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A nonlinear stability analysis of an incompressible elastic plate subjected to an all-round tension
\
Pergamon
J[ Mech[ Phys[ Solids\ Vol[ 35\ No[ 00\ pp[ 1150Ð1171\ 0887 Þ 0887 Elsevier Science Ltd[ All rights reserved Printed in Great Britain 9911Ð4985:87 ,*see front matter
PII ] S9911Ð4985"87#99922Ð6
A NONLINEAR STABILITY ANALYSIS OF AN INCOMPRESSIBLE ELASTIC PLATE SUBJECTED TO AN ALL!ROUND TENSION YIBIN FU$ Department of Mathematics\ University of Keele\ Sta}ordshire ST4 4BG\ U[K[ "Received 1 September 0886#
ABSTRACT When an incompressible elastic plate is subjected to an all!round tension\ linear stability analysis predicts that it will become neutrally stable with respect to both extensional and ~exural modes and to all wave! numbers when the tension is twice the shear modulus[ In this paper a weakly nonlinear analysis is conducted to determine the post!buckling states when the tension deviates from its neutral value by a small amount[ It is found that such a prestressed plate with any given thickness can bifurcate into an in_nite number of post!buckling states\ and that only one such state is sinusoidal\ the rest being non!sinusoidal[ All the post! buckling states are found to be stable with respect to small perturbations[ Þ 0887 Elsevier Science Ltd[ All rights reserved[ Keywords ] A[ buckling\ B[ elastic material\ plates\ _nite strain\ C[ stability and bifurcation[
0[ INTRODUCTION When an incompressible isotropic elastic plate is subjected to an all!round pressure\ a linear stability analysis using the theory of _nite elasticity shows that it is only stable if −1 ³ p ³ f "kh# where p is the pressure scaled by the shear modulus\ k is the wavenumber\ h is the plate thickness and f "kh# takes di}erent forms for extensional and ~exural modes "see e[g[ Ogden and Roxburgh\ 0882#[ Nonlinear stability analyses have been conducted by Fu and Rogerson "0883# and Fu "0884# for values of p near the upper branch p f "kh# of the neutral stability curve ^ the _rst of these two papers is concerned with a single near!neutral mode whereas the second is concerned with a triad of modes[ In this paper we investigate the lower branch stability properties "i[e[ for values of p close to −1#\ thus completing the weakly nonlinear stability analysis for the type of plate mentioned above[ This lower branch is non!dispersive in the sense that the neutral value of p is independent of wavenumber "thus\ all modes become neutral at the same time#[ Also the lower branch is the same for both extensional and ~exural modes[ Because of the co!existence and strong interaction of extensional and ~exural near!neutral modes at all wavenumbers\ we expect to see non!sinusoidal as well as sinusoidal post!buckling states[ This is precisely what we shall _nd in this paper[ Although a negative pressure is really a tension\ we shall still $ Tel[ ] 9933 "9# 0671 472 549[ Fax ] 9933 "9# 0671 473 157[ E!mail ] y[fuÝkeele[ac[uk[ 1150
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use the term pressure in the rest of this paper in order to be consistent with the analysis of Fu and Rogerson "0883#[ It is well!known that for problems involving non!dispersive wave modes "such as the evolution of surface waves#\ evolution equations are usually derived by imposing a solvability condition at second!order of an in_nite hierarchy of boundary value problems resulting from an asymptotic expansion[ However\ although the lower branch is non!dispersive as explained above\ we _nd that the second!order problem can be solved without the need to impose any solvability conditions and that the evolution equations are obtained by imposing a solvability condition at third!order "we note however that with the use of the virtual work method we do not need to do so explicitly#[ In the following section\ we summarize the original governing equations from the theory of _nite elasticity and derive the resulting _rst and second order boundary value problems after an asymptotic expansion[ These two boundary value problems are solved in Sections 2 and 3\ respectively[ The evolution equations are then derived in Section 4 with the aid of the virtual work method\ and their static solutions\ which describe the post!buckling states\ are sought in Section 5[ In Section 6 we show that all the post!buckling solutions found in Section 5 are stable and in the _nal section we summarize the main results obtained in the present paper[
1[ GOVERNING EQUATIONS We consider a homogeneous elastic plate composed of a non!heat!conducting elastic material which possesses an initial unstressed state B9[ A homogeneous stress _eld is then imposed upon B9 to produce a _nitely stressed equilibrium con_guration denoted by Be[ It is the stability of Be that we wish to study[ To this end\ we superimpose a small amplitude time!dependent perturbation on Be[ The resulting con_guration\ termed the current con_guration\ is denoted by Bt[ The position vectors of a representative particle relative to a common coordinate system are denoted by XA\ xi "XA# and x½i "XA\ t# in B9\ Be and Bt\ respectively[ The origin is chosen to lie in the mid!plane of Be and the plate surfaces correspond to x1 2h[ We write x½i "XA \ t# xi "XA #¦ui "x j \ t#\
"0#
where ui "xj\ t# is a small time!dependent displacement associated with the deformation Be : Bt[ The plate will be assumed to be in a state of plane strain so that all subscripts range from 0 to 1[ To simplify analysis we assume that the plate is in_nite along the x0!direction[ The deformation gradients arising from the deformations B9 : Bt and B9 : Be are denoted by F and Þ F\ respectively\ and de_ned by FiA
1x½i 1xi \ Þ FiA [ 1XA 1XA
"1#
It is clear from "0# that ÞjA \ FiA "dij ¦ui\ j #F
"2#
Nonlinear stability of plate under tension
1152
where here and henceforth a comma indicates di}erentiation with respect to the implied spatial coordinate[ Furthermore\ the convention whereby upper case indices refer to coordinates in B9 and lower case indices to coordinates in Be will be observed[ "Since the coordinates x½j do not appear in the following analysis\ no ambiguity will arise in\ for example\ ui\j[# In the absence of body forces\ the equations of motion and the incompressibility constraint are given by piA\A rui \ det F 0\
"3#
where r is the "constant# density and p is the _rst PiolaÐKirchho} stress which\ in component form\ is given by piA
1W −0 −pF Ai \ 1FiA
"4#
where W is the strain!energy function "per unit volume# and p is a Lagrange multiplier associated with the incompressibility constraint and interpreted as a pressure[ In the subsequent stability studies\ it will be assumed that on the two plate surfaces the traction vector pN\ where N is the unit normal to the plate surfaces in B9\ is prescribed in B9\ and that this traction will be maintained at this value during the incremental deformation Be : Bt[ Such an assumption is usually referred to as a dead! load traction boundary condition and is represented by "piA −p¹ iA #NA 9\
"5#
where p¹ iA is the value of piA calculated from "4# with F replaced by Þ F and p by p¹[ It is convenient to introduce a tensor function with components xij through xij "piA −p¹ iA #F ÞjA
0
1
1W −0 −pF Ai −p¹ iA Þ F jA [ 1FiA
"6#
In terms of this new tensor\ "3a# and "5#\ respectively\ may be written simply as xij\j rui \
"7#
xij n j 9\
"8# −T
F where "ni# is the unit normal to the plate surfaces in Be "and is parallel to Þ We write
N#[
p p¹¦p \ where p¹ is the pressure in Be and p is the incremental pressure associated with the deformation Be : Bt[ It can be shown "see e[g[ Fu and Rogerson\ 0883 ^ Fu and Ogden\ 0887# that xij has the following expansion about "F Þ\ p¹# ] xij A0jilk uk\l ¦01 A1jilknm uk\l um\n ¦05 A2jilknmqp uk\l um\n up\q ¦p¹ "u j\i −u j\k uk\i ¦u j\k uk\l ul\i #−p "d ji −u j\i ¦u j\k uk\i #¦O"o3 #\
"09#
where o is a small parameter characterizing the amplitude of ui\j and p \ A0\ A1\ A2 are the _rst!\ second! and third!order tensors of instantaneous elastic moduli in Be[
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Expressions for these moduli can be found in Fu "0883# "with errata given in Fu\ 0884# for the case when the principal stretches are all unit and in Fu and Ogden "0887# otherwise[ A similar expansion of the constraint eqn "3b# leads to ui\i 01 um\n un\m −01 "ui\i # 1 −det"um\n #[
"00#
We shall assume from now on that the pre!stress is an all!round pressure[ Since the plate is incompressible\ no deformation is produced by the pressure[ Con_guration Be then coincides with B9 and A0jilk is given by A0jilk m"d jl dik ¦d jk dil # where m is the shear modulus[ Before proceeding further\ we _rst non!dimensionalize the governing equations and boundary conditions using L "to be de_ned# as the length scale " for ui\ xi#\ m as the stress scale "for p¹\ p \ A0jilk \ etc[# and Lzr:m as the time scale " for t#[ In order to avoid introducing additional notation\ we shall use the same symbols to denote the cor! responding non!dimensionalized quantities[ The non!dimensionalized forms of the governing equations and boundary conditions then remain unchanged except that r 0\ m 0[ Thus\ for instance\ we now have A0jilk d jl dik ¦d jk dil [
"01#
The length scale L is arbitrary at the moment[ A natural choice for L would be the half plate thickness h\ but to simplify the analysis\ it will later be chosen to be the inverse of the wave number of the fundamental instability mode[ It is known from Ogden and Roxburgh "0882# and Fu and Rogerson "0883# that the plate under consideration is stable with respect to travelling wave perturbations only if −1 ¾ p¹ ¾ f"kh# where k is the wavenumber and the function f takes di}erent forms for ~exural and extensional instability modes[ As mentioned in the introduction\ we shall be concerned with the post!buckling behaviour of the plate when p¹ deviates from its lower branch neutral value −1 by a small amount[ Thus we let p¹ −1¦oa p¹0 \
"02#
where p¹0 is an arbitrary O"0# constant and a is to be determined[ Since the evolution equations are derived from a solvability condition imposed at third order\ we require 11 ui ½ o2 1t1 and so the appropriate slow time variable describing the wave modulation is given by t ot\
"03# a
since ui ½ o and ui depends on t through t only[ The order o deviation of p¹ from −1 speci_ed by "02# will induce an O"oa:1# wave speed "or growth rate#[ In order to absorb the factor eikvt "in the usual travelling wave representation eik"x0 −vt# # into the wave amplitude which depends on t\ we choose v o which gives a 1[ We then look for the following form of asymptotic solution ] ui oui"0# "x0 \ x1 \ t#¦o1 ui"1# "x0 \ x1 \ t#¦o2 ui"2# "x0 \ x1 \ t#¦= = = \ p op"0# "x0 \ x1 \ t#¦o1 p"1# "x0 \ x1 \ t#¦o2 p"2# "x0 \ x1 \ t#¦= = = [
"04#
Nonlinear stability of plate under tension
1154
On substituting "02# and "04# into "00#\ "7# and "8# with xij given by "09# and then equating the coe.cients of o and o1\ we obtain the equations "0# "0# u0\0 ¦u1\1 9\
"05#
"0# −p\i"0# 9\ A0jilk uk\lj
"06#
0 "0# 1ilk k\l
A
"0# 1\i
"0#
u −1u −p d1i 9\
"07#
at order o\ and "1# "1# ¦u1\1 01 ui\"0#j u"0# u0\0 j\i \
"08#
"1# "0# "0# A0jilk uk\lj −p\i"1# −A1jilknm um\n uk\lj −p\j"0# u"0# j\i \ 0 "1# 1ilk k\l
A
"1# 1\i
0 1
"1#
1 "0# "0# 1ilknm k\l m\n
u −1u −p d1i − A
u u
"0# "0# 1\k k\i
"19# "0#
"0# 1\i
−1u u −p u \
"10#
at order o1[ Note that "07# and "10# are boundary conditions which must hold on the boundaries x1 2h[ These two sets of boundary value problems are solved in the next two sections[
2[ LEADING!ORDER SOLUTION The leading!order constraint relation "05# implies the existence of a {{stream|| function c"x0\ x1\ t# such that u0"0# c\1 \ u1"0# −c\0 [
"11#
With A0jilk given by "01#\ "06# and "07# reduce to 91 ui"0# −p\i"0# 9\
"12#
"0# "0# u0\1 −u1\0 9\ p"0# 9 on x1 2h[
"13#
Eliminating p"0# from "12# by cross!di}erentiation and making use of "11# we obtain 93 c 9[
"14#
The two boundary conditions "13# yield 91 c 9 and
"91 c# \1 9 on x1 2h[
"15#
The solution to the boundary value problem "14# and "15# is obviously 91 c 9[
"16#
It then follows from "12# with i 1 and "13b# that p"0# 0 9[
"17#
Let c H"x1\ k# eikx0 [ Equation "16# yields H"x1 \ k# n0 "k:=k=# sinh"kx1 #¦n1 cosh"kx1 #\
"18#
where n0 and n1 are arbitrary real constants and k:=k= has been inserted to make
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H"x1\ −k# H"x1\ k#[ In our later numerical calculations\ we normalize H such that n01 ¦n11 0[ That n0 and n1 are arbitrary re~ects the fact that ~exural and extensional modes "corresponding to n0 9 and n1 9\ respectively# are both neutrally stable at p¹ −1[ The general solution for c is then given by
k −
k −
c s Ak "t#H"x1 \ k# eikx0 s Ak "t#j j "k# eksj x1 eikx0 \
"29#
where j0 "k#
n1 ¦n0 k:=k= n1 −n0 k:=k= \ j1 "k# \ s0 0\ s1 −0\ 1 1
"20#
and summation on the repeated subscript j is observed[ The shape function H has been written in the alternative form shown in the last expression of "29# to facilitate later manipulations[ For c to be real\ we impose the condition A−k "t# A Þk "t#\ where an overbar signi_es complex conjugation[ We note that the representation "29# corresponds to the fundamental mode having unit wavenumber\ which is a consequence of choosing the lengthscale L to be the inverse of the dimensional wavenumber of the fundamental mode[ We also note that we could have represented c by a Fourier integral under appropriate assumptions[ In fact all the Fourier sums in this paper could be converted to Fourier integrals under the same assumptions[ It follows from "29# that
k −
k −
u0"0# s kAk "t#s j j j "k# eksj x1 eikx0 s Ak "t#W0 "x1 \ k# eikx0 \
k −
k −
u1"0# s "−ik#Ak "t#j j "k# eksj x1 eikx0 s Ak "t#W1 "x1 \ k# eikx0 \
"21#
"22#
where W0 "x1 \ k# ks j j j "k# eksj x1 \ W1 "x1 \ k# "−ik#j j "k# eksj x1 \
"23#
and throughout this paper we adopt a modi_ed summation convention whereby a su.x appearing in one term more than once is summed from 0 to 1[
3[ SECOND!ORDER SOLUTION The second!order solution is determined by "08# and "19# subject to the boundary condition "10# on x1 2h[ Following Ogden "0884#\ we rewrite the constraint relation "08# as "0# "0# "u0"1# −u0"0# u0\0 # \0 ¦"u1"1# −u0"0# u1\0 # \1 9[
It then follows that there exists a function f"x0\ x1\ t# such that "0# "0# u0"1# f\1 ¦u0"0# u0\0 \ u1"1# −f\0 ¦u0"0# u1\0 [
"24#
Nonlinear stability of plate under tension
With the use of the expression for A "19# reduces to
1 jilknm
1156
given in Fu "0883#\ it can be shown that
1 1 91 ui"1# −p\i"1# −"3¦n0 #"c\01 ¦c\00 # \i \
"25#
where n0 is a second!order material constant and use has been made of "16#[ Similarly\ the boundary condition "10# with i 0\ 1 reduces to "1# "1# u0\1 −u1\0 9\ on x1 2h\
"26#
1 1 ¦c\00 #\ on x1 2h[ p"1# "4¦n0 #"c\01
"27#
and
"1# "1# Eliminating p"1# from "25# gives 91 "u0\1 −u1\0 # 9[ Substituting "24# into this equation and "26#\ we obtain
93 f 9\
"28#
91 f 9 on x1 2h[
"39#
Another boundary condition for f can be obtained from "27#[ Di}erentiating "27# "1# \ we obtain with respect to x0 and using "25# with i 0 to eliminate p\0 "91 f# \1 9 on x1 2h[
"30#
We observe that as is usual with asymptotic expansions involving non!dispersive wave modes\ the linearized operators on the left hand sides of "28#Ð"30# are the same as those in "14# and "15#[ The unusual feature here is that the inhomogeneous terms on the right hands of "28#Ð"30# are identically zero[ This implies that no solvability conditions need to be imposed in order to solve the present second!order problem[ The solution to "28#Ð"30# is obviously given by 91f 9 which is the same as that satis_ed by the function c at leading!order[ Without loss of generality\ we may take f 9 since otherwise it can be absorbed into c[ Hence\ the second!order displacement _eld is given by "0# "0# u0"0# u0\i [ ui"1# u0"0# ui\0
"31#
"1# Equation "25# with i 1 gives an expression for p\1 [ Integrating this expression subject to the boundary condition "27# yields 1 1 "0# "0# ¦c\00 # "4¦n0 #u1\m u1\m [ p"1# "4¦n0 #"c\01
"32#
This completes our solution of the second!order problem[ For later use\ we de_ne two functions Gmn "x1\ k# and Umn "x1\ k# through
"0# s Ak "t#Gmn "x1 \ k# eikx0 \ um\n
"33#
k −
"0# u0\mn s Ak "t#Umn "x1 \ k# eikx0 [ k −
"34#
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It follows from "21# and "22# that G00 "x1 \ k# ikW0 "x1 \ k# ik1 s j j j "k# eksj x1 −G11 "x1 \ k#\ G01 "x1 \ k# W?0 "x1 \ k# k1 s1j j j "k# eksj x1 G10 "x1 \ k#\ Umn "x1 \ k# ikGmn "x1 \ k#[
"35#
Thus\ we have
"1# s um\n p −
6
7
s Aq Ap−q ðG0m "x1 \ q#G0n "x1 \ p−q#¦W0 "x1 \ q#Umn "x1 \ p−q#Ł eipx0 \
q −
"36# and
p"1# "4¦n0 # s p −
6
7
s Aq Ap−q G1m "x1 \ q#G1m "x1 \ p−q# eipx0 [
q −
"37#
De_ning v"p\ q\ s j # i"d0p d0q −d1p d1q #s j ¦"d1p d0q ¦d1q d0p #\ we may write Gpq "x1\ k# as Gpq "x1 \ k# k1 v"p\ q\ s j #j j "k# eksj x1 \
"38#
where we have made use of the fact that s01 s11 0 ðsee "20#Ł[
4[ THE EVOLUTION EQUATIONS We now use the virtual work method "see Fu\ 0884 ^ Fu and Devenish\ 0885# to derive the evolution equations for the amplitudes Am "m 9\ 0\ [ [ [#[ The traditional way of deriving evolution equations is to expand all the governing equations and boundary conditions to third!order and then to impose a solvability condition after some reductions[ The virtual work method is in essence equivalent to this procedure but does not require the expansion of governing equations and boundary conditions to third!order[ We consider the virtual work integral
G
xpq nq u¼p ds 9\
"49#
1D
where 1D is the boundary of the rectangular region D "9 ¾ x0 ¾ 1p\ −h ¾ x1 ¾ h# in Be\ "nq# is the unit normal to 1D and u¼p is a linear solution given by u¼p Wp "x1 \ −k# e−ikx0 [
"40#
Equation "49# holds because xpqnq 9 on x1 2h ðsee "8#Ł and the integrals along the two vertical paths x0 9\ 1p cancel due to the periodicity of the integrand[ Applying the divergence theorem to "49#\ we obtain
Nonlinear stability of plate under tension
g0
1
1158
1
1 up u¼p ¦xpq u¼p\q dx0 dx1 9\ 1t1
o1
D
"41#
where use has been made of the equation of motion "7# and the fact that the depen! dence on t is through t only[ Substituting "02# and "04# into "09# yields an asymptotic expansion for xpq[ On inserting this expansion and "04a# into "41# and equating the coe.cients of o\ o1 and o2\ respectively\ we obtain
g
h
dx1
g
1p
dx1
g
1p
h
−h
dx1
g 0
1
"43#
11 ua"0# "0# 0 "2# "2# u¼a ¦p¹0 ub\a u¼a\b ¦ðAbanm um\n −1ub\a ¦sba Łu¼a\b dx0 9\ 1t1
"44#
g 0 9
"42#
0 1 0 "1# "1# "0# "0# Abanm um\n −1ub\a ¦ Abanmsr ur\s um\n u¼a\b dx0 9\ 1
9
−h
0 "0# "0# "Abanm um\n −1ub\a #u¼a\b dx0 9\
9
−h
h
g
1p
1
where 1 "1# "0# 2 "0# "0# "0# "0# um\n uc\d ¦05 Abadcnmsr ur\s um\n uc\d ¦p"1# ub\a \ sba Abadcnm
"45#
and use has been made of the identities "0# "0# "0# "0# "0# "0# "1# "1# "0# u¼a\b ub\c uc\a 9\ u¼a\b ub\c uc\d ud\a u¼a\b "ub\c uc\a ¦ub\c uc\a #\
which can be veri_ed by carrying out the summations followed by the use of "05# and "08#[ As a consequence of up"0# and up"1# being solutions of "05#Ð"07# and "08#Ð"10#\ respectively\ the _rst two equations "42# and "43# should be automatically satis_ed[ To show this\ we _rst note that for any su.ciently smooth vector function vm\
g
0 "Aqpsr vr\s −1vq\p #u¼p\q dx0 dx1
g
0 "Aqpsr u¼p\q −1u¼s\r #vr ns ds
1D
D
g
−
0 Asrqp u¼p\qs vr dx0 dx1 9\
"46#
D
where the last equation follows from the fact that u¼i is a linear solution with wav! enumber −k so that it satis_es "05#Ð"07# with ui"0# replaced by u¼i and p"0# 9[ Equation "42# then follows immediately with vm replaced by ui"0# \ whilst "43# reduces to
g
1p
h
−h
dx1
g 0 9
1
0 1 "0# "0# Aqpnmsr ur\s um\n u¼p\q dx0 9[ 1
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This reduced equation can be established by contracting "10# with u¼p\ integrating both sides along 1D\ applying the divergence theorem and _nally making use of "08# and "46#[ We now return to the third eqn "44#[ With the use of "46#\ the two terms in "44# which involve ui"2# vanish[ Substituting "21#\ "22#\ "33#Ð"37# into the resulting simpli_ed form of "44# and evaluating the integrals with respect to x0\ we obtain a
d1 Ak bp¹0 Ak ¦ s s aK"p\ q\ k#Ak−p Aq Ap−q \ 1 dt p − q −
"47#
where
g
h
Wm "x1 \ −k#Wm "x1 \ k# dx1 \ b
a −
−h
g
h
Gmn "x1 \ −k#Gnm "x1 \ k# dx1 \
−h
aK"p\ q\ k# K0 "p\ q\ k#¦K1 "p\ q\ k#¦K2 "p\ q\ k#\
g
0 2 K0 "p\ q\ k# Abadcnmsr 5 1 K1 "p\ q\ k# Abadcnm
h
Gab "x1 \ −k#Grs "x1 \ k−p#Gmn "x1 \ p−q#Gcd "x1 \ q# dx1 \
−h
g g
h
Gab "x1 \ −k#Gcd "x1 \ k−p#G0m "x1 \ q#G0n "x1 \ p−q# dx1 \
−h h
K2 "p\ q\ k# "4¦n0 #
Gab "x1 \ −k#Gba "x1 \ k−p#G1m "x1 \ q#G1m "x1 \ p−q# dx1 [
−h
In obtaining the expression for K1 "p\ q\ k# we have made use of the identity 1 "0# "0# um\n u0\cd u¼a\b 9\ Abadcnm
which can be established by expanding the left hand side out with the aid of Mathem! atica and making use of properties of um"0# and u¼m[ On substituting the expressions for Wi\ de_ned by "23#\ "38# for Gmn and "35c# for Umn into the above expressions\ we obtain a −k sinh"1kh#\ b:a −1k1 \ K j "p\ q\ k# k1 "k−p# 1 "p−q# 1 q1 K
j "p\ q\ k#\ j 0\ 1\ 2\ where K
0 "p\ q\ k# "0:2#A2jivunmlt v"i\ j\ sa #v"t\ l\ sb #v"u\ v\ sc #v"m\ n\ sd # = ja "−k#jb "k−p#jc "q#jd "p−q#V"a\ b\ c\ d\ p\ q\ k#\ 1 v"t\ l\ sa #v"u\ v\ sb #v"0\ m\ sc #v"0\ n\ sd # K
1 "p\ q\ k# 1Altvunm
= ja "−k#jb "k−p#jc "q#jd "p−q#V"a\ b\ c\ d\ p\ q\ k#\ K
2 "p\ q\ k# 1"4¦n0 #v"t\ l\ sa #v"l\ t\ sb #v"1\ v\ sc #v"1\ v\ sd # = ja "−k#jb "k−p#jc "q#jd "p−q#V"a\ b\ c\ d\ p\ q\ k#\
Nonlinear stability of plate under tension
1160
with V given by V"a\ b\ c\ d\ p\ q\ k#
sinh ð"−ksa ¦"k−p#sb ¦qsc ¦"p−q#sd #hŁ −ksa ¦"k−p#sb ¦qsc ¦"p−q#sd
when the denominator in this expression is non!zero and equal to its limiting value h otherwise[ The numerical evaluation of the kernel is computationally intensive because of the many summations in K
1[ After some numerical experimentation\ we _nd
0 and K that in integrating the system "47# the kernel K"p\ q\ k# can be replaced by K Þ "p\ q\ k# de_ned by aK Þ "p\ q\ k#
8¦3n0 ¦n1 K2 "p\ q\ k#\ 4¦n0
"48#
2 where n1\ which appears in the expression for Abadcnmsr \ is a third!order material constant[ The two constants n0 and n1 appear in the constitutive relation in the form T"1# 1E¦n0E1¦n1E2¦higher order terms\ where T"1# is the constitutive part of the second PiolaÐKirchho} stress tensor and E "FTF−I#:1\ see Fu "0883#[ The much simpler expression "48# was discovered in two steps[ First\ it was dis! covered that for the NeoÐHookean material aK"p\ q\ k# can be replaced by K2 "p\ q\ k# in integrating "47#[ Next\ when we assumed that only A0 and A1 were non!zero and wrote down the two di}erential equations for A0 and A1 with the aid of "47# for a general material\ we found that all the coe.cients in the two equations were pro! portional to the special combination 8¦3n0¦n1\ which is equal to 4¦n0 for the NeoÐ Hookean material[ This led us to trying replacing 4¦n0 by 8¦3n0¦n1 in the expression for K2 "p\ q\ k# and hence to the discovery of "48#[ We remark that K Þ "p\ q\ k# and K"p\ q\ k# are not identical[ What we can say is that if "47# is replaced by a _nite system and similar terms "such as A01 A1 # collected\ the two kernels give the same coe.cients in the di}erential equations[ On carrying out the summations in the expression for K2 "p\ q\ k# we obtain from "48#
aK
"p\ q\ k#\ Þ "p\ q\ k# 05"8¦3n0 ¦n1 #k1 "k−p# 1 "p−q# 1 q1 K
"59#
where
"p\ q\ k# K
sinh"1h"q−k## ðj0 "−k#j0 "p−k#j0 "q#j0 "q−p# 1"q−k#
¦j0 "k#j0 "k−p#j0 "−q#j0 "p−q#Ł¦
sinh"1h"p−q−k## 1"p−q−k#
= ðj0 "−k#j0 "p−k#j0 "−q#j0 "p−q#¦j0 "k#j0 "k−p#j0 "q#j0 "q−p#Ł[ In the above expression the factor sinh"1h"q−k##:"1"q−k## should be replaced by its limiting value h when q k\ and similar modi_cation applies to the other factor[
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In the special case when n0 9\ we have j0 0 0 and so K Þ "p\ q\ k# −
1"8¦3n0 ¦n1 # k"k−p# 1 "p−q# 1 q1 sinh"1kh# =
6
7
sinh"1h"q−k## sinh"1h"p−q−k## \ ¦ 1"q−k# 1"p−q−k#
"50#
which should be useful when one wishes to investigate general properties of the in_nite system "47#[ Since the kernel K Þ "p\ q\ k# given by "59# is clearly pure real\ solutions in which all Ak |s are real exist[ Our attention will be focused on such solutions in the following calculations[ In this case "47# may be written as d 1 Ak −1k1 p¹0 Ak ¦ s Aq1 Ak ðK Þ "9\ q\ k#¦K Þ "9\ −q\ k#Ł 1 dt q 0
¦ s s Aq Ak−p ðAp−q K Þ "p\ q\ k#¦Ap¦q K Þ "p\ −q\ k#Ł p 0 q 0
¦Aq Ak¦p ðAp¦q K Þ "−p\ q\ k#¦A−p¦q K Þ "−p\ −q\ k#Ł[
"51#
We note that d 1A9:dt1 9 so that a mean term will not be induced by nonlinear interactions if it does not exist initially[ In our numerical calculation we set A9 0 9[
5[ STATIC SOLUTIONS We refer to a post!buckling solution as a non!sinusoidal or multiple!mode solution if it contains more than one mode[ Static multiple!mode solutions correspond to solutions of the in_nite system of algebraic equations obtained by setting d:dt 9 in "51#[ To solve this system\ we assume that Ak 9 for k × N\ thus reducing the in_nite system to a _nite system for N unknowns A0\ A1\ [ [ [ \ AN\ where N is a suitably large integer[ We may start with N 1 or 2 and increase N in unit steps[ When we take N M and try to determine A0\ A1\ [ [ [ \ AM\ we use the values of A0\ A1\ [ [ [ \ AM−0 determined at the previous step and AM 9 as our initial guess and use Nag Library subroutine C94NBF to _nd the correct solution[ We take a solution as a valid solution if each Ak "k 0\ 1\ [ [ [# converges as N : [ We start our solution with N 1[ If we assume that Ak 9 for k × 1\ "51# yields A0 ð1p¼0 ¦c"0\ 0#A01 ¦1c"0\ 1#A11 Ł 9\ A1 ð1p¼0 ¦1c"1\ 0#A01 ¦c"1\ 1#A11 Ł 9\
"52#
p¼0 p¹0 :"8¦3n0 ¦n1 #\
"53#
where
Nonlinear stability of plate under tension
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and c"i\ j# is de_ned by sinh"1kh#c"k\ k?# 1kk?3
60
1h¦
1
sinh ð1"k−k?#hŁ 1 "n0 −n11 # 1 k−k? ¦
0¦3n01 n11 sinh ð1"k¦k?#hŁ k¦k?
7
"54#
with sinh ð1"k−k?#hŁ:"k−k?# set to its limiting value 1h when k k?[ We note that material properties "i[e[ n0 and n1# appear in our analysis through the special combination p¼0 de_ned by "53#[ For the general class of strain!energy functions N
W"l0 \ l1 \ l2 # s mi "l0ai ¦l1ai ¦l2ai −2#:ai i 0
proposed by Ogden "0861#\ where l0\ l1\ l2 are principal stretches\ and N\ m0\ [ [ [ \ mN\ a0\ [ [ [ \ aN are parameters to be chosen\ it can be shown that N
n0 −3 s mm ¦ m 0 N
n1 7 s mm ¦ m 0
0 N s m "a −1#"am −3#\ 1 m 0 m m
0 N s m "a −1#"am −3#"am −5#[ 5 m 0 m m
"55#
For the NeoÐHookean material "N 0\ a0 1#\ we have n0 −3\ n1 7 and so 8¦3n0¦n1 0[ For a three!term material a0 0[2\ a1 4[9\ a2 −1[9\ m0 0[380\ m1 9[992\ m0 −9[9126\ which was found by Ogden "0861# to give good agreement with a variety of exper! imental data\ we have n0 −3[64\ n1 09[20 and 8¦3n0¦n1 9[2924[ Finally\ for the Varga material "N 0\ a0 0#\ we have n0 −4\ n1 00 and 8¦3n0¦n1 9[ Thus\ for the Varga material\ the present nonlinear analysis becomes invalid and nonlinear e}ects are pronounced at a higher order[ The equation system "52# has three sets of non!trivial solutions given by
III ] A01 −
I ] A01 −1p¼0 :c"0\ 0#\ A1 9\
"56#
II ] A0 9\ A11 −1p¼0 :c"1\ 1#\
"57#
1p¼0 1p¼0 ðc"1\ 1#−1c"0\ 1#Ł\ A11 − ðc"0\ 0#−1c"1\ 0#Ł\ D D
"58#
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where D c"0\ 0#c"1\ 1#−3c"0\ 1#c"1\ 0#[ It can easily be seen from "54# that c"0\ 0# and c"1\ 1# are both positive[ In addition\ it can be shown with the aid of "54# that 1c"0\ 1# × c"1\ 1#\ 1c"1\ 0# × c"0\ 0# and hence D ³ 9[ Thus\ all the above solutions are only possible when p¼0 ³ 9[ For the NeoÐHookean material and materials with the three!term energy function discussed above\ this requires p¹0 ³ 9 and so the bifurcation is super!critical[ Since p¼0 can be scaled out of "56#Ð"58# and if "A0\ A1# is a solution so is "−A0\ −A1#\ we shall assume that p¼0 −0 and consider only those solutions with A0 − 9[ As an illustration\ we take n0 0\ n1 9\ h 9[4[ Equations "56#Ð"58# then give three sets of solutions for "A0\ A1# ] I ]"9[4440\ 9#\ II ]"9\ 29[0022#\ III ]"9[2298\ 29[95442#[ When we use the above solutions as starting solutions and increase N in unit steps\ solution I leads to a multiple!mode solution with A0 × 9\ solutions II lead to two multiple!mode solutions with Ak 9 except A1k "k 0\ 1\ [ [ [#\ whilst starting with solutions III the Nag Library subroutine C94NBF fails to converge[ Some numerical experiments with random choices for the starting values of A0 and A1 show that multiple!mode solutions with Ak 9 except Amk "k 0\ 1\ [ [ [# exist for any positive integer m[ However\ we now show that such multiple!mode solutions with A0 9 are included in the set of solutions obtained by imposing A0 9 and varying h in "9\ #[ Suppose that we have a multiple!mode solution with Ak 9 except Amk "k 0\ 1\ [ [ [# for some positive integer m[ The corresponding displacement com! ponent u0"0# is then given by ðsee "21#Ł
u0"0# s mkAmk "t#s j j j "mk# emksj x1 eimkx0 [
"69#
k −
We now re!scale ui"0# and xi according to u½i"0# mui"0# \ x½i mxi[ Since we have pre! viously scaled our dimensional displacement components and co!ordinates using the inverse of the fundamental wavenumber which is arbitrary\ this additional scaling makes no di}erence[ In terms of u½0"0# and x½i\ "69# becomes
u½0"0# s km1 Amk "t#s j j j "k# eksj x½1 eikx½0 [ k −
This shows that the multiple!mode solution under consideration is in fact a solution for a plate with surfaces de_ned by x½1 2mh and the solution has the k!th Fourier amplitude equal to m1Amk[ It can easily be veri_ed that this is indeed a correct solution to "51#[ The above argument shows that we may focus our attention on solutions with A0 × 9[ Thus we shall only consider the solution generated from starting solution I[ For this solution\ Table 0 shows the rapid convergence of A1k−0 "k 0\ 1\ [ [ [# as N is increased[ The A1k "k 0\ 1\ [ [ [# are found to be zero[ The corresponding pro_les of
Nonlinear stability of plate under tension
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Table 0[ n0 0\ h 9[4 –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– — N A0 A2 A4 A6 A8 ÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐ * 1 9[4440E¦99 9[9999E¦99 9[9999E¦99 9[9999E¦99 9[9999E¦99 2 9[4727E¦99 9[0006E−90 9[9999E¦99 9[9999E¦99 9[9999E¦99 4 9[4772E¦99 9[0012E−90 9[6912E−92 9[9999E¦99 9[9999E¦99 6 9[4778E¦99 9[0017E−90 9[5795E−92 9[4297E−93 9[9999E¦99 8 9[4778E¦99 9[0018E−90 9[5719E−92 9[4942E−93 9[3501E−94 00 9[4789E¦99 9[0018E−90 9[5712E−92 9[4948E−93 9[3236E−94 08 9[4789E¦99 9[0018E−90 9[5713E−92 9[4950E−93 9[3241E−94 18 9[4789E¦99 9[0018E−90 9[5713E−92 9[4950E−93 9[3241E−94 28 9[4789E¦99 9[0018E−90 9[5713E−92 9[4950E−93 9[3241E−94 –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– —
Fig[ 0[ Pro_les of u1 "solid line#\ u0\0 "dotted line# and u0\1 associated with the post!buckling state when n0 0\ h 9[4[
u1\ u0\0 " −u1\1# and u0\1 " u1\0# evaluated at x1 h are shown in Fig[ 0 and are calculated according to
u1"0# 1 s kAk j j "k# eksj x1 sin"kx0 #\ k 0
"0# −1 s k1 Ak s j j j "k# eksj x1 sin"kx0 #\ u0\0 k 0
"0# u0\1 1 s k1 Ak j j "k# eksj x1 cos"kx0 #[ k 0
"60#
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Table 1[ n0 0\ h 9[0 –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– — N A0 A2 A4 A6 A8 ÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐ * 1 9[4656E¦99 9[9999E¦99 9[9999E¦99 9[9999E¦99 9[9999E¦99 2 9[5047E¦99 9[0776E−90 9[9999E¦99 9[9999E¦99 9[9999E¦99 4 9[5145E¦99 9[1090E−90 9[2796E−91 9[9999E¦99 9[9999E¦99 6 9[5183E¦99 9[1075E−90 9[3112E−91 9[0296E−91 9[9999E¦99 8 9[5202E¦99 9[1117E−90 9[3312E−91 9[0329E−91 9[4628E−92 00 9[5213E¦99 9[1140E−90 9[3420E−91 9[0383E−91 9[5060E−92 02 9[5220E¦99 9[1154E−90 9[3482E−91 9[0429E−91 9[5391E−92 04 9[5225E¦99 9[1162E−90 9[3520E−91 9[0440E−91 9[5418E−92 06 9[5228E¦99 9[1179E−90 9[3545E−91 9[0452E−91 9[5591E−92 08 9[5230E¦99 9[1173E−90 9[3563E−91 9[0461E−91 9[5536E−92 10 9[5232E¦99 9[1176E−90 9[3575E−91 9[0467E−91 9[5566E−92 12 9[5233E¦99 9[1178E−90 9[3583E−91 9[0471E−91 9[5587E−92 14 9[5234E¦99 9[1189E−90 9[3699E−91 9[0474E−91 9[5602E−92 16 9[5235E¦99 9[1180E−90 9[3693E−91 9[0476E−91 9[5613E−92 18 9[5235E¦99 9[1181E−90 9[3696E−91 9[0477E−91 9[5621E−92 20 9[5235E¦99 9[1181E−90 9[3697E−91 9[0478E−91 9[5627E−92 22 9[5235E¦99 9[1182E−90 9[3609E−91 9[0489E−91 9[5631E−92 24 9[5235E¦99 9[1182E−90 9[3609E−91 9[0489E−91 9[5633E−92 26 9[5236E¦99 9[1182E−90 9[3600E−91 9[0489E−91 9[5635E−92 28 9[5236E¦99 9[1182E−90 9[3600E−91 9[0480E−91 9[5636E−92 60 9[5236E¦99 9[1182E−90 9[3601E−91 9[0480E−91 9[5638E−92 –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– —
Table 2[ n0 0\ h 0 –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– — N A0 A2 A4 A6 A8 ÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐ * 1 9[3423E¦99 9[9999E¦99 9[9999E¦99 9[9999E¦99 9[9999E¦99 2 9[3593E¦99 9[0372E−91 9[9999E¦99 9[9999E¦99 9[9999E¦99 4 9[3594E¦99 9[0379E−91 9[0947E−93 9[9999E¦99 9[9999E¦99 6 9[3594E¦99 9[0379E−91 9[0940E−93 9[0967E−95 9[9999E¦99 8 9[3594E¦99 9[0379E−91 9[0940E−93 9[0969E−95 9[0126E−97 28 9[3594E¦99 9[0379E−91 9[0940E−93 9[0969E−95 9[0126E−97 –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– —
For comparison\ Tables 1 and 2 and Figs 1 and 2 show the results for h 9[0 and 0\ respectively\ with n0\ n1 having the same values as in Table 0 and Fig[ 0[ An inspection of Tables 0Ð2 shows that as h is increased\ energy becomes increasingly more and more con_ned to the _rst few modes and convergence becomes more and more rapid[ As a result\ the pro_les of u1\ u0\0\ u0\1 becomes closer and closer to being sinusoidal[ In the limit h : \ we expect to _nd only sinusoidal solutions[ This is precisely the result found by Devenish and Fu "0886# for a half!space[ Conversely\ as h is decreased\
Nonlinear stability of plate under tension
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Fig[ 1[ Pro_les of u1 "solid line#\ u0\0 "dotted line# and u0\1 associated with the post!buckling state when n0 0\ h 9[0[
Fig[ 2[ Pro_les of u1 "solid line#\ u0\0 "dotted line# and u0\1 associated with the post!buckling state when n0 0\ h 0[
energy spreads into higher modes and the three pro_les are further away from being sinusoidal[ We note that these pro_les are also very sensitive to the values of n0 and n1\ i[e[ to the shape function H"x1\ k#[ Figure 3 shows these pro_les for n0 9[4\ n1 z2:1\ h 9[4\ which may be compared with those shown in Fig[ 0[ We see that
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Fig[ 3[ Pro_les of u1 "solid line#\ u0\0 "dotted line# and u0\1 associated with the post!buckling state when n0 9[4\ h 9[4[
at the same value of h\ the pro_les in Fig[ 3 are closer to being sinusoidal than those in Fig[ 0[ Numerical calculations for a large number of "n0\ n1# pairs show that the smaller the n01 −n11 is\ the closer to being sinusoidal the pro_les of u1\ u0\0\ u0\1 become[ In the critical case n0 n1\ only sinusoidal solutions "i[e[ single!mode solutions# are found[ The non!trivial solutions found above all have A1k 9 "k 0\ 1\ [ [ [#[ Since in particular A1 9\ one may think that by starting with N 2\ new solutions might be found[ Truncating "51# at N 2\ we obtain A0 ð1p¼0 ¦c"0\ 0#A01 ¦1c"0\ 2#A21 Ł¦b0 A01 A2 9\ A2 ð1p¼0 ¦1c"2\ 0#A01 ¦c"2\ 2#A21 Ł¦b1 A02 9\
"61#
where b0 −43"n03 −n13 # cosh"1h#\ b1 −"0:2#"n03 −n13 # sinh"3h#:sinh"5h#[ If n0 n1\ then b0 b1 9 and the solutions of "61# are analogous to those of "52#[ If n0 n1\ solutions of "61# can be obtained easily with the aid of Mathematica[ It is again found that "61# has non!trivial solutions only if p¼0 ³ 9[ When n0 0\ n1 9\ h 9[4\ "61# have three pairs of solutions for "A0\ A2# with A0 × 9 given by I ]"9[4727\ 9[9006#\ II ]"9[2145\ −9[90669#\ III ]"9[1297\ 9[91773#[ When we use the above solutions as starting solutions and increase N in unit steps\ we _nd that starting solution I leads to the solution shown in Table 0 and Fig[ 0\ C94NBF fails to converge when starting solution II is used\ whereas starting solution
Nonlinear stability of plate under tension
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Fig[ 4[ Evolution of A0 when ] "a# Ak "9# 0[90Ak"s# "9#\ A þ k "9# 9 ^ "b# Ak "9# 0[0Ak"s# "9#\ A þ k "9# 9 ^ "c# þ k "9# 9 ^ "d# Ak "9# 0[4Ak"s# "9#\ A þ k "9# 0:k1\ where k 0\ 1\ [ [ [ \ 28 and the super! Ak "9# 0[4Ak"s# "9#\ A script "s# signi_es the static solution[
III leads to a solution with Ak 9 except A2m "m 0\ 2\ 4\ [ [ [#[ Hence starting with N 2 does not yield any new solutions[ Suppose that we start with an even larger truncation number\ say N 09\ and obtain a set of starting solutions by solving the resulting system of ten simultaneous algebraic equations[ It would seem possible that some of the starting solutions might lead to new solutions[ However\ _nding all the possible solutions of a system of more than three simultaneous nonlinear algebraic equations is a non!trivial matter\ and even if we can _nd all the possible starting solutions\ there is no guarantee that C94NBF will converge to new solutions[ Because of these di.culties\ we have not attempted to start with truncation numbers larger than three[ Hence we cannot claim that the solutions found here are the only solutions possible[
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6[ STABILITY OF POST!BUCKLING STATES We _rst consider the special case n0 n1[ It is found in the previous section that in this case multiple!mode static solutions reduce to a single!mode solution[ If initially there is only a single mode\ integrating the system of eqns "51# shows that other modes are not excited[ This implies that in this special case a single mode can evolve by itself[ If we assume that only Am is non!zero\ "51# gives d1 Am −1m1 p¹0 Am −gm5 Am2 \ dt1
"62#
where g ðK Þ "9\ m\ m#¦K Þ "9\ −m\ m#¦K Þ "1m\ m\ m#Ł:"m6 sinh"1mh## 3"8¦3n0 ¦n1 # cosh"1mh#[ Since eqn "62# for m 0 can be obtained from that for m 0 by a suitable rescaling\ we may focus our attention on m 0[ In this case the non!trivial post!buckling state with A0 × 9 is given by A0 A0"s# z−1p¹0 :g[ In the neighbourhood of this post! buckling state\ we write A0 A0"s# ¦a"t# where a is a small amplitude perturbation[ Substituting this expression into "62# and linearizing\ we obtain d 1a:d 1t 3p¹0a[ With! out loss of generality we set p¹0 −0[ Then a small perturbation in the neighbourhood of the post!buckling state evolves as a simple harmonic oscillation with period p[ It can easily be deduced from a phase portrait analysis that perturbations with larger amplitude in general are also oscillatory[ The post!buckling state is therefore stable[ In the general case when n0 n1\ only multiple!mode post!buckling solutions are possible[ The stability of such post!buckling solutions can be determined by inte! grating the system "51#\ after appropriate truncation\ with the initial condition being the static solution plus a perturbation[ In Fig[ 4"aÐd# we have shown the evolution of A0 for increasingly large perturbations[ The in_nite system is truncated at N 28 and the calculation is for the NeoÐHookean material with n0 0\ h 9[0[ In all cases it 28 is found that the graphs for zSi 0 Ai1 and =A0 = are graphically indistinguishable\ which implies that higher harmonics have relatively small amplitudes compared with the fundamental mode at all times[ The sensitivity to the truncation number N is found to increase with the amplitude of perturbations[ For the small perturbation considered in Fig[ 4"a#\ the results corresponding to N 08\ 28 are almost identical[ It is also found that the result shown in Fig[ 4"a# is graphically indistinguishable from the linearized single mode result A0 "t# A0"s# ¦9[90A0"s# "9# cos"1t# where A0"s# is the static solution given in the row N 28 in Table 1[ This shows that for small pertur! bations both nonlinearity and higher modes have a negligible e}ect[ Figure 4"bÐd# shows that as the amplitude of perturbation increases\ the e}ects of nonlinear and higher modes come into play and the evolution of A0 gradually deviates from the linear single mode behaviour shown in Fig[ 4"a#[ However\ for all the cases we have considered\ perturbations never blow up and are always oscillatory[ We thus conclude that all the post!buckling solutions found in the present paper are stable[
Nonlinear stability of plate under tension
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7[ SUMMARY In this paper we have carried out a weakly nonlinear stability analysis for an incompressible elastic plate subjected to an all!round tension[ Our aim was to _nd post!buckling states and to determine whether they are stable or not[ This stability problem has the following unusual features[ The neutral value of tension for instability is a constant "i[e[ two after scaled by the shear modulus#\ independent of wavenumber[ Thus\ all the modes participate in the post!buckling process[ However\ unlike the usual nonlinear analysis involving non!dispersive wave modes\ the evolution equations are derived not at second!order but at third!order of an asymptotic expansion[ Also\ the neutral value of tension for instability is independent of whether the modes are extensional or ~exural[ This gives rise to an arbitrary parameter "n0# in the shape functions and to the possibility of an in_nite number of post!buckling solutions[ With the aid of the virtual work method\ we derived an in_nite system of evolution equations for the various Fourier components[ We also discovered a simple expression for the kernel in the nonlinear terms\ which reduces considerably the computing time required[ This simple expression shows that material nonlinearity manifests itself through the special combination 8¦3n0¦n1\ whose sign determines if the post!buck! ling is supercritical "if it is positive# or subcritical "if it is negative#[ If it is zero\ the nonlinear analysis presented in this paper becomes invalid and the evolution equations can only be derived by carrying out the expansions to higher order[ For the NeoÐ Hookean material and the three!term material\ this expression was found to be positive whereas for the Varga material it is zero[ For a _xed plate thickness\ we _nd an in_nite number of post!buckling solutions\ one for each n0\ but only the one with n0 n1 0z1 is sinusoidal[ Small amplitude perturbations superimposed on each of these post!buckling states are found to evolve as oscillations and thus\ all the post! buckling states found are stable[
REFERENCES Devenish\ B[ J[ and Fu\ Y[ B[ "0886# The nonlinear evolution of near!neutral and near!body wave modes in a pre!stressed incompressible elastic half!space[ Part II ] lower branch near! neutral modes[ International Journal of En`ineerin` Science 24\ 0330Ð0344[ Fu\ Y[ B[ "0883# On the propagation of nonlinear travelling waves in an imcompressible elastic plate[ Wave Motion 08\ 160Ð181[ Fu\ Y[ B[ "0884# Resonant!triad instability of a pre!stressed elastic plate[ Journal of Elasticity 30\ 02Ð26[ Fu\ Y[ B[ and Devenish\ B[ J[ "0885# E}ects of pre!stresses on the propagation of nonlinear surface waves in an incompressible elastic half!space[ Q[ J[ Mech[ Appl[ Math[ 38\ 54Ð79[ Fu\ Y[ B[ and Ogden\ R[ W[ "0887# Nonlinear stability analysis of pre!stressed elastic bodies[ Nonlinear Wave Phenomena[ Birkhauser\ to appear[ Fu\ Y[ B[ and Rogerson\ G[ A[ "0883# A nonlinear analysis of instability of a pre!stressed incompressible elastic plate[ Proceedin`s of the Royal Society of London A335\ 122Ð143[ Ogden\ R[ W[ "0861# Large deformation isotropic elasticity ] on the correlation of theory and experiment for incompressible rubberlike solids[ Proceedin`s of the Royal Society of London A215\ 454Ð473[
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Ogden\ R[ W[ "0884# Nonlinear e}ects associated with waves in pre!stressed elastic solids near bifurcation points[ Proceedin`s of the IUTAM Symposium on Nonlinear Waves in Solids\ ed[ J[ E[ Wegner and F[ R[ Norwood\ pp[ 098Ð002[ ASME\ New Jersey[ Ogden\ R[ W[ and Roxburgh\ D[ G[ "0882# The e}ect of pre!stress on the vibration and stability of elastic plates[ International Journal of En`ineerin` Science 20\ 0500Ð0528[ Wolfram\ S[ "0880# Mathematica\ 1nd edn[ Addison!Wesley\ California[
Pergamon
J[ Mech[ Phys[ Solids\ Vol[ 35\ No[ 00\ pp[ 1150Ð1171\ 0887 Þ 0887 Elsevier Science Ltd[ All rights reserved Printed in Great Britain 9911Ð4985:87 ,*see front matter
PII ] S9911Ð4985"87#99922Ð6
A NONLINEAR STABILITY ANALYSIS OF AN INCOMPRESSIBLE ELASTIC PLATE SUBJECTED TO AN ALL!ROUND TENSION YIBIN FU$ Department of Mathematics\ University of Keele\ Sta}ordshire ST4 4BG\ U[K[ "Received 1 September 0886#
ABSTRACT When an incompressible elastic plate is subjected to an all!round tension\ linear stability analysis predicts that it will become neutrally stable with respect to both extensional and ~exural modes and to all wave! numbers when the tension is twice the shear modulus[ In this paper a weakly nonlinear analysis is conducted to determine the post!buckling states when the tension deviates from its neutral value by a small amount[ It is found that such a prestressed plate with any given thickness can bifurcate into an in_nite number of post!buckling states\ and that only one such state is sinusoidal\ the rest being non!sinusoidal[ All the post! buckling states are found to be stable with respect to small perturbations[ Þ 0887 Elsevier Science Ltd[ All rights reserved[ Keywords ] A[ buckling\ B[ elastic material\ plates\ _nite strain\ C[ stability and bifurcation[
0[ INTRODUCTION When an incompressible isotropic elastic plate is subjected to an all!round pressure\ a linear stability analysis using the theory of _nite elasticity shows that it is only stable if −1 ³ p ³ f "kh# where p is the pressure scaled by the shear modulus\ k is the wavenumber\ h is the plate thickness and f "kh# takes di}erent forms for extensional and ~exural modes "see e[g[ Ogden and Roxburgh\ 0882#[ Nonlinear stability analyses have been conducted by Fu and Rogerson "0883# and Fu "0884# for values of p near the upper branch p f "kh# of the neutral stability curve ^ the _rst of these two papers is concerned with a single near!neutral mode whereas the second is concerned with a triad of modes[ In this paper we investigate the lower branch stability properties "i[e[ for values of p close to −1#\ thus completing the weakly nonlinear stability analysis for the type of plate mentioned above[ This lower branch is non!dispersive in the sense that the neutral value of p is independent of wavenumber "thus\ all modes become neutral at the same time#[ Also the lower branch is the same for both extensional and ~exural modes[ Because of the co!existence and strong interaction of extensional and ~exural near!neutral modes at all wavenumbers\ we expect to see non!sinusoidal as well as sinusoidal post!buckling states[ This is precisely what we shall _nd in this paper[ Although a negative pressure is really a tension\ we shall still $ Tel[ ] 9933 "9# 0671 472 549[ Fax ] 9933 "9# 0671 473 157[ E!mail ] y[fuÝkeele[ac[uk[ 1150
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use the term pressure in the rest of this paper in order to be consistent with the analysis of Fu and Rogerson "0883#[ It is well!known that for problems involving non!dispersive wave modes "such as the evolution of surface waves#\ evolution equations are usually derived by imposing a solvability condition at second!order of an in_nite hierarchy of boundary value problems resulting from an asymptotic expansion[ However\ although the lower branch is non!dispersive as explained above\ we _nd that the second!order problem can be solved without the need to impose any solvability conditions and that the evolution equations are obtained by imposing a solvability condition at third!order "we note however that with the use of the virtual work method we do not need to do so explicitly#[ In the following section\ we summarize the original governing equations from the theory of _nite elasticity and derive the resulting _rst and second order boundary value problems after an asymptotic expansion[ These two boundary value problems are solved in Sections 2 and 3\ respectively[ The evolution equations are then derived in Section 4 with the aid of the virtual work method\ and their static solutions\ which describe the post!buckling states\ are sought in Section 5[ In Section 6 we show that all the post!buckling solutions found in Section 5 are stable and in the _nal section we summarize the main results obtained in the present paper[
1[ GOVERNING EQUATIONS We consider a homogeneous elastic plate composed of a non!heat!conducting elastic material which possesses an initial unstressed state B9[ A homogeneous stress _eld is then imposed upon B9 to produce a _nitely stressed equilibrium con_guration denoted by Be[ It is the stability of Be that we wish to study[ To this end\ we superimpose a small amplitude time!dependent perturbation on Be[ The resulting con_guration\ termed the current con_guration\ is denoted by Bt[ The position vectors of a representative particle relative to a common coordinate system are denoted by XA\ xi "XA# and x½i "XA\ t# in B9\ Be and Bt\ respectively[ The origin is chosen to lie in the mid!plane of Be and the plate surfaces correspond to x1 2h[ We write x½i "XA \ t# xi "XA #¦ui "x j \ t#\
"0#
where ui "xj\ t# is a small time!dependent displacement associated with the deformation Be : Bt[ The plate will be assumed to be in a state of plane strain so that all subscripts range from 0 to 1[ To simplify analysis we assume that the plate is in_nite along the x0!direction[ The deformation gradients arising from the deformations B9 : Bt and B9 : Be are denoted by F and Þ F\ respectively\ and de_ned by FiA
1x½i 1xi \ Þ FiA [ 1XA 1XA
"1#
It is clear from "0# that ÞjA \ FiA "dij ¦ui\ j #F
"2#
Nonlinear stability of plate under tension
1152
where here and henceforth a comma indicates di}erentiation with respect to the implied spatial coordinate[ Furthermore\ the convention whereby upper case indices refer to coordinates in B9 and lower case indices to coordinates in Be will be observed[ "Since the coordinates x½j do not appear in the following analysis\ no ambiguity will arise in\ for example\ ui\j[# In the absence of body forces\ the equations of motion and the incompressibility constraint are given by piA\A rui \ det F 0\
"3#
where r is the "constant# density and p is the _rst PiolaÐKirchho} stress which\ in component form\ is given by piA
1W −0 −pF Ai \ 1FiA
"4#
where W is the strain!energy function "per unit volume# and p is a Lagrange multiplier associated with the incompressibility constraint and interpreted as a pressure[ In the subsequent stability studies\ it will be assumed that on the two plate surfaces the traction vector pN\ where N is the unit normal to the plate surfaces in B9\ is prescribed in B9\ and that this traction will be maintained at this value during the incremental deformation Be : Bt[ Such an assumption is usually referred to as a dead! load traction boundary condition and is represented by "piA −p¹ iA #NA 9\
"5#
where p¹ iA is the value of piA calculated from "4# with F replaced by Þ F and p by p¹[ It is convenient to introduce a tensor function with components xij through xij "piA −p¹ iA #F ÞjA
0
1
1W −0 −pF Ai −p¹ iA Þ F jA [ 1FiA
"6#
In terms of this new tensor\ "3a# and "5#\ respectively\ may be written simply as xij\j rui \
"7#
xij n j 9\
"8# −T
F where "ni# is the unit normal to the plate surfaces in Be "and is parallel to Þ We write
N#[
p p¹¦p \ where p¹ is the pressure in Be and p is the incremental pressure associated with the deformation Be : Bt[ It can be shown "see e[g[ Fu and Rogerson\ 0883 ^ Fu and Ogden\ 0887# that xij has the following expansion about "F Þ\ p¹# ] xij A0jilk uk\l ¦01 A1jilknm uk\l um\n ¦05 A2jilknmqp uk\l um\n up\q ¦p¹ "u j\i −u j\k uk\i ¦u j\k uk\l ul\i #−p "d ji −u j\i ¦u j\k uk\i #¦O"o3 #\
"09#
where o is a small parameter characterizing the amplitude of ui\j and p \ A0\ A1\ A2 are the _rst!\ second! and third!order tensors of instantaneous elastic moduli in Be[
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Expressions for these moduli can be found in Fu "0883# "with errata given in Fu\ 0884# for the case when the principal stretches are all unit and in Fu and Ogden "0887# otherwise[ A similar expansion of the constraint eqn "3b# leads to ui\i 01 um\n un\m −01 "ui\i # 1 −det"um\n #[
"00#
We shall assume from now on that the pre!stress is an all!round pressure[ Since the plate is incompressible\ no deformation is produced by the pressure[ Con_guration Be then coincides with B9 and A0jilk is given by A0jilk m"d jl dik ¦d jk dil # where m is the shear modulus[ Before proceeding further\ we _rst non!dimensionalize the governing equations and boundary conditions using L "to be de_ned# as the length scale " for ui\ xi#\ m as the stress scale "for p¹\ p \ A0jilk \ etc[# and Lzr:m as the time scale " for t#[ In order to avoid introducing additional notation\ we shall use the same symbols to denote the cor! responding non!dimensionalized quantities[ The non!dimensionalized forms of the governing equations and boundary conditions then remain unchanged except that r 0\ m 0[ Thus\ for instance\ we now have A0jilk d jl dik ¦d jk dil [
"01#
The length scale L is arbitrary at the moment[ A natural choice for L would be the half plate thickness h\ but to simplify the analysis\ it will later be chosen to be the inverse of the wave number of the fundamental instability mode[ It is known from Ogden and Roxburgh "0882# and Fu and Rogerson "0883# that the plate under consideration is stable with respect to travelling wave perturbations only if −1 ¾ p¹ ¾ f"kh# where k is the wavenumber and the function f takes di}erent forms for ~exural and extensional instability modes[ As mentioned in the introduction\ we shall be concerned with the post!buckling behaviour of the plate when p¹ deviates from its lower branch neutral value −1 by a small amount[ Thus we let p¹ −1¦oa p¹0 \
"02#
where p¹0 is an arbitrary O"0# constant and a is to be determined[ Since the evolution equations are derived from a solvability condition imposed at third order\ we require 11 ui ½ o2 1t1 and so the appropriate slow time variable describing the wave modulation is given by t ot\
"03# a
since ui ½ o and ui depends on t through t only[ The order o deviation of p¹ from −1 speci_ed by "02# will induce an O"oa:1# wave speed "or growth rate#[ In order to absorb the factor eikvt "in the usual travelling wave representation eik"x0 −vt# # into the wave amplitude which depends on t\ we choose v o which gives a 1[ We then look for the following form of asymptotic solution ] ui oui"0# "x0 \ x1 \ t#¦o1 ui"1# "x0 \ x1 \ t#¦o2 ui"2# "x0 \ x1 \ t#¦= = = \ p op"0# "x0 \ x1 \ t#¦o1 p"1# "x0 \ x1 \ t#¦o2 p"2# "x0 \ x1 \ t#¦= = = [
"04#
Nonlinear stability of plate under tension
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On substituting "02# and "04# into "00#\ "7# and "8# with xij given by "09# and then equating the coe.cients of o and o1\ we obtain the equations "0# "0# u0\0 ¦u1\1 9\
"05#
"0# −p\i"0# 9\ A0jilk uk\lj
"06#
0 "0# 1ilk k\l
A
"0# 1\i
"0#
u −1u −p d1i 9\
"07#
at order o\ and "1# "1# ¦u1\1 01 ui\"0#j u"0# u0\0 j\i \
"08#
"1# "0# "0# A0jilk uk\lj −p\i"1# −A1jilknm um\n uk\lj −p\j"0# u"0# j\i \ 0 "1# 1ilk k\l
A
"1# 1\i
0 1
"1#
1 "0# "0# 1ilknm k\l m\n
u −1u −p d1i − A
u u
"0# "0# 1\k k\i
"19# "0#
"0# 1\i
−1u u −p u \
"10#
at order o1[ Note that "07# and "10# are boundary conditions which must hold on the boundaries x1 2h[ These two sets of boundary value problems are solved in the next two sections[
2[ LEADING!ORDER SOLUTION The leading!order constraint relation "05# implies the existence of a {{stream|| function c"x0\ x1\ t# such that u0"0# c\1 \ u1"0# −c\0 [
"11#
With A0jilk given by "01#\ "06# and "07# reduce to 91 ui"0# −p\i"0# 9\
"12#
"0# "0# u0\1 −u1\0 9\ p"0# 9 on x1 2h[
"13#
Eliminating p"0# from "12# by cross!di}erentiation and making use of "11# we obtain 93 c 9[
"14#
The two boundary conditions "13# yield 91 c 9 and
"91 c# \1 9 on x1 2h[
"15#
The solution to the boundary value problem "14# and "15# is obviously 91 c 9[
"16#
It then follows from "12# with i 1 and "13b# that p"0# 0 9[
"17#
Let c H"x1\ k# eikx0 [ Equation "16# yields H"x1 \ k# n0 "k:=k=# sinh"kx1 #¦n1 cosh"kx1 #\
"18#
where n0 and n1 are arbitrary real constants and k:=k= has been inserted to make
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H"x1\ −k# H"x1\ k#[ In our later numerical calculations\ we normalize H such that n01 ¦n11 0[ That n0 and n1 are arbitrary re~ects the fact that ~exural and extensional modes "corresponding to n0 9 and n1 9\ respectively# are both neutrally stable at p¹ −1[ The general solution for c is then given by
k −
k −
c s Ak "t#H"x1 \ k# eikx0 s Ak "t#j j "k# eksj x1 eikx0 \
"29#
where j0 "k#
n1 ¦n0 k:=k= n1 −n0 k:=k= \ j1 "k# \ s0 0\ s1 −0\ 1 1
"20#
and summation on the repeated subscript j is observed[ The shape function H has been written in the alternative form shown in the last expression of "29# to facilitate later manipulations[ For c to be real\ we impose the condition A−k "t# A Þk "t#\ where an overbar signi_es complex conjugation[ We note that the representation "29# corresponds to the fundamental mode having unit wavenumber\ which is a consequence of choosing the lengthscale L to be the inverse of the dimensional wavenumber of the fundamental mode[ We also note that we could have represented c by a Fourier integral under appropriate assumptions[ In fact all the Fourier sums in this paper could be converted to Fourier integrals under the same assumptions[ It follows from "29# that
k −
k −
u0"0# s kAk "t#s j j j "k# eksj x1 eikx0 s Ak "t#W0 "x1 \ k# eikx0 \
k −
k −
u1"0# s "−ik#Ak "t#j j "k# eksj x1 eikx0 s Ak "t#W1 "x1 \ k# eikx0 \
"21#
"22#
where W0 "x1 \ k# ks j j j "k# eksj x1 \ W1 "x1 \ k# "−ik#j j "k# eksj x1 \
"23#
and throughout this paper we adopt a modi_ed summation convention whereby a su.x appearing in one term more than once is summed from 0 to 1[
3[ SECOND!ORDER SOLUTION The second!order solution is determined by "08# and "19# subject to the boundary condition "10# on x1 2h[ Following Ogden "0884#\ we rewrite the constraint relation "08# as "0# "0# "u0"1# −u0"0# u0\0 # \0 ¦"u1"1# −u0"0# u1\0 # \1 9[
It then follows that there exists a function f"x0\ x1\ t# such that "0# "0# u0"1# f\1 ¦u0"0# u0\0 \ u1"1# −f\0 ¦u0"0# u1\0 [
"24#
Nonlinear stability of plate under tension
With the use of the expression for A "19# reduces to
1 jilknm
1156
given in Fu "0883#\ it can be shown that
1 1 91 ui"1# −p\i"1# −"3¦n0 #"c\01 ¦c\00 # \i \
"25#
where n0 is a second!order material constant and use has been made of "16#[ Similarly\ the boundary condition "10# with i 0\ 1 reduces to "1# "1# u0\1 −u1\0 9\ on x1 2h\
"26#
1 1 ¦c\00 #\ on x1 2h[ p"1# "4¦n0 #"c\01
"27#
and
"1# "1# Eliminating p"1# from "25# gives 91 "u0\1 −u1\0 # 9[ Substituting "24# into this equation and "26#\ we obtain
93 f 9\
"28#
91 f 9 on x1 2h[
"39#
Another boundary condition for f can be obtained from "27#[ Di}erentiating "27# "1# \ we obtain with respect to x0 and using "25# with i 0 to eliminate p\0 "91 f# \1 9 on x1 2h[
"30#
We observe that as is usual with asymptotic expansions involving non!dispersive wave modes\ the linearized operators on the left hand sides of "28#Ð"30# are the same as those in "14# and "15#[ The unusual feature here is that the inhomogeneous terms on the right hands of "28#Ð"30# are identically zero[ This implies that no solvability conditions need to be imposed in order to solve the present second!order problem[ The solution to "28#Ð"30# is obviously given by 91f 9 which is the same as that satis_ed by the function c at leading!order[ Without loss of generality\ we may take f 9 since otherwise it can be absorbed into c[ Hence\ the second!order displacement _eld is given by "0# "0# u0"0# u0\i [ ui"1# u0"0# ui\0
"31#
"1# Equation "25# with i 1 gives an expression for p\1 [ Integrating this expression subject to the boundary condition "27# yields 1 1 "0# "0# ¦c\00 # "4¦n0 #u1\m u1\m [ p"1# "4¦n0 #"c\01
"32#
This completes our solution of the second!order problem[ For later use\ we de_ne two functions Gmn "x1\ k# and Umn "x1\ k# through
"0# s Ak "t#Gmn "x1 \ k# eikx0 \ um\n
"33#
k −
"0# u0\mn s Ak "t#Umn "x1 \ k# eikx0 [ k −
"34#
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It follows from "21# and "22# that G00 "x1 \ k# ikW0 "x1 \ k# ik1 s j j j "k# eksj x1 −G11 "x1 \ k#\ G01 "x1 \ k# W?0 "x1 \ k# k1 s1j j j "k# eksj x1 G10 "x1 \ k#\ Umn "x1 \ k# ikGmn "x1 \ k#[
"35#
Thus\ we have
"1# s um\n p −
6
7
s Aq Ap−q ðG0m "x1 \ q#G0n "x1 \ p−q#¦W0 "x1 \ q#Umn "x1 \ p−q#Ł eipx0 \
q −
"36# and
p"1# "4¦n0 # s p −
6
7
s Aq Ap−q G1m "x1 \ q#G1m "x1 \ p−q# eipx0 [
q −
"37#
De_ning v"p\ q\ s j # i"d0p d0q −d1p d1q #s j ¦"d1p d0q ¦d1q d0p #\ we may write Gpq "x1\ k# as Gpq "x1 \ k# k1 v"p\ q\ s j #j j "k# eksj x1 \
"38#
where we have made use of the fact that s01 s11 0 ðsee "20#Ł[
4[ THE EVOLUTION EQUATIONS We now use the virtual work method "see Fu\ 0884 ^ Fu and Devenish\ 0885# to derive the evolution equations for the amplitudes Am "m 9\ 0\ [ [ [#[ The traditional way of deriving evolution equations is to expand all the governing equations and boundary conditions to third!order and then to impose a solvability condition after some reductions[ The virtual work method is in essence equivalent to this procedure but does not require the expansion of governing equations and boundary conditions to third!order[ We consider the virtual work integral
G
xpq nq u¼p ds 9\
"49#
1D
where 1D is the boundary of the rectangular region D "9 ¾ x0 ¾ 1p\ −h ¾ x1 ¾ h# in Be\ "nq# is the unit normal to 1D and u¼p is a linear solution given by u¼p Wp "x1 \ −k# e−ikx0 [
"40#
Equation "49# holds because xpqnq 9 on x1 2h ðsee "8#Ł and the integrals along the two vertical paths x0 9\ 1p cancel due to the periodicity of the integrand[ Applying the divergence theorem to "49#\ we obtain
Nonlinear stability of plate under tension
g0
1
1158
1
1 up u¼p ¦xpq u¼p\q dx0 dx1 9\ 1t1
o1
D
"41#
where use has been made of the equation of motion "7# and the fact that the depen! dence on t is through t only[ Substituting "02# and "04# into "09# yields an asymptotic expansion for xpq[ On inserting this expansion and "04a# into "41# and equating the coe.cients of o\ o1 and o2\ respectively\ we obtain
g
h
dx1
g
1p
dx1
g
1p
h
−h
dx1
g 0
1
"43#
11 ua"0# "0# 0 "2# "2# u¼a ¦p¹0 ub\a u¼a\b ¦ðAbanm um\n −1ub\a ¦sba Łu¼a\b dx0 9\ 1t1
"44#
g 0 9
"42#
0 1 0 "1# "1# "0# "0# Abanm um\n −1ub\a ¦ Abanmsr ur\s um\n u¼a\b dx0 9\ 1
9
−h
0 "0# "0# "Abanm um\n −1ub\a #u¼a\b dx0 9\
9
−h
h
g
1p
1
where 1 "1# "0# 2 "0# "0# "0# "0# um\n uc\d ¦05 Abadcnmsr ur\s um\n uc\d ¦p"1# ub\a \ sba Abadcnm
"45#
and use has been made of the identities "0# "0# "0# "0# "0# "0# "1# "1# "0# u¼a\b ub\c uc\a 9\ u¼a\b ub\c uc\d ud\a u¼a\b "ub\c uc\a ¦ub\c uc\a #\
which can be veri_ed by carrying out the summations followed by the use of "05# and "08#[ As a consequence of up"0# and up"1# being solutions of "05#Ð"07# and "08#Ð"10#\ respectively\ the _rst two equations "42# and "43# should be automatically satis_ed[ To show this\ we _rst note that for any su.ciently smooth vector function vm\
g
0 "Aqpsr vr\s −1vq\p #u¼p\q dx0 dx1
g
0 "Aqpsr u¼p\q −1u¼s\r #vr ns ds
1D
D
g
−
0 Asrqp u¼p\qs vr dx0 dx1 9\
"46#
D
where the last equation follows from the fact that u¼i is a linear solution with wav! enumber −k so that it satis_es "05#Ð"07# with ui"0# replaced by u¼i and p"0# 9[ Equation "42# then follows immediately with vm replaced by ui"0# \ whilst "43# reduces to
g
1p
h
−h
dx1
g 0 9
1
0 1 "0# "0# Aqpnmsr ur\s um\n u¼p\q dx0 9[ 1
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This reduced equation can be established by contracting "10# with u¼p\ integrating both sides along 1D\ applying the divergence theorem and _nally making use of "08# and "46#[ We now return to the third eqn "44#[ With the use of "46#\ the two terms in "44# which involve ui"2# vanish[ Substituting "21#\ "22#\ "33#Ð"37# into the resulting simpli_ed form of "44# and evaluating the integrals with respect to x0\ we obtain a
d1 Ak bp¹0 Ak ¦ s s aK"p\ q\ k#Ak−p Aq Ap−q \ 1 dt p − q −
"47#
where
g
h
Wm "x1 \ −k#Wm "x1 \ k# dx1 \ b
a −
−h
g
h
Gmn "x1 \ −k#Gnm "x1 \ k# dx1 \
−h
aK"p\ q\ k# K0 "p\ q\ k#¦K1 "p\ q\ k#¦K2 "p\ q\ k#\
g
0 2 K0 "p\ q\ k# Abadcnmsr 5 1 K1 "p\ q\ k# Abadcnm
h
Gab "x1 \ −k#Grs "x1 \ k−p#Gmn "x1 \ p−q#Gcd "x1 \ q# dx1 \
−h
g g
h
Gab "x1 \ −k#Gcd "x1 \ k−p#G0m "x1 \ q#G0n "x1 \ p−q# dx1 \
−h h
K2 "p\ q\ k# "4¦n0 #
Gab "x1 \ −k#Gba "x1 \ k−p#G1m "x1 \ q#G1m "x1 \ p−q# dx1 [
−h
In obtaining the expression for K1 "p\ q\ k# we have made use of the identity 1 "0# "0# um\n u0\cd u¼a\b 9\ Abadcnm
which can be established by expanding the left hand side out with the aid of Mathem! atica and making use of properties of um"0# and u¼m[ On substituting the expressions for Wi\ de_ned by "23#\ "38# for Gmn and "35c# for Umn into the above expressions\ we obtain a −k sinh"1kh#\ b:a −1k1 \ K j "p\ q\ k# k1 "k−p# 1 "p−q# 1 q1 K
j "p\ q\ k#\ j 0\ 1\ 2\ where K
0 "p\ q\ k# "0:2#A2jivunmlt v"i\ j\ sa #v"t\ l\ sb #v"u\ v\ sc #v"m\ n\ sd # = ja "−k#jb "k−p#jc "q#jd "p−q#V"a\ b\ c\ d\ p\ q\ k#\ 1 v"t\ l\ sa #v"u\ v\ sb #v"0\ m\ sc #v"0\ n\ sd # K
1 "p\ q\ k# 1Altvunm
= ja "−k#jb "k−p#jc "q#jd "p−q#V"a\ b\ c\ d\ p\ q\ k#\ K
2 "p\ q\ k# 1"4¦n0 #v"t\ l\ sa #v"l\ t\ sb #v"1\ v\ sc #v"1\ v\ sd # = ja "−k#jb "k−p#jc "q#jd "p−q#V"a\ b\ c\ d\ p\ q\ k#\
Nonlinear stability of plate under tension
1160
with V given by V"a\ b\ c\ d\ p\ q\ k#
sinh ð"−ksa ¦"k−p#sb ¦qsc ¦"p−q#sd #hŁ −ksa ¦"k−p#sb ¦qsc ¦"p−q#sd
when the denominator in this expression is non!zero and equal to its limiting value h otherwise[ The numerical evaluation of the kernel is computationally intensive because of the many summations in K
1[ After some numerical experimentation\ we _nd
0 and K that in integrating the system "47# the kernel K"p\ q\ k# can be replaced by K Þ "p\ q\ k# de_ned by aK Þ "p\ q\ k#
8¦3n0 ¦n1 K2 "p\ q\ k#\ 4¦n0
"48#
2 where n1\ which appears in the expression for Abadcnmsr \ is a third!order material constant[ The two constants n0 and n1 appear in the constitutive relation in the form T"1# 1E¦n0E1¦n1E2¦higher order terms\ where T"1# is the constitutive part of the second PiolaÐKirchho} stress tensor and E "FTF−I#:1\ see Fu "0883#[ The much simpler expression "48# was discovered in two steps[ First\ it was dis! covered that for the NeoÐHookean material aK"p\ q\ k# can be replaced by K2 "p\ q\ k# in integrating "47#[ Next\ when we assumed that only A0 and A1 were non!zero and wrote down the two di}erential equations for A0 and A1 with the aid of "47# for a general material\ we found that all the coe.cients in the two equations were pro! portional to the special combination 8¦3n0¦n1\ which is equal to 4¦n0 for the NeoÐ Hookean material[ This led us to trying replacing 4¦n0 by 8¦3n0¦n1 in the expression for K2 "p\ q\ k# and hence to the discovery of "48#[ We remark that K Þ "p\ q\ k# and K"p\ q\ k# are not identical[ What we can say is that if "47# is replaced by a _nite system and similar terms "such as A01 A1 # collected\ the two kernels give the same coe.cients in the di}erential equations[ On carrying out the summations in the expression for K2 "p\ q\ k# we obtain from "48#
aK
"p\ q\ k#\ Þ "p\ q\ k# 05"8¦3n0 ¦n1 #k1 "k−p# 1 "p−q# 1 q1 K
"59#
where
"p\ q\ k# K
sinh"1h"q−k## ðj0 "−k#j0 "p−k#j0 "q#j0 "q−p# 1"q−k#
¦j0 "k#j0 "k−p#j0 "−q#j0 "p−q#Ł¦
sinh"1h"p−q−k## 1"p−q−k#
= ðj0 "−k#j0 "p−k#j0 "−q#j0 "p−q#¦j0 "k#j0 "k−p#j0 "q#j0 "q−p#Ł[ In the above expression the factor sinh"1h"q−k##:"1"q−k## should be replaced by its limiting value h when q k\ and similar modi_cation applies to the other factor[
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In the special case when n0 9\ we have j0 0 0 and so K Þ "p\ q\ k# −
1"8¦3n0 ¦n1 # k"k−p# 1 "p−q# 1 q1 sinh"1kh# =
6
7
sinh"1h"q−k## sinh"1h"p−q−k## \ ¦ 1"q−k# 1"p−q−k#
"50#
which should be useful when one wishes to investigate general properties of the in_nite system "47#[ Since the kernel K Þ "p\ q\ k# given by "59# is clearly pure real\ solutions in which all Ak |s are real exist[ Our attention will be focused on such solutions in the following calculations[ In this case "47# may be written as d 1 Ak −1k1 p¹0 Ak ¦ s Aq1 Ak ðK Þ "9\ q\ k#¦K Þ "9\ −q\ k#Ł 1 dt q 0
¦ s s Aq Ak−p ðAp−q K Þ "p\ q\ k#¦Ap¦q K Þ "p\ −q\ k#Ł p 0 q 0
¦Aq Ak¦p ðAp¦q K Þ "−p\ q\ k#¦A−p¦q K Þ "−p\ −q\ k#Ł[
"51#
We note that d 1A9:dt1 9 so that a mean term will not be induced by nonlinear interactions if it does not exist initially[ In our numerical calculation we set A9 0 9[
5[ STATIC SOLUTIONS We refer to a post!buckling solution as a non!sinusoidal or multiple!mode solution if it contains more than one mode[ Static multiple!mode solutions correspond to solutions of the in_nite system of algebraic equations obtained by setting d:dt 9 in "51#[ To solve this system\ we assume that Ak 9 for k × N\ thus reducing the in_nite system to a _nite system for N unknowns A0\ A1\ [ [ [ \ AN\ where N is a suitably large integer[ We may start with N 1 or 2 and increase N in unit steps[ When we take N M and try to determine A0\ A1\ [ [ [ \ AM\ we use the values of A0\ A1\ [ [ [ \ AM−0 determined at the previous step and AM 9 as our initial guess and use Nag Library subroutine C94NBF to _nd the correct solution[ We take a solution as a valid solution if each Ak "k 0\ 1\ [ [ [# converges as N : [ We start our solution with N 1[ If we assume that Ak 9 for k × 1\ "51# yields A0 ð1p¼0 ¦c"0\ 0#A01 ¦1c"0\ 1#A11 Ł 9\ A1 ð1p¼0 ¦1c"1\ 0#A01 ¦c"1\ 1#A11 Ł 9\
"52#
p¼0 p¹0 :"8¦3n0 ¦n1 #\
"53#
where
Nonlinear stability of plate under tension
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and c"i\ j# is de_ned by sinh"1kh#c"k\ k?# 1kk?3
60
1h¦
1
sinh ð1"k−k?#hŁ 1 "n0 −n11 # 1 k−k? ¦
0¦3n01 n11 sinh ð1"k¦k?#hŁ k¦k?
7
"54#
with sinh ð1"k−k?#hŁ:"k−k?# set to its limiting value 1h when k k?[ We note that material properties "i[e[ n0 and n1# appear in our analysis through the special combination p¼0 de_ned by "53#[ For the general class of strain!energy functions N
W"l0 \ l1 \ l2 # s mi "l0ai ¦l1ai ¦l2ai −2#:ai i 0
proposed by Ogden "0861#\ where l0\ l1\ l2 are principal stretches\ and N\ m0\ [ [ [ \ mN\ a0\ [ [ [ \ aN are parameters to be chosen\ it can be shown that N
n0 −3 s mm ¦ m 0 N
n1 7 s mm ¦ m 0
0 N s m "a −1#"am −3#\ 1 m 0 m m
0 N s m "a −1#"am −3#"am −5#[ 5 m 0 m m
"55#
For the NeoÐHookean material "N 0\ a0 1#\ we have n0 −3\ n1 7 and so 8¦3n0¦n1 0[ For a three!term material a0 0[2\ a1 4[9\ a2 −1[9\ m0 0[380\ m1 9[992\ m0 −9[9126\ which was found by Ogden "0861# to give good agreement with a variety of exper! imental data\ we have n0 −3[64\ n1 09[20 and 8¦3n0¦n1 9[2924[ Finally\ for the Varga material "N 0\ a0 0#\ we have n0 −4\ n1 00 and 8¦3n0¦n1 9[ Thus\ for the Varga material\ the present nonlinear analysis becomes invalid and nonlinear e}ects are pronounced at a higher order[ The equation system "52# has three sets of non!trivial solutions given by
III ] A01 −
I ] A01 −1p¼0 :c"0\ 0#\ A1 9\
"56#
II ] A0 9\ A11 −1p¼0 :c"1\ 1#\
"57#
1p¼0 1p¼0 ðc"1\ 1#−1c"0\ 1#Ł\ A11 − ðc"0\ 0#−1c"1\ 0#Ł\ D D
"58#
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where D c"0\ 0#c"1\ 1#−3c"0\ 1#c"1\ 0#[ It can easily be seen from "54# that c"0\ 0# and c"1\ 1# are both positive[ In addition\ it can be shown with the aid of "54# that 1c"0\ 1# × c"1\ 1#\ 1c"1\ 0# × c"0\ 0# and hence D ³ 9[ Thus\ all the above solutions are only possible when p¼0 ³ 9[ For the NeoÐHookean material and materials with the three!term energy function discussed above\ this requires p¹0 ³ 9 and so the bifurcation is super!critical[ Since p¼0 can be scaled out of "56#Ð"58# and if "A0\ A1# is a solution so is "−A0\ −A1#\ we shall assume that p¼0 −0 and consider only those solutions with A0 − 9[ As an illustration\ we take n0 0\ n1 9\ h 9[4[ Equations "56#Ð"58# then give three sets of solutions for "A0\ A1# ] I ]"9[4440\ 9#\ II ]"9\ 29[0022#\ III ]"9[2298\ 29[95442#[ When we use the above solutions as starting solutions and increase N in unit steps\ solution I leads to a multiple!mode solution with A0 × 9\ solutions II lead to two multiple!mode solutions with Ak 9 except A1k "k 0\ 1\ [ [ [#\ whilst starting with solutions III the Nag Library subroutine C94NBF fails to converge[ Some numerical experiments with random choices for the starting values of A0 and A1 show that multiple!mode solutions with Ak 9 except Amk "k 0\ 1\ [ [ [# exist for any positive integer m[ However\ we now show that such multiple!mode solutions with A0 9 are included in the set of solutions obtained by imposing A0 9 and varying h in "9\ #[ Suppose that we have a multiple!mode solution with Ak 9 except Amk "k 0\ 1\ [ [ [# for some positive integer m[ The corresponding displacement com! ponent u0"0# is then given by ðsee "21#Ł
u0"0# s mkAmk "t#s j j j "mk# emksj x1 eimkx0 [
"69#
k −
We now re!scale ui"0# and xi according to u½i"0# mui"0# \ x½i mxi[ Since we have pre! viously scaled our dimensional displacement components and co!ordinates using the inverse of the fundamental wavenumber which is arbitrary\ this additional scaling makes no di}erence[ In terms of u½0"0# and x½i\ "69# becomes
u½0"0# s km1 Amk "t#s j j j "k# eksj x½1 eikx½0 [ k −
This shows that the multiple!mode solution under consideration is in fact a solution for a plate with surfaces de_ned by x½1 2mh and the solution has the k!th Fourier amplitude equal to m1Amk[ It can easily be veri_ed that this is indeed a correct solution to "51#[ The above argument shows that we may focus our attention on solutions with A0 × 9[ Thus we shall only consider the solution generated from starting solution I[ For this solution\ Table 0 shows the rapid convergence of A1k−0 "k 0\ 1\ [ [ [# as N is increased[ The A1k "k 0\ 1\ [ [ [# are found to be zero[ The corresponding pro_les of
Nonlinear stability of plate under tension
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Table 0[ n0 0\ h 9[4 –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– — N A0 A2 A4 A6 A8 ÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐ * 1 9[4440E¦99 9[9999E¦99 9[9999E¦99 9[9999E¦99 9[9999E¦99 2 9[4727E¦99 9[0006E−90 9[9999E¦99 9[9999E¦99 9[9999E¦99 4 9[4772E¦99 9[0012E−90 9[6912E−92 9[9999E¦99 9[9999E¦99 6 9[4778E¦99 9[0017E−90 9[5795E−92 9[4297E−93 9[9999E¦99 8 9[4778E¦99 9[0018E−90 9[5719E−92 9[4942E−93 9[3501E−94 00 9[4789E¦99 9[0018E−90 9[5712E−92 9[4948E−93 9[3236E−94 08 9[4789E¦99 9[0018E−90 9[5713E−92 9[4950E−93 9[3241E−94 18 9[4789E¦99 9[0018E−90 9[5713E−92 9[4950E−93 9[3241E−94 28 9[4789E¦99 9[0018E−90 9[5713E−92 9[4950E−93 9[3241E−94 –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– —
Fig[ 0[ Pro_les of u1 "solid line#\ u0\0 "dotted line# and u0\1 associated with the post!buckling state when n0 0\ h 9[4[
u1\ u0\0 " −u1\1# and u0\1 " u1\0# evaluated at x1 h are shown in Fig[ 0 and are calculated according to
u1"0# 1 s kAk j j "k# eksj x1 sin"kx0 #\ k 0
"0# −1 s k1 Ak s j j j "k# eksj x1 sin"kx0 #\ u0\0 k 0
"0# u0\1 1 s k1 Ak j j "k# eksj x1 cos"kx0 #[ k 0
"60#
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Table 1[ n0 0\ h 9[0 –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– — N A0 A2 A4 A6 A8 ÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐ * 1 9[4656E¦99 9[9999E¦99 9[9999E¦99 9[9999E¦99 9[9999E¦99 2 9[5047E¦99 9[0776E−90 9[9999E¦99 9[9999E¦99 9[9999E¦99 4 9[5145E¦99 9[1090E−90 9[2796E−91 9[9999E¦99 9[9999E¦99 6 9[5183E¦99 9[1075E−90 9[3112E−91 9[0296E−91 9[9999E¦99 8 9[5202E¦99 9[1117E−90 9[3312E−91 9[0329E−91 9[4628E−92 00 9[5213E¦99 9[1140E−90 9[3420E−91 9[0383E−91 9[5060E−92 02 9[5220E¦99 9[1154E−90 9[3482E−91 9[0429E−91 9[5391E−92 04 9[5225E¦99 9[1162E−90 9[3520E−91 9[0440E−91 9[5418E−92 06 9[5228E¦99 9[1179E−90 9[3545E−91 9[0452E−91 9[5591E−92 08 9[5230E¦99 9[1173E−90 9[3563E−91 9[0461E−91 9[5536E−92 10 9[5232E¦99 9[1176E−90 9[3575E−91 9[0467E−91 9[5566E−92 12 9[5233E¦99 9[1178E−90 9[3583E−91 9[0471E−91 9[5587E−92 14 9[5234E¦99 9[1189E−90 9[3699E−91 9[0474E−91 9[5602E−92 16 9[5235E¦99 9[1180E−90 9[3693E−91 9[0476E−91 9[5613E−92 18 9[5235E¦99 9[1181E−90 9[3696E−91 9[0477E−91 9[5621E−92 20 9[5235E¦99 9[1181E−90 9[3697E−91 9[0478E−91 9[5627E−92 22 9[5235E¦99 9[1182E−90 9[3609E−91 9[0489E−91 9[5631E−92 24 9[5235E¦99 9[1182E−90 9[3609E−91 9[0489E−91 9[5633E−92 26 9[5236E¦99 9[1182E−90 9[3600E−91 9[0489E−91 9[5635E−92 28 9[5236E¦99 9[1182E−90 9[3600E−91 9[0480E−91 9[5636E−92 60 9[5236E¦99 9[1182E−90 9[3601E−91 9[0480E−91 9[5638E−92 –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– —
Table 2[ n0 0\ h 0 –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– — N A0 A2 A4 A6 A8 ÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐ * 1 9[3423E¦99 9[9999E¦99 9[9999E¦99 9[9999E¦99 9[9999E¦99 2 9[3593E¦99 9[0372E−91 9[9999E¦99 9[9999E¦99 9[9999E¦99 4 9[3594E¦99 9[0379E−91 9[0947E−93 9[9999E¦99 9[9999E¦99 6 9[3594E¦99 9[0379E−91 9[0940E−93 9[0967E−95 9[9999E¦99 8 9[3594E¦99 9[0379E−91 9[0940E−93 9[0969E−95 9[0126E−97 28 9[3594E¦99 9[0379E−91 9[0940E−93 9[0969E−95 9[0126E−97 –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– —
For comparison\ Tables 1 and 2 and Figs 1 and 2 show the results for h 9[0 and 0\ respectively\ with n0\ n1 having the same values as in Table 0 and Fig[ 0[ An inspection of Tables 0Ð2 shows that as h is increased\ energy becomes increasingly more and more con_ned to the _rst few modes and convergence becomes more and more rapid[ As a result\ the pro_les of u1\ u0\0\ u0\1 becomes closer and closer to being sinusoidal[ In the limit h : \ we expect to _nd only sinusoidal solutions[ This is precisely the result found by Devenish and Fu "0886# for a half!space[ Conversely\ as h is decreased\
Nonlinear stability of plate under tension
1166
Fig[ 1[ Pro_les of u1 "solid line#\ u0\0 "dotted line# and u0\1 associated with the post!buckling state when n0 0\ h 9[0[
Fig[ 2[ Pro_les of u1 "solid line#\ u0\0 "dotted line# and u0\1 associated with the post!buckling state when n0 0\ h 0[
energy spreads into higher modes and the three pro_les are further away from being sinusoidal[ We note that these pro_les are also very sensitive to the values of n0 and n1\ i[e[ to the shape function H"x1\ k#[ Figure 3 shows these pro_les for n0 9[4\ n1 z2:1\ h 9[4\ which may be compared with those shown in Fig[ 0[ We see that
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Fig[ 3[ Pro_les of u1 "solid line#\ u0\0 "dotted line# and u0\1 associated with the post!buckling state when n0 9[4\ h 9[4[
at the same value of h\ the pro_les in Fig[ 3 are closer to being sinusoidal than those in Fig[ 0[ Numerical calculations for a large number of "n0\ n1# pairs show that the smaller the n01 −n11 is\ the closer to being sinusoidal the pro_les of u1\ u0\0\ u0\1 become[ In the critical case n0 n1\ only sinusoidal solutions "i[e[ single!mode solutions# are found[ The non!trivial solutions found above all have A1k 9 "k 0\ 1\ [ [ [#[ Since in particular A1 9\ one may think that by starting with N 2\ new solutions might be found[ Truncating "51# at N 2\ we obtain A0 ð1p¼0 ¦c"0\ 0#A01 ¦1c"0\ 2#A21 Ł¦b0 A01 A2 9\ A2 ð1p¼0 ¦1c"2\ 0#A01 ¦c"2\ 2#A21 Ł¦b1 A02 9\
"61#
where b0 −43"n03 −n13 # cosh"1h#\ b1 −"0:2#"n03 −n13 # sinh"3h#:sinh"5h#[ If n0 n1\ then b0 b1 9 and the solutions of "61# are analogous to those of "52#[ If n0 n1\ solutions of "61# can be obtained easily with the aid of Mathematica[ It is again found that "61# has non!trivial solutions only if p¼0 ³ 9[ When n0 0\ n1 9\ h 9[4\ "61# have three pairs of solutions for "A0\ A2# with A0 × 9 given by I ]"9[4727\ 9[9006#\ II ]"9[2145\ −9[90669#\ III ]"9[1297\ 9[91773#[ When we use the above solutions as starting solutions and increase N in unit steps\ we _nd that starting solution I leads to the solution shown in Table 0 and Fig[ 0\ C94NBF fails to converge when starting solution II is used\ whereas starting solution
Nonlinear stability of plate under tension
1168
Fig[ 4[ Evolution of A0 when ] "a# Ak "9# 0[90Ak"s# "9#\ A þ k "9# 9 ^ "b# Ak "9# 0[0Ak"s# "9#\ A þ k "9# 9 ^ "c# þ k "9# 9 ^ "d# Ak "9# 0[4Ak"s# "9#\ A þ k "9# 0:k1\ where k 0\ 1\ [ [ [ \ 28 and the super! Ak "9# 0[4Ak"s# "9#\ A script "s# signi_es the static solution[
III leads to a solution with Ak 9 except A2m "m 0\ 2\ 4\ [ [ [#[ Hence starting with N 2 does not yield any new solutions[ Suppose that we start with an even larger truncation number\ say N 09\ and obtain a set of starting solutions by solving the resulting system of ten simultaneous algebraic equations[ It would seem possible that some of the starting solutions might lead to new solutions[ However\ _nding all the possible solutions of a system of more than three simultaneous nonlinear algebraic equations is a non!trivial matter\ and even if we can _nd all the possible starting solutions\ there is no guarantee that C94NBF will converge to new solutions[ Because of these di.culties\ we have not attempted to start with truncation numbers larger than three[ Hence we cannot claim that the solutions found here are the only solutions possible[
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6[ STABILITY OF POST!BUCKLING STATES We _rst consider the special case n0 n1[ It is found in the previous section that in this case multiple!mode static solutions reduce to a single!mode solution[ If initially there is only a single mode\ integrating the system of eqns "51# shows that other modes are not excited[ This implies that in this special case a single mode can evolve by itself[ If we assume that only Am is non!zero\ "51# gives d1 Am −1m1 p¹0 Am −gm5 Am2 \ dt1
"62#
where g ðK Þ "9\ m\ m#¦K Þ "9\ −m\ m#¦K Þ "1m\ m\ m#Ł:"m6 sinh"1mh## 3"8¦3n0 ¦n1 # cosh"1mh#[ Since eqn "62# for m 0 can be obtained from that for m 0 by a suitable rescaling\ we may focus our attention on m 0[ In this case the non!trivial post!buckling state with A0 × 9 is given by A0 A0"s# z−1p¹0 :g[ In the neighbourhood of this post! buckling state\ we write A0 A0"s# ¦a"t# where a is a small amplitude perturbation[ Substituting this expression into "62# and linearizing\ we obtain d 1a:d 1t 3p¹0a[ With! out loss of generality we set p¹0 −0[ Then a small perturbation in the neighbourhood of the post!buckling state evolves as a simple harmonic oscillation with period p[ It can easily be deduced from a phase portrait analysis that perturbations with larger amplitude in general are also oscillatory[ The post!buckling state is therefore stable[ In the general case when n0 n1\ only multiple!mode post!buckling solutions are possible[ The stability of such post!buckling solutions can be determined by inte! grating the system "51#\ after appropriate truncation\ with the initial condition being the static solution plus a perturbation[ In Fig[ 4"aÐd# we have shown the evolution of A0 for increasingly large perturbations[ The in_nite system is truncated at N 28 and the calculation is for the NeoÐHookean material with n0 0\ h 9[0[ In all cases it 28 is found that the graphs for zSi 0 Ai1 and =A0 = are graphically indistinguishable\ which implies that higher harmonics have relatively small amplitudes compared with the fundamental mode at all times[ The sensitivity to the truncation number N is found to increase with the amplitude of perturbations[ For the small perturbation considered in Fig[ 4"a#\ the results corresponding to N 08\ 28 are almost identical[ It is also found that the result shown in Fig[ 4"a# is graphically indistinguishable from the linearized single mode result A0 "t# A0"s# ¦9[90A0"s# "9# cos"1t# where A0"s# is the static solution given in the row N 28 in Table 1[ This shows that for small pertur! bations both nonlinearity and higher modes have a negligible e}ect[ Figure 4"bÐd# shows that as the amplitude of perturbation increases\ the e}ects of nonlinear and higher modes come into play and the evolution of A0 gradually deviates from the linear single mode behaviour shown in Fig[ 4"a#[ However\ for all the cases we have considered\ perturbations never blow up and are always oscillatory[ We thus conclude that all the post!buckling solutions found in the present paper are stable[
Nonlinear stability of plate under tension
1170
7[ SUMMARY In this paper we have carried out a weakly nonlinear stability analysis for an incompressible elastic plate subjected to an all!round tension[ Our aim was to _nd post!buckling states and to determine whether they are stable or not[ This stability problem has the following unusual features[ The neutral value of tension for instability is a constant "i[e[ two after scaled by the shear modulus#\ independent of wavenumber[ Thus\ all the modes participate in the post!buckling process[ However\ unlike the usual nonlinear analysis involving non!dispersive wave modes\ the evolution equations are derived not at second!order but at third!order of an asymptotic expansion[ Also\ the neutral value of tension for instability is independent of whether the modes are extensional or ~exural[ This gives rise to an arbitrary parameter "n0# in the shape functions and to the possibility of an in_nite number of post!buckling solutions[ With the aid of the virtual work method\ we derived an in_nite system of evolution equations for the various Fourier components[ We also discovered a simple expression for the kernel in the nonlinear terms\ which reduces considerably the computing time required[ This simple expression shows that material nonlinearity manifests itself through the special combination 8¦3n0¦n1\ whose sign determines if the post!buck! ling is supercritical "if it is positive# or subcritical "if it is negative#[ If it is zero\ the nonlinear analysis presented in this paper becomes invalid and the evolution equations can only be derived by carrying out the expansions to higher order[ For the NeoÐ Hookean material and the three!term material\ this expression was found to be positive whereas for the Varga material it is zero[ For a _xed plate thickness\ we _nd an in_nite number of post!buckling solutions\ one for each n0\ but only the one with n0 n1 0z1 is sinusoidal[ Small amplitude perturbations superimposed on each of these post!buckling states are found to evolve as oscillations and thus\ all the post! buckling states found are stable[
REFERENCES Devenish\ B[ J[ and Fu\ Y[ B[ "0886# The nonlinear evolution of near!neutral and near!body wave modes in a pre!stressed incompressible elastic half!space[ Part II ] lower branch near! neutral modes[ International Journal of En`ineerin` Science 24\ 0330Ð0344[ Fu\ Y[ B[ "0883# On the propagation of nonlinear travelling waves in an imcompressible elastic plate[ Wave Motion 08\ 160Ð181[ Fu\ Y[ B[ "0884# Resonant!triad instability of a pre!stressed elastic plate[ Journal of Elasticity 30\ 02Ð26[ Fu\ Y[ B[ and Devenish\ B[ J[ "0885# E}ects of pre!stresses on the propagation of nonlinear surface waves in an incompressible elastic half!space[ Q[ J[ Mech[ Appl[ Math[ 38\ 54Ð79[ Fu\ Y[ B[ and Ogden\ R[ W[ "0887# Nonlinear stability analysis of pre!stressed elastic bodies[ Nonlinear Wave Phenomena[ Birkhauser\ to appear[ Fu\ Y[ B[ and Rogerson\ G[ A[ "0883# A nonlinear analysis of instability of a pre!stressed incompressible elastic plate[ Proceedin`s of the Royal Society of London A335\ 122Ð143[ Ogden\ R[ W[ "0861# Large deformation isotropic elasticity ] on the correlation of theory and experiment for incompressible rubberlike solids[ Proceedin`s of the Royal Society of London A215\ 454Ð473[
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Ogden\ R[ W[ "0884# Nonlinear e}ects associated with waves in pre!stressed elastic solids near bifurcation points[ Proceedin`s of the IUTAM Symposium on Nonlinear Waves in Solids\ ed[ J[ E[ Wegner and F[ R[ Norwood\ pp[ 098Ð002[ ASME\ New Jersey[ Ogden\ R[ W[ and Roxburgh\ D[ G[ "0882# The e}ect of pre!stress on the vibration and stability of elastic plates[ International Journal of En`ineerin` Science 20\ 0500Ð0528[ Wolfram\ S[ "0880# Mathematica\ 1nd edn[ Addison!Wesley\ California[