Rods, flexion, torsion 367
curves. Let us also quote here the above mentioned work Arunakirinathar and Reddy, 1993 which deals with general non plane middle curve. The starting point of this work is the Timoshenko phenomenological model so
that it is essentially different from ours, which starts from three dimensional elasticity.
The paper is organized as follows. Classical statics of curves is recalled in Section 2. The asymptotic two- scale procedure is worked out in Section 3 for the elasticity problem in curvilinear coordinates, leading to
the Bernoulli’s structure and the inextensibility propertiy at order ε . In Section 4, we consider the somewhat
general case when the constitutive law is such that the traction effects are uncoupled from flexion and torsion ones; the role of the Lagrange multiplier is explained. Section 5 is devoted to the above duality method giving
the expressions of the E
2
, E
3
, E
4
. In Section 6, we give a short account of the theory for practical utilization as well as an example. The case of a general constitutive law is adressed in Section 7. Finally, Section 8 is devoted
to a special problem with very particular forces of order ε
3
by unit length. The required differential geometry reduces to a little space curve theory and equations in curvilinear
coordinates, which may be found in any treatrise of classical differential geometry let us quote Lichnerowicz, 1960 for instance.
2. Equations of the statics of a curve
In this section we consider a curve in the mathematical sense, that is to say without thickness; we shall write the classical equilibrium equations when considering it as a material system. We shall see later that they are, at
some asymptotic orders, the limit equations for thin rods.
The curve is parametrized by its arc s and the running point will be denoted by OP = rs. The orthonormal
Frenet frame at a point P is denoted by a
1
≡ t, a
2
≡ n, a
3
≡ b where t, n and b denote respectively the unit
tangent, principal normal and binormal vectors. For the sake of completeness, we recall the Frenet’s formulae
dt
ds
= ksn, dn
ds
= −kst + τ sb, db
ds
= −τ sn,
2.1
where ks and τ s denote respectively the curvature and the torsion at the point s. The equilibrium equations are derived according to the classical procedure: Let us denote respectively by
fs and ms
the linear densities and moments of the applied forces and by T and M the force and moment
describing the mechanical actions of the part s s upon the part s s
of the curve, then by writing the equilibrium equations of a part s
1
s s
2
and passing to the limit s
1
, s
2
→ s , we obtain
dT
ds
+ f = 0, dM
ds
+ a
1
∧ T + ms = 0,
2.2
368 J. Sanchez-Hubert, E. Sanchez Palencia
or, writing these equations in the Frenet frame,
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
, dM
1
ds − kM
2
= −m
1
s, dM
2
ds + kM
1
− τ M
3
= T
3
− m
2
s, dM
3
ds + τ M
2
= −T
2
− m
3
s 2.3
which will be called system of statics of curves. Clearly, the roles of the two components T
2
and T
3
and of the component T
1
are very different. In the sequel, it will prove useful to eliminate T
2
and T
3
. Using the last two equations, we obtain:
k dM
3
ds + τ M
2
+ dT
1
ds = −f
1
, −
d ds
dM
3
ds + τ M
2
− τ dM
2
ds + kM
1
− τ M
3
+ kT
1
= −f
2
+ dm
3
ds − τ m
2
, d
ds dM
2
ds + kM
1
− τ M
3
− τ dM
3
ds + τ M
2
= −f
3
− dm
2
ds − τ m
3
, dM
1
ds − kM
2
= −m
1
2.4
which we will called reduced system of statics of curves. Of course, the equilibrium equations may be written in others frames; this may be useful in cases where the
curve is not easily rectifiable. If the position of the points in a neighbourhood of the curve may be expressed in curvilinear ccordinates y
1
, y
2
, y
3
, with y
2
= y
3
= 0 on the curve , then we shall have ds
= |a
1
| dy
1
, a
1
=
dr
dy
1
and the equations of equilibrium of the forces are given by D
1
T
i
= f
i
, where D
1
denotes the covariant derivative in curvilinear coordinates which are expressed by D
1
T
i
≡ ∂
1
T
i
− Ŵ
i j k
T
k
,
Rods, flexion, torsion 369
Figure 1.
where Ŵ
i j k
are the Christoffel symbols. Analogously, for the moments we have the equilibrium equations:
D
1
M
1
= 0, D
1
M
2
− |a
1
|T
3
= 0, D
1
M
3
+ |a
1
|T
2
= 0, where T
2
and T
3
may be eliminated as before. In the sequel, we shall use the Frenet frame which gives easier formulas and provides a better insight of the geometric properties.
3. Modelling from the three-dimensional elasticity
