Investigation of a thermoelastic beam with method of lines

In this study, we began with the literature search on thermoelastic beam. The details of the outcome of this review is given in the introduction. Based on this review, we have decided to examine the cantilever beams, which is the most basic structure for the resultant spacecraft. The thermoelastic examination of the beam deals with the thermal stress that is generated by the heat effect on the beam and the resulting thermal moment. In our research, we define the thermal stress which only occurs when the material is under an obstacle. There are two types of obstacle that can be applied to the material. These are (i) the internal obstacle and (ii) the external obstacle. In the thesis, we discuss both the inner and the outer obstacles in detail and the conditions under which they can occur. The aim of the thesis is stated at the end of the introduction. We argued that the outcome of this research could be useful for new structural designs where the thermoelastic effects are to be considered. The chapter 2 deals with the Euler-Bernoulli beam theory that we used for modelling beams. We began with an overview of the Euler-Bernoulli beam theory. Later on, we mention the history of the Euler-Bernoulli beam theory and the scientists revealing this theory. We then present following assumptions that are made while obtaining the elasticity in the Euler-Bernoulli beam theory: 1. Beam is considered to be continuous and prismatic cross-section. In addition, it is made of flexible and homogeneous materials. The beam consists of materials with the same modulus of elasticity (E). At this point, the beam extends and contracts equally under the same tension. 2. The beam is considered isotropic and linear elastic. There is a mathematically proportional relationship between the stress and the strain. Hooke's law can be applied (σ = Eε) 3. The intersecting plane sections do not change before and after the bending state. 4. The axial length of the beam does not change due to deformation. In this section, we also describe the general equations of the Euler-Bernoulli beam theory and what the mathematical expressions in these equations are. Therefore, the main mathematical expressions in the Euler-Bernoulli beam theory are as follows: Beam deflection: u The slope of the beam: 𝜕𝑢/𝜕𝑥 Beam bending moment: 𝐸𝐼𝜕2𝑢𝜕𝑥2 Beam shear force: −𝜕𝜕𝑥(𝐸𝐼𝜕2𝑢𝜕𝑥2) Section 3 deals with how the elasticity equations are transformed into the thermoelasticity equations. For this purpose, how the bending that originates from a non-thermal load is explained. First, we give mathematical details of how the heat stress is generated based on the force balance principle. This allows us to define the heat moment. By using these definitions and the elasticity equations, we explain how the connection between the displacement and the thermal loading is established. Accordingly, there is a general relationship between thermal moment and displacement: 𝑑2𝑣𝑑𝑥2=−𝑀+𝑀𝑇𝐸𝐼 We also mention the historical development of the mathematical details of the curvature. The basic equations of thermoelasticity are introduced based on the hypotheses we propose. These equations can be listed as follows: a. Equations expressing strain in terms of displacement. b. Strain compatibility relationship. c. Stress equality relationship d. Stress-strain relations e. Temperature field equations. Our main purpose in this section is to show how the energy equation and tension become "coupled" during the removal of the temperature field equations. In the section 4, the details of how the thermoelasticity equations of the beam we examined are shown in detail. We use an euler-bernoulli beam and its boundary conditions are cantilever. We began with obtaining the equations of temperature of the beam. The temperature field equation took its final form after the possion ratio and the energy production are taken zero. Then we obtain the temperature distribution equation by means of our boundary conditions from this temperature field equation. Temperature distribution is considered to be 2D and time dependent. Since this situation makes it difficult to solve the equations, a transformation for the temperature distribution is made as follows: 𝜃(𝑥,𝑦,𝑡)=𝜃1(𝑥,𝑡)+(𝑦/ℎ)𝜃2(𝑥,𝑡) Based on this transformation, two temperature distribution equations are established. Then, we constructed a differential equation indicating the transverse displacement of the beam with respect to time. Finally, initial and boundary conditions are defined. In the Section 5, we give the solutions for our thermoelasticity equations in the Chapter 4. First, our main method, "method of lines", is described in detail. In addition, we provide information about the "mathematica" program in which the required code is written. In this section, the sequence and the temperature and displacement equations are explained in detail through the generated matrices. Finally, the main parts of the code we wrote at the end of the chapter are added. In the section 6, we present the outputs obtained by solving the thermoelasticity equations. Accordingly, we argue that the provided solution is appropriate with the literature. We then compare the solution that is produced by combining the energy and stress equations with the solution that is produced without combining these equations. We also discuss the reasons of differentiations. We describe how the thermal loading affected the transverse displacement of the beam by drawing graphs and explain how the heat load time has affected the displacement. We show that the progress of heat is much slower than elastic wave propagation. The thermal load changes the frequency of the beam. In this context, the load is harmonically charged at different frequencies and the results are shown in graphical form. Lastly, we deal with the equation that Boley introduced to describe the thermal loading characteristic of the material. Some assumptions are made in accordance with this equation and the proofs for these assumptions are presented by solving equations for different situations. The thesis ends with these results. The resources that are used throughout the research are appended.