On The Theory And Applications Of Distributed Constitutive Models Of Viscoelastie Materials

The study of viscoclastic material is known to have potential benefits nowadays, thus it is of great importance to describe the viscoelastic properties of those materials with accurate mathematical method. Applying the char-acteristics of distributed theory to investigate the viscoelastic materials, we finally address the distributed parameter Maxwell model and distributed or-der standard linear solid model. The model and the constitutive equation are derived and analyzed theoretically, and then validated by numerical methods. In this paper, we investigate the properties of viscoelastic materials in order to propose a new approach for the study of materials and the application of distributed theory.This master thesis consists of five parts. In chapter1, the background, significance and current developments of this thesis are provided, at the same time, our new ideas and research thinking are also described. In chapter2, there arc some preliminary of this paper. In section2.1. the definitions and some important properties of the fractional order calculus are listed. In section2.2, we make an introduction of fractional order viscoelastic material models. In section2.3, we offer some useful definitions and theorems used in the sub-sequent chapters.Both chapter3and4are the main part of this thesis, we mainly study the theory and applications of two kinds of distributed constitutive models of vis-coelastic materials. One is distributed parameter Maxwell model and the other is distributed order standard linear solid model. Our new idea is as follows: First, the two kinds of distributed models and relative constitutive equations are derived; Secondly, we discuss the properties of distributed operators and the distributed viscoelastic characteristics in time domain; Finally, we make a simulation to validate our results.In chapter3, the theory and applications of distributed parameter Maxwell model are discussed. In section3.1, we derive the fractional order weighted distributed parameter Maxwell model, and replace kernel function with Fourier series, then obtain the constitutive equation. In section3.2, we focus on the inverse Laplace transform of distributed parameter operators, including the inverse Laplace transform and the asymptotic properties. The relaxation pro-cess is discussed and simulated in the time domain in section3.3. Lastly, we make a conclusion.In chapter4, we focus on the theory and applications of distributed order standard linear solid model. In section4.1, based on the fractional order damper, we derive the distributed order damper. By applying the distributed order damper to the standard linear solid model, the distributed order standard linear solid model is derived, together with unified constitutive equations. In section4.2, the inverse Laplace transform and asymptotic properties of the distributed order operator are analyzed. In section4.3, the distributed order viscoelastic properties are discussed in time domain and the simulation are offered. Lastly, we make a conclusion.In chapter5, we make a summary of our research results and give future works.