{Os :2. ' A911. 3 . .nofiPAft u.-.’ . Inl;.l.n...l I .l..isérlwls., 1‘22... . 5D,. ‘ n . .t . a ‘9. :I. . I. 3.. p!” L33” ” «(RY Michigan State L_ University 760354795 UIN IVERSITY LIBRARIES IIIIIIIIIII III II III I III III 3 1293 00629 I This is to certify that the thesis entitled STUDIES IN COMPRESSIBLE AND INCOMPRESSIBLE MIXTURES presented by Devang Jayant Desai has been accepted towards fulfillment of the requirements for A MASTERS Mechanical degree in Engineering CI. \/- (farm-(Lu Major professor Date Feb. 23, 1990 0-7639 MS U i: an Aflirmative Action/Equal Opportunity Institution PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before due due. DATE DUE DATE DUE DATE DUE MSU Is An Affirmative Action/Equal Opportunity lnditution STUDIES IN COMPRESSIBLE AND INCOMPRESSIBLE MIXTURES by Devang Jayant Desai A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Mechanical Engineering 1990 27£)E314;1Q3:L_ / a... ABSTRACT STUDIES IN COMPRESSIBLE AND INCOMPRESSIBLE MIXTURES BY DEVANG JAYANT DESAI The work presented herein is focused on identifying qualitative and/or quantitative differences in the behavior of incompressible mixtures with solid constituents being various types of nonlinearly elastic incompressible solids in the context of the Theory of Interacting Continua. In particular, the response of mixtures with Neo- Hookean or Mooney-Rivlin solids with ideal fluid is studied for several cases of simple deformations. The results of these investigations could have a significant impact on material identification studies.Comparison of these results with the experimental results of Treloar are also presented. Furthermore, attention is also directed towards resolving issues which would permit the solution of boundary value problems involving compressible mixtures. This work also clarifies certain misconceptions pertaining to primitive concepts such as volume additivity and incompressibility of mixtures. In addition, the saturation equations of state for a mixture of a compressible solid and an ideal fluid and an appropriate free energy function for a compressible mixture is derived. DEVANG JAYANT DESAI Finally, a boundary value problem is solved to demonstrate that the presence of constraints such as inserts and inclusions which have differential constitutive characteristics can induce nonhomogeneous swelling characteristics and stress concentrations within the material. DEDIGKTED To My Grand-Uncle Dr. I. P. Desai for his unique vision and philosophy of life and absolute commitment towards understanding the complex social structure in India. And My parents and my wife for continuing patience and confidence in me. -iv- ACKNOWLEDGEMENT I sincerely thank Prof. M. V. Gandhi for giving me an opportunity to work with him and particularly for introducing me towards fundamental and challenging issues in the Theory of Interacting Continua. I am grateful to Prof. Gandhi for his continuous guidance and encouragement over the past three years. Working with him has not only been an intellectual and rewarding experience but also a pleasent one. Special thanks to Prof. B. S. Thompson for providing continuous financial assistance during the period of this research. I am also grateful Prof. P. K. Wong for accepting to be on my thesis committee and for making valuable suggestions and critical comments on my work. Special thanks to my senior colleague Dr. M. V. Usman for his helpful discussions during the completion of this work. -v- TABLE OF CONTENTS LIST OF FIGURES NOMENCLATURE II III IV INTRODUCTION PRELIMINARIES: NOTATIONS AND BASIC EQUATIONS OF THE THEORY OF INTERACTING CONTINUA VOLUME ADDITIVITY, INCOMPRESSIBILITY CONSTRAINT AND CONSTITUTIVE EQUATIONS a. VOLUME ADDITIVITY b. INCOMPRESSIBILITY CONSTRAINT c. CONSTITUTIVE EQUATIONS i. Mixture of an incompressible solid and an ideal fluid ii. Mixture of a compressible solid and an ideal fluid iii. Free energy function for a mixture of a nonlinearly elastic compressible solid and an ideal fluid MATERIAL CHARACTERIZATION USING SIMPLE DEFORMATIONS OF A CUBOID MIXTURE a. Free swelling of the unit cube -vi- viii 12 12 14 14 14 16 19 25 29 b. Uniaxial extension of the unit cube c. Biaxial extension of the unit cube d. Simple shear Numerical example and discussion V INTERACTION OF A CONSTRAINED NONLINEAR ELASTIC SOLIDS AND IDEAL FLUIDS The problem a. Swelling of a composite cylindrical mixture with differential core properties b. Swelling of a composite cylindrical mixture with rigid properties CONCLUDING REMARKS AND DISCUSSION ON APPLICATION OF THIS WORK TO DAMAGE IN COMPOSITE MATERIALS FIGURES BIBLIOGRAPHY -vii- 29 3O 30 32 36 36 36 43 45 47 62 LIST OF FIGURES Variation of stretch ratios for uniaxial extension of a cuboid mixture. Variation of the volume fraction of the solid with the stretch ratio for uniaxial extension of a cuboid mixture. Variation of the stretch ratios for bi-axial extension for a cuboid mixture. Variation of the nondimensional stress with shear angle of a cuboid Neo-Hookean type mixture (Cl/C2 - m ) when subjected to simple shear. Variation of the nondimensional stress with shear angle of a cuboid Mooney-Rivlin type mixture (Cl/C2 - 0.2) when subjected to simple shear. Variation of the nondimensional stress with shear angle of a cuboid Mooney-Rivlin type mixture (Cl/C2 - 0.6) when subjected to simple shear. Schematic of cylinder with differential core properties reference and current configuration. Variation of stretch ratios with the reference radial coordinate R. Variation of radial stress with the reference radial -viii- in A8 49 SO 51 S2 53 54 SS 10. 11. 12. 13. 14. coordinate R. Variation of circumferential stress with reference radial coordinate R. Variation of the volume fraction of the solid with the reference radial coordinate R Percentage volume changes of the swollen cylinder with reference radial coordinate R. Variation of radial and circumferential stretch ratios with the reference radial coordinate for rigid core. Variation of the radial stress with the reference radial coordinate for rigid core. .1x. 56 S7 58 S9 60 61 A A1, A2 A31' A52 A , A e m b. 1 EU c1, 02 ij’ fij F13. fi’ 51 I NOMENCLATURE Helmholtz free energy function of the mixture per unit mass of the mixture. Partial derivatives of Helmholtz free energy function with and I respect to invariants I1 2, 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 generalised constitutive equations for incompressible solids. Components of the rate of the deformation tensor for the solid and fluid, respectively. Components of the deformation gradient tensor. Components of the acceleration vector for the solid and fluid, respectively. Identity tensor. X,Z, x,z Invariants of the Cauchy-Green deformation tensor E. 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. 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-l,2,3 or r,0,z ). 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(Z) Ar, A0 ”1 p ”10' ”20 P1. P2 111' "ij 113 “i e, o 0... “H. 1J 1J oi, «i X Stretch ratio along the length of the cylindrical mixture. Stretch ratio in the thickness direction of the mixture slab. Radial and circumferential stretch ratios. 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-l,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- CHAPTER I INTRODUCTION Several phenomena involving the presence of two or more constituents are difficult or almost impossible to be mathematically modeled by conventional single constituent theories. Typically examples of such phenomena include the swelling of polymers or rubber-like materials, diffusion phenomena in blood vessels, moisture effects in composite materials. Classical diffusion theories model flow of a fluid through a solid assuming negligible deformation of the solid constituent. However, this assumption is violated in solid-fluid interactions where large deformations are involved. For example,it has been shown by Treloar [1] that a block of rubber can swell into several times its original volume when placed in a bath of an organic fluid. Furthermore the amount of swelling is strongly affected by subjecting the block to uniaxial or bi-axial extension, for example. The Theory of Interacting Continua, also known as Theory of Mixtures, has been successfully used to model phenomena involving two or more constituents. In this theory, the continua are assumed to be superimposed. Each spatial point in a 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 the neighborhood of a mathematical point and averaging them. The theory accounts for large deformations, dependence of material properties on all constituents and interaction between the constituents. In proof-of—concept studies it has been clearly shown that the Theory of Mixtures has substantially better predictive capabilities than traditional single constituent modeling concept. For example, results obtained by Gandhi [2] using Theory of Interacting Continua had better agreement with experimental results of Paul and Ebra-Lima [3] in modeling pressure-induced diffusion of organic liquids through a rubber layer than results obtained by using Ficks law ( Classical Diffusion Theory ). Gandhi, et a1.[4] have successfully used Mixture Theory to model behavior of a swollen cylinder under combined extension and torsion and, the results obtained show excellent correlation with the experimental results for global behavior of the swollen cylinder presented by Treloar [5]. They have also been able to predict the distribution of the fluid in the interior of the swollen cylinder and the effect of axial strain on the volume of the swollen cylinder. Currently experimental results pertaining to such detailed field information of these phenomena are not available. The mathematical basis of the general Theory of Interacting Continua has been well established for a long time. The historical development of the theory and comprehensive surveys of the progress in the field are presented in the review articles by Bowen [6], Atkin and Craine [7], Bedford and Drumheller [8] and Passman, et a1.[9]. 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. Shi, et a1. [10] and Rajagopal, et a1. [11] were the first to study equilibrium and steadyostate 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.+ Gandhi, et a1. [12,4,13,14] have not only used the above boundary condition to study boundary values problems, such as torsion of a cylindrical mixture, flexure of a mixture cuboid but have also contributed significantly to a better understanding of the saturation boundary condition by providing a strong mathematical basis to the same which was previously derived on an ad-hoc basis. In previous work reported in the literature, attention has been primarily focused on a mixture of an ideal fluid and a nonlinearly elastic incompressible solid. In particular, for solving the boundary value problems all investigators have assumed the solid constituent to be "Neo-Hookean" which is a nonlinearly elastic incompressible solid. Boundary value problems on mixtures with a nonlinearly elastic compressible solid or compressible mixture have not been addressed in the literature. In this work an ideal fluid is assumed to be one of the constituent for all the mixtures under consideration ,while the other constituent may be an incompressible or a compressible nonlinearly elastic solid . Depending on the type of the solid in the mixture, the mixture will be referred to as an incompressible mixture or a compressible mixture for convenience. The work presented herein is focused on identifying qualitative and/or quantitative differences in the behavior of incompressible + A saturated state represents an equilibrium state in which material elements of a solid-fluid mixture in a deformed and swollen state are in contact with the fluid with no fluid entering or leaving the mixture. mixtures with solid constituents being various types of nonlinearly elastic incompressible solids in the context of the Theory of Interacting Continua. In particular, the response of mixtures with Neo- Hookean solids and ideal fluids and mixtures with Mooney-Rivlin solids and ideal fluid is studied for several cases of simple deformations such as uniaxial extension, bi-axial extension and simple shear. The results of these investigations could have a significant impact on material identification studies, for instance solids exhibiting significant swelling characteristics, in their swollen states can be classified as Neo-Hookean-type or Mooney-Rivlin-type mixtures by performing simple experiments such as allowing solid specimens to swell freely and then, subjecting the specimens to uniaxial extension or shear, for example. Comparison of the results from this work with the experimental results of Treloar [l] is also presented. Furthermore, attention is also directed towards resolving issues which would permit the solution of boundary value problems involving compressible mixtures. The qualitative differences between the response characteristics of compressible and incompressible mixtures can be exploited in several practical applications. This work also clarifies certain misconceptions pertaining to primitive concepts such as volume additivity and incompressibility of mixtures. In addition, the saturation equations of state for a mixture of a compressible solid and an ideal fluid are derived. These equations are instrumental in solving boundary problems where saturation is assumed. A brief rationale for deriving an appropriate free energy function for a compressible mixture is presented herein. Such a function would permit the explicit description of constitutive equations for studying boundary value problems involving compressible mixtures. Finally, a boundary value problem focused on investigating the swelling characteristics of a composite cylinder is studied in detail. The composite cylinder features an inner core which exhibits material properties which are quite distinct from those of the surrounding concentric material. This work is motivated by the need for modeling the large deformations of polymeric composite materials under various under hygrothermal environments. This work partially addresses this need by characterizing the interaction of ideal fluids and idealized constrained nonlinear elastic solids in the context of Mixture Theory [15]. The deformations of the elastic solid may be restricted due to the presence of extensible/inextensible fibers, rigid inclusions and coatings on reinforcing fibers, for example. Such constraints can quantitatively and qualitatively alter the ability of these materials to undergo dimensional and constitutive changes very significantly. The results of these investigations for large deformations demonstrate that the constraints imposed by the core material induce nonhomogeneous swelling characteristics with significant gradients in the stretch ratios and severe stress concentrations at the fiber/matrix interface. These results could have important implications for a variety of fiber- reinforced composites featuring hygroscopic and temperature-sensitive matrix materials. The composite cylinder considered herein is an archetypal representative volume element of a unidirectional fiber-reinforced incompressible elastic composite. And study of such a representative volume element when subjected to complicated loads could provide vital information about the neighborhood of fiber. In chapter II the basic kinematic quantities are defined and basic postulates of the Theory of Interacting Continua are stated. Volume additivity, the incompressibility constraint, and the constitutive equations are stated in chapter III along with the derivation of saturation equations of state for a mixture of compressible solid-ideal fluid. Details of comparative studies between Neo-Hookean type solid-ideal fluid mixtures and Mooney-Rivlin type solid-ideal fluid mixtures are presented in chapter IV. In chapter V the boundary value problem of a cylinder with differential core properties is presented. Finally, concluding remarks and discussion on further application of this work to damage in composite material is presented. CHAPTER II PRELIMINARIES: NOTATIONS AND BASIC EQUATION A brief review of the notations and basic equations of the Theory of Interacting Continua are presented in this section for completeness and continuity. Let 0 and 0t denote the reference configuration and the configuration of the body at time C, 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 0t, respectively. Also 3: denotes 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 S1 and 82. Let g and X denote the reference positions of typical particles of 81 and 82. The motion of the solid and the fluid is represented by x - x1 (g, t), and y - x2 (X, t). (2.1) These motions are assumed to be one-to-one, continuous and invertible. The various kinematical quantities associated with the solid S and the fluid 8 are l 2 Velocity: u - , v - “‘ -— (2.2) Dmg D<2)~ Acceleration: f - Dt , g Dt , (2.3) 2 62 Velocity gradient: L - 32, M - 3E , (2.4) Rate of deformation tensor: 2-%(L+ET).§-% (2.5) where D(l)/Dt denotes differentiation with respect to t, holding g fixed and D(2)/Dt denotes a similar operation holding X fixed. The deformation gradient F associated with the solid is given by u; I Q.) Q) IXLLX (2.6) The total density of the mixture p and the mean velocity of the mixture w are defined by p - p1 + p2. (2.7) pg - p13 + 923. (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. (1) anaerxatign_2f_masa 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 Idet gl - p10, (2.9) and 6p __2 _ at + div (p2 y) 0, (2-10) where p10 is the mass density of the solid in the reference state. (2) O a e m Let g and 5 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 equations for the solid and fluid are given by div 2 - b - plf (2.11) div 2:! + 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 as I ' Z + I» (2.13) the equilibrium equations for the mixture may be written as div T - plf + ng. (2.14) 10 It may be 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. (3) se at'o o a u This condition states that (2.15) IQ + ta I I9 + l=l However, the partial stresses 2 and 5 need not be symmetric. (4) Surface conditions Let E and 3 denote the surface traction vectors taken by S1 and 82, 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 E - g n, and g - ng. (2.16) (5) Thermodynamical considerations The laws of conservation of energy and the entropy production inequality are not explicitly mentioned here for brevity. However, the relevant results are quoted. A complete discussion of these issues is presented in [14]. Let the Helmholtz free energy per unit mass of S1 and 52 be denoted by Aland A2, respectively. The Helmholtz free energy per unit mass of the mixture is defined by pA - plAl + p2A2. (2.17) Note that by setting 9- - grad ¢1+§-grad 432 +E, (2.18) IQ I E?‘ P! + Ifl I3! I N9- IH + Ifl where, ¢1 - p1(A1'A)r ¢2 - ll 92(A2-A), ¢1 + ¢2 - 09 equations (2.11), (2.12), (2.14) and (2.15) become divE - E - plf, divE-I-E- p25, div I - plf + ng, §+Z-?+2T The terms in g, I and b which contribute to the equations of motion depend on ¢l and ¢2 do not or the total stress. (2. (2. (2. (2. (2. (2. (2. 19) 20) 21) 22) 23) 24) 25) CHAPTER III VOLUME ADDITIVITY; INCOMPRESSIBILITY CONSTRAINT AND CONSTITUTIVE muons a) Volume additivity Volume additivity is an intrinsic property of mixtures which simply says, the total volume of the mixture at any state ( time or deformation ) is equal to the sum of the respective volumes of the superimposed continua at that state. In the present notation it can be stated as V - V + V (3.1) where VS and VS are the current volume of the respective constituents l 2 and Vm is the current volume of the mixture. (3.1) can also be written as l - v1 + u2 . (3.2) where ul, "2 are volume fractions of the respective constituents. Prior to this work volume additivity was misinterpreted as a constraint and was defined as volume of the mixture is the sum of the volumes of the respective continua in their reference state, which is true only if both the constituents of the mixture are incompressible. When two continua are superimposed to form a mixture, assuming no interconversion of mass, 12 l3 and both the constituents being chemically inert to each other the sum of the volumes of the constituents at any state (time or deformation) is equal to the total volume of the mixture at that state even though the individual constituents might undergo volume changes as in a case of compressible constituents, for example. Let the mixture under consideration constitute of a nonlinearly elastic solid 81’ and an ideal fluid 82. Consider V RS1 RS 3 4| p MS _l__v_l (3.3) m 910 s 1 m where VRS is the reference volume of the solid and V8 is the current 1 1 volume of the solid. Similarly v v RS 3 p_2.__z__2_ 34 p v v ”2 ( ' ) 10 m m where VRS is the reference volume of the fluid and VS is the current 2 2 volume of the fluid. As the fluid is ideal in nature VRS - V2 . 2 Substituting (3.3)-(3.4) in (3.2) results into V3 p p 5—1 -l + -3- - 1 (3.5) RS p10 p20 This relationship is valid for a mixture of any solid and an ideal fluid. 14 b) Ias2maressihili£1_ssns£raint Now if the nonlinearly elastic solid is incompressible that is Vsl- VRSIthen (3.5) yields 1+2.” (3.6) ”10 ”20 which can be considered as an incompressibility constraint relationship. This relationship was referred to as volume additivity constraint prior to this work [16]. C) QQHfiIIIHIIEE_EQHAIIQE§ i) Mixture of incompressible solid and an ideal fluid A mixture of an elastic solid and a fluid is considered. The solid is assumed to be nonlinearly elastic and incompressible, the fluid is assumed to be ideal. Thus all the constitutive functions are required to depend on the following variables: E, VE, p2, grad p2, T, grad T, g and y, where T denotes the common absolute temperature of the solid and the fluid. A lengthy but standard argument, based on the balance of energy, entropy production inequality, restrictions due to material frame indifference and the assumption that the solid is isotropic in its reference state, leads to the following results [10]. . 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 is given by A - A (11, I I 2. 3. p2. T). (3.7) 15 where 11’ 12, I3 are the principal invariants of B - EFT defined through 11 - tr g (3.8) 12 - % [(tr §)2 - tr g2], (3.9) 13 - det g - (det §)2. (3.10) Using (2.9), (2.26) and (3.4), I3 can be expressed in terms of p2 by the relation 1 2 -1 13 / - det g - (1 - p2/p20) . (3.11) Furthermore, on restricting attention to isothermal conditions equation (3.1) reduces to A - A (11’ I (3.12) 2: P2) The components of the partial stresses in the solid and fluid, and the interaction body force for isothermal conditions are given by E’Ii ' ‘ p :10 ski + 2” [(3%1 giz I1] Bki ' gIz Bkm Bmi]’ (3'13) ;ki ' ”Fig ski ’ ””2 ggzski' (3'14) Bk ‘ ‘ ZE— gfll” + ”1 ap2 a ‘ ”2 [[fi %I1]512 10 “k :k - 3%; 312] 312 k + a Si; Si; (uk - vk). (3.15) 16 In equations (3.7) - (3.9), p is a scalar which arises due to the volume additivity and incompressibility constraint. The constitutive parameter a accounts 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. It is also useful to record the representation for the total stress - ‘ ‘ _ - , 25. Q5. 2A. Tki ”k1 + “k1 pski ””2 ap2 5k1+2” [(61 + 312 II)Bki _ 2A. 312 BkmBmi]' (3.16) In the remainder of this thesis, only Q, and i and b, will be used. Hence, for notational convenience, the superposed bars are dropped. ii) ixture of om ressible d and a dea u d Green and Steel [17], Crochet and Naghdi [18] have presented the constitutive equations for a mixture of a nonlinearly elastic solid and a viscous fluid couple of decades ago and are not restated here for brevity. Rigorous verification of the constitutive equations given by the above researchers is not addressed here but at equilibrium of the solid and the fluid ( at saturation ) the constitutive equations derived by the above mentioned authors is in agreement with the constitutive equations obtained below. Consider a block of pure solid compressible material which a unit cube in its unstrained unswollen state. The unit cube is placed in a bath of an ideal fluid and is subjected to a triaxial extension due to tractions on its surfaces and the absorption of the fluid. Eventually, -. .01 m4“. . A I.- 1“” 17 the block, which is now a mixture of the solid and the liquid, reaches an equilibrium strained swollen state while in contact with the bath. This is what is meant by a saturation state. The mixture in the block has mass Mm and dimensions A1,A2,A3 in the direction of triaxial extension. In triaxial extension the triaxial directions are the principal directions. Hence, the derivation which follows would yield equations of state for principal directions only. Later, the equations of state will be tensorially transformed to general directions. The condition for the equilibrium of the block with the surrounding liquid and thus of saturation is 6 (MmA) - 6V (3.17) where, A - A(I1,12,I3,p2) is the Helmholtz free energy function per unit mass of the mixture. Equation (3.17) can be written as M 5A + A 6M - 5w -(3.18) m m where _ fi :14. 53A. QA— Also, for triaxial extension the invarients of Cauchy-Green tensor, 2 2 2 I1 - A1 + A2 + A3 (3.20) 2 2 2 2 2 2 12 - A122 + A223 + A3A1 (3.21) 2 2 2 I3 - A1A2A3 (3.22) The mass balance equation for the solid yields “math“??? 18 p (1 - III—T <3-23> 1 2 3 The variations of 11’ 12 and I3 from equations (3.20)-(3.23) are given by 611 - 2A16A1 + 2A26A2 + 2A36A3 (3.24) 612 - 2(Ag + Ag) 21521 + 2(Ai + Ag) A26A2 + 2(13 + if) A36A3 (3.25) 513 - lexgxgsxl + ZAZAiA§6A2 + 2A3A§Ai6x3 (3.26) The variation of p1 from equation (3.23) __;l__ 5p1 - p10 AiAzAZ (A A 6A + A A 6A + A A 6A (3.27) ) 2 3 2 3 1 1 3 2 l 2 3 The mass of the mixture is related to the density of the mixture by Mm ' ”(‘1‘2*3) - p1(A1A2A3) + p2(A1A2A3) (3.28) Hence, using (3.27) 6Mm - 6p (A1A2A3) + p (A2A36A1 + A1A36A2 + A1A25A3) (3.29) The tractions on the surfaces of the block are considered to be the total tractions. The expression for the virtual work done is given by SW - T11A2A36A1 + T22A1A36A2 + T33A1A26A3 (3.30) 19 Equation (3.19), (3.24)-(3.30) when substituted in equation (3.18) results [p(A1A2A3](A16II + A2612 + A3613 + Ap26p2) + A(A1A2A36p2 + 62(A2A36A1 + A1A36A2 + A1A28A3)) - T11A2A38A1 + T22A1A36A2 + T33A1A26A3 as _6A _26 611' A2 612 and' A3 a13 I On further simplification the resulting equation must be satisfied for I where A1 - all arbitrary variations of A1, A2, A3, and p2 the following saturation equations are obtained I 2 2 2 2 2 2 2 I11 - A,p2 + 2p [AIA1 + A2A1(A2 + A3) + A3A1A2A3] (3.31) 2 2 2 2 2 2 2 T22 - A,p2 + 2p [AIA2 + A2A2(A1 + A3) + A3A1A2A3] (3.32) 2 2 2 2 2 2 2 T33 - A,p2 + 2p [Alx3 + A2A3(A2 + A1) + A3A1AZA3] (3.33) and p(A1A2A3) A,p26p2 + A(A1A2A3)6p2 - 0 or pA,p2 + A - 0 (3.34) QA where A, - . 22 3p2 It is essential to note here that now four equations are available instead of three, as obtained incase of the mixture constituting only incompressible constituents [2]. iii) ee n un o tu o a o nea ela tic c m es b 'd d The free energy function for the mixture of a solid and an ideal fluid is obtained by adding the change in free energy due to mixing of 20 the solid and the fluid and the elastic free energy of the solid. The basis for adding up the two quantities is not clear but does seem reasonable. The elastic free energy function, that of the solid can be written as A5 - As(Il,Iz,I3) (3.35) where 11, 12, I3 are these invarients of the Cauchy Green tensor. The free energy function reduces to zero when the solid is stress free. The free energy of mixing Am of the solid and the fluid is complicated as compared to that of the solid and is obtained by considering the change in Gibbs free energy when ‘nl' moles of fluid are mixed with the solid as given by Flory-Huggins [l9] equations stated below. 3359 - RT {1n (1 - u ) + v + xuz} (3 36) an1 l 1 l ' where R - universal gas constant T - temperature AGm - change in the Gibbs free energy x - interaction parameter that depends on the combination of the specific solid and the fluid. If V1 is the molar volume of the fluid in the mixture, n1 number of moles present at any state and V volume of the solid at that state, S then volume additivity can be written as 21 vm - vS + n1V1 (3.37) From (3.4) p n V -2--‘1,—1 (3.38) ”20 m Using (2.9), (3.38) can be stated as p I 8V -2—3— Rs (3.39) n 1 v1”20 Differentiating (3.39) with respect to p2 yields an I 8 p 61 a—l-f— LEI—3‘1 <3”) ”2 1”20 3 ”2 Since the fluid is ideal and hence incompressible (3.5) can be written as p + ‘2‘ (3.41) 1 - u 1 ”20 Differentiating (3.41) with respect to p2 we have au 5-1 - il- (3.42) ”2 ”20 -—l -—2 (3.43) 22 Using (3.41) and (3.42), (3.43) yields BAG p El A '31 [111(1 11 ) + V + 2] ‘1 Z 3 _ - X” (I ) 1 - (3-44) aul V1 1 l 1 3 213p20 aul Considering mass balance (2.9) and rewriting yields, I3 - S (3.45) Differentiating with respect to ”1’ substituting in (3.44) yields ACm _ - KI 1n(l-v ) + l + xv Z§ ' (l'V )EX§ (3°46) v1 V1 6”1 Integrating with respect to v we get 1’ (1-u1) BI 8 By 213 3 _ -RI 2 5 AGm V1J-(1n(l-v1) + v1 + xul) [I3 + 1 ] avl + C (3.47) Where C is the constant of integration which is evaluated considering no change in Gibbs free energy in absence of solid constituent results in C - O. The thermodynamic system includes the mixture and the infinite bath surrounding of the fluid. Thus the total volume change of the thermodynamic system can be assumed to be zero and hence AG - AA 23 The free energy of mixing per unit mass of the mixture becomes, u 1n(1-u ) V A-AA-——']‘—RT __1+1+xv1 J- m V p V M v u l s 1 1 6V (1 ”11——§l 60 (3.48) avl 1 The total free energy per unit mass of the mixture is thus given as follows A-A +A , (3.49) s m where, As and Am are defined per unit mass of the mixture Now consider a mixture of an incompressible nonlinearly elastic solid such as a Neo-Hookean solid and an ideal fluid, the elastic free energy per unit mass is given by v p RT Ae - 1 10 (11- 3) (3.50) 2p The free energy of mixing is obtained by letting the volume of the solid VS - constant, 1. e. VS - VRS , equation (3.48) becomes Am - ”1 RT (1'"1)1“(1'”1) + (1 - v1)x (3.51) p V1 ”1 It should be noted that the total free energy per unit mass of the mixture obtained by substituting (3.50) and (3.51) in to (3.49) is the same as derived and employed by Gandhi et a1 [4] for an incompressible solid-fluid mixture. ‘2’. 7"! 24 It is clear that the free energy function in case of a mixture of a compressible nonlinearly elastic solid and an ideal fluid can only be obtained if the variation of the volume of the solid is known when a known amount of fluid has entered the mixture. Hence specific boundary value problem with compressible constituents cannot be solved unless experimental results are available. te t m x The free energy function of mixing can be obtained in the closed form if and only if the variation in volume of the mixture is known when a known quantity of fluid enters the mixture. In other words the change in the volume of the solid with change in the fluid content of the system. This relationship needs to be experimentally determined. Currently such experimental results are not available. ‘" m.... l'hI‘ ‘ l‘flr CHAPTER IV MATERIAL CHARACTERIZATION USING SIMPLE DEFORMATIONS 0F.A CUBOID MIXTURE The problem of swelling of an incompressible solid cube under simple deformations is reported in this chapter. A unit cube of nonlinearly elastic material is placed in an infinite bath of an ideal fluid and then is allowed to swell freely untill it attains saturation. The swollen cuboid is then subjected to uniaxial extension, equibiaxial extension and shear. The results obtained have a significant impact on material characterization or identification of incompressible solids. Let (X1, X2, X3) denote the position of a typical particle in reference configuration. The particle denoted by (X1, X2, X3) in the reference configuration may be represented in deformed configuration by the coordinates (x1, x2, x3). The problem is formulated in a generalized form and then reduced to the specific case of uniaxial, biaxial and shear deformation as follows, The general form of the deformation field is assumed to be x1 - A1 (X1 + CXZ) x2 ' A2X2 x3 - A3X3 (4-1) The deformation gradient F associated with the mixture is given by 25 26 The Cauchy-Green strain tensor B which is defined as takes the following form for the above deformation: Ai(l+C2) CA A o 21 2 g - cA A A o (4.4) 0 0 A 1 2 2 The principal invariants of B are given as 2 2 2 2 I1 - A1 (1 + C ) + A2 + A3 (4.5) 2 2 2 2 2 2 12 - AlA2 (l + C ) + A2A2 (1 + C ) (4.6) 2 2 2 I3 - A1A2A3 (4.7) The balance of mass equation for the solid constituent (2.9) may be expressed in terms of the stretch ratios as _ yl (4.8) where ”1 represents the volume fraction of the solid. The equilibrium equations are expressed in terms of the coordinates in the reference configuration for computational convenience. Assuming no external body 27 forces, the equations of equilibrium for the mixture and solid take the form 80 . _L1 -1_ 6x ij 0 (4.9) an -—11' F ‘1 - 0 (4.10) 6x pj 61 ——l1 F '1 - 0 (4.11) 6 XP The tensor B-1 that appears in these equations has the form given by 1/A1 -C/A2 0 -1 g - 0 1/A2 o (4.12) 0 0 1/A For the deformation field under consideration, (3.13)-(3.15) become, a -- 1+2 (A +AI)A2(1+C2) 11 P p10 ” 1 2 1 1 4 2 2 2 2 2 - A2[A1 (1 + c ) +c AlAzl] (4.13) _ - £1_ 2 _ 2 2 2 4 022 p p10 + 2p [ (A1 + A211)A2 A2 (C A1A2 + A2 )] (4.14) a - - ”1 + 2 (A + A 1 )A2 - A A“ (4 15) 33 P p ” 1 2 1 3 2 3 ' 10 3 3 3 012 - 2p [ (A1+A211)CA1A2 - A2[CA1A2(1+C ) + CA1A2]] (4.16) 28 013 - 023 - 0, (4.17) _ _ __£L, 26 “11 ”22 “33 P p20 ””2 ap2 (“'18) bk - 0 k - 1,2,3 (4.19) where, A - QA‘ and A - EA- 1 all 2 612 It is sufficient to satisfy any two of the three equilibrium equations (4.9)-(4.11) for the deformation field under consideration, the third equilibrium equation is automatically satisfied. By virtue of equation (3.16) total stresses are given as follows: _ 52L 2 2 T11 -p - pp2 apz + 2p [ (A1 + A211)A1(1 + C ) 4 2 2 2 2 2 - A2(A1(1 + C ) + C A1A2)] (4.20) T - - p - p QA_ + 2p [ (A + A 12)A§ 22 p2 6p2 1 2 2 2 2 4 - A2(C 2122 + A2)] (4.21) _ _ _ QA 3 _ 4 T33 p ppzap + 2p[(A1 + A211)A3 A2A3 ] (4.22) 3 2 3 T12 - 2p [ (A1 + A211)CA1A2 - A2(CA1A2(1 + C ) + CA1A2)] (4.23) T13 - T23 - O (4.24) 29 In the subsequent part of this section special cases of free swelling, uniaxial extension, biaxial extension and shear are considered. a. Free Swe n of U ube In the case of free swelling the cube of unit length is allowed to swell freely untill it achieves saturation and, at saturation the total stress on its boundary vanishes, hence c - 0, (4.25) Tij - 0, on boundary of swollen cuboid yields A1 - A2 - A3 - A (4.26) p - -pp2 34:: + 2p [(A1 + A211)A2- AzAl‘] (4.27) and T -T -T -o. (4.28) b. Uniaxial Extension The swollen cube is stretched in x1 direction which implies that the stretches in the other two directions are equal and hence, A - A - A, (4.29) C - 0, (4.30) and total stresses are given by _ _ _ QA_ 2 _ 4 T11 p pp2 892 + 2p [ (A1 + A2I1)A1 A2A1 ] (4.31) _ _ 5A. 2 p pp2 3P2 + 2p [ (A1 + A211)A and c. Egnibiaxinl Extension The case of equibiaxial extension is given by A - A C - 0 and total stresses are given by _ _ - _ EA. 2 T11 T22 P ””2 6,22 + 2” [ (A1 + ”211” 33" QA -A’ 30 2 T - T - T 12 d.§i_male_§hear In the case of simple shear the unit cube is allowed to swell 2 - A A 2 - A AA ] 34] and, - A A 2 “I (4. (4. (4. (4. (4. (4. (4. 32) 33) 34) 35) 36) 37) 38) freely untill it achieves saturation and then is sheared by an angle 1, i. e. shear is superimposed on a freely swollen cuboid. The stretch ratios ratios take form as follows, A - A - A - A, (4. 39) 31 C - tan(1) (4.40) and total stresses given by _ as. 2 2 I11 -p -pp2 apz + 2p [ (A1 + A211) A (l +tan 1 ) - A2A“ (1 + 3tan21 + tanay )] (4.41) _ _ _ §A_ 2 _ 4 2 I22 p pp2 3P2 + 2p [ (A1 + A211)A A2A (l +tan 1)] (4.42) 33 _ _ QA_ 2 _ 4 p pp2 692 2p [(A1 + A211)A A2A ] (4.43) 2 2 4 12 - 2p [ tan7(A1 + A211)A - A2(tan7(l + tany )A +tan1Aa)] (4.44) and T13 - T23 - 0. (4.45) Cases (a-c) cannot be solved for stretch ratios due to the presence of indeterminate scalar p. To eliminate the indeterminate scalar p saturation boundary condition is assumed where p becomes determinate [2]. The equations of state at saturation for a mixture of an incompressible solid and an ideal fluid were obtained by Gandhi et al are stated here for completeness 32 A + pA le- [P20 p2(P20' P2)] 611+ 2p [(A1+ A2I1)Bij- AZBimij] (4.46) Comparing (4.46) with (4.20-4.22) , it is observed that at saturation the indeterminate scalar p becomes determinate and is given by p - ° ppzoApz’ p20A (4-47) Now assuming that saturation is attained by the swollen cuboid when subjected to uniaxial extension, and equibiaxial extension. And then substituting equation (4.47) for p in equations (4.27), (4.32) and, (4.37) and the variation of stretch ratios, stresses and volume fractions is obtained. The numerical example and discussion on these results is presented next. NUMERICAL EXAMPLE AND DISCUSSION The explicit forms of equations (4.7), (4.8), (4.13) and (4.14) may be obtained for a specific choice of the Helmholtz free energy function A. The free energy function per unit mass of the mixture is assumed to be given [6] by l-V 11 El _1 A - _1 C1 (I1 - 3) +C2(I2 - 3) + V [V £n(l-V1) p l 1 + x(1-v1)]], (4.48) where, ‘uvn -d. 33 V1 is the molar volume of the fluid, x is the constant which depends on the particular combination of the solid and the fluid, R is the universal gas constant, T is the absolute temperature, MO is the molecular weight of the solid. The free energy function given by equation (4.48) is for a "Mooney-Rivlin type" nonlinear solid fluid mixture. When C - 0 the 2 free energy function represents a "Neo-Hookean type" nonlinear solid mixture. For numerical calculations the following material properties as given by Treloar [l] were used: Density of natural rubber in the reference state p10 - .9016 gm/cc Density of solvent (benzene) in the reference state p20 - .862 gm/cc Molar volume of the solvent V1 - 106 cc/mole The molecular weight of rubber between cross links Mc - 9151 gm/mole Natural rubber-benzene interaction constant x - .425 The numerical value of the universal gas constant R is given by 8.317 x 107 Dyne-cm/mole - °K, and the absolute temperature T was assumed to be 303.16’K. The constitutive co-efficients C1 and C2 are given as follows: 0 - -l-9 (4.49) C2 - 0C1 (4.50) 34 Numerical results presented in figures (1-6) for a cuboid mixture under simple deformations are for both Mooney-Rivlin and Neo- Hookean type mixture. The variation of the stretch ratios for uniaxial extension of a cuboid mixture is presented in figure 1. Corresponding experimental results obtained by Treloar [l] are also presented on the same graph for reference. The experimental results are in excellent agreement for tension and not quantitatively in agreement for the compression case. It I is observed that qualitatively there is no difference between Neo- Hookean type and Mooney-Rivlin type mixtures. Figure 1 clearly shows that Neo-Hookean mixtures admit more states as compared to Mooney-Rivlin i type mixtures. The variation of the volume fraction of the solid in the mixture with the stretch ratio is presented in figure 2. Figure 2 conveys that Neo-Hookean solids absorb more fluid to achieve saturation as compared to Mooney-Rivlin solids. These results have a substantial bearing on material characterization or material identification and, can be exploited as follows. Consider a block of rubber of unknown properties and known dimensions, place it in a fluid bath for a sufficient length of time so that the block is fully saturated. Then stretch the block in x1 direction by some amount. Measure the new dimensions of the swollen block and obtain the stretch ratios in the other two directions. Place the value of the proper stretch ratios in figure 1 and relate its position to the existing variation for Neo- Hookean and various Mooney-Rivlin mixtures and hence the material can be identified. Also figure 2 can be used for the same purpose if the amount of fluid required for complete saturation is measured. A parametric study can be done to incorporate the values of x which depends on the particular combination of the solid and the fluid. The variation of x for all known combinations is very small. 35 Similar results are obtained for Equi-Biaxial extension and the variation of the stretch ratios is presented in figure 3. The variation of the stresses for the Neo-Hookean and various Mooney-Rivlin mixtures subjected to simple shear with shear angle is presented in figures 4, 5 and 6. These results clearly satisfy Universal relationship for simple shear. Linear behavior of stresses in Neo-Hookean mixtures is observed as anticipated whereas nonlinear nature of variation of stresses is seen in case of Mooney-Rivlin mixtures. I CHAPTER V INTERACTION OF CONSTRAINED NONLINEAR ELASTIC SOLIDS AND IDEAL FLUIDS W The behavior of a constrained nonlinear elastic solid in the presence of an ideal fluid is investigated in this section. The constraints under consideration are those which result due to the presence of two or more elastic materials, rigid inclusions or i inextensible/extensible fibers, for example. The representative problem is presented in order to demonstrate the role of constraints in modifying the swelling characteristics of reinforced elastic solids. a. we 1 of a Com osite C nd c i tu e wit dif erential core W. Consider a solid cylinder composed of two different materials M1 and M2, with the material M1 occupying the region R e [O’Ri] and the material M2 occupying the region R e [Ri’Ro] in the reference configuration, such that, Ro > R1 > 0. Both cylinders are assumed to have a length LO. A schematic of the reference configuration and the current configuration is presented in figure 7. 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,6,Z). In the deformed swollen state the co-ordinates of the same particle are assumed to be described by 36 37 r - r(R), 0 - 9, and z - A2, (5.1) where A is a constant axial stretch ratio assumed to be unity. The deformation gradient associated with the mixture is, Ar 0 0 F - 0 A0 0 (5.2) 0 0 A The Cauchy Green tensor B which is defined as a - FFT (5.3) takes the following form for the above deformation: A2 0 o r 2 g - 0 A9 0 (5.4) 0 0 A2 where Ar - dr/dR and A0 - r/R denote the stretch ratios in the r and 0 directions. The principal invariants of B are then given as 2 2 2 I1 - Ar + A0 + A , (5.5) 2 2 2 2 2 I2 - A (A1. + A0) + AoAr and , (5.6) 2 2 2 I3 - AerA . (5.7) The balance of mass equation for the solid constituent (2.9) may be expressed in terms of the stretch ratios or the volume fraction of the solid "1 as p 4 Ti? - “-8) ”10 r o 38 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 Einr + or; ‘ ”00 - br - 0 (5.9) dr r where arr and 000 denote appropriate components of a , and br denotes the component of the interaction body force b in the radial direction. The equilibrium equations for the fluid constituent, namely (2.12), reduce to 32;; + :rr_;_:22 + br - 0 (5.10) dr r where “rr and «90 denote the components of n. Equations (5.9), (5.10) and (2.14) yield 33;; + Err_1_322 - 0 (5.11) dr r which is the equation of equilibrium for the mixture. For the deformation under consideration, it follows from (5.3) and equations (3.13) - (3.15) that the non-zero components of the partial stress tensors for the solid and fluid constituents are given by _ _ ”1 2 . 4 art P p10+ 2p[(A1 + A211)Ar A2Ar ] , (5.12) a - -P 31 + 2p (A + A I )A2 - A A 4 (5 13) 00 p10 1 2 l 0 2 0 ’ ' 39 _ _ £1 2 _ 4 azz P p10+ 2p[(A1 + A211)A A2A ] and, (5.14) £2 "rr - n00 - fizz - - P p20 - ppZAp2° (5.15) The only non-zero component of interaction body force is given by E dp] dpz d (A: + A?) _ br ' ' p A dr + ”1°p dr ”2 [(A1 + A211) A a: ' 3 10 r 2 r ; dA A3 dA L 2 __r .2. __2 2A2(Ar dr + 2: dr ) ] (5.16) -294 -24 _26 where, A1 611 , A2 812 and, Ap2 892 . It is sufficient to satisfy any two of the three equilibrium equations (5.9) - (5.11). Equations (5.12) - (5.15) are substituted into the equilibrium equations for the solid and the mixture (5.9) and (5.11), respectively, to get the following functional functional form of the equilibrium equations, which are stated in terms of the co-ordinates in the reference configuration for computational convenience: p 92 .1 dR p10 + g1 (A1, A2, Apz, A The mixture is assumed to be of a ”New-Hookean-type", that is, A is a R, Aé, A5, A) - O (5.17) r, A0! linear function of 11' Following this assumption the explicit forms of the equilibrium equations for the mixture and the solid for the deformation under consideration are given by 40 a -‘J— 2-2 - ‘oR A0 Ar 0 (5.18) and A 92 ...r - .r. _ _ .1. 2, 2 ' ”1 an + 2” AlAr [:2 AoR IA *0] A R [*9 AJ] A -A dA 1. .r__2 1....1 g_ 2_ 2 _ + A: } + p2A1 dR {Ar A0} 0, respectively (5.19) _ 2A. ”1 ”1”20 ap2 {A9 R dR In equations (5.18) and (5.19) the radial and the tangential stretch ratio Ar and A9, respectively, are related through the compatibility condition given by dfl# dR . (5.20) Equations (5.18) and (5.19) may be solved for p, Ar and A9 once the specific form of the Helmholtz free energy function for the mixture is known, and the appropriate boundary conditions are specified. The Helmholtz free energy function per unit mass of the mixture is assumed to be given by RTp l-u :1__19 81—1 A - p 2Mc (I1 - 3) + V1 V1 £n(1-v1) + x(l-u1)]] (5.21) where, R, T, V1, Mc’ and x are constants [l]. The appropriate boundary conditions for solving the set of ordinary differential equations (5.18) and (5.19) in regions [0,Ro] are given by 41 Ar(0) - Ao(0) , (5.22) (1) (2) Trr (R1) - rrr (R1) , (5.23) A0(1)(Ri) - A0(2)(R1) , and (5.24) Trr(Ro) - o . (5.25) The boundary condition given by equation (5.22) arises due to the compatibility requirement between the radial and tangential stretch ratios at the axis of the cylinder. The boundary condition (5.23) on radial stress tensor at R - R1 is due to the requirement that the radial stress be continuous at the interface. The boundary condition (5.24) arises due to the assumption of perfect bonding at the interface. The boundary condition on total traction vector represented by (5.25) is a consequence of the requirement that the outer surface of the composite cylinder be traction-free. Since a boundary condition for the partial traction vectors is not physically obvious, following the arguments presented in [10, 11, 12] 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 , (5.26) where, Srr represents the radial stress component for a saturated state, and is given by [15] _ 2L 2 Srr p (p2o - p2) 3P2 + p20A + 2 p AlAr . (5.27) For computational convenience, equations (5.18) and (5.19) may be combined to eliminate p, and for the Helmholtz free energy function given by (5.21) the resulting equation is given by (Ar-A0)[x[2x--}—]u1-AA] I.» L? 9 L9 where, (5.28) The set of ordinary differential equations given by (5.20) and (5.28) subjected to boundary conditions given by (5.22) - (5.25) were solved numerically. For the computational work the following properties were employed [1,6a]: Density of rubber in the reference state Density of solvent in the reference state Molar volume of the solvent The molecular weight of rubber between cross-links Rubber-solvent interaction constant M M l 2 p10 - .9016 .9016 mg/cc p20 - .862 .862 gm/cc v1 - 106.0 106.0 cc/mole Mc - 8891.0 4000 gm/mole x - .400 .400 The computational results are presented in Figures 8, 9, 10, 11 and 12 where it is clearly evident that the presence of two different elastic materials results in nonhomogeneous swelling characteristics, nonlinear distribution of the radial stress, discontinuity in the circumferential stress and the volume fraction of the solid and finally 43 a nonlinear volume change in the outer circumferential cylinder due to swelling. Figure 8 shows that the composite cylinder swells nonhomogeneously with the radial and tangential stretch ratios Ar and A0, varying nonlinearly with R in the interval [R1,RO]. In addition, there is a discontinuity in the radial stretch ratio Ar at the interface, where the material characteristics change abruptly. This is in sharp contrast to a cylinder composed of a single material, which would swell homogeneously with A1. and A0 equal and constant throughout the domain R e [0,R0]. Figure 9 shows the nonlinear distribution of the radial stress in the region R e [Ri'Rol’ which is induced by the presence of the material M1 occupying the region R e [O’Ri]° The nonlinear variation of the circumferential stress in the region R e [Ri’Ro] with the radial coordinate appears in figure 10. A discontinuity of the circumferential stress at the interface of the two different properties is observed. Figure 11 shows a discontinous but almost constant volume fraction of the solid along the radial coordinate. In figure 12 the percentage volume change of the cylinder with the radial coordinate is presented. This variation is particularly useful for experimentalist. b. well n o No near a t c de with a R 1 Core A special case of the problem considered in section (5a) is when the inner material M1 is assumed to be rigid. The governing equations for this problem are again given by equations (5.18) and (5.19). The appropriate boundary conditions are given by A0(R1) - l, and Srr(Ro) - 0 . 44 The computational results are presented in Figures 13 and 14. It is clearly evident from these results that the presence of the rigid core induces nonhomogeneous 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. A complete detail of this special case appears in [20]. CONCLUDING REMARKS AND DISCUSSION ON APPLICATIONS OF THIS WORK.TO DAMAGE IN COMPOSITE.MATERIALS The principal contributions of the work presented are a. Qualitative and/or quantitative differences in the behavior of incompressible mixtures with solid constituents being various types of nonlinearly elastic incompressible solids were identified. In particular, the response of the mixtures with Neo-Hookean solids and ideal fluids and mixtures with Mooney-Rivlin solids and ideal fluid is studied and compared with experimental results by Treloar for several cases of simple deformations such as uniaxial extension, bi- axial extension and simple shear. The results of these investigations could have a significant impact on material identification studies. b. This work presents sufficient basis to motivate experimentalist to conduct certain experiments with compressible solids to yield change in volume of the solid when a known amount of fluid enters in the solid. These results would provide the missing link to solve boundary value problems involving mixtures with compressible constituents. c. Certain misconceptions pertaining primitive concepts like volume additivity and incompressibility which apparently existed in previous literature were clarified. d. Finally, a boundary value problem focused on investigating the swelling characteristics of a composite cylinder is presented. The composite cylinder features an inner core which exhibits material properties which are quite distinct from those of the surrounding concentric material. This work is motivated by the need for 45 46 modeling the large deformations of polymeric composite materials under various under hygrothermal environments. This work partially addresses this need by characterizing the interaction of ideal fluids and idealized constrained nonlinear elastic solids in the context of Mixture Theory. The deformations of the elastic solid may be restricted due to the presence of extensible/inextensible fibers, rigid inclusions and coatings on reinforcing fibers, for example. Such constraints can quantitatively and qualitatively alter the ability of these materials to undergo dimensional and constitutive changes very significantly. The results of these investigations for large deformations demonstrate that the constraints imposed by the core material induce nonhomogeneous swelling characteristics with significant gradients in the stretch ratios and severe stress concentrations at the fiber/matrix interface. These results could have important implications for a variety of fiber— reinforced composites featuring hygroscopic and temperature-sensitive matrix materials. This problem also has significant implications in ability to predict local (in the neighborhood of the fiber/matrix interface in composites, for example) events occuring due to global (on the whole composite structure) effects. This is very critical in composites, particularly in damage studies for instance, from the boundary value problem it can be speculated that large gradients of stretch ratios and stresses in the neighborhoods of the fiber matrix interface due to smaller global effects occur. Damage in composite materials seemingly macroscopic phenomena is truely a microscopic phenomena and hence representative volume element considered in the above presented boundary value problem should be studied in damaged and undamaged configuration in detail and a continuum relationship should be developed to predict macroscopic response. This is taken up by the author as a part of Doctoral studies. 47 FIGURES 48 .8355. Bongo 6 Co c2236 _o_xo_c: to. mozot :mem Co comato> ”70¢ 7A 0E». IoEEm ooN cod cod ooé oofi. oo.N 00. F cod _ — - If - — n L _ b IP _ b 00.0 loo.— 4 [cod ...... of. . ....... r. I ..... I .. loos ... .. $me mo I. M so I No\ 5 I I 8 all: N0\ F0 .. ...... _ 00...... 3r OLLVH Her-Isis 49 .o..3x.E Bongo 0 Co coacouxo 3.6.5. .2 058 nouobm 05 5.; 28 05 Lo c0500.... 2:29 05 Lo co.uo...o> .N. or. 2 0.2.. IoEEm an: cod 006 00% 006.. I cod 00. P 00.0 mo No>o ........ mo ~o>o I 00 I. I8>o ..... i 00.0 I 0N6 oo. — 34 anos 3H1 :IO NOLLOVHJ awn-10A 50 9.3.5:. 20530 c .2 c2236 _o_xol_n tea more. sopobm mo co:oto> $.19... men: 0.2”. zobfim 00.0 00.0 00..v 006.. 00.N 00. F 00.0 _ . _ . _ . _ I _ _ L 00.0 .................. ..I..I..I.. ...... 18.4 109* lood I00.0 em; Eon”: I I .3 II. No)”. II... 18.2 so I No\ 5 I I 00 I .8\ B 00.N _. BY 0I1va HOBHIS 51 .coocm cacti 8. 0300.346 :05; ASHN \Fov 835E cab cooxOOIlooz Bongo 0 2.0 o_ co toozm 53> mmmbm OOMON 00.0? h h - _oco_mcoE_vcoc co cosoto> £4 .9... @589 Eoz< mfixm comm? oo..m - n IF _ n n b - 00...4 00.0 NN... II PC. I 070! 00.0 1070 TONd 00.0 SSHELLS CIEISI'IVNOISNEIWICINON 52 ...00 m 292m 3 08.0033. AN. 0" N0\Fov 0.302.... 0 b. c._>.ml lxmcoos. Bongo 0 co 20 co .60.? £5. mmobm _oco_mcoE_ccoc .6 co_u.o_..o> .m .o_I._ $9.89 302,. «firm O0.0N 006—. OO.N_. OO.m 003V 00.0 b I— b h n — b b L b b n b — L b h — h p L P I I I. O ....x.\........ . 1 ......III... 1 .............. 4 ....\\\\\ ...... \I... N \............... NIP .h. ......t .......... NN._. III I. F F ._. I... 883818 CEISITV’NOISNEWIGNON 53 ...oosm ofiaEfi 3 08.00.33 50...... A0.0HWO\P0V 0.3.5:. on o. c._>.m.l>ocoo§ Bongo 0 .6 o co Loocm 5.2. mmobm _oco.mcmE.vcoc .o co.o.o...o> ”0 .9... $9.88 m..oz< mfizm 00.0w 00.0w 00.NP 00.m 0044 00.0 . _ h . p L . n — p L _ . . L _ . b b 0*.OI p - ........ .. NJ. is... .. .......... NNH ill 9. :HIIV. SSEIELLS GEISI‘IVNOISNEIWIGNON 54 .moztomota .0. 66:6 U6 0.60 ...... .8266 9.8.6 ..o 5:66 6...... K .o: ZO_Ir mmmbm 658 or: “6 c050t0> ”m .9... m ml_. .O_‘ .OE ON mu m; p m m._. 9: 0o cozoto> H: .0_n._ m m.~ 003:8me ”NF .OE m mbqZEmOOO 1.3048 ON 0; 0; TL N4 0; 0.0 0.0 T0 N0 0.0. L _ L L L L L L L _ L — L L L — F _ L Ooo—‘P fi0.0NF fiodn— 0. 10.0.: a fiodm— a L000— 1 0.05L SESNVHO 3WfTWO/\ EOViNEIOézlEId 60 cm}. mzzamooo 0020:0000 220520251202 .0 0:00 29.. 00.. m 00.0:_0._000 .200: 00:0..000: 05 £25 0050.. £000.50 6050083006 0:0 6:00.. Lo :o_0.0_:0> ”m— .0_.._ 04 . . :0 N 00.0 00.0.0: E \ ovouoEE llllllll 050: 200000 _0::0._00E30.__0 0:8 1286 6500 0; v; m.— NN 0N 00 ¢.n 0.0 N6 Oll‘v’H HOE-His 61 0. 00.0... 0.60 29.. .00 m 3020.600 660.. 00:0..0..0.. 00:0..000. 0r: 5.; $80 .208 9.. ..o 802...; a: .0... om\m mquamooo mozmmmmmm :_:0lZOZ 0.0 — £5 00.0.0000 0.0 — n L b 00.0.om\_m 0.0 L L .v.0 . L N.0 L L L L b L J 000.0%: J 0.0.0”; .0 700.0 1 883818 'IVICJ‘V’EI WVNOISNEWlG-NON v BIBLIOGRAPHY 5a. 10. ll. 12. 62 BIBLIOGRAPHY Treloar, L.R.G., "The Physics of Rubber Elasticity," Oxford University Press, 1975. Gandhi, M. V., "Large Deformations of a Nonlinear Elastic Solid and the Diffusion of an Ideal Fluid treated as Interacting Continua", Doctoral Dissertation, U. of Michigan, Ann Arbor, (1984). Paul, D. R. and Ebra-Lima, 0. M., ”Pressure-Induced Diffusion of a Organic Liquid through a Highly Swollen Polymer Membrane", l._A221. Polyngr §c1,, ig,(1970) pp. 2201-2224 Gandhi, M. V., Usman, M., Wineman, A. S., Rajagopal, K. R., 5 "Combined Extension and Torsion of a Swollen Cylinder Within the 5 Context of Mixture Theory”,Anrn_ngnhnninn,12, (1989) pp 81 - 95. Treloar, L. R. G.,"Swelling of a Rubber Cylinder in Torsion: Part 1. Theory”, Polymer, 11, (1972) pp 195-202. Loke, K.M., Dickinson, M., and Treloar, L.R.G., "Swelling of a Rubber Cylinder in Torsion: Part 2. Experimental," Poiyner, 9 Vol. 13, pp. 203-207, (1972). .__'... -. Bowen, R.M., Continuum Physics, (ed. Eringen, A.C.) 3, Academic Press, 1975. Atkin, R.J. and Craine, R.E., ”Continuum Theories of Mixtures: Basic theory and Historical Development," Q,J, Mecn, Anni, Ma n,, XXIX, pt.(1976). Bedford A., Drumheller, D. 8., ”Theory of Immiscible and Structured Mixtures",int, J, Enginggring Sciengg, 21, (1983) pp 863. Passmen, 8., Nunziato, J. and Walsh, 8. R., "A Theory of Multiphase Mixtures. Sandia Report, SAND 02-2261, (1983). Shi, J.J., Wineman, A.S., and Rajagopal, K.R., "Applications of the Theory of Interacting Continua to the Diffusion of a Fluid Through a Nonlinear Elastic Medium,“ n;, J, Engineering Science, 9, pp. 871-889,1981. Rajagopal, K. R. Shi, J. J. and Wineman, A. 8.,"The Diffusion of a Fluid through a Highly Elastic Spherical Membrane", inn, J, Engineering Sgiengg _21 (1983) pp 1171. Gandhi, M. V., Rajagopal, K. R. and Wineman, K. R., "Some Nonlinear Diffusion problems within the Context of the Theory of Interacting continua". Ias._J._£nsinssrins_§sisnss§. 2: (ll/12). (1987) pp 1441. 13. 14. 15. l6. 17. 18. 19. 20. 63 Gandhi, M. V. and Usman, M., "The Flexure problem in the Context of the Theory of Interacting Continua", SES paper No. esp24-87016, presented at the Twenty-Fourth Annual Technical Meeting of the Society of Engineering Science, Salt Lake City, UT, September (1987). Gandhi, M. V. and Usman, M., ”Equilibrium Characterization of Fluid-Saturated Continua, and an Interpretation of the Saturation Boundary Condition assumption for Solid-Fluid Mixtures", Int, J, Ensinssrins.§sign22 (in preSS) (1989). Green, A.E., and Naghdi, P.M., "On Basic Equations for Mixtures", Q,J,Mecn, Anni, Matn,, XXII, (1969) pp. 427-438. Mills, N., "Incompressible Mixtures of Newtonian Fluids," nt J Enzinssrins_§sisnge. fl. (1966) pp- 97-112- Green, A. E. and Steel, T. R., "Constitutive Equations for Theory for Interacting Continua", Int, J, Engineering Science 4 (1966) pp 483-500 Crochet, M. J. and Naghdi, P. M., "0n Constitutive Equations for Flow of Fluid through an Elastic Solid”, Int, J, Engineering Sgience 4 (1966) pp 383-401. Flory, P. J., "Thermodynamics of High Polymer Solutions”, J, of ghssissl_£bxs12§. 19 (1942) pp 51-61- Gandhi, M. V. and Usman, M., "0n the Nonhomogeneous Finite Swelling of a Nonlinearly elastic cylinder with a rigid core", int, J, Engineering Science (in press), (1989). MICHIGAN STATE UNIV. LIBRARIES llIllWImWVI”Wll”IWIWIIHIIWIWHWWW 31293006299329