CCII mm an; N“. X r r .. x L by . 1 . ‘ .1 LL Wu LC. .v.‘ S l m «5 CL k. . ABSTRACT ON SOME PROBLEMS IN A THEORY OF THERMALLY AND MECHANICALLY INTERACTING CONTINUOUS MEDIA BY Yong Mok Lee Using a linearized theory of thermally and mechani- cally interacting mixture of linear elastic solid and viscous fluid, we derive a fundamental relation in an integral form called a reciprocity relation. This recipro- city relation relates the solution of one initial-boundary value problem with a given set of initial and boundary data to the solution of a second initial-boundary value problem corresponding to a different initial and boundary data for a given interacting mixture. From this general integral relation we derive reciprocity relations for a heat-con- ducting linear elastic solid, and for a heat-conducting viscous fluid. In this theory of interacting continua we pose and solve an initial—boundary value problem for the mixture of linear elastic solid and viscous fluid. We consider the mixture to occupy a half-space and its motion to be re— stricted to one space dimension. We prescribe a step function temperature on the face of the half—space where the face is constrained rigidly against motion. With the Yong Mok Lee aid of the Laplace transform and the contour integration, a real integral representation for the displacement of the solid constituent is obtained as one of the principal re- sults of this analysis. In addition, early time series expansions of the other field variables are given. ON SOME PROBLEMS IN A THEORY OF THERMALLY AND MECHANICALLY INTERACTING CONTINUOUS MEDIA BY Yong Mok Lee A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1971 ACKNOWLEDGMENTS I am deeply indebted to Professor C. J. Martin for suggesting this investigation and for his guidance during the preparation of the thesis. I would like to express my deep gratitude for his extreme patience and helpful advice during the preparation of the final draft. ii K‘\ k‘! h" Y» F“ TABLE OF CONTENTS INTRODUCTION . . . . . . . . . . . . . CHAPTER I. HISTORY . . . . . . . . . . 1.1. 1.2. CHAPTER 2.1. 2.2. 2.3. A. B. C. 2.4. A. B. CHAPTER 3.1. 3.2. 3.3. A. B. C. Darcy's Law . . . . . . . . . . Biot's Work . . . . . . . . . . II. THEORY OF INTERACTING CONTINUA Nonlinear Theory . . . . . . . . Linearized Theory . . . . . . . Summary of the Equations and Other Formulations . . . . . . . . . Fully Coupled Mixture Theory . The Mixture Theory of Green and Steel Uncoupled Theory . . . . . . . Single Constituent Theories . . Linear Thermoelasticity . . . . Linear Viscous Fluid . . . . . III. RECIPROCITY THEOREMS Introduction . . . . . . . . . . Reciprocal Relations for Mechanically and Thermally Interacting Mixture Special Cases . . . . . . . . . Reciprocity Relation for Heat Conducting Mixture of Linear Elastic Solid and Non— Newtonian Viscous Fluid . . . Reciprocity Relation for Mixture of Linear Elastic Solid and Newtonian Viscous Fluid in Isothermal Process . Reciprocity Relation for Heat Conducting Mixture of Linear Elastic Solid and Newtonian Viscous Fluid Occupying Infinite Region . . . . . . . iii Page 20 20 25 40 4O 42 42 43 43 45 48 51 65 65 66 68 93$ C?” H" ... TABLE OF CONTENTS (Cont.) Page D. Reciprocity Relation for Heat Conducting Elastic Solid . . . . . . . . . . . . . . 69 E. An Application of Reciprocity Relation in Mixture Theory . . . . . . . . . . . . . 73 CHAPTER IV. A FUNDAMENTAL ONE-DIMENSIONAL INITIAL- BOUNDARY VALUE PROBLEM . . . . . . . . 75 4.1. Introduction . . . . . . . . . . . . . . . . 75 4.2. Formulation of the Problem . . . . . . . . . 76 4.3. Solution by Integral Transforms . . . . . . 83 4.4. Inversion . . . . . . . . . . . . . . . . . 87 A. Location of Zeros of g:(p)g:(p) . . . . . . 87 B. Determination of Branches for gl(p) and 92 (p) . ’ ' 0 . O O O o o o o o o o o o o 89 C. Formulation of w(c,p) in Convolution Form . 91 D. Inversion of‘w 2(C,p) by Contour Integration 94 SUMMARY AND CONCLUSIONS . . . . . .A...~. . . . . . 120 REFERENCES . . . . . . . . . . . . . . . . . . . . . 122 APPENDIX . . . . . . . . . . . . . . . . . . . . . . 125 iv a: .\I 1., LIST OF FIGURES Figure Page 1. p-Plane . . . . . . . . . . . . . . . . . . . 92 2. p-Plane o o o o o o o~o o o o o o o o o o o o 102 3. u-Plane . . . . . . . . . . . . . . . . . . . 102 ela. INTRODUCTION Using a linearized theory of thermally and mechanically interacting mixture of linear elastic solid and viscous fluid, we derive a fundamental relation in an integral form called a reciprocity relation. This reciprocity relation relates the solution of one initial—boundary value problem with a given set of initial and boundary data to the solu- tion of a second initial-boundary value problem corresponding to a different initial and boundary data for a given inter— acting mixture. From this general integral relation, we derive reciprocity relations for a heat-conducting linear elastic solid, and for a heat—conducting viscous fluid. 2 In this theory of interacting continua we pose and solve an initial-boundary value problem for the mixture of linear elastic solid and viscous fluid. We consider the mixture to occupy a half-space and its motion to be restricted to one space dimension. We prescribe a step function temperature on the face of the half—space where the face is constrained rigidly against motion. With the aid of the Laplace transform and the contour integration, a real integral representation for the displacement of the solid constituent is Obtained as one of the principal results of this analysis. In addition, early time series expansions of the other field variables are given. Chapter I includes a historical survey of early works and the various descriptions on mixture theory. r- ,,,.. L .73. 7*. $4. 2 Chapter II presents modern mixture theories based on mathe- matically sound concepts of continuum mechanics. In Chapter III we derive the general integral reciprocity relation for a linearized version of an interacting mixture and in Chapter IV we pose and solve a basic one-dimensional prOblem using the linearized theory. It 5 L. si sue? ‘l‘ ‘ill E C f: n?.. V& r fifilu . n O h: Lu J3 . a. .2! \ . t . . A: why . . (K H e r d“ . a. 09 m9. QHiN It B ~ ‘ . t 0-H saw .2 \ . .2 T. f CHAPTER I . HISTORY 1.1. Darcy's Law The theoretical description of the dynamics of situations-in which one substance interpenetrates another has been a matter of interest to mathematicians, physicists and engineers for many years. The Case in which a fluid permeates a solid is appropriate to a wide range of prdblems such as soil mechanics, petroleum engineering, water purifi- cation, industrial filtration, ceramic engineering, diffu— sion problems, absorption of oils by plastics and the re- entry ablation process for spacecraft. A survey of earlier works on this subject up to 1959 is given by Scheidegger [l]*. An early work on this subject was the study of fluid flow through a porous solid with the assumption that the solid is undeformable. Intuitively, "pores" are void spaces which must be distributed more or less frequently through the solid if the latter is to be called "porous." Extreme small voids in a solid are called "molecular interstices," very large ones are called "caverns." "Pores" are void spaces intermediate be- tween caverns and molecular interstices: the limitation of their size is therefore intuitive and rather indefinite. Darcy [2] performed an experiment concerning the flow through a homogeneous porous solid.** A homogeneous filter bed of height h is bounded by horizontal plane areas of *Number(s) after name(s) refer to the list of references to be found at the end of this paper. **This experiment was originally performed by Darcy in 1856. 3 .___.__J (1‘ t] i: OI {11" 4 equal size A. These areas are congruent so that corres— ponding points could be connected by vertical straight lines. The filter bed is percolated by an incompressible liquid. If open manometer tubes are attached at the upper and lower boundaries of the filter bed, the liquid rises to the heights h2 and h1 respectively above an arbitrary datum level. By varying the various quantities involved, one can deduce the following relationship: - KA(h - h) Q = h2 1 (1.1) Here, Q is the total volume of fluid percolating in unit time, and K is a constant depending on the properties of the fluid and of the porous solid. The relationship (1.1) is known as Darcy's law. Darcy's law can be restated in terms of the pressure p and the density r of the liquid. At the upper boundary of the bed, whose height is denoted by 22, the pressure is p2 = rg(h2 - 22), and at the lower boundary, whose height is denoted by the zl, pressure is p1 = rg(hl - 2 Here g is the gravitational l)' constant. Inserting this statement into (1.1), one obtains Q = - KA((p2 - pl)/(rgh) + l) or, upon introduction of a new constant K', O = - K’A(p2 - P1 + rgh)/h (1.2) A constant of the type K',,'however, is not very satis— factory because one would like to separate the influence of 5 the porous solid from that of the liquid. By 1933, the empirical relationship K’ = k/u (1.3) was generally accepted where u is the viscosity of the fluid and k the "permeability" of the porous solid. Physically, permeability measurements are very simple. The experiments are performed whereby in a certain system a pressure drop and a flow rate are measured. The solution of Darcy's law corresponding to the geometry of the system and to the fluid employed is calculated, and a comparison between the calculated and the experimentally found results immediately yields the only unknown quantity k. DarcY's law (1.2), when accounting for the separation of the general constant into "permeability" and "viscosity," is expressible as follows: q Q/A = - (k/qu2 - p1 + rgh)/h (1.4) If the solid is isotr0pic and if we consider h as an infini- tesimal, then the expression (1.4) naturally extends to a vector form of Darcy's law: Q = - (k/u)(grad p - r9) (1.5) where g' is a vector in the direction of gravity. Engineering uses of Darcy's law are limited to flows exhibiting small pressure differentials and to constant viscosities and permeabilities. However, for liquids at high 'I! llFI Jpn.— f|.. r '41. n ‘3‘ I. c» r. 6 velocities or for gases.relation (1.1) is no longer valid. Further if k and u are variable then this law must be modified. The validity of Darcy's law has been tested on many occasions, and has been shown that it is valid for a wide domain of flows. For liquids, it is valid for arbitrary small pressure differentials. It has also been used to measure flow rates by determining the pressure drop across a fixed porous solid. For liquids at high velocities and for gases at very low and at very high velocities, Darcy's law becomes invalid. For given boundary conditions Darcy's law (1.5) is by itself not sufficient to determine the flow pattern in a porous solid because it contains three unknowns (q, p, r). Two further equations are therefore required for the complete specification of a problem. One is the connection between r and p of the fluid: r = r(p) (1.6) and the other a continuity equation, viz.: Br _ . — p E -- le (rq) (1-7) where t is the time and P is the porosity defined by the fraction of void to the total volume of the porous solid. A great variety of methods for the measurement of the porosity are described by Scheidegger[l].The physical conditions of flow for which solutions might be sought are (i) steady state flow, (ii) gravity flOW'With a free surface, and (iii) LL. #1. . Wag aSs 7 unsteady state flow. Of these, steady state flow solu— tions for incompressible fluids are most easily obtained; they are simply represented by solutions of Laplace's equation. Except for a few other special cases, Darcy's law leads to nonlinear differential equations. As an application of Darcy's law we will consider the steady state flow of an incompressible fluid. With the help of the equations (1.5) and (1.7), one may obtain Br P 3?": div ((rk/u)(grad P * r9)) (1.8) Due to the steady state condition, incompressibility and the porous solid being homogeneous, one has: As an example of a steady state solution we give the solu- tion for two-dimensional radial flow of an incompressible fluid into a well which is completely penetrating the fluid- bearing medium. Assuming that the well is a cylinder of radius R0,. with pressure p0,. and that the pressure at distance R1 from the well is pl, the required solution follows easily by considering equations (1.5) and (1.9) as 21rk u log (Rl/RO) where Q is the total discharge per unit time. A.major limitation in this theory is due to the assumption that the solid is rigid. In most applications 8 this is simply not true. To incorporate the effects that a deformable solid imposes upon the flow, it is necessary to develop some connection between the stresses and the corresponding strains of both the fluid and the solid. we will consider the case of soil consolidation. 1.2. Biot's‘Work A soil under load does not assume an instantaneous deflection under that load, but settles gradually at a variable rate according to the load variation as in clays and sands saturated with water. A simple mechanism to explain this phenomenon was proposed by Terzaghi [3] by assum- ing that the grains constituting the soil are bound together by molecular forces and constitute a porous material with elastic prOperties while the voids of the elastic skeleton are filled with water. A load applied to this system will produce a gradual settlement, depending on the rate at which the water is being squeezed out of the voids. Terzaghi applied these concepts to the analysis of the settlement of a column of soil under a constant load and prevented from lateral expansion. The remarkable success of this theory in predicting the settlement for many types of soils has led to the extension to the three-dimensional case and the establish— ment of equations valid for an arbitrary load variable with time. We will review extensive work done by Maurice A. Biot in this field. I“ In ('1- 9 Biot [4] assumed the following basic properties of the soil: (1) isotropy of the material, (2) reversibility of stress-strain relations under final equilibrium conditions, (3) linearity of stress-strain relations, (4) small strains, (5) the water contained in the pores is incompressible, (6) the water may contain air bubbles. (7) the water flows through the porous skeleton according to Darcy's law. we refer the points in this continuous medium to a rectangular cartesian system, xi, i = 1,2,3. Consider a small cubic element of the consolidating soil, its sides being parallel with the coordinate axes. This element is taken to be large enough compared to the size of the pores so that it may be treated as homogeneous, and at the same time small enough compared to the scale of the macrosc0pic phenomena in which we are interested, so that it may be considered as infini- tesimal in the mathematical treatment. Physically the stresses of the soil are composed of two parts: one which is caused by the hydrostatic pressure of the water filling the pores, the other caused by the average stress in the skeleton. They must satisfy the well—known equilibrium conditions of a stress field. Let Uij denote the stress components and let xi denote axes of the cartesian system. By Oij we shall mean the jth stress component of the skeleton acting on the face xi constant. Then according to the equilibrium for the infinitesimal element of volume we have lO 0.. . = O (1.10) and = 0.. . (l-ll) Denoting by 1.1.1 the component of the displacement in the xi direction, and assuming the strain to be small, the values of the strain components are (1.12) H NH4 2‘ + c: In order to describe completely the macrosc0pic condition of the soil, an additional variable giving the amount of water in the pores is considered. The increment of water volume per unit volume of soil is called the varia- tion in water content and is denoted by 9, and the incre— ment of water pressure is denoted by 0'. Let us consider a cubic element of soil. The water pressure in the pores may be considered as uniform throughout, provided either the size of the element is small enough or, if this is not the case, provided the changes occur at sufficiently slow rate to render the pressure differences negligible. Since it is assumed that the changes in the soil occur by reversible processes, the macrosc0pic condition of the soil must be a definite function of the stresses and the water pressure, i.e.: the seven variables e.., 6 must be definite func- 1] tions of the variables Oij and 0 . Furthermore if the it All subscripts run over values 1,2,3, and, when repeated indicate a sum on the index over 1,2,3. The notation fi j I denotes differentiation with respect to the jth independent variable, i.e., Bfi/ij . H‘ We r0; Cor Soj *1: 11 strains and the variations in water content are assumed to be small quantities, the relation between two sets of variables may be taken as linear. Consider the case where o = O. The six components of strain are then functions only of the six stress components Oij . Assuming the soil to have isotropic prOperties, these relations reduce to the well known expressions of Hookes' law for an isotropic elastic body in the theory of elasticity ** _.Al - __AL_. 5 eij ‘ 2G (Uij (1+v) Uzi ij) (1.13) where the constants G, v may be interpreted, respectively, as the shear modulus and Poisson's ratio for the solid skeleton. The effect of the water pressure 0 is now introduced. By reason of the assumed isotr0py of the soil, this effect is limited to a dependence upon the three strain components ell' e22, e33 and such dependence is uniform in each direction. Hence taking into account the influence of O, the relations (1.13) become _.J; _.__2__ .iL eij ‘ 2G (Uij (1+v) Ckkéij) + 3H-5ij (1°14) where H is an additional physical constant which plays the role of a bulk modulus. These relations express the six strain components of the soil as a function of the stresses in the soil and the pressure of the water in the pores. —.— *7: . The Kronecker delta, denoted by 6ij' is defined as 1 if i = j and o if 1 ¢ j. 3.. LL 12 To derive the dependence of the increment of water content 6 on these same variables Biot considers the general relation (1.15) O +aO +aO +aO +ao +aO +a 2 22 3 33 4 12 5 23 6 13 0' 7 and argues that because of the isotropy of the material a 012, 023, 013 cannot affect the water a4 = a5 = a6 = O and the effect of the shear stress components on 9 vanishes. Furthermore, change in sign of content. Therefore all three directions x1, x2, x3 must have equivalent properties so that a1 = a2 = a3. Relation (1.15) may be written in the form _ _l.._ _0_ 9 — 3H1 all + I! (1.16) where Hl are R are two new physical constants. To this point in the derivation Biot has used assump- tionS(l), (3), (4). He now uses (2) to show that the five constants can be reduced to four. This assumption, i.e., the existence of a potential energy, means that the work done to bring the soil from the initial state to its final state of strain and water content is independent of the way by which the final state is reached and is a definite function of eij' and 6 only. The potential energy of the soil per unit volume is _ 1. 1.17 U—2(oijeij+08) ( ) 13 As a result of some elementary manipulations, Biot shows that H = H and we may write the equation (1.16) as l I _3H kk R (1.18) Relations (1.14) and (1.18) are the fundamental relations describing completely in first approximation the properties of the soil, for strain and water content, under equilibrium conditions. They contain four distinct physical constants G, v, H and R. Solving equation (1.14) with respect to the stresses, then substituting into the equilib— rium conditions (1.10), one obtains 2 G Be Bo _ GV ui + 1—2v 5X1 — a Bxi — O (1.19) with _ 2(l+v) [g a _ 3(1—2v) H (1°20) There are three equations with four unknown ui, o. In order to have a complete system, one more equation is needed. This equation is derived from Darcy's law governing the flow of water in a porous medium. An elementary cube of soil is considered and the volume of water flowing per second per unit area through the face of the cube perpendicular to the xi-axis is denoted by vi . According to Darcy's law these three components of the rate of flow are related to the water pressure by the relations _ '5 Bo . '7 " an: 14 where the physical constant k is the coefficient of permeability of the soil, and u is the viscosity of the water. Since the water is assumed to be incompressible, one obtains 21__ i Bt — . (1.22) From equations (1.17), (1.20) and (1.21) one obtains that £2. 22 1.52 U V o — a t + Q Bt (1.23) where 1__1__Q Q—R H The four differential equations (1.18) and (1.22) are the basic equations satisfied by the four unknown ui, O. In a paper by Biot and Willis [5], methods of measure— ment for the four distinct physical constants G, v, H and R are described and the physical interpretation of the con- stants in various alternate forms is also discussed. In a later work by Biot [6] the stress-strain relations which are valid for the case of an elastic porous medium with nonuniform porosity, i.e., for which the porosity varies from point to point are derived and these relations lead to the six equations for the gix_components of the unknown displace- ment vector fields n for solid component, f for fluid component. The stress-strain relations are Oij = 2n eij + 6ij(le - abdg) (1.24a) T: H rfi 15 pf=-aMe +M§ (1.241)) where the coefficients M, l, n, g and pf of equations (1.24a) and (1.24b) are equivalent to the constants Q , 2Gv/(l—v), G, 9 and o of the equations of (1.19) and (1.23). The six equations for the six components of the unknown vector fields u and f are 2 3‘2: (11 eij) + 5X1 (1e - aMg) = 0 (1.25.1) Bf BE = (k/u) grad (OLMe — M5) = O (1.25b) We will consider a uniform porosity case, i.e., let us con- sider a particular case where the coefficients n, 1, (1, M and k/u are constants. In this case equations (1.25) become n Vzu + (n + l) grad e - abdgrad E = O (1.26a) Bf 3'; = (k/umd grad e - (km/u) grad 1;. (1.26b) These equations can be written in the form of equations (1.19) and (1.23) by the application of the divergence Operator to the equation (1.26b). With the aid of the general Papkovich— Boussinesq solution for Lame's equations of the theory of elasticity, the general solutions for the equations (1.26) are Obtained as _ . _ 23 + 1 _ cxM u — grad (to + l:.¢l) 2 n + 1 W1 §;r:-X-grad t (1.27a) = . _ m O, t—-———- > O. (2.4) ark ij Similarly, for a typical particle of s we have 2' 20 A.» . l \h... Aura 21 Yi(T) = Yi(YlIY20Y3oT)p Yi(t) = Yi (—oo<'1'_<_t) (2.5) or Yi(T) = yi(yl.y2.y3.t.¢) (2.6) together with >0 Byi(T) L______ > O . (2.7) BYA BYj We assume that the particles under consideration occupy the same poSition at time t so that y. = x. (2.8) Velocity vectors at the point xi = yi in S1 and 32 at time t are (1) (2) D xi D y. _ _ l i — Dt ' Vi ' Dt (2.9) where D(l)/Dt denotes differention with respect to t 1 and D(2)/Dt denotes holding Yj fixed. These holding Xj fixed in continuum s a similar operator for 52. Operators may also be written as (1) (2) D _.JL ._B_ D _.JL .B__ H1>> Acceleration vectors at time t are denoted by 1 and A A gi, where fi_ ..— ._-.. ___—g...__,-.l_.. 22 (2.11) The densities of S1 and 52 at time t are, respec- tively, p1 and p2, and the rate of deformation tensors at time t are defined to be 2d.. = u. . + u. . , 2f.. = v. . + v. . (2.12) where a comma denotes partial differentiation with respect to xk or yk. We also define a mean velocity wi be the equation wi = plui + pzvi , p = pl + p2 (2.13) and put 2_._ EL. .41. Dt ’ Bt + wm me ' (2'14) It then follows that (l) (2) D D _ ___ p1 Dt + p2 Dt ‘ p Dt ° (2.15) Let BB be an arbitrary fixed closed surface enclosing a volume B and let nk be the outward unit normal to BB. Let U be the internal energy per unit mass of the mix- ture. The externally applied body forces per unit masses of and s are denoted, respectively, by the vectors S1 2 Fi and Gi' And these vectors are defined through their 23 rate of work contributions Fiui and GiVi for arbitrary velocity fields u.l and vi . The surface force vector ti per unit area of BB is such that the scalar tiui' for arbitrary ui, is a rate of work per unit area of BB . And a similar definition can be made for the vector pi associated with the velocity vector vi . The scalar r is the heat supply function per unit mass of the two con- tinua due to radiation from the external world and heat sources. The flux of heat across BB is denoted by a scalar h per unit area and unit time. Theorem (Green and Naghdi) Let us postulate an energy balance at time t in the form B 1 .1 + I [ ( u + v )U + lp u u u + lp n V v v ]dA nkplk p2k Zlnkkii 22kkii A = J (pr + plFiui + pZGiVi)dV V + J (tiui + pivi)dA - JAh dA (2.16) A Then it follows that 122 = 2.17 Dt + pw’k'k O ( ) 24 which states that mass elements of the mixtures are con- served. The vectors ti' pi are defined with reference to an arbitrary surface A. When the surface at a point xi is perpendicular to the xk-axis, we denote the correspond- ing values by 0 ki' Wki and refer to these as stresses. Then it also follows from (2.16) that (Oki + T"1_ 0 (2.30) Energy equation: —-as 25.1 __ _BA — - — -p(T at+s + )+w(up vp) +0 d Bt Bt p (pq) pq +—— f 433 F - - + = 0, Tr(pq) pq [pq]( qp qu) qp.p pr in B , t.2 O. (2.31) In (2.31), 5-: 5i + E? is the total initial density of the mixture and IE., are the diffusive resistance 1 0ij' and the partial stress tensors in the equilibrium state. The linearized constitutive equations obtained from (2.19) are pA = p A + alepp + c12(p2 -p2) + (13(T-T) (2.32) pS =-(d3+ dgepp + d10(p2-p2) + d7(T-T)) (2.33) qi =--kT,i - K'(ui-vi) (2.34) *See notations used in Theorem on Page 31. 29 (2.35) _ _1 _ .1 _" _"* 0(ij) - (al-(E§ d1 d4)epp+(5_dl-+d8)(92 p2)+d9(T T) ‘txldpp't13fpp)6ij-+2(dl-+d5)eij +2u1di + 2u3fij (2.36) _ _ ‘1 5+3, _ _ ._pzalO(T-T)-+x4dpp-+xfpp)5ij-k2u4dij+2ufij (2.37) DE (u -vp)-D"(Fi.-A..) (2.38) o .. =-w .. = .. [11] [11] 13p p 3 11 We note that there are total of 24 constants d1....,d10, 11.1.13.x4.u1.u.u3,u4; o,a",D,D",k,K' 'which have to be determined by an experiment for the mixture. The entropy- production inequality [14] imposes restrictions upon the constitutive equations which, in the linearized theory, require the material constants to satisfy the following inequalities: *We use the notation A(ij) = 5(Aij‘kAji)'A[ij] = yAij - Aji). 3O 311+2u1 :0. ulzo. 31+2u20ou20:(u3+u4)254u1uv 2 (313+2u3+3)\4+2u4) _<_4(3)(1+2u1) (31+2u) (2.39) (130, D" _>_O,(a" -D)2_<_4dD",k3_O,K'254Tdk This linearized theory is well posed in the sense that the number of the field and constitutive equations equals the number of field quantities to be determined. We would ex- pect that the boundary conditions for the initial-boundary value problems for the mixture are similar to the classical boundary conditions for the elasticity problem in which the stresses, strain, and displacements are sought. We recall that the claSSical boundary conditions are: (a) the forces may be given on the surface of the body, (b) the displacements may be given on the surface of the body, (c) the forces may be given on some portions of the body surface, while the displacements are given on the other portions. Indeed, the proper form of the initial and boundary conditions which should be adjoined to the field and constitutive equations of the linearized theory of interacting continua so that the sufficiently smooth solu— tions of the field and constitutive equations are deter- mined uniquely are quite similar to the classical boundary conditions of the elasticity except that we have the temperature terms and we have to specify the boundary con- ditions to each component of the mixture. These conditions 31 are specified in the following theorem. Theorem (Atkin, Chadwick, Steel) [18] Let B be a bounded regular region of three-dimensional Euclidean space occupied by a mixture of an elastic solid and a viscous fluid undergoing a disturbance of small amplitude during the time interval t 2_0. We denote by BB the boundary and by B0 the interior of B. we use 1' BB2 and BB1, BB2 of BB and their complements with respect to BB and n notation BB for arbitrary subsets refers to the unit outward normal vector field on BB. II II Suppose that the constants 11,u1,l,13,03,14,u4,d,u,a ,D,D , k and K' satisfy the conditions (2.39) and that 01,02,04,05,d6,d7 and 08 satisfy the inequalities +0 0gd+d 00L <0 0(3EQ-l)+d +301 0 O‘152'-p—262'7—'1-5 3 4 352' _ _ (2.40) 29 1 p1 2 2 1 2 2 (€91 “'E_O‘2+ 0‘8) 3 (“1(- " 3)+O‘4+30‘5) (516+ 0‘6) Then there exists at most one set of functions Vi’ p2 of class C1 and wi, T of class C2 which satisfy equations (2025)! (2026), (2027)! (2029). (2030)! (2031)! (2034) to (2.38) and the subsidiary conditions (2.41) (2.42) u. = U., v. = V. on BB. for t 2_O , T = T + O on BB2, qpnp = F on B82 for t 2_0, (2.43) A A A where wi, u., v. A p p ' 6' Ru, 2.. U., Va, 9' F and F. p i i 2 i i i i i Gi' r are prescribed functions on the appropriate domains and Si, 55, T are given, strictly positive, constants. It is well known that for the dynamical motions of linear isotropic elastic solid, the displacement vector 'w may be represented as a sum of two components representing motions of dilatational and rotational types, i.e., w'= grad w + curl fl, where m and w satisfy the wave equations in which appear the speeds of propagation of dilatational and rota- tional body waves respectively. This representation is known to be complete in the sense that every sufficiently smooth solution ‘w of the equation of motion of linear isotrOpic elastic solid is expressible in the stated form where the scalar and vector functions m, t satisfy the above mentioned wave equations and in addition, div t = 0. 33 For the motions of the interacting continua of an elastic solid and a viscous fluid, Atkin [18] has estab- lished a decomposition of its motions into components representing motions of dilational and rotational types. Each part of the decomposition* is somewhat simpler in form than the original system of the differential equations (2.26). (2.29) to (2.31) and (2.35) to (2.38), but con- siderably more complicated than the wave equation. The main merit of this new formulation for the motions of the interacting continua is that it allows the investigation of the propagation of small amplitude plane waves in a non—heat conducting mixture of an isotrOpic solid and an inviscid fluid. We define new material constants by the following combinations of the material constants. Z _;_l _—2 2 + 0L +0Ll(2 _ ), K2—p2(<16+_(12). 1:“ 4 3 5 P 3 P K =-3(a +lCt -El"0.) G :0 +0 (2 44) 3 2 8 - 1 - 2 ' 1 1 5 ' P P _ __ '53., Bl=-TO.9, B2=p2TalO, B=D-a , Cd—-—E'_— Introducing the vector differential operator L[§,n] = g grad div - n curl2 (2.45) and supposing that body forces and heat sources are absent, the governing equations take the form 34 o * p2 + 52 div V = O (2.468.) I": A1+2ull ul+éD"]é+L[ A3 +2113. “-3 - éD'l]v - 4 - (0L +8 curl) (iv—v) +L[K1+§- G1,G1]w -grad(K3pz/52+Ble/T) = pl w (2.46b) I-[14+2u4. 04-;D'-]€v+1.[(2+202, “gimp +(a+B<' )-o (252) a at VP3 a 2 Cp1'sz ‘ ' t where ll. _ QL. 2 2 2_ 2 2 2 2 2 2 vPl — --(C1 + c2 + ((c1 c2) + 4c3c4) ) , .vQ ll _.3; 2 2 _ 2_ 2 2 2 2 2 2 vP2 —\/§(cl + c2 ((c1 c2) + 4c3c4) ) .1. _ 2 2 2 4 2 vP3 - (f(cl + c3) + (l-f)(c2 + c2)) It has been shown that if vPl and vP2 are real, vP3 is also real and the three dilatational wave speeds satisfy the inequalities vP2 g Vp3-S vP1 (2.53) 39 The form of equation (2.52) suggests that at high frequen- cies there are two modes of dilatational wave propagation associated with the speeds VPl' VPZ, while at low fre— quencies there is only one mode of wave propagation, associated with the speed v the second mode being a P3' diffused disturbance. From equations (2.51c) and (2.51d) one may obtain 2 -433) +a(fu2v2 "9—)”. = o (2.54) at at (€ng v2 The form of equation (2.54) suggests that rotational dis- turbances of the mixture comprise a single mode which has a wave—like character at all frequencies, the speed of prOpagation being of f%u in the limit when frequencies approach to zero and u in the limit when frequencies approach to infinity. So far we have reviewed the recent developments of the interacting continua of an isotropic elastic solid and a viscous fluid. Due to the complication of the system, comparatively little progress has so far been made concern- ing the application of the linearized theory to particular physical situations, or the properties and understandings of the character of the system of the partial differential equations. 40 2.3. Summary of the Equations and Other Formulations A. Fully Coupled Mixture Theory At this point we summarize the pertinent equations which govern the motion of a thermally and mechanically interacting continuous mixture according to the theory ex- pounded by Atkin, Chadwick and Steel [18]. We call this problem by the name "fully coupled mixture theory." strain-displacement equations rate of deformation-velocity equations d.. = . . , f.. = . . . 1) u(1.3) 1] V(1.3) (2 12) vorticity—velocity equations .. = u . . , .. = V . . 2.27 r1] [1.3] A13 [1.3] ( ) equations of motion ._ ._ au. ‘_ _. av. TTij'i + “’j + p2 Gj = p2 __lat (2.30) continuity equations apz _ 41 energy equations .62 _ - E a7 at + OL9dmm p2alO fmm + E-r k K _ * + 71,-: T'mm + -,I-;(um,m-Vm,m) _ O (2'31) constitutive equations =[al "(Ef'a1"a4)ekk'*(T:”*a8)(pz"pz) 0(ij) p '“a9(T"T)'**1dkk'*‘3fkk]5ij + 2(dl-+d5)eij4-2uldij4-2u3 fij , (2.35) ._ ._ '51 ._ ._ Tr(ij)“[“320‘2+pz(—:0‘2"‘18)ekk“‘32‘3‘10m‘m 5455 _. __ -( 5. a2-+92a6)(92-92)-+x4dkk-+xfkk]5ij +2u4<5tij + Zufi j , (2.36) 5; ‘5 “’1‘ -—_— lekk 1+—:a2 pz,i+a(ui-Vi) (2.37) + II a Eipq( I‘pq qu) o .. —D" .. 2.38 [13]= [ij]= Deijp(u p Vp) (r1 1j A13) ( ) The complete initial—boundary value problem is speci— fied by the above equations and: the initial conditions *Obtained from (2.31) by using (2.32) to (2.38) in (2.31). Usually the form so obtained is called heat equation. 42 (2.41), the boundary conditions (2.42). (2.45), and the material inequalities (2.39), (2.40). The problem is solved if one can obtain at each place xi and t > O the functions wi, Vi' p1, p2 and T. B. The Mixture Theory of Green and Steel. In a series of papers by Green and Steel [20], Green and Naghdi [14], Steel [21], a more tractable initial- boundary value prOblem than the fully coupled mixture theory of section 2.3 A has been presented. The major difference between the linearized version of the theory presented in [15] and that of A lies in the constitutive relations. Green and Steel's relations follow from A if one sets equal to zero X10 X3: X4! L11: U3: ”,4: (2.55) an, D, D". C. Uncoupled Theory. By the term uncoupled theory we shall mean the initial-boundary value problem as specified in section 2.3A using the constitutive equations of Green and Steel presented in section 2.3B and in addition, neglecting the time rate of change of the dilatational effects of the solid and fluid components in the energy equation. If this is done, then the heat equation is uncoupled from the equations of motion and the temperature may be treated asa avai cont tre; pro; Rent Sati 0ft the equa 43 as a known function of space and time. 2.4. Single Constituent Theories. So that we can make a comparison of the mixture theories presented in the previous sections with the clas- sical theories of elasticity and viscous fluids we record here the changes that must be effected. Our purpose is 'Umofold: it allows us to draw upon the vast literature available in the classical theories of single constituent continuous media and it allows us to give meaningful in- trepretation to the mechanical and thermal material properties used in the mixture theory of subsections 2.3. A. Linear ThermoelastiCity If the fluid component is not present then the body may be interpreted as a linear elastic solid undergoing thermal deformation in which the variation in temperature is small [22],[25],[24]. One seeks to obtain the compo- nents of displacement wi and the temperature T which satisfy equations (2.25), (2.26), (2.29), in the absence of the fluid component and.with certain modifications of the heat and constitutive equations, satisfy (2.31) and (2.35). These modifications are formally equivalent to employing (2.55) and setting d,al,d2,d6,d8,d10,x,u,K (2.56) equal to zero. line wher coef isot the Wher Spec rela' Elas Sati and 44 Equation (2.36) is then interpreted as the classical linear elastic stress-strain law if we identify (15 = “E' (14 = )(E, (19 = -YKE (2.57) where “E' XE are the Lame elastic constants, Y is the coefficient of linear thermal expansion and KB is the . _ l isothermal bulk modulus, KE — 3 (ZuE4-3XE). From (2.31) the modified heat equation becomes — fil_ — _§_ — _ kT,mm pce at YTKE at emm+pr — O (2.58) where 5' is the total density of the body, C8 the specific heat at constant strain. These coefficients are related to Q7 by d7 =- (2.59) The material inequalities (2.39) and (2.40) simplify to k _>_o, pEz 0, Ce 3 0, 2HE+3XE=3KE _>_o. (2.60) A properly posed initial—boundary value problem of elasticity consists of finding wi and T of class C2 which satisfy the modified equations and the following initial and boundary data: where iunct term is ig thQO] thee: 45 wl=vl>i, u1=Gi, T=E+T on B at t=O, ijnJ = 21 on 581, W1 = W1 on aBi, for t 2_O, (2.61) T = T + e on BB2, qknk - F on 5B2, for t _>_ O A A . where wi, ui, é, Z;i' Wi.efiF. and Fi are prescribed functions on the appropriate domains. We close this subsection with the remark that if the term -YTKE Sat—em (2.62) is ignored in (2.58), then the resulting thermoelastic theory is known as the classical uncoupled thermoelastic theory. [22], [23] B. Linear Viscous Fluid If the solid component is not present then the body may be interpreted as a fluid undergoing thermal deforma- tions in which the variation in temperature is small. One seeks to obtain the components of velocity vi and the temperature T which satisfy equations (2.12), (2.27), (2.30), (2.26), in the absence of the solid component, and with certain modifications of the heat and constitutive 46 equations (2.31), (2.36). These modifications are form- ally equivalent to employing (2.55) and setting a, d a a a d (2.63) 1' 4' 5' 8' 9 equal to zero. Equation (2.37) is then concerned with the unsteady linearized compressible flow about a state of rest of a heat-conducting viscous fluid, if we identify l: u, 302, 32 (£- d2 + d6), [5le of which )( and (4 P are the coefficients of viscosity of the fluid, [3&2 is the pressure of the fluid in the rest state, 32 (2- d + a ) is the isothermal bulk modulus and 3d a 2 6 10 is the product of the volume coefficient of thermal ex- pansion and the isothermal bulk modulus. From (2.31) the modified heat equation becomes 3c fl+YK Tf + kT, +51: = o (2.64) vat T pp pp where CV is the specific heat at constant volume, Y is the volume coefficient of thermal expansion, KT is the isothermal bulk modulus, and k is the thermal conductivity. The coefficient cV is related to Q7 by (2.65) o n I THIHI Q The material inequalities (2.39) and (2.40) simplify to 47 31 + 2u.2 0: U.Z 0' k 2.0'3% P KT_>_O and CV20. A properly posed initial-boundary value problem is then to find vi , p of class C1 and T of class C2 which satisfy the modified equations and the following initial and boundary data: A — A -— A Vi=vi’ p=p+p, T=T+T onBat t=O wpinp = 'i on BB1. v1 = Vi on 3B1, for t 3:0 (2.66) T=E+e on 5132, qknk = F on BB2, for t‘z O A A . where Vi' p , 'i' vi, 9, F Gi and r are prescribed functions on the appropriate domains. CHAPTER III. RECIPROCITY THEOREMS 3.1. Introduction Prior to considering the reciprocity relation in the mixture theory, we will review the well known reciprocity theorem of elasticity. Suppose that an elastic body is subjected to two systems of body and surface forces. The work that would be done by the first system's body and surface forces in acting through the displacements due to the second system's forces is equal to the work that would be done by the second system's body and surface forces in acting through the displacements due to the first system of forces. Mathe- matically this is incorporated in the Betti-Rayleigh re— ciprocal theorem. Theorem (Betti-Rayleigh) [25] Consider two equilibrium states of an elastic body: one with displacements ui due to the body forces F1 and surface forces Ti' and the other with displacements u! due 1 to body forces F; and surface forces Ti . Then it follows that I T.uf ds + I F.uf dv = I Tfu. ds + I Ffu. dv. 1 1 1 1 1 1 1 1 SE B BB B A generalization of the reciprocity relation to dynamic problems is given as follows. Theorem (Fung) [26] Consider two prOblems where the applied body force and the surface tractions and displacements are specified 48 49 differently. Let the variables involved in these two prdblems be distinguished by superscripts in parentheses such that the body force is Xéj)(xk:t), the specified surface traction is fij)(xk;t) on BB1, and the speci- fied displacement is g£j)(xk:t) on BB; where j = 1,2. Assuming that the action starts at t > O in each case, we have t f I x{1’(x.t-y)u[2’(x.y) dy dv B O t + I f fél)(xvt-Y)u£2)(X.Y) dy ds 8B1 0 t + f I 0[§)(Xat-Y)g£2)(X.Y)nj dy ds 5B1 . . t = J J X£2)(x,t-y)uil)(x,y) dY dv B O t + f I f£2)(x,t-y)u£1)(x,y) dy ds +£_ J 0(2) (xty)g( )(x y)nj dy ds As an illustration of this theorem consider the follow- ing prOblems. By problem 1 let us mean the displacement and stress field in an infinite region that results due to the body force system (1) _ (1) X1.- — Fi 5(p1) 6(t) 50 where pi is a fixed point in the medium and Fil) refers to a force magnitude. For prOblem 2 let us find the displacement and stress field in the infinite region due to the traveling impulsive force system (2) _ (2) f1 Xi — Fi 6(t - U) 6(x2) 6(x3) . Then substituting into the reciprocal theorem and using the properties of the Dirac delta function we find that (1) (2) Fi ui (p1.t) (3.1) (2) -- t x1 (1) = Fi f]]6(x2)6(x3) dxl dx2 dx3£)6(y- 7T)ui (x1,x2,x3,t-y)dy xl/U x = F£2)[ ui1)(xl,O,O,t — 1%) dxl. From this relation u§2)(p1,t) can be found when x (1) _ _1 . (l) _ ui (x1,O,O,t U) is known. When F1 — l and Fél) = Fgl) = O, Payton [27] has determined nil) by apply- ing the Laplace and Hankel transforms to the elastic equations of motion. The solution is (1) t (1) X1 2 11 (xot) = G(X :RIt) o 1.1 (xot) = ___—_— F(th) l 47rR2 1 2 4FR4 . (3.2) (I) _ £12 113 (Xpt) _ 4 F(th) 4WR where 51 F(R,t) = ’13:? H(t—R/c1) + 31:6(t—R/cl) -% H(t-R/c2) —El-6(t-R/c2) 2 G( R t) — (3353- - 1)l H(t—R/ )+ —-X-2-5(t R/ ) X1" " R2 R C1 ch ‘ C1 1 — 9-523 - ml H(t-R/c ) (5:- - 1);- 6(t—R/c ) R R 2 R c2 2 l R = (x2 + x2 + x2)2 - c and c are the s eeds of the 1 2 3 ' 1 2 P propagation of dilatational and equivoluminal waves. Payton used the equations(3.l) and (3.2) to compute the uiz). 1 3.2. Reciprocal Relations for Mechanically and Ther- mally Interacting Mixture We will investigate a reciprocity relation for the interacting continua of an elastic solid and linear viscous fluid using the theory derived in section 2. We put xi=xi+wi,p2=p2+n,T=T+e (3.3) where X1 is a reference position at time t = O, x is a position at time t, pz is the density of the fluid component at (xi,t), T is the temperature at (Xi't)' T is the initial temperature: then from (2.26) 8v an — i —+ --—-=o . at p2 axi (3 4) all quantities now being regarded as functions of X. and t. Since initially. the medium is in equilibrium 52 under zero total applied force it follows that a (3-5) 1 = pzaz' Let us consider the problems in which the body forces Fi(xi’t)’ Gi(xi't)' the specified surface tractions fi' gi, and the specified velocities ui, vi are given functions of time and space, respectively, for solid and fluid which starts its action at t > O, with the initial conditions = 0,. v. = O,n = 0,. 9 = O for t g_o, (3.6) Let the Laplace transform of a function u(xk,t) be written as 3(xk,P) where Huck») = ]' 6""t u(xk,t)dt. 0 We apply the Laplace transform with respect to the time t to every dependent variable. From (2.29), (2.30), (2.35), (2.36), (2.37), (2 55), (3.4) and (3 6) we Obtain “fil'p 3] + Big] — EiPG] (3.7a) ghi.p + $1 + 5251 = Eépgfi (3.7b) $1 =‘Eié2 h} _ 5341 35m i + a(P;; _ 3]) (3.7C) P p _ gik = 3P1- 51k + (<14 - §a1)gmm6ik+2(al+d5)gik+(a8+ %1-)T1’61k + 0‘9 9 51k (3.7d) 53 To complete the prdblem let us consider the boundary BB as the sum of the disjoint sets BB and BB' or as l 1 the sum of the sets (disjoint) BB2 and BB5. We specify for t 2.0, ui — vi = Ri on BBl (3.8a) (oij + 1rij)n.l =.Zk on BBl (3.8b) ui = (1.1.x).l = vi on 661 (3.8C) T = I + e on 6132 (3.8d) qpnp = F on BB? (3.8e) To aid in the computations we introduce the following combinations of material constants: p C1. 52:.4___1_1,Bl=68+3_1_ (3.9a) P P _ 5'2 + ‘5 _ 30L Y2 = p2 (-a'8 + l 2) (3°9C) P Now consider two problems specified by equations (2.26), (2.29), (2.30), (2.31), (2.35), (2.36), (2.37), (3.6) and (3.8). We identify one problem by a superscript one on all field variables and a second problem by a superscript two on all field variables. For example nil), vél), 9(1) will satisfy (3.6), (3.7) and (3.8) for R151), 21(1), U1”) , Vil)' G3(1), Fm 3(2), V(2), E,(2) , while u. 1 are solutions of (3.6), (2) 23(2) U<2) V<2) C9(2) F(2) I i I i I 0 (3.7), (3.8) for different Ri , 1 I 54 To derive the reciprocal theorem we begin with the equations of motion and the constitutive equations to which we have applied the Laplace transform and used the initial conditions as specified in equation (3.6). convenience these are For the sake of ~(3') _ ~(j _) - ~(j) = — ' ~(j) Opi p mi] '+ PlFi plPui (3.lOa) ~(j) ~4j) (j) _ —-'~4j) ”pi p + wi' -+ p2Gi — p2Pvi (3.10b) ~ . '_ ~,. 6' ~,- 1,- ~’. (1).”) = -_l- d 71(3) -_-_2- d eh) + a(Pwij) .(3)) (3.lOc) 1 2 ,1 1 mm,1 1 1 P P ~(j) O‘1 ~(j) 2341') ~(j) (1') 01k = -P_ 61k + Bzemm 51k + 3e ik + f317‘511: + 0‘93 61k (3.10d) ~(j) _ _ E2. _ ~(j) ~(j)_— (j) ~(j) 7Talk ' [ I? 0‘2 Y1“ + YZemm p2 OL10?" ]51k + xfrr 51k “43) (3.10e) + Zufik for j = 1,2 Multiplying equation (3.lOa) for j = 1 by 352) and again for j = then integrating over the region B, i ~(l) 2 by ui , we obtain, ~(l) ~42) - ~(l) Opi,p dv + I13 ui pl Fi dv mil) dv 342)~(1) - ~(2) °p1,p dv + f 111 plF i dv $42) dv 1 1 subtracting these two results and (3.11) 55 where dv is the element of volume of B. Consider the first integral on the left and let us apply the divergence theorem to it. In this way we obtain f312)0 Agi)‘, dv _J-Bu~1(2) ~(l)npds_J1"1’i(2:D 3;? dv (3.12) B B where ds is the element of surface area of BB. We note that the first integral on the right of (3.11) can be manip- ulated in the same way and the result is the same as (3.12) with the superscripts interchanged. Consider now the third integral on the left. From the constitutive equation (3.10c) we may write I 3:2) 3:1) dv = I §W(2) a(Pwél) - $§1))dv B B i + Pw(2)[ - :2- 0113(1) + Z; (1 3(1) ]n ds mm 2 5B p P - £3Pw£?; [ —E§-dl eém)+ Eg'az T“ )]dV (3.13) A similar result is Obtained after the superscripts are exchanged. Using the constitutive equation (3.10d), we have for the second integral on theright.of equation (3.12) 2 N1 - ~' ~ g L; 03:14:13; 3., _ I Pg(2) C(l)dv ip 0p1 = I Pe(2)[:2e 9é$)5p1 + 233ggi)]dv "’ip F P1 1 Pi 9 p1 B 56 A similar result is obtained for the middle integral on the right of equation (3.12) after the superscripts are inter- changed. When the equations (3.12), (3.13), (3.14) along with the equations after the superscripts are interchanged are substituted into equation (3.11), we have J‘fi’ (2) Pl~-(l)dV + fuimo 0(1)npds + J}; Pa 17917?) dv (3.15) U1 1 B BB p - J Pw.(2)[ -_-_g 0L e(1)+ -_l’- on n(1)]n.ds + f Pe(2)-—a n(1)dv 1 1 mm 2 1 mm flZ BB P P P - ~(2) 31 ~(1) ~(1) - .JBPe-1p [ P 5P1 + Bln 6pi + (199 5pi]dv = J'11~.(1)31Fi(2)dv + fai<1>3n ds + J' Pdwjfl)\7j(_2)dv B BB p p B _ ~ P P N _ J Pw(1)[- _Ed ale eéfi) + —l-a2n (2)]ni ds + I Peéi) _} d2n(2)dv BB P P B ~(1)[ ~(2) ~(2) _ J‘B pe —P-1 5pi + 611'] 6pi + (199 5pi]dv By the same process used to derive (3.15) from (3.10a), we find that from (3.10b) P1 P 1 f 91(2); 811).“; + gnaw. ds + Pd v$2>w $1>dv . 1 2 1 1 B B B + I91”)? 5} a g(1)+ 1 a 37$an as +f‘71(2)- f; a g(1)dv BB p p B '1 '5 ~(2) 32 ~(1) - ~(1) JBVi,p[( I? 0‘2 + YZe pZalo9 )(Dp1]dV W n ds + f Pdv§1)w§2)dv _ "*_(1)-2 ~i(?) ~(1)~(2) JOB Vi dv + £13 vi pi p B ~ '5 ~ 3 ~ . ,1 P ~ + J v§1)[— —%:ale(2) + j%-a2n(2)]n.ds + J v$13 jg-ale(2)dv BB p p 1 B 1,1 p m ~(1) 32 ~(2)_ — ~(2) .. Jl’Bvi'p[-. P (12 + Yzemm pzdloe 6pi]dv . (3.16) Before we add (3.15) and (3.16), we notice that the following volume integrals ~ P PP 1, My; 2 “gdepnm 1_ 2 Mg) B 5 le B - ~(2) £3 ~(1)_ ~ ~(1) - J Vi,p (- I> a2 + yzemm pzdloe )6pidv . ~(2) 0‘1 ~(1) ~(1) - J Peip [ —p—6pi + 611'] 6pi + c199 épi]dv may be recast by using (3.7) and (3.9) as , ~ P a ,1 ~(1 JB ”(mu-GB + 3:52 _ ?” erg) ?2 ' OL109 )Jdv (3'17) —'d a ~ +£pzéjM—a8.» %2_:_)n<1>__131_a99(1>]d, With the aid of (3.17) the sum of (3.15) and (3.16) can be written as (3.18) J vim-525:1)dv + Ju ui(2)_l~'(l) dv + J vj(_2)~(:jl_)n ds + ui2)~('}_)n ds ~ ~ P2 ~ P ~ +: J (vim- Pw.(2))(- (18151111) + —]-' a2n(1))nids Pa ~ ~ ~ ~ + g ( .25.; V1512) — alwngzhnmds + £9(1)(p2a10fé2)- agPeéufi))dv B 58 _ ~(1)- ~(2) (1 ) 2) 1 ~ 2 -'.'Bvi (32Gi dv+JBu.1 plFi dv+J‘a Bvi)1r1§i)n ds + ~Jl)~(2)n ds BB 1 p1 np +Jfiu)_~u5(_pza wm+pl .1 _ P ~(2) p alemm a n )nids 2 E'd 2 2 ~41) + I ( E’ vm BB M(l))n ds + f 6(2)(p2a10f(1)- a Pe(1))dv. W1 9 Let the flux of heat and heat supply function be incorporated into (3.18). The energy equation (2.31)reduces to a §§-+ a d 7 at 9 pp - pzalofpp r = O . (3.19) + qPIP alha HHTH Application of Laplace transform to (3.19) leads to ~ ~ _— 31~ E - Pa76 + Pagemm pzalofpp E'qppp + T_r — O . (3.20) ~ Consider the last integral on the left in (3.18). Using (3.20) it becomes -~1 ~2 B q(2) ~(1) and if we now apply the divergence theorem to the qp p term we may write (uSing (2.34) and (2.39)) J Pa7B(l)B(2)dv + EJB(1)?(2)dV - éJ” EWEQL ds B T B T BB P P -1 (nfim _B; wn~m)_qm E fBfif'p e’p dv E £643 (11 vp )dv. (3.21) 59 we note that a similar expression can be obtained for the last integral of (3.18) by interchanging indices. Substituting (3.21) into (3.18) and employing the transformed boundary conditions (3.8) leads to the general reciprocal theorem in the Laplace transformed state. Be- fore giving this expression we introduce one additional condition. We set Zij) = £1”) + 9i”) ‘ (3.22a) and require opinp = fij) (3.22b) Fpinp = gij) (3.22c) on the boundary, BB1. This introduction somewhat simplifies the notation but it must be recognized that only the total stress vector 2% is spebified on BB1 . Thus, by (3.8), (3.18), (3.21) and (3.22) we have IV£2)3265(_1)C3V + v; ui2)_l~j(_l)dv + fv .(2) ginds + ffii(2)%’i(1)ds B 5311 5B1 + I 512)E41)n ds + U(2)~(l)n ds _ 1 p1 p 313 1 p1 np 5131 1 . ~ 75 E + J <- Riz’n- :2- a1( 31- - plmwl + pla2(p(1’- %)/53nias as p 1 +1; (Vim) PW(2))[_E;OL (_p_1__~ 1(-1))/ + a (p(1)_ 53);] d 5 1 P Pi pl 2 p P “i S 60 o a ~ . E. Q ~ ~ + J (J52) (—Rn(12))nmds + J (-—-—2P 2 Vn(12) - alwnfiz)n nm ds 6131 BB1 + E J- 3<1);<2>dv - 1; any». a. - _1 r 3<1>3<2>ds E‘- B T BB p p T 55' 2 _ Eff 3(1) (3(2) __ ;;:(2)dV E? B IP I) P = [1‘3 v§1)328’§2)av + {3 ui(1>'51£~’i(2)dv + J' V§1)§i(2)as+j 3:51)???)(3 BB1 BB1 V<1)~ (_2)n d3 +J U(1)~<2)n + I i 7Tpl p1 np (38 B1 BB1 - ~(1) __2 511(2) ~(2) Ea - +J (-R’j_ )[- (11(P - )/p1 + P132(P2 ' P)/p]nids BB 1 - ~(1) ~(1) E_2_ p_1_ (2) ~_(2) p2 — + J (Vi - PWi )[- _ a1(P - pf )/pl + 9122(92 P)/P]nids BE P 1 5-0 E'd ‘~ + . ( —)(-2:R(1))nmds + J; (—%;2-Vél) — alW(l))nm ds 3131 5131 + E J' 3(2)}?(1) dv éj'o 5(2)~(1) npds - éj 3(2)F(1)ds T B T T -— 5132 BB2 _ 5' -~(2) ~(1) _ ~(1) 7f JB e'p (up vp )dv (3.23) Since the inverse transform of the product of two functions is the convolution of the inverses, we obtain: 61 Theorem (Reciprocity Relation for Heat Conducting mixture). Let B be a bounded regular region of three-dimensional Euclidean space occupied by a mixture of elastic solid and a Newtonian viscous fluid undergoing a disturbance of small amplitude during the time interval t 2.0. We denote by BB 0 the boundary and by B the interior of B and introduce regions 8, 80 of space-time defined by ° 0 B=[(p,t):pEB,tZO},/3={(p,t):p€B,tZO]. We use notation 6B1, BB2 and BBi‘, BB; fOr arbitrary sub- sets of BB and their complements with respect to 8B and 11 refers to the unit outward normal vector field on BB. Suppose that the constants X, p, a, k and K' satisfy 3x+2u20,u20,a20,k20,x’234¥a1<, andthat a a and a satisfy the inequalities Q‘2' 0‘ 6' 7 8 1! 4! a5! 0' d1 + as 2_O, 2 (Ea P 2 + d6). Let the mixture of elastic solid and viscous fluid be sub- jected to two systems which are distinguished by superscripts in parentheses. Let the functions vij), péj) ' uéj) of class C1 and T(j) of class C2 on B 'which satisfy equations (2.26), (2.29), (2.30), (2.31) on 6" , equations (2.25), (2.27) and (2.34) to (2.38) on B and the subsidiary conditions 62 wi=0,ui=0,vi=0,n=0, 6:0 on B at t=0, ui(j) _ Vfij) _ RU).O (j)n = fflj)’ (j)n = gfij) ' 1 0pi np 1 7TPi nP 1 O(3') (3') __ (j)* Opi np + Wpi np —,ZA on BB1 for t 2_O u. =‘U‘j), v. =‘V$j) on 55' for t 2_O , 1 1 1 1 l (J) _ - (J) (j) _ (j) — T T + 9 on BB2, qp nP — F ~ on 8B2 for t 2_O, where Rflj)'f.(j)'g§j)'ufij)'Vfij)'9(j)'F(j),F_(j) (3(3)::1nd r0') 1 1 1 1 1 1 are prescribed functions on the appropriate domains and 51, 55, E. are given, positive, constants. Then the work that would be done by the first system in acting through the velocities of the second system and the work that would be done by the second system in acting through the velocities of the first system satisfy t 32 J“ J' vi”) (x,t-Y)Gi(l) dy dv+31f J” ufiz’ (xx-12min (x.y>dy dv B o - t (2) (1) t (2) (1) +J f v.1 (Xpt-ymi (x.y)dyds+ I fui (X.t—y)fi (x,y)dyds 313 O 5310 + J" JO tvfz’ (x t- ym‘. ’o(1’ (x y>np dyds 81310 8B10 t -P - I I Riz) (X.t-y)[_ __2 01(P1-Pil) (X.y)) O P P BB 1 1 Pa ._ + _2(p‘1’1(x.y>-p2>] ni dy as + *Attention is called to the fact that onlyZ)i is known on 831 and hence only the sum fi + 91 is given. 63 t I J (Vi (th-Y)-Uj(- ) (Xot-Y) ) [- ___—'2_ al(Pl-P](_l) (x,y)) -' O 331 plp 31a _ ‘fg‘g (Pél)(X.Y) ‘ P2)]ni dy ds t —- (2) t J J p2a2Rm (x,t-y) nm dy ds + J J [52d2V(2) (x,t—y) o __ o m BB1 BB1 2 le$ )(x,t-y)]nm dy ds tilrbl t I {)r‘2’e‘l’ay dv B t J I [qé2)dy dv t .52 j. J; Vél) (XIt-Y)G1(_2) (XoY) dy dV B - t (1) Pl JJ; ui (X.t-y)Fi(2) (x,y)dy dv B t (1) I I Vi (X.t-Y)g£2)(x,y)dy ds 5 0 B1 t (1) 2 J I ui (X.t-y)f§ )(X.y)dy d8 53.0 1 64 t (1) (2) + J J v.l (x/c-yhrp.l (x.y)np dy ds t J {) Uil)(X.t-Y) Ogi)(x,y)np dy ds . .t __ J J R{1)) a B O plp :%- d2 (Pé2)(X.Y) - PJ] ni dy ds ° 't (1) .1(1) ‘52 -— 91(2) 331° p1p El-a { ‘2)(x > -" 1] n a ds 3 2 P2 J P2 i y J J: 2d 2R(l)(x, t— —y) nm dy ds B]. J J [Pzazvm (x,t-y) - alwm [:—— a1 —«i> 6(1) 0(1) .. ,'w , v , 13 ij 4 0 as [£1 4 m for i = 1,2. I Then the reciprocity theorem (3.24) reduces to: __ t P2 J J Vi2)(X.t-Y)Gil)(X.y) dy dV B O _. ,t + p1 J J ufi2’(x.t-y) FJ1)(x.y) dy dv B O 5' t (2) (1) K’ - -t (2) + z. J J r (x,t-y)9 (x,y)dy dv - :: J J [u (xot-y) 'T B O T I3 0 P _ (2) _ (1) vp (x,t y)]9'p (x,y) dy dv —- t (1) (2) = p2 J JO vi (x,t-y)Gi (x,y) dy dv B _. t + p1 I I0 u§1’(x.t-y) FJ2’(x.y) dy dv B " . t (1) (2) K’ - t (1) + E: J J r (x,t-y)e (x,y)dy dv — :3 J J [ u (x,t-y) 'r B o T 13 o P - vgl’(x.t-y) F;1>(x,y) dy a. B O t + J wéz)(x,t-y) f§1)(x,y) dy ds 0 BB1 t (2) (l) J) wi (x,t—y)opi (x,y)np dy ds Bl + alt—3 — t + 2: J J r(2)(x,t-y)9(1)(x,y) dy dv T B o 1 - t (2) (1) -:J I qp (x,t-YW (x,y)n dy 518 T BB 0 p 2 t I I F(2)(X:t-Y)9(1)(x.y) dy ds BB 0 2 Hind 71 — . t (1) (2) = p J J wi (x,t-y) Fi (X:Y) dY dV B O t f Wil)(X.t-y) f£2)(X.y) dY ds 0 + I I W£1)(X:t-Y)Oéi)(x,y)np dy ds t J r‘1), F_<__aw e -- ax: fX— BX, dX— at '— atBX' (405) P1(X,t) = 31(1 "' %)' (4.6) 77 (x,t) + E? %§-= O, (4.7) 5P2 at as the only non-vanishing kinematic relations, and we note that all other strain, rate of deformation and vorticity components are identically equal to zero. The constitu- tive equations (2.35) to (2.37) under the restriction (2.55) become p'a 2 —'d BP = 2_1 a V; + 1__2 5X2+a(—aa-W€-v).wy=wz=o (4.8) P 3X P _10‘1 aw ox(x.t) =(2(a1+a5) - (_E— - 0.4)) 3; _. 01 + agTe + (::-+ d8)n + a1 , (4.9) Oy(x,t) = OZ(X.t) = E1‘11 aw — O‘1 —(—:7--a4%$;+d9T9+(::+a8)n+al, ULlO) P P xy xz yz 1r (x t) = (2 +x)§-‘1+E>'(1a2-a)§-V--Ea Te x ' u ax 2 3 8 3x 2 10 3+—2 _ _ 1 _( E. 32+P2a6)n'P202 I (4. 2) _ ._ Ar - __1_ _ .231_- - TTy(X,t) —1rz(x,t) — )(ax+p2(E d2 01.8)BX pzleTe 3+"2 _ _ 1 -( a2+92a6)n-p2a2 , (4. 3) 78 w = w = w = 0. (4.14) xy xz yz The equationsof motion (2.29), (2.30) and the energy equa- tion (2.31),under the constraints (4.2) to (4.4),become ad 2 x —- a w __ _ = p -—-— ' (4.153.) ax x l at2 an x _ —- av ax" + “’x ‘ p2 at ' (4'15b) 2 a TTK' :2 E'a TTK' — 59 a Q 9 b w _ 2 10 av: (1.7T at+k 8X2+( _T_ )axat ( 7f )——-—BM 0. (4.16) To complete our formulation of the initial-boundary value problem we prescribe that for t :50 W(X.t) = EETX.t) = V(X.t) = O. at p2 = p20 e(X,t) = on (4.17) In addition, we require that on the boundary x = O, * 3N 6(O,t) = h(t) , B-E(O't) = O, V(O,t) = O, (4.18) while as xaw,‘we stipulate that B(X.t);W(X.t). p2(X.t). V(X.t). OX(X.t). WX(X.t). 0y(X.t) and wy(x,t) approach to zero. (4.19) At this point, we introduce dimensionless variables. For this purpose we use the notation introduced in (3.9) * h(t) is the Heaviside unit step-function defined to be zero for t < 0 and one for t > O. 79 wherein the relation (3.5) is used. A direct substitution of (3.9) into (4.9) yields 0 h(t)=(8 +28)§K+QLTB+B +d (420) x ' 2 3 8X 9 ln 1 ° which, if the material were elastic, would lead us to ex— pect 82 + 283 to play the role of the Lamé' constants (XE + 2LT? while agT' would play the role of (2LH3+ 31E)aE where dB is the coefficient of linear thermal expansion of an elastic material. With this in mind we choose a velocity cl , B -+ZB Ci ='—;:——41 (4.21) pl which would be the irrotational velocity of sound if the material were elastic. Since a7 530 by (2.40), we define w = ——— , (4.22) By a dimensional analysis we have that cl is a velocity while (4.22) has dimensions of length squared per unit time. Thus, if we take N N __LP— _L a — c , tO — C (4.23) then a dimensionless x-coordinate and time are given by 2 c x c t 1 _ t _ l c _—2—-,¢_——t __2. (4.24) w w 80 Proceeding further, we introduce non—dimensional partial stresses, solid displacement, fluid velocity, densities and diffusive force by G C! /\ A A OX=FTX$'W E'Ooza+12fs'Q=r'3"7:L2'B" 2 3 a 2 3 Y 2 3 I? = fix G—Vto 1’? = W x 82-+283 a y 82-F283 _pz'pz _pl'pl A_ awx n2 - -*::‘—— . n1 - -—::-—— : w - gf‘Ijfif‘ - (4°25) p2 pl 2 3 In addition, the following quantities are conveniently grouped: 82 __2.H..'_"_A.____ d =_.C_12'_r.__ d =w t0(62+263) l BZ+ZB3 2 BZ-+ZB3 p 2 3 2 3 p _ fa2 p2 Y1 ~ _ a9T+K _ p2alOT + K 6 -———'- __ . E -“——f:—- . E - ._ . (4.26) 2 2 C2 1 a T2 2 a T2! c=1 pl 1 7 7 Incorporating all of these changes leads us the following summary: Constitutive equations /\ /\ OX(C.T) = 80 Jig—TL + ale(t;.T)+[ (1—f)90- 61]n2(C.T) (4.27a) 81 A A A A A ww 10.52 1’31?“ (51+f0013Wa; T - d28(C.'r) +(42- (1—f)90>n2r >>- 5 >6 >> t>0, - 52—51 >> t>0,1 >>— 5 (4.37) 2 l 2' whe ef COi Se tC st of Si 83 onto '52 Where t =-::— and r =-:: . (4.38) Pl pl The assumption of (4.36) is analogous to neglecting the effect of the mechanical coupling term in the Fourier heat conduction equation in linear thermoelasticity theory [22], [23] and is justified by the conditions (a) and (b) of Section 2.2. Various material constants which appear in (4.37) have to be determined experimentally for the mixture. The re- strictions on the material constants, (4.37), are the results of (2.39), (2.40), and their interpretations in Section 2.4, Single Constituent Theory. For additional references consult [33] to [36]. With the aid of (4.35) to (4.38), the equations (4.28) to (4.31) are written 2A A an 2A Lg-t§vT-+to+ala—g-al—T2=a—g. <4-39) 3g 8T 2A A 2A 34) A 8 w 5w 2 B v A pg. .__2 _ .2! BC. BC 2 ag_%_e;___0, (4.41) 56 34) A 1?2+ 2%: o (4.42) 4.3. Solution by Integral Transforms We denote the Laplace transform of a function F(§,T) with respect to T by EXC,p), where 84 app) = J F(§.'r)e_pT d4 . (4.43) O c+im 2m F(§,T) = J F(g,p)ep'r dp. (4.44) c-iOD In (4.44) c is a positive real number such that the path of integration is any vertical line to the right of all singu- larities of E(§,p) in p-plane. The solution to the thermal prOblem, equations (4.41), (4.32) to (4.34), in the (Q,p) plane is Rap) = exp<-p1<*c). (4.45) l P Now transforming the equations of motion (4.39), (4.40) and the equation of continuity (4.42), then combining these equations with the solution to the thermal problem (4.45) gives 2 2 — 61 2 — d1 5 (D - pt - p )w+ (— D +t)v=— eXp(-p C) (4.46) p pt 6 d (51132 +Pt)V—V+ [ (82 -—2')D2 - t - pr];= - --2- exp(-p1~’t) (4.47) p pi where the differential operator is defined by __éL D — d: . Due to equations (4.32) to (4.34), the transformed boundary conditions and the regularity conditions are w(0,p) = O, 3(O,p) = 0 (4-48) 85 and as C-4m a(c IP) I 3(C0p) 40. (4.49) Solving for the displacement of the solid constituent and the velocity’ of the fluid constituent from equations (4.46) and (4.47) for the homogeneous solutions only which conform with (4.49), we have _ 1.? wh(z;,p) = B(p) exp(-121EI(91+92)C) 1.) + D(p) eXP(‘§‘f‘; (gl-g2)C)' (4.50) _ 9.9 vh = [p( (t+p) (ps2-52) + t 61}-211'P(91+92)2]B(p)exp(- 12137519142) 0 HM (t+p) (psZ-éz) + t 51} - %p(91-92)2]D(p) 2 exp<-§—f—l- (gl-gzm. where B(p) and D(p) are integration constants, and 9i(P)==(p82-62)(p+t)+(pr+t)+2t 61 1 + 2p%(psz-6i-52)§(pr+t+tr)é, 93(1)) = (1382-52) (p+t)+(pr+t)+2t 61 - 2p%(psz-6i-62)P(pr+t+tr)P, 2 ._ 2 2_ fl(P)—PS -61 52 o (4.51) (4.52) (4.53) (4.54) 85a Here we have assumed that 1/2 Re(§?fc> > o and Rag—fl (gl-gz)c> > o (4.55) for p such that Re(p) > M > 0, and we will show this to be so later. Let us indicate particular solutions to the equations (4.46) and(4.47) by a subscript "p". Then we have 6H: >=5-ex<-’/2c) (456) p lp A2 p p ° 5P(CIP) = 2 A1 .P{51+527952+t51+(P52-52)(P+t)%§_ [ <— + P(t52+r51)+t(51'52) pszdl‘égd1+éldj2_ p1/2 1 1exp(-p /2C) (4-57) p(tsz+r61)+t(61-52) . . where A1 = d1(P(32'r)‘é2't)+d2(51+t): (4°53) A2 = ppl’é(psz-él-62-t—pr- (t+p) (psz—éz)-2t51+p2r+pt+ptr). (4.59) The entire solutions are §(C:P) = Gh(C:P)r+ifip(C1P)o (4°60) 5(C.p) = 51102.9) + 590513). (4.61) To determine B(p) and B(p) we substitute equations (4.60) and (4.61) into the boundary conditions (4.48). Then 81(P) B(P) + D(P) +'X;T57' = 0 (4.62) 86 (91+92)2 B(p)p[(p+t)(p82-62)+t51- 4 1 + - )2 D(P)P[(p+t)(p52_52)+t51_ (91492 2 A1 + [p(61+62-p52+t61+(p52-62)(p+t»X;- (Psz’52)d1 51d2 + pf. +373] ‘0 ‘ .(4'63) Solving for the unknowns B(p) and B(p) from equations (4.62) and (4.63) simultaneously, we have B(p) = Eggtoiwz-psu awe-es.>(p+t>+(pr+t>+2ta.n {(P52‘52)d1 + 51d2} A1 + . pp1/291g2 — EX; ' (4.64) -A1 2 1 B(p) = EIEEX;'[61+62-psz+ 5((psz-62)(p+t)+(pr+t)+2t61}] {(P32-52)d1 + 51d2} A1 ‘ -' fix: ' (4.65) 1 pp /29192 The Laplace transformed displacement of the solid constitu- ent and velocity of the fluid constituent are now explicitly given by equations (4.50), (4.51), (4.56), (4.57), (4.60), (4.61), (4.64) and (4.65). These quantities constitute the completesolution, in the transform plane, to the initial- boundary value problem posed by equations (4.32) to (4.34), (4.39) to (4.42). Stresses of each constituent may beex- (pressed immediately in terms of the transformed displacement of the solid constituent and velocity of the fluid VF, r“ .k...J 87 constituent by means of the constitutive equations (4.27a) to (4.279). Now that the Laplace transformed displacements of the solid constituent and velocities of the fluid con- stituent are in their simplest form we proceed to invert these expressions. 4.4. Inversion A. Location of Zeros of gi(p)g§(p) As a first step toward the inversion of w(§,p), we examine the multiple valued functions appearing in equations (4.52) and (4.53). Set 2 '51 ’ 52 Then equations (4.52) and (4.53) become 91(p)=(psz-C2>(p+t)+(pr+t)+2t§1+2r1/28p1/2(p+€1)1/2(p+€2)1/2. (4.68) 92(P)=(PSZ-€2)(p+t)+(pr+t)+2t61—2r1/zsp1/2(p+el)1/2(p+€2)1/2. (4.69) 1 We take the domain of definition of p /2 as the entire p—plane cut along the negative real axis, and we choose a 1 ibranch of p /3 through the requirement that 1 * p /2 =nJ£ for p I z > 0. (4.70) ¥ .* ijrefers to the positive root for real positive 2. 88 o I I I 1 We take the domain of definition of (p+el) /2 as the entire p-plane cut from —61 to -a) along the negative real axis, 1 . and we choose a branch of (p+€1) /2 through the requirement that (P+€1)1/2 =~fZIEI for p = fl > -61 ** (4.71) We take the domain of definition of (p+o$2)1/2 as the entire p-plane cut from -62 to -oo along the negtive real axis, and we choose a branch of (p+62)1/2 through the re- quirement that ** (P+€2)1/2 = [fie—2 for p = Z > -62 (4.72) Now we will choose a domain in which g1(p) and g2(p) are single valued. In View of (4.70), (4.71) and (4.72), we find that gl(p) does not have branch points but 92(p) does. To see this we note that all possible branch points of g1(p) or g2(p) are the zeros of g:(p)g:(p) which is a fourth degree polynomial, 2(t32-r-62) 2t(-1+261-52-2r) 32 )+p2[ $2 91(p)9§(p) = s4[p4 + pS( [(t52+r"52 )2+4r((5:+<52 )] + S4 2 2 2t +p[(1+261-62)(ts +r-éz)+(2+2r)(61+52)];:- t2 + 3 (1+261-62)2] o (4.73) *tDue to (4.66), (4.67) and (4.37), we have 0 < e, < 61. 89 We now use (4.37), i.e. 0g§

= s4tp2+§.<-r-az+o(t>)p+ 2t[(1+261—62)(r-62)+(2+2r)(6i+62)«5(t)] [P2+ - ' 5% P 2 4r(61+62)+(r-62)2 t2(1+261-62)2 + 2 4r(51+52)+(r-62)2 and the zeros are easily found to be P1, P2, Pi, P: where r+52 'Vrél p1 = 32 Mt) +211 $2 +-o(t)] (4.74) _t[(1+251_52)(r-62)+(2+2r)(6i+62)+o(t)] P2 = ‘ ’* (4°75) 4r(6i+62)+(r-62)2 . +2 J‘Z—‘E-i-éz){r2-4r61+2r2<51‘25152(l‘r)+éi+5:+2r52+r26i+0(t)J ti ~ . I 4r(6i+62)+(r—6,)2 and p: is the conjugate of pi.+ Here we note that the expressions under the square root signs are positive due to (4.37). B. Determination of Branches for g1(p) and g2(p) All the zeros of gi(p)g§(p) are the zeros of g2(p) be- CENJse, due to (4.68), (4.70) to (4.72), (4.74) and (4.75), + * FWDI computational purposes it is desirable to have pi, pi 111 a series expansion of t. See Appendix 1. 90 we find that 2 4 2 Re glue.) = §§(r+éz-2c51) + o (4.76) 2 Im 91(p2) > % Std-E1 > 0. (4.77) Since g1(p) never vanishes on the entire p-plane with negative real—axis being deleted due to (4.70) to (4.72), g1(p) does not have branch points. Moreover, we find from equations (4.73), (4.74) and (4.75) 92(9) = 9—???) (p-pl)1/2(p-p’{)1/2(p-p2)1/2(p-p;)1/2. ' (4.73) and this shows that p1, pi, p2, p: are branch points of order one for g2(p). We define the domain of the g1(p) to be the entire p-plane with negative-real axis being deleted, and choose the branch g1(p) by the requirement that 91(9) = V (1252-62)(/z+t)+(£r+t)+2t<51+ J71:— W£+epl£+ez . for p = g > 0 (4.79) We define a domain in which g2(p) is single valued such that the domain is the entire p-plane out along the follow- ing curves; (a) the negative real axis (b) the line joining p1 and pi, that is, the locus of points p such that Re(p) = Re(pl) and mp?) _<_ Im(p) .1 Im(p1) (c) the curve joining p2 and p3, that is, the locus of points p along 91 (1) the line such that Re(p) = Re(pz) and Im(pz) _<_ Im(p) _<_ 1262022). (2) the portion of circular arc whose radius is equal to --J2 Re(pz) and whose argument lies -3v ‘ 3? be tween T and —4- , (3) the line such that Re(p) = Re(pz) and Re(pz).1 . Im(p) _<_ Im(P§)- We choose the single-valued branch of gz(p) by the require- ment that 92 (P) ="/ (1852-52 )(£+t)+(zr+t)+2tc51- J??? “J £+€1J£+€2 for p = E > 0. (4.80) Due to (4.79) and (4.80), we find rl/2 91(1)) = 1380 + 172— + 0(->) as p -> 00 (4-81) F S - 1 rl/z o1 4 92 92(9) '98( -m+ (5)) as p—> 00 (- ) We find that —62 < Re(pz) < Im(p;) < 0 with the aid of (4.37), (4.38),(4.67) and (4.75) as in the Figure 1. C. Formulation of w(§,p) in Convolution Form _ _ 2 It is expedient to define w(1)(§,p), w( )(C,p), w(1)(§,r) and w(2)(C,T) such that V7(C,p) : V7(1)(Clp)é(2)(C0P)l (4°83) cut for g2(p cut for g2(p) I' E' D' A' Ky Figure 1. The p-plane. 93 . C+i . w)1) = 2:1 I “’5<1> . . (C:P)eXp(pT) dp with i = 1,2 C’la’ (4.84) so that by the convolution theorem w(c.r) = I: w(1)(c.u)w(2)(c.T-u) du. (4.85) From (4.50), (4.56), (4.60), (4.64) and (4.65) set K a(1)(C:P) : 1" (4°86) fi‘z) = 1(r_sz?;(p_a) 9:92(54+az-ps2+§((ps2-é.>(p+t> (Ptb)I(P52‘92)d1+51d2) P9192 +(pr+t)+2t61)) + A1 1/2 - ] (_P__ } 2(r—sz)p(p-a) exp Zfl (91+gz)c A -t 1 1 Iéi+az-ps2+%((ps2-az)(p+t) “(r-s2)p(p-a> 9192 (P+b){(P52‘52)d1+51d2} ‘ P9192 +(pr+t)+2t61))+ - A1 ._ 21/2 _ 2(r'52)P(P-a)]expI 2f1 (91 92)C} + A1 eXP(-P1/2C) (4°37) (r-Sz)p(p-a) with a and b in (4.86) and (4.87) defined from (4.59) such that A2(p) = (r-s2)pp‘/2. (4.88) 94 Here a and —b are zeros of psz—éi-éz-t—pr-(p+t)(psz—éz)-2t51+pt+ptr = 0 and they are 1 a = -——————[r+t52~32-62—t-tr—[(sz-r-tsz+c32+t+tr)2 2(r-sz) _ 2 2 c 1/ “4(r-S )(-61_62-t+t02—2t61)] 2} (4 .89) b =-————1———{r+tsz-sz—c52-t-tr+[(sz-r-tsz+c52+t+tr)2 2(r-sz) 2 2 1/ “4(r—S )(‘51’52—t+t52-2t61)] 2] (4.90) and we find that a > 0, b > o by the relation (4.37). — 1 The inversion of w( )(C,p) of (4.86) is found with the aid of the table [37] J1 w(1)(C,T) = g-fo exp(-b(T-zz)) dz. (4.91) To obtain a real integral representation of w(2)(C:T) for T > C, consider the integral 1 2Wi f V7(2)(1:.1;>)exp(pr) dp (4.92) evaluated along the closed contour shown in Figure 1. . -(2) . D. InverSion of w (C,p) by Contour Integration _ 2 We express w( )(§,p) in a simpler form from (4.87); 5(2)(P' ) _ A3(P) 1 _ A1 )ex (_ 74(9 +9 )C) S p ‘(r-S?)p(p-a) 9192 2(r-sz)p(p-a) p- 2f1 1 2 A3 (P) 1 A1 i4— -((r-sz)P(P-a) 9192 + 2(r—32)P(P-a))eXPG-2f1(g1—gz)C) A1 exp(-pl/2C) (4-93) + (r-82)p(p-a) 95 where A3(P) = %d152(r'32)P3+(52(%Tdi+9152‘ ééidz) + r(éidz‘ %9152‘ édir)+t52(‘ %d182+%d1+%d1r+%d2))p2 + (r(_dléi' édléZ- %9251)+ édzéibz‘ %diég+t(32(%di“dz + dléz-dlél- %d261)- édlréz-d1r-d1r61+61d2+r61d2-d162 + %diéz' %9252+%Td2)+t2(%9252‘ %d152))P+'%t((51d2 “9152)(52-1-25i)+2(dz‘d1)(51+52)) + %t2(d2-d1)(-62+1+261). (4.94) With the aid of (4.81) and (4.82) we find that 1/2 exp(- 2f E-------(91+92)C +pT) 0(eXp(p(T-C)) as p -+ 00. / / / (4.95) 1 1 1 . exp<- 51—; (91-92): 1..) = o a), (4.96) and consequently as p-> a) _ 1/ 1/ w(2>(c.p>exp = o<§eXp>+ 0(geXp C. From (4.73) and (4.93) we see that the contributions from the small circles around the branch points of gz(p) go to zero as the radii tend to zero. Let us consider the contribution 96 to the integral from the small circle around the origin as the radius tend to zero. We may express w 2)(2‘;,p) of (4.93) as _(2) = A3(p) 1 _ 21/2 w (C19) (r-sz)p(p—a) 9192 [eXP[ 2f1 (91+92)C] 1/ - epr- (“filial-92m] A1 [ { 312% + )c) _ ex _ 2(r-sz)p(p-a) p 2E1 91 92 1/ + eXp(- Ere—12(91-92)Cl-ZeXp(-pl/2C)]- (4o98) We note that 1/ 1/ eXp(- 5;:3491+92)C)-exp(-45f;i(91-92)C) 1/2 = - Efi—Igzc +-O(p) as p -> 0, (4-99) 1/ 1/ eXp(- 55;:(91+92)C)+eXP(- 5g:3(91-92)C)-Zexp(-pl/2C) = 2p1/2C(1-‘;%1)+(3(p) as p —% O, (4.100) Combining (4.98), (4.99) and (4.100), the contribution to the integral (4.92) along the small circle around the origin goes to zero as the radius tends to zero. Also, from (4.93) we see that the contribution from the small circle around p = “£2 goes to zero as the radius tends to zero because the integrand is bounded around p = -€2. These considera- tions, with the aid of (4.89), (4.93) and Cauchy's integral theorem, lead to 97 lim (%EI'f w(2)(C.p) exp(pT)dp) E w(2)(§,1)+ 21. lim(f + f + f + f + f + f + f ”1 ON NM ML KJ HG1 G1G2 F2F1 + f + f + f + f + f + f + f DC BA N'O' M'N' L'M' J'K' + f + f + f + f + f ) GiH' 656; Fin; E'Fi C'D' A'B' A3(a) A1(a) al/z \ ) ~ ( - -—--—-)exp(- Zfl a (91(a)+92(a))c+ar) (r—sz)agl(a)g2(a) 2a(r-sz) a a a1/ A3( ) + A1( ) )eXp(_ 2f1 : (91(a)_92(a))§+at) -( (r-sz)ag1(a)g2(a) 2a(r-sz) + —§l£32-exp(— a1/2§+a1) (4-101) a(r-sz) in which the integrand of the integrals in the brackets is w(2)(§,p)exp(p1) and “lim” refer to the limit process such that the large radius tends to infinity and the small radii tend to zero. The values of pl/z, f1(p), g1(p), g2(p), ‘which are needed to evaluate the integrals in the parenthesis are to be determined consistently with the construction of the Riemann sheet described in (4.70), (4.71), (4.72), (4.79) and (4.80). D—l. Evaluation of gi(p) along the Contour For this purpose it is expedient to introduce new functions 21(9) = 9:(p) i = 1.2 (4-102) \vl 98 _ 1/2 . _ G(Zi) - Zi 1 - 1.2 (4.103) where we define a cut for G(Zi) to be the negative real axis on the Zi-plane such that G(£) =IJ7 for g > 0 and ‘ (4.103a) G(-£) = 172 for g > 0 . ' Then we shall choose proper signs along the contour in Fig. 1 in the expression so that gi(p) for i = 1,2 are consistent with the Riemann sheet described in (4.70) to (4.72), (4.79) and (4.80). Due to (4.79) and (4.80) we maylxiumsflt28chwarz reflection principle, and in this case we have gi(p*) = gi*(p)t (4‘104) and since the contour in Fig. 1 is symmetric with respect to the real line, we determine the value of gi(p) along the contour which lies in the upper half-plane only and utilize (4.104) for the lower half-plane. Considering the mapping of the contour in the half-plane of Fig. 1 into the Zi-plane, and then into the G-plane under the restriction (of (4.37), we find that g1(p) = G(z1(p)) along H6162, F2F1E, DC, BA, JK, LM (4.105) 91(9) = - G(ZI(P)) along NO (4-106) 92(p) = G(z,(p)) along HG1G2, BA, JK, LM (4.107) 92(9) = - G(Zz(P)) along FgFlE, DC, NO. (4.108) 99 We are now ready to evaluate the integrals in the parenthesis of (4.101). D-2. Integration along ON-N'O' for T Z.C and T < C* Along ON, p = -3 with 61 < 2 < a), and by (4.106) we find that (“r 2'"—'2 _ - “"2 where x = 5222 — (r+ts3-02),z+t+2tol-to2 - 1 1 Y = 2:1/252 /2(2-62) /’(£-e1)1/2. The contribution to the integral of (4.92) along the contour ON-N'O' is then 1 GD -A1(-E) - A3(-£) ‘— - . Sinaffc) 7T ftl (r—sz)g(g+a) ( + ((I'Sz)£(£+a))91|2 A1(-£) , .fZ _ 51“ m ex - d 4.110 2(r_sz)£(£+a) > (SJY:EI(I 91)C)) P( £1) 2 ( ) for 1.: C and T < C. D-3. Integration along the Arc NMeM'N' for T Z.C and T < C* We next consider the contribution to the integral of (4.92) along the arcs NMeM'N'. Near p = -€1, we have 1 that P /2(91‘92) and * For T < C, see Section D-7. 100 A3(P) + A1(P) (r-52)p(p-a)9192 2(r-sz)p(p-a) are analytic due to the fact that along ML, p = -£ and 91=ff:82£2+(r+t82-52)fi-(t+2t51-t52)+Zs/T\/7VE-€2J€1-£. (4.111) 92 = 9: and along ON, g2 = g: where g1 is given by (4.109). Relations (4.52), (4.53), (4.54), (4.66) and (4.67) lead to 9 2 1 1 91 “ 92 ' 4r1/2f1P /2(P + 62) /2 or 1 1 1 91 + 92 Zr /2P /2(P + 62) /2 (4 112) 2f1 _ 91 ‘ 92 i . and this last relation shows that the contribution to the integral (4.92) from the summand of the integrand, i.e., 1/2 A3(P) _ A1(P) ex _ P ((r-sz)p(p-a)9192 2(r-Sz)p(p-a)> p( 2f1 (91+92)C+pr) + A1(p) exp(-pl/2C+pT). (r-82)p(p-a) vanishes as the radius of the arc NMeM'N' approaches to zero. But the contribution to the integral (4.92) from the summand of the integrand, i.e., A3(P) _ A1(p) ex _ 2:1? _ (r-82)P(p-a)g1g2 2(r_sz)p(p_a)) P( 2f1 (91 92)C+pr) is not readily established because f1(p) has a branch point at p = -€1. We consider the mapping: u = (p + €1)1/2 (4.113) 101 which maps the contour in Fig. 2 onto the semi-circle C' in Fig. 3 with the direction clockwise in both cases. We define IR = 2;i f .I(u) exp(- zéEl-+ (u2 - €1)T) du (4.114) c . where I(u) = 2uA3(u2—€1) ‘ (r‘sz)(Hz-61)(u2‘€1‘a)91(u2’€1)92(u2‘€1) _ “A1(u2‘61) , (4.115) (r-s2>(u2-e1-a> v(u) = (‘12 ‘ €1)1/2(91(u2‘€1)-92(u2‘€1))- (4-116) ‘1 Now we consider the branch of (p + 61) /2 such that 1 (P+€1) /2 = -'J£+€1 for p = B > ~61, (4.117) with the cut from -€1 to —a> along the negative real axis. We consider the mapping u = (p + €1)1/2. (4.118) If we were to go over the circular arc twice in Fig. 2, then we see that the mapping (4.118) will map the circular arc into the semi-circle C“ shown in Fig. 3. As we did in (4.114), we define 1 v u IL = EET-fcul(u)exp(- 5%”) + (u2-61)T) du. (4.119) If we add (4.114) and (4.119), then by the residue theorem IR + IL = Residue (I(u)exp(- géEl-+ (u2-61)T)) (4.120) where u = 0 is the only singularity enclosed by the contours 102 C'-C". Consider Now IL as given in (4.119) where the branch defined by (4.116) is used. Due to (4.118), (4.119), 0 we see that the integration IL of (4.119) vanishes as the radius of the arc NM-M'N' approaches zero. p-plane u-plane ,//_C' c"./ Fig. 2 Fig. 3 D-4. Integration along KJ-J'K' for T > Along KJ, p = -2 with 0 < B < 62, and by (4.105) we find that 2 2 - 2 2 where x: 3222 -(r+tsz-02)£+t+2t01-t02 Y = 2.1/2571/2(.1-))1/2(22-))1/2, 103 For T Z.C, the contribution to the integral of (4.92) along the contour KJ-J'K' is found to be A1(’£) A3(‘E) 1 e2 . J2 _ 5111 e 1) ) W f0 ( (5761-£(R g C (2(r-s2)£(£+a) (r’52)£(£+a)19112) A1(-£) (r—sz)£(£+a) where A1(p), A3(p) and "a" are given by (4.58), (4.94) and - sinMic) ) exp(-,z~r) d) (4.122) (4.89) . *- D—5. Integration along ML-L'M' for T :.C and T < C Along ML, p = -E with EZ < 2 < El, and with the aid of (4.105) and (4.107) we find that the contribution to the integral of (4.92) along the countour ML-L'M' is E1 A1(‘E) . 5111 U 2;) d), (4.123) IE2 (Sz‘r)£(£+a) ( ale for T Z.C and T < C. D-6. Integration along DC-BA—A'B'-C'D' and HGl-Gle- FzFl-FlE-E'Fi—Fif‘g fifiGi-GiHi for r _>— C For T Z.C, the utilization of (4.104), (4.105) and (4.108) lead that the contribution to the integral (4.92) along the contour DC-BA-A'B'-C'D' and HGl-Gle-FzFl-FIE— E'Fi-FiFé-GéGi-GiHi vanishes. (4.124) *- For I < C, see this section D—7. 104 D—7. Modifications on Integrations for T < C When T < C, the contribution to the integral of (4.92) from the summand of the integrand, A3(P) A1(P) 1/2 ( _ _ )eX - E——'( +9 )C+ ) '(r-82)P(P'a)9192 2(r‘52)P(P“a) p( Zfl gl (2°1ZEE along the Bromwich contour vanishes because we let the contour for this summand be closed by the right arc of the circle which is shown as a dotted curve in Fig. 1. Then the contribution from this circular arc to the integral of (4.125) approaches to zero as the radius gets large. Hence for T < C, some modifications should be made on (4.122) and (4.124), but the results in (4.110), (4.120), (4.123) are not affected. D-8. Integration along DC-BA-A'B'-C'D' and HGl-Gle- szl—FlE—E'Fi-FiFg-G§61—G{Hi for T < C Along DC, p = Re p1 + i£ with 0 i. 2 2.1m p1 and we let 1 (p + 62) /2 = b1 + ib2 (4.126) A 1 (p+€1)/2=C1+iC2. With the aid of (4.52), (4.53), (4.102) and (4.103), we have 21 ’5 X1 + in (4.127) where 105 x1 = 82(Re2p1-£2)+r+t82—02)Re p1+t+2t01-t02 1 + 2r /zs(c1(a1b1-a2b2)—c2(a1b2+a2b1)) Y1 = ZESZRe p1+£(r+tsz-02)+2r1/zs(c1(a1b2-a2b1) +C2(a1b1-a2b2)) and Zz = X2 + iY2 (4.128) where x2 = 82(Re2p1—22)+(r+tsz-02)Re p1+t+2t01-t02 1 - 2r /25(c1(a1b1-a2b2)-C2(alb2 32b1)) 1 - 2£32Re p1+£(r+tSZ—02)-2r /25(C1(a1b2-a2b1) m N I + c2(a1b1—a2b2)). From (4.105) and (4.108), we have that 91 = G(Zi) 92 : ‘G(Zz) and the contribution to the integral of (4.92) along DC-BA —A'B'-C'D' when T < C, is 1/ 1 Im p1 exp(- 'ZEf—l'z 91:11:”) 1/2 —- R [ {A p)S'nh E—- 9 C) W f0 e (Sz-r)p(p-a) 1( l (Zfl 2 1 _ §?%£El cosh(§E-1--2 gZC)}]d£. (4.129) . . 1 Along F2F1, p = Re p2 + 12 With Im p2 j_£ 2.2 /2Re p2, and by (4.105) and (4.108) 91 =G(Zl) 106 92 = ‘G(Zz) where 21, Z2 are given by (4.127) and 4.128), and the con- tribution to the integral of (4.92) along Gle-FzFl-FiFé-GéGi for T < C is J'R ( pl/z C+ ) 1/ 2 e p exp -'-- 9 PT 2 %’f Re[ Zfl 1 {A1(P)Sinh(§f——'92C) Im P2 (Sz-r)p(p-a) 1 2A3(P) pl/2 — —§:§;—-cosh(§f: 92C)}] d2 . (4.130) Along G1H, p = 21/2Re pzexp(ie) with 0 fi.0 :.3v/4, and by (4.105) and (4.107) 91 ' G(Zi) 92 ‘ G(22) where Z1, Z2 are given by (4.127) and 4.128), and the contribution to the integral of (4.92) along HGl-FlE-E'Fi —GiH' for T < C is 1/2 1/ _ .p__ + 2 2 R exp( 2f 91C PT) 1/ ( e p2) f0 [sin0[Im( 1 ['A1(P)Sinh(§ér392C) W a” 4 (r-Sz)p(p-a) 1 1/2 ‘ 233:5) “mg—f: 92”“ 91/3 exp(- 2f1 91C+pT) 1/2 + cos 9[Re{ (A1(P)Slnh( 92C) (r-82)p(P-a) 2f cosh(§?—1—E- 92c) )] )de (4.131) 107 D—9. Integration along KJ-J'K' for T < C The contribution to the integral of (4.92) along KJ- J'K' for T < C is easily deduced from (4.122) as %-f0 exp(-£T)sin(JZt) A1(—£) dfi- (4°132) €2 (r—82)£(£+a) D-lO. W(2)(C:T) Obtained by Inversion of w(2)(fi,p) We consider two cases, i.e., T :.C and T < C. For T Z.C, we have that from (4.101) (2) = 1 A (a) A1(a) a1/2 w- “'T) a(r-sz)”91 ___)...“ _‘(TWIW A3(a) A1(a) a1/2 + 92(3))C+aT] —(91(:)92(a) + 2 )exp[ §——T“jig1(a) - 92(a))C+aT}+A1(a)eXp(-a1/2C+ar)] - I1(C:T) (4.133) 1 . where 11(C,T) = 2wi lim( f0 + f + f + fK N NM ML J +f +f +f +f ) N'O' M'N' L'M' J'K' whose integrals are evaluated in (4.110) ,(4.120) ,(4.122) and (4.123). For T < C, we have that from (4.101) and the vanishing of the integration of (4.125) “(a)“ z :fiE‘A1‘a>eXP<-a‘/ 2C+aT>-=hian +f+f +f +1 +1 +f ON NM ML KJ HG1 G1G2 F2F1 +f +f+f+f +f +f +f +f FIE DC BA N'O' M'N' L'M' J'K' GiH' + f ' .+ f ' I+ f .+ f + f ) G261 FIFZ E'F1 C'D' A'B' whose integrals are evaluated in (4.110) ,(4.120) ,(4. 123), (4.129), (4.132L (4.130) and (4.131). E. Inversion of w(§,p) in Real Integral Form Combining (4.133), (4.134), (4.85) and (4.86) leads to the complete description of the displacement field of the solid component w(C,T) T‘fa W(C, T) =J—fo f0 eXp(-b(u-zz)) dz W<2)(C,T-u) du. (4.135) From (4.133) ,(4.134) and (4.135) we see that w(C,T) satisfies the boundary and regularity conditions specified by (4.33) and (4.34). Since the material constants have to be determined by ex- periment for the mixture and such an experiment has not yet been devised, wesfimll not attempt any further investi- gation about the behavior of the displacement field of the solid component at this point, even though we have the exact solution given by (4.135) which may be evaluated numerically by computers. 109 4.5. Early Time Solutions The numerical evaluation of (4.135) does not seem to be an easy task. One way to avoid this difficulty is to represent w(§,p) in a power series with respect to ‘1 for sufficiently large p, and then invert the resulting expres- sion term by term. This procedure leads to an early time solution for W(C,T). As p -> a), we have that 2r1/Zsp1/2(p+€1)1/2 1 . M M M M M =2pp1/2r /zs[1+—1-+—?-+—§-+—1-+—§-+O(l—)] p 92 p3 p4 p5 96 (4.136) where 52(t+tr)-r(é§ + 62) M1 = . 2rs2 -(r2(éi+62)a2rsz(t+tr)(5i+éz)+s4(t+tr)2) M2 I . . . 8r284 1 (éi+52)3 (ai+32)(t+tr)2 M3 = _E" $6 + r252 2 2 3 _ (o1+62) (t+tr) + w ) rs4 r3 _ 1 5(t+tr)4 20(t+tr)3(6:+62) 2(t+tr)2(5i+52)2 ”4 =123( + ‘ r4 r352 r284 5(t+tr)(5i+52)3 + 5(6i+52)4 r36 53 I'll Algal I) I. 110 1 7(t+tr)5 5(t+tr)4(6§+62) 2(t+tr)3(5i+él)2 M5 = 256( r5 + ‘ r432 - ' r354 2(t+tr)2(6§+62)3 5(t+tr)(é§+52)4 7(6i+62)5) r256 r58 ‘ S1o With the aid of (4.68), (4.69), (4.79), (4.80) and (4.136), we have that as p —% oo g (p) - pS[1 + N1 + N2 + N3 + N4 + N5 + N6 + N7 1 — 1 -—_ 1 ___ 1 ___ 1 p—72 P pp )2 p2 pzp ;2 p4 94p 72 N N N N 1 +'_§ + "2—T7—'+ ‘l2'+ 11 + O(‘;fi] (4.137a) ( > [1 N1 N2 N3 N4 N5 N6 N7 929*PS -17 +"_- 1 +—- 1 +-——_ 1 ' P 2 p pp :2 p2 pzp 2 p3 p3p :2 N N N N + —_" i + "12' 11 + o(——)] (4.137b) where rl/2 N1 _ S N - (ts2 — 52) 2 232 M1r(tSZ-62) N3 254 2 N4 - 252 '- 1 3 _ 1? r /2M2 N5 = -—-—-—-- N1N4 - N2N3 111 N2 3 N6 —‘N1N5 ‘ N2N4 ”'7: 1 r /2M3 N7 = S " N1N6 - N2N5 - N3N4 2 N4 N8 ‘ "N1N7 ‘ N2N6 ‘ N3N5 ' 2 1 r /2M4 N9 = T— “ N1N8 " N2N7 - N3N6 ‘ N4N5 N3 N10 = ‘N1N9 ‘ N2N8 ‘ N3N7 ‘ N4N6 ‘ 2" 1 r /2M5 N11 = ‘—§———" N1N1o ‘ N2N9 ‘ N3N8 ‘ N4N7 ‘ N5N6' As p —+ 00, we have that 'f-%ET is, due to (4.54), (4.66), 1 1 _ 1 (1 _ 1.El.+ _.El._ §;§ .5: f1(P) p1;zs 2 P 8 pz 3'23 p3 3-5°7 4% 3.5-7-9 61 1 4 — _ ———5——— + o(-—-)). (4.133) 4'2 4 512 p5 p6 1/2 E 1 1 3 2 exp(- §§-(91+92)C) — expt-pC-(Nz- §—)C][1- %(N4- §€1Nz+§€1) N461 3N2€1 3'5 3 1 E1N2 3 2 2 + __g'[-N6+ — + 61 _(N4 — 61) C] P2 2 8 3:23 2. 2 8 N e BN 6 3-5 3-5-7 + £§{—N3+ ; 1 — 4 1 + ——-N263— e4 p 8 3123 4124 - :21 + 0(1— )1. (4.139) p4 is! 112 pl/z 1/ (N3'€1N1/2)C exp(- §§-'(91-92) C)=eXp[-p 2N1C][1- 1/ 1 p 2 . 2 2 3 2 1 1 3 3 (N3‘51N1/2) C [(§N1€1‘ §Ns€1 N5>C+1N3‘ §N1€1) C 1 + 2p ppl/2 1 (N3‘ §N1€1)(%N1€:’ %N3€1 + N5)C2 ( 1 ) (4 140) + + o o .2 010—257;] Let us consider A1(p) given in (4.56), (4.57) and (4.58). A2ZP) As p —> d) we have A1(p) d1 1 1 2 ——(—7=-——7—1+—R—R+——-R-R-RR 1 3 2 1 2 2 4 3 + '—3—(2R2R3 -R2 +R1R2 ”R1R3 )+ 7(R3 '3R2R3+R2 +2R1R2R3 -R1R2) P p 1 2 3 5 2 2 4 1 + —5—(-3R2R3+4R2R3-R2+R1R3—3R1R2R3+R1R2) + o(—-;)] (4.141) 'U *0 where {d2(t+él)/di}62-t R1 52-r R2 = sz—r+62+t(1+r-sz) R3 : -621-62-t(1-62+251)0 From (4.73) we immediately have 91(p)92(p) = p4s4(1 + -—-+ ——-+ ——-+ ——) (4.142) where 113 2t(-1+261-62—2r) (ts2+r-62)2+4r(6i+62) 52 S4 [(1 + 251-52)(tsz+r-52)+(2+2r)(éi +62)] m to hrT 2 C4 = §:(1 + 261-62)2. With the aid of (4.142), we find the asymptotic expansion 1 Of 91(p)gzzp) as p —+ 00 1 = 1 ( Cl 1__ C2 91(p)92(p) p252 29 p2 1 Ca 3 5 3 1 C4 3 2 +‘E§(‘ —§'+ ZC1C2 ‘ EZC1)+ ;z(- §—'+ §4C2+2C1C3) 15 2 3-5-7 4 1 — ——-c1c2 + Cl) + o(——)). (4.143) 24 4124 p5 With the aid of (4.143), we find the asymptotic expansion of B(p) and D(p), given in equation (4.50), is 4.) - —-/-<1+:°’-i . _ + i: + .4» (4.1.4) pzp '2 p P Pa P4 where aldz’dléz (t52+r-62—252) C1 S : — _.__ 1 dls2 232 - 4 C2 3 2 C1 2(61d2-d162) (tsZ+r-62-252) s = - + -—c - ——- - R +R - 2 "4' 421 4 dsz 1 2 82 1 (t(1+261’62)+2(éi+62)+(R1‘R2)(t52+r’52‘252) 252 114 (-RI+R2)(t(1+261-62)+2(6i+62)) $3 = 252 (R%+R3+R1R2)(t82‘02+r-282) C1 2 + 252 - -‘-1——(-R2 +R3+R1R2 (t(1+261-62)+2(6§+62)) (Rl-R2)(tsz—62+r—252) 82 . 82 —C2 3 2 2(51d2-d162) (t52+r"62"2'82) + 'f:‘ + 2 1)( 2 — R1+R2- 2 dls S C 3 5 3 +(--4—-+-8-C1C2 -§'5—C1). d1 D1 Dz 1 D = D +-—- +-—— + O-—— ) 4.145) (p) ——7—2p3pl 2 < o p p, (p3 ( where 2(51d2-d1 52) (tsz+r-52 -282) C D0 = 2R1-2R2- + + 5— 2 C2 3 2 C1 2(61d2-d152) D1 = 2R2-2R3-2R1R2+ T - g C1 + 2—— " R1+R2 dls2 (t52+r-62-252) 1 2 - 2 )+ ;§(t(1+251'52)+2(51+52) S + (Rl-R2)(tsz—62+r-232)) J1 «I'll-ill. Ill! [ll-ll. 115 (t(1+251-52)+2(ai+52)) 2 D2 = 2(2R2R3—R§+R1R2—R1R3)+(R1-R2) S2 (-R§+R3+r1R2)(tsZ+r—62—252) C1 2 - S2 + g—K-R2+R3+R1R2 (t(1+251‘52)+2(5i+52)) (RI-R2)(tsz+r-62-252) 52 S2 (3 2 C2)(2(51d2‘d152) (tsz+r-62-252) - —C1- ——' ‘ R1+R2 ' 8 2 dls2 52 C3 3 5 3 (- §—'+ Z'C1C2 - EZ'CI ). Combining (4.50), (4.56), (4.60), (4.139), (4.140), (4.141), (4.144) and (4.145) and inverting term by term, we have that for early time “ 5 W(C.T) = d1(—4T)3/2i3erfc 5%? +(R2-R1)(4T) /2i5erfc 2§% 7 +(R3+R1R2‘R:)(4T) /2 i7erfc ‘£_' + 2J2 61 (T‘C)3/2 r- 5 2 +exp(<-N2+ Yuma-cums)” ”#1757157 (s1 N 7/ _ €1N2 3 2 ‘C(N 4- _%:£ + g-ei))+ §(9/%) 2(52 51C(N4' 8 E1) (N -e N /2+ i 62)2 +C(—N6+ §%E1.- g-Nzei + gz-ei+ 4 1 22 8 1 §)+ . N g N 29(4T)5/2 i5erfc 1+D0C(2N161 - N3)(4T)3 16erfc 2;; T T Do N1E1 2 ‘ (D1+ _—(N3’ ___—Q 2) N 2J} \_/ 116 Here erfc(x) is the complimentary error function defined by 00 f exp(-m2) dm, x . erfc(x) = =1?!» and the repeated integrals of the complementary error func- tion are defined by ioerfc(x) = erfc(x), n on 5-1 i erfc(x) = f i erfc(t) dt, n = 1,2,°--. x . See [38] for example. A similar procedure may be used to find an early time solution for v(C,T). We begin by finding the asymptotic expansions of the factors in equations (4.51), (4.57) as p -9 00. The following factor is written as a series . . 1 expanSion in terms of (-) P 1 (P(¢+P)(P32‘52)+t51)- 3P(91+92)2) K1 K2 1 4T =Ko+—+—2—+O(—3-) (4.147) (P(tsz+r51)+t(51-éz)) P p p where 2 t(51'52)'52(2N4+N2) K0 = tsz+r61 2 -2$2(N6+N2N4) t(51‘@2)(t(51'52)“52(2N4+N2)) K1 = '- tsz+rol (tsz+r51)2 t2(51‘52)2 2t52(51'52) _ 2 K2 - 2(t(c31-c52)-s2(2N4+N2))+ (N6+N2N4) (tsz+r51) - (tsz+r61) 2 "' $2 (2N8 + 2N2N6 + N4). 117 The following factor is written as a series expansion in terms of (AJ 9 (p((t+p)(pSZ-éz)+t51)- %P(91-92)2) , = Lop2+L1p+L2+O(l) (ptsz+r51)+t(61-62)) p (4.148) where 2 L0 = __§____. ts2+r61 2 tSz’éz-Sle t52(51-52) L1: -’ tsz+r61 (tsz+rc51)2 tzsz (61-62)2 t(61-52)(t32-52-32Ni) t(61"52)-252N1N3 L2: -, + (tsz+r51)3 (tsz+r61)2 .(tsz+r61) The following factor is written as a series expansion in terms of (id P A1(p) (Psz‘ézidi 51d2 2 5 +6 - sz+té + 52-6 ) +t + + P( 1 2 P 1 (P 2.(P ))A2 p p1/2 ;T7;' p(tsz+rol)+t(61—62) -d1 J J 1 . = 17: (JO+ —l-+ —%-+ 0(—3fi) (4.149) pp 2(ts2+r<31) P P P where d Jo = 52(R1-R2)+t52-52" E:— 51 2 J1 = s2(R§-R3-R1R2)+(R1-R2)+ts2-éz-s2)+(c51+c52 +t61-t52) d2 t(51-62) - (52(R1-R2)+tsz—sz- a: 51 (t52+r51) 118 t2(61‘62 )2 d2 J2 = 2(52(R1-R2)+tsz-sz— a—-61) (tsz+rol) 1 t(51‘52) 2 - . (82(R2-R3-R1R2)+(R1-R2)(tsz-oZ-sz) (t52+r61) + (63+62+t61-t62))+sz(2R2R3-R3+R1Rg-R1R3) + (tsz‘éz‘sz)(Rg’Ra'R1R2)+(R1'R2)(5i+52+t(51‘52))- With the aid of (4.51), (4.57), (4.61), (4.139), (4.140), (4.141), (4.145), (4.147), (4.148) and (4.149), and with the elementary inversion process, we have that for early time v(c.r) = d1[ J0 (4.)1/2 ilerfc C (t32+r61) 2i?- _ J1 (4T)3/213erfc C - J2 (41*)5/2 i5erfc-£—-+ (tsz+r51) 2J;' (tsz+r51) T 1 mom-o3“ TAT-0V2 +exp{(-N2+ §€1)C}H(T’C){ P(5/2) + P(7/2) T2(T-C)7/2 Z N1C + fiwfz) + 1 + 72(4fl1/2 ilerfc 2f? 2 N C —;(4T)3/2 i3erfc 1 + °°°] , (4°150) T where To = K0 1 3 2 T1 : K1 + K0(Sl’C(N4‘ §'€1N2 + §'€1)) ill I I‘ll! III, I 119 _ 1 3 2 . , 1 3 2 T2 ‘ K2+K1[[31'C(N4‘ §€1N2+ §'€1)}]+Ko[32‘518(N4‘ §€1N2+ §€1) 1 3 ‘2 2 1 3 2 5 3 (N4' 5N2€1 + §'€1) , +(‘N6+ §N4€1' §N2€1)+‘§Z E1+ 2 C], Zo = D0L0 . - 1 Z]. - " CD0L0(N3- ‘2" €1N1), _ 1 2 1 2 Zz ‘ Lo{Di +'§DOC (Ns‘ 5N1E1) ] + DoLi- We note that these early time solutions of w(C,T) and 9(C,T) are in effect without any restrictions beyond the conditions of (2.40) and (2.41). And the early time solutions of all other field variables follow trivially from the equations (4.27), (4.146) and (4.150). SUMMARY AND CONCLUSIONS In this thesis, we have reviewed the major contribu- tions to the development of a theory of mechanically and thermally interacting continuous media. Beginning with the work of Darcy and Terzaghi we have traced the work of Biot, Truesdell and Toupin and the recent work of Green, Naghdi, Steel, Atkins and Chadwick. As is usual, the theoretical development has preceded the number of applications and in this thesis we have attempted to utilize a linearized version of the mixture theory to derive results which are readily applicable to practical boundary value problems. Our first result is in the form of an integral rela- tion commonly known as a reciprocal theorem. It relates the solution of one problem to that of another problem each of which is due to different boundary and initial data. We have indicated how this theorem reduces to a theorem applicable to a single constituent and we have shown how one might use such a theorem. Indeed, we intend to explore its uses in future research much along the lines used in classical elasticity. Our second major result consists of a solution of a fundamental initial boundary value problem using the linearized mixture theory. It is the first actual boundary 120 121 value problem to be solved using a mixture theory. Due to its complexity the results are given in integral form only. Further development must await experimental evidence con- cerning the size of the material properties. Such experi- ments, incidentally, are a second possible line of future research and it is our intention to attempt to devise simple analytical models which will lead to estimates of the material constants. We will be guided by those methods used in single constituent theories. The integral representation of the solution of the boundary value problem given in Chapter 4 is exact to terms of order t2 but, due to the complexity of the inte- grands in the integrals, not much can be inferred about the displacement field. For this reason we have given the starting solution, i.e., the early time approximation. This solution may prove more useful as far as actual compu- tation is concerned. REFERENCES 10. 11. 12. 13. 122 REFERENCES Scheidegger, A. E. The Physics of Flow Through Porous Media. New York: Macmillan Co., 1960. 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APPENDIX APPENDIX The location of the zeros of (4.73) may be given in a power series of t as follows, 2 1 1 2E11 Pi = §(-¢o-t(z¢oE1‘1)-t2(§£9(—§" 552) + 0(t3)) 1 (4k1+2¢oB1‘ §E1 ¢g'¢o¢1) i + '2‘“ 4WO-¢§+t 2 V4¢o ' $3 1 1 2 2 1 2 2((4k2 2¢0W2'(Z‘E1¢0+ §¢1) "(Do (‘2‘E2-E1/8)) + t 2 7717763 , 1 2 2 (4k1+2¢o¢1‘ §El¢0'¢o¢1) + 2 3f )+ 0(t3)) 8(4wo ‘ ¢o) 2 1 1 2 1 P2 = ‘ Z‘¢ot((§E1- 5;) + (§E2-Ei/8)t + 0(t2)) it 1 1 1 ‘"~2-—2 "' T(‘f‘sz‘ $00132)- 1% (5531' (ID—0') 1 1 2 1 2 1 2 4(k3' "¢033)‘ —¢o(—Ei‘ ——0(E2-'—E1) + 2 4 2 q)” 4 t + 0(t2)) 1 1 1 2 2 2 J4 0%" 511/032)“ 190 (5E1- 3;) and their conjugates p: and p:, where we used the fol- lowing abbreviations 2(r + 62) 2 (130:- S (Ir-62 )2+4r(éi + 52) $0 : S4 125 1.»!!yltn .. .le I‘U $1 ¢2 $3 $4 126 2(‘1+2 61 - Zéz-r) $2 inn-+2514.)(r-a.>+<2+2r><éi+62>> 2.. 2(1+2 61-62) S '§r(1+251-52)2 - 70/36 ¢o¢2/12 ‘ ¢o¢1/13 (¢o¢3+2¢2-4w4)/12 - (w? + 2¢o)/36 - wfi/lOS - wfiwl/36 + ¢owow2/24 ‘(¢: + ¢o¢i)/35+(¢o¢1¢2+¢o(¢o¢2+2¢2‘4¢4))/24 +(4¢o¢3’¢§¢3’¢§)/3 ((r—62 )2+4r(5i+52 ) )2 4 (;§((1+261-62)(r—62) 3326s8 2 +(2+2r)(éi+62))2 (§%(r+62)2-«33((r-62)2+4r(61+62)) 64:5: ‘2 1 ( 1 '3(2¢ w -23w ? +w 7 7+ 1 w2(6¢ w 7 2(‘20) 2333 ¢o o 4 194 2 3 2533 o o 1 4 .2 2 $0 2 2 2 +6¢1¢2’¢o¢2¢3'2¢o¢2+23¢0¢2¢4)‘ 2533(32¢o¢2W4+¢0¢1¢2) 1 . 3 3 + 5 (“32¢0¢1¢2 + 2¢3¢2)) 3 23 127 G1¢’o 6(H1'H0G1/(3Go)) __ _ + 1Q 1830 20 6 2 2 wo G2 (51“ o) 5 G1 G2 6_( 3G0 9G2 " Whiz-1'11““; ) H0 (3G 2 2(G1 ‘ 20) $2 9G0 4 37((b0'i'2k1-(l/1) ¢o 2 2k1 2¢2 1 —+———-—— -—E1 ¢o $0 ¢o¢o 2 2k2 4k1 2¢3 (1 (g__+ 2k1 _ 2¢2 $0 ¢o¢o ¢o¢o 2 1 90 W0 ¢o¢o