Schapery RA 1962 (PHD Dissertation)
Schapery RA 1962 (PHD Dissertation)
Schapery RA 1962 (PHD Dissertation)
t vJ..
.!.. Q.(t-T) \' ---2Le-
t
/T
s
I e
2 1 . L- T S . J
O
J
s
+ C .. It Q.(v) dV]
1) J 0 J ' .
(2.24a)
(2. 24b)
It should be noted th.a t equation 2. 24a is valid for all coordinates,
qi' while only those with index k (prescribed co'ordinates) are needed
to determine the unknown forces Qk. Therefore, on the basis of the
above relation betwee.n IQ and I we expect the Euler equations of
q .
the variation of 2. 23 to be
C
{s)
r '\' kj
qk = L 6l+T sP
. s
-76-
C
ts
)
kj
T
-tiT st v/T st
e s e sQ.(v.)dv + C
k
. Q . {v) dv
. 0 J J 0 J
S
s
The complementary force theorem can now be stated:
Considering a linear system as defined above and
assumed to be at rest in its reference state at t= 0,
the actual path followed by the forces Qk' which
are conjugate to the specified generalized coordi-
nates qk' is determined by making 1Q stationary
with respect to all small variations 6Q
k
from. the
actual values; that is, the integral equations 2.25
are the Euler equations of the variation
where 1Q is defined by equation 2.23.'
(2.25)
(2.26)
By completing this variation, the Euler equations are found to be
equations 2. 25.
It is clear that the converse theorem is also true', i. e., the
variation 2.26 vanishes as a result of 2.25. In addition, it can be
shown that the actual value of 16 is not an extremal, but is just a station-
ary point. However, as with Iq for p real and positive, IQ is an
absolute minimum for the actual path.
It is noted from equation 2.22 that the name "complementary"
principle is appropriate since
-77-
(2.27)
for the actual solution.
Another point of interest conc e rns the physical m eaning that
can be attached to the convolution functionals 2.l0a and 2.23 in con -
trast to the standard form used for reversible process es , e. g. Hamil-
ton's principle. A basic fact is, of course, that energy is d egraded
in any natural (irreversible) process. For example, if a viscoelastic
body is loaded by mechanical forces, but is not in thermodynamic
equilibrium, the motion of the state variables is such that mechanical
. energy is converted continuously into thermal energy, with an associ-
ated entropy increase. However, by using products of forward and
backward running coordinates and forces in energy functionals, this
irreversibility is removed in the sens.e that these functionals have the
stationary property during the entire process.
2.3. Application of the Basic Homogeneous Principl es to Vis coelastic
Solids
While the general variational principles developed in section
2.2 apply directly to a system defined by n thermodynamic state
variables, it is the purpose of this section to use them to formulate
variational principles for the thermal and mechanical behavior of
s:olids whose thermodynamic state is, in general, described by an
infinite number of these variables. As mentioned earlier, a differ-
ential mass element of a continuum can be interpreted as being a
uniform cell whose thermodynamic state is defined by m .variables,
say; and that this interpretation is valid as long as spacewise changes
-78-
of variables are small relative to characteristic atomic or molecular
distances. The total system is then defined by a set of m variables
which vary continuously throughout the body. Consequently, the
thermodynamic functions, V T and D
T
, and the external energy
supplied to the system are given by volume and surface integrals
with integrands consisting of a sum over the m variabl es of each
cell.
It will be seen that this method of deducing continuum princi-
pIes, in 'which thermodynamic f unctions are suggested b y the discre t e
cell analysis, has several advantages. For one, the determination of
.
variational principles is straightforward even though they are not at
all obv ious by simply examining the field (Euler) equations. In addi-
tion, this method provides functionals which are expres s e d in terms
of ,thermodynamic invariants, and conseque ntly the functionals are
independent of the particular coordinate system used. Another ad-
vantage is that the ' Euler equations are guaranteed to be consistent
with thermodynamics. '
We now proceed to determine both IIdisplacement" and "force"
(or stress) principles for anisotropic media; first for pure h eat con-
duction a.nd then for the combined thermo-mechanical behavior of
v iscoelastic =edia. It will be assu=ed that inertia effects due to
straining are negligible and that all variables vanish at the time origin.
If this is not the case in any particular problem, the appr opriate
functional can be modified as discussed in section 2.2. We also as-
sume that the geometric boundarie s of the solid do not change with
time except for small deformations and that all properties are inde-
-79-
pendent of temperature.
a. A simple example-,-heat conduction.
n) Principle for entropy displacement: The the rmody':'
namic state of a solid which is assumed to experience only thermal
energy changes is defined by the absolute temperature, T, at each
point. However, it has been shown that the variahonal principle 2.11
applies if entropy displacement, rather than temperature, is used as
the thermodynamic variable . That this can be done follows from the
conservation of energy statement
where
--
.
div h
. .
h. , = - c e
1, 1
(2.28)
c = specific heat, per unit volume (assumed independent
of temperature)
--
The relation' between entropy displacement, ' S, =, (S1' S2' S3)' and tem-
perature is then obtained by integration,
ce
S .. = - T
1, 1 r
where the entropy displacement is defined as ,
(2. 29)
(2. 30)
It is clear from equation 2.29 that S. completely specifies
, 1
the temperature, and hence the thermodynamic state of the body. It
is interesting to observe that the temperature is only a function of
-80-
div S and therefore is independent of curl S. This property is
analogous to the role of Inechanical displace:ment, u, since the rota-
->-
tion of :material ele:ments, curl u, has no effect on the ther:mody-
na:mic state of each ele:ment.
Let us now evaluate the ter:ms V; and D; In I
q
, equation
2.10, by using identity 2.19. Biot (2) calculated these functions fro:m
their ther:modyna:mic definitions; however, for our purposes it is
easier to use the identity
(2.31)
where n. is the outer unit nor:mal to the exterior surface A, and the
1
variables t and Tare i:mplied. In writing equation 2.31 we have
used the ther:mo-:mechanical analogy discussed in Part I, section 6.
In this, the boundary te:mperature 8 acts as a generalized force In
the sense that
-s
A
en. 6S. dA
1 1
= Q.Oq.
1 1
(2. 32)
is virtual work (or energy addition) done . on the body; the negative sign
,',
accounts for the definition that n
i
is an outer nor:mal. Since V ~
~ : c
and D Tare volu:me integrals of density functions, they can be deter-
mined by applying the divergence theore:m to the surface integral in
2. 31. This provide s the identity
S
en.S.dA =: S [es . . + 8,.$.] dB
All . B 1,1 1.. 1
(2. 33)
where B denotes volu:me integration.
Fro:m equation 2.29 we have
- 81-
T
8S . . = _ C
r
(S . . )2
1, 1 1, 1
(2.34 )
In order to express the second t e rm in the volume inte g ral in 2.33 a s
a function of entropy displaceme nt, we must utilize the l aw of heat
conduction given in P a rt I, equa tion 1.116,
K .. 8, . = - h . -
1J J 1
which can be inverted to find
8, . = - T A. . .S .
1 r 1J J
T S .
r 1
K . . = K ..
1J J1
where A. .. is the the rmal resi s tivity matrix defined as
1J
(2. 3 5)
(2. 36)
A. .. = [K. .]-l (2. 3 7)
1J 1J
which is assumed to be independent of 8. Using equation 2.36 we
now have
.
8, .S. = - T A. . . S.S.
1 1 r 1J 1 J
(2. 3 8 )
Substitution of equations 2.38 and 2.34 into identity 2.33 and
us ing relation 2. 31 y ields
~ ( ' ~ S Tr T
r
V
T
+ DT = {2C S . . (r)S .. (t .. r) + ~ 2 A. . . S.(r)S.(t-r)} dB
B 1, 1 J, } 1J 1 J
(2.39 )
On comparing this with equations 2 ~ 5, 2.7, and 2.10, the generalized
free energy density, V (per unit volume) is identified as
Tr 2
V = ..,..,.,C (S .. )
L.L 1, 1
(2.40)
which can also be expressed as a function of temperature by means
of the ene r gy equation 2. 29,
-82-
The dissipation density, D, is observed to be
D=
T r
-2 A .. S.S.
1J 1 J
or, in terTIlS of teTIlperature,
K ..
_ 1J
D - ZT e, ie, j
r
(2.41)
(2. 42)
(2. 43)
The functional I for entropy displaceTIlent, corresponding to
s
I in the general theory, can now be by substituting equation
q
2. 39 into equation 2. lOa,
S
ts . Tr T
r
.
I = {-2C S . . (r)S .. (t-r) +-2 A .. S.(r)S. (t-r)} dBdr
sOB " 1 ;., } 1J 1 J .
rts .
+ J 6(T)n.S.(t-r)dA dT
o A"
. e
(2.44) .
where Ae .is the portion of the surface on which e = e is .prescribed,
and S.n. TIlust satisfy the boundary conditions' on A where heat flow
1 1 S
is prescribed. This latter requireTIlent arises froTIl the fact that q.
. 1
in the general theory TIlust satisfy constraints, which are the heat flow
boundary conditions. In order to siTIlplify the notation, we shall use
the usual notation for convolution integrals,
(2.45)
where f and g are functions of tiTIle. Two useful properties of the
product f g are
>!c
f g = g f
>:< 1.1: )'e
(f + g) h:; f' h + g' h (2. 46)
-83-
In this simplified notation, the functional '2.44 becomes
S
Tr
Is::: B { 2C S. . S. . + -2 A. .. S . S.} dB
J, J 1, 1 IJ 1 J
+ S e':'n.S. dA
" A 1 1
8
(2. 47)
The variational principle for entropy displacement can be stated as
oI ::: 0
S
(2. 48)
for all arbitrary variations of S . consistent with its boundary condi-
1
tions. It can be shown that the Euler equations of 2.48 are the heat
conduction equations
1
[ C ,S .. ] ::: LS.
" 1, 1 'j J1 1
In "B (2. 49)
and the natural boundary condition is
ce
s . . :::--
1,1 T "
r
(2. 50)
That these are indeed the experimentally correct equations for entropy
displacement is verified by substitution of equation 2.29 into the law
The Euler equations of 2.48 can be obtained in the same form as
the general equations 2.1 if we first write
S. ::: S . : (x.)q. (t)
1 IJ 1 J
(2. 51)
In passing it may be recalled that even though the principle 2.48 was
derived from thermodynamics under the assumption of 8/T r 1,
it may have a wider of applicability; this follows from the fact
that the temperature range for which it is sufficiently accurate rests
ultimately on the accuracy of its Euler equations in satisfying experi-
mentally observed behavior.
-84-
where the quantities S .. (x.)
IJ 1
are as sumed functi ons of x., consistent
1
with the boundary conditions on S., and q.{t) are unspecified func -
1 J
tions of t ime. Substitution of 2.51 into the var i ational equations 2. 48
yields differential equations for t he q. which are identical in fo rm
J
with the basic relati ons 2.1.
13) Princi ple for t emperature: In numerical applications,
once the entropy di s place ment i s found, from e qua tion 2. 48, the tem-
perature distribution can be calculated from the ene rgy equat ion 2.29.
This procedure is similar to the one followe d with the potential
energy theorem of e lasticity, wherein displace ments are first calcu-
lated from the varia tional principle, and then stres s -strain equations
are us ed to f ind stresses. However, as in the e lasticity problem, in
many cas es it may be more des irabl e to obtain temperature ( "stress ")
directly from a variational principle. This can be done by using the
complementary function 2.23 for heat conduction, in which e co rre-
sponds to the generalized force Q.. Equation 2.41 provide s us i m-
1
mediate ly with V; as a function of temperature. The f unction D;
is found b y combining equati ons 2.38 and 2.35,
" _ S - ' 1 } _ 1 } e, j
DT - D dB - -""" e, .S. - 2T e, .K,"
B c. B 1 1 r B 1 IJ P ,
(2. 52)
The functional for t emperature, Ie' can now be ,written by making the
appropriate substitution in 2.23, thus
C ':' e, j
{Te e +Te,.[K. .-]}
r r 1 IJ P
dB
+ r e
J 1 1
As
(2. 53)
-85-
whe re AS is the portion of the boundary where heat flow is prescribed.
It will probably be more convenient to use the time 'derivative of Ie
in applications,
dB
(2. 54)
The temperature distribution is found from the variational equation
>;C .
iiI - 0
e -
(2. 55)
for all variations of e which satisfy the boundary conditions on Ae'
It can be easily verified that the Euler equation of 2.55 is
ce ::: [K. ,e,.J '
. , 1J J, i
in B (2. 56)
and the boundary condition is
.
n.K .. e, . ::: - h.n.
1 1J J 1 1
(2. 57)
Since these are also obtained by combining equations 2.29 and 2.35,
the variational principle is valid under the same conditions as is the
entropy displacement principle., It may be noted in passing that equa-
tion 2.56 for an isotropic, homogeneous solid takes the well -known
form
(2. 58)
,',
'This variational principle is similar to R.osen's (26); however, he
does not include a t i m ~ integral so that e must be held constant in
making the variation.
-86-
Let us now show that identity 2. 27 is valid for the application at
hand. Let e{x., t} be the actual tempe rature distribution, and S.
1 1
the corresponding entropy displacement for a body with n.S. pre -
1 1
scribed on AS arid e prescribed on Ae' Adding functionals 2.47
and 2.53,
{2. 59}
;;.'0:; :.:<
Since (y ~ + D
T
, satisfies equation 2.31 we arrive at the simple result
that for the "exact" solution,
{2.60}
The same complementary property can be expected in other appli-
cations as a consequence of the fact that the underlying thermodynamic
equations of motion are the same for all linear systems. In fact, this
. .
simple example of heat transfer illustrates well most of the essential
features found in other applications.
b. Thermo-viscoelasticity. In this section variational principles
for the linearized equations of thermo-viscoelasticity win be derived.
It should be recalled that the thermodynamic linearity assumption r e -
quires material properties to be independent of temperatur e' and the
effect of dissipation on temperature to be neglecte-i. , Even though
these assumptions are often too restrictive for thermal stress prob-
lems, several important sub-cases can be obtained from the general
linear analysis. For example thermoelasticity, viscoelasticity {with-
out temperature}, and heat conduction appear as special cases.
-87-
u ~ Principle for m ec hanical and entropy displacements:
We proceed as in the previous e xample by evaluating v; + D; in
terms of displacement with the aid of equation 2.19. For the case
of both mechanical and thermal perturbations we have
Q.q.
1 1
= r T.u.dA + \ . F.u. dB _ r en.S. dA
J
A
ll ' B
11
J
A
11
(2.61a)
where T. is the surface force p e r unit area, F. the pre scribed body
1 1
forc e per unit volume, u. the mechanical displacement, and the
1
variables t and .,. are implied. The entropy displacement integral
is written in terms of density functions by utilizing equation 2.33.
The integrals representing mechanical energy can also be expressed
as volume integrals by means of the divergence theorem. We find (13)
""S T.u.dA + S F.u.dB = S [cr ... + F.] u.dB + S cr .. e .. dB
All B . 1 " 1 B lJ, J 1 " 1 B lJ lJ
where e .. is the strain tensor,
lJ
" I
e .. = -2 (u .. + u .. )
1J l,J J,l
(2.6Ib)
(2. 62)
)'.:, ':<:
and cr
ij
the stress tensor. Since V ~ and DT are defined as functions
of state variables, they"must be evaluated under the condition of mechani-
cal equilibrium; this requires (neglecting inertia),
cr ... + F . = '0
1J,J 1
1n B (2.63)
Substitution of equation 2. 61a into equation 2.19, after making use of
equations 2.33, 2.61b, and 2.63 yields
= C {cr .. e .. - es . . - e, .S. }dB
J B 1J lJ 1, 1 1 1
(2. 64)
-88-
The last term in equa tion 2.64 can be expressed immediately
as a function of entropy displacement by using the heat conduction l aw,
2.36, to obtain equation 2.38; the assumption that the heat conduction
equation is unaffected by deformation was discussed in section 1.8.
Also, the operational stress -strain-temperature equation 1.100a,
is us ed to obtain
(f . . e ..
IJ IJ
e ] e. . - [ ( 3 ~ . 8] e ..
J.1v IJ IJ IJ
(2. 65)
In the previous example it was possible to evaluate the second
term in equation 2.64 , 8S .. , through the use of the energy equation
. 1, 1
2.29. However, we must now utilize the linearized version of equa-
tion 1.119, which can be written as
. H = - T S. . = d' 8 + T /3
0
. e:.
r 1, 1 e r IJ IJ
(2. 66)
or
e =
[
0] -1 [ 0] -1 0
T C S . . -T C (3 .. e ..
r e 1, 1. r e IJ ' 1J
(2.67)
where, in equation 1.119, we have set
I
P s = P s = constant,
D = 0, and
integrated with respect to time. The inverse of the specifi.c heat
operator,
[ Co] -1, is of the form
e
s
P +_1_
"PSH
(2. 68)
in which d and d(s) are positive constants; this property is ob-
e e
tained by comparing equation 2.67 and the earlier equation 1. 69. It
, 0 -1 0
is also seen from equation 1. 69 that [C] (3.. has this same form,
e 1J
but the coefficients are not necessarily positive.
-89-
Let us now substitute equations 2.65, 2.67, and 2.38 into
identity 2. 64 to find,
t . * .
2 (' (V
T
':' + S {[ Zfl.v e ] ':'e .. + a
T
-[ COa]
JOB 1J iJ. v 1J ' r e
. "
+ T A .. 5: S. }dB
r 1J 1 J
(2. 69)
in which the convolution notation 2,45 is indicated and we have used
the fact that
o )!( ':c 0
[ 13 a] e .. :: a [13 e.
J
.]
1J 1J 1J 1
(2. 70)
Thi s identity can easily be ve rifi e d by operating on it w ith the L aplace
transform. Elimina tion of the tempe rature from equation 2.69 and
s ubstituting the r es ult into equation (2.10a) yields the f u nctional to
be used in the variational principle:
I
US
1 (' . iJ.v 0 -1 0 0
:: -2J ([Z . . e ] e .. + T [(C) (5 .. + j3 .. e .. )] [5 . .. +j3 .. e .. ]
B 1J f.L v 1J r e . 1,1 1J 1J 1, 1 1J 1J
. );c
+ T A . . 5 . S.}dB
r 1J 1 J
S
,',
+ e'n.S.dA
All
. . a
S
':' S *
- . T. u . dA- F . u.dB
All " B 1 1
T
(2. 71)
where the force vector Ti is pres cribed on AT and temperature e
is prescribe d on Aa' Also, the strains must be expressed as functions
of displacement by using equation 2.62.
The equations for mechanical and entropy displacements are
determined by the stationary condition
6I :: 0
us
(2. 72.)
fo r all arbitrary. variations of u . and S. compatible with thei r
1 1
boundarY .conditions. By carrying out the variation we find that the
-90-
Euler equations of 2.72 are the three equilibrium equations;
in B
(2.73)
and the three heat conduction equations
o' 0
Cx. . . S . -S . .. -13 u .=0
e 1) 1 1,1) fJ.Y fJ.,Y)
in B (2. 74)
This same set of equations c an b e obtained directly from the stress-
strain- temperature equations, energy equation 2. 67, and heat con-
duction equation 2.36; on the basis of this remark as well as the fact
that the natural boundary conditions of the variation 2.72 are
= {[ fJ.v 0 0 0 -1 0 -1 _
n.CJ .. (u , S ) - Z .. + T 13 13 .. (C) ] e + T (C) 13 .. S }n.- T.
) 1) V fJ. 1) I' fJ. v 1) e fJ. v r e 1) fJ., fJ.) 1
S ..
1, 1
(2.75)
(2.76)
we conclude that the variational principle is valid within the region
of validity of the field equations 2.73. and 2.74.
Some special cases of the vari?-tional principle 2.72 will now
be c ons idered. For viscoelasticity without thermal effects, set e =0
in equation 2.64 and use the r es ult to write
IS * ' ~ ' ~ r *
I =...,. CJ .. e .. dB - F. u.dB - \ T. u.dA
u "- B 1) lJ B 1 . 1 .J All
T
(2. 77)
in which the stress tensor is to be expressed as a function of the
strains through the operator equation appropriate for the desired
de gree of anisotropy. In order to obtain the functional for the case
i n which there are strains due to temperature, but the tempe rature
-91-
field is prescribed, it is necessary to,first write equation 2.64 in
terrns of strains and temperature. This situation will be considered
later in connection with "non-homogeneous" variational principles.
The functional for adiabatic deformation'is obtained immediately
from equation 2.71 by setting S. = S .. = O. Furthermore, the vari-
1 1, 1
ational principle for thermoelasticity is deduced from 2.71 by simply
replacing the operators
zij , CO and
f.Lv e
o
13 ..
1J
with their corresponding
elastic constants. It is 2.1so clear that the' functional for pure heat con-
duction results by equating the displacement field to ze roo
In view of the remarks on the general principles in section 2.2,
the variational principle 2.72 can be formulated in terms of the
Laplace transforms of displacement. This is done by transfo rming
I with the help of the convolution theorem, thus
us
1
us
_ 1 S { f.Lv- - 0 -1- 0- 2 --}
- -2 Z.,. e e .. + T (C) is .. +I3 .. e .. ) +T "' .. pS.S. dB
B 1J f.L v 1J l' e 1, J. 1J 1J r 1J 1 J
S
-- S = - r--
- F.u.dB+ on.S.dA-j T.u.dA
B
11 A 11 All
e 'f
(2.78)
where all time derivatives in the operational coefficients are replaced
by the transform parameter p. It can be shown that the stationary
point of 1 is actually an absolute minimum for p real and positive.
us
Furthermore, it is clear that the Euler equations and natural boundary
conditions of
olus = 0 (2.79)
are the same equations 'one obtains by transforming equations 2.73,
2.74, 2.75 and 2.76.
-92-
Thus, it is seen that the thermo - viscoelastic variational prin-
ciple in t e rms of tra nsformed v aria bles is formally identi c al with the
one for thermo-elasticity. This leads to a correspondenc e rule in
which an approximate (or exact) transformed viscoelastic solution
can be obtained directly from a transfo rmed "associated" elastic
solution as calculated from a v a ria tional principl e; it is on ly necess a r y
to replace all material constants by the appropriate operators, e. g . ,
This same rule extends to all principles which we shall de-
rive from the basic thermodynamic ones. The correspondence rule
stated here is a generalization of the one shown by Biot for isothermal
viscoelasticity (2).
Before we discuss the complementary principle, l et us ex -
amine briefly an alternate method of calculating V; + D;, i. e ., use
of the thermodynamic definitions for the generalized free energy and
entropy production directly. This will show more clearly the close
correspondence between the basic form of the variational principle
gi ven i n section 2. 2 and the one ' derived in this section.
First, the generalized free energy density, V, (per unit volume)
is obtained by rel ating it to the Helmholtz free energy density, F.
We have the definitions
Reference to
V==U-TS
r
F == U - (e + T )S = V - es
. r
equations 1. 18a and 1. 26 shows that
' . . c e2
e (ElF) = [a . .
q
.
q
. + -T
q
]
ElT IJ 1 Jr '
qi
V = F -
(2. 80a)
(2. 80b)
(2.81)
-93-
where C is the specific he at (per unit v olume). It is impli c it In
q
the expression that the local thermodynamic state is defined by
e xcess. temperature, e, and the n coordinates q., six of which
1 .
(ql' q2"" , q6) are the me chanical strains
total free energy is
e ...
1J
Therefore, the
(2. 82)
Since the variational principle must be written in terms of e ntropy
displacement, rather than temperature, it is necessary to use the
linearized ene rgy equation 1. 66. Elimination of temperature differ ~
ence. 9, provides us with the appropriate formof the generalized
free energy,
IS Tr . 2
V
T
::; -2 {a .. q.q. + -C (S .. t f3 . q.) } dB
B
1J 1 J 1; 1 . 1 1 ..
q .
(2. 83)
The entropy production per unit volume separates into non-
, , ~
thermal and thermal components. The thermal component leads
to the dissipation per unit volume given by equation 2.42
T
_ r
D
t
= -2 A .. S.S.
1J 1 J
(2.84),
The non":therrnal contribution to dissipation is .obtained from equation
1. 55,
(2. 85)
Thus, we can write the total dissipation function as
':'This follows from the assumption that the thermal contribution to
entropy p;roduction is of a different degree of symmetry than the
other causes, as discussed in section L 8.
-94-
DT ::: ~ 2 r {b .. q.q. + T A .. 5.5. r dB
J
B
IJ 1 J l' 1J 1 J
(2. 86)
:.:::: ) : ~
It is now possible to substitute V T and D
T
, corresponding to
the functions given by equations 2. 83 and 2. 86, directly into the vari-
ational principle 2. 11. We thereby obtain a variational principle for
hidden coordinates as well as the observed variables of m.echanical
and entropy displacem.ent,
61 = 0
qs
with the definition
IS .,:' Tr , ~
I =-2 {(a .. q.+ b .. q.) q. + -c (S .. + i3.q.) (S .. + i3.q.)
q s B IJ J IJ J .'1. q I, 1 1 1 1, 1 1 1
+ T LS. S.} dB
l' 1J 1 J
S
,',
T: u.dA
All
T
(2.87)
(2. 88)
where the strains e.. [::: q . (i:= 1, 2, . . . , 6)] are to be expre s sed as
1J 1
functions of the m.echanical disp1acem.ents using equation 2. 62. It
can be shown that; the Euler equations of 2. 87 are not only the m.echani-
cal equilibrium. and heat conduction equations, but also (n-6) equations
for hidden coordinates as given by equation 1. 40. If, hQwever, we
initially elim.inate the hidden coordinates from. equation 2.88 by
solv ing the (n-6) equations, then the functional 2.88 becom.es identi-
cal with the first one we derived, equation 2.71.
13) Principle for m.echanical stresses and tem.perature:
The com.plem.entary principle for stresses and tem.perature will now
be deduced from. the basic functional 2.22. While it was possible to
use the sim.p1er form. 2. 23 in deriving the heat conduction principle,
-95-
it cannot be used now. This sterns from the fact that when we intro-
duce equation 2.19 . [2(V'T:' "" D'T:') == Q.(7)q.(t-7)). it is tacitly assumed
.11
that the internal stress field is in equilibrium. The additional term
which must be added if this is not the case is
S
[!<T.. .f F.] u. dB
B 1J. J 1 1
as seen by referring to equation 2. 61b. Thus we have
Q.q. - (V'T:' + D;T") :::: S [0" ... + F.] u.dB + (V;T" + D'T:') (2.89)
lIB 1J. J 1 1
where V; + D; is to be written as a function of stress and tempera-
ture with the help of equation 2. 64.
For the mechanical component, use is made of the stress-
strain-temperature equation 1. l03a derived in Part I. so that
" .. e .. =G .. [AJ:L.vq;- ] + !W ..
1J 1J 1J 1J fL v 1J 1J
(2. 90)
In order to express the first thermal integrand in equation
2. 64 in terms of temperature we use the energy e'quation 1. 63. After
linearizing and integrating with respect to time we find.
S .. 8 -
1,. 1
o
a .. G ..
1J 1J
(2. 91)
where CO is the constant-stress specific heat operator whose form
[(J"
is given by equation 1.64. Therefore
-8S ..
1. 1
== T8 -I- 8[ . .]
v 1J 1J
(2. 92)
r
,
The second thermal term is found immediately by using the heat con-
duction law 2. 35.
0, .
1
-0,.S. =-T
1 1
r
-96-
. . 0 .
[K.._J J
IJ P
(2. 93)
Now that . V; + D; is known as a function of t empe rature and
. stress, we can substitute equation 2. 89 into equation 2. 22 to obtain,
the complementary functiona l I!G"0'
S
U'\, .. n.cIA + S 0':'n.S.cIA
Au 1 IJ J ' As 1 1
~ : ( 0
20 [a .. Y J
IJ IJ
(2. 94)
where displacements U . are prescribed on A a nd the entropy. dis -
1 . U
p lacement (i. e., heat flow) is given on A. The me chanical dis-
. s
placement field, u i ~ ca.n b e e liminated from this functional by requir-
ing that all admis sible sta te s of stre s s in;the variationaI principle
satisfy the equilibrium equations. This is accomplishe d by expressing
the stresses as derivatives of certain s'tress fUpctions (31). How-
ever, an equivalent procedure is to interpret continuous functions
u . as Lagrange multipli:ers associated With side condition,
1
fIT . . + F. = 0
. in B
IJ, J 1
(2. 95)
This latter method will b e used since it leads to an easier proof.
As the compleme ntary principl e we state that actual stress
and temperature distributions are determined by the variational
e quation
-97':'
01 =; 0 '
luG '
(Z. 96)
where all a dmissible stresses satisfy the stre ss boundary conditions,
O'" n. =; T.
, 1J J 1
(Z. 97)
and all admissible temperature fields' satisfy the bounda r y conditions
on Ae' Performing the variation with stress es and displacements
varied independently we arrive at the' E u l ~ r equations, which are the
six compatibiiity conditions
I-lV 0 _ 1
A .. If5' + a .. e - -Z (u .. + u . . )
1J j-L V 1J I, J J, 1
in B (Z. 98)
the three equilibrium conditions,
<0' of F. =; 0
1J, J 1
in B
(Z. 99)
and the conservation of energy statement,
e, .
['K.. __ J], . =; cOe + T a?aT .. in, B
1J P 1 liT r 1J 1J ,
(Z.lOO)
In addition, the natural boundary conditions of the variation are th.e
conditions on the Lagrang.e multiplier
, u. = U.
1 1
on A
u
u.,
, 1
and the e ntropy ' displacement boundary condition,
e, .
n.K .. __
J
1 1J P
T S.n.
r 1 1
on
(Z. 101)
(Z.lOZ)
Equations 'z. 98 - Z. 102 are a complete set for calculation of
the stresses and temperature fi e l d. It isclear ,th<!.t these are the
- 98-
correct equations; the, compatibility condition 2. 9B relates str e ss-
depende nt strains to displacement -dependent strains, while equation
2.100 is simply the line a rized energy equation 2.91 combine d with heat
conduction law 2. 35.
It is interesting that if the equilibrium equations are not
identically satisfied in the varia tion, application of Lagrange multi-
pliers with the equilibrium equations gives a principle for displace-
ment s as well as stresses and temperature. In "fact, it will now be
shown that this principle is an extension of Reissner's principle for
elastic stresses and displacements (28) to thermo-viscoelasticity,
except for the constraint on boundary values. The functional 2.94
is cast in Reissner's form by using identity 2. 6lb in order to replace
the first integral in this functional. We find
= 1 r { ':' ':' [ fJ. v ) }
I -2 J -.,. .. (u. .1l- u .. ) + lIT A .. lIT dB
au . B 1J ' 1, J J, 1 1J 1J fJ. v
S
-',
+ ' u.dA
A 1 J.
T
(2.103)
where the thermal terms have been omitted for simplicity. As before,
Ti , is the stress vector on AT and' Fi is the prescribed
body force. ' Reissner's principle for viscoelasticity can be stated as
follows:'
Among all displac ements which satisfy the dis-
, placement boundary conditions and among all
stress states, the actual stresses and displace-
ments are determined by the variational
equation.
01 = 0
ou
-99-
in which the stresses and displacements are
varied independently.
(2. 104)
As Euler equations we find the six stress-displacement compatibility
c onditi ons,
1
=: -2 (u . . + u .. )
1, J J. 1
In B (2.105 )
and the three equilibrium equations
~ . -Ii' F. = 0
IJ, J 1
in B (2.106)
The natural boundary conditions are on the stresses,
1(J' n. =: T.
IJ J 1
on ( 2.107)
These field equations and boundary conditions are sufficien t for cal-
culation of the six stresses and three displacement components; It
should be noted that the admissible displacements in 2.103 must
satisfy their boundary conditions, while the stresses are not re-
stricted. However, just the converse is true in .the functional 2.94.
Nevertheless, from a practical standpoint, in using these principles
to generate approximate solutions it probably will be desirable to
choose stresses and displacements which satisfy their respectiv e
boundary conditions. In this case, the two principles are identical.
In concluding this section on the complementary principle,
it should be remarked that it has properties which are similar to
those of the displacement principle 2.72; 1. e . concerning special
-100-
lirnit cases, the correspondence rule, and an absolute minimum point.
As an example of an important limit case, consider the situation
occurring most frequently in practice in which the temperature change
due to stresses is negligible, so that the temperature is prescribed
variable in the functional 2.94 .(i. e., it is calculated prior to making
the stress analysis by using the equation for heat conduction). In
this case, the pure thermal terms in 2.94 do not affect the variation
and can therefore be omitted. Thus, the functionai becomes
S
-',
I 8 =: [u".+ F.]"u.dB
" B 1J, J 1 1
1 S . ,;, [p.v ] .
+ -2 {,(T.. A .. u . +
B IJ 1J fLY
-,-
U:" T . n. dA
1 1J J
(2.l08)
In which 8 is to be held constant when calculating the variation.
In regard to the minimum property, it can be shown that the
exact stresses and temperature make I,,8(P), with p real and
positive, an absolute minimum with respect to all stresses satisfying
the equilibrium equations 2.95 and stress boundary condition 2.97,
and all temperature fields satisfying the boundary conditions on tem-
perature. It should be added that in proving this minimum property, .
one must use the thermodynamic relations between AI;1.v, u?, and CO
, 1J IJ "
which are given in generalized coordinate notation in Part 1.
,;<
Thi s minimum character does not exist if the varied state s of stre s s
do not identically satisfy the equilibrium equations and stress boundary
conditions. Thus, the ,transform of the functional in Reissner's
principle, equation 2.103, is just stationary on the positive real
p-axis.
-101-
2.4. Determination of Non-Homogeneous Variational Pri nciples
from the Homogeneous Princ i p les for Linear Syst ems
So far in the applications we have utili zed t wo b asic variational
t he orems which are homogene ous in displacement and str e ss variables;
i. e . equation 2.11 for mechanical a nd thermal displacem ents, and
equation 2. 26 for mechanical and thermal stresses. In addition, it
was shown that the stress the ore m cou l d be derived dir ectly from
the original one for displace m e nts by making a change of variables.
Furthermore, in section 2. 3b thi s change , of variables led to a non-
hOl'llogene ous principle for mechanical displacements and stresses by
r etaining the equilibrium equations with Lagrange multipliers. An
analogous principle for t e mpe rature and entropy displacement could
h ave also been deduc e d in the heat conduction example; in this case
the he at conduction law 2. 36 would appear with a Lagrange multiplier.
Thus, it seern$ reasonable that a principle could be derived
fo r the calculation of all functions, U . -1:) e.and S.. Indeed,it
1 1J " 1
can b e shown that all of the above mentioned principles are special '
cases of this more general one .
We shall not h e re derive this general principie, but examine
brie fly t w o additional vari ational principles which are .also special
cas es . It will be seen that they are of practical importance and
r epresent companions to the homogeneous thermo-viscoe lastic
principles in section 2. 3b. The first one considered is for mechani-
cal displacements and temperatur e and the second one is for mechani--
cal stresses and entropy displacement.
-102-
a. Principle for m.echanical displacem.ents and tem.peratur e.
The functional is derived m.ost easily by m.eans of a heuristic argul"nent
using generalized variable s. Let Q,q, be divided into m.echanical
J J
variables and:therm.al variables qsQs' In analogy to the
derivation of equation 2. 22 we obtain the functional
(2.109)
where the index k indicates sum.m.ation over prescribed therm.a1
" coordinates, and index" i indicates sum.m.ation over prescribed
m.echanical forces. The" term. in the square" brackets is to be written
as a function of m.echanical coordinates and ther"m.al forces.
We now identify the therm.al energy Qsqs with its continuum.
representation
Q q == - S eniS, dA
s sA"
(2.110)
and fi .... st; use identity 2. 33 as well as 2.64 in determ.ining the disp1ace-
m.ent and tem.perature dependence of
, :.:::
(V
T
+ D
T
- q Q ); the procedure
s s .
for elim.inating the mechanical stresses and entropy displacem.ent is
sim.ilar to what we have followed with the previous cases. When
the results 'of this calculation are substituted into equation 2. 109 the
.I
functional for displacement and tem.pe rature is obtained.
I
" = 1 S .{ [ Zf-L v" ] *
ue
-2 .. e e .. -
B 1J f-L
v
" 1J
;,1 [K ,"" e, j] }"dB. S dA S d"B
iJ
P
- , u, - . , u.
A 1 B 1 1
r " T.
(2. lll)
-103-
where Tiis prescribed on AT and Si is specified on AS. The
displaceme nt and temperature fields are determined by the stationary
condition
6 I = 0
ua
(2.112)
in which all admissible displacements andtemperatur es must agree
with their respective boundary values on Au and Aa. The Euler
equations are the three equilibrium equations,
zI:L.Vu . - !3?a, . + F. = 0
IJ fL' vJ IJ J 1
and the energy equation
a, .
[K. ._J ]= cOa + T !3?e ..
IJ p . 'i e r IJ IJ
with the natural boundary conditions
[zI:L:"u - !3?a]n. = T .
. 1J fL' v 1J J 1
, a .
n.[ K.._J]
1 1J P
T S.n.
r 1 1
on
in B (2.113)
in B (2. 114)
(2. 115)
(2.116)
It is yasily verified that the se equations are the complete, correct set
for the description of the displace ment and temperature distributions.
An important limit case of 2.112 is when the t emperatu re fi e ld
is prescribed, and therefore is assumed independent of displacements.
The variational principle for this condition is .obtained by omitting the
pure thermal terms in equation 2.111 and holding the temperature fixed
in theva:dation 2.112. With the temperature prescribed, it can be
shown that all di,splacements satisfying the displacement
boundary which satisfy the equilibrium equations
-104-
make Iua an absolute minimum when p in on the positive real a xis.
However, if the temperature is a function of the displace m ent, Iua
is only stationary.
b. Principle for mechanical stresses and entropy displa ce-
ment. The functional for the mixed variables of mechanical stress
and entropy displacement is derived in a similar manne r. We find
that the field equations and natural boundary conditions for
S. are obtained from the variational equation
1
where
I
us
61 :: 0
us
:: r [lIT ... + F.]*u.dB + l2S' {[AI:".v" ]'\1' ..
J
B
. lJ, J 1 1 B 1J IL v 1J
<T
1J
and .
( 2.117a)
o -1 . o . . ':'
T [(C) (S .. +d .. IIT .. )] [S .. f 0I .. 1!1" .. ] - T S. [ A. .. S.]} dB
r iQ" 1, 1 1J 1J 1, 1 , 1J 1J r 1 1J J
-S U:!IT .. n.dA - J
r
@':'n.s.dA(2.117b)
A 1 1J J All
, u a
in wJ;tich U
i
is given on Au and @ is on AEl" The admis-
sible stresses and entropy displacements in 2.117 must satisfy the
boundary conditions on AT and AS' respectively. Furthermore,
as with the complementary principle. the displaceme,nts appearing
in the first integral are Lagrange multipliers which drop out if all
varied states of stress are chosen such that the equilibrium equations
are identically satisfied. ' We Lind that the Euler equations are the six
compatibility conditions,
(2.118)
-105-
and the heat c.onducti.on equati.on
with the natural b.oundary c.onditi.ons
.on A
u
(2. 119 )
( 2.120)
(2. 121)
which f.orrn the c.orrect, c.ornplete , set f.or the calculati.on .of s t ress and
'entr.oPy displacernent fields.
When the entr.oPy displac erne rit is pre scribed,the e x a ct s tr es s
state makes the fUIicti.onal Io-s(p) an abs.olut e rninirnurn ( with p r eal
and P.ositive) with respect t.o varied stress sta tes which s atisfy e qu ili-
briurn and stress c.oundary b.onditi.ons. As with the previ.ous n.on-
h.orn.ogene.ous principle, this minirnurnpr.ope rty d.oe s n.ot exi s t in the
general c.oupled therrn.o-rnechanical pr.oble m.. Furtherm..ore, it is
clear that the adiabatic limit case is .obtained by setting S. == 0 in the
1
functi.onal 2. 117.
2.5. C.ornm.ents .on Special Va ria ti.onal Principles when ' Di sSi pati.on
and Tem.perature Depende nt VisC.osity ar e C.onside r ed
It has been ernphasized tha t the pr evi.ous v ariati.ona l p rinciples
are valid .only f.or therm.odynam.ica11y linear s yst e m.s. a nd tha t this
assurnpti.op. requires the dissipati.on functi.on in the energy e qua ti.on
and the tem.perature dependence .of material pr.operties t.o b e n egle c t ed.
There d.oes not appear. at this tim.e, t.o be a variati.onal princi p l e
which can be used f.or the se n.online ar effects in the genera l the rm..o-
-106-
, mechanical problem. without imposing artificial constraints on
admissible variations. However. by slightly modifying the functionals
in the linear variational principles and imposing certain constraints, .
some useful results can be obtained. As illustrations of the method,
let us briefly consider two cases.
First, we shall indicate the necessary modifications of the
mechanical displacement-temperature principle presente d in section
2. 4a in order to include dissipation, but retain all other assumptions.
The only Euler equation which is changed is the energy equation 2.114.
It is seen from equation 1.119 that the energy equation with dissipation
,is
(2.122)
where D is the mechanical diss'ipation per unit volume, equation 2. 85,
m
which is to be expressed as a function of temperature and strains. Thus,
if the integral
D *
2S ) S dB
B . P ,
is added to the functional IuS' and
(2.123)
D
m is held constant in' the variation,
p
the energy e,quation 2.122 will be obtained as one of the Euler equations.
D
It is that this procedure treats 2 as if it were a known
p
thermal energy source.
Now, consider the case in which the temperature field is pre-
scribed and the viscosity is temperature dependent, but with the time-
superposition principle applicable. If the temperature
is in time, the two functionals , '2. 108 for stre s se sand
-107-
2. III for displacements, can be used directly without modification.
On the other hand, when the temperature varies with both time
and space, the se original principles must be modified in order to
obtain the correct Euler equations. The necessary modifications
are determined by examination of the effect of the temperature on
the Euler equations and boundary conditions. If the variable transfor-
mation 1. 121 is used to change from real time to reduced time, the
time dependence of material properties is removed; however new
terms with time-dependent coefficients ' are introduced thr ough the
spacial derivatives, according to equation 1. 122. Consequently, the
Euler equations and boundary conditions, as functions of reduced
time, will contain terms which are identical with those for a problem
without temperature dependent viscosity, as well as terms which arise
from the spacial variation of temperature. Thus, variational princi-
ples for mechanical variables can be obta,inedimmediately from the
previous ones, 2.108 and 2.111, if the necessary terms are added to
the functionals (as was done with the dissipation function considered
abov.e) and these te'rms are treated as known quantities in the variation;
of course,' these modified functionals depend on reduced time, rather
than physical time.
Another approach is, to use full operational-variational principles,
such as suggested by Biot (3). These can be employed for the most
general case of coupled thermo-mechanical behavior. The appropriate
functionals are obtained by simply removing the convolution integrals
, from the functionals in the previous sections, and adding the dis sipation
-108-
function,if desired . . Variation of these functionals will lead to the
correct Euler equations and boundary conditions if all time deriva-
tives, p, dissipation, and all temperature dependent properties are
treated as known quantities. Once the variation is carried out and
the Euler equations are obtained, the se fictitiously constrained
quantities are to take on their .actual significance.
Finally, as a practical point, it should be added that even
though the modified principles suggested in this section appear to be
quite artificial, it is possible to use them to calculate approximate
solutions. This follows frGm the fact that the stationary condition
makes the approximate Euler equations orthogonal to certain weighting
functionsj thus, the use of these principles is closely related to the
well-known methods of Galerkin and Kantorovich (13). Consider,
for example, the problem of calculating an approximate solution in
which the solution is assumed in the form of a series of prescribed
spacial functions with arbitrary, time dependent coefficients. By
the series into the appropriate functional and carrying
out the variation with respect to the coefficients, a set of integro-
differential equations for the coefficients is obtained. The ease with
which these resulting equations can be solved relies, of course, on the
particular problem, and the practicality of such a procedure will
depend upon further study.
-109 -
PART III
MATHEMATICAL PROPER TIES OF SOLUTIONS OBTAINED
FROM LAPLACE TRANSFORMED VARIATIONAL PRINCIPLES
3. 1. Introduction
-There are several ways in which the pr eceding variational
'principles for viscoelastic media can be used to obtain approximate
solutions. For example, the methods of Ritz and Kantorovich (13),
commonly used with the minimum principles of elasticity, could be
applied to the convolution principles; time dependence would be treated
just the same -as spacial dependence. However. for many cases, the
simplest .procedure will be to take the variation of a Laplace trans-
formed functional and thereby obtain an approximate or exact trans-
forme d solution; the time dependence is then found by inversion of
the transform. Of course, the final step of inverting the transform.s
may prove to be extremely difficult if standard. exact methods are
use_d. "T.\'lerefcire. it will often be expedient to use approximate inver-
sion tec:hniques; this subject is discussed later in Part IV. Discussion
of the convolution principles. in relation to their usefulness for the
calculation of approximate variational solutions. is als 0 deferred to
Part IV. where two examples are given to illustrate the essential
,
features.
In this Part. transformed variational principles will be studied.
First, we shall deduce the general form of time dependence of exact
and approximate solutions which are obtained from certain transformed
principles; all prescribedloads and displace;me nts (thermal and me-
chanical) are assumed to be :step functions of time applied at t = O.
-110-
This analysis is then used in a discussion of the relation b e tween
errors in approximate and exact viscoelastic and elastic solutions.
The conclusions reached in Part III apply to responses calculated
from those principles whose transform on the positive, real p-axis
attains an absolute minimum for the exact solution; namely, the
homogeneous principles 2.72 and 2.96 for both mechanical and ther-
mal variables, and the non-homogeneous principles 2.112 and 2.117
with the thermal variables prescribed throughout the body. The
remaining principles given in Part II do not fall within the scope of
the present discussion.
While all results in this Part apply directly to
bodies subjected to step inputs in time, they can be also used to
obtain the behavior for arbitrary time-dependent displacement and
load prescriptions;, one needs only to employ the Duhamel-super-
position integral. ,Furthermore, for simplicity, but without loss of
generalit"y, we have omitted all thermal variables in the actual
calculations.
Before summarizing the results, let us discuss briefly the
mQtivation for our study of time dependence. A clue to the fact that "
definite statements can be, made about the time dependence of solutions
' is provided by the general thermodynamic equations 1. 98
(3. 1)
With zero initial conditions, the transformed solution to these equations
is given by equation 2.24a
s
1 + 7 P
. s
-111-
+ C
ij
] TI.
p J
(3. 2)
If, for example, all 'forces are prescribed as step functions of time,
i. e.
r:;
t< 0
Q.
=
1
, t> 0
(3.3)
~
where the Q. are ,constants, the time' dependence of q. can be writt e n
1 1
immedia:tely as .
q
. =\ d ~ ) ( l
1 L 1J
S
-tiT
s ~
e ) Q.
J
+ C .. Q.t
1J J
or, if steady flow does not occur
-tiT
s ~
e ) Q
j
.
. (3. 4a)
(3. 4b)
Thus, as was initially observed by Biot (5), apart from the steady flow
term (C' .. Q.t) the time dependence of 'all coordinates is given by a
~ J . .
serie,s of decaying exponentials. By interpreting the observed coordi-
nates as mechanical displacements in viscoelastic media, it is expect e d
that the actual displacements would have the same time de p e nc;lence
if all applied loads are step functions of time.
In section 3. 2 it is shown that approximate displace ments
likewise have this property. Also, it is proved that this behavior
extends to mixed boundary conditions and to exact and approximate
stresses obtained from the above mentioned variational principles.
-112-
We should m.ention that a rigorous proof is m.adefor only those
responses which are r epr esent ed by a finit e sum. of spacial functions
with tim.e dependent coefficients. However. on the basis of certain
physically reasonable argum.ents. it is postulate d that this exponential
b e hav ior applies to all viscoelastic responses.
After establishing the tim.e dependence of solutions, we use the
results to exam.ine the error b e tween approxim.a te and exact v is co-
e lastic solutions. It is shown that uniform. convergence for certain
ranges of elastic constants, of a sequence of approxim.ate , associated,
elastic solutions to the exact solution, im.plies that the corresponding
sequence of approxim.ate viscoelastic solutions converges (in the m.ean)
to the exact function; this argum.ent rests on the assum.ption that certain
infinite series and im.proper integrals are absolutely .convergent or
else that the transient responses are quadratically integrable over
o .,; t < 00.
that the tim.e dependence of stress and displacem.ent
solutions is of the form. 3.4 has additional practical im.portance. For
exam.ple
t
it is known a priori that all of the singularities of Laplace
transform.ed s 'olutions are on the non-p 0 sitive real axis. Thus, in
orde r to invert a transform.ed solution exactly, it is only necessary
to study behavior on this axis rather than the entire com.plex p-plane.
Also, the form. 3.4 lends itself readily to an approxim.ate inversion
m.ethod which will be discussed in Part IV.
-113-
3.2. Time Dependence of Solutions for Step-Displaceme nt a nd Step-
.Load Inputs
a. Displacement respons e . In this section we sha ll u s e the
transformed displacement variational principle to calculate 'time de-
pendence of displacements which are given by the series
(a.) ~
u. = f. (x.)q (t) + U. (x.)H(t)
1 1 1 a 1 1
(3. 5)
In thi s expre s sion summation over o. (= I, 2, .. N) is implie d.
f ~ o . ) ( x . ) . are functions of the coordinates x. and vanish on the por-
1 1 1
tion of the boundary Au where displacements are prescribe d, qo.
are time-dependent generalized coordinates. and U. (= U . H) is the
. 11
prescribed displacement vector for which H(t) is the Heavis.ide step
function,
H(t) =
{
0,
I,
(3. 6)
t < 0
t> 0
It is not required that these displacements satis'fy the equili brium
equations or stress boundary conditions.
Let us now apply the Laplace transformed variational principle
2.79 in order 'to calculate q (p) when the f(o.) are given f unctions:
a 1
For simplicity thermal effects are neglected and zero initial con-
ditions are assumed. The appropriate displacement functional is
- - 1 S { flov- -} S --
I =." Zoo eooe dB - F.u.dB
u L. B 1J 1J flo v . B . 1 1
_ r T.ll.dA
J All
T
where T. ' and F. are ' prescribed forces given by
1 1
(3. 7)
T. =
1
-114-
T.(x.)
1 1
P
F. =
1
F . (x.)
1 1
P
and the transfonned displacement vector is
~
( )
U.
- 0. - 1
u = f. q +-
1 1 0. P
Also, from 1.100b
l
Dij(s)
p p.v
= 1
p+-
s p s
+ Dij + D'ij
p.v P P.v
(3. 8a)
(3.8b)
(3. 9a)
where each matrix in 3. 9a is positive semi-definite, i. e. for all e ..
1J
D ij{ s) e e..;::: 0;
fLv fJ.v 1J
D ij e e..;::: 0;
fLv fLv 1J
, ..
D 1J e e .. :> 0
fLv fLv 1J
(3. 9b)
but the matrix made up of the sum of those in 3. 9a is positive definite,
[
l
ij (s ) ij 'ij ]
D + D+ D . e e .. > 0; e . . e . . > 0
fJ.v fLv fLv fJ.v 1J 1J 1J
(3.9c)
s
We now write I as a function of q by using the transformed
u 0. .
strain-displacement relations
to find
where
1 _ .
e .. = -2 (u, . + u . . )
1J 1, J J. 1
(0.) _
e .. . =
1J
~
E .. ;:
1J
1 [ f ~ o . ! + f.{o.) ]
L: 1, J J, 1
1 ~ ~
.... [U .. +U . . ]
c. 1,J J,l
(3.10)
(3.12)
(3.13 )
-115-
The generalized coordinates can be found by mini mizing I
u
with resp'ect to each q. This leads to the following N linear
a
alge braic equa.tions,
in which the following definitior::s are employed,
D : ij{s) D1J
C =s {\ iJ.v
a/3 B.0 P +_1_ p
s Ps
Q = S 'Tj/3) dA + S dB
/3 A 11 B 11
T
Dij{S)
p iJ.v
1
P +-
P
s
(3.14)
(3.l5a)
(3.l5b)
It is noted that the singularities of Q/3 are simple poles (or
branch cuts if P = P (x,)) on the negative real p-axis, but if the
s s 1
boundary conditions are all on stress, Q/3
Also, froin symmetry of Zij we' have
iJ.v
is independent of p.
tional moduli in equations 3.15' have been left in the volume integrals
since properties inay be functions of x,.
1
We shall now establish the
dependence of Cia on p, and thereby the time dependence of
displacements.
First, the following theorem will be proveC::
Theorem I - The singularities of Ci occur only on the non-
a
positive real p-axis.
The proof 'will be made 'by showing that the determinant of C a/3( Ie a/3 l )
does not vanish when p is complex or real and positive. Let p=u+iv
-116-
and substitute this into 3.15 to find
where
and
I . ==
af3
1 2
(u+-)
Ps
D ij
+ fJ.v
2+ 2
u v
(3.1 6)
(3.17a )
(3.l7b)
(3.l7c)
It is noted by reference to equations 3.9 that Raf3 is positive definite
when u> 0, but it is indefinite when u < 0; also Iaj3 is positive s emi -
d e finite for all u and v.
Let us assume that I C AI = 0 and determine the permiss i bl e
at' '
values of u and v. This means that a non-trivial (real or compl ex)
solution y can be found such that
a
(R A - ivI A)y :;; 0
at' at' a
(3.18)
If the complex conjugate of Ya is denoted by Ya' multiplying equation
3.18 by Y f3 and summing yields
Since Raf3 and I a ~ are real symmetric matrices, Raf3Y aY f3 and
I Y YA are real numbers; in addition, the latter one is non-negative.
af3 a t'
-117-
First, assume that I
af3
Y
a
Y
f3 > O. But equation 3.19 cannot b e satisfied
unles s . v is zero since the left -hand side is rea l while the right-hand
side is imaginary. Now, suppose I y -y - 0 which can b e seen
af3 a f3-
from equation 3.17a to imply that
(3. 20)
But equation 3. 9c requires that this be a (non-zero) positive number
and therefore equation 3.19 cannot be satisfied. Thus, the determinant
of C
af3
cannot vanish unless p is real.
It only remains to show that there are no zeros of I C a(31 on
the positive real axis. That this is indeed the case follows immediatel y
from the fact that Raf3 is positive definite when u> O. Theor e m I
is therefore proved for the most general stress-strain relations whi ch
are thermodynamically admissible.
Further inforI:\'lation about the singularities of qa will now b e
obtained. However, in the following discussion we shall assume that
the relaxation times, p s' are independent of x .
1
This assumption
permits C af3 (defined by equation 3. 15a) to be written as
c =
af3
with the definitions
s
F(s) F I
P : ~ + paf3 + F af3
Ps
(3. 21)
(3. 22a)
(3. 22b)
-118-
and the equation for qo. becoIlles
. 1
p +-
s P s
The following theoreIll will be proved:
TheoreIll II - When the relaxation tiIlles, p s' are constant
the singularities of q are siIllple poles except
a.
at the origin where a double pole Illay occur.
(3. 22c)
(3. 23)
For the present let us aSSUIlle that Q/3 is constant. Also let
a. = 1,2, , Nand s = 1, 2, , M, so that we can write equation 3.23
as
G -q -
0./3 a. -
s=l
1
(p +-)
ps
in which each eleIllent of Go./3 is at IllOSt a polynoIllial of order
N(M+l). TheoreIll I iIllplies
R III
= G 1T (p + _1 ) r
Y
r
r::l
(3.24)
(3. 25)
where
a::
o
: relal)
is the :multiplicity
Y
r
L r
r=l
of the It follows that
qo. can be expressed as a ratio of pOlynoIllials in p given by
qa = ).}
II (p +....!... ) mr
. Y
r
r:::l
-119-
(3. 26)
The ratio multiplying Qj3/p can be written as a sum of partial fractions
if the order of the numerator is lower than that of the denominator;
if they are of equal order then q will contain an additional constant
a ..
term multiplying , Qj3A>. That either of these conditions is , always
satisfied can be shown to follow from equation 3.23 by letting p - 00.
ri" . .
If jD J [ > 0, then from equation 3. 22c we have
. !-Lv
tion 3.23 shows that qa must behave'like l/pZ.
fr (p +....!...) mr
r:::l Y r
1
as p- co
p
and eqtja-
as p - 00; hence
(3. 27 a)
If, however, [ F ~ j 3 [ ;:: 0 (which can only occur if
[ D ~ ~ [ ::: 0) then
constant as p - co (3. 27b)
Conseque'ntly, it is always possible to write qa as the partial fraction
series
where
q :::
a
- . R
~ [ I
r:::l
and Saj3 are real symmetric matrices.
(3. 28)
-120-
The order of the poles of qa can be determined by examining
the behavior of q in conjunction with equation 3.23 as p approaches
a
the roots of I Gar.>.l. Consider, then, p = E: - ~ and IE: 1 1; with
t-' '( t
P close to -'(t only the term in equation 3.28 which behaves like
-mi:
E: need be retained, thus
(3.29)
Multiplying equation 3.23 by q {3 and summing over (3, and then sub-
stituting equation 3.29 for qa yields
where p ==
\ ~ F(s)
+ F + F' _ . 1 Q- (t) m t
L +_1_
s p p
s
p - Ii (3g{3 E:
F
FI
and
F g(t)g(t) 2: 0
- a{3 a (3
F I (t) (t) >- 0
- a{3ga g{3
(3.30)
(3. 31a)
(3.31b)
(3. 31c)
The right-hand side of equation 3.30 h;;s a zero of .order m
t
, whose
value ~ u s t agree with the left-hand side. However, the latter can
have only simple zeros at all poles of:q since its first derivative
a
can never vanish in the finite p plane, which is
-[I
s
1 2
(p rf- - )
Ps
+Lz ] <0
p
(3. 32)
Thus m
t
= 1 for all finite and infinite values of '(t" It is to be noted
-121-
that this conclusion applies even when a root , '{t is equal to one of the,
relaxation time s p
, s
R e f e renc e to equa tion 3. 3 0 shows that this
equality can o c cur only if F{s) == o (this requires
I D ij (s) I == 0).
Also, e quation 3.23 indica tes tha t the d e terIllinant IG
a
(31 has a zero
at the origin i f and only if IF a(31 == 0 (which requires I I == 0).
If this latter condition e x i s ts q has a double pole at the orig in.
, a
TheoreIll II is thlis proved for the case in which Q(3 is
constant. FurtherIllore, the previous considerations lead to the follow-
/ ing tiIlle dependence of the generalized coordinates:
When
q =
a
3. When [ F a(3 [ == [F [ = 0
-1 -t/
-,[ L - (r) '{r
q == Q , '{ S (l- e )
a (3 r a(3
, r=l
(3.33a)
(3.33b)
(3.33c)
The restriction that the Q(3 are constant can be easily re-
Illoved. When Q (3 are of the general fo = given by equation 3.15b,
with p cons tant, then q
s a
1
contairusiIllple poles at - - as well as
Ps
1
at - - q (t) therefore has tiIlle dependence siIllilar to that shown
'{ r a
* '
This can be shown to be true even when (-lips) is a zero of I G
a
(3 I
by using the fact that F(s) vanishes in such a case.
-122-
in equations 3.33, except there will be additional exponentials with
tiIne constants p Also the. correspondence between the vanishing
. s
of a g iven deterIninant and the tiIne dependence indicated in cases 1,
2, and 3 above will not necessarily be the saIne. For exaInple, if
T . and F. in equation3.15bare z e ro and D
ij
::: 0, then q will
1 1 ~ v a
contain at Inost a siInple pole at the origin; hence it will not have the
terIn proportional to tiIne which is shown in equation, 3. 33co
b. Stress. response. Stresses which are derived froIn the
transforIned cOInp1eInentary principle can be shown to have tiIne de-
pendence siIni1ar to that of the disp1aceInents discussed above. We
consider .stresses that are given by
(J
1J
(a) -
::: f .. {x. )Q (t) + (J . (x.)H{t)
1J 1 a 1J ,1
in which a. is to be s wnIned out (a.::: 1, 2, 0 , N),
(3; 34)
given functions of the coordinates xi which vanish on AT wher_e
stresses are prescribed, Q
a
are tiIne dependent functions which we
shall call generalized stresses, and on AT the vector (J . n.H is equal
1J J
to the prescribed surface .force, T
io
It is further asswned that for
each a, the l ~ ) satisfy the equilibriwn equations
1J
l ~ ) . ::: (3.35)
1). J
.-
and t h ~ stresses (J . H(t) satisfy the equilibriwn equations with pre-
1J
scribed body forces F.H(t).
1
hence
(J + F '. =
1J, J 1
(3.36)
It is not required that the stres ses (J
1J
satisfy cOInpatibility or the
-123-
boundary conditions on displacement.
The Laplace transform of the complementary functional,
equation 2.94, is t with temperature neglected for . simplicity)
1
T<T- 2'
S
. . O: }dB - r u.o: .. n.dA
B IJ IJ I-'-v J A 1 IJ J
u
(3.37)
where U. is the transformed, prescribed surface displacement
1
u. =
1
"-'
U.{x.)
1 1
p
(3.38 )
d h . 1 l ' . AI-'- .. v
. an t e operatllZlna COITlP lance ITlatrlx
IJ
is given by equation 1.103a
s
l+TP
s
(3.39)
. and each ITlatrix cOITlposing A ij satisfies the same properties as .those
I-'-v
cOITlposing given by equations 3. 9b and 3. 9c.
J.J
The generalized. stresses are obtained just as ;the generalized
coordinates wer.e in the previous discussion. NaITlely, the transforITled
stresses,
<T = Q
IJ IJ a
<T
-i' ..2J.
p
(3.40)
are substituted into r and then the stationary condition or ::: 0
<T <T
provides :us with a Bet of N alg'ebraic equations for the IT , which is
. a
" B Q
a!3 .u
where we define
C ij( s)
I-'-v
1# p
. s .
(3.41)
(3. 42a)
-i24-
cij( s)
SB
{
'\' -;-;--!'fl::..:.-V
L 1+T P
s
s
(3. 42b)
The similarity between the present set of equations 3.41 and
equations 3. 14 which occur with the displaceme nt principle is evident.
However, an important difference is in the factors J:... and
p
the right-hand side which multiply the prescribed quantities
on
and
, ~
Q
p
; it is seen that TIp cannot have a double pole 'at the origin. Thus,'
in analogy with the previous theorems I and II we can state two com-
I
panion theorems:
Theorem III - The singularities of Q occur only on the
a.
non-positive real p- axis.
Theorem IV - When the retardation time s, Ts' are
constant the singularities of Qa. are
simple poles.
The time dependence of the generalized stresses is therefore
I
Q (t) = '\' T(r)e -t/A.
r
+ T
a. La. a.
r
(3. 43)
where T(r) an:d T are constants and the A. are positive constants .
a. a. r
Since a double pole at the origin does not occur there is no term
proportional to time.
* There is another point which should be mentioned concerning the
behavior of TIp at p = 00. It is noted from equations 3. 41 and 3.42
that if Ui'* 0 and [C ~ i J I = 0, then TIp is a non-zero constant at
p '" 00. Since this leads to an infinite (delta function) stre s s at t = 0
we shall rule this out in all of the follOwing work by as surning
Ie tij I > O. However, ' the analysis could be extended. if de sired, to
flv
include this singular behavior.
-125-
c. Generalizations. Strictly speaking, the theorem.s in
sections 3. 2a and 3. 2b apply to a restricted class of stresses and
displacements. To reiterate. theorems I and III are rigorously valid
for only those approximate and exact solutions which can be expressed
as finite sums of terms. each of which is a simple product of a space
dependent function and a time dependent function. Except for this
restriction, they are valid for the most general s6:ess-:strain rela-
tions consistent with thermodynamics; it should be noted that the
integral
rooo
is replaced by J,
(see first footnote in section 1. 7a) in which
can ' be used also. However, theorems II and
reg.uire, in addition to the re striction cited above, that the finite
sum I representation of the stress-strain relations be used.
the relaxation and retardation time s must be constant;
this requirement can actually be modified in that these time constants
need be, constant only over regions of a body, since the total
volume integrals in equations 3.15 and 3.42 then separate into inte-
grals each having constant values of 7s or Ps'
When the above mentioned conditions are satisfied, we showed
that the time dependence of all solutions is given by a series of de-
caying exponentials, with possibly a term proportional to time. In
"this section we shall discuss the time dependence of generalized
coordinates and stresses ,_when the restrictions are removed. The
most general behavior which is expected will be first postulated and
then the statements will be justified, to s orne extent, on physical
grounds and by showing that the correct result is obtained for special
-126-
cases.
Consider now the following postulates for stresses and dis-
p1aceme_nts in media which are subjected to loads and displacements
that are step functions of time. Exact solutions as well as approxi-
mate solutions obtained from the transformed variational principles
are applicable.
Postulate I - Thermodynamically admis sible displaceme nts
and stresses in bounded media ,can be repre-
sented by finite or uniformly convergent infinite
ser,ies as given by equations 3.5 and 3. 34,
respectively.
This postulate appears to be reasonable since one expects
that a thermodynamically admissible stress or displacement field
can always be approximated arbitrarily well by a finite number of
spacial functions (Fourier series, for example), each with a time
dependent coefficie'tl.t. Also, it is exp:cted that when the body is un-
bounded. the discrete sum is replaced by an integral (e. g. Fourier
integral).
Postulate, II - ' The most gene ra1 time dependence of the
generalized coordinates is
S
ao - tiT I
q = S (r)e - d'T + S + S t
a a .< a a
. 0 ,
(3. 44a)
' and the generalized stresses is
Q = seo T (T)e -tiT d'T + T
a 0 ' a a
(3. 44b)
where S (T) and T (7) are spectral distri-
a a,
-127-
butions of the variable '7 which ITlayconsist .
entirely or .partly of Dirac -delta functions.
It is clear that the finit e degree of freedoITl case s considered
in sections 3. 2a and 3. 2b are obtained iITlITlediately by setting
where 6( T)
S (7) =
u
I
S(r)6(T_'{ )
u r
r
T (7) =
u
I
T(r) 6(T-A. )
u r
r
is the Dirac delta function,
6( 7) = 0, ,. if. 0
s 00 6(T)dT = 1 .
-00
Thus, the forITls 3. 44a and 3. 44b for the generalized .variables
(3. 45a)
(3. 45b)
(3. 46) .
appear to be a natural :e.;xt e nsion of the earlier results to probleITls
for which finite SUITlS ar e r e place d by integrals and infinite SUITlS.
As additional justification of the postulate, let us exaITline
the case in which certain orthogonality pr operties exist; it will be
sufficient to consider just the stress principle. Suppose that the
spadal functions i?") in equation 3. 34 satisfy the orthogonality
. IJ . .
property (see equations 3.41 and 3.42)
6 uj3 is the Kronecker delta
6 =
uj3
[
1 "
o ,
u=j3
9- if. j3
(3. 47a)
(3.47b)
-128-
and the finite sum ( 2 ~ ) in Aiv is to be replaced by an infinite
integral (SOO). W i ~ h this orthogonality property the set of equations
. 0
3.41 reduces to the uncoupled set
(13 not summed) (3. 48)
The time dependence of QI3 is obtained immediately from Stieltjes
transform theory (32) (a,ssuming certain rea'E;onable convergence,
properties of the improper integrals) and it is found to be the same
/ as given by equation 3. 44b.
With the postulates appearing to be valid, we now turn to an
error analysis which makes direct use of them.
3. 3. Relation Between the Error in Approximate Viscoelastic and
Elastic Solutions
It will be assumed in the following calculations that all infinite
series and improper integrals are absolutely convergent in order to
perform the necessary operations, such as interchanging order of'
integrations. Strictly speaking, such an assumption should be verified
for each particular problem but we shall not pursue this point any
further. Use of Postulates I and II in connection with the series 3.5
and 3. 34 yields the most general time dependence of both approximate
and.exact displacements and stresses,
(3. 49a)
(3. 49b)
where, by assumption
u. = fT = 0
1 1J
-129-
for t < 0 (3.49c)
For the presentlet us omit the steady-flow term in u
i
' and
concern ourselves with a representative exact solution
S
oo ' /
-t.,. ,
ljJ == <,0 (T}e d.,. + ljJ (00)
0 e e
(3. 50a)
and approximate solution
r.OO /
-t .,.
tj; \ cp (7)e dT + ljJ (00)
a. J
O
a a
(3.50b)
where ljJ, ljJ(oo), and <,0(7) are implied functions of x.
1
which repre-
sent the corresponding functions in u. or 0"" ; i. e. , ,J, is to be
, , 1
interpreted as either a stress or displacement in order to simplify
the notation: It will also b,e convenient to define the t;ansient com-
ponent of the solutions as
square error, ; = (L::.ljJe
L::.ljJ == ljJ - ljJ(oo). Let us now integrate the
- L::.ljJ )2, over all positive time' to find,
a , '
2 [ , J [ - 1 - 1 J
= <,0 (7) - <,0 (T) L::.ij; (-) - L::.ljJ (-) d.,.
-0 e a e 7 a ,.
(3. 51)
where L::.ljJ ( .!..) denotes the transient component transformed with
, 7
respect to the transform parameter = p,and it is assumed that
2,< 00. Now with p on the real axis, p[ ljJ(p)] (=pL::.ljJ+ljJ(oo) }
is identical with an elastic solution whose, elastic moduli or com-
pliances are numerically equal to
fJ."
" A' ij
or .
fi"
Furthermore, ljJ( oo}
is elastic solution with moduli which are equal to Zij evaluated
, , fill
at p = O. Therefore, an important implication of equation 3. 51 is
-130-
- 1
if tVa(-r} is one solution out of a sequence of approxim.ate elastic
' solutions which is known to converge uniform.ly to the exact solution
(with respect to 'elastic constants whose values range over the values
taken by Z ij or Aij on the real interval 0,,;; F < oo), then the
lJ-v lJ-v
corresponding sequence of tim.e-dependent solutions converges (in the
m.ean) to the exact solution. An equivalent statem.ent is that the
integrated square error in the tim.e-:-dependent solution is "sm.all"
when the error on the real p axis is suffiCiently "small" for
o :;; p < 00.
The foregoing conclusion can be readily m.odified to allow for
the presence of a steady-flow term. in thedisplacem.ents, equation
3. 49a. It is only necessary to redefine tV as
(3. 52a)
and L.tj; as
L.ljJ == tV - 'tV( oo} - tV't (3. 52b)
, "
With the exception of tVa' the convergence of tV to tV is established
a, ,e
by the previous, results. However, sequence of values for
will also to the correct value if a sequence of associated
elastic solutions with com.pliances equal to c
ij
(see equation 3. 39) '
f1 v ,
has this behavior. : This follows from. the fact that the long-tim.e
. ' ,
(p - 0) value of , tV is the sam.e as an elastic solution with com.pli-
ances (pAiv)p=o.
The 'problem. of relating error in tim.e dependence to error
......
-131-
. on the positive real p-axis can be approached in another way.
Erdelyi (33,34) has derived Laplace inversion formulas from which
I it is possible to expand Li.lj;(t) in an infinite series of orthogonal
. .
functions, whos.e coefficients contain Li.ljj(p) 'evaluated at a discrete
number of positive, real values of p. The only assumption needed
is 'that Li.lJ;(t) be quadratically inte grable, i. e.
S; [Li.lJ;(t)) 2 dt < ro
(3. 53)
which is satisfied by.6.lJ; and Li.lJ; if the . integral$in equations 3.50
e a
are absolutely convergent. These inversion formulas. . lead to the same
conclusion deduced from equation 3.51; namely, a "small" error on
the positive real p-axis implies a "small" integrated square error in
time.
It is clear that the error estimate, equation 3.51, cannot
actually b,e used in practice since it ,requires knowledge of an exact
solution. Rather, it only indicates that a close approxi-
mation to an associated elastic solution leads to a good approximate
viscoelastic solution. It is felt that a stronger statement of what
is meant'by " sufficiently close" must cOD;le from a study of numerical
examples. nlustrations along these lines will be discussed in Part IV.
-132-
PART IV
APPROXIMA TE METHODS OF TRANSFORM INVERSION AND
NUMERICAL APPLICATIONS OF VARIATIONAL PRINCIPLES
4. 1. Intr oduction
Som.e num.erical exam.ple s are given in this Part which illustrate
the use of variational m.ethods in obtaining approxim.ate viscoelastic
solutions. It has already been m.entioned in Part II how a transform.ed
viscoelastic solution can be easily derived from. an associated elastic
problem. by applying the correspondence rule. However; as the re-
suIting transform. is often very difficult to invert, particularly when
realistic m.aterial properties are used, we shall first proceed to
develop two m.ethods of approxim.ate inversion, each of which can
often be used as a check on the other, followed by specific illustrations
of the variational process.
The first one considered in section 4. 2 is a m.odification of
Alfrey's rule (9), and the second one is a collocation m.ethod which
is based on the characteristic tim.e dependence deduced in Part III.
Withboth techniques it is necessary to know only an associated elastic
solution num.erically for certain ranges of elastic constants and nu-
m.erical values of the operational properties (e. g. m.oduli) for real,
positive v alues of p. It will be seen that a tim.e depende nt solution
can be calculated with very little effort once these num.erical quantitie s
have been determ.ined. Exam.ples in which the results of these approxi-
m.ate inversion m.ethods are com.pared to exact inversions are pre-
,
sented in an expanded version elsewhere (35).
-133-
In addition to the usefulness of the first ; method for inversion,
for those problems in which it is applicable, it also shows that if the
associated elastic solution is approximate, the "quality" of the visco-
elastic response is essentially the same as that of the elastic problem.
Section 4. 3 gives two numerical examples using transformed
variational principles. In the fir st we derive the approximate trans-
forme d displacementsin a thin plate with mixed boundary conditions.
The transform is deduced using an elastic solution obtained from
the potential energy principle. This example serves two purposes.
For one, the transform, though a very eomplicated function of an
operational Pois son's ratio, is easily i:riverted to illustrate the sim-
plicity of the approximate methods given in section 4. 2. Secondly,
the type and location of all singularities of the transformed solution
are examined. The findings are not used to calculate the exact in-
version (of the approximate transform) numerically, but to show that
the analytical form of time dependence agrees wi,th that predicted from
Postulates I and II in Part III.
As the next example, we use the compleme ntary energy prin-
ciple to calculate approximate stresses in a long, symmetrically
loaded cylinder. Two different app:roximate solutions are obtained
and both the elastic and viscoe lastic' responses are compared with
the exact stresses. This comparison serves to illustrate the relation
between elastic and viscoelastic errors which was discussed in Part III.
To conclude our discussion on numerical applications of vari-
ational principles, it is suggested in section 4.4 how convolution
-134-
principles can be em.ployed when it is not convenient or possible to
use the Laplace transform. m.ethod. This latter situation arise s when
assum.ed solutions are nonlinear functions of generalized coordinates
and is illustrated by a one-dim.ensional heat conduction problem..
4.2. Approxim.ate Methods of Laplace Transform. Inversion
a. Direct m.ethod. Of the two inver sion technique s to be dis-
cussed, the sim.plest one is calle d the "direct m.ethod" . It will be
shown to yield good results when the derivative of the tim.e de pendent
solution with respect to logarithm.ic tim.e, log t, is a slowly varying
function of log t. A m.odification of this m.ethod is also suggested in
order to handle functions for which the derivative of their. logarithm.
has this slowly varying property.
'The problem. which we pose is to find an approxim.ate repre-
sentation' of a viscoe lastic response, ljJ(t), from. the integral equation
. tjJ(p).= S: ljJ(t)e -ptdt (4.1)
where ljJ(p) is the Laplace eransform. of ljJ(t) and is known at least
num.e rically for all real, non-negative values of the transform.
param.eter, p. Let us represent pljJ(p) as a function of log p and
:'
define
/\ -
f( u) == pljJ(p) u == log p (4. 2)
f(v) == ljJ(t) v == log t (4. 3)
w=u-tv (4.4)
-135-
which renders 4.1 in the form.
1\ roo W _lOW
f{u) = In 10 \ f{w-u)lO e dw
J_
oo
(4. 5)
where In =: loge and log =: 10glO
w
The weighting function, 10 we -10
, ~
is drawn in figure 4. 1
which shows that it is practically a delta function if f{v) changes
slowly enough. This behavior im.plies that an approxim.ation to f{v)
A . . w _lOW
can be obtained directly in term.s of f{u) by replacing (In 10)10 e .
with. 6{ w -:: w 0)' i. e. a Dirac delta function located at the point
which will yield an approxim.ate inversion form.ula
w
o
'
(4. 6)
The point Wo is som.ewhat arbitrary in view of the spread of the
weighting function, which is about two decades. However we shall
now calculate the "best" value to use when f is closely approxi-
m.ated by a straight line in the two-decade interval, Iw-w I'<l.
o
To do so, we first expand f(v) in a Taylor series about the point
v 0 (=: Wo - u)
, . 1 . 2
f(v) = f{v 0) t f (vo)(v-v 0) t 2" f"{v o)(v-v 0) t ...
(4. 7)
where prim.es denote differentiation with respect to v. Substitution
of this expression into 4 . 5 yields
./'. , S 00 lOW
f{u) :::. f{v ) t (In 10)f (v) (w-w )lOw e - dw
o D 0
-00
(4. 8)
, ~
All figures for part IV follow the text.
*>!< !
It should be recalled that a delta function is defined such that
6{w-w
o
) ::: 0 if w*w
o
; SOO 6{w-w
o
)dw = 1
-00
-136-
where only the constant and linear t e r m in 4. 7 have been retained.
It is seen that the approxim.ate inversion form.ula 4 . 6 is obtained if
the integral in 4. 8 vanishes. This condition locates w at the
o
w
centroid of the area under the curve lOwe-lO , which is
w = SOO(lOg t)e -t
dt
Z ;n
o 0
where C is Euler's constant
S
oo ' -t
C "". - 0 (In t)e dt
0.58
(4. 9a)
(4. 9b)
When this value for w is substituted into 4. 6 and the re-
, 0
sulting expres sion is written in term.s ' of the original functions tj;(t)
* aI:'.d ptj;(p) by using 4. 2 and 4. 3, we find
tj;(t).!::! [p.p(p) 1 -c
p= e /t
(4. lOa)
.:.C
where e .::: O. 56. Due to the ske wed form of the weighting function
it has been found thfl,t a somewhat better formula is
.p(t)::::. [p.p(p) 1
p = O. 5/t
(4. lOb)
I
It is important to recognize that 4. 10 was derive d using an
about the exact inve rsion .p(t), rather than p.p(p). Since
i n practice this solution will not be known, one m.ust assume that if
* .
The fact that the weighting function is essentially ze ro except for a
two decade interval suggests that this inversion will be good at thos e
times in which the solution (as a function of log t) is linear, or nearly
.linear, for at least two' decades. If there is a strong curvature, but
it is somewhat rem.oved from this linear region., this curvature
should not pr oduce a significant error in th,e linear porti.on.
, -137-
d?(pljJ(p) )/ d(log p)2 is small for p ~ 0 then ljJ{t) behaves similarly.
In the event that log pljJ(p) is essentially linear for several
decades of p when plotted against log p, so that ' pljJ(p).::: Apm, it
can be easily verified for m < I that an approximate inversion for-
mula is
ljJ( t) (4. 11)
where
a=
S
OO -m -x'
x ' e dx
o
While thi s further approximation is valid for many practical situations,
it shouldbe recognized that the approximation is poor when m is
close to + 1.
Finally, we should remark that there does not' appear to be
an easy method of qua.ntitatively estimating the error involved in this
direct method. However, there are some qualitative techniques
which can be used. As one, the approximate solution could be
transformed numerically or analytically, and then compared to the
original transform for p ~ O. If the approximate solution is physically
acceptable, and its transform is relatively "close" to the original one,
it is reasonable. to assume that the error in time dependence is small.
Another check 'on the direct method can be made by comparing the in-
version to the result. of the method which will be discussed next.
b. Collocation method. We would now like to discuss a second
technique which is not as simple as the direct method, but it has other
,
advantages. For one, it is not restricted to functions whose derivative
-138-
is slowly varying with respect to logarithmic time. In addition, the
time dependence is given by a simple series of exponentials which
can be used readily in the Duhamel integral for the calculation of
responses to prescribed loads and displacements that are not step
functions. Thirdly, the accuracy of the inversion can be improved
by adding more terms to the series.
In Chapter III it was argued that with prescribed quantities
as step functions of time, the transient component of both exact and
approximate solutions obtained from minimum principles can be
expressed in terms of an integral
S
OO -tiT
.6tj; ::; <p( T)e d T
o .
(4. 12)
Assuming that this integral is absolutely convergent, a Dirichlet series
n
::;
o
l
i=l
-t/y.
1
S.e
1
(4. 13)
can be used as an approximation to the solution '.6tj;(t). The present
method makes use of this series for which the 'Y. are prescribed posi-
. 1
tive constants, -and the S. are unspecified coefficients to be , calculated
1
by minimizing the total square error between .6tj; and .6tj; .
o
The total square error is
* Actually, a neces sary and sufficient condition on .6tj;(t), in order that
it can be expanded in a Dirichlet series, is that it be quadratically
integrable i. e .
. (.6tj;)2dt< 00
which is less restrictive than requiring that it be given by the absolutely
convergent integral 4. 12.
-139-
(4. 14)
with the minimization yielding
i =- 1. 2, ... , n (4. 15a)
so that n relations are obtained between the Laplace transforms of
.6ljiD(l/y.) = .6lji(l/v.); i:::: 1.2, ... ,n
1 1
(4.15b)
-1
A more convenient form is obtained by multiplying these by Y
i
which yields
"fp.6ljiD (p)] / = [p.6lji(p) ] ; i = I, 2, ... ,n
p=l y. p=-l/y.
1 1
or explicitly
n
l
j=l
S.
_J_ =
y.
1+_1
Yj
[p.6lji(p) ] ; i = I, 2, ... , n
p=l/y.
1
(4.16a)
(4.1pb)
These equations are suffiCient for calculating the coefficients S. and
. . . J
hence dependence of the transient component .6lji(t). To obtain the
total solution, equation 3. 52.a, the constants lji' and lji(co) are
evaluated exactly by examining the behavior of pljJ(p) and p2 , ljJ(p) as
p tends to zero.
Thus, we see that the total square error is minimized by col-
locating the p-multiplied transform of the Dirichlet serie s 4.13 and
,
an as sociated elastic solution, p.6ljJ(p), at n points
p = l/y ..
1
With
,JAD-
this elastic solution given nUITlerically or graphically for 0 <: P < 00,
-,
suitable values of y. can be prescribed siITlply by inspection.
. 1 .
With regard to the er r or involved in this approximate ITlethod, .
it is of interest to calculate the total square error, E2, by using 4.14;
( 4.17)
which becoITles
2 seo . [1 1 } In [- 1 1 ~
E = <peT) 6.ljJ( -) - 6.ljJD( -) dT - S. 6.lj;(:-=-1 - 6.ljJD(- .. )
o . T '( 1 y. y .
. i=l 1 . 1
(4.18)
It is iITlportant to recognize that the sUITlITlation in 4.18 does not neces-
sarily vanish unless exact values of 6.lii(J....) ' are collocated with
y.
_ 1 _ 1 1
6.lj;D(-). Since, in practice, 6.lj;(-) can be evaluated only within
Yi . Yi
certain numerical accuracy , this sUITlITlation is generally not z e ro.
Consequently, if an approxiITlate inversion has been obtained with n
terITls, and itis desired to reduce the square error by using additional
terITls, it will be necessary in SOITle cases to evaluate the transforITls
6.lii{J....) and 6.ljJn(J....) with increased accuracy. When the transforms
Yi Yi .
are calculated with enough numerical accuracy, equation 4.18 shows
that if they are Il sufficiently close
ll
for OS p < eo, the total square
error of the approximate tiITle dependence is IisITlall. n
* .
Of course, it is possible to deterITline the paraITleters Yi by minimiz-
ing the square error 4.14 with respect to each one. However, this
procedure leads to a nonlinear set of equations in which the slopes of
6.liiD and 6.lii are collo<;:ated at the (initially) unk nown points l/Yi.
Thus, it g enerally will be desirable to choose enough values of Yi s o
that the slope condition is (closely) satisfied using only equation 4.16.
-141-
4.3. Numerical Applications of the Transformed Variational Principles
a. Displacement principle-Kantorovich method. We shall now
solve, approximately, a two-dimensional mixed boundary-value prob-
lem. The method of Kantorovich (13) will be used in conjunction with
the potential energy principle to first calculate approximate elastic
displacements u and v. The elastic constants will then be replaced
by the appropriate viscoelastic operators and the boundary pressure
by a transformed pressure. On the basis of the correspondence rule
this procedure yields the approximate, transformed viscoelastic solu-
tion. The inversion of this solution will be accomplished by using the
approximate methods in section 4.2.
Consider now the thin plate shown in figure 4.2, which is loaded
by a uniform tensile stress, . (T , on the edges x = + a, and clamped
o -
along the edges y = ..:!: b. It is assumed to be in plane stress and com-
. posed of homogeneous, isotropic, material.
a) Elastic solution: The elastic potential energy V E' is
WdB - (' T.u.dA
J A 1 1-
T
(4.19)
where W is the strain energy density, and all other symbols have been
defined in earlier The stationary condition, 6V E= 0, is used
to determine approximate displacements which are taken in the form
. . 2
u = b(l - " )f(p)
2
.v = b(l -" )"g(p)
where we define the dimensionless variables
" ;: Y..
b
p = x
a
(4. 20a)
(4. 20b)
(4.20c)
-142 -
It is seen that u and v bot h s.a tis fy the displacement boundary con-
ditions (as requir e d by the variational principle) and have physically
r easonable depende nce on y . f(p) and g(p) are unspecifi e d functions
which are calculate d from the Eul e r equations and natural boundary
conditions of OV E::; O. The s e equations for f and g coul d b e ob-
tained by substituting equations 4.20 directly into equation 4.19 and the n
taking the variation. Howeve r, the s ame set results from the simpler
procedure of substituting them into the general stationary condition (13)
::; -S {(T )ou.dB + S T .. n.-
B 1J, J 1 A . 1J J
T
T.)ou. dA
1 1
(4. 21)
in which the stresses are to be expressed iIi terms of the displacement
by using the stress-strain e quations,
(T ::; l2p.2
v
0 .. 1..9-'+ 2p.e ..
1J - V 1J 1J
For the problem at hand, the variations ou. are,
1
2
ou ::; bel - "l ) of
2 '
ov::; bel - "l )nog
the equilibrium equations, (J . . 'J
1J,J
are
(4. 22)
(4. 23)
(4. 24)
where A. is the aspect ratio, A. == -S' and the applied surface force,
is T ::; (T
x 0
Substitution of equations 4.23 and 4.24 into the
stationary condition 4.21, and noting that of and og are arbitrary,
yields the Euler equatiop.s
a(T aT 2 "
( at + A. a"lx
y
) (1 - "l ) d"l. ::; 0
(4. 2 5 ~ )
-143-
(4. 25b)
and natural boundary conditions
11 2
J
(0- I - 0- ) (1 - 'l1 )d'l1 = 0
o x p=l 0
(4. 26a)
S
l 3
'r I. ('l1-'l1 )d'l1 = 0
o xy p=l
(4. 26b)
By expressing the stresses in terrns of displacements, u and v,
and performing the indicated integrations, we obtain two differential
equations corresponding to equations 4.25
(_8_)f" _ ~ (1 - v)f + _2_(1 +V)gl = 0 (4. 27a)
15,,23 15"
(i-v) 4 ,,4 2 (1 + v)f' = 0 (4. 27b)
,,2 IQ5'"g - sg-BI"' -
where primes denote differentiation with respect to p, and th.e boundary
conditions at p = 1,
2
f _ 3 - 34v+3v
147A2(l+v)
,,(l-v) (4. 28a)
f" + S(l-v) fiv = 0
147A
4
(l+V) .
(4.2ab)
It is also required, on the basis of the symmetry in this problem, that
f is an odd function of p while g is even.
The solution of these equations leads to the displacements,
(4. 29)
v
0" b
fJ.
where
2
-3+34v-3v
42{1 +v)
2
ljJ ::: l7l-34v+3v
I-v
-144-
(4.30)
and the involved dependence upon the ratio is. especially to be
noted. As a typical result, u/(O" b/fJ.) is plotted in figure 4.3 against
o
v fo r O. 3 :5 v :5 O. 5 and T]::: 0, p ::: 1, A.::: 1.
13) Viscoelastic solution: To illustrate calculation of
the viscoelastic solution, it is sufficient to consider just the u-dis-
placeITlent with T]::: 0, P = 1, A.::: 1. The transforITled response is
found iITlITlediately by the correspondence rule. With the stress
as a step function of time, its transforITl is
u(p) ::: 0" b J(p) (p)
o p
0" /p and we have
o
0"0
(4.32)
where it is convenient to introduce the operational shear
J(p) = l/iJ,{p), and the function '(i(p), which depends on
p through only Poisson1s ratio according to equation 4.29, viz.,
/
(4. 33)
-145-
in which we have set A;::: P = 1, T) = O.
/\
Since u is numerically equal to the elastic displacem.ent shown
in figure 4.3, for engineering purposes it could be taken as a constant
and evaluated as som.e average value of Poisson's ratio. Then, recog-
nizing that J{p)/p is the transform. of the creep com.pliance in shear,
J (t), which can be m.easured experim.entally, the displacem.ent would
c
be sim.ply
A
u{t) .::::. (J buJ (t)
o c
(4.34)
Furtherm.ore, for typical polym.ers, J {oo)/J (0);;:' 1000 so that in
c c
this case the error due to taking ~ as a constant would be very sm.all
relative to the total variation of u{t).
However, since one of our objectives is to illustrate the sim.-
plicity of the approxim.ate inversion m.ethods by inverting a transform.
which is a very involved function of m.aterial properties, we shall con-
A
sider here the p-dependence of u. In order not to m.ask this dependence
by the large variation of J{p), 'G: will be inverted separately. Also,
it will 'be sufficient for our purposes to calculate just the tim.e dependence
A ,
due to the variation of u because the total solution can always be ob-
tained by using 'the inversion rule for transform. products, which pro-
vides the result
u{t)
(J b
o
where we have defined
Ii)p) =
dJ (T)
c u (t-7") dT'
dT v
(4.35a)
( 4 ~ 3 5 b )
-146-
which is the function that will be inverted.by the approximate methods.
These methods require that v(p) be calculated for 0::5 P < roo
We shall assume that the bulk modulus is constant and evaluate v(p)
from the well_known relation for isotropic bodies (13)
v(p)
= 3K- 2f1.(p) . =
2(3K +J.L(p) }
3KJ(p) -2
2{3KJ{p} +1}
(4. 36)
in which J(p) is to be calculated from exper.imental shear data obtained
on glass':filled polyisobutylene unde r constant frequency sinusoidal
,r loading (36) .! Such a test provides the frequency dependence of the so-
called complex shear complianc e , which is equal to the r atio of the
Fourier transformed strain to. the t'ransformed stress. The real part
of the complex compliance, J'(w), is plotted in figure 4.4.
One possible way of determining the operational compliance,'
J(p), is to express J(p) as an integral of J'(w). The integral is derived
by first writing the ' creep compliance as a Fourier sine transform of
,
J (w) (32) and then taking the p-multiplied Laplace transform of the
creep compliance. Omitting the details of this calculation we find
J(p)
Real p :> 0 (4. 37)
J
where J G == J (ro), which was numerically integrated for p real and
, ~
positive, and the result is shown in figure 4.4. It should be noted that
this integration is quite laborious because
J
J (w)' varies over many
decade s of frequency. In fact, it was found that an alternate model-
* . .
The similarity betweej:l the curves occurs b ecause the integrand in
equation 4 . '37, when expressed in terms of log w, contains a weighting
function which is similar to the one in the Laplace transform, figure
4.1.
-147-
fitting scheme (37) provided the same results with much less effort.
Substitution of J(p) from figure 4.4 into equa tion 4.36 yields the
operational Poisson1s ratio which is shown in figure 4.5, with K
chosen such that v(p =:: co) =:: 0.3.
The transformed displacement puv(p) is now readily obtained
for p real and positive by combining the curves in figures 4.3 and 4 . 5
and recognizing the fact that
(4. 38)
The direct method of inversion yields immediately
(4.39)
whose transient component, I::.u
v
== u
v
- uv(co), is plotted in figure 4.6.
An alternate approximation will now be calculated by the col-
location method. An approximate solution is assumed in the form
u (t) =::
v
D
where, by definition
and
S ::::
o
S
o
n
l
i=::l
-t/y.
S.e 1
1
-t/y.
S.e 1
1
0.285
(4.40 )
The coefficients, S., are calculated from the system of equations
1
4.l6b, with pI::. U
v
rep+acing pI::.ljj(p).
Examination of pI::. U
v
in figure 4.6 indicates that a five-term series (n::::5) will provide a
good approximation with
-148-
y . = 1O-(4+i)
1
1=1.2 . 5
Substitution of thes e time consta nt s a nd val ues for r pt; U
v
into 4.l6b yields the following system of equations:
I
"2
I
r.-l
I
T.OT
1
1. 001
1
1.0001
1
IT
1
"2
I
r.T
I
1. 01
1
1. 001
1
TOT
I
IT
1
"2
1
T.l
1
T.OT
1
TOOT
I
TOT
I
IT
1
"2
1
r.T
1
10001
I
1001
I
TOT
1
IT
1
"2
(4. 41)
J
p=l/y .
1
-+0. 20
+lg 20
+6.25
+8.20
Because of the relative magnitude of the mat rix elements, this system
can be easily solved by iteration to yield the time dependence
[
. 5 6 7
t;u ::: +10-
2
0.3le-
10
. t+i.24e-
10
t+
2
72e
-10 t
v
D
8 9 '
+4.698-
10
t_
0
60e
-10 tJ
::: U (t) -0.285
v
D
(4.43 )
which is plotted in figure 4. 6. It was found by adding two additional
terms to the series 4.40, with.
. -8.5 . -9.5
y ::: 10 and 10 ,only the short
time behavior of Llu
v
was affected slightly as shown in the same figure.
. D
In view of this, as well as the reasonably good agreement with the
direct method, it can be assurned:that the Dirichlet series 4.43 repr e -
sents a good to the exact inversion.
y) Behavior of the transformed Let us
now determine the form of time dependence of the u-displaceme nt
-149-
by examining the singularities of its transform, and thereby determine
whether or not it agrees with the behavior 3.50 deduced from Postulates
I and II. In order to simplify the analysis, we shall consider the semi-
infinite plate shown in figure 4.7.
For this geometry, the transformed displacement is found
from equation 4.29 and written as
u/rr b == J(p) ~ ( p )
o P
with 'J.{p) given by
where we have defined
<fJ
z
=
and all a., (i == 1, ,4) are given in equation 4. 31.
1
(4.44)
(4.45)
(4.46)
/\
First, the singularities of u{p) which are seen to depend on p
only through Poisson1s ratio,will be' found. The square roots appear-
ing in 0.
1
and 0.
2
give branch points at
v == -13. 9 ~ -2.13, 36.3, 2.33, 1, + 00
o
(4.47 )
" Also, after a considerable amount of algebra, it is found that u{v)
diverges only at the values
v == 2.33, ' -2.76, -1
o '
(4. 48)
which are simple poles in the v -plane.
-150 -
All of these singularities will now be shown to lie on the negat ive
real axis in the p-plane by using the relation 4.36 and analytical repre-
sentations of K(p) and f.L(p) as derived from thermodyna mics. Sub-
stitution of equations 1.108 and 1.112 into equation 4.36 yi e lds
s
(s') ( s')
3K (1-2
v
)- 2j-l (l+v) + 3K'(1-2V)-
1
(p + -)
P s
I
2j-l (l+v) = 0 (4.49)
where p , can be a positive real number or infinite and all coefiicien ts
. s
K(s), f.L (s'), Ki and j-l' are positive. This equation must hold for all
values of v, and in particular the real values given by equations 4.47
and 4.48. Setting p = r + is and equating the imaginary component of
4.49 to zero gives the restriction on s,
(4. 50)
s'
S
11 ' l ' f < 1 d 1. . 1 h
Ince a slngu ar pOInts occur or v,_ - an V> '2 It IS C ear t at
s must vanish at these points. Also, by setting the real part of 4.49
1\
to zero we find that r must be negative at the singularities of u(p).
" Thus, all poles and branch points of u(p) occur on the negative real
p-axis.
1\
In order to determine the behavior of u(p) in the nej.ghborhood
of each singularity (except v = .::: co) we calculate dv /dp f r o ~ equation
. "r' ".
4.49 to find
dv 1
=
dp '2
s'
3K(s') + j-l(s) r ,
---1;-'-- + 3 K + f.L
(p +-)
Ps
(4. 51)
-151-
which does not vanish at any of the values, v , ln equations 4.47 and
o
4.48. Thus, in the neighborhood of each of these values (except
v = ':::co)
+
(
d
V
) () d
V = V P - P + higher or er terrns
o dp v 0
(4. 52)
o
so that at each v the behavior of ';i wi th respect to p is the same
o
/'
as with respect to v; hence, all poles of u(p} are simple, and the
character of the branch points of is the same as for
j\
As indicated in equation 4.47, it is found that u has branch
points at v = + 00; furthermore, it can be that
A 1
u-
rv
as v.,..... + co
(4. 53)
To examine this in the p-plane, we solve equation 4.49 for v, which
yields
3K(s') _ 21: (s)
I
I I
1
+ 3K - 2fl
P +-
s'
P s'
v
=
3K(s') + fl (s')
(4. 54)
[I
I 1
2
+ 3K + fl
s'
p
Ps'
It is seeri that the poles of v(p) occur only when the denominator
vanishes, and that these are simple poles since the derivative of the
denominator does not vanish in the finite p-plane. Thus, at each of
the singularities of v, ps say,
as p - Ps
(4. 55)
In order to complete the examination of u, equation 4.44, we
have only to determine whether or not the poles of J(p) fall on the
-152 -
'"
poles of u(p). T h e form of J(p) is obtained from equation 1.103b,
J(p) = L
s
+!.
p
(4. 56)
where J(s), J, and 7 are non-negative. We shall also make use of
s
the inverse of the bulk modulus.
. -1
B(p) = K(p) ,as given by equation
1. 113 b,
. 1 \'
B (p) == KfPI::; ~
s
B
(s) .
+ B
l+7p P
s
(4. 57)
where B ( s) and B are non-negative. Substitution of equations 4.56
and 4.57 into Poisson's ratio, equation 4.36, yields
3J - 2B
v
::;
2(3J + B)
>.
3J(s) _ 2B(s)
+ 3J
- 2B
!-J 1 + 7 P
P
s
s
(4. 58)
=
3J(s) + B(s)
2[I
+ 3J ; B]
1 + 7 P
s
s
Now if p::; - 1/7 , which is a pole of J(p) if J(r)* 0, then Poissonis
r
ratio must satisfy
Here, at each pole of J(p), Poissonis ratio is restricted to the interval
-1 < v::S: ~ . which is free of the singularities of 4 (p).
To summarize, it has been shown that all singularities of u(p)
are on the non - negative ,real p - axis and that all poles are simple ex-
cept at the origin where a double pole occurs if J in equation 4. 56
does not vanish. With this information, the time dependence can
-153 -
be calculated by :means of the Bro:mwich- Wagner Inversion Integral (38)
given by
u{t) = -Zl . r eptu{p)dp
TTl .) BR
1
,r
6
+
ioo
t-
= ~ e
P
u{p) dp
6-ioo
(4.60)
where 6> O. Since it is clear fro:m the above exa:mination that u{p)
behaves like lip or I/pZ as p - 00, the Bro:mwich contour, indi-
cated by BRI in the inversion integral, can be defor:med to the contour,
BR
Z
' shown in figure 4.8 (38). Writing the integral 4.60 along BR
Z
leads to the ti:me dependent solution u(t);
-tu 1 I
S
oo L -U. t
u(t) = 0 e S{u)du + . Si e + S t
(4. 61)
1
where this integral results fro:m branch cuts along the negative real
axis and the series represents contributions fro:m the poles. Further-
:more, it can be shown that the -integral is absolutely convergent for
O:::S t < 00. Thus, the ti:medependence of u(t) is in agree:ment with
the general behavior predicted by Postulates I and II.
b. Stress principle - Ritz :method. As another illustration of
the correspondence rule, we shall apply the Ritz :method, in conj unction
with the co:mple:mentary energy principle, to co:mpute two approxi:mate
stress distributions in an elasti c cylinder and then use the results to
obtain viscoelastic stresses. In a ddition, the approxi:mations will be
co:mpared to the exact solution.
-154-
u} Elastic solution: Consider an isotropic, h omogeneous,
cylinder in plane strain with the o uter boundary rigidly supported and
the inner boundary loaded by a pressure, Pi' applied stepwis e at
t = 0, as shown in figure 4.9. The exact elastic stresses are easily
found to be
2
radial: cr
= - Pi
+A
(A
2
_ b 2 )
re e
(4. 62a)
r
;
tangential: p . + A (A
2 b
2
)
cree
= +2
1 e
(4. 62b}
r
where
A =
b
A
1 - 2v
-
=
+ A
2
(1 - 20
Pi
u e
1
and the shear stress, .,. re' is identically zero because of symmetry.
The approximate stresses are not required to satisfy com-
patibility, but they must fulfill the equilibrium condition,
dcr r cr r - cr e
dr + r =
and the stress boundary condition, . cr =
r
stres se s satisfying thes e conditions are:
cr " =
rU
Pi at r = a .
(4. 63)
Two sets of
(4. 64a)
(4. 64b}
(4. 65a}
(4. 65b)
with.,. re = in both sets. The coefficients,"a and (3, are determined"
-155 -
~ : (
. by m inimi z ing the c omple menta.ry e nergy V with respect to each
one. For a c y linder in plane strain, the compl e mentary e n e r gy is
,', Sb
V' = . W(21Tr)
a
dr
2 Sb
= 21T(1 - v} { ~ ( 2 + 2)
E 2 O'r 0'8
a
(4.66)
w.her e W is the strain energy d e nsity.
* Substitution of stres s e s 4.64 into V , carrying out the int egration,
and minimizing, i. e.
, ~
BV
au = 0
yields
( 1-2v }ex. - I) p .
I -v 1
0.= (4, 67a)
- -L.+ ( ~ - _v_}x.
2
_(1-2v }(2X.-l)
L. ' I-V I-v
4X. . .
Similarly, for approximation 4.65
(4. 67b}
Substituting these expressions for a. and f3 into the r e spe ctiv e stresses,
we obtain the elastic stresses which are, along with the exact solu tion,
shown in figure 4.10 for v = 0.45 an:d X. = 2.
f3} Viscoelastic solution: We turn now to calculation of
the vi s coelastic stresses. On the basis of the corresponde nce rul e ,
the time dependent stresses are obtained by Ireplacing Poi s son's ratio ,
stresses,and internal pressure in equations 4.64, 4.65, and 4.67 by
transformed quantiti e s and the n inve rting.
For specification of an operational Poisson's ratio, a constant
-156-
bulk rn.odulus, K, is assurn.ed along with an operational shear rn.odu-
Ius given by
fJ.(p)
fLrn.
P
:: --1
p+ T
(4. 68)
which corresponds to a Maxwell rn.odel. Substituting K and fJ.(p)
into equation 4.36, Poisson's ratio becorn.es
2v(p) ::
2fJ.
(3 - T )rp + 3
fJ.
(3 + ~ ) rp + 3
K
(4.69)
where we . shall take J.l. /K:: 0.6 in order that v (p :: (0) :: 0.25. The
rn. . .
pres sure is taken to be a step function, of rn.agnitude p., applied at
1
t :: 0 so that p. appearing in the elastic solutions is to be replaced
1
by p./p. Making these substitutions for v and p. and then inverting
11 '
yields the approxirn.ate tirn.e-dependent stresses:
0" /p.:: -1 + 0.416(2'::' _ lie -0.325 t/T
ru 1 . b
(4. 70a)
0" /p. :: -1 + 0.416(4 {- _ lie -0. 325t/T
Su 1 . 0
(4.70b)
0" /p.:: -1. + 0.980(1 _ .!. .!:)e -0. 284t,lr
r(3 1 2 r '
(4.71a)
/
:: -1 + 0 980 -0. 284t/r
O"e(3Pi . e
(4. 7lb)
Furtherrn.ore, it is easy to verify that the exact stresses are given by
2 .
, / :: _ 1 + 0.167(4 _ b
2
)'e -0. 278t/r ,
0" re Pi
(4.72a)
r
2
O"s /p.:: -1 + 0.167(4 + ~ )e-0.278t/r
e 1 2
(4.72b)
r
All of these stresses are plotted in figure 4.11 for r/b:: 0.7.
-157-
It is seen fr.om the equati.ons that the time dependence is in c.omplete
agreement with P.ostulates I aud II. " Furtherm.ore, " the graph sh.ows that
the quality .of appr.oximati.on (13) is better than that of (a) for bath the
visc.oelastic and elastic solutions, which is nat unexpected in view of
the remarks made in section 3.3.
4.4. Numerical Applicati ons .of C.onv.oluti.on Var iati.ona l Principles
The e.ase with which appr.oximate visc.oelastic s.oluti.ons were
calculated in the previous two secti.ons suggests that for mast linear
pr.oblems it is not essential to use ,the conv.olution principles. H.ow-
ever, we shall now discuss two classes .of proble"ms far which it may
be either mare convenient .or else necessary to use them instead of
the transf.ormed principles and related approximate (or: exact) inver:"
sian meth.ods.
a. C.omments on the dynamic ,Problem. " In wave pr.opagati.on
and vibrati.on problems the time dependence .of a resp.onse may be a
rapidly changing and .oscillating function of l.ogarithmic time. In
direct method is not expected t.o yield goad results.
The c.ollocation meth.od can still be used, in principle, if the transient
resp.onse is quadratic;ally integrable. H.owever, many exp.onential
terms may be needed t.o obtain a satisfact.ory solution, which will
require an extremel y accurate evaluation .of the transf.orms
(see equati.on 4.l8). Thus, while the appr.oximate transf.orm may be
,',
'This behavi.oris actually mare .of an excepti.on than a rule far vis c.o-
elastic wave propagati,on probl e ms. In fact Arenz (39) is currently
applying the direct and c.ollocati.on meth.ods to .one- and tw.o -dimen-
si.onal pr.oblems with success.
-158 -
relatively easy to obtain, inversion may be impractical to perform.
In viewof this possible complication, a simpler procedure
might be to assume approximate solutions which consist of prescribed
functions of time along with some arbitrary parameters. These
parameters could be calculated from the condition that the appropriate
convolution functional be stationary.
Let us briefly illustrate this with the displacement principle,
equation 2.111. With temperature neglected for simplicity, but with
kinetic energy included (s ee equation 2.14) we have
1 stlS {[o fJ.v ',l
I = 2." Z .. e e .. (tl-or) + p
u . 0 B lJ fJ.v lJr
dU
i
(t
l
-7)
d(tl-T)
S
tl S j'tlS
- F.(7)U.(t
l
-T)dBdT - T.(T)U.(tl-T)dA
OB
l
1 OA 1 1
. T.
dT
(4. 73)
where tl is written for the upper limit on the time integral to em-
phasize the fact that we must restrict ourselves to a definite interval
of time; 0 S; . t S; in the approximate method. As s urne displace-
ments represented by the series
(a. )
u. = C f. (x., t)+ U. (4.74)
lo.l . l 1
in which are functions of x. and t that vanish on A ,
l 1 U
the U
i
are prescribed boundary displacements, and the Co. are
arbitrary constants which are to be determined from the stationary
condition
iiI = 0
U
(4. 75)
-159-
for all arbitrary variations 5C. By carrying out the inte grations in
a
equation 4. 73 and the variations, a system of linear algebraic equa-
tions is obtained f or determination of the constants Ca' It can be
shown that the matrix of coefficients which multiply C is symmetric;
a
this is a consequence of the symmetry of Z f l ~ and the property of
ij
the convolution integral that
(4. 76)
where 0 is an operator ( e . g. an element of zJ:1.
v
). and f and g
lJ
are time dependent functions. It is important to reiterate that I
u
is made stationary for jus t the interval 0 ~ t ~ t
l
, and hence the r e -
sulting solution will not be valid for times larger than " t
l
b. Beat conduction example. There is another class of problems
in which the transformed prin"ciples (or correspondence rule) cannot be
used. Namely, that class for which the assumed solution is a non-
linear function of the unspecified generalized coordinates. This
application will be illustrated with a one-dimensional heat conduction
problem.
Gonsider the isotropic, homogeneous, semi-infinite solid
shown in figure 4.12. At t = 0, the face x = is suddenly brought to
a constant temperature 8 It is "assumed "that the temperature, 8,
o
is zero for t < 0.
For comparison purposes, let us first give the exact solution
which is derived by solving the heat conduction equation
with
-160 -
8 =' 0 for t < 0, x=:: 0
8 = 8 for x = 0, t > 0
o
8 - 0 as x - C X l ~ t > 0
(4. 77)
where C is the specific heat per unit volume and K is the heat con-
duction cQefiicient. The solution is well-known and is given by
(4.78)
For the approximate solution a simple form is assumed which
satisfies the temperature boundary conditions at x = 0 ' and x = 00:
e
8
, a.
=
-x/q(t)
e (4. 79)
o
where q(t) is a generalized , coordinate that is , to be determined by
using the variational principle, equation 2.55. For the particular
case considered here, the variational principle is '
(4. 80a)
which is' equivalent ' to
S
tlSOO (c 88(T) - K 8
25l
(T)) 68(t -'r)dx dT = 0
o 0 8T 8 xz. 1
Substitution of the temperature 4.79 and the variation,
,
, 8 x
o -x/q
= -- e 6q
2
(4. 81)
q
-161-
into equation 4. 80b arid r e cognizing that 6q is arbitrary yields
):.:
S
O:> [dq(T) 2 K ] -x + _ 0
o ---;:r.r-x -ex e -
(4. 82)
Performing the integration over x the E'uler equation for q('O
is obtained, thus
2C dq(T) _ 1 + 1
K dT - q(T) q('-'-t---T-'-)-
1
in which tl is to be treated as a constant parameter.
by first noting that it can also be written as
2C
K
dq(tl-T)
d(tl-T)
=
Taking the ratio of 4.83 and 4.84 we find
-dq(t
l
- T) = dq(T)
which integrates to
(4. 8 3)
This is solved
(4. 84)
(4.8Sa)
(4.8Sb)
When this is substituted into the differential equation 4.83 and the
result integrated we find
-"2 ( 2 A) K
q l-"3
q
. = C
(4. 86a)
in which the non-dimens ional variables are defined,
,',
. It is interesting to observe that the weighting function,
(
I + 1 )
-x q{T) q{tl -T}
e
is a symmetric function with respect to the point T = tl/2 as the time
variable T range S ove r the inte rval O;S; T ;s; t
l
,,_ q(T)
q = Ci\t,)
1
-162-
" T
T ::
tl
/' /' /\
(4. 86b)
so that O::S: T ::s: 1. Setting 7:: q :: 1 in equation 4.86 yields the value
(4. 87)
Using this value for q(t
l
) in equation 4. 86a provides an expression
directly relating q
A
and 7,
(4.88)
/\
which is plotted in figure 4.13. The values for q can now be used .in
the solution if the temperature 4 . 79 is written as
where
9 a ; e -X!ci'(T)
e-
o
"
x=
x
q(t
l
)
(4. 89a)
(4. 89b)
This approximate solution is plotted in figure 4.14 along with the exact
temperature, which, in terms of the variables ;; and r is
9
e
e=
o
2
rrr
(4. 90)
It is of interest to compare the approximate temperature given
above with one calculated using 09(7) as a weighting function in equa-
tion 4. 80b in place of 09(t
l
-7). Such a procedure is analogous to the
way in which a spacial weighting function is employed in Gale rkin IS
-163 -
method (13). Usi ng e quation 4.79 and 58(7) in equation 4. 8 0b we find
S
oo (d 2 K) -2
- -x e qdx=O
o d7 C
which yields the differential equation
,',
This is easily solved to find'
q =
so that the approximate temperature is
8
o
e
-1. 225
which is also shown in figure 4.14.
(4. 91)
(4. 92)
(4.93 )
(4.94)
It is observed that for T <: 0.8, 8
a.
is clo se r than e {3 to the
r..
exact solution, except at small values of x. Because of this,
appears to be the best approximation in the sense that
S
lr
OO
(8 -6 )2cJ.idT < r
l
r
OO
(6{3_ 6 )2dX d9
OJ
O
a. e JOJ
O
e
8
a.
(4. 95)
It should be added, in this regard, that the stationary condition on the
convolution functional, equation 4.80, minimizes the transfo rm, T 8'
with respect to. q . for p real and positive . . In contrast, condition 4.91
i s not associated with the minimization of a functional.
is of interest to mention that the same result is obtained by applying
the di r ect method of inversion to the exact transformed solution of
equation 4. 77.
-164-
Although further study on the use of convolution variational
principles is needed, this section does suggest that they are useful
tools for approximate analysis.
- 1 6 ~ -
REFERENCES
1. Biot, M.' A.: "Theory of Stres s -Strain Relations in Anisotropic
Viscoelasticity' and Relaxation Phenomena, ." Journal of
Applied Physics, (1954) ~ pp . 1385-91.
2. Biot, M. A.: \ "Linear Thermodynamics and the Mechanics of '
Solids, " Proceedings of the Third U. S. National Congress
of Applied Mechanics (1958) pp. 1-18.
3. Biot, M. A.: "Therrnoelasticity and Irreversible Thermodynamics, "
Journal of Applied Physics, (1956)!:2 pp. 240-253.
4. Biot, M. A.: "Theory of Deformation of a Porous Viscoelastic
Anisotropic Solid, " Journal of Applied Physics (195 6 ) 27 ,
pp. 459-467. -
5. Biot, M. A.: "Variational Principles in Irreversible Thermo-
dynamics with Application to Viscoelasticity, " Physical
R eview (1955) J,2, pp. 1463 -69.
6. Hunt er, S. C.: "Tentative Equations for the Propagation of Stress,
Strain and T empe rature Fi e lds in Viscoelastic Solids, II
Journal of Mechanics and Physics of Solids, (1961) 9,
pp. 39-51. . -
7. Chu, Boa-Teh: "Deformation and Thermal Behavior of Linear
Viscoelastic Materials," Brown Unive rsity, NOrd 1 7838,
Report No.1, (October 1957); Report No.2 (June 1958).
8. Eringen, A. C.:
Mechanics, "
"Irreversible Thermodynamics and Continuum
Physical R eview (1961) 117, pp. ,1174-1183.
9. Ferr y, J. D.: Viscoelastic Properties of Polymers, John Wiley
and Sons, Inc., New York (1961).
10, Ke, T.: "Experimental Evide nce of the Viscous Behavior of
Grain Boundaries in Metals," Physical Review (1947) 2.!.,
pp. 533-546. .
11. Treloar, L. R. G.: The Physics of Rubber Elasticity, Oxford
Press (1958).
12. Theocaris, P. S.; Mylonas, D:: "Viscoelastic Effect s in
Birefringent Coatings," Journal of Applied Mechanics (1961)
~ , pp. 601-607.
13. Sokolnikoff, 1. S.: Mathematical Theory of Elasticity, Second
Edition, McGraw-Hill Book Co., Inc., New York (1956).
14. DeGroot, S. R.: Thermodynamics of Irreversible Processes,
North Holland Publishing Co., Amsterdam (1951).
-166-
15. Prigogine, 1.: "Le Domaine de Validit.f de la Thermodynamique
des Phenomenes Irreversib1e s," Physica, No . 1-2 (1949),
pp. 272-289.
16 . Lee, J. F.: Sears, F. W., Thermodynamics, Addison- Wesley
Pub. Co. Reading, Mass. (1956).
17. Williams, M. L.; Landel, R. F.; Ferry, J. D.: "The Temperature
Dependence of Relaxation Me chanisms in Amorphous Polymers and
Other Glass Forming Liquids," Journal of AInerican Chemical
Society, (1955) 77, pp. 3701-07.
18. Frazer; Duncan; Collar: Eleme ntary Matrices, The Mac millan
Company, New York (1947).
19. Prigogine, 1.: Introduction to Thermodynamics of Irreversible
Processes, Charles C. Thomas Pub., Springfield, Illinois (1955 ).
/ 20 .. Morland, L. W.; Lee, E. H.: "Stress Analysis for Linear Viscoelastic
Materials with Temperature Variation," Transactions of the
Society of Rheology (1960) IV, pp. 233-263.
21. Davies, R. 0.; Jones, G. 0.: "Thermodynamic and Kinetic Properties
of Glasses," Advances in Physics, (1953) pp. 370-410.
22. Biot, M. A.: "On the Instability and Folding Deformation of a
Layered Viscoelastic Medium in Compression," Journal of
Applied Mechanics, (1959) pp. 393-400.
23. Knauss, W.G.: Unpublished current research, Graduate Aeronautical
Laboratories, California Institute of Technology.
24. Hemp, W. S.: "Fundamental Principles and Theorems of Thermo-
ElastiCity," The Aeronautical Quarterly (1956) 2 . .' pp. 184-192.
25. Timoshenko, S., and Young, D. H.: Advanced Dynamics, McGraw-
Hill BookCo., Inc., New York (1948 ).
26. Rosen, P.: "On Variational Principles for Irreversible Processes, "
Proceedings of the Iowa Thermodynamics Symposium, (April 27-
28, 1953), pp. 34-42.
27. Besseling, J . F.: "A Theory of Small Deformations of Solid Bodies, "
Stanford University Report SUDAER No. 84, AFOSR TN-59-605,
(February 1959). .
28. Reissner, E.: "On a Variational Theorem in Elasticity, " Journal
of Mathematics and Physics, pp. 90-95.
-167-
29 . Reissner, E.: "On Variational Principles in Elasticity, "
Symposium of the American Mathematical Society on the
Calculus of Variations and Its Applications (April 12, 1956).
30. Lee, E. H.: "Stress Analysis in Vis coelastic Bodies," Quarterly
of Applied Mathematics (1955).22, pp. 183-190.
31. Washizu, K.: "A Note on the Conditions of Compatibility, "
Journal of Mathematics and Physics, (1957-58) 2.., pp. 306-312.
32. Gross, B.: Mathematical Structure of the Theories of Visco-
elasticity, Hermann, Paris (1953).
33. Erdelyi, A.: "Note on An Inver sion F.ormula for the Laplace
Transformation," Journal of the London Mathematical Society,
(1943) XVIII, pp. 72-77.
}4. Erdelyi, A.: "Inversion Formulae for the Laplace Transformation, "
Philosophical Magazine and Journ<j.l of Science, (194 3) XXXIV,
pp. 533-537.
35. Schapery, R. A.: "Approximate Methods of Transform Inversion
for Viscoelastic Stress Analysis," GALCIT SM 62-2,
California Institute of Technology (1962) (To be presented at
the Fourth U. S. National Congress of Applied Mechanics,
June 1962, Berkeley, California.)
36. Landel, R. F.: "The Dynamic Mechanical Properties of a Model
Filled System," Transactions of the Society of Rheology (1958)
~ , pp. 53-75.
37 . Schapery, R. A.: "A Simple Collocation Method for Fitting
Viscoelastic Models to Experimental Data," GALCIT SM 61-23A,
California Institute of Technology (1961).
38. McLachlan, N. W.: Complex Variable and Operational Calculus,
Macmillan Company, New York (1944).
39. Arenz, R. J.: Unpublished current research,", Graduate .Aero-
nau.tical ,Laboratories, California Institute of Technology.
! '
-168-
0.4
0 .2
CENTROID
AT IN :: Wo
-3 -2
Figure 4.1.
..
-I
o 2
w
Weighting Fun<;tion in Laplace Transform
with Logarithmic Time Sc.ale
3 .
x . u
1--------+--P :; a'
Figure 4.2. Clamped Plate with Uniform Tensile
Stres s on the Ends
4
U
0-
0
b/jJ-
-169 -
0.40
= 0
p = 1
0.35
A -
I .
0.30
0.3 0.4
1/
Figure 4.3. De pendence of Displacement
on Poisson' s Ratio
0.5
-8
LOG J'(w)
LOG J (p) -9
(CM
2
/DYNE)
-10
LOG J'(w)
(EXPER 1 MENTAL)
J(p)
______ ____ -L ______ L-____ ____ ____
-2 o 2 4 6 8 10
LOG w; LOG p, (sec-I)
,
Figure 4.4. Real Part of C omplex Shear Compliance
and Ope rational Compliance for Glas 5 -Fille d
Polyisobutylene at 12. SoC
-17 0 -
v(p) 0.4
0.3 L-______ L-____ ____ ______ ______ ____
-2 O . 2 4 6 8
LOG p, (sec-I)
Figure 4.5. Operational Poisson's Ratio
4
2
t:.u v 0 (n = 7)
r;=O
P= I
A = 1
0
---
4
1
4
Figure 4.6.
5 6 7 8 9
-LOG t (sec)
I I I I
5 6 7
LOG P (sec-I)
8
Time Dependence of L is placement Due t o
Change,in Poisson' s Ratio
10
10
I
9
I
10
-171-
.IF _ X
'I'-b' u
Figure 4. 7 . Semi-Infinite Clamped Plate with Uniform
Tensile Stres s on the End
SINGULARITI ES OF u
LOCATED
r 0,
ON AX I S
s=O\
..
s
8 + i Q)
p-PLANE
(p=r+is)
(CONTOUR
EQUIVALENT
TO SRI)
8- i Q)
Figure 4.8, Bromwich Contour, BR l' and Equivalent
Contour, BR
Z
0-
8
p.
I
(Jr
p.
I
- 17 2-
BOU NDA RY
F igure 4 .9. Inte rnall y Pressurized Cylinder
- 0.4 ,..,------,------,------,--.:.-----r-----.
-0.6
-0.8
-1.0
,
0.5
0-
.8e
p.
I
-Lae[3
p.
I
(J8a
/'"
p.
I
~ / '
/'
(Jr[3
/"
/"
../
/'
---
--
--
--
-
0.,6
0.7
r
b
0.8
../
'11=0.45
A= 2
/"
-/
/"
--.
--
\..- O-ra
p.
I
0.9 (.0
Fi g ure 4.10. Radial Dependence of Elastic S t r e ss e s
in Cylinder
ae
p.
I
0" '
r
p.
I
- ~
- L73-
0 . 2
r
0.7 =
b
0
A 2 =
0"8e
-0.2
' p.
I
O"ef3
p.
I
-0.4
0" ea
--'
p.
I
- 0 .6
-0.8
-
--- --
-- -
-
-
- 1. 0 '--_ _ --'-___ ~ __ _'_ ___ .l.__ __ __"_ ___ _'____"
o 23 4 5
t
T
F igur e 4.11. Ti me Dependence of Vi scoela stic Stres ses
in Cylinder
Figure 4.12. Semi- Infini t e Solid f o r T hermal Analysis
6
/I. q
q: q(td
-174-
. I. 0 ,----,----,----r-------,---,
0.5
o L-___ -L ____ ~ ____ ~ L - ____ -L ____ ~
o 0.2 0.4 0.6 0.8 1. 0
"
T :;
Figure 4.13. Time Dependence of Generalized Coordinate
1.0 ~ - - - - - - r - - - - - - - r - - - - - ~ - - - - - - ~ - - - - - - _ .
8
e
e=
o
0.8 8
a
8
0
:;---
0.4
0.2
8f3
-:--
o ~ ~ ~ ~ ~ ~ ____ L-__ -L __ ~
o 0.2 0.4 0.6 0.8 1.0
1\
T =..!...
t,
Figure 4. 14. Time De pendence of Temperature