HE HERMOELASIC SQUARE A mnemonic for remembering thermodynamic identitie he tate of a material i the collection of variable uch a tre, train, temperature, entropy. A variable i a tate variable if it integral over any cloed path in the other tate variable i zero. Clearly, thi definition i circular unle we aume there exit a minimal et potulated tate variable. For example, the zeroth law of thermodynamic potulate the exitence of a temperature. he preure i uually potulated to be a tate variable. For typical application involving invicid fluid, only thee two tate variable temperature preure are needed in order to determine the other tate variable. For olid, more tate variable are needed. he tate potulate of thermodynamic ay that the minimum number of tate variable needed to determine all other tate variable i equal to one plu the number of quaitatic work mode. For a ga, there i only one work mode, namely pdv where preure p work to produce a volume change dv. For a more general material, there are a total of ix work mode, correponding to the ix independent component of tre cauing change in the ix component of train. In addition to the ix tate variable that characterize tre or train, the tate potulate ay there mut be one more variable uch a temperature or entropy that characterize the thermal tate of the material. Suppoe we take train entropy to be our primitive tate variable. Such a material i aid to be thermoelatic, the firt law of thermodynamic implie that the pecific tre tenor (i.e., tre divided by denity) i given by a derivative of the internal energy function with repect to train, while the temperature i the derivative of energy with repect to entropy. he relationhip between tre train doe not have to be linear the principal characteritic of thermoelaticity i that tre mut be expreible a a true function of train entropy. In what follow, we peak of the tre tenor a a ingle entity rather than a collection of ix individual component (jut a, ay, velocity i regarded a a ingle vector rather than a collection of three component). Likewie, the train tenor i a ingle entity. hu, the tate potulate ay, given a tenor variable (tre or train) a thermal variable (temperature or entropy), then there exit equation of tate uch that all of the other tate variable may be computed from thee two primitive independent tate variable. hermodynamical identitie uch a the Legendre tranformation the Maxwell Gibbian relation have nothing to do with thermodynamic they are imply relationhip that hold whenever you work with any function of two variable. When one of thee variable i a tenor, ome identitie take lightly modified form involving the tenor inner product intead of ordinary calar multiplication.
Notation baic thermodynamical concept. o emphaize the connection with ordinary PdV identitie, the pecific tre tenor (i.e., tre divided by denity) i denoted the train tenor i denoted. he numerical plu/minu ign in our identitie may differ from what you are accutomed to becaue our tre our train are both poitive in tenion. For invicid fluid, the increment in mechanical work per unit ma i given by Pdv, where P i the preure v i the pecific volume (i.e., volume per unit ma, which i imply the invere of the denity ρ). he mot ueful meaure of volumetric train i ε v ln( v v o ), for which the material time rate i ε v v v. With thi logarithmic train meaure, derivative of the form vd( [ ) dv] become imply d( ) dε v the pecific work increment become P Pdv vpdε v --- dεv ρ Note that the pecific work increment i a pecific tre (i.e., preure divided by denity) time a train increment. When generalizing to olid vicou fluid which have nonzero hear tree, the increment in work i given by, where i the econd Piola Kirchhoff (PK2) tre divided by reference denity :d ρ, i Lagrange o train. he double dot i the tenor inner product defined between any two tenor,, a A ij B ij. Incidentally, a different calar product defined by A à hould be B avoided à :B ij B ji becaue it i not an inner product (it fail the poitivity axiom). he work increment i dw. Here, the ymbol i an inexact differential, indicating that the integral of work over :d d a cloed path in tate pace i not necearily zero. hu work W i not a tate variable. If dq denote the heating increment, then firt law of thermodynamic tate that the um of the work heat inexact differential i itelf an exact differential. Hence there mut exit a tate variable the internal energy u uch that du dw + dq he entropy i defined uch that d dq + dd, where i temperature dd i the increment in diipation. A reverible proce ha zero diipation, o the firt law become du :d + d he form of thi equation ugget (but doe not require) that the internal energy u may be regarded a a function of train entropy. If thi thermoelatic aumption i adopted, then it follow that
Subcript indicate which variable i being held contant in the partial derivative. he indicial form of the firt equation i P ij ( u v ij ). Material propertie are frequently defined in term of econd derivative of the energy function. For example, the (pecific) fourth-order ientropic elatic tiffne tenor i the derivative of (pecific) tre with repect to train holding entropy contant, o it ha component ( P ij v kl ) ( 2 u v ij v kl ), which we write in ymbolic notation a ------- 2 u --------. he (pecific) iothermal elatic tiffne i the derivative of with repect to holding temperature contant. It relationhip to the ientropic tiffne may be determined through application of the chain rule: ------- ------- + ------- By uing combination of Maxwell Gibbian relation, the lat term may be expreed in term of the Gruneien or thermal expanion tenor the contant train pecific heat, both of which are regarded a meaurable propertie. hu knowledge of the Maxwell Gibbian relation i eential whenever one wihe to find connection between one material property another. he thermoelatic quare help you remember thee relationhip.
Irreverible procee. Recall that the firt law may be written in the form du only for reverible procee. Many author claim that thi equation :d + d hold even in the preence of diipation. In thi cae, the tre train are not the real tre real real real train. Intead o-called thermodynamic tre train mut be defined o that. Conider, for example, platic flow. he :d real real dd :d real thermodynamic tre i choen to equal the real tre:. he train increment i broken into elatic plu platic part: real e + p. For platicity, the diipation i d d d dd. Conequently, the thermodynamic train increment mut be defined uch that :d p. hu, for platic deformation, the equation contained in the thermoelatic :d :d e quare are valid o long a the train i regarded a the elatic train, not the total train! hi make ene intuitively ince the mechanical part of internal energy correpond to the tretching of the elatic lattice (which i characterized by e ). he thermal part of the internal energy come from ordinary external heating from diipation aociated with irreverible platic flow (characterized by p ) of material through the lattice. d Other energy meaure. If we define a new variable h u, called : enthalpy, then the firt law ( du ) may be alternatively written a :d + d dh + d. By uing enthalpy, we have tranformed the firt law into a form :d where pecific tre entropy appear to be the natural independent variable, which might be more convenient for certain application uch a tre-controlled loading. Other energy meaure uch a Gibb energy Helmholtz free energy are more helpful in other ituation. he thermodynamic quare provide a mean of () remembering how thee alternative energy meaure are related to each other (2) recalling the natural independent variable aociated with the energie, (3) recalling how derivative of the energy meaure are related to other tate variable or to other derivative.
How to ue the thermoelatic quare. he thermoelatic quare (hown on the next page) contain key thermoelatic relationhip in recognizable pattern. he eaiet way to learn the quare i to deduce for yourelf how the identitie all obey conitent geometrical pattern in the quare. he numerical plu/minu ign are alway related in ome manner to the arrow on the thermoelatic quare. he paragraph below provide further detail in cae the pattern are not evident to you. he edge of the thermoelatic quare tell you the four energie their natural independent variable. For example, internal energy u i naturally a function of entropy train. Enthalpy h i naturally a function of entropy,, pecific tre,. he contact (Legendre) tranformation that relate one energy to another run parallel to the arrow in the thermoelatic quare. (For example, u a i parallel to the arrow connecting, o u ). o find the derivative of an energy with repect to one of it natural variable (holding the other one contant), imply travel from the differentiation variable (which alway lie at a corner near the energy) to the diagonally oppoite point acro the quare. he plu/minu i aigned according to whether you move with (+) or againt ( ) the arrow. For example, note that i oppoite the arrow point from to. herefore we have h +, u + g ------. Similarly, h ------- a ------. Maxwell relation involve derivative of a variable at a corner of the quare with repect to an adjacent corner variable. On the thermoelatic quare, the derivative of one corner variable with repect to another i equal to plu or minu the derivative of the mirror image of thee two variable. he mirror image of acro the horizontal i wherea the mirror image of acro the vertical i. he numerical ign ued in the Maxwell relation i poitive if the two arrow on the thermoelatic quare are ymmetric in the ame mirror image direction. he variable held contant in the derivative i found by continuing the equence to the next corner. For example, ( ) involve variable on the far left edge, o the contant i then next corner after travelling from to. he mirror image derivative ( ) involve variable on the far right edge. he two arrow on the thermoelatic quare are not ymmetric acro thi horizontal mirror direction, o the ign mut be negative. hat i, ( ) ( ). he formula for the econd derivative obey omewhat more complicated pattern, a you can deduce yourelf. With a little bit of practice, you will find the thermoelatic quare to be an extremely valuable memory aid. We hope you find it helpful!
hermoelatic quare Energie iotropic tate tenor tate u(, ), internal v, pecific volume --, train tenor* ρ a(, ), Helmholtz p, tenile preure, pecific tre* u a g g(, ), Gibb, temperature, temperature h(, ), enthalpy, entropy, entropy h *For iotropic,, poitive in tenion. 3 -- ln( v v o ) Ĩ pvĩ Otherwie, i Lagrange train i PK2 tre divided by ρ o. Conjugate relation a g ------- h ------- u : a ------ g ------ Gibbian relation ȧ : + ṡ Ṫ ġ Ṫ : h ḣ + ṡ : + ------ ------- + ------- contact (Legendre) tranformation u a h g u h a g : Maxwell relation 2 u ------------ + ------- 2 a -------------- + ------- 2 g --------------- ------ 2 h ------------- -6 of 7- Rebecca Brannon, rmbrann@ia.gov /home/rmbrann/each/hermo/hermosquare
hermoelatic quare (cont d) 2 u ------- Ẽ h : ------- Contact (Legendre) tranformation u a h g 2 a 2 g - 2 h - ------- Ẽ ------- Ẽ ------- Ẽ g a ------- ------- : ------- + : ------- + : ------- u h : a g Don t forget: i pecific tre (i.e., tre divided by denity) it i poitive in tenion. hu, the tiffne (below) i conventional tiffne divided by denity. he double dot i the tenor inner product (i.e., ). Ã :B A ij B ij 2 u 2 2 a - 2 2 g - 2 2 h 2 + ------ -- a ----- ------ --- ------ u ------ --- ------ h + ------ -- g ----- c ---- v c ---- p hermal expanion tenor: ------ β pec. thermal tiffne: B ------- Ẽ :β Grüneien tenor: --- γ 2 u ------------ ---- B c v Specific tiffne: + c. Ẽ Ẽ v γ γ -7 of 7- Rebecca Brannon, rmbrann@ia.gov /home/rmbrann/each/hermo/hermosquare