NONLOCAL DEFLECTION OF MICROTUBULES UNDER POINT LOAD

Year 2015, , 33 - 39, 01.09.2015

Çiğdem Demir Ömer Civalek

https://doi.org/10.24107/ijeas.251254

Cited By: 2

Abstract

Bending of microtubules due to a point load has been investigated by using Euler Bernoulli beam theory.The governing equations are derived based on Hamilton’s principle. The size effect is taken into consideration using the Eringen’s nonlocal elasticity theory. Some parametric results have been presented for nonlocal beam

Keywords

Nonlocal beam, microtubules, Dirac-Delta function

References

• [1] Eringen, A.C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54, 4703–4710, 1983.
• [2] Peddieson, J., Buchanan, G. R., McNitt, R. P., Application of nonlocal continuum models to nanotechnology. International Journal of Engineering Science, 41, 305-312, 2003.
• [3] Sudak, L.J., Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. Journal of Applied Physics, 94, 7281–7287, 2003.
• [4] Civalek, Ö., Akgöz, B., Free vib ration analysis of microtubules as cytoskeleton components: Nonlocal Euler-Bernoulli beam modeling. Scientia Iranica, Transaction B- Mechanical Engineering, 17(5), 367-375, 2010.
• [5] Civalek, Ö., Demir, Ç., Akgöz, B., Free vibration and bending analyses of cantilever microtubules based on nonlocal continuum model. Mathematical and Computational Applications, 15, 289-298, 2010.
• [6] Civalek, Ö., Demir, Ç., Bending analysis of microtubules using nonlocal Euler-Bernoulli beam theory. Applied Mathematical Modeling, 35, 2053-2067, 2011
• [7] Hawkins, T., Mirigian, M., Yasar, M. S., Ross, J., Mechanics of microtubules. Journal of Biomechanics, 43(1), 23-30, 2009.
• [8] Howard, J., Hyman, A. A., Dynamics and mechanics of the microtubule plus end. Nature, 422, 753-758, 2003.
• [9] Jiang, H., Zhang, J., Mechanics of microtubule buckling supported by cytoplasm. Journal of Applied Mechanics, 75, 061019-1-9, 2008.
• [10] Lu, P., Lee, H.P., Lu, C., Zhang, P.Q., Application of nonlocal beam models for carbon nanotubes, International Journal of Solids and Structures, 44(16), 5289-5300, 2007.
• [11] Reddy, J.N., Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science, 45, 288- 307, 2007.
• [12] Reddy, J. N., Pang, S. D., Nonlocal continuum theories of beams for the analysis of carbon nanotubes. Journal of Applied Physics, 103(2), 023511, 2008.
• [13] Wang, Q., Shindo, Y., 2006. Nonlocal continuum models for carbon nanotubes subjected to static loading. Journal of Mechanics of Materials and Structures, 1(4), 663- 680, 2006.
• [14] Wang, Q., Liew, K.M., Application of nonlocal continuum mechanics to static analysis of micro-and nano structures. Physics Letters A, 363: 236-242, 2007.
• [15] Gao, Y., Lei, F-M., Small scale effects on the mechanical behaviors of protein microtubules based on the nonlocal elasticity theory, Biochemical and Biophysical Research Communications, 387(3), 467-471, 2009.
• [16] Kasas, S., Dietler, G. Techniques for measuring microtubule stiffness. Current Nanoscience, 3(1), 79-96, 2007
• [17] Kis, A., Kasas, S., Babic, B., Kulik, A.J., Benoit, W., Briggs,G.A.D., Schonenberger, C., Catsicas, S., Forro, L., Nanomechanics of microtubules. Physical Review Letters, 89(24), 248101-1-248101-4, 2002.
• [18] Lu, P., Lee, H.P., Lu, C., Zhang, P.Q. , Application of nonlocal beam models for carbon nanotubes. International Journal of Solids and Structures, 44(16), 5289-5300, 2007
• [19] Lu, P., Dynamic analysis of axially prestressed micro/nanobeam structures based on nonlocal beam theory. Journal of Applied Physics, 101: 073504, 2007.
• [20] Pradhan, S. C., Nonlocal finite element analysis and small scale effects of CNTs with Timoshenko beam theory, Finite Elem. Anal. Des., 50 , 8–20, 2012.
• [21] Civalek, Ö., Demir, Ç., Akgöz, B., Static analysis of single walled carbon nanotubes (SWCNT) based on Eringen’s elasticity theory. Int. J. Eng. Appl. Sci., 2, 47–56, 2009.
• [22] Akgöz, B., Civalek, Ö., Modeling and analysis of micro-sized plates resting on elastic medium using the modified couple stress theory. Meccanica, 48, 863–873, 2013.
• [23] Akgöz, B., Civalek, Ö., A new trigonometric beam model for buckling of strain gradient microbeams. International Journal of Mechanical Sciences, 81, 88–94, 2014.
• [24] Civalek Ö., Akgöz B., Vibration analysis of micro - scaled sector shaped graphene surrounded by an elastic matrix. Computational Materials Science, 77:295 -303, 2013.
• [25] Gürses M, Akgöz B, Civalek Ö. Mathematical modeling of vibration problem of nanosized annular sector plates using the nonlocal continuum theory via eight-node discrete singular convolution transformation. Appl Math Comput, 219, 3226–40, 2012.
• [26] Shi, Y.J., Guo, W.L., Ru, C.Q. , Relevance of Timoshenko-beam model to microtubules of low shear modulus. Physica E, 41(2), 213-219, 2008.
• [27] Li, C., Ru, C.Q. , Mioduchowski, A., Length-dependence of flexural rigidity as a result of anisotropic elastic properties of microtubules. Biochem. Biophys. Res. Commun. 349 (3), 1145–1150, 2006.
• [28] Wang, C.Y., Ru, C.Q., Mioduchowski, A., Vibration of microtubules as orthotropic elastic shells. Physica E, 35 (1), 48–56, 2006.
• [29] Gu, B., Mai, Y.-W., Ru, C.Q., Mechanics of microtubules modeled as orthotropic elastic shells with transverse shearing. Acta Mech., 207,195-209, 2009.
• [30] Wang, C.Y., Zhang, L.C., Circumferential vibration of microtubules with long axial wavelength. J. Biomech., 41 (9), 1892-1896, 2008.
• [31] Li, C., Ru, C.Q., Mioduchowski, A., Torsion of the central pair microtubules in eukaryotic flagella due to bending-driven lateral buckling. Biochem. Biophys. Res. Commun., 351 (1), 159-164, 2006.
• [32] Emsen, E., Mercan, K., Akgöz, B., Civalek, Ö., Modal analysis of tapered beam column embedded in winkler elastic foundation. IJEAS, 7(1), 25-35, 2015.
• [33] Huang, G.Y., Mai, Y.-W., Ru, C.Q., Surface deflection of a microtubule loaded by a concentrated radial force. Nanotechnology, 19, 125101-125106, 2008.
• [34] Tuszyński, J. A., Luchko, T., Portet ,S., Dixon, J. M., Anisotropic elastic properties of microtubules. Eur. Phys. J. E., 17, 29-35, 2005.
• [35] Wang, C.Y., Ru, C.Q., Mioduchowski, A., Orthotropic elastic shell model for buckling of microtubules. Phys. Rev. E, 74, 052901-052905, 2006.
• [36] Demir, Ç., Bending and free vibration analysis of nano and micro structures based on nonlocal elasticity theory. MSc thesis. Graduate School of Natural and Applied Sciences, Akdeniz University, 2012.
• [37] Shen, H.-S., Nonlocal shear deformable shell model for bending buckling of microtubules embedded in an elastic medium. Phys. Lett. A, 374 (39), 4030-4039, 2010.
• [38] Demir, Ç., Civalek, Ö., Torsional and longitudinal frequency and wave response of microtubules based on the nonlocal continuum and nonlocal discrete models. Appl. Math. Modell. 37(22), 9355, 2013.