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I; 2 fit. an x: Rhianna. 53.... u a 3.4.3 0‘? 41:79.70 a. or} ~ . z. . . g .35.“ .3 .révvnunrves J... 3.. )8 3‘ .1?- .311?! :fi-Lsill 1..., 911.35.519.96 .3! 9 A!» . I? 5!; \ k that... \x .33.”: 1)....5I57 Q. .74....) .‘gttézn .1 .fi. . 1.1L. 5?!!;:w\3 5. .z! 3.! v: v I E .C I}‘|7!I.A I... x 13 . [$2.34 .3... wigs. y...) ‘t A? THESIS SUTATE ilillislllllllllzlflll "HUIHI IIHIHIHWII 23 01390 7849 This is to certify that the dissertation entitled Adiabatic Propagation of Phase Boundaries in a Thermoelastic Bar presented by Ralph Worthington has been accepted towards fulfillment of the requirements for Ph . D . degree in Mechanics M an 1A-, professor Date H" “I " 61g MSUiJ an Affirmariw Action/Equal Opportunity Institution 0-12771 LIBRARY Mlchigan State University PLACE N RETURN BOX to roman this checkout from your rocord. TO AVOID FINES rotum on or bdoro doto duo. DATE DUE DATE DUE DATE DUE I I I MSU Is An Afflnnotivo Action/Equal Oppolttmlty lnotltwon W m1 ADIABATIC PROPAGATION OF PHASE BOUNDARIES IN A THERMOELASTIC BAR By Ralph Worthington A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Materials Science and Mechanics 1995 so In: der of a “hit Phrs knou in 3 st.- farm].V HOUSlt the inte. Phise b< hm)“ 0: “I “mix, . Sjus‘DUhe ABSTRACT ADIABATIC PROPAGATION OF PHASE BOUNDARIES IN A THERMOELASTIC BAR By Ralph Worthington A Helmholtz free energy function is introduced for a one dimensional two phase solid. The free energy accounts for both thermal and mechanical effects. Phase transfor- mations can occur between two distinct phases, one in which the stress response is depen- dent on the both the deformation and current temperature, while the other is independent of any thermal effects. The model admits a thermomechanical coupling parameter (12 which can be associated with a coefficient of thermal expansion in the thermomechanical phase. If the coupling parameter a2 is set to zero, then one retrieves results from what is known as the separable theory. A set of initial conditions are proposed such that a single phase boundary is present in a stable equilibrium configuration. Such configurations are shown to be a two parameter family of states. The initial configuration is disturbed by a set of dynamic boundary condi- tions that give rise to a wave pulse, the wave propagates from one of the boundaries into the interior of the bar. This travelling wave eventually encounters and interacts with the phase boundary, and it is shown that the encounter is characterized by a one parameter family of solutions. The fully thermomechanical theory is treated analytically in the small 002 limit, and thermal effects are shown to play a major role in the interaction. The Clau- sius-Duhem inequality is shown to restrict the family of solutions. To anc fESC [hm I Wis tract PJEUQ ACKNOWLEDGMENTS It has been my privilege to have had the following Professors serve on the my doc- toral committee: Professor Melissa Crimp Professor David Gmmmon Professor Thomas Pence Professor David Yen To my major professor, Dr. Pence, I wish to express my sincere appreciation for his guid- ance, infinite patience, and constant sense of humor. I found all three an invaluable resource during my research. I also wish to thank my parents, for their constant support throughout my University education. I wish to acknowledge financial support from the US. Army Research Office under con- tract DAALO3-89-0089, and from the MSU Graduate Schools for the Dissertation Com- pletion Fellowship. 1.] 2. B. 3. M 1. Introduction ........................................................... l 1.1 Overview ...................................................... 1 1.2 Literature Review ............................................... 2 1.3 Problem Statement .............................................. 10 2. Balance Laws ........................................................ 15 3. Material Model ....................................................... 20 3.1 Construction of the Material Response Functions ...................... 20 3.2 Separable Energy ............................................... 31 3.3 Characteristics and Riemann Invariants ............................. 34 3.3.1 Separable Materials ........................................... 36 3.3.2 Nonseparable Materials ........................................ 39 3.4 Multiphase Materials ............................................ 43 iv 3.4.1 The ground-state equivalence temperature .......................... 48 3.4.2 The Latent Heat .............................................. 53 3.5 Summary of the Two Phase Material Model .......................... 54 4. The Initial Boundary Value Problem ...................................... 66 4.1 Governing Equations ............................................ 66 4.2 Initial Configurations ............................................ 67 4.3 Static Configurations ............................................ 68 4.4 Equilibrium Configurations ....................................... 70 4.4.1 Maxwellian Configurations ..................................... 71 4.4.2 Mechanically Neutral Configurations ............................. 75 4.4.3 Entropically Neutral Configurations .............................. 78 4.4.4 Omnibalanced Configurations ................................... 81 4.5 Initial Disturbance .............................................. 85 5. The Interaction of the Initial Wave Pulse with the Phase Boundary .............. 92 5.1 The Wave Pulse-Phase Boundary Interaction ......................... 93 5.2 The Purely Mechanical Problem ................................... 95 V \J m 8. En. 5.3 New Features of the Fully Therrnomechanical Interaction .............. 104 6. Analysis of the Initial Interaction Region .................................. 110 6.1 A Method for a Solution to the Initial Boundary Value Problem ......... 110 6.2 The Master Equation for the Initial Interaction ....................... 116 6.3 Construction of the Centered Simple Wave Fan ...................... 121 7. Construction of a Solution for Small Coefficient of Thermal Expansion ......... 127 7.1 Perturbation Analysis of the Master Equation ........................ 128 7.2 Small 0Q Decomposition of the Field Quantities ..................... 134 7.3 Existence of Fan ............................................. 142 8. Entropy Production and Dissipation ...................................... 146 8.1 The Second Law of Thermodynamics. ............................ 146 8.2 The Entropy Restriction for Separable Materials .................... 152 8.3 The Entropy Restriction for Fully Thermal Materials ................. 156 9. Solutions Obeying a Thermal Version of the Kinetic Relation ................. 160 9.1 Driving Traction ............................................. 160 9.2 Driving Traction for Separable Materials with OB Initial Conditions . . . . 165 vi 10C. Appent Appenc List of E 9.3 Kinetic Relation for Separable Materials .......................... 168 9.4 Kinetic Relation for Fully Thermal Materials ....................... 169 10. Conclusions and Recommendations for Future Work ....................... 171 10.1 Conclusions ................................................ 171 10.2 Recommendations for Future Work ............................. 172 Appendix A. Algorithm for Reduction of Equations (6.5)-(6.l3) ................. 173 Appendix B. Transition in phase-2 is a shock ................................ 176 List of References ...................................................... 183 Ta Ta! Tab. Tabl Table Table 1 Table 2 Table 3 Table 4 Table 5 Table 6 LIST OF TABLES Material parameters ........................................... 29 Temperature intervals for preferred phase .......................... 50 Material parameters for phase-1 for \y (y, 0) ....................... 59 Material parameters for phase-2 for w (y, 9) ....................... 60 Lower bound temperature of transformed material 68. when lower bound C71 = C72 = r ........................................... 151 Lower bound temperature 0;. for separable material when lower bound C71 = C72 = 7 ........................................... 153 viii Figure Figure , Figure 3. Figure 1.1 Figure 3.1 Figure 3.2 Figure 3.3 LIST OF FIGURES Displayed is the nonmonotonic stress strain response for the material model describing phase transformations. Note for the prescribed level of stress in rm 5. r 5 1M that there are three possible values for the strain 7 ........ 14 This figure is a schematic representation of the Helmholtz free energy function u: (y, 0) plotted against the strain 7 at a constant temperature 0. Shown are the two free energy functions w] and W2 , each represents a distinct phase of the material.'l‘he arrows, which are shown at each of the vertices, acknowledge that the location of these vertices shift as the temperature changes. At each temperature there exists a level of strain for which the values of the free energies are equivalent, this strain is designated 71 . ........................................................ 63 This schematically shows how the two free energy functions interrelate for the three temperature intervals. In the intervals 9 < 9: and 0 > 0; the phase- 2 free energy has a lower vertex and in this sense is the preferred phase. For the interval a; < o < e; phase-1 is the preferred interval ............... 64 The stress-strain response for fixed temperature for the two phase material considered in this document. Both phases have a linear stress strain response, but the overall stress strain response is non-monotonic. From this figure one observes that the strain is not unique for a prescribed stress 1 e (rm, 1M] , and thus the bar can accommodate a multitude of different deformed configu- rations ...................................................... 65 - 5. FlguI Figure Figure 4.1 Figure 4.2 Figure 4.3 Figure 4.4 This figure is the Helmholtz free energy function \v (y, 9) at a constant temperature 0. Shown is one pair of Maxwell strains 71;“ and 72“ . These strains are determined from the requirement that an equilibrium configura- tion satisfy the criterion [ [w]] — ((1)) [ [y] ] = 0. Schematically this cri terion is shown by the line of slope ((1)) which is tangent to both free energy functions, the points of tangency identify the locations of the Maxwell strains ............................................... 88 Shown is the two phase stress strain response at constant temperature. The 72 Maxwell criterion, j t (y, e) d'Y — ((t» [ [y] ] = 0 , can be interpreted Tr graphically as the equal area rule. This rule states for the Maxwellian config- uration that the area below the phase-1 stress-strain curve but above the line er is equal to the area above the phase-2 stress strain curve but below the line er. On this figure the equal area rule identifies one pair of Maxwell strains 7:“ and 7;“. ......................................... 89 This is a schematic representation for the three canonical phase-l strains for different temperatures. The mutual intersection of all three strains is the loca- .. on tion of the omnibalanced temperature 0 . Here the acoustic speeds are the same cl = c2 = 2 , and the values for the material parameters are p = 1 , e =1,y =2,c72—c71=3,a2.=1,x.,=5 ................. 90 This is a schematic representation of the three canonical phase-2 strains for different temperatures. The mutual intersection of all three strains is the loca- .. on tion of the omnibalanced temperature 0 . This diagram is the complement of Figure 4.3 in that all the material parameters remain the same as in that figure ....................................................... 91 Fis' Figure 5.1 Figure 5.2 Figure 5.3 This is a graphical representation in the (xt)-plane showing the initial interac- tion of an incoming pulse with a stationary phase boundary for the purely mechanical problem. The wave speed of the phase boundary is assumed pos— itive S (t) > 0 in this figure. There exist six distinct regions during this inter action: E1 & E2 are the equilibrium configurations in phase-l and phase-2 respectively, 1W represents the region carrying the incoming wave pulse traveling through phase-1, region R is that region in phase-1 where the reflected wave travels, while S arises from the interaction of the incoming wave and the reflected wave; finally region T represents that phase-2 region containing a transmitted wave. The forward movement of the phase bound- ary transforms phase-2 material into phase-1 material ................ 107 This is a schematic representation of the solution region in the (Av, S) -plane for the purely mechanical problem with Maxwellian initial conditions.This region is defined by the criterion of positive dissipation, D 2 O , thus the lines 2:0 and s = 0 provide the boundaries for the admissi- bility region. Also shown are curves of constant dissipation given by (5.16). The values for the material parameters were chosen to be cl = c2 = 2 , p= 142—71:5 .......................................... 108 This is a schematic representation of the linear kinetic relation with various mobilities x for the purely mechanical problem with Maxwellian initial con- ditions. The solid line 2:0 is the line of zero dissipation,where 2 is given by (5.18), which divides the plane into regions of positive and negative dissipation. Therefore, under the criterion of positive dissipation, the line 2:0 also restricts the solution space in the (A7, 3') -p1ane. The values for the material parameters were chosen tobe c1 = c2 = 2 , p = 1, 72-71: 5. ............................................... 109 Figur. Figure Figure Figure 6.1 Figure 6.2 Figure 7.1 This is a graphical representation in the (xt)-plane of the initial interaction of a wave pulse with a stationary phase boundary in the fully thermodynamical theory. Regions E1 and F4 are the initial equilibrium states separated by the phase boundary at x=s(t). The incoming wave (IW) strikes the phase bound- ary setting it into motion, where s > 0 is assumed. The lW-phase boundary interaction gives rise to the regions S, 80 and R in phase-1, and T and the centered simple wave fan in phase-2. The region R represents a reflected wave, while S arises from the interaction of the 1W and the reflected wave. So is that material which has undergone a phase transformation from phase-1 to phase-2. The IW striking the phase boundary also produces a transmitted wave in phase-2, this is designated by the letter T. Finally the transition from the E2 state to the T state is a centered simple wave, which requires CT < 652. The other possibility, that of a shock transition from the E; to the T—state, is not considered here but is discussed briefly in Appendix B. Comparing this diagram to figure 5.1 shows the additional complexity inherent in the fully thermodynamical theory. ...................................... 125 This is a representation of the fan transition in phase-2. The point (x1,t*) is located on the contiguous line between T and the fan, where 31—: = A1 = CT. The point (x2,t*) is on the line between the fan and the EQ2 state, where 1* dt CE: > CT , which becomes an admissibility condition on the fan solution investigated in Section 7 .3. .................................... 126 = A2 = c132. Correct ordering of the speeds in the fan requires that This is a schematic plot which shows the admissible region for a self cen- tered wave for the case where cl = c2. The region that lies between the line S = 0 and the curve 2 = 0 is the admissible region for the purely mechani- cal case with Maxwellian initial conditions as previously encountered in Fig- ure 5.2. The region above the curve a) = 0 is the region in which the self centered wave may exist. Therefore the region of existence for the centered simple wave is the area between the two curves. Values for the material parameters were chosen to be c = 2 and 72 - 71 = 5 ............... 145 Figure 8.1 This is a graphical representation in the (xt)-plane where the transition in the phase-2 regions gives rise the formation of a shock. Regions El and E; are the initial equilibrium states separated by the phase boundary at x=s(t). The incoming wave (IW) strikes the phase boundary setting it into motion, where s‘ > 0 is assumed. The IW-phase boundary interaction gives rise to the regions S, So and R in phase-l, and T and the shock in phase-2. The region R represents a reflected wave, while S arises from the interaction of the 1W and the reflected wave. So is that material which has undergone a phase transfor- mation from phase-1 to phase-2. The IW striking the phase boundary also produces a transmitted wave in phase-2, this is designated by the letter T. Finally the transition from the E; state to the T state is a shock, and so involves 3 shock conditions, and the introduction of an yet unknown shock speed Ii, whose value must be between (:132 and CT , where CT > 6:132 .This fig- ure, in conjunction with figure 6.1, give the two essential ways in which the purely mechanical situation displayed in Figure 5.1 are complicated by ther- mal effects in the adiabatic limit ................................. 182 1.1 1.1 deal topic resea can b tally aCOll‘ E. an "0 Stress vex. [1 {CSan Strains ”lather. 1. Introduction 1.1 Overview Recently, the topic of solids able to undergo phase transitions has received a great deal of attention from the mechanics community. In this section we will briefly introduce topics pertaining to phase transformations in order to familiarize the reader with the research to be presented. It is well known that certain materials which behave elastically can be modeled with a stress-strain response derivable from a potential function. Classi- cally this potential function is defined as the material’s strain energy density and is usually a convex function, resulting in the equilibrium equations being elliptic in nature. Typically this gives rise to a stress-strain response which is one to one, i.e. associated with each stress there exists a unique strain. However, if the materials potential function is not con- vex, then the equilibrium equations may lose ellipticity. This may result in the stress-strain response losing the one to one behavior, and for certain levels of stress the associated strains may not be unique. It is this non-determinacy which makes these problems both mathematically challenging and representative of phase transformation phenomena Figure 1.1 is a schematic diagram of a hypothetical material in which the strain energy density W is not a convex function of some strain component 7, say tensile strain. If one were to consider a tensile test of a specimen composed of such a material, with a load being applied to the specimen resulting in the stress being contained in the range cm 5 o (y) S cm , then it is seen from this figure that the material need not be in a state of homogeneous deformation. It may be that the specimen contains regions of strain that dif- fer radically, some regions being in a state of “low” strain adjacent to one in a different 1 sure that s fers 2 on El phase mum t abilit} of int: morin It‘Seafc then Lh. . 0f lllCrc‘ 1‘2 Lite U65 that I Value. Ex Solid [0 s 1“ Spectra 2 state of “high” strain. Thus, the possibility exists that within the body there exist surfaces that separate regions of low strain from high strain, and across such surfaces the strain suf- fers a discontinuity. 'Ihese surfaces are called phase boundaries, while the material itself on either side is said to occupy a particular phase. A region of a material undergoes a phase transition when it transforms from one phase into another. For materials capable of undergoing these types of transitions the study of equilib- rium configurations, quasi-static and dynamic motions is possible. In all three areas, the ability of the body to transform between different phases and generate phase boundaries is of interest. For quasi-static and dynamic problems one must consider the possibility of a moving phase boundary, and the interaction of such a boundary with other surfaces. Further complicating the issue is the addition of temperature effects. In general, research into the purely mechanical motions of such materials has received more attention then the modeling of thermal motions. The latter, however, is now also becoming an area ' of increasing research activity. 1.2 Literature Review In general a material can be thought of undergoing a phase transition when proper- ties that characterize the material change in response to a state variable reaching a critical value. Examples of such transitions are between liquid and gas, solid and liquid, as well as solid to solid and other combinations. The model for the van der Waals fluid captures such behavior, where at constant temperature it is either a fluid or vapor depending the level of its specific volume. In the case of solids, the generic ausenite/martensite transition comes IOl Slat cuss disc [[3115 a gre. quite of pit H101? 3 to mind, where depending on the level of temperature and stress the solid may be in either state, the states having different material properties. Furthermore this type of solid to solid transition is the mechanism for the shape memory effect. In what is to follow we will dis- cuss an elastic solid in which the stress strain response is not monotone, and allows for discontinuous field quantities which may be interpreted as a solid undergoing a phase transformation. The study of elastic solids experiencing discontinuous field variables has received a great deal of attention in previous decades and the literature concerning this topic is now quite extensive. From a mechanistic point of view, one may think of three different classes of problems: equilibrium, quasi-static, and the fully dynamical problems. Some of the more relevant literature which addresses these types of problems will be explored below. In (1975), Ericksen discussed an elastic bar in equilibrium having a non-mono- tonic stress-strain relation. For a range of stresses and displacements this type of response allows for more than one strain to be realized. When the bar is subjected to the boundary conditions of a controlled load or diSplacement (soft or hard device) it is shown that infi- nitely many equilibrium configurations are possible. In an attempt to rectify this defi- ciency, the author studies an energy minimization criterion in a attempt to determine a unique and stable solution. Thus he demonstrated that for certain nonlinear elastic materi- als, a bar composed of such a material subjected to specific load/displacement boundary data may not support an unique solution. The author states that one may interpret this abil- ity of the solid elastic material to accommodate a range of strain for the associated speci- fied stress as a solid undergoing a solid to solid transformation. This ability to model a phase transformation in a solid via the nonmonotone stress-strain response has greatly CORK dam. prop: UOllS. diiltlr class'u tions ; solutil 4 contributed to research efforts in this phenomena. Knowles and Stemberg (1978), in the setting of plane non-linear elastostatics, demonstrate that a certain set of deformations having continuous displacements posses the property that the displacement gradients may be discontinuous In elastostatics, the equa- tions of equilibrium which govern the deformation fields are classified as elliptic partial differential equations, and when discontinuities in the deformation gradients arise, the classification of these governing equations changes from elliptic to hyperbolic; the equa- tions are said to lose their ellipticity. Thus within a hyperelastic solid one can construct solutions such that the displacement field is everywhere continuous, yet the deformation gradients may be piecewise continuous as long as the equilibrium equations suffer a loss of ellipticity. Therefore surfaces may exist such that the deformation gradients suffer a jump in value when passing from one side of the surface to the other, and these jumps in field quantities may be attributed to the material undergoing a phase transformation. The authors refer to such a singular surfaces as an “equilibrium shock”, however now the term phase boundary is more widely used. The existence of such a discontinuous surface leads to an additional system of equations which connect the field quantities adjacent to the sur- face, these equations are the Rankine-Huginot equations. If one considers a body capable of containing these types of discontinuities, then families of equilibrium states parameterized by time may be constructed so as to yield the quasi-static evolution of a phase boundary. During such quasi-static motions the total energy may change, thus the passage of a phase boundary may dissipate the energy within the body. Knowles and Stemberg explore the quasi-static evolution of a phase boundary through an elastic body and the dissipation of energy which occurs during such processes. The) inqui static law 0 demui the A tensor em 00' found mullit [he ph‘ have b, 5 They demonstrate that the equations which govern such processes are similar in form to those of steady irrotational flows in a compressible inviscid fluid. The dissipation of energy during the motion of a phase boundary directs one to inquire as to the use of certain evolution criteria during these types of motions. For quasi- static motions Knowles (1979) investigates the dissipation of energy, and from the second law of thermodynamics derives a dissipation inequality for a three dimensional body. He demonstrates for all motions of a phase boundary that the dissipation should be nonnega- tive. A dissipation function is derived and shown to be a function of the energy momentum tensor, also known as the driving traction or force on the defect, whose effects in a differ- ent context where previously studied by Eshelby (1975). For such quasi-static evolution problems, the lack of uniqueness of a solution is found to be even more extreme than the equilibrium problem. The reason being that each equilibrium configuration is indeterminate for the reasons outlined above, and the speed of the phase boundary through the material inherits this indeterminacy. Various procedures have been proposed to resolve this issue of determining a unique solution among the infi- nite number of admissible candidates. One type of selection criterion is to impose an energy minimization condition to each equilibrium configuration during the evolution of a phase boundary. This results in the driving traction being equal to zero for each equilib- rium state. Another approach requires all motions to occur with maximum dissipation of energy. Instead of using an energy platform, one may introduce an additional set of consti- tutive relations, a nucleation criterion and kinetic relation. The former imposes conditions on the emergence of phase boundary, while the later governs the actual evolution of the boundary. Abeyaratne and Knowles (1988) use this method to resolve the issue of nonu- niuuei non-1i differs ies 0ft agauu; preble lhfil [hi nuclea author: the bar gates tl ary and lIlIEma' present then pn speed. i Spfied. 1 m'i‘ract Baillie. I OfEHErg Shmm U 6 niqueness for a one-dimensional, isothermal, elastic bar whose stress strain response is non-linear. In the fully dynamical motion of a phase boundary, the governing equations are different from the quasi-static case due to the additional inertia terms. Within the dynam- ics of elastic bars, James (1980) studies some general properties of solids containing prop- agating phase boundaries. Abeyaratne and Knowles (1990a), investigate the Riemann problem of a bar which has a non-monotonic, tri-linear stress strain response. They show that the lack of a unique solution can be rectified by the use of a kinetic relation and a nucleation criterion, as for the quasistatic case. With these two additional criteria the authors study the solution for the propagation and interactions of phase boundaries within the bar. For a body consisting of a elastic layer of finite thickness Pence (1991a) investi- gates the initial interaction of a incoming shear pulse with a single stationary phase bound- ary and the subsequent ringing of shear waves between the external boundaries and the internal phase boundary. The author specifies that a single phase boundary is initially present, and pursues a treatment that excludes additional phase boundary nucleations. He then proceeds to investigate the family of solutions parameterized by the phase boundary speed. Various impedance criteria are used as a selection technique for the phase boundary speed. It is shown for the special case of a completely reflecting or transmitting wave interaction with the phase boundary, that the motion of the phase boundary is periodic in nature. In another study, Pence (1991b) examines the same problem from the perspective of energy and dissipation. Using a criterion based on the dissipation within the layer, it is shown that there are exactly two motions which permit no dissipation, and one motion that IDEAL". ilii p: to the I wuiur. lOl isc: fora Iii OH 501 ested in heusixe problem dinami Occur u prOpagE Warm are C105 lug equ filters ‘ llll‘egu'g IESGIVES (199 1) t Chamca] 7 maximizes the dissipation rate. In Abeyaratne and Knowles (1992a), (1992b) a set of sim- ilar problems of the interaction of a wave pulse and a stationary boundary are treated. Another approach for resolving the issue of nonuniqueness is to attribute structure to the phase boundary, essentially giving it a finite thickness and matching conditions within the boundary to the conditions at the interface. Slemrod (1983) uses this approach for isothermal motions of a van der Waals fluid. Although the material model is nominally for a fluid, the equations which govern the processes are mathematically similar to those of a solid and thus the associated ideas apply to solid-solid modeling. The reader inter- ested in all the above techniques should see Truskinovsky’s (1991) paper which compre- hensively discusses the formulation and results using these methods for a broad class of problems. So far all of the above papers cited concerned motions, whether quasi-static or dynamic, which were assumed to proceed isotherrnally. The literature for motions which occur without the isothermal constraint is more limited. James (1983) considers the steady propagation of a phase boundary within a therrnoelastic bar, allowing for changes in tem- perature during a dynamical process. Under the assumption that all motions within the bar are close to an equilibrium state, he is able to show for adiabatic motions that the govern- ing equations are unable to determine a unique solution, and thus the addition of thermal effects does not resolve the issue of uniqueness.Truskinovsky (1985) is one of the first to investigate thermal effects for motions within a heat conducting nonlinear elastic bar. He resolves the issue of nonuniqueness by attributing structure to the phase boundary. Gurtin (1991) explicitly formulates the general laws which govern all motions for the thermome- chanical propagation of a phase boundary. In Abeyaratne and Knowles (1993a) a stress- ..w.a sunr- transit I‘D mat SUUCR nun at mcuen for salt uun. ll tuneuo 8 strain-temperature response function is introduced for a material able to undergo phase transitions. This response function is piecewise continuous for the different phases of the material. From the appropriate stress response function the free energy density is con- structed for either phase using thermodynamic arguments. They introduce a kinetic rela- tion and then study the hysteretic response of the material for quasi-static motions. The Clausius-Duhem inequality is a thermodynamic restriction on all admissible motions for the above mentioned problems. It can be used to eliminate possible candidates for solutions produced from the other field equations, but it will not provide a unique solu- tion. This inequality may be reformulated by the construction of an entropy production function for processes occurring within a body, and for all admissible motions the entropy production must be nonnegative. Abeyaratne and Knowles (1990b) show for a three dimensional body with a continuous temperature field that the rate of entropy production occurring during the motion of a phase boundary consists of three parts: one from the material dissipation away from phase boundary, a second part which arises from heat con- duction, and a contribution from the moving phase boundary. Restricting the class of materials to that which is therrnoelastic they show that the rate of entropy production is due only to heat conduction and the motion of the phase boundary. Mathematically, the above equilibrium problems give rise to a system of elliptic partial differential equations (PDE). When these equations admit solutions which have continuous displacements, but displacement gradients which are discontinuous, then the form of these equations changes from elliptic to hyperbolic. In the case of fully dynamic phase boundary motion one has the additional inertia terms and the classification of the governing equations is now normally hyperbolic. With respect to engineering applications 9 and applied mathematics, hyperbolic PDE’s were studied extensively in the context of gas dynamics, see Courant and Friedrichs (1956) or Landau and Lifshitz (1987). The class of boundary value problems presented in this document are typically found in the studies of systems of hyperbolic equations. Lax (1973) considers a very gen- eral form for systems of hyperbolic PDE’s or conservation laws, and displays certain aspects of their nature and also addresses the admissability criteria for weak solutions of such systems. Hattori (1986) considers the Van der Waals fluid, governed by a specific system of conservation laws, for which he proposes a maximum entropy rate admissability criterion for all solutions. Truskinovsky (1991) also works with discontinuous solutions which occur in these types of systems, and explores the implications of various physical models which may be taken as a basis for the conservation laws. Therrnodynamically, a phase transformation is classified according to the type of discontinuity present in the materials free energy (Rao and Rao 197 8). A first order phase ' transformation is characterized by a material free energy that is continuous, but whose first derivatives are not. A second order transformation is similarly described by a continuous first derivative, but a discontinuous second derivative. One may also speak of mixed order transformations. However in this document the problems considered will be exclusively first order. Various authors have proposed models for the material in which the materials free energy possesses the required continuity, and yet still allows for transitions to occur. Falk (1980) proposes a function for a Helmholtz free energy to model the phase transition between Austenite/Martensite. The function is of the Landau-Devonshire type and is capa- ble of supporting first order transitions. The author derives the fundamental properties of Her-:9- - sue} mu; em . mud cou; tion C» ity of : transit Which 1-3 Pro Enjoy D; We be..- e“ Confider Since We Separated 10 such a material, and discusses the phenomena of shape memory and hysteresis which the model allows. Niezgodka and Sprekels (1988) derive the necessary equations which gov- ern a thermomechanical dynamical phase transition. Introducing a Landau-Devonshire model for the material’s free energy into the governing equations results in a system of coupled non-linear partial differential equations. The free energy function introduced in this document will not be of the Landau- Devonshire type. Since we wish to focus our attention on that subset of the material’s response for which a transformation can take place, and not the entire spectrum, we utilize an efficient method of constructing a material model. By constructing a free energy func- tion composed of a set of discrete functions, one for each phase, and requiring the continu- ity of this function at transition points, the model is able to capture first order phase transitions. This discreteness allows us to use a lower order polynomial for the free energy, which in turn simplifies the mathematics. 1.3 Problem Statement The material within this document can be thought of as being composed of two major parts. The first part consists of Chapters 1-4, the second consists of Chapters 5-9. We begin by first introducing the field equations for a one dimensional continuum, we consider a bar, where the equations are specialized to account for adiabatic motions only. Since we wish to consider the motion of phase boundaries, the Rankine-Hugoniot equa- tions are presented. These being jump conditions for the field variables between regions separated by a discontinuitys surface. Here this surface is initially motionless and so con- hide; natur. is the temp-e not Un ill of ( in deta liuh ha- Statgs a When it Uniqlm _ ”fiction [hair are 1 1 stitutes an equilibrium phase boundary. Eventually it is set in motion via the introduction of a wave pulse. In Chapter 3 we develop a 1-D constitutive model for a solid able to undergo phase transformations. A model for the Helmholtz free energy model is developed, it is a func- tion of the strain and temperature, and can be thought of as a “potential well”. The free energy is constructed in such a manner so as to accommodate two distinct phases. One of the major differences between the two phases is that only one possesses a shape strain. The stress response is derived from the free energy, one phase has a stress response that is independent of temperature while the other is temperature dependent. From it’s two phase nature, the stress-strain curve is nonmonotonic. One feature of the nonmonotonic behavior is the lack of uniqueness involved in a equilibrium configuration, even. by specifying the temperature and stress, or temperature and elongation, the state of strain within the bar is not unique. We show that the equations of equilibrium characterize a two parameter fam- ily of configurations. In Chapter 4 we investigate this lack of uniqueness in equilibrium configurations in detail. There we introduce three canonical equilibrium configurations, each configura- tion having a separate criterion in addition to the equations of equilibrium.These canonical states are families of one parameter equilibrium configurations. We then demonstrate that when any two of the three states coincide, then the resulting equilibrium configuration is a unique state. These unique equilibrium states play a central role in understanding the con- nection between the present fully thermomechanical description, and simpler descriptions that are purely mechanical in nature To begin the second part of this document we formulate and impose a set of initial “‘0 pt; bance Wave f del’eloj U011 Th than Wu We u. 12 conditions such that a single phase boundary is present in an equilibrium configuration. This initial configuration is then disturbed by a set of dynamic boundary conditions which give rise to a wave pulse that propagates into the bar from one of the boundaries. This travelling wave eventually encounters and interacts with the phase boundary, it is this interaction that we later investigate in detail. In general, this interaction causes the phase boundary to move, leading to phase transformations as particles pass through the moving phase boundary. A temperature-independent version of this problem was considered by Pence (1991a, 1991b). In his work a layer was composed of a two phase elastic material, but the stress response was independent of thermal effects in both phases. The mathematical equations which compose Pence’s problem are identical to ours. In Chapter 5 we modify this purely mechanical problem so that we may compare any solutions from our problem with those from the purely mechanical problem. One of the major goals of this research is to extend this previous work so as to encompass thermal effects, and demonstrate any new features of our more complete physical theory. The interaction of the incoming wave pulse with the phase boundary gives rise to two possible scenarios regarding the transmitted wave that becomes the leading distur- bance after the interaction. The first involves a shock, the other involves a centered simple wave fan. We only consider the latter case in this document In Chapters 6 and 7 we develop a system of algebraic equations which completely characterizes the initial interac- tion. This system of equations is indeterminate, there being more unknown field quantities than equations. Considering the phase boundary speed as a parameter, we are able to reduce the system of equations to a singe master equation, a nonlinear algebraic equation her? (I) H. H, 1;. —‘ (D [/1 , . >41 13 for a single field variable. A solution to this equation not being evident, we develop a solu- tion, albeit a family of solutions, for the unknown field variable via a perturbation from the purely mechanical state. We show once this perturbation solution is constructed that we can than calculate all the remaining field quantities. By comparing our results with those from the purely mechanical problem we are able to determine the leading order thermal effects, which is one of the major objectives of this research. In Chapters 8 and 9 we demonstrate how the second law of thermodynamics restricts this family of solutions. Finally, we consider an additional constitutive relation, a kinetic relation between the phase boundary speed and the driving traction acting on the phase boundary. We show that this additional criterion singles out a unique solution to the interaction. 14 1(7) Figure 1.1 Displayed is the nonmonotonic stress strain response for the material model describing phase transformations. Note for the prescribed level of stress in rm 5 r 5 I” that there are three possible values for the strain 7 2. Balance Laws Consider a body 3 which is a bar of length L and of constant unit cross section. Let el denote the triad of mutually orthogonal unit vectors associated with the reference configuration, where 91 is parallel to the rod’s axial direction. Denote the position vector of a particle in the reference configuration by x , then B = { (x1, x2, x3) II x1 6 [0, L] }. Let the position of the particle x at time t be y (x, t) , the defamation of the bar, and con- sider motions of the type: y(x,t) = x+u(xl,t)el, (2.1) where the function 11 (x1, I) is the displacement of the particle. Deformations of the form (2.1) describe longitudinal deformations of the bar. The problem is essentially one dimen- sional, therefore let x1 = x for notational convenience. In what is to follow the displacement u (x1, t) is assumed to be a continuous func— tion almost everywhere with first and second derivatives which are piecewise continuous. Denote the strain in the bar as y and the velocity v , then by definition — IY _ ’ (2.2) - v —t . (2.3) 15 L'sir. for u of line: “antics. 16 The strain 7 is required to satisfy 7 > —1 in order for the deformation (2.1) to be invert- ible. Introduce the following notation, let 1: denote the stress, 8 the internal energy per unit volume, n the entropy per unit volume, q the heat flux, 0 the absolute temperature, r the heat supply per unit mass, and p the mass density. Using a Lagrangian description, the local equations of motion (Dunn and Fosdick 1988) forthebarare: it -91 8t - Bx’ iv _ at pat "a? (2.4) de _ 3v _Bq at ' ‘5?“ 5r 992:_l_?_(9) t 6 pax 9 The first of these,(2.4)1, ensures compatibility of the displacements, (2.4)2 is the balance of linear momentum. Equation (2.4)3 is the balance of energy or the first law of thermody- namics, while (24),; is the second law of thermodynamics. face: adia,L re9th: quick, Uansi Occur.” ”him. 17 If within the bar an interface exists, x = s (t) , where the fields suffer a disconti— nuity then these field quantities must satisfy certain jump conditions across such an inter- face. These jump conditions are the Rankine-Hugoniot equations, which in this one dimensional setting are [[V]] = -S[[Y]]. [[1]] -$P[[V]]. (2.5) [[tvll + [[qll -stteii-§[[v2]]. Spllnll + [[3]] so. Here it denotes the speed in which the surface of discontinuity propagates, s = gis (t) . The square brackets denotes the jump in the enclosed quantity, say f, across s(t): [ [f] ] = f (s+) - f (s') ; while ((1’)) is the average of the function f across the inter- face: ((f)) = %[f(s+) +f(s')] . The class of problems to be investigated is now restricted to those which describe adiabatic motions, in so doing the internal energy production r and the heat flux q are required to vanish. One may regard adiabatic motions as idealized processes which occur quickly with respect to the continuum thermodynamic time scales associated with the transfer of heat by diffusion and radiation. Isothermal motions may be considered to occupy the other end of the spectrum, where events occurs so slowly that the body has enough time for the transfer of heat such that the temperature can equilibrate with an external ambient temperature. Under the adiabatic assumption, the equations of motion l6. wher them 18 and jump conditions simplify into the final form which will be used throughout the remainder of this document: (2.6) [[V]] +S[[Y]] = 0. [ltll+Sp[[Vll =0. (2.7) Stilell—<>l[rll) =0. Spllflll-SO- When working within the context of the mechanics of solids it is often convenient to uti- lize the Helmholtz free energy function \v (y, 9) . The internal energy and Helmholtz free energy are not independent, but are related through the expression 8 = timer]. (2.8) 6h Fit the the LA Will. 19 From standard thermodynamic definitions (Ziegler 1988, Ericksen 1993) the stress and entropy may be derived from the Helmholtz free energy _ 8v _ 8v 1-3—y, n —-56. (2.9) . dc dw d . From (2.8) the left hand srde of (2.6)3 can be expressed a = d? + a{(971) . which from the definitions (2.9) and (2.6)1 is simplified 31—: = 13-;- + 03—1]. Therefore the first law of thermodynamics (2.6)3 in the adiabatic setting can be expressed in the alternative fashion 93:32 II .o (2.10) which requires a particle’s entropy to remain constant ph. P“ the; der uns tain the. nut but [101: [an]; 3. Material Model 3.1 Construction of the Material Response Functions The purely mechanical problem of a packet of shear waves interacting with a phase boundary has been previously studied (Pence 1991a, 1991b). However during such processes a more complete description necessitates accounting for thermal as well as mechanical effects (James 1983,Truskinovsky 1985). Recently, a material model has been developed which may be used to describe a multi-phase therrnoelastic solid (Abeyaratne and Knowles 1992c). The model considered in this document is similar in form to theirs, but we more fully use its ability to model thermomechanical motions by less simplifica- tion of material parameters. One major goal of this research is to introduce this model and demonstrate its ability of capturing nonlinear elastic adiabatic motions. We also will dem- onstrate its ability to collapse into a form which is purely mechanical, and for which cer- tain previously determined results are shown to fall under a more complete thermomechanical framework. The model to be presented intuitively seems more realistic than a description which allows only mechanical motions, and the use of this more complex energetic description should enable one to gain a better understanding of observed physical phe- nomena such as thermal softening and the Austenite/Martensite transformation, the later being more in the focus of this research. This section explains the model while latter sec- tions modify it, the final form being the basis for the remainder of this work. Assume the internal energy to be a state function with thermodynamic variables of temperature and strain. For a linear elastic stress strain response, the internal energy a 20 and pera den 10 a den “I”. 21 should be a quadratic function in strain 7 at constant temperature 9: e=é. (3.1) Here and throughout this manuscript the tilde superscript will be used to indicate a func- tional dependance on strain 7 and temperature 6. From this initial form of the internal energy we will construct the Helmholtz free energy from which the stress and entropy may be derived. An alternative approach is to begin with the internal energy using the strain and entropy as the appropriate thermodynamic variables from which the stress and tem- perature follow. In this setting, any material whose internal energy can be additively decomposed into a function of strain alone and a function of entropy alone will be referred to as separable. This special form of the internal energy ensures that the stress is indepen- dent of the entropy and results in the mechanical jump conditions (2.5)13 being indepen- dent from the energy jump condition (2.5)3. Knowledge of this special form for the internal energy function will prove useful in future development within this document. From (2.9) the Helmholtz free energy u! can be expressed \II = WM?) = é—efi. (3.2) where 1] (y, 6) , the entropy of the material, is also a function of strain and temperature. The entropy can be developed by using the following results from (3.2) and (2.9) " — 91". (3.3) [“—-2—(°’tr— 21(9)] 41(9) [7 a(9)1a".(e)+b(e)] (3.4) here the superimposed prime denotes differentiation with respect to the function’s lone variable. Equation (3.4) may be integrated thereby giving the entropy to within an arbitrary function of the strain 9 titre) =j [— Lg—m— a(§)l -u(§)lr— a(§)1a(§)+b'(§)]d—§+Ftv) ..(35> where F (7) resulting from the integration is a function of the strain 7 only. Combining the internal energy (3.1) and the entropy (3.5) to construct the free energy (3.2) one may then calculate the stress via definition (29)] fit. 9) = M9) [r—a(9)l- (3.6) 9 95 0] [we [Y-a(§)] -u(§)a’(§)l g -eF'm. or in the alternative form this by u math IA] I IAQI Form.- U511] g ‘ 23 H19) = €1(9)7+C2(9) —6F’(r). (3.7) where (31(6) = M9) 49? Mac—lg (3.8) - , d§ , d5 C2(9) — -u(9)a(9) +0? 11 (ONO-5+9? u(§)a (i)? (3.9) The following restrictions are now placed on the response functions so that certain desir- able features may be incorporated into the material model. These assumptions are guided by the desire for a model with enough generality to capture the phenomena of interest in a mathematically tractable fashion. (Al) The slope of the stress-strain curve is independent of temperature. (A2) The internal energy has a linear dependance on the temperature at constant strain. 2 Formally the restriction (A1) states: —a—7r' (y, 6) = O. Carrying out this differentiation 893v using equations (3.7) and(3.8) leads to d2 —2 F(y) = Cl’(0) = x = constant. (3.10) dt Aunt; Equau' 1'an FrunlIE ASSllmp [hill the thl’C 3 equation 24 After performing the necessary integration, the expression for F (7) is found to be: 2 F(y) = gamut. (3.11) Equations (3.8) and (3.10) yield: )5 = (1(9) = 314 11(6) -9Iu' (5) %),where upon 9 rearranging one finds a x = -j u’(§)d—§. (3.12) From (3. 10)-(3. 12) x = 0. 11(9) = u = constant. Ftv) = Knit. (3.13) 2 Assumption (A.2) states that 125-EH, 0) = 0, so that (3.13) coupled with (3.1) require 86 that the functions a (6) and b (6) satisfy a(e) = a, 13(9) = 59+6, (3.14) where a , 5 , and b are material constants. Inserting expressions (3.13) and (3.14) into equations (3.1), (3.5), and (3.6) for the internal energy, entropy and stress yields e=é(y,0)=%u(y—a)2+50+b, (3.15) than imphc mung mutt; Thuteh aIllhmic amblgum.- 25 Tl = Time) = 51MB) +ky+k, (3.16) 'c=‘t'(y,6) =u(y—a)—k6, (3.17) where a, 5, b, u, k, k are constants which characterize the material. The non-dimensionalization of the argument within the logarithmic function is implicit in the integration of the function F(y) of (3.13)3. This can be achieved by recog- nizing that the integration constant k may be redefined in any convenient manner. To make this explicit, let k = - 61MB.) + k , where the normalization temperature 0‘ is E’ E). (3.18) 9. 5 6Xp(—6- This temperature will be defined more precisely in later sections of the document. The log- arithmic function in the entropy (3.16) can now be normalized using (3.18) 1301.9) = 51n( °.|C,1n(e )+uay+k, (3.31) fire) = u(r—r')—ua(9—9'). The reader is reminded that 9' is essentially a free parameter in this description since any redefinition of the value for 0‘ can be compensated for by a redefinition of the values k and y. by . . .. - 9;", - eold -) enew => kold _> kold + prln :- 5 knew ’ old Yold —) Yold - (1( 9old - 6new) 5 Ynew ‘ (3.32) 3.2 St relerar sis one Thus by St fmuted 0n 31 3.2 Separable Energy In the above analysis we have chosen to use the stain y and temperature 0 as the relevant thermodynamic variables, whereas in a more traditional thermomechanical analy— sis one would work with the strain 7 and entropy n as the independent variables. We now recast the above model in terms of the strain and entropy, then the internal energy, (3.15) may be expressed é(r.9) = 5(Y.é(Y.Tl)) = mm. (3.33) assuming that one may invert the entropy function f] (y, 6) for the temperature: 0 = 9 (y, 11) . Using (3.16) one finds that the temperature as a function of strain and entropy is given by ln(%) = Tl-lla'Y-k 9 pCY where upon inverting é (y,n) = supp—19935). (3.34) PC, Thus by substituting from equation (3.34) into (3.33), the internal energy may be expressed as a function of the strain and entropy. An analogous procedure can be per- formed on the stress, the results of these operations gives 32 it: t 2 t __ ._~ A e = é(y,n) =L2-1(y—(y —0t9 )) +pCYO exp(n—E%Lk)+b, (3.35) ‘Y I = tan) = u(r—r‘)—ua9’[exp(%)— 1]. (3.36) ‘Y Equation (3.35) demonstrates that if the coefficient of thermal expansion or vanishes (or —> 0) then the internal energy can be written as the additive combination of a function of the strain alone and a function of the entropy alone, i.e. é (7,11) = 3(7 cf")2 + pC e'exp('li‘) +13 (or = 0). (3.37) 2 7 pC7 Recall that (3.37) is the special form of the internal energy that we termed separable in Section 3.1. If the internal energy is separable then the equations governing the mechani- cal evolution of the body are independent of those controlling the thermal evolution (Cou- rant and Freidrichs 1956). Since this decoupling between thermal and mechanical processes will play a significant role in this research, we temporarily proceed by develop- ing the associated material response functions under this specialization. Setting or = 0 the stress and temperature reduce to 3m = 3%étm) = tun—v"). _ 3~ .. : 11:3 9(11) - Eaten) - 0 exp( pC‘Y ) In th: Since it and (3.; WIllicit. u 33 In this specialized form the equations of motion (3.23)-(3.25) reduce to By _ 3v _ - — p'a‘i - “-3; (a - 0) 9 (3-40) " 11;]2 3'1 _. _ 9 exp( PC, )at — 0 (0t — 0). (3.41) Since the normalization temperature 9. need not vanish, one draws Otexp( 123%) at 0 r and (3.41) reduces to @ = o (a = 0). (3.42) at which, when coupled with (3.38), gives rise to 39 _ - B—t _ 0 (or — O). (3.43) Thus, in the absence of discontinuities such as shocks, (3.43) shows that boundary value problems which have isothermal initial conditions will proceed isotherrnally. Furthermore, considering this special (or = 0) case, the entropy for each particle will persist from the 34 initial state. From (3.39)-(3.41), it is seen that if the internal energy is separable, then the gov- erning equations for the body simplify into two distinct sets of governing equations, one being (3.39) and (3.40), which are a purely mechanical in nature, i.e. they only involve the velocity and strain fields, the other set is (3.42), which govems the thermal evolution within the body. For materials with such a separable energy one could envision boundary value problems in which a specific thermomechanical problem is posed and from the above results we see that the mechanical fields develop independently of the thermal fields. For such boundary value problems, the mechanical fields being independent of the thermal fields, the investigation can completely ignore the thermal evolution which occurs and concentrate on the purely mechanical problem. 3.3 Characteristics and Riemann Invariants It is observed that the governing equations(3.23)-(3.25) are a system of homoge- neous quasi-linear first order partial differential equations in terms of the three indepen— dent field quantities: v, y and 6. One may derive a different yet equivalent set of governing equations by use of the method of characteristics (Renardy and Rogers 1992). This technique yields an alternative set of governing equations which are linear combina- tions of the original system of equations, such that the dependent variables in each result- ing equation are differentiated in the same direction in the (x,t)-p1ane. The directions of differentiation are called the characteristic directions or characteristics of the system, and the alternative equations are the characteristic equations of the system. In general this whet. andC Where I are the . ibflhe( FOTEaCh equaIIOD 35 method of construction (Renardy and Rogers 1992) proceeds as outlined in the next para- graphs. Consider a system of first order partial differential equations written in the form AUX+BUt = C. (3.44) where the column vector U consists of the unknown functions to be determined, and A, B, and C are the coefficient matrices, which may be functions of x, t, and the components of the vector U, but not UK or Ut. In this study U = [7, v, 6] T. The characteristic directions and equations for (3.44) are obtained from the solutions to the eigenvalue problem AT(A-}.B) = 0, (3.45) where the scalars A are the eigenvalues for the above problem, and the column vectors A are the left eigenvectors for the system. The eigenvalues it. are the characteristic directions for the original system of equations §=i. e%) For each eigenvalue the associated eigenvector A is used to determine the characteristic equation by forming linear combinations of the original system in the form 36 ATAUX+ ATBUt = ATC, (3.47) where the dependent variables in the resulting equation are all differentiated in the same direction, the directional derivatives being the characteristic directions of the system. 3.3.1 Separable Materials We begin by considering the method of characteristics for a material having a separa- ble energy. We use our original set of PDE’s (3.23)—(3.25), setting or = O , the governing equations are then 17:?! at 8x EV- 3)! = 4 95? .pa (a 0) (3. 8) r 39 _ * 8v utr-r )fi +13%; — 110-7 )5; Analyzing the system of equations (3.48) it is not immediate that the mechanical field quantities are decoupled form the thermal field quantities. However, by recasting the sys- tem of equations using the method of characteristics, the temperature is shown to decouple from the strain and velocity. Proceeding with the method of characteristics for the above case in which or = 0, produces an alternative system of governing equations, the charac- teristic equations and the characteristic directions of the original system. Written in matrix form as in (3.44), where the vector U = [7, v, 0] T , the governing equations are now Perfo: tic dU The $33: the left a lion. Equ (342m equillOn 1 direCUOn 37 1 0 0 Vx O —l O vt O O u 0 YX + —p 0 O 7t = O (a = 0) - (it-7*) 0 0 pa _0 -ll(Y-Y )-pC1 _e,_ 0 Performing the calculations described above, the characteristic equations and characteris- tic directions are found to be: d6 dx — = _ = .4 (it 0011 dt 0, (3 9) ((1:0) FL" HEY- 2‘- fl (“fl/gm — Oon dt — :t p' (3.50) The system of equations (3.49)-(3.50) is composed of a characteristic equation, which is the left equation of either set, and the associated characteristic direction, the right equa- tion. Equation (3.49) was to be expected since it was derived in an alternative manner, see (3.42). Along each characteristic direction it follows that the associated characteristic equation may be integrated, providing an algebraic relationship along the characteristic direction 9 = constant ong—J: = O, (3.51) (0:0) 3 = 91‘: J9 2 vxfiy constant on dt :t p’ (3-5) ”241K lnm» whit? 38 the constants of integration in the above equations are known in the literature as the Rie- mann Invariants of the system (of. Renardy and Rogers 1992). For ease of notation we introduce the material constant C = .lll/p, (3.53) which is the acoustic wave speed of the material, this allows (3.39) to be expressed as v 2!: c7 = constant on 31—: = :|:c. (or = O) (3.54) Through the use of (3.52) one can calculate the strain and velocity fields independently of the temperature field, and vice-versa. In fact from the above results it is seen that all motions which begin from a constant temperature proceed isothermally, if the formation of shocks are excluded. Thus if or = 0 then the mechanical and thermal evolution of the material proceeds independently of one another. Consider an initial-boundary value problem such that the above set of characteris- tic equations hold. If the initial-boundary data is given then the Riemann invariants can be calculated along the associated characteristic directions. One could then proceed with the initial-boundary value problem and formulate a set of algebraic equations relating the Rie- mann invariants in different regions within the domain, where the different regions are connected along one or more of the associated characteristic directions. Pence in his inves- tigations (1991a,b) demonstrated the use of this technique in formulating families of solu- tions for a set of problems under this purely mechanical framework. 39 3.3.2 Nonseparable Materials Consider now the more general problem in which the thermal-mechanical responses are coupled, from the earlier development this occurs when the coefficient of thermal expansion does not vanish, or at 0. The method of characteristics will now be developed for this case. The equations of motion in terms of field variables y ( x, t) , 9 (x, t) and v (x, t) are found by inserting the constitutive response,(3.15)-(3.21), into (3.23)-(3.25), and using (3.53) i=9: at 8x' c237 2 86_3v “ax cad—x-Et’ co— (7— —oze))g"+ca =c2((v—v’)-a(e-e’))g"—x. (3.55) (3.56) (3.57) The first step in finding the characteristic directions and equations is to cast (3.55)-(3.57) inmauixform r 1 _ .. _ 1 O O vx 0 _1 0 c2 -czor Y, + ‘1 0 t a. 2 It at _c2((7—v )—a(e—o )) o o _ _6,_ _0 —c (y—(y —or9 )) 4:1 0 0 - ll CO .58) as before the characteristic directions and equations as well as the eigenvalues and eigen- 40 vectors for are to be determined. Performing the necessary calculations one finds that the characteristic equations and directions are d" 0, (3.59) 4 2 2 2 ’2 00¢de 2dy 2d9 dx / core i —. —— —= _= —. 3.60 c + C1 dt+cdt cordt 00ndt ti 1+ C1 ( ) Equation (3.59) is equivalent to requiring that a particle’s entropy remain unchanged dur- ing smooth processes. This can be shown by differentiating equation (3.16) with respect to time and comparing the result with _-__ it- d_x_ — +pc ordt —0 on dt —0. (3.61) One can integrate the expression (3.61) along its characteristic direction and arrive at the conclusion that a particle’s entropy is conserved along its characteristic direction. This constant would be one of the Riemann invariants for this thermomechanical problem, i.e. )+ pczoty + E = constant on _d_x = 0. (3.62) 2 ' (it n = fi (7. 9) = 9Cyln( OD The wave speed in (3.60) is no longer constant, as in (3.52), but now is dependent on the material’s temperature. It is seen that the wave speed is a monotonically increasing func- 41 tion of the temperature for the model under consideration here. In general equations (3.60) and (3.46) cannot be integrated, and hence the other two Riemann invariants are not known in advance. However, it is possible to calculate these two Riemann invariants if the initial data are specified in a particular manner. Since the knowledge of these invariants will enable us to construct a solution to the problem defined in Chapter 6, the discussion of this special case will be pursued in the following paragraphs. Consider the problem just outlined above and in addition suppose that the entropy is initially constant on a region contained within the domain. As a result of equation (3.62) the entropy within this region is constant for all time. Under these assumptions we may manipulate (3.61) and find a relationship between 7 and 6 in this same region d7 _ _ C1 99 dx - a _ czotedt on dt — 0. (3-63) Since 1] does not vary in the region for all time we can substitute from (3.63) into equa- tions (3.60), resulting in 4 2 C 4 2 2 2 i’cz+coregv_ 27 [C2+c_or 6:lg—0=O ond—x==|:c ’1+c__or6. (3.64) C1 t 9C a C7 I (1! C1 The two equations (3.64) may now be integrated along their characteristic directions to produce two additional Riemann invariants, the results of which are shown below: 42 d>(9) —(O) (O) iV'JE,[2¢ (9) + d) (0) ln( II = constant, (3.65) on 2 2 dx_ core a —=Fc ’1+ C1 , (3.66) where I C O. This can be viewed as the adiabatic correction to a purely mechanical iso- thermal theory (in which the sound speed is formally temperature independent). Presum- ably an adiabatic correction to an isothermal theory with a temperature dependent sound speed would behave similarly. A familiar example is provided by flow in a compressible fluid (Landau and Lifshitz (1987), Whitharn (1974)), where the sound speed is, in general, given by Jg—E where p is pressure and p is density. For a polytropic gas, the sound speed in an isothermal setting follows from the ideal gas law p = Rpe , where R is the ideal I p where l = CP/Cv = l + R/Cv , Cp and Cv give the specific heats at constant pressure gas constant, as c a ./R_9 . For adiabatic conditions, the entropy Cvln( 3) is constant, 43 and volume. This gives the adiabatic sound speed as J? = JIRO .=. cadiabatic . Thus 2 = c 1 + 33—9 , which displays a similar temperature dependance to the adia- Cvc batic correction of isothermal sound speed. cadiabatic 3.4 Multiphase Materials We now broaden the class materials considered to those which allow for the exist- ence of different phases and the transformation between phases. At each point within the body the material is said to occupy a phase, this being determined by the value of strain and temperature at that location. In light of the previous derivations, consider a multiphase material for which each distinct phase can be modeled using the Helmholtz free energy function (3.21) developed earlier in this chapter. For first order phase transitions it is _ required that the free energy be a continuous function of strain and temperature, but its derivatives may suffer discontinuities. Begin the construction of the multiphase model by assuming that the internal energy, stress, entropy, and free energy functions in any phase has the form as provided by equations (3.15)-(3.21). Assume for generality that there are n possible phases and that from (3.21) the free energy in phase i is .2 .. PC? a t 2 v,(r,e) = T[y-(yi-mie )] +pC7i9[1-ln(e )] — pcizorie'y — 12,0 + 51- (3.68) Thus each phase is characterized by material constants ci, 7:, bi, bi, Iii, ii, and the normal- ized temperature 6‘ is chosen to be the same for all phases. The relations (3.32) between 44 the material parameters k , y‘ and 6*, ensures that there is a no loss in generality in requiring 6‘ to be the same for all phases. A slightly modified form of thefree energy (3.68) can be written which displays its “potential well” structure 2 2 2 _ pCi t t 2 0 pCi (Xi * 2 Wee) = -2—[7-(yi+a,(e—e ))] +pCYi9[l—ln(?)]——2—(6—O) (3.69) 2 at 2 pc,(a,e) +5 2 -= ~ -(pci0ti'yi +ki)6+ 2 i. At a fixed temperature 6 it is seen from (3.69)that the free energy is a quadratic function in strain 7, and the associated energy {iii (7, 9) may be thought of as a “potential well”, whose strain-vertex is located at y = 7: + ori(9 - 9.) . The location of each of these verti- ces changes as the temperature varies (the wells movie up and down). By definition a first order phase transformation process that advances through equilibrium states requires that the free energy be a continuous function during the phase transition. Therefore for a first order equilibrated transformation, say between phase—i and phase-j, it is required that Vi - ‘I’,- = 0- Consider a material which has n phases, then at each set of temperature and strain, (7, 6) , there are n values of the free energy iii (7, 6) i=1,...,n. The minimum value of the collection of free energies {iii defines the energy minimal phase-i associated with the pair (7, 9) . This energy minimum implies a phase indicator function I, where (IE [Ln-.11] =1(Y.9)), 1(7: 9) 3 $1”, 0) (Y: 9) = mini=1,uqni’i(yv 6) r 45 From this discussion it is evident that a particular set (7, 0) exists for which two distinct phase’s free energies will have equal values. The collection of these sets may be thought of as curves in the (y, 6) plane where the different phase’s free energy functions inter- sect. This gives rise to an intersection strainfy'u = s)“. (6) , the strain in]. (9) at temper- ature 6 where the free energy between phase i and j have the same value. One may think of this as that level of strain where the potential wells intersect. This intersection of free energies naturally leads to a method of determining which phase a material would inhabit given a value of (y, 9) . If one considers the principle of energy minimization as the crite- rion for selection between two phases, the intersection strain indicates that point in (7, 6) where two phases would exchange favorably, and the possibility of a phase change exists. For the material represented by (3.69) two distinct possibilities for i, j (6) exist: one in which ci ¢ cj and that for which ci = C]. . When such intersections exist, the former case, in general, yields two roots for the intersection strain, while the later yields a single root. This discrepancy between the number of roots is simple to understand. Recall for a fixed temperature that the free energy is a quadratic function of strain, and when the wave speeds are not equal then the curvatures of the two free energy functions are not equal. Because these curvatures are not the same, the quadratic nature of the two free energies functions gives rise to intersections which occur at two different locations (real roots) or which do not intersect at all (imaginary roots). However, when the wave speeds are the same, and thus the curvatures are equivalent, then there is at most one location where the two free energies intersect. 46 A direct calculation yields the two intersection strains for ci ¢ cj 1 am (252(9) 9(9))2 ?,,,-(e)=‘; 1:9“ "“ , (3.70) (ci —cj) 2p(ci2-cj2) where a 2 * 2 * 2 2 *- .-.-.(6) = (cl/yj -ci'yi -t-(0tjcj —orici)(0—6 )), c: 2 2 . .. 2 s 2 * 2 r 2 r '- .-.-. (9) = 4p(ci —cj)(2(bi—bj) +p(ci7i -cjy )—2p(oriciyi -0tjcjyj )6 + muffin _ “12“,?7'39‘2 + 2p (c1, — Cu) 9( 1 411(2))— 2(12,- 121-)9 ). 0 The other case ci = cj produces the single root A A 2 $2 .2 ~ ~ _, 2 b-—b- + c . — . -2 k-—k. 0 Yu<9>= (. ,) 9(7, 7,) (, ,)+ 2 ,_ . . (3.71) 2pc (7, —7,- + (ai-ajHO—G )) 6 2 e c t 2 2 $2 2 t2 :2 2p(Cyi—Cyj)0(l-ln(;))-2pc (oriyi -otjyj)0 +pc (oti'yi —orjyj )0 2pc2(Y: 4: + (oni —aj) (9 - 9') ) For future reference we define phase-l, index i=1 in equation (3.69), as the parent phase. The material model will now be restricted such that phase-l has a free energy that is sym- metric with respect to the strain for a fixed temperature. Formally, this assumption is stated: 47 (A3) Phase-l, the parent phase, has a free energy {[11 which is symmetric with respect to the strain 7 at fixed temperature 9. Satisfaction of assumption (A.3) gives 7: + «1(9— 9*) = 0. (3.72) Since the temperature 9 may vary, this requires that or 1 = 0 and 7: = 0. Therefore, in the parent phase the coefficient of thermal expansion vanishes and the reference state is strain free. Furthermore, since a zero value for the coefficient of thermal expansion repre- sents a material which has been deemed separable, implying that the mechanical equations of motion decouple from the equations of thermal evolution, the parent phase material response is referred to as separable. Henceforth it will be assumed that the bar is composed of a two phase material, one being the separable parent phase defined in (A3), the other phase being the more gen- eral material, i.e. of the nonseparable type, also called fidly thermal. Formally we express this assumption: (AA) The number of phases will be restricted to two, which we will call phase-1 and phase-2 (phase-l being the parent phase). Together (A3) and (AA) state that the bar consists of a two phase solid, the two phase’s having different thermomechanical properties. The reference state for the parent phase 48 was defined to be strain free, and for notational convenience the offset strain in phase-2 is designated 7" a 7;. Figure 3.1 displays the two phase material’s free energy functions for constant temperature, as mentioned earlier these two functions can be visualized as dis- tinct potential wells. Each vertex can be thought of as that energy’s ground state, since it is the minimum value for that energy. Shown in Figure 3.1 are arrows located at the vertex of each well, these arrows are to indicate that the vertices are not stationary but may change positions depending on the value of the temperature 6. 3.4.1 The ground-state equivalence temperature We define the ground-state equivalence temperature as that temperature for which the w- value of the free energy vertices, or ground states, have the same value. At all other temperatures the vertices involve different \y- values, and thus one phase has a free energy ground state whose w- value is less than the other. Thus a ground state equivalence tem- perature separates temperature intervals associated with a natural change in stability of the ground states. To inquire further into this issue we construct the function AV (6) which is defined as the difference between the free energy vertex in phase-2 and that in phase-l AV (9) 5 “’2' (3.73) vertex 1 IVGI'IBX In phase-2 the free energy vertex is located at y = 7. + a2(6 — 0.) , while in phase-l it is at 7 = 0, hence AV (6) isexpressed 49 2 a2 AV(9) = —pc 2 “2 (9 9'2) +p(C2 —C22)e[1—1n(-9,)] (3.74) o c ore) ,. (—pc2ot2'y +k2-k1)9+ p—i(—2—2-——--+62—b1. From the freedom inherent in selecting the normalization temperature 9‘ (3.18), the nor- malization temperature is now chosen as the ground-state equivalence temperature. This choice of 6* requires that AV(0‘) .-:- 0 , which in turn provides a quadratic equation that 9 must satisfy *2 6+ it ~ ~ It 2 A A 2(p(C72—C71) ~pc§or2y —k2 +k1)6 + 2 2(b2-b1) = O. (3.75) pc2or2 pc20t2 Since, in general, there are two roots for equation (3.75), there exists two ground-state equivalence temperatures. Solving (3.75) for 6. yields e__1 3 ~ ~ e 6 = 2 ——i(p (C22—C .21) - pc§a2y — k2 + k1) :l: 9 , (3.76) pc2or2 where l at 1 e ~ ~ 6 = ——((p(C72—C C72) —pc§or2y —k2+k1)2 -2pC2(12(62— 61))2 . pc2or2 The model naturally gives rise to phase transformations provided that 9. is a real quantity and we henceforth only consider parameter values for which this is the case. For conve- 50 nience, the two ground—state equivalence temperatures are distinguished from one another based on their relative values, i.e. 9: < 6; . This ordering depends on the relative values of the material parameters. From the ordering of the ground-state equivalence temperatures, three distinct temperature intervals naturally arise: 9 < 6: , 6: < 9 < 0; , and 9; < 6. One of the phases will have the lower energy ground state in the two intervals 0 < 0: and 9; < 0 , while the other phase will have the lower energy ground state for the interval 0: < 9 < 6;. Table 2 summarizes the preferred phase for each of these intervals. Table 2: Temperature intervals for preferred phase Temperature interval Phase with lower energy ground state 9 < 9: phase-2 e: < e < 92 Phase'l * phase-2 92 < 6 To verify this table note from an asymptotic analysis of AV (9) for "6 — d.“ » 0 , that the quadratic component of AV (0) is the dominant term. Under such conditions AV (6) behaves like a parabola, and the value of this function is either positive or negative depending on the coefficient of the quadratic term, which according to (3.74) is —pc§or§/ 2 < 0. Therefore phase-2 has the lower ground state for temperatures which are 51 much greater or less than both 9: and 6; , which is precisely the two intervals 9 < 9: and e2<9. Figure 3.2 schematically shows how the two free energy functions behave for the different temperature intervals. Note that the ground-state equivalence temperature 9: separates a low temperature stable phase-2 material with a shape strain 7‘ + a2(6 — 9:) from a high temperature stable phase-1 material with no shape strain. This type of material behavior is similar to Austenite/Martensite systems. It will prove useful for latter purposes to display the ground-state equivalence tem- peratures for two special cases. The first case being when the specific heats are equal. C72 = C the ground-state equivalence temperatures (3.76) then simplify to 71’ o' = _;_2(pc§a2y'+122—El)io‘, (3.77) 2 2 l(C22=C,“) @3_l(2w~~2 22~~)2 — _2§ pczaz’y +k2—k1 -2pC2a2(b2—bl) . pc20t2 Note that 9‘ is real if 52 — 51 is negative, or if 52 — 51 is positive but sufficiently small. The other case is where both phases are separable and thus a2 = O. Multiplica- tion of (3.7 5) by or: followed by letting a2 —) 0 yields a first order equation for 0‘ . Thus one of these two roots 6“ becomes infinite as a result of the singular perturbation. For- mally expressing (3.76) in the form 52 2 * - ~ 4 C —C — c or -k +k .. 9 = (N 72 7‘) 22227 2 1)(—1i9), (3.78) pc20t2 2 2 l (I). _ l- 2pc20t2(b2—b1) 2 2 t ~ ~ 2 (p (C72 — C11) - pc2a2y — k2 + k1) performing a Taylor series expansion for O about a2 = O, and collecting similar powers * of a2 , gives the two different series expansions for 0 : .. 2 12 —ii — C —C " " _" 9 ___ [2 1 92(272 71)]+2l+ b2 191?. ~ +O(or2), pc2a2 “2 P(C72-C71)-k2+k1 and :- 6 —b e =- 2 11 T+0(a2) Here, as is standard, O(z) denotes a quantity that, after division by z , is finite as z —-) O. In the limit as a2 -) 0 there are two distinct cases depending on the relative values of material parameters, the results are 91 = —oo («2 = 0) (3.79) .. B -3 91 = 2 bl; Q 62:00 53 Application of this result would in general restrict attention to a limited temperature range, and the material parameters entering the model would then be chosen as the basis of this temperature range. In particular, temperatures 6 are regarded as positive on some absolute temperature scale. However in this section, and from time to time in what follows, it is convenient to treat 9 as an arbitrary real number purely for the purpose of clarifying the global mathematical structure of a physical description that would certainly be localized in an application setting. This is the sense in which results like 9: = —oo should be consid- ered. 3.4.2 The Latent Heat Any heat produced or absorbed from the transformation between two equilibrated phases is the latent heat of transformation AT. The latent heat of transformation from phase-2 to phase-l is expressed AT = %[fi2(y', e')—fi1(o,e’)]. (3.80) In particular for (It > O , p > 0 it follows that the transformation from phase-2 to phase-1 is exothermic if L]. > O and is endothermic if AT < O. From definition (3.80) and the existence of two ground-state equivalence tempera- tures we conclude the existence of two latent heats, one for each ground-state equivalence temperature. The general expression for the latent heat can be computed from (3.19) 54 9‘ 2 ' ~ -- M = 3[Pczazv +k2-k1] . which motivates the definition A.“ and 1.1.2 where 9i 2 .. - _, . 7m = flpczazv +k2—kl] (1:1,2). (3.31) The difference between the two latent heats is 29‘ 2 * ~ ~ AAT = 11.2—le = T[pc2a27 +k2 -k1] , which does not, in general, vanish. Note however, that A)».r = 0 when 6“ = 0 which corresponds to the existence of a single ground-state equivalence temperature. We hence- forth use the generic terms 9’ and 7hr , where the specific ground-state equivalence tem- perature and latent heat is inferred, when necessary, from the appropriate temperature interval under consideration. 3.5 Summary of the Two Phase Material Model For future reference the free energy, entropy and stress response for both phases are given below in terms of the more familiar thermodynamic variables, these relation- ships between the different forms having just been developed. These forms will utilized throughout the remainder of this document. 55 Phase—2 material: 2 .. C t a: 2 W2(Y,6) = 922[Y_(y +oc2(0-6 ))] +529[1—1n( ‘DJCD )] (3.82) pczaz 2 pc2(a 9:32 — 32(9-9 ) —(pc§azy +k2)6+——2—§2——+52, 132(7) 6) = bzln(§) + pciaz'y +122 , (3.83) 762 (y, 6) = pcgw - 7') - pciazw —- 9‘). (3.84) Phase-1 material: pc2 in (v. 9) = 7172+518[1-1n(%)]—E19+61, (3.85) 6 .. 9 .. 111(9) = Blln(e—;)+k1. (3.86) 761(7) = pcfy. (3.87) 56 In the more familiar thermodynamic variables p, hi, 7., 9., C a2, 27,121, 5, these func- Yi’ tions are Phase-2 material: ~ '1 t t 2 2 W2(Y.9)'= {(7—(7 +a2(9—9 ))) +pC729[1-1n(-.)] 2 (9—9) .2 .. a6 . _[kl+pA—r]e+u_2_(_2__2_+b If 2 2’ 31. (3.88) - .. p - 71201.9) = PC721n(2;)+l12%(Y-Y )+ a: +kl’ 6 9 home) = uz(v—7')-uza2(e-e') . In the Phase-l material: {4'11 (7, 0) = E2372+pC716[1—ln(-:;)]—E10+51, (3.89) {11(0) = pCYlln(%)+El , Q i, (7) = 1117. Although we have not as yet defined a phase selection criterion, given some (7, O) , we now state that the intersection strains 71 (6) (3.70)-(3.71) define the upper/lower limits of strain that the material can support in a particular phase given a temperature 0 . This 57 assumption is based on the principle of energy minimization as was discussed earlier in calculating (3.70)-(3.71). We now turn our attention to the stress response for the two phase solid and restrict attention to a single intersection strain 71. Then from (3.84) and (3.87) one obtains i1(y,6) = pcfy forphase 1:O0 fa’ a f — r° (3.94) In general, f is a function representing the fully coupled material, which collapses to f 0 when considering a separable material. The function f “2 is seen to provide the bridge between these two materials. It is to be emphasized that this so far generic function f could represent quantities that are known a—priori, such as the free energy function. Alter- natively f could represent quantities that are determined as part of the solution to a prob- lem, such as the field variables (7, v, 6) that describe physical processes. In the latter case, an exact determination of the function may not be possible in a fully coupled mate- rial, but the exact determination may be possible in the reduced problem involving a sepa- rable material. In such cases f is unobtainable. whereas f° is obtainable. This, in turn, renders f “2 unobtainable. In such instances it will be extremely useful to understand the leading order effects of thermomechanical coupling by investigating f0L2 in the small az-limit. In such instances, the perturbation expansion for f “2 becomes a natural object of study. 63 A W089) fixed temperature 9 V1 W2 Figure 3.1 This figure is a schematic representation of the Helmholtz free energy function \v (7, 9) plotted against the strain 7 at a constant temperature 0. Shown are the two free energy functions w] and \v2 , each represents a distinct phase of the material.The arrows, which are shown at each of the vertices, acknowledge that the location of these vertices shift as the temperature changes. At each temperature there exists a level of strain for which the values of the free energies are equivalent, this strain is designated 71 6 > 9*2 Mme) W2 is preferred ‘Vl W2 TY> 0‘1? 6 (6‘2 A W (Y 6) w, rs preferred W2 W1 > Y W(Y.9) W1 + . W2 6<9 1 1112 is preferred > 'Y Figure 3.2 This schematically shows how the two free energy functions interrelate for the three temperature intervals. In the intervals 6 < 0: and 6 > 6; the phase-2 free energy has a lower vertex and in this sense is the preferred phase. For the interval 0; < 6 < 0; phase- 1 is the preferred interval. i 65 1(7, 6) Fixed temperature 6 Figure 3.3 The stress-strain response for fixed temperature for the two phase material con- sidered in this document Both phases have a linear stress strain response, but the overall stress strain response is non-monotonic. From this figure one observes that the strain is not unique for a prescribed stress I 6 [1w 1M] , and thus the bar can accommodate a multi- tude of difl'erent deformed configurations. 4. The Initial Boundary Value Problem Thus far we have derived a specific constitutive model for a two phase solid, and stated a set of assumptions which we wish to work under. Certain features of the model were then analyzed in order to gain insight into its nature. In this section of the document we define a specific initial boundary value problem for the material with this two-phase constitutive response. The problem to be described is similar to the one considered by Pence (1991a, 1991b), who investigated a purely mechanical problem involving a set of two equations for the two unknown field quantities y and v , the temperature field being of no concern to that investigation. However in this thesis the major thrust is the consider- ation of thermal effects.This motivates a more detailed study of two-phase equilibrium ini- tial conditions, which becomes a major focus of this chapter (Section 4.4). A wave pulse is then introduced into this system by imposing an end displacement (Section 4.5). In later chapters we explore the interaction of the pulse with the phase boundary. 4.1 Governing Equations To begin we state the governing equations of motion for the body in each of the two phases: (4.1) 9.4-3’ 9| 2’ 67 ; = of?- cfotig—g (4.2) 89 c.12—[y (y:— —i"ote)]gY+C a— =ef[(y—y;’)—oti(e-e*)]g—: (4.3) (3336+ 2 3y e‘a “i“‘atzo “-4) where i=1 or 2 and in the parent phase (21 = O and y: = 0 , and in the second phase a2 at o and y' = 7;. The set of equations (4.1)-(4.3) for the unknown field variables 7, v, 6 is a system of quasi-linear partial differential equations, and in general is hyper- bolic in nature. It is well known that hyperbolic systems with initially smooth fields may at later times break into solutions that are discontinuous (Lax (197 3), Renardy and Rogers (1992)). Thus given a set of smooth initial conditions the system (4.1) - (4.3) may admit a solution for the (7, v, 6) fields which is discontinuous at later times. Here a set of initial] boundary data will proposed and the subsequent initial boundary value problem will be investigated. If solutions can be determined then inequality (4.4) is used to certify admis- sibility which may either eliminate or place restrictions on the range of the solutions. 4.2 Initial Configurations Attention is restricted to an initial state with uniform temperature 6 that contains a single phase boundary, the initial fields 7 (x, 0) , v (x, O) are taken to be piecewise homoge- neous. The initial position of the phase boundary is designated to be so. In the initial con- 68 figuration the material to the left of the phase boundary (0 < x < so) is in phase- 1, and the material to the right of the phase boundary (so < x < L) is in phase-2. For the two phases the constitutive response is not the same, and thus they have different sets of governing equations; the significant difference is that the coefficient of thermal expansion or1 vanishes in phase-1, which decouples the mechanical evolution of the fields from the thermal evolution. Mathematically the two distinct phases will interact across the phase boundary through the Rankine-Hugoniot equations (2.7). In what is to follow the initial temperature and displacement field within the bar are prescribed, this type of initial boundary data is referred to as a hard device. 4.3 Static Configurations Consider the initial configuration where the strain, velocity and temperature are prescribed. We define an initial configuration to be a static configuration if the initial velocity everywhere within the bar vanishes and if the initial temperature field is constant throughout the bar. Thus an initial piecewise-homogeneous configuration that is static and contains a single phase boundary is summarized by 71 O 712 where 71 = 71(9) is the intersection strain introduced in Section 3.4. The strain field (4.5) is compatible with the displacement boundary conditions u(0,t) = —80 u(L,t) = 0 for t < 0 , (4.7) provided that 50 is suitably restricted. To obtain this restriction, consider the average strain 70 (4.8) From (4.5) and (4.7) the initial displacement field is then u(x,0) = 1100!) = Y1x+ (72-71)so-72L on0 0 in (4.15), this process yields . _ C —C " . Y¥x0(e) = (52 26.1) _( 722 ,, yl)[[ A"I” *_1]+ 111(2)]9 9 PC 7 C 7 ‘ (Cw—€709 9 (4.16) MXO e MXO A “ 72 (9) = 71 (9) +7 - From (4.16) it is interesting to note that, when the MX equilibrium state is specified, the difference in the two strains as a2 —) 0 is just the transformation strain 7'. Both strains ‘ display a complicated logarithmic temperature dependance. The second part of the two part decomposition (3.92), for the el = c2 case, is to determine the a2 dependance for the Maxwell strains. Using results (4.15) and (4.16) in “‘2 MK“: (3.93), one finds the expressions for 7;“ and '71 are: (12(6 — 9')[pé[:—T— (C12-C11) [I - ln[§)]) - (62- 61)] 2 t I A ‘ l A pc 7 (y +a2(9-9 )) 012600 —e) + MX A 71 a2 (9) = 2(7’ + ot,(é — e')) (4.17) MX A MX A a a 72 “’(6) =7, “’(6)+a2(9-6) The strains (4.17) also show a complicated logarithmic temperature dependance. The dif- ference in these strains is seen to be the value a2(§ — 0.). For future reference it will also be useful to obtain the Maxwell strains for the c1 = c2 case with the additional condition that the specific heats of the two phases are 75 equal. Letting C71 = C72 = C7 in (4.15) leads to A . .. A . 2(b2—b1)+2p[c2a§0 ——T]G-pc aze 6 2pe2(y' + a2(0 — (3')) (4.18) MX " MX A 'F A It 72 (9) = 71 (9) +7 +a2(9-0 ). For this special case, the temperature dependance of the strains is no longer logarithmic, in fact the numerator is quadratic while the denominator is linear in the temperature (5. 4.4.2 Mechanically Neutral Configurations The second canonical equilibrium configuration is the mechanically neutral state, for notational purposes it will be abbreviated (MN). Along with the condition of equilib- rium (4.12), a mechanically neutral state must satisfy the additional requirement [[8]] = ((1))[[7]1 (MN) . (4.19) Note from (2.5)3 that this condition must be satisfied for dynamical processes whenever s a: 0. However if s = 0 then (4.19) need not hold, this accounts for the possibility of equilibrium configurations that are not mechanically neutral. One would thus anticipate that an initial state satisfying (4.19) would allow for a relatively smooth transition from the initial configuration into a dynamic state. From its definition in terms of an additional restriction on the two parameter fam- 76 ily of initial equilibrium states, one anticipates that a mechanically neutral state also defines a one parameter family of initial configurations. Namely, by specifying any one of the triplet (71, '72, 9) the criteria (4.19) coupled with the conditions of equilibrium should determine the other two. Thus, analogous to a Maxwellian configuration, for an initial equilibrium temperature 0 the mechanically neutral criteria (4.19) determines the two strains '71 and 72. An alternative definition of a mechanically neutral state, in terms of the free energy and entropy, can be derived via (3.2) and (4.19) [[W]]-((T)>[[Y]]+é[[11]] = 0 (MN). (4.20) If we assume the initial configuration to be mechanically neutral then the triplet ('71, '72, 0) must satisfy both PC pczz .2301, (7 —a26 )) - 71+p(C.fl—Cyl)0+62-fil= 1 2 * 2 * 2 (4°21) 5(P°2(72 " 7 )‘ Pczo‘zm ’ 9 )4” 93171) (72 ’ 71) and condition (4.12). If the initial temperature 0 is specified, then calculating the roots for produces two pairs of roots for the case c1 at c2. Like the 1le state, the MN state has two roots because of the quadratic nature of the free energy function and the difference in cur- vatures cl at c2. Furthermore, the general results for the MN strains (cl at c2) will not be required in what is to follow and thus are not presented. Turning our attention to the more tractable case of cl = 02 , we calculate the 77 mechanically neutral strains when the two phases have a common acoustic wave speed, and like the MX case this calculation yields only a single root for both phases 2pc2(y*—a26*) (4.22) MN A MN A 9' A It: 72 (9) = 71 (0)+7 +a2(6—6 ). From (4.22) both MN strains are seen to vary quadratically with the temperature 8. It is interesting to note that the difference in these two strains is '7‘ + a2(0 - 0‘) , this differ- ence changing linearly with the temperature 6. Continuing with the case cl = c2 , it will prove useful to determine the two part decomposition (3.92) for the MN strains (4.22). Proceeding with the decomposition, let 0t2 —9 0 in (4.22) and simplify the resulting expressions to find . 8—6 + C —C 6 Y1114mm) = ( 2 1) 920.72 71) ’ pc 7 (4.23) 72"”(6) = 7'1"”(6) +7'. As in the MX case (4.16), the difference between the two strains (4.23) is the transforma- tion strain 7. . Also note that the temperature dependance in both (4.23)” is linear. Continuing with the two part decomposition for the MN strains (cl = c2) , that part Of the strains which depend on the (12 coefficient is found by using definition (3.93) with results (422) and (4.23). Carrying out this calculation yields 78 . ote’B—6+C—Cé (xzé yllt’lNa2(e)= 2 [(2 1) P( 72 71) ]+ 2 9027‘” 4129‘) 2(7 ‘0‘29 ) (4.24) 171er2 a MNor2 . a t- 72 (0) = 71 (9) +or2(0-0 ). It is seen that the difference between these two strains is a quantity which is linearly dependent on the temperature. Like the Maxwellian case it will also prove useful to calculate the MN strains for the c1 = c2 case when the specific heats of the two phases are equal. By requiring that C71 = C72 = C7 in (4.22) the MN strains simplify into MN . _ 2 (132 — 131) + otipezé‘)2 71 (e) "" 2 . . 9 2pc (7 —or20 ) (4.25) 72‘" (6) = 7'1“" (6) + y" + «2(6 - e'). For this case the MN strains (4.25) are seen to differ by the temperature function 7' + a2(0 — 0‘). 4.4.3 Entropically Neutral Configurations The third type of canonical type of equilibrium states is the entropically neutral state, Which will be referred to with the abbreviation (EN). An initial equilibrium configuration is defined to be entropically neutral if it satisfies the additional criteria Fro: and call)" tying the or CHSEC EN such a Med 1115 in“ 79 [[11]] = 0 (EN) . (4.26) From (4.26) an entropically neutral configuration requires that ('71, 72, 0) satisfy .. .. p .. pCYzln[§] + pefiotzw2 — 7 )+ (T)? = pCYlln[§-] (4.27) and (4.12). Like the Maxwellian and mechanically neutral initial configurations, an entropi- cally neutral initial configuration defines a one parameter family of initial states, by speci- fying any one of the triplet (71, '72, 0) the conditions of equilibrium and (4.27 ) determines the other two. However, unlike the other two canonical equilibrium configurations, for the case cl at c2 the entropically neutral criterion yields a single root for the two strains ('71m (8) , '75" (6)) for a given initial temperature 9. Once again we consider the simpler case of equal wave speeds cl = c2. Under such an assumption (4.27) gives rise to an algebraic equation for which YEN can be deter- mined, this result coupled with the equilibrium equation (4.11)2 allows for the calculation EN 0f 2’1 . The expressions for the entropically neutral strains are: 7f" (6) = ——21—[(Cn—Cyl)ln[§]+g—I)-a2(é—e‘), c (12 (4.28) EN .. EN .. t a '- 72 (9) = 71 (6)+'Y +(12(9—9 ) It is interesting to note that these equilibrium strains are of a different a2 dependency than 80 either the Maxwellian and the mechanically neutral cases. From (4.28) it is seen that the entropically neutral strains are singular in the a2 —> 0 limit, whereas for both the mechan- ically neutral and Maxwellian cases a finite quantity results from the limit process. The EN strains also display a logarithmic dependance on the temperature 0 , and thus have a complex temperature dependance. For the entropically neutral configuration the two part decomposition (3.92) yields some interesting results. Still considering the cl = c2 case, we first investigate the case of a separable material. Thus let a2 —-) 0 in (4.27), and note that this process removes all dependency on the deformation in condition (4.27). Carrying out the details of this limit- ing process provides a specific temperature for the entropically neutral configuration . .. 7. BEN = 9 exp[ , T ]. (4.29) e (Cw-C72) First, note that the temperature 0WD depends only on specific values of material parame- ters, and thus is a constant value. Second, when a2 = 0 , the entropically neutral strains are found via the equations of equilibrium, criterion (4.27) plays no part in their determi- nation. Together these strains forrn a one parameter family. Thus for a separable material the entropically neutral case is quite different than either of the other two canonical con- figurations. Recall that the MN and MX configurations generated a one parameter family of strains based on specifying an initial temperature, whereas for the EN configuration the initial temperature is specified via (4.29), while the accompanying two strains form an one parameter family of solutions. 81 Continuing with the focus on materials where cl = c2 , we consider the case when the two phases have identical values for their respective specific heats, then (4.28) simpli- fies to M 2 t c a20 7f”<é) = - -a,(é—e'), (4.30) 7§N(é) = 7fN(0) +2; +a2(0-0‘). Under this restriction the strain 7?“ (0) has a linear temperature behavior, while if one inserts 7f” (0) into (4.30)2 the strain yEN is seen to be the constant value an :- AT 72 = Y _ 2 at ‘ c «20 4.4.4 Omnibalanced Configurations All three of the canonical equilibrium states just introduced (MX, MN, EN) are characterized by a set of strains ('71, '72) once the temperature 0 is specified. Thus the ini- tial temperature 6 pararneterizes three equilibrium states. In addition since (4.13), (4.19) and (4.26) are distinct equations, these special types of equilibrium states will not, in gen- eral, coincide. However there may exist special temperatures for which these states do coincide. These special temperatures, if they exist, will be called omnibalanced (OB). Here it is significant to note that an omnibalanced initial state implies any two of (4.13), (4.19) and (4.26) which in turn requires satisfaction of the third. Thus at the special omni- balanced temperatures (if they exist) there exist equilibrium states that are simultaneously MX, MN and EN. Finally, an omnibalanced state, being the intersection of two one 82 parameter families, is an unique initial state ('71, 72, 0). To formulate the equations which define this configuration, recall that the OB state must simultaneously satisfy the conditions for the MX, MN and EN configurations. Thus, one can choose any pair of strains for either phase-1 or phase-2, say MX and MN, and the difference between theses two strains is required to vanish. Such equations define the OB temperature.This process of obtaining an equation for the OB temperature can proceed using six different pairsrygdx = vim , 7'1“ = 7?" , '71:“ = 7113" plus the three others that are generated under 1 -) 2. Once an OB temperature is determined from one of the six equations, the strains (vim, '7?) can be calculated by inserting this temperature into one of the three equations (MX, MN, EN) for the strains (71, 72). When considering the fully thermal material the calculation of the OB state is algebraically intractable due to the fact the temperature is involved in a logarithmic manner in the strains for both the MX and EN configurations. In fact this algebraic problem persists to the case when the acoustic wave speeds are the same for both phases. Figures 4.3 and 4.4 are graphical representa- tions of all three canonical equilibrium strains versus temperature for both phase-l and phase-2 respectively. These figures demonstrate that for both phases the three strains inter- sect at two locations, and it is precisely these locations that represent the omnibalanced state. However there are two specific cases for which this state can be found, the first being when the acoustic wave speeds and the specific heats are identical for both phases, the second case is when the acoustic wave speeds are identical and both materials are sep- arable. Consider first the case when cl = c2 and C71 = C72 , then one may determine the OE temper rion Wt Where 83 the OB state from the strains (4.18), (4.25) and (4.30) and the knowledge that at an OB temperature all three sets of strains must be equivalent. Using this criterion for the calcula- tion we find the OB criterion provides two sets of initial states, this OB configuration is - .03 603 = e'-7—iY—, 0‘2 0‘2 t?” = t’- ’2 .. (4.31) azc 0 on on -on 71 = 72 'Y 2 where 7GB _ jothe' [- Zpkrw‘ — aze‘) + 20t29t (51 - $2) + orzempczwm2 — ailing orzpcem From (3.76) and result (4.31), we see that there exists two OB temperatures for each of the ground-state equivalence temperatures 0. . Since, in Chapter 3, it was shown that there exists two ground-state equivalence temperatures 0: and 0; , given by (3.76), this implies the possibility of four OB temperatures. Consider now the second case, when both phases are a separable material so that (12 = O with cl = oz. The OB temperature is now the entropically neutral temperature 0“ (4.29), since by definition the OB state must satisfy all criteria which define the three canonical equilibrium configurations. The strains are found using this OB temperature and the expression for either the MX strains (4.16) or the MN strains (4.23), since both MX and M Slates. M and 03 t. Which mi lence 16m mOUOHS 2) here. 84 and MN criteria must be satisfied. To summarize, this OB state consists of .. . A 90130 = 9 exp[ ‘ T J 9 (CH—C72) 52—bl+p0‘(C72—Cyl)exp[e, CAT C ) 7?“ = ( “— 72) .(ot,=0) (4.32) This case differs from the previous OB results (4.31), because now there exist only. two OB temperatures, one for each value of the ground-state equivalence temperatures (3.76). With result (4.32) we end any further analysis and development of the OB state. However, there remains a number of open issues concerning the canonical equilibrium states. Most notably would be a study of the correspondence between the transformation and OB temperatures, which should include an analysis of any symmeu'y relationships which might exist between the four OB temperatures and the two ground-state equiva- lence temperatures. However, the main focus of this research topic concerns dynamical motions and not equilibrium states, and therefore our study of the equilibrium states ends here. 4.5 l minin- introd' into th pUlSC ( ()5215 displac. lhcrmor fixed. T ment am deuce. FlOm 1th bOUlldan'eJ 85 4.5 Initial Disturbance So far we have required the bar to be in an initial equilibrium configuration con- taining a single phase boundary at so. A special set of boundary conditions will now be introduced such that a wave pulse will emerge from the left boundary (x = 0) and travel into the body so as to eventually reach and interact with the phase boundary. The wave pulse originates in phase-1, the phase in which the energy is separable. The dynamic boundary conditions to be described are active for the period 0 S t 5 To. During this interval the left boundary (x = 0) undergoes a smooth ramp-type displacement to a final value 5,; , while the right boundary (x = L) remains fixed. Fur- thermore, for all time t > To it will be required that the displacement at each end remains fixed. This set of initial and boundary conditions corresponds to controlling the displace- ment and temperature of the ends of the bar, and are commonly referred to as a hard device. Mathematically this set of boundary conditions is expressed I -60- (8 —50)— for OStSTO, u(0,t) = F To —5F for t>To, (4 33) u(L,t) = 0 for t20. From the prescribed displacement field (4.33) the corresponding velocities on the two boundaries during the interval 0 5 ts T0 are 86 -[5F—50]/T05Av OSISTO for , v(0,t) =[ () t>T (4.34) v(L,t) = 0 fortZO. The conditions described by equations (4.33) and (4.34) are such that during the interval 0 < ts To the left boundary x = 0 undergoes the ramp deformation u (0, t) , while simultaneously the boundary x = L remains fixed. The deformation along the lower boundary generates a wave pulse of width cho , which subsequently propagates into that part of the bar which is in phase-l. Turning attention to the entrance of the initial wave pulse into the body, the wave pulse’s velocity and strain can be mathematically related to the adjacent equilibrium con- ditions using the Riemann invariants in phase-l. For the period 0 5 ts T0 , equations (3 .49) and (3.50)2 must be satisfied between the initial equilibrium state and the dynamic state within the wave pulse. More precisely, equation (3.49) restricts changes in entropy, and states that the temperature within the region occupied by the incoming wave packet remains equal to that in the equilibrium state 0 . The'second equation (3.50)2 produces a relationship between the initial equilibrium conditions and the strain and velocity fields in the incoming wave. Writing out this second equation gives: c171 = Av-I-cl'yw ong-:=-cl. (4.35) Here Av is the velocity and 7“, is the strain carried by the wave pulse, the velocity Av is define the it from («‘1 present Here [he Conditio l. the initla equlllbm 87 defined by (4.34). The strain 7w can be determined by using (4.35) since the strain 71 and the velocity within the wave pulse Av are prescribed. For future convenience we define the driving strain increment A7 to be: A7 5 7w — 7l , (4.36) from (4.35) and (4.36) we find the relationship between the driving strain increment and prescribed velocity Av is A7 = $31. (4.37) Here the driving strain increment A7 can be thought of as the forcing input to the initial conditions. Using a similar analysis as that leading to result (4.35), it can be shown that once the initial wave has passed through a particular point in the bar, that point retums to its equilibrium configuration, and remains in that state until another disturbance occurs. Fit He 4 Pei tture mi; ed in l l 7]] - ( (‘ >) W} 100 [10718 88 Fixed temperature 0 747,6» “,2 ' > Y1Mx 72m 7 Figure 4.1 This figure is the Helmholtz free energy function \y (7, 0) at a constant tem- perature 0. Shown is one pair of Maxwell strains 7:“ and 7;“. These strains are deter- mined from the requirement that an equilibrium configuration satisfy the criterion [ [w] ] — ((1)) [ [7] ] = 0. Schematically this criterion is shown by the line of slope ( (1)) which is tangent to both free energy functions, the points of tangency identify the locations of the Maxwell strains. 89 1(7,9)A Stress response for constant temperature phase 2 Figure 4.2 Shown is the two phase stress strain response at constant temperature. The 72 Maxwell criterion, It (7, 0) d7 — ( (1)) [ [7] ] = 0 , can be interpreted graphically as the 71 equal area rule. This rule states for the Maxwellian configuration that the area below the phase-l stress-strain curve but above the line tMX is equal to the area above the phase-2 stress strain curve but below the line 1'“. On this figure the equal area rule identifies one pair of Maxwell strains 7?“ and 7;“. '71 ‘V‘f‘r' — Ell Strain — til Strain ""- HX Strain l 3.. O Figure 4.3 This is a schematic representation for the three canonical phase-1 strains for different temperatures. The mutual intersection of all three strains is the location of the .03 omnibalanced temperature 0 . Here the acoustic speeds are the same cl = c2 = 2 , and thevaluesforthematerialparametersarep = 1,0‘ =1,7. = 22C72’C7l = 3, a2 = 1,21. = 5. Figure ferem 1 balance . mated; 91 5“.- it £5. — 2! Strain — I‘m Strain "nu“ HI Strain io' $-10 . i Figure 4.4 This is a schematic representation of the three canonical phase-2 strains for dif- ferent temperatures. The mutual intersection of all three strains is the location of the omni- .03 balanced temperature 0 . This diagram is the complement of Figure 4.3 in that all the . material parameters remain the same as in that figure. EEI bOI let? be CO tOl 5. The Interaction of the Initial Wave Pulse with the Phase Boundary In Chapter 4 a set of initial and boundary data were described, these conditions generate a wave pulse from the left boundary. The pulse travels at speed cl from this boundary into the purely mechanical phase (phase-l). At time t = so/cl the pulse will reach the phase boundary and interact with it in some way. In general, the initial interac- tion of the wave and the phase boundary will set the phase boundary in motion while giv- ing rise to the possibility of a wave being transmitted into the phase-2 region and a wave being reflected back into the phase-l region. The complexity of such a problem may be understood if the reader recalls similar problems occurring in elastic materials involving the reflection and transmission of a wave striking a boundary, and the subsequent genera- tion and interaction of reflected and transmitted waves (Achenbach 1990). Pence (l991a,1991b) proposed a similar initial boundary value problem in a two phase elastic solid. In his study Pence considered an infinite layer of material which under- went simple shearing motions. Furthermore, the material model Pence studied was purely mechanical in nature and thus did not consider any contributions of thermal effects. Although geometrically a bar undergoing longitudinal deformations is different than shearing within a layer, mathematically the goveming equations for the two different problems are identical. Like Pence we assume that the interaction of the wave pulse and the stationary phase boundary will set the phase boundary in motion, and that the phase boundary will come to rest when the encounter is over. In the following subsection we will compare and contrast the problem Pence investigated with the one under study here, so as to clarify the role that thermal effects play in such a problem.The intention is to identify 92 93 the similarities and differences between the purely mechanical problem and the fully ther- mal problem to be considered later. 5.1 The Wave Pulse-Phase Boundary Interaction In order to explore the temperature effects in the fully thermal problem, while try- ing to keep the problem under consideration here somewhat similar to Pence’s, the follow- ing assumptions are made: (A5) The initial wave pulse is generated before any interaction with the phase boundary. (A.6) The interaction of the initial wave pulse with a phase boundary will not lead to the creation of additional phase boundaries. (A.7) The phase boundary will not encounter either of the boundaries x = 0 or x = L during the interaction process. (A.8) During the interaction the phase boundary will move with a constant speed ds 8:... dt' (A.9) After the interaction the phase boundary comes to rest. 94 Assumption (A5) is satisfied if the time period in which the pulse is generated To is restricted such that To < so/c1 . Assumption (A.6) requires that the material to left of the phase boundary (0 < x < s (t) ) always remains in phase-l, while the material to the right of the phase boundary (s (t) < x < L) is in phase-2. It then follows that the strain field must satisfy: 7(x, t) <7I for 07I for s(t)[[7ll}. (5.9) 1(8 (t) at) In this purely mechanical setting the function F(t) represents the work being performed on the external boundaries, while the function D(t) represents the rate change of energy due to phase boundary motion. The expression inside the braces of (5.9) is commonly referred to as the mechanical driving traction f (t) nub (viz. Abeyaratne and Knowles ( 1991)): adja', exprc If F l" moth Satisfy cal set 99 h 7(s(t)+.t) mm“ = j 2(7)d7— <>> 11711. (5.10) 7 (s (t) at) As seen from (5.10) the driving traction provides a source of information on the fields adjacent to the phase boundary. From the definition of the mechanical driving traction the expression (5.9) may be expressed D(t) = so)f“‘°°“ (5.11) If F (t) = 0 , then the second law of thermodynamics in this purely mechanical setting motivates the requirement E < 0 . This gives the requirement that admissible motions must mech satisfy D (t) 2 0, which in turn restricts s (t) f (t) 2 0. Thus in this purely mechani- cal setting the quantity D (t) represents the dissipation rate. For the initial encounter calculating the dissipation rate via. (5.9) one obtains D = {—8 [c§(c§ — $2) (71~ — 72) 2 — of“: - 8.2) (75 - 71) 2] (5.12) 72 + Item-H7241) 71 Furthermore, when the initial conditions are Maxwellian (4.14), as considered by Pence, the second bracketed term vanishes and (5.12) reduces to With 6pr ' ing s lhes Where Require 100 D = -2118, [e§(o§—s2) (7T—72)2—cf(cf—s'2)(73—71)2] (MX). (5.13) With significant algebraic manipulation, via. the computer program MATHEMATICA, we express the field quantities 7T and 7S on the left hand side of (5.13) in terms of the driv- ing strain A7 , the initial conditions, and the phase boundary speed s via. (5.3) and (5.5). These operations give "2% [C§(C§ — s2) (7T-72) 2 - 030:? - $2) (75 -71) 2] = (5.14) pScfZ (A7,S) “1* c2) (c1 + 8) (c2 - S)’ where the function 2‘. (A7,s’) is defined 2mm) ss‘ztzmtc, -c,)A7—c,<71 -72))1 —s[2 (cl - c2) 2A72— 2c2(c1—c2) (71—72)A7+ c§(71—72)2] (5.15) “[291C2A7( (C1—c2)AY ‘Cz(71‘72))] - This operation allows the dissipation (5.13) to be expressed _ [)8sz (A7,S) ‘ (c1+c2) (cl-t8) (cz—S) (MX) . (5.16) Requiring D 2 0 restricts s to an interval of values for each A7. From (5.16) and the [€51 that and give: ”em ion 1 101 (’ \ hich ’7 Julie (11 ”Hill 02 Con “ilk re k ((16ng 101 restriction on the phase boundary speed s’ < min (C12 c2) , the requirement D 2 0 implies that 8'2 (A7, 3’) 2 0 (MX) , (5.17) and thus an analysis of $2 (A7, 8) is sufficient to determine the admissible values 8' for given A7 , that is the solution region in the (A7, s') -plane. Following the analysis in Pence, it is seen that along the line 8 = 0 and locus of points 2‘. (A7, 8) = 0 the dissipa- tion vanishes, and thus bounds regions in the (A7, S) -plane to one sign. Figure 5.2 is a plot of the admissability region in (A7, 8) -plane for the special case where c1 = c2 = c. Under these conditions the function 2 (A7,S) reduces to 2(A7,s’) = —2s'2cA7(71-72) —$c2(7l —72)2+2c3A7(71—72) , (5.18) which allows the dissipations function (5.16) to be expressed D _ pcs [—2$2cA7 (‘7l - 72) - 302(71- 72) 2 4’ 203137 (71 ‘ 72)] (5.19) 2(c2 - 82) Figure 5.2 also illustrates curves of constant dissipation, i.e. D = constant. However, the requirement of positive dissipation does not yield a unique solution for the various field quantities, but only reduces the range of possible values for the parameter s‘ for given ini- tial conditions. In particular, if A7 > 0 then s is confined to a range of nonpositive values where both the extreme values s = 0 , s = s (A7) |min give D = 0. Similarly if A7 < 0 then s is confined to a range of nonnegative values where given the boundary values 102 s' = 0, s’ = s'(A7) 1m" givesD = 0. To find a unique solution for the results (5.3)-(5.6) one must introduce some addi- tional criterion. Pence (l99la, 1991b) uses a variety of different of selection criteria to determine a unique solution. In the first paper this is achieved by enforcing various requirements on the reflectivity versus transmissivity (a phase boundary impedance), while in the later paper motions are determined under the extremum principle that the dis- sipation, defined in (5.11), is maximized at each instant. Another method for selecting physically meaningful solutions is the introduction of a kinetic relation, this being an additional constitutive relation which relates the speed of the phase boundary to the various field quantities on either side of the phase boundary. This information is typically provided by the driving traction (5.11). In this purely mechanical setting a standard functional form for a kinetic relation is s = firm"). where ,‘F is a functional form motivated by the phase boundary kinetics.The simplest form of a kinetic relation is that of a linear kinetic relation, which implies that the phase bound- ary speed s is a linear function of the driving traction fun“. Mathematically this can be expressed as s = xfm°°", (5.20) where K is the phase boundary mobility, a material parameter that is here assumed to be dish; hneti posed eql C0nd If: {01. 103 constant. Since D 2 0 implies 1c(fmch)2 2 0, via (5.20), this requires that the mobility K be a nonnegative quantity. In a similar setting to that of Pence (199la,b), Lin and Pence (1993a) utilize such a linear kinetic relation and are able to construct an implicit relation between the phase boundary speed and the initial conditions. Furthermore, they are able to show for infinitesimal wave pulses that the maximally dissipative solution is equivalent to the linear kinetic relation for one value of the mobility, and that, in general, this maximally dissipative solution is quantitatively similar to the criterion based on the use of a linear kinetic relation. For the purely mechanical problem, the use of a linear kinetic relation is now pro- posed. From (5.11) and (5.16) the driving traction during the initial encounter is 2 mech p012(A7,S) = MX . 5.21 “0 (el +c2) (cl +s) (cz-s') ( ) ( ) Equations (5. 16) and (5.20) give rise to 2 . Kpc‘z (M’s) (MX) , (5.22) = (c1+c2) (c1+S) (cz-S) which in turn provides an implicit expression for the phase boundary speed. Equation (5.22) admits a unique solution ( A7, S) for the initial encounter, i.e. given a set of initial conditions and the mobility it, one can determine the phase boundary speed s via (5.22). If, following Lin and Pence (l991a), one assumes cl = c2 = c then this implicit equa- 1L tier Fig‘. mot COIll “'an asa; them 5.3 N be dew limit 33 {mm m 104 tion takes the simplified form . g _ A7 = S + cm 72) (MX). (5.23) Icpc2 (71 - 72) 2(02 - 92) Figure 5.3 is a graph in (A7, S) -space of the linear kinetic relation (5.20) for a range of mobilities K . .The graph also shows the criterion D 2 O which describes the region that contains solutions to (5.23). For the purely mechanical problem, we have shown for the initial interaction of the wave pulse with the phase boundary that a solution exists if the phase boundary is treated as a parameter. Furthermore, the linear kinetic relation singles out a unique solution. At this point we end our discussion of the purely mechanical problem and return to the fully thermal theory we have developed. 5.3 New Features of the Fully Thermomechanical Interaction We now turn our attention to the initial interaction of the incoming pulse with the phase boundary under the framework of a fully thermomechanical theory. In this setting, the problem increases in complexity from the purely mechanical theory in essentially three ways. First, at each point in the domain there are three field variables (7, v, 0) that must be determined, rather than simply the two mechanical field variables (7, v) ; only in the limit as the coefficient of thermal expansion vanishes does the temperature field decouple from the mechanical fields. Second, the new family of characteristic directions :1: = 0 obtained in (3.63), 105 which are essentially particle paths, introduces a new region into the solution. Recall for the mechanical theory that the interaction gives rise to 5 regions: incident wave, incident! reflection interaction zone, transmitted wave, and the undisturbed initial equilibrium states. In particular, a single region will, in general, encompass both transformed and untransformed material (interaction zone if s‘ > 0, transmitted wave if s’ < 0 ). In the ther- momechanical theory this single region bifurcates into two regions (untransformed inter- action zone, transforrned interaction zone if s' > 0; untransformed transmitted wave, transformed transmitted wave if s < 0 ). Third, in the second phase the wave speed associated with Riemann invariant (3.64) is no longer a constant, but instead is a monotonically increasing function of the temperature (1 (220.29 x 2 2 — = =Fc 1+ . dt 2 C12 Thus, in phase-2 the interface between two wave regions supporting different tempera- tures need not be a contact discontinuity]. In particular, this is the case for the interface between the T-region, arising from a transmitted wave, and the initially equilibrated phase-2 state, henceforth referred to as the TIE; interface (Figure 5.1). Besides the contact discontinuity, the two additional possibilities are a classical shock and the centered simple wave fan (Whitham (1974)). A centered simple wave fan is also known as a rarefaction 1. To avoid confusion, we define a contact discontinuity as a discontinuous surface separating same phase regions that travels at one of the characteristic speeds of the material.'l'his definition is commonly used by mathematicians (Smoller (1983) page 334), investigators in gas dynamics would likely call this a Chap- man-Jouget wave (Dunn and Fosdick (1988)). In the problem under study here, the wave speed in phase-l is given by (3.53) and so any interface between the various regions are again contact discontinuities. Across such contact discontinuities the integration of the characteristic equations yields the Riemann invariants, which can be used to help formulate a solution. 106 wave in gas dynamics. A shock occurs if the characteristic speed associated with the right propagating Riemann invariant is greater in the T-region than that in the phase-2 equilib- rium configuration. In view of the temperature dependance of the speed (3.64), the shock occurs if 9T > 0. The case of a centered simple wave fan, henceforth simply a fan, occurs when the characteristic speed associated with the right propagating Riemann invariant is less in the T-region than that in the phase-2 equilibrium configuration. By use of (3.64) the case of a fan occurs when 61. < 6. 107 \ T/FQ interface IW t W Riemann invariant x 3 m... Jump condition l E1 . E; Figure 5.1 This is a graphical representation in the (xt)-plane showing the initial interac- tion of an incoming pulse with a stationary phase boundary for the purely mechanical problem. The wave speed of the phase boundary is assumed positive 3 (t) > O in this fig- me. There exist six distinct regions during this interaction: E1 & E2 are the equilibrium configurations in phase-1 and phase-2 respectively, 1W represents the region carrying the incoming wave pulse traveling through phase-l, region R is that region in phase-1 where the reflected wave travels, while 8 arises from the interaction of the incoming wave and the reflected wave; finally region T represents that phase-2 region containing a transmitted wave. The forward movement of the phase boundary transforms phase-2 material into phase-l material. figu. ford} defin! fidei I Pfifior 108 Figure 5.2 This is a schematic representation of the solution region in the (Av, S) -plane for the purely mechanical problem with Maxwellian initial conditions.’I'his region is defined by the criterion of positive dissipation, D 2 O , thus the lines 2:0 and s = 0 pro- vide the boundaries for the admissibility region. Also shown are curves of constant dissi- pation given by (5.16). The values for the material parameters were chosen to be c1 =c2 =2,p = 142—11: 5. 109 Figure 5.3 This is a schematic representation of the linear kinetic relation with various mobilities K for the purely mechanical problem with Maxwellian initial conditions. The solid line 2:0 is the line of zero dissipation ,where 2 is given by (5. 18), which divides the plane into regions of positive and negative dissipation. Therefore, under the criterion of positive dissipation, the line 2:0 also restricts the solution space in the (A7, 8) - plane.The values for the material parameters were chosen to be cl = c2 = 2 , p = 1 . 72-71: 5- 6.1 will SUE him the side Cases: a fan, face is Phase. We. I 11mm EL 6. Analysis of the Initial Interaction Region 6.1 A Method for a Solution to the Initial Boundary Value Problem The previous development of the characteristic equations and Riemann invariants will now be utilized in an attempt to analyze the changes in the temperature, strain, and stress fields in the proposed initial boundary value problem. Since this method of solution hinges on using the Riemann invariants, any restrictions necessary for the integration of the characteristic equations to produce the Riemann invariants in phase-2 need to be con- sidered. In view of (A9) we note that the phase boundary may move with either positive or negative velocity: s > O or s < 0. For those cases where the phase boundary has negative speeds s < 0 ,the characteristic equation (3.60) may no longer be integrated to give the Riemann invariant (3.65). Since it is desired to use the integrated form (3.65) of the char- acteristic equations to find a solution, the problems investigated are restricted to those for which the phase boundary has a nonnegative value. Recall for the purely mechanical case in which a2 = 0 that this requires that Ay S 0 if the initial state is Maxwellian. Thus, the s 2 O investigation of the initial interaction consists of two distinct cases: (i) the phase boundary moves with positive velocity s > O , and the TIE/2 interface is a fan, and (ii) the phase boundary moves with positive velocity s > O , and the TI52 inter- face is a shock. For either case, the characteristic directions and Riemann invariants in phase-l are given by equations (3.51) and (3.52), while in phase-2 they are given by (3.62) and (3.65). In what is to follow we investigate the case of a fan, with emphasis on how the thermal effects contribute to differences between the purely mechanical theory and the 110 111 separable and fully thermal theories. The case where the interface is a shock is discussed briefly in Appendix B. The solution procedure we consider consists of utilizing the jump conditions across the phase boundary along with the Riemann invariants to produce a set of algebraic equations. The field during the initial interaction may be subdivided into five distinct regions centered at (x, t) = (so, 50/ c1) . Locally each is wedge shaped. Each region is characterized by a triplet of field values (7, v, 9) . These five regions are given as follows: S°- The incoming wave in which (7, v, e)I = (y1 + A7, —c1A'y, é) . This region occupies the wedge _oo< t — so/c 1 < c1 ' The region to the left of the initial position of the phase boundary in which the initial pulse and the wave that has reflected back from the phase boundary are interacting. The triplet values (7, v, 0) s are as yet not known. This region occupies the wedge , <0. cl < The region to the right of the initial position of the phase boundary in which the incoming pulse and the reflected wave are interacting. This region represents the material that changes from phase-2 to phase- 1. The triplet of values (7, v, 6) 5° are as yet not known. The region occupies the 112 wedge / (0) vT_J&72[2¢(eT) +¢(O)ln[¢(6T) +(0) H, where C ch (9) = e + 712—2. c202 Region So and T: 4011-73.) = vT- vso. -s'p (vT — vs.) = pcim -r') - pciazwr - 9‘) — pcfvso. l 2 ' ‘ 2 ~ 1 2 2 . 1 * a §[PCirso + 903(71- r )— pciazw-r- 9 )] (7T- 75.) . (6.9) (6.10) (6.11) (6.12) (6.13) The equations, (6.5) through (6.13), are a system of nine equations for the ten quantities 116 73’ VS, 05, 78., vs., 05., 71., VT, 01., s' . As it is not clear as to whether a solution for this problem exists, we propose treating s as a parameter and investigating the family of solu- tions resulting from the system of equations and field quantities. 6.2 The Master Equation for the Initial Interaction By treating the phase boundary speed s as a parameter the problem presented by (6.5)-(6.13) reduces to a system of nine equations and nine unknowns. Although a simple algebraic elimination of the various field quantities is not immediate, it can be shown that a laborious elimination process leads to a single master equation for the temperature at which is independent of the other eight unknown field quantities. The other unknown field quantities can then be written in terms of the temperature 9.1. and the phase boundary speed 3. Therefore, if a solution to this master equation can be found then we have deter- mined one family of solutions to the system of equations. The reader is directed to Appen- dix A for a discussion of the actual reduction technique. The master equation for 61 is the nonlinear algebraic equation 2 C (Sc -c) 6 /C72(c1—S)13(0T) 72 l 21n[-;f)+c§or201.= czar2 9 c (. 2) (6.14) .. sc -c ‘ .. ./C72(c1- s) e (e) “'2 2‘ 2 ln[§;] + c§ot29+(scl — cf)2Ay + s'c1(71-72) . C20‘2 117 where s(e) = 24>(e) +o(0)ln[d’(°)"¢(0)]. <1>(0) +d>(0) (6.15) (0) = 9 + sop N98). N A closed form solution for this equation is not obvious. Equation (6.14) may be viewed as an algebraic equation for the temperature 91.. Values 91. satisfying this equation will be a function of the incremental forcing strain A7 and the phase boundary speed s. Thus a complete knowledge of the initial and boundary conditions does not provide sufficient information for the determination of a unique tem- perature 91.. Once a value 0.1. satisfying (6. 14) is obtained, it can be shown that the other eight field quantities (vs, vs, 98, 75., Vs” 08., YT, VT) will satisfy the nine governing equations (6.5)-(6.13) if and only if they are given by the following expressions: C A - _rz 3 'YT - Y2+c§azln(er)’ (6.16) VT = 10,2[e(eT)-e(é)]. (6.17) _ .. C " 73. = (91" S) 1[JG—7203(6T) — 13 (9) ) + cl (71+ 2A7) + {72 + 7131n(§-)]], czoc2 T (6.18) 118 C a vs, = Sffijc—nwwp —13(9))+s[72+ é -—72-1n(9—)]- s (71 + 2&0], (6.19) c2012 95. = Tl (9T) +T2(6T) + [T3 (9..) +T4(9T)]T5 (9T), (6.20) vs = vs... (6.21) rs = 75.. (6.22) as = e. (6.23) The expression (6.20) for OS. makes use of the auxiliary functions: 2 pc C “ T1(6T)5T1_[_2_2[72+_2L1m(§)_(y-a26)]2+pC720T+62-51]. P 11 c2012 T2(0T)= (fl[o(eT)- 9(6)]+c (71-1-2117) 2Cyl(cl+s s) C126 7 +—ln — , “[2 czar: (0 TN T(6)= pcf j—fi(6)— 13(6) +c( +2A)+s +E—zln(-§-), 3 r -m 2[ r J 71 7 72 c202 e C a . ‘ . T4 (9.1.) ape: [72+ jfl-ln(§-)-(y —a26 )]—pc§or26.r, c2012 119 T5 (0T) = 1 (@[flwfi —fi(0)] +cl (S+2Ay) - 2pC71(cl+ S) C 6 —c1[72 + Tflln(9—):| ]. c2012 T From the master equation (6.13) it is seen that A7 is a variable which acts as a forcing parameter. For each A7, the freedom to vary S is anticipated to generate a family of solu- tions, satisfying all the mathematical balance laws, for each initial-boundary value prob- lem characterized by the forcing parameter A7. For future discussion of the master equation it is convenient to rewrite (6.14) in the form ‘I’(6T,S,cl,c2,C e',a2)—‘r(é,s,cl,cz,c o',ot,) (6.24) 72’ '0) (A‘Y’ s! 71—729 cl) = O 72’ This uses the functions .. . c we, s, c1, c2, cyz, e , 012) -=- /c1,2 (cl - s) t) (e) ——21£(scl — c§)1n(%) + 02029 , c2012 6 (D(Ar.s'.71- 72. c1) '='(Sc1 — cf)2AY + $010!, — 72). (6.25) Since m (0, O, 71 - 72, cl) = 0 it follows that at = 0 provides a solution to (6.24) whenever A7 = O and S = O. This is simply the persistence of the initial state in the 120 absence of a disturbance A7. However, neither A7 = O with S at O , nor S = 0 with A7¢O, gives a) = 0.Thus while 91. = 6 is asolution of (6.14) when both A7 = o and S = 0, in general 01. at 0 if either A7 at 0 or S ¢ 0. Solutions 6.1. to (6.14) with A7 = O and S at 0 correspond to spontaneous motion of the phase boundary in the absence of an initial disturbance. Solutions 6T to (6.14) with A7 at O and S = 0 generate dynamical motions with an immobile phase boundary. Once any solution to the master equation is determined, the strains 75:78.,7T must comply with all restrictions on the transformation of new phases as given in (6.4), which in turn will place restrictions on the range of the initial strain increment A7. An explicit solution of (6.14) is not as yet known. However, later in this research we construct a solu- tion for (6.14) under the constraint of a2 « l , for which more quantitative information on the admissibility of A7 is presented. It is interesting to note that (6.24) is satisfied when 0,. = o and a) = 0, which corresponds to phase boundary motion in the absence of a transmitted wave. This is seen by letting 9.1. = 0 in equations (6.16) and (6.17). For this case the speed of propagation of the phase boundary is determined from the satisfaction of a) = O S c12A7 - 2A7 2471-72). (6.26) 121 6.3 Construction of the Centered Simple Wave Fan Recall that the TIP/2 interface here is a centered simple wave fan, and within this region it is required that the three field quantities 7, v, and 0 exist in self-similar form. In order to construct the self-similar solution it is required that the slope 93‘ of the Riemann dt invariant (3.65) change smoothly from one edge of the fan to the other. Mathematically this condition can be expressed as A = (if: , where A may vary throughout the fan region. The fan is the wedge shaped region depicted in Figure 6.2, this region being contiguous with the region T and the phase-2 equilibrium configuration. In Figure 6.2 it is shown that at the time t* the two outer edges of the fan are located at x = x1 , for which the fan’s reciprocal slope is A1 = 3—1: , and at x = x2 for the reciprocal slope A2 = 7% ; the order- ing of these quantities being xl < x2 and Al < A2. One may express the three field quan- tities as a function of the coordinate A, this yields 7 = 7(A) , v = i? (A) and e = 6 (A) . Any self similar solution must satisfy the appr0priate boundary conditions at the edges of the fan envelope, these conditions are HA1) = 7,. Wm) =vT, 6(A,) =eT. (6.27) 7 (A2) = 7,. fir/1,) = 0. 5(A2) = 6. (6.28) To determine the temperature 0 (A) the unknown temperature function 0 (A) is substi- tuted into the positive characteristic direction associated with the Riemann invariant (3.65) 122 4 2“ A — a-t — JC2+——(-:-Y—2—, (6.29) solving (6.29) for 6 (A) yields 9(A) = 77—2“ 42). (6.30) c2012 The two coordinates A1 and A2 can be expressed in terms of the temperatures 9T and 0 via (6.27)3 and (6.28)3. From Figure 6.2 it is seen at x = xf,ror which A = A1 , that the temperature is 6 = 6.1.. Thus Al must satisfy (:72 22 c402 C A1 = _2 2 —72 +9.r . (6.31) c2112 At the other edge of the fan x = x5, A = A2, and the temperature 6 = 0 , thus c‘tct2 C A2 = é—2[——7—2§+e]. (6.32) 12 6202 This knowledge of the two slope coordinates A1 and A2 now allows for the determina- tion of the other two unknown functions 7 (A) and i} (A) . To determine the strain 7 = 7 (A) the Riemann invariant condition (3.62) is uti- 123 lized, where within the fan envelope it requires A (3,21%?) + c§ot272 = Cyzln[ 2?) J + c2627 (A) . (6.33) 2 2 O! Inserting the expression (6.30) into (6.33) allows for the strain 7 (A) to be expressed C 610120 7(4) = y,+—,’—2-1n[ 2 22 2 J. (6.34) CYZ(A -c2) c2012 The boundary conditions (6.27) 1 and (6.28)1 require that 7 (A1) = 7T and 7 (A2) = 72. Inserting (6.31)-(6.32) into (6.34) one can demonstrate that both these conditions are satis- fied. Finally the velocity field v = {I (A) can be determined through the use of the Rie- mann invariant condition (3.65)2 «gnu-MO) (O)ln( )] = constant. (6.35) Utilizing the initial conditions allows this condition to be written -.a-rc:.izw>>+rm<::::::::::::ll= (6.36) 46:42am) +¢(0)ln(::g: 12(3)] 124 From (6.36) the velocity v = v(A) is d)(0)—(O) ~ 6.37 _ (9(A))—¢(O) ( ) / 2th (0)ln - . 7 ¢(e(4))+¢(0) where .. .. c C <1>(9(A)) = e(A) +7112 = 4—722A2. 02(12 c2012 Inserting the expression (D (0 (A) ) into the equation the velocity is 2C C A -c A—c v(A) = —2—72(A2—A) +—E[ln[—2—2-]—1n[ 2]]. (6.38) 0292 c2012 A2 + c2 A + c2 The boundary conditions (6.27)2 and (6.28) require that \7 (A1) = vT and 7 (A2) = 0 inserting into (6.28) both these conditions are shown to be satisfied. Thus, in summary, within the centered simple wave fan the strain, the velocity and temperature field are given by (6.34), (6.38), and (6.30) for all A obeying CT S A S c132. 125 Phase 1. Phase 2. / fan IW El E; t W Riemann invariant x=s(t) x W Imp condition Figure 6.1 This is a graphical representation in the (xt)-plane of the initial interaction of a wave pulse with a stationary phase boundary in the fully thermodynamical theory. Regions E1 and E; are the initial equilibrium states separated by the phase boundary at x=s(t). The incoming wave (IW) strikes the phase boundary setting it into motion, where S > 0 is assumed. The IW-phase boundary interaction gives rise to the regions S, 8° and R in phase-l, and T and the simple centered wave (fan) in phase-2. The region R represents a reflected wave, while S arises from the interaction of the IW and the reflected wave. 8" is that material which has undergone a phase transformation from phase-l to phase-2. The IW striking the phase boundary also produces a transmitted wave in phase-2, this is desig- nated by the letter T. Finally the transition from the E2 state to the T state is a simple cen- tered wave, which requires CT < CE .The other possibility, that of a shock transition from the E2 to the T-state, is not considered here but ls discussed briefly in Appendix B. Com- paring this diagram to figure 5.1 shows the additional complexity inherent in the fully thermodynamical theory. 126 Figure 6.2 This is a representation of the fan transition in phase-2. The point (x1,t*) is located on the contiguous line between T and the fan, where g = A1 = CT. The point (x2,t*) is on the line between the fan and the E; state, where 3t = A2 = CEZ. Correct ordering of the speeds in the fan requires that c132 > CT , which becomes an admissibility condition on the fan solution investigated in Section 7.3. 7. Construction of a Solution for Small Coefficient of Thermal Expan- sion In this the chapter we examine the significance of the coefficient of thermal expan- sion for the initial interaction. Recall that the coefficient of thermal expansion, a2 , is a material constant responsible for the free energy being of the nonseparable form, and when a2 vanishes the mechanical field variables 7, v decouple from the thermal field variable 6 in the governing equations. In Chapter 4 the role of this material constant was analyzed for the initial equilibrium configurations, using the two part 012 decomposition (3.92)-(3.94). This two part decomposition will continue to be utilized in what is to follow. The purely mechanical problem was introduced in Chapter 5, and explicit results (5.3)-(5.6) were constructed for the initial interaction of a wave pulse with a phase bound- ary. In the fully thermomechanical problem posed in Chapter 6, fully explicit solutions are not obtained by virtue of the complexities of analyzing the master equation (6.14). For this problem we now wish to investigate the effects for small non- zero values of 012 , in order to garner insight into the first order temperature effects. Recall that the two-phase model developed was for adiabatic motions, the possibil- ity for heat transfer being excluded. Within this framework a finite value for the coefficient of thermal expansion results in the thermal and mechanical field quantities being coupled, and when this coefficient vanishes this coupling no longer exists. Ngan and Truskinovsky (1994) investigate problems for phase transforming solids in which the role of heat con- duction (the Fourier model) is also accounted for. It is interesting to note in their develop- ment that the thermal conductivity is a coupling parameter similar to our (12. Whereas here a2 was the parameter enabling one to examine the link between adiabatic and purely 127 128 mechanical motion, in their paper the thermal conductivity provides a heat conduction- adiabatic link. Beginning with the equations which mathematically describe the initial interac- tion, (6.14)-(6.23), the leading order temperature effects are to be extracted via the assumption 012 « 1 . Furthermore, it must be determined what factors, if any, ensure the retrieval of the previous results (5.3)-(5.6) when 012 —> 0. Accordingly, various perturba- tion and asymptotic procedures will be used to determine the leading order temperature effects]. Recall that once a solution for 9.1. is found satisfying equation the master equation (6.14), then the other field quantities relating to the initial interaction may be determined explicitly from (6.16)-(6.23). Since it is not evident that a solution exists for the master equation, we propose to investigate solutions to (6.14) under the additional assumption that a2 is small. In particular since a2 = 0 yields the separable theory for which 6.1. = 0 , our interest is in solutions for which 0.1. —) 0 as a2 —> O. 7 .1 Perturbation Analysis of the Master Equation The master equation (6.14) may be expressed in a more useful form, one in which all terms which have a temperature dependance are written on the left side of the equality sign 1. It should be pointed out since or; has units of reciprocal temperature, that these procedures for- mally require nondimensionalization of a; via multiplication by some characteristic temperature in the problem. It is in this sense that we operate, and which statements like a2->0 need to be under- stood. 129 . C (Sc- c2) .. @(Ct-Sflfiwfl-MOH *2 2‘ 2 ln(:"T)+c§a,(e -6): c2012 (7.1) (so1 —cf)2A7 + s'c1(71—72) . The difficulty in finding a solution for this equation lies primarily in the non-linearity con- tained within the function 13 (9) , which from (6.15)1is 19(6): 2¢(8) +(0)ln[¢(e) 4(0)]. d>(0) +d>(0) The function 19 (6) contains (b (9) in a linear and logarithmic manner, therefore the 012 analysis will begin with d) (0) . Expanding the function (P (0) about a2 = 0 produces 6 . .2 (320.2 C c2112 2C,” 8C2 + O(az) (7 ) 1 22- 22 244 /C Ocor2 /C Gca Oca (Me) = 12[1+ 2 __2] = 12[1+ 2 2_ 2 2 72 72 The logarithmic term of 19 (0) contains (1) (6) both in the denominator and in the numer- ator. Use of (7 .2) permits the expansion of the numerator and denominator about «2 = 0 which leads to ,/C 72 Ociai Sign: cor 2C72 2 c2 2 8C72 <1> (e)— <1> (0): + 0(62), (7.3) 130 2 2 2 4 4 2 /C Oca 0c 01 ¢(e)+<1>(0) = ——72[1+ 2 2- c2 2 +O(a ) (7.4) c2‘22 4C72 16C 2 i 72 upon which division of (7.3) by (7.4) produces ¢(6)— ¢(O)_ 1 90 2(12 62C4a 4 eczaz 02c4a4 _1 _ 2 2_ 2 0L2 6 _2_ 2 _2_ 2 42(9) “12(0) =2 2C1!2 2 +O(°‘2) 1+ 4—_C2 +——O(cr2) .05) 8C72 16C72 Expanding the denominator of (7.5) and collecting terms of similar order in 012 yields me) (0)_ 9° “2 Ozc‘a‘ ec2a2 ec2o2 - __2__ 2_ __2-2 2 2 2 2 2 ¢(0) +¢(0) 4T2" 8C2 +O(a a2) = 4(312 __[1 +0(°‘2)] (7 6) 72 Utilizing this result allows for the logarithmic part of the function 13 (6) to be expressed l(¢(9)—¢(0))_ 902a: "_Ciai “ <1>(e)+(0) '1“ F721 2—c2 +0(a “2) 2 2 2 2 — In 6620‘2 +ln1— 9°22“ ——2+o.(ot) ' 4c72 2c—_2 2 The second logarithmic expression in (7.7) may be expanded about 012 = 0 using a Taylor (7.7) series to the fourth order 131 2 + O(a 4=)] Ocza 022 2C72 2 2 2 2 Scion 4 1n 1— 2-C—2 +O(or2). (7.8) Consolidating results (7.2), (7.7) and (7.8) permits the function 13(9) to be written to the order O(ag): 2 2 C C QC2 on +6c 0t 13(9) = 72+ 4211142 ———z 2 ——ZC+0(a ). (7.9) c20‘2 c2°‘2 a2 Equation (7.9) is the final form for 13 (6) expansions, and will be used in what is to fol- low. Note that 13 (6) appears twice in equation (7.1), i.e. in the difference 13 (6.1.) - 13 (6) . From the expansion (7.9), this diflerence in 13(6) results in the a: sin- gularity canceling out 13(0T)—t3(9)= c—Jfflln(e:+) 02a 2.1.(0 —9)+O(a2). (7.10) co‘22 2J—2 Use of (7.10) in (7.1) gives C72(c2(c1- S) + c2 — SCI) 2 c20‘2 C —S A 3 1n( :J)+a2[-;—E;-+ l](6T—9) = m+0(a2), (7.11) where a) -=— (Scl — cf)2Ay + Sc1 (‘yl — 72) , as defined previously in (6.27), depends upon 132 the initial conditions, the material properties, and the phase boundary speed.To further simplify (7.11) let a = §(s',c1,c2) = (c2(cl—s') +ci—s'cl) = (c1+c2) (cz—s'), (7.12) which allows (7. 1 l) to be expressed more simply C 9 c —s .. 212§1n(;)+a2( 1 - + l](9T—O) = m+0(a:). (113) 2 2 Since a solution for 0.1. to (7.13) is not obvious, we propose the following representation for 0;: e —é[1+ ciazm '1'" 1p]exp C é . (7.14) 72 2 ,. c a (0 . Thus (p by definition is (p = (6.1./9)exp[-(23 : )- 1 . Hence if 012 -> 0 and 6T—> 9, 7 then (p -> 0. Thus q: is expected to be a small quantity whose small 012 representation remains to be determined. If both 012 and 1p vanish, then (7.14) demonstrates that the tem- perature 0T —> 6 , the equilibrium temperature. 133 Inserting the proposed representation (7.14) into master equation (7.13) yields 5C7; .. 2 cl—s pc§a2m Ex—ln(1+(p) +6pc20t2[—Zc2 +1] exp Ty;— q): 2 2 . 2 (7.15) , c —s c a to 6pc§a2[—;—C2— +1][1—exp[pc:zé D + O(az). Equation (7 .15) is now regarded as an equation for the quantity (p. The leading order a2 effect is determined by expanding (7.15) for small a2, and since (p is expected to be small, the logarithm term is written using the expansion In (1 + (p) z (p + O((pz) . From these operations one can determine (p from (7.15) 5A —a)c 6 c +2c -s q) = 2 ( 1 22 )a2+o(a:). (7.16) Here, as is standard, o(z) denotes a quantity, that after division by z , vanishes as 2 —) O. From (7.16) it is observed that the quantity (p is of the third order in 012 , thus, for a2 « 1 , the logarithmic expansion for small q) is valid. 134 7.2 Small or, Decomposition of the Field Quantities Inserting result (7.16) into (7.14) indicates that the temperature 9T admits the expansion 2 5A can) mc9(c +2c -S) (2:2: )[1 2 1 i ai+o O , C72 > O and a2 tends to zero through positive values, one may draw that (7.39) is equivalent to the requirement that %’s 0 , where we recall that (o and § are given by (7.23) and (7.12). The latter, § = E,(s, c1, c2) = (C1 + c2) (c2 — s') , shows that § 2 0 since 5 < c2 in the 012 —) 0 limit for fan existence. Therefore from (7.39) the criterion for the existence of a fan is m<0 or (so1 - cf)2Ay + Sc1(71—72) < o. (7.40) This condition may be interpreted in various ways, in particular it may be used to obtain a resuiction on 8, namely -2Aycl S < (72 T 71) - 2A7 -2A*{cl > if — (72 - 7,) — 2A7 11 71 if 72—71-2Ay<0 , (7.41) S —2A7>0. Now 72 > 71 and since the driving strain increment A7 would normally be much less than 72 — ‘71 , the standard case in (7.41) would in general be (7.41)2. Since it is already 144 required that O < s < c2 it follows that (7.41)2 may or may not further restrict the phase boundary velocity. In particular if A)! > 0 then (7.41); provides no additional restrictions. However if A'y < 0 then (7.41)2 restricts the lowest phase boundary speed to a finite posi- tive value. Figure 7.1 is a schematic diagram in the (A7, 8') -plane for the region satisfying the fan criterion (7 .41), this figure only considers positive values of s . This region lies above the curve to = 0, where the equation for the curve is given by (6.26). Also shown in Figure 7 .1 is the admissibility region for the purely mechanical problem defined by (5.17), this region was previously displayed in Figure 5.2. In Figure 7.1 this admissability region is below the curve 2 = O and above the line s = 0, and from Figure 7 .1 we see that the curve to = 0 is contained within the admissibility region. Therefore the area between the two curves 2 = O and to = 0 defines the region for which the construction of a centered simple wave fan solution is admissible for the a2 —-> 0 limit. 145 Figure 7.1 This is a schematic plot which shows the admissible region for a centered sim- ple wave for the case where cl = oz. The region that lies between the line s = O and the curve 2 = 0 is the admissible region for the purely mechanical case with Maxwellian irri- tial conditions as previously encountered in Figure 5.2. The region above the curve to = 0 is the region in which the centered simple wave may exist. Therefore the region of existence for the centered simple wave is the area between the two curves. Values for the material parameters were chosen to be c = 2 and 72 - 71 = 5. 82.4; ~|-- "___ ‘Ilw 8. Entropy Production and Dissipation In Chapter 7 it was shown that if the phase boundary speed is treated as a free parameter then it parameterizes a family of solutions involving a centered simple wave fan. It was the case that an explict solution, albeit a family of solutions, was determined only when considering the small 012 limit, which in turn provided insight into the leading order thermal effects. Recall that the second law of thermodynamics (2.6),, was not uti- lized in determining the family of solutions, although it is one of the four governing field requirements in each phase of the material.'Ihus, it is natural to inquire as to what restric- tions this inequality places on the set of possible solution candidates in (A7, 8') -space. It is acknowledged that such a requirement is not directly given in the purely mechanical theory, but is provided indirectly through the positive dissipation criteria requirement, more specifically the dissipation function (5.9) must be nonnegative D (A7, 8) 2 0. In this chapter we investigate any relationships that may exist between the positive dissipation requirement for a purely mechanical material and the restrictions imposed by the second law of thermodynamics for the fully thermal materials, with emphasis on the separable material limit. 8.1 The Second Law of Thermodynamics The second law of thermodynamics in the absence of heat flux and internal energy sources states that the rate of change in entropy for a system must be nonnegative during all processes, globally this requires 146 147 Ef—Jndvzo. (8.1) V The domain under consideration is a bar which contains a propagating surface of disconti- nuity, the appropriate form for the above equation is written 5 (t) h 911 $1 _. > £dtdx+£1>dtdx s[[n]]_0. (8.2) S! From (8.2) it is observed that the time rate of change in entropy arises from two distinct sources, those resulting from local thermomechanical processes in the bulk material, and that contribution from the movement of the surface of discontinuity through the domain. Considering (8.2) in its local form one retrieves (2.6),, and (2.7),. Note that the jump in entropy across a surface of discontinuity is not required to vanish, and thus this jump con- dition must be applied across all phase boundaries, and if present, any shocks within the domain. However, by use of the Riemann invariants (3.62), the jump in entropy vanishes across all contact discontinuities. Consider the initial interaction of the wave pulse with the stationary phase bound- ary, and focus on the period of time for which the phase boundary is still in motion. To implement equation (8.2) during this interval requires the consideration of 7 separate regions within the bar: the parent phase equilibrium state, the region containing the incom- ing wave, the regions labelled S, T and S°(the material having undergone a phase transfor- mation), the phase-2 equilibrium state, and the region containing the centered simple wave fan. Figure 6.1 depicts the initial interaction and shows the seven regions. Invoking 148 restriction (8.2) to the various regions and the surfaces separating adjacent regions, one recognizes that all the integral terms on either side of the phase boundary vanish as do the terms involving jumps across all contact discontinuities. Therefore the only nonvanishing contribution to (8.2) is that associated with the jump in entropy across the phase boundary, thus (8.2) reduces to —s [n (7T, 9T) — 71(Yso,95o)] 2 o, (8.3) where the entropy function (3.19) is used to express es. “(730, 080) = pCYlln ?' +k1, (8.4) 6 .. p .. “(715%) = pC721n[§]+pc§az(yT—y )+ 3+k1- Inequality (8.3) implies that us. 2 111. since s > O , indicating as the material transforms, from phase-2 in the T-region to phase-l in the S°-region, that the transforming material’s field variables must change in a manner such that the difference in entropy is nonnegative, i.e. n was.) —n (1,. 0,) 20. (85) Through the use of the Riemann invariant (3.62) between regions T and the material in the phase-2 equilibrium configuration one may show that 11 (71* 9.1.) = n (72, 62) , which written out in its full form is 149 6 pic]. 2 it: ~ pC721n( ,) + pc2a2(yT— y )+ —e;- + k1: C1’I—t (8.6) .. pk ]+pc§a2(yz—y )+ O‘T‘Fkl. °.l 0» pC721n[ Equation (8.6) provides a relationship between the entropy in region T and the initial con- figuration within the bar, while (8.5) restricts the difference in entropy between T and 8°. Therefore, instead of using the entropy 111. in inequality (8.5) we choose to use that from the phase-2 equilibrium configuration, via equality (8.6), and thus write restriction (8.5) as an equation between the regions S0 and the phase-2 equilibrium state. Therefore, condi- tion (8.5) is equivalent to 9 o " . Cylln[—S7] — C721n[%]—c§a2(72 — y )— :—'f 2 o. (8.7) The inequality (8.7) is a restriction on the temperature OS. , all other field quantities being prescribed by specification of the initial conditions and material parameters. This restric- tion (8.7) is equivalently expressed c c _L2 1-4—1 2 ' AC‘ t C C a — 63.29 10 1"exp{ 2 2:2 Y)+ AT. . 71 C719 (8.8) Thus, given an admissible initial data set (71, 72, 6) , the second law of thermodynamics states that all allowable motions must adhere to condition (8.8), where (8.8) restricts the 150 possible range of values of the temperature 98.. It is interesting to note that since 9so is the temperature of the material which has undergone a phase transformation, (8.8) is a restriction on the transformed material’s temperature. It is also interesting to consider that the second law of thermodynamics, a dynamic balance requirement without analogue in the purely mechanical setting, places restrictions on a thermal field variable, a field vari- able not having any mechanical analogue. We now focus our attention on the special case when the different phases have identical specific heats, C71 = C then (8.8) simplifies to 72’ czar L1. 95. 2 éexp {ZTZWZ —y )+ } a 93410wa d (c71 = (:72) . (8.9) 1 C76 1‘ Table 5 summarizes the relation between the lower bound value of OS. and the initial tem- perature 6 as a function of initial strain 72. 151 Table 5: Lower bound temperature of transformed material OS. when .. - lower bound Value of 72 Temperature 650 1 be ad OWCT ll . AT . < - 6 . <9 72 Y 2 '1 S llower bound czazfl . L, . = —- e o = 9 72 Y 2 2 S |lowerbound czaze .. AT .. 72>? — 2 1‘ es°lowerbound>e czotze As shown in Chapter 6, to explicitly calculate the temperature 68. one uses (6.20) after determining a solution for the master equation (6.14) for the temperature 9.1.. In order to further quantify the restriction (8.7 ) we now consider the special case where the coeffi- cient 012 « l . Inserting the decomposition Gs. = 6;. + 6:: into (8.7) yields 93° 9:: 6 2 * A"r Cylln ? +C711n 1+O—o— -Cyzln —, —cza2(12—y)—g;20. (8.10) so All further discussion of (8.10) appears in the following sections for the case a2 -> 0 and a2«1. 152 8.2 The Entropy Restriction for Separable Materials Recall that the limit a2 —) 0 reduces the free energy to its separable form, in which case (8.10) reduces to 9°. .. cylln[—%]—Cyzln[9;]—§—Izo. (8.11) Once again considering the special case where the two phases have the same specific heats then inequality (8.11) simplifies into the form 9°. 2 éexp {37—} 2 9°. S t S C79 = . 8.12 lower bound (C72 C71) ( ) The inequality (8.12) demonstrates that if the latent heat vanishes (Ll. = 0) then the transformed material’s temperature can not decrease below the initial equilibrium temper- determines ature 8. For the case of a separable material, the value of the quantity C79 the magnitude of the temperature 0;. . As was the case for 012 —) O a table is lower bound constructed which outlines the range of values for (8.12) which the temperature 6;. l bound may achieve, these values being listed in Table 6. OWCI‘ Table 6: Lower bound temperature 9;. for separable material when lower bound C71 = C72 = C7 M 'tud r). /c 9" T t 6°. 38“! e 0 T Y empera ure 8 lower bound AT... < 0 6;. < 6 C79 lower bound ATr > 0 6;. > 6 C79 lower bound Recall from (3.80) that if 21 > 0 then the 2 —-> 1 transformation is exothermic, whereas if 2.1. < 0 then the transformation is endothermic. If we consider C79. > O , then from Table 6 an exothermic transformation coincides with the transformed region’s lower bound tem- perature being greater than its initial equilibrium temperature, which represents a heating of the transformed material. Similarly an endothermic reaction corresponds to transformed region’s lower bound temperature being less than its initial equilibrium temperature, a cooling of the transformed material. Consider the results found for the special case of mechanically neutral initial con- ditions (4.19)-(4.25), this allows 9;. to be expressed in terms of the function 2 (A7, 8') by means of (7.36): cf}; (Ay,S) (MN). 6;. = 6+ C71(c1+ c2) (c1 + 8) (c2 — S) 154 Substituting this expression into (8.11) yields 2 " c 2 A ,s C ‘ A 1n[%+ ‘ 1 ( Y ) ]—(-:—721n[9;)— 1,20 (MN), 6 6 C71(cl+c2) (c1+S) (cz-S) 71 6 C716 which in turn reduces to 2 . clE(Ay,s) > C71(c1+c2) (c1+8) (CZ—S) ' s C " 2. . (8.13) e [CXp{C-:;131n[-67.]+ T } —3,] (MN). 71 6 C716 6 The inequality (8.13), along with the condition 8 > O, resuicts the set of solutions in (A7, 3') -space. Recall from Chapter 5, in a purely mechanical setting, a similar restriction was derived, e. g. 82 (A7, 8) 2 O. This was a consequence of the assumption that all motions must have nonnegative dissipation, and like (8.13), reduced the set of solutions in 1 the (A7, 8) -space. If we now consider the inequality (8.13) for the particular case C72 = C71 and AT = O , then since s 5 c2 we find that 2 (Ave) 2 0 ((272 = C71, 2., = 0) . (8.14) which is essentially the same restriction found for the purely mechanical problem (sec (5.17 )). However it is noted that one cannot directly compare the two conditions, because result (5.17) was derived for the Maxwellian initial state while (8.14) is for the mechani- cally neutral initial state. 155 A more fundamental and less restricted result occurs for the case in which the ini- tial state is omnibalanced. In Chapter 7 it was shown for an omnibalanced initial configu- ration that 62. can be written in terms of the purely mechanical dissipation function D (Ay,s') via (7.37). This reduces (8.11) to C C AOB A. AOB D—MS—Yélzracfle {exp{J—21n[9 ]+ T,}—9 } (OB). (8.15) Recall however that an OB-initial state must involve initial temperatures as given by (4.32)1. Substitution from (4.32)1 and invoking pCYls > O in (8.15) gives D (137,8) 2 0 (OB) . (8.16) Thus the positive dissipation criterion in the purely mechanical description has a strict thermodynamic basis in terms of (2.7 )4 in the separable material limit. In view of the com- plexity encountered thus far in interpreting second law issues, any further consideration into these issues would most profitably restrict attention to omnibalanced states. In partic- ular, the first order 012 -correction to (8.16) for the omnibalanced state is addressed in the next section. 156 8.3 The Entropy Restriction for Fully Thermal Materials Our previous discussion has established that the a2 —) 0 admissibility region coin- cides with the admissibility region for the purely mechanical problem provided that the thermomechanical problem had omnibalanced conditions. It is of interest to determine how the admissability region changes with the consideration of thermal effects. In what follows we address this issue and concentrate on OB initial states when cl = c2. To determine how the 012 —> O admissibility region changes for small but finite a2 , we evaluate (8.10) along all of the boundaries of the (12 = O admissibility region. Recall that these boundaries are defined by (5.17), which gives the line 8 = 0 and the curve 2 = 0. We here evaluate (8.10) along 2 = 0. By definition (5.17) the quantity 90 so C7114?) - C721n[ ]. vanishes along 2 = 0. Therefore to determine the thermal correction to the admissible ‘1’.| G I >’ “-1 region we need only evaluate the remaining expression in (8.10) along 2 = 0. Define the function 2 as "'2 zaCYlln l+—S—° —c§ot2(vz—y'), (8.17) O 65. whereby evaluating z along 2‘. = 0 provides the thermal correction. Locations where z > 0 are points on the boundary where the purely mechanical admissibility criterion and the thermal admissibility criterion are satisfied. Similarly at those points where z < 0 the 157 purely mechanical admissibility criterion is satisfied, but the thermal admissibility crite- rion is not satisfied. In this manner one can show where addition of the thermal correction (8.17) causes the admissibility region to “grow” (2 > 0) and “shrink” (z > 0) in a point wise fashion. From (7.35) the temperature 6:: can be expanded to the first order in a2 , which in turn allows for the logarithmic expression in (8. 17) to be expanded 6a. 6a, Cylln 1+ e—j— = C117;- + O(otfi), (8.18) so where we evaluated (7.37) along 2 = o to write 9;. = 6. Using (7.35) and (8.18) in (8.17) the function 2 can be expressed c201 033(7 -Y ) z = —72[c22A + 22 21 -2$2(72—y )]+0(ot§). 2s (c -S ) where we have used (7.27) and the definitions for (n and § which are found in Chapter 7. 2 c or Focusing on the leading order 012 effect, we note the coefficient -—23 is positive when 28 considering 0 < 012 « 1 , hence to determine the sign of z we need only consider the expression 3. C S - e c22A7+ 1: 2 271) —25‘2(72-'y ) (8.19) (c -S ) evaluated along the curve 2 = 0. Using (5.18) the curve 2 = 0 yields the expression for .. - -—'-,.—~ 158 the strain increment A7 _gc _ 247 = (272 27,) , (c -S ) which when inserted into (8.19) yields . . 2 '- srgn[z|2=0] = srgn [-28 (72—7 )J. (8.20) Using the OB phase-2 strain (4.31)2 in (8.20) yields sign[z|2=0] = sign[-2-S::'—1;:l. (8.21) (12¢ 6 From (8.21) we see that the thennal correction depends entirely upon the sign of AT/(f, since we have been operating under the assumption that 012 is a small but finite quantity. Since 6* > 0 it follows that (8.21) shows that it is the latent heat KT which determines how the admissible region changes in a global fashion near the boundary 2 = O. Namely if X1. > 0 , corresponding to a 2 —> 1 transfonnation being exothermic, then the function 2 > O and globally the admissibility region expands beyond the curve 2‘. = 0. Similarly if M < O , an endothermic transformation, then the admissibility region contracts inward from the boundary 2 = 0. Attention is now focused on the admissibility boundary s = O. The previous anal- ysis for the boundary 2‘. = O is general up to (8.19), hence to determine how the boundary S = 0 shifts we need to determine where z > O and z < 0 along s = 0. Evaluating (8.19) 159 along 8' = 0 and inserting the result into (8.18) yields sign[z|s=o] = sign [c22Ay] . (8.22) Recall that for our problem 8' > 0 => 137 < O , thus (8.22) yields that z < 0 everywhere along 3 = 0. Hence the admissibility region contracts inward from the boundary 3 = O. 9. Solutions Obeying a Thermal Version of the Kinetic Relation 9.1 Driving Traction The driving traction was defined in the purely mechanical setting in (5.10). In the fully thermomechanical setting the driving traction f (t) is defined by f(t) =}(Y.6)E[[£ll—<(t))[[7]]-((9)>llnll. (9.1) Using the definition of the free energy, w = e—Bn , the jump in internal energy can be expressed [[8]] = [[111]] +((9))[[11]] +(<11))[[9]]. where the jump in the product 911 has been expressed [[971]] =((9))[[11]]+((Tl))[[9]]. Substituting from these results into (9.1) produces the driving traction in the more useful form f(t) = [[v]]—<[[7]l +<>[[9]l. (9.2) The driving traction (9.2) is not what is commonly seen in the literature, the conventional 160 161 definition of the driving traction (Abeyaratne and Knowles 1990b, Fried 1992) in terms of the free energy and stress is f(t) = [M] — ((1)) [[7]]. (9.3) The difference between these driving tractions arises from assumptions on the temperature fields. Form (9.3) assumes that the temperature fields are smooth, analogous to the dis- placement field. This smoothness implies that across all interfaces the jump in temperature vanish, including across any phase boundary. The form of the driving traction defined in this document (9.2) was proposed to account for a temperature field with a discontinuous nature. Note that (9.2) reduces to the form (9.3) if [ [6] ] = 0. It is also recognized that for isothermal problems, and thus all purely mechanical problems, that the correct form of the driving traction is given by (9.3). However, (9.3) is not necessarily correct for those - problems with a separable energy, which allows for such temperature jumps. A useful form of the driving traction may be obtained upon writing the first term on the right hand side of (9.2) as '0 ‘1’ [[v1] = v1—v’ = jdv. (9.4) C. By invoking the fundamental definitions for stress and entropy the free energy differential dth can be expressed 162 dry = g—lldy-t-g—‘gde = tdy—nde, (9-5) and thus the jump in free energy may be written 7. 9. [[v1] = jrdy— jnde. (9.6) 7' 9' which allows the driving traction (9.2) to be written 7. 9. f6) = jrdr-<(t)>[[r]]— jndG—((n>)[[9]] . (9.7) 1' 0' Once again for smooth temperature fields the term inside the brackets vanishes, retrieving the more commonly used definition of the driving traction (9.3). In Chapter 5 the driving traction (5.10) for the purely mechanical problem was defined f(t)m°°h = jitvldv-<>ttvll. To understand relationship between the purely mechanical form (5.10) of the driving trac- tion and the thermal/separable energy form (9.2), some consideration of the temperature field within the body is required. In a purely mechanical setting the temperature is of no 163 consideration, it is assumed that the problem is isothermal. Furthermore, in hyperelasticity the stress is assumed to be derivable from a potential function, the strain energy function, here denoted W (7) , and thus 1(7) = %W (7) . Recall in working with the Helmholtz free energy that the stress is ‘C (y, 0) = 887‘" (y, 6) . Thus if the free energy does not depend on temperature (as in the purely mechanical case), then the Helmholtz free energy is equivalent to the strain energy function. Working under the assumption of hyperelastic- ity, the integral term of the driving traction (5. 10) is ‘1’2 _ 728 _ [fitment - jyfiwom - 11w11. the driving traction (5. 10) is h . mom“ = 11W]l-<>tlvli. (9.8) Thus the definition of the driving traction (9.2) reduces to the proper form in the limit of the purely mechanical case (9.8). Comparing (9.7) with definition (9.8) for the special case of isothermal motions recovers the familiar form of the driving traction commonly pre- scribed for those problems which are purely mechanical in nature § 1 f (t) Imm mm, = j tdy — ((1)) 1 [vi] a f (t) ””11. (9.9) 7- Returning to definition (9.1) the local balance law (2.7)3 permits the driving traction to be 164 expressed in its simplest form f0) = —(<9>> [ [11]]. (9.10) which is the form of the driving traction which we choose to use in the rest of this docu- ment. Since ((0)) > 0 , the second law restriction (8.5) can be written f(t) 20. (9.11) Therefore, we conclude that the second law of thermodynamics states that the driving trac- tion acting on the interface must be positive. Writing out the driving traction for initial interaction via (9. 10) gives 1‘ = -%(9-r+98.)(nT—ns.). (9.12) or since 11.1. = 112, f = -%(9T+OS.)(n2—ns.). (9.13) Since all field quantities can be decomposed into two parts via (3.92)-(3.94), the first bracketed quantity in (9.13) may be written 165 0., + 08, = 031+ 0:2 + 0:, + 0:3. (9.14) The jump in entropy using (8.4) and (8.5) is 1'12 - "so = A 0 or, A (9.15) 6 2 e pl 9 o + 9 o 9 pC721n[?] + pc2012(')(2 — y )+ f—pc1lm[£fi]_pcllm[g] From this jump in entropy we foresee one of the major problems of this research t0pic, the driving traction has field variables, which are themselves complex functions, in a logarith- mic fashion. To avoid this difficulty, we limit our attention to the OB equilibrium state. 9.2 Driving Tl’action for Separable Materials with OB Initial Conditions We conclude our analysis of the driving traction by considering separable materi- als with OB initial states. Guided by the consideration of Chapter 8 it seems clear that con- sistency with the purely mechanical theory only occurs in the case of an omnibalanced initial state. In this case (7 .19)1 and (7 .37) give that —1 A 1 mech . 3(0T+08.)-—(0+ZSPCYID (47.8)) (OB). (9.16) 166 while (7.37), (8.4), (8.6) yield ,. p . 112-71$. — przln[§]+ 2:1— Cylln[:;] (9.17) 1 mech C ln(l+—,——D (A .80) (on). p 1” Sp0C71 Y Recall for C1 = c2 that omnibalanced states involve initial temperatures 6 obeying (4.32)1 in the 012 -> 0 limit. In this case (9.17) further simplifies to 1 mech n— .=—C 1n(1+ . D (A ,9) (c =c,OB). (9.18) Thus for an omnibalanced initial state, the 012 —> 0 limit of the driving u'action is given by o _ . D111°°11(Ay,s) D'“”"(Av.8) _ f - pC71[0+ 259C“ ln 1+ SpC716 (c1 —c2, OB). (9.19) Recall that the omnibalanced temperature (4.32)1 is 60130 = 0'exp[ , AT ]. 0 (C71 — C72) Note if the latent heat is nonvanishing then the limit C71 —-> C72 gives that the tempera- .ono ‘ ture 0 -> oo . This limit permits the logarithmic expression in (9.19) to be expanded via a Taylor series 167 mech . mech . ln[l+D _ (Arm) = D . (Al/'5) +o(é’2). (9.20) spC710 spC710 Using result (9.20) simplifies (9.19) C A mech mech .— f" = ._EE[20+D (M’s) )[D (M’s) +019 2)] (OB) ’ 2 Spcyl Sprlé collecting like powers of 6 gives mech 1'0 = D 3017.8) +0094) (OB). (9.21) Thus for the case where the specific beats are the same and the initial state is omnibal- anced we find that the separable form of the driving traction is mech f° = D 31“") (C,, = C7,. (013)). (9.22) Note that (9.22) is equivalent to the driving traction that was found for the purely mechan- ical problem (5.11). Again we have demonstrated the close correspondence between the purely mechanical and the separable theories. 168 9.3 Kinetic Relation for Separable Materials We now wish to consider the use of a kinetic relation to single out a particular phase boundary speed for separable materials with OB initial conditions. In Chapter 5 the use of a linear kinetic relation was investigated, and from (5.20) it is natural to assume that the kinetic relation for the case of a separable material is s = K f°. (9.23) Driven by result (9.22), we examine (9.23) for the same conditions. First, we note that for these initial conditions the dissipation function is given by (5.16), then with a specified mobility x and initial conditions, (9.22) and (9.23) give an implicit equation for 8 A7 = 2 S +SC (721-722) (OB) . (9.24) 1:99 (71—72) 2(c —S) This is result (5.23) found for the purely mechanical case, which was to be expected. Therefore Figure 5.3 being a graph in the (Av, 8) -plane of the linear kinetic relation (5.23) also describes the linear kinetic relation for a separable material. This figure shows the curve (9.24) for different values of the mobility x , from the figure it is seen that as the mobility decreases the phase boundary speed decreases. 169 9.4 Kinetic Relation for Fully Thermal Materials We now wish to include thermal effects in the linear kinetic relation. This change might yield a different value for the phase boundary speed from that determined in the separable theory Just as the change in the admissibility region could be determined by examining first order a2 effects in the 2nd law as discussed in section 8.3, we expect that the change in the phase boundary speed can be accomplished by a similar analysis of the driving traction. The linear kinetic relation is s = rc(f°+ fa”), (9.25) where in the (12 —-) 0 limit we retrieve the separable case (9.23). Thus the term 1(ch2 par- ticipates in the thermal correction to the separable phase boundary speed (9.23). For OB initial states we find the leading order thermal correction to the driving traction can be expressed 6% 0. A * 0 f’=—p 9+ , D c§a2(yz-y)+C,flln 1+-—i— 2spc71 9+ . D 89C): (9.26) 9:“ a. + 9 f e “ _ —2_S_ [c721n[3,] + ;‘—I— C111n[9;]-Cylln( 1 + 43—7)] + O(ai). 9 9 Spc'yle Q 170 Note in (9.26) that the mechanical field variable 9:3. and the dissipation function D, both complicated functions, appear in the a logarithmic manner. Thus to determine the global behavior of f a2 a numerical study might be in order. Since such an approach is not in the a 0 same vein as the rest of this investigation, a detailed analysis of f 2 remains to be explored. 10. Conclusions and Recommendations for Future Work 10.1 Conclusions From our analysis we have shown a number of results for the problem considered, most of these demonstrated how the purely mechanical theory falls under a more complete thermomechanical framework. Some of the more significant results are listed below. 1. For the mechanical field quantities of strain and velocity, a one to one correspondence exists between the purely mechanical and separable theories. 2. The positive dissipation criterion for the purely mechanical theory is a direct conse- quence of the second law of thermodynamics for the separable theory provided that the initial configuration is omnibalanced. Thus, the purely mechanical criterion has a sound thermodynamic foundation. 3. For a phase transformation occurring in a separable material, our model predicts the possibility for a temperature change within the transformed material. If the initial configu- ration is omnibalanced, then this change correlates directly with the dissipation function in the separable theory limit. 4. The separable theory, a theory which was shown to account for temperature effects, does not remove the nonuniqueness present in problems concerned with phase boundary motion. Thus, a higher order theory is required to resolve this issue. A reasonable resolu- 171 172 tion involves a separate kinetic relation. 10.2 Recommendations for Future Work As with most research problems, there are a number of issues which we choose not to address here, of these some of the more significant ones are listed below. 1. Investigation of the shock instead of the centered simple wave fan needs to be com- pleted. This may provide a simplification in analysis. 2. For the case a2 « 1 , a more complete analysis of the admissible solution region needs to be performed to determine how it changes in comparison to the solution region from the separable theory. This may prove untenable due to the presence of complex functions con- tained within logarithms expressions. 3. Although this was a purely analytic study, a numerical study of the transformed materi- als temperature 98,, might prove fruitful. The results of such a study could be directly incorporated into a study of the admissible solution region. Recall that the growth of this region depends solely on the temperature 65,. 4. For the case a2 « l , a study of the incorporation of thermal effects into the linear kinetic relation needs to be performed. The results of such a study would determine if the inclusion of thermal effects would cause the phase boundary speed to increase or decrease from the speed found for a separable material. Appendicies Appendix A. Algorithm for Reduction of Equations (6.5)-(6.13) The algorithm outlined below expresses the eight field quantities Vs , Vs , 65, vso , 73° , OS0 , VT, 7T as functions in terms of the initial conditions, temperature 9.1., and the phase boundary speed. From (7 .3) the temperature in region S as = 6. (M) For later use we use (6.7) and (6.8) to relate the velocity and strain in regions S and So v8 = vSo , (A.2) 78 = 'Yso. (A3) From (6.9) the strain 71 is found to be C A 71.:72 + —72—1n(6-9-),(A.4) c20‘2 while from (6.10) the velocity vT is vT = @[mep —o(é)], (A5) @(9) -(O) (D(O) +(O) where 19(9) = 24> (9) + d) (0) ln( ).Using results (A.2) and (A3) we may rewrite (6.5) 173 174 v8.) = clyso—c1('yl+ 2A7) . (A.6) From (6.11) the velocity vSo vs. = vT+s' (YT—75..) , (A7) and from results (AA) and (A5) the expression for the velocity . C “ vs. = @(mep—me))+s[yz+—2i1n(é%)]—sys.. (A.8) c20‘2 Combining results (A.6) and (A.8) to find expressions for both the velocity and the strain in region S0 in terms of the unknown temperature 6.1. one finds c ,. C 6 vs. = ‘71-}[jC—fl(fi(6T)—fi(9))+ {72 + —2—Tlln(6;)]-S (71 + 2A7) ], (A9) C2% 1 . C 6 73° = Cl+S[@(fi(6T)—fi(9))+s[yz+-?7—2-1n(6:r)]+c1(71+2A7)]. (A.10) Czaz To express the field variable 98. in terms of the temperature 61. , insert results (A.4), (A.10) into equation (6.13), and by simplifying one finds 95. = T1 (0,.) +T2(6T) + [T3(6T) +T4(9T)]T5(6T), (A.11) 175 where we have used the auxiliary functions 2 1 pc C 6 ¥ * 2 A A 71 020.2 T 2 T2 (9.1.) E (@[fi(91~) —13(é)] +cl (y1 +2Ay)+ C 6 2 . 72 r , 2 2C71(C11’S) czaz T e - 9c: 9 “ 2 C721“) 3( T)=-(-61—+—S_)- @[fi( T)-fi(6)]+C1(Yl+ A'Y)+S 72+?azn '6; , C 6 .. . T4 (er) a pc§[yz + 7111n(-9—)— (y — 0:29 )J-pciazeT, czot2 T T5 (or) = l A c 39971 (cl+s)(~/E—12[‘9(9'r) “WU +0. (sumo- C 2 6 c1[72++1n(§;)] ]. C20‘2 Finally the master equation for the temperature 9.1. is found by inserting results (A4), (A5), (A9), (A.10) into equation (7.9) and simplifying 2 . C (SC —C ) 9 2 czar2 0 C72(s'cl—c§)l [] (A.12) /c,2(c1— g) o (6) 2 —; + ciafia-(Scl —- cf)2Ay + s'cl (71—72) . czaz Appendix B. Transition in phase-2 is a shock As discussed in Chapter 6, the characteristic curves in the phase-2 material are not globally parallel as is the case in phase-l. Therefore transitions between different states can occur through two types of mechanisms, a centered simple wave fan and a shock. As the case of the centered simple wave fan was analyzed in detail in Chapter 6 we now turn our attention to the case where the transition is a shock. The purpose of this appendix is to formulate the necessary equations which mathematically describe the shock problem. Recall that if the transition was a centered simple wave fan then the Riemann invariants imposed a set of conditions between the field variables on adjacent sides of the fan. For the case of a shock these conditions need not be satisfied, instead a new set of constraints, the Rankine-Hugoniot equations, now must be satisfied. Recall that these equations were the jump conditions (2.7) across the phase boundary. This is because the phase boundary and the shock are discontinuity surfaces.The Rankine-Hugoniot equations across the shock are: [M] Hillvll = 0. [[1]]+lip[[vll=0. (13.1) li([[e]]—<[[v]]) =0. fipllnll $0. 176 177 where Ii is the speed of the shock, which we assume to be constant, in view of the self- similar nature of the square-wave pulse that initiates the process. We wish to reconsider a boundary value problem outlined in Chapter 6, except now where the T/Ez interface is a shock, all other assumptions are assumed the same as the case of the centered simple wave fan. Thus, the dynamic boundary conditions (4.33)- (4.37 ) and assumptions A.5-A.9 (Chapter 5) still hold. These conditions give rise to an ini- tial wave pulse, originating in the phase-1 material, striking the phase boundary and set- ting it in motion. Once again the pulse striking the phase boundary can generate a reflected wave, giving rise to regions R, S, 8°, and a transmitted wave, creating region T. However, now the TIP/z interface is a shock, across which equations (B.1) are required to be satis- fied. Figure BI is a graphical representation in (xt)-space of the temporal changes within the body where the phase boundary speed is positive. During the initial encounter there are four regions of interaction which are of interest, these four have been labeled R,S,S° and T in Figure B. 1. One can think of these regions as follows: S is the region to the left of the initial position of the phase boundary in which the initial pulse and the wave which has been reflected from the phase boundary are interacting; 8" is that region to the right of the initial position of the phase boundary in which the incoming pulse and the reflected wave are interacting, R is the result of initial incoming pulse being reflected from the phase boundary and clear of any further interactions, finally the region T is that region in which some part of the initial pulse has transmitted through the phase boundary. The shock is to occur between the phase-2 equilibrium state and the T region. As was the case for the fan, region R, which contains the final reflected wave, 178 occurs later than either S or SO and thus decouples from the other three regions. Therefore, when analyzing the initial interaction we only need to consider the regions S, 80, and T. We again use the method of characteristics to relate the various regions of the domain, between S and the incoming pulse-the Riemann invariants (3.51) and (3.52); between regions S and S°-the two Riemann invariants (3.52), across the moving phase boundary between T and the S°-the 3 jump conditions (2.7 ), between T and the initial phase-2 equi- librium state-the three shock conditions (B. 1). In Figure B.1 the various characteristics and Riemann invariants are shown, as well as the jump conditions across the phase bound- ary, and the shock conditions in the phase-2 region. From the procedure outlined above we generate ten equations between the nine field quantities: GS, 630, 61-, 75, 130, yr, vs, v30, vT, the shock speed Ii , and the phase boundary speed 8. Guided by Figure BI and the above outline, the ten equations which ovem the interactions between the S, 8°, and T are written out below. g Region S and the incoming pulse Vs—Cr'Ys = —c1(yl+2Ay) , (B.2) 9s .. é .. pCYlln .91 +k1 = pCYlln ? +k1. (B.3) Region S and S0 vso "' Cl‘YSo = VS — CIYS 9 (B°4) 179 vs..+cl')(So = vs+clys. (35) Region T and the phase-2 equilibrium state, characterized by the formation of the shock wave 46(72—71.) = vz— vT, (B6) 5va = pain, _y’) _ pc§a2(é _ e‘) — pcim- y’) + pciasz— 9"). (13.7) 2 2 pC a t 2 A DC a It: 2 [73(72—7 + aze ) + pCflO + 62] — [73(yT—y + aze ) + pC120T+ 52] = (BS) 1 " A * t a 5(PC§(72 - 'Y )- pciazw — 9 )+ pciflT— y )—pc§a2(6T— 9 )) (72 .41.) , Region So andT —S (YT " lYso) = VT "' vso 9 (3'9) 2 r 2 r 2 —Sp(vT—vs.) = pc2(yT—y )—pc2a2(6T-6 )—pcl'ys., (BID) 2 2 pc2( ' 9')2 C e 8 pc‘ 2 C e B — 11 —2"’ 71""7 +02 +9 72 T+ 2 - TYS°+p 11 S"+ 1 " (B ) l :- . §( pcfys. 4' Pciw'r " Y )-pc§a2(6T - 9 )) (1T — 78°) . The above equations are a system of ten equations for the eleven quantities Vs , vs, BS, 75., Vso, 95° . 7r , vT , 0T, Ii, 8' this system of equations completely characterize the ini- 180 tial interaction. Comparing this set, to that of (6.5)-(6.13) for a fan, it is to be noted that we have obtained an extra unknown 5, and an extra equation. The extra equation is due to the fact that 3 shock conditions are given across a T/FQ shock interface, whereas only 2 character- istic equations held across a T/Ez fan. On the other hand, the equations obtained here, (B.9)-(B.11), are simpler than those which describe the fan, (6.9) and (6.10), because those for the shock have no unknown field quantities appearing in a logarithmic fashion. Preliminary studies indicate that an elimination procedure similar to that employed in Chapter 6 yields a master equation which would be the analogue of (6.14). This master equation is in terms of the strain VT, and is given by the following equation: 181 —c2 s‘c 8+0 (‘YT- (v -oc29 ))+S -—c—IYT- c (72+2M)+ 1 2 2 at t 2 It: at 2 A czaz veins—(v -229 >> 472—9 «129 >> ]—2:>C.29+ (S-Cr) (pciazm—yT) —2pC72) 0:92 [99in - (7‘ — 9293) + 963% - (7‘ - 929'» — 993929] (72 -w )2 ' _ 2 (8 C1) (Pczaz (72 ’ 7T) ’ ZPCyz) —[pc§(v2 — (7' - a26'))—pc§a2é—pc§(v-r — (7‘ — a26‘))] (72 - YT) + [pcim - (t!m - «129'» + 992(72 - (7' - «293) — 902029] (72 - YT) (pc§a2 (v2 — 72) — 29C”) 2 [3020.2 (72 - YT)( pc2[(YT- (7— cat—29)) -(YT-(Y a—2—9))] —29C229]_ o (992a2(v2— 72)- 29C”) ‘ One would anticipate that this equation can be analyzed in a fashion similar to the case of a fan. 182 Phase 1. Phase 2. shock 52 M Riemann invariant x=s t ( ) W Jump conditions Figure B.1 This is a graphical representation in the (xt)-plane where the transition in the phase-2 regions gives rise the formation of a shock. Regions E1 and 152 are the initial equi- librium states separated by the phase boundary at x=s(t). The incoming wave (IW) strikes the phase boundary setting it into motion, where s > 0 is assumed. The IW-phase bound- ary interaction gives rise to the regions S, 8° and R in phase-l, and T and the shock in phase-2. The region R represents a reflected wave, while S arises from the interaction of the IW and the reflected wave. 8" is that material which has undergone a phase transfor- mation from phase-1 to phase-2. The IW striking the phase boundary also produces a transmitted wave in phase-2, this is designated by the letter T. 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