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Energy Decay Estimates For the Von Karman Plate Equations
In Nonlinear Elasticity l
presented by

Peter Vafeades

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ENERGY DECAY ESTIMATES FOR THE VON KARMAN
PLATE EQUATIONS IN NONLINEAR ELASTICITY

By

Peter Vafeades

A DISSERTATION

Submitted to
Michigan State University
in partial fulfilllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Metallurgy, Mechanics and
Materials Science

1987

ABSTRACT

ENERGY DECAY ESTIMATES FOR THE VON KARMAN
PLATE EQUATIONS IN NONLINEAR ELASTICITY
By

Peter Vafeades

This dissertation is concerned with the analysis of Saint-Venant edge
effects for nonlinear elastic plates. The model used is based on the von
Karman plate equations: a coupled system of two nonlinear elliptic partial
diflerential equations with the biharmonic operator as the principal part.
Energy methods are used to establish a nonlinear integro—diflerential ine~
quality for a quadratic functional. Arguments based on comparison
theorems are then used to establish exponential decay of end effects. The
results constitute a version of Saint-Venant’s principle for nonlinear elastic

plates.

iii

Emu Kain) Amer)

ACKNOWLEDGMENTS

-.

I would like to express my sincere and deeply felt appreciation towards
my advisor Professor Cornelius O. Horgan for his guidance, assistance,
persistence and patience without which this research would not have been
possible. I should also like to mention that Professor Horgan inspired me to
study mechanics and in particular questions related to Saint-Venant’s prin-
ciple, for which I am ever so grateful. I would also like to thank the

members of my committee, Drs. Nicholas Altiero, Thomas Pence and David

Yen.

iv

Table of Contents

List of Tables ........................................................................................................ vii
List of Figures ................................................................... . .................................. viii
CHAPTER 1: INTRODUCTION .................................................................... 1

CHAPTER 2: STATEMENT OF THE BOUNDARY VALUE
PROBLEM ............................................................................................................ 4

CHAPTER 3: THE LINEARIZED PROBLEM: THE BIHAR-
MONIC PROBLEM ............................................................................................ 10

CHAPTER 4: ENERGY DECAY ESTIMATES 16

Section 4.1: Derivation of an Integro—Difl'erential Inequality for the

energy for the von Ka’rman equations. .......................................................... 16

Section 4.2: A Comparison Theorem for Integro-difl'erential Ines

qualities. ............................................................................................................... 20

Section 4.3: An Improved estimate for the Biharmonic Problem.

.................................................................................................................................. 22
Section 4.4: A First Result for the von Karman equations. ..................... 23
Section 4.5: An Improvement on the Result (4.42) for E(0)>0.209.
.................................................................................................................................. 26
Section 4.6: An Improvement on Result (4.42) for E(0)<O.529. ............ 31
Section 4.7: An Estimate with z-Dependent Decay Rate. ......................... 35
CHAPTER 5: DISCUSSION OF RESULTS ................................................. 39
Section 5.1: A Summary of the Results of Chapter 4 for the von

Karman Equations. ............................................................................................ 39

Section 5.2: A Comparison of All Results. ................................................... 41

Section 5.3: Remarks on the Constants k, u, and the Total

Energy E(0). ........................................................................................................ 57
Section 5.4: Questions for Further Investigation. ....................................... 58
APPENDIX A: VERIFICATION OF (4.17). ............ , .................................. 59
REFERENCES .................................................................................................... 65

vi

List of Tables «

Table 4.1: Values of 2. ...................................................................................... .29
Table 4.2: Values of 20. ..................................................................................... 33
Table 4.3: Values of E. ...................................................................................... 37
Table 5.1: 295 values. ............................ . ............................................................ 43
Table 5.2: 29,, values. ......................................................................................... 44

vii

List of Figures

Figure 2.1: Semi-infinite isotropic elastic plate. ....................................... 5
Figure 3.1: Semi-infinite strip R. .............................................................. 11
Figure 4.1: Ic/2k vs. total energy. ................................................................. 25
Figure 4.2: Results I, II for E(0)=1.0. ............................................................ 30
Figure 4.3: Results 1, III for E(0)=0.10. ......................................................... 34
Figure 4.4: Results 1, IV for E(O)=1.0. .......................................................... 38
Figure 5.1: Results 1, III, IV for E(0)=0.05 (Case A). .................................... 45
Figure 5.2: Results 1, III, IV for E(O)=O.10 (Case A). .................................... 46
Figure 5.3 Results I, III, IV for E(0)=0.15 (Case A). ..................................... 47
Figure 5.4: Results 1, III, IV for E(0)=0.20 (Case A). .................................... 48
Figure 5.5: Results I, II, III, IV for E(O)=0.25 (Case B). ................................ 49
Figure 5.6: Results I, II, III, IV for E(0)=0.30 (Case B). ............................... 50
Figure 5.7: Results I, II, III, IV for E(0)=0.40 (Case B). ............................... . 51
Figure 5.8: Results I, II, III, IV for E(0)=0.50 (Case B). ................................ 52
Figure 5.9: Results I, II, IV for E(0)=l.0 (Case C). ....................................... 53
Figure 5.10: Results I, II, IV for E(0)=2.0 (Case C). ..................................... 54
Figure 5.11: Results I, II, IV for E(O)=5.0 (Case C). ..................................... 55

viii

Figure 5.12: Results I, II, IV for E(0)=—10.0 (Case C). ................................... 56

ix

CHAPTER 1

INTRODUCTION

Saint-Venant’s principle in elasticity theory has had a long history. For long
prismatic bars with traction-free lateral surface and subjected to end loads only,
SAINT-VENANT suggested that the detailed mode of application and distribu-
tion of forces over the end of the cylinder is immaterial so that it is always possi-
ble, to some degree of approximation, to replace the applied system of forces by a

statically equivalent one having the same resultant force and moment.

Much work has been done to justify this engineering principle, which lies at
the foundations of elasticity theory and strength of materials. For an account of

the major developments, up to 1972, concerning this issue for the linear theory of

elasticity, see GURTIN [1].

In 1965 renewed eflorts began on establishing a principle of Saint-Venant
type for the original cylinder as well as for thin elastic bodies. In the framework
of the linear theory of elasticity, the problem of comparing two statically
equivalent systems of traction is reduced, by applying superposition, to the study
of the stresses in the same body subject to arbitrary sclf-cquilibrated tractions on
part of the boundary with the remainder of the boundary traction-free. TOUPIN
[2] and KNOWLES [3] made use of the connection between Saint-Venant’s princi-
ple and the decay of strain-energy away from the loaded end of the body to pro-
vide estimates of the rate of stress decay. For a comprehensive review of work on

Saint-Venant’s principle up to 1983 see the survey by HORGAN and KNOWLES

[4]-

As is discussed in [4], and also is clear from earlier work, the issues underly-
ing the validity of a Saint-Venant principle are mathematical questions for the
elliptic partial differential equations involved. Thus the exponential stress decay
established in [2], [3], for example, demonstrate clearly the boundary-layer charac-

ter of the solutions corresponding to a self-equilibrated load.

Compared to the amount of research carried out on the analysis of Saint-
Venant’s principle for linear elasticity, very little has been done to examine such
questions for nonlinear elasticity. Clearly the issues are much more complicated
here. A major difliculty arises since superposition no longer holds and so consider-
ing self-equilibrated loads is not sufficient. Furthermore, instabilities might have
to be taken into account. Also the decay rate for end effects, even if exponential,
might depend on the overall loading as well as on geometry. Early work on the
nonlinear elasticity problem was carried out by ROSEMAN [5] and BREUER and
ROSEMAN [6]. A review of this research and of other work up to 1983 on
second-order nonlinear partial differential equations is given in [4]. HORGAN and
KNOWLES [7] considered a Saint-Venant principle for finite anti-plane shear and
obtained exponential decay rates depending on loading and on material. Since
then BREUER and ROSEMAN [8] have considered the plane problem and
KNOPS and PAYNE [9] the three-dimensional cylinder problem. See also [10] for
results on small deformations superimposed on large for plane incompressible elas-
ticity. Results for second-order quasilinear partial difierential equations in two

and three dimensions have been obtained recently in [11 - 14].

In this thesis, we are concerned with a nonlinear elasticity problem for plates
modeled by the v. KARMAN [15] equations. The exponential decay of an

energy-like quadratic functional involving the stresses and second derivatives of

the deflections due to self-equilibrated tractions and prescribed deflections on part
of its boundary can be regarded] as a Saint-Venant principle type of result. The
only previous work on questions like this for the von Karman equations is that of
KALANTAROV [16]. He established an upper bound for such an energy func-
tional which is inversely proportional to the square of the distance from the
loaded end for sufficiently large distances. We obtain stronger results involving

an exponential decay of energy here.

Here in Chapter 2 we formulate the boundary value problem to be studied.
The stresses and deflections in a semi-infinite thin elastic plate are described by
the von Karman system. The plate is clamped and is rendered traction-free along
its long sides while self-equilibrated tractions and prescribed displacements are
assigned to its left end. In Chapter 3 we discuss the linearized version of the
problem posed, the biharmonic problem for a semi-infinite strip with traction-free
lateral sides and self-equilibrated end loads. We review the technique of
KNOWLES [3] to derive exponentially decaying upper bounds for an energy func-
tional. These results are used by us in Chapter 4. In Chapter 4 we present the
main results of this thesis. We establish an integro—diflerential inequality for an
energy-like quadratic functional for the von Karman system. By means of a com-
parison theorem for such inequalities a series of upper bounds demonstrating
exponential decay of this energy functional are established. In Chapter 5 these

results are compared to one another and discussed.

CHAPTER 2

STATENIENT OF THE
BOUNDARY VALUE PROBLEM

Consider the rectangular semi-infinite isotropic elastic plate of Figure 2.1.
The plate has thickness 2h, width 2H and extends to infinity in the positive 6
direction. The two semi-infinite lateral sides described by the planes n = -H and
n = H, are clamped, that is, no displacement is allowed along these sides and the
plate’s slope there is equal to zero. An arbitrary self-equilibrated traction is

applied at the end 5 = 0, while the stresses are assumed to vanish as E-voo .

In 1910, v. KARMAN [15] introduced a coupled system of two fourth-order
quasilinear elliptic partial differential equations to describe the large deflections
and stresses in a thin elastic plate. We use v. KARMAN’S model for a geometri-

cally nonlinear elastic plate here.

Let c} be the Airy stress function, whose second derivatives yield the stresses
in the plate. Then for fixed g in [-h,h] 346,17) satisfies the following inhomogene—

ous boundary conditions

 

 

ire—H) = 91(8). axe—H) = 92(6). l

346.”) = 93(5). MU!) = 94(6), 0<E<°°.

l&(0’n) = [11(71): @409”) = h2(fl), -H<T]<H, ’ (2'1)
and

@5460), (35,4617), $,,,,(€,n) ->0 (uniformly in 17) as E-+ 00,)

where the subscript on it indicates partial differentiation and 91, 92, 93, 9,, 12,, I12

né-H,C:—h
///////// —-——____.

 

clamped

7/////// -—---

 

 

 

 

n=H ,C =-h

/////r//// ‘--—-__.

 

 

clamped
,/<:¢//////7////////// -‘—-—-

I.

 

FIGURE 2.1 SEMI-INFINITE ISOTROI’IC ELASTIC PLATE

are prescribed functions. Let <73 be the Airy stress function associated with the
stresses necessary to cancel out any deformation of the plate (d is the Airy stress
function for the so-called bending stresses, see STOKER [17] and KNIGHTLY
[18]); then it satisfies the same inhomogeneous boundary conditions as 6 along the
clamped sides and homogeneous boundary conditions elsewhere. Thus

‘

'Re-H) = ale—H) = 91(5).
ire—H) = ale-H) = 92(5),
‘ 345,11) = 345,”) = 93(5),

 

 

area) = Merl) = yrs), o<e<oo. ’ (2'2)
340.72) = 0. $40.71) = 0. —H <n<H.
[5555(6”) ,agnGJI) Fenlan) #0 (uniformly in U) as €—+OOJ
We now define
¢ = R-a, (2.3)

the Airy stress function for the so-called membrane stresses. Note that o satisfies
the original inhomogeneous boundary conditions on 6 =0, and homogeneous

boundary conditions on the lateral clamped sides

¢(O,fl) = h1(fl), ¢€(Oin) = h2(’l)r —H<17<H,
(“5"”) = ¢n(€:-H) = 02 (2'4)
¢(£,H) = ¢,,(E,H) = O, 0<E<oo,

and that the corresponding stress field tends to zero as E-voo , that is,

45546.17). «46.71), ¢.,.,(€.n) -+0 (uniformly in n) as E—voo. (2-5)

The von Karman system of equations for the midplane reads [15]:

A(2£.n)¢ = E(wtzn-wtfwnn)
Aliar“ = (h /D )(wace‘2¢£nw£n+¢eewnn) (2-6)
where A62”) is the biharmonic operator in terms of the (5,17) variables, w(€,17) is the

midplane deflection, E is Young’s modulus, D=Eh3/ 12(1—1/2) is the plate’s modulus
and u is Poisson’s ratio. The functions ¢(E,r7) and (46,77) are assumed to be four

times continuously diflerentiable on the open semi-infinite strip and twice

continuously differentiable on its boundary.

The boundary conditions on 41 are given in (2.4), (2.5). The functions h, and
112 must satisfy some additional conditions in view of the fact that the tractions
on the 5 =0 face are self-equilibrated. If the tractions on the 6 =0 face are T

«-

=(T‘, T"), where the superscript indicates the corresponding component, we have

I!
den=o, (2.7a)
-H
H
7' Mid" = o. (2.7b)
-H

In view of the fact that on the 6=0 face the unit outward normal is n =

(-1,0) we have
T5 = —¢,,,, and T" = —¢.,, on 5:0. (2.8)

On integration by parts, using the boundary conditions and the assumed smooth-

ness of (11 we deduce that

th)=h7—H)=o, as)
and
hl(—H)=hl(H)=h,’(-H)=h1’(H)=0. (2.10)

By virtue of the smoothness assumptions on ¢ and the boundary condition (2.4),

the boundary condition (2.5) may be integrated to yield

“51’”. 0545,”), $45.17) ->0 (uniformly in 17) as 6—»00. (2.11)

The boundary conditions on w are:

lie—H) = wn(€.-H) = 0.

 

 

w(€,H) = w,,(E,H) = 0, 0<€<Oo,
* * (2.12)
045.17). 0045.77), wn(€.n)-*0
(uniformly in 17) as E-roo for any 3‘ in [-I1,h],)
and
«(0.12) = 1.01) w.(o,n) = 1m) ' (2.13)

on the {=0 face where the prescribed functions j1(17), jg(17)are such that

1.1(iH) = jl’ (iH) = jeliH) = 0- (2-14)

We now introduce dimensionless spatial variables by setting

 

 

= .5- = 17. 2 15
I H) y H . ( ' )
We also introduce a dimensionless midplane deflection
w
= — 2.16
z 1. _ ( )
In terms of these dimensionless quantities (2.6) becomes:
45
Alamlfiz' = 4211-213 Zr»!
¢ ¢ ¢
Agmyz = 12(1—u2)( Eli; z” -2 E22 2,, + E12112 2,”), (2.17)

where Ag”) is the biharmonic operator in terms of the (2,3)) variables. Hen-
ceforth, we drop the subscript notation on the biharmonic operator and so
A2=A(2z.r)'

We now introduce the dimensionless Airy stress function 2,0(z,y) by

1R2.y)= 91% (2.18)

and employ the following scale change in z and t0:

1
z = u,
Van—u")
12(1—12')
Thus we obtain
A212 = 2(u13—u” 11””) = —[u,u],
Agu = (an on -2u,y v," +uW v”) = [u,v] on R, (2.20)
where the bilinear form [..,] is defined as
[1.9] = [saw-211,9.y-rfwgu (2.21)

and R is the semi-infinite strip 0<x <00, -1<y <1.

The boundary conditions on v, the dimensionless Airy stress function, are

”(01y)=’;l(y)1 ”1(01y)=£2(y)1 _l<y<11
v(:r,:l;l) = vy(z,j:l)=0, 0<:1: <00, (2.22)
um). v.(z.y). vy(z.y) -»0 (uniformly .-.. y) as 2 —+oo.

where the functions 51, 52 are such that

Mil) =iil'(;1:1) = 52cm) = o. (2.23)
The boundary conditions on u, the dimensionless displacement out of the
plane, are
u(0.11)=J~'1(y), "3(01y) =iz(y). -1<y<1.
u(z,:hl)=uy(z,:l:l)=0, 0<z<oo, (2.24)

u(:1:,y), u,(:1:,y), uy(z,y) ->0 (uniformly in y) as :1: -+oo.

The functions 31. 5'2 satisfy

31(i1)=31’(i1) =32(:1:1) = 0. (225)
By a classical solution to the boundary value problem posed above we mean

a pair of functions (u, u) that are four times continuously differentiable in the
interior of R and twice continuously diflerentiable on its boundary, that are solu-
tions to the system (2.20) and satisfy the boundary conditions (2.22) and (2.24)

for prescribed functions III, 112, 5'1, 5'2, assumed to be sufliciently smooth.

CHAPTER 3

THE LINEARIZED PROBLEM:
THE BIHARMONIC PROBLEM

In 1966, J. K. KNOWLES [3] formulated and proved a version of Saint-
Venant’s principle appropriate to the plane strain and generalized plane stress
solutions of the equations of the linear theory of isotropic elastic equilibrium in
bounded simply-connected plane domains of general shape. In [3] an explicit esti—
mate (lower bound) is obtained for the rate of exponential decay of the energy
with distance from a portion of the domain boundary carrying a self-equilibrated
load. This result for biharmonic functions was established in [3] using difl'erential
inequality arguments. The special case of the biharmonic equation in a semi-
infinite strip subject to self-equilibrated loads on the near end only constitutes a
linearized version of the problem described in Chapter 2 here. In the present
chapter we provide a brief description of the methods and results of [3] for the
semi-infinite strip (see also KNOWLES [19]). Subsequent improvements on these
results obtained by other authors will also be summarized.‘] Our treatment in
Chapter 4 of the nonlinear problem described in Chapter 2 will be seen to make

use of energy decay estimates of the type used in [3].

Thus we consider a semi-infinite strip R of width 2 in the (2,31) plane whose

 

T A comprehensive review of work on Saint-Venant’s principle wu given in 1983

by HORGAN and KNOWLES [4).

10

11

t

 

 

 

 

 

 

 

SEMI-INFINI'I‘I‘. STRIP R

FIGUR t 3.!

long sides are traction free and whose end :r=0 carries a self-equilibrated load (see
Fig. 3.1). The stress field is assumed to vanish at infinity. \Ne are concerned with

solutions (23 of the biharmonic equation

A243 =0 on R, (3.1)

( 45 is the Airy stress function), subject to the boundary conditions:

¢(0.y)=k(y). ¢.(O.y)=l(y), -1<y<l.
¢(z,;l;1) = ¢,(z,;1:1)=0, 0<z<oo, (3.2)
¢n(z,y), ¢w(z,y), ¢yy(x,y)->0 (uniformly in y) as z-+oo,

where k and l are prescribed functions such that

k(:l:l) = H (i1) = ((4:1) = o. (3.3)

We introduce the notation:

Rz={(z,y)inR I122},

L,={(z,y)inR Ix=z}, (3.4)
so that z is a running variable along the z-axis. Clearly, R0512.

Following [3], [19] we define the function e(¢5) by

e(¢) = ¢§+2¢fy+¢fy (3.5)
so that the energy stored in R, is given byi

E(z) = {emu/1. (3.6)
Thus, the total energy is:
E(O) = [mad/1. (3.7)

By repeatedly applying Green’s theorem and using the boundary conditions (3.2)

it is shown in [3] that

E(z) = _f(¢z¢zz —¢¢nz +2¢y ¢zyldy- (3'8)
L:

 

1 It is shown in [19] that E(z) is finite.

13

We recall from [3] the following difierentiation formula for functions f continuous

on the closure of R,
.31. f 71.1 = 41.3. (3.9)
21?, L, .
Thus we may write

E(z) -_- §;J(¢,¢n—¢¢m +2¢,¢,,)dA (3.10)

which implies that

fE(s)ds = —f(¢,¢,,—¢¢m+2¢,¢,,)dA. (3.11a)

3.

It is convenient for subsequent purposes in Chapter 4 to introduce the Volterra

integral operator

FU E jU(s)ds, (3.11b)
0

on continuous scalar valued functions. Thus the left-hand side of (3.11a) can be

on

written as fE(s)ds -FE. Applying Green’s theorem and the boundary condi-
0 V

tions (3.2) we can now write

[E(s)ds = fE(s)ds -FE = f(¢3+¢3—¢¢,,)dy, 0<z<oo. (3.12)
z 0 L,

\Ne can now construct as in [3] the integro—diflerential functional

f(z,E’ ,FE) a E' (z)+41c‘-’(fE(s)ds — FE)= E’ (z) + 4k2fE(s)ds, (3.13)
0 z
for any real constant k. The result (3.12) and the expression for E’ (2) given by

(3.5),(3.6), (3.9) may be combined to yield
f (z,E’ ,FE) = —f[¢f,+¢;-’,+2¢3,—4k‘-’(¢3+¢3—¢¢n)1dy, 0<z <00. (3.14)
L:

We now seek, as in [4], to find a positive value of k for which

14

f(z,E' ,FE) g 0, 0 g z<oo. (3.15)
In [3] Knowles made use of VVirtinger-type inequalities (see Appendix A here) to

establish (3.15) with the value of k given by
k = 0.70. (3.16)
In 1974, FLAVIN [20], by using a sharper \Nirtinger‘ inequality, established (3.15)

with the larger value of k given by

k = 1.11. (3.17)
Using arguments based on first-order difierential inequalities Knowles esta-
blished in [3] that (3.15) yields the exponential decay estimate
E(z) _<_ 2E(0)e'2’“, 0 g z<oo, (3.18)
with the value of k given by (3.16). Such a value of k, which provides a lower
bound for the actual decay rate we shall call an "estimated decay rate". FLAVIN
[20] employed the same arguments as Knowles to proceed from (3.15) to (3.18)
and so obtained (3.18) with the improved value given by (3.17) for the estimated

decay rate.

We shall show in Chapter 4 (see Section 4.3) that direct employment of a
comparison theorem for integro-diflerential inequalities (see Section 4.2) applied to
(3.15) yields the result

E(z) g E(0)e‘2’“, z 2 0, (3.19)
for any value of k for which (3.15) holds. Thus we obtain an improvement on

(3.18) by eliminating the multiplicative factor of two in that estimate.

By using a diflerent argument, OLEINIK and YOSIFIAN [21], [22] esta-
blished in 1978 that

E(z) < _LEIQL = M (3.20)

-’ coslz(2kz) 1.1.3-“: '
with the value of k given by (3.17). Earlier, in 1975, MIETH [23] had obtained

(3.20) with the value of k given by (3.16). Observe that (3.19) is a sharper result

15

than (3.20).

Recently, KNOWLES [19] has used arguments based on higher-order energies
to obtain exponential decay estimates of a more elaborate nature than (3.18). His
results do yield an exponential decay of the energy E (z) with estimated decay rate

given by

k = 1.35. (3.21)
As is discussed in [4], (see also [19]) all of the preceding results underestimate

the exact decay rate which is given by
k = 2.10. (3.22)
This result may by arrrived at through consideration of eigenfunction expansions
for biharmonic functions in the semi-infinite strip ([4],[19]), the completeness of

which has been established by GREGORY [24].

In the sequel, we shall invoke directly the inequality (3.15) for the integro—
differential functional (3.13), with the best constant k obtained to date, namely

(3.17).

CHAPTER 4
ENERGY DECAY ES TIL/IATES

4.1 DERIVATION OF AN INTEGRO-DIFFERENTIAL INEQUALITY
FOR THE ENERGY FOR THE VON KARMAN EQUATIONS

On the semi-infinite strip R we have the system of partial differential equa-

tions (2.20):

A21) = 2(ué—uuu = —[u,u],

w)
Azu = (un v” —2u,y v,,+uw v3) = [u,v], (4.1)

subject to the boundary conditions (2.22) and (2.24). It is convenient to rewrite
the system (4.1) in divergence form (BERGER [25], p. 692) and so
Ago = ("12)1111 —2(u, uy )W+(uy2)n, (4.2)
Agu = (“1m v),,-2(u,y 12),, -l-(u:I v)W on R. (4.3)
We now introduce an energy for the von Karman system. Observing that the
principal part of the von Karman system of equations is the biharmonic Operator,

we define, for solutions (11, v) of (4.1),

é(u,v) = e(u) + e(v) (4.4)
where
e(u) = u; + 2 11,: + “vii: (4.5)

We note that the quadratic function (4.5) has exactly the same form as the energy

density for the biharmonic problem (see (3.5)).

Using the same notation as that introduced in Chapter 3, we define the

16

17

energy functional E(z) by
E(z) = jé(u,v)dA, 052 <00. (4.6)
R1

Adopting the arguments of [19] it can be shown that E(z) is finite for z in [0,00).

The total energy then is
E(O) = fé(u,v)dA. (4.7)
R
Clearly, E(z) can be decomposed into two parts

E(z) = fé(u,v)dA =hfe(u)dA +kfe(v)dA =E,,(z) +E,,(z). (4.8)

I

We now seek to establish an integro-diflerential inequality for E (2). First, we

multiply (4.3) by u and integrate over R, to get

gquudA = Jaw, v]dA. (4.9)

Applying Green’s theorem to the left-hand side of (4.9) and using the boundary
conditions (2.24) we obtain

IquudA = —fuum dy + f(u,un+2uy ufl)dy + Eu(z). _ (4.10)
R L. L.

I

Applying Green’s theorem to the right-hand side of (4.9) using the boundary con-
ditions (2.22), (2.24) and equation (4.2) we get

I!“ [11, v]dA = {[-(uwv),+u,uwv-2uy unv]dy
+ [(vvm—v, 1133—212y v”)dy

L.

— E,(z). (4.11)

On equating (4.10) and (4.11), combining terms and integrating by parts, making
use of the boundary conditions (2.22), (2.24) we get

18

E(Z) = j_l( uz2+uy2_uuu)+(vz2+vy2_wu )lz dy.

+ [(1521), +2u, "w v +uuy 11,” +11, u, v, +uu,y v” )dy. (4.12)
L

I

We are now ready to construct the functional

E ’(z) + 4k2fE(s)ds. (4.13)
Using the differentiation result (3.9), and the boundary conditions (2.22), (2.24) we

obtain, for any constant k, the identity

on
E ’(z) + 4k2fE(s)ds
1
={—[[u,2, +11]: +2uz§-4k2(ug2+uy2—uuzz)ldy}

+ {—f [123+vy3 +2'Uzg-4k2(v¢2+vy2—sz )]dy}
L

+ 4k2].(uy212, +2u, "1w v-i-uuy v,y +u, uy v” +uuzy vy )dA. (4.14)
R.
The first two integrals in (4.14) have exactly the same form as their counterparts
for solutions of the biharmonic equation (see equation (3.14)). ‘ Thus, if we choose

1: =1.11 (4.15)
it follows from (4.14) and the results of Chapter 3 that

E '(z) + 4k27E(s)ds

S4k2k['(u,2v, +2u, “w v +uuy v,” +u, u” 1),, +uuzy v” )dA

54k2[11+ 12 + I3 + I, + 15]. (4.16)
It is shown in Appendix A that

I, + 12 + 13 + I, + 15 g 71E3/2(z) (4.17)

where the constant a has the value

p = 0.619. (4.18)
Thus, on using (4.17) and (4.16) we obtain the integro-diflerential inequality

19

E '(z) — 4k2pE3/2(z) + 41:2 f E(s)ds g 0, (4.19)
where k and u are given by (4.15) and (4.18) respectively. The inequality (4.19)
allows us to establish several difl'erent exponential decay estimates for the energy

E(z) defined in (4.8). These results follow from a comparison theorem for

integro-difierential inequalities which we now describe.

20

4.2 A CONIPARISON THEOREM FOR
INTEGRO-DIFFERENTIAL INEQUALITIES

We state here a comparison theorem for integro-difierential inequalities of
which we will make repeated use in the sequel. For a complete discussion and
proof see LAKSHMIKANTHAM and LEELA [26], pp. 350-1, and WALTER [27]
p. 122.

THEOREM 1

Let F .‘CfLRj—vC/LR] be an integral operator mapping continuous scalar valued
functions on J, a subset of R, into continuous scalar valued functions on J.

Assume that

(i) f (2, U', U F U ) maps continuously C(JxRa) into R and that f is nondecreasing

in U’ for fixed (2, U ,FU) and nonincreasing in FU for fixed (2, U’ ,;U) (4.20)
(ii) For any 111, 112, in C(J R)

111(2) S u2(z) implies that Ful S F112, for all z in J = (0,00); (4.21)
(iii) 1‘ (z,V' ,V,FV) g 0 and (4.22)
f(z,W’,W,FW) 20, for all z in (0,00),whereV, W are 01(J,R). (4.23)

Then

V(0) _<_ W(0) implies that (4.24)
V(z) _<_ W(z), 220. (4.25)

We introduce the notation
oo
f(z,E’ ,E,FE) at" (z) — 4121113229) + 4k2(fE(s)ds — FE)
o

= E’ (z)- 4k2uE3/2(z) + 4k27E(s)ds (4.26)

for the left-hand side of our inequality (4.19), where the Volterra integral operator

21

F has been defined in (3.11b). It may be readily verified that f satisfies the
hypotheses (4.20), (4.21), and (4.22) of Theorem 1. Thus it remains for us to

determine a comparison function H (3)2 0, such that

f (2.}! ’ 1H J71 ) .>_ 0. ' (4.27)
and
11(0) 2 E (0). (4.28)

in order to conclude from (4.25) that

E(z) S H(z), z _>_ 0. (4.29)
In what follows we show that several such choices of H (2) lead to exponentially

decaying estimates for E (z).

22

4.3 AN INIPROVED ESTIIVIATE FOR THE
BIHARMONIC PROBLEM

First we consider the application of Theorem 1 to the integro-difierential ine-
quality (3.15) obtained for the biharmonic problem discussed in Chapter 3. We
recall that (3.15) reads:

f(z,E’ ,FE) = E' (z) + 4k2(7E(s)ds — FE) = E' (z) + 4k27E(s)ds S 0. (4.30)
Since E does not appear explicitly on the left-hand sides ’of (4.30), we have used
the notation f (z,E’ ,FE) instead of f (z,E' ,E,FE). It may again be readily
verified that f satisfies the hypotheses (4.20), (4.21) of Theorem 1. Furthermore,
the hypothesis (4.22) is satisfied by virtue of (4.30) if the constant k is chosen as
in (3.17). Now the function H (2) defined by

11(2) = E(0)e"2"‘ (4.31)
is such that

f (z,H’ ,FH) = H' (z) + 41sz119)“ = 0. (4.32)
Furthermore, E(O) = H(0) and so Theorem 1 yields the result

5(2) S E(0)e'2"‘, .220, (4-33)
where k is given by (3.17). This provides a sharper estimate than (3.18) or (3.20).

23

4.4 A FIRST RESULT FOR THE
VON KARMAN EQUATIONS

Returning to the nonlinear problem, we seek a comparison function Hl(z)
satisfying (4.27), (4.28), where f is given by (4.26). Our first choice is to consider
a function similar to (4.31), which was employed for the biharmonic problem.
Thus we consider

Hl(z) = E(0)e”“ (4.34)
where K is a positive constant, as yet undetermined. Clearly,

H1(0) = E(O) (4.35)
and so (4.28) is satisfied. We now seek the largest value of R. such that (4.27)

holds. We have

H1' (2:) = —1cE(0)e-“ = —IcHl(z), (4.36)
and
i; H1(s)ds —- FH, = jH,(s)ds = 59%;: = gig-51, (4.37)

and so recalling the definition of f in (4.26) we obtain
“ 1 _ 2 3/2 4k2
f(Z:H11H1,FHI)—-’°H1(Z)—4kflH1 (Z)+TH1(Z)

2
= H,(z)[—1c — 4k2uH11/2(z) + %}. (4.38)
Since 0 S [ill/2(2) g [ill/2(0) for all z in (0,00), and fill/2(0) = E1/2(0) by virtue of
(4.34) it follows that f (2,111' ,H1,FH1) is non-negative for all z in (0,00) if n>0 is

such that

2
—x — 4k2uE1/2(O) + % 2 0. (4.39)

The largest value of 1c is obtained by taking the equality sign in (4.39) and so we
find

24

1c = 2k(V M2+l—M) (4.40)

where
M = ka173'(0_) (4°41)
and k, p are given by (4.15) and (4.18) respectively. Thus (4.27), (4.28) are
satisfied and so from (4.29) and (4.34) we obtain the exponential decay estimate
E(z) S E(0)e"", 220. (4.42)
A decay estimate similar in form to (4.40), (4.41), (4.42) was obtained by HOR-

GAN [28] in his investigation of plane entry flows for the Navier-Stokes equations.

We observe that the decay rate It given by (4.40) is a monotonically decreas-
ing function of M for 0<M<0o and so 1c<2k. Thus the estimated decay rate for
the von Karman equations is slower than that predicted by (4.33) for the bihar-
monic problem. Moreover, as the total energy E(O) increases, M increases and so 11:
decreases. The dependence of It on E(O) is shown graphically in Figure 4.1. In the

sequel, we shall refer to the estimate (4.42) as result I.

We should also note that in deducing the condition (4.39) from (4.38), we
used the inequality H11/2(z) _<_E1/2(0) for OSz<oo. Recalling the definition of
H1(z) in (4.34), we see that this bound deteriorates as 3 increases. In the next sec-
tion, we present an argument which does not haye this shortcoming and which

leads to a sharper decay estimate than (4.42) for sufliciently large values of z.

SEE .59. .m> cams 3 on

SE SEE .59. .
2 s m s s m 1.. m m __ cos
id
nmd
1nd
14.0
ind
10.0
1nd
1nd
imd

 

I:

42/,1

26

4.5 AN IMPROVEMENT ON THE RESULT (4.42)
FOR E(O) > 0.209

In Section 4.4 we have established the decay estimate (4.40)-(4.42). On using
the result (4.42) in our basic integro-difierential inequality (4.19), we obtain the
weaker integro—diflerential inequality

—3x1 00

E’ (z) —4k2pE3/2(0)e 2 +41:2 [15(3)
3

-3x1

= E’ (z) — 4k2uE3/2(0)e 2 + 4k2(_[E(9)ds — FE) g 0. (4.43)
o
Denoting the middle term of (4.43) by §(z,E' ,FE) it may be readily verified that

6 satisfies the hypotheses (4.20), (4.21) and (4.22) of Theorem 1. Thus, it remains

for us to determine a comparison function H 2(2) such that

9(21H2'1FH2) Z 0 and H2(0) Z E(0). (4-44)
in order to conclude that

E(z) _<_ H2(z), z_>_0. (4.45)
Consider the function
:12
112(2) = 0,33/2(0)pe 2 — or?” (4.46)

where C, and D are constants to be determined. We have

-3x2

112' (z) = —§2-’EC,E3/2(0)peT + acne-2'“, (4.47)

and

27

{H2 2(s )ds —FH2= IH2(s)ds

 

 

-3xz
__ atEtmuvue 2 .oe-uv
_ _. (4.48)
3_s 2*
2
Thus,
1—3“ 31: C +8k20
To ensure that g(z,H2’ ,FHQZO, the constant 01 is chosen to satisfy
01(— - —)_ > 4k2 (4.50)

where 1:, k are given by (4.40), (4.15) respectively. The smallest constant 01 is
obtained by taking the equality sign in (4.50), and so we get
24k2n

161112—9102,
provided that l6k2—9K2 > 0. On using (4.40), (4.41) we see that this condition is

, = (4.51)

satisfied if

4

E0 >— , 4.52
(o ) 45112 ( )

that is, if

E(O) > 0.209. (4.53)
It remains to satisfy the second inequality in (4.44). Thus we set

0,E3/2(0)u — D = E(0), (4.54)
thereby satisfying the second of (4.44) with equality, and so

D = o,E3/2(0)p — E(0). (4.55)

By virtue of (4.51) and (4.55), the result (4.45) reads:

28

—31cz
E(z) g E(0)e’2‘” + 0,)1E(3/2)(0)[e 2 44'“), z_>_0, (4.56)
provided that E(O) > 0.209, where
24k2n

= 4.57

’ 16k2—9n2’ . ( )

n = 21W M2+1—M), (4-58)
M = kin/E(0), (4.59)

where k=1.11 and u=0.619.
Note that the estimate is made up of two terms: the first is the estimate for

the linear problem, i.e. the biharmonic problem, while the second is an exponen-
tially decreasing correction which depends on the magnitude of E(0). In the

sequel, we shall refer to (4.56) as result II.

The result is not valid for E(0)<0.209 but for such E(O) the estimated decay
rate 1: as given by (4.40) is quite close to the estimated decay rate 2k for the
biharmonic problem. In fact for E(0)=0.209 we get It: 0.734(2k). Moreover, It is

even closer to 2k for smaller energies (see Figure 4.1).

The result II provides a sharper estimate than I for values of z greater than
2, where 2 depends on the total energy E(O) as indicated in Table 4.1 (see also

Fig. 4.2).

Table 4.1: Values of E

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

153(0) 2
0.25 0.54
0.50 0.63
1.00 0.77
1.50 0.89
2.00 0.98
2.50 1.06
3.00 1.14
3.50 1.21
4.00 1.27
4.50 1.33
5.00 1.39
6.00 1.49
7.00 1.59
8.00 1.68
9.00 1.76
10.0 1.84

 

 

 

30

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3qu5 SEE .58.
N 825%

 

 

 

 

(2)3 A083N3

31

4.6 AN IMPROVEMENT ON RESULT (4.42)
FOR E(O) < 0.529

The approach we used in- the previous section was to simplify our basic

integro—diflerential inequality

00
f(z,E’ ,E,FE) = E' (z) - 4k2pE3/2(z) + 41:2 j E(s)ds g 0, (4.60)
z
by using the estimate (4.42) for the nonlinear term E3/2(z). Here we consider a

difl'erent method which also involves simplification of (4.60).

Recall from the end of Section 4.2 that the estimate

150) 3 He) (4.61)
holds for comparison functions H (2) such that

f(z.H' ,H,FH) 2 o, o _<_. < oo, (4.62)
and
11(0) 2 E(0). f (4.63)

We seek a comparison function H3(z)>0 which satisfies

H3’ (2) — 4k2pH3/2(2) = —2kH3(z), z 2 0, (4.64)
so that f(z,H3' ,H3.FH3) reduces to

f(z,H3’ ,H3,FH3) = —2kH3(z) + 411-2 j H3(s)ds

oo

= —2kH3(z) + 4k2({H3(s)ds — FH3) a f(z,H3,FH3). (4.65)

The first order ordinary differential equation (4.64) has the solution

173(2) = (0.21:5 + 2kp)-2, (4.66)

where 02>0 is a constant yet to be determined. It remains to establish that the

choice H(z) = H3(z) satisfies (4.62) and (4.63).

TO show that (4.62) is satisfied, it suffices, by virtue Of (4.65), to show that

f (2,113,171,) 2 0, 0 g z <00. (4.67)

A direct verification of (4.67) is diflicult due to the complexity in integration Of
the function H3(z) defined in (4.66). We will establish (4.67) by using a contradic-
tion argument. Thus, suppose that

f(z,H3,FH3) < 0, 0 g z <00. (4,68)
Consider the function H .,(z) defined by

mm = [(02+2kp)e"‘]"2. (4.69)
It may by readily verified that

f (z,H,,FH,) = 0, z 2 0, (4.70)
and that
H3(0+) < H,(0+). (4.71)

We now employ a stricter version of Theorem 1 (WALTER [27] p. 122)
wherein (4.22) is strict and (4.24) is replaced by V(0+)< W (0+). Now the operator
f defined in (4.65) satisfies the hypotheses (4.20), (4.21) of Theorem 1. Further-
more, in view of (4.68), (4.70) and (4.71) the remaining hypotheses of the stricter
version of Theorem 1 are satisfied for the choices V = H 3, W = H 4. Thus, we con-
clude that

173(2) < 114(2), 2 Z 0, (4.72)
which is a contradiction by virtue Of the definitions Of H3, H4 in (4.66), (4.69)

respectively. Thus, (4.68) cannot hold and we deduce that (4.67) holds.

It remains for us to choose the constant 02 > 0 such that (4.63) is satisfied
with H 5 H3. The largest such constant is Obtained by taking the equality sign
in (4.63) and so

0, = E(0)-1/'-’ — 21:71. . (4.73)

TO ensure that 02 is positive, we require that

33

 

q
C

E(O) < i, as 0.529. (4.74)
4km
Thus for sufiiciently small total energies E(O) satisfying (4.74), we conclude from

(4.61), (4.66), and (4.73) the decay estimate

 

13(2) s 1 1 . ‘ (4.75)
l E(O) c u(e )l
where k, p are given by (4.15) and (4.18) respectively. In what follows we refer to

(4.75) as result III.

The result 111 is an improvement over the estimate I of Section 4.4 for z 220

(see Fig. 4.3). The value of 20 depends on the total energy E(O) as indicated in
Table 4.2 below.

Table 4.2: Values of 20

 

 

 

 

 

 

E(O) z,

0.10 2.41
0.20 3.13
0.30 3.99
0.40 5.28
0.50 8.58

 

 

 

 

34

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ofioion SEE .58
N 825%

 

 

 

 

(2)3 ASHBNB

35

4.7 AN ESTIMATE WITH Z-DEPENDENT
DECAY RATE

We now describe an argument which is more elaborate than the preceding
and which leads to a decay estimate with a z-dependent decay rate. We seek

comparison functions H (z) of the form

175(2) -.- E(0)c-41):,J'(%]—, (4.76)
where x(z)20 is a function to be determined. The function x(z) must be such that
175(2) satisfies (4.27), (4.28). The particular form of the right-hand side Of (4.76)
has been chosen so that H5(2) may be readily integrated. This integration is
necessary in the evaluation of f (2,1-15' ,H5,FH5). Furthermore, the results Obtained
already suggest that we seek a decay estimate with decay rate 1: (given by (4.40))
for small values Of z and decay rate 2k (k is the decay rate for the biharmonic
problem) for large values Of 2. Thus, the function x(z) appearing in (4.76) will

be required to satisfy

X(0) = 0. X’ (0) = K. (4.77)

x(z)-+<>0. x’ (z)-+2k as z-wO. (4.78)
and

X' ' (2) Z 0. z _>_ 0. (4.79)

Observe that the first Of (4.77) ensures that H5(0) = E(O) and so (4.28) holds.

By direct calculation from (4.76) we Obtain

j (2,11; ,H5,FH5) =

-2)

 

E 0 :42 [X"(2) — X'2(z) " 4kg” E(:)e 2 X' 3”(3) + 4’52]. (4'80)

We now seek to find a x(z) satisfying conditions (4.77)2-(4.79) and which renders

the right-hand side Of (4.80) nonnegative.

36

Consider the choice

XIV) = It + (ilk—”)2 (4.81)

Z+C3 ’
where C3>0 is a constant tO be determined. It is readily verified that (4.77)2,

(4.78); and (4.79) are satisfied by x’ (z) for arbitrary Values Of 03. Upon integra-

tion and use of (4.77)l we obtain

 

x(2) = £x’(s)ds = 2k[z — 03(1 — A)ln( 2 +303”, (4.82)
where
x=fiu<1 um)

21:
Clearly x satisfies (4.78), for arbitrary values of 03. It remains to satisfy (4.27),

that is fZO, in order to conclude that

 

E(z) S H5(z), z 2 0. (4.84)
Thus by virtue Of (4.80), we choose the constant 03 in (4.81) such that
.2 (/—- ——)-""l’
gun—mflpy-“JQPWR 2X”Wd+4H2fi, 220 (4%)
K .

The largest such value Of 03, depending on the value of E(0), is determined

numerically and the results are shown in Table 4.3.

Thus we have established the estimate

 

I x-IZ + C
.E <E0-W_———i. >0 .6
(0. He 2+C3, z_. M8)
where
Z + 03 K _y
4"."I = Z — 03(1— A)ln( C3 ), A = E—k- < I. (4.81)

This result will be called result IV.

The result IV is sharper than estimate I for sufliciently large z(see Fig. 4.4).
The values of 2 beyond which (4.86) provides the sharper estimate, 2 2 .7. are

shown in Table 4.3.

37

Table 4.3: Values of C3 and E

 

E(O) C3 3'

 

0.05 1 .00 0.85

 

0.10 1.13 0.90

 

0.15 1.25 0.94

 

0.20 1.35 0.97

 

0.25 1.45 1 .00

 

0.30 1 .55 1.03

 

0.35 1 .64 1 .06

 

0.40 1.73 1 .08

 

0.45 1.82 1.11

 

0.50 1.91 1.13

 

1.00 2.75 1.33

 

1 .50 3.59 1.50

 

2.00 4.43 1.65

 

2.50 5.26 1.78

 

3.00 6.10 1.90

 

3.50 6.95 2.02

 

4.00 7.81 2.13

 

5.00 9.54 2.33

 

6.00 11.27 2.51

 

7.00 13.03 2.69

 

8.00 14.81 2.85

 

10.00 18.38 3.15

 

 

 

 

 

38

>_ oz< _ mbammm no 285.4028 4.4 0....
.osuflovm SEE .59
N 82585

 

 

 

 

 

(2)3 ASHENE

CHAPTER 5
DISCUSSION OF RESULTS

5.1 A SUNIMARY OF THE RESULTS OF CHAPTER 4
FOR THE VON KARMAN EQUATIONS

In Chapter 4 we have established four upper bound results on the energy E(z)
associated with the the von Karman system for our problem. Of these, result I Of
Section 4.4 and result IV of Section 4.7 hold for all total energies E(0), whereas
result 11 of Section 4.5 holds for sufficiently large total energies (E(O) > 0.209) and
result 111 of Section 4.6 holds for sufliciently small total energies (E(O) < 0.529).
We now present a summary of these results:

Result 1: For any total energy E(O):

E(z) S E(0)e"", z 2 0, (5.1)
where

x = 210/1142 + l-M), (5-2)

M = kuVE(0), (5.3)
and

k = 1.11, p = 0.619. (5.4)

Result 11: For sufliciently large energies, E(0) > 0.209:

-3xz

E(z) g E(0).-2" + o,pE3/2(o)(. 2 — e'2h ), z _>_ 0, (5.5)
where
24131:
a —— > o, 5.6
1 161:2 - 91:2 ( )

39

40

and 1:, k are still given by (5.2)-(5.4).

Result III: For sufiiciently small total energies, E(O) < 0.529:

 

15(2) s 1 1 . z 2 0.
bl _ 2k 88-1 2
[We #(6 )l
where k, p are given by (5.4).

Result N: For any total energy E(O):

1 >142 "l" C
< '2’“ —_——3 >
E(z)_E(0)e 2+03 , z_0,
where
0
Z, = Z — 03(1 — X)In(z_+—3),
03
X = .5... 1’
2]: <

(5.7)

(5.8)

(5.0)

(5.10)

where 1:, k are given by (5.2)-(5.4) and 03 is a constant that depends on E(O) (see

Table 4.3).

41

5.2 A CONIPARISON OF ALL RESULTS

In Chapter 4 after establishing result I of Section 4.4 we proceeded to estab-
lish a series of improvements on this first result. Once presented, all new results
were compared to result I. We now compare all results to one another and con-
struct the best upper bound for E (z) for difierent total energies E(0). Due to the
fact that results H and III are not valid for all energies we break up the discussion

into three parts:

A. For E(O) < 0.209 where results I, III and IV are valid.
B. For 0.209 < E(O) < 0.529 where all results are valid.
C. For E(O) > 0.529 where the results I, II, and IV are valid.

A. E(O) < 0.209:

As can be seen from Figures 5.1-5.4 the best upper bound for E(z) is provided by

the function:

E(0)e"“, for 0 S 2 < F
)5“: + 03 (5'11)

, forF<z<oo
2+C3

UIIZ) =
E(0)e"2""

where z' is as defined in (5.9) and ‘2' depends on the total energy E(O) (see Table
4.3).

Note that the estimate III does not enter into the composition of U1(z), and
that for this range of total energies result I is the sharpest available for small 2,

whereas IV is the sharpest for larger 2.
B. 0.209 < E(O) < 0.529:

As can be seen from Figures 5.5-5.8 the best upper bound for E (z) is provided by

42

the function:

E(0)e"“, {010 < z < E
U2(Z) = -3x: (5'12)

E(0)e'2’" + CluE3/2(0)(eT - e'zb), for z > 2
where 2 depends on the total energy E(O) (see Table 4.1).
Note that results III and N do not enter into the composition of U2(z), and
that for this range of energies result I is the sharpest available for small 2,

whereas II is the sharpest for larger z.
C. E(O) > 0.529:

As can be seen from Figures 5.9-5.12 the best upper bound for E(z) is provided by
the function U2(z) defined in (5.12). Qualitatively, for this range of E(O) we have
exactly what held for the range of energies in B with the only exception being that

the estimate III is not valid here.

Another way to compare the bounds Obtained is to determine the distance 295
from the left end of the plate 2 = 0, at which 95% of the total energy E(O) has
dissipated (see Table 5.1) and the distance 299 from the end, at which 99% of the
total energy E(0) has dissipated (see Table 5.2). These distances are called
"characteristic decay lengths". Note that a characteristic decay length of 2.00

corresponds to one width of the plate.

43

Table 5.1: :95 values

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

E(0) I 11 III Iv

0.05 1.58 - 1.62 1.53
0.10 1.68 - 1.78 1.61
0.15 1.76 - 1.92 1.67
0.20 1.83 - 2.08 1. 73
0.25 1.90 1.70 2.25 1.78
0.30 1.95 1.74 2.45 1.82
0.40 2.06 1.82 2.99 1.91
0.50 2.16 1.88 4.35 1.98
1.00 2.57 2.16 - 2.30
2.00 3.20 2.60 - 2.79
3.00 3.71 2.96 - 3.20
4.00 4.15 3.28 - 3.55
5.00 4.55 3.57 - 3.87
6.00 4.92 3.84 - 4.16
7.00 5.26 4.09 - 4.44
10.00 6.16 4.76 - 5.17

 

 

 

 

 

 

44

Table 5.2: 29,, values

 

 

 

 

 

 

 

 

 

 

 

 

 

E(0) I II 111

0.05 2.42 - 2.38 2.32
0.10 2.58 - 2.55 2.40
0.15 2.70 - 2.71 2.48
0.20 2.81 - 2.88 2. 55
0.25 2.91 2.52 3.06 2.61
0.30 3.00 2.57 3.27 2.67
0.40 3.17 2.67 3.83 2.77
0.50 3.32 2.76 4.35 2.87
1.00 3.95 3.17 - 3.28
5.00 6.99 5. 20 - 5.33
10.00 9.47 6. 96 - 7.04

 

 

 

 

 

 

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57

5.3 REMARKS ON THE CONSTANTS k, u AND
THE TOTAL ENERGY E(0)

As already discussed in Chapter 3, we have used. the largest available value of
k for which (3.15) holds, it = 1.11, which underestimates the exact decay rate, R
= 2.10. If one could establish (3.15) with a value of k larger than 1.11, the
results of Chapter 4 would still hold and would become sharper. One would only

need to re-evaluate the values of 2, E, 20 and the constant 03.

The constant p in (4.17) is probably greatly overestimated because of the
repeated use of some weak inequalities in its derivation (see Appendix A). Any
improvement in the value of u would result in an immediate improvement of all

results in Chapter 4.

All results presented in Chapter 4 involve the total energy E(0) which
depends on the geometry and the boundary data at the z = 0 end. In [3] Knowles
established an upper bound for the total energy E(0) of the biharmonic problem in
terms of the applied traction for a certain class of finite domains with the help of
variational arguments (see also [4]). The results of Chapter 3 remain valid when
E(0) is replaced by an upper bound. It is reasonable to anticipate that by using
variational arguments an upper bound for the total energy E(0) of our problem
can be established in terms _of the traction applied at the end and the displace-

ment there. We shall not pursue this issue here.

58

5.4 QUESTIONS FOR FURTHER INVESTIGATION

We can now raise a number of interesting questions for further investigation.
Firstly, in the context of the semi-infinite strip, one could ask how our decay esti-
mates depend on the particular boundary conditions used. Secondly, one could
introduce a distributed lateral load on the plate and investigate its efl‘ect.
Thirdly, interesting bifurcation and associated stability questions arise when non
self-equilibrated compressive loads are introduced. Such stability questions have
already been investigated by many authors for specific sets of boundary condi-
tions. It would be very interesting to attempt to extend these already existing
results by admitting wider classes of boundary conditions through the application
of a Saint-Venant principle type of argument. Finally, our problem as well as any
of the above mentioned open questions can be posed for a "long" thin plate of

finite size.

APPENDIX A

VERIFICATION OF (4.17)

To verify (4.17) we follow an approach used by HORGAN [28]. In this
development we make use of the following Wirtinger-type inequalities for

sufiiciently smooth functions w(y) defined on the interval (-1,1) of length 2:

(i)If w(y) is Cl(-1,1) and w(—l) = w(1) = 0, then1
I ”2 1
f wfdy 2 ?[ wzdy. (A.1)
-1 -1

(ii) If w(y) is C’(-1,1) and w(—l) = w’ (-1) = w(l) = w'(l) = 0, then

413

11w; dy > -2-— wzdy, . (A.2)
l

and

Iron“; dy > —_{w2dy, (A.3)

\

where #0 is the smallest positive root of the transcendental equation cospcoshu = l
and so #0 = 4.73 which is slightly larger than 31r/2. For convenience, we use the

latter value in (A.3) and thus we obtain

 

1 4 ‘ I
I wédy > 32: Iwzdy. (AA)
-1 -1

For a more detailed discussion of the inequalities (A.l) - (A.3), see [28] and the

 

‘ The multiplicative constant is written in a manner displaying the interval
length explicitly.

59

60

references cited therein.

We also make use of the following one-dimensional Sobolev inequality:

1 1 1

.1. .4... s. 511.2... "19””) (A5)
for sufficiently smooth functions w(y) such that w(-‘—l) = w(l) = 0. Inequalities of
the form (A.5) in two and three dimensions have been widely used in investiga-
tions of uniqueness and stability for the Navier-Stokes equations. A direct proof
of (A.5) is given by HORGAN [28], where it is shown that (A.5) holds with o=4.
A modification of the argument given in [28] may be. used to show that (A.5)

holds with a=l. This value of a is taken in (A25). below.

Finally here, we state the following simple algebraic inequality

(0 + (2)3/2 2 Vain/F for a, b > o, (A.6)
which we also use in the sequel.
We now employ some of the above inequalities to establish a series of inter-
mediate results:

A. We first establish an upper bound for

_[m‘(z,y)dA (A1)
3.
where m(z,y) is a function that vanishes along the semi-infinite clamped sides of

the plate and that tends uniformly to zero as :r—voo. In view of (2.22) and (2.24)

v u can be sub-

the functions u, v as well as their first partial derivatives 11,, av, ,, ,

stituted for In. On applying the Sobolev inequality (A.5) we get
co

IM‘(=.y)dA S %I[Im2(s.yldylllm.2(s.y)dylds- (A3)
R, x L, L,

Green’s theorem, the regularity conditions and the Cauchy-Schwarz inequality

allow us to write, for such .922,

61

Im2(.,,,)dy = —2Im(s.y)m.(s.y)dA

I mm )"iImfm
.<_ 2(I m2...) (I my... ‘2.’ (A.9)'
On combining (A.8) and (A.9) we get
ImwA _<_ dim2M)1/2(Im.’M)1/2(Im.’dA)- (A.10)
R, R, R, R.

B. Suppose now that the function m in (A.10) is chosen to be u (or equivalently

v). Then we have
‘dA < 2M ”2 2dA ”2 24A A11
I" _dI" ) (In. ) (In, )- (- )

We now apply inequality (A.3) with w = u , in an obvious way in (A.11) to get
W 2
Iu‘dA <3°227(Iw .dA) (— “In 34A)” (— ,;IuILdA)
9 2

0’2 / 2
< —— E, . A.l2
- 9.5 [ (2)1 ( )
Note that (A.12) is also valid when we write v in place of u.
C. Now suppose that the function In in (A.10) is chosen to be uy (or equivalently

v”). Wehave
Iu‘dA <0(Iu2M)1/2(Iu2dA)1/2(Iu2dA) (A13)
R v — R v R 3v R w ' '

We now apply inequality (A.2) with w=u to the first integral on the right-hand
side of (A.13) to get

 

Iu‘dA <—(Iu, 334.4) ”(In 3,4,4)” _<_21/2 E2(z). (A.14)
Again, we note that (A.14) Is also valid when we write v in place of u.

D. Now, suppose that the function In in (A.10) is chosen to be u, (or equivalently

v, ). We have

Iu:dA g e(fufdA )1/2(Iu,§dA)‘/2(Iu,§dA). (A.15)

62

Now apply inequality (A.l) with w=u, to the first integral on the right-hand side
of (A.15) to get

2 3/2 1/2
qu‘dA g—I—(jugu) (Inga)
R, R: R!

0 2
S mE. (Z). , , .- (A.16)

Again, we note that (A.16) is also valid when we write v in place of u.

E. In view of inequality (A2) we have

.,.“!sz S LIuJ‘LdA
8. "2 8.
Ed?)

fl ’

which is also valid when we write v in place of u.

 

S (A.17)

F. In view of inequality (A.1) we have

lufdA S ifuédA
a, #2 a,
2Eu(z)
<
-— "2 i

which is also valid when we write v in place of u.

 

(A.18)

We now proceed to obtain upper bounds for the five integrals I 1 — 15 defined
in (4.16) and thereby establish (4.17). The Cauchy-Schwarz inequality, (A.14),
(A.18) for v, and (A.6) yield:
I 1 = I u,2v3dA
R.
S (IuyidA )1/2(1.!”22M )1/2

01/221/4 32
_ WE / (Z) (A.19)

The Cauchy-Schwarz inequality, (A.16), (A.12) for v and (A.6) yield:

..

63

I2 = 2111,11” vdA
3.
s 2(IufidA)1/2(.I.u,‘dA)1/4(_[v‘dA)1/4
R, R, R,
g 2(%%317/:-)E3/2(z). (A20)

Similarly, the Cauchy-Schwarz inequality, (A.12), (A.14) for v and (A.6)

yield:
13: qu’vndA
3.
1/2 1/4 1/4
s<Iv3dA) (Iuw) (IuIdA)

01/221/2
- an”?

The Cauchy Schwarz inequality, (A.17), (A.16), (A.14) for v and (A.6) yield:

E3/2(z). (A21)

I4 = 111,11,va
’ 1/2 1/4 1/4
SqudA) (fuzidA) (HIM)
R, R, R,
01/2

3 2
g 21/431/219/215/ (2). (A22)

 

The Cauchy-Schwarz inequality, (A.12), (A.14) for v and (A.16) yield:
15 = fun,” v, M
R’
o 1/2 1/4 1/4
SUuzidA) (Hid/1) livid/4)
R, R R
———E3/2(z). (A23)

Combining the estimates (A.19)-(A.23) we obtain

 

II + 12 + [3 + 1, + 15 s pE3/2(z) (A.24)
where
- 21/4 4 21/2 1 21/2 01/2 ~
" ‘ (3172' + 3' + T .+ W + T) Ira/2 "’ ”‘9' (”5)

and the numerical value for u in (A25) has been obtained on taking 0:1. This

establishes (4.17) as desired.

64

REFERENCES

[1] GURTIN, M. E., The linear theory of elasticity, Handbuch der Physik, C.
TRUESDELL ed., Vol. VIa/2, pp. 1-295. Springer-Verlag, Berlin, 1972.

[2] TOUPIN, R. A., Saint-Venant’s principle. Arch. Rational Mech. Anal. 18,
83-96 (1965).

[3] KNOWLES, J. K., On Saint-Venant’s principle in the two-dimensional theory
of elasticity. Arch. Rational Mech. Anal. 21, 1-22 (1966).

[4] HORGAN, C. 0., & KNOWLES, J. K., Recent developments concerning
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[5] ROSEMAN, J. J., The principle of Saint-Venant in linear and nonlinear plane
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66

(1982).

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[10] ABEYARATNE, R., HORGAN, C. 0., & CHUNG, D.-T., Saint-Venant end
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materials. J. Appl. Mech. 52, 847-852 (1985).

[1 l] HORGAN, C. 0., & PAYNE, L. E., Decay estimates for second order quasil-
inear partial difierential equations. Advances in Appl. Math. 5, 309-332 (1984).

[12] HORGAN, C. O., & PAYNE, L. E., Decay estimates for a class of second-
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279-289 (1984).

[13] BREUER, S., & ROSEMAN, J. J., Phragmen-Lindelfif decay theorems for
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[14'] BREUER, S., & ROSEMAN, J. J., Decay theorems for nonlinear Dirichlet
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67

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68

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