We propose a whole numerical framework to study the buckling problems of film-substrate systems: from 2D to 3D modeling, from classical to multi-scale perspective. The main objective is to apply advanced numerical methods (path-following techniques, bifurcation indicators, bridging techniques, multi-scale analyses, etc.) for multiple-bifurcation analyses of film-substrate buckling problems. Through incorporating these numerical methods with FEM, it can predict the occurrence and post-bifurcation evolution of wrinkling patterns in various boundary and loading conditions as well as complex geometry configurations

Potier-Ferry, Michel,

Xu, Fan

I-Revues

22ème Congrès Français de Mécanique Lyon, 24 au 28 Août 2015
A numerical framework for pattern
formation modeling of
film-substrate systems
F. XU, M. POTIER-FERRY
Laboratoire d’Etude des Microstructures et de Mécanique des Matériaux,
LEM3, UMR CNRS 7239, Université de Lorraine, France.
{fan.xu, michel.potier-ferry}@univ-lorraine.fr
Résumé :
Nous proposons un cadre général pour étudier le problème de flambage de
systèmes film-substrat de manière numérique : de la modélisation 2D ou 3D,
d’un point de vue classique ou multi-échelle. L’objectif principal est d’appliquer
aux systèmes film-substrat des méthodes numériques avancées (des techniques
de cheminement, des indicateurs de bifurcation, des méthodes de couplage de
modèle, des analyses multi-échelle, etc.) pour des analyses de bifurcations mul-
tiples. En intégrant ces méthodes numériques avec la méthode des éléments
finis, il peut prédire l’événement et l’évolution de post-bifurcation des modes
d’instabilité dans diverses conditions aux limites et de chargements ainsi que les
configurations géométriques complexes.
Abstract :
We propose a whole numerical framework to study the buckling problems
of film-substrate systems: from 2D to 3D modeling, from classical to multi-
scale perspective. The main objective is to apply advanced numerical methods
(path-following techniques, bifurcation indicators, bridging techniques, multi-
scale analyses, etc.) for multiple-bifurcation analyses of film-substrate buckling
problems. Through incorporating these numerical methods with FEM, it can
predict the occurrence and post-bifurcation evolution of wrinkling patterns in
various boundary and loading conditions as well as complex geometry configu-
rations.
Mots clefs : Wrinkling; Post-buckling; Bifurcation; Film-
substrate; Path-following technique; FEM.
1 Introduction
22ème Congrès Français de Mécanique Lyon, 24 au 28 Août 2015
Surface morphological instabilities of a stiff thin layer attached on a soft
substrate have been widely observed in nature such as wrinkles of hornbeam
leaf and human skin, and these phenomena have raised considerable research
interests over the last decade [1, 2, 3]. Besides, in modern industry, surface
wrinkles can be widely applied ranging from micro/nano-fabrication of flexible
electronic devices with controlled morphological patterns [4, 5] to the mechanical
property measurement of material characteristics [6]. In terms of stability study,
several theoretical and numerical works have been devoted to linear perturbation
analysis and nonlinear buckling analysis [4, 7, 8]. However, most previous works
have been mainly constrained to determine the critical conditions of instability
and the corresponding wrinkling patterns near the instability threshold. The
post-buckling evolution and mode transition of surface wrinkles are only recently
being pursued [9, 10, 11, 12].
Pattern formation modeling and post-buckling analysis deserve new numer-
ical investigations, especially through finite element method that can provide
the overall view and insight into the formation and evolution of wrinkle patterns
in any condition. Therefore, we propose a whole numerical framework to study
surface wrinkling of film-substrate systems: from 2D to 3D modeling, from
classical to multi-scale perspective [13, 14, 15, 16, 17]. The main objective is
to apply advanced numerical methods for multiple-bifurcation analyses of film-
substrate systems, especially focusing on the tracing of post-buckling evolution
and surface mode transition. These advanced numerical approaches include
path-following techniques, bifurcation indicators, bridging techniques, multi-
scale analyses, etc. Through incorporating them with finite element method, it
can predict the occurrence and whole evolution of wrinkling patterns in vari-
ous boundary and loading conditions as well as complex geometries. The point
of this framework lies in, but is not limited to, the application and improve-
ment of the following numerical methods to the instability pattern formation of
film-substrate systems:
• Finite element method to be able to deal with all the geometries, behaviors
and boundary conditions;
• Path-following technique for nonlinear problem resolution and post-buckling
tracing;
• Bifurcation indicator to detect bifurcation points and the associated in-
stability modes;
• Reduction techniques of models by multi-scale approaches;
• Bridging techniques to couple full models and reduced-order models con-
currently.
22ème Congrès Français de Mécanique Lyon, 24 au 28 Août 2015
Within this framework, both 2D and 3D models incorporate Asymptotic
Numerical Method (ANM) [18, 19] as a robust path-following technique and
bifurcation indicators well adapted to the ANM, so as to predict a sequence of
multiple bifurcations and the associated instability modes on their post-buckling
evolution path as the load is increased. The tracing of post-bifurcation response
is an important and difficult numerical problem. The ANM gives interactive
access to semi-analytical equilibrium branches, which offers considerable advan-
tage of reliability compared with classical iterative algorithms. The underlying
principle of the ANM is to build up the nonlinear solution branch in the form
of relatively high order truncated power series. The resulting series are then
introduced into the nonlinear problem, which helps to transform it into a se-
quence of linear problems that can be solved numerically. In this way, one
gets approximations of the solution path that are very accurate inside the ra-
dius of convergence. Moreover, by taking the advantage of the local polynomial
approximations of the branch within each step, the algorithm is remarkably ro-
bust and fully automatic. Furthermore, unlike incremental-iterative methods,
the arc-length step size in the ANM is fully adaptive since it is determined a pos-
teriori by the algorithm. A small radius of convergence and step accumulation
appear around the bifurcation and imply its presence (see Fig. 1). To our best
knowledge, it appears to be the first time that addresses the post-bifurcation
instability problems of film-substrate from the quantitative standpoint, through
applying these advanced numerical approaches.
Typical 3D post-bifurcation patterns include sinusoidal, checkerboard and
herringbone shapes (see Fig. 2), with possible spatial modulations, boundary
effects and localizations. The post-buckling behavior often leads to intricate
response curves with several secondary bifurcations that were rarely studied
in the literature and only in the case of periodic unit cell [20]. In conven-
tional finite element analysis, the post-buckling simulation may suffer from the
convergence issue if the film is much stiffer than the substrate. However, the
proposed advanced finite element procedure allows accurately describing these
bifurcation portraits by taking into account the effect of boundary conditions
and localizations, without any convergent problem. The occurrence and evolu-
tion of sinusoidal, checkerboard and herringbone patterns (see Fig. 2) as well
as period-doubling mode (see Fig. 1) will be highlighted. The results are ex-
pected to provide insight into the formation and evolution of wrinkle patterns in
film-substrate systems and be helpful to control the surface morphology, which
could be used to guide the design of film-substrate systems to achieve desired
wrinkling patterns for micro/nano-fabrication [21].
Acknowledgement
The authors acknowledge the financial support of French National Research
22ème Congrès Français de Mécanique Lyon, 24 au 28 Août 2015
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
−0.07
−0.06
−0.05
−0.04
−0.03
−0.02
−0.01
0
Load F (N/mm)
T
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e
 d
is
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e
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(m
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0 0.5 1 1.5
−1
0
1
0 0.5 1 1.5
−1
0
1
0 0.5 1 1.5
−1
0
1
1st bifurcation
2nd bifurcation
3rd bifurcation
1st mode
2nd mode
3rd mode
Figure 1: Bifurcation curve and the associated wrinkling modes with respect
to the incremental compression. Period-doubling mode appears at the third
bifurcation point.
(a) (b)
(c) (d)
Figure 2: Numerical results on different types of 3D wrinkling patterns: (a)
sinusoidal mode, (b) checkerboard mode, (c) herringbone mode, (d) top view of
(c).
22ème Congrès Français de Mécanique Lyon, 24 au 28 Août 2015
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A numerical framework for pattern formation modeling of film-substrate systems

International audienceWe propose a whole numerical framework to study the buckling problems of film-substrate systems: from 2D to 3D modeling, from classical to multi-scale perspective. The main objective is to apply advanced numerical methods (path-following techniques, bifurcation indicators, bridging techniques, multi-scale analyses, etc.) for multiple-bifurcation analyses of film-substrate buckling problems. Through incorporating these numerical methods with FEM, it can predict the occurrence and post-bifurcation evolution of wrinkling patterns in various boundary and loading conditions as well as complex geometry configurations.Nous proposons un cadre général pour étudier le problème de flambage de systèmes film-substrat de manière numérique : de la modélisation 2D ou 3D, d’un point de vue classique ou multi-échelle. L’objectif principal est d’appliquer aux systèmes film-substrat des méthodes numériques avancées (des techniques de cheminement, des indicateurs de bifurcation, des méthodes de couplage de modèle, des analyses multi-échelle, etc.) pour des analyses de bifurcations multiples. En intégrant ces méthodes numériques avec la méthode des éléments finis, il peut prédire l’événement et l’évolution de post-bifurcation des modes d’instabilité dans diverses conditions aux limites et de chargements ainsi que les configurations géométriques complexes

Potier-Ferry, Michel

Hal-Diderot

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A numerical framework for pattern formation modeling of film-substrate systems

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