376 J. Sanchez-Hubert, E. Sanchez Palencia
which amounts to a clamping condition at the leading order. As for the new arbitrary functions, we have
ˆu
1
= 0, θ 0
= 0. 3.32
The results of this section may be summed up in the next proposition: P
ROPOSITION
1: The introduction of the asymptotic expansion 3.17 into the variational formulation 3.15 shows that:
1 u s is an inextensional displacement of the curve C, satisfying the clamped conditions 3.30.
2 u s
+ εu
1
s has a Bernoulli’s structure, the new unknown functions θ s and ˆu
1
s coming from the local integration satisfy the bou ndary conditions 3.32.
3 The previous results are consistent with f
ε
= oε
2
, i.e. the total applied force by unit length tends to zero as ε
ց 0 faster than ε
2
.
4. The rod problem in the case when the traction is not coupled with the moments
4.1. Local constitutive relation In this section, we shall follow a procedure analogous to that of Koiter in shell theory. In other words, we
shall admit that the local problem, which furnishes the strain-stress relation is analogous to that of the theory of straight rods see Sanchez-Hubert and Sanchez Palencia, 1992, Ch. VII and 1991; let us recall that properties.
We saw in Section 3.5 that the leading terms γ
ij
of the expansion of the strain tensor γ
ε ij
vanish, so that the corresponding expansion is
γ
ε ij
= εγ
1 ij
+ · · · and, accordingly, we have for the stress tensor
σ
ε ij
= εσ
1 ij
+ · · · . Then see Sanchez-Hubert and Sanchez Palencia, 1992, the components σ
1 ij
are determined as functions of of four functions E
i
, i
= 1, . . . , 4, which depend on u
s,
ˆu
1
s and θ s. Roughly speaking, E
1
is the lengthening at order ε, E
2
, E
3
the kinematic flexions and E
4
the kinematic torsion. Of course, as we shall
see later, here the corresponding expressions in terms of u s,
ˆu
1
s and θ s are different since they involve the curvature and the torsion of C. Moreover, when σ
1
is determined, we may compute the components of
the moment M on the section which are of order ε
4
and the traction on the section of order ε
3
, the other components of the resultant force on the section vanishing at this leading order. In the sequel, it will prove
Rods, flexion, torsion 377
useful to denote these four components by T
2
, T
3
, T
4
, T
1
respectively, indeed ε
3
T
1
= ε
3
Z
6
a
11kh
γ
1 kh
u
ε
dz
2
dz
3
, traction
ε
4
T
2
≡ −ε
4
M
3
= ε
4
Z
6
a
11kh
γ
1 kh
u
ε
z
2
dz
2
dz
3
, ε
4
T
3
≡ ε
4
M
2
= ε
4
Z
6
a
11kh
γ
1 kh
u
ε
z
3
dz
2
dz
3
,
flexion moments ε
4
T
4
≡ ε
4
M
1
= ε
4
Z
6
a
13kh
γ
1 kh
u
ε
z
2
− a
12kh
γ
1 kh
u
ε
z
3
dz
2
dz
3
torsion moment 4.1
Components T
i
are thus defined in terms of the functions E
i
: the corresponding relation is the local constitutive relation which only depends on the local elasticity coefficients and on the section 6.
In this section, we only consider the case when the local relation is such that T
1
only depends on E
1
and M
1
, M
2
, M
3
on E
2
, . . . , E
4
. This is classically satisfied in the case of constant coefficients under symmetry properties.
Let us now consider the rod fixed by its extrmities and submited to given forces f
ε
by unit of length. We saw
Proposition 1 that f
ε
must be small with respect to ε
2
. Moreover, by analogy with the case of straight rods see Sanchez-Hubert and Sanchez Palencia, 1991 and 1992 we shall suppose
f
ε
= ε
4
fs 4.2
by unit of length on C. For the sake of completeness and for a better insight on the phenomena, we shall prescribe also a given
moment
m
ε
= ε
4
m
1
a
1
. 4.3
As the given forces are of order ε
4
, the equilibrium equations 1.1 show that T
ε
is of the same order and we shall write
T
ε
= ε
4
T + · · · .
As for the resultant moment M
ε
, it follows from 2.3 that
M
ε
= ε
4
M + · · ·
the equilibrium equations write
dT
1
ds − kT
2
= −f
1
, dT
2
ds + kT
1
− τ T
3
= −f
2
, dT
3
ds + τ T
2
= −f
3
, 4.4
378 J. Sanchez-Hubert, E. Sanchez Palencia
dM
1
ds − kM
2
= −m
1
, dM
2
ds + kM
1
− τ M
3
= T
3
, dM
3
ds + τ M
2
= −t
2
, 4.5
where the factor ε
4
desappears. Now by eliminating T
2
and T
3
as in Section 2, we obtain the reduced system of statics for the curve C:
k −
dM
3
ds − τ M
2
− dT
1
ds = f
1
, τ
dM
2
ds + kM
1
− τ M
3
+ d
ds dM
3
ds + τ M
2
− kT
1
= f
2
, τ
dM
3
ds + τ M
2
− d
ds dM
2
ds + kM
1
− τ M
3
= f
3
, −
dM
1
ds + kM
2
= m
1
. 4.6
In order to continue our study, we must use some properties of the Lagrange multipliers. 4.2. The constraint of inextensibility and the Lagrange multiplier
At the present state, in order to identify some of the terms in the previous equations, it will prove useful to introduce some elements of the general theory of constrained variational problems. We refer to Brezzi and
Fortin 1991 for the general theory but we shall rather follow the presentation of Sanchez-Hubert and Sanchez Palencia 1997, Sect. VIII.1 which is closer to our present context.
As the inextensibility constraint dv
1
ds − kv
2
= 0 4.7
only involves the two variables v
1
s and v
2
s, we develop here the the theory for
v
= v
1
, v
2
∈ V = H
1
0, l × H
2
0, l. 4.8
Indeed, we shall see later that those spaces are the appropriate ones for the energy bilinear form and the kinematic boundary conditions. Moreover, the results of the present section will be used later in a slightly more
complex framework when other unknowns, in addition to u
1
and u
2
, are involved; but this point is irrelevant. We start with a “preliminary problem” the variational formulation of which is
Find u
∈ V such that
au, v = hF, vi ∀v ∈ V ,
4.9
where a is the energy bilinear form, supposed to be positive and coercive on V , F
∈ V
′
is the given forcing term and where the brackets denote the duality product between V
′
and V . Here V
′
is the dual of V , usually defined in association with another “pivot” space H which is identified to its dual, prescribing that V is dense
in H . We then have V ⊂ H ⊂ V
′
with dense and continuous embeddings. The pivot space is generally chosen
Rods, flexion, torsion 379
in order to express the equations and boundary conditions with the help of integrations by parts in a suitable context, usually it is L
2
here L
2 2
as we have two components. Then, problem 4.9 is equivalent to
Au = F,
4.10 where A is the operator continuous from V in V
′
defined by
au, v = hAu, vi.
4.11 Now let us introduce a constraint, more precisely, we replace V by its closed subspace
G
= {v ∈ V ; Bv = 0},
4.12 where B is some continuous operator from V to another auxiliary pivot space H
m
Bv = 0 is the expression of
the constraint. The constrained problem is then:
Find u ∈ G such that
au, v = hF, vi ∀v ∈ G.
4.13 As the test function only runs in G which is only a subspace of V , this problem cannot be written under the
form 4.10. To write the “modified equation”, let us introduce the space M
′
which is the image of V by B. It is a subspace of H
m
, supposed to be dense in H
m
H
m
is chosen to fulfil this condition. Moreover, M
′
is a Hilbert space for the norm transported by B from G
⊥
orthogonal of G in V . Identifying H
m
with its dual, we define the dual M of M
′
and we have M
′
⊂ H
m
⊂ M. 4.14
We note that B is also a continuous operator from V into M
′
. Let B
∗
be its adjoint, it is a continuous operator from M into V
′
then the equations equivalent to 4.13 are expressed with the aid of a Lagrange multiplier λ
∈ M:
Au + B
∗
λ = f,
Bu
= 0. 4.15
In our problem, 4.7, 4.8, we shall choose as pivot space for the Lagrange multiplier H
m
= L
2
0, l and we shall prove that the image coincides with L
2
0, l. It means that for a given ϕ ∈ L
2
0, l we may find v
1
, v
2
∈ V defined in 4.8 such that ϕ
= dv
1
ds − kv
2
. 4.16
Indeed, let us consider v
2
as known, then v
1
s =
Z
s
ϕσ + kv
2
σ dσ
4.17 belongs to H
1
0, l and satisfies to the prescribed condition v
1
= 0. It must also satisfy = v
1
l =
Z
l
ϕσ + kv
2
σ dσ
4.18
380 J. Sanchez-Hubert, E. Sanchez Palencia
which implies
Z
l
kv
2
σ dσ = −
Z
l
ϕσ dσ. 4.19
Consequently, v
2
must be chosen as an element of H
2
0, l satisfying 4.19. This is obviously possible: we may find v
2
with compact support in an interval where k keeps the same sign and we continue v
2
with vanishing values out of that interval. Then, the general hypotheses of the theory are satisfied with
V
′
= H
−1
0, l × H
−2
0, l 4.20
and M
′
= H
m
= M = L
2
0, l. 4.21
The adjoint of B is defined by hB
∗
p, v i
V
′
V
= hp, Bvi ∀p, v ∈ L
2
× V that is to say
B
∗
p
1
, v
1 H
−1
H
1
+ B
∗
p
2
, v
2 H
−2
H
2
=
Z
l
p dv
1
ds − kv
2
ds then, taking v
1
and v
2
in the space DO, l of test functions, we have
B
∗
p
1
= − dp
ds ,
B
∗
p
2
= −kp. 4.22
Moreover, the dualities H
−1
, H
1
and H
−2
, H
2
are those of distributions more exactly, continuations or restrictions, as D is dense in H
1
and H
2
, so that B
∗
p =
− dp
ds ,
−kp .
4.23 It should be noted, for ulterior utilization, that the image of B
∗
coincides with the subspace G of V
′
which is the polar set of G, i.e. the subspace formed by the elements of V
′
such that hf, vi
V
′
V
= 0 ∀v ∈ G. 4.24
Physically, V
′
is the “space of forces” and G the subspace formed by the forces with vanishing work in the
inextensional displacements. Coming back to the system 4.6, we see that T
1
appears exactly under the form − dT
1
ds in Eq. 4.6
1
and −kT
1
in Eq. 4.6
2
which are precisely the components of B
∗
p in 4.22. As a consequence, the unknown T
1
appears as the Lagrange multiplier when we take the product of the Eqs 4.6 by respectively v
1
, v
2
, v
3
and ϕ which are the components of a test function kinematically admissible, i.e. satisfying the constraint of inextensibility 4.7. and the boundary conditions 3.31 and 3.32.
Rods, flexion, torsion 381
It is then convenient to write the system 4.6 in terms of the components T
i
defined in 4.1 and to denote by λ the Lagrange multiplier T
1
, i.e. to write the system under the form:
k dT
2
ds − τ T
3
− dλ
ds = f
1
, τ
dT
3
ds + kT
4
+ τ T
2
+ d
ds −
dT
2
ds + τ T
3
− kλ = f
2
, τ
− dT
2
ds + τ T
3
− d
ds dT
3
ds + kT
4
+ τ T
2
= f
3
, −
dT
4
ds + kT
3
= m
1
. 4.25
The unknowns are thus u
1
, u
2
, u
3
, ϕ ∈ H
1
× H
2
× H
2
× H
1
and the Lagrange multiplier λ ∈ L
2
0, l ; the
equations are 4.25 and the constraint dv
1
ds − kv
2
= 0. 4.26
The system 4.25, 4.26 is the specific form of the system 4.15. Moreover, the bilinear form of energy is
Z
l
T
2
E
2
+ T
3
E
3
+ T
4
E
4
dy
1
4.27
which is the analogous of the form au, v of the general theory 4.11 or 4.13. The space of the u
1
, u
2
, u
3
, θ ∈ V is here H
1
× H
2
× H
2
× H
1
and the subspace G is G
= u
1
, u
2
, u
3
, u
4
≡ θ ∈ V ; du
1
ds − ku
2
= 0 .
4.28
5. Expression of the functions E
