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Application of the cosserat spectrum theory to viscoelasticity
\
Pergamon
J[ Mech[ Phys[ Solids\ Vol[ 35\ No[ 09\ pp[ 0858Ð0879\ 0887 Þ 0887 Elsevier Science Ltd[ All rights reserved Printed in Great Britain 9911Ð4985:87 ,*see front matter
PII ] S9911Ð4985"87#99934Ð2
APPLICATION OF THE COSSERAT SPECTRUM THEORY TO VISCOELASTICITY XANTHIPPI MARKENSCOFF\$% WENSEN LIU\%& and MICHAEL PAUKSHTO' % University of California\ San Diego\ Department of Applied Mathematics and Engineering Sciences\ La Jolla CA 81982\ U[S[A[ & San Diego State University\ College of Engineering\ San Diego CA 81071\ U[S[A[ ' Saint Petersburg State University\ Institute of Mathematics and Mechanics\ Saint Petersburg\ Russia "Received 19 December 0886 ^ in revised form 10 February 0887#
ABSTRACT The Cosserat Spectrum theory is applied in the theory of linear viscoelasticity[ The solution for the Laplace transform of the displacement of the viscoelastic problem is expressed in a series of the Cosserat eigenfunctions\ which are dependent only on position\ and the coe.cients are expressed as convolutions of the time dependent body force or surface loading provided that the inverse Laplace transforms of the viscoelastic moduli are known[ This renders the Cosserat Spectrum theory advantageous for the solution of viscoelastic problems[ Several examples are shown[ Þ 0887 Elsevier Science Ltd[ All rights reserved[ Keywords ] A[ creep\ B[ viscoelastic material\ C[ Cosserat Spectrum[
0[ INTRODUCTION The homogeneous Navier equations Duł ¦v99 = uł 9\ ðv "l¦m#:m "0:"0−1n#\l and m being the Lame|s constants\ n the Poisson|s ratioŁ\ with homogeneous boundary conditions of displacement or traction admit nontrivial solutions when v takes values in a set of points lying outside the physical range of Poisson|s ratio called the Cosserat Spectrum[ The Cosserat Spectrum theory was introduced by Cosserat and Cosserat "0787# and then fully developed by Mikhlin "0862#\ who proved the completeness and orthogonality of the Cosserat eigenfunctions and represented the displacement _eld uł as summation of the Cosserat eigenfunctions for the boundary value problems of displacement or traction[ Pobedria "0869# applied the Cosserat Spectrum theory to 1!D viscoelastic problems[ In a recent paper\ Markensco} and Paukshto "0887# applied it to problems in elasticity and thermoelasticity[ They also developed a variational principle in thermoelasticity within the frame of the Cosserat Spectrum theory[ In the present article we develop a theory of linear viscoelasticity based on the Cosserat Spectrum theory[ We show that in the Laplace transformed space the Navier equations hold and the Cosserat Spectrum theory can be applied[ The solution for $ To whom correspondence should be addressed[ E!mail ] xmarkensÝames[ucsd[edu 0858
0869
X[ MARKENSCOFF et al[
the Laplace transform of the displacement function of viscoelastic problems is expre! ssed in series of the Cosserat eigenfunctions\ which are functions of geometry only "and not the transform variable s#\ and the coe.cients are functions of s that can be easily inverted for rational functions\ which is the case in the commonly used vis! coelastic models "Christensen\ 0860#[ Thus\ once the inverse Laplace transforms of these functions that depend on the viscoelastic moduli are known\ the solution may readily be expressed as convolutions of the time dependent body force or surface loading[ In this respect\ this method of solution has an advantage over the standard methods as in Lee "0844#\ Radok "0846#\ and Lee et al[ "0848#[
1[ FORMULATION OF THE GOVERNING EQUATIONS OF VISCOELASTICITY IN TERMS OF THE COSSERAT SPECTRUM THEORY The equations of quasi!static equilibrium are given by sij\j "t#¦Fi "t# 9 in V
"0#
with boundary conditions of displacement uł uł "t#
on 1V
"1a#
or traction łtn "t# sł "t# = nł on 1V
"1b#
while the stress sij "t# and strain oij "t# 01 ðui\ j "t#¦u j\i "t#Ł
"2#
are related by the constitutive equations "Christensen\ 0860# sij "t#
g
t
ð1m"t−t?# doij "t?#¦l"t−t?#dij dokk "t?#Ł
"3#
9
with m"t# and l"t# being the time dependent Lame|s moduli[ By applying the Laplace transform on the above equations\ denoting by {{ || the transformed quantities\ and combining the transformed eqns "0#\ "2# and "3#\ we obtain the Navier equations for the transformed displacements g
¼ u¼ j\ji − u¼i\ jj ¦v
Fi
m¼
in V
"4#
where ¼ v
l¼ ¦m¼ m¼
"5#
with boundary conditions of displacement uł¼ uł¼ b or traction
on 1V
"6a#
Application of the Cosserat Spectrum theory to viscoelasticity
0860
¼ łt sł¼ = nł on 1V
"6b#
We now observe that the displacement uł¼ satisfying eqns "4# and "6# may be expressed in terms of a series of the Cosserat eigenfunctions\ which satisfy the homogeneous Navier equations 91 u½n ¦v ½ n 99 = u½n 9 in V
"7#
with homogeneous boundary conditions of displacement u½n 9 on 1V
"8a#
łtn "u½n # sł "u½n # = nł 9 on 1V
"8b#
or traction
As presented by Mikhlin "0862#\ the Cosserat eigenfunctions are complete in the Sobolev space H 0\ and thus\ the solutions of the inhomogeneous eqn "4# with bound! ary conditions of displacement ðeqn "6a#Ł or traction ðeqn "6b#Ł admit the Mikhlin "0862# representation theorems as follows ] The solution for the boundary value problem of displacement g
:
F ¼ 99 = uł¼ − Duł¼ ¦v m¼
in V
"09a#
uł¼ 9 on 1V
"09b#
admits the representation g
:
v 0 "F\ u½n"−0# # "−0# "# "# ½n : ł \ u½n #u½n ¦ uł¼ s u½n ¦"F "F\ u½n #u½n m¼ n 0 0¦v v ¼ ¼ ½ n −v
6
g
7
"00a#
where g
:
g
:
"F\ u½# 0 Ð F = u½ dV
"00b#
Duł¼ ¦v ¼ 99 = uł¼ 9 in V uł¼ uł¼ b on 1V
"01a#
The solution of
"01b#
admits the representation v ¼v ½n uł¼ uł¼ 9 ¦ s "div uł¼ 9 \ div u½n #u½n ¼ v −v ½n n 0
"02a#
Duł¼ 9 9 in V
"02b#
where uł¼ 9 satis_es
0861
X[ MARKENSCOFF et al[
uł¼ 9 uł¼ b
on 1V
"02c#
The solution for the boundary value problem of traction g
:
F ¼ 99 = uł¼ − Duł¼ ¦v in V m¼ sł¼ "uł¼ # = nł ł¼t on 1V
"03a# "03b#
can be represented as 0 uł¼ s m¼
n 0
¼ 1" łf\ u½n"−0# # "−0# 0−v ½n ¼ ¼ " łf\ u½n #u½n u½n ¦" łf\ u½n"# #u½n"# ¦ 0¦v v ¼ ¼ −v ½n
6
7
"04a#
where g
: ¼ " łf\ u½# 0 Ð F = u½ dV¦Ð ł¼t = u½ dS
"04b#
By u½n"−0# in the above equations is indicated the in_nite orthogonal subspace of eigenfunctions corresponding to the eigenvalue of in_nite multiplicity v ½ −0\ by u½n"# is indicated the in_nite orthogonal subspace of eigenfunctions corresponding to the eigenvalue of in_nite multiplicity v ½ \ while by u½n is indicated the orthogonal subspace of eigenfunctions corresponding to the eigenvalues of _nite multiplicity of the discrete spectrum v ½ n[ We may note here that the Cosserat eigenfunctions are functions of the space variables only "and not the transform variable s#[
2[ INVERSION OF THE LAPLACE TRANSFORMS An advantage of expressing the solutions of the viscoelastic problem in terms of the Cosserat eigenfunctions is that the inversion of the Laplace transform may be obtained by convolution\ provided that inverse Laplace transforms of the terms involving the viscoelastic moduli can be easily obtained[ Let us consider the representation eqn "00# for the boundary value problem of displacement with homogeneous boundary conditions and denote 0 ¼# m¼ "0¦v
"05a#
0 m¼
"05b#
v ½n m¼ "v ¼# ½ n −v
"05c#
G
"−0# "s#
G
"# "s# G
"v ½ n \ s#
If for a given viscoelastic material\ eqn "05# can be inverted to obtain G"−0# "t#\ G "t# and G"v ½ n\ t#\ then the solution of the viscoelastic problem in the time domain may be obtained by convolution as follows "#
Application of the Cosserat Spectrum theory to viscoelasticity
uł "t# s
n 0
g
t
ðF ł "t?#\ u½n"−0# ŁG "−0# "t−t?# dt?u½n"−0# ¦ s n 0
9
¦s n 0
g
g
0862
t
ðF ł "t?#\ u½n"# ŁG "# "t−t?# dt?u½n"#
9
t
ðF ł "t?#\ u½n ŁG"v ½ n \ t−t?# dt?u½n
"06a#
9
where ł "t#\ u½Ł 0 ÐF ł "t# = u½ dV ðF
"06b#
Similarly\ the inversion of the representation eqn "02# for the boundary value problem of displacement with inhomogeneous boundary conditions\ and eqn "04# for the boundary value problem of traction may be obtained[ Thus for any time dependent body force loading or boundary loading\ the solution of viscoelastic problems may be written immediately in the form of eqn "06# if the Cosserat eigenfunctions are known for the geometry of the problem[ The functions G"−0# "t#\ G"# "t# and G"v ½ n\ t# for the commonly used viscoelastic models are presented in the Appendix[
3[ APPLICATIONS AND EXAMPLES "0# Time dependent temperature loadin` We consider viscoelasticity problems with thermal loading that may\ in addition\ be time dependent[ As shown by Markensco} and Paukshto "0887#\ the temperature loading is equivalent to a body force loading that results in the representation of the displacement _eld in terms of a series of the Cosserat eigenfunctions according to ½n v "T\ div u½n #u½n uł "2v−0#a s v ½ n −v n 0
"07#
for the boundary value problem of displacement with zero boundary displacement\ and
uł "2v−0#a s n 0
6
1 0−v ½n "T\ div u½n #u½n "T\ div u½n"−0# #¦ v¦0 v−v ½n
7
"08#
for the boundary problem of traction with traction free boundary condition[ The displacement _eld in the Laplace transform space s in thermoviscoelasticity is\ accord! ingly\ represented by the series v ½n
\ div u½n #u½n ¼ −0#a s "T uł¼ "2v ¼ ½ n −v n 0 v
"19#
for the boundary value problem of displacement with zero boundary displacement\ and by
0863
X[ MARKENSCOFF et al[
uł¼ "2v ¼ −0#a s
n 0
$
1 0−v ½n
\ div u½n"−0# #u½n"−0# ¦
\ div u½n #u½n "T "T ¼ ¦0 ¼ −v v v ½n
%
"10#
for the boundary value problem with traction free boundary condition[ By applying the convolution theorem\ according to the analysis of the previous section\ we have the representation of the displacement for the solution of the ther! moviscoelastic problem
uł s
n 0
g
t
ðT"t?#\ div u½n ŁG"v ½ n \ t−t?# dt?u½n
"11a#
$
"11b#
9
where G"v ½ n \ t# v ½ n aL−0
2v ¼ −0 v ¼ ½ n −v
%
for the boundary value problem of displacement\ and
uł s
n 0
6g
t
ðT"t?#\ div u½n"−0# ŁG "−0# "t−t?# dt?u½n"−0#
9
¦
g
t
ðT"t?#\ div u½n ŁG"v ½ n \ t−t?# dt?u½n
"12a#
9
where G "−0# "t# 1aL−0
$
% $ %
¼ −0 2v ¼ ¦0 v
G"v ½ n \ t# "0−v ½ n #aL−0
2v ¼ −0 v ¼ −v ½n
"12b#
for the boundary value problem of traction[ If the material is initially undisturbed and the temperature T is not time dependent\ then the convolution integrals are easily performed[ "1# The Lame problem under time dependent loadin` This is the problem of a spherical\ viscoelastic shell under time dependent internal and external pressure p0 and p1\ respectively[ The classical elastic solution "Lure\ 0853# is ur −
"p1 r12 −p0 r02 #r 0 0 "p1 −p0 #r02 r12 − 2 2 m"2v−0# 3m "r12 −r02 #r1 r1 −r0
uu u8 9
"13#
where r0 and r1 are the internal and external radius\ respectively[ It can be veri_ed that the _rst term in eqn "13# is an eigenfunction corresponding to v ½ 0:2\ and the
Application of the Cosserat Spectrum theory to viscoelasticity
0864
second term is an eigenfunction corresponding to v ½ [ In the Laplace transform space s\ the displacement _eld will be u¼r −
"p¼1 r12 −p¼0 r02 #r 0 0 "p¼1 −p¼0 #r02 r12 − ¼ −0# m¼ "2v 3m¼ "r12 −r02 #r1 r12 −r02
u¼u u¼8 9
"14#
If the viscoelastic material is Hookean solid in dilatation and Kelvin solid in distortion\ the material parameters represented by eqns "A2#Ð"A4# in the Appendix are now substituted into eqn "14# u¼r −
"p¼1 r12 −p¼0 r02 #r 0 "p¼1 −p¼0 #r02 r12 − 1"q9 ¦q0 s# "r12 −r02 #r1 2k"r12 −r02 #
u¼u u¼8 9
"15#
The inverse Laplace transform of eqn "15# gives the solution for any general time dependent loading ur − −
ðp1 "t#r12 −p0 "t#r02 Łr 2k"r12 −r02 # r02 r12 1q0 "r12 −r02 #r1
g
t
0
ðp1 "t#−p0 "t#Ł exp −
9
1
q9 "t−t?# dt? q0
uu u8 9
"16#
"2# The `ravitatin` sphere Let the body force acting on a viscoelastic sphere be gravity according to the Newton|s law of gravitation[ It is known from potential theory that the resultant attraction is directed along the radius to the center of the sphere and that its magnitude is proportional to radial distance[ Consequently the body force is given by ł − F
rgrł r9
"17#
where r is density\ g the gravitational acceleration\ r9 the radius of the sphere and łr position vector measured from the center of the sphere[ The displacement _eld of an elastic gravitating sphere is presented by Lure "0853#[ We rewrite it as follows ur −
0 rgr9 r2 1r r− 1 ¦ 09m v¦0 2v−0 r9
uu u8 9
$ 0 1
% "18#
so that the _rst part is an eigenfunction corresponding to v ½ −0\ and the second part is an eigenfunction corresponding to v ½ 0:2[ If the spherical shell is of viscoelastic material\ then the Laplace transform of the displacement _eld will be
0865
X[ MARKENSCOFF et al[
u¼r −
0 rgr9 r2 1r r− 1 ¦ ¼ ¦0 ¼ −0 09sm¼ v 2v r9
$ 0 1
%
u¼u u¼8 9
"29#
Suppose that the material behaves as a Hookean solid in dilatation and Maxwell ~uid in distortion\ then the material parameters are represented by eqns "A03#Ð"A05#[ The inverse Laplace transform of eqn "29# will be ur −
$
r2 rgr9 r − 1 04k r9
0 1%0 1
rgr9 0 t 1q0 ¦ 1 exp − 09 k 2k tm tm
r−
uu u8 9
"20#
where tm ð"2kp0¦1qi#:2kŁ[ When t : \ eqn "20# reduces to ur −
rgr9 r2 rgr9 r r− 1 − 09k 04k r9
0 1
uu u8 9
"21#
It is interesting to note that ur is _nite as t : [ We have veri_ed that both eigenfunctions satisfy 9×uł 9\ that is\ both are distortion!free vector[ Furthermore\ the second eigenfunction associated with v ½ 0:2 produces uniform normal stress components and no shear components[ Physically\ since both eigenfunctions are of zero distortion\ the displacement _eld remains _nite even though the material is of the Maxwell ~uid type in distortion[ "3# Heat ~ow past a thermally insulated spherical cavity Suppose the material behaves as a Hookean solid in dilatation and a Maxwell ~uid in distortion\ then the material parameters are represented by eqns "A03#Ð"A05#[ The temperature _eld is given by T"t# T0 "t#¦T1 "t#
"22a#
where T0 "t# is a linear function of position z r cos u\ i[e[\ T0 "t# t"t#r cos u\ T1 "t#
t"t#r91 cos u 1r1
"22b#
where r9 is the radius of the spherical cavity[ The thermoelastic solution to this problem was given by Florence and Goodier "0848# who treated the temperature _eld with constant t[ Here we view this problem in thermoviscoelasticity with arbitrary t"t#[ The Laplace transform of the displacement _eld corresponding to free thermal expansion caused by T0 "t# is given by
Application of the Cosserat Spectrum theory to viscoelasticity 0 1
0866
1
u¼r at¼ r cos u u¼u 01 at¼ r1 sin u
"23#
The inverse Laplace transform of eqn "23# is given by ur 01 at"t#r1 cos u uu 01 at"t#r1 sin u
"24#
The Laplace transform of the displacement caused by the temperature _eld T1 "t# is a Cosserat eigenvector corresponding to the eigenvalue v ½ −0\ i[e[\ u¼r
¼ −0#at¼ r91 "2v r92 − 2 cos u 1"v ¼ ¦0# 2r
u¼u
¼ −0#at¼ r91 r9 "2v r92 − 2 sin u ¼ ¦0# 1"v 1r 5r
0 1 0 1
"25#
The inverse Laplace transform of eqn "25# gives ur
uu
r91 r92 − 2 cos u 3 2r
t
0 1 g 0 1 g
t"t?#G "−0# "t−t?# dt?
9
r91 r9 r92 − 2 sin u 3 1r 5r
t
t"t?#G "−0# "t−t?# dt?
"26a#
9
where G−0 "t# de_ned by eqn "12b# is written as G "−0# "t# 1a
$
0 1%
2p0 1q0 t d"t#¦ 1 exp − tm tm ktm
"26b#
If t t9H"t#\ then eqn "26# becomes ur
at9 r91 1q0 t 2− exp − 1 ktm tm
−
uu
at9 r91 1q0 t 2− exp − 1 ktm tm
r92 r9 − 2 sin u 1r 5r
$ $
r92 cos u 2r2
0 1%0 1 0 1%0 1
"27#
where tm ð"2kp0¦1q0#:2kŁ is the relaxation time of the viscoelastic material[ When t : \ eqn "27# reduces to ur
2at9 r91 r92 − 2 cos u 1 2r
uu
r92 2at9 r91 r9 − 2 sin u 1 1r 5r
0 1 0 1
"28#
It is interesting to note that ur is _nite when t : [ We have veri_ed that the
0867
X[ MARKENSCOFF et al[ "−0#
eigenfunction satis_es 9×u½ 9[ Physically\ since the eigenfunction is of zero distortion\ the displacement _eld remains _nite even though the material is of the Maxwell ~uid type in distortion[
4[ CONCLUSIONS The Cosserat Spectrum theory is applied to the linear theory of viscoelasticity[ The Mikhlin representation theorems in the form of Laplace transform are written in such forms that the inverse Laplace transforms are easily carried out for some frequently used viscoelastic models in terms of convolutions[ Also the physical meaning associ! ated with the Cosserat eigenfunctions corresponding to the eigenvalues v ½ −0 "distortion free# and v ½ "divergence free# ties naturally with viscoelastic behavior that may di}er in the dilatation and distortion part of the constitutive relation[
ACKNOWLEDGEMENT The support of the National Science Foundation through the Institute of Mechanics and Materials at University of California\ San Diego to Michael Paukshto is gratefully acknowledged[
REFERENCES Christensen\ R[ M[ "0860# Theory of Viscoelasticity an Introduction[ Academic Press\ New York[ Cosserat\ E[ and Cosserat\ F[ "0787# Sur les equations de la theorie de l|elasticite[ C[ R[ Acad[ Sci[ Paris 015\ 0978Ð0980[ Florence\ A[ L[ and Goodier\ J[ N[ "0848# Thermal stress at spherical cavities and circular holes in uniform heat ~ow[ Journal of Applied Mechanics 15\ 182Ð183[ Lee\ E[ H[ "0844# Stress analysis in viscoelastic bodies[ Quarterly of Applied Mathematics 02\ 072Ð089[ Lee\ E[ H[\ Radok\ J[ R[ M[ and Woodward\ W[ B[ "0848# Stress analysis for linear viscoelastic materials[ Transactions of the Society of Rheolo`y 2\ 30Ð48[ Lure\ A[ I[ "0853# Three!dimensional Problems of the Theory of Elasticity[ Interscience\ New York[ Markensco}\ X[ and Paukshto\ M[ V[ "0887# The Cosserat spectrum in the theory of elasticity and applications[ Proceedin`s of the Royal Society of London A343\ 036Ð043[ Mikhlin\ S[ G[ "0862# The spectrum of a family of operators in the theory of elasticity[ Russian Math[ Surveys 17\ 34Ð77[ Pobedria\ B[ E[ "0869# On solution of contact!type problems in the linear theory of visco! elasticity[ Doklady Akad[ Nauk SSSR 089\ 186Ð299[ Radok\ J[ R[ M[ "0846# Viscoelastic stress analysis[ Quarterly of Applied Mathematics 04\ 087Ð 191[
Application of the Cosserat Spectrum theory to viscoelasticity
0868
APPENDIX The functions of G"−0# "t#\ G"# "t# and G"v ½ n\ t# for the commonly used viscoelastic models are as follows[ Model 0*Hookean solid in dilatation and Kelvin solid in distortion The constitutive equations take the form skk "t# 2kokk "t# sij "t#
g
"A0#
t
ðq9 ¦q0 d"t−t?#Ł deij "t?#
"A1#
9
where k is the buck modulus in dilatation and q0:q9 the retardation time in distortion\ sij and eij are the stress deviator and strain deviator\ respectively[ The complex Lame moduli m¼ \ l¼ and the spectrum parameter v ¼ are m¼ 01 "q9 ¦q0 s#
"A2#
l¼ 02 "2k−q9 −q0 s#
"A3#
v ¼
5k¦q9 ¦q0 s 2"q9 ¦q0 s#
"A4#
The functions of G"t# for the boundary value problem of displacement\ described by eqn "09#\ are now written as G "−0# "t#
0
2 1q9 ¦2k exp − t 1q0 1q0
G "# "t#
G"v ½ n \ t#
0
1 q9 exp − t q0 q0
1
"A5#
1
"A6#
0
5v "0−2v ½ ½ n #q9 ¦5k exp − t "2v "0−2v ½ n −0#q0 ½ n #q0
1
"A7#
The functions of G"t# for the boundary value problem of traction\ described by eqn "03#\ are now written as G "−0# "t#
1
"A8#
0
"09#
G "# "t#
½ n \ t# G"v
0
2 1q9 ¦2k exp − t q0 1q0 1 q9 exp − t q0 q0
1
0
5"v "0−2v ½ n −0# ½ n #q9 ¦5k exp − t "2v "0−2v ½ n −0#q0 ½ n #q0
1
"A00#
Model 1*Hookean solid in dilatation and Maxwell ~uid in distortion The constitutive equations take the form skk "t# 2kokk "t# sij "t#
g
t
9
0
"A01#
1
t−t? q0 exp − deij "t?# p0 p0
"A02#
0879
X[ MARKENSCOFF et al[
where k is the buck modulus in dilatation\ p0 the relaxation time and q0 the viscosity in distortion[ The complex Lame moduli m¼ \ l¼ and the spectrum parameter v ¼ are m¼
q0 s 1"0¦p0 s#
"A03#
2k¦"2kp0 −q0 #s l¼ 2"0¦p0 s#
"A04#
5k¦"5kp0 ¦q0 #s 2q0 s
"A05#
¼ v
The functions of G"t# for the boundary value problem of displacement\ described by eqn "09#\ are now written as G "−0# "t#
$
G "# "t#
G"v ½ n \ t#
0
2p0 1q0 2kt d"t#¦ exp − 2kp0 ¦1q0 p0 "2kp0 ¦1q0 # 2kp0 ¦1q0
$
1p0 0 d"t#¦ q0 p0
1%
"A06#
%
"A07#
5v ½ n p0 "2v ½ n −0#q0 −5kp0
$
× d"t#¦
0
"2v 5kt ½ n −0#q0 exp − p0 ð"2v "0−2v ½ n −0#q0 −5kp0 Ł ½ n #q0 ¦5kp0
1%
"A08#
The functions of G"t# for the boundary value problem of traction\ described by eqn "03#\ are now written as G "−0# "t#
$
G "# "t#
G"v ½ n \ t#
0
5p0 1q0 2kt d"t#¦ exp − 2kp0 ¦1q0 p0 "2kp0 ¦1q0 # 2kp0 ¦1q0
$
1p0 0 d"t#¦ q0 p0
1%
"A19#
%
"A10#
½ n −0#p0 5"v ½ n −0#q0 −5kp0 "2v
$
d"t#¦
0
½ n −0#q0 "2v 5kt exp − ½ n −0#q0 −5kp0 Ł ½ n #q0 ¦5kp0 p0 ð"2v "0−2v
1%
"A11#
Pergamon
J[ Mech[ Phys[ Solids\ Vol[ 35\ No[ 09\ pp[ 0858Ð0879\ 0887 Þ 0887 Elsevier Science Ltd[ All rights reserved Printed in Great Britain 9911Ð4985:87 ,*see front matter
PII ] S9911Ð4985"87#99934Ð2
APPLICATION OF THE COSSERAT SPECTRUM THEORY TO VISCOELASTICITY XANTHIPPI MARKENSCOFF\$% WENSEN LIU\%& and MICHAEL PAUKSHTO' % University of California\ San Diego\ Department of Applied Mathematics and Engineering Sciences\ La Jolla CA 81982\ U[S[A[ & San Diego State University\ College of Engineering\ San Diego CA 81071\ U[S[A[ ' Saint Petersburg State University\ Institute of Mathematics and Mechanics\ Saint Petersburg\ Russia "Received 19 December 0886 ^ in revised form 10 February 0887#
ABSTRACT The Cosserat Spectrum theory is applied in the theory of linear viscoelasticity[ The solution for the Laplace transform of the displacement of the viscoelastic problem is expressed in a series of the Cosserat eigenfunctions\ which are dependent only on position\ and the coe.cients are expressed as convolutions of the time dependent body force or surface loading provided that the inverse Laplace transforms of the viscoelastic moduli are known[ This renders the Cosserat Spectrum theory advantageous for the solution of viscoelastic problems[ Several examples are shown[ Þ 0887 Elsevier Science Ltd[ All rights reserved[ Keywords ] A[ creep\ B[ viscoelastic material\ C[ Cosserat Spectrum[
0[ INTRODUCTION The homogeneous Navier equations Duł ¦v99 = uł 9\ ðv "l¦m#:m "0:"0−1n#\l and m being the Lame|s constants\ n the Poisson|s ratioŁ\ with homogeneous boundary conditions of displacement or traction admit nontrivial solutions when v takes values in a set of points lying outside the physical range of Poisson|s ratio called the Cosserat Spectrum[ The Cosserat Spectrum theory was introduced by Cosserat and Cosserat "0787# and then fully developed by Mikhlin "0862#\ who proved the completeness and orthogonality of the Cosserat eigenfunctions and represented the displacement _eld uł as summation of the Cosserat eigenfunctions for the boundary value problems of displacement or traction[ Pobedria "0869# applied the Cosserat Spectrum theory to 1!D viscoelastic problems[ In a recent paper\ Markensco} and Paukshto "0887# applied it to problems in elasticity and thermoelasticity[ They also developed a variational principle in thermoelasticity within the frame of the Cosserat Spectrum theory[ In the present article we develop a theory of linear viscoelasticity based on the Cosserat Spectrum theory[ We show that in the Laplace transformed space the Navier equations hold and the Cosserat Spectrum theory can be applied[ The solution for $ To whom correspondence should be addressed[ E!mail ] xmarkensÝames[ucsd[edu 0858
0869
X[ MARKENSCOFF et al[
the Laplace transform of the displacement function of viscoelastic problems is expre! ssed in series of the Cosserat eigenfunctions\ which are functions of geometry only "and not the transform variable s#\ and the coe.cients are functions of s that can be easily inverted for rational functions\ which is the case in the commonly used vis! coelastic models "Christensen\ 0860#[ Thus\ once the inverse Laplace transforms of these functions that depend on the viscoelastic moduli are known\ the solution may readily be expressed as convolutions of the time dependent body force or surface loading[ In this respect\ this method of solution has an advantage over the standard methods as in Lee "0844#\ Radok "0846#\ and Lee et al[ "0848#[
1[ FORMULATION OF THE GOVERNING EQUATIONS OF VISCOELASTICITY IN TERMS OF THE COSSERAT SPECTRUM THEORY The equations of quasi!static equilibrium are given by sij\j "t#¦Fi "t# 9 in V
"0#
with boundary conditions of displacement uł uł "t#
on 1V
"1a#
or traction łtn "t# sł "t# = nł on 1V
"1b#
while the stress sij "t# and strain oij "t# 01 ðui\ j "t#¦u j\i "t#Ł
"2#
are related by the constitutive equations "Christensen\ 0860# sij "t#
g
t
ð1m"t−t?# doij "t?#¦l"t−t?#dij dokk "t?#Ł
"3#
9
with m"t# and l"t# being the time dependent Lame|s moduli[ By applying the Laplace transform on the above equations\ denoting by {{ || the transformed quantities\ and combining the transformed eqns "0#\ "2# and "3#\ we obtain the Navier equations for the transformed displacements g
¼ u¼ j\ji − u¼i\ jj ¦v
Fi
m¼
in V
"4#
where ¼ v
l¼ ¦m¼ m¼
"5#
with boundary conditions of displacement uł¼ uł¼ b or traction
on 1V
"6a#
Application of the Cosserat Spectrum theory to viscoelasticity
0860
¼ łt sł¼ = nł on 1V
"6b#
We now observe that the displacement uł¼ satisfying eqns "4# and "6# may be expressed in terms of a series of the Cosserat eigenfunctions\ which satisfy the homogeneous Navier equations 91 u½n ¦v ½ n 99 = u½n 9 in V
"7#
with homogeneous boundary conditions of displacement u½n 9 on 1V
"8a#
łtn "u½n # sł "u½n # = nł 9 on 1V
"8b#
or traction
As presented by Mikhlin "0862#\ the Cosserat eigenfunctions are complete in the Sobolev space H 0\ and thus\ the solutions of the inhomogeneous eqn "4# with bound! ary conditions of displacement ðeqn "6a#Ł or traction ðeqn "6b#Ł admit the Mikhlin "0862# representation theorems as follows ] The solution for the boundary value problem of displacement g
:
F ¼ 99 = uł¼ − Duł¼ ¦v m¼
in V
"09a#
uł¼ 9 on 1V
"09b#
admits the representation g
:
v 0 "F\ u½n"−0# # "−0# "# "# ½n : ł \ u½n #u½n ¦ uł¼ s u½n ¦"F "F\ u½n #u½n m¼ n 0 0¦v v ¼ ¼ ½ n −v
6
g
7
"00a#
where g
:
g
:
"F\ u½# 0 Ð F = u½ dV
"00b#
Duł¼ ¦v ¼ 99 = uł¼ 9 in V uł¼ uł¼ b on 1V
"01a#
The solution of
"01b#
admits the representation v ¼v ½n uł¼ uł¼ 9 ¦ s "div uł¼ 9 \ div u½n #u½n ¼ v −v ½n n 0
"02a#
Duł¼ 9 9 in V
"02b#
where uł¼ 9 satis_es
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uł¼ 9 uł¼ b
on 1V
"02c#
The solution for the boundary value problem of traction g
:
F ¼ 99 = uł¼ − Duł¼ ¦v in V m¼ sł¼ "uł¼ # = nł ł¼t on 1V
"03a# "03b#
can be represented as 0 uł¼ s m¼
n 0
¼ 1" łf\ u½n"−0# # "−0# 0−v ½n ¼ ¼ " łf\ u½n #u½n u½n ¦" łf\ u½n"# #u½n"# ¦ 0¦v v ¼ ¼ −v ½n
6
7
"04a#
where g
: ¼ " łf\ u½# 0 Ð F = u½ dV¦Ð ł¼t = u½ dS
"04b#
By u½n"−0# in the above equations is indicated the in_nite orthogonal subspace of eigenfunctions corresponding to the eigenvalue of in_nite multiplicity v ½ −0\ by u½n"# is indicated the in_nite orthogonal subspace of eigenfunctions corresponding to the eigenvalue of in_nite multiplicity v ½ \ while by u½n is indicated the orthogonal subspace of eigenfunctions corresponding to the eigenvalues of _nite multiplicity of the discrete spectrum v ½ n[ We may note here that the Cosserat eigenfunctions are functions of the space variables only "and not the transform variable s#[
2[ INVERSION OF THE LAPLACE TRANSFORMS An advantage of expressing the solutions of the viscoelastic problem in terms of the Cosserat eigenfunctions is that the inversion of the Laplace transform may be obtained by convolution\ provided that inverse Laplace transforms of the terms involving the viscoelastic moduli can be easily obtained[ Let us consider the representation eqn "00# for the boundary value problem of displacement with homogeneous boundary conditions and denote 0 ¼# m¼ "0¦v
"05a#
0 m¼
"05b#
v ½n m¼ "v ¼# ½ n −v
"05c#
G
"−0# "s#
G
"# "s# G
"v ½ n \ s#
If for a given viscoelastic material\ eqn "05# can be inverted to obtain G"−0# "t#\ G "t# and G"v ½ n\ t#\ then the solution of the viscoelastic problem in the time domain may be obtained by convolution as follows "#
Application of the Cosserat Spectrum theory to viscoelasticity
uł "t# s
n 0
g
t
ðF ł "t?#\ u½n"−0# ŁG "−0# "t−t?# dt?u½n"−0# ¦ s n 0
9
¦s n 0
g
g
0862
t
ðF ł "t?#\ u½n"# ŁG "# "t−t?# dt?u½n"#
9
t
ðF ł "t?#\ u½n ŁG"v ½ n \ t−t?# dt?u½n
"06a#
9
where ł "t#\ u½Ł 0 ÐF ł "t# = u½ dV ðF
"06b#
Similarly\ the inversion of the representation eqn "02# for the boundary value problem of displacement with inhomogeneous boundary conditions\ and eqn "04# for the boundary value problem of traction may be obtained[ Thus for any time dependent body force loading or boundary loading\ the solution of viscoelastic problems may be written immediately in the form of eqn "06# if the Cosserat eigenfunctions are known for the geometry of the problem[ The functions G"−0# "t#\ G"# "t# and G"v ½ n\ t# for the commonly used viscoelastic models are presented in the Appendix[
3[ APPLICATIONS AND EXAMPLES "0# Time dependent temperature loadin` We consider viscoelasticity problems with thermal loading that may\ in addition\ be time dependent[ As shown by Markensco} and Paukshto "0887#\ the temperature loading is equivalent to a body force loading that results in the representation of the displacement _eld in terms of a series of the Cosserat eigenfunctions according to ½n v "T\ div u½n #u½n uł "2v−0#a s v ½ n −v n 0
"07#
for the boundary value problem of displacement with zero boundary displacement\ and
uł "2v−0#a s n 0
6
1 0−v ½n "T\ div u½n #u½n "T\ div u½n"−0# #¦ v¦0 v−v ½n
7
"08#
for the boundary problem of traction with traction free boundary condition[ The displacement _eld in the Laplace transform space s in thermoviscoelasticity is\ accord! ingly\ represented by the series v ½n
\ div u½n #u½n ¼ −0#a s "T uł¼ "2v ¼ ½ n −v n 0 v
"19#
for the boundary value problem of displacement with zero boundary displacement\ and by
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uł¼ "2v ¼ −0#a s
n 0
$
1 0−v ½n
\ div u½n"−0# #u½n"−0# ¦
\ div u½n #u½n "T "T ¼ ¦0 ¼ −v v v ½n
%
"10#
for the boundary value problem with traction free boundary condition[ By applying the convolution theorem\ according to the analysis of the previous section\ we have the representation of the displacement for the solution of the ther! moviscoelastic problem
uł s
n 0
g
t
ðT"t?#\ div u½n ŁG"v ½ n \ t−t?# dt?u½n
"11a#
$
"11b#
9
where G"v ½ n \ t# v ½ n aL−0
2v ¼ −0 v ¼ ½ n −v
%
for the boundary value problem of displacement\ and
uł s
n 0
6g
t
ðT"t?#\ div u½n"−0# ŁG "−0# "t−t?# dt?u½n"−0#
9
¦
g
t
ðT"t?#\ div u½n ŁG"v ½ n \ t−t?# dt?u½n
"12a#
9
where G "−0# "t# 1aL−0
$
% $ %
¼ −0 2v ¼ ¦0 v
G"v ½ n \ t# "0−v ½ n #aL−0
2v ¼ −0 v ¼ −v ½n
"12b#
for the boundary value problem of traction[ If the material is initially undisturbed and the temperature T is not time dependent\ then the convolution integrals are easily performed[ "1# The Lame problem under time dependent loadin` This is the problem of a spherical\ viscoelastic shell under time dependent internal and external pressure p0 and p1\ respectively[ The classical elastic solution "Lure\ 0853# is ur −
"p1 r12 −p0 r02 #r 0 0 "p1 −p0 #r02 r12 − 2 2 m"2v−0# 3m "r12 −r02 #r1 r1 −r0
uu u8 9
"13#
where r0 and r1 are the internal and external radius\ respectively[ It can be veri_ed that the _rst term in eqn "13# is an eigenfunction corresponding to v ½ 0:2\ and the
Application of the Cosserat Spectrum theory to viscoelasticity
0864
second term is an eigenfunction corresponding to v ½ [ In the Laplace transform space s\ the displacement _eld will be u¼r −
"p¼1 r12 −p¼0 r02 #r 0 0 "p¼1 −p¼0 #r02 r12 − ¼ −0# m¼ "2v 3m¼ "r12 −r02 #r1 r12 −r02
u¼u u¼8 9
"14#
If the viscoelastic material is Hookean solid in dilatation and Kelvin solid in distortion\ the material parameters represented by eqns "A2#Ð"A4# in the Appendix are now substituted into eqn "14# u¼r −
"p¼1 r12 −p¼0 r02 #r 0 "p¼1 −p¼0 #r02 r12 − 1"q9 ¦q0 s# "r12 −r02 #r1 2k"r12 −r02 #
u¼u u¼8 9
"15#
The inverse Laplace transform of eqn "15# gives the solution for any general time dependent loading ur − −
ðp1 "t#r12 −p0 "t#r02 Łr 2k"r12 −r02 # r02 r12 1q0 "r12 −r02 #r1
g
t
0
ðp1 "t#−p0 "t#Ł exp −
9
1
q9 "t−t?# dt? q0
uu u8 9
"16#
"2# The `ravitatin` sphere Let the body force acting on a viscoelastic sphere be gravity according to the Newton|s law of gravitation[ It is known from potential theory that the resultant attraction is directed along the radius to the center of the sphere and that its magnitude is proportional to radial distance[ Consequently the body force is given by ł − F
rgrł r9
"17#
where r is density\ g the gravitational acceleration\ r9 the radius of the sphere and łr position vector measured from the center of the sphere[ The displacement _eld of an elastic gravitating sphere is presented by Lure "0853#[ We rewrite it as follows ur −
0 rgr9 r2 1r r− 1 ¦ 09m v¦0 2v−0 r9
uu u8 9
$ 0 1
% "18#
so that the _rst part is an eigenfunction corresponding to v ½ −0\ and the second part is an eigenfunction corresponding to v ½ 0:2[ If the spherical shell is of viscoelastic material\ then the Laplace transform of the displacement _eld will be
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X[ MARKENSCOFF et al[
u¼r −
0 rgr9 r2 1r r− 1 ¦ ¼ ¦0 ¼ −0 09sm¼ v 2v r9
$ 0 1
%
u¼u u¼8 9
"29#
Suppose that the material behaves as a Hookean solid in dilatation and Maxwell ~uid in distortion\ then the material parameters are represented by eqns "A03#Ð"A05#[ The inverse Laplace transform of eqn "29# will be ur −
$
r2 rgr9 r − 1 04k r9
0 1%0 1
rgr9 0 t 1q0 ¦ 1 exp − 09 k 2k tm tm
r−
uu u8 9
"20#
where tm ð"2kp0¦1qi#:2kŁ[ When t : \ eqn "20# reduces to ur −
rgr9 r2 rgr9 r r− 1 − 09k 04k r9
0 1
uu u8 9
"21#
It is interesting to note that ur is _nite as t : [ We have veri_ed that both eigenfunctions satisfy 9×uł 9\ that is\ both are distortion!free vector[ Furthermore\ the second eigenfunction associated with v ½ 0:2 produces uniform normal stress components and no shear components[ Physically\ since both eigenfunctions are of zero distortion\ the displacement _eld remains _nite even though the material is of the Maxwell ~uid type in distortion[ "3# Heat ~ow past a thermally insulated spherical cavity Suppose the material behaves as a Hookean solid in dilatation and a Maxwell ~uid in distortion\ then the material parameters are represented by eqns "A03#Ð"A05#[ The temperature _eld is given by T"t# T0 "t#¦T1 "t#
"22a#
where T0 "t# is a linear function of position z r cos u\ i[e[\ T0 "t# t"t#r cos u\ T1 "t#
t"t#r91 cos u 1r1
"22b#
where r9 is the radius of the spherical cavity[ The thermoelastic solution to this problem was given by Florence and Goodier "0848# who treated the temperature _eld with constant t[ Here we view this problem in thermoviscoelasticity with arbitrary t"t#[ The Laplace transform of the displacement _eld corresponding to free thermal expansion caused by T0 "t# is given by
Application of the Cosserat Spectrum theory to viscoelasticity 0 1
0866
1
u¼r at¼ r cos u u¼u 01 at¼ r1 sin u
"23#
The inverse Laplace transform of eqn "23# is given by ur 01 at"t#r1 cos u uu 01 at"t#r1 sin u
"24#
The Laplace transform of the displacement caused by the temperature _eld T1 "t# is a Cosserat eigenvector corresponding to the eigenvalue v ½ −0\ i[e[\ u¼r
¼ −0#at¼ r91 "2v r92 − 2 cos u 1"v ¼ ¦0# 2r
u¼u
¼ −0#at¼ r91 r9 "2v r92 − 2 sin u ¼ ¦0# 1"v 1r 5r
0 1 0 1
"25#
The inverse Laplace transform of eqn "25# gives ur
uu
r91 r92 − 2 cos u 3 2r
t
0 1 g 0 1 g
t"t?#G "−0# "t−t?# dt?
9
r91 r9 r92 − 2 sin u 3 1r 5r
t
t"t?#G "−0# "t−t?# dt?
"26a#
9
where G−0 "t# de_ned by eqn "12b# is written as G "−0# "t# 1a
$
0 1%
2p0 1q0 t d"t#¦ 1 exp − tm tm ktm
"26b#
If t t9H"t#\ then eqn "26# becomes ur
at9 r91 1q0 t 2− exp − 1 ktm tm
−
uu
at9 r91 1q0 t 2− exp − 1 ktm tm
r92 r9 − 2 sin u 1r 5r
$ $
r92 cos u 2r2
0 1%0 1 0 1%0 1
"27#
where tm ð"2kp0¦1q0#:2kŁ is the relaxation time of the viscoelastic material[ When t : \ eqn "27# reduces to ur
2at9 r91 r92 − 2 cos u 1 2r
uu
r92 2at9 r91 r9 − 2 sin u 1 1r 5r
0 1 0 1
"28#
It is interesting to note that ur is _nite when t : [ We have veri_ed that the
0867
X[ MARKENSCOFF et al[ "−0#
eigenfunction satis_es 9×u½ 9[ Physically\ since the eigenfunction is of zero distortion\ the displacement _eld remains _nite even though the material is of the Maxwell ~uid type in distortion[
4[ CONCLUSIONS The Cosserat Spectrum theory is applied to the linear theory of viscoelasticity[ The Mikhlin representation theorems in the form of Laplace transform are written in such forms that the inverse Laplace transforms are easily carried out for some frequently used viscoelastic models in terms of convolutions[ Also the physical meaning associ! ated with the Cosserat eigenfunctions corresponding to the eigenvalues v ½ −0 "distortion free# and v ½ "divergence free# ties naturally with viscoelastic behavior that may di}er in the dilatation and distortion part of the constitutive relation[
ACKNOWLEDGEMENT The support of the National Science Foundation through the Institute of Mechanics and Materials at University of California\ San Diego to Michael Paukshto is gratefully acknowledged[
REFERENCES Christensen\ R[ M[ "0860# Theory of Viscoelasticity an Introduction[ Academic Press\ New York[ Cosserat\ E[ and Cosserat\ F[ "0787# Sur les equations de la theorie de l|elasticite[ C[ R[ Acad[ Sci[ Paris 015\ 0978Ð0980[ Florence\ A[ L[ and Goodier\ J[ N[ "0848# Thermal stress at spherical cavities and circular holes in uniform heat ~ow[ Journal of Applied Mechanics 15\ 182Ð183[ Lee\ E[ H[ "0844# Stress analysis in viscoelastic bodies[ Quarterly of Applied Mathematics 02\ 072Ð089[ Lee\ E[ H[\ Radok\ J[ R[ M[ and Woodward\ W[ B[ "0848# Stress analysis for linear viscoelastic materials[ Transactions of the Society of Rheolo`y 2\ 30Ð48[ Lure\ A[ I[ "0853# Three!dimensional Problems of the Theory of Elasticity[ Interscience\ New York[ Markensco}\ X[ and Paukshto\ M[ V[ "0887# The Cosserat spectrum in the theory of elasticity and applications[ Proceedin`s of the Royal Society of London A343\ 036Ð043[ Mikhlin\ S[ G[ "0862# The spectrum of a family of operators in the theory of elasticity[ Russian Math[ Surveys 17\ 34Ð77[ Pobedria\ B[ E[ "0869# On solution of contact!type problems in the linear theory of visco! elasticity[ Doklady Akad[ Nauk SSSR 089\ 186Ð299[ Radok\ J[ R[ M[ "0846# Viscoelastic stress analysis[ Quarterly of Applied Mathematics 04\ 087Ð 191[
Application of the Cosserat Spectrum theory to viscoelasticity
0868
APPENDIX The functions of G"−0# "t#\ G"# "t# and G"v ½ n\ t# for the commonly used viscoelastic models are as follows[ Model 0*Hookean solid in dilatation and Kelvin solid in distortion The constitutive equations take the form skk "t# 2kokk "t# sij "t#
g
"A0#
t
ðq9 ¦q0 d"t−t?#Ł deij "t?#
"A1#
9
where k is the buck modulus in dilatation and q0:q9 the retardation time in distortion\ sij and eij are the stress deviator and strain deviator\ respectively[ The complex Lame moduli m¼ \ l¼ and the spectrum parameter v ¼ are m¼ 01 "q9 ¦q0 s#
"A2#
l¼ 02 "2k−q9 −q0 s#
"A3#
v ¼
5k¦q9 ¦q0 s 2"q9 ¦q0 s#
"A4#
The functions of G"t# for the boundary value problem of displacement\ described by eqn "09#\ are now written as G "−0# "t#
0
2 1q9 ¦2k exp − t 1q0 1q0
G "# "t#
G"v ½ n \ t#
0
1 q9 exp − t q0 q0
1
"A5#
1
"A6#
0
5v "0−2v ½ ½ n #q9 ¦5k exp − t "2v "0−2v ½ n −0#q0 ½ n #q0
1
"A7#
The functions of G"t# for the boundary value problem of traction\ described by eqn "03#\ are now written as G "−0# "t#
1
"A8#
0
"09#
G "# "t#
½ n \ t# G"v
0
2 1q9 ¦2k exp − t q0 1q0 1 q9 exp − t q0 q0
1
0
5"v "0−2v ½ n −0# ½ n #q9 ¦5k exp − t "2v "0−2v ½ n −0#q0 ½ n #q0
1
"A00#
Model 1*Hookean solid in dilatation and Maxwell ~uid in distortion The constitutive equations take the form skk "t# 2kokk "t# sij "t#
g
t
9
0
"A01#
1
t−t? q0 exp − deij "t?# p0 p0
"A02#
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X[ MARKENSCOFF et al[
where k is the buck modulus in dilatation\ p0 the relaxation time and q0 the viscosity in distortion[ The complex Lame moduli m¼ \ l¼ and the spectrum parameter v ¼ are m¼
q0 s 1"0¦p0 s#
"A03#
2k¦"2kp0 −q0 #s l¼ 2"0¦p0 s#
"A04#
5k¦"5kp0 ¦q0 #s 2q0 s
"A05#
¼ v
The functions of G"t# for the boundary value problem of displacement\ described by eqn "09#\ are now written as G "−0# "t#
$
G "# "t#
G"v ½ n \ t#
0
2p0 1q0 2kt d"t#¦ exp − 2kp0 ¦1q0 p0 "2kp0 ¦1q0 # 2kp0 ¦1q0
$
1p0 0 d"t#¦ q0 p0
1%
"A06#
%
"A07#
5v ½ n p0 "2v ½ n −0#q0 −5kp0
$
× d"t#¦
0
"2v 5kt ½ n −0#q0 exp − p0 ð"2v "0−2v ½ n −0#q0 −5kp0 Ł ½ n #q0 ¦5kp0
1%
"A08#
The functions of G"t# for the boundary value problem of traction\ described by eqn "03#\ are now written as G "−0# "t#
$
G "# "t#
G"v ½ n \ t#
0
5p0 1q0 2kt d"t#¦ exp − 2kp0 ¦1q0 p0 "2kp0 ¦1q0 # 2kp0 ¦1q0
$
1p0 0 d"t#¦ q0 p0
1%
"A19#
%
"A10#
½ n −0#p0 5"v ½ n −0#q0 −5kp0 "2v
$
d"t#¦
0
½ n −0#q0 "2v 5kt exp − ½ n −0#q0 −5kp0 Ł ½ n #q0 ¦5kp0 p0 ð"2v "0−2v
1%
"A11#