Vibration Analysis of Micro-Damaged Plates with Riesz-Caputo Fractional Derivative
Year 2022,
, 989 - 997, 27.10.2022
Soner Aydınlık
,
Ahmet Kırış
Abstract
In this study, with the help of Riesz Caputo fractional derivative definition, non-local analysis of micro-damaged plates are investigated by micro-elongation theory without defining the nonlocal kernels. The frequency spectrum and mode shapes of the microelongated plate with four clamped edges for different values of the fractional continua order and the material length scale parameter are carried out. 3-dimensional vibration analysis are done using the Ritz energy method. The main contribution of the study to the scientific literature is the demonstration that the nonlocal vibration analysis modeled with the concept of fractional derivative is a more suitable model than the classical theory and it fits better with the experimental results.
References
- Aydinlik, S., and Kiris, A., 2020. Fractional Calculus Approach to Nonlocal Three-Dimensional Vibration Analysis of Plates. AIAA Journal, 58, 355–361.
- Aydinlik, S., Kiris, A. and Sumelka, W., 2021a. Nonlocal vibration analysis of microstretch plates in the framework of space-fractional mechanics—theory and validation. The European Physical Journal Plus, 136, 169.
- Aydinlik, S., Kiris, A. and Sumelka, W., 2021b. Three-dimensional analysis of nonlocal plate vibration in the framework of space-fractional mechanics — Theory and validation. Thin-Walled Structures, 163, 107645.
- Cosserat, E. and Cosserat F., 1896. Sur la théorie de l’élasticité, Ann. Toulouse, 10, 1-116.
- Cottone, G., Di Paola, M. and Zingales, M., 2009. Elastic waves propagation in 1D fractional non-local continuum. Physica E: Low-dimensional Systems and Nanostructures, 42(2), 95-103.
- Eringen, A.C., 1968. Theory of micropolar elasticity, New York, Fracture, an advanced treatise (Ed. Leibowitz H.): Academic Press.
- Eringen, A.C., 1983. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54(9), 4703-4710.
- Eringen, A.C., 1984. Plane waves in nonlocal micropolar elasticity. International Journal of Engineering Science, 22, 1113-1121.
- Eringen, A.C., 1990. Theory of thermo-microstretch elastic solids. International Journal of Engineering Science, 28, 1291-1301.
- Eringen, A.C. and Edelen, D.G.B., 1972. Nonlocal elasticity. International Journal of Engineering Science, 10(3), 233–248.
- Khurana, A. and Tomar S.K., 2017. Rayleigh-type waves in nonlocal micropolar solid half-space. Ultrasonics, 73, 162-168.
- Low, K.H., Chai, G.B., Lim, T.M. and Sue, S.C., 1998. Comparisons of experimental and theoretical frequencies for rectangular plates with various boundary conditions and added masses. International Journal of Mechanical Sciences, 40, 1119-1131.
- Mindlin, R.D., 1964. Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis, 16, 51–78.
- Mindlin, R.D., 1965. Second gradient of strain and surface-tension in linear elasticity. International Journal of Solids and Structures, 1(4), 417-438.
- Mindlin, R.D. and Eshel, N.N., 1968. On first strain-gradient theories in linear elasticity. International Journal of Engineering Science, 4(1) , 109–124.
- Odibat, Z., 2006. Approximations of fractional integrals and Caputo fractional derivatives, Applied Mathematics and Computation, 178, 6527–6533.
- Sumelka, W., 2015. Non-local Kirchhoff–Love plates in terms of fractional calculus. Archives of Civil and Mechanical Engineering, 15, 231-242.
- Zhou, D., Cheung, Y.K., Aub, F.T.K. and Lo, S.H. 2002. Three-dimensional vibration analysis of thick rectangular plates using Chebyshev polynomial and Ritz method. International Journal of Solids and Structures, 39, 6339–6353.
Mikro Hasarlı Plakların Riesz-Caputo Kesirli Türevi ile Titreşim Analizi
Year 2022,
, 989 - 997, 27.10.2022
Soner Aydınlık
,
Ahmet Kırış
Abstract
Bu çalışmada Riesz Caputo kesirli türev tanımı yardımıyla, nonlokal çekirdekler tanımlamadan, mikrogenleşme teorisi ile modellenen mikro hasarlı plakların nonlokal titreşim analizi yapılmıştır. Dört ucu ankastre-“clamped” (CCCC) mikro hasarlı plağın frekans spektrumu ve mod şekilleri kesirli türev mertebesinin ve birim uyum katsayısının farklı değerleri için elde edilmiştir. 3-boyutlu titreşim analizi Ritz enerji yöntemi ile gerçekleştirilmiştir. Çalışmanın bilimsel literatüre temel katkısı, kesirli türev kavramıyla modellenen nonlokal titireşim analizinin klasik teoriye göre daha uygun bir model olduğunun ve deneysel sonuçlarla daha iyi örtüştüğünün gösterilmesidir.
References
- Aydinlik, S., and Kiris, A., 2020. Fractional Calculus Approach to Nonlocal Three-Dimensional Vibration Analysis of Plates. AIAA Journal, 58, 355–361.
- Aydinlik, S., Kiris, A. and Sumelka, W., 2021a. Nonlocal vibration analysis of microstretch plates in the framework of space-fractional mechanics—theory and validation. The European Physical Journal Plus, 136, 169.
- Aydinlik, S., Kiris, A. and Sumelka, W., 2021b. Three-dimensional analysis of nonlocal plate vibration in the framework of space-fractional mechanics — Theory and validation. Thin-Walled Structures, 163, 107645.
- Cosserat, E. and Cosserat F., 1896. Sur la théorie de l’élasticité, Ann. Toulouse, 10, 1-116.
- Cottone, G., Di Paola, M. and Zingales, M., 2009. Elastic waves propagation in 1D fractional non-local continuum. Physica E: Low-dimensional Systems and Nanostructures, 42(2), 95-103.
- Eringen, A.C., 1968. Theory of micropolar elasticity, New York, Fracture, an advanced treatise (Ed. Leibowitz H.): Academic Press.
- Eringen, A.C., 1983. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54(9), 4703-4710.
- Eringen, A.C., 1984. Plane waves in nonlocal micropolar elasticity. International Journal of Engineering Science, 22, 1113-1121.
- Eringen, A.C., 1990. Theory of thermo-microstretch elastic solids. International Journal of Engineering Science, 28, 1291-1301.
- Eringen, A.C. and Edelen, D.G.B., 1972. Nonlocal elasticity. International Journal of Engineering Science, 10(3), 233–248.
- Khurana, A. and Tomar S.K., 2017. Rayleigh-type waves in nonlocal micropolar solid half-space. Ultrasonics, 73, 162-168.
- Low, K.H., Chai, G.B., Lim, T.M. and Sue, S.C., 1998. Comparisons of experimental and theoretical frequencies for rectangular plates with various boundary conditions and added masses. International Journal of Mechanical Sciences, 40, 1119-1131.
- Mindlin, R.D., 1964. Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis, 16, 51–78.
- Mindlin, R.D., 1965. Second gradient of strain and surface-tension in linear elasticity. International Journal of Solids and Structures, 1(4), 417-438.
- Mindlin, R.D. and Eshel, N.N., 1968. On first strain-gradient theories in linear elasticity. International Journal of Engineering Science, 4(1) , 109–124.
- Odibat, Z., 2006. Approximations of fractional integrals and Caputo fractional derivatives, Applied Mathematics and Computation, 178, 6527–6533.
- Sumelka, W., 2015. Non-local Kirchhoff–Love plates in terms of fractional calculus. Archives of Civil and Mechanical Engineering, 15, 231-242.
- Zhou, D., Cheung, Y.K., Aub, F.T.K. and Lo, S.H. 2002. Three-dimensional vibration analysis of thick rectangular plates using Chebyshev polynomial and Ritz method. International Journal of Solids and Structures, 39, 6339–6353.