GALCIT REPORT NO 1-IS Studies Relating To the...propellants, solid rocket grains, and solid rocket...
63
GALCIT REPORT NO (, 1- IS REPORT---- PASADENA, CALIFORNIA FORM AL · !I SM 7·H
Transcript of GALCIT REPORT NO 1-IS Studies Relating To the...propellants, solid rocket grains, and solid rocket...
GALCIT REPORT NO (, 1-IS
REPORT----
PASADENA, CALIFORNIA
FORM AL · !I SM 7 · H
FUNDAMENTAL STUDIES RELATING TO THE
MECHANICAL BEHAVIOR OF SOLID PROPELLANTS,
ROCKET GRAINS AND ROCKET MOTORS
GALCIT SM 61-15
GALCIT 11 8 - PROGRESS REPORT NO. 1
Aerojet Contract S-416394-0P
April 11, 1961 - June 30, 1961
FUNDAMENTAL STUDIES RELATING TO THE
MECHANICAL BEHAVIOR OF SOLID PROPELLANTS,
ROCKET GRAINS AND ROCKET MOTORS
P. J. Blatz
W. L. Ko
A. Zak
June 196 1
This program is being supported by t he Aeroj et-General Corporation, Sacramento Division, under t echnical cognizance of Dr. F. J. Climent
to provide t echnical support to the Polaris Project
Guggenheim Aeronautical Laboratory
California I nstitut e of Technology
Pasadena, California
FOREWORD
During the past thr ee years, the mechanical testing of solid
propellants, solid rocket grains, and solid rocket motors unde r id eal
ized conditions has bee n receiving increase d attention. Today it is not
uncommon to see a multitude of new techniques and analyses being
investigated. O ne may expect to see dummy propellant prepare d with
glass bead filler to observe its dilata tion to rupture; to ink circles,
rectangular g rids at various critical areas on a g rain surface, and to
observe the distortion of these grids as a result of thermal cycling
and/or slump; to subj e ct rectangular parallel-opipedal-shaped specimens
to both torsion and bia xial tension as well as hydrostatic compression
and parallel-plate t ension; to apply theories of large elastic stra in, and
non-linear viscoelasticity; to search for an isotropic failure crite rion
as well as a crack propagation criterion. I n short the mechanics of
propellant beha vior from small d e formation a ll the way to fr a ctur e
initiation and propagatio n has become quite sophisticated. Gradually
the results of this testing and their thinking a re being integrat e d i n a
logica l scheme of analysis which is being passed along to the engineer
and being used in predicting performance of rocket motors.
This p a rticular program will p e rta in to four areas:
1) The cha r a cterization of polyurethane propellant behavior
out to fracture initiation in terms of lar ge strain theory.
2) The development of a failur e criterion and crack propagation
criteria for said materials.
3) The gener a tion, where possible, of macroscopic mechanical
parameters in terms of molecula r p a r amete rs .
4} The solution of certain stress proble ms, in both linear and
non-linear theory, which are prerequisite to engineering
applica tions.
A s such it is part of a continuing research study of structural inte grity
problems in solid propellant rocket motors being conducted under the
general direction of Dr . M. L. Williams in the Guggenheim Aeronautical
Laboratory.
This preliminary report is i n tended as an i nt e rim working document
to be circulated for the purpose of stimulating discussion.
iv
TABLE OF CONTENTS ,:,
FOREWORD
I MECHANICAL BEHA VICR OF RUBBERY MATERIALS
(Continuum Rubbers, Foams, Propellants)
A. I ntroduction
B. Large Strain Theory
C. Experimental
D. Conclusion
II FRACTURE BEHAVIOR OF RUBBERY MATERIALS
III MOLECULAR STATISTICS
IV STRESS A NALYSIS
A. Introduction
B. Elastic Field in a Clamped-Free Sector Space
'~ The contents of this quarterly progress report comprise only
Chapters I and IV. Material for the other chapters will be included
in subsequent reports also to be organized as indicated.
v
I. MECHANICAL BEHAVIOR OF RUBBERY MATERIALS·
A. Introduction
The term rubbery materials include s both foams and filled
rubbers (which may be propellants) as well as continuum rubbers .
Experience dictat e s that the mechanical behavior of foams and filled
rubbers (up to 90 percent by weight solids loading) is controlled mainly
by the mechanical properties of the rubber binde r, and for this reason,
it is prudent to start this investigation with a review of the known
behavior of rubbery materials.
One typical feature of rubbery or hyper-elastic materials is
high extensibility under relatively small load. The deformation s are
so large (ultimate extension ranges from 500 percent to 1000 percent)
that the classical theory of small strain is no longer adequat e to expre ss
the behavior of such materials. The modern approach initiated by
Rivlin, Murnaghan, et al, is to find a strain-energy function from
which the constitutive stress-strain law can be established.
For a homogeneous, isotropic e lastomeric continuum" the work
done in deformation which is stored as strain energy, depends only on
certain invariant characteristics of the state of deformation. These
invariants in turn are functions only of the proper or diagonal value s of
the deformation t e nsor. The purpose of this introduction is to present
experimental data on a number of rubbers and show how, from these
data one can determine the relation of the strain energy function to the
invariants of the d e formation state.
Obvious! y the algebra involved in effecting this data reductio n is
simplified in the case of homogeneous deformations. For this reason,
specimens were tested in uniaxial tension and biaxial tension. Data were
procured on
a)
b) c)
0 SBR - 1500 rubber ( 1. 75 7o S )
Polyurethane foam rubber (47 fJo by volume voids - 40.M dia.)
SBR - 1500 rubber ( 3 fJo S)
-2-
B. Theory of Large Homogeneous Strains
For homogeneous deformations, it is convenient to define three
invariants as follows:
< r. 1)
( I. 2)
( I. 3)
where the invariants t I, , I 2 , I3
} are the set orig inally used by Rivlin.
Setting , the constitutive law reduces to :
~=I, 2, 3 ( I. 4)
where Q'. are the homogeneous components of the engineering stress L.
(load on undeformed cross-section)
Q'. ... are the homogeneous components of the true stress
A.. are the homogeneous components of the deformation gradients L.
(new length I unstretched length)
w is the strain energy I unit volume undeformed material.
For simple tension, the deformation-stress field is defined as:
A.J·- A.~c - A lat '
CJ.~. = O:.ni ) Oj = <J'k = 0
where A is the logitudinal extension ratio
)\lat is the lateral contraction ratio
Substitution into (I. 4) yields :
W 3 may be e liminated by subtraction :
( I . 5)
( I . 6 )
( I . 7 )
( I. 8 )
-3-
( I . 9)
Assuming w 1 and w 2 are constant, a plot of ~~/(A- 2 - A.~at) vs l/A.2 A.~at
may be used to d e termine the se material paramete rs.
For biaxial tension, the defor mation-stress field is d e fine d as:
A.i.= A.' J3 = AAtlt ( I . 10 )
OJ= Olat )
where A.th is the thickness contraction ratio
<J.lat is the lat e ral stress appli e d to maintain constant width.
Substitution into (I. 4) yie lds:
( I . 11)
0 ( I. 12 )
Again W 3
is e liminated by subtractio n:
(1. 13 )
Again (I. 13) m ay be plotte d to fit the assum ption that W 1
and W 2
are
constants . A ctually, constancy of the l e ft hand members of (I. 9) alone is
insufficient to guarantee that W 1 and W 2 are constants, si n c e they could
vary in such a way to maintai n the ri ght-hand member con sta nt . For this
r eason a doubl e che ck is needed and is provided by the type of biaxial
t e st d e scribed in (I. ll)and(I. 12).
We can anticipate two simple types of r e sults:
a) W 2 = 0, W 1 = constant = 'i (by comparison with linear theor y)
b) W 2 = W 1 = constant
-4-
I n the case a), it follows that:
( I. 14 )
W 3
= W 3
(J 3 ) independent of J 1 ( 1.15)
For this case, (1.4) becomes :
( I. 16 )
And from comparison with linear theory, the character of the leading
terms in the expansion of J 3
W 3
is established to be:
(I. 17 )
It will be important to establish the radius of convergenc e of the
entire section which this power serie s represents analytically. For, if it
turns out that
a) the ultimate dilatation J 3 ~" falls within the circle of convergence,
and
b) C(J~' -1 )2 <<-
3 (K- ~ M ) (J'~ -1) , then a very useful
3 3 constitutive equation develops, namely:
(I. 18 )
Consider now the case in which both W 1 and W 2 are constan t:
(!.19)
(I. 20 )
(I. 21 )
i nd e pendent of J 1 and J 2 {I. 22 )
-5-
For this case, (I. 4) b e comes:
cri-1\i.. = cJ.. J3 = .M L-!A~-+ ~~t 1 + J3~ I. 23 ...
Before contin uing, l e t us che ck the theory to this point.
C. Experimental Adductions to the The ory as Deve loped
1. Materials t e sted or to be t e ste d
a. Polyur e tha ne Foam Rubber (47 9o void)
Com p'Jsition : PPG:MTDA ~TDI = 70:30:107
The polyur e thane elastomer was fill e d with 47"o by volume of water
soluble sodium-chloride-salt and then caste d . The foam was obtained
by e xtracting the salt with boiling wate r, then dried. The ave rage diameter
of the holes is 40M.
b. SBR-1500( 1 . 75fJo Sulfur) = cured at 307°F for 45 m i n.
c. SBR-1500(39o Sulfur) = cured at 307°F for 45 m i n.
d. Natural Rubber(2fJo Sulfur)= cur ed at 307°F for 20 m i n.
e. Natural Rubber(4'Jo Sulfur)= cured at 307°F for 20 min.
£. Cis - 4 = cured at 307°F for 20 min.
Paracril cured 0 45 min. g. B = at 307 F for
h. Ne oprene - GNA = cur e d at 307°F for 20 min.
i. Butyl-21 7 = cured at 307°F for 45 min.
The mat e rials from (b) to (i) were cur e d fro m the gum stocks at
307°F i n a hot-press-mold for certain cure-time as shown above, then the
m old which contain s a speci men was r emoved from the pr ess and cooled
in the atmosphe r e.
2 . Specimen Geometries
. a. Uniaxial Tensile Test Specimens:
As shown in the ske tch (cf. Figure I. 1) two types of t est
specimens were used. The working length remains the same ( .Q. = 5 11)
for both types of specimen, and uJ= 1 11 for type I, uJ = 1/2 11 for type
II r es p ective ly. The thickne ss t for e ach t e st is listed on Tabl e I . 1 •
I n order to r educe the end e ff e ct both sides of both ends of a
spec i men we r e glued o n thin aluminum sheets. The met al sheets (or blocks)
-6-
which are thinner than the specimen were put in between the metal
sheets to protect the ends from being squeezed too far when gripped.
It is especially good for a foam rubber specimen.
The markings were made at the rniddl e edges of the specimen
by ink.
b . Biaxial Tensile Test Specimens:
" The working length for this case is ~= 1 for Type I and
~ = 1 I 2" for Type II, (and uJ >> .t >> t) as shown in Figure I. 2. The
thickness t and width .u for each test are listed on Table I. 1 .
A thin sheet of specimen is glued both sides of both ends on thin
steel plates to prevent the lateral contraction.
The markings were made at the middle edges of the working
length.
3. Test Procedures
All the test specimens were prestressed at constant temperature
up to 40'}o so'}o elongation few times to eliminate hysteresis. I n
orde r to read X.lat or X.th corre sponding to the equilibrium state in
uniaxial and biaxial tension, the crosshead was stoppe d and the chart
allowed to continue on until the load-time curve leveled off. For every -4
X., the corresponding X.lat • X.th were then measured to 10 in. accuracy.
In uniaxial tension, for extensions up to 1", the measurements of X.lat
were made for every 0. 25" interval of extension, and beyond that every
0. 5" interval. In biaxial tension, X.th were measured for every 0. l ''
intervaL
From the test data the following curves were plotted (cf Figures
3 ..-._... 26 ):
i) cruni v.s. l\. <rbi v.s. ;..,.
ii) o'uni A O'bi "-A2-A2 v.s.
(\'l)\:tiOit ' v.s. /\2/\2 Az.-A-z. I at th th
iii) "-iat v.s. A. ' Atn v.s. A.
iv} J-a(=l\~ ) vs A ' J3(= A\ ) '<S. A I at th
-7-
4. Results and Discussion
A complete set of derived information obtained from Figures
3 26 is shown in Table I. We note the following:
a) W 1 and W 2 are both non-zero, positive constants for
both continuum rubbe rs and foam.
b) The continuum rubber dilates enorrnously in biaxial tension.
c) The foam's behavior i n biaxial is comparable to its behavior
in uniaxial tension.
Since item b) is an u nusual but not unexpected feature of rubber, hitherto
unr e ported i n the open lit e rature, we plan to procure more biaxial data to
double check this result. Following this an effective value of Poisson's ratio
will be determined, using (I. 18 ). A compl e te discussion of this and the
isotropic failure criterion will be presented in the following quarterly.
.Q.=s"
\at A. lat.
METAL BLOCKS
TYPE I TYPE JL
DIMEf.JSIO~S t ARE LISTi=D 0"-1 TABLE I.1
FIG. I. 1. Ut-JIA).(\AL T~NSILE TeST SPECIMEI-JS
\
0 0
TYPE 1
IYPt=: 1I
0 0 0
Q. =: 1" , W AND t
l= Y2' ,
-9 -
~SPt=:ClMI:N
THI~ STEEL PLATES
GLUED ON A SPECIMEt--1
ARE LISTED 0~ TABLE I. 1
FIG. I. 2. BIA~IAL IE:t-JSILE TEST SPECIMEN
TABLE=. I. 1. LIST OF EXPERIMEI-JTAL RESULTS - 10-
Ut-JIAXIAL. TENSION
SB~- lSOO SBR- l SOO PU FOAM PU FOAM MAIERIALS
(1.75% S) (1.75~.5) (47% VOID) (47% VOID)
SAHPL...E 011"\ENSIONS 3 29'"64 >< Eo~64 X o/64 329-'64 "3;764" 5/64 40/B x~g x 3/S 4o/a ><4/e ><- 3/8
R. " w " t l : w : t 64 : 12.8 : I b4 : 6-4 : 1 13-3 : 2 -7 : 1 13."3 : ).3 : 1
AMBIENT TEMPERATUI<:E 75 75 75 75
OF 75 75 75 75 75 7S
• 107.2 loo.5 2-4.0 25.5 (J'uni ) psi.
96.9 30-5 30.5 107.0 115.0
A.. 2.15 2.40 \.90 2.00
2.23 2..60 2.20 2.34 2 . 40
• 0.&78 0 . &65 0.8(,5 o.854
Alat 0.657 0.598 0.682 0.818 0.816
J: (= t\,\~at) 0.':!90 1.020 1.420 1.460
0.992 0.931 1.020 \.565 1.598
• in-lbs 78.5 85.7 14.2 17.6 w, 1n3
82.2 115.6 73.0 28.8 32.8
~' psi. 17 14 3.0 2 . 9
17 12. 16 3 .2 3.0
psi. 23 2.2 \3 . 3 14.8
w2.' 20 2.7 19 15.8 15.2
psi. 80 72 32.6 35.4
_,(,( ) 74 78 70 37.6 36.8
dJYdA-0 0 0-474 0 . 471
0 0 0 0 .462 0 . 452
BI~IAL. TE~S\Ot-J
SAM_ft...~ DIMENSIOt..IS X \JJJW. t &4/<;.4 X 34%4 ~ %4 3~ x340k ~s/e 64 6+ b4
8/e,.. 5118 x 3/s i6JI6 X I02/I6 X 3 /j 6
1. : lJJ : t 12. 8 : 68 : I '-.4 : 68 : I 2.7 : 17 : 1 5.3 ; 34 : 1
AMBIENT TEMPERATURE 75 75 7S 76
op 75 7? 75 78 76
"' 114-4 101.7 22.6 19.7
crbi )
-ps1. \15. 7 too.s 113.2 18.5 24.1
~ 2 . 2.0 1.90 2.10 2.10
2.00 2.00 2.40 2.10 2.10
• o.e.so 0.741 0.848 0.850
.A:th 0.749 0.700 0.652 o.891 0.813
J: ( = " >--th) 1-43 \ .4- J \.78 \.79
1.50 \.40 \.56 1.87 l. 71
w"' in -lbs 86.0 57.2 19.2 16.0
) in~ 72.7 61.7 lo4-.o \7.2 19.1
\6 13 3 1. 5
w, '
psi. 13 16 II 2 2.5
34 35 14 13.5 w2.
' ps.i.
47 25 36 14 13.0
86 92 34 30 ~ ) ps.i.
118 82. 94 32 31
dJ% 0.421 0.457 0.75 0.7o
d/\, 0.525 0 .400 0.459 0.73 o.~s
' ·
-1 1-
U t-JIA X I A L T E N S ION
S BR- 1500 S BR - 1500 MAT!:RIALS ( 3% S) ( 3% S)
SAMP\....E DIMENSIONS 32~64" 6""64" 5/64 3 2<y64 ,.37G4>< s 16+ .Q. x w- " t Q. : ur : t G4 : 12.8 : 1 64 : 6.4 : 1
AMBIENT TEMPERATURE 77 76 OF 77 77
• 12 9 . 0 120.0
<J'..,ni ' psi . II B . o 129. 8
A.~ 2.00 2. 0 0
2.0 0 2 .00
• o . G9 8 0 . 711
A lat 0-73 0 0 .711
• ( 2 0 . 975 l . o 1
J 3 = .A .\1,.t) l .o s 1. o I
• in- lbs 79 . 1 7 7 . 4 WJ ih 3
74. 1 78. 5
w,, psi. 2 6 19
22. 2.4
-psi. 2 1 31 Wz, 24 2.5
-psi. 9 4 100 M ) 98 92.
dJYc 0 0 d .>-.. 0 0
B IA X IAL TE"--SION
SAMPL-E DIMENSIOI-JS 6~4 x3+o/64 x s/64 3%4.)(3~4" 5/64 1 x ur 'X t .2.. : u.r : t 12. 8 : 68 : 1 6 .4 : 68 : 1
AMBIEI-IT TEMPeRATURE 7~ 7 7
OF 7b 77 .. I 03. 2. I 2. 7 . .3
O'bi '
t>si. 9 6.5 11 0.0
". \.7 1.7
1- 5 1.7
• 0 .7 7 6 0. 8 0 4
A th 0 . 91 4 0 - 8 04
J: (= .\ /\th) 1.3 2. \. 370
I· 22. 1. 370
w• in - lbs 44. 2 5 1.5
) in:! 2.8. 2 48. 1
2.1 15
-w; '
psi . 13 15
3 0 5 1
w2. ' psi. 52. 5 9
102 148 ~ ) -psi. 1.30 I 3 2.
dJ% 0 .45 0.5 8
c.\ 0.51 o.se
O'u
ni
(~si
.) IZ
O
100 80
60
40
20
0 l.
o
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t =~
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A
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'0
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od
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psi I VJ
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FIG
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: 1
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urx
t =
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)
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----------t-~~=--+--+-----~~~
1.1
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I I
I I I I
l.o
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I
1.0
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1.2
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-36-
IV. STRESS ANALYSIS
A. I ntroduction
As a result of elaborate studies of the restrictions imposed
upon interior ballistic design by exterior ballistic considerations, most
large case-bonded solid propellant rocket motors possess a srnall
aspect ratio, of the order of three to one. This means that the assumptions
made in a plane strain analysis generate a solution which is valid only
over a very small fraction of the length of the grain. Thus the problem
of obtaining useful solutions near the ends of a grain becomes important.
In effect this is being done experimentally by photoelastic technique and
analytically by the r e laxation technique. Unfortunately in both these
cases, no estimate of the error i nvolved at the corner is generated.
The corner problem entails its own unique analytic features which are
incompatible with the behavior of real materials and i n compatible with
the assumptions of relaxation.
Thus two problems demand attention. In ord e r for the enginee r
to predict yield or some form of failure due to high stresses at a corner
he must first have the elastic stress field for a perfe ctly elastic material.
But in order to obtain this rapidly with the aid of a relaxation t e chnique,
he must first solve the particular analytic problem close to the corne r .
The complexity of this corner problem depends intimately on the geometry
of the body. We propose to study this problem in four steps . /t-i'-
First we shall consider the plane strain anqlplane stress
deformation of an e lastic sector-space, subjected to the condition of
clamped-free boundaries . Se~ondly, in a subsequent report we shall ,.1 ....
consider the plane strain an9 plane stress deformation of an elastic slab.
Subsequent treatments will deal with the e lastic column subjected to
radial tension or compression, free axial face; and lastly subjecte d to
axial tension or compression, free radial face. The last problem refers
to the so called parallel-plate test used for det e rmination of the dilatatio nal
component of the strain energy.
-37-
B. Elastic Fie ld in a Clampe d-Fre e S e ctor-Space
1. G e ometry
For the purpos e of this discussion, the e lastic sector-spac e
will be bounde d by two edges - one free locate d at e = 0 • and one
clamped located at e = ()( I n the case cJ.. > rr , the sector d e ficie n c y
of the total space will be termed a crack of flank angle ( 2rr - o/.. ). And
as ol.... - 2rr , the crack degenerate s to a radial line. In the generalized.
sense, the term space encompasses the two limiti ng structur e s of infinite
a nd zero thicknesses. I n the form e r case, the bounderies are planes;
in the latte r , edges. The language of the associated analysis will be
couched in terms of the paramete r C) which, for plane strain, is
identical with -tJ , and which, for plane stress, is replaced by V / ( 1 + v ).
It is convenient to refe r the corner of the spac e to the origin of a system
of polar coordinates; the two edges are then geometrized as rays .
2. The Biharmonic Function
Under the influence of arbitarily imposed surface tractions a nd
displacements (no body forc e s) equilibrium in the plane e lastic fi eld is
represented by the solution of the followi ng e quation s :
Cl?::. re --~ +.
d l
I d 'L:ye -+--y:- "de = o
-+ _ 1_ aO'e r "de-
C1 O"e ----= 0
Lrz =- Lez = 0
-0
(IV. 1)
(IV. 2)
(IV. 3)
(IV. 4)
Application of the method of characte ristics to (IV. 1 ), (IV. 2) i n troduces
the stress fun c tion p from .which the stresses are g enerat ed by:
-3 8 -
() ~ ... + q, E:'J9 -r r !2 (IV. 5)
0::= e t,.. ... (IV. 6)
LYe= ~- ~re r-2. -y-
(IV. 7)
The subscripts affixed to <f' denot e the appropriate partial differ entiation.
From Hooke' s Law, we have:
2)-<Er= (1-0'J CJ-r- (}()s (IV. 8 )
(IV. 9)
2;..< E~ = ~ - cr ( CJy + CJe-) - 0 i n plane strai n (IV. lOa)
2)-A. E-z = - o- l ().,_ -T (j~) in plane stress (IV. lOb)
(IV. 11)
E:.,...~ = E e~ = o (IV. 12)
The strains are generated from three displacement components
u. for radial
for tangential
for axial, as follows:
-39-
(IV. 13)
U I "d ..r--Ee=---:y:-+ Y'~
(IV. 14)
I ( I "dU "d\r _ ~ \ Ere= 2 y de -;- -en- I )
(IV.l5)
"dVJ E-e= "dz =o, l.V=O ,for plane strain (IV. l6a)
- harmonic function • for plane str e ss (IV. l6b)
Since Er, E9
, and Ere are generated from only two of the three
displacement components, their compatibility with simply-connected
Eucledean space must be guaranteed. Elimination of U a nd IJ from (IV. 13),
(IV. 14), (IV. 15) yields
(IV.l7)
Returning to (IV. 5), (IV. 6), (IV. 7), via (IV. 8 ), (IV. 9), we find that the
compatibility of the strain components specifies the characte r of <f' :
or (IV. l 3b)
Equation (IV. 18b) is termed polar biharrr1onic because it expresses
the ite rate d application of the polar Laplacian o n the stress function .
Because of th e de gene racy of the roots of the characteristic e quation,
representing (IV. l 8b). care must be exercised in expressing the complete
-40-
set of solutions which span the biharmonic fie ld. Noti ng first that
(IV. l8a) is differentially homogeneous i n r 4 , we set r = e s, and write:
,+, -4.-t. --t-4..+.. -t-2~ -4..+-. + 4,f, +,f., -0 'Tssss '::tsss 'I'ss 'I's:seQ 't'see 'I'ee 1e~ee-
Separation of variables is accomplished by writing <f> = S(s)
so that:
(IV.l9)
Equation (IV. 20) is the characteristic representation of (IV. 1 8a) a nd is
meaningful only for the substitutions:
® o, k
'2.
+ ) 2. -\ e~n~
Let us consid e r each of five substitutions in turn:
a) II
ij ::: 0, @ = A +B e
s /11/ - 4- sIll + 4- s " = 0
+ 3 2. ,._ ( \'2.. D- 4 D + 4- D ==. D D-2) =o
2S 2.5 C-+- D s + t= -e. --t- Fse
® = A -s inh ke --t- B Cosh 'r<.e
(IV. 21)
(IV. 22)
(IV. 23)
(IV. 24)
(IV. 25)
-41-
'= (D-+i.k)( D- ik') ( D-2+ tK-)(D-2-tk)= o (IV. 26)
(IV. 27)
(IV. 29)
D 4 - 403 -t- 2 (2.- k2 )"D 2 +4- k,_D-+ k 4 - 4- k:z. (IV. 30)
-= (D+k)(D-k)(D-Z+k)(D-2-k) = o
(IV. 31)
)( \<:~r - k~Y" 2 k~r 2 - k..k r cp =(A-sin ke -t BCoS K6 Ce +D-e. -+ E=. T" -e. -t r y -€... ) (IV. 32a)
1<. -k 2+k '2.-k) <f' = (A sin \<6 -+ B Co-s \<e) ( C y + D "'r -+ E T -+ F Y (IV. 32b)
d) II ( 2. - \) 0 =- ® \ et;ane ., ® = e sine
s""- 4 s ''' + s" ( 2 + 4 '\- 4 s' (-2 -- 1) + s (----=!__ -3\ = o
El "taneJ et;ane et;ane ') (IV. 33)
-42-
s'"'- 4 s"'+ 2 S" + 4 S '- 3 :3 :::: ( D- \ 'l4
( D 4 \) ( D- 3) = 0 (IV. 34a)
4 s"- s s' + 4 s - ( D- \)2
= o (IV. 34b)
s s s- A€.-+- BS€.. (IV. 35)
q_, = e -sin t7 (A r + BY k r) (IV. 36)
e} 11 11'11 ( 2 i:ane\ ®=-\1}1 I+ e), @ = 6 CoS~
(IV. 37)
~11 ''- 4 s'"-+2S"--+4-S 1 - 3 ~ =. (D-1)2
(D+t")(D- ~) = 0 (IV.38a)
-4S"-+ e $'- 4- S - - ( D- \)2 - o (IV. 38b)
S - A e. s + B s-e.. s (IV. 39)
(IV. 40)
Equations (IV. 21)- (IV. 40) repr esent the complete set of known
analytic solutions to the polar biharmonic. I n addition to the s e, there are
presumably a large class of non-analytic functions which represent the
application of a biharmonic fi e ld to a si ngular geometry. A complete analysis
of these singular fields has not been carried out.
-43-
Roughly speaking, solutions of the form (IV. 28) are used when
it is desired to expand arbitrary boundary conditions along the boundary
rays . In doing this, care must be exercised in approaching the limit
r = 0 ; which rneans in the problem at hand that a small sector of
radius E must be cut out of the corne r and then be allowed to shrink to
zero. By the same token, solutions of the form (IV. 32b) are used when
expanding boundary conditions along tangential trajectories, such as
when the sector- space is replaced by a finit e sector of radius R. In
this case care must be exercised in approaching the limits e:: 0 and
e = o1. For the most gene ral type of problem - a segment of sector-
space bounded by rays 9 = 0 and 8 = ot.. and by arcs r = E and r = R, it
is nece ssary to compound linearly solution s of the form (IV. 28) and
(IV. 32b) i n order to expand arbitrary boundary conditions on all four edges.
This becomes quite complicated and tedious . Very often the Fourier
sums conve r ge extremely slowly. A great d eal of funda1nental analysis
remains to be done in this problem area.
3. The eigenvalue approach
It is co nveni ent to regroup the t e rms of (IV. 32b ) i n orde r to
factor out a single term i n r, as follows:
4 =!»+'[A-sin (n+l) 6 + B Cos (YI-+1) e-+ C <s;n (n-C) 6-+ ""[)COS (."n-l)e 1 (IV. 4la)
Since it will b e shown late r on that \n\< i , for the problem at hand, we
shall rewrite (IV. 4la) as:
(IV.4lb)
Note that n may be positive or negative . I n the latter case, t he pre
exponential factor becomes r l- n while the post-trigonometric factor
retains its character. The importance of t his statement will become
apparent.
-44-
Since i n the problem of sector- space bounded by two rays, only
four homogeneous boundary conditions are involved. only four constants
A, Br C,. D, are needed, and it will be assumed that the space can be
spanned by a single eigenvalue. The legitimacy of this assumption remains
to be established. I n a finite bound ed spac e, at least six or eight boundary
conditions are involve d, whether homogeneous or not, and in this case n
is construed to represent a set of eigenvalues. Furthermore, it is usually
necessary to determine two sets of e igenvalues for both radial and tangential
expansions. Thus we can attempt to get by with a single eigenvalue only
because a sector-space is involved.
4. Eigenvalue solution for a clamped-free quarter space
Sketch 1 reveals the analytic nature of the boundary conditions
at the free and clamped edges of a quarter-space .
e= lT 2
U=O
'j
t.T=-~
O'e = ?:'re = 0 ' e=o
-00
Sketch 1 Clamped-Fr~e Quarter Space
Formally speaking, ate= 0, the free condition d err1ands that au.x
(J'J = ?:..,'/ = 0, and ate = ~ ' the clamped condition demands that u)' = d 'j = o. The listed boundary condition follow from:
a:= ()'r sir{e + ere c.os"'e + Lra Sin 29 - ()e I ..; e=o
<Yr- <Je 2. 'Cre\
e== o
(IV. 42)
(IV.43~
Uj = U Sin 9 + V" Cos E1 = ul , \ 9=2.
<:lUx- (sine_:.,..+ c.os~ ~\(u.cos& -v-SinG\ =- d'V \ ~- "'' -y- "d9) J ~r e=~
2.
-45 -
(IV. 4 4 )
(IV. 45)
5. The str e ss e s and displacements i n clampe d-fr e e quarte r space,
We r e writ e (IV. 4lb) as :
1> = -r'-+'n F(e) , whe r e
F (9) = A c;>in ( 1-+ Yl ') & -t- B cos ( 1+ n) 9 -+ C. Sin ( 1-"Yl') 9-+ P CoS (l-"Yl) 9
From (IV. 5) (IV. 6 ) and (IV. 7) evolve the str e sse s:
CJr -
o'e =
(I+Y1}F-.F 11
l \ -YI
(1-t-YI)Ylf= r ,_.,..
(IV. 4 6)
(IV. 47)
(IV. 4 8 )
(IV. 4 9 )
(IV. 50)
N ote that if inde e d IY\\ < 1 , then all the stress com ponents are singular a t t he
corne r, a nd a fortiori , if n is negative . Combining ((IV. 8 ), (IV . 9 ), a nd (IV . 11))
with ((IV.l3~ (IV.l 4 ), a nd (IV. 15)), we obtain:
(IV. 51 ) y 1-Y\
- ( u. 1 'dV") (1-+n)[(H·Y\) o--n1 F-t- <J F'' 2 ;U \y + y ¢G = - --'--- ---'---"-.._:___.:_ __ ~---r ,_...,
v- _ (1-+n)(I-Y\2
)(1-o-) ~ d 1- (1-l'\) cr- =' I fde ~ 2}-<--- Fe- .- -~-+--
~'"' 11. y 1-YI • Y\ y' n ""'r y
(_rde-+- c & + I - 0"' r Ill+ ( Q 1 _ _L \ -+ J ) J Y1Y' - "" \ d y) y
- 46-
(IV~ 52)
(IV. 53 )
(IV. 54)
(IV. 55)
(IV. 56)
(IV. 57)
(IV. 58)
(IV. 60)
(IV.6 1)
(IV. 62)
In (IV, 62) the t e rms i nvolving F satisfy identically as can be
observed by usi ng (IV. 47 ). It follows that :
~( 7 <1}) - ~"" .J):.,"
~
g=Kr
f ::: I si n 8 + J cos8, whe r e I, J, K, are constants. (IV. 64)
From c ondition s (IV. 42) and (IV. 43) w it h (IV , 49),(IV. 50} it follows
that :
F (0) = F 1 (0);; 0, frorr. which
D = - B
C = - A ( I+Y\'\ \ l- Y\ -)
, from which
F(e) =A [sin (l+n)e- \~~sin (1-n)e]+ e>[ c.os(1-+-ni e- cos ( 1-n) e]
(IV. 65)
(IV. 66)
(IV. 67)
(IV. 68)
F(El) = A (I+ n) L Cos ( I+Y\) e- c.os (1-'r\) e] - B [(I+ n) 5 in (I+ Yl) e- (1-Y\)Sin (1-n)e} (IV. 69)
FC9) =: -A (1-+Yl)l(I-+Yl)Sin (I+Y\) e- (1-YI)Sin (1-n) e 1 - (IV. 70)
- B[ (1+)1)2.CoS ll+YI) 6- (l -Y1)4
CoS (1-YIJ 6 J
\Fd = -A[cos(l+n)6 _ (l+Yl.) CoS(I-n)e]-tsfsin(1+-n)6 _-sin (l-n)e] (lV. 71) j e 1-+n (1-Yl.)2. L 1-tn 1-n J
Fr o m condi t ions (IV . 4 4) and (IV. 45) with (IV. 52), (IV. 57~, (IV. 63),
(IV. 64), it follows t hat
-tee) = I - J - o (IV. 72)
-48-
d-'=k=O (IV.73)
t=\-!-) = _ 2A (l+n) -sin n ~- 2 B Yl Cos Y1 ~ (IV. 7 5 )
I.
Fde= A 'S1n n-- 2 B cos "n--;::;-~ 2 2 ( 1 + n:1) . '1T 2. n "lr
(i+n)(l-n)l.. 2 1-n ~ (IV. 77}
0 =A ( l+n)(2.-n-2.cr) cosni -+B(1-n)(t-Y1-2CT)sinn1 (IV. 7 8 )
o= A(I+Y1)(1-tn-2<r)'Sinni- B ( l-n)(2..+Yl-20")C.oSh1_ (IV.79)
E quations (IV. 7 8 ) and (IV. 79) possess a n o n -trivial solution only i f
the d etermi nant of the coe fficie n ts of A and B is z e ro. This leads to :
r 2. 1..} 2. 1T" r z < )2. ] . 2. or L4(1-o-)-YI c.osn-z=tn- 1-20" s1nn2 , or (IV. 80'
1- 2cr =- cos2n-; -+V Y\
1- CoS
2Y1 ~-:;in"-"'~ (IV. t'. l}
Figure 1 shows the b ehavior of the magnitu d e of the e i genvalue n as a function
of P oisson's ratio 0" • Note that as cY- o , n must a rproach and is bounded
by unity, whereas when cr- 1 /2, n ~cos n i , so that the magnitude of the
minimum e i genvalue is 0 . 595. Si n c e ( 1-2 lJ) is an even functio n of n, it
- 49 -
follows that b oth positive a nd negative value s of the e i g e nvalue are
e qually acc e ptable. w e will decide from physical co nsideration s wh ich
values are extrane ous. Finally obs e rve t h at, for 0' = 1 I 4, n ........_ 3/4 ,
which might b e c h ose n a s a typically repres e ntative e igenvalue for
materials of inte rrnediate dilatan c y ( 1) .
From (IV . 7 8 ) w e have :
A 1-n B l+Y'l
n-1+2U' ta ""rr > 0 n••-'2.-20"-n 2
_Q_- - 1 B -
c ~=-
n- (1-z.o-") 1T --------'--""t -q n -n 2 2-2.()-n <.O
T he str e ss function can n ow be writt e n in the followi ng way
<f> - Bn rn..,., Fn (e)
(IV. 8 2 )
(IV. 83 )
(IV. 84)
(IV. 85 ~
w he r e for each e i ge nvalue n the re e x ists a con stant B a n d a fun ction n
F n ( e ). The above expr e ssio n for <p is a complimentary solution,
potentially comp l e x d e pen ding upon the potentiall y comp l ex n atur e of
the e i ge nvalue s. In orde r to pr e scribe a d e fi n ite ta ngential di splac eme n t
( 1) Max Kne i n . Z u r Theorie d e s Dr u ckve rsu chs. Abhandlunge n aus d en> Ae rodynam i s c hen I nstitut a n d e r T echnischen H o chs chule Aac he n, H e ft 7, 43 (1 9 27)
-50-
on the ray e = rr/2 a particular solution has to be found which
generates this displacement. The unknown coefficients Bn in (IV. 85)
have to be evaluated by c o n sideratio n of appropriate boundary conditions
along a circumferential arc far fron1 the corner.
Similar arguments apply immediately to the free-free and
clamped-clampe d spaces. These cases were all discussed by M . L.
Williams(2
) who, recognized and pointed out that n must be regarded
as complex if one is to fit non-trivial boundary conditions. In addition,
Williams writes down the general for mulae for the eigenvalues in
spaces with non-normal corners. Knein restricts his discussio n to
normal corners.
5. Complex Variable Approach
One of the difficulties i nhe r ent in the e i genvalue method restricted
to the real domai n is that it operates o nly with one set of a pair of
degenerate set of functions which satisfy the biharrnonic equation.
For the general contour one must use both s e ts -the circular and the
hype rbolic functions. These sets may be combined by the introduction
of an appropriate variable.
The apparent difficulty connected with the use of (IV. 2 8 ) arises
from the presence of the logari thmic argument. This however can be
disposed of very neatly by mapping the sector of flank angle ex: (we
now generalize to the non-normal corner) onto a strip of width o< •
This is accomplished by the mapping (cf Sketch 2)
or (IV. 8 6)
(2} M. L. Williams: Stress Singularities Resulting from Various Boundary Conditions in Ang ular Corners of P lates in Extension, Jour n al of Applied Me c ha nics , 52 6, ( 1 9 52 )
-51-
8=o<.
;:nr u=o ~=o,
e=o ()a=o
Sketch 2
Note that this mappi ng allows parallel lines, which r e place the rays, to meet
at infinity. I n order to specify the boundary condition s on the paralle l lines,
the expressions for the stresses and displacements i n terms of complex
pote ntials must be i ntroduced.
The basis for this technique l ies i n the identification of the g eome try
of the elastic body with that of the complex plane. This i mmediate ly bestows
a large number of useful and simplifying features upon the field e quation .
mappin g :
The polar plane is related to the complex plane by the a nalytic
i.e z=re
'
-i.e z = ye (IV . 8 7 ~
(IV. 88 )
Treati n g z and z as two new i nd ependent variables a nd followi ng the usual
rules of calculus of transformation of var iables , we can co nvert the e quilibrium
equations, and the compatibility equatio n to the new variables as follows :
. - d t z "dz
(IV. 89)
(IV. 90 )
-52-
(IV. 9 1)
Note the e l ega nt form of the Laplacian. Now i ntroduce ne w stress components
and r e write the equilib rium equatio n s :
s - <Yy + 06 (IV. 92)
"t= O"r-O'a +
2 1 ""L re (IV. 93)
t= O"r- O"'e - i Lre 2 (IV. 94)
(IV. 95)
r:. sz + 2 t + 2 ~ ~ z = o (IV.96)
Now i ntroduc e the stress fun ction:
4 <p -Zl
(IV. 97)
(IV.98)
(IV. 99)
New strains are i ntroduced a .1d then Hooke 1 s Law ·
'0= E..,..- E:a + 1, E: re 2
Ey- Ee i, Er-a 0 - z
2M1.9- = (1- 2.CJJ S
2)-{ y-= t
2M o = t
New displacements are i ntroduced and related to the strain s :
U+U. --2 + z Uz-+ z u~
19-=---=---------vz z'
¥- ZVIz - U/2.
v ~z.' ~ u .... - Vl/2. T = --=__::_:._"'-!:.__-=--"--='--
1/~~
Elimination of u and u from (IV . 108 ), (IV. 109), (IV. 110) establishes
compatibility:
-53-
(IV . 100)
(IV.10l)
(IV. 102)
(IV. 10 3)
(IV.l04)
(IV. 10 5)
(IV. 106)
(IV. 107)
(IV. 10 3 )
(IV.l09)
(IV.110)
(IV. 111)
- 5·:1:-
I ntroduction of the str e sses from (IV. 103), (IV.104), (IV. 105) and then the
stress function from (IV. 97), (IV. 98), (IVo 99) e stablishes the biharmonic
character of ~ :
(IV. 112)
I n a purely formal fashion, if <f> is assumed to be analytic, (IV. 112) can be
i n teg rated to expr ess 1' i n tern1s of two arbitrary holomorphic fu nc tions
~ and 'X, bot h of which must be harmonic:
(IV. ll 3)
The derivatives are us eful
(IV. 114)
(IV. 115)
(IV. 11 G)
Note that cy - 9 (z) depe nds only on z and so has no derivative w ith respect
to z. This is significance of harmonic behavior. Note also that
(IV.ll7)
since~ may be a complex fu n ction.
The stress and displacements may be expressed in terms of t and
(IV" 118)
(IV. 11 9)
(IV. 120)
-55 -
(IV . 121)
(IV . 122)
R e .. memberin g that z2 =r, it is r e adily observe d that C g e n e r ates a pure
totation and B a li near displace m e r.t along the x-axis. Thus Band C m ay be
s e t e qual to zero with no loss of gene rality.
6. Boundary co nditio n s i n the Complex Plane
From (IV . 92), (IV. 93), a nd (IV. 94), w e have ;
0 9=0
"t- t - 2 1- ?:.,..6 - 0 e-=o
T herefor e t must b e real.
Along the other ray, w e have from (IV. 10 6 ), a nd (IV. 107)
U+ U = 2 U. =- 0 r -at e = o~.
The r e fore 'U. rnust be i maginary. Al so from (IV. 89\ w e have:
"dU.\_ (fT _2__ +"'~ ~ \ ( u- V.) - o <n· - ~ ~~ f~ "dZ )
9=~
which b e comes.
(IV. 123'1
(IV. 124)
(IV. 125)
(IV . l2 6)
(IV.l27 )
The problem n ow is to choose 9 or ~ and X so as to most co nv enientl y fit
the s e boundary conditio n s. I n additio n to the set (IV . 123), -(IV . 127) w e s h ail
exp e ct to impos e some restriction , at r ~ m, t hat e ithe r the stresses or
displacements g o to zero.
- 56 -
As already i ndicated, the problerr1 of choosing is tremendously
simplified by mappi ng the sector onto an i nfinite strip. This transformation
introduces new variables:
_2_ - ~ ~
d~--e. ~
d -~ d
"d'i = -e ~ :s so that the Laplacian becomes:
The biharmonic loses its sirnple one-term character:
A n d the complex stresses, displacements and potentials become
On the boundaries of the strip, we now have
_2_=t=t 2
(IV.l28 )
(IV. 129)
(IV. 130)
(IV. 131)
(IV. 132)
(IV. 133)
(IV. 134)
(IV. 135 }
(IV. 136\
(IV. 137)
(IV. 138 )
-57-
at )- $ = Zi.ol. J (IV. 139)
Equations {IV . 138 ) and (IV. 139) may be solve d by first taking
Fourier transforms a nd then solvi ng a pair of simultane ous algebraic e quations
and then inverting the Fourier i nt egrals . The exact d e tails of this procedure
are b eing worked out and will be r e ported i n the next m o n thl y report. I n
addition, the sector-space will also be mapped o nto a half-plane by the mapping
(IV. 140}
This reduces the i n tegral equations to a Hilbert problem, which will be
discussed i n d e tail in the subseque nt r e port.
0
~ Ci
,l!l 2 0 CfJ CfJ 0 (l_
7_ 0
ill :J J
~ 7. w ~ ill
lL 0
ill u 7_ ill 0 z w 0. ill D
0
\
\
\..0 ~ · ,....,
I
I
I
i\..
-Sa-0
co
,.,.
\[") o ·
