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Cellular instabilities analyzed by multi-scale Fourier series: A review
1. | LEM3, Laboratoire d'Etudes des Microstructures et de Mécanique des Matériaux, UMR CNRS 7239, Université de Lorraine, Ile du Saulcy, 57045 Metz Cedex 01, France, France |
2. | Department of Mechanics and Engineering Science, Fudan University, 220 Handan Road, 200433 Shanghai, China |
3. | Laboratoire d'Ingénierie et Matériaux LIMAT, Faculté des Sciences Ben M'Sik, Université Hassan II de Casablanca, Sidi Othman, Casablanca, Morocco, Morocco, Morocco |
4. | School of Civil Engineering, Wuhan University, 8 South Road of East Lake, 430072 Wuhan, China, China |
5. | Université de Montpellier, Laboratoire de Mécanique et Génie Civil, UMR CNRS 5508, CC048 Place Eugène Bataillon, 34095 Montpellier Cedex 05, France |
References:
[1] |
S. Abdelkhalek, Un Exemple de Flambage Sous Contraintes Internes: Étude des Défauts de Planéité en Laminage à Froid Des Tôles Minces, Doctoral dissertation, École Nationale Supérieure des Mines de Paris, 2010. |
[2] |
S. Abdelkhalek, P. Montmitonnet, M. Potier-Ferry, H. Zahrouni, N. Legrand and P. Buessler, Strip flatness modelling including buckling phenomena during thin strip cold rolling, Ironmaking and Steelmaking, 37 (2010), 290-297.
doi: 10.1179/030192310X12646889255708. |
[3] |
J. C. Amazigo, B. Budiansky and G. F. Carrier, Asymptotic analyses of the buckling of imperfect columns on non-linear elastic foundations, Internat. J. Solids Structures, 6 (1970), 1341-1356. |
[4] |
K. Attipou, H. Hu, F. Mohri, M. Potier-Ferry and S. Belouettar, Thermal wrinkling of thin membranes using a Fourier-related double scale approach, Thin-Walled Structures, 94 (2015), 532-544.
doi: 10.1016/j.tws.2015.04.034. |
[5] |
A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North Holland Publ, Amsterdam, 1978. |
[6] |
N. N. Bogolyubov and Y. A. Mitropolski, Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York, 1963. |
[7] |
N. Bowden, S. Brittain, A. G. Evans, J. W. Hutchinson and G. M. Whitesides, Spontaneous formation of ordered structures in thin films of metals supported on an elastomeric polymer, Nature, 393 (1998), 146-149. |
[8] |
F. Brau, H. Vandeparre, A. Sabbah, C. Poulard, A. Boudaoud and P. Damman, Multiple-length-scale elastic instability mimics parametric resonance of non- linear oscillators, Nature Physics, 7 (2011), 56-60. |
[9] |
M. C. Cross, P. G. Daniels, P. C. Hohenberg and E. D. Siggia, Phase-winding solutions in a finite container above the convective threshold, J. Fluid Mech., 127 (1983), 155-183.
doi: 10.1017/S0022112083002670. |
[10] |
M. C. Cross and P. C. Hohenberg, Pattern formation out of equilibrium, Rev. Modern Phys., 65 (1993), 851-1112. |
[11] |
N. Damil and M. Potier-Ferry, Amplitude equations for cellular instabilities, Dynamics and Stability of Systems, 7 (1992), 1-34.
doi: 10.1080/02681119208806124. |
[12] |
N. Damil and M. Potier-Ferry, A generalized continuum approach to describe instability pattern formation by a multiple scale analysis, Comptes Rendus Mecanique, 334 (2006), 674-678.
doi: 10.1016/j.crme.2006.09.002. |
[13] |
N. Damil and M. Potier-Ferry, A generalized continuum approach to predict local buckling patterns of thin structures, European Journal of Computational Mechanics, 17 (2008), 945-956. |
[14] |
N. Damil and M. Potier-Ferry, Influence of local wrinkling on membrane behaviour: A new approach by the technique of slowly variable Fourier coefficients, J. Mech. Phys. Solids, 58 (2010), 1139-1153.
doi: 10.1016/j.jmps.2010.04.002. |
[15] |
N. Damil, M. Potier-Ferry and H. Hu, New nonlinear multiscale models for membrane wrinkling, Comptes Rendus Mecanique, 341 (2013), 616-624. |
[16] |
N. Damil, M. Potier-Ferry and H. Hu, Membrane wrinkling revisited from a multi-scale point of view, Advanced Modeling and Simulation in Engineering Sciences, 1 (2014), p6.
doi: 10.1186/2213-7467-1-6. |
[17] |
A. Eriksson and A. Nordmark, Instability of hyper-elastic balloon-shaped space membranes under pressure loads, Comput. Methods Appl. Mech. Engrg., 237 (2012), 118-129.
doi: 10.1016/j.cma.2012.05.012. |
[18] |
F. Feyel and J. L. Chaboche, FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials, Comput. Methods Appl. Mech. Engrg., 183 (2000), 309-330.
doi: 10.1016/S0045-7825(99)00224-8. |
[19] |
G. Geymonat, S. Muller and N. Triantafyllidis, Homogenization of nonlinearly elastic materials: Microscopic bifurcation and macroscopic loss of rank-one convexity, Arch. Ration. Mech. Anal., 122 (1993), 231-290.
doi: 10.1007/BF00380256. |
[20] |
R. Hoyle, Pattern Formation, An Introduction to Methods, Cambrige University Press, Cambridge, 2006.
doi: 10.1017/CBO9780511616051. |
[21] |
H. Hu, N. Damil and M. Potier-Ferry, A bridging technique to analyze the influence of boundary conditions on instability patterns, J. Comput. Phys., 230 (2011), 3753-3764.
doi: 10.1016/j.jcp.2011.01.044. |
[22] |
Q. Huang, H. Hu, K. Yu, M. Potier-Ferry, N. Damil and S. Belouettar, Macroscopic simulation of membrane wrinkling for various loading cases, Internat. J. Solids Structures, 64-65 (2015), 246-258.
doi: 10.1016/j.ijsolstr.2015.04.003. |
[23] |
G. W. Hunt, M. A. Peletier, A. R. Champneys, P. D. Woods, M. A. Wadee, C. J. Budd and G. J. Lord, Cellular buckling in long structures, Nonlinear Dynamics, 21 (2000), 3-29.
doi: 10.1023/A:1008398006403. |
[24] |
G. Iooss, A. Mielke and Y. Demay, Theory of steady Ginzburg-Landau equation in hydrodynamic stability problems, Eur. J. Mech. B Fluids, 8 (1989), 229-268. |
[25] |
Y. Lecieux and R. Bouzidi, Experimentation analysis on membrane wrinkling under biaxial load - Comparison with bifurcation analysis, Internat. J. Solids Structures, 47 (2010), 2459-2475. |
[26] |
B. Li, Y. P. Cao, X. Q. Feng and H. J. Gao, Mechanics of morphological instabilities and surface wrinkling in soft materials: A review, Soft Matter, 8 (2012), 5728-5745.
doi: 10.1039/c2sm00011c. |
[27] |
Y. Liu, K. Yu, H. Hu, S. Belouettar and M. Potier-Ferry, A Fourier-related double scale analysis on instability phenomena of sandwich beams, Internat. J. Solids Structures, 49 (2012), 3077-3088. |
[28] |
K. Mhada, B. Braikat and N. Damil, A 2D Fourier double scale analysis of global-local instability interaction in sandwich structures, 21ème Congrès Français de Mécanique, Bordeaux, France, 2013. |
[29] |
K. Mhada, B. Braikat, H. Hu, N. Damil and M. Potier-Ferry, About macroscopic models of instability pattern formation, Internat. J. Solids Structures, 49 (2012), 2978-2989.
doi: 10.1016/j.ijsolstr.2012.05.033. |
[30] |
H. Moulinec and P. Suquet, A numerical method for computing the overall response of nonlinear composites with complex microstructure, Comput. Methods Appl. Mech. Engrg., 157 (1998), 69-94.
doi: 10.1016/S0045-7825(97)00218-1. |
[31] |
R. Nakhoul, P. Montmitonnet and M. Potier-Ferry, Multi-scale method modeling of thin sheet buckling under residual stresses in the context of strip rolling, Internat. J. Solids Structures, 66 (2015), 62-76.
doi: 10.1016/j.ijsolstr.2015.03.028. |
[32] |
A. H. Nayfeh, Perturbation Methods, John Wiley and Sons, New York, 1973. |
[33] |
A. C. Newell and J. A. Whitehead, Finite band width, finite amplitude convection, J. Fluid Mech., 38 (1969), 279-303.
doi: 10.1017/S0022112069000176. |
[34] |
S. Nezamabadi, J. Yvonnet, H. Zahrouni and M. Potier-Ferry, A multilevel computational strategy for handling microscopic and macroscopic instabilities, Comput. Methods Appl. Mech. Engrg., 198 (2009), 2099-2110.
doi: 10.1016/j.cma.2009.02.026. |
[35] |
Y. Pomeau and S. Zaleski, Wavelength selection in one-dimensional cellular structures, Journal de Physique, 42 (1981), 515-528.
doi: 10.1051/jphys:01981004204051500. |
[36] |
R. Rossi, M. Lazzari, R. Vitaliani and E. Onate, Simulation of light-weight membrane structures by wrinkling model, Internat. J. Numer. Methods Engrg, 62 (2005), 2127-2153.
doi: 10.1002/nme.1266. |
[37] |
E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, Lecture Notes in Physics, 127, Springer-Verlag, Heidelberg, 1980. |
[38] |
L. A. Segel, Distant side walls cause slow amplitude modulation of cellular convection, J. Fluid Mech., 38 (1969), 203-224.
doi: 10.1017/S0022112069000127. |
[39] |
R. J. M. Smit, W. A. M. Brekelmans and H. E. H. Meijer, Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling, Comput. Methods Appl. Mech. Engrg., 155 (1998), 181-192.
doi: 10.1016/S0045-7825(97)00139-4. |
[40] |
P. Suquet, Plasticité et Homogénéisation, Doctoral dissertation, Université Pierre et Marie Curie, 1982. |
[41] |
M. A. Wadee and M. Farsi, Cellular buckling in stiffened plates, Proc. R. Soc. A, 470 (2014), 20140094, 17 pp.
doi: 10.1098/rspa.2014.0094. |
[42] |
J. E Wesfreid and S. Zaleski editors, Cellular Structures in Instabilities, Lecture Notes in Physics, 210, Springer-Verlag, Heidelberg, 1984.
doi: 10.1007/3-540-13879-X. |
[43] |
Y. W. Wong and S. Pellegrino, Wrinkled membranes-Part1: Experiments, Journal of Mechanics of Materials and Structures, 1 (2006), 3-25. |
[44] |
F. Xu, H. Hu, M. Potier-Ferry and S. Belouettar, Bridging techniques in a multi-scale modeling of pattern formation, Internat. J. Solids Structures, 51 (2014), 3119-3134.
doi: 10.1016/j.ijsolstr.2014.05.011. |
[45] |
F. Xu and M. Potier-Ferry, A multi-scale modeling framework for instabilities of film/substrate systems, J. Mech. Phys. Solids, 86 (2016), 150-172.
doi: 10.1016/j.jmps.2015.10.003. |
[46] |
K. Yu, H. Hu, S. Chen, S. Belouettar and M. Potier-Ferry, Multi-scale techniques to analyze instabilities in sandwich structures, Composite Structures, 96 (2013), 751-762.
doi: 10.1016/j.compstruct.2012.10.007. |
show all references
References:
[1] |
S. Abdelkhalek, Un Exemple de Flambage Sous Contraintes Internes: Étude des Défauts de Planéité en Laminage à Froid Des Tôles Minces, Doctoral dissertation, École Nationale Supérieure des Mines de Paris, 2010. |
[2] |
S. Abdelkhalek, P. Montmitonnet, M. Potier-Ferry, H. Zahrouni, N. Legrand and P. Buessler, Strip flatness modelling including buckling phenomena during thin strip cold rolling, Ironmaking and Steelmaking, 37 (2010), 290-297.
doi: 10.1179/030192310X12646889255708. |
[3] |
J. C. Amazigo, B. Budiansky and G. F. Carrier, Asymptotic analyses of the buckling of imperfect columns on non-linear elastic foundations, Internat. J. Solids Structures, 6 (1970), 1341-1356. |
[4] |
K. Attipou, H. Hu, F. Mohri, M. Potier-Ferry and S. Belouettar, Thermal wrinkling of thin membranes using a Fourier-related double scale approach, Thin-Walled Structures, 94 (2015), 532-544.
doi: 10.1016/j.tws.2015.04.034. |
[5] |
A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North Holland Publ, Amsterdam, 1978. |
[6] |
N. N. Bogolyubov and Y. A. Mitropolski, Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York, 1963. |
[7] |
N. Bowden, S. Brittain, A. G. Evans, J. W. Hutchinson and G. M. Whitesides, Spontaneous formation of ordered structures in thin films of metals supported on an elastomeric polymer, Nature, 393 (1998), 146-149. |
[8] |
F. Brau, H. Vandeparre, A. Sabbah, C. Poulard, A. Boudaoud and P. Damman, Multiple-length-scale elastic instability mimics parametric resonance of non- linear oscillators, Nature Physics, 7 (2011), 56-60. |
[9] |
M. C. Cross, P. G. Daniels, P. C. Hohenberg and E. D. Siggia, Phase-winding solutions in a finite container above the convective threshold, J. Fluid Mech., 127 (1983), 155-183.
doi: 10.1017/S0022112083002670. |
[10] |
M. C. Cross and P. C. Hohenberg, Pattern formation out of equilibrium, Rev. Modern Phys., 65 (1993), 851-1112. |
[11] |
N. Damil and M. Potier-Ferry, Amplitude equations for cellular instabilities, Dynamics and Stability of Systems, 7 (1992), 1-34.
doi: 10.1080/02681119208806124. |
[12] |
N. Damil and M. Potier-Ferry, A generalized continuum approach to describe instability pattern formation by a multiple scale analysis, Comptes Rendus Mecanique, 334 (2006), 674-678.
doi: 10.1016/j.crme.2006.09.002. |
[13] |
N. Damil and M. Potier-Ferry, A generalized continuum approach to predict local buckling patterns of thin structures, European Journal of Computational Mechanics, 17 (2008), 945-956. |
[14] |
N. Damil and M. Potier-Ferry, Influence of local wrinkling on membrane behaviour: A new approach by the technique of slowly variable Fourier coefficients, J. Mech. Phys. Solids, 58 (2010), 1139-1153.
doi: 10.1016/j.jmps.2010.04.002. |
[15] |
N. Damil, M. Potier-Ferry and H. Hu, New nonlinear multiscale models for membrane wrinkling, Comptes Rendus Mecanique, 341 (2013), 616-624. |
[16] |
N. Damil, M. Potier-Ferry and H. Hu, Membrane wrinkling revisited from a multi-scale point of view, Advanced Modeling and Simulation in Engineering Sciences, 1 (2014), p6.
doi: 10.1186/2213-7467-1-6. |
[17] |
A. Eriksson and A. Nordmark, Instability of hyper-elastic balloon-shaped space membranes under pressure loads, Comput. Methods Appl. Mech. Engrg., 237 (2012), 118-129.
doi: 10.1016/j.cma.2012.05.012. |
[18] |
F. Feyel and J. L. Chaboche, FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials, Comput. Methods Appl. Mech. Engrg., 183 (2000), 309-330.
doi: 10.1016/S0045-7825(99)00224-8. |
[19] |
G. Geymonat, S. Muller and N. Triantafyllidis, Homogenization of nonlinearly elastic materials: Microscopic bifurcation and macroscopic loss of rank-one convexity, Arch. Ration. Mech. Anal., 122 (1993), 231-290.
doi: 10.1007/BF00380256. |
[20] |
R. Hoyle, Pattern Formation, An Introduction to Methods, Cambrige University Press, Cambridge, 2006.
doi: 10.1017/CBO9780511616051. |
[21] |
H. Hu, N. Damil and M. Potier-Ferry, A bridging technique to analyze the influence of boundary conditions on instability patterns, J. Comput. Phys., 230 (2011), 3753-3764.
doi: 10.1016/j.jcp.2011.01.044. |
[22] |
Q. Huang, H. Hu, K. Yu, M. Potier-Ferry, N. Damil and S. Belouettar, Macroscopic simulation of membrane wrinkling for various loading cases, Internat. J. Solids Structures, 64-65 (2015), 246-258.
doi: 10.1016/j.ijsolstr.2015.04.003. |
[23] |
G. W. Hunt, M. A. Peletier, A. R. Champneys, P. D. Woods, M. A. Wadee, C. J. Budd and G. J. Lord, Cellular buckling in long structures, Nonlinear Dynamics, 21 (2000), 3-29.
doi: 10.1023/A:1008398006403. |
[24] |
G. Iooss, A. Mielke and Y. Demay, Theory of steady Ginzburg-Landau equation in hydrodynamic stability problems, Eur. J. Mech. B Fluids, 8 (1989), 229-268. |
[25] |
Y. Lecieux and R. Bouzidi, Experimentation analysis on membrane wrinkling under biaxial load - Comparison with bifurcation analysis, Internat. J. Solids Structures, 47 (2010), 2459-2475. |
[26] |
B. Li, Y. P. Cao, X. Q. Feng and H. J. Gao, Mechanics of morphological instabilities and surface wrinkling in soft materials: A review, Soft Matter, 8 (2012), 5728-5745.
doi: 10.1039/c2sm00011c. |
[27] |
Y. Liu, K. Yu, H. Hu, S. Belouettar and M. Potier-Ferry, A Fourier-related double scale analysis on instability phenomena of sandwich beams, Internat. J. Solids Structures, 49 (2012), 3077-3088. |
[28] |
K. Mhada, B. Braikat and N. Damil, A 2D Fourier double scale analysis of global-local instability interaction in sandwich structures, 21ème Congrès Français de Mécanique, Bordeaux, France, 2013. |
[29] |
K. Mhada, B. Braikat, H. Hu, N. Damil and M. Potier-Ferry, About macroscopic models of instability pattern formation, Internat. J. Solids Structures, 49 (2012), 2978-2989.
doi: 10.1016/j.ijsolstr.2012.05.033. |
[30] |
H. Moulinec and P. Suquet, A numerical method for computing the overall response of nonlinear composites with complex microstructure, Comput. Methods Appl. Mech. Engrg., 157 (1998), 69-94.
doi: 10.1016/S0045-7825(97)00218-1. |
[31] |
R. Nakhoul, P. Montmitonnet and M. Potier-Ferry, Multi-scale method modeling of thin sheet buckling under residual stresses in the context of strip rolling, Internat. J. Solids Structures, 66 (2015), 62-76.
doi: 10.1016/j.ijsolstr.2015.03.028. |
[32] |
A. H. Nayfeh, Perturbation Methods, John Wiley and Sons, New York, 1973. |
[33] |
A. C. Newell and J. A. Whitehead, Finite band width, finite amplitude convection, J. Fluid Mech., 38 (1969), 279-303.
doi: 10.1017/S0022112069000176. |
[34] |
S. Nezamabadi, J. Yvonnet, H. Zahrouni and M. Potier-Ferry, A multilevel computational strategy for handling microscopic and macroscopic instabilities, Comput. Methods Appl. Mech. Engrg., 198 (2009), 2099-2110.
doi: 10.1016/j.cma.2009.02.026. |
[35] |
Y. Pomeau and S. Zaleski, Wavelength selection in one-dimensional cellular structures, Journal de Physique, 42 (1981), 515-528.
doi: 10.1051/jphys:01981004204051500. |
[36] |
R. Rossi, M. Lazzari, R. Vitaliani and E. Onate, Simulation of light-weight membrane structures by wrinkling model, Internat. J. Numer. Methods Engrg, 62 (2005), 2127-2153.
doi: 10.1002/nme.1266. |
[37] |
E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, Lecture Notes in Physics, 127, Springer-Verlag, Heidelberg, 1980. |
[38] |
L. A. Segel, Distant side walls cause slow amplitude modulation of cellular convection, J. Fluid Mech., 38 (1969), 203-224.
doi: 10.1017/S0022112069000127. |
[39] |
R. J. M. Smit, W. A. M. Brekelmans and H. E. H. Meijer, Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling, Comput. Methods Appl. Mech. Engrg., 155 (1998), 181-192.
doi: 10.1016/S0045-7825(97)00139-4. |
[40] |
P. Suquet, Plasticité et Homogénéisation, Doctoral dissertation, Université Pierre et Marie Curie, 1982. |
[41] |
M. A. Wadee and M. Farsi, Cellular buckling in stiffened plates, Proc. R. Soc. A, 470 (2014), 20140094, 17 pp.
doi: 10.1098/rspa.2014.0094. |
[42] |
J. E Wesfreid and S. Zaleski editors, Cellular Structures in Instabilities, Lecture Notes in Physics, 210, Springer-Verlag, Heidelberg, 1984.
doi: 10.1007/3-540-13879-X. |
[43] |
Y. W. Wong and S. Pellegrino, Wrinkled membranes-Part1: Experiments, Journal of Mechanics of Materials and Structures, 1 (2006), 3-25. |
[44] |
F. Xu, H. Hu, M. Potier-Ferry and S. Belouettar, Bridging techniques in a multi-scale modeling of pattern formation, Internat. J. Solids Structures, 51 (2014), 3119-3134.
doi: 10.1016/j.ijsolstr.2014.05.011. |
[45] |
F. Xu and M. Potier-Ferry, A multi-scale modeling framework for instabilities of film/substrate systems, J. Mech. Phys. Solids, 86 (2016), 150-172.
doi: 10.1016/j.jmps.2015.10.003. |
[46] |
K. Yu, H. Hu, S. Chen, S. Belouettar and M. Potier-Ferry, Multi-scale techniques to analyze instabilities in sandwich structures, Composite Structures, 96 (2013), 751-762.
doi: 10.1016/j.compstruct.2012.10.007. |
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