Y ;"

 

 

iNVESTEGATIQEd SF A DYNAMIC
SAINT-VENANT REGEON N A
SEMI-ENFINSTE STRIP 7

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
HARTLEY T. GRANDIN, JR.
1972

“ems

This is to certify that the
thesis entitled

INVESTIGATION OF A DYNAMIC SAINT-VENANT
REGION IN A SEMI-INFINITE STRIP

presented by

Hartley T. Grandin, Jr.

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Mechanics

 

    

Major professor
;

\

MW
‘ I

. 1

0-169

 

LIBRARY

MlChlgan State
University

 

 

ABSTRACT

INVESTIGATION OF A DYNAMIC SAINT-VENANT
REGION IN A SEMI-INFINITE STRIP

By

Hartley T. Grandin, Jr.

The investigation examines the steady state response of a
semi-infinite strip with stress-free edges to time-harmonic self-
equilibrated shear and normal stresses on the finite edge. The
mathematical analysis is based on the equations of linear
elasticity for generalized plane stress and involves a biorthogonal
eigenfunction expansion of a four component stress vector.

Solutions for three different boundary stress distributions
at one frequency are examined in detail and reveal significant non-
decaying stress modes. The shapes of these modes are shown
graphically.

The eigenvalues are tabulated for seven different frequencies

between 100 and 100,000 cycles per second.

INVESTIGATION OF A DYNAMIC SAINT¥VENANT
REGION IN A SEMI-INFINITE STRIP

By

Hartley T. Grandin, Jr.

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Metallurgy, Mechanics and
Materials Science

1972

ACKNOWLEDGMENTS

I wish to express my gratitude to Dr. Robert Wm. Little
for his encouragement to enter upon a formal graduate degree
program and his guidance throughout my course of study. His
interest in this research endeavor and seemingly unlimited
capacity to sort order out of chaos have contributed immeasurably
to the tangible results embodied in this paper and the intangible
personal inspiration of its author. I wish to thank Committee
members Dr. Gary L. Cloud, Dr. J. Sutherland Frame, and Dr.
George E. Mase for their help and suggestions. Finally I would
like to express my indebtedness to Dr. Donald J. Montgomery for
his support during my course of study at Michigan State University
and to the Division of Engineering Research for the financial

support during a portion of this research.

ii

Chapter

I.

II.

III.

TABLE OF CONTENTS

ACKNOWLEDGMENTS
LIST OF TABLES
LIST OF FIGURES

LIST OF PRINCIPAL SYMBOLS

INTRODUCTION

1.1 Historical Background
1.2 Problem Statement
1.3 Assumptions and General Solution Outline

FORMULATION 0F PROBLEM FOR FIRST MIXED BOUNDARY
CASE

2 1 Equations of Elasticity

2 2 Boundary Conditions

2.3 Equations for the First Mixed Case

2 4 Solution of the Equations for First

Mixed Case

2.5 Definition of the Biorthogonal Operator
and the Adjoint Equation

2.6 Solution of the Adjoint Differential
Equation

2.7 Summary

FORMULATION 0F PROBLEM FOR SECOND MIXED
BOUNDARY CASE

3 1 General Remarks

3 2 Boundary Conditions

3.3 The Auxiliary Variable, Q

3 4 Equations for the Second Mixed Case

3 5 Solution of the Equations for Second

Mixed Case

3.6 Definition of the Biorthogonal Operator
and the Adjoint Equation

3.7 Solution of the Adjoint Differential
Equation

3.8 Summary

iii

Page

ii

vii

viii

bur—I

14

17

19

21

21
21
22
23

25

29

33
3S

Chapter Page

IV. FOUR-VECTOR FORMULATION FOR THE PURE STRESS CASE 37
4.1 The Odd Problem Four4Vector Formulation 37
4.2 The Even Problem Four-Vector Formulation 38
4.3 Four4Vector Form of the Biorthogonality
Operator 40
V. SATISFACTION OF BOUNDARY CONDITIONS ON THE
SURFACE X = 0 43
5.1 Procedure Outline 43
5.2 The Adjoint Function Arbitrary Constants 44
5.3 Satisfaction of Boundary Conditions for
the Odd Problem 45
5.4 Satisfaction of Boundary Conditions for
the Even Problem 50
VI. DETERMINATION OF THE EIGENVALUES 54
6.1 General Remarks 54
6.2 Determination of the Real Eigenvalues for
the Odd Problem 54
6.3 Determination of the Real Eigenvalues for
the Even Problem 57
6.4 Determination of the Complex Eigenvalues
for the Odd and Even Problems 58
VII. SPECIFIC BOUNDARY VALUE PROBLEMS 64
7.1 General Remarks 64
7.2 The Boundary Term FS 65
7.3 Results and Conclusions 66
BIBLIOGRAPHY 8O
APPENDICES
APPENDIX A. TABULATED EIGENVALUES 82

APPENDIX B. ASYMPTOTIC SOLUTION OF THE ODD
PROBLEM TRANSCENDENTAL EQUATION 89

iv

Table

VI-l

VI-Z

VI-3

VI-4

VII-l

VII-2

VII-3

A-l

A-2

A-4

LIST OF TABLES

Real Eigenvalues and Phase Velocities for the
First Four Modes of the Odd Problem

Real Eigenvalues and Phase Velocities for the
First Four Modes of the Even Problem

Comparison of Low Frequency, Odd Problem,
Complex Eigenvalues with the Static Problem
Eigenvalues

Comparison of Low Frequency, Even Problem,
Complex Eigenvalues with the Static Problem
Eigenvalues

Convergence of Eigenfunction Constants for
Problem 1

Convergence of Eigenfunction Constants for
Problem 2

Approximation of Boundary Function, Problem 1

Roots of Transcendental Equations (6.1-1) and
(6.1-2)
Frequency = 100 cycles per second

Roots of Transcendental Equations (6.1-1) and
(601-2)
Frequency = 1000 cycles per second

Roots of Transcendental Equations (6.1-l) and
(601-2)
Frequency = 6380 cycles per second

Roots of Transcendental Equations (6.1-1) and
Frequency = 10000 cycles per second

Roots of Transcendental Equations (6.1-1) and
(601-2)
Frequency = 31600 cycles per second

Page

59

6O

62

62

70

71

72

82

83

84

85

86

Table

Page
A-6 Roots of Transcendental Equations (6.1-1) and
(6.1-‘2)
Frequency = 63800 cycles per second 87
A-7 Roots of Transcendental Equations (6.1-l) and
(601 -2)
Frequency = 100000 cycles per second 88

vi

Figure

VI-l

VI-2

VI-3

VII-1
VII-2
VII-3
VII-4
VII-5
VII-6

VII-7

LIST OF FIGURES

Phase Velocity Variation for the First Three

Modes of the Odd Problem

Phase Velocity Variation for the First Three

Modes of the Even Problem

Eigenvalues in First Quadrant of Complex Plane
for Frequencies of 1000 and 31600 cycles per

second; a = X + iY

Boundary Stresses on Surface x
The Non-Decaying Wave Modes at
Stress 0 at x = 0, Problem

YY

Boundary Stresses on Surface x
The Non-Decaying Wave Modes at
Stress ny at x = 0, Problem

Boundary Stress Oxyb

Modes at x = 0, Problem 3

vii

X

X

2

and the Non-Decaying Ox

0, Problem 1

= 0, Problem 1

0, Problem 2

= 0, Problem 2

y

Page

61

61

63
73
74
75
76
77

78

79

LIST OF PRINCIPAL SYMBOLS

stress components in Cartesian coordinates.
auxiliary "stress like" variable.

displacement components in Cartesian coordinates.
boundary stress and displacement functions on

the finite edge x = 0.

angular frequency.

time.

density.

Young's modulus.

Poisson's ratio.

propagation constant in the x direction equal
to Zn divided by the wavelength.

2
= 1%2 _ EL (1 _ V2)
a E

d 2
='l\/;2 - £QQ_ (1 + v)
a E

 

 

= (1 + n2)/(\)n2 - v - 2)

= (vn2 + 2n2 - v)/(1 +~n2)

 

v _
1 . . .
= , eigenvector of the first mixed
v2 boundary case.
z7
1
= z , adjoint problem eigenvector for the
2J first mixed boundary case.

viii

= , eigenvector of the second mixed
WZJ boundary case.
L

= , adjoint problem eigenvector for
u2 the second mixed boundary case.
J

 

 

eigenfunctions of the four-vector formulation.

normalization constants.

arbitrary constants of the adjoint functions.

w/a , phase velocity.

\IE/p , phase velocity of infinitely long
waves in a cylinder.
2
1:— (1 - v2).

represents complex conjugate transpose.

represents complex conjugate.

CHAPTER I

INTRODUCTION

1.1 Historical Background

 

In 1853, Barre de Saint-Venant presented his solution for
torsion in long prismatic bars of various cross sectional shapes
(1). In this solution, he was able to satisfy the end conditions
producing the twist in the bar only up to a resultant force and
couple, the distribution of which, he assumed, could differ from
the required distribution in the body of the bar. He postulated
that variations in the distribution of statically equivalent end
loadings must have little effect on the twisted bar except near the
ends. This declaration became the basis for what is now known as the
Saint-Venant principle. In a footnote of his memoir, Saint-Venant
went on to say that the influence of forces in equilibrium acting
on a small portion of a body extends very little beyond the parts
upon which they act (2).

The importance of this principle in the subsequent develop-
ments of the theory of elasticity cannot be over-stated, and the
problem of providing mathematical clarification of the principle and
justification of its use has been the subject of serious consideration
since its enunciation. As a practical matter, analytical stress
analysis of equilibrium systems using classical theory constantly

resorts to the principle by replacing one force system with a

statically equivalent one with the assumption that the resulting
errors exist in a region extending very little beyond the surface
of application. Indeed, without the use of this principle, many
problems would be too complicated to solve.

The earliest solution for a system of forces in equilibrium
on the edge of a plate is attributed to Thomson and Tait (1867),
followed by a more complete solution by Maurice Levy (1877), (3).
M.J. Boussinesq (1885), (4), one of the most distinguished pupils
of Saint-Venant, was able to define the region of local perturbation
resulting from the application of statically equivalent systems of
loads normal to the infinite half Space. This work has been a
standard reference in textbooks as proof of the Saint-Venant principle.
As recently as 1945, von Mises (5) argued that Boussinesq's solution
was for a particular loading, and that Saint-Venant's principle in
its traditional form does not hold true if equilibrant force systems
are introduced tangent to the plane surface. He prOposed to modify
the principle with the introduction of the concept of astatic equili-
brium which requires forces to remain in equilibrium when rotated
through any angle. E. Sternberg (1953), (6) supplied a precise formula-
tion and proof of the SainEVenant principle as modified by von Mises.

The von Mises-Sternberg papers define the Saint-Venant boundary
region for bodies having very general geometries. Recent engineering
requirements have necessitated the determination of the magnitude of
the stresses in this boundary region for specific geometries. Tech-
niques now exist to investigate the stress distribution in the Saint-
Venant boundary region for particular time-independent boundary con-

ditions on the semi-infinite strip (7), cylinder (8), wedge and cone (9).

The investigation of the possible existence of a dynamic
Saint-Venant region is limited. B.A. Boley (1954), (10), using a
simple model, found that a region similar to the static boundary
region exists for slowly applied loads; but as the rate of loading
is increased, the region extends to longer portions. L.w. Kennedy
and O.E. Jones (1969), (11) examined the effect of altering the
radial distribution of statically equivalent pressure step loads on
a semi-infinite cylinder having zero lateral displacement on the end.
They concluded that any differences in time-average dynamic stresses
and strains are negligible at distances greater than five times the
cylinder diameter, and differences in peak values are small at dis-
tances greater than twenty diameters. The effect of a self-equili-
brated load representing the difference between the two statically
equivalent applied loads remains undetermined.

The Kennedy and Jones investigation, as well as other of the
most recent studies in wave propagation, employed transform techniques
which permit asymptotic solutions to the equations of motion at large
propagation distances only for the mixed end boundary conditions of
one stress and one displacement (12). The important Saint-Venant
boundary region with pure end stress conditions specified still re-

quires investigation.

1.2 Problem Statement

 

A semi-infinite strip with stress free edges is end loaded
with time-harmonic self-equilibrated shear and normal stresses. The
decay characteristics of the resulting stress distribution and pro-

pagation modes are investigated through the analytic solution of the

equations of linear elasticity.

1.3 Assumptions and General Solution Outline

The homogeneous, isotropic, linearly-elastic, semi-infinite
strip occupies the region -1 s y s l and x S m. The solution is
based on the linearized equations of elasticity for generalized
plane stress and is obtained by use of an extension of a technique
of biorthogonal eigenfunction expansion developed by M.W. Johnson and
R.W. Little (1965), (7). The boundary conditions for this problem are:

a bounded solution as x approaches infinity

(1.3-l)
stress free infinite boundaries, y = + l

oyy(X, :,1) = oxy(x, :_l) = 0 (1.3-2)

and one of the following conditions on the finite boundary x = 0:

the first mixed boundary condition

Oxx = 0*Xb(y) cos mt
(1.3-3a)
u = u ( ) COS wt
y yby
the second mixed boundary condition
Oxy = okyb(y) cos wt
(1.3-3b)
ux = qu(y) COS wt
the pure stress boundary condition
Oxx = oxxb(y) COS wt
(1.3-3c)

o (y) cos wt

oxy xyb

the pure displacement boundary condition

u = uxb(y) cos wt

(1.3-3d)
u = uyb(y) cos mt .

This problem is formulated to investigate the Saint—Venant
boundary region with pure stress end conditions. In the course of
the development, formulations are obtained which accept the Specifica-
tion of each of the mixed boundary conditions. The mixed boundary
cases are not examined separately because these boundary conditions
are not of immediate interest to this problem. Their development,
however, is a vital step in the formulation of the pure stress problem.

The formulation of the pure stress case involves three major
steps, each of which is discussed separately in Chapters 11, III,
and IV.

In Chapter II, a second-order partial differential vector
equation is deve10ped in terms of the normal stresses Oxx and Oyy'
This development is called the first mixed boundary case because
specification of Oxx and Oyy on the boundary x = 0 is equivalent
to specification of the normal stress Gxx and the displacement u
in the y direction. The stress Oyy by itself is not a boundary
stress on the surface x = 0, but in combination with Oxx allows
specification of uy. The homogeneous boundary conditions on the
infinite edges leads to an eigenvalue problem which is solved for the
eigenvalues and eigenfunctions. A biorthogonality operator is found
which permits the direct calculation of the eigenfunction constants
from the prescribed mixed boundary conditions. As was indicated

earlier, the investigation of the Saint-Venant boundary region is to

be made for pure stress end conditions, not conditions involving

one boundary stress and one boundary displacement. Thus, for this
study, the solutions developed in this chapter will not be used to
solve a particular mixed boundary value problem, but rather, they
constitute an intermediate step in the pure stress boundary formula-
tion continued in Chapters III and IV.

’In Chapter III, a second-order partial differential vector
equation is developed in terms of the shear stress Oxy and a new
function Q. This deveIOpment is called the second mixed case be-
cause specification of Oxy and Q on the surface x = 0 is equi-
valent to specification of the shear stress Oxy and a displacement
ux in the x direction. The solution proceeds in the same manner
described for the first mixed case. Again, the solutions obtained
for the second mixed case are treated as elements of the final
formulation.

In Chapter IV, the results of Chapters II and III are com-
bined to write a series eXpansion of the vector [0 ,o ,0 ,Q].
XX yy xy
The two biorthogonality Operators are combined to produce a
biorthogonality operator which can be applied to the four component

vector.

In Chapter V, boundary functions a

and o
xxb x

are s ecified
yb p ’
and Oyy and Q are written as a series expansion on the surface
x = 0. The eigenfunction constants are obtained by applying the

four-vector biorthogonal Operator. Chapter VII includes a discussion

of three examples.

CHAPTER II

FORMULATION OF PROBLEM FOR FIRST MIXED BOUNDARY CASE

2.1 Equations of Elasticity

 

The governing equations for the first mixed case are derived
from the following linearized equations of elasticity using the gen-
eralized plane stress assumption.

Equations of Motion

 

 

2
BOXX + 3331 = p i_3§ (2.1-1)
ax ay 3:2

2
23:1.+ 3311.: p 2_3x. (2.1-2)
ax ay atz

Stress-Displacement Relations

 

Bux l

g; - E— (oXX - voyy) (2°1‘3)
All

__x.= l _ . -
By E (oyy egxx) (2.1 4)
3:§.1.i:x = Ziktal (2 1 5)
by ' ax E Oxy ' '

2.2 Boundary Conditions

For the first mixed case, the boundary conditions specified

on the finite boundary x = 0 are

oxx = oxxb(y) cos wt (2.2-l)

U =11
y yb

(y) cos wt . (2.2-2)
As has been indicated earlier in Section 1.3, equation (2.1-4) shows

that the condition (2.2-2) is equivalent to the specification of

(2.2-1) and ny on the boundary x = 0.

= cos mt (2.2-3

The infinite boundaries, y = :pl, are taken to be stress free,

O'yy(xs : 1) = OXY(X’ i 1) = 0’ (202-4)

and the solution is required to be bounded as x approaches infinity.

2.3 Equations for the First Mixed Case

The general equations given in Section 2.1 can be reduced
to two equations for Oxx and oyy. Differentiating equation
(2.1-5) once with respect to x and once with respect to y yields
the following expression
3 3 2

a U A U A o
X + ———1—-= Zglfyl» XY . (2.3-1)

2 2
axsy ayax E axay

 

Substituting equations (2.1-3) and (2.1-4) gives

 

 

2 2 2 2 2
5 0 B 0 a G A o a o

ZXX - v -—§11-+ ——§11 - v ZXK = 2(l+v) —;—51-. (2.3-2)
ay ay ax ex 3 5y

Differentiating equation (2.1-l) with respect to x, (2.1-2) with

reSpect to y and adding gives

 

2 2 2
a a 2 BU an A o a o
2_Xt=pa_(x+__r)___>_<1___1x . (2.3-3)
axay 2 ex ay 2 2

at 5x BY

Substituting equation (2.3-3) into (2.3-2) and using equations

(2.1-3) and (2.1-4) yields the first equation.

 

2 2 (1_ 2) 2
a—2(oxx+o)+a—3(oxx+oyy)=gEv EL2(oxx-to)
3x yy By at yy

(2.3-4)

The second equation is obtained by differentiating equation
(2.1-l) with respect to x, (2.1-2) with respect to y, subtracting

and substituting equations (2.1-3) and (2.1-4).

 

 

 

 

2 2
a o a o 2
xx _ yy = Q£1+V) A _ -
2 2 E 2 (OXX ny) (203 5)
BX BY at
Equations (2.3-4) and (2.3-5) can be written in matrix form
28 28 28
A5—2+a—2=N5—2, (2.3-6)
Bx By at
where
2 1 r0 1
xx
A = ’ S =
-l O o
1 yys
2 u
2+v-v -\)(1+\))
= 2
N E
-(1+v) l+v

This matrix equation is the desired first mixed boundary case
formulation.
2.4 Solution of the Equations for First Mixed Case

The boundary stresses (2.2-1) and (2.2-3) are time-harmonic

of frequency w. Thus, a time—harmonic solution form is assumed

10

using separation of variables

3 = V(y)X(x)eiwt , (2.4-1)
where
V1(y)
V(y) =
v2(y)

Substituting this assumed form of solution into equation (2.3-6)

yields
2X 2V 2
Ava—5+Xa‘7'l'w vi=0. (2.4-2)
BX BY
This suggests the possible form X(x) = emX which when sub-
stituted into equation (2.4-2) yields
dZV 2
——-—2 + (sz - a m = 0 , (2.4-3)
dy
or
2
g—%-+-HV = 0 , (2.4-4)
dy
where
2 2 2 '2 2 27
P—,‘§’—-(2+v-v)-2a -1;3’—(v+v)-a
H ='.
2 2 ‘2
- 53%— (1+v) + 01 9-9— as.)
E E J

 

 

Equation (2.4-4) is an ordinary second-order matrix differential
equation with constant coefficients. EXpanding equation (2.4-4)

yields

 

ll

 

 

2
d v1
2 + hllvl + h12v2 = 0 (2.4-5)
dy
2
d v2
+ + = .-
2 h21V1 h22"2 0 (2 4 6)
dy
These two equations may be written in the form
d4 d2 v1
——""l" 'l' —-+ - = . .-
4 (h11 h22) 2 h11h22 h12h21 O (2 4 7)
dy dy v2

The roots of

the characteristic equation obtained from (2.4-7) are:

_ _l”__2 _“w
r = + w/az _ EQ_.(1-VZ) = + am (2.4-8a)
1,2 —- E ._
2 2 7"? _-._ "
r3 = +1 a - —E£— (1+ ) = + an , (2.4-8b)
,4 -' E '-
where
1 \/ 2. ”Quasi—m7“ 2 A
m = 3' a - E (l-v ) (2.4—9)
1 2 lwé 2-”- _..
n = —’ a --—9£— (1+v) . (2.4-10)
a E
The solution of equation (2.4-4) has the following form:
v1 = C1 Slnh amy +C2 cosh amy + C3 sinh any + C4 cosh any
v2 = C5 Slnh amy + C6 cosh amy + C7 Sinh any + C8 cosh any
Substituting these expressions for v1 and v2 into either of
the equations (2.4-5) or (2.4-6) defines four of the constants.
v1 = C1 sinh amy +C2 cosh amy + C3 sinh any + C4 cosh any (2.4-ll)

12

= KC1 sinh amy + KC cosh amy - C sinh any - C cosh any , (2.4-12)

2 3 4

1 + n2

vn2 - v - 2

 

The remaining constants are determined from the boundary conditions.
Satisfaction of the boundary conditions on y = i_l.

(a) On the surfaces y = :.l, the normal stress Oyy is zero
for all x and time t. This implies, from equation

(2.4-1), that v2(:_l) = 0. Thus from equation (2.4-12):

ll
O

KC1 sinh am - C3 sinh an (2.4-13)

I
O

KC2 cosh am - C cosh an - (2.4-l4)

4

Two equations appear as a result of separating the solution
into even and odd functions of y.

(b) 0n the surfaces y = :_l, the shear stress Oxy is zero
for all x and t. The governing equations do not
explicitly contain ny’ so that a relationship between
Oxx’ Oyy’ and Oxy must be constructed. Differentiating

equation (2.1-2) with respect to x and substituting equa-

tion (2.1-5) yields

 

2 2
32a=paz 2am, 3:. 1.15m
5x2 8,2 E xy ay axay

Differentiating this equation with respect to x and sub-

stituting equation (2.1-3) yields

 

2 2 2 2
[5—_20(1+\D az]aoxv=_[a__2__g_\;a_z_]§fzx_
E

2
ex E at 5x ax at ay
2 ac
- 9 3— i. (2.4-15)
E 2 by
at
50
If C is constant at y = + 1, then -—§l-— 0 and
xy BX
equation (2.4-15) becomes
2 2 BO 2 50
a.— _ E2.a__. __11.+.Q.a__.__§£.= 0 at y = + 1 .
2 E 2 By E 2 By —
ax at . at

Using the assumed solution form from equation (2.4-1) gives

(2.4-16)

E

d +
[ 2 + u 2] v2(il) - U)2 dv1(_l)
y E dy

Substituting equations (2.4-11) and (2.4-12) into equation

(2.4-16) and separating into even and odd functions of y

 

 

yields
2 ’2 K 2
C am(a K.+ E£_ - —2EQ—)cosh am
1 E E
2 2
+ C3an(-a + ”w + 39w )cosh an = 0 (2.4-17)
E E
and
2 .2 K 2
C am(a K + Efl— - —22£—)sinh am
2 E E
2 2
+ Caan(-a +19; + vgw )sinh an = o . (2.4-18)

The solution of equations (2.4-13) and (2.4-17) yields the

transcendental equation for the odd eigenvalues.

tanh am _ 4mn

_ . (2.4-l9)
tanh an (1+n2)2

14

In a similar manner the equation for the even eigenvalues

is obtained by solution of equations (2.4-14) and (2.4-l8).

tanh an 4mn
.__..___ = _—_———— 2. -2
tanh am ( A 0)

(1+1?)2

The transcendental equation, (frequency equation), (2.4-l9)
is used to define the eigenvalue a when the eigenfunctions
v1 and v2 are odd in y, and equation (2.4-20) is used
when these eigenfunctions are even.

The eigenfunctions divide into even and odd functions of

the form:

K sinh am

Ov1 = C1 (Sinh amy + —-;E;E—;;'31nh any)
_ . K sinh am .
Ov2 C1 (K Sinh amy -—;E;E—;;-51nh any)
(2.4-21)
e 1 2 amy cosh an COS any)
_ _ K cosh am
ev2 — C2 (K cosh amy -—:;;E—EE-cosh any) .

The constants C1 and C2 are determined from the boundary con-

ditions on the remaining surface x = 0.

2.5 Definition of the Biorthogonal Operator and the Adjoint Equation

 

Consider Vr as the solution vector of the differential
t
equation (2.4-3) associated with the r h eigenvalue ar'

2

d V
2

"55 = (arA - wZN)Vr (2.5-1)
dY

Premultiplying equation (2.5-1) by the complex conjugate transpose

of some arbitrary vector function of y, ZS, and integrating from

15

 

 

         

y = -l to y = l yields(1)
2 z
1 d V 1 31
+ + 2
I Z 21‘ dy = by Z (a A - sz)V dy , Z =
_1 s d _1 s r r s z
y 82
Integrating the left side by parts gives
+ avr dz:v1 1 .122: 1 2
__ _ __ = A _ . _
s dy dy _1 dyz [182+ (a u) 2% dy. (2 5 2)

 

. . . . th
ConSider the arbitrary vector function ZS as being the 3

solution vector of the following differential equation, termed

the adjoint equation(2)
2
d Z
+ +
-—-25 = (azA - wZN )z , (2.5-3)
dy s s

where the eigenvalue a8 is determined from the same transcendental
equation obtained for the equation (2.5-l). It remains to be shown
that this condition is satisfied.

+
Premultiplying equation (2.5-3) by Vr and integrating

from y = -l to y = 1 produces
2
l d Z
+( A+
J‘v+——dy=f1r(av-2 -wZN+)Zdy
r
-l dy

Taking the complex conjugate transpose of this equation,

1 d22:1

2 y=J‘1::2r2(aA-wN)de,

     

 

-1 dy

 

+
(1) () represents complex conjugate transpose

(2) "'

( ) represents complex conjugate

l6

and subtracting from equation (2.5-2) yields

 

 

+ l
+:v—rufiv =(2- 2)IZ+AVd (25-4)
8 dy dy r _1 01r 0[s [I s r y ' °

 

Expanding the left side of equation (2.5-4) gives

 

 

 

- - l
d d
E vrl + ; dvr2 _ dzsl v _ 232 v =
$1 dy $2 dy dy rl dy r2 _1
( 2 2)1 Z+AV d 2 5 5
0’r - 0[s [l s r y ° ( ° - )

The boundary conditions on the surface y = i_l in Section

2.2 for the original differential equation give:

(a) vr2(;t 1) = 0

(b) From equation (2.4-l6)

dvr1(:l) 2 E dvr2(tl) dvr2(il)

_.____—_=_ +V
d a 2
y r pm dy dy

where it is noted that the eigenvalue ar appears in the second

boundary condition. Substituting these values into equation (2.5-5)

 

 

yields
dv dv dv d2 1
[5 -a2 —§—'-—£g + v r2 + g -——E; -._4§l v =
2
81 r pm dy dy 32 dy dy r1 _1
2 2 1 +
(”r - “5)i ZSAVrdy . (2.5-6)
-1

Imposing the following boundary conditions on the adjoint problem:

d2 (+1)
31" _
(8) §;—"“‘ - 0
(b) 2 (+1)-2L5 (1)-2 (+1)
2 -' as 2 51 V 31 — ’

l7

and substituting these into equation (2.5-6) yields

1

 

 

dv l
2 2 - E r2 _ 2 2 +
(as - ar)zsl 2 dy — (Ur as)j ZsA'Vrdy ’
pm -1 -l
or
1 dv
2 2 + - E r2
- z A + '———-——-— = . 2. -
(“r as)[j1 S vrdy 231 2 dy ] 0 < 5 7)
- pm -1

Thus, biorthogonality is defined by equation (2.5-7) for a2 # a:.
r

2.6 Solution of the Adjoint Differential Equation

The adjoint differential equation (2.5-3) can be written as

2
§—%-+ n+2 = 0

dy

: (2.6-l)

where the matrix H is defined with equation (2.4-4). The solution
of equation (2.6-l) is obtained by the same procedure used in the

solution of equation (2.4-4). EXpanding equation (2.6-l) gives

 

 

2
d 21 _ _
+ + = -
2 h1121 h2122 O (2'6 2)
dy
dzz2 _ _
dyz + hlzzl + hzzz2 = o . (2.6-3)

These equations can be written in the following form

d4 _ d2 21

— + . —— + - - - - - = . . _"
4 (hll + h22) 2 hllh22 h12h21 z 0 (2 6 )

dy dy 2

The roots of the characteristic equation obtained from equation

(2.6-4) are:

 

V/ 2 2 2
r12=i a -9-‘”—(1-v) =-_e_a (2.6-5a)

l8

 

 

 

2
= ‘2 2 0‘) = — -
r3,4 :\/a - i—E (1+v) i an , (2.6 5b)
where
- l 2 Q 2 2
m = :' a - w (1‘V ) (2°6‘6)
a E
- 1 2 29 2
n = a a - Eu) (1+\)) . (2°6-7)

The complex conjugate of the adjoint functions representing the

solution of the adjoint equation are:

1 Blsinh amy + B cosh amy + B sinh any + B cosh any

NI
II

2 3 4

B

sinh amy + B cosh amy + B sinh any + B cosh any .

5 6 7 8

Substitution of these functions into either equation (2.6-2) or

(2.6-3) defines four of the constants.

sinh any - KB cosh any (2.6-8)

N
II

B sinh amy + Bzcosh amy - KB

1 1 3 4

22 Blsinh amy +'B2cosh amy + B3sinh any + Bacosh any (2.6-9)

Satisfaction of the Adjoint Problem Boundarngonditions.

 

In defining orthogonality in Section 2.5, the following two
boundary conditions were imposed on the adjoint problem:

délen
“0 37— : O
2 —E—2- 210:1) - v£1(:1) .
pw

(b) 5,011) = a

Substitution of equations (2.6-8) and (2.6-9) into (a) and (b) and
separating into even and odd functions of y yields the following

two transcendental equations which are identical to those defining

19

the eigenvalue a for the original governing equation (2.4-4).

For the adjoint functions odd in y,

tanh am _ 4mn

2
tanh an (1+n2)

For the adjoint functions even in y,

tanh an = 4mn
tanh am (1+n2)2

Similar to the eigenfunctions, the adjoint functions divide into

even and odd functions of the form:

- m cosh am

0 1 B1 (Sinh amy - n cosh an Sinh any)

N
ll

- , m cosh am
= + _ _.____.. °
022 B1 (Sinh amy Kn cosh an Sinh any)
(2.6-10)
- __ fl sinh am
e21 B2 (cosh amy - n EEEE—Eg'cosh any)
5 = B (cosh amy + EL- filflh_gfl cosh any)
e 2 2 Kn siM1an
2.7 Summary

The y-dependence of the solution is carried by the eigen-
functions (2.4-21) of which there are an infinite number correspond-
ing to the roots, a, of the transcendental equations. The correct
solution of the governing differential equations requires the
summation of the distinct solutions. Separating the solution into

even and odd functions of y, this sum may be expressed as

 

 

20

 

 

co
'- 1< ' h
sinh amy + __§%EE_QE sinh any
= C1 K sinh e O r
C r K sinh amy - —_;IHH€32 sinh any
yy __ j (r)
r=l
(I)
_
Fcosh amy + E_EEEE_QE cosh any
COS an 1(Q’X+UJC)
+ C e e r _
2r K cosh m
K cosh amy -----41- cosh any
cosh an j .
L - (r)
r=1 (2.7-1)
The constants C1r and C2r are obtained formally by the specifica-
tion of Oxx and oyy on the surface x = 0 and the application

of the biorthogonality operator (2.5-7).

This solution and the

solution deve10ped in Chapter III will be combined in Chapter IV

in a formulation which allows specification of pure stress boundary

conditions, a and o , on the surface x
xx xy

0.

CHAPTER III

FORMULATION OF PROBLEM FOR SECOND MIXED BOUNDARY CASE

3.1 General Remarks

In this chapter, the equations given in Section 2.1 are used
to derive governing equations for specifications of Oxy and the
displacement ux on the finite surface x = 0. To permit formulation
of the problem in this manner, a new variable Q is introduced, and
the governing equations involving only the stress oxy and the
"stress-like" variable Q are obtained. It will be shown in Section
3.3 that the specification of oxy and Q on the surface x = 0
is equivalent to specification of ny and uX up to a rigid
body displacement.

As in Chapter II, the solutions of the second mixed case
developed in this chapter will not be used to solve a particular
mixed value problem, but will be combined with those obtained in
Chapter II for specification of the stresses ka and Oxy on

the surface x = 0.

3.2 Boundary Conditions

 

For the second mixed case, the boundary conditions specified

on the finite boundary x = 0 are

qu = okyb(y) COS wt (3.2-l)

21

22

ux = uxb(y) cos wt (3.2-2)

It will be shown in Section 3.3 that condition (3.2-2) is equiv-

alent to specification of (3.2-1) and Q on the surface x = 0.

D
II

Qb(y) cos wt (3.2-3)

The infinite boundaries, y = i_l, are stress free,

:+1
o y(X _ )

x + l = 0 , 3.2-4
y o y( , _ ) ( )

X

and the solution is required to be bounded as x approaches

infinity.

3.3 The Auxiliary Variable, Q

 

To construct equations which do not involve o x and Oyy’
x

the variable Q is introduced and is defined

 

2 .
so a u

a9.= XX - vp._.J£ (3.3-1)

ax ay 3:2

with the requirement that it be bounded as dimension x approaches
infinity.
Differentiating equation (2.1-3) with respect to y and

substituting equation (2.1-2) yields

 

 

2 2
a ux _ 1. 80xx 8 ”1 BOXX
- - v p 2 - . (3.3-2)
BXBY E 8y at ax

Substituting equation (3.3-l) into equation (3.3-2) gives

ax By E ax AX

 

 

 

 

Integrating with respect to x gives

23

_ l
ST-E (Q +voxy) + f(y) ,

and substituting equation (2.1-5) yields

Slum,
E

l
ax - E (Q + vsxy) — f<y> . (3.3-3)

xy

If uy is bounded as x approaches infinity, f(y) may be taken
to be zero. Equation (3.3-3) may be written

au
——§-— 1 (Q + vo )
Xy

ay — E (3.3-z.)

Specification of Q and oxy on the surface x = 0 is equivalent

to specification of Oxy and uX up to a rigid body displacement.

3.4 Equations for the Second Mixed Case

 

Differentiating equation (3.3-3) with respect to y and

substituting equation (2.1-4) into the result gives

 

BC 60 an
59+—¥1-v x"-<2+\»>—-l" =0-
By BX BX 8y

Differentiating again with respect to y and using equations (2.1-1)

and (2 .1 -2) gives

2 2
2 2 u
a.9._ 9—331._ 2 §_EEX.+ p 5.. (iii V EQE) = 0
2 2 2 5X ay
ay ax sy at

Substituting equations (2.1-5)

2 2
2 2 u 2 2 711
3.9.- 2.331._ 2 2_3:1.+ a__.2_i - a__.211:22. + a__.l_x = 0
\J O' \)
2 2 2 P 2 P 2 E 9 2
BY ax ay 5t 5X at Xy at 9x

and (3.3-l) yields

 

 

2 2
2 2
a.9.-2&1-22__°_xy_+pa_a_ui_va 2(1+v)
2 2 2 25x 9 2 E xy
ay ax ay t at
2
2
ECXX_E_02.=O
axay ax

Using equations (2.1-l) and (2.1-5) yields the first equation for

d .
Oxy an Q

2 2 2
O' 2 2 2 0
§_§§Y.+ 3 §_;EY.+ BJ% _ aJ% =.E§LZX_L p i_§§1. (3.4-1)
E
Bx BY BX BY Bt

The second equation for Oxy and Q is obtained by adding
the derivative of equation (3.3-l) with respect to x to the deriv-
ative of equation (2.1-l) with respect to y and substituting equa-

tions (2.1-5) and (3.3-4) into the resulting equation.

2
2 B o 2
1+ ~
B_92.+_§>LY.=ELE__ELB__2(Q _vOX) (3.4-2)
ax by at y

Equations (3.4-l) and (3.4-2) may be written in matrix form

 

 

2 2 2
D 9—%-+ a_%.= M a—%-, (3.4-3)
BX BY Bt
tflmre
O l1 0
xy
D = , T =
-1 2 Q
--V 1-
-(2+\)) 3 .

 

 

This matrix equation is the desired second mixed boundary case

formulation.

25

3.5 Solution of the Equations for Second Mixed Case

 

The boundary conditions (3.2-1) and (3.2-3) are time—har-

monic of frequency w. A time—harmonic solution form is assumed

1'.ch iwt
T = W(y)e e , (3.5-1)
where
w1(y).
W(y) =
w2(y)

Substituting this assumed form of solution into equation (3.4-3)

gives
2
(13+ (032M " 021))“, = O 9 (3.5-2)
dy
or
2
__dV2’+Gw=o , (3.5-3)
dy
where
P 2 2 T
2
-%—E<v+v2> saw-a
G =
2 2 2 2 2
_w_2E (2+3\)+\))+Q/ w—JE (3+3v)-201

 

 

Expanding equation (3.5-3) yields

2
d w1
-——§—-+ gllw1 + glzw2 = O (3.5-4)
dy
dzw2
;;§—-+ g21w1 + gzzw2 = O . (3.5-5)

These two equations may be written

26

d4 d2 w1
+ - = -
4 + (311 + g22) 2 g115322 g12’321 0 ' (3'5 6)

The roots of the characteristic equation obtained from equation

(3.5-6) are identical to the roots of the first mixed problem of

Chapter II:

 

 

J2 032 2

11,2": 6* "LE(1"’) =ia‘“
2 22m2

‘34:: 0‘ ' E 0+") =i0’“

The solution of equation (3.5-3) has the following form:

ll

w1 Clsinh amy + Czcosh amy + C3sinh any + C cosh any

4

w2 Cssinh amy + C6cosh amy + C sinh any + C cosh any

7 8

Substituting these expressions for w1 and w2 into either of

the equations (3.5-4) or (3.5-5) defines four of the constants

yielding:

w1 = Clsinh amy + Czcosh amy + C3sinh any + C4cosh any (3.5-7)
w2 = Clsinh amy + Czcosh amy + RC3sinh any + RCacosh any , (3.5-8)
where

 

The remaining constants are determined by specification of the

boundary conditions on the surfaces y = i_1 and x = 0.
Satisfaction of the boundary conditions on y = :.l.

(a) On the surfaces y = i;l, the shear stress Oxy is zero

for all x and t. This implies, from equation (3.5-1),

27
that w1(:l) = 0. Thus, from equation (3.5-7):

(3.5-9)

ll
0

C sinh am‘+ C3sinh an

1
(3.5-10)

I
O

Czcosh am + C4cosh an -

(b) On the surfaces y = i.l, the normal stress gyy is zero
for all x and t. Because the equations do not explicitly
contain 0 , a relation between 0' , Q, and 0 must
YY YY XY
be constructed. Differentiating equation (3.3-3) with
respect to y and substituting into equation (2.1-4) gives
a9. 5" a"x
+ 41— —’9$ - (2+v)—l = 0 . (3.5-11)
BY Bx vBX BY
Differentiating equation (2.1-l) with respect to x and substitut-

ing from equation (2.1-3) yields

 

2 2
a o B o 2

2x“ + -——§1-= 9-5—5 (0 - v0 ) . (3.5-12)
ax axay E at xx yy

Multiplying the derivative of equation (3.5-12) with respect to

x by Poisson's ratio, v, and substituting equation (3.5-11) gives

2 ad 2 2
a..- _ 2a..__u a.._ a__a9.

ax at2 5x .5x at “y
+ [-257+§ (2+v) EL—z ——’$l=o . (3.5-13)
Bx BC BY
50'
If 0 is constant on the surfaces y = i_l, then g;11 = 0 and

equation (3.5-13) becomes

Using the assumed solution form from equation (3.5-1) gives

28

2

2932.
[m

dw (:1)
+ [2a2 - §'(2+v)w2] E;l-—- = 0 . (3.5-14)

2
dy

(W (‘11)

 

Substituting equations (3.5-7) and (3.5-8) into equation (3.5-14)

and separating into even and odd functions of y yields

 

 

2 2
C3am.[(-a2 + 99-")R + 2a2 - (2+v) QQ- cosh an
E E
2 2
+ Clam[:a - (1+v) QfiL. cosh am = 0 (3.5-15)
and
2 2 2 '2
C40n [‘-a +-QEL‘)R + 2a - (2+v)'2£— ] sinh an
E E
2 -2
+ Czam [a - (l+\)) Li:— J sinh am = O . (3.5-16)
Solution of equations (3.5-9) and (3.5-15) yields
tanh an 4mn
—-———— = —-——-— . 05"
tanh am 2 2 (3 17)
(1+n)
Solution of equations (3.5-10) and (3.5-16) yields
tanh am 4mn
'—--- = --—-—-. (3.5-18)
2
tanh an (1+n2)

The transcendental equation (3.5-17) is used to define the eigen-
value a when the eigenfunctions w1 and w2 are odd in y.
Note that this equation is identical to equation (2.4-20) which
defines the eigenvalue a for the first mixed problem when the
eigenfunctions v1 and v2 are even in y. The transcendental
equation (3.5-18) defines a when the eigenfunctions w1 and w2
are even in y. Likewise, this equation defines the eigenvalue
for odd functions of v1 and v2. This relation is equivalent to

the fact that if Oxx and Oyy are even functions of a given

variable, 0*y is odd in that variable.

29

It is desirable at this point to define as the even problem
the one which has Oxx and ny in terms of functions even in the
y-variable and Oxy in terms of functions odd in the y-variable.
The odd problem is defined as the one with oxx and Oyy odd in
the y —variable and Oxy even in the y-—variable. The presub-
scripts I'o" and "e" used to designate odd and even will now be
used to identify quantities related to the odd and even problems
respectively. Thus, for example, for the odd problem the presub-
script "0" is applied to the eigenfunctions of the first mixed case
which are odd in the y..variable, and to the eigenfunctions of the
second mixed case which are even in the y..variable.

The eigenfunctions of the second mixed case divide into

sets for the even and odd problems of the form:

_ . sinh am .
ew1 — C1 (Sinh amy EEKE‘EE'Slnh any)
w =C (sinh _R_§_£1£_gm_sinh )
e 2 1 amy sinh an any
(305-19)
_ cosh am
Ow1 — C2 (cosh amy BEEF—EH cosh any)

R cosh am
cosh an

S
I

o 2 — C2 (cosh amy cosh any)

The constants C1 and C2 are determined from the boundary con-

ditions on the surface x = O.

3.6 Definition of the Biorthogonal Operator and the Adjgint Equation

Consider wr as the solution vector of the differential

(2
equation (3.5-2) associated with the r h eigenvalue ar.

3O

 

         

r = 0 (3.6-l)

Premultiplying equation (3.6-l) by the complex conjugate trans-
pose of some arbitrary vector function of y, Us’ and integrating
from y = -l to y = 1 yields

1 dzwrd 1 Usl

+
f U fly (a: D - w 2M)w dy . U =
S S
-l dy2 - u

 

82
Integrating the left side by parts gives

1 l d2U+ l

+ 2 2
+ j‘ —— ZS--wrdy J" Us(ozrD - w M)Wrdy - (3-6‘2)
-1 -1 dy2 -1

+dwr an:
UsF-er

 

 

. . . th .
ConSider the vector function US as being the 5 solution vector

of the following differential equation, termed the adjoint equation

 

dZU
2+ 2+
28 = (& D ‘ w M )U 3 (3°6‘3)
dy s s
where, as will be shown later, the eigenvalue a is determined

L

from the same transcendental equation obtained for the equation
(3.6-l).
. . . + . .
Premultiplying equation (3.6-3) by wr and integrating

from y = -l to y = 1 gives

1 d U 1
+ s _ +_2+ 2+
j w 2 dy —-£lwr(asn - w M )Ude

Taking the complex conjugate transpose of this equation,

1 dzd: 1

+ 2 2
28 - jIUS(aSD - w M)wrdy ,

 

     

 

-1 dy

and subtracting it from equation (3.6-2) yields

31

1

+ 2 2 +

——E- S = (a -a)funw dy. (3.6-4)
r S '18 r

dw dU )
r

 

 

s dy dy _1

Expanding the left side of equation (3.6-4) yields

 

 

- - l
d
a wr1 + a dwrZ _ dusl w _ dusZ w ) =
51 dy $2 dy dy r1 dy r2 _1
l
2 2 +
(qr - as)§lUSDwrdy . (3.6-5)

The boundary conditions on the surface y = :_1 may be

written in the form:

(a) ”ti“- 1) = o

2 dw (+1)
2 DLlL __r£:_ 2 _ 9. 2 __ =
(b) [-ar '1" E de + [201' E (2+v)u) O ,

where it is noted that eigenvalue ar appears in the second

boundary condition. Substituting these values into equation (3.6-5)

 

 

gives
dw 2a2 - 2-(2+v)w2 dw dd 1
[ - r1 J r' E rl 52 w =
u ' 2 "“‘ "“-‘
sl dy 32 - 2 + w dy dy r2 _1
0‘,r E
2 21 +
(a - oz )jU D w dy. (3.6-6)
r s _1 s r

The following boundary conditions are imposed on the adjoint func-

 

tions:
( ) —-—-dusz(i1) = o
a dy
_ 201S - gnaw,2 _
(b) u (+1) = u (+1)
81 -' 2 32 —

Substituting these boundary conditions into equation (3.6-6) gives

32

 

 

 

 

 

2 Q 2 2 Q 2 1
20/3 - E (2+vhn _ dwr1 - 2ar - E (2+v)w a dwrl —
2 $2 dy $2 dy _
'02 + QQ_ -a2 + E'wz -l
s E r E
l
2 2 +
(qr - as)f UsDwrdy ’
-1
or
2
2 2 1 + 29$- dw 1
(a -a)jUDwdy- E 6 —-El =0.
r s s r 2 2 52 dy
-l 2 gm 2 gm -1
(‘Q8 + E )(-ar + E )
(3.6-7)

Biorthogonality is defined by (3.6-7), but it is not in a usable

form because the eigenvalue ar appears in the Operator. Multiply-
2

ing equation (3.6-7) by (-a: + 2%L2

 

 

l
l 2 2mg— dw
2 2 + 2 g E -
(a - a ) I U (-a + w )Dw dy - '—-—-————— u r1 = O ,
r s _1 s r E r 2 2 52 dy
-a +.E£_ _1
s E
and recalling from equation (3.6-l)
dzwr 2
aerr = 2 + U.) er ,
dy

the term under the integral sign can be written in the following

form

2
2 dw 2+

2
+ + +
U(-a2+9—‘”)Dw=9——‘”UDw -U-—-r-wUMw
s r E r E s r s dyz s r

After these substitutions, the biorthogonality can be defined as

 

 

2
l d w
J‘ (U+Dw --§—-U+——5-E-U+Mw dy
_1 r 2 s 2 p s
l
dw
- v u r1 = 0, a2 f 02 .
2 2 $2 dy 1 r s
.22. _ a '
E S (3.6-9)

33

3.7 Solution of the Adjoint Differential Equation

The adjoint differential equation (3.6-3) can be written as
+
d U + c U = o (3.7-1)

where the matrix C is defined with equation (3.5-3). The solution
of equation (3.7-l) is obtained by the same procedure used in the

solution of equation (3.5-3). Expanding equation (3.7-1) gives

2

d ”1 _ _
——--2 + gllu1 + g21u2 = O (3-7‘2)
dy

2

d u2 _ _
;;§-'+ g12”1 + g22”2 = O ' (3'7’3)

These equations can be written in the following form

d4 - - d2 — - - - u1
( 4 + (g11 + gzz) 2 + g11322 ' 812821 = 0 '
dy dy u2

The roots of the characteristic equation are identical to those

obtained for the adjoint equation of the first mixed problem:

2 2 2 -——
_ + - _ BE. _ _
1,2 ._ V<y E (1 v )

\/ 2 2ng —-
r3,4 i- a ‘ E (1+V) i-Q“

 

H'-
3

 

The complex conjugate of the adjoint functions representing the

solution of the adjoint equation are:

1 " BISInh amy + B

C
I

2cosh amy + B3sinh any + Bacosh any

112 BSSinh amy + B6cosh amy + B7Sinh any + B8COSh any

34

Substitution of these functions into either equation (3.7-2) or

(3.7-3) defines four of the constants, thus:

C
l

1 “ -RBlsinh amy - RBzcosh amy - B33inh any - B cosh any (3.7-4)

4

C
II

2 Blsinh amy + Bzcosh amy + B3sinh any + B cosh any . (3.7-5)

4

‘_atisfaction of the Adjoint Problem Boundary Conditions.

In developing the biorthogonality operator in Section 3.6,
the following two boundary conditions were imposed on the adjoint
functions:

c182 (:1) _

(a) - 0

dy

2

2
(b) (-oz2 + %—)Gl(:1) = [2a -EP- <2+v>w2352(:1>

Substitution of equations (3.7-4) and (3.7-5) into (a) and (b) and
separating into even and odd functions yields the following two
transcendental equations which are identical to those defining the
eigenvalue a for the original governing equation (3.5-3). For

the even problem, the adjoint functions are odd in the y..variable,

and the corresponding eigenvalues are the roots of

tanh an 3 4mn
tanh am (1+n2)2

For the odd problem, the adjoint functions are even in the y-

variable, and the eigenvalues are the roots of

tanh am = 4mn
2 2
tanh 0m (1411 )

35

The adjoint functions divide into sets corresponding to
the even and odd problem in a similar manner as the eigenfunctions.

They are of the form:

- . m cosh afl .
= .. + —
eu1 B1 ( R Sinh amy n cosh an Slnh any)
5 = B (s inh am - fl SEE—L” sinh )
e 2 l y n cosh an any
(3.7-6)
- m sinh am
= .. + -— __
Ou1 B2 ( R cosh amy n sinh an cosh any)
m sinh am

cosh any) .

C. I
I

o 2 - B2 (COSh amy - n sinh an

3.8 Summary

The y-dependence of the solution is carried by the eigen-
functions (3.5-l9) of which there are an infinite number correspond-
ing to the roots, a, of the transcendental equations (3.5-17) and
(3.5-18). The solution of the differential equation requires the
summation of the distinct solutions. Separating the solution

into even and odd functions of y, this sum may be expressed as

 

 

 

 

co
'— h
oxy cosh amy - EEEE—gg'cosh any
' t
= C el(oarx+w )
Q 1r cosh amy - B—EEEflLiflfl cosh any
COSh an (r)
r=l
m r“sinh amy - Eiflh—gm sinh any
sinh an i( a x+wt)
+ C2r R sinh am e e r
Slnh amy - sinh an Slnh any (r)
_' (3.8-l)

36

The arbitrary constants C11. and C2r are obtained formally by
the specification of Oxy and Q on the finite edge x = O and
the application of the biorthogonality operator (3.6-9). This
solution will now be combined with the first mixed case to form

a general solution for any acceptable boundary conditions. As

indicated earlier, the stress boundary problem is the one of most

importance.

CHAPTER IV

FOUR-VECTOR FORMULATION FOR THE PURE STRESS CASE

4.1 The Odd Problem Four-Vector Formulation

In Chapters II and III, eigenfunction expansions were
developed for Oxx and Oyy’ and for Oxy and Q, respectively.
These solutions are given by equations (2.7-1) and (3.8-1),

and the functions in the odd problem have the following form:

m . i( a x + wt)
_ , K Slnh am . o r
okx - rE£C1r(Slnh amy + —-;EEE—;E Slnh any)(r)e
m K sinh QE i(oarx + wt)
Oyy = rElclr(l< Sinh amy - sinh an Sinh any)(r)e
. (401-1)
00 cosh am 1(oat-x + wt)
Oxy - rEIC2r(COSh amy - m cosh any) (r)e
00 l( a X + LDC)
_ R cosh am 0 r
Q rEICZr(cosh amy -":;;;:7;H cosh any)(r)e

In this chapter, the four solutions are combined to form
a four-vector [o ,o ,0' ,Q]. Substituting the rth term of the
XX yy xy

solutions (4.1-1) into the coupled differential equation (3.5-11)

60 50 go
a9+7u - v.33}: _ (2 + v) __JLX = 0 (3.5-11)
By a a by repeated
yields the following relation between the constants C11. and C21.

37

38

= ( 1 2m ) C
2r \m2 - v-Z (r) lr

The desired odd eigenfunction expansion of the four-vector becomes

 

 

 

 

 

o Foul
xx
0‘ on2 i( arx + mt)
Y)’ = C n e O , (4-1'2)
a o r o 3
XY
Q oné
L .J
o b A r=1 (r)
where
K sinh am .
= ' +---—--——- h
onl Slnh amy sinh an Sin any
_ , K sinh am .
GHQ — K Slnh amy - sinh an Slnh any
_ 1 2m cosh_gm
on3 ——§—-— (cosh amy ESEE-an cosh any)
vn -v-2
_ i 2m R cosh am
onh ‘——§—-- (cosh amy -—;3;E7;; cosh any)
vn -v-2
_ 1 + n2 _ vn2 + 2n2 - v
R ‘ “—2—— ’ R - 2
vn -v-2 1 + n
a are the roots of %EEE_QE = __3E%_§. .
0’ <1+n >

4.2 The Even Problem Four-Vector Formulation

The even problem formulation involves the following variable

forms:
no 1(ax+wt)
_ K cosh am e r
Oxx - r£1C3r(cosh amy -—:;;E-;;-cosh any)(r)e
m i( a X + wt)
Kcosh am e r
= -—>_-——-————— h
0yy 2 C3r(K COSh amy cosh an COS any)(r)e

r l

(4.2-1)

39

co . l( 0/ X + wt)
0‘ = 2C (sinh amy -w—sinh any) e e r
xy r=1 4r sinh an (r)
w R sinh am i(earx + mt)
Q = rEIC4r(S lnh amy - m Slnh any)(r)e

Substituting the rth term of the solutions into equation (3.5-11)

yields

C

_ i 2m
C _ ( 3r

4r vnz-v-Z )(r)

The desired even problem eigenfunction expansion becomes

 

 

 

 

(I) ..
- l
okx {enl
a enz i( a x + wt)
yy = c en3 e e r , (4.2-2)
e r n
ny e 4
Q j (r)
e - r=1
where
= cosh +-1L£§EEL£EE cosh n
enl amy cosh an a y
_ K cosh am
eflz — K cosh amy -—ES;E—;;'cosh any
_ i 2m , sinh am .
en3 ~ :;§:;:; (Slnh amy sinh—an Sinh any)
_ i 2m , R sinh am .
€114 “ an-v-Z (3 1nh amy - -—_—'—S inh om Sinh any)

K and R are defined with equation (4.1-2), and a are the

roots of

tanh an = 4mn
2
tanh am (1+n2)

40

The general solution is the sum of the expansions (4.1-2)
and (4.2—2). However, every function can be replaced with the sum
of an even and odd function, and the series expansions will be

used separately for convenience.

4.3 Four4Vector Form of the Biorthogonality Operator

 

The biorthogonality for the first mixed case is given by
(2.5-7).

1 dv
+ —

I 2 AV dy + 4&5 z
S r

 

2 2
—0,ar#as
-1

 

The eigenfunctions nri of the four-vector form equal the eigen-
functions vri of the two-vector form divided by a constant, so

that the biorthogonality is not violated.

1
1 2 1 an
‘ - E - 2 2 2
I [231’282] “r1 dy +‘__§ zsl d;£_ = 0’ Ur # as
-1 -1 0 “r2 pw -1
- (4.3-1)

The biorthogonality for the second mixed case is given by

(3.6-9).
2
l d w
J" (”:Dwr ‘ "E? U: 3;): ' E ”:er>dy
-1 pm
v _ dwr1 l — 2 2
_-—TE—-——— u 2 E_-— - 0, 0 # a
w 2 s y 1 s r
£__ - a _
E s

The functions “ti equal the two-vector form eigenfunctions wri

multiplied by a constant, and the biorthogonality is not violated.

41

l 2
I[JJ]DW3_E[u-]d Tlr3
_ 31’ 52 2 81’ 52 d 2
r4 pw y nr4
(E - - an v - dan 1
- EDEUSI’USZ] M dy - 2 2 ”32 E;——' - O,
nr4 2%L'- a -l
2 2
as = or (4.3-2)

Combining the biorthogonality relations (4.3-1) and (4.3-2) yields

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 F 2 1 o o" .an-T
I1 [zsl’ZSZ’usl’u32] '1 0 0 O nr2
0 0 O 1 “r3
L O O -l 2 nrh
r 1 r
“New
- 2 [zsl’ZSZ’usl’GSZJ 2 “r2
pm 0 o 1 o dy an
_o 0 o 1 “r4
’- _ ' q)
0 0 0 0 “r1
- - - - o o o o
‘ [231’232’usl’u32] 2 “r2 ) dy
0 0 -(v:v ) (1+v) “r3
0 0 -(v +3v+2) (3+3v) “r4 /
1 - -
+ E 5 d,an _ v - (”113 = 0 2 if 2 (4 3.3)
pwz sl dy wz 2 U32 dy ’ 0s 0[r ° °
E - Q’s ‘1

The adjoint functions 231, defined by equations (2.6-10),
and usi, (3.7-6), are determined up to an arbitrary constant for
each set. The determination of these constants is explained in
Section 5.2.

The four-vector form of the biorthogonality operator Q

can be written as

42

'0 0 0 0
+ + E + +
- [Zs’Us] [O F] - (;)[Zs’Us] [O M_

 

 

J 0" J 0
+ + + +
+ [28413] [0 PJ 6(y-1) - [28418] [0 P] 5(y+1) , (4.3-4)

where matrices A, D, F, J, M, and P are defined as

 

 

2 ..
2 1 O 1 E 51—2- O
A = _1 0 , D = _1 2 , F = ——2- dy 2
am _d__.
O dyzg
0 §_ -v 1 O O
J = £— y M - ”V P - V
,2 0 0 ’ E “(2+V) 3 , 2 2 £1... 0
94’ L: - as dy

CHAPTER V

SATISFACTION OF BOUNDARY CONDITIONS ON THE SURFACE X = O

5.1 Procedure Outline

The eigenfunction constants are determined in terms of the

time harmonic stresses Oxx and Oxy on the surface x = 0. On

this boundary, the general eigenfunction expansion has the form

 

 

 

 

 

 

 

 

 

r 1 °°
, .. oxxb r ‘
O m R
xx 1
C. .
Oyy 1’21 3ng “2
= cos wt = Cr cos wt ,
o’xy Oxyb 1‘3
co
' Q "x=0 Z C.n.4 r—l __n4u (r)
j=1 J J
or
F. .. co
Oxxb -
m n1
C. .
E 1ng
J “2
= Cr (5.1-1)
Oxyb “3
2 C.“. r=1 fl,
j=1334 N-(r) .

 

Multiplying both sides of equation (5.1-1) by the biorthogonality
operator as, (4.3-4), and integrating from y = -l to y — 1

yields the following equations

43

44

co
F + 2 B .C =NC , S =1,2,...,oo . (5.1-2)

N8 is a normalization factor, FS involves the boundary stresses

0 and O , and the summed term arises from the series expansion
xxb xyb

of o and Q. The system of equations is solved for the 3 con-

YY

stants by truncating the series.

5.2 The Adjoint Function Arbitrary Constants

The arbitrary constants of each set of adjoint function
vectors are chosen such that the adjoint function vectors and the
eigenfunction vectors of each mixed case are a biorthonormal set.
The necessity of this requirement becomes evident in the actual de-
velopment of equation (5.1-2).

Consider the solution of either of the mixed boundary cases.
For the first mixed case, the eXpansion (2.7-1) can be used, and
the biorthogonality operator of (2.5-7) applied in the manner des-
cribed in Section 5.1, or the four-vector expansion can be used by
substituting the series expansion for Oxy and Q. In the latter
procedure, an inner biorthogonality condition yields the same
relation developed using the two-vector expansion. The series term
of equation (5.1-2) does not appear in this mixed boundary case,
thereby yielding an explicit solution for the eigenfunction con-
stants. Also, each term of the resulting equation for the eigen-
function constants is multiplied by a single adjoint function
arbitrary constant which can be taken equal to unity. These same
characteristics are true of the second mixed case. In the four-

vector formulation for the pure stress end condition, the equations

45

(5.1-2) involve a sum of terms, some of which are multiplied by
the adjoint function arbitrary constant of the first mixed case and
others multiplied by the constant of the second mixed case. This
condition requires the evaluation of each constant or at least a
determination of the ratio of the constants. The second alternative
is dismissed because equations coupling the two adjoint problems
are not evident.

The normalization constant N8 of equation (5.1-2) for the
four-vector form is the sum of the normalization constants for the

first and second mixed cases

N = k(1),,(1) + k(2>N(2>

s s s s s ’

2
where kél) and k; ) are the adjoint function arbitrary constants
of the first and second mixed cases respectively. In constructing

a biorthonormal set for each case, these constants are chosen as

k<1) = 1,Nm k(2) = 1/N<2>
S S S S

,

with the result that NS = 2.

5.3 Satisfaction of Boundary Conditions for the Odd Problem

The eigenfunction expansion for the odd problem is given
by (4.1-2). Substituting the stress boundary conditions at x = O,
premultiplying both sides by the odd problem biorthogonality

operator, OQS, and integrating from y = -l to y - l

46

 

 

 

 

Oxxb
_ Cjnjz n1]
1 j—l 1
02 dy = CS I Oi T‘2 dy
-1 S °xyb -1 S n3
°° m
2 C “j, _ _
j_ _1 J' o (8)
yields
F + SB 3 _ (k(1>N(1>+ k<2)N<2))C ,
3 i=1 31 5 k3 s s
where

 

 

 

 

1
_ 2 - 2 _
Fs ’ [1 {(2281 z 2)Gxxb + [FV+V )usl + (V +3V+1)“32
1
d
E - d2 d _ v 5 kab
- 2 L131 Oxyb y 2 2 32 dy
9w dy 22.. a _1
E s
1 E - d2
Bsx =1. { slniZ [vu 1 + (1+3v)u 2 _—2- u32 7] “14 dy
1 Pw dy
1
+2.; dT‘12
2
pm 51 dy _1
1 an 1
k(1)N (1) ‘ ‘ _J;_ - $2
=I1 [(2% z$2)nsl 31n321dy + 2 zsl dy
w -1
k(2)N (2)_ 2 - 2 - E .. d2
I [NW )“31 + (V +3v+1)u32 ‘ 2 ”s1 2] “s3
-1 pm dy
- - E - d2
- [vusl + (1+3v)u82 + ‘7 U82 ——2] “54 dy
pm d
1
_ v - d“s3
m2 - a2 32 dy _1

(5.3-l)

(5.3-2)

(5.3-3)

(5.3-4)

(5.3-5)

(5.3-6)

47

The odd problem adjoint functions zi are given by equations
(2.6-10), the functions ui by (3.7-6), and the eigenfunctions
“1 by (4.1-2).
The evaluation of the coefficients Bsi requires care.
The arguments of the hyperbolic functions are related to the eigen-

value a through the following expressions

arm =\/oz2 -'Q—-(1-v2) , om =2_\/Q_P_ 20”)

The eigenvalues occur in pairs, one of the pair being the negative

 

 

of the complex conjugate of the other, and some of the eigenvalues
are real. Bsi must be determined independently for each of the

following relations between the roots 01:

2 2
(l) 01 1‘ 018
(2) 01 = as
(3) ai = -as and:
(a) O[imi = O[sms’ C1lini = asns’ mi = dms’ ni = -ns
(b) aimi = asms, aini = -aSnS, m1 = -ms, ni = nS
(C) aimi = -asms, aini = -aSnS, m1 = ms, ni = nS

BSi for Case (1)

Substitution of the eigenfunctions and adjoint functions

into (5.3-4) and performing the required operations yields

 

 

 

 

= (1) . 2 _ 1 1
Bsi 2kg COS“ “smssmh aimi “smSKi 2 2 2 2 2 2 2 2 2 2 2 2
0’ 1'1 "O’Jl. 0’ “1 'Q’.n 0/ n ’O’.m
S 1 1 S S 1 1 S S l 1
tanh a SmS aimiKi E ( 2n2 - 2 2 _ 1
tanh a. m. 2 2 2 2 2 C1Is s 0[in i
1 i as ns'a. 1ni pm

1 S

2 2
-a;m. s
1 1

2 2 2 2
*§§'(a n - a,m.) - 1]
s s 1 1

pm

 

tanh a n a.m,Kr m [
s s

tanh a.m,
1 1 a n

m use
m use

1
tanh a 2
-a0ni

tanh asms a,n,K, E 2 2 2 2
2 (asms - aini) - 1
pm

53H
m KJH
to

.n.
11am

(0
I'"

 

tanh a n a.n.K m
s s i _§_ _§__ 2 2 2 2 _
+‘ 2 2 n [ 2 (asns aini) 1

tanh a.n.
1 1 a n -aini 8 pm

CDNH
CDNH

. E 2 2

2 1+3v-vR + ———'a.m.
(2) Zaim'K' S w2 1 1

+ i 2k cosh a m sinh a,m. 1 1 p
s s s 1 1 2 2 2 2 2

1 +'n. a n - a.n.

1 s s 1 1

 

- E 2 2 E 2 2
u _ _ 2
a.(1+n?)R.K, 1+3v vRS + 2 aini l+2v +- 2 aini 4m
_ 1 1 1 1 pm _ gm 3
2 2 2 2 2 2 2 2 2 2 2
O’sms ' aini O’sms. " “1‘“1 (I‘ms)

 

 

E 2 2 E 2 2
.— —— + —
2a m m.K. H3" VRs + 2 aimi 1+2“ 2 “1‘“1 tanh a m
_ s s 1 1 gm _ 11w __$___§
1+1n2 2 2 2 2 2H2 2 tanh aim,
i asms aimi aS s aim 1

H-PO

E
9w
2 2 2 2 2

l +'n m - .n.
i as s 0{1 1

2a m m,K,R,
s s 1 1 1

 

+

pwz tanh aSmS
- 2 2 2 2 tanh a.m, ' (5.3-7)
a n - a.n. 1 1
s s 1 1

1+2v + “—E— a2n2
1 1

BSi for Case (2)

 

 

' 1 1
B = k(1)K -1 _IE Slnh am COSh am + am.sinh am cosh am +
ss 3 n Sinh an cosh an 2 2 2 2
a m a n
2E n tanh am m tanh an
+_ 2-—”-_—
2 m tanh an n tanh am

49

 

 

 

 

 

 

+ 2 n_tanh am _ m tanh an
2 2 2 2 m tanh an n tanh am
O’m'O’
1 2mk(2)K sinh osh
' __2__ (1+3v-vR) [1 + 0‘“ C 0"“
1+n ‘3‘“
ZamR sinh am cosh am 1 n_tanh an
' 22 22 -mtanham
am -an
+ (1+2v) QLR sinh am cosh am 1 +.81nh an cosh an
n s1nh an cosh an an

+ 2am sinh am coshm 1 __m_tanh@”

2 2 2
a m? _ a n n tanh an

 

+201m Sinhamcosham 1 _mtanhcxm

am 22 22 ntanhan
am

E am
+—
2 'an

2 2 '
[1 +_31nh am cosh am

 

1

pm

1 +

 

n sinh an cosh an

+_ E2 a2n2 [95 sinh am cosh am
an

 

sinh an cosh an )

pm

2amR sinh am cosh am 1 _ E.£§ED_QE;] (5 3-8)

O[ZmZ _ aZnZ m tanh am

BSi for Cases (3a) and (3b)

 

Eliminating the i subscript from the functions in (5.3-4)
through the substitution of the relations given in (3a) gives the
same expression for BSi as with the substitution of the relations
given in (3b). The resulting expression is almost identical to
(5.3-8) except for one sign change.

Consider (5.3-8) written as

B = k(1)K I - —— . (5.3-8)
SS 8

BSi for the cases (3a) and (3b) are

50
B . _—_ kinx + —-—-—-———— . (5.3—9)

BSi for Case (3c)

 

Substituting the relations listed in (3c) again yields a
relation similar to (5.3-8) except for a sign change on the first
term

(1) i 2mk(2)K
1331 = -k K - ————5—-§-— . (5.3-10)
S l + n

Substituting the adjoint functions into (5.3-S) and (5.3-6),

 

 

 

and factoring out the arbitrary constants kél) and kiz), the
following is obtained for the normalization constants Nil) and
N(2):
s
2 .
N(1) = (1+v)gn -l) m S1nh am cosh am _ 1
s 2 n sinh an cosh an
vn - v - 2
fig-D 'tl h 4 2
+ 251“ 20”“ C03 0““ [(2-2v)n + (l+5v)n - (1+v)] (5.3-11)
2amm (1+n )

 

 

N(2) _ i 2m(l+gxl+2v+n2) m_ sinh am cosh am _
_ 2 .
s (vn -v-2)(1+n2) n Sinh an cosh an

 

2 .
+ (n -1):mh gm COSh 0““ [ (2-2\J)n4 + (l+5v)n2 - (l+v)] .(5.3-12)
2amn (l+n )

2 2
In the derivation, biorthogonality was defined for ai * aS
but examination of the operator reveals biorthogonality holds for

a I
ai # aS 130

5.4 Satisfaction of Boundary Conditions for the Even Problem

The even problem is handled in exactly the same way using

the eigenfunction expansion (4.2-2) and the even problem biorthogonality

Sl

operator. Substituting the even problem eigenfunctions and adjoint
functions into (5.3-3) for Fs’ (5.3-4) for Bsi’ (5.3-5) and

(5.3-6) for the normalization constants yields the following:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(1) . 2 1 1
= 2 K. _....____ - -
Bsi ks 31““ O’s,"‘sC°‘”’h “imi 0’s“"53 1 2 2 2 2 2 2 2 2 2 2 2 2
a n -aun. a m -a n a -a.
S S S S S l 1
+ tanh aimi “1m1K1 E ( 2 2 _ 2 2 1
tanh a m 2 2 2 2 2 asns “1“1) '
S S OIsns-Q’1ni pm
_ tanh aimi aimiKi T;_ _E__( _ 2 2 _ 1
tanh a n 2 2 2 2 n 2 a “ 91mi)
s s a n -a.m. 3 pm
5 s 1 1
- tanh aini ainiKi E ( m _ 2 2) - 1
tanh asms 2 2 2 2 2 a O[ini
oIsms O[in1 pm
tanh a.n, a,n,K, m
+ 11111 i—(an-2n2)-l
tanh asns 2n2_ 2 2 2 “i 1
as s CYin1 S pm
20’ m2K 1+3\)-\)RS + —'2- (rim,
2 o o
+ i 2k( )sinh a m cosh q,m, 1 1 1 pm
3 s s 1 1 1+“2 2 2
. CY n " CY n.
1 1 1
2 2
+2 ——— 2
1 v't 2 afni 4m
2 2 2 2
a n - a.m, (1+n )
1 1 s
E 2 2 E 2 2
2 1+3 - +-——— 1+2 + ——— 2
C311(1+ni)RiKi V VRs pUL)2 aini V N2 “1% 4mS
' 2 2 2 2 2 ' 2 2 2 2 2 2
- - +
OIsms aini Cvsms aimi (1 n5)
E 2 2 E 2 2
+ _. —— __
2asmsmiKi 1 3v vRs + pr aimi 1+2v + 9&2 aimi tanh aimi
- 2 2 2 2 2 - 2 -
1+ni asm - a.m. a n2 - agm? tanh O(sms
s 1 1 s s 1 1
E 2 2 E 2 2
+ _ __ __
2a m m,R.K 1 3V vRs + 2 mini 1+2V + 2 aini tanh cum,
+ s s 1 1 i ppm _ 49w 1
1+112 2 2 2 2 2 2 2 2 tanh a m
1 asms ' O’i“1 asns " aini S S

52

Bsi for Case (2)

 

1
+ am sinh am cosh cym[—;—‘§+ “-2—2"

am an

 

1 o
B = k( )K 1 + m sinh am cosh am
33 s n Slnh an cosh an

 

+

22 _________

2 n tanh an m tanh am
a m -a2n2 m tanh am n tanh an

i 2mk(2)K si h sh
+————§-—- (1+3v-vR)[l- “ 0‘7"“ 0”"

2 am

l+n

+ 2am R sinh am cosh am 1 _ n_tanh am
2 2 2 2 m tanh an
(I’m an

 

E.R sinh am cosh am 1 sinh gn cosh an
n sinh an cosh an an

+ (l+2v)[

2am sinh am cosh am 1

2 2

.E
2 2 n
a m - a n

tanh an
tanh am

 

+ _§__ 2 2 1 _ sinh am cosh am _ Zam sinh am cosh am
20’m Om] 22 22
pm a m - a n

m tanh an
1_—_._._____...
n tanh am

 

 

E
+ 20
pm

2n2 m,R sinh am cosh gm
(1’11

sinh an cosh an
n sinh an cosh an

 

:3

2am R sinh am cosh am
2 2n2

+ 2
am '0’

tanh an

3

1 - — Lam—0"“ J . (5.4-2)

 

Bsi for Cases (3a), (3b), (3c)

For the three cases of condition 3, the expression for B81
differs from (5.4-2) by only one sign change. Consider (5.4-2)

written as

53

 

(1) 1 2m k(2)K
B S = k K + Z
s 3 1+“
Bsi for the case (3) are
(1) i 2mk(2)K
B . =k K - ————S§— . (5.4-3)
31 3 1+“

The normalization constants for the even problem are:

 

 

 

N(l) = gliv)§n2-l) 1 _ m_sinh am cosh am
3 2 n sinh an cosh an
vn -v-2
2 ' 4 2
+ Q“ "”31““ “‘2“ C03“ m“ [ (2-2v)n + (l+5v)n - (1%)] (5.4-5)
(1+n )2qmn

N(2) __ i 2m(l+\)) Q+2v+n2) 1 _ Q sinh arm cosh am
— 2 2 °
3 (vn -v-2)(1+n ) n Sinh an cosh an

2
+ in -l)sinh am cosh am

2 2 [(2-2v)nA + (1+SV)n2 - (1+v)] .(5.4-6)
2amm (1+n )

CHAPTER VI

DETERMINATION OF THE EIGENVALUES

6.1 General Remarks

The eigenvalues a are the roots of the transcendental

equations

tanh am 4mn

-— = —— 6.1-1)
2 2 (

tanh an (1+n )

tanh an 4mn

-—-——-- = ---- . (6.1-2)

tanh am (1+n2)2

Equation (6.1-l) is used for the odd problem, and equation (6.1-2)
is used for the even problem. Similar equations occur in the in-
vestigation of wave prOpagation in an infinite plate. This prob-
lem has received much attention, and the solutions of these equa-
tions have been made, but, as far as this writer has been able to
determine, not for the complex values of a associated with the
decaying wave modes close to the finite edge of the plate. This
study is concerned with the investigation of all bounded modes for
each frequency of propagation. Equations (6.1-1) and (6.1-2) de-
fine the eigenvalues on the real axis and the upper half of the

complex plane.

6.2 Determination of the Real Eigenvalues for the Odd Problem

The transcendental equations are solved with the frequency

w as a parameter and the Poisson ratio v taken to be equal to

54

55

1/3. Recalling that

2
2 2 2 2
a m = a - Q§_'(1'V )
2
2 2 2 2
(1117-01 ~“5‘UL(1+V) ,

it is convenient to substitute constants "a" and "b" such that

2 2 2

2 2 2

For v = 1/3, the constant b equals 3a. Equation (6.1-l) can now

be written as

tanh'2 a =4q2Q2—aJa2-3a

a -

f——————- 2 2
tanh 2 3a (20 - 3a)

a -

 

 

(6.2-l)

Consider the solution of equation (6.2-1) when a is very large

compared to "a" and the hyperbolic terms approach unity.

a6 - 5.25q4a + 6.75q2a2 - 2.53a3 = o (6.2-2)

Only one root of equation (6.2-2) yields a real argument for the

hyperbolic functions. This root is

a = 1.884 V a . (6.2-3)

Consider the solution of equation (6.2-l) when a and "a" are very

small and the hyperbolic functions can be replaced by their arguments.

40,4 - 1202a + 9a = 40,4 - 1203.1 (6.2-4)

For this equation to be satisfied, "a" (or w) must equal zero.

56

The phase velocity, c, of a given mode equals w/a, and the
wave length A is defined in terms of a by a = Zn/A. When a

is very large (small wave length), the limiting phase velocity is

C

w/a = w/l.884‘/a , c = 0.563 E/p

When a approaches zero the phase velocity approaches zero.
Intermediate values of a have been obtained numerically
with the computer for seven values of frequency. For each frequency,
the root yielding the lowest phase velocity is identified as the
first mode root, and the mode with the next highest phase velocity
is mode two, and so on.
Table VI-l lists the odd problem real eigenvalues of the
first four modes for the frequencies examined and the correSponding

\/ E- , the velocity of

prOpagation of plane longitudinal waves along a rod. Tabulation

phase velocity ratio c/cO where c0

of all the eigenvalues for these frequencies is found in the
appendix.

Figure VI-l is presented to graphically illustrate the
phase velocity-wave length dependence. Phase velocity variation
with changes in wave length is termed dispersion stemming from the
fact that this characteristic causes distortion of the shape of a
prOpagating pulse. The first mode curve is in close agreement with
the dispersion curve for flexural waves in cylindrical bars, and
the trend of the higher mode curves indicated by the few data points
plotted is in agreement with published phase velocity curves for
flat plates and cylindrical bars. The intent here is not to present,

a complete solution of the transcendental equations, but to attach

57

some physical significance to the real eigenvalues and make some
comparisons of the values used to solve the problem with published

results (13)-(15).

6.3 Determination of the Real.Eigenvalues for the Even Problem

Following the same procedure as in article 6.2, equation

(6.1-2) can be written as {‘5

tanh “(12 - 3a - hazvgz - a {(12 - 3a
2 2
tanh '02 - a (2d - 3a)

Considering the solution of (6.3-1) when a is very large relative

 

 

 

(6.3-1)

 

to "a" yields the result (6.2-3). Considering the solution when a 2.;

and "a" are very small yields

-8a2+9a=0,

Thus, when a is large the phase velocity approaches 0.563 co.

or

0
II
Q l8

When a approaches zero, the phase velocity approaches co.
Intermediate values of a were obtained numerically, and

Table VI-2 lists the even problem real eigenvalues of the first

four modes and the corresponding phase velocities for the frequencies

examined. Figure VI-2 illustrates the dispersive nature of the

first three modes. The flmplied shape and values of the curves are

close to those found in the literature for flat plates and

cylindrical bars. For the first mode, the velocity ratio c/cO

approaches 1.062 for the flat plate and approaches one for the

cylindrical bar as a approaches zero.

58

6.4 Determination of the Complex Eigenvalues for the Odd and Even
Problem

Writing equation (6.2-1) in the following form

 

(20,2 - 3a)2tanhV 012 - a - Aazfaz - aJaZ - 3a tanh V012 - 3a = 0,

(6.4-l)

and (6.2-2) as

 

(20/2 - 3a)2tanhVa2 - 3a - 4(12 {0:2 - a V02 - 3a tanh (:2 - a = 0,
(604-2)
and letting a = x + iy, solutions were made numerically by use of

a digital computer. An asymptotic solution of (6.4-1) for a very

 

large compared to "a" aided the search for the roots by isolating
the roots in the complex plane and giving very good approximations
of the roots at low frequencies.

It is of interest to note the comparison of the complex
roots of (6.4-1) and (6.4-2) with the roots obtained by Johnson
and Little in their solution of the static problem. This comparison
is given in Tables VI-3 and VI-4.

The first quadrant complex roots for two frequencies, 1000
and 31600 cycles per second, are plotted in Figure VI-3. The roots
for all frequencies examined are tabulated in the appendix. The
calculations were made for a material density of 15.1 slugs per
cubic foot and a Young's modulus of 30(106) pounds per square inch.

All dimensions are in units of feet.

59

Table VI-l -- Real Eigenvalues and Phase Velocities for the First
Four Modes of the Odd Problem

 

Eigen- Velocity Wavelength
Frequency w Constant value Ratio Strip Width
cps "a" Cox c/cO

100 0.00123 0.259 0.144 12.130
1000 0.123 0.969 0.384 3.242
6380 5.0 4.30 0.552 0.731
2.42 0.980 1.300
10000 12.3 6.63 0.561 0.474
5.38 0.691 0.584
3.29 1.131 0.955
1.44 2.583 2.182
31600 123.0 20.89 0.563 0.150
19.12 0.615 0.164
18.38 0.640 0.171
16.90 0.696 0.186
63800 500.0 42.12 0.563 0.074
38.69 0.613 0.081
38.39 0.618 0.082
37.78 0.628 0.083
100000 1230.0 66.07 0.563 0.048
60.18 0.618 0.052
59.63 0.624 0.053

58.89 0.632 0.053

Table VI-2 -- Real Eigenvalues and Phase Velocities for the First
Four Modes of the Even Problem

 

Eigen- Velocity Wavelength
Frequency w Constant value Ratio Strip Width
cps "a" ea c/cO

100 0.00123 0.0372 1.000 84.451
1000 0.123 0.373 0.997 8.422
6380 5.0 4.056 0.584 0.774
2.238 1.060 1.404
1.131 2.097 2.778
10000 12.3 6.580 0.565 0.477
3.926 0.947 0.800
3.316 1.120 0.947
31600 123 20.89 0.563 0.150
18.84 0.624 0.167
17.74 0.663 0.177
15.85 0.742 0.198
63800 500 42.12 0.563 0.074
38.58 0.615 0.081
38.12 0.622 0.082
37.36 0.635 0.084
100000 1230 66.07 0.563 0.048
60.65 0.619 0.052
60.38 0.616 0.052
59.93 0.621 0.052

61

 

 

 

 

1»
1.5 .
c/c
0 C924
h\
1.0 ~ b X
\ \
®‘H \\ \
\ \ \ \ “
‘YL \ ‘ ~ ~.1
\"‘ _____________ ‘1i5kc;:; r5
0" E
O°5 _ Mode 3
2
L
o 5 10 15 20 a L

Figure VI-l -- Phase Velocity Variation
of the Odd Problem

for the First Three Modes

 

 

 

1.5 i
3/
c/co S;
c2 Q
YEP \
1.0‘<D~\ \ \
. n x
\ \., \
\ \\ \
‘ \\\\
\ \\ é
\ :-;2‘f ‘ _ ____¢
\ “~-—~Q—:$::“_
‘o « f 0
0°5 ‘ Mode(l(
O 5 10 15 20 a

Figure VI-2 -- Phase Velocity Variation for the First Three Modes

of the Even Problem

62

Table VI-3 -— Comparison of Low Frequency, Odd Problem, Complex
Eigenvalues with the Static Problem Eigenvalues*

Static, Odd Problem 100 cps, Odd Problem

i1.384339 + 13.748838 :1.384371 + 13.748499

1.676105 6.949980 1.676120 6.949802
1.858384 10.119259 1.858393 10.119137
1.991571 13.277274 1.991577 13.277181
2.096626 16.429872 2.096630 16.429796
2.183398 19.579409 2.183401 19.579346
2.257320 22.727036 2.257323 22.726982
2.321714 25.873384 2.321716 25.873337
2.378758 29.018831 2.378759 29.018789
2.429959 32.163617 2.429960 32.163579

 

0""- .- I:
1

Table VI-4 -- Comparison of Low Frequency, Even Problem, Complex
Eigenvalues with the Static Problem.Eigenva1ues*

Static, Odd Problem 100 cps, Odd Problem

:11.125365 + i2.106196 i_1.125419 + i2.105550

1.551575 5.356269 1.551595 5.356036
1.775544 8.536683 1.775555 8.536538
1.929405 11.699178 1.929412 11.699072
2.046853 14.854060 2.046858 14.853977
2.141891 18.004933 2.141895 18.004865
2.221723 21.153414 2.221726 21.153355
2.290553 24.300342 2.290555 24.300292
2.351048 27.446203 2.351050 27.446158
2.405013 30.591295 2.405014 30.591255

From Johnson and Little (7)

63

 

 

 

L
30 4 9
49
Y o
O
o
9
25 d O
O
<9 9
o O
20 - a
e
O
0 9
15 ~
9 O
('3 9
‘9 O
10 ~ 0 ® 9—1000 cps, Odd Problem
0 O— 1000 cps, Even Problem
9 9 9— 31600 cps, Odd Problem
5 ~ C) CF‘31600 cps, Even Problem
9
O
0 1 2 3 4 x

Figure VI-3 -- Eigenvalues in First Quadrant of Complex Plane for
Frequencies of 1000 and 31600 cycles per second;
a = X +»iY

 

CHAPTER VII

SPECIFIC BOUNDARY VALUE PROBLEMS

7.1 General Remarks F‘i

The general solution of the problem is in the form of an

eigenfunction expansion of the four-vector

 

 

 

 

 

.oxx '1 n1 :7
o n i(a x + wt)
yy = C 2 e r , (7 .1-1)
oxy r n3
11
.. Q .. r=1 .. 4.1 (r)

where the constants Cr are determined from the particular

bo ndar stresses a d h fa e = hro h
u y oxxb n oxyb on t e sur c x 0 t ug

the application of the biorthogonality operator 63, (4.3-4). This

procedure was detailed in Chapter V, and it was seen that the

evaluation of the constants requires the solution of the infinite

set of equations (5.3-2)

8 ._ 31 1 s s s s s
1-1
repeated

The FS term is the only term containing the boundary functions.

1
- - 2 - 2 -
FS — [1 {(2281 - zsz)okxb + [(v + 30 + l)u82 + (v + v )uS

1
1
_.E__- 12.. __v - Elisa 33
- 2 usl 2 oxyb dy - 2-—._; u$2 dy (5. - )
pm dy 9%L-- as -1 repeated

65

In this chapter, solutions are made for the following

self-equilibrated end loadings:

 

Problem oxxb Ox b
1 -3y +10y3 - 7y5 0
2 cos E- - £_ 0
2 11
3 0 -sin fly

7.2 The Boundary Term FS

 

-3y + 10y3 - 7yS

Problem 1 Oxxb =
oxyb = 0

where

_ ( ) m.S cosh asms
k (Slnh qsmsy - Eg'EBEF—Egfig Slnh asnsy)

m cosh a m

 

5 = k(1) (sinh a m y‘+
s S S

o 52 Sinh asnsy) .

K n cosh a n
s s s 3

Performing the required integration yields

tanh a n
_(1) ms 33 44 22
FS kS RS Eg'cosh asms -;Fr:3;-- ( 16asnS 7200/SnS 1680)
s s

 

1 2 2
+ as 5 (160asnS + 1680)]

n
S S
tanh a m
s S 4 4 2 2
+ COSh asms[:-—-g-—€- (lOaSmS + 7ZOaSms + 1680)
a m
S S
1 2 2
+ 5 5 (-16008ms - 1680)]

a m
S S

1

- IO§HIZW

p.11 “A“

 

I. 5" .5'.

66

_ 11 _2
Problem 2 oxxb ‘ 003 2 Y n
Oxyb = 0

Following the same procedure using the even adjoint functions gives

 

 

(1) ‘m 4 tanh a m
F =k R—gcosham tanham TI SS
8 s s s aim n s s 2 2 2 tanh a n
s s s n +4asns s s
4tanh a m
+ cosh asms [72112—5 - -———§-§-] .
n +4asms “3“5“
Problem 3 o = 0

xxb

Oxyb -s in rry

Substitution of these functions into (5.3-3) and integrating gives

2
F =k()2nsinham R(V+V2+-LIT2)
s s s s 2 s 2
a m +fl pm
8 s
2 2 2
-(v +3v+1)-—-2—V—'—(a:ms+n)
$1-012
E
m tanhan
1 _§_ 3 s 2 2 E 2
+222ntanham [(V+3V+1)'(V+v+ 2n)
an‘l'n S SS pm
3 s
2 2 2
++——(asns+n)] .
&.a2
E

7.3 Results and Conclusions

A detailed study of the steady state solution was made for
the intermediate frequency, 6380 cycles per second, for each of the
time—harmonic boundary loadings defined by problems 1, 2, and 3.
The eigenvalues used for this frequency are tabulated in Table A-3

in the appendix. Tables VII-1 and VII-2 illustrate the convergence

 

III. ' 'hu:c ”’11er

67

of the eigenfunction constants for problems 1 and 2; Table VII-3
gives the comparison of the prescribed boundary functions with the
truncated eigenfunction expansion approximation for problem 1.
These examples are representative of the solutions for the three
problems at this frequency. The boundary functions and the non-
decaying stress wave modes of problems 1, 2, and 3 have been plotted
in Figures VII-1 through VII-7. The decaying modes at x = 0, the
sum of which is the difference between the boundary functions and
the sum of the non-decaying modes at x = 0, are not shown as the
major concern is the stress magnitude and distribution beyond the
decay region. The phase velocities and wave lengths of each mode
are given in Tables VI-l and VI-2.

Examination of the results shown in Figures VII-l through

VII-7 leads to the following conclusions:

1. A Saint-Venant boundary region does not exist for the
dynamic problem. The self equilibrated, time-harmonic
boundary stresses of this problem produce stress waves
which do not decay. The existence of non-decaying
stress waves at this frequency insures the existence
of non-decaying stress waves for any time—dependent
boundary loading which includes this frequency component.

2. Application of the time-harmonic, self-equilibrated

stress 0 with zero shear stress 0 in problems

xxb xyb

1 and 2 produces significant non-decaying ckx, Oyy’
and oxy stress modes. These modes are the only

significant modes remaining beyond a distance of 2

strip widths for the odd problem and l strip width for

1 --.:¢-’

1% 7'.

 

cFP'H'E”_'.‘

68

the even problem from the loading surface X = 0.
3. The time-harmonic, self-equilibrated shear stress Oxyb

with zero normal stress 0 in problem 3 yields a

xxb
non-decaying 0*y stress mode very similar in magnitude
and shape to the applied stress without producing Oxx
and 0 modes.

YY

4. The shape and magnitude of the non-decaying stress modes
are very sensitive to the boundary stress distribution.
Problems 1 and 2 each involve self-equilibrated normal

boundary stress distributions of approximately

oxxb
the same peak magnitude as shown in Figures VII-l and
VII-4. Figure VII-2 of problem 1 shows a resulting mode
1 oxx stress wave having a peak magnitude 5.7 times
greater than the peak magnitude of the applied stress.
Figure VII-5 of problem 2 shows three non-decaying

modes of Oxx stress waves, all of which have peak

values less than the peak value of the boundary stress.

Solutions of these problems were attempted for other fre-

quencies, but the results were not conclusive because of un-

satisfactory convergence of the eigenfunction constants. At the

lower frequencies, round-off error is suspected as the main source

of difficulty. At the higher frequencies, the equations (5.3-2)

become highly ill-conditioned. Examining problem 1 at the lower

frequency of 1000 cycles per second, the trend of the convergence

of the single non-decaying eigenmode constant appeared to be toward

zero.

The higher frequency solutions indicated the existence of

 

C

69

more non-decaying modes, but the convergence of the eigenfunction

constants was too unstable to permit conclusive statements.

 

 

70

H

«Hm.o

0H

ooq.H

'H

mqm.m

0H

wwH.o

'H

wON.H

muoom Ho

u mmH.ou H wom.o I HmH.ou
+ wom.o: H 0mm.H + mom.ou
+ Num.m H mHm.N + omm.m
+ HON.o H qu.o + mmH.o
u omo.o H mow.H n mmo.o
Hme 0H muoom Ho “Ham NH
o u D%Mb
nxx
% u x m n u b
m n m OH + m

H mHm.o moH.Ou H wqm.o

H Nm¢.H + Ham.o| H omw.H

H mmm.N + «Hm.m H nmm.m
H qu.o + ooN.o H omm.o
H an.H u mNo.o H dom.o
muoom Ho pHmm OH muoom mo

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+ mwN.o

mHo.o

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H EmHnoum How mucmumcoo COHuocswcmem mo oucowuo>cou In HuHH> eHan

ucwumcou

71

-H

«00.0 + «00.0:

~H

000.0 + Nq0.0

-H

mNH.0 mHH.0

-.-I

000.N Nm0.01

w-l

m00.0 + mHo.on

mecca mo pHcm HH

.C

N

‘H

m00.0 + m00.0:

~r-Il

0H0.0 + mm0.0

w-i

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H

N00.0 + mH0.0:

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muoom Ho uHmm m

N EoHpoum HoH museumcou coHuucsucome Ho mucmwpo>coo nu NuHH> oHan

k)
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L)

ucwumooo

72

Table VII-3 -- Approximation of Boundary Function, Problem 1

Boundary Function: 0 = -3y + 10y3 - 7y5

= 0

Series Solution with 15 pairs of Eigenvalues

Prescribed Stresses

Cxxb
0.00000
-0.29007
-0.52224
-0.64701
-0.63l68
-0.46875
-0.18432
0.15351
0.42624
0.45657

0.00000

nyb
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000

0.00000

Series Solution

0
XX

0.00000
-0.29013
-0.52222
-0.64696
-0.63174
-0.46878
-0.18422
0.15348
0.42612
0.45662

-0.00051

0'
xy

0.00000
0.00002
0.00006
-0.00005
-0.00004
0.00009
-0.00001
-0.00011
0.00008
0.00010

0.00000

73

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r Moo-
b 0.0 0.0 «.0 ~.0 N.0n «.0: 0.0: 0.0:
0 I phxb lll
xx . m.0
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0

 

74

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76

 

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77

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78

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BIBLIOGRAPHY

 

1?

10.

BIBLIOGRAPHY

Todhunter, I. and Pearson, K., A History of the Theory of
Elasticity and of the Strength of Materials, Vol. 2,
pt. 1, Cambridge Univ. Press, London, 1893, Chapter 10. Hr?

Todhunter, I. and Pearson, R., A History of the Theory of
Elasticity and of the Strength of Materials, Vol. 2,
pt. 1, Cambridge Univ. Press, London, 1893, Chapter 10,
pp. 11-12.

 

Todhunter, I. and Pearson, K., A History of the Theory of
Elasticity and of the Strength of Materials, Vol. 2, p;
pt. 2, Cambridge Univ. Press, London, 1893, Chapter 13,
p. 273.

 

 

Todhunter, I. and Pearson, R., A History of the Theory of
Elasticity and of the Strength of Materials, Vol. 2,
pt. 2, Cambridge Univ. Press, London, 1893, Chapter 13,
p. 271.

 

von Mises, R., "0n Saint4Venant's Principle," Bull Am. Math.
Soc., vol. 51, 1945, pp. 555-562.

 

Sternberg, E., "On SaintJVenant's Principle," Quart. Appl.
Math., vol. 11, 1954, pp. 393-402.

 

Johnson, M.W., Jr. and Little, R.W., "The Semi-Infinite
Elastic Strip,"(1uart. Appl. Math., vol. 22, 1965,
pp. 335-344.

Klemm, J.L. and Little, R.W., "The Semi-Infinite Elastic
Cylinder under Self4Equilibrated End Loading," SIAM J.
Appl. Math., vol. 19, 1970, pp. 712-729.

Thompson, T.R. and Little, R.W., "End Effects in a Truncated
Semi-Infinite Wedge and Cone," Quart. J. Mech. Appl.
Math., vol. 23, no. 2, 1970, pp. 185-196.

Boley, B.A., "The Application of Saint4Venant's Principle
in Dynamical Problems," J. Appl. Mech., vol. 22, no. 2,
1955, pp. 204-206.

 

80

ll.

12.

l3.

14.

15.

81

Kennedy, L.W. and Jones, 0.E., "Longitudinal Wave Propagation
in a Circular Bar Loaded Suddenly by a Radially Dis-
tributed End Stress," J. Appl. Mech., vol. 36, no. 3,
1969, pp. 470-478.

Folk, R., Fox, 0., Shook, C.A. and Curtis, C.W., "Elastic
Strain Produced by Sudden Application of Pressure to
One End of a Cylindrical Bar. 1. Theory," J. Acoust.
Soc. Am., vol. 30, 1958, pp. 552-558.

Kolsky, H., Stress Waves in Solids, Dover, 1963, Chapter 3.

Redwood, M., Mechanical Waveguides, Pergamon Press, 1960,
Chapter 5.

Jones, O.E. and Ellis, A.T., "Longitudinal Strain Pulse
Propagation in Wide Rectangular Bars, Part 1 -
Theoretical Consideration," J. Appl. Mech., vol. 30,
no. 1, 1963, pp. 51-60.

[5:

 

 

APPEND ICE S

APPENDIX A

TABULATED EIGENVALUES

82

TABLE A’I --ROOTS OF TRANSCENDENTAL

PU N

OOQNO‘U‘

FREQUENCY

EQUATION
000 PR

oaN

1.384371
1.676120
1.858393
1.991577
2.096630
2.183401
2.257323
2.321716
2.378759

2.429960

(601-1)
OBLEM

.259341 +10.000000

3.748499

6.949802
10.119137
13.277181
16.429796
19.579346
22.726982
25.873337
29.018789
32.163579

= 100 CYCLES
POISSON RATIO = 1/3

H-

EQUATIONS (6.1-1)

PER SECOND

EQUATION (6.1-2)
EVEN PROBLEM

eaN
.0371997 +-10.000000
1.125419 2.105550
1.551595 5.356036
1.775555 8.536538
1.929412 11.699072
2.046858 14.853977
2.141895 18.004865
2.221726 21.153355
2.290555 24.300292
2.351050 27.446158
2.405014 30.591255

83

TABLE A-2 --ROOTS OF TRANSCENDENTAL EQUATIONS (6.1-1)
AND (601-2)

0 w A)

O o: '4 0 U1

10
11
12
13
14
15
16
17
1s

19

FREQUENCY

EQUATION

(601-1)

ODD PROBLEM

oaN

.969392+-i 0.000000

1.387431
1.677591
1.859280
1.992178
2.097068
2.183736
2.257588
2.321933
2.378940
2.430112
2.476535
2.519015
2.558169
2.594481
2.628334
2.660042
2.689860

2.717999

3.714782

6.932145
10.107082
13.268010
16.422389
19.573132
22.721629
25.868635
29.014596
32.159796
35.304422
38.448604
41.592435
44.735984
47.879302
51.022428
54.165395

57.308225

= 1000 CYCLES PER SECOND
POISSON RATIO = 1/3

EQUATION

(6.1-2)

EVEN PROBLEM

a

e

.3730024-i 0.000000

1.130700
1.553632
1.776676
1.930134
2.047367
2.142276
2.222023
2.290794
2.351247
2.405180
2.453862
2.498226
2.538975
2.576654
2.611694
2.644439
2.675173

2.704127

2.040549

5.332933

8.522219
11.688658
14.845783
17.998107
21.147604
24.295285
27.441726
30.587278
33.732171
36.876562
40.020558
43.164241
46.307669
49.450887
52.593930

55.736826

 

If

84

TABLE A-3 --ROOTS OF TRANSCENDENTAL EQUATIONS (6.1-1)

10
11
12
13
14
15
16
17

18

AND

FREQUENCY = 6380 CYCLES PER SECOND

(6.1-2)

POISSON RATIO

EQUATION

(6.1-1)

ODD PROBLEM

oaN

2.421033
1.355817
1.702163
1.877602
2.005841
2.107627
2.192156
2.264475
2.327682
2.383821
2.434316
2.480197
2.522237
2.561030
2.597039
2.630638

2.662129

: 4.295876+ 1 0.000000

0.000000

1.992252

6.179568

9.609329
12.893890
16.122062
19.322040
22.505794
25.679321
28.845969
32.007763
35.165999
38.321548
41.475016
44.626840
47.777341

50.926763

1/3

EQUATION

1601-2,

EVEN PROBLEM

eO‘N

4.0559994-i 0.000000

2.237781
1.130842
1.577832
1.798075
1.945888
2.059326
2.151671
2.229616
2.297072
2.356535
2.409702
2.457780
2.501657
2.542008
2.579357
2.614120

2.646631

0.000000

0.000000

4.304994

7.924157
11.261712
14.512666
17.724623
20.915484
24.093585
27.263356
30.427379
33.587264
36.744066
39.898511
43.051111
46.202239

49.352174

'ulkx
F. '.

 

LE

85

TABLE A-4 --ROOTS OF TRANSCENDENTAL EQUATIONS (6.1-11

N 0‘ L“ k U h)

m

10
11
12
13
14
15
16
17
18

19

AND

(601-2)

FREQUENCY = 10000 CYCLES PER SECOND

POISSON RATIO = 1/3

EQUATION

(601‘1)

ODD PROBLEM

oaN

5.378575
3.286863
1.436153
1.517139
1.826559
1.983445
2.095982
2.185563
2.260571
2.325330
2.382416
2.433512
2.479787
2.522091
2.561062
2.597194
2.630877
2.662425

i 6.630079 + 1 0.000000

0.000000

0.000000

0.000000

4.818572

8.800626
12.306705
15.658226
18.937701
22.177246
25.392200
28.590880
31.778207
34.957288
38.130183
41.298318
44.462708
47.624099

50.783048

EQUATION

1601‘2)

EVEN PROBLEM

eaN

3.926267
3.315675

.805028
1.710236
1.913085
2.043406
2.142976
2.224562
2.294043
2.354712
2.408631
2.457193
2.501392
2.541960
2.579457
2.614321
2.646901

2.677480

1 6.579686 + 1 0.000000

0.000000

0.000000

1.973710

6.917090
10.582463
13.994701
17.304375
20.561275
23.787170
26.993212
30.185738
33.368631
36.544408
39.714775
42.880930
46.043740
49.203849

52.361744

an. ‘41 ~43'..j
W

 

86

TABLE A-S --ROOTS OF TRANSCENDENTAL EQUATIONS (6.1-1)

\OGNO‘U'I

10
11
12
13
14
15
16
17
1e
19

20

AND

FREQUENCY =

(601’?)

POISSON RATIO

EQUATION (6.1-1)
ODD PROBLEM

oaN

19.116382
18.381749
16.903148
14.554569
11.561352
10.709123
9.690573
8.071672
5.638132
1.109372
0.000000
.821718
1.574184
1.705364
1.665112
1.493388
1.058952
0.000000
1.174472

at 20.893342 + i 0.000000

0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
6.327763
11.332493
14.202922
18.660664
22.653411
26.420677
30.065269
33.689863
35.977305

38.321627

31600 CYCLES PER

1/3

SECOND

EQUATION (6.1-2)
EVEN PROBLEM

(I

e N

18.839587
17.740090
15.850131
11.088598
10.700487
9.681430
7.950855
4.312302
1.216463
.382669
0.000000
1.365751
1.669688
1.701246
1.597806
1.333585
0.000000
.363617

1.461126

i 20.893334 +i.0.000000

0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
9.825665
12.971777
16.518870
20.694499
24.557542
28.253380
31.868434
35.246044
36.386089

40.076479

TABLE A-6 --ROOTS OF TRANSCENDENTAL EQUATIONS

87

AND (601-2)

(601-1)

FREQUENCY = 63800 CYCLES PER SECOND

POISSON RATIO = 1/3

EQUATION (6.1-1)
ODD PROBLEM

oaN
:t42.125041+-1 0.000000
38.691841 0.000000
38.387766 0.000000
37.777907 0.000000
36.855312 0.000000
35.603678 0.000000
33.993858 0.000000
31.980117 0.000000
29.496923 0.000000
26.477242 0.000000
23.205347 0.000000
22.171072 0.000000
21.657992 0.000000
20.824644 0.000000
19.670625 0.000000
18.256867 0.000000
16.281783 0.000000
14.064993 0.000000
11.648308 0.000000
4.201674 0.000000
0.000000 11.271708
1.311928 17.237081
.292534 23.194187
0.000000 25.992610

EQUATION (6.1-2)

EVEN PROBLEM

eaN

i 42.125041-11 0.000000
38.577849 0.000000
38.121319 0.000000
37.356459 0.000000
36.272106 0.000000
34.845962 0.000000
33.041099 0.000000
30.802011 0.000000
24.795354 0.000000
22.382882 0.000000
22.202007 0.000000
21.644669 0.000000
20.797816 0.000000
19.682077 0.000000
18.195303 0.000000
16.543159 0.000000
14.153948 0.000000
10.156298 0.000000
7.960167 0.000000
.802532 13.652395
1.161230 20.331783
0.000000 24.405508
.904466 27.419767
1.521996 32.102359

88

TABLE A-7 --ROOTS OF TRANSCENDENTAL EQUATIONS (6.1-1)

OGNO‘UTPUNW

AND

1601-2)

FREQUENCY = 100000 CYCLES PER SECOND

POISSON RATIO = 1/3

EQUATION (6.1-1)
ODD PROBLEM

60.177115
59.629017
58.893544
57.965518
56.837344
55.498506
53.934958
52.128313
50.054807
47.684164
44.979451
41.904074
38.478920
35.378636
34.900756
34.463530
33.836205
33.033031
32.012504
30.800500
29.420589
27.730862
24.000258
20.984984
17.915348
15.191106

6.107614

0.000000

1.230817

i 66.070545+ 1 0.000000

0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
13.641388
21.138086

EQUATION (6.1-2)
EVEN PROBLEM

60.654611
60.381975
59.926245
59.284975
58.454013
57.427032
56.195045
54.745858
53.063368
51.126645
48.908797
40.226163
36.751606
34.853644
34.455069
33.829152
32.991782
31.995858
30.813558
29.399656
27.860995
23.511384
21.533973
18.332334
12.807290
10.498085
.825374
.553100
0.000000

1 66.070545 + 1 0.000000

0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
16.772158
24.873496
28.918273

APPENDIX B

ASYMPTOTIC SOLUTION OF THE ODD PROBLEM TRANSCENDENTAL EQUATION

The transcendental equation for the odd problem is

tanh am 4mn

tanh an = (1+n2)2 (3'1)
where
2
0,2612 = 82 - 9%— (1 - 02) (8.2)
2
a2n2 = oz - Z§Q_.(1 + v) (B.3)

The eigenvalue 0 is found for a given frequency w and Poisson

ratio equal to 1/3. Thus, (B.2) and (B.3) can be written in terms

of the constant "a"

2 2
a Jug—(1H1)
22 2
am =QI-a
qznz = 02 - 3a .

Substituting these two expressions into equation (B.1) yields

 

tanh £2 - a = 4072 {0’2 - a «(i172 - 3a (B a)

2 2
tanh o2 - 3a (2a - 3a)

 

The asymptotic solution yields values of a in the complex plane
for a very large relative to the constant "a". Substituting

0 = x + iy gives

89

90

Vaz-a= fxz-yZ-a+2ixy =x+€1+i(y+32), (8.5)

 

where 91, 52 << 1.
Squaring both sides of (8.5), cancelling like terms, and neglecting

the higher order terms of 61 yields

.3

Xel - ye2 = - 2

yel '1’ X62 = O .

Solving these two equations for 31 and 32 gives
c1 = - fix—2 (8.6)
2y
a
62 - 2y . (8.7)

The right hand side of equation (B.4) becomes

 

2 . .
4612 1/82 - a V82 - 3a = 40’ [0’ + ‘31 + 162]“ + 3(31 + 1‘32)]

(282-4602 (2,2..3.)2

 

 

Substituting 6c = 31 + iez, the right side is

2
4o (a + ec)(a + see) 484 +16chc + 1282.:

 

 

(2a2 - 3a)2 404 - 12a2a + 9&2

As 0 becomes very large relative to "a", this ratio approaches 1.

The left side of (B.4) can be written as

tanthz - a _ tanh” + 61 + i(y + 6’2” _ tanh(p + 16)
r-—--- tanh[x +13; + i(y + 3e )] - tanh(r + is)
tanh a2 - 3a 1 2

 

 

 

tanh p +'i tan_gr . l + i tanh r tan 8
1 +'i tanh p tan q tanh r + i tan 3

 

, (B-8)

where

91

p=M1-3fi . r—Ml-Efi
2y 2y
' 3a
q=y(1+—32) , 3=y(1+—2) .
2y 2y

As X and Y become very large,

tanh p a tanh x a l
tanh r a tanh x a l

and (8.8) may be written as

tanh x[1 - tan q tan s] + i[tan q + tanhzx tan s]

 

(8.9)
tanh x[1 - tan q tan s] + i[tan s + tanhzx tan q]

The ratio (8.9) must approach unity for large a. This requires

that tan q equal tan 3
tan q = tan 8

a 3a
tan (y + 2y) tan(y 2y)

Using the identity for the tangent of the sum, and replacing the

tangent of the small argument with the argument itself yields

 

 

a 3a
tan y +~2y = tan y + 2
a 3a
1 - 2y tan y 1 -2y tan y

As y increases without limit, tan y should remain of the order

of magnitude 1. Thus, tan q and tan s approach unity. Assume

 

 

for y:
= (424-511 '1’ 6 , 6 << 1 . (8.10)
a tan y +1;-
tan q = tan(y + E;- = ,y = l (8.11)

1 -'- tan
y Y

92

For y defined by (8.10),

 

_ 4n+l _ 1+6

tan y — tan[( 4 )n + 5] — {:3
Substituting into (8.11) yields

1:61 a..- 2.12:9.

1-0+2Y 1 2y (1-6) ’
or 6 = - %;' which is negligible for large values of y. Like-
wise, requiring tan 5 to approach unity and solving for 6 gives

3a

6 = - E;' which is negligible for large values of y. Thus, for

large values of y, the imaginary part of a is approximated by

_ 4n+1
y (“Z—‘9n .

Having isolated the imaginary part of a, the real part, x, is

determined from equation (8.4) numerically with the compUter.

        

LIB

1111'“

99

 

 

 

 

MIITIIITIWINIWILIIEW1111011151111|
3 1293 0306171