Bu çalışmada, yerel olmayan elastisite çerçevesinde başlangıç değerleri yöntemi kullanılarak bir çubuğun burkulması araştırılmıştır. Bilindiği gibi nanoteknoloji, moleküler boyutta (1-100 nm) fonksiyonel sistemlerin mühendisliğidir. Atom ve moleküller ölçeğinde özel yöntem ve tekniklerle yapıların, materyallerin ve araçların inşâ edilmesini, bu ölçekte ölçme, tahmin etme, izleme ve yapım faaliyetlerinde bulunmayı, benzeri görülmemiş özelliklerde yeni nanoteknolojik aygıtlar üretmeyi hedefler. Nanoteknolojiyi uygulanabilir kılan şey, atomların yapısı ve aralarındaki olağanüstü organizasyon özelliği olduğundan atomların yapısı ve davranış biçimlerinin çok iyi bilinmesi gerekir. Nanoteknolojide ilk uygulamalar karbon nanotüp yapısı kullanılarak gerçekleştirilmiştir. Karbon nanotüpler hem yapısal, hem de mekanik özellikleri bakımından nano ölçekteki malzemelere en güzel örneklerden biri olup, sahip oldukları olağanüstü özelliklerden dolayı bilinen en sert ve en güçlü liflerdir. Ayrıca karbon nanotüpler, moleküler boyutta grafit karbonların içi boş silindirik çubukları olarak düşünülür ve geniş çapta nanoteknolojik uygulamalarda kullanılırlar. Çalışmada başlangıç değerleri yönteminin uygulanabilirliği için gerekli olan taşıma matrisi verilmiştir. Bir çubuğa ait taşıma matrisinin bilinmesinin en önemli avantajlarından biri, çubukların kuvvetler etkisi altındaki davranışlarını sistematik olarak incelemeyi mümkün kılmasıdır. Taşıma matrisi elde edildikten sonra çeşitli türlerde desteklenmiş çubuklar için kritik yükler hesaplanmıştır. Uygulamalarda elde edilen sonuçlar, yerel olmayan etkilerin nanoyapıların mekanik davranışlarını anlamada klasik elastisiteye göre çok daha güçlü olduğunu ve karbon nanotüplerin burkulmasında da yerel olmayan etkilerin önemli olduğunu göstermektedir.

Nanotechnology is a field of applied science and technology covering a broad range of topics. The main unifying theme is the control of matter on a scale smaller than 1 micrometer, normally between 1-100 nanometers, as well as the fabrication of devices on this same length scale. Nanotechnology cuts across many disciplines, including colloidal science, chemistry, applied physics, materials science, and even mechanical and electrical engineering. Carbon NanoTube (CNT) is a new form of carbon, configurationally equivalent to two dimensional graphene sheet rolled into a tube. It is grown now by several techniques in the laboratory and is just a few nanometers in diameter and several microns long. They are the stiffest and strongest known fibers and have unique electrical properties. An ideal nanotube can be thought of as a hexagonal network of carbon atoms that has been rolled up to make a seamless cylinder. In classical elasticity theory the stress tensor at a given point depends linearly on the strain tensor of the same point. Thus, local elastic theory contains no information about the long range forces between atoms i.e., there is no internal length scale. On the other hand, the theory of nonlocal continuum mechanics assumes that the stress state at a given reference point is considered to be function of the strain states of all points in the body. Properties of materials at the nanoscale differ fundamentally from those of their bulk counterparts. Carbon nanotubes are one of the structural elements that are used in nanotechnological applications widely. In this study, they are envisioned as hollow cylindrical bars of graphitic carbon at the nanoscale. For these reasons nonlocal theory is more capable to display the mechanical behaviour of materials at the nanoscale. Carbon nanotubes hold substantial promise for the development of nanotechnology. However, thorough understanding of the mechanical behavior of carbon nanotubes is essential. Currently, theoretical treatment of buckling of carbon nanotubes has relied on the use of classical continuum mechanics models, such as the elastic shell model or the Bernoulli- Euler beam bending model. Although classical continuum models are relevant to some extent, and efficient in computation for models at large length scales, the applicability of these classical continuum models at small length scales is questionable. Multiple recent experimental results have shown a significant size-effect in mechanical properties when the dimensions of the specimen or the probed material volume become small. The classical continuum theories lack the capability of representing such effects since they do not include any internal length scale. Consequently, these theories are expected to fail when the specimen size become comparable with the internal length scale(s) of the material. Several modifications of the classical elasticity formulation have been proposed to address this deficiency. They are of nonlocal or gradient type and, as a common feature, include one or several intrinsic length scales. Nonlocal elasticity theory was proposed to account for the scale effect in elasticity by assuming the stress at a reference point to be a function of strain field at every point in the body. When the structure considered falls into the nanometer range, nonlacality parameter γ has a significant influence on the outcomes of this study. In some cases, the ratio of local and nonlocal buckling loads can be 1.4. The buckling of carbon nanotubes has been studied by molecular mechanics in literature. In this paper the method of initial values is used in the frame of nonlocal elasticity. This method gives the values of the displacements and stress resultants throughout the rod once the initial displacements and initial stress resultants are known. A priority of this method is that the high degree of statical indeterminancy adds no extra hardship to the solution of the problem. It is interesting to note that the size of the matrix (2x2) from which the buckling determinant obtained in the presented method is the half of its classical counterpart. Local modeling of nanomaterials can be inaccurate, inadequate and misleading. Proposed nanotechnology devices are envisioned to have lengths on the order of nanometers (10-7 cm < L < 10-6 cm). It is clear that for the devices of this size nonlocal effects could be significant. The results are used to display that nonlocal effects could be significant in buckling of carbon nanotubes.