Çift Duvarlı Tüplerde Burkulmanın Başlangıç Değerleri Yöntemiyle İncelenmesi

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Tarih
2012-05-24
Yazarlar
J. Torkan, Reza
Süreli Yayın başlığı
Süreli Yayın ISSN
Cilt Başlığı
Yayınevi
Fen Bilimleri Enstitüsü
Institute of Science and Technology
Özet
Bu çalışma da tek ve çift duvarlı çubukların burkulma yükleri başlangıç değerleri yöntemiyle hesaplanmıştır. Nanoteknoloji de kullanılan karbon nanotüpler genellikle tek duvarlı veya çift duvarlı çubuklardır. Özellikle çift duvarlı tüplerin mekanik davranışlarının incelenmesi çok önemlidir. Çift duvarlı tüplerde iki tüp arasındaki etkileşim van der waals kuvveti ile göz önüne alınır. Başlangıç değerleri yöntemi, probleme ait bilinmiyenlerin başlangıçtaki değerlerinin bilinmesi halinde, problemin çözümünün sistematik olarak veren bir yöntemdir. Bu yöntem ile tek duvarlı bir çubuğun burkulma yükü 2x2 lik bir determinant yardımıyla olmaktadır. Aynı problemin klasik yöntem ile çözümünde karşımıza çıkan determinant 4x4 mertebesindedir. Benzer şekilde çift duvarlı çubuğun burkulma yükleri başlangıç değerleri yöntemiyle 4x4 mertebesindeki determinantlarla klasik yöntemde ise 8x8 mertebesindeki determinantlarla yapılmaktadır. Tek ve çift duvarlı çubuklara ait burkulma yükleri çizelgeler halinde verilerek, elde edilen sonuçlar karşılaştırmıştır. Çift duvarlı çubukların burkulma yükleri, tek duvarlı çubukların burkulma yüklerinden daha büyüktür.
In this study, the buckling load of single and double walled beams are calculated by the method of initial values.In generally, Single and Double walled nanotubes are widely used in nanotechnology.The study of mechanical behavior of double walled carbon nanotubes has great importance.The interaction between the inner and outer tubes are denoted by van der waals force.The method of initial values gives the values of the displacements and stress resultants throughout the rod once the initial displacments and initial stress resultants are known. The buckling loads of a single-walled tube can be obtained by the determinant of 2x2 matrix in this method. 4x4 determinant appears in the solution of same problem with classical method.Similarly the buckling loads of a double-walled tube are calculated by the determinant of 4x4 matrix. 8x8 determinant appears in the solution of same problem.The buckling loads of single and double walled tubes are presented in tables. The buckling loads of double walled tubes are greater than the buckling loads of single walled tubes. The equations of equilibrium,the constitutive relations and geometrical compatibility. Conditions of a rod in the plane are Where v is deflection, is rotation around the binomial, M is bending moment and T is shear force. Aboue system can be written in matrix form as below Or Where, ; The solution of about system of differential equations is, Where, Example: The buckling load of a simple beam (figure 3.2), Figure 3.2: Single-walled beam In this case, two of initial values are known, v(0)=M(0)=0 The other initial values can be abtained by using the boundary conditions, v(L)=0, M(L)=0 The boundary conditions reach us to the following system, The bucking determinant be comes, det The elements of Carry- Over matrix can be obtained analytically. In this thesis Carry-Over matrix will be calculated opproximately. The series expansion of is (equation 3.10). (3.10) The elements of Carry- Over matrix is calculated by using about relation. The governing equations for a double-walled beam are (figure 3.3), Figure 3.3: Double-walled beam Where v1 is the deflection in outer tube, v2 is the deflecton in inter tube, is the rotation about the binormail in outer tube, is the rotation about the binormail in inner tube, M1 is the bending moment in outer tube, M2 is the bending moment in inner tube,T1 is the shear force in outer tube, T2 is the shear force in inner tube. The tream c(v1-v2) displays the in teraction between the inner and outer tubes and called ad Van der waal force. This system can be written as, Example: The buckling loads of a double-walled simple beam (figure 3.4). Figure 3.4: Two-hinged, double-walled beam The coefficient matrix A for a double walled tube is, The Carry-Over matrix can be found by using ,(equation 3.10). (3.10) In this case four of initial values are known, v1(0)=M1(0)=v2(0)=M2(0)=0 The other initial values can be obtained by the help of end conditions, v1(L)=M1(L)=v2(L)=M2(L)=0 About conditions give the following relation, Then the buckling determinan becomes, The buckling loads of single and double walled simple bars are presented in the following table (table 3.1). Table 3.1: Two-hinged single and double walled tip of the rod bucklingresults Number of Terms Buckling load single-walled beam Buckling load double-walled beam 6 9.0000 9.9804 8 9.4780 9.6669 10 9.9142 10.1029 12 9.8668 10.0553 14 9.8697 10.0582 16 9.8696 10.0580 18 9.8696 10.0580 20 9.8696 10.0580
Açıklama
Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2012
Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 2012
Anahtar kelimeler
Tek duvarlı nanotüpler, Çift duvarlı nanotüpler, SWNTs, MWNTs
Alıntı