STRESS ANALYSIS OF rnPIcAL FLAWS IN AEROSPACE STRUCTURAL COMPONENTS US WG
3-D HYBRID DISPLACEMENT FINITE ELEMENT METHOD
Satya N. Atluri* and K. Kathiresan**
Georgia I n s t i t u t e of Technology
A t l a n t a , Georgia
Abs rrac t
known asymptotic s o l u t i o n f o r
An assumed hybrid displacement f i n i t e element
method f o r the solution of modes I, I1 and 111
s t r e s s i n t e n s i t y factors which vary along a n d r b i trary curved three-dimensional crack front i n s t r u c t u r a l components was developed f o r linear l y e l a s t i c
materials, The present method can be applied t o
three-dimensbmal mixed mode f r a c t u r e problems with
Special
complex crack and s truc tura 1 geometries
crack front singular elements were developed Where
proper asymptotic solutions f o r displacements and
s t r e s s e s a r e embedded. The above f i n i t e element
method i s presently used to analyze some important
and typical flaw (fracture) problems which a r e commonly encountered i n aerospace s t r u c t u r a l component
applications.
d i s p ~ a c m e n t s(which prcd*
I ace
singular s t r a i n s and s t r e s s e s )
unknown independent parameters
vector representing * d e a 1,II
and 1x1 s t r e s s tntenstty f a c t o r s
assumed i n t e r p o l a t ion functfons
for boundary displacements
assumed boundary t r a c tiona which
contain r e g u l a r a s well as s i n 6
gular modes
clement nodal displacements
.generalized nodal displacements
.
,
global amtrieea a f t e r the assembly of element matrices
iooparame tr LC nondhene iona 1 coordinate eye tern
Cartesian coordinate rystem
displacements i n x,y,z d i r e c t i o n s
respectively
Poisson's r a t i o
coordinates a s defined i n Ref.
27
Maxwell's stress functions
Nomenclature
t o t a l p o t e n t i a l energy of the hybrid-displacement model
volume of mth f i n i t e element (m =
1,2,3,. .,N)
t o t a l number of f i n i t e elements of
the s t r u c t u r e
e n t i r e boundary of mth element
that portioil of aVm where T a r e
i
spec i f ied
t h a t portion of a V where Gi a r e
m
specified
independently assumed i n t e r i o r d i s placements f o r each element
(ui + u
) / 2 within each e l e ment vm j a i
independently assumed inter-element
boundary displacements which a r e
inherently compatible
Lagtange mu1 t i p l i e r terms which
a r e phys i c a l l y t h e independently
assumed a r b i t r a r y inter-element
boundary t r a c t i o n s
body forces
.
s t r e s s tensor < i l x , y , e ; j l x , y , z )
outward normal vector a t the
boundary
complete e l l i p t i c i n t e g r a l of
the second kind
e l l i p t i c a l angle
(edge)crack length and dimension
of the e l l i p s e (crack)
dimension of t h e e l l i p s e (crack)
radius of the hole
dimensions of the p l a t e
thickness of p l a t e ia f a s t e n e r
hnle problems and surface flaw
problems and thicknese of adhes i v e i n bonded p l a t e problem
t h i r h e s s e s of adherents
specified s u r f a c e t r a c t i o n s on Sa
m
specified boundary displacements
on Su
elast'Peity tensor
shear modulii of adherents
shear modulus of adhesive
cltt, (i=1,2)
elemental volume
elemental s u r f a c e area
regular polynomial basis functions ,
which do not include any r i g i d
body modes, for i n t e r i o r displacemen t s
regular polynomial bas i s functions
which contain only r i g i d body modes
for i n t e r i o r displacements
Y1
- + -Y2
h
p2
t e n s i l e load appf ied to the
plate
modes I, 11, and 111 s t r e s s i n t e n s i t y factortl,=(KI*KIIdCIII 1
I.
*Professor, School of Engineering Science and
Mechanics, Member A I M .
MPos t-Doc t o t a l F e l l o w , School of Engineering
S cience and Mechanics.
Introduction
The importance of determining t h e s t r e s s fntens i t y f a c t o r s f o r the investigation of f r a c t u r e a f
s truc t u r a l components need not be o v e r e m ~ ! ~zed,
n~f
e s p e c i a l l y s o i n the case of aerospace s t r u c t u r a l
components where accurate estimates of s t r e s s e s and
.
CrS.ri.:. tn the n - o f - d p cade, meny s t r u c t u r a l engineers and s c i e n t i e t s ha*
fscwsed t h e i r a t t e n t i o n t o f r a c t u r e problem s o
that t h e e t r u c t u r a l components may be analyzed and
designed f o r t h e i r intended purpose without c a t a strophic f a i l u r e due to t h e presetace of flaws o r
imperfections which may be present i n the s t r u c t u r e
due t o manufacturing d i f f i c u l t i e s , tool. marks, e t c .
In applying f r a c t u r e m c h a n i c s theory t o problem
with such flawe and imperfections i n s t r u c t u r a l
components, the analyst must know the stress intens i t y f a c t o r s a s well a s m a t e r i a l c h a r a c t e r i s t i c s
l i k e f r a c t u r e toughness, i n order t o assess the
a c t u a l s i t u a t i o n . While experiments ard needed t o
find m a t e r i a l charac t e r i s t i c s l i k e fractu'ke toughness, exact and cloeed f o ~ , ~ o l u t i o la ur e atrpilable
for some f r a c t u r e problems
, b u t they a r e I h i t e d
to p o b l a ~ u with
,
simple geometries of s t r u c t u r e ,
cracks and simple logding conditions, Even in the
case of two-dimensional problems when the geometries
of s t r u c t u r e and crack a r e no longer simple, the
closed form s o l u t i o n become extremely tedious,
forcing t h e inves t i g z t o r t o make sbilplifying assumptions which only approximate the a c t u a l condition.
In the ca :e of three-dimensional problem with a r b i t r a r i l y curved crack f r o n t , the mathematics Is so
complex and cumbersome, t h e c lased form s o l u t i o n is
a lmos t Lmpoes i b l e
r t e s i m arc! mcre crftfral
.
Severa 1 approximate me thods have been f o m la ted
and developed t o solve two-dimensional f r a e t u r e
problemj accurately i n r e c e n t years a s s u m a r i z e d
by Rice
A most t e c e n t survey on the a v a i l a b l e
methods of s o l u t i o n f o r f r a c t u r e problems was done
by pian4. Evan though a considerable amount of
research work has been done on two-dimensional fracture problems, the work on t h r e e - d h n s i o n a l fracture problems is very limited. Currently, f o r
analyzing problems of p r a c t i c a l significance i n
aerospace s truc t w a 1 a p p l i c a t i o n s such a s corner
cracks o r i g i n a t i n g from f a s t e n e r holes, bonded p l a t e
problems with cracks and s u r f a c e flaw problems i n
p i a t e s , e t c . , only highly approximate solt*tions
whish involve various ad hoc "correction f a c t o r s "
are available.
extended h i s two-dimensions 1
approach and solved three-dimensional problems of
buried, surface and corner ( c i r c u l a r ) cracks.
Three-dimensional s t r e s s a n a l y s i s of a f i n i t e s l a b
containing a transverse c e n t r a l crack f o r mode I
problems was c a r r i e d out by S i h e t a16. Recently,
Pian and Ploriya7 developed s p e c i a l crack il a t e n t s
for t h r e e - d b n s ional f r a c t u r e ana lys is using a
hybrid-stress f i n i t e element method. Several c i r cular an3 e l l i p t i c a l crack problems were solved by
cruse8 by using boundary-integral equation method.
Bergan e t a19 used the concept of s t r a i n energy
r e l e a s e r a t e to canpuf e t h e s t r e s s i n t e n s i t y f a c t o r s
for three-dimensional problems. Special quadratic
isoparametric elements which possess the proper
s i n g u l a r i t y of 1/fi i n s t r a i n s with the mid-side
nodes of the faces, which a r e perpendicular t o the
crack f r o n t , placed a t the quarter points were developed by I3arsounlo f o r three-dimens iona 1 l i n e a r
f r a c t u r e problems. Such elements have been developed and used f o r f r a c t u r e analysis of p l a t e s ,
s h e l l s and two-dimensional problemslOP. Hsu and
~ i u 1 2investigated t h e v a r i a t i o n of s t r e s s intens i t y f a c t o r s along t h e crack f r o n t f o r problems of
corn r cracks t?i:amting from holes i n f i n i t e plates.
Shahf3 a l s o studied the s t r e s s i n t e n s i t y f a c t o r s
for through and p a r t throu* cracks o r i g i n a t i n g a t
fastener holes. A parametric type of study on the
qua+t e r - e l l i p t i c a l cracks l y n a t i n g from holes i n
p l a t e s was done by Ganong
using Schware-Nermrann
.
race^^
e l l i p t i c a l corner Cracks orf gina ting from holes i n
~
p l a t e s were conducted by Hal1 and ~ i n ~ e f ro rl aluminum and titanium a l l o y spaciaens. Experimental
i n v e s t i g a t i o n of f r a c t u r e and fatigue crack growth
behavior of surface flaws and flaws o r i g i n a t i n g
from f a s t e n e r holes f o r s t e e 1 specimens was dcne by
Hall e t a l l 6 . A s t r e s s freezing photoelasticity
technique was used by McGowan and smith17 t o study
the s t r e s s intensity f a c t o r s for deep corner cracks
emanating from a hole i n a plate.
Considerable a t tention is being paid, recently,
t o the problem of surface f Xaws in p l a t e s i n tension
and bending due t o t h e f a c t that in the l i t e r a t u r e
s a l u t l o n s of e t r e s s i n t e n s i t y factors vary vide la
when the plate thickness i s mumller d e n compared
with other d b e n s i o n s of the specimen and crack
pentrates deeper i n the t h f c h s s d i r e c t i o n . Anal y t i c a l s o h tions being unavailable, sewera P inves
t i g a t o r s studied t h i s problem using approximate
semis o l u t i o n procedures. ~ o b a ~ a s h i lstudied
8
e l l i p t i c a l and semi-circular surface flaw problems
subjected t o t e n s i l e and bending loads using SchuareNeumann a l t e r n a t i n g technique. Schroedl and Smith
19 used three-dimensional photoelastieity and extens i v e l y studied the surface flaw problems a Approximate solutions usEng alternating technique f o ~
surface f ws in bending were obtained by Shah and
is Lmprwing his :.olutionu
K ~ b a ~ a s h i ' ~ Kobayashi
.
by considering more area on t?e boundary where the
s t r e s s e s a r e erased i n the a l t e r n a t i n g technique.
-
Adhesively bonded s truc turcs conts ining m e t a 11 i c
laminates possess several well known fea tures vhirh
make them highly a t t r a c t i v e from a standpoint r f
f r a c t u r e control. Thus, from an analysis point of
viex there 1; s been a considerable i n t e r e s t i n
assessing the influence of material, geometrical aad
o t h e r s t r u c ~ u r a lvariables on the s t r e s s - i n t e n s i t y
f a c t o r s for cracks in S U C ~s t r u c t u r e s . In e a r l i e r
work Erdogan and ~ r i n * ' considered a sandwich plate
c o n s i s t i n g of an isoptropic sheet and an orthotropic
sheet held together by an adhesive layer, and the
sandwich p l a t e was assumed t o have a p a r t - t h r ~ u g h
and a deb nding crack. Wore recectly, Keer, Lin
and Hura2' considered t h ~
praklem of two e l a s t i c ,
i s o t r o p i c sheets, one of which is cracked, which
a r e bonded together by an adhesive of f i n i t e thickness. They assumed i n t h e i r analysis t h a t no debonding w i l l occur around the crack. It should 3e
noted here that the above works on adhesively bonded p l a t e with crack a r e a£fec t i v e ly two-dimes ional
analyses i n nature and l i n e a r shear spring elements
representing the adhesive layer were used to join
the f r o n t and back sheets i n the analyses. The
abo~
2 b r i e f review indicates that a more thorou*
and a rigorous three-dimens ional analysis of pract i c a l l y s i g n i f i c a n t problems in aerospace s t r u - t u r a l
applications l i k e quar ter-e :1i p t i c a l corner cracks
emanating from holes i n p l a t e s and surface flaws i n
plates under unfaxial tension and adhesively bonded
plates with crack i n one p l a t e subjected to t e n s i l e
loading i s needed and such a task i s c a r r i e d out
using the developed and tested three-dimensfma1 assumed hybrid displacement f f n i t e element method.
11. Formulation
The present f i n i t e element method is based on a
modified v a r i a t i o n a l p r i n c i p l e of p o t e n t i a l energy
with relaxed continuity requirements f o r displacements a t the inter-element boundary. The v a r i a t i o n a l
p r i n c i p l e is a three f i e l d p r i n c i p l e , with a r b i t r a r y
i n t e r i o r displacements
element, i n t e r - e l e -f o r---the
-a
- 7
-. - - r -.... - - - . ...--..------.. .- 7 - - r d- -..
-The
v
a
r
i
a
t
i
onal principle
a s variables
'3f5Bt'yis
t h e s t a t i o n a r y c o n d i t i o n of a modif ied v a r i a t i o n a l p r i n c i p l e of p o t e n t i a l energy f o r
which the functional t o be v a r i e d is
---..-
me
L Y - C
.
j-,
"C.U.YY-
J
i n t e r i o r displacements u
assumed w i t h i n
the i n t e r ePement boundary displacement vi, assumed independ e n t l y on avm, is continuous $cross aVm and subIhe f i r s t
jected to the c a d i t i o n vi = u on Su
v a r i a t i o n of t h e modified t o t a i poted!ial energy,
6
h
,would y i e l d t h e following E u l e r equation and
na r a l boundary conditions: a ) The i n t e r i o r d i s placements u i n V generate s t r e s s e s which s a t i e i
f y l o c a l equilibr'Pum; b) The i n t e r i o r displacements u. coincide with bound 7ry displacemeuts v
on the ilement boundary a V ; c ) The t r a c t i o n s , *i,
jenerated by t h e i n t e r i o r $isplacements ui coinc ide with the independent l y assumed boundary t r a c t i o n s TLi on t h e element boundary aVm. The
s t a t i o n a r y condition of the functional
for
a r b i t r a r y admissible v a r i a t i o n s of ui, v and T
a l s o yield the e s s e n t i a l boundary eondit fon thak
the independently assumed boundary t r a c t i o n s T
coincide with t h e prescribed boundary t r a c t i 0 n b 4 ~
on the boundary Scm.
o need noc be c o n t ~ n u o u sa c r o s s
a6m and
.
I n analyzing the three-dimens iona 1 e l a s t i c f racture problems by the hybrid displacement model, t h e
domain of the problem i s divided i n t o two d i s t i n c t
regions : 1 ) A small f i n i t e region near the crack
f r o n t where the dominant s i n g u l a r behavior of s t r a i n s
and s t r e s s e s i s present and the elements i n t h i s
rr ion a r e r e f e r r e d a s " s i ~ , u l a r elements"; 2 ) khe
gion away from the crack f r o n t *ere the e f f e c t s
3, s i n g u l a r i t y a r e no longer f e l t and the behavior
of s t r e s s e s and s t r a i n s may be approximated by regular polynomial bas i c functions. h e elements i n
t h i s region a r e r e f e r r e d t o a s "regular elements . ' I
E s s e n t i a l l y , i n c o n s t r u c t i n g t h e hybrid d i s p l a c e ment f i n i t e element model f o r t h e s i n g u l a r elements
the following t h r e e f i e l d functions a r e assumed: 1 )
i n t e r i o r displacement f i e l d w i t h i n each element which
need not s a t i s f y the c o m p a t i b i l i t y condition a t t h e
inter-elemen t boundary a p r i o r i and includes the
proper asymptotic displacement s o l u t i o n s ; 2 ) boundary displacement f i e l d on t h e boundary of each
element which is inherently compatible with the
boundary displacement f i e l d of neighboring elements
a t the inter-element boundary; 3) t h e Lagrange m u l t i p l i e r s , which a r e physically t h e independently assumed boundary t r a c t i o n s , assumed on the boundary
of each element t o match t h e independently assumed
i n t e r i o r displacement f i e l d and t h e boundary d i s placement f i e l d on the boundary o f each element.
The above three f i e l d v a r i a b l e s a r e assumed a s
shown below.
hi?, = rvR~[aIi+ CU,I[~,,I +
biIs
hLils
rus~{ksl
(2)
=
[L,I 1sl
(3)
=
CR~I~YI
(41
t h r e e f i e l d functions a s shown
BY assuming t h ~
above, the proper form nf s i n g u l a r i t y i n s t r e s s e s
and s t r a i n s a r e embedded i n t h e f i n i t e element
nrc)ced~q--
Nnw,
PIC
<'?
-
t':
--a
('*'
'- -
'
s t i t u t e d i n ~ q (1)
.
t o o b t a i n t h e td&l p o t e n t i a l
energy i n terms o f t h e nodal displacements {q},
s t r e s s - i n t e n s i t y f a c t o r v e c t o r [ks] and the undeterIt can be
mined parameters [a],{PI] and [BI1].
e a s i l y s h m t h a t , f o r no body f o r c e , the c o n t r i bution by t h e r i g l d body modes t o t h e t o t a l potent i a l energy (terms corresponding t o {BZI]) would
become zero. As t h e undetermined parameters {a)
and (PI] a r e independent f o r each of t h e elements,
they need t o be eliminated from t h e expression of
t o t a l p o t e n t i a l energy and expressed only i n terms
of {q] and {k 1 which a r e a s s o c i a t e d with the ent i r e system o r elements and segments along t h e
crack f r o n t , r e s p e c t i v e l y . This can be done as
follows. The f i r s t v a r i a t i o n of the t o t a l p o t e n t i a l
energy f u n c t i o n a l w i t h r e s p e c t t o t h e parameters
(a] and [ $ 1 is taken and equated t o zero t o o b t a i n
From these equations ( a ) and {PI]
two e q u a t d n s
can be found i n terms of {q] and [k 1. Then t h e s e
a r e s u b s t i t u t e d i n t h e equation of @he t o t a l potent i a l energy t o o b t a i n t h e t o t a l p o t e n t i a l energy
Using the v a r i a t i o n only i n teams of [q] and {k
a l p r i n c i p l e t h e following f i n a l a l g e b r a i c sys tern
of equations a r e obtained where t h e unknowns a r e
the nodal displacements and t h e t h r e e e l a s t i c s t r e s s
i n t e n s i t y f a c t o r s along the crack f r o n t .
.
3.
Further d e t a i l s of the t h e o r e t i c a l and nmeric a l d e t a i l s of t h e formulations a r e not provided
here f o r want of space and can be found i n Ref. 24
and 25. I n t h e following, a b r i e f d e s c r i p t i o n of
the t h r e e independently assumed f i e l d functions
namely t h e i n t e r i o r displacements, t h e boundary
displacements and the boundary t r a c t i o n s is given.
These f i e l d f u n c t i o n s a r e t h e modified and improved
version f o n d i n Ref. 26.
Assumed I n t e r i o r Displacements f o r Singular Elemen t
A.
-
The form of t h e assumed i n t e r i o r displacement
f o r s i n g u l a r elements i s given i n equation (2).
The f i e l d f u n c t i o n [U 1 which c o n t a i n s pure & r a i n ing modes i s assumed In terms of isoparamctric coo r d i n a t e sye tern and c a r t e s i a n coordinate sys tern.
The geometry of t h e b a s i c element considered i s a
2 0 node i s o p ~ r a m e t r i cb r i c k element with 60 degrees
of freedom ( l e e F i d . 1 ) .
F i g l 1 Three Dimensional Mapping "of S ingular Ele-
. _-- -
[U
]{B
a r e cannon for elements t h a t share the cornman
? r r - -3no *n Pi*. 411. wit\ AE betap the
crack f r o n t , the boundary displacement can be assumed as follows on six faces of the elerent.
hm*nrls**
is then w r i t t e n i n the c a r t e s i a n component
foBm a4
Face
U
-
+
Face
u =
+
+
+
u
-
a1
+ a25 + a3{ + a4r + a 4r
Face ADHE (IS
f4
s i n I(2-2v-cos 28
i)
-1)
The r e s t r i c t i o n tha t [U 2 should contain only
pure s t r a i n i n g modes forcesR the terms corresponding
8 8 t o be assumed i n terms of
to B B B 8
physEa? m c a ~ r K a $ & * s ~ f t e S
mi .m i l a r l j , [U 1 wt- 'ch
contains the pure r i g i d body term. can be & i t t e n
13 component form as
The singular displacement f i e l d function [us] ,
which plays the predominant r o l e i n the singular
element, has the i d e n t i c a l form as the asymptotic a l l y c o r r e c t near f i e l d displacement t30lution*~
with the s t r e s s i n t e n s i t y f a c t o r s being used
d i r e c t l y a s undetermined parameters {ks]
.
B
Assumed Boundary DLsplacements for S ingular
Eleiinent
A s mentioned e a r l i e r , the boundary displacement f i e l d vi on the boundary of each e l e r e n f is
assumed i n such a way t h a t i t i s inherently compat i b l e w i t h the boundary displacement f i e l d of
neighboring elements a t the inter-element bound.ries
This is accomplished by using the same and
unique interpolation func tims on the inter-elenrent
boundary. This assures the inter-element d i s l lacemcnt campatibility since the
nodal displacements
.
The parameters (a ,a2* ..-, a8) i n Eqs. (91,
(11). and (I)
are determined uniquely in t e r m of
the relevant u-displacement coordinates on faces
AWE, ABCD & EFGH, and ADHE, respectively. For
example, ( a , a
.*.,
a8) in Eq. ( 9 ) for face ABFE
a r e uniquely found i n terms of the relevant nodal
d i d p l a c m t s q sq2.q
. q 0 ~ q 1 3 ~ q l + a n d q i 5 * It
should be n o t e d h e r e
s h l a r in erpola ea a r e
used f o r the other two displacements v and w corresponding t o a l l six faces of the element and the
other three elements which contain the crack f r m t
(rt.12 S 6 s n; -n/2 s ;8 s 0 ; -n s 0 r - ~ ~ 1 2 ) .
B:>
C.
Assumed Boundary Tractions f o r Singular G l e r e n t
The boundary tra; tians { T ~ ~ ]"thematically
,
.
interpreted as
m u l t i p l i e r s i n the assume6
. Lagrange
- ..
-.
-*-;
.
r *
1
.
Aab-LA.h
L.:-L
+;eweuL
I I I ~ L ~ I V Gb
C~
can be assumed i n any convenient form. Hmever ,
one of the relevant n a t u r a l conditions of the present v a r i a t i o n a l principle is t h a t t h e boundary
tractions generated by the assumed element i n t e r i o r
displacements {ui] mus t ma fch the independentlp
assumed boundary tractions [T 1 a t the i n t e r element boundary. ~ l s o ,the kfekent i n t e r i o r d i s placement f i e l d f u e l s i s forced t o s a t i s f y the equil i b r ium equation t h o u g h the var i a t i o n a l principle
Thus for b e t t e r numerical accuracy of the formulation
the assumed boundary tractions IT ] a r e obtained
from an equilibrated s t r e s s f ieldbi Booreover, s i n c e
the assumed i n t e r i o r displacement f i e l d contains a&
v a r i a t i o n in displacements, the assumed t r a c t i o n s
{ T ~ must
~ ] a~l s o contain a l/& v a r i a t i o n i n tractions f o r the s ingular element. Thus the assumed
boundary t r a c t i o n s a r e e s s e n t i a l l y assumed i n two
p a r t s namely regular and singular p a r t s , For t h e
regular p a r t , the equilibrated s t r e s s f i e l d can be
obtained from three-dimensional Maxwell I s s t r e s s
functions
The r e l a tionships between Maxwell 's
s t r e s s fuactions and the s t r e s s e s a r e given as follows :
.
.
Once the- stresses a r e c a l c u l a t e d by
, the
-
usin Eq.
t r a c t i o n s a r e obtained a s (T ) @!
1
I n the t r a c t i o n modes, assumed
theij
si&gulsr stress f i e l d , the stress i n t e n s i t y f a c t o r s
k ,k ,k a r e replaced by the unknown parameters
1 2 3
us28 Qtj3 and a S 4 m
f?].
111. Problem
n e above hybrid dieplacement f i n i t e elaneat
method was used t o solve f o r the e o l u t i o a o f otress
intenuity f a c t o r s along the crack f r o n t f o r three
problems namely q u a r t e r - e l l i p t i $ a l corner cracks
emanating from f a s t e n e r holes subjected t o tension,
p l a t e s with s u r f a c e flaws subjected t o tension and
bending, and bonded p l a t e s with holes and crack i n
one p l a t e under tensile load.
A.
Corner Cracks Near Fastener Holes
For s h p 1 Z c i t y of modelling, and the knowledge
that the stress jnceneity values for s i n g l e and
double (symmetric) corner cracks origfna ting from
holes d i f f a r anly by about ten
the prob lem of double (symmetric) q u a r t e r - e l l i p t i c a l corn e r cracks o r i g i n a t i n g from f a s t e n e r holes was considered f o r the analysie by the present threedimens iona 1 hybrid displacement f b i t e element
model. Fig. 2 i l l u s t r a t e s the nomenclature of the
double corner cracked hole problem. The loading
The singular p a r t of the stress f i e l d can be
assumed from the asymptotic near . f i e l d solution f a r
stressest7. The assumed regular s t r e s s functions
which produce the needed s t r e s s modes a r e given below.
k"T+
- AA
SECTION
Fig, 2
Corner Cracks Emanating from Holes i n P l a t e s
f o r this problem would be uniaxial tension a t
I z l = H.
Two d f f f e r e n t geometries have been cons idered and they a r e given i n the following t a b l e ,
a long with corresponding ~ o i s s o n ' sr a t i o s .
h e dimensions W and H a r e given by the followi n g equations
2w = 6 ( 2 ~+ c )
Due to symmetry of t h e problem, only, a q u a r t e r
of the s t r u c t u r e needs t o be considered f a r t h e
f i n i t e element breakdown. The ficf te e l m a t breakdown of quarter of t h e problem i s given i n F i g . 3;
the t o t a l number of f i n i t e elements and the t o t a l
number of degrees of freedom f o r t h i s b r e a k d m are
156 and 2670 respectively. The r e s u l t s and discussions £0; this prebl& a r e presented i n t h e next
section.
Fig. 4 Surface Flaws in Plates in Tension and Bending
Fig. 3
B o
F i n i t e Element Breakdown of Quarcer of the
Corner Cracked Hole Problem
P l a t e s with Surface Flaws
The problems of semi-circular and s e m i - e l l i p t i c a l s u r f a c e flaws i n thin p l a t e s with ddCfferent
crack depth t o thickness r a t i o s under bendfng and
t e n s i l e loads were a l s o considered f o r the a n a l y s i s
by the present f i n i t e element model. The problea
is schematically s h a m i n F i g . 4 and Fig. 5 i l l u s t r a t e s the f i n i t e element breakdown of q u a r t e r of
s t r u c t u r e with 156 t o t a l number of f i n i t e elements
and 2670 t o t a l number of degrees of freedom. In
Fig. 5, the thickness d i r e c t i o n (A0 o r 3%)
is shown
elongated t o show the f i n i t e element breakdown
c l e a r l y . The geumetric parameters of the problems
considered a r e given below.
a /c
a h
W/c
H/c
Load in%
0 .2
0.6
4.0
4.0
Bending
0.2
0.8
4.0
4.0
Bending
Fig. 5
F i n i t e Element B r e a k d m of Quarter
Surface F l a w Problem Ln Plates
dB
the
The Poisson's r a t i o i s assumed t o be 1I3 for the
surface flaw problems i n plates. The results and
discussions for these problems a r e also given in
the next sectfon.
In the solution procedure f o r surface f l a w problems and corner cracks near fastener hole problems,
i n order t o eliminate the transleitima1 and r o t a t i o n a l ( r i g i d body) movements of the p l a t e s , a t
least one displacement i n each d i r e c t i o n was sup-
pressed. D.e Eyxmetry conditions of the problems
were obtained- by .property
suppressing t h e d i s p l a c e *
- _ .. _
- a
,
.
- ,.,, .
-,
rLi
;‘ef - A G U A"&
LU
~tle
plane of symmetry, 'i.t?er r r a r u - ~ ? ? - ! . g s e(or c i r c l e )
was divided i n t o s kx segwa t s f o r the; , ?robletus r 3
obtain the v a r i a t i o n of m e s t r e s s i n t e n s i t y f a c t o r
along the crack f r o n t .
C.
a l s o assumed t o be 7.0, The loading i s t h e unlaxin'*
tpnsj-oy which 'e e l - ? l l n J -T '--&pa
r.?.i* ,
ana r e adhesive. The non-dimensional parameter
(W/fi)
s was taken t o be 10.0, Thea t h e problem was
solved f o r c r a c k length t o h o l e r a d i u s r a t k o s of
0.35, .?,1;4, 2.1 and 3.5.
The r e s u l t s of t h e
s t r e s s i n t e n s i t y f a c t o r v a r i a t i o n a r e given i n the
next section.
~~~~
' - A '
L
C
4
Cracks i n Adhesively Bonded P l a t e s with Holes
IV.
Resul t s and Discussions
The present method was a l s o used t o solve t h e
problem of two i s o t r o p i c adhesively bonded p l a t e s
containing a through-the-thickness h o l e , and a
s ingle crack near t h e h o l e i n one o f the p l a t e s .
The schemacic diagram of the problem is given i n
Figs. 6 and Fig. 7 i l l u s t r a t e s the f i n i t e element
breakdown of t h e problem with 2 1.6 t o t a l number of
f i n i t e elements and 3822 t o t a l number of degrees of
freedom. The Young's Modulii of the l a t e s and
adhesive were assumed t o be 1.O3 x 10 ps i and 2.8 x
A.
Corner Cracks Near Fastener Boles
The s o l u t i o n s of t h e s t r e s s i n t e n s i t y factors
f o r the symmetric corner cracks emanating from, h o l e s
i n f i n i t e p l a t e s under tension f o r the two different
geometries a r e given i n F i g s . 8 and 9. Il-ese stress
7
/--
Adhesive
7.4
Fig. 6 Bonded P l a t e s Containing a Through-theThickness Hole and a Single Crack Near the Hole
ELLIPTICAL ANGLE
, (P
F i g . 8 V a r i a t i o n of S t r e s s I n t e n s i t y Factor f o r
Corner Crack Emanating from Hole ( a / c = l . l ; C / R =
0.99142; a / t = 0.48)
No. of D.O. F. = 3822
No. of Elements = 216
F i g . 7 F i n i t e ~lementCreakdown of Half of the Problem of Bonded P l a t e s Coutaining a Through- theThickness Hole and a Single Crack Near the Hole
.
1 0 5 p i respectively
The ~ o i s s o n ' sr a t i o s of tke
p l a t e s and the adhesive v e r e assumed t o h e 0.33 and
0.4, respectively. The r a t i o t l / t w a s assumed a s
1.0. Thc r a t i o of adherent
thickness t o
adhesive thickness was assumed t o be 7.0. 'X'he r a t i o
of f i e p l a t e width Ztl t o the h o l e d j w e t e r was
I
1.e !
0
a
, * J psi
I
I
't4
ELLIPTICAL
ANOLE
, dj)
%
'
Fig. 9 V a r i a t i m of Stress I n t e n s i t y Factor f o r
Corner Crack Emanating from Hole ( a / c = ~ . s ; c / R=
i n t e n s i t y f a c t o r s a r e normalized with r e s p e c t t o
PRESENT
the t h e o r e t i c a l value o f the s t r e s s i n t e n s i t y faco KOBAYA..IW'S CUI
tor of the point where the crack f r o n t and the m i A REFERENCE [a13
ncc axis fsltersect when the complete e l l t p t i c a l
1-67
crack is embedded i n a n i n f i n i t e medium and i s subjected t o remote uniform tension go. The normelizing f a c t o r is given by ~ f i / ~ ( k ) where
,
b i s the
semi-mhor a x i s ( m i n h of a and c ) , A s a r e s u l t ,
~ i g s .8 and 9 depict t h e a c t ~ a vl a r i a t i o n of t h e
s t r e s s i n t e n s i t y f a c t o r s , but f o r a constant. Solutions of s t r e s s i n t e n s i t y f a c t o r s f o r the problem
of c o m e r cracks emanating from hole w r t h geometric
parameters a / c
1,1, C/R
0.99142 and a f t
0-48
by t h e present method and by s t r e s s f q f i n g ' photoe i a s t f c i t y method by McGowaa and Smith are prssexted iri Fig. 8. Present soSuticm by the f b i t e elenent model f o r the problem with geometric parametes
a / c = 1 5 C/R = f.0 and a / t r 0 75, along w i t h
~ a n o n ~ ' s ls4o l u t i o n f o r s i n g l e corner crack, is given
i n Fig. 9. Though t h e present s o l u t i o n canno
compared d i r e c t l y with the s o l u t i o n of Ganong
Fig. 9 because of the d i f f e r e n c e i n the probXem, the
present s o l u t i o n s e w to dLffer s l i g h t l y from Ref.
14, The eoluticm of t h e f i r s t problem in Fig, 8
F i g . 10 Variation of S t r e s r I n t e n s i t y Factor for e
a k f f e r s from t h a t of the r e g u l t e bteined from the
Setnl-Circular Surface Crack i n e P l a t e Subjected to
experfment. by HcCaran and
where only the
Tension and Bending
end values of the a t r u s s i n t e n s i t y f a c t o r 8 w r e
a v a i l a b l e f o r comparison. The reaaons f o r t h e d i f ferences and discrepancies of theee r e s u l t s can be
enumerated as follows,
I
anon^'^
eatirmrtss t h a t h i s r e s u l t s a r e about
20 percent higher than t h e previous estimates. Due
t o inadequate experimental r e s u l t s , i t was n o t poss i b l e t o v e r i f y the present r e e u l t s with them. The
end values of str s i n t e n s i t y f a c t o r s cbtained by
McGovan and Smithe9 through experiments in Fig. 8
show t h a t they agree a t t h e f r o n t s u r f a c e and not
a t t h e hole surface. Ilheozetically one would expect the s t r e s s i n t e n s i t y f a c t o r t o be higher a t
the h o l e s u r f a .e due t o the s t r e s s concentrat Lon
e f f e c t along the 'iole s u r f a c e . Further coaments on
the reasons f o r a - crepancies are given l a t e r i n
this s e c t i o n .
B.
P l a t e s With Surface Flaws
The s o l u t i o n of stress i n t e n s i t y f a c t o r s along
the crack f r o n t for semi-circular surface flaws i n
p l a t e s subjected t o bending a s well a s tension Tor
crack depth t o thickness r a t i o s 0.6 and 0.8 are in
Figs. 10 and 11. F i g . 1 2 g i v e the s o l u t i o n £or the
s t r e s s i n t e n s i t y f a c t o r s along the crack f r o n t f o r
the s e z 5 - e l l i p t i c a l s u r f a c e flaws i n p l a t e s subjected
t o bending f o r crack depth to thickness r a t i o s 0.6
and 0 .a. Th2y a r e normalized with respect to t h e
values of s t r e s s i n t e n s i t y factor of e i t h e r a comp l e t e l y embedded c i r c ke or a t the minor a x i s of the
e l l i p s e , when the c i r c l e o r e l l i p s e i s completely
embedded i n an i n f i n i t e s u l i d and subjected t o remote tension. Thus, Figs. 10, 11 and 12 represent
the a c t u a l v a r i a t i o n of s t r e s s i n t e n s i t y f a c t o r s
but f o r a constant. The igrresponding s o l ions oband Raju e t alYQ f o r
tained by Kobsyashi e t a 1
semi-circular surface crack problems a r e a l s o ktven
i n Figs. 10 and 11. For t h e semi-ciicular s u r f a c e
problems i n Figs. 10 and 11, i t can be seen t h a t the
s o l u t i o n of s t r e s s i n t e n s i t y f a c t o r s f o r ben6ing
lbads by t h e present procedure agrees w e l l with the
r e s u l t s estimated by ~ o b a ~ a s h i ~ e x fcoer ~a t 15 per
cent d i f f e r e n c e a t the i . ~ t e r s e c t i o nof t h e crack
f r o n t and the f r o n t f r e e s u r f a c e . However, t h e r e
Fig. 11 Variation of S t r e s s I n t e n s i t y Factor f o r
a Deep Semi-Circular Surface Crack i n a P l a t e Subjected t o Tens ion and Bending.
i s a large difference i n the s o l u t i o n fo tensile
load between t h e present and Kobqashi 'sf8 procedure.
A recent s o l u t i o n by Raju and ~ e w m a a ~given
*
i n Figs, 10 and 11 for t e n s i l e loads compares
wSsh t h e present solutton b e t t e r than t h a t of KobayaIt should he noted here that t h a t degrees
shi'sta.
~ ~the semiof freedom used by Reju and ~ e w m a nfor
c i r c u l a r surface flaw problems are 4317. As the
solutions corresponding t o t h e problem of semi-ellipt i c a l s u r f a c e flaws i n p l a t e s a r e being revised by
the author of Ref. 18 ( p r i v a t e communication), w e
choose n o t t o campare the present solution a t the
presene tfme
Fig. 13 V a r i a t i o n of S t r e s s Concentration Factor
ELLlPTlCIL
AWL€
,
@
F i g . 12 Variation of S t r e s s I n t e n s i t y Factor f o r a
S e m i - e l l i p t i c a l Surface Crack i n a P l a t e Subjected
t o Bending.
In the a l t e r n a t i n g technique used by Kobayashi18 for semi-circular and s e m i - e l l i p t i c a l s u r f a c e
flaw problems, one of the s t e p s involved i s the
removal of t r a c t i o n s on the boundary of the s truct u r e . These t r a c t i o n s a r e removed only over a smell
a r e a near the crack i n Ref. 18. A s mentioned
before, the s o l u t i o n s a r e b e i r ? ~r e v i s e d and improved
i . e . , the t r a c t i o n s a r e now being removed over a
larger area near the crack which would r e s u l t i n
b e t t e r convergence of the s o l u t i o n . This may be one
of the reasons f o r the discrepancies i n s o l u t i o n of
s t r e s s i n t e n s i t y f a c t o r s by t h e p r e s e n t and Kobayashi ' s l 8 procedures. I n analyzing t h e above prob lerns ,
the previ 3us i n v e s t i g a t o r s have used highly approximate s o l u t i o n procedures o r two-dimensiona 1 analogs
and various ad hoc correction f a c t o r s . But the pres e n t hybrid displacement f i n i t e element model i s a
rigorous t h r e e - d h n s i o n a l f i n i t e element procedure
where a l l the boundary conditions, such as f r e e
s u r f a c e s , a r e automatically taken i n t o account and
s a t i s f i e d i n an i n t e g r a l average sense.
C.
f o r a S i n g l e P l a t e Containing a Through- the4hickness Hole.
of the p l a t e . A f t e r making s u r e t h a t t h e s t r e s s
concentration e f f e c t s a r e w e l l accounted f o r , t h e
problems of cracks i n bonded p l a t e s with holes were
solved.
The s o l u t i o n of the s t r e s s i n t e n s i t y f a c t o r s
for'various c r a c k length to hole r a d i u s r a t i o s f o r
t h e problem of two i s o t r o p i c adhesively bonded plates
containing a through- the- thickness h o l e and a s i n g l e
crack near t h e h o l e i n one of t h e p l a t e & is given
i n Fig. 14. The s t r e s s i n t e n s i t y f a c t o r s a r e normalized by using the value of the s t r e s s i n t e n s i t y
Cracks i n Adhesively Bonded P l a t e s w i t h Holes
In solving the crack problems i n adhesively
bonded plates with holes, the f i n i t e element grid
was arrived a t a f t e r a c a r e f u l study of the expected
s t r e s s concentration near the hole and a f t e r v e r l f y ing that the s t r e s s e s near the h o l e without the
crack a r e accurate enough so t h a t the obtained res t i l t s f o r s t r e s s i n t e n s i t y f a c t o r s f o r cracks near
the hole may be meaningful. Before s o l v i n g the
problem of cracks i n bonded plat- with holes, the
same f i n i t e element grid was used t o s o l v e a simple
problem of a p l a t e with a hole i n tension. Thiswas
c a r r i e d out t o make s u r e t h a t t h e f i n i t e element
g r i d was f i n e enough to generate a s t r e s s concentrat i o n f a c t o r of about 3.0 a t the hnle. The s o l u t i o n
of the s t r e s s concentration f a c t o r f o r the above
problem a t mid-thickness of the p l a t e i s given i n
Fig. 13. As can be seen from t h e f i g u r e , the s t r e s s
c o n c e ~ t r a t i o na t the hole i s 3.213.
Also, the
s t r e s s concentration is found t o vary along the
thickness dFrection of the p l a t e . The value of the
s t r e s s concentration decreased from 3.213 a t t h e
m i d - t h i c k ~ ~ e stso 3.134 a t the f r o n t and backsurfaces
F i g . 14 V a r i a t i o n of S t r e s s I n t e n s i t y Factor for
Bonded P l a t e s Contair ing a Through- the-Thickness
Hole and a S i n g l e Crack Near the Hole f o r Various
Crack Lengths
.
f a c t o r s of a s i n g l e i n f i n i t e p l a t e w i t h t h e crack.
I n analyzing t h e s e problems, the adhesive was cons i d e r e d as a continuum and was assumed t o debond with
t h e crack along t h e plane of t h e crwk u n t i l i t
reaches the p l a t e w i t h no c r a c k . : !.-cia1 crack elements were used o n l y i n the cracked plate and a l l
t h e elements f o r t h e adhesive were modeled using
r e g u l a r isoparametric b r i c k elements. I n Fig. 14,
t h r e e d i f f e r e n t s e t s of s t r e s s i n t e n s i t y f a c t o r s a r e
presented. The middle curve r e p r e s e n t s the v a r i a -
a
t i o n of e t r e e a ?ntms&?yf a c t o r s computed direetlv
L L U W LAW I I ~ ~ LA )UL- L L ~ S ~ A Y C ; ~ ~ r~rUnLl t e element method
in which t h e st- a s intmsity f a c t o r s are assumed
as unknowns and computed d i r e c t l y . These values
may be considered a s t h e average values of t h e s t r e s s
i n t e n s i t y factors over the length of the crack
front. The upper and lower curves represent t h e
v a r i a t i o n of s t r e e a i n t e n s i t y f a c t o r s a t the o u t e r
and inner t i p s of t h e c r a c k f r o n t . The o u t e r t i p
corresponds t o the crack t i p a t the f r e e s u r f a c e ol
the cracked p l a t e and the fnner t i p c o r r c ~ p o n d st o
the crack t i p where the cracked p l a t e j o i n t h e adhesive. The s t r e s s i n t e n s i t y f a c t o r s f o r t h e o u t e r
and inner t i p s a r e cornfluted fraa ehe d i s p l s c ~ t
s o l u t i o n of t h e f i n i t e element me od u s h g k r a c k
opening displacement (COD) method
5?3.
F i n i t e Elements i n Lfrtear Fracture Hecbanics
-- - "
Inti. J Numerical Methods in Engrp., 1/01 10,
pp. 25-37, 1976.
.
.
-
Barsoum, R . S . , "A Degenerate Solid Element f o r
Linear Fracture Analysis of P l a t e Bend i n n al:d
--General Shells," inti. J . A-rical
Wethods i~
Engrg., Vol. LO, pp. 551-564, 1976.
LI
Hsu, T.M. and AoFo Liu, "Stress I n t e n s i t y Fact o r s for Truncated E l l i p t i c a l Cracks, Seventh
Maticma1 Symposium on Fracture Mechanir , Gollege Park, MD, August 1973.
Shsh, R o c * , "Stress I n t e n s i t y Factors far
Through and Par t-Through Cracks OriginaDhg a t
Pas t e n e r Holes ,'I Ei@ th National Symposium on
F r a c t u r e Mechanics, Brown Univ , Providence,
R I , August 1974.
.
This work was supported by AFOSR under g r a n t
No. 74-2667. This a ~ l dthe supplemental support from
the Georgia I n s t i t u t e of Technology are gre t e f u l l y
acknwledged. The authors express t h e i r apprec iation t o W i l l knm J . Walker f o r continued encouragecnent
thin research e f f o r t .
References
Tada, E., P.C. P a r i s and G.R. Irwin, The
S t r e s s Analysis of Cracks Handbook, Del Research
Corp., Hellertawn, PA, 1972.
S i h , C.C., ~andbooko f S t r e s s - I n t e n s i t y F a c t o r s ,
Lehigh University, Bethlehem, PA, 1973.
Rice, J .R., "Mathema t i c a l Analysis in Mechanics
of F r a c t u r e ," Fracture, P.LI Advanced 'I'rea t i s e ,
e d i t e d by H. L i e b w i t e , Vol. X I , Academic P r e s s ,
New York, pp. 192-308, 1968.
0
Pian, T.H.H.,
"Crack Elements," Proceedinns ,
World Congress on F i n i t e Element Methods 4-n
S t r u c t u r a l Mechanics , Bournemouth, Dorset , Eng
. -.-_
1975.
.
Tracey, D.M., "3-D E l a s t i c S i n g u l a r i t y Element
f o r Evaluation of K along an A r b i t r a r y Crack
Front ,'I I n t J * Fracturfe, Vol. 9 , pp, 340-343,
1973.
.
S i h , G X . , P.D. H i l t o n , R.J. Hartranft and R.V.
Kiefer , "Three-Dimensional S t r e s s Analysis of a
F i n i t e Slab Containing a Transverse C e n t r a l
Crack," Tech. Report No. NOOl4-67-A-O37O-OOl2,
Office ' of Naval Research, Arlington, VA, 1975.
Pian, T.H .H. and K. Moriya, "Three-D Lmensima 1
F r a c t u r e Analysis by Assumed S t r e s s Hybrid Elements," ( t o be presented) I n t e r n a t i o n a l Conf. on
Numerical Methods i n F r a c t u r e Mechanics, Univ.
College, Swansea, England, Jan. 1978.
.
Cruse, T.A , "Boundary-Integral Equation Method
f o r Three-Dimens i o n a l E l a s t i c Fracture Mechanics
Analysis , I ' Interim Report, A i r Force Office of
S c i e n t i f i c Research, Arlington, VA, 1975.
Bergan, P.G, and B. Aamodt, ' F i n i t e Element An,olys is of Crack Propagation i n Three-Dimens i o n a l
S o l i d s Under Cyclic Loading, " Nuclear R n ~ i n e e r a
and Design, Vol. 29, pp. 1.80-188, 1974.
Ganong , G .P., 'quarter-Ellip t i c a l Cracks E m n a t i n g From Holes i n P l a t e s ,"P h o D o Thesis,
Colorado S t a t e Univ., F o r t Collins, Colorado,
July 1975.
H a l l , L.R. and R. W . Finger, "Investigation of
Flaw Geometry and Loading Effects on Plane
S t r a i n Fracture i n Metallic S t r u c hures ," Tech.
Report NASA-CR-72659, Boeing Gaarpany, Dec 1971.
.
Halll, L.R., R.C. S b h and W.L. Engstorn, "Fract u r e and Fatigue Crack Growth Behavior of Surface Flaws Originating at Fastener Holes,"
AFFDL-TR-74-47, A i r Force F l i g h t F ynamics Lab ,
Ohio, 1974.
McCaran, J.J. and C.U. Smith, "stress I n t e n s i t y
Factors f o r Deep Cracks Emanating f r m the
Corner Formed by a Hole I n t e r s e c t i n g a Plate
Surface," Tech. Report, VPI-E-74-1, Virginia
Polytechnic I n s t i t u t e d S t a t e University, Blacksbure, VA, January 1974.
Kobayashi, A.S., N. Polvanish, A.F. Emery and
W J. Love, "Surface Flaws i n a P l a t e i n Bending," ~ r o c e e d i n g s ,Twelfth Anaual Heeting of
Society of Engtneerirrg Science, Austin, TX,
Schroedl, MA. and C.G. Smith, *'Local S t r e s s e s
Meat Deep Surface Flaws under Cylindrical Bending F i e l d s ," Pmgress i n Flaw Growth and Fract u r e Toughness t e s t i n g , edited by J.G. Uufman,
ASW STP 536, pp. 45-63, 1953.
Shah, R.C. and A.S. Kobayashi, ' S t r e s s Intens i t y Factors for an E l l i p t i c a l Crack Approaching the Surface of a Beam i n Bending," S t r e s s
Analysis and Crack Growth, edited by W.T.
Corten and J . P. Gallagher, A S W STP 513, pp.
3-21, 1972.
Erdogan, F. and K. A r b , "A Sandwich P l a t e with
a Par:-Through and a Debonding -Crack,'' Engineering Frac. Mech., Vol, 4, pp, 449-458,1972.
Keer, L.M,, C.T. Lin and T. Mura, "Fracture
Analysis of Adhesively Bonded Sheets, " ASME
paper No. 764A-APM-12, presented a t the ASME
Wmter Annual Meeting, New York, Dec. 5-10,
1976 ( t o appear i n the J. of Awl. Mech.).
Tong, P.,
"New Displacement Hybrid Finite Ele-
.
24
A t l u t i , S O N m , A.S Kobayashi and M. Nakagaki,
'$AnAssumed Displacement Hybrid Finite Element
Method f o r Fracture Mechanics," I n t l . 3. Fracture Mech., Vol. lZ,*pp. 257-271, A p r i l 1975.
25
Kathiresan, K., "Three-Dimens ional Linear
E l z s t i c Fracture Mechanics Analysis by a Displacement Hybrid F i n i t e Element Model," Ph.D.
Thesis, S c b o l of Engineering Science and
Mechanics , Georgia I n s t i t u t e of Technolagy ,
Atlanta, GA, Sept. 1976.
26
A t l u r i , S .N., K. Kathiresan and A.S., Kobayas h i ; ''!3'hree-~imens i o n a l Linear Fracture Mechan i c s Analysis by a Disy lacement-Hybrid F i n i t e
Element Moddl, *' Paper L-713, Transactions of
Third International Conference on S t r u e t u r d
Mechanics i n Reactor Technology, CECA, CEE ,
CEEA, Luxemburg, September 1975.
27
Kassir, M.K. and G. C. S i h , "Three-Dimensional
S t r e s s Distribution Around Crack Under A r b i t r a r y Loading,". J, Apple Mech., Vol. 33, pp.
601-611, 1966.
28
Raju, I.S. andJ.C.Newman, Jr., "Improved
Stress-Intensity Factors f o r Semi-Elliptical
Surface Cracks i n Finite-Thickness P l a t e s , I 1
Transact ions of the Fourth I n t e r n a t i a n a l Conference on Structuraf Mechanics in Reactor
Technology, San Francisco, CA, August 1977.