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Universal formulae for the limiting elastic energy of membrane networks. (English) Zbl 1244.74080

Summary: We provide universal formulae for the limiting stretching and bending energies of triangulated membrane networks endowed with nearest neighbor bond potentials and cosine-type dihedral angle potentials. The given formulae account for finite elasticity and solve some deficiencies of earlier results for Helfrich-type bending energies, due to shape-dependence and sensitivity to mesh distortion effects of the limiting elastic coefficients. We also provide the entire set of the elastic coefficients characterizing the limiting response of the examined networks, accounting for full bending-stretching coupling. We illustrate the effectiveness of the proposed formulae by way of example, on examining the special cases of cylindrical and spherical networks covered with equilateral triangles, and discussing possible strategies for the experimental characterization of selected elastic moduli.

MSC:

74K15 Membranes
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[1] Angelillo, M.; Babilio, E.; Fortunato, A., Folding of thin walled tubes as a free gradient discontinuity problem, J. Elasticity, 82, 243-271 (2006) · Zbl 1094.74021
[2] Baillie, C. F.; Johnston, D. A.; Williams, R. D., Nonuniversality in dynamically triangulated random surfaces with extrinsic curvature, Mod. Phys. Lett. A, 5, 1671-1683 (1990)
[3] Dao, M.; Li, J.; Suresh, S., Molecular based analysis of deformation of spectrin network and human erythrocyte, Mat. Sci. Eng., 26, 1232-1244 (2006)
[4] Davini, C.; Pitacco, I., Relaxed notions of curvature and a lumped strain method for elastic plates, SIAM J. Numer. Anal., 35, 677-691 (2000) · Zbl 0928.74099
[5] Discher, D. E.; Boal, D. H.; Boey, S. K., Phase transitions and anisotropic responses of planar triangular nets under large deformation, Phys. Rev. E, 55, 4, 4762-4772 (1997)
[6] Discher, D. E.; Boal, D. H.; Boey, S. K., Simulation of the erythrocyte cytoskeleton at large deformation. II: micropipette aspiration, Biophys. J., 75, 1584-1597 (1998)
[7] Ericksen, J. L., On the Cauchy-Born rule, Math. Mech. Solids, 13, 199-220 (2008) · Zbl 1161.74305
[8] Espriu, D., Triangulated random surfaces, Phys. Lett. B, 194, 271-276 (1987)
[9] Evans, E.; Waugh, R.; Melnik, L., Elastic area compressibility modulus of red cell membrane, Biophys. J., 16, 6, 585-595 (1976)
[10] Evans, E.; Waugh, R., Osmotic correction to elastic area compressibility measurements on red cell membrane, Biophys. J., 20, 3, 307-313 (1977)
[11] Evans, E., Bending elastic modulus of red blood cell membrane derived from buckling instability in micropipet aspiration tests, Biophys. J., 43, 1, 27-30 (1983)
[12] Fedosov, D.A., Caswell, B., Karniadakis, G.E., 2009. General coarse-grained red blood cell models: I. Mechanics. ArXiv:0905.0042; Fedosov, D.A., Caswell, B., Karniadakis, G.E., 2009. General coarse-grained red blood cell models: I. Mechanics. ArXiv:0905.0042
[13] Fraternali, F., Lorenz, C.D., Marcelli, G. Curvature estimation of membrane networks via a local maximum-entropy approach. J. Comput. Phys., doi:10.1016/j.jcp.2011.09.017; Fraternali, F., Lorenz, C.D., Marcelli, G. Curvature estimation of membrane networks via a local maximum-entropy approach. J. Comput. Phys., doi:10.1016/j.jcp.2011.09.017 · Zbl 1238.92008
[14] Gompper, G.; Kroll, D. M., Random surface discretization and the renormalization of the bending rigidity, J. Phys. I France, 6, 1305-1320 (1996)
[15] Hale, J. P.; Marcelli, G.; Parker, K. H.; Winlowe, C. P.; Petrov, G. P., Red blood cell thermal fluctuations: comparison between experiment and molecular dynamics simulations, Soft Matter, 5, 3603-3606 (2009)
[16] Hartmann, D., A multiscale model for red blood cell mechanics, Biomech. Model. Mechanobiol., 9, 1-17 (2010)
[17] Helfrich, W., Elastic properties of lipid bilayers: theory and possible experiments, Z. Naturforschung C, 28, 693-703 (1973)
[18] Helfrich, W.; Kozlov, M. M., Bending tensions and the bending rigidity of fluid membranes, J. Phys. II France, 3, 287-292 (1993)
[19] Hess, S.; Kröger, M.; Hoover, W., Shear modulus of fluids and solids, Physica A, 239 (1997)
[20] Kroll, D. M.; Gompper, G., The conformation of fluid membranes: Monte Carlo simulations, Science, 255, 5047, 968-971 (1992)
[21] Kühnel, W., Differential Geometry. Curves-Surfaces-Manifolds (2002), American Mathematical Society: American Mathematical Society Providence, RI
[22] Lidmar, J.; Mirny, L.; Nelson, D. R., Virus shapes and buckling transitions in spherical shells, Phys. Rev. E, 68, 051910 (2003)
[23] Lipowsky, R.; Girardet, M., Shape fluctuations of polymerized or solidlike membranes, Phys. Rev. Lett., 65, 23, 2893-2896 (1990)
[24] Marcelli, G.; Parker, H. K.; Winlove, P., Thermal fluctuations of red blood cell membrane via a constant-area particle-dynamics model, Biophys. J., 89, 2473-2480 (2005)
[25] Mofrad, M.R.K., Kamm, R.D. (Eds.), Cytoskeletal Mechanics: Models and Measurements. Cambridge University Press, 2006.; Mofrad, M.R.K., Kamm, R.D. (Eds.), Cytoskeletal Mechanics: Models and Measurements. Cambridge University Press, 2006.
[26] Müller, M.; Katsov, K.; Schick, M., Biological and synthetic membranes: What can be learned from a coarse-grained description?, Phys. Rep., 434, 113-176 (2006)
[27] Naghdi, P. M., The theory of shells and plates, (Trusdell, C., S. Flügge’s Handbuch der Physik, vol. VIa/2 (1972), Springer Verlag: Springer Verlag Berlin, Heidelberg, New York), 425-640 · Zbl 0807.73001
[28] Nelson, D., Piran, T., Weinberg, S. (Eds.), Statistical Mechanics of Membranes and Surfaces, second ed. World Scientific, Singapore, 2004.; Nelson, D., Piran, T., Weinberg, S. (Eds.), Statistical Mechanics of Membranes and Surfaces, second ed. World Scientific, Singapore, 2004. · Zbl 1059.82002
[29] Nettles, A., 1994. Basic mechanics of laminated composite plates. Number NASA Reference Publication, 1351. Marshall Space Flight Center, National Aeronautics and Space Administration, MSFC, Aalabama 35812.; Nettles, A., 1994. Basic mechanics of laminated composite plates. Number NASA Reference Publication, 1351. Marshall Space Flight Center, National Aeronautics and Space Administration, MSFC, Aalabama 35812.
[30] Schmidt, B., A derivation of continuum nonlinear plate theory from atomistic models, SIAM Mult. Model. Simul., 5, 664-694 (2006) · Zbl 1117.49018
[31] Schmidt, B., On the passage from atomic to continuum theory for thin films, Arch. Ration. Mech. Anal., 190, 1-55 (2008) · Zbl 1156.74028
[32] Seung, H. S.; Nelson, D. R., Defects in flexible membranes with crystalline order, Phys. Rev. A, 38, 1005-1018 (1988)
[33] Tu, Z. C.; Ou-Yang, Z. C., Elastic theory of low-dimensional continua and its application in bio- and nano-structures, J. Comput. Theor. Nanosci., 5, 422-448 (2008)
[34] Waugh, R.; Evans, E., Thermoelectricity of red blood cell membrane, Biophys. J., 36, 115-132 (1979)
[35] Zhou, Z.; Joós, B., Stability criteria for homogeneously stressed materials and the calculation of elastic constants, Phys. Rev. B, 54, 6, 3841-3850 (1996)
[36] Zhou, Z.; Joós, B., Mechanisms of membrane rupture: from cracks to pores, Phys. Rev. B, 56, 2997-3009 (1997)
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