A new Fourier-related double scale analysis for wrinkling analysis of thin films on compliant substrates

Abstract In this paper, a new Fourier-related double scale approach is presented to study the wrinkling of thin films on compliant substrates. By using the method of Fourier series with slowly variable coefficients, the 1D microscopic model proposed by Yang et al. (2015) is transformed into a 1D macroscopic film/substrate model whose mesh size is independent on the wrinkling wavelength. Numerical tests prove that the new model improves computational efficiency significantly with accurate results, especially when dealing with wrinkling phenomena with vast wavenumbers. Besides, we propose a strategy to efficiently trace the wrinkling pattern corresponding to the lowest critical load by accounting for several harmonics of Fourier series in this new model. The established nonlinear system is solved by the Asymptotic Numerical Method (ANM), which has advantages of efficiency and reliability for stability analyses.

[1]  Heng Hu,et al.  New nonlinear multi-scale models for wrinkled membranes , 2013 .

[2]  Khadija Mhada,et al.  A 2D Fourier double scale analysis of global-local instability interaction in sandwich structures , 2013 .

[3]  George M. Whitesides,et al.  Spontaneous formation of ordered structures in thin films of metals supported on an elastomeric polymer , 1998, Nature.

[4]  Salim Belouettar,et al.  Multi-scale nonlinear modelling of sandwich structures using the Arlequin method , 2010 .

[5]  L. Mahadevan,et al.  Geometry and physics of wrinkling. , 2003, Physical review letters.

[6]  J. Hutchinson,et al.  Herringbone Buckling Patterns of Compressed Thin Films on Compliant Substrates , 2004 .

[7]  Salim Belouettar,et al.  Macroscopic simulation of membrane wrinkling for various loading cases , 2015 .

[8]  Erasmo Carrera,et al.  Analysis of thickness locking in classical, refined and mixed multilayered plate theories , 2008 .

[9]  Heng Hu,et al.  About macroscopic models of instability pattern formation , 2012 .

[10]  Heng Hu,et al.  A bridging technique to analyze the influence of boundary conditions on instability patterns , 2011, J. Comput. Phys..

[11]  Z. Suo,et al.  Nonlinear analyses of wrinkles in a film bonded to a compliant substrate , 2005 .

[12]  Salim Belouettar,et al.  A new Fourier-related double scale analysis for instability phenomena in sandwich structures , 2012 .

[13]  Willi Volksen,et al.  A buckling-based metrology for measuring the elastic moduli of polymeric thin films , 2004, Nature materials.

[14]  J. Rogers,et al.  Finite width effect of thin-films buckling on compliant substrate : Experimental and theoretical studies , 2008 .

[15]  Guillaume Rateau,et al.  The Arlequin method as a flexible engineering design tool , 2005 .

[16]  Huajian Gao,et al.  Surface wrinkling of mucosa induced by volumetric growth: Theory, simulation and experiment , 2011 .

[17]  Hachmi Ben Dhia,et al.  Global-local approaches: the Arlequin framework , 2006 .

[18]  J. Reddy.  A Simple Higher-Order Theory for Laminated Composite Plates , 1984 .

[19]  Olivier Polit,et al.  High-order triangular sandwich plate finite element for linear and non-linear analyses , 2000 .

[20]  Salim Belouettar,et al.  Multi-scale techniques to analyze instabilities in sandwich structures , 2013 .

[21]  Erasmo Carrera,et al.  Radial basis functions collocation and a unified formulation for bending, vibration and buckling analysis of laminated plates, according to a variation of Murakami’s zig-zag theory , 2011 .

[22]  Huajian Gao,et al.  Mechanics of morphological instabilities and surface wrinkling in soft materials: a review , 2012 .

[23]  Michel Potier-Ferry,et al.  A generalized continuum approach to predict local buckling patterns of thin structures , 2008 .

[24]  Michel Potier-Ferry,et al.  A New method to compute perturbed bifurcations: Application to the buckling of imperfect elastic structures , 1990 .

[25]  Michel Potier-Ferry,et al.  Influence of local wrinkling on membrane behaviour: A new approach by the technique of slowly variable Fourier coefficients , 2010 .

[26]  Hui‐Shen Shen.  A Two-Step Perturbation Method in Nonlinear Analysis of Beams, Plates and Shells , 2013 .

[27]  Salim Belouettar,et al.  A novel finite element for global and local buckling analysis of sandwich beams , 2009 .

[28]  Michel Potier-Ferry,et al.  Méthode asymptotique numérique , 2008 .

[29]  I. Bizjak,et al.  Measurement of branching fractions for B-->eta(c)K(*) decays. , 2002, Physical review letters.

[30]  Erasmo Carrera,et al.  Multi-scale modelling of sandwich structures using hierarchical kinematics , 2011 .

[31]  Michel Potier-Ferry,et al.  Membrane wrinkling revisited from a multi-scale point of view , 2014, Adv. Model. Simul. Eng. Sci..

[32]  Wei Hong,et al.  Evolution of wrinkles in hard films on soft substrates. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[33]  Gaetano Giunta,et al.  A new family of finite elements for wrinkling analysis of thin films on compliant substrates , 2015 .

[34]  E. Carrera.  Theories and Finite Elements for Multilayered Plates and Shells:A Unified compact formulation with numerical assessment and benchmarking , 2003 .

[35]  Olivier Polit,et al.  A family of sinus finite elements for the analysis of rectangular laminated beams , 2008 .

[36]  Bruno Cochelin,et al.  Asymptotic-numerical methods and pade approximants for non-linear elastic structures , 1994 .

[37]  Ramesh Talreja,et al.  Modeling of Wrinkling in Sandwich Panels under Compression , 1999 .

[38]  John A. Rogers,et al.  Buckling of a stiff thin film on a compliant substrate in large deformation , 2008 .

[39]  H. G. Allen.  Analysis and design of structural sandwich panels , 1969 .