Nonlinear vibrations and post-buckling behavior of microtubes containing current based on the theories of Couple stress and strain changes.

Master's thesis

in the field of mechanical engineering, applied design trend

Abstract

In this research, an analytical solution is presented to show the dependence on size and the effect of temperature difference on nonlinear free vibrations and the behavior after buckling of microtubes containing current. Based on Euler-Bernoulli model, strain gradient theory, Couple's stress theory and von-Karman nonlinear geometry, mathematical equations were expanded in terms of three length parameters. Also, Hamilton's principle has been used to obtain the governing equation and its related boundary conditions. Using the Galerkin method, the governing equation is written in the form of the Dauphin equation. After that, a powerful solution method called homotopy analysis method has been used to obtain analytical relations for nonlinear natural frequency at different flow speeds. In order to investigate the behavior after buckling of microtubes containing current, the Galerkin method has been used to solve the static governing equation. To complete the research, nonlinear vibrations and post-buckling behavior of targeted microtubes containing flow are also discussed. In this type of microtubes, the properties of the material change continuously along the thickness, according to the law of power distribution. For a comprehensive and inclusive comparison between different theories, linear and non-linear analysis has been done using the theories of strain gradient, Couple stress and classical mechanics. The results show that the non-classical length parameters, temperature difference and power law index have a significant effect on the nonlinear vibrations, critical speed and buckling amplitude of microtubes containing current. Nonlinear Vibrations, Postbuckling Behavior, Target Materials, Size Effects

1-1. Introduction

Nanoscience and technology (nanoscience and nanotechnology) is the ability to gain control of matter in nanometer (molecular) dimensions and exploit the properties and phenomena of this dimension in new materials, tools and systems. In fact, nanotechnology is the technology of changing the properties of the molecules that make up the materials, and for this reason, the change in the nano scale is the best definition for this technology. From this definition, it follows that nanotechnology is not a field, but a new approach for all fields. The main goal of most researches in this field is to form new compounds by changing the existing materials and also analyzing their behavior.

1-2. History of nanotechnology

Throughout history and since ancient Greece, people and especially scientists believed that materials can be divided into small parts to reach particles that cannot be crushed and these particles form the basis of materials. Perhaps the Greek philosopher Democritus [1] can be considered the father of nanoscience because around 400 years before Christ, he was the first to use the word atom, which means indivisible in Greek, to describe the particles that make up materials.  The starting point and initial development of nanotechnology is not precisely known. The first spark of nanotechnology (although it was not yet known at that time) was struck in 1959. In this year, Richard Feynman[2] proposed the idea of ??nanotechnology during a speech entitled "There is a lot of space at low levels". He presented the theory that in the near future we will be able to directly manipulate molecules and atoms. The term nanotechnology was first used by Noriyo Tainguchi[3], a professor at Tokyo University of Science, in 1974. He used this term to describe the construction of materials whose dimensional tolerance is in nanometers. Minsky was able to strengthen Feyman's thoughts. Marvin Minsky[4], the father of artificial intelligence, and his student K.Eric Drexler[5], gathered a group of computer students in an association. He occupied the minds of younger people with ideas that he named nanotechnology. In 1986, this term was recreated and redefined by Drexler in a book entitled: "The Engine of Creation: The Beginning of the Nanotechnology Era". Drexler received his doctorate in nanotechnology from MIT in 1991.The prosperity of many important technologies, including information technology and biotechnology, as two huge achievements of the 20th century, will be disrupted without the use of nanotechnology. The importance of nanotechnology

Experience has shown that the properties of a pure substance are reasonably stable and this allows us to identify substances based on their properties. But scientists' findings show that a nanometer-sized material will have different properties than its larger particles. However, shrinking the particles is a physical change and we expect that with this physical change, the main properties of the material will not change.

1-2. Applications of nanotechnology

Nanotechnology is the ability to produce new materials, tools and systems with superior properties on the scale of one to one hundredth of a nanometer (one billionth of a meter), which in the last decade, this multidisciplinary field has been able to occupy a special place in the research and industry sector in various fields of engineering and medical sciences. The basis of nanotechnology is working at these levels to create larger structures and new molecular organization. These nanostructures, which are made from the smallest known building blocks, are the smallest man-made objects and have new physical, chemical and biological properties and behavior. The goal of nanotechnology is to know and take advantage of these properties and use them effectively. Currently, controlling the properties of nanoscale objects has an important role in various fields such as: physics, chemistry, materials science, biology, medicine, nuclear engineering and computer simulation. It has been proven that carbon nanotubes are ten times more resistant and stronger than steel, while its weight is one-sixth of steel, and with nanoparticles, cancer cells can be targeted and destroyed. Nanoscale systems have the potential to make hypersonic travel less expensive and increase the efficiency of computers millions of times. Therefore, researchers are looking for systematic methods to produce products based on the nanometer scale. The basis of all natural materials and systems is based on the nanometer scale. The control and changes of materials at molecular levels means that by determining new properties for materials on this scale, the production of all man-made objects, from cars, tires, computer circuits to medicines and replacement of tissues, can be influenced and cause the invention and creation of new objects. In the 21st century, nanotechnology will be a strategic branch of science and engineering, which will renew the technologies currently used in the manufacture and production of many products in different branches and in all angles and different economic, cultural, social, political, military, etc. It penetrates and will overshadow human life widely.

Abstract

In this investigation, an analytical solution is presented to show the size effect and temperature difference on nonlinear vibration and postbuckling behavior of micro-pipes conveying fluid. On the basis of Euler-Bernoulli beam model, strain gradient theory, and Von K?rm?n geometric nonlinearity, the mathematical formulations are developed in terms of three length scale parameters. The Hamilton's principle is employed to obtain the differential equation of motion and the corresponding boundary conditions. The governing equation is written in the form of Duffing equation by using Galerkin method. Subsequently, a powerful analytical technique called the homotopy analysis method (HAM) is employed to determine the explicit expressions for nonlinear fundamental frequency and critical velocity in different fluid velocities. In order to study postbuckling behavior of micro-pipes conveying fluid, Galerkin method is used to solve static governing equation. To complete this investigation, nonlinear vibration and postbuckling behavior of micro-pipes conveying fluid made of functionally graded materials (FGMs) are also conducted.