Beyond N, V, and E in Equilibrium Thermodynamics [Archive]

Erik

May20-04, 01:24 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nIf understand things correctly, systems whose equilibrium states are\nsatisfactorily characterized by the number N(k) of particles of\nspecies k, the total volume V, and the total energy E are known to\nthermodynamicists as "simple systems". Furthermore, the textbooks on\nequilibrium thermodynamics that I\'ve seen do not venture very far away\nfrom a treatment devoted exclusively to such "simple systems".\n\nCan anyone recommend a good textbook on equilibrium thermodynamics\nwhich goes beyond "simple systems" and treats systems whose\nequilibrium states need to be characterized by more macrovariables\nthan N(k), V, and E? I would prefer a textbook on only\n_thermodynamics_ over a textbook on only _statistical mechanics_,\nalthough a textbook which treats both would do fine (my reason for\nthis preference is simply that I am, relatively speaking, much more\nfamiliar with statistical mechanics than thermodynamics and would\ntherefore like to see a treatment which doesn\'t use partition\nfunctions and other tools available to statistical mechanics but not\nto thermodynamicists).\n\nOn a related matter, at equilibrium the energy contributions from\ntemperature-entropy, pressure-volume, and velocity-momentum are TS,\n-PV, and vp, respectively. Why is the last contribution vp? I would\n(naively?) expect the contribution to be a factor of 2 smaller, i.e.\nvp/2 = mv*v/2.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>If understand things correctly, systems whose equilibrium states are
satisfactorily characterized by the number N(k) of particles of
species k, the total volume V, and the total energy E are known to
thermodynamicists as "simple systems". Furthermore, the textbooks on
equilibrium thermodynamics that I've seen do not venture very far away
from a treatment devoted exclusively to such "simple systems".

Can anyone recommend a good textbook on equilibrium thermodynamics
which goes beyond "simple systems" and treats systems whose
equilibrium states need to be characterized by more macrovariables
than N(k), V, and E? I would prefer a textbook on only
_thermodynamics_ over a textbook on only _statistical mechanics_,
although a textbook which treats both would do fine (my reason for
this preference is simply that I am, relatively speaking, much more
familiar with statistical mechanics than thermodynamics and would
therefore like to see a treatment which doesn't use partition
functions and other tools available to statistical mechanics but not
to thermodynamicists).

On a related matter, at equilibrium the energy contributions from
temperature-entropy, pressure-volume, and velocity-momentum are TS,
-PV, and vp, respectively. Why is the last contribution vp? I would
(naively?) expect the contribution to be a factor of 2 smaller, i.e.
vp/2 = mv*v/2.

Arnold Neumaier

May21-04, 10:14 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nErik wrote:\n> If understand things correctly, systems whose equilibrium states are\n> satisfactorily characterized by the number N(k) of particles of\n> species k, the total volume V, and the total energy E are known to\n> thermodynamicists as "simple systems". Furthermore, the textbooks on\n> equilibrium thermodynamics that I\'ve seen do not venture very far away\n> from a treatment devoted exclusively to such "simple systems".\n>\n> Can anyone recommend a good textbook on equilibrium thermodynamics\n> which goes beyond "simple systems" and treats systems whose\n> equilibrium states need to be characterized by more macrovariables\n> than N(k), V, and E? I would prefer a textbook on only\n> _thermodynamics_ over a textbook on only _statistical mechanics_,\n\nTry\nL.E. Reichl,\nA Modern Course in Statistical Physics.\nUniv. of Texas Press, Austin 1980.\n\nThe first 4 chapters are about thermodynamics; later comes statistical\nmechanics. The two are in fact indivisible.\n\n\n> although a textbook which treats both would do fine (my reason for\n> this preference is simply that I am, relatively speaking, much more\n> familiar with statistical mechanics than thermodynamics and would\n> therefore like to see a treatment which doesn\'t use partition\n> functions and other tools available to statistical mechanics but not\n> to thermodynamicists).\n>\n> On a related matter, at equilibrium the energy contributions from\n> temperature-entropy, pressure-volume, and velocity-momentum are TS,\n> -PV, and vp, respectively. Why is the last contribution vp? I would\n> (naively?) expect the contribution to be a factor of 2 smaller, i.e.\n> vp/2 = mv*v/2.\n\nV and P are not velocity and momentum but volume and pressure. Hence\nthere is no relation to mv^2/2.\n\n\nArnold Neumaier\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Erik wrote:
> If understand things correctly, systems whose equilibrium states are
> satisfactorily characterized by the number N(k) of particles of
> species k, the total volume V, and the total energy E are known to
> thermodynamicists as "simple systems". Furthermore, the textbooks on
> equilibrium thermodynamics that I've seen do not venture very far away
> from a treatment devoted exclusively to such "simple systems".
>
> Can anyone recommend a good textbook on equilibrium thermodynamics
> which goes beyond "simple systems" and treats systems whose
> equilibrium states need to be characterized by more macrovariables
> than N(k), V, and E? I would prefer a textbook on only
> _thermodynamics_ over a textbook on only _statistical mechanics_,

Try
L.E. Reichl,
A Modern Course in Statistical Physics.
Univ. of Texas Press, Austin 1980.

The first 4 chapters are about thermodynamics; later comes statistical
mechanics. The two are in fact indivisible.

> although a textbook which treats both would do fine (my reason for
> this preference is simply that I am, relatively speaking, much more
> familiar with statistical mechanics than thermodynamics and would
> therefore like to see a treatment which doesn't use partition
> functions and other tools available to statistical mechanics but not
> to thermodynamicists).
>
> On a related matter, at equilibrium the energy contributions from
> temperature-entropy, pressure-volume, and velocity-momentum are TS,
> -PV, and vp, respectively. Why is the last contribution vp? I would
> (naively?) expect the contribution to be a factor of 2 smaller, i.e.
> vp/2 = mv*v/2.

V and P are not velocity and momentum but volume and pressure. Hence
there is no relation to mv^2/2.

Arnold Neumaier

Erik

May22-04, 04:50 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<40ADD652.3070305@univie.ac.at>...\n> Erik wrote:\n> > If understand things correctly, systems whose equilibrium states are\n> > satisfactorily characterized by the number N(k) of particles of\n> > species k, the total volume V, and the total energy E are known to\n> > thermodynamicists as "simple systems". Furthermore, the textbooks on\n> > equilibrium thermodynamics that I\'ve seen do not venture very far away\n> > from a treatment devoted exclusively to such "simple systems".\n> >\n> > Can anyone recommend a good textbook on equilibrium thermodynamics\n> > which goes beyond "simple systems" and treats systems whose\n> > equilibrium states need to be characterized by more macrovariables\n> > than N(k), V, and E? I would prefer a textbook on only\n> > _thermodynamics_ over a textbook on only _statistical mechanics_,\n>\n> Try\n> L.E. Reichl,\n> A Modern Course in Statistical Physics.\n> Univ. of Texas Press, Austin 1980.\n>\n> The first 4 chapters are about thermodynamics; later comes statistical\n> mechanics. The two are in fact indivisible.\n\nThanks! My local library has a copy so I\'ll check it out.\n\n> > although a textbook which treats both would do fine (my reason for\n> > this preference is simply that I am, relatively speaking, much more\n> > familiar with statistical mechanics than thermodynamics and would\n> > therefore like to see a treatment which doesn\'t use partition\n> > functions and other tools available to statistical mechanics but not\n> > to thermodynamicists).\n> >\n> > On a related matter, at equilibrium the energy contributions from\n> > temperature-entropy, pressure-volume, and velocity-momentum are TS,\n> > -PV, and vp, respectively. Why is the last contribution vp? I would\n> > (naively?) expect the contribution to be a factor of 2 smaller, i.e.\n> > vp/2 = mv*v/2.\n>\n> V and P are not velocity and momentum but volume and pressure. Hence\n> there is no relation to mv^2/2.\n\nI used upper-case letters for volume and pressure and lower-case\nletters for velocity and momentum. Table 1 in the preprint ("On the\nfoundations of thermodynamics") you sent me gives the (equilibrium)\nenergy contribution from some pairs of thermodynamic variables. Table\n1 gives the contribution from the velocity-momentum pair as v*p. Why\nis it v*p and not v*p/2? The energy contribution from the angular\nvelocity-angular momentum pair is similarly different by a factor of 2\nfrom my uninformed expectations.\n\nErik\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<40ADD652.3070305@univie.ac.at>...
> Erik wrote:
> > If understand things correctly, systems whose equilibrium states are
> > satisfactorily characterized by the number N(k) of particles of
> > species k, the total volume V, and the total energy E are known to
> > thermodynamicists as "simple systems". Furthermore, the textbooks on
> > equilibrium thermodynamics that I've seen do not venture very far away
> > from a treatment devoted exclusively to such "simple systems".
> >
> > Can anyone recommend a good textbook on equilibrium thermodynamics
> > which goes beyond "simple systems" and treats systems whose
> > equilibrium states need to be characterized by more macrovariables
> > than N(k), V, and E? I would prefer a textbook on only
> > _thermodynamics_ over a textbook on only _statistical mechanics_,
>
> Try
> L.E. Reichl,
> A Modern Course in Statistical Physics.
> Univ. of Texas Press, Austin 1980.
>
> The first 4 chapters are about thermodynamics; later comes statistical
> mechanics. The two are in fact indivisible.

Thanks! My local library has a copy so I'll check it out.

> > although a textbook which treats both would do fine (my reason for
> > this preference is simply that I am, relatively speaking, much more
> > familiar with statistical mechanics than thermodynamics and would
> > therefore like to see a treatment which doesn't use partition
> > functions and other tools available to statistical mechanics but not
> > to thermodynamicists).
> >
> > On a related matter, at equilibrium the energy contributions from
> > temperature-entropy, pressure-volume, and velocity-momentum are TS,
> > -PV, and vp, respectively. Why is the last contribution vp? I would
> > (naively?) expect the contribution to be a factor of 2 smaller, i.e.
> > vp/2 = mv*v/2.
>
> V and P are not velocity and momentum but volume and pressure. Hence
> there is no relation to mv^2/2.

I used upper-case letters for volume and pressure and lower-case
letters for velocity and momentum. Table 1 in the preprint ("On the
foundations of thermodynamics") you sent me gives the (equilibrium)
energy contribution from some pairs of thermodynamic variables. Table
1 gives the contribution from the velocity-momentum pair as v*p. Why
is it v*p and not v*p/2? The energy contribution from the angular
velocity-angular momentum pair is similarly different by a factor of 2
from my uninformed expectations.

Erik

Arnold Neumaier

May24-04, 04:31 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nErik wrote:\n> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<40ADD652.3070305@univie.ac.at>...\n\n>>>On a related matter, at equilibrium the energy contributions from\n>>>temperature-entropy, pressure-volume, and velocity-momentum are TS,\n>>>-PV, and vp, respectively. Why is the last contribution vp? I would\n>>>(naively?) expect the contribution to be a factor of 2 smaller, i.e.\n>>>vp/2 = mv*v/2.\n\n> I used upper-case letters for volume and pressure and lower-case\n> letters for velocity and momentum. Table 1 in the preprint ("On the\n> foundations of thermodynamics") you sent me gives the (equilibrium)\n> energy contribution from some pairs of thermodynamic variables.\n\nActually you should not publicly discuss a draft of a paper that I sent\nyou personally. You are free to do whatever you like with manuscripts\nthat I put online. But all my drafts are incomplete (since they are\ndrafts!) and might have many inaccuracies, which you should discuss only\nwith me.\n\n> Table\n> 1 gives the contribution from the velocity-momentum pair as v*p. Why\n> is it v*p and not v*p/2? The energy contribution from the angular\n> velocity-angular momentum pair is similarly different by a factor of 2\n> from my uninformed expectations.\n\nOf course, v and p are not equilibrium quantities; my table is more\ngeneral.\n\nNote that the Euler equation, which contains the term v dot p, looks\nlike an equation for the energy, but in fact it is just a definition of\nthe entropy in terms of the energy and other contributions. It looks\nlike an energy balance only because the defining equation is multiplied\nby T to get rid of the beta and have things in traditional units.\nBut since S is undefined, this formal balance has no contents apart from\ndefining S.\n\nThe energy balance is given by the first law, and is about _changes_ in\nenergy. And the change of kinetic energy is\nd(mv^2/2)= mv dot dv = v dot dp,\nwhich is exactly what you get from the v dot p in the Euler equation.\nThe same argument applies to the kinetic energy of a rigid body,\nrelating angular velocity and angular momentum.\n\n\nArnold Neumaier\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Erik wrote:
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<40ADD652.3070305@univie.ac.at>...

>>>On a related matter, at equilibrium the energy contributions from
>>>temperature-entropy, pressure-volume, and velocity-momentum are TS,
>>>-PV, and vp, respectively. Why is the last contribution vp? I would
>>>(naively?) expect the contribution to be a factor of 2 smaller, i.e.
>>>vp/2 = mv*v/2.

> I used upper-case letters for volume and pressure and lower-case
> letters for velocity and momentum. Table 1 in the preprint ("On the
> foundations of thermodynamics") you sent me gives the (equilibrium)
> energy contribution from some pairs of thermodynamic variables.

Actually you should not publicly discuss a draft of a paper that I sent
you personally. You are free to do whatever you like with manuscripts
that I put online. But all my drafts are incomplete (since they are
drafts!) and might have many inaccuracies, which you should discuss only
with me.

> Table
> 1 gives the contribution from the velocity-momentum pair as v*p. Why
> is it v*p and not v*p/2? The energy contribution from the angular
> velocity-angular momentum pair is similarly different by a factor of 2
> from my uninformed expectations.

Of course, v and p are not equilibrium quantities; my table is more
general.

Note that the Euler equation, which contains the term v dot p, looks
like an equation for the energy, but in fact it is just a definition of
the entropy in terms of the energy and other contributions. It looks
like an energy balance only because the defining equation is multiplied
by T to get rid of the \beta and have things in traditional units.
But since S is undefined, this formal balance has no contents apart from
defining S.

The energy balance is given by the first law, and is about _changes_ in
energy. And the change of kinetic energy is
d(mv^2/2)= mv dot dv = v dot dp,
which is exactly what you get from the v dot p in the Euler equation.
The same argument applies to the kinetic energy of a rigid body,
relating angular velocity and angular momentum.

Arnold Neumaier

baffledMatt

Jun14-04, 03:14 PM

I used upper-case letters for volume and pressure and lower-case
letters for velocity and momentum. Table 1 in the preprint ("On the
foundations of thermodynamics") you sent me gives the (equilibrium)
energy contribution from some pairs of thermodynamic variables. Table
1 gives the contribution from the velocity-momentum pair as v*p. Why
is it v*p and not v*p/2? The energy contribution from the angular
velocity-angular momentum pair is similarly different by a factor of 2
from my uninformed expectations.

Erik

Isn't it because when a particle collides with the walls of the container the change in momentum is 2p? Then we say that the average force on the walls per particle is F = \frac{\Delta p}{\Delta t}, where \Delta t is the time between collisions which is \frac{L}{v}. Turn the handle a bit and I think you get the right result.

This is all explained much better on the scienceworld page on kinetic theory:
http://scienceworld.wolfram.com/physics/KineticTheory.html

Hope that helps.

Matt

Igor Khavkine

Jun16-04, 05:30 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Thu, 20 May 2004 14:24:50 -0400, Erik wrote:\n\n>\n> If understand things correctly, systems whose equilibrium states are\n> satisfactorily characterized by the number N(k) of particles of\n> species k, the total volume V, and the total energy E are known to\n> thermodynamicists as "simple systems". Furthermore, the textbooks on\n> equilibrium thermodynamics that I\'ve seen do not venture very far away\n> from a treatment devoted exclusively to such "simple systems".\n>\n> Can anyone recommend a good textbook on equilibrium thermodynamics\n> which goes beyond "simple systems" and treats systems whose\n> equilibrium states need to be characterized by more macrovariables\n> than N(k), V, and E? I would prefer a textbook on only\n> _thermodynamics_ over a textbook on only _statistical mechanics_,\n> although a textbook which treats both would do fine (my reason for\n> this preference is simply that I am, relatively speaking, much more\n> familiar with statistical mechanics than thermodynamics and would\n> therefore like to see a treatment which doesn\'t use partition\n> functions and other tools available to statistical mechanics but not\n> to thermodynamicists).\n\nTry this book\n\nThermodynamics and an introduction to thermostatistics\nCallen, Herbert B. (Wiley, 1985)\n\nIt treats thermodynamics quite abstractly and has some good examples.\nIt even has a few chapters on stat mech, but they are at the end of the\nbook and are not relevant to the rest of it.\n\n> On a related matter, at equilibrium the energy contributions from\n> temperature-entropy, pressure-volume, and velocity-momentum are TS,\n> -PV, and vp, respectively. Why is the last contribution vp? I would\n> (naively?) expect the contribution to be a factor of 2 smaller, i.e.\n> vp/2 = mv*v/2.\n\nLet v be the independent variable. Start with a simple particle with\nvelocity v and energy E = mv^2/2. If you increase velocity from v to\nv + dv, the energy changes to\n\nE + dE = m(v+dv)^2/2 = m(v^2 + 2v dv)/2 = mv^2/2 + mv dv,\n\nwhere I\'ve neglected terms of order (dv)^2. You see that dE = p dv, where\nwe\'ve called p = mv. Then by homogeneity arguments (as in the case of\npressure or temperature), the term that has to be added to the energy\nis pv.\n\nAlso, Landau seems to be quite fond of thermodynamics, from my reading of\nhis books. Treatments of several kinds of non-simple systems are sprinkled\nthroughout his books. For example Chapter II of volume 5 (Statistical\nPhysics I) treats the cases of momentum-velocity as well as angular\nmomentum-angular velocity. Also, in volume 7 (Elasticity Theory) he treats\nthe case of stress-strain.\n\nHope this helps.\n\nIgor\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Thu, 20 May 2004 14:24:50 -0400, Erik wrote:

>
> If understand things correctly, systems whose equilibrium states are
> satisfactorily characterized by the number N(k) of particles of
> species k, the total volume V, and the total energy E are known to
> thermodynamicists as "simple systems". Furthermore, the textbooks on
> equilibrium thermodynamics that I've seen do not venture very far away
> from a treatment devoted exclusively to such "simple systems".
>
> Can anyone recommend a good textbook on equilibrium thermodynamics
> which goes beyond "simple systems" and treats systems whose
> equilibrium states need to be characterized by more macrovariables
> than N(k), V, and E? I would prefer a textbook on only
> _thermodynamics_ over a textbook on only _statistical mechanics_,
> although a textbook which treats both would do fine (my reason for
> this preference is simply that I am, relatively speaking, much more
> familiar with statistical mechanics than thermodynamics and would
> therefore like to see a treatment which doesn't use partition
> functions and other tools available to statistical mechanics but not
> to thermodynamicists).

Try this book

Thermodynamics and an introduction to thermostatistics
Callen, Herbert B. (Wiley, 1985)

It treats thermodynamics quite abstractly and has some good examples.
It even has a few chapters on stat mech, but they are at the end of the
book and are not relevant to the rest of it.

> On a related matter, at equilibrium the energy contributions from
> temperature-entropy, pressure-volume, and velocity-momentum are TS,
> -PV, and vp, respectively. Why is the last contribution vp? I would
> (naively?) expect the contribution to be a factor of 2 smaller, i.e.
> vp/2 = mv*v/2.

Let v be the independent variable. Start with a simple particle with
velocity v and energy E = mv^2/2. If you increase velocity from v to
v + dv, the energy changes to

E + dE = m(v+dv)^2/2 = m(v^2 + 2v dv)/2 = mv^2/2 + mv dv,

where I've neglected terms of order (dv)^2. You see that dE = p dv, where
we've called p = mv. Then by homogeneity arguments (as in the case of
pressure or temperature), the term that has to be added to the energy
is pv.

Also, Landau seems to be quite fond of thermodynamics, from my reading of
his books. Treatments of several kinds of non-simple systems are sprinkled
throughout his books. For example Chapter II of volume 5 (Statistical
Physics I) treats the cases of momentum-velocity as well as angular
momentum-angular velocity. Also, in volume 7 (Elasticity Theory) he treats
the case of stress-strain.

Hope this helps.

Igor