Vol. 14, No. 4 (2004) 1–22
c World Scientific Publishing Company
MATHEMATICAL ANALYSIS OF NONLINEAR BONDED JOINT MODELS
FRANCOISE KRASUCKI Laboratoire de M´ ecanique et de G´ enie Civil,
UMR 5508 Universit´ e Montpellier II,
Place Eug` ene Bataillon, 34695 Montpellier Cedex 5, France [email protected]
ARNAUD M ¨ UNCH
INRIA-Rocquencourt, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France
[email protected]
YVES OUSSET
Office National d’Etude et de Recherches A´ erospatiales, B.P. 72, 92322 Chatillon Cedex, France
[email protected]
Received 23 October 2002 Revised 23 September 2003 Communicated by P. Podio Guidugli
Within the framework of nonlinear elasticity, we consider the problem of two adherents joined along their common surface by a thin soft adhesive. Two stored energy functions are considered: the stored energy function of Saint Venant–Kirchhoff and the stored energy function of Ciarlet–Geymonat. Using the asymptotic expansion method, the limit energy associated to each of these stored energy functions is obtained. The aim of this paper is to give a rigorous mathematical analysis of the formally derived limit problem.
We show that the limit problem associated to the Saint Venant–Kirchhoff case admits at least one solution and the limit problem associated to the Ciarlet–Geymonat case admits exactly one solution. An analytical comparison in the one-dimensional case and a three-dimensional numerical application are also presented.
Keywords: Bonded joint; nonlinear elasticity; asymptotic development.
1. Introduction
A pioneering paper presented at the Annual Meeting of the American Society of Mechanical Engineers by Goland and Reissner in 1943, deals with the determination of the stresses in cemented lap joints. This problem is formulated as one in plane strain and the authors obtain explicit solutions for two limiting cases. “(a) The case where the cement layer is so thin that its effects on the flexibility of the joint may
1
be neglected; (b) the case in which the joint flexibility is mainly due to the cement layer. Case (b) is significant for the analysis of cemented metal sheets.” (Ref. 1, p. A-17). In this case “the role played by the cemented layer is analogous to that of a system of infinitesimal coil springs positioned between the two plates.” (Ref. 1, p. A-24). Since that time the modelling of bounded structure has progressed at least in two directions:
(1) In order to predict the delamination and the propagation of interlaminar cracks, more sophisticated models (nonlinear elasticity, plasticity, damage, . . .) h ave been introduced for the behavior of the adherents and of the adhesive.
(2) Formal asymptotic developments and mathematically assessed convergence theorems have been used in order to obtain a rational justification of the assumption on linear displacements and constant stresses in the adhesive.
The present paper aims to contribute mainly to the second direction under the following fundamental mechanical assumption:
(A) “The thickness of the adhesive, the flexibility of the adhesive and the external loads are small of the same order (under a natural adimensionalization).”
When all the materials are linearly elastic the assumption (A) corresponds to the case (B) of Goland and Reissner. For general linearly elastic materials, the rigorous convergence theorems of Refs. 2 and 3, prove that the most interesting results are obtained under the assumption (A).
In Sec. 2, we give a precise formulation of assumption (A), in terms of the small adimensional parameter ε, as far as it is concerned the thinness and the flexibility of the adhesive. For this, we at first introduce the domain obtained inserting the adhe- sive of thickness εh between the two adherents and then, we make explicit (see (2.8)) the assumption on the flexibility of the adhesive for two general nonlinear hypere- lastic models: the Saint Venant–Kirchhoff model and a Ciarlet–Geymonat model.
Bothcoincide withthe usual Hooke law for small displacements (see e.g. Refs. 4 and 5) and then the assumption 2.8 is the most interesting (as has been proved e.g. in Refs. 2 and 3).
In Sec. 3, in order to complete the formal statement of assumption (A), we define at first an equivalent problem in a fixed domain and then introduce the assumption (3.2) on the external loads. Under the previous assumptions we look for a formal development of the critical points of the mechanical energy I(ε, E(v) and we identify the first term as a critical point of a suitable energy I
SV0, given by (3.7), for the Saint Venant–Kirchhoff model and I
CG0, given by (3.10), for Ciarlet–Geymonat model.
The next sections of the paper are devoted to a mathematical analysis of the two models. In Sec. 4, we study the Saint Venant–Kirchhoff model. The energy I
SV0is continuous and differentiable for the strong topology of V (defined in (4.3)).
However, the energy is non-convex. In order to prove the existence of at least
one critical point, we introduce th e energy ˆ I
SV0defined on ˆ V (defined by (4.12)),
where the adhesive has been replaced by the surface S and the corresponding part
of the energy by a surface energy. To a critical point of ˆ I
SV0one can associate a regular critical point of I
SV0, i.e. a critical point linear in the thickness variable of the adhesive. In order to prove that ˆ I
SV0has at least one critical point, we remark that the compact imbedding of H
1/2(S) into L
2(S) implies the lower semicontinuity for the weak topology of ˆ V . We also prove that, for a suitable choice of the loads, I
SV0may have many (non-regular, i.e. only piecewise linear in the thickness variable of the adhesive) critical points. We conjecture that only a regular critical point of I
SV0corresponding to a critical point of ˆ I
SV0may be a minimum of I
SV0. The situation is completely different for the Ciarlet–Geymonat model I
CG0. Indeed, sucha functional is convex and positive on a convex subset U of W (see (5.1) and (5.2)). Following the approach of J. Ball (Ref. 7), we can prove that there exists a unique minimum of I
CG0. As in Sec. 4 we also introduce a functional ˆ I
CG0defined on ˆ U (see (5.14) and deduce that u is linear in the thickness variable of the adhesive and it is associated to the unique minimum ˆ u of ˆ I
CG0.
The last two sections are devoted to the comparison of the models. In Sec. 6 such a comparison is done in a one-dimensional situation where the structure is composed of three bars. When the solution is linear in the adhesive one can explicitly compute the energy in the adhesive as a function of the strain. At last in Sec. 7 we test the asymptotic models for the single lap joint structure, one of the most studied in the literature. As in the one-dimensional case, the Ciarlet–Geymonat model is slightly softer than the Saint Venant–Kirchhoff model. For this last case a comparison is also done with the complete three-dimensional model. The numerically obtained rate of convergence is in good agreement with the assumption on the asymptotic expansion of u(ε).
2. Problem Statement
In the three-dimensional Euclidean space E
3referred to the orthonormal frame (0; e
1, e
2, e
3), let Ω
−and Ω
+be two disjoint domains withpiecewise-C
2boundaries
∂Ω
+and ∂Ω
−. Let S = ∂Ω
+∩ ∂Ω
−be the common part of the external surfaces
which is assumed to have a positive 2D measure and which is assumed to be pro-
jectable onto the plane { x
3= 0 } . To “insert the adhesive”, let Ω
+(resp. Ω
−) be
moved in the e
3(resp. − e
3) direction of an amount equal to the half-thickness, εh/2
of the joint. ε is a small dimensionless parameter and h is a global characteristic
length (for example the diameter of Ω). Then, let Ω
±ε= { x
ε:= x ± εh/2e
3, x ∈ Ω
±} ,
Ω
mε= { x
ε:= x + εz/2e
3, − h < z < h, x ∈ S } , S
ε±= { x
ε:= x ± εh/2e
3, x ∈ S }
and Ω
ε= Ω
+ε∪ Ω
−ε∪ Ω
mε∪ S
ε+∪ S
ε−be the physical reference configuration of
the assembly (see Fig. 1(a)). Following the usual terminology, Ω
+εand Ω
−εare
filled by the adherents, while Ω
mεis filled by the adhesive. The structure is fixed
on the parts Γ
+u⊂ ∂Ω
+ε, resp. Γ
−u⊂ ∂Ω
−ε. The external boundary of the adhe-
sive is traction-free. The complementary part of the boundary Γ
fis submitted to
surface loads f
εd. Obviously, one can consider other types of boundary conditions
.
h
+
f
f S
++
m S
~e
3~e
2~e
1u
+
u
.
(a)
.
+
f
+
u
S
+h S
u f
m S
+
tr
+tr
(b)
Fig. 1. Bonded assembly: (a) the physical problem (Ω
ε), (b) the rescaled problem (Ω
tr≡Ω).
(e.g. a complementary combination of components of the stress vector and of the displacement as in the numerical example of Sec. 7).
The study is within the framework of nonlinear elasticity and we consider that eachbody is made of an isotropic h yperelastic material. A Lagrangian descrip- tion is used to describe the deformation of the structure: we design by E the Green–Lagrange strain tensor defined by
E(u) = 1
2 (F
t(u)F(u) − I
3) ; F(u) = ∇ u + I
3, (2.1) where F is the gradient of the mapping x
ε→ x
ε+ u
ε(x
ε), u
εthe displacement field, I
3is the identity tensor of E
3and ∇ u
ε= (
∂u∂xεε). Finally, we denote by σ
ε(x
ε) the second Piola–Kirchhoff stress, a symmetric tensor-valued field. The hyperelastic behavior is given by the relation
σ
ε(x
ε) = ∂ W ˘
ε∂E (x
ε, E(u
ε(x
ε))) , (2.2) where ˘ W
εdesigns the stored energy function. In this work, we are concerned with two stored energy functions. These are
• the stored energy function of Saint Venant–Kirchhoff (Ref. 4):
W ˆ
SVε(F) = ˘ W
SVε(E) = λ
ε2 (Tr E)
2+ µ
εTr E
2= − 3λ
ε+ 2µ
ε4 F
2+ λ
ε+ 2µ
ε8 Tr((F
tF)
2) + λ
ε4 cof F
2+ 6µ
ε+ 9λ
ε8 , (2.3)
( F
2= Tr(F
tF), cof F
2=
12( F
4− Tr(F
tF)
2) where Tr designs the trace operator), which is the simplest hyperelastic stored energy function valid for small strains;
• the stored energy function of Ciarlet–Geymonat (Ref. 5):
W ˆ
CGε(F) = a F
2+ b cof F
2+ Γ(det F) + e , (2.4) withΓ(δ) = cδ
2− d log(δ) (δ > 0) and withthe coefficients a, b, c, d ∈ R
∗+and e ∈ R satisfying the relations
3a + 3b + Γ(1) + e = 0 , 2a + 4b + Γ
(1) = 0 , 2b + 1
2 Γ
(1) + 1
2 Γ
(1) = λ
ε2 , − 2b − Γ
(1) = µ
ε,
(2.5) leading to:
W ˘
CGε(E) = λ
ε2 + µ
εTr E + λ
ε2 [(Tr E)
2− Tr E
2] + λ
εdet E − λ
ε+ 2µ
ε4 log(det(I + 2E)) . (2.6) Remark 1. For small strains, the stored energy function of Ciarlet–Geymonat W ˘
CGεcoincides, for a suitable choice of the coefficients as in (2.5), with the stored energy function ˘ W
SVε(see Ref. 5):
W ˘
CGε(E) = ˘ W
SVε(E) + o( E ) . (2.7) Remark 2. The stored energy function of Ciarlet–Geymonat is defined for det F > 0, which corresponds to the local orientation-preserving condition.
As in Ref. 6, the adhesive is assumed to be soft, i.e. its elastic stiffness is small with respect to the ones of the adherents. This is stressed by introducing λ
m, λ
+, λ
−(resp. µ
m, µ
+, µ
−) independent of ε and of same order, suchthat:
λ
ε= λ
±, µ
ε= µ
±in Ω
±ε,
λ
ε= ελ
m, µ
ε= εµ
min Ω
mε. (2.8) Finally, the mechanical energy is defined as
I
ε(E(v)) =
Ωε
W ˘
ε(x
ε, E(v)) dx
ε− L
ε(v) , (2.9) L
εbeing a linear form associated to the surface load:
L
ε(v) =
Γf
f
εd· v dΓ . (2.10)
In the case of ˘ W
SVε, the corresponding energy, seen as a function of the displacement field u
ε, is defined on the functional space
V
SV(Ω
ε) = { v ∈ (W
1,4(Ω
ε))
3, v = 0 on Γ
±u} , (2.11)
whereas in the case of ˘ W
CG, the corresponding space is:
V
CG(Ω
ε) = { v ∈ (W
1,6(Ω
ε))
3, det F(v) > 0, v = 0 on Γ
±u} . (2.12) Remark 3. The parameter ε being fixed, the solution u
εis, at least formally, a critical point of the mechanical energy. In the case of the stored energy function of Ciarlet–Geymonat, thanks to J. Ball’s results, we know that there exists a minimum (Ref. 9). This point is still an open question for the energy associated to ˘ W
SVε(Ref. 2).
The aim of this paper is to give a rigorous mathematical analysis of the limit problem obtained formally in Refs. 6 and 8. In the text, the repeated summation convention is used: Latin i, j, k, . . . (resp. Greek α, β , . . .) indices take their values in the set { 1, 2, 3 } (resp. { 1, 2 } ).
3. Rescaling and Asymptotic Expansion
(i) In order to make apparent the dependence on ε of the problem, we define, following the approach of Ciarlet and Destuynder (Ref. 9), an equivalent problem in the fixed domain Ω
tr(see Fig. 1(b)). For this purpose, we set π
ε: x = (x
1, x
2, x
3) ∈ Ω
tr→ x
ε= (x
ε1, x
ε2, x
ε3) ∈ Ω
εdefined by:
π
ε(x
1, x
2, x
3) =
x
1, x
2, x
3− h 2 (1 − ε)
∈ Ω
+ε, for x ∈ Ω
+tr, π
ε(x
1, x
2, x
3) = (x
1, x
2, εx
3) ∈ Ω
mε, for x ∈ Ω
m, π
ε(x
1, x
2, x
3) =
x
1, x
2, x
3+ h 2 (1 − ε)
∈ Ω
−ε, for x ∈ Ω
−tr,
(3.1)
withΩ
±tr= { x ±
h2e
3, x ∈ Ω
±} , Ω
m= { x +
z2e
3, − h < z < h, x ∈ S } , S
±= { x ±
h2e
3, x ∈ S } . In order to simplify the notations, we identify Ω
+trwithΩ
+and Ω
−trwithΩ
−. At last, we set ¯ Ω = ¯ Ω
+∪ Ω ¯
−∪ Ω ¯
m. The displacement field and the elastic properties of the bodies are then defined without rescaling: u(ε, x) = u
ε(x
ε) = u
ε◦ π
ε(x), λ(ε, x) = λ
ε(x
ε) and µ(ε, x) = µ
ε(x
ε) whereas the following rescaling of the load is performed:
f
εd(x
ε) = εf
d(x) . (3.2)
Let us also recall the usual relations
∂x∂εα
(φ
ε(x
ε)) =
∂x∂α
(φ(x)),
∂x∂ε3
(φ
ε(x
ε)) = ε
−1∂x∂3
(φ(x)) and
Ωmε
φ
ε(x
ε) dx
ε= ε
Ωmφ
ε◦ π
ε(x) dx ≡ ε
Ωmφ(x) dx for any differentiable function φ
εdefined in Ω
mε. Then the equivalent problem is to look for a critical point of the following energy defined in Ω:
I(ε, E(v)) =
Ω
W ˘ (ε, x, E(v)) dx − L(ε, v) , (3.3) where L(ε, v) = ε
Γf
f
d· v dΓ.
(ii) In the Saint Venant–Kirchhoff case, we obtain
Ω
W ˘
SV(ε, x, E(v)) dx =
Ω±
λ
±2 (Tr E(v))
2+ µ
±Tr(E(v))
2dΩ +
Ωm
(ε
−2W
−2(v) + ε
−1W
−1(v) + W
0(v) + ε
1W
1(v) + ε
2W
2(v)) dΩ , (3.4) with
W
−2(v) = 1
8 (λ
m+ 2µ
m)(v
k,3v
k,3)
2, W
−1(v) = 1
2 (λ
m+ 2µ
m)v
k,3v
k,3v
3,3, W
0(v) = 1
2 (λ
m+ 2µ
m)v
23,3+ λ
m2 v
k,3v
k,3v
α,α+ 1
2 v
p,αv
p,α+ µ
m2 (v
α,3+ v
k,αv
k,3)
2, W
1(v) = λ
mv
α,α+ 1
2 v
k,αv
k,αv
3,3+ µ
mv
3,α(v
α,3+ v
k,αv
k,3) , W
2(v) = λ
m2
v
α,α+ 1
2 v
k,αv
k,α 2+ µ
m2 v
3,αv
3,α+ µ
m4 (v
α,β+ v
β,α+ v
k,αv
k,β)(v
α,β+ v
β,α+ v
l,αv
l,β) . This suggests to look formally for a solution u(ε, x) such that:
u(ε, x) = u
0(x) + εu
1(x) + · · · . (3.5) One can easily show that, thanks to (3.2), u
0≡ 0 in Ω leading to th e expansion
I
SV(ε, E(u(ε))) = ε
2I
SV0(u
1) + O(ε
3) . (3.6) As a consequence, the initial problem is, for the leading term, formally reduced to the search of the critical points of the following energy:
I
SV0(v) =
Ω
W ˘
0(x, E(v)) dx − L
0(v) , (3.7) with
Ω
W ˘
0(x, E(v)) dx
=
Ω±
λ
±2 (Tr e(v))
2+ µ
±Tr(e(v))
2dΩ +
Ωm
W
−2(v) + W
−1(v) + 1
2 (λ
m+ 2µ
m)v
3,32+ µ
m2 v
α,3v
α,3dΩ , (3.8)
where L
0(v) =
Γf
f
d· v dΓ and e(v) =
12( ∇ v + ( ∇ v)
t) th e linear part of th e strain tensor E.
(iii) In a similar way for the Ciarlet–Geymonat case, we assume u(ε, x) = εu
1(x) + · · · and we obtain
I
CG(ε, E(u(ε))) = ε
2I
CG0(u
1) + O(ε
3) , (3.9) with
I
CG0(v) =
Ω±
λ
±2 (Tr e(v))
2+ µ
±Tr(e(v))
2dΩ +
Ωm
µ
m+ λ
m2
1
2 v
23,3+ v
3,3− log(1 + v
3,3)
dΩ +
Ωm
µ
m2 v
α,3v
α,3dΩ −
Γf
f
d· v dΓ . (3.10)
In the following, the quantity u
1will be simply noted as u.
4. Study of the Limit Problem: The Saint Venant Kirchhoff Case (i) From the relation
W
−2(v) + W
−1(v) + 1
2 (λ
m+ 2µ
m)v
23,3+ µ
m2 v
α,3v
α,3= 1
8 (λ
m+ 2µ
m)((v
1,32+ v
22,3) + (v
3,3+ 1)
2− 1)
2+ µ
m2 (v
1,32+ v
2,32) , (4.1) the limit energy becomes simply:
I
SV0(v) =
Ω±
λ
±2 (Tr e(v))
2+ µ
±Tr(e(v))
2dΩ + 1
8
Ωm
(λ
m+ 2µ
m)((v
1,32+ v
2,32) + (v
3,3+ 1)
2− 1)
2dΩ + 1
2
Ωm
µ
m(v
21,3+ v
22,3) dΩ − L
0(v) . (4.2) Then, assuming f
d∈ (L
2(Γ
f))
3, the functional I
SV0is defined and continuous on the space:
V = { v
−∈ (H
1(Ω
−))
3, v
+∈ (H
1(Ω
+))
3, v
m∈ (L
4(Ω
m))
3, v
k,3m∈ L
4(Ω
m) , v
+|S+= v
m|S+; v
−|S−= v
m|S−, v
±= 0 on Γ
±u} , (4.3) where v
+, v
−and v
mare the restrictions of v to Ω
+, Ω
+and Ω
m, respectively. V is endowed withthe natural norm v
Vsuchthat:
v
2V= v
+2(H1(Ω+))3+ v
−2(H1(Ω−))3+ v
m2(L4(Ωm))3+ v
m,32(L4(Ωm))3. (4.4)
Proposition 1. I
SV0is continuous for the strong topology of V and we have, for all w ∈ V :
DI
SV0(u)(w) =
Ω±
(λ
±e
kk(u)e
kk(w) + 2µ
±e
ij(u)e
ij(w)) dΩ + 1
2
Ωm
(λ
m+ 2µ
m)((u
21,3+ u
22,3) + (u
3,3+ 1)
2− 1)
× (u
1,3w
1,3+ u
2,3w
2,3+ (u
3,3+ 1)w
3,3) dΩ +
Ωm
µ
m(u
1,3w
1,3+ u
2,3w
2,3) dΩ −
Γf
f
d· w dΓ . (4.5) Remark 4. The functional I
SV0is not convex; as a consequence, the continuity for the strong topology does not imply the lower semi-continuity for the weak topology.
(ii) Thanks to Proposition 1, the critical points u ∈ V of I
SV0are solutions of the following variational problem: ∀ w ∈ V ,
( P )
Ω±
(λ
±e
kk(u)e
kk(w) + 2µ
±e
ij(u)e
ij(w)) dΩ + 1
2
Ωm
(λ
m+ 2µ
m)((u
21,3+ u
22,3) + (u
3,3+ 1)
2− 1)
× (u
1,3w
1,3+ u
2,3w
2,3+ (u
3,3+ 1)w
3,3) dΩ +
Ωm
µ
m(u
1,3w
1,3+ u
2,3w
2,3) dΩ =
Γf
f
d· w dΓ .
(4.6)
The corresponding local equations are:
σ
±ij= λ
±e
kk(u
±)δ
ij+ 2µ
±e
ij(u
±) in Ω
±,
σ
±ij,j= 0 in Ω
±,
(4.7)
σ
mα3= µ
mu
mα,3in Ω
m,
σ
m33= 1
2 (λ
m+ 2µ
m)[(u
m1,3)
2+ (u
m2,3)
2+ (u
m3,3+ 1)
2− 1] in Ω
m, (σ
i3m+ σ
33mu
mi,3)
,3= 0 in Ω
m,
(4.8) u
±= u
mon S
±,
σ
±i3= σ
i3m+ σ
33mu
mi,3on S
±, (4.9) u
±= 0 on Γ
±u,
σ
±· n = f
don Γ
f. (4.10)
It appears that, at the first order, the behavior in the adherents is that of linear elas-
ticity whereas it remains nonlinear in the adhesive. In order to prove the existence
of at least one solution to problem ( P ), we introduce the following functional I ˆ
SV0(v) = 1
2
Ω±
(λ
+(e
kk(v
±))
2+ 2µ
±e
ij(v
±)e
ij(v
±)) dΩ + 1
2h
S
λ
m+ 2µ
m4h
2([v
1]
2+ [v
2]
2+ ([v
3] + h)
2− h
2)
2dΩ + 1
2h
S
µ
m([v
1]
2+ [v
2]
2) dΩ −
Γf
f
d· v dΓ , (4.11) where [v](x
1, x
2) = v
+(x
1, x
2,
h2) − v
−(x
1, x
2, −
h2) defined on S , is the jump of the displacement field across Ω
m. ˆ I
SV0is defined on the space
V ˆ = { v
−∈ (H
1(Ω
−))
3, v
+∈ (H
1(Ω
+))
3; v
±= 0 on Γ
±u} , (4.12) endowed withthe norm v
Vˆsuchthat v
2Vˆ= v
+2(H1(Ω+))3+ v
−2(H1(Ω−))3. Proposition 2. If u ˆ ∈ V ˆ is a critical point of I ˆ
SV0, then
u =
ˆ
u in Ω
±,
u ˆ
+h
h 2 + x
3+ u ˆ
−h h
2 − x
3in Ω
m, (4.13)
is a regular solution of problem ( P ).
Proof. Thanks to the continuous imbedding (H
1/2(S))
3⊂ (L
4(S))
3, we verify that u defined by (4.13) belongs to V . Furth ermore, if ˆ u is a critical point of ˆ I
SV0, then for all ˆ w ∈ V ˆ ,
Ω±
(λ
±e
kk(ˆ u)e
kk( ˆ w) + 2µ
±e
ij(ˆ u)e
ij( ˆ w)) dΩ + 1
h
S
λ
m+ 2µ
m2h
2([ˆ u
1]
2+ [ˆ u
2]
2+ ([ˆ u
3] + h)
2− h
2)
× ([ˆ u
1][ ˆ w
1] + [ˆ u
2][ ˆ w
2] + ([ˆ u
3] + h)[ ˆ w
3]) dS + 1
h
S
µ
m([ˆ u
1][ ˆ w
1] + [ˆ u
2][ ˆ w
2]) dS =
Γf
f
d· w dΓ . (4.14) Using (4.5) and (4.13), we have for w ∈ V ,
DI
SV0(u)(w) =
Ω±
(λ
±e
kk(ˆ u)e
kk( ˆ w) + 2µ
±e
ij(ˆ u)e
ij( ˆ w)) dΩ
+ 1 2
Ωm
(λ
m+ 2µ
m) 1
h
2([ˆ u
1]
2+ [ˆ u
2]
2) + [ˆ u
3]
h + 1
2− 1
× [ˆ u
1]
h w
1,3+ [ˆ u
2] h w
2,3+
[ˆ u
3] h + 1
w
3,3dΩ +
Ωm
µ
m[ˆ u
1]
h w
1,3+ [ˆ u
2] h w
2,3dΩ −
Γf
f
d· w dΓ . (4.15)
Then, since w ∈ V implies ˆ w ≡ w
|Ω+∪Ω−∈ V ˆ , we deduce, identifying S
+and S
−with S, and thanks to (4.14) and to the Fubini’s theorem that DI
SV0(u)(w) = 0 for all w ∈ V .
(iii) Theorem 1. The functional I ˆ
SV0defined on the space V ˆ has at least one critical point.
Proof. To obtain the result, we show that ˆ I
SV0is lower bounded on ˆ V and reaches its infimum. Thanks to the Korn’s inequality, there exist two reals α > 0, β > 0 suchthat
I ˆ
SV0(v) ≥ α v
2Vˆ− β f
d(L2(Γf))3v
Vˆ. (4.16) As a consequence, we have to show only the lower semi-continuity for the weak topology. From the relation:
([v
1]
2+ [v
2]
2+ [v
3+ h]
2− h
2)
2= ([v
1]
2+ [v
2]
2+ [v
3+ h]
2)
2+ h
4− 2h
2([v
1]
2+ [v
2]
2+ [v
3+ h]
2) , (4.17) I ˆ
SV0appears to be the sum of a convex and continuous terms for the strong topology, a continuous linear form and an additional term: −
4h1 S(λ
m+ 2µ
m)([v
1]
2+ [v
2]
2+ [v
3+ h]
2) dS. The compactness of the injection H
1/2(S) ⊂ L
2(S) implies that this additional term is weakly continuous. Since a convex and continuous function is also weakly lower semi-continuous, ˆ I
SV0is lower semi-continuous. The minimum is then reached.
(iv) Identifying S
+and S
−with S, the transmission conditions on S become, for any critical point of ˆ I
SV0:
σ
α3+= σ
α3−= µ
mh [u
α] + λ
m+ 2µ
m2h
3([u
1]
2+ [u
2]
2+ ([u
3] + h)
2− h
2)[u
α] , σ
33+= σ
33−= λ
m+ 2µ
m2h
3([u
1]
2+ [u
2]
2+ ([u
3] + h)
2− h
2)([u
3] + h) .
(4.18)
From (4.13) and (4.18) it follows that any critical point of ˆ I
SV0gives rise to a solution of the problem with linear displacements and constant stresses in the adhesive.
The structure of the system (4.18) suggests that, for some choice of f
d, th e functional ˆ I
SV0could have many critical points. For f
d≡ 0, ˆ v = 0 is the unique minimum. Furthermore, we have:
D
2I ˆ
SV0(v)(w, w) = ˆ
Ω±
(λ
±e
kk(w)e
kk( ˆ w) + 2µ
±e
ij(w)e
ij( ˆ w)) dΩ + 1
h
S
µ
m([w
1][ ˆ w
1] + [w
2][ ˆ w
2]) dS
+ 1 2h
3S
(λ
m+ 2µ
m)([v
1]
2+ [v
2]
2+ ([v
3] + h)
2− h
2)
× ([w
1][ ˆ w
1] + [w
2][ ˆ w
2] + [w
3][ ˆ w
3]) dS + 1
h
3S
(λ
m+ 2µ
m)([v
1][ ˆ w
1] + [v
2][ ˆ w
2] + ([v
3] + h)[ ˆ w
3])
× ([v
1][w
1] + [v
2][w
2] + ([v
3] + h)[w
3]) dS , (4.19) leading to
D
2I ˆ
SV0(0)(w, w) = ˆ
Ω±
(λ
±e
kk(w)e
kk( ˆ w) + 2µ
±e
ij(w)e
ij( ˆ w)) dΩ + 1
h
S
µ
m([w
1][ ˆ w
1] + [w
2][ ˆ w
2]) dS + 1
h
S
(λ
m+ 2µ
m)[w
3][ ˆ w
3] dS , (4.20) which is a definite positive isomorphism. This implies that ˆ I
SV0is strictly convex at 0 and so, for f
d(L2(Γf))3small enough, the functional ˆ I
SV0has a unique absolute minimum. Similar arguments can be used on the functional I
SV0to prove
Proposition 3. There exists δ > 0 such that for f
d(L2(Γf))3< δ, the problem ( P ) has a unique solution, which is the absolute minimum of I
SV0on V.
(v) In order to study the non-uniqueness of the solution of ( P ), let us consider the application G : R
3→ R
3; (x
1, x
2, x
3) → (G
1, G
2, G
3) defined by:
G
1(x
1, x
2, x
3) = µ
mx
1+ λ
m+ 2µ
m2 x
1(x
21+ x
22+ (x
3+ 1)
2− 1) , G
2(x
1, x
2, x
3) = µ
mx
2+ λ
m+ 2µ
m2 x
2(x
21+ x
22+ (x
3+ 1)
2− 1) , G
3(x
1, x
2, x
3) = λ
m+ 2µ
m2 (x
21+ x
22+ (x
3+ 1)
2− 1)(1 + x
3) .
(4.21)
Then, Eqs. (4.8) and (4.9) become simply:
u
±= u
mon S
±,
G(u
m,3) = σ
+· e
3= σ
−· e
3in Ω
m. (4.22)
If, f
dbeing fixed, this system determine u
m,3in a unique way, then u
mis linear
withrespect to x
3in Ω
mand it is so completely determined by [u]. This solution
corresponds to the regular one obtained from Proposition 2. If, on the contrary,
the system (4.22) possesses several solutions, then, we can take u
mpiecewise linear
withrespect to the variable x
3on arbitrary subintervals of ] −
h2,
h2[. These solutions
do not correspond to critical points of ˆ I
SV0. An example of this situation is given
by the case σ
+· e
3= 0. In this case, the system (4.22) has the solutions
u
mi,3= 0 , (4.23) u
mα,3= 0 , u
m3,3= − 1 , (4.24) (u
m1,3)
2+ (u
2,3)
2= λ
mλ
m+ 2µ
m, u
m3,3= − 1 . (4.25) If f
d≡ 0, the unique solution is u
mi,3= 0, which corresponds to the regular solution of Proposition 2, and so I
SV0has a strict minimum. Besides, thanks to a straight- forward computation of the second derivative, we can easily show that the other solutions do not correspond to a strict minimum of I
SV0. We conjecture th at th is situation is general: the minima of I
SV0are (regular) critical points associated to the minima of ˆ I
SV0.
5. Study of the Limit Problem: The Ciarlet Geymonat Case (i) The functional I
CG0(3.10) is defined on the set:
U = { v = (v
−, v
m, v
+); v ∈ (L
2(Ω))
3, v
−∈ (H
1(Ω
−))
3, v
+∈ (H
1(Ω
+))
3, v
mk,3∈ L
2(Ω
m), (1 + v
m3,3) > 0 p.p. in Ω
m, log(1 + v
3,3m) ∈ L
1(Ω
m) ,
v
|+S+= v
m|S+, v
−|S−= v
|mS−, v = 0 on Γ
±u} . (5.1) By virtue of the convexity of the function x → x − log(1 + x), U is a non-empty (0 ∈ U ) convex set of the Hilbert’s space:
W = { v = (v
−, v
m, v
+); v ∈ (L
2(Ω))
3, v
−∈ (H
1(Ω
−))
3, v
+∈ (H
1(Ω
+))
3, v
mk,3∈ L
2(Ω
m), v
+|S+= v
m|S+, v
−|S−= v
m|S−, v = 0 on Γ
±u} , (5.2) endowed withthe natural norm v
Wsuchthat
v
2W= v
+2(H1(Ω+))3+ v
−2(H1(Ω−))3+ v
m2(L2(Ωm))3+ v
m,32(L2(Ωm))3. (5.3) Following the approach of J. Ball (see Ref. 7), we extend the domain of definition of I
CG0to W . To this end, let us define the function g : R → R ¯ ≡ R ∪ {±∞} :
g(ζ) =
(λ
m+ 2µ
m) 1
2 ζ
2+ ζ − log(1 + ζ)
if 1 + ζ > 0 ,
+ ∞ if 1 + ζ ≤ 0 .
(5.4) Remark 5. g is a Carath´ eodory’s function.
Lemma 1. The functional
v → F (v) =
Ωm
g(v(x)) dx , (5.5)
is convex, positive and weakly lower semi-continuous from L
2(Ω
m) to R ¯ .
Proof. Since g(ζ) ≥ 0 and is a Carath´ eodory’s function, the functional F is positive and lower semi-continuous from L
2(Ω
m) to ¯ R (see Ref. 10). Furthermore, ζ → g(ζ) is convex and so v → F(v) is convex.
(ii) Theorem 2. There exists at least an element u ∈ U such that:
I
CG0(u) = Inf
v∈U
I
CG0(v) . (5.6)
Proof. From 0 ∈ U and I
CG0(0) = 0, we deduce that:
Inf
v∈U
I
CG0(v) ≤ 0 . (5.7)
We then extend the definition of the functional I
CG0to the space W by defining:
I ¯
CG0(v) =
Ω±
λ
±2 (e
kk(v
±))
2+ µ
±e
ij(v
±)e
ij(v
±)
dΩ + 1 2 F (v
m3,3) + 1
2
Ωm
µ
m((v
1,3m)
2+ (v
2,3m)
2) dΩ −
Γf
f
d· v dΓ . (5.8) We remark that ¯ I
CG0(v) < ∞ ⇒ v ∈ U . Indeed, if ¯ I
CG0(v) < ∞ , th en F (v
3,3m) =
Ωm
g(v
3,3m(x)) dx < ∞ and so 1 + v
3,3m> 0 p.p. in Ω
m. Besides, log(1 + v
m3,3) = 1
2 (v
m3,3)
2+ v
3,3m− 1
λ
m+ 2µ
mg(v
3,3m(x)) ∈ L
1(Ω
m) , (5.9) thanks to v
3,3m∈ L
2(Ω
m), g(v
m3,3) ∈ L
1(Ω
m) and
λm+2µ1 m∈ L
∞(Ω
m). We deduce then
I ¯
CG0(v) = I
CG0(v) < ∞ ⇐⇒ v ∈ U . (5.10) Then, thanks to Lemma 1, ¯ I
CG0is convex, weakly lower semi-continuous on W in R ¯ . Since g(ζ) ≥
12(λ
m+ 2µ
m)ζ
2, for ζ > − 1, we deduce, like in Ref. 2, that there exist two reals α > 0, β > 0 such that:
I ¯
CG0(v) ≥ α v
2W− β f
dL2(Γf)v
±(H1(Ω±))3. (5.11) Hence, there exists u ∈ W suchthat ¯ I
CG0(u) = Inf
v∈WI ¯
CG0(v) ≤ 0. From (5.10), u ∈ U showing that ¯ I
CG0reaches (and so I
CG0) its minimum on u ∈ U :
I
CG0(u) = inf
v∈U
I
CG0(v) = inf
v∈W
I ¯
CG0(v) . (5.12)
Proposition 4. The minimum u of I
CG0is unique.
Proof. The result is a direct consequence of the strict convexity of the application ζ → g(ζ) for 1 + ζ > 0.
When the solution u satisfies the regularity condition (1 + u
m3,3)
−1∈ L
∞(Ω
m),
then ¯ I
CG0admits a directional derivative for all v ∈ W
∞= { v ∈ W ; v
m3,3∈
L
∞(Ω
m) } . Moreover the minimum u satisfies the following variational equation
for all v ∈ W
∞:
( Q )
Ω±
(λ
±e
kk(u
±)e
kk(v
±) + 2µ
±e
ij(u
±)e
ij(v
±)) dΩ + 1
2
Ωm
(λ
m+ 2µ
m)
1 + u
m3,3− 1 1 + u
m3,3v
3,3mdΩ +
Ωm
µ
m(u
m1,3v
m1,3+ u
m2,3v
2,3m) dΩ −
Γf
f
d· v dΓ = 0 .
(iii) As in the Saint Venant–Kirchhoff case, it is possible to define the functional I ˆ
CG0(v) = 1
2
Ω±
(λ
±(e
kk(v
±))
2+ 2µ
±e
ij(v
±)e
ij(v
±)) dΩ + 1
2h
S
(λ
m+ 2µ
m) 1
2 [v
3]
2+ h[v
3] − h
2log
1 + [v
3] h
dS + 1
2h
S
µ
m([v
1]
2+ [v
2]
2) dS −
Γf
f
d· v dΓ , (5.13)
on the set U ˆ =
v
−∈ (H
1(Ω
−))
3, v
+∈ (H
1(Ω
+))
3, 1 + [v
3]
h > 0 p.p. on S , log
1 + [v
3]
h
∈ L
1(S), v
±= 0 on Γ
±u, (5.14)
which is a nonempty convex subset of the Hilbert space:
W ˆ = { v
−∈ (H
1(Ω
−))
3, v
+∈ (H
1(Ω
+))
3; v
±= 0 on Γ
±u} ,
endowed withthe natural norm v
2Wˆ= v
+2(H1(Ω+))3+ v
−2(H1(Ω−))3. Th en, in a similar way to Lemma 1, we have:
Lemma 2. Let ˆ g be the Carath´ eodory’s function defined as:
ˆ g(ζ) =
(λ
m+ 2µ
m) 1
2 ζ
2+ hζ − h
2log
1 + ζ h
if 1 + ζ h > 0 ,
+ ∞ if 1 + ζ
h ≤ 0 .
(5.15)
Then, the functional v → F(v) = ˆ
Sˆ g(v(x)) dS is convex positive and weakly lower semi-continuous from L
2(S) to R ¯ .
As in the proof of Theorem 2, we extend the definition of ˆ I
CG0(v) to the space W ˆ with:
I ˜
CG0(v) = 1 2
Ω±
(λ
±(e
kk(v
±))
2+ 2µ
±e
ij(v
±)e
ij(v
±)) dΩ + 1
2h
S
µ
m([v
1]
2+ [v
2]
2) dS + 1 2h
F ˆ ([v
3]) −
Γf
f
d· v dΓ . (5.16)
Then, in a similar approach to Theorem 2 and Proposition 4, we obtain
Theorem 3. There exists a unique element u ˆ ∈ U ˆ such that : I ˆ
CG0(ˆ u) = inf
v∈Uˆ
I ˆ
CG0(u) = inf
v∈Wˆ
I ˜
CG0(v) . (5.17) When the solution ˆ u satisfies the regularity condition (h + [ˆ u
3])
−1∈ L
4/3(S), then I ˆ
CG0is Gˆ ateaux-differentiable at ˆ u. Moreover, the minimum ˆ u can be characterized by the following variational equation ∀ v ∈ W ˆ :
( ˆ Q )
Ω±
(λ
±e
kk(ˆ u
±)e
kk(v
±) + 2µ
±e
ij(ˆ u
±)e
ij(v
±)) dΩ + 1
2h
S
(λ
m+ 2µ
m)
h + [ˆ u
3] − 1 h + [ˆ u
3]
[v
3] dS + 1
h
S
µ
m([ˆ u
1][v
1] + [ˆ u
2][v
2]) dS −
Γf
f
dv dΓ = 0 . (iv) Proposition 5. If ˆ u ∈ U ˆ verifies (5.17) then
˜ u =
u ˆ in Ω
±,
ˆ u
+h h
2 + x
3+ u ˆ
−h h
2 − x
3in Ω
m,
(5.18) is the minimum of I
CG0.
Proof. (i) Let us verify that ˜ u ∈ U. Indeed, if ˜ u ∈ U ˆ , th en ˜ u defined by (5.18) verifies ˜ u
m=
uˆh+(
h2+x
3)+
uˆh−(
h2− x
3) ∈ L
2(Ω
m), ˜ u
m,3=
[ˆu]h∈ (L
2(Ω
m))
3, 1+ ˜ u
m3,3= 1 +
[ˆuh3]> 0 p.p. in Ω
mand
Ωmlog(1 + ˜ u
m3,3) dΩ
m= h
Slog(1 +
[ˆuh3]) dS < ∞ .
(ii) Finally, let us show that
I ˜
CG0(˜ u) ≤ I ˜
CG0(v) ∀ v ∈ U . (5.19) We associate to v the function ˜ v as follows:
˜ v =
˜
v
±= v
±in Ω
±,
˜
v
m= v
+h
h 2 + x
3+ v
−h h
2 − x
3in Ω
m. (5.20) Then, using the equality ˜ v
m,3=
[v]h=
1h h2−h2
v
,3dx
3and the convexity of ζ → g(ζ), we write
g(˜ v
m3,3) ≤ 1 h
h2−h2
g(v
m3,3) dx
3. (5.21)
For v ∈ W , we note ˆ v ∈ W ˆ the restriction of v to Ω
+and Ω
−. An explicit computation leads, with(5.20) and (5.21), to ˜ I
CG0(ˆ v) ≤ I ˜
CG0(˜ v) ≤ I ˜
CG0(v) and so, thanks to (5.17), we get for any v ∈ W :
I ˆ
CG0(ˆ u) = I
CG0(˜ u) ≤ I ˜
CG0(ˆ v) ≤ I ˜
CG0(v) . (5.22)
.
0 1
–3 –2 –1 1
(CG) (SV)
[u3]
h 2
h(m+2m) I
0
m
[u3
]=h
Hooke
.
Fig. 2. Strain energies
IΩ0massociated to ˘
WSV(E) [(SV )], to ˘
WCG(E) [(CG)] and to ˘
WSV(e) [Hooke].
6. Comparison in the One-Dimensional Case
In the one-dimensional case, we consider a structure composed of three bars. The adherents have a length h and the adhesive bar has a length εh. We consider only the most significant case when the solution is linear in the adhesive. In the Saint Venant–Kirchhoff case, thanks to Proposition 2, the energy of the solution in the adhesive is:
I
Ω0m(u
3) = h
2 (λ
m+ 2µ
m) [u
3]
h + 1 2
[u
3] h
22. (6.1)
The transmission conditions (4.9) become simply σ
±33= (λ
m+ 2µ
m)
[u
3] h + 1
2 [u
3]
h
21 + [u
3] h
. (6.2)
In the Ciarlet–Geymonat case, we obtain in a similar way, by virtue of Proposition 5 I
Ω0m(u
3) = h
2 (λ
m+ 2µ
m)
[u
3] h + 1
2 [u
3]
h
2− log
1 + [v
3] h
. (6.3)
The transmission conditions can be written as:
σ
±33= λ
m2 + µ
m1 + [u
3]
h − 1
1 +
[uh3]. (6.4)
–4 –2 0 2 4
–2 –1 1
(CG)
[u
3 ]
h 1
m+2m33
Hooke (SV)
[u3
]=h
.
Fig. 3. Stress
σ±33associated to ˘
WSV(E) [(SV )], to ˘
WCG(E) [(CG)] and to ˘
WSV(e) [Hooke].
Figure 2 represents the limit strain energy associated to the adhesive, up to the constant
h(λm+2µ2 m)as a function of the dimensionless quantity
[uh3], for th e two stored energy functions. The figure depicts also the limit energy obtained with the stored energy function ˘ W
SV(e) =
λ2(Tr e)
2+ µ Tr(e
2): I
Ω0m(u
3) =
h2(λ
m+ 2µ
m)(
[uh3])
2. As expected with the relation (2.7), the three energies coincide for small strains
[uh3]. Furthermore, the graphs clearly show that the strain energy associated to ˘ W
SVis not convex (Remark 4) and that the strain energy associated to ˘ W
CGhas a vertical asymptote: the flattening of the adhesive ([u
3] = − h) would require an infinite strain energy.
Similar conclusions can be obtained from the evolution of σ
33±(up to the constant
1
λm+2µm
