DYNAMIC GENERALIZED THERMO-COUPLE STRESSES IN ELASTIC MEDIA

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1 DYNAMIC GENERALIZED THERMO-COUPLE STRESSES IN ELASTIC MEDIA THESIS Submitted to the Faculty of Science Alexandria University In Partial Fulfillment for the Degree of M.Sc. (Applied Mathematics) By Amany Mohammed EI-Sayed B.Sc. in Mathematics 1992 Alexandria University Supervised by Professor Darwich M. Hamatto Professor of Applied Mathematics Professor Farid A. Hamza Professor of Applied Mathematics Professor Salem M. Salerrl Professor of Applied Mathematics Faculty of Science - Alexandria University 1998

2 Acknowledgement I would like to thank Professor Darwich Hamatto for helping in the last stages of the thesis. I would like also to thank Professor Farid A. Hamza for suggesting the topic of the thesis and for his help throughout all stages of research. My thanks are also offered to the late Professor Salem M. Salem. I would like also to thank Professor Hany H. Sherief for his help.

3 (ii) NOTE This thesis is submitted to the Faculty of Science, University of Alexandria in partial fulfillment for the degree of "Master of Science in Applied Mathematics. The Candidate Amany Mohammed EI-Sayed has also Passed the following post graduate courses: 1 - Fluid Dynamics 2 - Quantum Mechanics 3 - Aerodynamics 4 - Methods of Applied Mathematics 5 - Computer Science 6 - Gennan Language and has satisfied the examiners in these courses. Prof. Dr. Khayreyya EI-Nady Head of Department of Mathematics Faculty of Science University of Alexandria

4 (iii) Table of contents Acknowledgement Note Table of Contents INTRODUCTION CHAPTER I GENERAL REVIEW 1.1 Basic Equations of The linear Theory of Elasticity 1.2 Basics of the Theory of Thermodynamics First Law of thermodynamics Entropy and The second law of thermodynamics 1.3 Classical Theory of Uncoupled Thermoelasticity 1.4 Coupled Theory of Thermoelasticity 15 Generalized Theory of Thermoelasticity CHAPTER II THEORY OF ElASTICITY WITH COUPLE STRESS 2.1 The Theory of Elasticity with Couple Stress 2.2 The Theory of Generalized Micropolar ll1errnoelasticity 1 11 III Uniqueness of Solution 57 CHAPTER III AN AXISYMMETRIC THERMAL SHOCK PROBLEM FOR A HALF SPACE 3.1 Formulation of the Problem 3.2 Solution in the Laplace Transform Domain 3.3 Inversion of the Laplace Transforms 3.4 Numerical Results REFERENCES

5 INTRODUCTION The theory of elasticity is concerned with the study of the response of elastic bodies to the action of forces. A body is called elastic if the deformation of the body disappears with the removal of the forces [1]. The elastic property of material is characterized mathematically by certain functional relationships connecting forces and deformations. Among such relationships, a linear law stemming from a generalization of Hooke's law states in effect, that the extensions of spring like bodies, produced by the tensile forces, are proportional to the forces [1]. During the 150 year period following the discovery of Hooke's law in 1676, the growth of the science of elasticity proceeded from a synthesis of solutions of special problems [1]. The first attempt to deduce general equations of motion of elastic solids was made by Navier in 1821 [2]. Navier's work attracted the attention of A. Cauchy [3], who gave a formulation of the linear theory of elasticity that remains unchanged to the present day [1]. Cauchy showed that the state of stress at an interior point of a deformable body is determined by a set of nine functions that satisfy three partial differential equations and their number reduces to six due to certain symmetry relations. The theory of thermoelasticity deals with the effect of mechanical and thermal disturbances on an elastic body. In the nineteenth century, Duhamel [4] was the first to consider thermoelastic problems. In IH55 Nellmann [51 rcdcrivcd the equations obtained by Duhamel using a different approach. Their theory, the theory

6 (2 ) of uncoupled thermoelasticity consists of the heat equation which is independent of mechanical effects and the equation of motion which contains the temperature as a known function. There are two defects in this theory. First, the fact that the mechanical state of the elastic body has no effect on the temperature. Second, the heat equation being parabolic, implies that the speed of propagation of the temperature is infinite which contradicts physical experiments. The history of the development of the field of thermal stresses forms an interesting study of growth of a scientific discipline [6]. A glance at the number of publications in the field (Table 1 ) [6] shows that a remarkable growth has taken place in the field, but that it has taken place relatively recently. For example, one may mote that only 17 papers were published in the first 65 years of the subject ( i.e., from ), and an equal number in the next 20 years. But an admittedly incomplete listing for the period alone contains more than ten times this number of papers, and there is no doubt that the rate of publications has continued to increase ever since that it has become impossible to count the number of publications accurately. In 1956, M. Biot [7] introduced the coupled theory of thermoelasticity. In this theory the equations of elasticity and of heat conduction are coupled which is in accord with physical experiments since any change of the temperature leads to the presence of strain, in the elastic body and vice versa. In most cases, the solutions which are obtained by the classical theory differ little from that obtained by using the theory of coupled thermoelasticity. This theory is useful in many problems. The equations of this theory consist of the equation of motion which is hyperbolic partial

7 ( 3 ) Period No. of publications Cumulative No. of papers During the period before TABLEt THERMAL STRESS PUBLICATIONS

8 ( 4 ) differential equation and of the equation of energy conservation which is parabolic. There is a defect in this theory. The second equation being parabolic implies that if an elastic medium extending to infinity is subjected to a thermal or a mechanical disturbance, the effect will be felt instantaneously at infinitely distant points which contradicts physical experiments Among the contributions in the subject of coupled thermoelasticity are the works of Nowacki who solved a problem for a half-space with heat sources in [8] and Ignaczak who solved a one dimensional problem for a spherical cavity in [9]. Hetnarski who solved a one-dimensional thermal shock problem in [10] and obtained the fundamental solution of the coupled problem in [11]. Bahar and Hetnarski have obtained the state space approach to the theory in [12] while Takeuti and Tsuji have solved a problem for a plate due to rolling in [13]. Uniqueness of solution was proved by Weiner in [14] and some variational principles were obtained by Nickell and Sackman in [15]. In 1967, Lord and Shulman [16] introduced the theory of generalized thermoelasticity with one relaxation time for the special case of an isotropic body. This theory was extended by Sherief [17] and by Dhaliwal and Sherief [18] in 1980 to include the anisotropic case. In this theory a modified law of heat conduction including both, the heat flux and its time derivative replaces the conventional Fourier's law. The heat equation associated with this theory is hyperbolic and hence eliminates the paradox of infinite speeds of propagation inherent in both uncoupled and the coupled theories of thermoelasticity. Among the contributions to the subject of generalized thermoelasticity are the works of Ignaczak who proved uniqueness for this theory in [19] and [20]. Sherief also proved a uniqueness theorem under less

9 ( 5 ) conditions and studied stability of this theory in [21]. State space fonnulation for one - dimensional problems was done by Anwar and Sherief in [22] and by Sherief in [23] while that for two-dimensional problems was done by Sherief and Anwar in [24]. Sherief in [25] has obtained the fundamental solution of this theory. Sherief and Anwar have solved some one and two dimensional problems in this theory in [26]-[29]. Sherief and Ezzat has obtained the solution for a problem in this theory in the fonn of a series of functions in [30]. Sherief and Hamza has solved some two dimensional problems and studied wave propagation in this theory in [31] and [32]. The classical theory of elasticity does not explain certain discrepancies that occur in the case of problems involving elastic vibrations of high frequencies and short wave lengths, that is vibrations due to the generation of ultra sonic waves. The reason for these discrepancies lies in the microstructure of the material which exerts special influence at high frequencies and short wave lengths [33]. w. Voigt in 1837 [34] attempted to eliminate these discrepancies by suggesting that the transmission of interaction between two particles of a body through an elementary area lying within the material was affected not solely by the action of a force vector but also by a moment (couple) vector [33]. This led to the existence of couple stress in elasticity. Later, brothers E. and F. Cosserat [35] in 1909, gave a unified theory in which every material particle is capable of both a linear displacement and rotation during the deformation of the material. Thus, in this theory, called the theory of Cosserat continuum or the theory of elasticity with couple stress, the defonnation of the body is detennined by a displacement vector

10 (6) and, independently of this, by a rotation vector. The Cosserat continuum went unnoticed for a long time. In the sixties it gained a considerable attention by researchers. This is due to its utility in investigating deformation properties of solids for which the classical theory is inadequate [33]. This elastic model is considered to be more realistic than the classical elastic model in studying earth science problems. Eringen and Suhubi [36], [37] and Eringen [38] gave modem formulation of Cosserat medium equations which became known as the equations of the micropolar theory of elasticity or the theory of asymmetric elasticity. These equations were also developed by Truesdell and Toupin [39]. Micropolar elasticity was further extended to include the thennal effects by Eringen [40], Nowacki [41] and Iesan [42]. Among the contributions to the subject of micropolar thermoelasticity are the works of Shanker and Dhaliwal [33] who have solved several plane strain problems for an infinite body. E. Soos proved a uniqueness theorem for thermoelastic materials having a microstructure in [43]. Shanker and Dhaliwal solved some dynamic thermoelastic problems in micropolar theory in [44]. Chi rita proved the existence and uniqueness for the equations of linear coupled thermoelasticity with microstructure, in [45]. Chandrasekharaiah obtained the equations for a generalization of these equations which he calls the heat flux dependent micropolar thermoelasticity and proved variational and reciprocal principles for his equations in [46] and [47], respectively. This thesis consists of three chapters.

11 (7 ) The first chapter consists of five sections. Section 1 contains a review of the classical theory of elasticity. Section 2 contains a review of the essentials of the theory of thennodynamics. Section 3 contains the derivation of the basic equations of the theory of uncoupled thennoelasticity. Section 4 contains the derivations of the basic equations of the coupled theory of thennoelasticity. Section 5 contains the derivations of the basic equations of the generalized theory of thennoelasticity with one relaxation time The second chapter consists of three sections. Section 1 contains a derivation of the governing equations of the theory of micropolar Elasticity. Section 2 contains a derivation of the governing equations of the theory of generalized micropolar Thennoelasticity. Section 3 contains a statement and proof of a uniqueness theorem for these equations. Chapter 3 contains a sol ution of a problem in the context of the theory of generalized micropolar thennoelasticity. The problem is concerned with the problem of a half space whose boundary is rigidly fixed and subjected to an axisymmetric thennal shock. There are no body forces, body couples or heat sources affecting the medium. The deformation of the medium is due solely to the thermal shock. Laplace and Hankel transfonn techniques are used to solve the problem. The inverse Laplace transfonns are obtained using a numerical technique. The results are represented graphicall y.

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