Photoelasticity was introduced as a quantitative tool in the experimental stress analysis of anisotropic birefringent composite model materials. The investigation was divided into three parts: (1) separation of principal stresses or strains, (2) photoelastic calibration, and (3) development of model-prototype relations

Prabhakaran, R.

English

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DEPAROMENT OF MECHANICAL ENGINEERING AND MECHANICS
SCHOOL OF E:lGINEERING
OLD DOMINION UNIVERSITY.
NORFOLK, VIRGINIA
I 79-32575
(NA5A -CP-162316)	 PHOTOE'LASTIC STUDIES OF
ADVANCED LAP JOT,NT CONCFPT Final Report, 15
Mar. 1 97 8 - 3 D SpP• 19 ?q (01(1 Dominion Univ.
Research Foundation)	 1U	 219
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PHOTOELASTIC STUDIES OF ADVANCED LAP
JOINT CONCEPT
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R. Prabhakaran
Principal Investigator
Final Report
For the period March 15, 1978 - September 30, 1979
Prepared for the
National :aeronautics and Space administration
Langley Research Center
Hampton, Virginia
Urder
Research Grant NSG 1507
P.A. Cooper, Technical Monitor
Structures and Dynamics Division
October 1979
r
DEPARTMENT OF MECHANICAL ENGINEERING AND MECHANICS
SCHOOL OF ENGINEERING
OLD DOMINION UNIVERSITY
NORFOLK, VIRGINIA
PHOTOELASTIC STUDIES OF ADVANCED LAP
JOINT CONCEPT
By
R. Prabhakaran
Principal Investiagtor
Final Report
For the period March 15, 1973 - September 30, 1979
Prepared for the
National. Aeronautics and Space Administration
Langley Research Center
Hampton, Virginia 23665
:ender
Research Grant NSG 1507
P.A. Cooper, Technical Monitor
Structures and D ynamics Division
Submitted by the
Old Dominion Universit y Research Foundation
P.O. Box 6369
.Norfolk, Virginia 23508
October 1979
A
PHOTOELASTIC STUDIES OF ADVANCED LAP JOINT CONCEPT
By
R. Prabhakaranl
INTRODUCTION
This report summarizes work done in lap joint analyses under the Composites
for Advanced Space Transportation Systems (CASTS) program during the period from
March 15, 1978 to September 30, 1979. 	 The purpose of this research was to u
introduce photoelasticity as a quantitative tool in the experimental stress I,
'	 analysis efforts of the CASTS program. 	 Work was directed towards two goals: j
f	 (1) developing experimental techniques and confirming experimentally the analysis
of various single lap joint configurations and (2) developing the transmission
photoelastic methods as applied to anisotropic birefringent composite model
materials.
The results of the first part of the investigation, dealing with photo-
elastic analysis of lap joint configurations, are given in detail in "An
Investigation of the Effect of Adhesive Stiffness on the Adherends of Single
Lap Joints," a proposed NASA TP by Paul A. Cooper. and Stephen A. Hann. 	 This
report will elaborate the work relating to birefringent anisotropic models.
.	 The work done in this phase of the investigation can be divided into three
parts:
(1) separation of principal stresses or strains,
(2) photoelastic calibration, and
(3) development of model-prototype relations.
The following section will describe each of the above studies.
l Associate Professor, Department of Mechanical Engineering and Mechanics,
Old Dominion University, Norfolk, Virginia 23508.
WORK PERFORMED
Separation of Principal Stresses or Strains
In orthotropic birefringent models, the isochromatic fringes do not give
the principal stress (or strain) difference. For a meaningful photoelastic
analysis of such models, the individual values of principal stresses or strains
hw a to be determined. The methods suggested so far are not satisfactory
because they require either successive integration (shear difference method, a
method requiring numerical solution of the compatibility equation), leading to
an accumulation of error, or a not-so-easy experimental technique (holography).
It was desired, therefore, to consider methods which were completely
experimental and relatively simple. The first method examined was the possi-
bility of using oblique incidence on birefringent composite models. When
light is incident normally on such a model, the isochromatic fringe order is
related to the principal strains as
	
1\	 2
	
No	
feL	
Lrcos 2
 6 + E  sin g
 S) - ^sin2 d +	 cost S
J fETI
2 1/2
	+ rr I
 - ^1 fEL sin 2S1 i	 ( )
L`\	 x//I yLT	 JJ !
where e 
	
is the major principal strain, 
c  
is the minor principal strain,
$ is the angle between e x and the material symmetry axis L, and feL'
feT, fyLT are the principal material-strain-fringe values. If the light
beam is incident at an angle B Y , the isochromatic fringe order for oblique
incidence is given by
	
1	 1	 //	 \
	
N B	 cosI	 f^ [{cos 2 a + E Y
 sin 6) - Isin2 S + ey cos 2 SJ f 
2
EL
	
x	 x EL	 \	 x	 /	 \\	 x	 / eT
21/2
E \ f
+ C\1 - 
x 
.1 EL sin 2S^
ex) fyLT (2)
2
^r
where sY is the secondary principal strain and is given by
y	 Y	 ze^, = s CO5 2 0 + e sinz B C
For a model under a state of plane stress, it is possible to show that
the strain normal to the plane of the model is related to the in-plane prin-
cipal strains through the following relation:
cos z
 8(v L3 
+ 'T3'LT) + sin z s(vT3.+ vL3yTL)^
ez = 
-Ex 	 1 - vLTvTL
Fs
	 B(VL3 + VT3yLT)' + cosz S(VT3 + VUVTL)1e 	
1 - vLTvTL	 j
Substituting the expression for e  from equation (4) into equation (3) and
again substituting the expression for eY into equation (2) and simplifying,
it is possible to write
zNp =
	
h	 1	 J [Ce
x
 + De 
Y ] 
z + [Ee
x
 + Fe
Y 
]2
0 x	 cos z 0C 
f&L 
l
where
f
C = cos z B - A sin e B sinz 6 cos z p - B sin z e sin 4 a - fEL sin e`	 x 	 B
ET
(3)
(4)
(5)
+ fEL A sin
z a cos 4
 9 + fEL B sinz q ^ sinz a cos z
 B
ET	 ET
(6)
(Cont Id)
'^	 3
D = cos2 8  sin2 0 - A sin g 0x sin4 S - B sing 0x sin2 B cos t a
BL cos t
 0 cost	
f
g = EL A sin 2
 0 sin2 a cos t S + f EL B sin2 0 cos 4
 BfET	 x	 fET	 x	 fET	 x
E f LT sin 2S (1 + A sin2 
6  cos 2 S + B sin Bx - sin2 B)yL
P	 fEL sin 2$ (-cos ? Bx + A sin 0x sin2 g + B sin2 Bx cos 2 S)
yLT
(6)
(ConclId)
In equations (6), A and B are given by
A = 
vL3 + vT3vLT
1 - vLTvTL
B 
= 
VT3 + vL3vTL
1 - vLTvTL
It is possible to solve for the principal strains e x and e 	 from
equations (1) and (5), knowing the normal incidence-isochromatic fringe
order, the oblique incidence-isochromatic fringe order, and the isoclinic
parameter. It is, of course, necessary to measure the principal material
properties, elastic and photoelastic. These are the Poisson's ratios
V
LT' VTL' and the strain-fringe values fEL' fET, fyLT; the constants
A and B can be obtained by a calibration procedure which also yields
the angle of oblique incidence, 8x , for the combination of the available
oblique incidence adapter (using a chopped prism) and the model material.
C7)
4
Three calibration specimens, for instance tensile specimens, are chosen with
the angle between the L-axis and the specimen axis (the latter being the
[i
loading direction) equal to 0 0 , 90 0 , and an intermediate value. The
angle between the major principal strain direction and the L-axis,
is known for each specimen; it is 0 0 for the first two and can be computed 	 k
for the third. For each specimen, the normal and oblique incidence fringe
orders are measured and substituted in equations (1) and (2). The result- 	 a
ing set of three equations for normal incidence can be solve_' for the three
principal strain-fringe values feL' feT, and fYLV The set of three
equations for oblique incidence can be solved for A, B, and 8x.
The procedure for obtaining e
x
 and ey from equations (1) and (5)
1
can be simplified if the angle 8 x can be chosen so that D = F = 0 in
equation (5). In this case, equation (5) redu!::s to
N 8 cos ex fEL
x
•	 Ex =	 CB)
h 02 + E2
The above value of E x
 can be substituted in equation (1) to determine
ey . The angle ex needed to make D = F = 0 is given by
tan'- e Y
	
	 1 --	 (9)
A sing B + B coS 2
 a
Thus, the angle for oblique incidence will vary from point to point,
depending upon the isoclinic parameter B.
A second method that is proposed to be tested is based on the dispersion of
birefringence in composite models. If the variations of the fundamental photo-
elastic constants with the wavelength are different, then the principal stresses
or strains can be determined from the isochromatic fringe patterns obtained for
two different wavelengths. Specimens have been designed and fabricated for
testing the feasibility of this method.
5
Photoelastic Calibration
Beam, compression and tension specimens have been conventionally used as
photoelastic calibration specimens. In the case of orthotropic model materials,
the circular disk specimen has not been used as a calibration specimen because
a closed form solution is not available. However, in view of the difficulty
and expense in fabricating such model materials, it is preferable to minimize
the amount of material required for calibration.
A numerical solution for the stresses in an anisotropic circular disk under
diametral compression has been utilized in an attempt to employ the circular
disk as the calibration specimen. A nonlinear least-squares method is being
applied to a single isochromatic fringe pattern in order to determine the
three fundamental photoelastic constants and the residual isochromatic bire-
fringence due to the fabrication procedure.
The isochromatic fringe order can be written as
(al cos t $ + a2 sine
 a) - 1 (a l
 sin 2
 B + 0 2 cos t 6)]2fT
	
+ [ fLT 
C a l - 02) sin 201 2 ) 1 / 2 + Nr
	 (l0)
where Nr is the residual fringe order, known to be distributed uniformly
throughtout the model; h is the model thickness
let
gk = [ 1 Ca l k cost a  + a2k sin2
 0 k] - 1 (o lk sin 2 0 k + a2 COS 2
 ak)]2
	
fL	 fT
	
+ [ 1	 (a lk 
- 
a2k ) sin 2
 Bk ] 2 - [N-Nr] 2	(11)
fLT	 h
6
N = h ^[1
f 
where	 k = 1, 2, 3,	 4 .	 ............M CM > 4) {
'
1
s
It is possible to write
Igy	 =	 [b] {Dky (12)
where
( g }	 -	 -gl
-g2
-gM (13}
r
[b]	 = ag1 ag, agl akl
3f afT 2f LT Mr
agm agm agm
afL afT @f 
LT aNr (14)
{Dk{	 = Dft
DfT
+DfLT
IDNr (15}
then, if
[d] = [b] T [b ]	 (16)
7
.'I
is evaluated, we can write
( k) = L d] -1 I b] T (g)
	
(17)
Assuming initial values of f L , fT , fLT , Nr, by an iterative procedure,
the photoelastic constants and the residual birefringence can be determined.
The method is being applied to a unidirectionally reinforced circular disk.
Development of Model-Prototype Relations
The glass-fiber reinforced polymeric model materials being used as photo-
elastic materials incorporate commerical fibers in a commercial matrix. Thus
they can be considered as structural composita materials. However, many of
the structural composite materials incorporate graphite fibers, boron fibers,
?	 etc., in various matrices, such as aluminum. Therefore, an attempt was made
in this investigation to treat the glass fiber-polymeric composite as a
material to model another composite.
CONCLUSIONS
It appears that once the stresses or strains in the photoelastic composite
are determined, the corresponding stresses in any other structural composite
(having the same axes of symmetry) can be found by employing scaling factors
which are functions of the elastic constants. It is proposed to test these
scaling relationships by studying a suitable configuration.
8
40,


Photoelastic studies of advanced lap joint concept

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