# American Institute of Mathematical Sciences

March  2012, 17(2): 553-574. doi: 10.3934/dcdsb.2012.17.553

## Adhesive flexible material structures

 1 Dipartimento di Matematica Politecnico di Bari, via Re David 200, 70125 Bari, Italy 2 Dipartimento di Ingegneria della Produzione Termoenergetica e Modelli Matematici, Università di Genova, Piazzale Kennedy, Fiera del Mare, Padiglione D, 16129 Genova, Italy 3 Dipartimento di Matematica "Francesco Brioschi", Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

Received  November 2010 Revised  June 2011 Published  December 2011

We study variational problems modeling the adhesion interaction with a rigid substrate for elastic strings and rods. We produce conditions characterizing bonded and detached states as well as optimality properties with respect to loading and geometry. We show Euler equations for minimizers of the total energy outside self-contact and secondary contact points with the substrate.
Citation: Francesco Maddalena, Danilo Percivale, Franco Tomarelli. Adhesive flexible material structures. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 553-574. doi: 10.3934/dcdsb.2012.17.553
##### References:
 [1] S. Antman, "Nonlinear Problems of Elasticity,", Second Edition, 107 (2005). Google Scholar [2] B. Aksak, M. P. Murphy and M. Sitti, Adhesion of biologically inspired vertical and angled polymer microfiber arrays,, Langmuir, 23 (2007), 3322. doi: 10.1021/la062697t. Google Scholar [3] K. T. Andrews, L. Chapman, J. R. Fernández, M. Fisackerly, M. Shillor, L. Vanerian and T. VanHouten, A membrane in adhesive contact,, SIAM J. Appl. Math., 64 (2003), 152. doi: 10.1137/S0036139902406206. Google Scholar [4] G. Bellettini and L. Mugnai, Characterization and representation of the lower semicontinuous envelope of the elastica functional,, Ann. Inst. Poincaré Anal. Non Lin., 21 (2004), 839. doi: 10.1016/j.anihpc.2004.01.001. Google Scholar [5] R. Burridge and J. B. Keller, Peeling, slipping and cracking--Some one dimensional free boundaries problems in mechanics,, SIAM Review, 20 (1978), 31. doi: 10.1137/1020003. Google Scholar [6] R. Courant and H. Robbins, "What Is Mathematics? An Elementary Approach to Ideas and Methods,", Oxford Univ. Press, (1979). Google Scholar [7] G. dal Maso, A. DeSimone and F. Tomarelli, eds., "Variational Problems in Materials Science,", Proceedings of the International Conference held in Trieste, 68 (2004), 6. Google Scholar [8] A. A. Evans and E. Lauga, Adhesion transition of flexible sheets,, Physical Review Letters, 79 (2009). Google Scholar [9] M. Frémond, Contact with adhesion, in, Topic in Nonsmooth Mechanics, (1988), 157. Google Scholar [10] A. Ghatak, L. Mahadevan, J. Y. Chung, M. K. Chaudhury and V. Shenoy, Peeling from a biomimetically patterned thin elastic film,, Proc. Royal Soc. London A, 460 (2004), 2725. doi: 10.1098/rspa.2004.1313. Google Scholar [11] O. Gonzalez, J. H. Maddocks, F. Schuricht and H. von der Mosel, Global curvature and self-contact of nonlinearly elastic curves and rods,, Calc. Var. Partial Differential Equations, 14 (2002), 29. Google Scholar [12] W. Han, K. L. Kuttler, M. Shillor and M. Sofonea, Elastic beam in adhesive contact, Int. J. Solids, Structures, 39 (2002), 1145. doi: 10.1016/S0020-7683(01)00250-5. Google Scholar [13] N. Israelachvili, "Intramolecular and Surface Forces,", Academic Press, (1992). Google Scholar [14] K. L. Johnson, Adhesion and friction between a smooth elastic spherical asperity and a smooth surface,, Proc. Royal Soc. London A, 453 (1997), 163. doi: 10.1098/rspa.1997.0010. Google Scholar [15] K. Kendall, "Molecular Adhesion and its Applications,", Kluwer Academic/Plenum, (2001). Google Scholar [16] J. Lee, C. Majidi, B. Schubert and R. Fearing, Sliding-induced adhesion of stiff polymer microfiber arrays. I. Macroscale behavior,, J. R. Soc. Interface, 5 (2008), 835. doi: 10.1098/rsif.2007.1308. Google Scholar [17] F. Maddalena and D. Percivale, Variational models for peeling problems,, Int. Free Boundaries, 10 (2008), 503. doi: 10.4171/IFB/199. Google Scholar [18] F. Maddalena, D. Percivale, G. Puglisi and L. Truskinowsky, Mechanics of reversible unzipping,, Continuum Mech. Thermodyn., 21 (2009), 251. doi: 10.1007/s00161-009-0108-2. Google Scholar [19] F. Maddalena, D. Percivale and G. Puglisi, Energy minimizing states in adhesion problems for elastic rods,, Appl. Mathematical Sciences, 4 (2010), 3749. Google Scholar [20] F. Maddalena, D. Percivale and F. Tomarelli, Scaling deduction of curvature elastic energies,, to appear., (). Google Scholar [21] F. Maddalena, D. Percivale and F. Tomarelli, Elastic structures in adhesion interaction,, to appear in Proc., (2010). Google Scholar [22] C. Majidi, On formulating an adhesion problem using Euler's elastica,, Mechanics Research Communications, 34 (2007), 85. doi: 10.1016/j.mechrescom.2006.06.007. Google Scholar [23] C. Majidi, Shear adhesion between an elastica and a rigid flat surface,, Mechanics Research Communications, 36 (2009), 369. doi: 10.1016/j.mechrescom.2008.10.010. Google Scholar [24] X. Oyharcabal and T. Frisch, Peeling off an elastica from a smooth atractive substrate,, Physical Review E, 71 (2005), 036611. doi: 10.1103/PhysRevE.71.036611. Google Scholar [25] D. Percivale and F. Tomarelli, Scaled Korn-Poincaré inequality in BD and a model of elastic plastic cantilever,, Asymptotic Analysis, 23 (2000), 291. Google Scholar [26] D. Percivale and F. Tomarelli, From special bounded deformation to special bounded Hessian: The elastic-plastic beam,, Math. Models Methods Appl. Sci., 15 (2005), 1009. doi: 10.1142/S0218202505000650. Google Scholar [27] D. Percivale and F. Tomarelli, A variational principle for plastic hinges in a beam,, Math. Models Methods Appl. Sci., 19 (2009), 2263. doi: 10.1142/S021820250900411X. Google Scholar [28] P. Podio-Guidugli, Peeling tapes,, in, 11 (2005), 253. doi: 10.1007/0-387-26261-X_25. Google Scholar [29] M. C. Strus, L. Zalamea, A. Raman, R. B. Pipes, C. V. Nguyen and E. A. Stach, Peeling force spectroscopy: Exposing the adhesive nanomechanics of one-dimensional nanostructures,, NANO LETTERS, 8 (2008), 544. doi: 10.1021/nl0728118. Google Scholar [30] M. Shillor, M. Sofonea and J. J. Telega, "Models and Analysis of Quasistatic Contact,", Lecture Notes in Physics, 655 (2004). Google Scholar [31] M. Sofonea, W. Han and M. Shillor, "Analysis and Approximations of Contact Problems with Adhesion or Damage,", Pure and Applied Mathematics (Boca Raton), 276 (2006). Google Scholar [32] Y.-P. Zhao, L. S. Wang and T. X. Yu, Mechanics of adhesion in MEMS-a review,, J. Adhesion Sci.Technol., 17 (2003), 519. doi: 10.1163/15685610360554393. Google Scholar

show all references

##### References:
 [1] S. Antman, "Nonlinear Problems of Elasticity,", Second Edition, 107 (2005). Google Scholar [2] B. Aksak, M. P. Murphy and M. Sitti, Adhesion of biologically inspired vertical and angled polymer microfiber arrays,, Langmuir, 23 (2007), 3322. doi: 10.1021/la062697t. Google Scholar [3] K. T. Andrews, L. Chapman, J. R. Fernández, M. Fisackerly, M. Shillor, L. Vanerian and T. VanHouten, A membrane in adhesive contact,, SIAM J. Appl. Math., 64 (2003), 152. doi: 10.1137/S0036139902406206. Google Scholar [4] G. Bellettini and L. Mugnai, Characterization and representation of the lower semicontinuous envelope of the elastica functional,, Ann. Inst. Poincaré Anal. Non Lin., 21 (2004), 839. doi: 10.1016/j.anihpc.2004.01.001. Google Scholar [5] R. Burridge and J. B. Keller, Peeling, slipping and cracking--Some one dimensional free boundaries problems in mechanics,, SIAM Review, 20 (1978), 31. doi: 10.1137/1020003. Google Scholar [6] R. Courant and H. Robbins, "What Is Mathematics? An Elementary Approach to Ideas and Methods,", Oxford Univ. Press, (1979). Google Scholar [7] G. dal Maso, A. DeSimone and F. Tomarelli, eds., "Variational Problems in Materials Science,", Proceedings of the International Conference held in Trieste, 68 (2004), 6. Google Scholar [8] A. A. Evans and E. Lauga, Adhesion transition of flexible sheets,, Physical Review Letters, 79 (2009). Google Scholar [9] M. Frémond, Contact with adhesion, in, Topic in Nonsmooth Mechanics, (1988), 157. Google Scholar [10] A. Ghatak, L. Mahadevan, J. Y. Chung, M. K. Chaudhury and V. Shenoy, Peeling from a biomimetically patterned thin elastic film,, Proc. Royal Soc. London A, 460 (2004), 2725. doi: 10.1098/rspa.2004.1313. Google Scholar [11] O. Gonzalez, J. H. Maddocks, F. Schuricht and H. von der Mosel, Global curvature and self-contact of nonlinearly elastic curves and rods,, Calc. Var. Partial Differential Equations, 14 (2002), 29. Google Scholar [12] W. Han, K. L. Kuttler, M. Shillor and M. Sofonea, Elastic beam in adhesive contact, Int. J. Solids, Structures, 39 (2002), 1145. doi: 10.1016/S0020-7683(01)00250-5. Google Scholar [13] N. Israelachvili, "Intramolecular and Surface Forces,", Academic Press, (1992). Google Scholar [14] K. L. Johnson, Adhesion and friction between a smooth elastic spherical asperity and a smooth surface,, Proc. Royal Soc. London A, 453 (1997), 163. doi: 10.1098/rspa.1997.0010. Google Scholar [15] K. Kendall, "Molecular Adhesion and its Applications,", Kluwer Academic/Plenum, (2001). Google Scholar [16] J. Lee, C. Majidi, B. Schubert and R. Fearing, Sliding-induced adhesion of stiff polymer microfiber arrays. I. Macroscale behavior,, J. R. Soc. Interface, 5 (2008), 835. doi: 10.1098/rsif.2007.1308. Google Scholar [17] F. Maddalena and D. Percivale, Variational models for peeling problems,, Int. Free Boundaries, 10 (2008), 503. doi: 10.4171/IFB/199. Google Scholar [18] F. Maddalena, D. Percivale, G. Puglisi and L. Truskinowsky, Mechanics of reversible unzipping,, Continuum Mech. Thermodyn., 21 (2009), 251. doi: 10.1007/s00161-009-0108-2. Google Scholar [19] F. Maddalena, D. Percivale and G. Puglisi, Energy minimizing states in adhesion problems for elastic rods,, Appl. Mathematical Sciences, 4 (2010), 3749. Google Scholar [20] F. Maddalena, D. Percivale and F. Tomarelli, Scaling deduction of curvature elastic energies,, to appear., (). Google Scholar [21] F. Maddalena, D. Percivale and F. Tomarelli, Elastic structures in adhesion interaction,, to appear in Proc., (2010). Google Scholar [22] C. Majidi, On formulating an adhesion problem using Euler's elastica,, Mechanics Research Communications, 34 (2007), 85. doi: 10.1016/j.mechrescom.2006.06.007. Google Scholar [23] C. Majidi, Shear adhesion between an elastica and a rigid flat surface,, Mechanics Research Communications, 36 (2009), 369. doi: 10.1016/j.mechrescom.2008.10.010. Google Scholar [24] X. Oyharcabal and T. Frisch, Peeling off an elastica from a smooth atractive substrate,, Physical Review E, 71 (2005), 036611. doi: 10.1103/PhysRevE.71.036611. Google Scholar [25] D. Percivale and F. Tomarelli, Scaled Korn-Poincaré inequality in BD and a model of elastic plastic cantilever,, Asymptotic Analysis, 23 (2000), 291. Google Scholar [26] D. Percivale and F. Tomarelli, From special bounded deformation to special bounded Hessian: The elastic-plastic beam,, Math. Models Methods Appl. Sci., 15 (2005), 1009. doi: 10.1142/S0218202505000650. Google Scholar [27] D. Percivale and F. Tomarelli, A variational principle for plastic hinges in a beam,, Math. Models Methods Appl. Sci., 19 (2009), 2263. doi: 10.1142/S021820250900411X. Google Scholar [28] P. Podio-Guidugli, Peeling tapes,, in, 11 (2005), 253. doi: 10.1007/0-387-26261-X_25. Google Scholar [29] M. C. Strus, L. Zalamea, A. Raman, R. B. Pipes, C. V. Nguyen and E. A. Stach, Peeling force spectroscopy: Exposing the adhesive nanomechanics of one-dimensional nanostructures,, NANO LETTERS, 8 (2008), 544. doi: 10.1021/nl0728118. Google Scholar [30] M. Shillor, M. Sofonea and J. J. Telega, "Models and Analysis of Quasistatic Contact,", Lecture Notes in Physics, 655 (2004). Google Scholar [31] M. Sofonea, W. Han and M. Shillor, "Analysis and Approximations of Contact Problems with Adhesion or Damage,", Pure and Applied Mathematics (Boca Raton), 276 (2006). Google Scholar [32] Y.-P. Zhao, L. S. Wang and T. X. Yu, Mechanics of adhesion in MEMS-a review,, J. Adhesion Sci.Technol., 17 (2003), 519. doi: 10.1163/15685610360554393. Google Scholar
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