Nous donnons une déduction purement géométrique des équations cinématiques de Kirchhoff-Love et de celles de Reissner-Mindlin dans le cas d'une plaque simplement connexe. Cette déduction est obtenue uniquement à partir des relations de compatibilité de Saint Venant et de la représentation intégrale de Cesàro-Volterra. Aucune information concernant le matériau constitutif de la plaque ou le chargement n'est utilisée

GEYMONAT, Giuseppe

KRASUCKI, Françoise

SERPILLI, Michele

Geymonat, Giuseppe

Krasucki, Françoise

Serpilli, Michele

HAL Descartes

A Geometrical Deduction of the Kinematics of Some Plate's Models

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18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
A Geometrical Deduction of the Kinematics of Some Plate’s Models
Giuseppe Geymonat, Françoise Krasucki & Michele Serpilli
CNRS and Université Montpellier II
Laboratoire de Mécanique et Genie Civil
case courrier 048, Place Eugène Bataillon, 34695 Montpellier Cedex 5, France
krasucki@lmgc.univ-montp2.fr
Abstract :
We present a deduction of Kirchhoff-Love and Reissner-Mindlin kinematics of a simply-connected plate by using
the formal asymptotic developments method applied to the compatibility conditions of Saint Venant and the formula
of Cesàro-Volterra, without the use of any information coming from the loading or the constitutive behavior.
Résumé :
Nous donnons une déduction purement géometrique des équations cinématiques de Kirchhoff-Love et de celles
de Reissner-Mindlin dans le cas d’une plaque simplement connexe. Cette déduction est obtenue uniquement à
partir des relations de compatibilité de Saint Venant et de la représentation intégrale de Cesàro-Volterra. Aucune
information concernant le matériau constitutif de la plaque ou le chargement n’est utilisée.
Key-words :
Saint Venant relations; Cesàro-Volterra representation; plate models
1 Introduction
The justification of the usual models of thin plates has been a subject largely discussed in the last
twenty years. In particular the Kirchhoff-Love’s kinematical assumptions have been obtained
by formal asymptotic developments from the 3D equilibrium equations taking into account the
loading and the peculiar geometry of the structure: indeed, the reference configuration of the
plate is Ωε = ω×] − ε, ε[ where ω is a domain of R2 and the thickness is 2ε > 0. The small
parameter ε describes the thinness of the plate and will tend to zero. Let us recall that, under
suitable assumptions on the loads, the formal asymptotic expansions have been justified by
different arguments (see e.g. ref.(1) and (2))
In this paper we give, when ω is simply-connected, a different purely geometrical formal
deduction of the Kirchhoff-Love’s kinematical assumptions and we apply the same method to
obtain the Reissner-Mindlin kinematics.
2 The Saint Venant Compatibility Conditions and the Formula of Cesàro-Volterra for a
3D Plate-like cylindrical domain
Let be given a deformation matrix field eε(xε) = (eεij)(xε) ∈ M2sym, defined into the variable
(simply-connected) domain Ωε and satisfying the Saint Venant compatibility conditions:
∂εlje
ε
ik(x
ε) + ∂εkie
ε
jl(x
ε)− ∂εlie
ε
jk(x
ε)− ∂εkje
ε
il(x
ε) = 0 in Ωε. (1)
Then, there exists a displacement field uε = (uεi ) whose (linearized) strain tensor field is eε(xε)
( Saint Venant stated this result in 1864, but it was rigorously proved by Beltrami only in 1886).
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18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
V. Volterra in 1906 provided a formula giving explicitely the displacement as a line integral on
a path contained in the domain of a certain function of the symmetric matrix field. In 1907
Cesàro re-wrote the formula of Volterra in a more refined way. Precisely, let γε(xε0, xε) denote
a (regular) path from a fixed point xε0 to xε completely contained in Ωε, then the representation
of Cesàro-Volterra of the displacement field uε = (uεi ) takes the following form:
uεi (x
ε) =
∫
γε(xε
0
,xε)
[eεij(yε) + (xεk − yεk)(∂εkeεij(yε)− ∂εi eεkj(yε))]dyεj (2)
In order to study the asymptotic behavior of the deformation field eε and of the displacement
field uε when the thickness ε goes to zero, we introduce the usual change of variables that
allows to transform the problem posed on a variable domain (dependent of ε) onto a problem on
a fixed domain (independent of ε), (see ref.(1)). Let Ω := ω×] − 1, 1[ and let x = (xi) denote
the generic point in the set Ω; at each point x ∈ Ω, we associate the point xε ∈ Ωε through the
bijection (Fig. 1):
Πε : x = (xi) ∈ Ω → x
ε = (xεi ) = (x1, x2, εx3) ∈ Ω
ε
. (3)
Consequently,
Figure 1: The reference configuration
∂εα = ∂α and ∂ε3 =
1
ε
∂3.
and we denote eij(ε)(x) = eεij(xε), resp. ui(ε)(x) = uεi (xε), the deformation, resp.the displace-
ment, transformed by Πε. In the sequel, we will omit the explicit dependence of eij(ε) from x.
By applying the change of coordinates (3) to the Cesàro-Volterra formula (2) we obtain for the
in-plane displacements uα and the transverse displacement u3:
uα(ε)(x) =
∫
γ(x0,x)
[eαβ(ε)(y) + (xσ − yσ)(∂σeαβ(ε)(y)− ∂αeσβ(ε)(y))+
+ε(x3 − y3)(
1
ε
∂3eαβ(ε)(y)− ∂αe3β(ε)(y))]dyβ
+ε[eα3(ε)(y) + (xσ − yσ)(∂σeα3(ε)(y)− ∂αeσ3(ε)(y))+
+ε(x3 − y3)(
1
ε
∂3eα3(ε)(y)− ∂αe33(ε)(y))]dy3,
u3(ε)(x) =
∫
γ(x0,x)
[e3β(ε)(y) + (xσ − yσ)(∂σe3β(ε)(y)−
1
ε
∂3eσβ(ε)(y))]dyβ
+ε[e33(ε)(y) + (xσ − yσ)(∂σe33(ε)(y)−
1
ε
∂3eσ3(ε)(y))]dy3.
(4)
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18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
In the sequel, we will omit the dependence on y of the integrand function and the dependence
of γ from the starting and ending points x0 and x.
3 Deduction of the Kirchhoff-Love kinematics
By applying the previous coordinates change in (1) we obtain the following equations on the
fixed domain Ω:
∂11e22(ε) + ∂22e11(ε)− 2∂12e12(ε) = 0,
1
ε2
∂33e22(ε) + ∂22e33(ε) =
2
ε
∂23e23(ε),
1
ε2
∂33e11(ε) + ∂11e33(ε) =
2
ε
∂13e13(ε),
1
ε
∂23e11(ε) = ∂1(−∂1e23(ε) + ∂2e13(ε) +
1
ε
∂3e12(ε)),
1
ε
∂13e22(ε) = ∂2(∂1e23(ε)− ∂2e13(ε) +
1
ε
∂3e12(ε)),
1
ε
∂12e33(ε) =
1
ε
∂3(∂1e23(ε) + ∂2e13(ε)−
1
ε
∂3e12(ε)).
(5)
Since in the re-scaled equations (5) the parameter of smallness ε appears explicitly in a poly-
nomial form, we assume that there exists a formal asymptotic expansion of the deformations
eij(ε):
eij(ε) = e
0
ij + εe
1
ij + ε
2e2ij + ε
3e3ij + ... (6)
By substituting the asymptotic development (6) in (5) and by isolating the zeroth order term
e0ij , we find:
∂33e
0
αβ = 0,
∂23e
0
11 − ∂13e
0
12 = 0,
∂23e
0
12 − ∂13e
0
22 = 0,
∂11e
0
22 + ∂22e
0
11 − 2∂12e
0
12 = 0.
(7)
From equation (7)1 we derive immediately the characterization of the components of the leading
term eKLαβ := e
0
αβ:
eKLαβ = cαβ(x1, x2) + x3rαβ(x1, x2). (8)
Thus, we can notice that the deformation field is decomposed in two parts, which are well-
known in theory of plates: the former cαβ, the so-called in-plane strain tensor, governs the in-
plane deformations; the latter rαβ, the so-called curvature tensor, characterizes the deformations
outside the middle-plane of the plate.
By substituting (8) in (7)2,3,4, we obtain the compatibility conditions for the fields cαβ and rαβ
in the case of a plate:
∂22c11 + ∂11c22 − 2∂12c12 = 0,
∂1r12 − ∂2r11 = 0,
∂1r22 − ∂2r12 = 0.
(9)
No conditions have been found for the remaining components eKLi3 of the deformation. They
follow from the formula of Cesàro-Volterra (2). Indeed inserting the asymptotic development
(6) in (4) we obtain the two formal asymptotic expansions:
uα(ε) = u
0
α + εu
1
α + ε
2u2α + ε
3u3α + ... (10)
u3(ε) =
1
ε
u−13 + u
0
3 + εu
1
3 + ε
2u23 + ε
3u33 + ... (11)
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18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
whose leading terms are:
u0α(x) =
∫
γ
[e0αβ + (xσ − yσ)(∂σe
0
αβ − ∂αe
0
σβ) + (x3 − y3)∂3e
0
αβ ]dyβ,
u−13 (x) = −
∫
γ
(xσ − yσ)∂3e
0
σβdyβ,
(12)
and represent the formula of Cesàro-Volterra in the case of a plate. Naturally, by applying the
independence from the path to the line integral (12), we obtain once more the equations (7) and
hence (8)-(9). The substitution of (8) into (12) suggests the following definitions uKLα := u0α,
uKL3 := u
−1
3 and hence eKLα3 := 12(∂αu
KL
3 + ∂3u
KL
α ) and eKL33 := ∂3uKL3 . Moreover, by deriving
the formulae (12) along the through-the-thickness coordinate x3, we obtain that eKLi3 = 0.
Let us stress explicitely the difference between eKLi3 and e0i3:
• eKLi3 are calculated starting from the formula of Cesàro-Volterra and they represent the
strain functions associated to the displacements uKLα and uKL3 ;
• e0i3 are only the zeroth order terms of the asymptotic development of the matrix field e(ε).
In order to recover from (12) the classical Kirchhoff-Love kinematics of a plate:{
uKLα (x1, x2, x3) = uα(x1, x2)− x3∂αw,
uKL3 (x1, x2) = w(x1, x2),
(13)
one has only to consider the Airy function w(x1, x2) ∈ C3(ω) (i.e. such that rαβ := −∂αβw)
and to choose a path γ̂ all contained in the middle plane of the plate (i.e. γ̂ ⊂ ω). One then
obtains (13) with uα(x1, x2) :=
∫
γ̂
[cαβ + (xσ − yσ)(∂σcαβ − ∂αcσβ)]dyβ.
It is important to remark once more that in order to obtain (13) we did not make any as-
sumption on the constitutive behavior of the material or on the equilibrium conditions of the
system.
4 Deduction of the Reissner-Mindlin kinematics
All plate theories obtained by asymptotic analysis and based on a limit process when the thick-
ness tends to zero give rise to non-shearable kinematics, as in Kirchhoff-Love plate model. It
has been remarked (see e.g. ref. (3)) that the the engineering derivation of equations of shearable
plates, like Reissner-Mindlin plate model, is based on the following set of internal constraints:
∂33uα = 0 and ∂3u3 = 0. (14)
Let now be given a deformation matrix eε satisfying the compatibility conditions (1) and the
constraints
eεα3(x
ε) = eεα3(x1, x2)
eε33(x
ε) = 0
(15)
which are the analogous of (14). It then follows that eεαβ is a linear function of the transverse
coordinate xε3:
eεαβ(x
ε) = cεαβ(x1, x2) + x
ε
3r
ε
αβ(x1, x2). (16)
and that cεαβ and rεαβ satisfy the following set of Reissner-Mindlin compatibility conditions in
Ωε:
∂ε11c
ε
22 + ∂
ε
22c
ε
11 − 2∂
ε
12c
ε
12 = 0,
∂ε11r
ε
22 + ∂
ε
22r
ε
11 − 2∂
ε
12r
ε
12 = 0,
∂ε2r
ε
11 − ∂
ε
1r
ε
12 = ∂
ε
12e
ε
13 − ∂
ε
11e
ε
23,
∂ε1r
ε
22 − ∂
ε
2r
ε
12 = ∂
ε
12e
ε
23 − ∂
ε
22e
ε
13.
(17)
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18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
By applying the change of coordinates (3), cεαβ , resp.rεαβ , resp.eεα3 turns into cαβ(ε), resp. rαβ(ε),
resp. eα3(ε) and one can define the corresponding asymptotic expansions.
The representation of Cesàro-Volterra of the displacement field u(ε) = (ui(ε)) takes the form:
uα(ε)(x) =
∫
γ
[cαβ(ε) + (xσ − yσ)(∂σcαβ(ε)− ∂αcσβ(ε))+
+εy3(rαβ(ε) + (xσ − yσ)(∂σrαβ(ε)− ∂αrσβ(ε)))+
+ε(x3 − y3)(rαβ(ε)− ∂αe3β(ε))]dyβ+
+ε[eα3(ε) + (xσ − yσ)(∂σe3β(ε)− ∂αe3σ(ε))]dy3,
u3(ε)(x) =
∫
γ
[e3β(ε) + (xσ − yσ)(∂σe3β(ε)− rσβ(ε))]dyβ.
(18)
The asymptotic expansions of cαβ(ε), rαβ(ε) and eα3(ε) induce an asymptotic development for
the displacement functions ui(ε):
ui(ε) = u
0
i + εui + ε
2u2i + ε
3u3i + ... (19)
with :
u0α(x) =
∫
γ
[c0αβ + (xσ − yσ)(∂σc
0
αβ − ∂αc
0
σβ)]dyβ,
u1α(x) =
∫
γ
[c1αβ + (xσ − yσ)(∂σc
1
αβ − ∂αc
1
σβ) + y3(r
0
αβ + (xσ − yσ)(∂σr
0
αβ − ∂αr
0
σβ))+
+(x3 − y3)(r
0
αβ − ∂αe
0
3β)]dyβ + [e
0
α3 + (xσ − yσ)(∂σe
0
3β − ∂αe
0
3σ)]dy3,
u03(x) =
∫
γ
[e03β + (xσ − yσ)(∂σe
0
3β − r
0
σβ)]dyβ,
(20)
We can easily prove that u0α is independent from the transverse coordinate x3. Thus,
u0α(x) =
∫
γ
[c0αβ + (xσ − yσ)(∂σc
0
αβ − ∂αc
0
σβ)]dyβ := u
0
α(x1, x2). (21)
We notice that u0α governs only the in-plane deformation state, while we are looking for a
Reissner-Mindlin kinematics which takes into account not only membranal and flexural but
also shear behaviors. In order to obtain the missing information we intensively use (20).
At first, we derive (20)3 with respect to x3 and we have ∂3u03 = 0. Hence:
u03 := w(x1, x2). (22)
Secondly, from (20)2 and from (20)3, by deriving u1α with respect to x3 and u03 with respect
to xα and by summing up these two derivatives, we find:
∂3u
1
α + ∂αu
0
3 = 2e
0
α3(x1, x2). (23)
and hence one has the complete characterization of the displacement functions u1α and u03:{
u1α(x1, x2, x3) = u
1
α(x1, x2) + x3(2e
0
α3 − ∂αw),
u03(x1, x2) = w(x1, x2).
(24)
Thirdly, from (20)2, if we derive u1α with respect to xβ and u1β with respect to xα, we have:
c1αβ(x1, x2) + x3r
0
αβ(x1, x2) =
1
2
(∂βu
1
α + ∂αu
1
β), (25)
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18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
and hence using (24)
c1αβ =
1
2
(∂βu
1
α + ∂αu
1
β) and r0αβ =
1
2
(2∂βe
0
α3 + 2∂αe
0
β3 − ∂αβw). (26)
We can now define the Reissner-Mindlin kinematics:{
uRMα (x1, x2, x3) := u
0
α(x1, x2) + u
1
α(x1, x2) + x3(2e
0
α3 − ∂αw),
uRM3 (x1, x2) := w(x1, x2),
(27)
and consequently, we can calculate the deformation field as follows:
eRMαβ := c
0
αβ(x1, x2) + c
1
αβ(x1, x2) + x3r
0
αβ(x1, x2),
eRMα3 := e
0
α3(x1, x2),
eRM33 = 0.
(28)
As we can notice from the definition (27) of the Reissner-Mindlin displacements, the quan-
tity 2e0α3−∂αw clearly represents the rotation of the transverse fiber of the plate. This fiber does
not remain perpendicular to the middle plane after the deformation, as in the Kirchhoff-Love
plate: hence, this is an example of a plate which can shear. The function e0α3, which appears in
the rotation of the transverse segment, measures precisely the deviation from a Kirchhoff-Love
strain state and identifies a new deformation state of a shearable plate, namely the Reissner-
Mindlin strain state.
5 Conclusions
The previous formal deduction of the Kirchhoff-Love kinematics (12) or (13) and of the Reissner-
Mindlin kinematics (27)for simply-connected plates is purely geometrical since we have used
no information about the material of the plate or about the loading. We have only used the
Saint Venant compatibility conditions and the Cesàro-Volterra integral representation. Both are
deeply connected with the metric structure of the deformation and of the manifold to whom
this deformation is applied. Hence one can develop an analogous approach in other situations :
beams, shells, ...
References
[1] P. G. Ciarlet, Mathematical Elasticity, vol. II: Theory of Plates, North-Holland Publishing
Co., Amsterdam, 1997.
[2] B. Miara and P. Podio-Guidugli, Deduction by scaling: a unified approach to classic plate
and rod theories Asymptotic Analysis (2006) in press
[3] P. Podio-Guidugli, An exact derivation of the thin plate equation, Journal of Elasticity 22
(1989) 121-133.
6


A geometrical deduction of the kinematics of some plate's models

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A geometrical deduction of the kinematics of some plate's models

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