An Asymptotic Expansion for Perturbations in the Displacement Field due to the Presence of Thin Interfaces

0.25 MB·

by  Jihene Lagha

An Asymptotic Expansion for Perturbations in the
6 Displacement Field due to the Presence of Thin Interfaces
1
0
2
Jihene Lagha ∗ Habib Zribi †

n
a
J

4
Abstract
2
We derive an asymptotic expansion for two-dimensional displacement field associ-

] ated to thin elastic inhomogeneities having no uniform thickness. Our derivation is
P
rigorousandbasedonlayerpotentialtechniques. Weextendthesetechniquestodeter-
A
minearelationship betweentraction-displacement measurementsandtheshapeofthe
. object.
h
t
a Mathematicssubject classification(MSC2000): 35B30,35C20,31B10
m Keywords: Thin inhomogeneities, small perturbations, Lam´e system, asymptotic expansions, boundary
[ integralmethod

1
v 1 Introduction and statement of main results
4
7 Let D be a bounded C2,η domain in R2 for some η > 0. For a given ǫ > 0, let D be an
ǫ
7
ǫ−perturbation of D,i.e., there is h∈C1(∂D) such that ∂D is given by
6 ǫ
0
. ∂Dǫ := x˜:x˜=x+ǫh(x)n(x),x∈∂D ,
1
0 n o
where n(x) is the outward normal to the domain D. We assume that h(x) ≥ C > 0 for all
6
1 x∈∂D.
: Consider a homogeneous isotropic elastic object occupying R2. Suppose that the thin
v
layerD \D liesinsideR2. Letthe constants(λ ,µ ),(λ ,µ ), and(λ ,µ )denotetheLam´e
i ǫ 0 0 1 1 2 2
X coefficients of R2\D , D, and D \D, respectively. It is always assumed that
ǫ ǫ
r
a µ >0, λ +µ >0 for j =0,1,2.
j j j
We also assume that
(λ −λ )(µ −µ )≥0, (λ −λ )2+(µ −µ )2 6=0 for j =1,2.
0 j 0 j 0 j 0 j
(cid:16) (cid:17)
LetC ,C ,andC betheelasticitytensorsforR2\D ,D,andD \D,respectively,which
0 1 2 ǫ ǫ
are given by
C =λ I⊗I+2µ I, m=0,1,2, (1.1)
m m m
∗Department of Mathematics, Faculty of Sciences, University of Tunis El Manar, Tunis 2092, Tunisia
([email protected])
†DepartmentofMathematics,CollegeofSciences,UniversityofHafrAlBatin,P.o. 1803,HafrAlBatin
31991,SaudiArabia([email protected])
1
where I is the identity 4-tensor and I is the identity 2-tensor (the 2×2 identity matrix).
We define
C :=C χ +C χ +C χ , C:=C χ +C χ ,
ǫ 0 R2\Dǫ 2 Dǫ\D 1 D 0 R2\D 1 D
where χ is the indicator function of D.
D
Let u denote the displacement field in the presence of the thin D \D, that is, the
ǫ ǫ
solution to
∇· C ∇u =0 in R2,
ǫ ǫ
(1.2)
( uǫ(x(cid:0))−bH(x(cid:1))=O(|x|−1) as |x|→∞,
where∇u = 1 ∇u +∇uT isthestraintensorandHisavector-valuedfunctionsatisfying
ǫ 2 ǫ ǫ
∇· C ∇H = 0 in R2. Here and throughout the paper MT denotes the transpose of the
0 (cid:0) (cid:1)
matrixbM.
(cid:0) (cid:1)
Thebelastostatic system corresponding to the Lam´e constants (λj,µj) is defined by
L w:=µ ∆w+(λ +µ )∇∇·w. (1.3)
λj,µj j j j
∂w
The corresponding conormal derivative associated with (λ ,µ ) is defined by
j j
∂ν
j
∂w
:=λ (∇·w)n+µ ∇w+∇wT n=λ (∇·w)n+2µ ∇wn. (1.4)
j j j j
∂ν
j
(cid:0) (cid:1)
The problem (1.2) is equivalent to the following problem b
L u =0 in R2\D ,
λ0,µ0 ǫ ǫ
 L u =0 in D \D,
λ2,µ2 ǫ ǫ
 uLǫλ1−,µ1=uǫuǫ=+0 inoDn,∂D, uǫ − =uǫ + on ∂Dǫ, (1.5)
∂u(cid:12)ǫ ∂(cid:12)uǫ ∂u(cid:12)ǫ ∂(cid:12)uǫ
(cid:12) = (cid:12) on ∂D, (cid:12) = (cid:12) on ∂D ,
The notation uǫ|±u∂oǫνn(1x∂(cid:12)(cid:12)(cid:12))−D−Hde∂(nxνo2)t(cid:12)(cid:12)(cid:12)e=+tOhe(|lxim|−i1t)s from∂aνos2u|xt(cid:12)(cid:12)(cid:12)−s|id→e ∞a∂nν.d0(cid:12)(cid:12)(cid:12)i+nside of D,ǫrespectively. We
will sometimes use ue for u | and ui for u | .
ǫ ǫ + ǫ ǫ −
The first achievement of this work, a rigorous derivation of the asymptotic expansion
of u | as ǫ → 0 where Ω is a bounded region away from ∂D, based on layer potential
ǫ Ω
techniques.
Theorem 1.1 Let u be the solution to (1.5). Let Ω be a bounded region away from ∂D.
ǫ
For x∈Ω, the following pointwise asymptotic expansion holds:
u (x)=u(x)+ǫu (x)+o(ǫ), (1.6)
ǫ 1
where the remainder o(ǫ) depends only on (λ ,µ ) for j=0,1,2, the C2-norm of X, the C1-
j j
norm of h, and dist(Ω,∂D), u is the unique solution to
∇· C∇u =0 in R2,
(1.7)
( u(x(cid:0))−H((cid:1)x)=O(|x|−1) as |x|→∞,
b
2
and u is the unique solution of the following transmission problem:
1
L u =0 in R2\D,
λ0,µ0 1
 Lλ1,µ1u1 =0 in D,
 u∂1u(cid:12)(cid:12)1−−−u1∂(cid:12)(cid:12)+u1=h(cid:16)=(K∂0,1−h K(M2,1)∇b−uiM(cid:17)n )∇ui τ oonn ∂∂DD,, (1.8)
with τ is thetaun∂1gν(e1xn(cid:12)(cid:12)(cid:12))t−i=alOv∂e(cν|xt0o|(cid:12)(cid:12)(cid:12)−r+1t)o ∂∂Dτ,(cid:16) (cid:2) 2,1 0,1 b (cid:3) (cid:17) as |x|→∞,
λ (λ +2µ ) 4(µ −µ )(λ +µ )
M := l k k I⊗I+2µ I+ l k l l I⊗(τ ⊗τ),
l,k k
λ +2µ λ +2µ
l l l l
µ (λ −λ )+2(µ −µ )(λ +µ ) µ
K := l k l l k l l I⊗I+2 k −1 I
l,k
µ (λ +2µ ) µ
l l l l
(cid:0) (cid:1)
2(µ −µ )(λ +µ )
+ k l l l I⊗(τ ⊗τ).
µ (λ +2µ )
l l l
By taking λ = λ and µ = µ , we reduce our problem to that proposed in [17]. So it
2 1 2 1
is obviousto obtainthe asymptotic expansionofthe displacementfieldresulting fromsmall
perturbations of the shape of an elastic inclusion already derived in [17, Theorem 1.1]. Our
asymptotic expansion is valid in the cases of thin interfaces with high contrast parameters
and can be derived similarly to [16], for more details on behaviors of the leading and first
order terms u and u , we refer the reader to [2, Chapter 2]. In [9, 10] the authors derive
1
asymptoticexpansionsfor the boundary displacementfield inthe case λ =λ andµ =µ
0 1 0 1
in both cases of isotropic and anisotropic thin elastic inclusions, the approach they use,
based on energy estimates, variational approach, and fine regularity estimates for solutions
of elliptic systems obtained by Li and Nirenberg [20]. Unfortunately, this method does not
seem to work in our case.
Let v be the solution of the following problem:
L v=0 in R2\D,
λ0,µ0
 Lλ1,µ1v=0 in D,
 v∂|v− =v=|+∂v oonn ∂∂DD,, (1.9)
∂ν1 − ∂ν0 +
As a consequenceofthethevor(xem)(cid:12)(cid:12)(cid:12)−1.F1,(xw)e=(cid:12)(cid:12)(cid:12)obOta(|ixn|−th1e)folalosw|xin|g→re∞lat.ionshipbetweentraction-
displacement measurements and the deformation h.
Theorem 1.2 Let u , u, and v be the solutions to (1.5), (1.7), and (1.9), respectively.
ǫ
Let S be a Lipschitz closed curve enclosing D away from ∂D. The following asymptotic
3
expansion holds:
∂F ∂u ∂u
u −u · dσ− ǫ − ·Fdσ
ǫ
∂ν ∂ν ∂ν
ZS 0 ZS 0 0
(cid:0) (cid:1) (cid:0) (cid:1)
=ǫ h [M −M ]∇ui τ ·∇viτ + [K −K ]∇ui n·(C ∇vi)n dσ+o(ǫ),
0,1 2,1 2,1 0,1 1
Z∂D (cid:18) (cid:19)
(cid:0) (cid:1) (cid:0) (cid:1) (1.10)
b b b b
where the remainder o(ǫ) depends only on (λ ,µ ) for j=0,1,2, the C2-norm of X, the C1-
j j
norm of h, and dist(S,∂D). The dot denotes the scalar product in R2.
The asymptotic expansion in (1.10) can be used to design algorithms to identify cer-
tain properties of its thin elastic like location and thickness based on traction-displacement
measurements. To do this, we refer to asymptotic formulae related to measurements in the
samespirit,far-fielddata,currents,generalizedpolarizationtensors,elasticmomenttensors,
multistatic response at single or multiple frequencies, and modal measurements that have
been obtained in a series of recent papers [1, 3, 6, 7, 13, 18, 19, 22].
Our techniques in this paper seem simpler to implement and have the big advantage
to derive high order terms in the asymptotic formulae and allow a generalization to 3-
dimensional interface problems by using [15, 16].
This paper is organizedas follows. In section 2, we review some basic facts on the layer
potentials of the Lam´e system and derive a representation formula for the solution of the
problem(1.5). Insection3,wederiveasymptoticexpansionsoflayerpotentials. Insection3,
based on layer potentials techniques we rigorously derive the asymptotic expansionfor per-
turbationsinthe displacementfieldandfindthe relationshipbetweentraction-displacement
measurements and the deformation h (Theorem 1.1 & Theorem 1.2).
2 Representation formula
Let us review some well-known properties of the layer potentials for the elastostatics. The
theory of layer potentials has been developed in relation to boundary value problems in a
Lipschitz domain.
Let
Ψ:= ψ :∂ ψ +∂ ψ =0, 1≤i,j ≤2 .
i j j i
n o
or equivalently,
1 0 x
2
Ψ=span θ (x):= ,θ (x):= ,θ (x):= .
1 2 3
0 1 −x
( (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) 1(cid:21))
Introduce the space
L2(∂D):= f ∈L2(∂D): f ·ψ dσ =0 for all ψ ∈Ψ .
Ψ
n Z∂D o
In particular, since Ψ contains constant functions, we get
fdσ =0
Z∂D
4
for any f ∈L2(∂D). The following fact is useful later.
Ψ
If w∈W1,23(D) satisfies Lλ0,µ0w =0 in D, then ∂∂νw0 ∂D ∈L2Ψ(∂D). (2.1)
(cid:12)
TheKelvinmatrixoffundamentalsolutionΓj fortheLam´esyste(cid:12)(cid:12)mLλj,µj inR2 isknown
to be
A B x⊗x
Γ (x)= j log|x|I− j , x6=0, (2.2)
j 2π 2π |x|2
where
1 1 1 1 1 1
A = + and B = − .
j j
2 µ 2µ +λ 2 µ 2µ +λ
j j j j j j
(cid:16) (cid:17) (cid:16) (cid:17)
The single anddouble layer potentials of the density function φ on L2(∂D) associatedwith
the Lam´e parameters (λ ,µ ) are defined by
j j
S [φ](x)= Γ (x−y)φ(y)dσ(y), x∈R2, (2.3)
j,D j
Z∂D
D [φ](x)= λ ∇ ·Γ (x−y)⊗n(y)
j,D j y j
Z∂D
+µ ∇ Γ (x−y)n(y) T +∇ ΓT(x−y)n(y) φ(y)dσ(y)
j y j y j
!
(cid:16)(cid:2) (cid:3) (cid:17)
:= K (x−y)φ(y)dσ(y), x∈R2\∂D. (2.4)
j
Z∂D
The followings are the well-known properties of the single and double layer potentials due
to Dahlberg, Keing, and Verchota [11]. Let D be a Lipschitz bounded domain in R2. Then
we have
∂S [φ] 1
j,D (x)= ± I+K∗ [φ](x) a.e. x∈∂D, (2.5)
∂νj ± 2 j,D
(cid:12) (cid:16) (cid:17)
(cid:12)
D [φ](cid:12) (x)= ∓ 1I+K [φ](x) a.e. x∈∂D, (2.6)
j,D ± 2 j,D
(cid:12) (cid:16) (cid:17)
where Kj,D is defined by (cid:12)
K [φ](x)=p.v. K (x−y)φ(y)dσ(y) a.e. x∈∂D,
j,D j
Z∂D
and K∗ is the adjoint operator of K , that is,
j,D j,D
K∗ [φ](x)=p.v. KT(x−y)φ(y)dσ(y) a.e. x∈∂D.
j,D j
Z∂D
Here p.v. denotes the Cauchy principal value.
5
LetD♯ be the standarddoublelayerpotentialwhichis definedforanyφ∈L2(∂D)by
j,D
∂Γ (x−y)
D♯ [φ](x)= j φ(y)dσ(y), x∈R2\∂D. (2.7)
j,D ∂n(y)
Z∂D
The following lemma holds, for more details see [17].
Lemma 2.1 Let D be a bounded Lipschitz domain in R2. For φ∈L2(∂D)
1
D♯ [φ] (x)= ∓ I±B n⊗n+K♯ φ(x) a.e. x∈∂D, (2.8)
j,D ± 2µ j j,D
j
(cid:12) (cid:16) (cid:17)
∂S [φ](cid:12) 1
j,D (x)= ± I∓B n⊗n+ K♯ ∗ φ(x) a.e. x∈∂D, (2.9)
∂n ± 2µj j j,D
where K♯ is define(cid:12)(cid:12)(cid:12)d by (cid:16) (cid:0) (cid:1) (cid:17)
j,D
∂
K♯ [φ](x)=p.v. Γ (x−y)φ(y)dσ(y) a.e. x∈∂D,
j,D ∂n(y) j
Z∂D
and K♯ ∗ is the adjoint operator of K♯ , that is,
j,D j,D
(cid:0) (cid:1) K♯ ∗[φ](x)=p.v. ∂ Γ (x−y)φ(y)dσ(y) a.e. x∈∂D.
j,D ∂n(x) j
Z∂D
(cid:0) (cid:1)
Moreover, for φ∈C1,α(∂D),
∂D♯ [φ] ∂D♯ [φ] ∂ λ
j,D − j,D = (φ·τ)n+ j (φ·n)τ on ∂D. (2.10)
∂νj + ∂νj − ∂τ 2µj +λj
(cid:12) (cid:12) (cid:16) (cid:17)
Note that we dr(cid:12)(cid:12)op the p.v. in (cid:12)(cid:12)this stage; this is because ∂D is C2,η.
Denote by
X(∂D):=L2(∂D)2, X (∂D):=L2(∂D)×L2(∂D), Y(∂D):=W2(∂D)×L2(∂D),
Ψ Ψ 1
where W2(∂D) is the first L2-Sobolev of space of order 1 on ∂D.
1
Thefollowingtheoremisofparticularimportancetousforestablishingourrepresentation
formula.
Theorem 2.2 Suppose that (λ −λ )(µ −µ ) ≥ 0 and 0 < λ ,µ < ∞. For any given
0 2 0 2 2 2
(f ,f ,g ,g )∈Y(∂D)×Y(∂D ), there existsa uniquesolution (ϕ ,ϕ ,ψ ,ϕ )∈X(∂D)×
1 2 1 2 ǫ 1 2 2 0
X(∂D ) to the following integral equations
ǫ
S [ϕ ] −S [ϕ ] −S [ψ ]=f on ∂De, e (2.11)
1,D 1 − 2,D 2 + 2,Dǫ 2 1
∂S1,D[ϕ1] −(cid:12)(cid:12) ∂S2,D[ϕ2] (cid:12)(cid:12)− ∂S2,Dǫ[ψe2] =f on ∂D, (2.12)
2
∂ν1 − ∂ν2 + ∂ν2
(cid:12) (cid:12) e
S2,D[ϕ(cid:12)(cid:12)2]+S2,Dǫ[ψ2](cid:12)(cid:12)−−S0,Dǫ[ϕ0] + =g1 on ∂Dǫ, (2.13)
∂S2,D[ϕ2] + ∂S2,Dǫ[ψ2e] (cid:12)(cid:12)− ∂S0,Dǫ[ϕe0](cid:12)(cid:12) =g on ∂D . (2.14)
2 ǫ
∂ν2 ∂ν2 − ∂ν0 +
Moreover, if (f ,g )∈L2(∂D)×L2(e∂D(cid:12)(cid:12)), then (ϕ ,eϕ )(cid:12)(cid:12)∈L2(∂D)×L2(∂D ).
2 2 Ψ Ψ (cid:12)ǫ 2 0(cid:12) Ψ Ψ ǫ
e
6
Proof. The unique solvability of the system of integral equations (2.11)-(2.14) can be
done in exactly the same manner as in [21, Theorem 5.1] by using [12], see also [4]. By
using (2.1), ∂S [ϕ ]/∂ν ,∂S [ψ ]/∂ν ∈ L2(∂D). It then follows from (2.12) that
1,D 1 1 − 2,Dǫ 2 2 Ψ
∂S [ϕ ]/∂ν ∈L2(∂D). Since
2,D 2 2 + Ψ (cid:12)
(cid:12) e
(cid:12) ∂S [ϕ ] ∂S [ϕ ]
(cid:12) ϕ = 2,D 2 − 2,D 2 ,
2 ∂ν2 + ∂ν2 −
(cid:12) (cid:12)
with ∂S2,D[ϕ2]/∂ν2 − ∈L2Ψ(∂D), we conclud(cid:12)(cid:12)e that ϕ2 ∈L2Ψ(cid:12)(cid:12)(∂D). For any η ∈Ψ, we have
(cid:12) ∂S [ϕ ] ∂S [ϕ ]
(cid:12) 2,D 2 ·ηdσ = 2,D 2 ·ηdσ =0.
Z∂Dǫ ∂ν2 Z∂D ∂ν2 (cid:12)+
(cid:12)
Inordertojustifythelastequality,weuse∂S [ϕ ]/∂ν ∈(cid:12) L2(∂D). Then∂S [ϕ ]/∂ν ∈
2,D 2 2 + Ψ 2,D 2 2
L2(∂D ). It then follows from (2.14) that ∂S [ϕ ]/∂ν ∈L2(∂D ). Thus
Ψ ǫ 0,Dǫ 0 (cid:12)0 + Ψ ǫ
(cid:12)
∂S [ϕ ] ∂S [ϕ ] (cid:12)
ϕ = 0,Dǫ 0 − 0,Dǫe 0 ∈(cid:12) L2(∂D ).
0 ∂ν0 + ∂ν0 − Ψ ǫ
(cid:12) (cid:12)
This completes the preoof. e (cid:12)(cid:12) e (cid:12)(cid:12)
We now prove a representation theorem for the solution of the transmission problem
(1.5) which will be the main ingredient in deriving the asymptotic expansion in Theorem
1.1.
Theorem 2.3 The solution u to the problem (1.5) is represented by
ǫ
H(x)+S [ϕ ](x), x∈R2\D ,
0,Dǫ 0 ǫ
uǫ(x)= SS2,D[[ϕϕ2]]((xx)),+eS2,Dǫ[ψ2](x), xx∈∈DDǫ,\D, (2.15)
1,D 1 e
where (ϕ ,ϕ ,ψ ,ϕ )∈X(∂D)×X (∂D ) is the unique solution to the following integral
1 2 2 0 Ψ Ψ ǫ
equations:
e e S [ϕ ] −S [ϕ ] −S [ψ ]=0 on ∂D, (2.16)
1,D 1 − 2,D 2 + 2,Dǫ 2
∂S1,D[ϕ1] −(cid:12)(cid:12) ∂S2,D[ϕ2] (cid:12)(cid:12)− ∂S2,Dǫ[ψe2] =0 on ∂D, (2.17)
∂ν1 − ∂ν2 + ∂ν2
(cid:12) (cid:12) e
(cid:12) (cid:12)
S [ϕ(cid:12) ]+S [ψ ](cid:12) −S [ϕ ] =H on ∂D , (2.18)
2,D 2 2,Dǫ 2 − 0,Dǫ 0 + ǫ
∂S2,D[ϕ2] + ∂S2,Dǫ[ψ2e] (cid:12)(cid:12)− ∂S0,Dǫ[ϕe0](cid:12)(cid:12) = ∂H on ∂D . (2.19)
ǫ
∂ν2 ∂ν2 − ∂ν0 + ∂ν0
e (cid:12)(cid:12) e (cid:12)(cid:12)
(cid:12) (cid:12)
Proof. Let(ϕ ,ϕ ,ψ ,ϕ )be the unique solutionof (2.16)-(2.19). Thenclearlyu defined
1 2 2 0 ǫ
by (2.15) satisfies the transmission conditions (the conditions on the fourth and fifth lines
in (1.5)). It is knownethaet
S [ϕ ](x)=Γ (x) ϕ dσ+O(|x|−1) as |x|→∞.
0,Dǫ 0 0 0
Z∂Dǫ
e e
7
Since ϕ ∈ L2(∂D ), which gives ϕ dσ = 0, and then u defined by (2.15) satisfies
0 Ψ ǫ ∂Dǫ 0 ǫ
u (x)−H(x)=O(|x|−1) as |x|→∞. This finishes the proof of the theorem.
ǫ R
Weenowintroducesomenotation. LeteΦǫ bethediffeomorphismfrom∂Donto∂Dǫ given
byx˜=Φ (x)=x+ǫh(x)n(x), wherex=X(t)∈∂D. Definethe operatorsQ andR from
ǫ ǫ ǫ
L2(∂D)×L2(∂D ) into Y(∂D) by
ǫ
∂S [ϕ] ∂S [ψ]
Q (ϕ,ψ):= S [ϕ] −S [ψ]◦Φ , 1,D − 0,Dǫ ◦Φ , (2.20)
ǫ (cid:18) 1,D (cid:12)− 0,Dǫ ǫ(cid:12)+ ∂ν1 (cid:12)− ∂ν0 e ǫ(cid:12)+(cid:19)
e (cid:12) e (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12)
R (ϕ,ψ):= S [ϕ]◦Φ −S [ϕ] +S [ψ]◦Φ −S [ψ],
ǫ 2,D ǫ 2,D + 2,Dǫ ǫ − 2,Dǫ
(cid:18)
(cid:12) (cid:12)
e ∂S [ϕ] (cid:12) ∂S [ϕ]e ∂(cid:12)S [ψ] e ∂S [ψ]
2,D ◦Φ − 2,D + 2,Dǫ ◦Φ − 2,Dǫ ,
ǫ ǫ
∂ν2 ∂ν2 (cid:12)(cid:12)+ ∂ν2 e (cid:12)(cid:12)− ∂ν2 e(2.(cid:19)21)
(cid:12) (cid:12)
and the matrix-valued function H on ∂D by
ǫ
∂H
H (x):= H x+ǫh(x)n(x) , x+ǫh(x)n(x) . (2.22)
ǫ
∂ν
(cid:18) 0 (cid:19)
(cid:0) (cid:1) (cid:0) (cid:1)
By adding (2.16) to (2.18) and (2.17) to (2.19), we get the following lemma.
Lemma 2.4 Let (ϕ ,ϕ ,ψ ,ϕ ) ∈ X (∂D)×X (∂D ) be the unique solution of (2.16)-
1 2 2 0 Ψ Ψ ǫ
(2.19); then (ϕ ,ϕ ) and (ϕ ,ψ ) satisfy the following system of integral equations:
1 0 2 2
e e
e Qǫ(ϕ1e,ϕ0)=Hǫ−Rǫ(ϕ2,ψ2) on ∂D. (2.23)
Inthenextsection,wewillpreovideasymptoticexepansionsofthelayerpotentials,which
are appeared in the system of integral equations (2.23) with (ϕ ,ϕ )∈L2(∂D)×L2(∂D )
1 0 ǫ
and (ϕ ,ψ ) ∈ C1,η(∂D)×C1,η(∂D ). These asymptotic expansions will help us to derive
2 2 ǫ
the asymptotic expansion of the displacement field uǫ. e
e
3 Asymptotic expansions of layer potentials
Let a,b∈R, with a<b, and let X(t):[a,b]→R2 be the arclength parametrization of ∂D,
namely, X is an C2-function satisfying |X′(t)|=1 for all t∈[a,b] and
∂D :={x=X(t),t∈[a,b]}.
Then the outwardunit normalto D, n(x), is givenby n(x)=R−π2X′(t), where R−π2 is the
rotation by −π/2, the tangential vector at x, τ(x) = X′(t), and X′(t) ⊥ X′′(t). Set the
curvature κ(x) to be defined by
X′′(t)=κ(x)n(x).
We will sometimes use h(t) for h(X(t)) and h′(t) for the tangential derivative of h(x).
Then, X˜(t) = X(t)+ǫh(t)n(x) = X(t)+ǫh(t)R−π2X′(t) is a parametrization of ∂Dǫ.
By n(x˜), we denote the outward unit normal to ∂D at x˜.
ǫ
8
Let x˜=x+ǫh(x)n(x)∈∂D for x∈∂D. The following asymptotic expansions of n(x˜)
ǫ
and the length element dσ (x˜) hold [6]:
ǫ
n(x˜)=n(x)−ǫh′(t)τ(x)+O(ǫ2), (3.1)
and
dσ (x˜)= 1−ǫκ(x)h(x)+O(ǫ2) dσ(x). (3.2)
ǫ
Here,the remaindertermO(ǫ2)isb(cid:0)oundedbyCǫ2 forsom(cid:1)econstantC whichdependsonly
on C2-norm of ∂D and C1-norm of h.
Letφ(x)andφ(x)beavectorfunctionandscalarfunction,respectively,whichbelongto
C2([a,b]) for x = X(·) ∈ ∂D. By d/dt, we denote the tangential derivative in the direction
of τ(x)=X′(t). We have
d ∂φ d ∂φ
φ(x) =∇φ(x)X′(t)= (x), φ(x) =∇φ(x)·X′(t)= (x).
dt ∂τ dt ∂τ
(cid:0) (cid:1) (cid:0) (cid:1)
TherestrictionoftheLam´esystemL inDtoaneighborhoodof∂Dcanbeexpressed
λj,µj
as follows [17]:
∂2φ
L φ(x)=µ (x)+λ ∇∇·φ(x)·n(x)n(x)+µ ∇(∇φ)T(x)n(x)n(x)
λj,µj j∂n2 j j
∂φ d
−κ(x) (x)+ C ∇φ(x) τ(x) , x∈∂D. (3.3)
j
∂ν dt
j
(cid:16)(cid:0) (cid:1) (cid:17)
b
We have from [17] the following lemma.
Lemma 3.1 Let φ ∈ L2(∂D ), we denote by φ := φ◦ Φ . For j = 0,2, the following
ǫ ǫ
asymptotic expansions hold:
e e
∂S [φ]
S [φ]◦Φ =S [φ]−ǫS [κhφ]+ǫ h j,D +D♯ [hφ]
j,Dǫ ǫ ± j,D j,D ∂n j,D ±
(cid:12) (cid:16) (cid:17)(cid:12)
e (cid:12) +O1(ǫ2) on ∂D, (cid:12)(cid:12) (3.4)
∂S [φ] ∂S [φ] ∂S [φ] ∂S [κhφ]
j,Dǫ ◦Φ = j,D +ǫ κh j,D − j,D
ǫ
∂νj e (cid:12)(cid:12)± ∂νj (cid:12)(cid:12)± (cid:18) ∂νj ∂νj (cid:19)(cid:12)(cid:12)±
(cid:12) ∂D♯(cid:12) [hφ] ∂ (cid:12)(cid:12)
+ǫ j,D − h C ∇S [φ] τ +O (ǫ2) on ∂D,
∂ν ∂τ j j,D 2
(cid:18) j (cid:16) (cid:0) (cid:1) (cid:17)(cid:19)(cid:12)(cid:12)± (3.5)
b (cid:12)
(cid:12)
where kO1(ǫ2)kW12(∂D),kO2(ǫ2)kL2(∂D) ≤ Cǫ2 for some constant C depends only on λj,µj,
the C2-norm of X, and the C1-norm of h.
Now, we are going to derive asymptotic expansions of S [φ](x˜) and ∂S [φ]/∂ν (x˜)
2,D 2,D 2
for φ ∈ C1,η(∂D), x˜ = x+ǫh(x)n(x) ∈ ∂D , and x ∈ ∂D. Because ∂D is C2,η, S [φ] is
ǫ 2,D
9
C2,η(R2\D). We have
∇S [φ](x˜)−∇S [φ](x) −ǫh(x)∇2S [φ](x)n(x)
2,D 2,D + 2,D +
(cid:12) (cid:12)
(cid:12)(cid:12) = 1 ∇2S [φ] x+l(x˜(cid:12)(cid:12) −x) −∇2S [φ](x) dl(cid:12)(cid:12)(x˜(cid:12)(cid:12)−x)
2,D 2,D +
≤C(cid:12)(cid:12)(cid:12)Z0x˜−h x 1+η S (cid:0)[φ] (cid:1) (cid:12)(cid:12) i (cid:12)(cid:12)(cid:12)
2,D C2,η(R2\D)
≤Cǫ(cid:12)(cid:12)1+η φ(cid:12)(cid:12) (cid:13)(cid:13) . (cid:13)(cid:13) (3.6)
C1,η(∂D)
Thus (cid:13) (cid:13)
(cid:13) (cid:13)
∂S [φ]
2,D (x˜)= ∇S [φ](x) +ǫh(x)∇2S [φ](x)n(x) +O(ǫ1+η)
∂τ 2,D + 2,D +
(cid:16) (cid:12) (cid:12) (cid:17)
(cid:12) × τ(x)+ǫh′(t)(cid:12)n(x)+O(ǫ2)
(cid:16) (cid:17)
∂ ∂S [φ]
= S [φ](x)+ǫh(x) 2,D (x) +O(ǫ1+η), x∈∂D, (3.7)
∂τ 2,D ∂n +
(cid:16) (cid:12) (cid:17)
where kO(ǫ1+η)kL2(∂D) is bounded by Cǫ1+ηkφkC1,η(∂(cid:12)(cid:12)D). Similarly to (3.6), we get
∂S [φ]
S [φ](x˜)=S [φ](x)+ǫh(x) 2,D (x)+O(ǫ2), x∈∂D, (3.8)
2,D 2,D ∂n +
(cid:12)
where kO(ǫ2)kL2(∂D) is bounded by Cǫ2kφkC1,η(∂D). I(cid:12)(cid:12)n conclusion, we get from (3.7) and
(3.8) that
∂S [φ]
S [φ](x˜)=S [φ](x)+ǫh(x) 2,D (x)+O(ǫ1+η), x∈∂D, (3.9)
2,D 2,D ∂n +
(cid:12)
where kO(ǫ1+η)kW12(∂D) is bounded by Cǫ1+ηkφkC1,η((cid:12)(cid:12)∂D).
It follows from (1.4), (3.1), (3.3), and Taylor expansion that, for x˜∈∂D,
∂S [φ]
2,D (x˜)=λ ∇·S [φ](x˜)n(x˜)+µ ∇S [φ](x˜)+∇ST [φ](x˜) n(x˜)
∂ν 2 2,D 2 2,D 2,D
2
(cid:16) (cid:17)
= λ ∇·S [φ](x)+µ ∇S [φ](x)+∇ST [φ](x) n(x)
2 2,D 2 2,D 2,D
+
h (cid:16) (cid:17)i(cid:12)
(cid:12)
−ǫh′(t) λ ∇·S [φ](x)+µ ∇S [φ](x)+∇ST [(cid:12)φ](x) τ(x)
2 2,D 2 2,D 2,D
+
h (cid:16) (cid:17)i(cid:12)
(cid:12)
+ǫh(x) λ ∇∇·S [φ](x)·n(x)n(x)+µ ∇∇S [φ](x)n(x)n(cid:12)(x)
2 2,D 2 2,D
h
+µ ∇∇ST [φ](x)n(x)n(x) +O(ǫ1+η)
2 2,D
+
i(cid:12)
∂S [φ] ∂S [φ(cid:12)]
2,D 2,D (cid:12)
= (x)+ǫκ(x)h(x) (x)
∂ν2 + ∂ν2 +
(cid:12) (cid:12)
(cid:12) (cid:12)
−ǫ d h(x(cid:12)) C ∇S [φ](x) τ(x) (cid:12) +O(ǫ1+η), x∈∂D, (3.10)
2 2,D
dt
(cid:18) (cid:16) (cid:17) (cid:19)(cid:12)(cid:12)+
b (cid:12)
(cid:12)
10