Comptes Rendus Mécanique (CRM)

Amina Remadi

Ahmed Bahloul

Chokri Bouraoui

Prediction of fatigue crack growth life under variable-amplitude loading using finite element analysis

Xiang Fan

Hang Lin

Hongpeng Lai

Rihong Cao

Jie Liu

Numerical analysis of the compressive and shear failure behavior of rock containing multi-intermittent joints

Xu Wang

Peter Schiavone

C. R. Mecanique 347 (2019) 967–972Contents lists available at ScienceDirect
Comptes Rendus Mecanique
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A screw dislocation near a non-parabolic open inhomogeneity 
with internal uniform stresses
Xu Wang a,∗, Peter Schiavone b,∗
a School of Mechanical and Power Engineering, East China University of Science and Technology, 130 Meilong Road, Shanghai 200237, China
b Department of Mechanical Engineering, University of Alberta, 10-203 Donadeo Innovation Centre for Engineering, Edmonton, Alberta T6G 
1H9, Canada
a r t i c l e i n f o a b s t r a c t
Article history:
Received 21 September 2019
Accepted 23 October 2019
Available online 6 November 2019
Keywords:
Non-parabolic inhomogeneity
Uniform field
Screw dislocation
Anti-plane elasticity
Conformal mapping
We use conformal mapping techniques and analytic continuation to prove that the stress 
field inside a non-parabolic open inhomogeneity embedded in a matrix subjected to 
uniform remote anti-plane stresses can nevertheless remain uniform despite the presence 
of a screw dislocation in its vicinity. Furthermore, the internal uniform stresses inside the 
inhomogeneity are found to be independent of both the shape of the inhomogeneity and 
the presence of the screw dislocation. On the other hand, we find that the existence of the 
nearby screw dislocation exerts a significant influence on the non-parabolic shape of the 
inhomogeneity.
© 2019 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
1. Introduction
It has been established that the internal stresses inside a non-elliptical inhomogeneity embedded in a matrix subjected 
to uniform remote anti-plane shear stresses can nevertheless remain uniform despite the additional influence of a screw 
dislocation in the vicinity of the inhomogeneity [1–4]. In all of the aforementioned analyses, the inhomogeneity is bounded 
by a closed curvilinear interface and referred to as ‘non-elliptical’ to signify that the action of the nearby screw dislocation 
causes a departure of the shape of the inhomogeneity away from the classical ellipse. Very recently, Wang and Schiavone 
[5] demonstrated the remarkable result that the stresses inside a parabolic inhomogeneity with an open interface are un-
conditionally uniform when the surrounding matrix is subjected to uniform anti-plane and in-plane stresses at infinity. It 
is of interest to note that the uniformity of stresses inside a parabolic inhomogeneity in the case of anti-plane elasticity is 
consistent with the observation that the flow within a parabolic inclusion is constant under a uniform incident flow [6,7].
In this paper, we investigate the uniformity of stresses inside a non-parabolic open inhomogeneity when the matrix is 
subjected not only to uniform remote anti-plane stresses but also to the action of a screw dislocation in the vicinity of 
the inhomogeneity. The term “non-parabolic open inhomogeneity” is adopted here to indicate that the open shape of the 
inhomogeneity is a departure from that of a conventional parabola due to the action of the nearby screw dislocation. Using 
the techniques of conformal mapping and analytic continuation [8,9], we show that the internal stresses remain uniform if 
the non-parabolic shape of the inhomogeneity and the location of the screw dislocation are chosen judiciously. In contrast 
to the case of a non-elliptical inhomogeneity interacting with a screw dislocation [1–4], the uniform stresses inside the 
* Corresponding authors.
E-mail addresses: xuwang@ecust.edu.cn (X. Wang), p.schiavone@ualberta.ca (P. Schiavone).https://doi.org/10.1016/j.crme.2019.10.006
1631-0721/© 2019 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
968 X. Wang, P. Schiavone / C. R. Mecanique 347 (2019) 967–972Fig. 1. A screw dislocation interacting with a non-parabolic open inhomogeneity with internal uniform stresses when the matrix is subjected to uniform 
remote anti-plane stresses.
non-parabolic open inhomogeneity are independent of the specific shape of the inhomogeneity. Detailed numerical results 
are presented to demonstrate the theory. As a final point, we mention that the practical importance of the uniformity 
property lies in the fact that a uniform interior stress distribution within the non-parabolic inhomogeneity is optimal since 
it eliminates the possibility of stress peaks which are well-known to be responsible for the failure of the inhomogeneity 
[10].
2. Complex variable formulation for anti-plane elasticity
We first establish a Cartesian coordinate system {xi} (i = 1, 2, 3). Under anti-plane shear deformations of an isotropic 
elastic material, the two shear stress components σ31 and σ32, the out-of-plane displacement w and the stress function φ
can be expressed in terms of a single analytic function f (z) of the complex variable z = x1 + ix2 as [11]
σ32 + iσ31 = μ f ′(z), φ + iμw = μ f (z) (1)
where μ is the shear modulus, and the two stress components can be expressed in terms of the stress function as [11]:
σ32 = φ,1, σ31 = −φ,2 (2)
3. Uniform stresses inside a non-parabolic inhomogeneity
As shown in Fig. 1, we consider a screw dislocation with Burgers vector b located at z = z0 in the vicinity of a non-
parabolic open inhomogeneity (denoted by S1) which is perfectly bonded to the surrounding matrix (denoted by S2) through 
the interface L when the matrix is additionally subjected to uniform remote anti-plane stresses (σ∞31 , σ∞32 ). Our objective 
is to investigate whether the internal stresses inside the inhomogeneity can still remain uniform through a judicious de-
sign of the non-parabolic open shape of the inhomogeneity and the location of the screw dislocation. In what follows, the 
subscripts 1 and 2 are used to identify the respective quantities in S1 and S2.
The boundary value problem for the two-phase composite has the following form:
f2(z) + f2(z) =  f1(z) +  f1(z)
f2(z) − f2(z) = f1(z) − f1(z), z ∈ L
(3a)
f2(z) ∼= b
2π
ln(z − z0) + O (1), z → z0 (3b)
f2(z) ∼= σ
∞
32 + iσ∞31 z + O(z 12 ), |z| → ∞ (3c)μ2
X. Wang, P. Schiavone / C. R. Mecanique 347 (2019) 967–972 969Fig. 2. The problem in the ξ -plane.
where  = μ1/μ2. Equation (3a) describes the continuity of traction and displacement across the perfect interface L; 
Eq. (3b) characterizes the logarithmic singular behavior of f2(z) at the location of the screw dislocation; Eq. (3c) speci-
fies the asymptotic behavior of f2(z) at infinity due to the remote anti-plane loading.
We introduce the following conformal mapping function for the matrix region
z = ω(ξ) = ξ2 + p ln(ξ + ξ̄0 − 2H 12
)
, ξ = ω−1(z), H > 0, Re{ξ} ≥ H 12 (4)
where ξ0 = ω−1(z0) or conversely z0 = ω(ξ0), and p is a complex constant to be determined. The branch cut for the 
logarithmic function appearing in Eq. (4) is taken as a semi-infinite line connecting ξ = 2H 12 − ξ̄0 and ξ = ∞ within the 
left half-plane Re{ξ} < H 12 . Thus, the mapping function in Eq. (4) is single-valued in Re{ξ} ≥ H 12 . Indeed, using the mapping 
function in Eq. (4), the matrix S2 is mapped onto the right half-plane Re{ξ} ≥ H 12 in the ξ -plane; the inhomogeneity-matrix 
interface L is mapped onto the vertical straight line: Re{ξ} = H 12 , −∞ < Im{ξ} < +∞ in the ξ -plane (see Fig. 2) and the 
screw dislocation located at z = z0 is mapped onto the point ξ = ξ0 (see Fig. 2). The information characterizing the screw 
dislocation has been embedded in the construction of the mapping function in Eq. (4). When p = 0 or when the screw 
dislocation is far from the interface, the interface L becomes a parabola described by
L : x1 = H − x
2
2
4H
(5)
Thus when p 	= 0 and when the screw dislocation is in the vicinity of the inhomogeneity, the mapping function in 
Eq. (4) with Re{ξ} = H 12 , −∞ < Im{ξ} < +∞ describes a non-parabolic open interface. There are in total three parameters 
ξ0, p and H in Eq. (4). However, through normalization, we find that ξ0 H−
1
2 and p/H can be treated as two dimensionless 
non-trivial parameters describing the non-parabolic shape of the inhomogeneity.
In order to ensure that the mapping function in Eq. (4) is one-to-one for the matrix region, it is necessary that
ω′(ξ) 	= 0 for Re{ξ} > H 12 (6)
which is equivalent to the following condition
Re{ξ0} − Re
{√(
ξ0 − 2H 12
)2 − 2p̄} ≥ 0 (7)
The stresses inside the non-parabolic inhomogeneity are uniform if
f1(z) = kz, z ∈ S1 (8)
where k is a complex constant to be determined.
By enforcing the continuity condition across the perfect interface L in Eq. (3a) using Eq. (8) and with the aid of analytic 
continuation [8,9], we arrive at
f2(ξ) = f2
(
ω(ξ)
) =  + 1kω(ξ) +  − 1 k̄ω̄(2H 12 − ξ), Re{ξ} ≥ H 12 (9)
2 2
970 X. Wang, P. Schiavone / C. R. Mecanique 347 (2019) 967–972which can be written more explicitly in the following form
f2(ξ) = k( + 1) + k̄( − 1)
2
ξ2 − 2H 12 k̄( − 1)ξ
+ k̄ p̄( − 1)
2
ln(ξ − ξ0) + kp( + 1)
2
ln
(
ξ + ξ̄0 − 2H 12
)
, Re{ξ} ≥ H 12
(10)
Using Eq. (10) to satisfy the logarithmic singular behavior at the location of the screw dislocation in Eq. (3b) and the 
remote asymptotic behavior in Eq. (3c), we obtain the following relationships:
k̄ p̄( − 1)
2
= b
2π
,
k( + 1) + k̄( − 1)
2
= σ
∞
32 + iσ∞31
μ2
(11)
through which the two complex constants k and p can be uniquely determined as
k = σ
∞
32 + iσ∞31
μ1
, p = μ1b
π( − 1)(σ∞32 + iσ∞31 )
(12)
which clearly indicates that the constant p has the dimension of length. It is seen from Eq. (12) that the constant p is real 
valued only when σ∞32 	= 0 and σ∞31 = 0; it is purely imaginary only when σ∞31 	= 0 and σ∞32 = 0.
Thus, the internal uniform stress field inside the non-parabolic open inhomogeneity is explicitly given by
σ31 = σ∞31 , σ32 = σ∞32 , z ∈ S1 (13)
It can be seen quite clearly from the above that the internal uniform stress field is independent of the non-parabolic 
shape of the inhomogeneity and is also independent of the existence of the screw dislocation. However, the existence of the 
nearby screw dislocation exerts a significant influence on the non-parabolic open shape of the inhomogeneity described by 
the mapping function in Eq. (4) via the two complex parameters ξ0 and p.
In addition, the stresses are non-uniformly distributed in the surrounding matrix as follows
σ32 + iσ31 = σ∞32 + iσ∞31 +
(σ∞32 − iσ∞31 )(1 − )
[
2H
1
2 − p̄2(ξ−ξ0) +
p
2(ξ+ξ̄0−2H
1
2 )
]

(
2ξ + p
ξ+ξ̄0−2H
1
2
) , Re{ξ} ≥ H 12 (14)
The prescribed uniform remote stresses in the matrix can be observed quite clearly from the above expression. In fact, 
the non-parabolic shape of the inhomogeneity and the location of the screw dislocation in Fig. 1 are obtained from Eq. (4) by 
choosing ξ0 = H 12 (1.05 + 0.8i) and p = H . In this case, the mapping function for the matrix in Eq. (4) is indeed one-to-one 
by considering the fact that
Re{ξ0} − Re
{√(
ξ0 − 2H 12
)2 − 2p̄} = 0.5157H 12 > 0 (15)
More non-parabolic shapes of the inhomogeneity can be found in Figs. 3 and 4. The internal stress field inside the 
inhomogeneity shown in Fig. 3 with real p is σ31 = 0, σ32 = σ∞32 ; whilst that in Fig. 4 with purely imaginary p is σ31 = σ∞31 , 
σ32 = 0. There is a sharp corner on the interface when choosing: (i) ξ0 = H 12 (1.05 + 0.8i), p = −0.15805H in the left and 
upper subplot of Fig. 3; (ii) ξ0 = 1.05H 12 , p = −0.1H in the left and lower subplot of Fig. 3; (iii) ξ0 = 1.05H 12 , p = 0.47H i
in Fig. 4. In all of these plots with a sharp corner on the interface, condition (7) is met in the case of equality. It can be 
easily verified from Eqs. (1) and (9) that the stresses in the matrix are bounded at these sharp corners.
When both ξ0 and p are real valued, it is deduced quite simply from Eq. (7) that p and ξ0 should satisfy the following 
restriction in order to ensure that the mapping in Eq. (4) is one-to-one in the matrix region
ξ0 = ξ̄0 > H 12 , p = p̄ ≥ 2
(
H − H 12 ξ0
)
(16)
In addition, when p = p̄ = 2(H − H 12 ξ0) in Eq. (16), there is a sharp corner on the interface L. When the parameter ξ0 is 
real valued but the parameter p is purely imaginary, the permissible range of Im{p} for different real values of ξ0(> H 12 ) is 
illustrated in Fig. 5. Im{p} and ξ0 should lie within the region enclosed by the curve shown in Fig. 5. When Im{p} is chosen 
just as the value on the curve in Fig. 5, the interface L will possess a sharp corner. In addition, the permissible range of 
Im{p} increases as ξ0 increases, as shown in Fig. 5.
The present method can be extended in a straightforward manner to the more general scenario in which there is an 
arbitrary number of screw dislocations in the matrix.
X. Wang, P. Schiavone / C. R. Mecanique 347 (2019) 967–972 971Fig. 3. The non-parabolic shape of the inhomogeneity and the location of the screw dislocation for different values of ξ0 and p = p̄. The star in each subplot 
indicates the location of the screw dislocation.
Fig. 4. The non-parabolic shape of the inhomogeneity and the location of the screw dislocation with ξ0 = 1.05H 12 and p = 0.47H i. The star indicates the 
location of the screw dislocation.
4. Conclusions
We have demonstrated for the first time that the internal stresses inside a non-parabolic open inhomogeneity can indeed 
be maintained uniform when the matrix is simultaneously subjected to the action of a screw dislocation and uniform remote 
anti-plane stresses. Through the introduction of the conformal mapping function in Eq. (4), the internal uniform stress field 
in the inhomogeneity is quite simply determined by Eq. (13) and the exterior non-uniform stress distribution in the matrix 
is obtained in Eq. (14).
972 X. Wang, P. Schiavone / C. R. Mecanique 347 (2019) 967–972Fig. 5. The permissible range of a purely imaginary parameter p for different real values of ξ0(> H
1
2 ).
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant No. 11272121) and through a Discov-
ery Grant from the Natural Sciences and Engineering Research Council of Canada (Grant No: RGPIN – 2017 - 03716115112).
References
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(2019), https://doi .org /10 .1016 /j .crme .2019 .08 .002, in press.
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Mech. A, Solids 70 (2018) 1–7.
[4] M. Dai, P. Schiavone, C.F. Gao, Nano-inclusion with uniform internal strain induced by a screw dislocation, Arch. Mech. 68 (2016) 243–257.
[5] X. Wang, P. Schiavone, Uniformity of stresses inside a parabolic inhomogeneity, submitted for publication.
[6] J.R. Philip, Seepage shedding by parabolic capillary barriers and cavities, Water Resour. Res. 34 (1998) 2827–2835.
[7] A.R. Kacimov, Yu.V. Obnosov, Steady water flow around parabolic cavities and through parabolic inclusions in unsaturated and saturated soils, J. Hydrol. 
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A screw dislocation near a non-parabolic open inhomogeneity with internal uniform stresses

Jiangyu Wu

Guansheng Han

Meimei Feng

Hailing Kong

Bangyong Yu

Luzhen Wang

Yuan Gao

Mass-loss effects on the flow behavior in broken argillaceous red sandstone with different particle-size distributions

Conor J. Maguire

C. R. Mecanique 347 (2019) 550–553Contents lists available at ScienceDirect
Comptes Rendus Mecanique
www.sciencedirect.com
Hommage à Abraham de Moivre
Abraham de Moivre : ses traces dans les mathématiques. 
Une analyse bibliométrique
Conor J. Maguire
Société des lettres, sciences et arts du Saumurois, 41–43, rue de la Croix-Verte, 49400 Saumur, France
i n f o a r t i c l e r é s u m é
Historique de l’article :
Reçu le 23 avril 2019
Accepté le 23 avril 2019
Disponible sur Internet le 2 juillet 2019
Mots-clés :
Abraham de Moivre
Probabilités
Nombres complexes
Bibliométrie
Culturomics
Big data
Nous analysons dans cet article, de manière quantitative, les traces d’Abraham de Moivre 
que l’on retrouve dans les littératures française et anglaise, en utilisant la technique de 
la bibliométrie. Nous analysons des dizaines de millions de livres et d’articles numérisés 
par Google Books, qui constituent une source riche de big data. Nous déployons le moteur 
de recherche N-gram. Les résultats montrent statistiquement l’influence de ce grand 
mathématicien français sur la théorie des probabilités et sur les nombres complexes. Bien 
qu’il ait passé la plus grande partie de sa vie en exil à Londres, nous suggérerons qu’il 
serait approprié qu’une rue soit baptisée du nom d’Abraham de Moivre dans la ville de 
Saumur, en France, où il a étudié pendant deux ans à l’Académie protestante.
© 2019 Académie des sciences. Publié par Elsevier Masson SAS. Tous droits réservés.
Un poème1 d’Henry Wadsworth Longfellow (1807–1882) saisit avec élégance à la fois le caractère transitoire de nos vies 
et la possibilité, par les œuvres d’un grand esprit, de laisser des traces durables (empreintes de pas) sur « les sables du 
temps », c’est-à-dire sur notre culture :
Lives of great men all remind us
We can make our lives sublime,
And, departing, leave behind us
Footprints on the sands of time ;2
Dans cet article, nous quantifions les traces laissées dans la mémoire des hommes par le grand esprit que fut Abraham 
de Moivre (1667–1754). Les imprimés recèlent un parcours important de notre culture et c’est dans ce fonds documentaire 
que l’on peut trouver des « empreintes de pas », des influences et des influenceurs sur l’évolution de la civilisation humaine 
[1]. L’exploitation de ce type d’informations est maintenant appelée culturomics [2]. Ce domaine est bien facilité par le fait 
qu’une grande partie des connaissances humaines est en train d’être numérisée. Roser [3] montre l’extraordinaire croissance 
Adresse e-mail : cm@chateaudebeaulieu.fr.
1 A Psalm of Life.
2 Une traduction littérale en français.
Les vies des grands hommes nous le rappellent
Nous pouvons rendre nos vies sublimes,
Et en partant, laissez derrière nous
Des empreintes de pas sur les sables du temps ;https://doi.org/10.1016/j.crme.2019.06.006
1631-0721/© 2019 Académie des sciences. Publié par Elsevier Masson SAS. Tous droits réservés.
C.J. Maguire / C. R. Mecanique 347 (2019) 550–553 551Fig. 1. Vagues d’influence du livre The Doctrine of Chances dans le corpus anglais, 1700–2008. Lissage des données = 30 ans. Noter que la courbe « probability »
était beaucoup plus grande que les deux autres vagues de chances ; donc, les courbes « Doctrine of Chances » et « doctrine of chances » ont été multipliées par 
un facteur heuristique de 6 pour rendre les trois vagues plus facilement comparables.
Source des données : Google Books.
du secteur du livre en Europe occidentale sur une période de 1300 ans. Par conséquent, il existe de plus en plus de données 
à exploiter. Jusqu’en l’an 700, environ 120 livres seulement étaient produits chaque année ; mais en 1790, mille ans plus 
tard, 20 millions avaient été imprimés. Selon l’analyse de Google, il existe aujourd’hui un total de 129 864 880 livres [4], 
dont un quart du total « Google » a déjà été scanné : environ 30 millions de livres [5].
Nous déployons la technique de la bibliométrie pour analyser les sources imprimées. La bibliométrie est devenue une 
approche reconnue pour extraire de gros volumes de données (big data) sous forme imprimée [6,7]. La technique est main-
tenant bien établie et est utilisée comme outil de prise de décision [8]. Nous comparons les références à Abraham de Moivre 
et à ses œuvres (occurrences ou hits) dans le corpus3 de la littérature numérisée publiée dans Google Books, et nous uti-
lisons le générateur N-gram de Google [9] pour effectuer une analyse bibliométrique de séries chronologiques de 1700 à 
2008.4 L’ensemble de données examiné couvre des milliards de mots et de phrases dans des millions de livres écrits en 
sept langues (www.culturomics .org). Nous analysons également la base de données Google Scholar, qui regroupe la plupart 
des revues et ouvrages universitaires en ligne et examinés par des pairs, des documents de conférence, des thèses, etc., 
et contient environ 160 millions de documents [11]. Nous faisons des comparaisons entre Abraham de Moivre et d’autres 
grands mathématiciens ; étant donné qu’il a passé la majeure partie de sa vie à Londres, nous montrons comment il a été 
référencé à la fois dans la littérature anglaise et dans celle en sa langue maternelle, le français.
1. Probabilités
L’un des domaines des mathématiques auxquels Abraham de Moivre a grandement contribué est la théorie des proba-
bilités. Il a été l’un des pionniers du développement de techniques quantitatives destinées à structurer la prise de risque 
(qui fait souvent partie intégrante de la vie) et du calcul des résultats probables des processus stochastiques (omniprésents 
dans les systèmes physiques). La capacité à estimer le risque, ou les probabilités, a toujours été importante pour la sur-
vie ; cependant, tout au long de l’histoire, cela s’est fait simplement par instinct, tant chez l’homme que chez l’animal [12]. 
Fontanari et al. [13], affirment que « l’esprit humain possède une connaissance probabiliste de base », mais ils avertissent que 
cette connaissance est faillible.5 Daniel Kahneman [15], psychologue qui a reçu le prix Nobel d’économie en 2002 pour son 
développement de l’économie comportementale, a expliqué pourquoi une telle prudence est nécessaire : « Nous cherchons des 
modèles, croyons en un monde cohérent [. . .] Nous ne gérons pas bien le caractère aléatoire et voyons plutôt des modèles où il n’y en 
a pas. Nos cousins, les macaques rhésus, le font aussi » ! Ainsi, le développement, aux xviie et xviiie siècles, d’outils techniques 
permettant de calculer les probabilités constituait un progrès critique pour l’avancement de la société.
Le terme doctrine of chances d’Abraham de Moivre s’appelle de nos jours en anglais the theory of probability ou probability 
theory. La terminologie de la probabilité a évolué au fil du temps. La Fig. 1 suggère que le choix du titre Doctrine of Chances
par Moivre pour son livre [16] est probablement une cause majeure de la vague de références génériques ultérieures à 
la doctrine of chances6 telle que trouvée dans le corpus anglais. Vers 1750, les références à Doctrine of Chances sont plus 
3 Un ensemble de documents, artistiques ou non (textes, images, vidéos, etc.), regroupés dans une optique précise.
4 Noter les limitations du générateur de N-gram [10].
5 Un exemple familier de partialité humaine est ce que l’on appelle « l’erreur du parieur » (the Gambler’s Fallacy), qui peut se produire dans une situation 
très familière : la décision d’un gardien de but, qui pense qu’il voit une tendance, dans les tentatives précédentes, de se déplacer à droite ou à gauche dans 
une situation de pénalité [14].
6 À noter qu’une recherche N-gram dans Google Books permet de distinguer les lettres majuscules des minuscules. Par conséquent, les recherches « Doc-
trine of Chances » et « doctrine of chances » donnent des résultats différents.
552 C.J. Maguire / C. R. Mecanique 347 (2019) 550–553Tableau 1
Nombre de résultats (hits) pour les noms de mathématiciens sélectionnés dans Google Books, Google 
Scholar, Internet et les catalogues de la Bibliothèque nationale de France (BNF) et de la British Library 
(BL).
Mathématicien Google Books Google Scholar Internet BNF BL
Jean–Robert Argand (1768–1822) 6 790 639 95 300 2 1
Caspar Wessel (1745–1818) 15 900 2 500 205 000 34 20
Abraham de Moivre (1667–1754) 28 200 4 610 142 000 15 21
Carl Friedrich Gauss (1777–1855) 182 000 22 700 1 350 000 125 158
Leonhard Euler (1707–1783) 408 000 29 000 1 710 000 315 272
Fig. 2. Le nom d’Abraham de Moivre dans le corpus anglais, 1800–2008, comparé à ceux de certains mathématiciens dont les noms sont inscrits sur la tour 
Eiffel, à Paris. Lissage des données : 3 ans.
Source des données : Google Books.
nombreuses, alors que doctrine of chances atteint son apogée autour de 1840–1850. À partir des années 1850, le terme 
doctrine of chances a décliné et a été dépassé, dans le corpus anglais, par une vague des termes modernes, the theory of 
probability ou probability theory.
2. Nombres complexes
Il est instructif de comparer le travail d’Abraham de Moivre à celui d’une sélection de mathématiciens renommés qui ont 
également apporté une contribution importante au développement de nombres complexes ; en particulier, une comparaison 
avec un vrai géant : le mathématicien et physicien suisse Leonhard Euler (1707–1783). Bien entendu, il convient de rappeler 
qu’une comparaison avec Euler revient à un affrontement avec le Mozart des mathématiques [17]. Le Tableau 1 indique 
le nombre de résultats (hits) de Moivre par rapport à une sélection de mathématiciens qui ont apporté des contributions 
majeures au développement des nombres complexes : Euler, Wessel, Argand et Gauss. La recherche dans Google Books a 
généré environ 28 200 résultats (hits) pour Abraham de Moivre et environ 408 000 pour Leonhard Euler. Les hits de Moivre 
ne représentent que 7,8 % du total de ceux d’Euler ; cependant, étant donné le statut historique de celui-ci, la position 
relative, ou le classement, de celui-là est néanmoins impressionnante. Il est également important de comprendre que la 
citation relative (le terme est utilisé ici dans un sens très général pour inclure les hits dans différents référentiels) n’est 
pas nécessairement linéaire [18]. Ce phénomène non linéaire est connu en linguistique comme la loi de Zipf [19,20]. Une 
variante de cette distribution est appelée « loi de Price » [21,22].
3. Les corpus français et anglais – et autres
Une analyse de la fréquence des références à, par exemple, « formule de Moivre » dans le corpus français et à « de Moivre’s 
formula » dans le corpus anglais de 1700 à 2008 montre que Moivre était en fait beaucoup plus référencé dans la littérature 
C.J. Maguire / C. R. Mecanique 347 (2019) 550–553 553française par rapport à ce qu’il en était dans la littérature anglaise. Aujourd’hui, la célèbre formule de Moivre est référencée 
en anglais presque dans les mêmes proportions qu’en français. En outre, elle est référencée partout dans le monde ; en 
espagnol, par exemple, « Fórmula de Moivre » est référencée à près de 40 % du total français. En allemand, « Satz von de Moivre »
est référencé à 2,5 fois le total français.
De nos jours, le nom d’Abraham de Moivre est référencé à peu près dans les mêmes mesures en français, en anglais, 
en allemand, en néerlandais, en polonais et en suédois. En italien, il est référencé à 30 % de ce qu’il l’est en français et en 
espagnol à environ 16 %. Son influence est clairement internationale.
4. Un panthéon national – la tour Eiffel, Paris
Nous comparons ici les hits dans Google Books (le corpus anglais) du nom « Abraham de Moivre » avec ceux d’un échan-
tillon de mathématiciens français renommés qui ont été honorés par une inscription apposée sur la tour Eiffel à Paris.
Bien que le nom d’Abraham de Moivre ne soit pas inscrit sur la tour, il ressort clairement de la Fig. 2 qu’il occupe depuis 
longtemps une position honorable, importante, dans la littérature anglaise par rapport à une liste sélectionnée parmi ceux 
dont les noms sont inscrits.
5. Conclusions
L’influence d’Abraham de Moivre sur les mathématiques, sur notre culture en fait, persiste. Ses traces, ses « empreintes 
de pas sur les sables du temps », ne se sont pas estompés : ils resteront pérennes.
En prenant tout en compte, il apparaît à l’évidence que notre homme mérite de prendre sa place dans le panthéon 
des grands mathématiciens – et d’être reconnu dans un lieu d’importance national. Peut-être qu’une rue nommée en son 
honneur à Saumur, où il a étudié pendant deux ans, serait un geste approprié de la part de la ville ?
Références
[1] E.W. Hulme, Statistical Bibliography in Relation to the Growth of Modern Civilization: Two Lectures Delivered in the University of Cambridge in May 
1922, Butler & Tanner Grafton, London, 1922.
[2] J.-B. Michel, E. Lieberman-Aiden, Quantitative analysis of culture using millions of digitized books, Science 331 (6014) (2010) 176–182, https://doi .org /
10 .1126 /science .1199644.
[3] M. Roser, Books, publié en ligne sur www.OurWorldInData .org, 2017.
[4] L. Taycher, Books of the World, Stand up and Be Counted! All 129,864,880 of You, Google Inc., 2010.
[5] T. Wu, What ever happened to Google books?, New Yorker (11 septembre 2015).
[6] E. Lieberman-Aiden, J.-B. Michel, Uncharted: Big Data as a Lens on Human Culture, first edition, Riverhead Books, New York, ISBN 1594487456, 2013.
[7] C.M. Stützer, M. Welker, M. Egger, Computational Social Science in the Age of Big Data, Neue Schriften zur Online-Forschung, vol. 15, Herbert von 
Halem Verlag, 2018.
[8] OECD et SCImago Research Group, Compendium of Bibliometric Science Indicators, OECD, Paris, 2016, http://oe .cd /scientometrics.
[9] D. Jurafsky, J.H. Martin, Speech and Language Processing, Stanford University, 2014, https://lagunita .stanford .edu /c4x /Engineering /CS -224N /asset /slp4 .
pdf.
[10] S. Zhang, The pitfalls of using Google Ngram to study language, www.wired .com, 2015, N-grams: for fun (and profit?), TheBetterEditor of New England.
[11] E. Orduña-Malea, J.M. Ayllón, A. Martín-Martín, E. Delgado López-Cózar, Methods for estimating the size of Google Scholar, Scientometrics 104 (3) 
(2015) 931–949.
[12] A. Hintze, R.S. Olson, C. Adami, R. Hertwig, Risk sensitivity as an evolutionary adaptation, Sci. Rep. 5 (2015) 8242.
[13] L. Fontanari, M. Gonzalez, G. Vallortigara, V. Girotto, Probabilistic cognition in two indigenous Mayan groups, Proc. Natl. Acad. Sci. USA 111 (48) (2014) 
17075–17080.
[14] A. Sarkar, The Gambler’s Fallacy and Hot Outcome: Cognitive Biases or Adaptive Thinking for Goalkeepers’ Decisions on Dive Direction During Penalty 
Shootouts, Bowling Green State University, OH, USA, 2017.
[15] D. Kahneman, Thinking, Fast and Slow, Farrar, Straus and Giroux, New York, 2013.
[16] A. de Moivre, The Doctrine of Chances: Or, a Method for Calculating the Probabilities of Events in Play, first edition, W. Pearson, London, 1718.
[17] D. Stipp, A Most Elegant Equation: Euler’s Formula and the Beauty of Mathematics, Hachette Book Group, New York, 2017.
[18] M. Brzezinski, Power laws in citation distributions: evidence from Scopus, Scientometrics 103 (2015) 213, https://doi .org /10 .1007 /s11192 -014 -1524 -z.
[19] G. Zipf, Human Behavior and the Principle of Least Effort, Addison-Wesley, 1949.
[20] I. Moreno-Sánchez, F. Font-Clos, Á. Corral, Large-scale analysis of Zipf’s law in English texts, PLoS ONE 11 (1) (2016) e0147073, https://doi .org /10 .1371 /
journal .pone .0147073.
[21] P.T. Nicholls, Price’s square root law: empirical validity and relation to Lotka’s law, Inf. Process. Manag. 24 (4) (1988) 469–477.
[22] C. Petersen, J. Grue Simonsen, C. Lioma, Power law distributions in information retrieval, ACM Trans. Inf. Syst. 34 (2) (2016) 8.


Abraham de Moivre : ses traces dans les mathématiques. Une analyse bibliométrique

Christian Licht

Thibaut Weller

C. R. Mecanique 347 (2019) 555–560Contents lists available at ScienceDirect
Comptes Rendus Mecanique
www.sciencedirect.com
Asymptotic analysis of a thin linearly elastic plate equipped 
with a periodic distribution of stiffeners
Christian Licht a,b,c, Thibaut Weller a,∗
a LMGC, Université de Montpellier, CNRS, Montpellier, France
b Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
c Centre of Excellence in Mathematics, CHE, Bangkok 10400, Thailand
a r t i c l e i n f o a b s t r a c t
Article history:
Received 13 June 2019
Accepted 19 July 2019
Available online 23 August 2019
Keywords:
Hard abutting and rigidification of plates
Asymptotic modelling
Periodic homogenization
Reduction of dimension
Variational convergence
We derive several models of thin plates equipped with a periodic distribution of stiffeners. 
Depending on the orders of magnitude of the different parameters involved, diverse 
situations arise, from classical Kirchhoff–Love behaviour with additional energy term to 
full rigidification.
© 2019 Académie des sciences. Published by Elsevier Masson SAS. This is an open access 
article under the CC BY-NC-ND license 
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
To mathematically derive the Reissner–Mindlin model of thin linearly elastic plates (see [1]), we considered a periodic 
distribution of plates abutted together through thin and soft adhesive layers. So, it is natural now to examine the case when 
the adhesive layers are stiff. One of the main motivations is the rigidification of plates through a distribution of parallel 
stiffeners. But it is also a first step in the study of the optimization of plates. In that respect, a more distant goal lies in 
the relation between the design of the plate and the improvement of some selected aspects of its mechanical performances, 
without altering the total quantity of material employed (see [2] for example). We will present here six models indexed by 
p = (p1, p2) in {1, 2, 3} × {1, 2}, where p1 is a geometric parameter linked to the stiffeners layout, while p2 accounts for 
the order of magnitude of the rigidity of the stiffeners.
More precisely, as usual we make no difference between the Euclidean physical space and R3 with orthonormal basis 
{e1, e2, e3} and, for all ξ = (ξ1, ξ2, ξ3) in R3, we define ξ̂ := (ξ1, ξ2) and denote the standard Euclidean distance by dist . Let 
(τ k, νk) := (e3−k, ek), for k = 1, 2 and (τ 3, ν3) := 1√
2
(e1 + e2, e1 − e2), (h, ε, η) three small positive real numbers and ω a 
domain of R2 with a Lipschitz-continuous boundary ∂ω. For all k in {1, 2, 3}, we define:
	ki :=
{
iενk +Rτ k
}
∩ ω, i ∈Z, Ik :=
{
i ∈Z;	ki = ∅
}
, 	k,h :=
( ⋃
i∈Ik
	ki
)
× (−h,h), 	hp1 :=
⋃
k≤p1
	k,h
* Corresponding author.
E-mail addresses: christian.licht@umontpellier.fr (C. Licht), thibaut.weller@umontpellier.fr (T. Weller).https://doi.org/10.1016/j.crme.2019.07.001
1631-0721/© 2019 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license 
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
556 C. Licht, T. Weller / C. R. Mecanique 347 (2019) 555–560The following two subsets of 
h := ω × (−h, h) denoted by
Bhη,ε,p1 :=
{
x ∈ 
h;dist(x,	hp1) < ηε
}
, P hη,ε,p1 := 
h \ Bhη,ε,p1
are occupied by linearly elastic materials with strain energy densities μWl and W respectively, where μ is a large stiffness 
parameter. Introducing SN as the space of all N × N symmetric matrices equipped with the usual inner product and norm 
denoted as for RN by · and | · |, W and Wl are two positive quadratic forms on S3.
Fig. 1. The thin plate, the triplet of geometric parameters (η, ε,h) and the stiffeners’ layout in the case p1 = 3.
The structure made of these two parts perfectly bonded together is clamped on hD := ∂ω × (−h, h) and subjected to 
body forces and surface forces on h± := ω × {±h} of densities f h and gh . Hence, the equilibrium of the structure involves 
a quadruplet s := (μ, η, ε, h) of data and leads to:(
P sp
)
Min
{
J sp(v); v ∈ H1hD (

h;R3)
}
where, classically, for all domain G in RN and all smooth part γ of its boundary ∂G , H1γ (G; RN) denotes the subspace of 
the Sobolev space H1(G; RN ) made of the elements with vanishing trace on γ ,
J sp(v) :=
∫
P hη,ε,p1
W (e(v))dx + μ
∫
Bhη,ε,p1
Wl(e(v))dx − Lh(v)
Lh(v) :=
∫

h
f h · v dx +
∫
h+∪h−
gh · v dx̂
e(v) being the strain tensor associated with the displacement field v .
Clearly, if ( f h, gh) belongs to L2(
h × (h+ ∪ h−); R3), 
(
P sp
)
has a unique solution usp and, considering the data s as a 
parameter, we are interested in the asymptotic behaviour of usp when s takes values in a countable set of (0, +∞)4 with 
s̄ := (+∞, 0, 0, 0) as a unique limit point. As in the mathematical derivation of Kirchhoff–Love theory of plates (cf. [3,4]), it 
is convenient to introduce the linear mappings h and Sh:
ξ = (ξ̂ , ξ3) ∈R3 → hξ = (ξ̂ ,hξ3) ∈R3
v ∈ L2(
h;R3) → Sh v ∈ L2(
;R3) s.t. (Sh v)(x) = 1h 
h(v(hx)), ∀x ∈ 
 := ω × (−1,1)
We make the following assumption on the loading:
(H1)
{
∃ ( f , g) ∈ L2(
 × (+ ∪ −);R3) s.t.
f h(hx) = hh f (x) a.e. x ∈ 
, gh(hx) = h2h g(x) a.e. x ∈ ±
therefore, us,p := Shusp is the unique solution to(
Ps,p
)
Min
{
J s,p(v); v ∈ H1 (
;R3)
}
D
C. Licht, T. Weller / C. R. Mecanique 347 (2019) 555–560 557where
J s,p(v) :=
∫
Pη,ε,p1
W (e(h, v))dx + μ
∫
Bη,ε,p1
Wl(e(h, v))dx − L(v)
L(v) :=
∫


f · v dx +
∫
+∪−
g · v dx̂
eαβ(h, v) =eαβ(v), eα3(h, v) = 1
h
eα3(v), 1 ≤ α,β ≤ 2, e33(h, v) = 1
h2
e33(v)
with D the reciprocal image by h of hD and, similarly, index h is dropped for the image by (
h)−1 of h± , 
h , Bhη,ε,p1 , 
P hη,ε,p1 , 	
k,h and 	hp1 .
2. A convergence result
We assume that
(H2)
⎧⎪⎨⎪⎩
∃ μ̄ ∈ (0,+∞] s.t. μ̄ := lim
s→s̄
(2μη), μ̄ ∈ (0,+∞) if p2 = 1, μ̄ = +∞ if p2 = 2
lim
s→s̄
ηε
h
= 0, lim
s→s̄
h2
ηε2
= 0
and introduce the space V K L(
) of Kirchhoff–Love displacements vanishing on D :
V K L(
) :=
{
v ∈ H1D (
;R3) s.t. ei3(v) = 0 in 
, 1 ≤ i ≤ 3
}
and the positive definite quadratic form on S2 defined by
W K L(q) := Min{W (e); e ∈ S3 s.t. ê = q}
where êαβ = eαβ , 1 ≤ α, β ≤ 2, for all e in S3.
Let (τ , ν) in 
{
(τ k, νk), k = 1, 2, 3
}
. We perform the change of coordinates
x = (x1, x2, x3) ∈R3 θ→ R3  y = (yτ , yν, y3) := (x · τ , x · ν, x3)
and, for all v in H1(
; R3), define vτ by vτ (y) = v(θ−1(y)) · τ . To shorten notations, we write ∂τ vτ for the derivative in 
the sense of distributions ∂yτ vτ . Note that ∂τ 1 vτ 1 = e22(v), ∂τ 2 vτ 2 = e11(v), ∂τ 3 vτ 3 =
1
2
(e11(v) + 2e12(v) + e22(v)). For all 
k in {1, 2, 3}, we define the real convex quadratic function W kl by:
W kl (t) := Inf
{
Wl
(
(Q k)ᵀ e Q k
)
; e ∈ S3, e11 = t
}
where Q k = τ k ⊗ e1 + νk ⊗ e2 + e3 ⊗ e3 and (Q k)ᵀ denotes the transpose of Q k .
Let
V p :=V K L(
) if p2 = 1, V (p1,2) := V K L(
) ∩
( ⋂
k≤p1
{
∂τ k vτ k = 0
})
J̄ p(v) :=
∫


[
W K L(ê(v)) + (2 − p2) μ̄
∑
k≤p1
W kl (∂τ k vτ k )
]
dx − L(v)
Then we have the following result.
Theorem 2.1. Under assumptions (H1) and (H2), as s goes to s̄, us,p converges strongly in H1(
; R3) toward the unique solution up
to (
P̄p
)
Min
{
J̄ p(v), v ∈ V p
}
and
558 C. Licht, T. Weller / C. R. Mecanique 347 (2019) 555–560J̄ p(up) = lim
s→s̄
J s,p(us,p) (1)∫


W K L(ê(up))dx = lim
s→s̄
∫
Pη,ε,p1
W (e(us,p))dx (2)
μ̄
∫


∑
k≤p1
W kl (∂τ k (up)τ k )dx = lims→s̄ μ
∫
Bη,ε,p1
Wl(e(us,p))dx when p2 = 1 (3)
The elementary proof is achieved in two steps through a standard method of variational convergence.
Step 1 (asymptotic behaviour of us,p )
Proposition 2.1. When s goes to s̄, us,p (up to a not relabelled subsequence) weakly converges in H1(
; R3) toward some up in V p
such that
J̄ p(up) ≤ lim
s→s̄
J s,p(us,p)
Proof. As, clearly, us,p is bounded in H1(
; R3), we deduce that there exists some up in H1D (
; R3) such that, up to a 
not relabelled subsequence, us,p weakly converges in H1(
; R3) toward up , which does belong to V K L(
). Moreover, the 
very definitions of W K L , W kl , 1 ≤ k ≤ 3, and Jensen’s inequality imply:∫
Pη,ε,p1
W (e(us,p))dx ≥
∫
Pη,ε,p1
W K L(ê(us,p))dx
μ
∫
Bη,ε,p1
Wl(e(us,p))dx ≥ 2μη
∫
R2×(−1,1)
∑
k≤p1
W kl (∂τ k (< u >
k
s,p)τk )dx
where
• < u >ks,p :=
∑
i∈Ik
1
2ηε
ηε∫
−ηε
ũs,p
(
(x · τ k)τ k + (iε + t)νk
)
dt χkε,i
• χkε,i is the characteristic function of 
{
(i + t)ενk, 0 < t < 1
}
×R × (−1, 1)
• ṽ is the extension by 0 to H1(R2 × (−1, 1); R3) of all v in H1D (
; R3)
As < u >ks,p has the same strong limit ũp in L2(
; R3) as ũs,p , a standard lower semi-continuity argument yields the 
result. 
Step 2 (identification of up )
Proposition 2.2. For all v in V p, there exists a sequence vs in H1D (
; R3) such that
lim
s→s̄
J s,p(vs) ≤ J̄ p(v)
Proof. It is straightforward by using test functions like
ws(x) = hh ρ(x) + ε
∑
k≤p1
(
δη(x · νk/ε)hρ(x) + ϕη(x · νk/ε)(ψk,0)
)
with
(ρ, ψk) in C∞(
; R3 ×R2) vanishing on D ,
δη, ϕη the 1-periodic functions such that
δη(t) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
0 if t < η
(t − η)/η if η ≤ t ≤ 2η
1 if 2η ≤ t ≤ 1/2
δ (1 − t) if 1/2 ≤ t ≤ 1
, ϕη(t) =
⎧⎨⎩t if |t| < ηη(1 − 2t)
1 − 2η if η < t < 1 − η
η
C. Licht, T. Weller / C. R. Mecanique 347 (2019) 555–560 559to check that vs defined by
vs ∈ H1D (
;R3) ;
∫
Pη,ε,p1
DW (e(h, vs)) · e(h, w)dx + μ
∫
Bη,ε,p1
DWl(e(h, vs)) · e(h, w)dx =
=
∫


DW K L(ê(vs)) · ê(w)dx + (2 − p2) μ̄
∫


∑
k≤p1
DW kl (∂τ k (vs)τ k ) · ∂τ k wτ k dx, ∀w ∈ H1D (
;R3)
satisfies the assertion.
Thus up is the unique minimizer in V p of J̄ p and satisfies (1) and, consequently, (2) and (3). Hence, the whole sequence 
us,p weakly converges in H1(
; R3), but also strongly because
lim
s→s̄
∫
Pη,ε,p1
W K L(ê(us,p))dx ≤ lim
s→s̄
∫
Pη,ε,p1
W (e(us,p))dx =
∫


W K L(ê(up))dx ≤ lim
s→s̄
∫
Pη,ε,p1
W K L(ê(us,p))dx
and ∫
Bη,ε,p1
|ê(us,p)|2 dx ≤ C
μ
, lim
s→s̄
∫


|ei3(us,p)|2 dx = 0, 1 ≤ i ≤ 3 
As quoted in [5,6], to make more precise the asymptotic behaviour of usp , we develop a variant of Theorem 2.1. As no 
ambiguity ensues, we use the same symbol ê for an element e of S3 such that its non vanishing entries are eαβ = êαβ , 
1 ≤ α, β ≤ 2, and let e⊥ := e − ê. Then we have the following theorem.
Theorem 2.2. There exists a unique zp in L2(
; R3) such that e(h, usp) converges strongly in L2(
; S3) towards ê(up) + zp ⊗s e3 . 
Moreover,
•
(
DW (ê(up) + zp ⊗s e3)
)⊥ = 0
•
∫


W
(
ê(up) + zp ⊗s e3
)
dx =
∫


W K L(ê(up)) dx
• (up, zp) is solution to(
Qp
)
Min
{∫


[
W
(
ê(v) + z ⊗s e3
)
dx + (2 − p2) μ̄
∑
k≤p1
W kl (∂τ
k vτ k )
]
dx − L(v) ; (v, z) ∈ V p × L2(
;R3)
}
Proof. As e(h, us,p) is bounded in L2(
; S3), it converges weakly toward some ê(up) + zp ⊗s e3 up to a not relabelled 
subsequence. Moreover (up, zp) appears as the unique solution to 
(
Qp
)
because for all element z of C∞(
̄; R3) vanishing 
on D , e(h, Zh), with Zh = hh Z , Z(x̂, x3) :=
x3∫
0
z(x̂, t) dt , converges strongly in L2(
; S3) toward z ⊗s e3. Hence, the whole 
sequence e(h, us,p) converges weakly in L2(
; S3), but also strongly because∫
Bη,ε,p1
|e(h, us,p)|2 dx ≤ C
μ
,
lim
s→s̄
∫
Pη,ε,p1
W (e(h, us,p))dx =
∫


W K L(ê(up))dx ≤
∫


W
(
ê(up) + zp ⊗s e3
)
dx ≤ lim
s→s̄
∫
Pη,ε,p1
W (e(h, us,p))dx 
Now these mathematical results can immediately be rephrased in terms related to the genuine physical problem 
(
P sp
)
, 
which will supply our asymptotic model. Let
V K L(

h) :=
{
v ∈ H1
hD
(
h;R3) ; ei3(v) = 0,1 ≤ i ≤ 3
}
V hp :=V K L(
h) if p2 = 1, V h(p1,2) := V K L(
h) ∩
( ⋂
{∂τ k vτ k = 0}
)
k≤p1
560 C. Licht, T. Weller / C. R. Mecanique 347 (2019) 555–560ūhp :=S−1h up
z̄hp(x) :=zp
(
(h)−1x
)
a.e. x ∈ 
h
then we have
Theorem 2.3. The fields ūhp and z̄hp are solutions to(
P̄hp
)
Min
{∫

h
[
W K L(ê(v)) + (2 − p2) μ̄
∑
k≤p1
W kl (∂τ k vτ k )
]
dx − Lh(v) ; v ∈ V hp
}
(
Q̄hp
)
Min
{∫

h
[
W
(
ê(v) + z ⊗s e3
)
+ (2 − p2) μ̄
∑
k≤p1
W kl (∂τ k vτ k )
]
dx − Lh(v) ; (v, z) ∈ V hp × L2(
h;R3)
}
and
lim
s→s̄
1
h3
∫

h
|ê(usp) − ê(ūhp)|2 dx = 0,
∫

h
|e⊥(usp)|2 dx ≤ Ch3 (4)
∫

h
|e⊥(usp) − hz̄hp ⊗s e3|2 dx = o(h3) (5)
3. Physical interpretation
The second line of hypothesis (H2) refers to the design of the stiffeners but also to their layout. The condition 
ηε
h
→ 0
is clear: the stiffeners have to be slender. As to 
h2
ηε2
→ 0, it encloses various information. On the one hand, because 
h2
ηε2
= h
ε
/
ηε
h
, it says that the slenderness of the microscopic plates constituting the genuine plate 
h (see Fig. 1) is lesser 
than the one of the stiffeners. On the other hand, the thickness h of the plate being given, it yields that the distance 
between two nearest parallel stiffeners has to be large enough (more precisely, the condition is ε  h
ηε
).
Theorem 2.3 tells us that, when the order of magnitude of the rigidity of the stiffeners is 
1
2η
, the asymptotic behaviour 
of the structure is the one of Kirchhoff–Love type. The stiffeners supply an additional term μ̄
∑
k≤p1
W kl (∂τ k vτ k ) to the classical 
term W K L(ê(v)) stemming from the sole W . When the rigidity is of an order of magnitude larger than 
1
2η
, the periodic 
distribution of stiffeners of direction τ k implies a vanishing stretch in this direction. Hence, to get a full rigidity, it suffices 
to use three families of stiffeners, in our case a fourth direction like 
e1 − e2
2
is not necessary.
To go to the essential we assumed that the stiffness of each family of layers was the same, it is easy if not tedious to 
consider pk2 ∈ {1, 2}, k = 1, 2, 3!...
References
[1] C. Licht, T. Weller, An asymptotic Reissner–Mindlin plate model, C. R. Mecanique 346 (2018) 432–438.
[2] P. Villaggio, Mathematical Models for Elastic Structures, Cambridge University Press, Cambridge, UK, 1997.
[3] P.G. Ciarlet, Mathematical Elasticity, vol. II: Theory of Plates, North-Holland, Elsevier, 1997.
[4] O. Iosifescu, C. Licht, G. Michaille, Nonlinear boundary conditions in Kirchhoff–Love plate theory, J. Elast. 96 (2009) 57–79.
[5] C. Licht, T. Weller, Approximation of semi-groups in the sense of Trotter and asymptotic mathematical modeling in physics of continuous media, Discrete 
Contin. Dyn. Syst., Ser. S 12 (2019) 1709–1741.
[6] Y. Terapabkajornded, S. Orankitjaroen, C. Licht, Asymptotic model of linearly visco-elastic Kelvin–Voigt type plates via Trotter theory, Adv. Differ. Equ. 
186 (2019), https://doi .org /10 .1186 /s13662 -019 -2104 -6.


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