Klasik ve mikrogermeli ortam teorisiyle modellenen plaklarin caputo kesirli türevi yardimiyla nonlokal titreşim analizi

Abstract

Bu çalışmada dikdörtgen plakların yerel olmayan üç boyutlu titreşim analizi Caputo kesirli türevi yardımıyla incelenmiştir. Kesirli türev son yıllarda mühendislik, fizik, finans, biyoloji gibi birçok alanda yaygın bir şekilde kullanılmaktadır. Kesirli türevin sürekli ortamlar mekaniğine uygulamaları ise, nonlokal problemlerin ve bellekli malzemelerin modellenmesinde literatürde var olan yöntemlere göre yeni bir bakış açısı getirmektedir. Klasik sürekli ortamlar mekaniğinin gelişmesine büyük katkı sağlayan gerilme tansörü kavramı 19. yy başlarında Cauchy tarafından ortaya konulmuş ve böylece lineer elastisite teorisi için hareket denklemi üç boyutlu duruma genelleştirilmiştir. Ancak bu modelde malzemenin iç karakteristik uzunluğu hesaba katılmadığı için 1960'larda yerel olmayan elastisite teorisi geliştirilmiştir. Son yıllarda kesirli analizin yaygınlaşmasıyla birlikte yerel olmayan yeni modeller geliştirilmiştir. Bu yeni modellerin temel avantajı, klasik sürekli ortamlar mekaniğinin genel yerel olmayan yapısına benzer olmasının yanı sıra, kesirli türevin tanımından kaynaklanan bazı eklemeler sayesinde fiziksel gerçeklere daha uygun olmasıdır. Ayrıca, kesirli türev kullanımı nedeniyle fiziksel büyüklüklerde meydana gelen birim uyuşmazlığı da birim uyum katsayısı tanımlanarak ortadan kaldırılabildiği için genelleştirilmiş kesirli yer değiştirme gradyanları ve kesirli gerilme büyüklükleri gibi büyüklükler, klasik olanlarla aynı fiziksel birimlere sahip olurlar. Mekanikte gerek uzaysal değişkenler ve gerekse de zaman değişkeni üzerinde kesirli analiz yapmak çok daha gerçekçi olduğundan, bu alandaki çalışmalar giderek yaygınlaşmaktadır . Bu tez çalışmasında nonlokal etkileri yansıtmak için klasik yer değiştirme gradyanları yerine, Caputo kesirli türevi yardımıyla tanımlanan kesirli yer değiştirme gradyanları kullanılmıştır. Burada kabul edilebilir fonksiyonlar olarak sınır fonksiyonlarıyla çarpılmış Chebyshev polinomları alınmıştır. Sınır fonksiyonları plağın temel geometrik sınır koşullarını sağlayacak şekilde seçilmiş, ancak gerilme sınır koşulları dikkate alınmamıştır. Kesirli türevlerin nonlokalite üzerindeki etkisini göstermek için, farklı sınır koşullarına sahip bazı dikdörtgen plakların titreşim analizi incelenmiştir. Sonuçlar, beklentilere uygun olarak kesirli türevin mertebesi klasik türevin mertebesine yaklaştıkça, nonlokal etkinin azalarak elde edilen frekans değerlerinin klasik durum için elde edilen frekans değerlerine yaklaştığını göstermektedir. Tez çalışmasında klasik titreşim probleminin Caputo kesirli türeviyle incelenmesinin yanı sıra mikrogermeli ortam teorisiyle modellenen dikdörtgen plakların titreşim analizi de Caputo kesirli türevi yardımıyla elde edilmiştir. Mikrogermeli ortam parçacağın klasik şekil değiştirmesinin yanı sıra, bu klasik şekil değiştirmeden bağımsız mikro hacimsel genleşme ve mikro dönme yapabildiği kabulüne dayanmaktadır. Kesirli türev yardımıyla mikrogermeli ortam teorisiyle modellenmiş plakların titreşim problemini nonlokal teoriyle incelemek hesapları basitleştirmesiyle beraber klasik teoriye göre daha iyi sonuçlar vermektedir.

In this study, nonlocal (3-D) vibration analysis of rectangular plates are investigated within the framework of fractional calculus in the sense of Caputo fractional derivative. Fractional derivative has been widely used in many areas such as engineering, physics, finance, biology in recent years. The applications of fractional derivative to the continuum mechanics bring a new perspective in the modeling of nonlocal problems and memory materials according to the existing methods in the literature. The concept of stress tensor, which plays a major role in the development of continuum mechanics, has been demonstrated by Cauchy in the early 19th century and thus many mechanical problems have been solved. However, in the 1950s, for small scale materials in one-dimensional solids, some deficiencies of the theory in nonlinear displacements under surface stresses in acoustic wave propagations were observed. The concept of non-local elasticity was developed in the 1960s to overcome this handicap due to the Cauchy model not taking into account the internal characteristic length of the material. Nonlocality is that the behavior of the material at one point depends not only on the quantities of the specified point but on the properties of all points in the neighborhood of the specified point and is used widely in the solution of many challenging problems in mechanics. With the popularity of fractional analysis in recent years, new non-local model definitions are introduced with the help of differential equations of fractional order Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real or complex number powers of the differentiation operator. The main advantages of these new models are that they resemble the general nonlocal structure of classical continuum mechanics and they are more suitable for physical realities. Besides, unit inconsistency in physical quantities due to using fractional derivative can be overcome by introducing the length scale parameter, thus the generalized fractional displacement gradients or fractional strain quantities have same physical units as the classical ones. Some non-local approaches in the classical theory, such as Eringen's nonlocal approach become equal to take the order of the derivative of the displacement function as fractional order. Fractional analysis can also be used over time variable rather than on spatial variables. This corresponds to the materials with memory in mechanics, indicating that the behavior of the material at a given specified time depends also the mechanical behavior of the material in the past. As it is much more realistic to do fractional analysis on both spatial and time variables, studies in this area are becoming increasingly popular. With the developing technology of today, the need for materials requiring super mechanical, thermal, electrical and magnetic properties increases rapidly compared to the previous half century. Accurate calculation of stress and deformation in composites are major concern in design of such materials using in bio medical, electronic, automotive, civil, marine and aerospace engineering. In contrast to these superior features, it is another challenge to make their weight and dimensions are as small as possible. But experiments show that the size effects become dominant in such small dimensional materials. Although Eringen's micropolar and microstretch theories conclude better results than classical continuum theory, they still reflect only contribution of the specified point unless the nonlocal behavior is also taken into account. Applications of non-local theory help to estimate and clarify the behavior of small-scale materials. Eringen used nonlocal kernels to study screw dislocation and Rayleigh surface waves and the dispersion relations for transverse plane waves in linear isotropic nonlocal micropolar elastic solids. Since both micropolar and microstretch theories contain more unknown material constants and high number of complex equations, it is quite difficult to use nonlocal kernels. Using fractional calculus instead of defining nonlocal kernels may overcome these drawbacks. Studies on the plates with classical theory initiated by Kirchhoff's classical plate theory and based on the primary assumption; the normal of the mid plane remains normal and straight during deformation. Most works about bending, buckling and vibration of plates can be found on the reviews. Reissner introduced the effect of transverse shear deformation for thick plates by using complementary energy principle. Mindlin proposed a first order shear deformation theory with shear correction factors. Liew et al studied the vibration and bending of anisotropic thick plates. Then finite element models are used by many researchers. Comprehensive overviews on buckling and vibration of composites plates are given by Noor and Peters and Leissa. The exact solution for free 3D vibration analysis of thick rectangular plates with simply supported boundary conditions are given by Srinivas et al. It is necessary to understand vibration characteristics in order to ensure reliable design of micro structured materials. To overcome the deficiencies of the classical or first order shear deformation theories, many theories have been developed for correct prediction of response of the materials having inner structures. Taskin et al. used generalized differential quadrature method to obtain the natural frequencies and modal loss factors for the vibration and damping analysis of three-layered sandwich cylindrical shells with stiff composite faces and a viscoelastic core. Challamel et al. generalized Eringen's non-local elastic model with the help of fractional analysis. This fractional model is perfectly compatible with the dispersive wave properties of the Born-Kármán Lattice model. The elastic wave propagation in the non-local continuum medium is studied by Cottone et al. and Sapora et al. for different values of fractional order derivative. In the present study, instead of displacement gradients, Caputo fractional derivative is used to reflect nonlocal effects. Due to the symmetry of Chebyshev polynomial, the vibration modes can be classified as symmetric and antisymmetric modes. In such a case, each class can be examined as a different case. Thus, the problem is simplified because a smaller set of eigenvalue equation is dealt with while accuracy remains same. To demonstrate the efficiency of fractional derivative on nonlocality, we consider the rectangular plates with different boundary conditions; (CCCC), (SSSS), (FFFF), (CFSF), (CFFF), where C, S and F denotes clamped, simply-supported and free boundaries, respectively. The frequency spectrum of rectangular plates for different boundary conditions are given in tables and figures for fractional derivative of different orders ( ) and different values of length scale parameter ( l ). This analysis shows that as the order of the fractional derivative approaches to the classical derivative, the nonlocal effect decreases and the results are become very close to the values of classic ones as expected. Nonlocal vibration analysis of plates modelled by microstretch theory is also examined by using Caputo fractional derivative in this study. The difficulty in modeling of nonlocality in microstretch materials due to the presence of new material constants and high number of complex equations can be overcome by using fractional calculus. The frequency spectrum of plates with different boundary conditions are obtained by Ritz energy method for the different orders of fractional derivative and different values of the length scale parameter. The results are very similar to the values of classic ones as the fractional derivative approaches to the classical derivative. The results indicate also that the non-local impact is getting dominant when the order of the fractional derivatives moves away from the classical derivative order.