goal of the present paper is the study of unilateral frictionless contact problems for hemitropic elastic material, their mathematical mod- elling as unilateral boundary value problems of Signorini type and their analysis with the help of the spatial and boundary variational inequality technique.

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R. Gachechiladze, J. Gwinner, D. Natroshvili

In classical elasticity similar problems have been considered in many monographs and papers (see, e.g., [8], [11], [12], [13], [15], [16], [17], [18], [19], [20], [21], [36], and the references therein).

Due to the complexity of the physical model of a hemitropic continuum we need a special mathematical interpretation of the unilateral mechanical constraints related to the microrotation vector and the couple stress vector.

We note that the displacement vector and the force stress vector are subject to the usual classical unilateral conditions. However, we have to pay special attention to the fact that the force stress and couple stress vectors depend on the displacement and microrotation vectors.

Here in this paper we can present a reasonable mathematical model for the unilateral constraints that apply to hemitropic material in contact. We transform the unilateral boundary value problem to the corresponding spatial variational inequality (SVI) equivalently. Furthermore by boundary integral techniques, we can reduce the SVI to an equivalent boundary variational inequality (BVI). Applying the potential method we establish some coercivity properties of the boundary bilinear forms involved in the BVI and thus prove uniqueness and existence for the original unilateral problems.

2. Basic Equations and Green Formulae

2.1. Field equations. Let Ω⁺⊂R³ be a bounded domain with a smooth connected boundary S := ∂Ω⁺, Ω⁺ = Ω⁺∪S; Ω⁻ = R³\Ω⁺. We assume that Ω∈ {Ω⁺,Ω⁻}is occupied by an elastic material possessing the hemitropic properties.

Denote byu= (u1, u2, u3)^> andω= (ω1, ω2, ω3)^> the displacement vector and the microrotation vector, respectively; here and in what follows the symbol (·)^> denotes transposition. Note that the microrotation vector in the hemitropic elasticity theory is kinematically distinct from the macroro- tation vector ¹₂ curl u.

In the linear theory of hemitropic elasticity we have the following con- stitutive equations for the force stress tensor {τpq} and the couple stress tensor{µpq}

τpq=τpq(U) : = (µ+α)∂uq

∂xp + (µ−α)∂up

∂xq +λδpqdivu+ +δ δpqdivω+ (κ+ν)∂ωq

∂xp + (κ−ν)∂ωp

∂xq −

−2α X3 k=1

εpqkωk, (2.1)

µpq=µpq(U) : =δ δpqdivu+ (κ+ν)h∂uq

∂xp

− X3 k=1

εpqkωk

i+

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+βδpqdivω+ (κ−ν)h∂up

∂xq

− X3 k=1

εqpkωk

i+

+(γ+ε)∂ωq

∂xp

+ (γ−ε)∂ωp

∂xq

, (2.2)

where U = (u, ω)^>, 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 (see [1]).

The components of the force stress vector τ⁽ⁿ⁾ = (τ₁⁽ⁿ⁾, τ₂⁽ⁿ⁾, τ₃⁽ⁿ⁾)^> and the coupled stress vectorµ⁽ⁿ⁾= (µ⁽ⁿ⁾₁ , µ⁽ⁿ⁾₂ , µ⁽ⁿ⁾₃ )^>, acting on a surface ele- ment with the normal vectorn= (n1, n2, n3), read as

τ_q⁽ⁿ⁾= X3 p=1

τpqnp, µ⁽ⁿ⁾_q = X3 p=1

µpqnp, q= 1,2,3. (2.3) Let us introduce the generalized stress operator (6×6 matrix differential operator)

T(∂, n) =

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

6×6

, T^(j)= T_pq^(j)

3×3, j= 1,4, (2.4) where∂= (∂1, ∂2, ∂3),∂j=∂/∂xj,

T_pq⁽¹⁾(∂, n) = (µ+α)δpq

∂

∂n+ (µ−α)nq

∂

∂xp

+λnp

∂

∂xq

, Tpq⁽²⁾(∂, n) = (κ+ν)δpq

∂

∂n+ (κ−ν)nq

∂

∂xp +δnp

∂

∂xq−

−2α X3 k=1

εpqknk, T_pq⁽³⁾(∂, n) = (κ+ν)δpq ∂

∂n+ (κ−ν)nq ∂

∂xp

+δnp ∂

∂xq

, T_pq⁽⁴⁾(∂, n) = (γ+ε)δpq ∂

∂n+ (γ−ε)nq ∂

∂xp

+βnp ∂

∂xq

−

−2ν X3 k=1

εpqknk.

(2.5)

In view of (2.1), (2.2), and (2.3) it can easily be checked that (τ⁽ⁿ⁾, µ⁽ⁿ⁾)^>=T(∂, n)U.

The static equilibrium equations for hemitropic elastic bodies are written as

X3 p=1

∂pτpq(x) +%Fq(x) = 0,

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X3 p=1

∂pµpq(x) + X3 l,r=1

εqlrτlr(x) +% Gq(x) = 0, q= 1,2,3,

where F = (F1, F2, F3)^> andG= (G1, G2, G3)^> are the given body force and body couple vectors per unit mass. Using the relations (2.1)–(2.2) we can rewrite the above equations in terms of the displacement and microrotation vectors:

(µ+α)∆u(x) + (λ+µ−α) grad divu(x) + (κ+ν)∆ω(x) + +(δ+κ−ν) grad divω(x) + 2αcurlω(x) +%F(x) = 0, (κ+ν)∆u(x) + (δ+κ−ν) grad divu(x) + 2αcurlu(x) + +(γ+ε)∆ω(x) + (β+γ−ε) grad divω(x) + 4νcurlω(x)−

−4αω(x) +%G(x) = 0,

(2.6)

where ∆ is the Laplace operator.

Let us introduce the matrix differential operator corresponding to the system (2.6):

L(∂) :=

L⁽¹⁾(∂), L⁽²⁾(∂) L⁽³⁾(∂), L⁽⁴⁾(∂)

6×6

, (2.7)

where

L⁽¹⁾(∂) : = (µ+α)∆I3+ (λ+µ−α)Q(∂),

L⁽²⁾(∂) =L⁽³⁾(∂) : = (κ+ν)∆I3+ (δ+κ−ν)Q(∂) + 2αR(∂), L⁽⁴⁾(∂) : =

(γ+ε)∆−4α

I3+(β+γ−ε)Q(∂)+4νR(∂).

(2.8)

HereIk stands for thek×kunit matrix and R(∂) :=





0 −∂3 ∂2

∂3 0 −∂1

−∂2 ∂1 0





3×3

, Q(∂) :=

∂k∂j

3×3. (2.9) Note that obviously

R(∂)u=



∂2u3−∂3u2

∂3u1−∂1u3

∂1u2−∂2u1



= curlu, Q(∂)u= grad divu . (2.10)

Due to the above notation, the equations (2.6) can be rewritten in matrix form as

L(∂)U(x) = Φ(x),

U = (u, ω)^>, Φ = (Φ⁽¹⁾,Φ⁽²⁾)^> := (−%F,−%G)^>. (2.11) Let us remark that the operatorL(∂) is formally self-adjoint, i.e.,L(∂) = [L(−∂)]^>.

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2.2. Green’s formulae. For real–valued vectors U := (u, ω)^>, U⁰ :=

(u⁰, ω⁰)^>∈[C²(Ω⁺)]⁶ there holds the following Green formula [29]

Z

Ω⁺

L(∂)U·U⁰+E(U, U⁰) dx=

Z

∂Ω⁺

T(∂, n)U+

·[U⁰]⁺dS, (2.12) where nis the outward unit normal vector toS =∂Ω⁺, the symbols [·]^± denote the limits onS from Ω^±,E(·,·) is the bilinear form associated with the potential energy density defined by

E(U, U⁰) =E(U⁰, U) =

= 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 (2.13) with

upq=upq(U) =∂puq− X3 k=1

εpqkωk, ωpq=ωpq(U) =∂pωq, p, q= 1,2,3,

(2.14) andu⁰_pq=u⁰_pq(U⁰) andω⁰_pq=ω⁰_pq(U⁰) represented analogously by means of U⁰. Here and in what followsa·b denotes the usual scalar product of two vectorsa, b∈R^m: a·b=

Pm j=1

ajbj.

The expressionsupq(U) and ωpq(U) are calledgeneralized strainscorre- sponding to the vectorU = (u, ω)^>.

From (2.13) and (2.14) we get

E(U, U⁰) =

= 3λ+ 2µ 3

divu+ 3δ+ 2κ

3λ+ 2µ divω

divu⁰+ 3δ+ 2κ

3λ+ 2µ divω⁰ + +1

3

3β+ 2γ−(3δ+ 2κ)² 3λ+ 2µ

(divω)(divω⁰) +

+µ 2

X3 k,j=1, k6=j

∂uk

∂xj

+∂uj

∂xk

+κ µ

∂ωk

∂xj

+∂ωj

∂xk

×

× ∂u⁰_k

∂xj

+∂u⁰_j

∂xk

+κ µ

∂ω_k⁰

∂xj

+∂ω⁰_j

∂xk

+

+µ 3

X3 k,j=1

∂uk

∂xk

−∂uj

∂xj

+κ µ

∂ωk

∂xk

−∂ωj

∂xj

×

× ∂u⁰_k

∂xk

−∂u⁰_j

∂xj

+κ µ

∂ω_k⁰

∂xk

−∂ω_j⁰

∂xj

+

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+ γ−κ²

µ

X³

k,j=1, k6=j

1 2

∂ωk

∂xj

+∂ωj

∂xk

∂ω⁰_k

∂xj

+∂ω_j⁰

∂xk

+

+1 3

∂ωk

∂xk

−∂ωj

∂xj

∂ω_k⁰

∂xk

−∂ω_j⁰

∂xj

+

+α

curlu+ν

αcurlω−2ω

·

curlu⁰+ν

αcurlω⁰−2ω⁰ + +

ε−ν² α

curlω·curlω⁰. (2.15) The potential energy densityE(U, U) is positive definite with respect to the variablesupq(U) andωpq(U) (see (2.14)), i.e. there existsc0>0 depending only on the material constants such that

E(U, U)≥c0

X3 p,q=1

u²_pq+ω_pq²

. (2.16)

From (2.16) it follows that the material constants satisfy the inequalities (cf. [2], [29])

µ >0, α >0, 3λ+ 2µ >0, µγ−κ²>0, α ε−ν²>0,

(3λ+ 2µ)(3β+ 2γ)−(3δ+ 2κ)²>0, (2.17) whence we easily derive

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.

(2.18)

Due to [29] we can characterize the kernel of the energy bilinear form as follows.

Lemma 2.1. Let U = (u, ω)^> ∈[C¹(Ω)]⁶. Then E(U, U) = 0in Ω is equivalent to

u(x) = [a×x] +b, ω(x) =a, x∈Ω, (2.19) where a and b are arbitrary three-dimensional constant vectors and where the symbol×denotes the cross product of two vectors.

We call vectors of the type U(x) = ([a×x] +b, a)^> generalized rigid displacement vectors. It is evident that if a generalized rigid displacement vector vanishes at one point then it is zero vector, i.e.,a=b= 0.

Throughout the paper L2, W^s =W₂^s, and H^s =H₂^s with s∈ Rstand for the well-known Lebesgue, Sobolev–Slobodetski˘ı, and Bessel potential spaces, respectively (see, [37], [38], [27]). Note thatH^s=W^sfors≥0. We denote the associated norm byk · kH^s.

From the positive definiteness of the energy density it easily follows that there exist positive constants c1 and c2, depending only on the material

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constants, such that B(U, U) : =

Z

Ω⁺

E(U, U)dx≥

≥c1

Z

Ω⁺

X³

p,q=1

(∂puq)²+ (∂pωq)² +

X3 p=1

[u²_p+ω²_p]

dx−

−c2

Z

Ω⁺

X3 p=1

[u²_p+ω_p²]dx (2.20)

for an arbitrary real–valued vectorU∈[C¹(Ω⁺)]⁶, i.e., for anyU∈[H¹(Ω⁺)]⁶ the following Korn’s type inequality holds (cf. [11], Part I, § 12, [5], Sec- tion 6.3)

B(U, U)≥c1kUk²_[H¹_(Ω⁺_)]⁶−c2kUk²_[H⁰_(Ω⁺_)]⁶, (2.21) wherek · k[H^s(Ω⁺)]⁶ denotes the norm in the space [H^s(Ω⁺)]⁶.

Remark 2.2. From (2.12) it follows that Z

Ω⁺

L(∂)U·U⁰−U·L(∂)U⁰ dx=

Z

∂Ω⁺

n[T U]⁺·[U⁰]⁺−[U]⁺·[T U⁰]⁺o

dS (2.22) for arbitraryU := (u, ω)^>,U⁰:= (u⁰, ω⁰)^> ∈[C²(Ω⁺)]⁶.

Remark 2.3. By standard arguments, Green’s formula (2.12) can be extended to Lipschitz domains and to the case of vector functions U ∈ [H¹(Ω⁺)]⁶ andU⁰∈[H¹(Ω⁺)]⁶ andL(∂)U ∈[L2(Ω⁺)]⁶ (cf. [32], [27])

Z

Ω⁺

L(∂)U·U⁰+E(U, U⁰) dx=

[T(∂, n)U]⁺,[U⁰]⁺

∂Ω⁺, (2.23) whereh·,·i_∂Ω⁺ denotes the duality between the spaces [H^−1/2(∂Ω⁺)]⁶ and [H^1/2(∂Ω⁺)]⁶, which extends the usual “real” L2-scalar product , i.e., for f, g∈[L2(S)]⁶ we have

hf, giS = X6 k=1

Z

S

fkgkdS=: (f, g)L2(S).

Clearly, in the general case the functionalT(∂, n)U∈[H^−1/2(∂Ω⁺)]⁶is well defined by the relation (2.23).

3. Formulation of Unilateral Problems and Main Existence Results

3.1. Mechanical description of the problem. Mathematical aspects of the mechanical unilateral problems (Signorini type problems) in the frame- work of the classical elasticity theory are well described in many works (see, e.g., [12], [19], [20], and the references therein). Here we will apply the

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analogous arguments and construct a mathematical model for the unilateral problems in the theory of elasticity of hemitropic materials. The main diffi- culties appear in the reasonable interpretation of the unilateral restrictions for the microrotation vector and/or the couple stress vector.

We assume that a hemitropic elastic body, in its reference configuration, occupies the closure of a domain Ω⁺ ∈ R³. The boundary S = ∂Ω⁺ is divided into three partsS =SD∪ST ∪SC with disjointSD, ST, andSN. For definiteness and simplicity in what follows we assume (if not otherwise stated) thatSDhas a positive surface measure,SD∩SC=∅and, moreover, S, ∂SC, ∂ST,∂SD areC^∞-smooth.

The hemitropic elastic body is fixed along the subsurface SD, i.e., the displacement vectoru and the microrotation vectorω vanish onSD. The surface force stress vector and the surface couple stress vector are applied to the portionST, i.e., [τ⁽ⁿ⁾(U)]⁺=ψ⁽¹⁾ and [µ⁽ⁿ⁾(U)]⁺=ψ⁽²⁾ onST, where ψ^(j)= (ψ₁^(j), ψ^(j)₂ , ψ₃^(j))^>, j= 1,2,are given vector functions onST.

The motion of the elastic body is restricted by the so calledfoundation F which is a rigid and absolutely fixed body. We are interested in the deformation of the hemitropic body brought about due to its motion from its reference configuration to another configuration when some portion of the material surface of the body comes in contact with the foundationF. Note that in the case of statics we ignore the dependence of all mechanical and geometrical characteristics involved on timet. The actual surface on which the body comes in contact with the foundation is not known in advance but is contained in the portionSC ofS.

We confine our attention to infinitesimal generalized deformations of the body (see (2.14)). Moreover, we assume that the foundation surface∂F is frictionless and no force and couple stresses are applied onSC. Therefore, by the standard arguments (for details see, e.g., [20], Chapter 2) we arrive at the followinglinearized conditions on SC for the displacement vectoru and the force stress vectorτ⁽ⁿ⁾(U):

(i) the so called non-penetration condition,

[u·n]⁺≤ϕ, (3.1)

where nis the outward unit normal vector to∂Ω⁺ and ϕis a given scalar function characterizing theinitial gap between the foundation and the elastic body;

(ii) the conditions describing that a compressive normal force stress must be developed at the points of contact, while the normal component of the force stress vector is zero if no contact occurs,

τ⁽ⁿ⁾(U)·n+

≤0,

τ⁽ⁿ⁾(U)·n+

[u·n]⁺−ϕ

= 0; (3.2) (iii) the condition showing that the tangential components of the force stress vector vanish onSC,

τ⁽ⁿ⁾(U)+

−n

τ⁽ⁿ⁾(U)·n+

= 0. (3.3)

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Further, we make a fundamental observation: since the surface ∂F is frictionless, the microrotation vectorωis not restricted by any condition on SC even at the points where the contact with the rigid foundationFoccurs and, therefore, it is very natural to require that the couple stress vector µ⁽ⁿ⁾(U) is zero onSC,

µ⁽ⁿ⁾(U)+

= 0 on SC. (3.4)

Now we are in a position to formulate mathematically the unilateral problem that corresponds to the equilibrium state of a hemitropic body for given data: body force and body couple vectors Φ⁽¹⁾ = (Φ⁽¹⁾₁ ,Φ⁽¹⁾₂ ,Φ⁽¹⁾₃ )^>

and Φ⁽²⁾ = (Φ⁽²⁾₁ ,Φ⁽²⁾₂ ,Φ⁽²⁾₃ )^> in Ω⁺, surface force stress and surface couple stress vectorsψ⁽¹⁾ andψ⁽²⁾ onST, and initial gapϕonSC.

Problem (UP). We have to find a vectorU = (u, ω)^>) ∈ [H¹(Ω⁺)]⁶ satisfying the following system of equations and inequalities:

L(∂)U=−Φ in Ω⁺, (3.5)

[u]⁺ = 0, [ω]⁺= 0 on SD, (3.6) [τ⁽ⁿ⁾(U)]⁺=

T⁽¹⁾u+T⁽²⁾ω+

=ψ⁽¹⁾ [µ⁽ⁿ⁾(U)]⁺=

T⁽³⁾u+T⁽⁴⁾ω+

=ψ⁽²⁾



 on ST, (3.7) [u·n]⁺≤ϕ

τ⁽ⁿ⁾(U)·n+

≤0 r_SC

τ⁽ⁿ⁾(U)·n+

, r_SC[u·n]⁺−ϕ

SC = 0 [τ⁽ⁿ⁾(U)]⁺−n

τ⁽ⁿ⁾(U)·n+

= 0 [µ⁽ⁿ⁾(U)]⁺= 0











on SC, (3.8)

whererΣ denotes the restriction operator to Σ,

Φ = (Φ⁽¹⁾,Φ⁽²⁾)^>∈[L2(Ω⁺)]⁶, Φ^(j)= Φ^(j)₁ ,Φ^(j)₂ ,Φ^(j)₃ >

, (3.9)

equation (3.5) is understood in the weak sense, i.e., Z

Ω⁺

E(U, V)dx= Z

Ω⁺

Φ·V dx (3.10)

for arbitrary infinitely differentiable functionV ∈[C₀^∞(Ω⁺)]⁶with compact support in Ω. Due to the well-known interior regularity results for solutions of strongly elliptic systems (see, e.g., [12]) we conclude that the equation (3.5), with Φ as in (3.9), holds pointwise almost everywhere in Ω⁺.

The condition (3.6) and the first inequality in (3.8) are understood in the usual trace sense, while (3.7) and the fourth and fifth equations in (3.8) are

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understood in the generalized functional sense described in Remark 2.3, Ψ : = (ψ⁽¹⁾, ψ⁽²⁾)^> ∈[L2(ST)]⁶,

ψ^(j)= ψ₁^(j), ψ^(j)₂ , ψ₃^(j)>

, j= 1,2, (3.11)

ϕ∈H¹²(SC). (3.12)

Throughout the paperH^s(Σ) :=

rΣf : f ∈H^s(S) is the space of restrictions to Σ ⊂ S of functions from the space H^s(S), while He^s(Σ) :=

f ∈ H^s(S) : suppf ⊂ Σ for s∈ R. We recall that He^s(Σ) and H^−s(Σ) are mutually adjoint function spaces and L2(Σ) is continuously embedded in He^−1/2(Σ) (for details see, e.g., [27], [37]). In the third equation of (3.8) the symbolh ·,· iSC denotes the duality brackets between the spacesHe^−1/2(SC) and H^1/2(SC), which is well-defined due to the embedding (3.11) and the boundary condition (3.7) implying that

rΣ[τ⁽ⁿ⁾(U)·n]⁺∈rΣ[He^−1/2(Σ)]³ for Σ∈ {SD, ST, SC}.

The second inequality in (3.8) means that r_SC[τ⁽ⁿ⁾(U)·n]⁺, ψ

SC ≤0 for all ψ∈H^1/2(SC), ψ≥0.

Note that due the unilateral conditions (3.8) the problem (UP) is nonlinear since the coincidence set of the foundationF and the elastic body (i.e., the subset of SC where [u·n]⁺ =ϕ) is not known in advance. We have the following uniqueness result.

Theorem 3.1. The problem (UP)possesses at most one solution.

Proof. LetU⁽¹⁾= (u⁽¹⁾, ω⁽¹⁾)^> andU⁽²⁾= (u⁽²⁾, ω⁽²⁾)^> be two solutions of the problem (UP) and denote U = (u, ω)^> := U⁽¹⁾−U⁽²⁾. It is evident thatU satisfies the homogeneous differential equation (3.5) with Φ = 0, the homogeneous Dirichlet boundary conditions (3.6), the homogeneous Neu- mann boundary conditions (3.7) with Ψ := (ψ⁽¹⁾, ψ⁽²⁾)^>= 0. Moreover, the tangential components of the force stress vector [τ⁽ⁿ⁾(U)]⁺−n[τ⁽ⁿ⁾(U)·n]⁺ and the couple stress vector [µ⁽ⁿ⁾(U)]⁺vanish onSC due to the fourth and fifth conditions in (3.8). Therefore, by Green’s formula (2.23) and using the third equality in (3.8) we get

Z

Ω⁺

E(U, U)dx=

[T(∂, n)U]⁺_S,[U]⁺_S

S=

[τ⁽ⁿ⁾(U)]⁺_S,[u]⁺_S

S =

=D

τ⁽ⁿ⁾(U⁽¹⁾)·n+ SC−

τ⁽ⁿ⁾(U⁽²⁾)·n+

SC,[u⁽¹⁾·n]⁺_S_C−ϕ−[u⁽²⁾·n]⁺_S_C+ϕE

SC

=

=−D

τ⁽ⁿ⁾(U⁽¹⁾)·n+

SC,[u⁽²⁾·n]⁺_S_C−ϕE

SC

−

−D

τ⁽ⁿ⁾(U⁽²⁾)·n+

SC,[u⁽¹⁾·n]⁺_S_C −ϕE

SC

≤0,

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due to the first and second inequalities in (3.8). Thus, E(U, U) = 0 in accordance with (2.16) and by Lemma 2.1 it then follows thatU = 0 in Ω⁺

sinceSD6=∅. This completes the proof.

Below, in the subsequent subsections we will give some different formu- lations of the above unilateral problem (UP) with the help of variational inequalities.

3.2. Spatial variational inequality formulation. Let us put H(Ω⁺;SD) :=n

V = (v, w)^>∈[H¹(Ω⁺)]⁶: r_SD[V]⁺= 0o

(3.13) and forϕ∈H^1/2(SC) introduce the convex closed set of vector functions

K_ϕ:=n

V = (v, w)^>∈H(Ω⁺;SD) : r_SC([v]⁺·n)≤ϕo

. (3.14) Let us consider the followingspatial variational inequality (SVI):

FindU = (u, ω)^>∈Kϕsuch that Z

Ω⁺

E(U, V −U)dx≥

≥ Z

Ω⁺

Φ·(V −U)dx+hΨ, r_ST[V −U]⁺iST for all V ∈Kϕ, (3.15) whereE(·,·) is given by (2.15) and Φ, Ψ andϕ are as in (3.9), (3.11) and (3.12). Note that the duality relation in the right-hand side of (3.15) can be written as a usual Lebesgue integral over the subsurfaceST due to (3.11) and Remark 2.3.

Further we show that the SVI (3.15) and the unilateral problem (UP) are equivalent.

Theorem 3.2. If a vector functionU solves theSVI (3.15), thenU is a solution to the unilateral problem (3.5)–(3.8), and vice versa.

Proof. First, we assume thatU = (u, ω)^> ∈K_ϕ solves the SVI (3.15) and prove that it solves then the unilateral problem (3.5)–(3.8). It is evident that

V =U±V⁰∈Kϕ for arbitrary V⁰∈[C₀^∞(Ω⁺)]⁶. (3.16) From (3.15) it then follows that

Z

Ω⁺

E(U, V⁰)dx= Z

Ω⁺

Φ·V⁰dx for all V⁰ ∈[C₀^∞(Ω⁺)]⁶,

which shows thatU is a weak solution of the equation (3.5). Due to the interior regularity results the equation (3.5) holds point wise almost everywhere in Ω⁺ since Φ∈[L2(Ω⁺)]⁶.

The Dirichlet type condition (3.6) is automatically satisfied due to the inclusionU ∈Kϕ sinceKϕ⊂H(Ω⁺;SD).

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For the solution vectorU and for anyV ∈Kϕin accordance with Green’s formula (2.23) we have

Z

Ω⁺

E(U, V−U)dx= Z

Ω⁺

Φ·(V−U)dx+

[T(∂, n)U]⁺,[V−U]⁺

∂Ω⁺. (3.17) From (3.15) and (3.16) we conclude

[T(∂, n)U]⁺,[V−U]⁺

∂Ω⁺≥ hΨ, r_ST[V−U]⁺iST for all V ∈Kϕ. (3.18) In virtue of (3.11) and

rΣ[T(∂, n)U]⁺∈rΣ[He^−1/2(Σ)]⁶ rΣ[V −U]⁺∈[H^1/2(Σ)]⁶



 for Σ∈ {SD, ST, SC}, (3.19) and since elements from K_ϕ vanish on SD we can decompose the duality relations in (3.18) as follows

Dr_ST[T(∂, n)U]⁺, r_ST[V−U]⁺E

ST

+D

r_SC[T(∂, n)U]⁺, r_SC[V−U]⁺E

SC

≥

≥ hΨ, r_ST[V −U]⁺iST for all V ∈K_ϕ. (3.20) Further, let us choose the vector V as in (3.16) but now with arbitrary V⁰∈[H¹(Ω⁺)]⁶such that [V⁰]⁺_S ∈[He^1/2(ST)]⁶. It is clear thatV ∈K_ϕand from (3.20) we easily derive

Dr_ST[T(∂, n)U]⁺, r_ST[V⁰]⁺E

ST

=

=hΨ, r_ST[V⁰]⁺iST for all [V⁰]⁺_S ∈[He^1/2(ST)]⁶, (3.21) whence the conditions (3.7) onST follow immediately.

Note that the first condition in (3.8) is satisfied automatically sinceU = (u, ω)^>∈K_ϕ.

Now, from (3.20) we conclude Dr_SC[T(∂, n)U]⁺, r_SC[V −U]⁺E

SC

≥0 for all V ∈K_ϕ, (3.22) i.e.,

Dr_SC[τ⁽ⁿ⁾(U)]⁺, r_SC[v−u]⁺E

SC

+D

r_SC[µ⁽ⁿ⁾(U)]⁺, r_SC[w−ω]⁺E

SC

≥0 (3.23) for allV = (v, w)^>∈K_ϕ.

If we takev=uandw=ω±χwith arbitraryχ∈[H¹(Ω⁺)]³ such that [χ]⁺_S ∈ [He^1/2(SC)]³, then V = (v, w)^> ∈K_ϕ and from (3.23) we see that [µ⁽ⁿ⁾(U)]⁺= 0 onSC, i.e., the fifth condition in (3.8) is satisfied.

From (3.23) then we have

Dr_SC[τ⁽ⁿ⁾(U)]⁺, r_SC[v−u]⁺E

SC

≥0 (3.24)

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for allV = (v,0)^> ∈Kϕ. To show the remaining three conditions in (3.8) we proceed as follows. First we rewrite (3.24) in the form

Dr_SC[τ⁽ⁿ⁾(U)]⁺_n, r_SC[v−u]⁺_nE

SC

+D

r_SC[τ⁽ⁿ⁾(U)]⁺_t , r_SC[v−u]⁺_t E

SC

≥0, (3.25) where the subscriptsnandtdenote the normal and tangential components of the corresponding vectors defined on the boundary S: an = a·n and at=a−n anfor arbitrarya∈R³. We setV = (v,0)^>withv=u±gwhere g ∈ [H¹(Ω⁺)]³, [g]⁺_S ∈He^1/2(SC), and [g]⁺_n = 0 on SC. These restrictions imply that V = (v,0)^> ∈ K_ϕ and substitution into the inequality (3.25) leads to the equation

Dr_SC[τ⁽ⁿ⁾(U)]⁺_t , r_SC[g]⁺_t E

SC

= 0. (3.26)

Since [g]⁺_t = [g]⁺ is arbitrary we conclude that the fourth condition in (3.8) is satisfied. Therefore, (3.25) yields

Dr_SC[τ⁽ⁿ⁾(U)]⁺_n, r_SC[v−u]⁺_nE

SC

≥0, (3.27)

for allV = (v,0)^>∈K_ϕ.

Let us putV = (v,0)^> withv=u−gwhere g∈[H¹(Ω⁺)]³, [g]⁺_S =nϑ, ϑ ≥ 0, and ϑ ∈ He^1/2(SC). Then V = (v,0)^> ∈ K_ϕ and therefore from (3.27) we get

D

r_SC[τ⁽ⁿ⁾(U)]⁺_n, r_SCϑE

SC

≤0, (3.28)

which shows that the second condition in (3.8) holds.

Finally, if in (3.27) we substituteV = (v,0)^> ∈K_ϕ withr_SC[v]⁺_S =n ϕ onSC, we arrive at the inequality

Dr_SC[τ⁽ⁿ⁾(U)]⁺_n, ϕ−r_SC[u·n]⁺E

SC

≥0. (3.29)

On the other hand, sinceϕ−r_SC[u·n]⁺≥0 and [τ⁽ⁿ⁾(U)]⁺_n ≤0 onSC we

have D

r_SC[τ⁽ⁿ⁾(U)]⁺_n, ϕ−r_SC[u·n]⁺E

SC

≤0, (3.30)

which along with (3.29) implies that the third condition in (3.8) is fulfilled as well.

Now, we prove the inverse assertion. LetU = (u, ω)^>be a solution to the unilateral problem (3.5)–(3.8) with data satisfying the assumptions (3.9), (3.11), and (3.12). We have to show that thenU solves the SVI (3.15). It is clear that for the solution vectorU and for anyV ∈K_ϕ the formula (3.17) holds in accordance with Green’s formula (2.23). Note that the assumption (3.11) implies by embeddingL²(Σ)⊂He^−1/2(Σ) thatrST[T(∂, n)U]⁺= Ψ∈ rSTHe^−1/2(ST) and by the trace theorem thatrΣ[V −U]⁺∈[H^1/2(Σ)]⁶ for

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Σ ∈ {ST, SC}. Therefore in view of the Dirichlet homogeneous condition (3.6) we can rewrite (3.17) as follows

Z

Ω⁺

E(U, V −U)dx= Z

Ω⁺

Φ·(V −U)dx+D

Ψ, r_ST[V −U]⁺E

ST

+ +D

r_SC[T(∂, n)U]⁺, r_SC[V −U]⁺E

SC

. (3.31)

With the help of unilateral conditions (3.8) we derive D

r_SC[T(∂, n)U]⁺, r_SC[V −U]⁺E

SC

=D

r_SC[τ⁽ⁿ⁾(U)]⁺_n, r_SC[v−u]⁺_nE

SC

+ +D

r_SC[τ⁽ⁿ⁾(U)]⁺_t , r_SC[v−u]⁺_t E

SC

+D

r_SC[µ⁽ⁿ⁾(U)]⁺, r_SC[w−ω]⁺E

SC

=

=D

r_SC[τ⁽ⁿ⁾(U)]⁺_n, r_SC[v−u]⁺_nE

SC

=

=D

r_SC[τ⁽ⁿ⁾(U)]⁺_n, r_SC[v]⁺_n −ϕE

SC

−D

r_SC[τ⁽ⁿ⁾(U)]⁺_n, r_SC[u]⁺_n −ϕE

SC

=

=D

r_SC[τ⁽ⁿ⁾(U)]⁺_n, r_SC[v]⁺_n −ϕE

SC

≥0

due to the second inequality in (3.8) and sincer_SC[v]⁺_n−ϕ≤0 in accordance

with (3.14). Now, (3.31) completes the proof.

Thus we have shown that the unilateral problem (UP) (3.5)–(3.8) and the SVI (3.15) are absolutely equivalent.

Let us remark that the bilinear form B : [H¹(Ω⁺)]⁶×[H¹(Ω⁺)]⁶ → R with

B(U, V) :=

Z

Ω⁺

E(U, V)dx (3.32)

is bounded on [H¹(Ω⁺)]⁶×[H¹(Ω⁺)]⁶ and strictly coercive onH(Ω⁺;SD) (for details see [32]), i.e., there are positive constantsc3and c4 such that

B(U, V)≤c3kUk²_[H¹_(Ω⁺_)]⁶kVk²_[H¹_(Ω⁺_)]⁶ for all U, V∈[H¹(Ω⁺)]⁶, (3.33) B(U, U)≥c4kUk²_[H¹_(Ω⁺_)]⁶ for all U ∈H(Ω⁺;SD). (3.34) It is easy to see that the linear functionalP : [H¹(Ω⁺)]⁶→Rwith

P(V) :=

Z

Ω⁺

Φ·V dx+hΨ, r_ST[V]⁺iST for all V ∈K_ϕ, (3.35) where Φ and Ψ are as in (3.9) and (3.11), is bounded due to the Schwarz inequality and the trace theorem.

Therefore, due to the general theory of variational inequalities in Hilbert spaces (see, e.g., [12], [8], [14]) we have the following uniqueness and existence results for the variational inequality (3.15) which can be written now as

B(U, V −U)≥ P(V −U) for all V ∈K_ϕ. (3.36)

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Theorem 3.3. TheSVI (3.15) (i.e., the variational inequality(3.36))is uniquely solvable.

As an easy consequence of Theorems 3.2 and 3.3 we have

Corollary 3.4. The unilateral problem (UP) (3.5)–(3.8) is uniquely solvable.

It is also well-known that, in turn, the variational inequality (3.36) (that is, (3.15)) is equivalent to the followingminimization problem: Find a min- imum on the convex closed setKϕ of the energy functional (see, e.g., [12]) J(V) := 2⁻¹B(V, V)− P(V), (3.37) i.e., findU ∈Kϕsuch that

J(U) = min

V∈Kϕ

J(V). (3.38)

It is evident that the minimization problem is also uniquely solvable.

Let us remark that we can reduce equivalently the above unilateral problem (UP) to the case when the right-hand side vector function in (3.5) vanishes. To this end, consider the auxiliary boundary value problem (BVP)

L(∂)U0=−Φ in Ω⁺,

[U0]⁺= 0 on SD, [T(∂, n)U0]⁺= 0 on S\SD, (3.39) whereU0= (u0, ω0)^>∈[H¹(Ω⁺)]⁶ and Φ is as in (3.5).

This mixed BVP is uniquely solvable (for details see [29]). Therefore, if we denoteU^∗:=U−U0, whereU solves the unilateral problem (3.5)–(3.8) and U0 is the solution vector of the auxiliary BVP, then we see that the vector U^∗ is a solution to the unilateral problem (UP) with homogeneous differential equation (3.5) (i.e., with Φ = 0) and withϕ^∗:=ϕ−r_SC[u0]⁺_n in the place ofϕ. Therefore, in what follows, we assume that Φ = 0 without loss of generality.

3.3. Boundary variational inequality formulation. Here we reduce the unilateral problem (UP) (3.5)–(3.8) (with Φ = 0) to an equivalent boundary variational inequality (BVI). To this end, for a given vector f = (f⁽¹⁾, f⁽²⁾)^> ∈[H^1/2(S)]⁶,f^(j)= (f₁^(j), f₂^(j), f₃^(j))^>,j= 1,2,let us consider the vector function

U(x) =V(H⁻¹f)(x), x∈Ω⁺, (3.40) where V(·) is the single layer potential operator and H is the boundary integral operator onS =∂Ω⁺ generated by the single layer potential (see the Appendix, formulas (A.2), (A.11), and Theorems 4.2 and 4.3). It can easily be verified that the vector (3.40) solves the Dirichlet BVP

L(∂)U = 0 in Ω⁺, U = (u, ω)^>∈[H¹(Ω⁺)]⁶,

[U]⁺=f on S. (3.41)

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Let us introduce the so called Steklov–Poincar´e operator A relating the Dirichlet and Neumann data for a vector (3.40) (see the Appendix, Theo- rem 4.2)

Af :=

T(∂, n)V(H⁻¹f)+

= [−2⁻¹I6+K]H⁻¹f. (3.42) With the help of the equalities stated in the Appendix, Theorem 4.3.iii, it can easily be shown that

A=H⁻¹[−2⁻¹I6+K^∗] =L −[−2⁻¹I6+K]H⁻¹[−2⁻¹I6+K^∗], (3.43) whence it follows that the operator

A: [H^1/2(S)]⁶→[H^−1/2(S)]⁶ (3.44) is an elliptic, self-adjoint pseudodifferential operator of order 1.

Denote by XS{Λ⁽¹⁾,Λ⁽²⁾, . . . ,Λ⁽⁶⁾} the linear span of generalized rigid displacement vectors onS (for more details we refer to the Appendix A.1).

Theorem 3.5. The Steklov–Poincar´e operator (3.42) possesses the following properties:

hAf, fiS ≥0 for all f ∈[H^1/2(S)]⁶, (3.45) hAf, fiS = 0 if and only if f =rS([a×x] +b, a)^>, a, b∈R³, (3.46)

i.e., f ∈XS{Λ⁽¹⁾,Λ⁽²⁾, . . . ,Λ⁽⁶⁾},

hAf, giS =hAg, fiS for all f, g∈[H^1/2(S)]⁶, (3.47) kerA=XS

Λ⁽¹⁾,Λ⁽²⁾, . . . ,Λ⁽⁶⁾ , (3.48) i.e., for an arbitrary generalized rigid displacement vectorχ(x) = ([a×x] + b, a)^>, x∈S, we haveAχ= 0 onS.

Proof. The relations (3.45) and (3.46) follow from inequality (2.16), Lem- ma 2.1, and Green’s identity

Z

Ω⁺

E(U, U)dx=hAf, fiS with U =V(H⁻¹f). (3.49) The equality (3.47) is an easy consequence of (3.43), while (3.48) can be established with the help of Theorem 5.3.ii in the Appendix.

Further, let H(S, SD) : =n

g= (g⁽¹⁾, g⁽²⁾)^>, g^(j)∈[H^1/2(S)]³, j= 1,2 : r_SDg= 0o

≡

≡ eH^1/2(S\SD)6

, (3.50)

and forϕ∈H^1/2(SC) introduce the convex closed set of vector functions K_ϕ:=n

g= (g⁽¹⁾, g⁽²⁾)^>∈H(S, SD) : r_SC(g⁽¹⁾·n)≤ϕo

. (3.51) Let us consider the followingboundary variational inequality(BVI):

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Findf = (f⁽¹⁾, f⁽²⁾)^> ∈K_ϕsuch that Af, g−f

S ≥ hΨ, r_ST(g−f)iST for all g= (g⁽¹⁾, g⁽²⁾)^> ∈K_ϕ, (3.52) where A is the Steklov–Poincar´e operator, Ψ and ϕ are as in (3.11) and (3.12). Note that the duality relation in the right-hand side of (3.52) can be written as a usual Lebesgue integral over the subsurfaceST due to (3.11) and Remark 2.3.

First we establish the following strict coercivity property of the opera- torA.

Theorem 3.6. The Steklov–Poincar´e operator (3.42)is strictly coercive onH(S, SD), i.e., there is a positive constantC1 such that

hAf, fiS ≥C1kfk²_[H1/2(S)]⁶ for all f ∈H(S, SD). (3.53) Proof. The proof coincides word for word with the proof of Lemma 4.2

in [13].

Thus, the bilinear form hAf , giS is strictly coercive on the subspace H(S, SD) and bounded on [H^1/2(S)]⁶×[H^1/2(S)]⁶, i.e. there is a positive constantC2 such that

hAf, giS ≤C2kfk[H^1/2(S)]⁶khk[H^1/2(S)]⁶ for all f, g∈[H^1/2(S)]⁶. (3.54) Therefore, the BVI (3.52) is uniquely solvable due to the general theory of variational inequalities in Hilbert spaces (see, e.g., Theorems 2.1 and 2.2 in [14]).

Further we show that the boundary variational inequality (3.52) is equivalent to the unilateral problem (UP).

Theorem 3.7. (i) If f ∈K_ϕ solves the BVI (3.52) then the vector U given by(3.40)is a solution to the unilateral problem(UP) (3.5)–(3.8)with Φ = 0.

(ii) If U ∈ K_ϕ is a solution of the problem (UP) with Φ = 0, then f = [U]⁺_S is a solution of the BVI (3.52).

Proof. Letf ∈K_ϕbe a solution of the BVI (3.52) and construct the vector U by (3.40). It is evident that U ∈ [H¹(Ω⁺)]⁶ and L(∂)U = 0 in Ω⁺ in accordance with Theorem 4.2 in the Appendix. Moreover, since [U]⁺ = ([u]⁺,[ω]⁺)^> =f ∈K_ϕ on S we see that the conditions (3.5) with Φ = 0, (3.6), and the first inequality in (3.8) are satisfied.

Note that

Af = [T(∂, n)U]⁺= [τ⁽ⁿ⁾(U)]⁺,[µ⁽ⁿ⁾(U)]⁺>

∈[H^−1/2(S)]⁶. (3.55) Further, letg=f±hwhere h= (h⁽¹⁾, h⁽²⁾)^>∈[He^1/2(ST)]⁶. Evidently, g∈K_ϕand from (3.52) we get

Af, h

S =hΨ, r_SThiST for all h= (h⁽¹⁾, h⁽²⁾)^>∈[He^1/2(ST)]⁶. Consequently,

r_STAf =r_ST[T(∂, n)U]⁺= Ψ on ST (3.56)