Analysis of buckling, vibrations and wave propagation in twisted nanobeam using strain gradient and non-local Eringen theories.

dissertation

to obtain a master's degree

in the field of mechanical engineering (applied design)

abstract

In this research, the analysis of vibrations, buckling and wave propagation of a wrapped nanobeam under axial load on a substrate Pasternak is paid. First, the displacement and displacement field of the wrapped beam is obtained. Then, using the obtained displacement field, strain-displacement relationships are obtained. Eringen's strain gradient and non-local theories are used to apply the size effects caused by the nano scale. Finally, the equations of motion of the wrapped nanobeam are obtained using the energy method and Hamilton's principle. Using the analytical method, the natural frequencies, critical buckling load and wave propagation speed of the wrapped nanobeam are calculated. Finally, the natural frequency, phase speed, cutoff frequency, wave number and critical buckling load of the wrapped beam are obtained under the influence of three small scale parameters: length, non-local Eringen parameter, torsion angle rate, thickness, length of the wrapped nanobeam and elastic bed. The results of this research show that the phase speed in the wrapped nano beam increases with the increase in the twist angle rate in the wrapped nano beam. Also, the wave number has an inverse relationship with the thickness of the nanobeam, while it has a direct relationship with the wave propagation frequency. Increasing the twist angle rate increases the natural frequency of the system, which is more evident in higher thicknesses. The critical buckling load has a direct relationship with the Winkler and Pasternak coefficients, and with the increase in the length of the twisted nano beam, the effect of these coefficients on the critical buckling load increases. The effect of changes in the twist angle rate on the group speed is noticeable at low frequencies, but with the increase of the emission frequency, the graphs converge. Also, with the increase of twist angle rate, the escape frequency increases slightly. The values ??of phase velocity and group velocity using the strain gradient theory are much higher than the modified and classical Couple stress theory.

Key words: buckling analysis, vibrations and wave propagation, wrapped nanobeam, strain gradient theory, Eringen's theory of nonlocal elasticity, twist angle

1st chapter

Theoretical debates

1-1- Introduction to nanotechnology

Nanotechnology is a general term that refers to all advanced technologies in the field of nanoscale work. Nano is a Greek word that means dwarf, which is mathematically equivalent to one billionth, and nanotechnology includes dimensions ranging from 1 to 100 nm. Nano science and technology is the art and ability to take control of matter in nano dimensions and the science of manipulating and rearranging atoms to make materials and tools in nanometer scale.  In this technology, tools and objects are made in atomic sizes and molecule by molecule is done by programmed robots in nanometer scale. In this technology, new properties of materials affected by the dominance of quantum properties over classical properties are used. Nanotechnology is actually a new approach in all fields and it is a transdisciplinary science that includes all sciences and it can be said that it is the connecting point of science in the future. In expressing the importance of this technology, it is said that it is not a part of the future, but the whole future. The Lycurgus Cup, which is kept in the British Museum in London, is an example of the use of this technology in the past, dating back to the 4th century AD. The interesting thing about this cup is that the light shining on the cup from the outside turns it green and it turns red when the light shines from the inside. Microscopic studies have uncovered the secret of this cup and it has been found that inside the glass of this cup, there are nano particles made of gold and silver, and nano particles have different properties than non-nano particles.

The advancement of nano technology entered a new phase with the invention of electron microscopes.In 1931, the German scientist Maxnat and Ernst Roeske invented the first type of these microscopes. Penn State University physics professor Warwin Mueller invented the ion field electron microscope and was the first person in history to observe atoms uniquely and their arrangement on the same surface. Despite the efforts made, Feynman, a physicist and winner of the Nobel Prize in Physics, is known as the founder of nanotechnology. In 1959, he published an article about the capabilities of this technology in the future. In his speech at the banquet after receiving the Nobel Prize, he revealed the idea of ??nanotechnology to the public and believed that there is a very large space in very small sizes. He believed that in the near future, motors as big as the head of a needle will be made.

After this year, the activity in the nano field started to grow significantly. In 1980, at the IBM research center in Switzerland, a technique was invented that magnified the image of the atom. In 1990, for the first time, scientists moved atoms and wrote the first sentence with atoms. With nano technology, man can now make the material world as he wants. It is enough to put the materials of the material world together once again, atom by atom and molecule by molecule. In the words of Hurst Stommer, winner of the Nobel Prize: "The emergence of nanotechnology can give mankind the necessary control over the material world in an unprecedented and unique way." Resources, aerospace, national security, electronic industry, etc., are used.

Nano technology, with its unique features, has created many capabilities in various fields, examples of which are mentioned below.

The combination of three fields of information technology, nano and molecular biology has led to the creation of molecular electronics, in which biological processor wires can be prepared using DNA molecules and DNA-based genes are used to transmit and store information.

In nano structures, such as nanoparticles and nano layers, the ratio of surface area to volume is very high. Therefore, they are ideal components for use in composite materials, chemical reactions, drug transfer and storage. Nano catalysts increase the efficiency of chemical reactions and combustion and on the other hand reduce waste materials and pollution. Nanostructured ceramics are harder and stronger than ceramics made at the micron scale. By using this technology, drugs can be delivered to the same target point in the body, without affecting other parts of the body. Therefore, the effect of the drug is more and its side effects are greatly reduced. Also, more than half of the drugs on the micron scale cannot be dissolved in water, while this is possible on the nano scale and there will be a chance to find new drugs with greater efficiency. Firstly, displacement field and deflection of twisted beam is demonstrated. Then strain-displacement relations are calculated using obtained displacement field. Strain gradient and nonlocal Eringen theories are used to implement the effect of size due to nano scale. Finally, the governing equations of motion for twisted nanobeam are obtained using Energy method and Hamilton's principle. Natural frequencies, critical buckling load, and wave propagation speed of twisted nanobeam are calculated by employing analytical method. Also, natural frequency, phase speed, cut of frequency, wave number and critical buckling load of twisted nanobeam are obtained by considering three length scale parameter, nonlocal Eringen's parameter, rate of twist angle, thickness, the length of twisted nanobeam and elastic medium. The results of this work indicate that phase speed in twisted nanobeam increases with an increase in the rate of twist angle.