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V "11" (a I” i r WK »¢.{“‘. ’ffiti‘ #2,; yum?" ‘ 51-”- 2:153}- z ,r. 2.01.»: JCW" ma ff wright [glwi gr UNIVERSITY LIBRARIE 111111111111111111111111| 11 1 11111211111111 4'“ 2mg W .1 3 1293 00781 This is to certify that the dissertation entitled LARGE DEFORMATIONS AND SOLID_FLUID INTERACTIONS IN IDEALIZED COMPOSITES MODELED IN THE CONTEXT OF MIXTURE THEORY presented by MOHAMMAD USMAN has been accepted towards fulfillment of the requirements for DOCTORAL degree in MECHANICAL ENGINEERING (\1 - \. / - /C\ 71:1 1 u Major professor Date September 11, 1989 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 LIBRARY Michigan State University PLACE It RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before duo due. DATE DUE DATE DUE DATE DUE MSU is An Atflnndive ActiorVEqual Opportunity institution amna-nt LARGE DEFORMATIONS AND SOLID-FLUID INTERACTIONS IN IDEALIZED COMPOSITES MODELED IN THE CONTEXT OF MIXTURE THEORY by Mohammad Usman A Dissertation Submitted to Michigan State University, East Lansing, MI in partial fulfillment of the requirements of Doctor of PhiIOSOphy in The Department of Mechanical Engineering 1989 ‘2‘) 9043 ABSTRACT LARGE DEFORMATIONS AND SOLID-FLUID INTERACTIONS IN IDEALIZED COMPOSITES MODELED IN THE CONTEXT OF MIXTURE THEORY MOHAMMAD USMAN This thesis is focussed on the phenomenological treatment of mixtures of non-linearly elastic solids and ideal fluids in the context of Mixture Theory (Theory of Interacting Continua), and this work has relevance to several areas of technical interest. Classical theories do not adequately account for the interaction between a highly deformable solid and a fluid in a diffusion process. The limitations of the classical theories point out the need to use an appropriate theory which is capable of realistically taking into account the interaction of solids and fluids undergoing large deformations. The mathematical framework of the Theory of Interacting Continua has been well established, however, some problems concerning the definition of total stresses in the mixture continuum, the constitutive formulation of entropy flux at the boundary, saturation characterization of a solid-fluid mixture and specification of partial stresses at the boundary are still issues which are currently being debated. It is not the intent of this thesis to resolve the first two issues. Instead this thesis addresses the last two issues only, and these issues have a significant bearing on solving boundary value problems within the context of Mixture Theory. In this work the Euler equations for three-dimensional characterization of equilibrium states of non-homogeneously deformed continua are derived for the case when not only the boundary of the mixture but also the entire domain of the mixture is in a saturated state. An interpretation of the saturation boundary condition assumption for equilibrium problems is presented, and this conclusion validates the assumption of a saturation condition only at the mixture boundary for equilibrium boundary value problems, which has been traditionally employed in previous work. Furthermore, a specific form of the Helmholtz free energy function for the mixture is presented which features the interaction of the elastic energy and the energy of mixing. A rigorous treatment of the issues discussed above renders the application of Mixture Theory tractable. Three fundamental boundary value problems are presented which serve as test beds for evaluating the applicability and predictive capability of Mixture Theory. Finally, a linearized version the Mixture Theory is presented to model deformations of composite-fluid mixtures where the fluid content is not necessarily small. It has been demonstrated that the theory could predict results which could not be predicted by single constituent structural theories. Copyright by MOHAMMAD USMAN 1989 To My Wife, Safia Khalid, and Children, Assad and Omar whose patience and encouragement made this work possible ACKNOWLEDGEMENTS I express my sincerest gratitude to Professor Mukesh V. Gandhi; Mechanical Engineering Department; for his continuous inspiration, guidance and support over the past five years at Michigan State University. Professor Mukesh V. Gandhi has not only been a source of intellectual and rewarding experience in research and academics but also he has been a source of education in all walks of life. I am grateful to Professor Roy V. Erickson; Statistics and Probability Department, Professor John J. McGrath; Mechanical Engineering Department and Professor Brian S. Thompson; Mechanical Engineering department, who served as members on my dissertation committee for their time and efforts devoted during my doctoral research work, and for their comments and suggestion during the completion of this document. Special thanks to my friend Mr. Devang J. Desai for his invaluable discussions on research issues, and morale support in the pursuit of my goals. I thank all my professors and friends I have interacted with during my long stay of six and a half years at Michigan State University for all the enjoyment and learning growth I experienced. I would like to thank my parents, brothers and sisters for their concern and encouragement to pursue my graduate studies. Finally, I express my deep gratitude to the members of my family living with me at Michigan State University: my wife; Safia , children; Assad and Omar, brother-in-law; Zubair, sister-in-law; Qaisra, nieces; Amna and Isma, ii and nephew; Rehan for their love and everlasting support, and whose presence during my school years made my graduate studies an enjoyable and rewarding experience. iii TABLE OF CONTENTS LIST OF FIGURES vii NOMENCLATURE x I INTRODUCTION 1 II REVIEW OF THE GENERAL THEORY OF INTERACTING CONTINUA 13 III CONSTITUTIVE EQUATIONS 22 3.1 CONSTITUTIVE ASSUMPTIONS 22 3.2 REDUCED FORM OF CONSTITUTIVE EQUATIONS 27 3.3 SPECIFIC FORM OF THE HELMHOLTZ FREE ENERGY FUNCTION 28 IV EQUILIBRIUM CHARACTERIZATION OF FLUID-SATURATED CONTINUA AND AN INTERPRETATION OF THE SATURATION BOUNDARY CONDITION ASSUMPTION FOR SOLID-FLUID MIXTURES 31 4.1 HISTORICAL BACKGROUND 31 4.2 EQUILIBRIUM CHARACTERIZATION OF NON-HOMOGENEOUSLY DEFORMED FLUID-SATURATED CONTINUA 33 iv 4.3 AN INTERPRETATION OF THE SATURATION BOUNDARY CONDITION ASSUMPTION FOR SOLID-FLUID MIXTURES V LARGE DEFORMATION ANALYSIS OF NON-LINEAR MATERIALS WITHIN THE CONTEXT OF MIXTURE THEORY 5.1 MOTIVATION 5.2 FLEXURE OF A MIXTURE CUBOID 5.3 NON-HOMOGENEOUS FINITE SWELLING OF A NON-LINEARLY ELASTIC CYLINDER WITH A RIGID CORE 5.4 FINITE EXTENSION AND TORSION OF CYLINDRICAL MIXTURE WITH A RIGID CORE VI MODELING OF HYGRO-THERMAL EFFECTS IN POLYMERIC COMPOSITE MATERIALS WITHIN THE CONTEXT OF MIXTURE THEORY 6.1 MOTIVATION 6.2 CONSTITUTIVE MODELING OF AN ANISOTROPIC POLYMERIC COMPOSITE MATERIAL UNDER HYGRO-THERMAL ENVIRONMENTS 6.3 EFFECT OF MOISTURE AND TEMPERATURE ON THE VIBRATIONAL CHARACTERISTICS OF COMPOSITE MATERIALS 38 42 42 45 57 67 77 77 81 85 6.4 CONSTITUTIVE MODELING OF COUPLED HYGRO-THERMO-ELASTIC PHENOMENA IN CONSTRAINED ISOTROPIC POLYMERIC MATERIALS 96 CONCLUDING REMARKS 106 FIGURES 109 BIBLIOGRAPHY l3l vi LIST OF FIGURES Flexure of a cuboid mixture 46 Variation of the radial coordinate with the reference coordinate X1 109 Variation of the volume fraction of the solid with the reference coordinate X1 110 Variation of the non-dimensional radial stress with the reference coordinate X1 111 Variation of the non-dimensional circumferential stress with the reference coordinate X1 112 Variation of the current volume with degree of bending 113 Percentage change in the swollen volume with the degree of bending 114 Swelling of elastic cylinder with a rigid core 115 Variation of the radial and circumferential stretch ratios with the reference radial coordinate 116 vii 10. ll. 12. 13. 14. 15. 16. 17. 18. 19. 20. Variation of the radial stress with the reference radial coordinate 117 Variation of the circumferential stress with the reference radial coordinate 118 Variation of the volume fraction of the solid ratio with the reference radial coordinate 119 Variation of the swelling ratio Q with ratio of rigid core radius to outer radius of cylinder 120 Variation of the radial stress with the ratio of rigid core radius to outer radius of cylinder 121 Variation of the circumferential stress with the ratio of rigid core radius to outer radius of cylinder 122 Variation of the radial stretch ratio with the non-dimensional radial reference coordinate 123 Variation of the circumferential stretch ratio with the non-dimensional radial reference coordinate 124 Variation of the volume fraction of the solid with the non-dimensional radial reference coordinate 125 Variation of the radial stress with the non-dimensional radial reference coordinate 126 Variation of the circumferential stress with the non-dimensional radial reference coordinate 127 viii 21. 22. 23. Variation of the radial stress with the non-dimensional radial reference coordinate 128 Variation of the radial displacement with the radial coordinate 129 Variation of the radial stress with the radial coordinate 130 ix A A1, A2 A31' A32 A , A e m b1 Bij °1, c2 dij' f1: Fij {(2) fi’ 51 h .1. NOMENCLATURE Helmholtz free energy function of the mixture per unit mass of the mixture. Partial derivatives of Helmholtz free energy function with respect to invariants 11 and I2, respectively. Helmholtz free energy function of the solid and fluid per unit mass of the solid and fluid, respectively. Helmholtz free energy function of the elastic deformation and mixing per unit mass of the mixture, respectively. Components of the interaction body force. Components of the Cauchy-Green deformation tensor. Coefficients appearing in the dynamical part of the constitutive equations. Components of the rate of the deformation tensor for the solid and fluid, respectively. Components of the deformation gradient tensor. Stretch ratio as a function of Z coordinate. Components of the acceleration vector for the solid and fluid, respectively. Thickness of the mixture slab. Identity tensor. I1, I2, I3 Invariants of the Cauchy-Green deformation tensor g. LU, M13 l5 xi’ 3'1 x,z, x,z Components of the velocity gradient tensor for the solid and fluid, respectively. Unit outer normal vector. Scalar appearing in the constitutive equations due to the incompressibility constraint. Heat flux vector. Components of the reference and current radial coordinates, respectively. Labels denoting a solid and a fluid particle, respectively. Components of the total stress characterizing the state of the mixture in a saturated state (i,j-1,2,3 or r,0,z ). Time variable. Total surface traction vector. Absolute temperature of the mixture continua. Components of the total stress tensor for the mixture. Components of the velocity vector for the solid and fluid. Components of the mean velocity vector for the mixture. Functions defining the current configuration of the solid and fluid, respectively. Reference position of the solid and fluid particle, respectively. Current position of a solid and fluid particle, respectively. Components of the reference and current coordinates in the cartisian coordinate system, respectively. xi A A(2) Ar, A0 'I ‘’1 p ”10' ”20 P1. P2 Pij' A13 11’ pi e, a ”13' '13 01, x1 x Rotation of the cylindrical mixture per unit current length. Stretch ratio along the length of the cylindrical mixture. Stretch ratio in the thickness direction of the mixture slab. Radial and circumferential stretch ratios. Entropy of the system. Volume fraction of the solid in the mixture. Density of the mixture. Density of the pure solid and fluid, respectively. Mass per unit volume of the mixture for the solid and fluid, respectively. Components of the vorticity tensor for the solid and fluid, respectively. Coefficients appearing in the dynamical part of the constitutive equations (i-l,2,3,4). Components of the reference and current coordinate in the radial coordinate system. Components of the partial stress tensor for the solid and fluid, respectively (i,j-1,2,3 or r,0,z). Components of the partial surface traction for the solid and fluid, respectively. A constant which depends on the particular combination of the solid and the fluid. xii GHAPTERONE INTRODUCTION Several physical problems in engineering involve the interaction of more than one chemically inert constituents. Typically, such interactions take place between two or more solids, solids and fluids, and between a variety of fluids, for example. When large deformations are encountered in problems of this kind involving mixtures, traditional single constituent continuum mechanics is clearly inadequate to model the phenomenological aspects of such interactions. Specifically, problems where significant interaction between the constituents is encountered are amenable to treatment in the context of Mixture Theory (Theory of Interacting Continua) [1,2]. The large deformations of solid- fluid mixtures, where interaction between the solid and fluid is very significant and the swelling of the solid may be several times it original volume, is one such example. This thesis is focussed on the treatment of solid-fluid mixtures in the context of Mixture Theory, and this work has relevance to several areas of technical interest.fmua diffusion of a fluid in a solid [3,6], the swelling and saturation of [elastomers [5,6], flow of fluid through solids [7], mechanical behavior of fiber-matrix mixtures [8], hygro-thermal effects in equivalent composite materials [9], and swelling of constrained materials [10] are but a few examples of these areas of technical interest. The Theory of Interacting Continua models the mixture as a superimposition of individual continua. Each spatial point in the mixture is assumed to be simultaneously occupied by material particles from each constituent. This essentially amounts to taking into account contributions from each constituent in a neighborhood of the point and averaging them. The theory accounts for large deformations, dependence of material properties on both constituents and interactive forces. The application of the Theory of Interacting Continua to model the phenomenological behavior of solid-fluid mixtures has been historically motivated due to limitations in classical single constituent continuum mechanics. Classical approaches which have been used to study the diffusion of fluids through solids such as Fick's Law [3,11-13] and Darcy's Law [14], for example, assume that the solid is rigid. However, this assumption is violated in solid-fluid interactions where the mixture undergoes large deformations [15,16]. Furthermore, the dependence of swelling on strain and kinematical constraints has been demonstrated by Treloar [17] , and Paul and Ebra-Lima [18]. Classical theories do not adequately account for the interaction between a highly deformable solid and a fluid in the diffusion process. The limitations of classical theories point out the need to use an appropriate theory which is capable of realistically taking into account the interaction of solids and fluids. The study of the mechanical response of the mixture of a non- linearly elastic solid and an ideal fluid has been of particular interest to several researchers [19,20]. The application of Mixture Theory to study problems involving large deformations, swelling and diffusion of fluid through non-linearly elastic solid has been very limited due to the lack of physically obvious ways for specifying the partial tractions, which are an integral part of the theory, and the lack of experimentally determined constitutive functions which are - currently unavailable in the literature. The purpose of this work is to investigate the mechanical response of the mixture of a constrained/unconstrained non-linearly elastic solid and an ideal fluid undergoing large deformations and resolve some problems associated with the applicability of Mixture Theory. This investigation, in turn, will serve as a test-bed for evaluating the validity and predictive capability of Mixture Theory in modeling the interaction of elastic solids and ideal fluids. The mathematical framework of the Theory of Interacting Continua has been well established, however, some problems concerning the definition of total stresses in the mixture continuum [21], the constitutive formulation of entropylflux at the boundary [22] , saturation characterization of a solid-fluid mixture and specification of partial stresses at the boundary [23,24] are still issues which are currently being debated. The first two issues concerning the definition of total stress and the constitutive formulation of entropy flux are theoretical in nature, and these issues contribute towards the formulation of different versions of the theory [21,22,251 . It is not the intent of this thesis to resolve the first two issues. Instead Green and Naghdi's approach which has been extensively employed by several investigators [4,6,7,23] to solve boundary-value problems in Mixture Theory has been adopted. This thesis addresses the last two issues, namely, saturation characterization of a solid-fluid mixture and specification of partial stresses at the boundary. The resolution of these issues has a significant bearing on solving boundary-value problems within the context of any of these various approaches [21.22.25] . A critical review of the field makes it clearly evident that the applications of the Theory of Interacting Continua to solve boundary- value problems of physical interest have been very limited. One of the difficulties .in these problems arises due to the lack of physically obvious ways for specifying the partial tractions, which are an integral part of the Theory of Interacting Continua. Rajagopal, et al. [4] and Shi, et a1. [23] were the first to study equilibrium boundary-value problems by employing auxiliary condition at the boundary of solid-fluid mixtures in an effort to bypass the difficulties associated with specifying partial tractions at the boundary. The use of these auxiliary conditions rendered a whole class of boundary-value problems tractable where the boundary of the mixture could be assumed to be saturated+. However, these auxiliary conditions were scalar in nature and derived on an ad hoc basis, which was not necessarily thermodynamically consistent. By employing a rigorous thermodynamic criterion for closed systems, Rajagopal, et al. [24] provided a systematic rationale for characterizing saturated states of homogeneously deformed and swollen cuboid. In particular, they obtained tensorial equations relating the total stresses with the stretch ratios and the volume fraction of the solid in the saturated mixture. These equations could then be used to prescribe additional boundary conditions by assuming material elements at the boundary of the mixture continuum to be in a saturated state, and +. A saturated state represents an equilibrium state in which material elements of a solid-fluid mixture in a deformed and swolen state are in contact with the fluid with no fluid leaving or entering the mixture. thereby bypass the difficulty associated with prescribing the partial traction conditions at the boundary. This approach has been adopted by Gandhi, et a1. [6-7] to study equilibrium boundary-value problems of technical interest, and this work has yielded analytical results which are quantitatively and qualitatively consistent with experimental observations [17,18]. The assumption of a saturated state at the boundary of the mixture continuum proposed by Rajagopal, et al. [24] was a valuable contribution in resolving the difficulties associated with the specification of traction boundary conditions in solving boundary-value problems. However, this assumption yields additional boundary conditions, which could conceivably result in an overdetermined system of equations representing the boundary-value problem. Thus, it is not clear whether the additional constraints imposed due to the assumption of a saturated state at the boundary are compatible with the equations of motion in the mixture domain, and the associated boundary conditions in the context of the Theory of Interacting Continua. This is a major unresolved issue which has not been addressed in previous work. This work addresses this issue in addition to answering several major questions, which have been left unanswered previously: a) How can the equilibrium states of non-homogeneously deformed interacting continua be characterized when not only the boundary of the mixture but also the entire domain of the mixture is in a saturated state? b) Is it possible to have only material elements at the boundary of the mixture in a saturated state and equilibrium in the domain of the mixture, which is not necessarily saturated? c) When the boundary is assumed to be in a saturated state does it necessarily imply that the mixture domain, which is in equilibrium, is also saturated? All these questions pertain to the case of a solid-fluid mixture in an infinite fluid bath where the mixture and the bath are considered to be a thermodynamically closed system at a uniform constant temperature To. In an attempt to answer question (a), Rajagopal, et a1. [24] suggested that the global analysis pertaining the the case of the whole mixture being a saturated closed system may be undertaken by minimizing a functional. However, they concluded that for mixtures in three- dimensions the general problem was too complicated to be amenable to analysis. Furthermore, in their attempt to answer questions (b) and (c), they incorrectly concluded their discussion on this class of equilibrium problems by stating that the use of the saturation boundary condition permits the mixture in the interior to be not necessarily saturated. In this work, the Euler equations for the three-dimensional characterization of equilibrium states of non-homogeneously deformed continua are derived for the case when not only the boundary of the mixture but also the entire domain of the mixture is in a saturated state. The global analysis addresses question (a). Furthermore, questions (b) and (c) are addressed by showing that the use of the saturation boundary conditions does not permit the interior of the mixture to be in any state other than the saturated state for equilibrium boundary-value problems, there the mixture boundary is continuously in contact with an infinite fluid bath. This result is in sharp contrast to arguments presented in previous work, which had allowed the possibility of the interior of the mixture to be in a state, which was not necessarily saturated. Finally, by answering questions (a), (b) and (c), it is shown that the assumption of a saturated state at the boundary represents a natural boundary condition for equilibrium states of non-homogeneously deformed fluid-saturated mixtures, and is, therefore, compatible with the equations of motion and the associated boundary conditions in the context of the Theory of Interacting Continua. This conclusion validates the assumption of a saturation condition only at the mixture boundary for equilibrium problems, which has been employed in previous work [4,6,7,23] . Saturation characterization of fluid-saturated continua and an interpretation of the saturation assumption at the boundary of the mixture discussed above has been treated rigorously in this thesis for the the first time, and these results provide capabilities to address any equilibrium and steady state boundary-value problem within the context of Mixture Theory. Furthermore, it may be pointed out that the Helmholtz free energy function used in these applications [4,7,24] was based on Flory-Huggins equation and Neo-Hookean elastic material characterization [17]. The resultant Helmholtz free energy function employed in previous work [4,7,23] was incorrect due to subtle inaccuracies in the interpretation of several mathematical terms which resulted in a lack of coupling between the elastic energy and the energy of mixing. These inaccuracies in the specific form of Helmholtz free energy function have been addressed in this work. In principle, Mixture Theory can be employed to to solve a wide variety of problems involving solid-fluid mixtures undergoing large deformations with complex geometries and loading characteristics. However, for simplicity the work presented in this thesis will be focussed on equilibrium problems. Furthermore, attention will be focussed on problems with simple geometries and loading characteristics which would serve as test-beds for evaluating the validity and predictive capabilities of Mixture Theory in modeling the interaction of elastic solids and ideal fluids undergoing large deformations. This philosophy is similar to the developments in finite elasticity of single constituent continuum mechanics where fundamental problems with simple geometries and loading characteristics were addressed during the infancy of continuum mechanics. In this thesis, the following three fundamental boundary-value problems in Mixture Theory have been investigated: 1. The problem of flexure of a cuboid mixture presented herein has been motivated by the published work of Rivlin [26] which was focussed on single constituent cuboids. The computational results of finite flexure of a swollen cuboid demonstrate that the fluid gets non- homogeneously redistributed in the domain of the deformed cuboid compression, neutral and tension zones, and the stretch ratios and stresses in the mixture domain are non-linearly distributed in a complex fashion . 2. The second problem of non-homogeneous finite swelling of a non- linearly elastic cylinder with a rigid core is presented within the context of Mixture Theory. The first treatment of the problem of constrained swelling of bonded-rubber cylinders appears to be due to Treloar [10], where the cylinder is assumed to be saturated with the fluid, and the saturated solid-fluid mixture is treated as an equivalent homogenized continuum. In the approach presented herein the interaction of the solid and the fluid is treated by considering the heterogeneous mixture within the context of Mixture Theory. This formulation permits the analysis of the individual motion of the solid constituent and the fluid constituent by incorporating the interaction between the two. It may also be pointed out that Treloar's approach of treating the saturated solid-fluid mixture as an equivalent homogenized continuum is restricted to equilibrium problems only. However, the formulation within the context of Mixture Theory would permit the investigation of problems where the state of materials elements in the domain could range from being completely dry to fully saturated as in the study of time- dependent diffusion problems, for example. 3. The third problem presented herein is an extension of second problem discussed above. In this problem a constrained cylindrical mixture of a non-linearly elastic solid and an ideal fluid undergoing combined extension and twist is considered. The results demonstrate that the presence of the constraint significantly affects the swelling characteristics of the cylinder. Furthermore, the radial stress at the interface of the rigid core and the elastic cylinder changes significantly. This stress reversal, which is a function of the radius of the rigid core and the angle of twist; has never been demonstrated in earlier work. The awareness of this result is very significant since it could have tremendous impact on damage characterization of fiber re-inforced polymeric composite materials In the problems discussed above an attempt has been made to address following questions for answering key issues associated with the large deformation analysis of solid-fluid mixtures: 10 a) How does the presence of the fluid affect the deformations in the three - dimens ional mixture continuum? b) How is the fluid distributed in the mixture continuum? c) How do the partial stresses in the solid and fluid, and the total stress in the mixture vary in the mixture domain? d) How are the volume changes related to the external loading? In particular, does the fluid leave or enter the system under a specific loading condition? The discussion on swelling of a non-linearly elastic cylinder with a rigid core under no loads, and under finite extension and twist presented earlier is very useful in understanding the micromechanical behavior of polymeric composite undergoing finite deformations. However, the fiber-reinforced polymeric composite materials do not, in general, undergo large deformations but contain fluid where volume fraction of fluid may not be necessarily small. The mechanical response of fiber re- inforced polymeric composite materials under hygro-thermal environments is amenable to treatment in the context of Mixture Theory. In this work, a linearized version of Mixture Theory is presented to model the interactions between the fiber-reinforced polymeric composites and fluids where the fluid content may not be necessarily small. This approach not only permits the explicit incorporation of moisture and its effects but also allows the possibility of deriving constitutive equations to model the coupled hygro-thermo-elasto-dynamic response of idealized fiber-reinforced composite plates undergoing small deformations and large rotations. The hygro-thermal dimensional changes 11 along the thickness of the composite mixture are explicitly incorporated in the assumed form of the defamation field. The non-linear vibration of moderately-thick laminated polymeric composite plates is considered by incorporating the effects of transverse shear and rotary inertia. It may be emphasized that the mathematical structure and physical meaning of the quantities involved in constitutive equations and equations of motions presented in the work discussed in the above paragraph are very different from the corresponding quantities in single constituent classical laminated plate theories [27]. This unique structure of the constitutive equations and the explicit form of the equations of motion are precipitated by the current approach whereby moisture is explicitly incorporated as a second constituent. This approach is necessitated whenever the moisture content in the polymeric composite materials is of more than infinitesimal amount. This work is the first attempt at addressing the phenomena of non-linear vibration of laminated composite plates by incorporating hygro-thermal environmental effects in an explicit sense and this work is anticipated to be significanlty relevant to applications in defense, aerospace and manufacturing environments, where significant variations in the moisture and temperature conditions may be encountered. In summary, the principal contributions of this dissertation work may be documented as follow: 1) The Euler equations for the three-dimensional characterization of equilibrium states of non-homogeneously deformed continua are derived for the case when not only the boundary of the mixture but also the entire domain of the mixture is in a saturated state. 12 2) An interpretation of the saturation boundary condition assumption for equilibrium problems is presented, and this result validates the assumption of a saturation condition only at the mixture boundary for the equilibrium boundary-value problems, which has been employed in previous work [4,7,23]. 3) A specific form of the Helmholtz free energy function for mixture is presented which features the interaction of the elastic energy and the energy of mixing. 1 4) Three fundamental boundary-value problems are presented whirflm serve as test beds for evaluating the applicability and predictive capabilities of Mixture Theory. 5) Finally, a linearized version of Mixture Theory is presented on model composite-fluid interactions where the fluid content in the mixture may not be necessarily small. It has been demonstrated that the theory could predict results which could not be predicted by single constituent structural theories. A brief review of the notation and basic equations relevant to a mixture of interacting continua is presented in chapter two. The constitutive equations for the mixture of a nonolinearly elastic solid and an ideal fluid are discussed in chapter three. Chapter four is focussed on the equilibrium characterization of non-homogeneously deformed fluid-saturated continua, and on the interpretation of saturation boundary condition. The application of Mixture Theory in modeling finite deformations of solid-fluid mixture is demonstrated by presenting three problems of interest in chapter five. In chapter six a linearized version of Mixture Theory is presented for the case when anisotropic/isotropic equivalent polymeric composite materials undergo small strains and large rotations, and have vulnerability of absorbing fluids in amounts which are not necessarily small. CHAPTER THO REVIEW OF THE GENERAL THEORY OF INTERACTING CONTINUA 2.1 PRELIMINARIES: NOTATIONS AND BASIC EQUATIONS A brief review of the notations and basic equations of the Theory of Interacting Continua is presented in this section for completeness and continuity. The historical development and a detailed exposition of the theory are succinctly presented in the comprehensive review articles by Atkin and Craine [l] and Bowen [2]. Let 0 and 0t denote the reference configuration and the configuration of the body at time t, respectively. For a function defined on 0 x R.and Otx R, V and grad are used to represent the partial derivative with respect to 0 and 0:, respectively. Also 3; denotes the partial derivative with respect to t. The divergence operator related to grad is denoted by div. The solid-fluid aggregate will be considered a mixture with S1 representing the solid and $2 representing the fluid. At any instant of time t, it is assumed that each place in the space is occupied by particles belonging to both 81 and 82. Let g and X denote the reference positions of typical particles of 51 and $2. The motion of the solid and the fluid is represented by 13 l4 §-§1(§’ t)» andz'§2 (X: t)- (2-1) Where the subscript ~ denotes a quantity in an orthogonal coordinate system. These motions are assumed to be one-to-one, continuous and invertible. The various kinematical quantities associated with the solid 8 and the fluid 82 are 1 (1) (2) D x D x - . _ ____:l _ ____:2 Veloc1ty. u Dt , v Dt , (2.2) Du)L1 Du)! Acceleration: f - Dt , g - Dt , (2.3) 63 a! Velocity gradient: L - 5g , M - 5E , and (2.4) Rate of deformation tensor: Q - % (L + L?), E - %(n + MT), (2.5) where D(1)/Dt denotes differentiation with respect to t, holding x fixed, and D(z)/Dt denotes a similar operation holding 2 fixed and the subscript underscore (_) denotes a tensorial quantity in an orthogonal coordinate system. The deformation gradient E associated with the solid is given by E - —"1. (2.6) The total density of the mixture p and the mean velocity of the mixture 3 are defined by P 'PI+P2: (2-7) 15 and pg - 912 + p22. (2.8) where p1 and p2 are the densities of the solid and the fluid in the mixed state, respectively, defined per unit volume of the mixture at time t. The basic equations of the Theory of Interacting Continua are presented next. 2.1.1 W Assuming no interconversion of mass between the two interacting continua, the appropriate forms for the conservation of mass for the solid and the fluid are P1 ldet El - P109 (2-9) and 69 __2 _ at + div (p2 z) o, (2.10) where p10 is the mass density of the solid in the reference state. Let g and 1 denote the partial stress tensors associated with the solid S1 and the fluid 82, respectively. Then, assuming that there are no external body forces, the balance of linear momentum for the solid and fluid are given by div g - b - plf (2.11) and 16 div 5 + b - ng. (2.12) In equations (2.11) and (2.12), 2 denotes the interaction body force vector, which accounts for the mechanical interaction between the solid and the fluid. By defining the total stress tensor as I ' z + z : (2.13) the equilibrium equations for the mixture may be written as div I - plf + ng - p a. (2.14) where a is the acceleration vector for the mixture. It may pointed out that it is sufficient to satisfy any two of equations (2.11), (2.12) and (2.14) to satisfy the balance of linear momentum. Equation (2.14) may be rewritten in terms of the coordinates in the undeformed configuration denoted by xi to yield .2. 331. T - . 2.15 axj [ jk axk] P a1 ( ) By employing the following definitions, i1 - x1 + "1 , (2.16) an 8U _l_J. __1 813 2 ax I ax ' (2'17) j i 8U 6U w. - l ‘—1 + ——1 , and (2.18) lj 2 8x1 8x1 17 :13 - eij + % [ekJ + wkj] [e1k - wik] , (2.19) where, U1, eij’ wij and £13 are the components of the displacement vector, the linear stretch tensor, the linear rotation tensor and the total strain tensor, respectively. The equations of motion represented by equation (2.15) may be re-written as 53.. 6x1 [Tjk [61k + e1k + wik]] - p a1. (2.20) 2.1.3 W This condition states that g + g - g + g . (2.21) However, the partial stresses g and 5 need not be symmetric. 2.1.4 W Let g and 5 denote the surface traction vectors taken by S and S l 2' respectively, and let 9 denote the unit outer normal vector at a point on the surface of the mixture region. Then the partial surface tractions are related to the partial stress tensors by and (2.22) HI I la 1:: l8 In anticipation of presenting constitutive equations, the balance of energy and entropy production inequality will be stated in this subsection. Let U1 and U2 be the internal energy per unit mass of S1 and $2 , respectively. Let r1 and r2 denote the heat supply per unit mass of (1) K S and 82. The heat flux from $1 and 82 are represented by q and l qéz). The energy balance for the mixture may be written as 6q .—1. £111- - - pr arj p dT ¢ + bi(ui vi) + aiiji + ‘iiji 0, where pr - plr1 + pr2 , (1) (2) qj-qj +qj 9 ”U " P1U1 + ”2U2 ' 5L (1) (2) ¢ - ax (Pluj U]. + quj U2) 9 (1) _ _ uj u.J wj , and (2) _ _ uj Vj wj. The entropy production inequality will be documented next. Let "1 and "2 be the entropy per unit mass of S1 and 82. For a common temperature T of 19 S1 and 82 such that T>0 the entropy production inequality may be written 88 S pT fig + T ¢ - pr + T div [1; ] z o , where F» - p101 + 2202 Aa - Ua - Tna, a - 1,2 pA - plA1 + p2A2 - p(U - Ta), and w - div [9101 (g - g) + p202 (z - 3)] Physically u represents change in entropy of the mixture due to the interaction between the two constituents 81 and 82. Note that n, q and r are defined per unit mass of the mixture. The laws of conservation of energy and the entropy production inequality are explicitly mentioned above. Next, the relevant results for the derivation of constitutive equations are quoted. A complete discussion of these issues is presented in [21]. Let the Helmholtz free energy per unit mass of 81 and 82 be denoted by Asiand A82, respectively. The Helmholtz free energy per unit mass of the mixture is defined by 20 pA - p1 A51 + p2 A82. (2.23) Note that by setting 2 - grad ¢1 + E - grad p2 + E, (2.24) a - ¢1 1 + E. (2.25) 1-%1+2. new where, ¢1+¢2-0, equations (2.11) - (2.13) become div g - E - plg, (2.27) div 5 + b - ng, (2.28) g + i - ET + ET. (2.29) The terms in g, 1 and b which depend on ¢1 and d2 do not contribute to the equations of motion or the total stress. Attention is restricted to a mixture of an incompressible solid and an ideal fluid. It is assumed that the volume of the mixture in any deformation state and at any given time is the sum of the volumes occupied by the solid and fluid constituents at that time [28]. This implies that the motion of the interacting continua is such that it satisfies the following relationship 21 f1_+£2__1 ”10 P20 , (2.30) where p20 is the true mass density of the fluid in the reference state. It may be emphasized that this assumption has a significant bearing on the form of the constitutive equations, and renders the constitutive equations to be more tractable due to the elimination of the density of one of the constituents as an independent variable by virtue of equation (2.30). CHAPTER III CONSTITUTIVE EQUATIONS 3.1 CONSTITUTIVE ASSUMPTIONS A mixture of an elastic solid and a fluid is considered. The solid is assumed to be non-linearly elastic, and the fluid is assumed to be ideal. Thus all the constitutive functions A, n, b, g, «, d1, ¢2 and q are required to depend on the following variables: E, VE, p2, grad p2, T, grad T, u and v, where A is the Helmholtz free energy for the mixture defined per unit mass of the mixture, n denotes the entropy of the system, 3 represents the heat flux vector, T denotes the common absolute temperature of the solid and the fluid and rest of the variables are defined in previous part of the text. Following Crochet and Naghdi [20] and Shi, et al. [4] , The partial stress tensors and diffusive body force vector for the solid and fluid constituents may be written as the sum of static and dynamic part as follows I9] I IQI + IQI (3.1) 22 23 I’ll I I’ll + I)” m :8 O. (3.2) ”Ti -I 10'" + (3.3) where superscript s denote the static part and superscript d denote the s s s dynamical part of the constitutive equations and g g , b depend upon d d d statical variables and E , i , 2 together with A, n, f and the heat flux vector depend on all variables. The energy balance law and the application of the Clausius-Duhem inequality yield following constitutive relations _ _ 26 0 8T . (3-4) -5 P1 ”k1 ' P arij ij ' P p10 5ki’ (3'5) -3 M p2 «k1 - - p p2 3p: ski - p p20 Ski’ and (3.6) OF __S 1.1 15 QA 8P2 P 3p; b (3.7) -- + —_—— k ”2 axk arij ”1 ap2 axk p,, axk ' where the Helmholtz free energy function A is assumed to depend on E , p, and T. In equations (3.5)-(3.7), p is an indeterminate scalar arising from the use of volume additivity assumption/incompressibility constraint equation (2.30). The dynamical part of the partial stress tensors and diffusive body force vector satisfy the reduced entropy inequality 24 _d _d _d _d ”(ki) dik + 1r(k1) fik + "[ki] (rik ' Aik) + bk (“k ' Vk) ' % [qk + T( P1"1( “k ' wk ) + P2"2( vk ' wk ))] a: Z 0' (3'8) where () or [] around the subscripts denote the symmetric and skew symmetric parts of the tensors, respectively. Following the arguments based on the restrictions due to the principle of material objectivity, as presented by Crochet and Naghdi in [20] it may be concluded that the constitutive functions may depend upon the velocities of the constituents only through the relative velocity ui-vi, upon the velocity gradient only through rate of deformation tensors fij and dij and the relative vorticity tensor rij - Aij’ and upon the deformation gradient only through Bij - Fki ij . Furthermore, it is assumed that both the solid and fluid are initially isotropic with a center of symmetry, hence as a consequence of this assumption, the constitutive functions depend on F through C13. ij, where g - ET. E . 13 It is assumed that dynamical parts of the partial stress tensors and diffusive body force vector depend linearly on the dynamical variables given by _d _d -d "d . . a[1j] - -«[ij] - -c1 (rij - Aij)' and (3.11) _d bk - c, (uk- vk). (3.12) 25 The coefficients appearing in the equations (3.9)-(3.12) are function of p1, p2 and T. From equation (3.8) it may be concluded that #1 Z 0. 11 + § #1 2 0. #. z 0. 1. + § #4 Z 0» 2 (#3 + #2) S 4 #1 #.. (3.13) 2 (#2 + #3)] S 4(71 + % #1)(73 + % #3). and who [ (12+73) + O p N O O I) W O The constitutive equations are written in terms of the Helmholtz free energy function A per unit mass of the mixture, and the form of this function, under the assumption of isotropy, is given by A - A (11: 12: 13: 92: T): (3-14) where 11, I2, I3 are the principal invariants of B - E . ET defined through 11 - tr B , (3.15) I2 -12‘ Ht: 3)2 - tr 12]. (3.16) and 13 - det g - (det 32. (3.17) 26 Using (2.9), (2.30) and (3.17), I3 can be expressed in terms of p2 by the relation 1/2 13 - det z - (1 - pz/p20)'1. (3.13) Hence, The Helmholtz free energy function A is assumed to depend on II, 12, p2 and T, so equation (3.14) reduces to A - A (I1, 12, p2,T). (3.19) Substitution of equations (3.4)-(3.7), (3.9)-(3.12) and (3.lS)-(3.18) along with the functional form of the free energy function, given by equation (3.19), into equations (3.l)-(3.3) yields the constitutive equations as follows -s -d ”k1 ' 5k: ¢1 + ”k1 + ”k1 ' - -51. 213211 _aA ”k1 ski f1 9 p10 5k1 + 2” {[31, + 31, I1 Bki ax, Bkm Bmi + 71djjski + 2;.,dk1 + 12131st1 + 2p2£ki-c, (rk1 - Aki), (3.20) -s -d "ki ¢1 + ”k1 + “k1 ' “k1 __ ,flz__2A ”k1 6ki 81 P p20 ski ””2 apzski + 73d335k1 + 2“3dk1 P1 P2 + 7‘fjj8ki + Zfl‘fki‘l’ 5(1‘1‘1 ' Aki), and (3'21) P10P20 27 ad, ___ -s -d bk - axk + bk + bk , if: _P_"" mafia 2A. 2A. bk ' axk ' p10 axk + ”1 ap, axk ' p2 {[31, + a12 111512 -213. £1.52. _ 312 311} 312 k + a p10 p20 (uk vk). (3.22) It is to be noted that c1 and c2 have been redefined, and instead in equations (3.21,3.22) two new constitutive parameters a, 6 aprear which account for a contribution to the interaction body force due to relative motion between the solid and the fluid. The interaction between the solid and the fluid is evident in these equations, where the partial stress of each constituent is affected by the deformed state of both the constituents. 3.2 REDUCED FORM OF CONSTITUTIVE EQUATIONS Steady state and equilibrium formulation of the problems where dynamical parts of the constitutive equation for aij and xij do not contribute to the complete analysis of the solid-fluid mixtures may further simplify the constitutive equations (3.20)-(3.22). Furthermore, for isothermal condition the components of the partial stress tensors for the solid and fluid, and the interaction body force vector may be written as - __flL 2A 3A .23 ”k1 Pp ski + 2”{[31 + 31 11]Bki a1 Bkm Bmi}’ (3'23) 10 1 2 2 - -- 1’2. , 2A xki p p20 ski pp2 6p26ki , and (3.24) 28 a a P P p ' ___]. 3211....2 (EA. 3.43. b" +9 - {[ + I]6 k ”10 axk 1 3P2 axk 2 311 612 1 ii 23.. 51.1 ' 612 B1:} Bil,k + a p10 p20 (“k ° Vk)- (3.25) It is also useful to record the representation for the total stress Tki ”k1 + “k1 p51:1 “’2 a);2 51:1 + 2” {( 011 + 612 I1 )Bki _ .65. 31 BkmBmi}. (3.26) 2 In the remainder of this paper, only 2, and E and E, will be used. Hence, for notational convenience, the superposed bars are dropped. 3.3 SPECIFIC FORM OF THE HELMHOLTZ FREE ENERGY FUNCTION The application of the Theory of Interacting Continua to study diffusion and swelling phenomena of non-linearly elastic solids requires a particular form of the Helmholtz free energy function A for the solid- fluid mixture. Ideally, a broad experimental program should be setup to determine the Helmholtz free energy function for a given solid-fluid mixture. Due to the lack of experimental data for determining the specific form of A, Treloar's work [17] has been modified to suit the constitutive equations defined per unit mass of the mixture in the previous section. 29 The specific form of the Helmholtz free energy function is derived by assuming that the mixture is of "Neo-Hookean type," that is, A is a linear function of 11. The free energy function for a mixture of this type may be written as: A - Ae + Am , (3.27) Where, Ae is free energy of deformation and Am is the free energy of mixing for the solid in the uncross-linked state both defined per unit mass of the mixture. The first term on the left hand side of the equation (3.27) represents the strain energy function for a Neo-Hookean material per unit mass of the mixture and may be given as A :1 RTpJQ Ae - p [ZMC (I1 - 3) , (3.28) The second term Am in equation (3.27) is derived from Flory-Huggins relation [17] and is given by: A 1-u 1121.4. Am — p V1 [ v1 2n(l-v1) + x(l-v1)], (3.29) where, V1 is the molar volume of the fluid, x is a constant which depends on the particular combination of the solid and the fluid, R is the universal gas constant, T is the absolute temperature, 30 MO is the molecular weight of the polymeric solid between the cross-links. The specific form of the free energy function given by equations (3.27)-(3.29) may be used to get an explicit form of the components of the partial stress tensors for solid constituent, for fluid constituent, and interaction body force vector, and are given, respectively, as: - p 011 - p :13 811 + 2p 3%1 B11 , (3.30) «11 p p20 511 pp2 ap2611 , and (3.31) Pan 39 p p b __ +pdA._Z_p§.A_B +a,_1.__2_( _v). k p10 axk lap2 axk 2a11 2£,k p10 p20 “k k (3.32) CHAPTER FOUR. EQUILIBRIUM CHARACTERIZATION OF FLUID-SATURATED CONTINUA AND AN INTERPRETATION OF THE SATURATION BOUNDARI’CONDITION ASSUMPTION FUR SOLID-FEUID MIXTURES 4.1 W The mathematical basis of the general Theory of Interacting Continua has been well established for a long time. A critical review of the field makes it clearly evident that the applications of the Theory of Interacting Continua to solve boundary-value problems of physical interest have been very limited. The main difficulty in these problems arises due to the lack of physically obvious ways for specifying the partial tractions, which are an integral part of the Theory of Interacting Continua. Rajagopal, et a1. [4] and Shi, et a1. [23] were the first to study equilibrium boundary-value problems by employing auxiliary conditions at the boundary of solid-fluid mixtures in an effort to bypass the difficulties associated with specifying partial tractions at the boundary. The use of these auxiliary conditions rendered a whole class of boundary-value problems tractable where the boundary of the mixture could be assumed to be saturated. However, these auxiliary conditions 31 32 were scalar in nature and derived on an ad hoc basis, which was not necessarily thermodynamically consistent. By employing a rigorous thermodynamic criterion for closed systems, Rajagopal, et al. [24] provided a systematic rationale for characterizing saturated states of homogeneously deformed and swollen cuboids. In particular, they obtained tensorial equations relating the total stresses with the stretch ratios and the volume fraction of the solid in the saturated mixture. These equations could then be used to prescribe additional boundary conditions by assuming material elements at the boundary of the mixture continuum to be in a saturated state, and thereby bypass the difficulty associated with prescribing the partial traction conditions at the boundary. This approach has been adopted by Gandhi, et al. [6,7] to study equilibrium boundary-value problems of technical interest, and this work has yielded analytical results which are quantitatively and qualitatively consistent with experimental observations [17,18,29]. In this chapter, the Euler equations for the three-dimensional characterization of equilibrium states of non-homogeneously deformed continua are derived for the case when not only the boundary of the mixture but also the entire domain of the mixture is in a saturated state. This global analysis addresses the question that How can the equilibrium states of non-homogeneously deformed interacting continua be characterized when not only the boundary of the mixture but also the entire domain of the mixture is in a saturated state. Furthermore, it has been shown that the use of the saturation boundary conditions does not permit the interior of the mixture to be in any state other than the saturated state for equilibrium boundary value problems, where the mixture boundary is continuously in contact with an infinite fluid bath. This result is in sharp contrast to arguments presented in previous work, which had allowed the possibility of the interior of the mixture 33 to be in a state, which was not necessarily saturated. Finally, it is shown that the assumption of a saturated state at the boundary represents a natural boundary condition for equilibrium states of non- homogeneously deformed fluid-saturated mixtures, and is, therefore, compatible with the equations of motion and the associated boundary conditions in the context of the Theory of Interacting Continua. This conclusion validates the assumption of a saturation condition only at the mixture boundary for equilibrium problems, which has been employed in previous work [4,7,23]. Next section is focused on the equilibrium characterization of non- homogeneously deformed fluid-saturated continua. An interpretation of the saturation boundary condition assumption for equilibrium problems is presented in section 4.3. 4.2 EQUILIBRIUM.CHARACTERIZATION OF NON-HOMDGENEOUSEY DEFORMED FLUID-SATURATED CONTINUA This section is focused on deriving the equilibrium equations for interacting continua undergoing large non-homogeneous deformations where the entire domain and the boundary of the mixture are in a saturated state. Consider a non-linearly elastic solid which is immersed in an infinite bath of an ideal fluid. The solid-fluid mixture is assumed to be non-homogeneously deformed and swollen in the presence of external forces, and the mixture and the bath are assumed to be at the same constant temperature To. A finite time after the solid is immersed in the fluid bath, the solid-fluid mixture is assumed to attain an 34 equilibrium state which is herein called a saturated state+. The system (i.e., the mixture and the bath) is assumed to be brought from a first state (denoted by superscript l), to a second state (denoted by superscript 2), by means of a reversible process in which W is the work done and Q the heat received by the system. If A(1) and A(2) denote the total Helmholtz free energy of the system at states 1 and 2, then A<2) _ A<1) _ (0(2) , 0(1)) , (T(2)H<2) _ T(1>H<1)), _ (0(2) _ 0(1)) _ T00,(2) , H(1>). Here U and H denote the internal energy and the entropy of the system, respectively. Since U(2) - 0(1) - Q + W, and 10(H(2) - H‘l’) - Q. for a reversible process, it follows that A”) - Am - w. (4.1) Thus, in the case of an infinitesimal change in the state of the system which is caused by a reversible process at constant temperature, the variation in the Helmholtz free energy of the system equals the infinitesimal work done on the system [30,31]. + It may be emphasized that the various states attained between the original pure solid state and the final saturated state of the mixture are non-equilibrium states, and the problems pertaining to these intermediate states may be investigated as time-dependent problems in interacting continua. 35 The thermodynamic arguments presented above can now be generalized [24] to characterize the saturated states of non-homogeneously deformed interacting continua in the absence of external body-forces by obtaining the stationary value of the functional x - I pA dv - I ti xi ds, (4.2) v as subject to the constraint (2.26). In equation (4.2) x1 are the components of the position vector of a typical particle in the deformed configuration, v represents the mixture domain in the deformed configuration and as represents the boundary of this domain. The stationary value of the functional « is given by GK - I 6(JpA) dvo - I t1 6u1 ds - 0, (4.3) v as where vo represents the solid domain in the reference configuration, and J is the Jacobian of the deformation gradient tensor F. It may be emphasized that the same kinematical description which maps the solid constituent from the reference configuration to the deformed configuration also maps the solid constituent from the reference solid to a deformed mixture configuration. The use of equations (2.7), (2.26) and (3.6) in equation (4.3) yields AP pp Am _LL _ZLZ 1/2 I [ [ 1/2 + 3/2] 513 + 13 #(A1 611 + A2 512)] de - I t1 Sui ds - O, (4.4) as _éA. _flA. _flA. where A1 611’ A2 812' and A,p2 apz. 36 The variations in I1, 12 and I3 appearing in equation (4.4) may be written in terms of the variation in the displacement gradients with respect to the reference coordinates as follows: 611 - 2 Fik Sui'k (4.5) ‘12 " 2 (Pin F26. F11: ‘ Fm F21: Fm) Ml1,1: (4'5) 51-21'15 16u (a7) 3 jk ij 3 i,k° ' Substitution of equations (4.5) - (4.7) in equation (4.4) yields -1 I; J ij[{p20A + (p20 - p2) p A,p2} 613 + 2p {(A1 + I1 A2) 0 Bij - A2 Bu Bij}] 6ui’k dvo - {as t1 Sui ds - 0. (4.8) By introducing a tensor whose components TIj are defined by 3 T1] ' {”20A + (”20 ‘ P2)” A'Pz} 511 + 2” {(A1 + I1 A2) Bij - A2 BU Blj}’ (4.9) equation (4.8) may be re-written as -l 8 IV J ij T11 6“i,k dvo - Ias ti 6u1 ds - 0. (4.10) 37 It may be emphasized that the partial derivatives of the displacements appearing in equations (4.5)-(4.10) are taken with respect to the reference coordinates. In equation (4.10) the first term may be expressed as an integral over the mixture domain in the deformed configuration to yield. '13 6(u )dv-I c an ds L 13 1.1 as 1 1 3 3 IV [(Tij Sui)’j - T13:j Sui] dv - I33 t1 6ui ds 0, (4.11) where the partial derivatives are taken with respect to the coordinates in the deformed configuration. Applying the divergence theorem to the first term in equation (4.11) yields s s 0 — Ias (Tij nj - ti)6ui ds - Iv Tij,j 6ui dv. (4.12) For arbitrary variations in the displacement components u1 in the domain, and prescribed tractions and/or displacements on the boundary, equation (4.12) yields the following equations: Equilibrium equations TIj j - 0 in v, (4.13) Natural boundary conditions TIj nj - ti on 651, (4.14) Essential boundary conditions 111 - u1 on 632, (4.15) A where u1 are the prescribed displacements, and as - as1 + 852. Equations (4.13) - (4.15) characterize the equilibrium states of interacting continua undergoing large non-homogeneous deformations where 38 the entire domain and the boundary of the mixture are in a saturated state. In these equations Tij may be interpreted as components of the total stress tensor for interacting continua when the solid-fluid mixture is in a saturated state. Equations (4.14) represent the natural boundary conditions for prescribed tractions t1, and equations (4.15) represent the essential boundary conditions for prescribed displacements 31. Equations (4.13) - (4.15) address question (a) raised in the introduction. The next section is devoted to answering questions (b) and (c). 4.3 AN INTERPRETATION OF THE SATURATION BOUNDARX'CONDITION ASSUMPTION FOR.SOLID-FEUID MIXTURES A broad class of equilibrium boundary-value problems in solid-fluid mixtures [4,6,7,23] have been treated by assuming the equilibrium equations of the Theory of Interacting Continua to hold in the mixture domain, and by assuming the material elements at the boundary of the mixture continuum to be in a saturated state. For this class of boundary-value problems, the equilibrium equations for the solid and the mixture based on the Theory of Interacting Continua may be written as a - bi - 0 in v, and (4.16) 11.1 1' - 0 in v. (4.17) 1.1.3 The assumption pertaining to the material elements at the boundary of the mixture continuum being in a saturated state may be prescribed as Tij nJ - t1 on asl. (4.18) 39 Similarly, the displacements on a part of the boundary 332 may be prescribed as u1 - u1 on 652, (4.19) where as - as1 + 682 represents the total boundary of the mixture domain. In this section, it is demonstrated that the boundary-value problem represented by equations (5.1) - (5.4) is equivalent to assuming both the mixture domain and the mixture boundary to be in a saturated equilibrium state. The equilibrium equations (5.1) and (5.2) may be expressed by substituting the constitutive equations (3.7) - (3.10) to yield a 11.3., p10 3x _L J 11* 2 ax) ““1 + A2 11’ ”11 ' A2 ”1k 31:11} an EA. _2 (4.20) L 1211. L 3X1 ['13 - PP2 3P2]6ij + 2 axj P[(A1 4" A2 11) 81:] - A2 811‘ Bkj]}o (421) The scalar p may be eliminated from equations (5.5) and (5.6) to yield ELML :16. {”20 [(A1 + A2 11) 6k! ‘ A2 Bkz] Bk£,j + p10 p2 axJ [” ”2 ap2] um 33. 321 , p2 apz 6x3 ij +21. 8x 1 p [(A1 + A2 I1) Bij - A2 Bik Bkj]} - 0. (4.22) 40 The functional form of the Helmholtz free energy function of the mixture A defined in equation (3.6) may be used to yield 8A 6A 61 6A 81 6A a ___1 __2 __p __ + + , (4.23) axj 611 6x3 612 axj apz 8xj Equation (5.8) may be substituted into equation (5.7) to give L. EL axj [”20 A + (”20 ‘ ”2) P 662] 51j + 2p[(A1 + 11 A2) Bij - A2 Bik Bkj]} - 0. (4.24) By virtue of the definition of the components of the total stress tensor for a saturated state presented in equation (4.9), equation (5.9) may be rewritten as . - 0. (4.25) Equations (5.10) along with the boundary conditions (5.3) and (5.4) are an alternative representation of the boundary-value problem presented by equations (5.1) - (5.4). Furthermore, equations (5.10), (5.3) and (5.4) are identical to equations (4.13) - (4.15), respectively, which represent a boundary-value problem where both the domain of the solid- fluid mixture and the boundary are in a saturated state. Thus it may be concluded that when the boundary of a mixture domain is continuously in contact with an infinite fluid bath, and assumed to be saturated, for equilibrium, the entire domain of the mixture must be necessarily saturated. This result is physically reasonable, for if the boundary of the mixture domain is saturated, then the material elements adjacent to the boundary ought to be saturated for equilibrium problems. An 41 iterative use of this argument would result in the entire mixture domain being saturated. This conclusion addresses questions (b) and (c) which were raised in Section 1, and thereby validates the use of the saturation condition only at the boundary, which has been employed in previous work [4,6,7,23]. This work addresses several issues related to the equilibrium of solid-fluid mixtures. In particular, the equilibrium states of non- homogeneously deformed interacting continua have been characterized for cases where not only the boundary of the mixture but also the entire domain of the mixture is in a saturated state. This work generalizes previous work which was restricted to homogeneously deformed cuboids and focused on specification of auxiliary saturation boundary conditions. The equilibrium characterization has been employed to demonstrate that the assumption of the saturation boundary condition in the context of the Theory of Interacting Continua necessarily requires the mixture domain to be saturated for equilibrium problems, where the mixture boundary is continuously in contact with an infinite fluid bath. Thus the assumption of the saturation boundary condition for a broad class of problems in solid-fluid mixtures has been shown to be compatible with the equations of motion and the associated boundary conditions in the context of the Theory of Interacting Continua. Furthermore, the use of the saturation boundary condition in previous work has been interpreted, clarified and validated. CHAPTER FIVE LARGE DEPORMATION’ANALXSIS OP NON-LINEAR MATERIALS WITHIN THE CONTEXT OF’MIXTURE THEORY 5.1 MOTIVATION The issues of saturation characterization of fluid-saturated continua and an interpretation of the saturation boundary condition at the mixture boundary addressed in chapter four render the application of Mixture Theory tractable. In particular, these results resolve rigorously the difficulties associated with the specification of traction boundary conditions in solving equilibrium boundary-value problems. In principle, the theory can be employed to solve a wide variety of problems involving solid-fluid mixtures, and undergoing large deformation with arbitrary geometries and external loads. However, for simplicity the boundary-value problems presented herein will be focussed on equilibrium problems with simple geometries and loading characteristics. The primary interest in the development of Mixture Theory is the evaluation of the applicability and predictive capabilities of the theory to address the phenomena which could not be addressed by the single constituent continuum theory. This philosophy is similar to the development in finite elasticity of single constituent continuum mechanics where fundamental problems with simple geometries and loading characteristics were addressed during the infancy of the 42 43 continuum mechanics. Three fundamental problems of interest within the context of Mixture Theory are presented in this chapter. The problem presented herein pertains to the investigation of finite flexure of a cuboid mixture. The first treatment of the this problem in finite elasticity of single constituent appears to be due to Rivlin [26]. The same problem is studied herein for a cuboid which is a mixture of a non-linearly elastic solid and an ideal fluid. The study of this problem would help understanding the physical phenomenon of redistribution of the fluid in the mixture domain when the mixture cuboid is under finite flexure. Furthermore, the neutral plane in the classical problem of flexure of the rubber cuboid is fixed i space whereas the location of the neutral plane in flexure of a cuboid mixture is a function of the degree of flexure and the fluid content of the mixture. The second boundary-value problem investigated herein pertains to the non-homogeneous finite swelling of a non-linearly elastic cylinder featuring a rigid core. The results of this investigation for large swelling demonstrate that the constraint imposed by the rigid core induces non-homogeneous swelling characteristics with significant gradients in the stretch ratios and stress concentration at the bond interface. The effect of finite loading on the behavior of the same non- linearly elastic cylindrical mixture featuring a rigid core is addressed in problem 3. In particular, finite extension and twisting of the cylindrical mixture featuring a rigid core is considered. It has been demonstrated that not only significant gradients in the stretch ratios and the fluid content could be observed but also radial stress at the interface of the rigid core and the elastic cylinder changes significantly. This stress reversal, which is a function of the radius of the rigid core and the angle of twist; has never been demonstrated in 44 earlier work. The awareness of this result is very significant since it could have tremendous impact on damage characterization of fiber re- inforced polymeric composite materials The three boundary-value problems with relatively simple geometries and loading conditions presented in this chapter will be a valuable contribution to the body of literature on Mixture Theory, and the solutions to these boundary value-problems will help evaluate the applicability and predictive capability of the theory. Furthermore, it is anticipated that the results presented herein could be used to guide and motivate experimental work. The results of such experiments will help validate the applicability of the theory to more complicated geometries and loading conditions. 45 5.2 FLEXURE OF A.MIXTURE CUBOID 5.2.1 21.9212! The understanding of the deformation characteristics of solid- fluid mixtures subjected to a variety of external loads is important in several problems of technical interest in engineering. In this work, the pure flexure of a cuboid mixture of an incompressible isotropic elastic solid and an ideal fluid is studied in the context of Mixture Theory. Recently, this approach has been employed [6] to predict experimental results [29] for the finite extension and torsion of a cylindrical mixture, where it has been demonstrated that the volume of swollen cylinder reduces with twisting when the axial stretch ratio is held constant. Boundary-value problems of this kind involving the mixture of an elastic solid and an ideal fluid have been considered previously [6- 7] in an analogous manner. In the present work, the problem of finite flexure of a cuboid mixture of an elastic, incompressible solid and an ideal fluid is considered. The problem is formulated in the context of Theory of Interacting Continua to account for the interaction between the solid and fluid. For simplicity, the Helmholtz free energy function for the solid-fluid mixture is assumed to be of a ”Neo-Hookean" type. The non- uniform distribution of the fluid within the mixture and variation of the radial stresses in the swollen, deformed cylindrical configuration is investigated computationally. It is anticipated that these computational results will guide and motivate future experimental work. The classical results for the flexure of a cuboid of an incompressible 46 non-linearly elastic material due to Rivlin [26] are also derived by considering the special case of a mixture with zero fluid content. 5.2.2 finite flexure of a cuboid mixture Consider a cuboid of a non-linearly elastic material shown in Figure l” X (*2 max X FIGURE 1: FLEXURE OF A CUBOID MIXTURE Let (X1, X2, X3) denote the position of a typical particle in the reference configuration. The cross-section of the cuboid is defined by the planes X1 - a1, X1 - a2, X2 - h and X2 — -h. The cuboid is immersed in an infinite bath of an ideal fluid, and the swollen cuboid is subjected to tractions on the faces corresponding to X2 - i h. The particle denoted by (X1,X2,X3) in the reference configuration may be represented in the deformed configuration by the coordinates (x1, x2, x3). 47 The specific form of the deformation field considered herein is based on the assumption that each plane of the undeformed body which is normal to the Xl-axis becomes, in the deformed state, a portion of the curved surface of a cylinder whose axis is the X3-axis. Furthermore, each plane which is normal to the Xz-axis in the undeformed state becomes, in the deformed state, a plane containing the X3-axis; and material points in the body suffer no displacement parallel to the X - 3 axis. Thus, x1 - f(X1) cos ¢(X2) x2 - f(X1) sin ¢(X2) , (5.2.1) and x3 - X3 . For notational convenience the following abbreviations will be employed in the subsequent part of the paper: C - cos ¢(X2) S = sin ¢(X2) , (5.2.2) and f - f(X1) The deformation gradient E associated with the solid is given by f'C -£s¢- 0 g - f'S fed. 0 , (5.2.3) 0 0 1 48 where the prime denotes differentiation with respect to X1 and the dot denotes differentiation with respect to X2. The Cauchy-Green tensor B - E IT can now be represented as 1'202+£232¢? (f'z-fzd-2)sc 0 g - (f'z-f2¢-2)SC 1'252+£202¢-2 0 . (5.2.4) 0 0 1 The equilibrium equations are expressed in terms of the coordinates in the reference configuration for computational convenience. Assuming no external body forces, the equations of equilibrium for the mixture take the form 13 _1 _ ij - O . (5.2.5) The tensor 3-1 that appears in these equations has the form given by L .5 o f' f' -1 _S_ _C_ E -¢.f ¢.f 0 (5.2.6) _ 0 0 1. The principal invariants of B are then given as 2 2 -2 I1 - f' + f ¢ + 1 , (5.2.7) 2 2 2 2 2 -2 12 - ¢ f + f' + f f ¢ , and (5.2.8) 49 2 13 - f2 f' $2 . (5.2.9) The balance of mass equation for the solid constituent (2.9) may be expressed in term of the stretch ratios as 51 1 (5.2.10) P10 ff'¢ ° The equilibrium equations which are appropriate for the deformation being considered are documented next. Since assumed form of the deformation implies that the stresses depend only on the coordinate X1 and X2, the equations of equilibrium for the solid constituent (2.11), fluid constituent (2.12) and mixture (2.14), respectively, reduce to: 60 2d -1- _ _ 6X7 F15 ba 0 (a, fi, 1 1,2) , (5.2.11) an ad -1 _ 6X F75 + ba 0 (5.2.12) 1 M -l _ 6X1 F73 O , (5.2.13) For the deformation under consideration, it follows from equation (5.2.4) and equations (3.23-3.25) that the non-zero components of the partial stress tensor for the solid and fluid constituents are given by 2 _ - ”1 2 , 2 2 -2 all p p + 2p A1 (C f + S ¢ ) , (5.2.14) 50 p 2 022 - -p —l + 2p A1 (32 f' + c2 £2 $2) , (5.2.15) ”10 _ _ 51 033 p p10 + 2 pA1 , (5.2.16) 2 2 2 a12 - 2p A1 (f - f $ ) sc , (5.2.17) and - - - - pl — «11 «22 «33 p p20 pp2 Ap2 , (5.2.18) the non-zero components of the interaction body force vector are given by .1352 . P3P , b1 ' ‘ , ”1 + p ¢ f + ' ”1 ”2 A”2 ' ”1 ”2 A”2 p f 10 10 u .2 co - 2p2 A1 0 (f + f ¢ ) + 2p2 A1 Sf¢ , and (5.2.19) - c' s c ' b _ -£§ p' + 5.51 + .51 p' AP , .5152 AP 2 p f' 1 - f' 2 2 - 2 10 ploo f I. .2 C. -2p2A1 S (f + f ¢ ) + 2 p2 A1 C F ¢ . (5.2.20) It is sufficient to satisfy any two of the three equilibrium equations (5.2.11-5.2.l3). Equations (5.2.14-5.2.20) are substituted in equilibrium equations for the solid (5.2.11) and mixture (5.2.13), respectively, and after trigonometric manipulations yield the following equations: 51 51.21; .2. .2 21.2-22 p10 6X1 + 1 6X1 (Vlf ) + 1 V1 f (f é f ) _ gi- ' p y ' n g - p1 3P2 p2 + ‘fi—l 1 f (f + f ¢ ) 0 . (5.2.21) ”1 a? + 2 6A. - - 1 ¢'f (2v 4’" + v °¢-) - p P p10 3X2 1 1 1 apz 2 p u 1 72—3: 1 4;. f2 45.. - 0. (5.2.22) and 2.13 .fi_ QA— -3_ 2 .f_'. .2 2 2 - - (pp + 1 (V f' ) + 1 V (f ° ¢° f ) ' 0! 6X1 6X1 2 6p2 6X1 1 f 1 (5.2.23) 2 2pA Q where 1 -—]‘ , and A - A . 1 8I V1 1 The explicit forms of equations (5.2.21-5.2.24) may be obtained for a specific choice of the Helmholtz free energy function A. The specific form of the Helmholtz free energy function per unit mass of the mixture has been discussed in chapter three, and is being repeated here for convenience as A v RTp A l-u A - J —1-Q (I - 3) + E —-l 2n(l-u) + x(1-u) . (5.2.25) p 2M 1 V v 1 c 1 1 Two of the appropriate boundary conditions for solving the set of partial differential equations (5.2.21-4.24) are given by 52 Trr (r1) - O , and (5.2.26) Trr (r0) - O . (5.2.27) The boundary condition on the total traction vector represented by equation (5.2.26) and (5.2.27) are the consequence of the requirement that the outer cylindrical surface of the flexed cuboid by traction- free. Since a boundary condition for the partial traction vector is not physically obvious, following the arguments presented in chapter four, it is assumed that the outer surface of the cylinder is in a saturated state. This assumption results in the boundary condition represented by Srr (r1) - 0 , (5.2.28) Srr (r0) - O , (5.2.29) where Srr represents the radial stress component for a saturated state, and can be explicitly written by using equation (4.9) as - M '2 S r p (p20 - p2) apz f p20A + 2 p A1 f . (5.2.30) r The governing equations (5.2.21-5.2.24) for the combined extension and torsion of a swollen cylinder are highly non-linear and coupled and may be solved numerically for the variables f, ¢, and p. For computational convenience, equations (5.2.21) and (5.2.23) may be combined to eliminate p, and for the Helmholtz free energy function given by (5.2.25) the resulting equation is given by By K1111 (2x l‘ul) 6X1 + 1 3X1 (ylf ) + 1u1 f (f ¢ f ) 0 . (5.2.31) 53 Similarly, the scalar P is eliminated from equations (5.2.22) and (5.2.24) to yield 2 [-x (2x - f1: ) + £3w9] - 0. (5.2.32) ff'¢. . 1 In equations (5.2.31) and (5.2.32) M K - -—-£§— . (5.2.33) ”10 1 Equations (5.2.31) and (5.2.32) govern the equilibrium of the swollen deformed cylindrical configuration attained due to the flexure of the original cuboid mixture. The special case of a mixture with zero fluid content will now be considered to recover the results for the flexure of a cuboid of a non-linearly elastic incompressible material due to Rivlin [26]. For this case p 1_ 1 , (5.2.34) ”10 whereby equations (5.2.31) and (5.2.32) are identically satisfied. In addition, by virtue of equation (2.9) and (5.2.34) p - —l- - f'f¢. - 1 , (5.2.35) V 1 ”10 On differentiation of (5.2.35) with respect to X yields 1 ¢- 3x1 (f'f) - o . 54 which has a solution ¢~ - constant - m , (5.2.36) and f(X1) - (2m x + n)1/2 , (5.2.37) 1 where m and n are constants of integration. The solution represented by equations (5.2.36) and (5.2.37) corresponds to the classical solution given by Rivlin [26]. In the general case of a mixture, equation (5.2.32) admits the solution, ¢~ - constant - C1 , (5.2.38) which reduces (5.2.31) to 2 3 _ _J— _ I v _ o [Ku1(2x l'Vl) f ] fu1 ulf + fC1 . (5.2.39) The ordinary differential equation given by (5.2.39) subjected to boundary conditions given by (5.2.28) and (5.2.29) was solved numerically for a cuboid, whose dimensions in the reference configuration were assumed to be 1 unit in the Xl-direction and 2 units in the Xz-direction. The following material properties [29] were used for the numerical calculations: Density of rubber in the reference state p10 - .9016 gm/cc Density of solvent in the reference state p20 - .862 gm/cc Molar volume of the solvent V - 106.0 cc/mole l 55 The molecular weight of rubber between MC - 8891.0 gm/mole cross links Rubber-solvent interaction constant x - 0.400 The computational results for two prescribed value of C1 corresponding to ¢max - 30° and 60° (see Figure l) are presented in Figures 2-5. The computational results for change in volume with curvature are presented in Figures 6 and 7. Figure 2 shows the variation of the radial coordinate f(X1) with reference coordinate X1 for two different values of ¢max’ For the case of no flexure (¢max- 0) the cuboid is homogeneously swollen whereby the stretch ratio f(X1) is equal and constant throughout the domain. However, when the swollen cuboid undergoes finite flexure (¢max - 30° or 60°) gradients in the stretch ratio are evident. These gradients become significant as the degree of flexure increases from ¢max- 30° to ¢max- 60°, for example. Furthermore, even in the case of finite flexure, the deformation near the inner and outer radial surface is relatively homogeneous, and the gradients in the stretch ratio increase from inner radial surface towards the neutral axis and then become almost constant near the outer radial surface. Figure 3 shows the variation of the radial stresses for two different values of curvatures (¢max- 30, 60). It can be seen from Figure 3 that the non-dimensional radial stress is compressive and approaches zero at the inner and outer radial surface due to the boundary conditions given by equations (5.2.28) and (5.2.29) which require the inner and outer radial surface to be traction free. It is also observed that the neutral plane shifts with the degree of flexure, and the maximum radial stress always occur at the neutral plane. 56 The corresponding variation of the non-dimensional circumferential stress is shown in Figure 4. The radial and circumferential stresses plotted in Figures 3 and 4 denoted by Trr and A RIP T00 have been non-dimensionalized with respect t0'-ji-'lg. c The variation of the volume fraction of the cuboid along the reference coordinate X1 is shown in Figure 5 for two different values of ¢max' It is evident from Figure 5 that fluid enters the swollen deformed cuboid as the curvature ¢max is increased. Furthermore, the inner portion of the flexed cuboid retains less fluid than the outer portion. It is clear from Figures 2,3,4 and 5 that the variation of stretch ratio, stresses and volume fractions in the neighborhood of the neutral axis are maximum non-homogeneous, and the non-homogeneity increases with the increase of flexure. The non-dimensional ratio of the current volume V (in the swollen flexed state) to the volume of the original unswollen rubber cuboid Vu is presented in Figure 6. It is clear from these results that as the curvature of the cuboid is increased the fluid leaves the cuboid resulting in the reduction of the current volume of the swollen cuboid. Finally, the ratio of the change in the volume AV (V - V0) to the saturated volume swollen unflexed volume V0 is plotted in Figure 7. It is anticipated that the computational results presented herein will be useful for motivating and guiding future experimental work in this area. 57 5.3 NON-BOHOGENEOUS FINITE SWEL'LING OF A NON-LINEARLY ELASTIC CYLINDER WITH A.RIGID CORE 53-1 2:22:12! In this section, the constrained equilibrium swelling of a non- linearly elastic cylinder bonded to a rigid core is investigated in the context of Theory of Interacting Continua. In particular, the effect of the constraint due to the presence of the rigid core on the swelling characteristics of the non-linearly elastic cylinder is studied. Whereas a solid cylinder of a non-linearly elastic material swells homogeneously, the presence of a rigid core constrains the ability of the surrounding elastic material to swell, and thereby induces non- homogeneous swelling characteristics. The first treatment of the problem of constrained swelling of bonded-rubber cylinders appears to be due to Treloar [10], where the cylinder is assumed to be saturated with the fluid, and the saturated solid-fluid mixture is treated as an equivalent homogenized continuum. In the approach presented herein the interaction of the solid and the fluid is treated by considering the heterogeneous mixture within the context of Theory of Interacting Continua. This formulation permits the analysis of the individual motion of the solid constituent and the fluid constituent by incorporating the interaction between the two. It may also be pointed out that Treloar's approach of treating the saturated solid-fluid mixture as an equivalent homogenized continuum is restricted to equilibrium problems only. However, the formulation within the context of Theory of Interacting Continua would permit the investigation of problems where the state of material elements in the domain could 58 range from being completely dry to fully saturated as in the study of time-dependent diffusion problems, for example. In the problem considered herein, both the solid and fluid constituents are at rest. However, the fluid can be non-homogeneously dispersed throughout the mixture domain, which gives rise to gradients in the fluid density. The physical mechanism for the existence of such gradients is provided by the presence of an interaction body force which each constituent exerts on the other. The results of this investigation for large deformations demonstrate that the constraint imposed by the rigid core induces non-homogeneous swelling characteristics with significant gradients in the stretch ratios and severe stress concentrations at the bond interface. These results could have important implications for a variety of fiber-reinforced composites featuring hygroscopic polymeric matrix materials, for example. Consider a hollow cylinder of a non-linearly elastic material described by internal and external radii R1 and R0, respectively, and a length L0 in the reference configuration as shown in Figure 8. This hollow cylinder is assumed to be perfectly bonded to a rigid cylindrical core of radius R1. The co-ordinates of a typical material particle in the reference configuration will be denoted by cylindrical co-ordinates (R, 6, Z). In the deformed swollen state the co-ordinates of the same particle are assumed to be described by r - r(R), 0 - 9, and z - AZ, (5.3.1) 59 where (r, 0, 2) denote the co-ordinates of the particle at (R, 9, Z) in the deformed swollen configuration, A being a constant axial stretch ratio assumed to be unity. The Cauchy-Green strain tensor n which is defined as 3 ' E E (5.3.2) takes the following form for the above deformation: r 1 12 o o r g- o 12 0 , (5.3.3) o 2 Lo 0 1 , where Ar - dr/dR and A9 - r/R denote the stretch ratios in the r and 0 directions, respectively. The principal invariants of Q are then given as 2 2 2 I1 - Ar + 20 + A , (5.3.4) 2 2 2 2 2 12 - A (Ar + A0) + lair, and (5.3.5) 2 2 2 The balance of mass equation for the solid constituent (2.9) may be expressed in terms of the stretch ratios as p 'l- - :—§_: - v1, (5.3.7) ”10 r0 60 where v1 represents the volume fraction of the solid. The equations of equilibrium which are appropriate for the deformation being considered are documented next. Since the assumed form of deformation implies that the stresses depend only on the radial co- ordinate r, the equations of equilibrium for the solid constituent, namely (2.11), reduce to do J + M - b - 0, (5.3.8) where arr and 000 denote the appropriate components of g, and br denotes the component of the interaction body force 2 in the radial direction. The equilibrium equations for the fluid constituent, namely (2.12), reduce to dx x - x 3.11 + -II----11 + b - 0, (5.3.9) r r r where 'rr and «00 denote the components of z. The equilibrium equation for the mixture (2.14) reduce to dT T - 1: $33+JLr—fl-o (5.3.10) For the deformation under consideration, it follows from (5.3.3) and equations (3.23) - (3.25) that the non-zero components of the partial stress tensors for the solid and fluid constituents are given by - a + + o 09 P p ( 1 2 1) 0 2 9' ( ° ° ) 10 61 __ 1L 2, 4 022 p p10 + 2p(Al + A211)A 2pA2A and (5.3.13) pi. «tr - K00 - fizz - c p p20 - ppz Apz. (5.3.14) The only non-zero component of interaction body force is given by P d”1 ”2 g 2 2 br" p A dR +”1530 AdR '”2{(A1+A2I1)Acm(‘r+*o) 10 r 2 r r cu 13 cu 2 -2A2 (Ar 55+ 1251)} (5.3.15) r _QA. _QA. _QA. where, A1 611, A2 312 and Ap2 apz. It is sufficient to satisfy any two of the three equilibrium equations (5.3.8) - (5.3.10). Equations (5.3.11) - (5.3.15) are substituted into the equilibrium equations for the solid and the mixture (5.3.8) and (5.3.10), respectively, to get the following functional form of the equilibrium equations, which are stated in terms of the co- ordinates in the reference configuration for computational convenience: 6P p - __L— o r _ dR p10 + 81 (A13 A2) Ang Ar, A0, R, Ar, A0, A) 0, (5.3.16) and dp .. a + 32 (A1, A29 Apzy Ar, A9, R, A}, A5, A) - 0. (5.3.17) In equations (5.3.16) and (5.3.17) the prime denotes differentiation with respect to the reference radial co-ordinate R, and the radial and 62 the tangential stretch ratio Ar and A0, respectively, are related through the compatibility condition given by 3511—1 dR - . (5.3.18) Subsequently, the mixture is assumed to be of a "Neo-Hookean-type," that is, A is a linear function of I1. For this case the explicit forms of the equilibrium equations for the mixture and the solid are given by dp A A - _ - 91... 521. __r _ _r_ _L_ 2_ 2 _ dR dR [””2 apz] + 2” 5‘1": [:11 A R ["r ’0] A R [*0 NJ] 0' 0 0 (5.3.19) and dp A A .. __r__r_ _ __L 2,2 "’1dR+2” A1"r[:R 10R [*r ‘9] 1011903.” - .1.__r._£ L31: V1 plp20 3P2 A9 +Ar dR + p2A1 dR L2{A +A2 . (5.3.20) Equations (5.3.18) - (5.3.20) may be solved for p, Ar and A0 once the specific form of the Helmholtz free energy function for the mixture is known, and the appropriate boundary conditions are specified. The specific form of the Helmholtz free energy function per unit mass of the mixture which will be used here has been discussed in chapter three, and is being repeated here for convenience as RTp l-v ”a _19 21 _1 63 Two of the appropriate boundary conditions for solving the set of ordinary differential equations (5.3.18) - (5.3.20) are given by A0(Ri) - l, and (5.3.22) Trr(Ro) - 0. (5.3.23) The boundary condition on the tangential stretch ratio given by equation (5.3.22) arises due to the fact that the inner surface of the elastic cylinder is bonded to the rigid core. The boundary condition on the total traction vector represented by equation (5.3.23) is a consequence of the requirement that the outer surface of the cylinder be traction- free. Since a boundary condition for the partial traction vectors is not physically obvious, following the arguments presented in chapter four, it is assumed that the outer surface of the cylinder is in a saturated state. This assumption results in the boundary condition represented by Srr(Ro) - 0, (5.3.24) where, Srr represents the radial stress component for a saturated state, and can be explicitly written by using equation (4.9) as 2 _ _ 2A. Srr p (p20 p2) apz + p20A + 2 p A1 Ar. (5.3.25) The governing equations (5.3.18) - (5.3.20) for the finite swelling of a non-linearly elastic cylinder are highly non-linear and coupled, and may be solved numerically for the variables Ar, A and p. 0 However, for computational convenience, equations (5.3.19) and (5.3.20) 64 may be combined to eliminate p, and for the Helmholtz free energy function given by (5.3.21) the resulting equation is given by .1— (Ar - A0)[K[2x - l-u1]v1 - Aer] R A dA --—l “I - - (5.3.26) A dR L 2 r K 2x - 1-” ”l - Ar 1 where, M K - -—§—— V 0 ”1o 1 The set of ordinary differential equations given by (5.3.18) and (5.3.26) subjected to boundary conditions given by (5.3.22) and (5.3.24) were solved numerically. For the computational work the material properties which are used here are same as given in section 5.2. Computational results are presented in Figures 9-15. Figure 9 shows the variation of the radial and circumferential stretch ratios with respect to the non-dimensional reference radial co- ordinate for various values of Ri/Ro' It may be pointed out that Ri/Rorepresents the ratio of the radius of the rigid core to the outer radius of the elastic cylinder in the unswollen configuration, and is numerically bounded by 0 and 1. a 0 represents a "thick” elastic ”I” 0'“ O R cylinder bonded to a relatively thin fiber-like rigid core, and R1 a l 0 represents an elastic "membrane" cylinder bonded to a relatively thick rigid core. In the absence of a rigid core the unconstrained solid cylinder would swell homogeneously with the radial and tangential stretch ratios equal (Ar — A0 a 2.08) and constant throughout the domain. It is clearly evident from Figure 9 that the presence of the 65 rigid core induces significant non-homogeneous swelling characteristics in the cylinder with large gradients in stretch ratios in the neighborhood of the rigid core. Figure 10 presents the variation of the non-dimensional radial stress with respect to the non-dimensional reference radial co-ordinate for various values of Ri/Ro‘ The radial stresses are tensile and considerably higher in the immediate neighborhood of the rigid core where the elastic cylinder is bonded, and zero at the outer radial surface which is assumed to be traction-free. Similarly, very large gradients in the non-dimensional circumferential compressive stresses arising due to the presence of the rigid core are shown in Figure 11. The radial stress and circumferential stresses plotted in Figures 10 and 11 denoted by Trr and T00 have been non-dimensionalized with respect to A R T __519 . M c Figure 12 clearly shows that the presence of the rigid core constrains the swelling ability of the surrounding elastic material resulting in a higher volume fraction of the solid in the vicinity of the rigid core. This constraining effect of the rigid core on the ratio of the overall swollen volume to the unswollen volume is presented in Figure 13. Figure 14 shows the variation of the non-dimensional radial stress at the outer and inner surface of the elastic cylinder with respect to the ratio Ri/Ro' Whereas the radial stress at the outer surface is zero for all values of Ri/Ro due to the assumption that the outer surface is traction-free, the radial stress at the bond interface exhibits an interesting non-monotonic variation with respect to the ratio Ri/Ro' A similar non-monotonic variation in the circumferential stress at the interface is evident as shown in Figure 15. 66 The constrained swelling of a non-linearly elastic cylinder bonded to a rigid core has been presented in the context of Theory of Interacting Continua. Whereas a solid cylinder of a non-linearly elastic material swells homogeneously, it is clearly evident from the computational results presented herein that the presence of the rigid core induces non-homogeneous swelling characteristics. Furthermore, it has been demonstrated that the rigid core initiates significant gradients in the stretch ratios and severe stress concentrations at the bond interface. These results could have important implications in a variety of commercial applications featuring reinforcing fibers embedded in hygroscopic polymeric matrix materials. 67 5.4 FINITE EXTENSION AND TORSION OF.A.CYLINDRIGAL SOLID-FEUID HIXIURE'UITH.A.RIGID GORE. 5.4.1 m In section 5.3, the constrained equilibrium swelling of a non- linearly elastic cylinder bonded to a rigid core in absence of external load has been investigated in the context of Theory of Interacting Continua. In this work, finite extension and torsion of a swollen elastic cylinder perfectly bonded to a rigid cylinder is investigated in the context of interacting continua. The results of these investigations not only demonstrate significant gradients in the stretch ratios and severe stress concentrations but also stress reversals in the neighborhood of the interface of the inner and outer cylinders. The general problem of finite extension and torsion of a non- linearly elastic swollen cylinder bonded to a rigid core is formulated and discussed within the context of Theory of Interacting Continua in the following subsection. In addition, the computational results demonstrating the combined effect of the rigid core and finite extension and twist on the spatial variation of the radial and tangential stretch ratios and the distribution of the fluid in the swollen deformed state are presented. The role of the rigid core in inducing non-homogenous swelling characteristics, stress concentrations, and stress reversals at the bond interface undergoing finite extension and twist is also discussed. 68 Consider a solid circular cylinder composed of two materials M1(rigid) and M2 (nonlinearly elastic) occupying the region R 5 [R1, R2] in the reference configuration, such that R2 > R1 > 0. Both cylinders are assumed to have a length L0 in the reference configuration (see Figure 1). It is assumed that both materials are perfectly bonded to each other at the interface. The co-ordinates of a typical material particle in the reference configuration will be denoted by cylindrical co-ordinates (R, 9, Z). The cylinder is assumed to be subjected to the following deformation: r - r(R), o - e + $12, and z - AZ, (5.4.1) where (r, 0, 2) denote the co-ordinates of the particle at (R, 9, Z) in the deformed swollen configuration, A and ¢ being constants. The above deformation corresponds to a finite elongation (with an associated stretch ratio A) along the z-co-ordinate direction, followed by a rotation of p per unit current length. The Cauchy-Green tensor g which is defined as g - {ET (5.4.2) takes the following form for the above deformation: 69 (Q12 1 [an] o o r 2 2 ' o [a] + (¢Ar)2 ¢A2r ' (5.4.3) 0 ¢A2r A2 L J 12 o o r ' o A: + (¢R110)2 pAonR ' (5'4°4) Lo ¢A2A0R 12 , where Ar - dr/dR and A - r/R denote the stretch ratios in the r and 0 0 directions, respectively. The principal invariants of g are then given as I - 12 + 12(1 + ¢22212) + 12, (5.4.5) 1 r 9 2 2 2 2 2 2 2 2 I2 - 1 (Ar + 10) + 101r(1 + w R A ), and (5.4.6) 2 2 2 I3 - AerA . (5.4.7) The balance of mass equation for the solid constituent (2.9) may be expressed in terms of the stretch ratios as ”1 -—— - K'i-K - ”1' (5.4.8) ”10 r o where v1 represents the volume fraction of the solid. The equations of equilibrium which are appropriate for the deformation being considered are documented next. Since the assumed form 70 of deformation implies that the stresses depend only on the radial co- ordinate r, the equations of equilibrium for the solid constituent (2.11), reduce to do ..rr.+.JaL___1£.- b - 0, (5.4.9) where arr and 000 denote the appropriate components of g, and br denotes the component of the interaction body force 2 in the radial direction. The equilibrium equations for the fluid constituent (2.12), reduce to dx x - w 3:11 + —£1—;——11 + br - 0, (5.4.10) where 'rr and «00 denote the appropriate components of g. The equilibrium equation for mixture (2.14) reduce to dT T - T __LI + _££____ii _ 0, (5.4.11) dr r which is the equation of equilibrium for the mixture. For the deformation under consideration, it follows from equation (5.4.4) and equations (3.23) - (3.25) that the non—zero components of the partial stress tensors for the solid and fluid constituents are given by _ _ £1_ 2 _ 4 arr p p10 + 2p(A1 + A2I1)Ar 2p(A2)Ar, (5.4.12) _ _ £1_ 2 2 2 2 a p p + 2p(A1 + A2I1)A0(1 + p R A ) 00 10 71 -2pA2{A:(l + ¢2R2A2)2 + ¢2R2A4A3}, (5.4.13) _ _ f1_ 2 - 4 2 2 2 ”22 p p10 + 2p(A1 + A2I1)A 2pA2{A (l + 0 R Aa)}, (5.4.14) 2 2 2 2 2 2 2 092 - 2p¢R {(A1 + A211)A A9 - A2[A 19((1 + ¢ R A )A0 + 1 )1}, (5.4.15) and p «rr - «00 - "zz - - p p20 - pp2 Ap2 , respectively. (5.4.16) The only non-zero component of the interaction body force vector is given by P do dp b - - 1 + A -———Z r p10 Ar dR ”1 p2 ArdR - p2 A1 I;§E A: + A? + ¢2A2R2A§ + A2} - p2A2 :;§§ [1:12 + A§A2 + A:A3(l + ¢2R2A2)] , (5.4.17) where A1 - 3%1’ A2 - 3%; and AP2 - 3%;. It is sufficient to satisfy any two of the three equilibrium equations (5.4.9) - (5.4.11). Equations (5.4.12) - (5.4.17) are substituted in to the equilibrium equations for the solid and the mixture (5.4.9) and (5.4.11), respectively, to get the following functional forms of the equilibrium equations which are stated in terms of the co-ordinates in the reference configuration for computational convenience: 72 d? p ' — _1— ! p 2 2 - dR p10 + 81 (A1, A2, A”2' Ar. A9. R, Ar, A0, A, ¢ R ) 0, (5.4.18) and dp _ v I 2 2 _ .. dR + 82 (A1, A29 Apz’ Ar, A0, R, Ar, A0, A, ¢ R) 0. (5.4.19) In equations (5.4.18) and (5.4.19) the prime denotes differentiation with respect to the reference radial coordinate R, and the radial and tangential stretch ratio Ar and A0, respectively, are related through the compatibility condition given by -£-———£. (5.4.20) Subsequently, the mixture is assumed to be of a 'Neo-Hookean-type,” that is, A is a linear function of 11‘ For this case the explicit forms of the equilibrium equations for the mixture and the solid are given by _2_1 pp 51. +2”, 9.2-5:. , ,, dR dR 2 ap2 1 r dR AaR r 0 _ 1 2- 2 2 2 2 2 _ 20R ( 0 Ar + p A R A0)] 0, (5.4.21) and dp dA A - —— -—I - -—1 - ”1 dR + 2” Alxr [ dR A R [Ar *9] 0 A R 01 (1 -1 ) 1 2 2 2 2 2 2 28. .1 __r .l_.__I__fl_ ' (*o‘Ar + ” A R ‘0’] ' ”1”20 ap2 ”1[ r dR + A0 ] (A -A ) I Q 2 2 _ r dR 0 R + p A AoArR] 0. + N 'b N > H r—fi )4 + y (5.4.22) 73 Equations (5.4.20) - (5.4.22) may be solved for p, Ar and A0 once the specific form of the Helmholtz free energy function for the mixture is known, and the appropriate boundary conditions are specified. The specific form of the Helmholtz free energy function per unit mass of the mixture has been discussed in chapter three, and is being repeated here for convenience as V RTp A l-u A - -1 —1-9 (I1 - 3) + 31 [Tl £n(l-u1) + x(l-V1)]], (5.4.23) 1 p 2Mc V 1 Two of the appropriate boundary conditions for solving the set of ordinary differential equations (5.4.20) - (5.4.22) are given by A0(Ri) - 1.0 (5.4.24) Trr(Ro) - 0 (5.4.25) The boundary condition given by equation (5.4.24) arises due to the compatibility requirement between the radial and tangential stretch ratios at the axis of the cylinder. The boundary condition on the total traction vector represented by equation (5.4.25) is a consequence of the requirement that the outer surface of the cylinder be traction-free. Since a boundary condition for the partial traction vector is not physically obvious, following the arguments presented in chapter four, it is assumed that the outer surface of the cylinder is in a saturated state. This assumption results in the boundary condition represented by Srr(Ro) - 0, (5.4.26) 74 where Srr represents the radial stress component for a saturated state, and can be explicitly written by using equation (4.9) as 25. 2 Srr - p (p20 - p2) apz + p20A + 2 p A1 Ar . (5.4.27) The governing equations (5.4.20) - (5.4.22) for the combined extension and torsion of a swollen cylinder are highly non-linear and coupled, and may be solved numerically for the variables Ar, A0 and p. For computational convenience, equations (5.4.21) and (5.4.22) may be combined to eliminate p, and for the Helmholtz free energy function given by (5.4.23) the resulting equation is given by _ - 1 _ 2 2 2 2 (Ar A9)[K(2x 1_V1)v1 Aer] + w A R A Ar 9 RA dA 3’1 ER: ‘ ' .;L__ 2 (5.4.28) r K (2x - 1_y1)u1 - Ar where, MC K - . ”10V1 The set of ordinary differential equations given by (5.4.20) and (5.4.28) subjected to boundary conditions given by (5.4.24) and (5.4.26) were solved numerically. The material properties used are the same as given in section 5.2 The computational results are presented in Figures 16-21 for a value of the axial stretch ratio A - 1.00. The computational results presented in Figures 16-20 are for a fixed interface location Ri/Ro- .001 which is the ratio of the reference rigid core radius to the outer radius of the cylinder. 75 The variation of the radial stretch ratio with the non-dimensional reference coordinate R/RO for three different values of twist p is presented in Figure 16. It is observed from this plot that the radial stretch has steep gradients and are significant in the neighborhood of the constraint. It is very evident that the severity of radial stretch and its gradient increases with the increase of twist. A similar plot is presented for the variation of circumferential stretch ratio in Figure 17. The gradients in the stretch ratio away from the constraint are small whereas in the neighborhood of the constraint singularities in the stretch ratios are observed, as anticipated. . The variation of the volume fraction of the solid with the non- dimensional coordinate R/Ro for different values of the angle of twist is presented in Figure 18. It is clearly evident that fluid leaves the system when subject to twist, and also the rigid core restricts the swelling characteristics of non-linear material surrounding it. The variations of the non-dimensional radial stress with the non-dimensional radial reference coordinate is presented in Figure 19. Stress reversals are observed for larger angle of twist. These stress reversals may have significant impact on the design of the bonding adhesive for the polymeric composites. The variation of the circumferential stress with non-dimensional radial coordinate is presented in Figure 20. In this case, it is observed that the stress reversal does take place but at at a location R/Ro which is away from the rigid core interface as contrast to the radial stress reversal. It may also be noted that the circumferential stress reversals are independent of the degree of twist, and take place at a location R/Ro - .575 for a fixed value of Ri/Ro-.OOl. It is concluded that the circumferential stress reversals in the context of the present work are 76 not of important character. The effect of radius of rigid core on the mechanical behavior of the constrained polymeric material is investigated by plotting the variations of the radial stress for different core radii. Clearly gradients in the radial stress are significant for larger core radii. This result could be vital in selecting the diameter of the fiber for the fabrication of fiber- reinforced polymeric composite materials. CHAPTER SIX MODELING OF'HYGRO-THERHAL EFFECTS IN POLIHEHIC COMPOSITE MATERIALS WITHIN THE CONTEXT OF MIXTURE THEORY 6.1 MOTIVATION The investigation of the swelling of a non-linearly elastic cylinder featuring a rigid core under no loads, and under finite extension and twist presented in chapter five is useful in understanding the micro-mechanical behavior of polymeric composites undergoing finite deformations. However, fiber-reinforced polymeric composite materials db not, in general, undergo large deformations but contain volume fractions of fluid which may not be necessarily small. The mechanical response of polymeric composite materials under hygro-thermal environments is amenable to treatment in the context of Mixture Theory. In this work, a linearized version of Mixture Theory is presented to model the composite-fluid interaction where the fluid content is not necessarily small. The extensive utilization of polymeric composite materials in engineering practice has triggered focused attention, both experimental and theoretical, on understanding the response of these materials in order to ensure safe and efficient designs for engineering applications. 77 78 Significant progress has been made in modeling the constitutive characteristics of polymeric composite materials and the dependence of these characteristics on moisture and temperature environments [32-35]. In this chapter the the deficiencies of traditional modeling techniques are addressed by treating the fluid/moisture as a second constituent along with the equivalent polymeric composite, and the non-linear hygro- thermo-elasto-dynamic response of polymeric composite plates is modeled rigorously in the context of Mixture Theory. The mechanical vibrational characteristics of a structure depend on the stiffness, mass and damping properties of the material. In polymeric composite materials the stiffness, mass and damping characteristics are all functions of moisture and temperature; and strictly, the mechanical response, the hygroscopic response and the thermal response are non- linearly coupled. Furthermore, the presence of moisture and heat also produces undesired dimensional changes in polymeric composite materials. Whenever moisture is present in a composite material in amounts which may not be infinitesimal, it is imperative that the moisture be explicitly treated as another constituent in order to realistically address moisture-induced effects in composite materials. The presence of moisture in polymeric fiber-reinforced composite materials is treated rigorously as another constituent within the context of Mixture Theory. This approach has been employed recently to study the interaction of fluids and non-linearly elastic polymers for finite deformations in order to address several phenomena such as swelling, saturation and diffusion [4,6,7,23]. Furthermore, the author in last chapter has employed Mixture Theory to study constrained I polymers in order to demonstrate that the constraint due to the presence 79 of the fibers restricts the ability of the surrounding polymeric material to swell, thereby inducing non-homogeneous swelling characteristics. In this work the deficiencies of traditional modeling techniques are addressed by treating the moisture-composite aggregate within the context of Mixture Theory. This approach not only permits the explicit incorporation of moisture and its effects, but also allows the possibility of deriving constitutive equations to model the coupled hygro-thermo-elasto-dynamic response of idealized fiber-reinforced composite plates undergoing small deformations and large rotations. The hygro-thermal dimensional changes along the thickness of the composite mixture are explicitly incorporated in the assumed form of the deformation field. The non-linear vibration of moderately-thick laminated polymeric composite plates is considered by incorporating the effects of transverse shear and rotary inertia. It may be emphasized that the mathematical structure and physical meaning of the quantities involved in constitutive equations and equations of motions presented in this work are very different from the corresponding quantities in single constituent classical laminated plate theories [27]. This unique structure of the constitutive equations and the explicit form of the equations of motion is precipitated by the current approach whereby moisture is explicitly incorporated as a second constituent. This approach is necessitated whenever the moisture content in the polymeric composite materials is of more than infinitesimal amount. The authors believe that this work is the first attempt at addressing the phenomena of non-linear vibration of laminated composite plates by incorporating hygro-thermal environmental effects in an explicit sense and this work is anticipated to be significanlty relevant to applications in defense, aerospace and manufacturing environments, 80 where significant variations in the moisture and temperature conditions may be encountered. The constitutive equations for the mixture of an anisotropic polymeric solid and an ideal fluid are discussed in Section 6.2. Equations of motion for the hygro-thermo-elasto-dynamic response of polymeric composite plates undergoing small deformations and large rotations are presented in section 6.3. Furthermore, the theoretical developments are employed to model the non-linear vibration of moderately-thick laminated composite plates by incorporating hygroscopic effects, rotary inertia and shear deformations. A model for micro- mechanics of an isotropic polymeric matrix with a constraint(fiber) for the infinitesimal hygro-thermo-elastic response is presented in section 6.4, and an illustrative example is presented in order to demonstrate the applicability of the proposed model. 81 6.2 CONSTITUTIVE MODELING OF AN ANISOTROPIC POLYMERIC MITE MATERIAL.UNDEH.HYGRO-THEHMAL ENVIRONMENTS In this section a linearized version of constitutive equations in Mixture Theory are presented. These constitutive equations feature large rotations and infinitesimal strains, and incorporate hygroscopic effects in an explicit sense. This approach of formulating constitutive equations, where fluid in the equivalent polymeric composite material is treated as another constituent, could predict dimensional and constitutive changes due to the presence of moisture which may not be predicted by single constituent approach. The linearized version of constitutive equations from finite Mixture Theory will be derived next. In the reference configuration the solid(equivalent anisotropic polymeric composite) and fluid(moisture) are assumed to have reference densities 31 and 22, both defined with respect to the volume of the reference solid, respectively. Both constituents are assumed to have a constant common temperature T. Subsequent to the interaction of the solid and fluid, the displacements, velocities of the solid and fluid, and temperature changes; together with their space derivatives are assumed to be small. Only first-order terms in displacements and second order terms in rotations are systematically retained in strain- displacement relations and hence quantities of the same order are retained in the constituents equations for aij’ «11, and b1. Similarly, only linear and quadratic terms in Helmholtz free energy function are retained. Furthermore, each spatial point in the composite-fluid mixture 82 is assumed to be simultaneously occupied by material particles from each constituent. These assumptions leads to the following equations p2 - Z2 + 3 , and (6.2.1) where U1 is the displacement of a solid or fluid particle which was initially at the reference coordinate X1, 5 is the small perturbation in the reference fluid density p2 and 0 is the small perturbation in the common mixture temperature T. Assuming that the strains to be much smaller than rotations, the strain-displacement relations and mass balance for solid and fluid components of moderately-thick polymeric composite mixture may be written, respectively as ,1 ers - ers 2 U3,r U3,s , r,s-(l,2,3) (6.2.2) p1 - 31 (1 - em), and m-l,2,3 (6.2.3) %€+V1%&;'°' (6.2.4) where ers ul- (uma + 08;). r,s-(l,2,3) (6.2.5) It is further assumed that the perturbation in the density of the fluid 5 differs slightly from the reference density 32, so that 8 together with I—i—I and‘n—i-n are small, and of the same order as e , where T T :1 ij 83 and a are the reference absolute temperature and entropy of the system. The partial stresses and other variables in the constitutive relations (expressed in suitable non-dimensional form) are also of the same order. The Helmholtz free energy function defined per unit mass of the mixture may be written in the this context as - l l 2 PA ' 2 Eijkl ‘kl ‘13 + 313 ‘13 5 + 2 “c 5 + Q 6 o + 1 h 92 + c o p (6 2 6) 13 13 2 c c ' ° - Where 3 - $1 + 32 , and Eijkl’ BH and Qij are anisotropic elastic, hygro-elastic and thermo-elastic moduli, respectively, and me, hc and cc are moisture coefficient, thermal coefficient and hygro-thermal coefficient, respectively. The components of the partial stresses in the solid and fluid, the interaction body force, and the explicit form of the entropy may be given by 01J - - p sij + Eijkl ekl + Bij 5 + Qij 0 , (6.2.7) nij - - [p + :2 (31‘1 ckl + me p + Cc0)] 513 , (6.2.8) bi - a (ui - vi) , and (6.2.9) a I o [Qij ‘13 + hco + cc 3]. (6.2.10) 84 It is also useful to record the representation of the components of the total stress tensor Tij - - [2p + 32 (Bk1 ekl + mc 3 + cc 0)] 61J (6.2.11) + Eijkl ‘kl + B11 5 + Qij 0 . If the polymeric composite is assumed to be saturated with the surrounding fluid at all times, the indeterminate p may be determined from the saturation boundary condition discussed in chapter four, and is given as p - - 32 (Bij ‘13 + mc 3). (6.2.12) Equation (6.2.12) when substituted in equation (6.2.11) yield T13 ' Eijkl ‘k1 + (313 + ”2 “c 513) 5 + ”2 Bkl ‘kl 513 (6.2.13) + cc 0 sij + Qij 0 , which are the constitutive equations for the anisotropic equivalent polymeric composite-moisture mixture. 85 6.3 THE EFFECT OF'HDISTURE AND TEMPERATURE ON THE VIBRATIONAL CHARACTERISTICS OF COMPOSITE HATERIALS In this section a model is proposed to predict the non-linear vibrational response of equivalent anisotropic composite materials within the context of Mixture Theory. The mechanical response of the polymeric composite materials under hygro-thermal environments is modeled by treating the mixture as composed of an anisotropic solid (equivalent composite) and an ideal fluid (moisture). This approach not only permits the explicit incorporation of moisture and its effects, but also allows the possibility of deriving constitutive equations to model the coupled hygro-thermo-elasto-dynamic response of idealized fiber- reinforced composite materials undergoing small deformations and large rotations. In particular, the non-linear vibration of moderately-thick laminated polymeric composite plates under hygro-thermal conditions is considered by incorporating the effects of transverse shear and rotary inertia. Furthermore, the hygro-thermal dimensional changes along the thickness of the composite mixture are explicitly incorporated in the assumed form of the deformation field. The mathematical structure of the constitutive equations and the equations of motion presented in this work, and the physical interpretation of the quantities involved in these equations are significantly different from corresponding equations and quantities in classical laminated composite plate theories. It is anticipated that this work will be significantly relevant to applications in defense, aerospace and manufacturing environments, where significant variations in the moisture and temperature conditions may be encountered. 86 This section is followed by the derivation of equilibrium equations for moderately-thick plates, and in section 6.3.2 a model for hygro- elastic vibration of moderately-thick laminated composite plates is presented. For a moderately-thick plate, by assuming the strains to be much smaller than the rotations, the equilibrium equations in the absence of external body forces may be written as +h/2 +h/2 _ N11 1 + N21,2 + 113 I - I p a1 dz , (6.3.1) 'h/z -h/z +h/2 +h/2 _ N21,1 + N22’2 + T23 |-h/2 - p a2 dz , (6.3.2) -h/2 +h/2 ' d. +h/2 . Q1'1 + Q2'2 + T33 I + ax I (T11 U3,1 + T12 U3,2) dz -h/2 -h/2 +h/2 L + ay (T21 U3,1 + T22 "3.2) dz -h/2 +h/2 +h/2 + ('1‘31 U3,1 + T32 U3,2) |-h/2 - p a3 dz (6.3.3) -h/2 +h/2 M11,1 + M12,2 - Q1 + m1 - I p 2 a1 dz , (6.3.4) -h/2 87 +h/2 M21’1 + M22’2 - Q2 + m2 - I p 2 a2 dz , (6.3.5) -h/2 and +h/2 +h/2 M31,1 + M32,2 + m3 ' T33dz ' I (T32 U3,1 + T32 ”3.2)dz -h/2 -h/2 +h/2 +h/2 + z T33 ’_h/2 + z [(T11 U3,1 + T12 U3,2),1 —h/2 +h/2 + (T21 U3,1 + T22 U3’2),2] dz - I p 2 a3 dz . (6.3.6) -h/2 In equations (6.3.1) - (6.3.6), +h/2 Nafi - I Tafi dz , 0,8 - 1,2 (6.3.7) -h/2 +h/2 Qa - I Ta3 dz , a - 1,2 (6.3.8) -h/2 +h/2 Mafl - I z Tafi dz , 6,6 - 1,2 (6.3.9) -h/2 +h/2 m - z T I , (6.3.10) 1 13,3 -h/2 +h/2 m2 - z T23,3 I , and (6.3.11) 88 +h/2 - z(T U + T U ) . (6.3.12) 31 3,1 32 3,2 -h/2 m3 6.3.2 HYGRD-ELASTIC vxnnarrou or'uonnnarxnx>rnlcx.LauInarzn courosxrn PLATES The equations of motion and the constitutive equations for a moderately-thick plate composite plates in hygro-thermal environments have been discussed previously. Herein, the problem of vibration of moderately-thick laminated composite plates in isothermal environments is formulated in order to account for mid-plane stretching due to large deflections, large rotations, and dimensional changes due to moisture- induced swelling effects. It may be emphasized that isothermal conditions have been assumed for convenience only, and this assumption does not result in any loss of generality. The hygroscopic dimensional changes along the thickness of the plate are explicitly incorporated in the assumed form of the deformation field. A deformation field of the following form is considered in order to accommodate these effects: x (X,Y,Z,t) - i (X,Y,t) + AZ 5 (X,Y,t) , y (X,Y,Z,t) - § (X,Y,t) + AZ 0 (X,Y,t) , and (6.3.13) 2 (X,Y,Z,t) - E (X,Y,t) + AZ . In equations (6.3.13) the reference (undeformed) position of a typical particle is represented by the coordinates (X,Y,Z), and the coordinates of the same particle in the deformed and swollen state are denoted by (x,y,z). The mid-plane of the plate in the undeformed state is assumed 89 to be at Z - 0, and (§,§,E) represent the deformed position of a typical particle on the mid-plane of the plate in the undeformed position. A denotes the stretch ratio in the z direction which accounts for hygrosc0pic dimensional changes along the thickness. 6 and 0 represent the slopes in the X2 and Y2 planes, respectively, due to bending only. The displacement field corresponding to equations (6.3.13) is given U - U(X,Y,Z,t) - U°(X,Y,t) + AZ€(X,Y,t), U - U(X,Y,Z,t) - V°(X,Y,t) + AZ€(X,Y,t),and (6.3.14) U - W(X,Y,Z,t) - W°(X,Y,t) + (A-l)Z , where U°, V‘, W° are mid-plane displacements given as U°-§-x, V° - y - Y, and (6.3.15) w-E All equations will now be stated in a form applicable to the analysis of moderately-thick laminated composite plates. For notational convenience, the standard contracted notation will be employed, where stress and strain components are represented by single subscripts. Thus the single subscripts l,2,3,4,5,6 represent the double subscripts ll,22,33,l3,23,12, respectively. With this notation, and using equations 90 (6.2.5), (6.3.14) and (6.3.15) into equations (6.2.2) yield the following set of strain-displacement equations: 6 - 6° + AZk , a - 1,2,6 a a 63 % (AZ-l) , (6.3.16) A(£ + U3,l)’ and A(n + U§,2). where l U‘2 5 - 1,2 (no sum on p) "5.19 + 2 3.5’ + U’ 2.1 + U U“ ' U° 3,1 3,2' 1,2 (6.3.17) k1 - 691 3 k2 - n,2, and k6 -632 +0910 It is also useful to record constitutive equations (6.2.13) for isothermal conditions in the contracted form as T. - E 1 ij ej + (31 + 32 me) 3 + $2 Bj ej , and 1,3 - 1,2,3 (6.3.18) TI - EIJ eJ + 316 . I,J - 4,5,6 (6.3.19) 91 where g and g are anisotropic elastic and hygro-elastic moduli in contracted notations, and fl is related to the dillitational strain (61 + £2 + 63) through equations (2.30) and (6.2.3), and yields 8 - p20 (61 + 62+ 63) - 32 . The equations of motion (6.3.1) - (6.3.6) and the strain-displacement relations given by equations (6.3.16) represent a general three- dimensional problem and can not be solved in a two-dimensional (plate- like) sense. The problem may be considered in two steps. In the first step, the anisotropic laminated plate may be assumed to be deformed to an intermediate state due to out-of-plane moisture and temperature effects. In this intermediate state the thickness is assumed to be changed by a stretch ratio A. In the second step, the deformation from the intermediate state to the final state is assumed to be due to mechanical loads and due to in-plane hygroscopic effects. Once the hygroscopic environment is specified, A, and, therefore, 63 can be determined; and for a constant value of A the equation of motion (6.3.6) is trivially satisfied, which reduced the problem to a plane stress problem. Now the strain-displacement relations are given by £1 - 6i + 2 k1, e6 - £6 + z k (6.3.20) - (f + U3,l)’ and 92 £5 - (q + U3’2), and the equations of motion reduce to + N - P U 6,2 + R5 1,tt ’tt’ + N - P U 2,2 2,tt + Rn’tt’ + N(U Q1,1 + Q2’2 3'Ni) ' P U3,tt’ (6.3.21) M1,1 + M6,2 - Q1 - Ie’tt + R Ul,tt’ and 6,1 + M2,2 ' Q2 ' In’tt + R U2,tt' In equations (6.3.21), m-n Z (P, R, I) - E:II:+1(1, 2,22) pm dz, (6.3.22) h/2 . . Na, Ma - . (1, 2) Ta d2, (6.3.23) -h/2 a-l,2,6 /2 . QC - T dZ, and C - 4,5 (6.3.24) 93 N(U N L L 3' 1) ax (N1 03,1) + ay (N6 U3,1) + 33; (N6 um) + $5,012 113,2). (6.3.25) The definitions given by equations (6.3.22) - (6.3.24) are based on the assumption that the laminated composite plate is composed on n layers, and 2m and 21M1 are the coordinates of the lower and upper boundary of each individual layer. In addition, h - Ah, and Z - AZ . (6.3.26) For one plane of symmetry parallel to the plane of the layer, the constitutive equations in terms of the force and moment resultants which appear in the equations of motion may be written as - ‘ f” + ‘ m E2 0 ' 1'2 (6 3 27) ”2 Ba ”2 c h /2 ' fl - 1,2,6 ° ° - K6 - p2 E , and fl - 1,2,6 (6.3.28) 94 X ‘B-C C,D - 4.5 (no sum on D) (6.3.29) Where 5 is the swollen thickness of the composite and for notational convenience Q1 - Q4 and Q2 - Q5 and A m-n Z l A A A - (m) 2 - Aafi' Bafi’ Dafl Z I” Eafi (1,2,2) d2, 02,13 1,2,6 (6.3.30) m-l 2 m m-n z — _ _ 1 A A2 A A , B , D - E: Im+ 8 (1,2,2 ) d2, and a-l,2,6 a a a a mp1 2 m m-n Z _ 1 (m) A ACD - E: Im+ KC KD CCD d2. C,D - 4.5 (6.3.31) A (no sum on D) mpl 2 m In equation (6.3.31) KC and KD are the shear correction coefficients. The authors would like to emphasize that the mathematical structure and physical interpretations of the quantities involved in the constitutive equations and the equations of motion presented in section 5 are significantly different from the corresponding quantities in single constituent classical laminated plate theories ( for example compare equations (40-45), reference [27] with equations (6.3.27-6.3.29) in this sections). This unique structure of the constitutive equations and the explicit form of the equations of motion is precipitated by the 95 current approach whereby moisture is explicitly incorporated as a second constituent. This approach is necessitated whenever the moisture content in the polymeric composite materials is of more than infinitesimal amount. Furthermore, in this approach the theory is developed in terms of the total strains, which can not be conveniently decoupled into mechanical and non-mechanical strains, whereas the classical laminated plate theories are traditionally formulated by superposition of the mechanical strains and non-mechanical strains for the overall response of the composite plates under hygro-thermal environments. It may be noted from the mathematical expression for g, g and 9 that in the absence of hygroscopic effects the theory reduces to classical laminated plate theory. A methodology has been presented to address the phenomena of non- linear vibration of moderately-thick laminated composite plates in hygro-thermal environments. The methodology is based on the linearization of Mixture Theory in order to incorporate moisture content as another constituent, and to accommodate mid-plane stretching due to large deflections, large rotations, and dimensional changes due to hygro-thermal effects. It is anticipated that this work will be significantly relevant to applications in the defense, aerospace and manufacturing environments, where significant variations in the moisture and temperature conditions may be encountered. 96 6.4 CONSTITUTIVE MODELING OF COUPLED mono-.m-msnc WA IN CONSTRAINED ISOTROPIC POEXMERIC MATERIALS The understanding of deformation characteristics of a matrix material around a fiber (micro-mechanics of composite materials) in the presence of moisture, temperature and external loading is important for several areas of technical interest, the damage characterization of fiber-reinforced polymeric composite materials is one such example. In this section, a model for the infinitesimal hygro-thermo-elastic response of representative isotropic matrix constrained by a fiber is presented within the context of Mixture Theory. Numerous experimental and theoretical studies have highlighted the vulnerability of polymeric fiber-reinforced composite materials to severe hygro-thermal environments. Traditionally, single constituent theories have been employed to model the constitutive response of idealized polymeric composite materials in hygro-thermal environments. Typically, these theories do not incorporate moisture in an explicit sense, and the moisture and temperature effects are uncoupled from the mechanical response of the composite material. In this chapter the deficiencies of traditional modeling techniques are addressed by treating the composite-moisture aggregate within the context of Mixture Theory. This approach, micro-mechanics of composite materials within the context of Mixture Theory, not only permits the explicit incorporation of moisture and its effects, but also allows the possibility of deriving constitutive equations to model the coupled hygro-thermo-elastic response of idealized fiber-reinforced composite 97 materials undergoing small deformations. An illustrative example is presented in order to highlight the significant role of the fibers in modifying the hygro-thermo-elastic characteristics of the overall composite material. A representative cylindrical polymeric matrix-moisture mixture with a rigid core(fiber) is considered. The polymeric matrix is assumed to be isotropic. Furthermore, it is assumed that only the matrix is sensitive to changes in ambient temperature and moisture and fiber is insensitive to these changes. The explicit form of the Helmholtz free energy function given by (6.2.6) for the above mentioned assumption take the form NIH 3A - A0 + a1 emm + azfi + a30 + + a5 emn eInn + % asfi + % a7 0 + aaemmfi (6.4.1) +a9em0+aloflfl, where a1 - a10 are material constants. The components of the partial stresses in the solid and fluid, the interaction body force, and the explicit form of the entropy may be given by aij - (-p + 31 emm+ szfl + 330 + 54) 611+ sseij , (6.4.2) 98 «11 - -(p + flemm + £25 + £30 + f4) Sij , (6.4.3) bi - dlemm’i - (12 8,1 , and (6.4.4) n -- -l [a + a a + a e + a ] (6 a 5) p 3 7 9 mm 10” ° ° ' In equations (6.4.6 - 6.4.4) the material constants are redefined as - - £1 __1_ 31 a4 p a1, 32 p a1 + a8, 33 - a9 , sh - a1, 35 - 2(a1 + a5) - n+3 - - - 5152 _ _____2 - f1 as ”2 p 32 ' f2 p 32 + a6 ”2' - - - .2 f3 310 , f4 a2p2 , (11 p a1, and - 21 d2 p a2 . In the following section, attention is focused on the equilibrium formulation only, hence, the part of the interaction body force in equation (6.4.4) which arises due to relative motion of the constituents has already been dropped. It is also useful to record the representation for the components of the total stress tensor Tij- [-2p+(sl-f1)emm+ (32-f2)fi + (SB-£3)0 + (s4'f4)]6ij + 5 (6.4.6) 5°13 99 6.4.2 ILLUSTRATIVE EXAMPLE The problem of swelling of a cylindrical polymeric material with a rigid core is considered within the context of infinitesimal Mixture Theory. In this section, the swelling characteristics of an idealized polymeric composite representative element are investigated by assuming that: (i) The fiber is located at the cylindrical axis and perfectly bonded to the surrounding polymeric matrix material. (ii) The polymeric matrix material surrounding the fiber is isotropic. (iii) The representative element is exposed to an ideal fluid bath with isothermal conditions. (iv) The cylindrical representative element is free from initial stresses . Consider a hollow cylinder of a non-linearly elastic material described by internal and external radii R1 and R0, respectively, and a length L0 in the reference configuration as shown in Figure 1. This ho1low cylinder is assumed to be perfectly bonded to a rigid cylindrical core of radius R1. The co-ordinates of a typical material particle in the reference configuration will be denoted by cylindrical co-ordinates (R, 9, Z). In the deformed swollen state the co-ordinates of the same particle are assumed to be described by 100 r - r(R), 0 - 9, and z - AZ , (6.4.7) where (r, 0, 2) denote the co-ordinates of the particle at (R, 9, Z) in the deformed swollen configuration, A being a constant axial stretch ratio assumed to be unity. For the assumed deformation field (6.4.7) the infinitesimal strain tensor may be written as w,R O 0 g - O w/R O . (6.4.8) 0 0 0 where w is the radial displacement defined as w - r(R) - R . (6.4.9) It is useful to record that the volume additivity assumption and the incompressibility constraint (2.30) along with the mass balance of the solid constituent yield 8 - p20 emm - p2 . (6.4.10) It may be pointed out that due to volume additivity assumption and incompressibility constraint equation (6.4.10), 8 may be eliminated from governing equations in favor of emm The equations of equilibrium which are appropriate for the infinitesimal deformation being considered are documented next. Since the assumed form of the deformation implies that the stresses depend 101 only on the radial co-ordinate R, the equations of equilibrium for the solid constituent (2.11), reduce to (6.4.11) where arr and 090 denote the appropriate components of g, and br denotes the component of the interaction body force 2 in the radial direction. The equilibrium equations for the fluid constituent (2.12), reduce to dx x - I _rr JILL—ii _ dR + R + br 0 , (6.4.12) where 'rr and «00 denote the components of 5. The definition of the total stress given by equation (2.13) along with equations (6.4.11) and (6.4.12) yield (6.4.13) which is the equation of equilibrium for the mixture, and Trr and T00 denote the total radial and total tangential stresses, respectively. It is sufficient to satisfy any two of the three equilibrium equations (6.4.11)-(6.4.13). Equations (6.4.2)-(6.4.4), along with the strain tensor (6.4.8) are substituted in equilibrium equations (6.4.11) and (6.4.13),to yield 102 :3 w— -2 p,R + (31- f1+ 55)w’RR + (31- f1+ 35) R - (sl- f1+ sS) 2 R +(32- f2) fi’R - 0, and (6.4.14) 0. - p,R+ (31+ ss)w,RRf(sl+sS)-§g-(s1+ss)w/R2+ s2fi’R' (6.4.15) The indeterminate scalar p may be eliminated to yield the governing equation 0) _'B_ “an" R 95 - o . (6.4 16) R Equation (6.4.16) has a solution given by 6(a) - 61R + c2/R . (6.4.17) where c1 and c2 are integration constants and are to be evaluated from the boundary conditions. Two of the appropriate boundary conditions for evaluating the integration constants are w(Ri) - 0, and (6.4.18) Trr(Ro) - 0 . (6.4.19) 103 For the equilibrium problem considered herein the indeterminate scalar p as discussed in chapter four is given by p p - -p20(f2 + £36m + $2.52 . (6.4.20) The total radial stress equation may be written by using equation (6.4.20) in equation (6.4.6) as T (6.4.21) rr ' ' “16mm 613 + “2313 ' ”2(32+f2)513' where a1 - - [(31+f1)+p20(f2+32)], and a2 - s5 . The integration constants c1 and c2 may be determined using equations (6.4.17) and (6.4.21) and the boundary conditions (6.4.18)-(6.4.19), and are given by - 2 - p2(s 2+f 2)Ro c1 2 , (6.4.22) 2 2 a2(Ro +Ri )-2alRo 2 c2 - -c1R1 . (6.4.23) For the computational work the following material properties were employed [36]. Density of the polymeric - 1590 Kg/m3 ”1 104 matrix in the reference state Density of the fluid in the 32 - 27.49 Kg/m3 reference state True density of the fluid p20 - 997 Kg/m3 in the reference state Poisson's ratio of the matrix u - 0.493 Shear Modulus of the matrix G - 24.166 GPa Swelling coefficient 1 - 3.56le3 of the matrix cm/cm/wt%H20 The following relationships are employed for the calculation of the material constants: _LK. _ a4 1-2v ' 85 G ' G +V 2 11:21 a6 ' 1-2u 7 ' as ' '2 1-2u 7' The computational results are presented in Figures 22 and 23. Figure 22 shows the variation of the non-dimensional radial displacement A g with respect to the non-dimensional reference radial R/Ro 0 w- coordinate for various values of Ri/Ro' It may be pointed out that Ri/Ro represents the ratio of the radius of the rigid core to the outer radius of the polymeric composite cylinder, and is numerically bounded by 0 and 1. When the ratio Ri/Ro is in the neighborhood of 0.0, then it represents a ”thick" polymeric cylinder bonded to a relatively thin fiber-like rigid core. Similarly, when the ratio Ri/Ro is in the neighborhood of 1.0, then it represents a polymeric "membrane" cylinder 105 bonded to a relatively thick rigid core. In the absence of the rigid core the swelling of the unconstrained polymeric cylinder would exhibit a linear relationship between the radial displacement w and the radial coordinate R. It is clearly evident from Figure 22 that the presence of the rigid core induces non-homogeneous swelling characteristics in the immediate neighborhood of the core. Figure 23 presents the variation of the non-dimensional radial “ T _ .1: stress Trr G with the non-dimensional radial coordinate R/Ro for various values of Ri/Ro' The radial stresses are tensile and considerably higher in the immediate neighborhood of the rigid core where the polymeric cylinder is bonded, and zero at the outer radial surface which is assumed to be traction-free. CONCLUDING REMARKS CONCLUDING REMARKS This dissertation resolves the issue of specification of saturation boundary conditions and three-dimensional characterization of a fluid- saturated solid continua within the context of Mixture Theory. In particular, the equilibrium states of non-homogeneously deformed interacting continua have been characterized for cases where not only the boundary of the mixture but also the entire domain of the mixture is in a saturated state. This work generalizes previous work which was restricted to homogeneously deformed cuboids and focused on specification of auxiliary saturation boundary conditions. The equilibrium characterization has been employed to demonstrate that the assumption of the saturation boundary condition in the context of the Theory of Interacting Continua necessarily requires the mixture domain to be saturated for equilibrium problems, where the mixture boundary is continuously in contact with an infinite fluid bath. Thus the assumption of the saturation boundary condition for a broad class of problems in solid-fluid mixtures has been shown to be compatible with the equations of motion and the associated boundary conditions in the context of the Theory of Interacting Continua. Furthermore, the use of the saturation boundary condition in previous work has been interpreted, clarified and validated. Three classical boundary-value problems featuring relatively simple geometries and simple loading characteristics have been presented to study the solidofluid interactions under large deformations. The 106 107 effect of constraints on the swelling characteristics of non-linearly elastic solids under no loading and under simple loading conditions is also investigated. It is anticipated that the results presented herein could be used as the basis for experimental study. The results of such experiments will help validate the applicability of the theory to more complicated geometry and loading conditions. In addition, a linearized version of Mixture Theory is presented to model the interactions between the fiber-reinforced polymeric composites and fluids where the fluid content may not be necessarily small. This approach not only permits the explicit incorporation of moisture and its effects but also allows the possibility of deriving constitutive equations to model the coupled hygro-thermo-elasto-dynamic response of idealized fiber re-inforced composite structures undergoing small deformations and large rotations. This work is the first attempt at addressing the phenomena of non-linear vibration of laminated composite structures by incorporating hygro-thermal environmental effects in an explicit sense and this work is anticipated to be significanlty relevant to applications in defense, aerospace and manufacturing environments, where significant variations in the moisture and temperature conditions may be encountered. The work presented in this thesis represents a quantum leap in pushing the frontier of Mixture Theory for modeling equilibrium problems involving solid-fluid mixtures. 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I 3III1II IIIQI (III II LIILIIIIII]II|2|IIIIIIIES