Thermostatic derivative recursion table (expressing derivatives in terms of material properties)

Transcription

1 See dicuion, tat, and author rofile for thi ublication at: htt:// hermotatic deriatie recurion table (exreing deriatie in term of material roertie) Reearch July 205 DOI: 0.340/RG CIAIONS 0 READS 89 author: Rebecca Brannon Unierity of Utah 89 UBLICAIONS 679 CIAIONS SEE ROFILE Some of the author of thi ublication are alo working on thee related roject: utorial View roject Damage Mechanic JHU Sabbatical View roject All content following thi age wa uloaded by Rebecca Brannon on 6 July 205. he uer ha requeted enhancement of the downloaded file.

2 Rebecca Brannon (c) July 6, 205 Recurion table for thermotatic deriatie Author: Rebecca M. Brannon. Coyright i reered. Coie may be made for indiidual ue, but not for rofit. Lat udated July 6, 205 0:59 am. COMMENS? Send your feedback (eecially tyo alert) to comutationalolidmechanic@gmail.com Any eron working on thermodynamic mut eentually mater the kill of conerting thermodynamic deriatie into exreion inoling only meaurable material tate ariable and tabulated material roertie. Howeer, a ytematic mean of doing o i rarely reented in thermodynamic textbook. Mot tudent take a random walk through ariou nebulou identitie, hoing to eentually tumble uon the anwer. hi document make the roce ytematic (guaranteed ucce). Claical reerible thermotatic i motly math, not hyic. It i a grand alication of the calculu of function of two ariable, founded on the contitutie * aumtion that the internal energy er unit ma u i exreible a a function of the ecific olume (i.e., olume er ma, which i /denity) and the entroy. hi function u (, ) i called a fundamental otential becaue all other quantitie of interet in thermodynamic can be determined from it. For examle, a exlained in any undergraduate thermodynamic textbook, the temerature and reure can be found by u and u o be a fundamental otential, the internal energy mut be exreed a a function of and, o we call thee two indeendent ariable the natural ariable for the internal energy. Equation (a) imlie that i a function of and. In rincile, thi equation could be oled for a a function of and. You could then ubtitute (, ) into u (, ) to obtain u (, ). Sadly, the function u (, ) i not a fundamental otential. It ha lot information content. Haing only u (, ), you will no longer be able to uniquely determine all thermodynamic quantitie. he fact that u (, ) embodie le information than u (, ) i not obiou the roof and further information can be found in mot adanced thermodynamic textbook. Fortunately, all i not lot. Een though u (, ) i not a fundamental otential, there i a different energy meaure that i a fundamental otential when it i exreed a a function of and. If you eek a fundamental otential for which and are the indeendent natural ariable (intead of and ), then you mut ue a contact (Legendre) tranformation to introduce an alternatie energy meaure, called the Helmholtz free energy: a u A outlined in any good thermodynamic textbook, thi new ariable i a fundamental otential function when it i exreed in term of and. With it, you may comute entroy and reure by a and a Similar trategie and introduction of new energie (otential) can be alied to ermit contruction of fundamental otential in term of any conenient air of thermomechanical tate ariable. he roce leae u with o many formula, that mnemonic and executie ummary table are needed to kee eerything organized and ueful. * he word contitutie mean relating to a articular material or cla of material, oibly further retricted to certain contraint on erice condition. For examle, many material can be regarded to be elatic under trict condition on, ay, the magnitude and/or duration of loading. In claical thermotatic, we conider material and erice condition for which only two indeendent ariable, erha entroy and olume, are needed to fix the alue of all other quantitie of interet (uch a reure and temerature). hi contitutie aumtion i broadly alicable to gae and inicid fluid and alo alicable to iotroic olid when the deformation are contrained to allow change in ize but not change in hae. e.g. hermodynamic and an Introduction to hermotatitic (985) by H.B. Callen. () (2) (3)

3 hermodynamic quare he thermodynamic quare i a mnemonic deice that hel you recall the natural ariable aociated with the energie, a well a many other thing uch a the Legendre tranformation and Maxwell relation. he quare i contructed by lacing the tate ariable on the corner and the energie on the edge, along with two arrow a hown. State ariable (corner of the thermodynamic quare): ecific olume ( --, where ρ i ma denity) m 3 /kg ρ temerature K reure N/m 2 J/m 3 ecific entroy J/(kg K) Energie * (edge of the thermodynamic quare): a a Helmholtz free energy g Gibb free energy natural function of and J/kg natural function of and J/kg h enthaly natural function of and J/kg u ecific internal energy natural function of and J/kg he Legendre tranformation (i.e., the relationhi between the energie) are inferred from the thermodynamic quare by ubtracting energie in the off-diagonal in the ame direction a the arrow: u a h g and h u g a (4) In the thermodynamic quare, the energie are urrounded by their natural ariable. Looking at the quare, for examle, u (, ) i a fundamental otential becaue u i urrounded by and. Gibb free energy i a fundamental otential when it i written in the form g (, ). Similarly, according to the thermodynamic quare, a (, ) and h (, ) are fundamental otential. With a fundamental otential, you can get eerything ele. For examle, the Gibbian relation gie u u a a g g hee equation aly when differentiating with reect to an energy natural ariable, holding the other natural ariable contant. he final reult i found by moing diagonally acro the quare, etting the ± ign baed on whether you moe with or againt the arrow. If e denote any of the energie ( uahor,,, g) and if x and y are the natural ariable aociated with e, then e x y x where x i the ariable diagonally ooite from x on the thermodynamic quare multilied by + if traering the diagonal moe with the arrow, or if ooing the arrow. Secifically,,,, and. (7) he formula lited exlicitly in Eq. (5) are ecific intance of the generic Eq. (6) h h u g h (5a) (5b) (6) Author: Rebecca M. Brannon. Coyright i reered. Coie may be made for indiidual ue, but not for rofit. Lat udated July 6, 205 0:59 am. COMMENS? Send your feedback (eecially tyo alert) tocomutationalolidmechanic@gmail.com * he energie are alo tate ariable in the ene that they return to their original alue for any cloed ath of ariation in the other tate ariable. 2

4 Author: Rebecca M. Brannon. Coyright i reered. Coie may be made for indiidual ue, but not for rofit. Lat udated July 6, 205 0:59 am. COMMENS? Send your feedback (eecially tyo alert) to comutationalolidmechanic@gmail.com he thermodynamic quare can alo be ued to recall Maxwell relation: hee formula all inole deriatie of a tate ariable with reect to another tate ariable on the ame edge of the thermodynamic quare. he ariable held contant can be either of the two tate ariable on the other ide of the quare. An edge deriatie like thi i equated to it mirror-image edge deriatie acro the quare with the ± ign aigned to be + if the ymmetry of the edge i the ame a that of the arrow or otherwie. For examle, in Maxwell relation, inole ariable and, which are on a ertical edge ue the correonding mirror-image ariable on the other ertical edge In thi cae, we are talking about ertical edge. he arrow are alo ymmetric about the ertical. herefore a oitie ign i ued in the Maxwell relation inole ariable and, which are on a horizontal edge ue the correonding mirror-image ariable on the other horizontal edge In thi cae, we are talking about horizontal edge, but the arrow are ymmetric about the ertical. herefore a negatie ign i ued in the Maxwell relation. IMORAN: what i being held contant in the deriatie matter. Maxwell relation inole only tate ariable (corner of the thermodynamic quare). Moreoer, not only are the major layer in the deriatie following edge ymmetry, but the thing being held contant i too. For examle , but a In thermodynamic, artial deriatie are almot alway of the form ( A B) C, which quantifie how ome ariable A change in reone to change of B during rocee that hold C fixed. Changing the contraint (i.e., changing the thing held contant) will change the meaning of the artial deriatie. A artial deriatie A B that fail to indicate what i being held contant i meaningle. here are 336 way to form deriatie of the form ( A B) C uing the eight thermodynamic ariable ( uahg,,,,,,, ). Of thee, ome will hae reaonably intuitie hyical meaning. A a rule, deriatie that inole only corner tate ariable (,,, ) can be readily interreted hyically, and are mot likely to be meaured in the lab and tabulated in handbook. We call thee material roertie (they are function of the thermodynamic tate, not contant). For examle, the deriatie ( ) rereent the olume change roduced in a ga if the temerature i increaed while holding the reure contant. Similarly, ( ) i the local loe of a reureolume cure meaured under iothermal condition. Of the 336 oible deriatie, the one that inole energie often lack aarent meaning. For examle, ( h) a i bizarre and incomrehenible. We need a way to tranform the deriatie inoling energie into exreion that inole more eaily interreted element. Ditilling i the roce of conerting any deriatie into a form inoling only tate ariable (,,, ) and material roerty deriatie (which are meaningful and often tabulated in handbook). Simlifying the ditilling roce i the goal of thi document u g h (8) 3

5 hermotatic material roertie: he tate ariable (,,, ) are regarded a eaily meaurable or eaily controllable in the lab. In any roce, only two tate ariable can be indeendently controlled at any time. Standard exeriment will ary one tate ariable while holding a econd tate ariable contant (thu controlling exactly two ariable). he reulting ariation of the other two tate ariable i recorded. Suoe, for examle, that an exeriment i conducted in which the olume i aried under thermally inulated (AKA, adiabatic, contant entroy, ientroic * ) condition. hen the data record how the deendent tate ariable (reure and temerature) change in reone to thi adiabatic olume change. he ientroic bulk modulu (a material roerty) i determined from the adiabatic reure-olume cure. he adiabatic temerature-olume cure lead to a leer-known material roerty called the Grüneien arameter. You can erform different exeriment that ary different tate ariable, holding different tate ariable contant. In eery cae, the loe of the reone function are roortional to material roertie. roertie that can be meaured in thi way are lited below: K K κ κ c c B B α γ Bulk modulu at contant temerature J/m 3 N/m 2 bulk modulu at contant entroy J/m 3 N/m 2 comreibility at contant temerature K m 3 /J comreibility at contant entroy K m 3 /J ecific heat at contant olume J/(kg K) ecific heat at contant reure J/(kg K) change in reure with reect to temerature at contant olume J/(m 3 K) change in reure with reect to temerature at contant entroy J/(m 3 K) olumetric thermal exanion coefficient at contant reure /K the Grüneien arameter dimenionle Material roertie are defined equal to (or roortional to) the deriatie of one tate ariable (,,, ) with reect to a econd tate ariable, holding a third tate ariable contant. In light of the relationhi lited in Eq. (5), only three of the aboe ten material roertie are indeendent all of the other can be comuted from them. A goal of thi document i to how you how to erform thee conerion between roertie. Secifically, if you hae a handbook that lit three roertie but you really want a different roerty, then you can comute it. Material roertie are ometime defined in term of econd deriatie of the energie with reect to their natural ariable. For examle, ince the ientroic bulk modulu i defined to be roortional to the loe of the ientroic reure-olume cure, we know it i roortional to ( ), which (uing the firt exreion in Eq. 5a) i equialent to ( 2 u 2 ). Each energy i exreible a a function of it two natural ariable. Any function of two ariable ha exactly three indeendent econd-artial deriatie. herefore natural grouing of three indeendent material roertie correond to the econd-artial deriatie of an energy. Since there are four energie ( uahg,,, ), material roerty trilet found in handbook uually correond to econd-artial of one of the energy function. * In general, adiabatic mean no heat flow i ermitted into our out from the ytem, while ientroic mean no entroy i generated. For general material, thee term mean different thing becaue, een under adiabatic condition, it i till oible to generate entroy ia irreerible material diiation (which i like internal heating from friction, a ooed to external heating ulied directly from an outide ource). hi effect can be modeled only by ermitting the energy function to deend on more than jut two tate ariable it mut additionally deend on other internal tate ariable. Een without material diiatie mechanim, you can till generate entroy under adiabatic condition by alying the load dynamically. hi document coer only claical thermotatic in which material diiation i zero and load are alied ery lowly. In thi cae, adiabatic and ientroic are ynonymou. Author: Rebecca M. Brannon. Coyright i reered. Coie may be made for indiidual ue, but not for rofit. Lat udated July 6, 205 0:59 am. COMMENS? Send your feedback (eecially tyo alert) tocomutationalolidmechanic@gmail.com 4

6 Author: Rebecca M. Brannon. Coyright i reered. Coie may be made for indiidual ue, but not for rofit. Lat udated July 6, 205 0:59 am. COMMENS? Send your feedback (eecially tyo alert) to comutationalolidmechanic@gmail.com Material roertie are roortional to (not alway identical to) deriatie of one tate ariable holding a third tate ariable contant. he roortionality factor are introduced merely a a conenience. For examle, the deriatie that define material roertie often contain negatie ign to enure that the definition will be oitie for mot material. Material roerty definition inoling differential of ecific olume uually contain a normalizing factor of the ecific olume itelf, which alter the meaning lightly from being an increment in olume to an increment in olumetric train, defined ε * ln( o ). Here, o i any contant reference olume; which goe away in differential form (i.e., dε d i indeendent of o ). With thi logarithmic definition of olumetric train, the exreion ( ) i equialent to ε. Material roerty definition inoling the differential of entroy are likewie uually multilied by becaue d i the heat increment (for reerible rocee). hi aid, the mathematical definition of the aboe thermodynamic roertie are lited below, categorized according to whether they characterize mechanical effect, thermal effect, or thermo-mechanical couling effect. Mechanical material roertie (reure-olume relationhi): K K a u κ K κ K unit J/m 3 2 J/m m 3 /J m 3 /J hermal material roertie (temerature-entroy relationhi): c c hermomechanical (couling) material roertie: ( 2 g 2 ) ( 2 h 2 ) a g α B B γ J/(kg K) u 2 ( ) J/(kg K) h g h ( ) /K J/(m 3 K) a J/(m 3 K) -- 2 u dimenionle * For mall olume change, and therefore o o ε ln( o ) o hu, the logarithmic train reduce to engineering train in thi cae. he logarithmic train i o ideal for generalization to large olume change becaue it goe to + at full exanion and at full comreion (zero olume). Engineering train doe not obey thi nice roerty. 5

7 hi roce ha eliminated the energy e from being exlicitly reent. If and q are tate ariable, then the remaining deriatie in thi exreion can be equated to material roertie (or the deriatie can be imlified uing Eq. 9 if two ariable haen to match each other), and you will hae ucceeded in fully ditilling your original deriatie into a form inoling readily meaurable quanti- hee exreion how how each material roerty i related to econd deriatie of energy otential. he hyically meaningful (ractical) definition, which are cited firt, would be ued to et u laboratory exeriment to meaure the roertie. he olumetric exanion coefficient α i the increment in olumetric train [ dε ( d) ] induced er unit change in temerature, meaured while holding the reure contant. Similarly, ince d equal the increment dε in olumetric train, the iothermal bulk modulu K i the loe of the reure. train cure that i meaured under contant temerature condition. he definition of K ue a negatie becaue, for mot material, an increae in olume uually correond to a decreae in reure, making K > 0. For a reerible roce, heat flow i roortional to the entroy roduction. Conequently, any deriatie that hold contant may be regarded a a meaurement taken under quaitatic inulated condition. For examle, K i the (negatie) loe of the reure-train cure that i meaured without ermitting heat to flow into or away from the ytem. For reerible thermoelaticity, the increment of heat (er unit ma) added to a ytem equal Δ ; therefore, the ecific heat ( c and c ) can be regarded a the amount of heat needed to induce a unit change in temerature in a unit ma the reult deend on whether the heat i added at contant olume or at contant reure, which i why there are two ecific heat. Ditilling deriatie he goal of thi document i to outline a neer-fail rocedure for you to conert any artial deriatie of the form ( A B) C into an exreion that inole only tate ariable and material roertie. If you encounter a deriatie ( A B) C in which one of the letter of the alhabet ( ABor,, C) i reeated, then you would aly one of the following A A C A B You will neer ee all three letter in a deriatie reeated. Secifically, A i meaningle. Mot of the time, you won t be o lucky to hae the ame ariable aearing twice in a deriatie, o imlification i more arduou. Recall that all of the material roerty definition inole the deriatie of one tate ariable with reect to a econd tate ariable, holding a third tate ariable contant. In other word, none of the material roerty deriatie exlicitly inole an energy ( uagh,,, ). Conequently, the firt tak i to eliminate energie from a artial deriatie. Gien a deriatie of the form ( A B) C, uoe that A i an energy. If the indeendent ariable (B and C) haen to be the ame a the energy natural ariable, then you can imly aly Eq. (6), and you re done. Equation (6) hold only when differentiating with reect to one of the energy natural tate ariable, holding the other natural ariable contant. Suoe that the natural ariable aociated with an energy e (i.e., either a, g, h or u ) are x and y, and you wih to imlify a deriatie of the form ( e ) q, where and/or q are not natural ariable for that energy. In thi cae, you would ue the chain rule o that you can imlicitly introduce the natural ariable function e e( x, y). Secifically e q e x y x q e y x y q Now that we hae deriatie inoling natural ariable, Eq. (6) may be ued to write e q x x q + y y q 0 A B B ( A A) A (9) (0) () Author: Rebecca M. Brannon. Coyright i reered. Coie may be made for indiidual ue, but not for rofit. Lat udated July 6, 205 0:59 am. COMMENS? Send your feedback (eecially tyo alert) tocomutationalolidmechanic@gmail.com 6

8 Author: Rebecca M. Brannon. Coyright i reered. Coie may be made for indiidual ue, but not for rofit. Lat udated July 6, 205 0:59 am. COMMENS? Send your feedback (eecially tyo alert) to comutationalolidmechanic@gmail.com tie. On the other hand, if and/or q i an energy, then more work remain. For each of the remaining deriatie, you mut ue multiariable calculu to recat them into a form that ut the energy in the numerator, at which oint you can aly a erion of Eq. () for each energy you wih to remoe. Gien a generic artial deriatie of the form ( A B) C two key calculu identitie are ued at thi tage. o change thing o that B (intead of A ) i differentiated, ue A B C ( B A) C o change thing o that C (intead of A ) i differentiated, ue A B C ( C B) A ( C A) B. (2). (cyclic identity) (3) Sidebar: he reence of the negatie ign in Eq. 3 might be confuing to reader who are ruty in multiariable calculu. After all, for ordinary ingle ariable calculu, eeryone know that dy dz dx. here i no negatie in thi equation, o why i there one in Eq. 3? he anwer dx dz dy reole around what i being held contant in deriatie. Suoe that y i a function of a econd ariable α o that y y( x, α). It i definitely true that y ( z x). hi equation ha no x α α ( z y) a negatie ign becaue α i held contant in all deriatie, o thi formula i effectiely making a tatement about a world in which α i alway contant in eery deriatie (o your faorite formula from ingle-ariable calculu till aly). Contrat thi reult with Eq. 3, which ha different thing held contant in all three deriatie. o get Eq. 3, you firt note that the ery act of writing ( A B) C imlie that, at leat in ome local neighborhood, A A( B, C). Imagine locally inerting thi relationhi to obtain C C( A, B) o that dc C C da db. In a world where C i AB BA contant, we know that dc 0 and thi incremental equation may be oled for da db to gie the right-hand-ide of Eq. 3, negatie ign and all. o emhaize that the reult alie when C i held da contant, you mut note that i really A db B C o reiterate, if an energy i the thing being differentiated, your firt tak i alway to aly Eq. () to get rid of it! If the remaining deriatie inole no energie, then you are done becaue they mut be exreible in term of material roertie and/or tate ariable. Otherwie, if you till hae artial deriatie inoling energie, then you need to ue Eq (2) or (3) to moe thoe energie o that they become the thing being differentiated, after which Eq. () can be alied to eliminate them. Alway follow the equence (firt eliminate from numerator, then denominator, then the held contant art) to roduce an exreion that inole only meaurable quantitie (material tate ariable and material roertie). Deiating from the equence will get you nowhere. he deriatie ditilling roce i eentially recurie. o exedite thi tage of the work, we hae roided comuter-generated recurion table that how you which artial deriatie identitie [Eq. 9,, 2 or 3] you need to ue. he recurion table roide formula for eery oible deriatie of the form ( A B) C that can be made uing the eight thermodynamic ariable ( uagh,,,,,,, ). 7

9 Intruction for uing the recurion table: In the table to follow, any three-character ymbol of the form ABC i a hort-hand notation for ( A B) C, which i the deriatie of A with reect to B holding C contant. he three-column comuter-generated recurion table (anning age 9 and 0) ermit you to reformulate any thermodynamic deriatie of the form ( A B) C o that it i ultimately hraed only in term of tate ariable and material roerty deriatie (i.e., in term of meaurable thing ). he firt te i to recuriely aly thi three-column table until it roide no further imlification. hen the Material roerty recurion table on age i alied to exre the reult in term of whateer et of material roertie you hae aailable in a handbook. A wort-cae cenario. Recall that ditilling deriatie require uing identitie and thermodynamic relationhi to re-write a deriatie into a form that inole no exlicit reence of energie. herefore, the mot difficult deriatie to ditill would be the artial deriatie of an energy with reect to an energy, holding an energy contant. Suoe, for examle, you wih to exre the deriatie ( a g) h in term of tate ariable and material roertie. Firt write thi deriatie in our horthand notation a agh. he firt table tell you that agh equal - gh - gh, which tranlate into more conentional notation a ( a g) h ( g) h ( g) h. hi entry i imly alying Eq. () to eliminate the Helmholtz free energy a from being exlicitly reent. Your new exreion till inole ome different energie ( g and h ) in the deriatie gh and gh, o you mut go back to the table and look them u. You will find that the table cite entrie that aly Eq. (2) to moe the energy g o that it become the thing being differentiated. Again alying the table lead to yet another alication of Eq. (), thi time to remoe the Gibb function g from being exlicitly reent. At thi tage, you will hae deriatie inoling the enthaly h held contant. Looking u thee deriatie in the table gie entrie that aly Eq. (3) to moe h o that it become the thing differentiated. Alying the table one lat time gie entrie that aly Eq. (), after which the three-column comuter-generated table roduce no further change. By back ubtitution, you will hae ucceeded in exreing the original deriatie in term of tate ariable and rimitie material roerty deriatie. Examle Suoe that you wih to ditill the deriatie of the Helmholtz free energy with reect to entroy holding reure contant, ( a ). In other word, uoe that you deire to exre thi deriatie in term of meaured material roertie and the thermodynamic tate. hi deriatie, ( a ), i denoted a in our notation. he firt table (tarting on age 9) ay a - - he firt table roide no alteration of either or. Hence, they are rimitie material deriatie, and they may be looked u in the econd table on age, which ay that c B hu, by back ubtitution, a c B With thi, we hae achieed our goal of exreing the original deriatie in term of tate ariable (,, and ) and material roertie ( and ). c B (4) Author: Rebecca M. Brannon. Coyright i reered. Coie may be made for indiidual ue, but not for rofit. Lat udated July 6, 205 0:59 am. COMMENS? Send your feedback (eecially tyo alert) tocomutationalolidmechanic@gmail.com 8

10 Author: Rebecca M. Brannon. Coyright i reered. Coie may be made for indiidual ue, but not for rofit. aag aah aa aa aa aau aa aga 0 agg Infinity agh - gh - gh ag - g - g ag - g - g ag - g agu - gu - gu ag - g aha 0 ahg - hg - hg ahh Infinity ah - h - h ah - h - h ah - h ahu - hu - hu ah - h aa 0 ag - g - g ah - h - h a Infinity a - - a - au - u - u a - aa 0 ag - g - g ah - h - h a - - a Infinity a - au - u - u a - aa 0 ag - - g ah - - h a - - a - - a Infinity au - - u a - aua 0 aug - ug - ug auh - uh - uh au - u - u au - u - u au - u auu Infinity au - u aa 0 ag - g - ah - h - a - - a - - a - au - u - a Infinity gaa Infinity gag 0 gah - ah + ah ga - a ga - a + a ga a gau - au + au ga - a + a gga ggh gg gg gg ggu gg gha - ha + ha ghg 0 ghh Infinity gh - h gh - h + h gh h ghu - hu + hu gh - h + h ga - a + gg 0 gh - h + g Infinity g - + g gu - u + g - + ga - a + a gg 0 gh - h + h g - g Infinity g gu - u + u g - + ga - + a gg 0 gh - + h g - g - + g Infinity gu - + u g - + gua - ua + ua gug 0 guh - uh + uh gu - u gu - u + u gu u guu Infinity gu - u + u ga - a + a gg 0 gh - h + h g - g - + g gu - u + u g Infinity haa Infinity hag ag + ag hah 0 ha a ha a ha a + a hau au + au ha a + a hga ga + ga hgg Infinity hgh 0 hg g hg g hg g + g hgu gu + gu hg g + g hha hhg hh hh hh hhu hh ha a + hg g + hh 0 h Infinity h h + hu u + h + ha + a hg + g hh 0 h h Infinity h + hu + u h + ha a + a hg g + g hh 0 9 h h h Infinity hu u + u h + hua ua + ua hug ug + ug huh 0 hu u hu u hu u + u huu Infinity hu u + u ha a + a hg g + g hh 0 h h h + hu u + u h Infinity aa Infinity ag / ag ah / ah a 0 a / a a / a au / au a / a ga / ga gg Infinity gh / gh g 0 g / g g / g gu / gu g / g ha / ha hg / hg hh Infinity h 0 h / h h / h hu / hu h / h a g h u a - a / a g - g / g h - h / h 0 Infinity / u - u / u a - a / a g - g / g h - h / h 0 / Infinity u - u / u ua / ua ug / ug uh / uh u 0 u / u u / u uu Infinity u / u a - a / a g - g / g h - h / h 0 u - u / u Infinity

11 Author: Rebecca M. Brannon. Coyright i reered. Coie may be made for indiidual ue, but not for rofit. aa Infinity ag / ag ah / ah a / a a 0 a / a au / au a / a ga / ga gg Infinity gh / gh g / g g 0 g / g gu / gu g / g ha / ha hg / hg hh Infinity h / h h 0 h / h hu / hu h / h a - a / a g - g / g h - h / h Infinity 0 u - u / u / a g h u a - a / a g - g / g h - h / h 0 Infinity u - u / u ua / ua ug / ug uh / uh u / u u 0 u / u uu Infinity u / u a - a / a g - g / g h - h / h / 0 u - u / u Infinity aa Infinity ag / ag ah / ah a / a a / a a 0 au / au a / a ga / ga gg Infinity gh / gh g / g g / g g 0 gu / gu g / g ha / ha hg / hg hh Infinity h / h h / h h 0 hu / hu h / h a - a / a g - g / g h - h / h Infinity 0 u - u / u / a - a / a g - g / g h - h / h Infinity 0 u - u / u a g h u ua / ua ug / ug uh / uh u / u u / u u 0 uu Infinity u / u a - a / a g - g / g h - h / h / 0 u - u / u Infinity uaa Infinity uag ag - ag uah ah - ah ua a - a ua - a ua a - a uau 0 ua a uga ga - ga ugg Infinity ugh gh - gh ug g - g ug - g ug g - g ugu 0 ug g uha ha - ha uhg hg - hg uhh Infinity uh h - h uh - h uh h - h uhu 0 uh h ua a - a ug g - g uh h - h u Infinity u - u - uu 0 u ua - a ug - g uh - h u - u Infinity u - uu 0 u ua a - a ug g - g 0 uh h - h u - u - u Infinity uu 0 u uua uug uuh uu uu uu uu ua a - ug g - uh h - u - u - u - uu 0 u Infinity aa Infinity ag / ag ah / ah a / a a / a a / a au / au a 0 ga / ga gg Infinity gh / gh g / g g / g g / g gu / gu g 0 ha / ha hg / hg hh Infinity h / h h / h h / h hu / hu h 0 a - a / a g - g / g h - h / h Infinity u - u / u 0 a - a / a g - g / g h - h / h Infinity / u - u / u 0 a - a / a g - g / g h - h / h / Infinity u - u / u 0 ua / ua ug / ug uh / uh u / u u / u u / u uu Infinity u 0 a g h u

12 Author: Rebecca M. Brannon. Coyright i reered. Coie may be made for indiidual ue, but not for rofit. Lat udated July 6, 205 0:59 am. COMMENS? Send your feedback (eecially tyo alert) to comutationalolidmechanic@gmail.com Material roerty recurion table Helmholtz Gibb Enthaly Energy (, ) (,) (,) (,) K, c, B κ, c, α κ, c, B K, c, γ / / -/ -*/ / / -/ γ u */ -/ / / -/ / / B 2 a */ / / K u / / -*/ K a / -/ / α g / -*/ -/ / -*/ / / c g */ / / c a */ / / -/ / / -/ B 2 a / -/ h / B / -/ -*/ / / / -*/ c h / / -*/ c u / / / -*/ -/ / / γ u / -*/ / κ h / -*/ / κ g / / h -/ B / / -*/ -/ -/ / / α 2 g / -*/ / / u internal energy ecific olume Remember thee relationhi: a Helmholtz free energy temerature κ K g Gibb free energy h enthaly entroy reure κ K

13 Examle 2 Suoe you wih to ditill the deriatie of the temerature with reect to internal energy, holding olume contant, ( u), or u in our table notation. he firt table (tarting on age 9) ay u / u u Back ubtitution gie u / ( ) he line trigger moing to the material roerty table on age, which lit. c hu, back ubtitution gie the final reult: u Examle 3 Suoe you wih to ditill the deriatie of the temerature with reect to internal energy, holding enthaly contant, ( u) h, or uh in our table notation. he firt table (tarting on age 9) ay Uing the table again gie Uing the table again gie One more time gie c uh /uh. uh h - h. h - h / h and h - h / h. h, h+, h +, and h + he firt table now gie no further imlification becaue all energie (u and h) hae been remoed from all deriatie. Back ubtitution gie uh /(- / (+ ) + ( + )/( + )) hi reult i rather ugly becaue the tarting deriatie, uh, inoled two energie. Neerthele, by uing the firt table, we hae conerted to a form that inole no energie. Uing the econd table, each of the energy-free deriatie may be exreed in term of material roertie. For examle, / B. (5) Author: Rebecca M. Brannon. Coyright i reered. Coie may be made for indiidual ue, but not for rofit. Lat udated July 6, 205 0:59 am. COMMENS? Send your feedback (eecially tyo alert) tocomutationalolidmechanic@gmail.com Similarly alying the table on age for the remaining energy-free deriatie lead to the final exreion of ( u) h in term of tate ariable and roertie. 2

14 Author: Rebecca M. Brannon. Coyright i reered. Coie may be made for indiidual ue, but not for rofit. Lat udated July 6, 205 0:59 am. COMMENS? Send your feedback (eecially tyo alert) to comutationalolidmechanic@gmail.com Relationhi between roertie Recall that the reult in Eq. (4) wa exreed in term of the material roertie c and B. he ecific heat at contant reure might be aailable from ome material handbook, but other handbook are likely to lit alue for c intead. he ientroic thermal tre coefficient B i a quirky material roerty that i not likely to be lited in any handbook. Een though we hae defined a total of ten material roertie in thi document, only three are indeendent. Material handbook will tabulate three roertie, and it will be your job to comute other (deendent) roertie a needed. How do you aign alue to the material roertie c and B K, that aear in Eq. (4) if you hae a handbook that tabulate only the iothermal bulk modulu, the ecific heat at contant olume and the linear exanion coefficient αlinear? he anwer i c c c + 9( αlinear ) 2 K B 3α linear c K (7) 3αlinear he roce ued to derie uch formula i the ubject of thi ection. More often than not, the three roertie lited in your faorite handbook will all belong to one of the four grouing in the material roerty recurion table. If, for examle, your handbook lit alue for the exanion coefficient α, the contant reure ecific heat c, and the iothermal comreibility κ, then your handbook uort the Gibb grou of roertie (econd column in the roerty recurion table). If you eek the alue for a roerty in a different column, then you need to equate the entry for that roerty to the entry in the Gibb column of the roerty recurion table. hereafter, you tay in the Gibb column, recuriely imlifying until your non-gibb roerty i exreed in term of the Gibb roertie aailable in your handbook. Suoe, for examle, you eek the alue of, exreed in term of Gibb roertie. Firt go to the material roerty recurion table and locate any exreion inoling c. hen equate it to the exreion in the Gibb column. For examle, the roerty recurion table ay in the Helmholtz column (8) c -*/ in the Gibb column. (9) Staying in the Gibb column (becaue Gibb roertie are reumed to be aailable), the roerty table tell u c α α κ. Back ubtituting thee four reult into Eq. (9) gie c α from the Gibb column. (24) κ Equating thi reult with Eq. (8) and oling for c c gie (6) (20) (2) (22) (23) α2 c c κ J/kg K (25) 3

15 NOE: ecific heat ha been defined in thi document to equal the amount of heat needed to induce a unit temerature change in a unit ma. Similarly, the ecific olume i the olume er unit ma. For a finite olume V of ma M, V M, o you might ee the aboe reult exreed a Vα2 c c (26) Mκ CAUION: Many book define ecific heat to be the amount of heat needed to induce a unit temerature increae in a unit mole of material, not a unit ma a we hae done. Let x be one of our er unit ma roertie. Let an aterik denote the analogou molar (er unit mole) roerty. hen x * x( M N), where N i the number of mole and M i the ma. Multilying both ide of Eq. (26) by M N conert the ecific heat er unit ma to ecific heat er unit mole, and the lat term that inoled diiion by M change to diiion by N. Hence, Eq. (26) might aear in ome textbook a, J/mol K (27) where, a mentioned, an aterik denote the er mole erion of the roerty. he key i to ay ery cloe attention to how your reference book define a material roerty. You might need light adjutment like thee to ue the handbook roertie. Checking unit i eential. he thermal exanion coefficient i another examle of a differently defined roerty. In thi document, we defined the olumetric thermal exanion coefficient α to be the olumetric train reulting from a unit temerature change, holding reure contant. Many book will intead tabulate the linear exanion coefficient αlinear, which i the length train er unit temerature change at contant reure. Conider a cube with dimenion L o L o L o that i then heated under contant reure o that it exand to new cube dimenion L L L. he olumetric train i the log of the olume ratio he linear train i the log of the length ratio herefore, Vα2 c * c * Nκ ε ε lin ε ln----- V ln ln----- L V o ln----- L. L o. L 3 L3 o. (28) L o In other word, a gien linear train will roduce a olumetric train that three time a large if the ame linear train occur in all three atial direction. herefore, if you hae a handbook that lit the linear exanion coefficient, you can conert it to the olumetric exanion coefficient by triling: α 3ε lin 3α linear (29) (30) /K (3) Some other ueful roerty relationhi can be readily deduced from the roerty recurion table. he comreibilitie κ and κ are not a oular a the bulk moduli K and K. How are thee related? hi quetion i again anwered by finding κ and K in the roerty recurion table and erforming cro-correlation. hat table ay Author: Rebecca M. Brannon. Coyright i reered. Coie may be made for indiidual ue, but not for rofit. Lat udated July 6, 205 0:59 am. COMMENS? Send your feedback (eecially tyo alert) tocomutationalolidmechanic@gmail.com κ in the Gibb column (32) 4

16 Author: Rebecca M. Brannon. Coyright i reered. Coie may be made for indiidual ue, but not for rofit. Lat udated July 6, 205 0:59 am. COMMENS? Send your feedback (eecially tyo alert) to comutationalolidmechanic@gmail.com / in the Helmholtz column (33) Staying in the Helmholtz column the table ay. hu, by back ubtitution, the aboe three equation imly Similarly, Another le-common material roerty i the thermal reure coefficient B. According to it definition, thi arameter meaure the reure increae induced by a unit temerature change erformed under contant olume condition. How i B related to more commonly aailable roertie? he roerty table ay in the Helmholtz column (37) -/ in the Gibb column (38) Staying in the Gibb column, the table ay α in the Gibb column (39) κ in the Gibb column (40) Back ubtituting thee relationhi lead to the formula or κ B K κ α K -----, κ B K B K α. (42) In other word, the thermal reure coefficient i imly the olumetric exanion roerty time the iothermal bulk modulu. hi make a lot of ene becaue, under a unit temerature change, you can imagine letting a amle firt exand at contant reure (roducing a train α ), and then recomreing iothermally (o that the temerature change will be the ame) back to the original olume. he reure required to do thi i the iothermal bulk modulu time the train, a indicated in Eq. (42). Going back to the original olume i needed becaue the B i defined to be the reure change holding olume contant. he Grüneien arameter might be new to you. hi material roerty and the other energetic roertie (i.e., thoe lited in the energy column of the roerty table) are often ued in acoutic. he energetic roertie are imortant in acoutic wae motion becaue ound wae trael o fat that there i not ufficient time for heat to conduct away from the ytem (i.e., entroy i contant * ). wo different roerty et (enthalic and energetic) both hae entroy a a natural ariable. hen (34) (35) (36) (4) * Acoutic wae are low amlitude wae. Hence, een though they are dynamic, they diturb the material only ery lightly and the aociated entroy roduction i negligible. High amlitude (hock) wae, on the other hand, roduce coniderable entroy een though they are adiabatic. 5

17 why are energetic roertie more commonly ued in acoutic? he anwer i that olume i treated a the indeendent ariable in mot wae mechanic code. In other word, mot material model take the olume change a inut and redict the reure change a outut. hu, the controlled tate ariable are the internal energy natural ariable, and. Getting back to the meaning of the Grüneien arameter γ, note that the roerty table tell u that γ in the (internal) energy column (43) / in the Helmholtz column (44) Staying in the Helmholtz column, the roerty table ay in the Helmholtz column (45) in the Helmholtz column (46) hu, thee equation imly γ B c, dimenionle (47) or, noting that ρ, where ρ i the ma denity, and alo uing Eq. (42), γ K α ρc Note from the roerty table that. dimenionle (48) γ ρ [ ln( 0 )] , (49) ρ [ ln( ρ ρ 0 )] where ρ i the ma denity, while 0 and ρ 0 are reference alue (at the beginning of an exeriment or at a tandard tate). he Grüneien arameter quantifie enitiity of temerature to olume change under ientroic condition. he fact that the Grüneien arameter i defined in term of logarithm ugget that, for real material, the ientroic temerature-denity relationhi tend to be a traight line on log-log cale. If the relationhi i not a traight line, it merely mean that the Grüneien arameter (i.e., the local loe in thi log-log lot) in t a contant. he alue of the Grüneien arameter i tyically in the neighborhood of.0. Uing the recurion roerty table, you can roe the following mixed roerty relationhi: K K B B c ---- c ρc α B c K B B K K α (mnemonic: ubcrit alhabetical in each ratio) (50) (5) (52) (53) Author: Rebecca M. Brannon. Coyright i reered. Coie may be made for indiidual ue, but not for rofit. Lat udated July 6, 205 0:59 am. COMMENS? Send your feedback (eecially tyo alert) tocomutationalolidmechanic@gmail.com For quick reference, the following age ummarize formula that allow you to comute all ten material roertie if you hae a handbook citing three roertie. 6

18 Author: Rebecca M. Brannon. Coyright i reered. Coie may be made for indiidual ue, but not for rofit. Lat udated July 6, 205 0:59 am. COMMENS? Send your feedback (eecially tyo alert) to comutationalolidmechanic@gmail.com Gien Helmholtz roertie K, c, B : K c B Bulk modulu at contant temerature J/m 3 ecific heat at contant olume J/kg K change in reure with reect to temerature at contant olume J/m 3 K he other (non-helmholtz) roertie are found by B 2 K K K κ ρc κ K K B K ρc K c c K B K α ρc B B K γ α ρc Gien Gibb roertie κ, c, α : κ c α comreibility at contant temerature m 3 /J ecific heat at contant reure J/kg K olumetric thermal exanion coefficient at contant reure /K he other (non-gibb) roertie are found by Gien enthalic roertie κ, c, B : comreibility at contant entroy m 3 /J ecific heat at contant reure J/kg K change in reure with reect to temerature at contant entroy J/m 3 K he other (non-enthalic) roertie are found by Gien energetic (internal energy) roertie K, c, γ : bulk modulu at contant entroy J/m 3 ecific heat at contant olume J/kg K the Grüneien arameter dimenionle he other (non-energetic) roertie found by K K α κ κ κ K K ρc ---- α κ K ρc κ c c K ρc B K K α B B γ α ρc κ c B K K ρc ---- κ κ κ K K B ρc κ K B 2 κ c c K ρc K α B B B K α γ ρc K c γ K K ργ 2 c κ κ K K K K ργ 2 c K c c K K B ργc B B α γ K K 7

19 hermodynamic conitency and inconitency hermodynamic conitency mean that a theoretical model (good or bad) i imlemented in uch a way that all thermodynamic deriatie identitie hold. o illutrate the concet, we will conider a contried examle of a model that i thermodynamically inadmiible. Suoe that laboratory exeriment conducted at contant temerature ugget that reure i a linear function of train, where (recall) train i defined a ε ln( 0 ). Further uoe that the loe of the reure-train lot deend on the temerature at which the exeriment i conducted. In other word, the iothermal bulk modulu, K , (54) ε i a function of temerature, but not a function of train. If a numerical thermoelaticity model i already aailable in which the all material roertie ( K, α, c ) are contant, handling a temerature-deendent bulk modulu might eem to be a imle matter of modifying the model to et the alue of K according to the current temerature. It might eem that no further code reiion would be required. Howeer, uch a model would be thermodynamically inadmiible, a we will now how. Uing Eq. (54), the deendence of the bulk modulu on temerature i quantified by the following mixed artial deriatie: where dk d, (55) B (56) ε Noting from the material roerty table that B K α, and recalling that K deend only on temerature in our contried examle, Eq. (55) become Equialently, ε ε dk α K d ε α ε dk d K B ε. (for thi contried examle) (57). (for thi contried examle) (58) he right hand ide i nonzero by remie. herefore, the left hand ide mut be nonzero a well. In other word, temerature deendence of the bulk modulu require train deendence of the thermal exanion coefficient. Uing a contant α would reult in a thermodynamically inadmiible model. he diiion by K in Eq. (58) might make it eem that ( α ε) i negligible, but one mut inect the goerning equation, where it can be een that term inoling train deendence of α are comarable in order of magnitude to term inoling temerature deendence of. ε reure-train reone at ariou temerature. K Author: Rebecca M. Brannon. Coyright i reered. Coie may be made for indiidual ue, but not for rofit. Lat udated July 6, 205 0:59 am. COMMENS? Send your feedback (eecially tyo alert) tocomutationalolidmechanic@gmail.com 8

20 Finding fundamental energy otential Author: Rebecca M. Brannon. Coyright i reered. Coie may be made for indiidual ue, but not for rofit. Lat udated July 6, 205 0:59 am. COMMENS? Send your feedback (eecially tyo alert) to comutationalolidmechanic@gmail.com We hae mentioned that u (, ) i a fundamental otential function. Determining thi function from laboratory data uually entail erforming exeriment in which tate ariable (not energie) are controlled. Change in tate ariable (not energie) are meaured. In other word, u (, ) i neer meaured directly intead, it mut be inferred from tractable data. You might, for examle, meaure how reure arie with ecific olume under inulated condition. You might additionally hae meaurement of the temerature hitory induced by heating in contant olume condition, which (becaue the heat increment d for reerible thermoelaticity) i eentially a family of relationhi between entroy and temerature for each fixed olume. hee indiidual laboratory-meaured relationhi between tate ariable are called equation of tate (EOS). For generalized material model that include hear tre, ractitioner often ue EOS to mean the relationhi between reure, olume, temerature, and entroy, while relationhi between hear tre and hear train (a well a yield, fracture, etc.) are referred to a the contitutie model. hi i an unfortunate corrution of terminology becaue it aume that deiatoric (hear) reone can be decouled from iotroic (reure-olume-temerature) reone. If an aniotroic material uch a a fiber-reinforced comoite i ubjected to an iotroic increae in ize (with no change in hae), the tre change i not iotroic there i a larger tre required in the fiber direction. We refer that EOS mean any relationhi between meaurable tate ariable, with no exlicit reence of an energy. For inicid fluid, a fundamental otential function can be found wheneer you hae two indeendent EOS equation inoling the four tate ariable (,,, ): Gien a ytem of two indeendent equation (uually laboratory data), inoling the four tate ariable (,,, ), the fundamental otential are found a follow: o get u (, ), ole the ytem for and a function of and. hen integrate u (, ) and u (, ) o get a (, ), ole the ytem for and a function of and. hen integrate a a and (, ) (, ) o get g (, ), ole the ytem for and a function of and. hen integrate g g and (, ) (, ) o get h (, ), ole the ytem for and a function of and. hen integrate h h and (, ) (, ) Kee in mind: when integrating a artial deriatie, the integration contant i actually a function of the quantity held contant in the artial deriatie. Once one of the energy otential i found, the other energie may be found by uing Eq. (4). You mut exre the reult in term of natural ariable for the function to be a fundamental otential. 9

21 he entroic fundamental otential Recall that ( u ). herefore, auming that temerature i alway oitie *, the loe of u lotted a a function of (for any fixed alue of ) i eerywhere oitie, imlying that the relationhi i globally inertible for a a function of u (and ). Conequently, not only i u (, ) a fundamental otential, o i u (, ). When u (, ) i a fundamental otential, internal energy become a natural indeendent ariable. herefore, a different thermodynamic quare alie a hown. In the reiou ection, we conjectured that two indeendent equation (reumably lab oberation) were aailable inter-relating (,,, ), in which cae you could immediately obtain the fundamental otential for internal energy u. Now uoe that you hae two indeendent equation [meaured or theoretical] aailable inoling ( u,,, ). In thi cae, you hould eek u (, ) a a fundamental otential. When ( u,,, ) are the ariable, the two indeendent equation are called entroic equation of tate (not becaue entroy aear anywhere but becaue thee ariable imly that it i the entroic fundamental otential i mot releant). In thi cae, you mut ole the ytem for and a function of u and. hen integrate and u (, ) (59) u u (, ) u u (, ) Once u (, ) i found, it may (if deired and if tractable) be inerted to obtain the energetic fundamental otential u (, ). In ractice, engineer need a thermodynamically conitent model when only a a reure-olume cure i aailable. hi i one equation inoling and, but finding a fundamental otential require a econd equation. When faced with a dearth of data like thi, it i common for the contitutie modeler to imly hyotheize that the internal energy arie in roortion to temerature, where the contant of roortionality i regarded a a material roerty (to be determined by tuning the model a data later become aailable, although it would be unethical to tune the model differently for each different exeriment, a getting different alue would inalidate the hyotheized equation). EXAMLE: IDEAL GAS. For an ideal ga, the entroic equation of tate are Here, c i a material contant and R nr u, where n N M i the number of mole er unit ma and R u i the unieral ga contant [ R u 8.3 J/(mol K)]. hi i a ytem of two equation inoling ( u,,, ). What i the entroic fundamental otential? Soling the ytem for and a function of u and, and then ubtituting the reult into Eq. (59) gie Integrating the econd equation with reect to gie and R --. (ideal ga) (6). (ideal ga) (62) * hi i ometime regarded a an aumtion rather than immutable truth becaue [a mentioned in a footnote of hyic: art II by Halliday and Renick] ome material can be laced into an excited tate where the quantum definition of temerature gie negatie alue. hi tate i not reached by aing continuouly through zero. Intead, temerature jum from oitie to negatie ia an inerion in the quantum tructure. In thi tate, the other equation of macrocale thermotatic continue to hold if they are roerly rehraed to allow for negatie temerature. In articular, the econd law inequality need to hae temerature in the denominator multilying both ide by temerature to get require changing the direction of the inequality. Δ u a u g -- g/ R and u cr. (ideal ga) (60) u cr u Rln+ fu ( ) u -- Author: Rebecca M. Brannon. Coyright i reered. Coie may be made for indiidual ue, but not for rofit. Lat udated July 6, 205 0:59 am. COMMENS? Send your feedback (eecially tyo alert) tocomutationalolidmechanic@gmail.com 20