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Thermoelasticity and irreversible thermodynamics
Maurice A. Biot
To cite this version:
Maurice A. Biot. Thermoelasticity and irreversible thermodynamics. Journal of Applied Physics,
American Institute of Physics, 1956, 27 (3), pp.240-253. �10.1063/1.1722351�. �hal-01368671�
Thermoelasticity and Irreversible Thermodynamics
M. A. Biot
Citation: Journal of Applied Physics 27, 240 (1956); doi: 10.1063/1.1722351
View online: http://dx.doi.org/10.1063/1.1722351
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/27/3?ver=pdfcov
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240
PARKIN.
PERRY.
AND
WU
tion of such theory to hydrofoils is still in its infancy,
and there are a large number of practical problems on
which
it
should be brought to bear. Further, the
experience of workers in aerodynamics has long since
proved the value of basic experimental work to compare
with the theory and to suggest new assumptions and
approaches. The writers hope
that
the effort along these
lines made here will prove of value to those who work
on these interesting problems in the future.
VII.
APPENDIX. NOTATION
c hydrofoil chord length
C
p
pressure coefficient, defined
by
the equation
preceding Eq.
(4)
C p
min
minimum value of pressure coefficient on upper
surface of hydrofoil
F Froude number,
F=
Uj(gc)l
g acceleration due to gravity
h submergence of trailing edge below undisturbed
free surface
n curvilinear coordinate measured normal to
stream filament
p pressure anywhere in fluid
P
..
pressure
at
some point (x
..
,y
..
) on upper surface
of
hydrofoil
p~
pressure
at
upstream infinity
[=
pg(h-
y
..
) for
points on profile upper surface]
q velocity anywhere in fluid
q, velocity along free surface
q
..
velocity
at
some point (x",y
..
) on upper surface
of hydrofoil
r radius of curvature of streamline
R radius of curvature
at
some point (x",y
..
) on
upper surface of hydrofoil
s curvilinear coordinate measured along a stream
filament
U uniform
flow
velocity
at
infinity
x rectangular coordinate
y rectangular coordinate
y
..
y-coordinate of upper surface of hydrofoil
a geometrical angle of
attack
TJ
depth of
flow
over hydrofoil
p fluid density
JOURNAL
OF
APPLIED
PHYSICS
VOLUME
27.
NUMBER
3
MARCH.
1956
Thermoelasticity and Irreversible Thermodynamics
M.
A.
BIOT*
Cornell Aeronautical Laboratory, Inc., Buffalo, New York
(Received September 24, 1955)
A unified treatment is presented of thermoelasticity
by
application and further developments of
the
methods of irreversible thermodynamics.
The
concept of generalized free energy introduced in a previous
publication plays the role of a "thermoelastic
potential"
and
is used along with a new definition of
the
dissipation function in terms of
the
time derivative of an entropy displacement.
The
general laws of thermo-
elasticity are formulated in a variational form along with a minimum entropy production principle. This
leads to equations of the Lagrangian type,
and
the concept of thermal force is introduced
by
means of a
virtual work definition.
Heat
conduction problems can then be formulated
by
the methods of matrix algebra
and mechanics. This also leads to the very general property
that
the entropy density obeys a diffusion-type
law. General solutions of the equations of thermoelasticity are also given using the Papkovitch-Boussinesq
potentials. Examples are presented and
it
is shown how
the
generalized coordinate method may be used to
calculate the thermoelastic internal damping of elastic bodies.
1. INTRODUCTION
S
OME new methods and concepts in linear irre-
versible thermodynamics have been developed in
previous
publications
1
•
2
by
the writer.
In
particular,
variational principles were introduced in conjunction
with a generalization of the definition of free energy
to
include systems with nonuniform temperature. Our
purpose
is
to develop more extensively this line
of
approach for problems of thermoelasticity and heat
conduction.
The
expression for the generalized free energy is
developed for thermoelasticity and is referred to as a
* Consultant.
1 M.
A.
Biot, Phys. Rev. 97, 1463-1469 (1955).
2 M.
A.
Biot,
].
App!. Phys. 25, 1385-1391 (1954).
thermoelastic potential. A new expression for the dis-
sipation function
is
defined in terms of a vector field
representing
the
rate of entropy
flow.
The
term thermoelasticity encompasses a large cate-
gory
of
phenomena.
It
includes
the
general theory of
heat conduction, thermal stresses, and strains set
up
by
thermal
flow
in elastic bodies and the reverse effect
of temperature distribution caused
by
the elastic de-
formation itself leading to thermoelastic dissipation.
It
is
well known
that
the
latter
is
an
important cause of
internal damping in elastic bodies.
The
use of generalized coordinates and the varia-
tional method leads to concepts of generalized forces
of
the Lagrangian type which are applicable to both the
nu~chll.nical
and
the purely- thermal. problems.
Thil?
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THERMOELASTICITY
AND
THERMODYNAMICS
241
unified treatment is a result
of
some very basic physical
laws obeyed
by
all irreversible phenomena.!
As
a
flexible
and
compact formulation of a wide class of
problems
it
brings
out
hidden relationships with other
phenomena and situates the particular field
of
thermo-
elasticity in a broader frame.
After two introductory sections to establish the basic
equations in the usual way,
Sec.
4 points out the
identity
of
these equations with those of a porous
elastic body containing a compressible viscous fluid.
The
analogy provides a useful intuitive model for
thermoelasticity and heat conduction. Section 5 es-
tablished general solutions of these equations. A rather
significant result is obtained, namely,
that
the distribu-
tion of entropy satisfies a diffusion equation. Instead of
deriving the thermoelasticity equations directly from
thermodynamics
we
prefer to follow the inverse process
of showing
that
they lead to a variational formulation
which coincides with a general principle of irreversible
thermodynamics. This
is
done in Secs. 6
and
7.
It
requires the use of
an
entropy
flow
field which plays
the role
of
a coordinate conjugate
to
the temperature.
The
concept of thermoelastic potential
is
introduced
and the dissipation function
is
discussed in terms of the
rate of entropy production.
It
is
also shown how the
general principle of minimum entropy production
applies to thermoelasticity and heat conduction. Gen-
eralized coordinates and admittance matrices are dis-
cussed in
Sec.
8.
This leads to the treatment of practical
thermal problems
by
matrix algebra in analogy with
stress and vibration analysis methods.
The
concept
of
generalized thermal force
is
defined by a virtual dis-
,placement method as in mechanics.
The
virtual dis-
placement in this case
is
the variation of entropy
flow.
The matrix equation also indicates
that
the propagation
of
entropy as a diffusion process is quite general and
applies to the most general case
of
anisotropy. Exten-
sion to anisotropic media and to dynamics
is
briefly
treated in Sees. 9
and
10.
As
an illustration, a classical
problem in pure
heat
conduction is solved
by
the present
methods in
Sec.
11.
An interesting aspect of this solution
is
shown
by
its formulation in terms of a continuous
relaxation spectrum in analogy with
the
treatment
of
viscoelastic media
2
and the appearance of a steady-state
variable which
is
a linear function of time.
The
last
section deals with thermoelastic damping.
It
illustrates
on the simple example of a cantilever rod how vibration
problems with thermoelastic damping may be treated
by
the generalized coordinate method. Such
an
approach
can be used, for instance, to evaluate the thermoelastic
damping of crystals in piezoelectric oscillators.
Thermoelastic damping and its experimental veri-
fication were the object of extensive work
by
Zener.
3
•
4
The present treatment brings these phenomena into
3 C. Zener, Phys. Rev. 53, 90-99 (1938).
4 C. Zener, Elasticity and Anelasticity
of
Metals (The University
of
Chicago Press, Chicago, 1948).
the general frame of irreversible thermodynamics
and
its variational formulation.
2.
THE
ENTROPY OF
AN
ELASTIC ELEMENT
The
thermodynamic variables defining the
state
of
an
element are its absolute temperature T+O
and
the
strain components
eij.
We are dealing here with the
linear problem where
0 is an increment of temperature
above a reference absolute temperature
T for the
state
of
zero stress and strain.
The
strain components are
defined in terms of the displacement field
U",UIIU
z
of
the medium, by,
au", au",
aU
II
e"",,=-
e"'I1=-+-'
ax ay ax
(2.1)
These equations of
state
are the relations between
the
stress, the strain,
and
the temperature. We write
U",,,=
2fJ.e"",,+Xe-{30
U
IIII
=
2fJ.
e
llll+
Xe
-{30
u
..
= 2fJ.ezz+Xe-{38
U"lI
=
fJ.e"l<
(2.2)
In
these expressions X and
fJ.
are the Lame constants, for
isothermal deformation,
(2.3)
and
at
is
the coefficient of thermal expansion. Consider
an
element
of
unit
size. Denote
by
U its internal energy
and
h the heat absorbed
by
it. Conservation of energy
requires
".
dh=dU-
L
u"vde"
•.
(2.4)
This
is
the first law of thermodynamics.
The
entropy is
dh
dU
1,..
ds=-=---
L u,..de,.. (2.5)
Tl
Tl Tl
with T
l
=
T+8.
The
summation L is extended to all
distinct pairs of
fJ.,
II.
This may also be written
1
au
1
,..
[au
]
ds=--dT
l
+-
L
--u,..
de,.
•.
Tl
aT
l
Tl
ae,..
(2.6)
The
second law of thermodynamics requires
that
ds
be
an
exact differential in T
land
ep.
••
This implies
(2.7)
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242
M.
A.
BlOT
which is verified by the equations of state (2.2).
It
also implies
that
au
au".
-=u"v-T
1
-.
ae"v
aT
1
From (2.2)
we
find for
J.I.=
II
and for
p.¢v
au
--u"v=Tl/3
ae"v
au
--u"v=O.
ae"v
Also
because
of
(2.4)
we
have
dh
au
-=-=c
dT! aT!
(2.8)
(2.9)
(2.10)
(2.11)
where c is the specific heat of the unit volume for
e".=O, i.e., in the absence of deformation. Since
we
consider a linear theory, c
is
assumed independent
of
the temperature in the vicinity
of
the equilibrium
temperature. Introducing
(2.9), (2.10), and (2.11) into
the expression for the entropy gives
dT
l
ds=
c-+{3de.
T!
(2.12)
Integrating,
we
find,
s= c
log(
1 +
~
)+{3e
(2.13)
with a constant
of
integration such
that
s=o
for
T1=T1e=0. For small changes
of
temperature this is
limited
to
the linear term in 0
s=
(cO/T)+{3e.
(2.14)
We
note
that
Ts represents the amount
of
heat h ab-
sorbed
by
the element for small changes
of
volume and
temperature.
h=cO+T{3e.
(2.15)
We
obtain the well-known result
that
under adiabatic
transformation positive dilatation produces a cooling.
In
this case
cO=
-
T{3e.
(2.16)
Substitution in the stress-strain relations (2.2) shows
that
for adiabatic deformations the Lame constant A
is
replaced
by
A+{32(T/c)
and
J.I.
remains unchanged.
3.
THE
EQUATIONS OF
THE
TEMPERATURE
AND DEFORMATION
FmLDS
We shall
now
derive differential equations for the
displacement field
of
the solid and the distribution
of
temperature
o.
The
equations
of
equilibrium
of
a stress field with a body
force
X,
V,
Z,
per unit mass in a continuum
of
mass
density
pare
au
xx
au
XI/
au
zz
-+-+-+pX=O
ax ay
az
(3.1)
Substituting expressions (2.2) for the stress com-
ponents and assuming the body force to be zero for
simplicity,
we
find
J.l.V'2
U +(A+J.I.) grade-{3 gradO=O. (3.2)
These are three equations for U and
O.
The fourth
equation
is
provided
by
the law
of
heat conduction.
The heat absorbed
by
an
element is
h,
per unit volume.
Hence with a coefficient
k
of
heat conduction
we
find
ah
-=k
div gradO=kV'
2
0.
at
Introducing the expression (2.15) for h yields
ao
ae
kV'
2
0=c-+T{3-.
at
at
(3.3)
(3.4)
This equation, together with the three equations (3.2),
determine the time history
of
the deformation and
thermal fields.
As
expected because
of
the cooling and
heating associated with a change
of
volume, the two
fields are coupled through the coefficient
{3.
As
may be
seen from the derivation this coupling
is
a consequence
of
the laws
of
thermodynamics.
Equations
(3.2) and (3.4)
may
also be expressed in
terms
of
the local entropy density s instead
of
the
temperature
0
by
using relation (2.14).
4.
THE
ANALOGY
WITH
THE
THEORY OF
ELASTICITY OF POROUS MATERIALS
In
earlier papers
5
,6
we
have established the equations
for the deformation and fluid seepage
of
an
elastic
porous solid whose pores are filled with a compressible
viscous fluid. We will show
that
these equations are the
same as the thermoelasticity Eqs.
(3.2) and (3.4) with
the temperature playing the same role as the fluid
pressure. Equations relating the stress field to the
6 M.
A.
Biot,
J.
Appl. Phys. 12, 155-164 (1941).
• M.
A.
Biot,
J.
Appl. Phys. 26, 182-185 (1955).
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