Quasi-static and dynamic analysis of viscoelastic plates

The behavior of real bodies under external loads constitutes a significant part of solid mechanics. Rheology is the science, which studies the behavior of the material under the influence of applied stress. The mathematical form of the rheology is called the constitutive equations or stress-strain relations. This is, of course, an ideal case since all materials change shape, namely to undergo deformation when the self-weight of the materials is considered as a factor that causes stress. The amount of deformation changes depending on the amount of applied stress, environmental conditions (such as ambient temperature), rate and duration of loading and type of material. The behavior of materials can be divided into three main groups under different stress and temperature conditions: (1) Elastic behavior (2) Plastic behavior (3) Viscous behavior. Elastic behavior is time-independent. An elastic body undergoes an instantaneous strain, with stress application and the strain stays constant under the constant stress. The strain in the elastic body only depends upon the final value of stress and not upon its history. Plastic behavior is also time-independent. When a body that shows plastic behavior is stressed beyond the yield point, it experiences instantaneous deformation followed by plastic deformation. Plastic deformation, regardless of time and rate of loading, depends only on the maximum value of the stress in the past. In viscous behavior, stress is proportional to strain rate. When the body that exhibits viscous behavior is loaded, the deformation is not instantaneous but delayed and it increases with time. When the load is removed, there is a permanent deformation. There are also transition regions between the main groups of behavior. Numerous materials exhibit both viscous and elastic behavior under the applied stress. Viscoelastic behavior is commonly composed of combinations of elastic and viscous behavior. There are some phenomena that occur in viscoelastic bodies such as instantaneous elasticity, creep, stress relaxation, instantaneous elastic recovery, delayed recovery and permanent or viscous deformation. The rate of loading and duration of applied load affect deformation of a viscoelastic material. The resulting strain will be the same regardless of the rate of loading for the elastic and plastic behavior; however, in a viscoelastic substance, the slow loading results in higher strain than the faster loading. The strain value at any time depends upon the whole stress history. The values of strain should be limited in addition to the values of stress of structural systems in practice. For example, creep phenomenon occurs in metals at elevated temperatures and in concrete, wood and plastics at normal (room) temperatures. Special precautions that minimize the effects of creep should be taken especially in high-rise buildings. Moreover, machine components that operate at high temperatures, asphalt pavements deform under stress and these deformations are mostly irreversible. Another important phenomenon, which is actually in direct relation to the creep, is relaxation. The relaxation phenomenon should also be considered in some application areas. For instance, it is expected that the time dependent relaxation of stress in steel, which is used in prestressed reinforced concrete elements should be low. This and other similar examples that represent the time dependence of the mechanical behavior of materials necessitate the need for considering time as a variable. A constitutive equation of viscoelastic behavior includes time as a variable in addition to the stress and strain variables. It is very important to model the time-dependent behavior, in other words, viscoelastic behavior properly for the accurate analysis of structures made of viscoelastic materials. The most commonly used method by the science of rheology is to develop some rheological models and to make laws of deformation with the help of these models. With this purpose, starting with standard rheological models in literature (such as Kelvin, Maxwell, etc.) behavior of plates represented by different mechanical models is discussed within the scope of thesis for a more realistic approach. Plates are important structural elements having more than one function. One of the main functions of plates is to transfer the applied loads to the supporting members of load-bearing system properly and the other one is to form a convenient basis for its purpose by providing continuity between members of load-bearing system. The fact that plates are one of the most commonly used structural elements in engineering makes the plate theories and applications as an interesting research topic. In this thesis study, quasi-static and dynamic analysis of plates modeled with viscoelastic constitutive equations is studied under different types of time dependent loads considering the Kirchhoff plate theory that ignores the transverse shear stress. Due to the difficulty of obtaining closed-form solutions for the problems which have complex geometries, loading conditions and constitutive relations, numerical solution techniques are employed. The finite element method is widely used because it is systematic and suitable for programming and it can be easily applied to solve complex problems in engineering disciplines. In this study, which examines linear viscoelastic plate problems, a mixed finite element formulation is developed. By applying the Gâteaux differential approach, a new functional that includes the boundary conditions of the problem is proposed through the potential operator condition is satisfied. Hereditary integral form of the constitutive law is used in order to represent the viscoelastic behavior. The solutions obtained in the Laplace-Carson domain are transformed into the real-time domain using various numerical inverse Laplace transform methods. With this study, the performance of the numerical inverse transformation techniques such as Maximum Degree of Precision (MDOP), Dubner & Abate and Durbin are compared and, the detailed information is given about the content, parameter, precision, advantages/disadvantages and so on of transform methods. A special computer program is developed in Fortran programming language in order to perform analysis. The unique aspects of this thesis study and the possible contributions of the proposed method to the literature can be summarized as follows: by using this new functional, moment and shear force values that are important for engineers can be obtained directly without any mathematical operation. In addition, geometric and dynamic boundary conditions can be obtained easily and a field variable can be included to the functional systematically. Moreover, shear locking problem can be eliminated. The effectiveness of the mixed finite element formulation developed for the analysis of viscoelastic Kirchhoff plates is shown by several numerical examples. In addition, it is aimed to form the base for the analysis of viscoelastic plate problems by considering the original problems that have not been discussed in the literature yet. The studies that use the formulation proposed within the scope of the thesis for the analysis of viscoelastic structural elements with arbitrary geometries still continue. Thus, the gap in the literature related to the analysis of viscoelastic structural elements with complex geometries will be able to fill and viscoelastic analysis, which is a more realistic approach, will be widely used for the investigation of engineering structures in the future.