A review of the theory of photoelasticity

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A REVIEW qF THE THEORY OF PHOTOELASTICITY

by

Bibliotheek TU Delft

R. C. T

ennys on

Faculteit der Luchtvaart- en Ruimlevaart\El(:lV'i Kluyverweg 1

2629 HS Delft

DECEMBER, 1962

UTIA REVIEW NO. 23

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(3)

A REVIEW OF THE THEORY OF PHOTOELASTICITY

by

R. C. Tennyson

"

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•

ACKNOWLEDGEMENT

The author wishes to acknowledge the significant contribution to the review by Mr. R. C. Radford. and express his appreciation to Dr. G. N. Patterson •. D~rector of the Institute of Aerophysics. for permitting this review to be published .

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•

SUMMARY

The theory of both the plane and circular polariscope employ-ing the reflected light techn-ique is developed. The solution for the principal stresses using goniometrie compensation in conjunction with normal and

oblique incidence readings for a two dimensional analysis is outlined in detail, with a suggested technique for plotting principal stress traject'ories. Also discussed, are correction factors for the preceding analysis, such as the index of refraction and the reinforcing effect of birefringent coatings .

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TABLE OF CONTENTS

NOTATION iv

IMPORTANT DEFINITIONS 1

EXPERIMENTAL STRESS ANAL YSIS: Photoelasticity 4

THE PLANE POLARISCOPE 4

THE CIRCULAR POLARISCOPE 8

WHITE LIGHT ANAL YSIS 11

OBLIQUE INCIDENCE FORMULAE 11

REINFORCING EFFECT OF BIREFRINGENT COATINGS 14

PLANE STRESS PROBLEMS 14

FLEXURE PROBLEMS 16

GONIOMETRIC COMPENSATION 17

CALIBRATION OF PLASTIC: A Suggested Experimental

Technique 18

A SUMMARY OF IMPORTANT POINTS 19

COLOUR-STRESS CONVERSION T ABLE 21

REFERENCES 22

APPENDIX I: Derivation of Principal Stresses in a Plane 23 APPENDIX II: Isostaties - To Determine Principal Stress

Directions from Isoclinics for Non-Axial

Loading 26

FIGURES 1-18

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a o A E f i K n p r s t

v

NOTATION

amplitude of vibration vector Angstrom Unit (10- 8 cms.) Young's Modulus of Elasticity fringe constant of plastic angle of incidence

sensitivity factor of plastic fringe number 0, 1, 2, etc. 211 x. frequency of light wave angle of refraction

vibration vector thickness

resultant light vector passing to camera

Greek Symbols

J

À

principal normal stresses shear stress

principal normal strains shear strain

analyzer angle of rotation (degrees) retardation of light waves

wavelength of light wave

(12)

9 )) (3 Subscripts X, y, z 1, 2 max. p, m c. p. s. n, 0 c,

s

o NOTATION (continued)

phase angle, radians Poisson's ratio

angle of inclination of fast axes to principal stress

Cl

2

general orthogonal axes

maximum, minimum principal axis directions maximum

plastic, metal cycles per second norm al, oblique

coating in composite member composite structure

initial structure with no coating

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Important Definitions

Surface Principal Stresses,Uj

,cr;

at a 'point':- (see Appendix I for derivation of Principal Stress es)

The maximum and minimum norm al stresses at a point. They are at 900 to each other. A surface point in equilibrium is subjected to two principal stress es (the third principal stress is always norm al to a free sur- . face and its value is always zero).

Plane

Stress:-Stresses in a part which is thin with respect to transverse dimensions, having loads acting in the plane of the part. The

CG

stress, act-ing normal to the plane of the part is either zero, or constant throughout the thickness. Q\ and

\12

acting in the plane of the part are constant in magni-tude and direction throughout the thickness.

Uniaxial Stress:

-The condition when q-2

=

( J 3

=

0;

Cl

₁~ 0 Biaxial

Stress:-Condition when

\J

3 = 0

\J

1 ~ \J2 ~ 0 (generally surface stress es)

Maximum Shear Stress,

' l

max, acting in the plane of the free

surface:-'L"

max = 1/2 (

CJt -

<Ti)

acts at 450 from

q-;

and

'G.

Surface Principal Strains

é

l' E.. 2 at a

'point':-The maximum and minimum normal strain at a point, and act at 900 from each other.

Maximum Shear Strain

-y

max in the plane of a Free

Surface:-and acts at 450 from ~, ) El. . Isoclinic:- (see Appendix II)

A black line of equal inclination of principal stress (or principal strains)

y

e. g.

1

observed black line, 450 (0() analyzer angle

cr

l'

Cl

2 are orthogonal

but 0(. they make with H, say

(14)

Isostatic:- (see Appendix IJ)

A line to which one of the principal stresses is tangent. Two orthogonal families of isostatics exist on a surface of a part.

Isochromatic:-A line of constant colour - in normal incidence observations; it is also a line of constant magnitude of ( \ j 1 -

cr

2) or

(e

1 - E.. 2

_h

whence it represents lines of maximum

'L

max.

Tint of

Passage:-A sharp (narrow width band of colour) isochromatic located between the red and blue isochromatics. A calibration colour because of its narrow band width.

Fringe

Order:-The number of fringes (tints of passages. generally) passing a

-given point when the part is loaded from zero to a -given load. n + 0'/180 where n = O. 1, 2 • . . . and 0' = analyzer angle of rotation.

Fringe Value or Fringe Constant:- 'fr

The mangitude of (~ 1 -

€.

2) necessary to produce a shift of one fringe in a given plastic (depends on plastic. and thickness>.

Sensitivity or K factor of

Plastic:-A constant expressing the sensitivity of a given plastic (of any thickness) to produce a given number of fringes per unit strain and per unit of thickness of plastic.

Double Refraction or Birefringence:- (Fig. 1)

Certain transparent materials such as crystals of calcite and mica or certain strained plastics demonstrate this phenomenon of birefringence. They divide an incident ray of light into two beams which travel at different speeds through the material and which are vibrating in orthogonal planes to each other. i. e. they are polarized at right angles to each other.

Relative Retardation:- (Fig. 2)

When an incident ray of light traverses either a permanent or temporarily birefringent substance, the two beams into which it splits travel through at different speeds. whence they are said to be retarded by a phase tag of

cf •

or retardation

<f';t ;

birefringence. where t = thickness of material thru which light passes in retardation.

(15)

Circular Polarization:- (Fig. 6)

Isochromatic lines can be examined much easier when black isoclinics are removed. This can be done by placing a quarter-wave retarda-tion plate

0.

e. it splits incident light up into two components,

-L

to each other, with phase lag

d

= /\ /4) between each polarizer and the part coated with plastic,'

The light emerging from a combination polaroid - quarter-wave system with their optical axis àt 450 _{is known as circularly-polarized light. A quarter}

wave plate is permanently birefringent material. À is usually yellow wave-length.

Free

Boundary:-On a free boundary (unloaded) the principal stresses lie along and normal to the edge. This norm al principal stress is always zero by

definition of a free boundary. Hence one can always calculate the direct stress along the edge.

Colour and Wave

Length:-( a)

(b)

White light:-

=

day light, is made up of the entire spectrum or range of colours viz:

390 x 10 12

~

frequency of.:::: 770 x 10 2 cps

cps radiation

À

=

3 x 1010

and wave length cms.

frequency

Monochromatic light:- light made up of one

>-..

only. Full Wave

Plate:-A permanently birefringent plate which increases the fringe order oof the incident light by one wave length. In our equipment )\.

=

Ó :;

5750 A (yellow wavelength) . It is used to increase the sensitivity of the circular polariscope at low stress levels in which colours are not distinct enough for accurate measurements.

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Experimental Stress Analysis: PHOTOELASTICITY

Previous methods of experimental stress analysis, such as strain gauges, have never been entirely satisfactory. The fact that all stram gauges have a finite length perrnitted only mean values of strain to be obtained over some intervaL In regions of high stress-strain gradients, the gauge readings are difficult to relate to the actual state of strain existing in the specimen.

The photoelastic technique however, provides in effect, a

continuous distribution of strain gauges of virtually zero gauge length. Using this method, the maximum shear stress and the directions of the principal

stresses as well as their separate magnitudes can be determined directly.

In early applications of the photoelastic technique, models of

the actual test specimen to be analyzed were made from a birefringent material.

Such materials as bakelite, glass, cellulose and plastics exhibit the pro-perty of double refracUon or bi-refringence when strained. The stress dis-tributions were then revealed by shining polarized light directly through the

model and reflecting the incident light from a silvered or reflective . interface,

or receiving the light as it travelled out the back surface of the specimen.

Attempts were then made to relate the stresses in the model to those which

would be found in the actual specimen.

Recently, with the development of good bonding cements, it

has now becom e possible to fix the birefringent material directly to the test

specimen. By providing a shiney reflective interface between the specimen

and the plastic coating, the incident polarized light can be reflected and the

strain-induced birefringence .rnay be observed. Since the plastic follows the

deformation of the test piece, providing the bond is very good, the strains

in the plastic and those of the specimen (neglecting 3-D effects) can be

assumed identicaL Whence by computing the strains in the birefringent

material through colours, the corresponding stresses in the specimen can

.. be calculated, taking into account if need be, a correction factor for the

reinforcing effect of the plastic coating.

The Plane

Polariscope:-In the following analysis monochromatic light (of one wavelength) is assumed to be the source. although white light is usually employed witp very little error resulting. Whence virtually no modification in the theory will be necessary.

I \

(Refer to Fig. 3) Light is radiated from a source s passes

through a Polarizer P, traverses the birefringent plastic M and af ter being

reflected from the silvered surface of the metal specimen, passes through the

analyzer A to the eye (or camera). It is assumed that the angle between the

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incident and refleeted I'ays is small (say <: 50). However the index of refractian of current birefringent coatings is of the order 1. 59 and this would reduce the angle to almost 00 (Fig. 4). Thus the rays may be eonsidered to act normal to the surface of both the plastic and the specimen.

Figure 5 represents a small element of the face of the plastic coated specimen, viewed from tb.e direction of A. The plastic is aeted upon by the principal stresses \J"" 1 and ~ 2' considered to be horizontal and vertical for eonvenience.

A ray of ligh. polarized in the plane OA from the direction of P is incident on the plastic. The vihration is simple harmonie and the

transverse "displacement" at au angle /I e,.( , to the vertical rnay he represented

by the equation:

(1) where: p is 2

7T'

times the frequeney of the

light and t is the tiro e .

This displacement may be resolved into two compQnents in the directions of 1 and 2:

(2)

oB

=

S:z.

=

a.

c..co

I>

t

c.c;:)

e>t.

(3)

Now the principal straius t!:. 1 and

C

2 change the velocities with which these two cornponents traverse the plastic*, thus introducing a relative retardation

cf"

between tb.e two waves:

viz:

dn

=

:2./<-t"

(2.

1 -

E2.)

where

,<

is the

(4)

strain-optic coefficient (sensitivity

constant) of the plastic; 2tp is used

because the light passes through the thiekness of the plastic (tp) twice .

The phase differenee in radians is given by:

.-en

-=

41f'

J<

1:, (

EI

-E2)

À

(5)

* The quantitative law of photoelasticity is: the intensity of the principal

strain difference(~ 1 -

!

2)is directly proportional to the value of

cf'

ft.

d

being the relative retardation of polarized light traversing the plastic under normal incidence and measured in terms of entire or fractional wave lengths,and t is the thickness of the plastic coating.

Further. it ean be shown experimentally that the incident plane polarized light (or circu.larly polarized light) is divided into two component .... vibrations in the direction of the principal stresses (whence strains)

cr

_{l &}

cr2 )

and the retardation of these vibrations being

oe

magnitude of difference (<rl -

0-

_{2 )}

(& also l. 1 -

e

2) by the ~irefringent material under load. 5

(18)

-Let the times required for the two wave components to tra-verse the plastic (incident

+

reflected path) be tl and t2.

On emerging from the plastic, the two wave forms are given by the equations:

0511 =. CL

~~o<

CcO

(f"t -

ftl )

(6)

(7) Thus the phase difference in radians is:

(8) It should be noted here that each wave component is also re-tarded by an amount

'1\/2

corresponding to a phase shift of

1T'

radians upon reflection at the plastic metal interface. Since this shift affects both wave forms equally, it wil! have no resultant effect on the phase change at

emergence. Hence it wil! not be inc1uded in the analysis.

lf the axes of polarization of the analyzer and polarizer are now "crossed" (i. e. at right angles to each other) then the light emerging from the analyzer can be represented vectorally by OE and OD. where:

00 ::

5,1 Cc:)

~

=-

a.

~~

CUO

oi ar-:!

(pt -

f

tl )

oe -

.s

~I

sJY....

0(

==

CL

~o<

C4:) 0/.. Cd:J

(~t

-

P

-tZ )

(9)

(10) The resultant vibration V is the vectorial sum of the two com-ponents OE and OD.

\} =

C O - o t :

9:

~Wt:b<' [~(pt-pt.)

-

~()~t-J1lz)J

(11)

2

which upon simplification becomes

.. S p

(t

I -

t2 )

2 'J

=

0... ~ 2.-< s..tM.- (

2 J

The factor

~

p [

+ -

4:-

I

~

1.:1

J

represents the simple harmonic variation with time.

when

The amplitude is

( constant for given material of given thiCkness.) It can be seen that some light will reach the observer except

(i) ~:l~ =-0

(19)

In case (i), Q = n

1i'" /

2 when n is an integer, this is the condition

that a minimum intensity of light will emerge. Referring to Fig. 5 it can be seen that this occurs when either the polarizer or analyzer axis is aligned parallel to one of the principal stresses. The locus of all such dark points for one particular angular setting of the polarizer-analyzer combination defines the isoclinics for that angular parameter. By mapping the isoclinics (see Ref. 1) for increments of angle of combination from zero to 900, a complete set of stress traj ectories can be obtained.

For the remaining case (ii), P(tl - t2) /2 =

nit is the condition

for a minimum intensity of light to emerge from analyzer.

41j'-But from equation (8) P(t1 - t2) = en =

T

K tp (e 1 - t!:. 2)· Therefore the condition for a minimum bf intensity is:

(12)

:l.tp

K

The principal strain difference may be related to the principal stress difference by Hookè's Law i. e.

(13) (-see from plane stress-strain equations) viz.

(JK =- E

(~x

+-

V

~y)

(14)

I-J}2

(15)

t«dimensions of specimen in x, y directions Replace x and y by subscripts 1 and 2

From (14) subtraci. eqliatioh~ ~ (15) and get: or or

(<rx -

<T-y)=-

E

(~x

+ vE. y - z:.,/ -y

~>()

1_)12-( q-X - IJ'

y)=-

E

C

~

x (\ -V ) -

~

'I ( 1 - )J ) ) ~_v2. ( Zo

x -

<i..

'I )

(c:rx -

Q""

y)

(I

1-

Y )

E

Therefore the condition for minimum intensity can now be ex-pressed as: r'\ À

(Cii

-<;S"l.)=

~

(16)

(20)

which defines a new stress-optie coefficient.

If n

=

O.

(CT

1 -

cr-

2)= 0 therefore the principal stresses are

equal; points where this occurs are called isotropie points .and willofcourse

be dark.

Points at which n

=

1 form a dark band or fringe of the first

order. points for n = 2. a fringe of the second order. etc. It can be seen that

(t;j

1 - ~2)for a fringe of order 2 has twice the value of (~1

-

~ Vfor a fringe

of order 1. Thus to determine the principal stress difference at a point. it is

necessary to know the order of the fringe passing through the point and the

stress difference represented by the fringe of the first order.

These fringes are known as isochromatics because. in white

light analysis they correspond to the extinction of some particular wavelength

and hence appear as a uniformly coloured band. It must be noted th at when

monochromatic light is used. these lines are black. When white light is used

the patterns appear coloured.

The most important use of white light with the plane polariscope

is in the study of isoclinics. The Circular

Polariscope:-Using aplane polariscope, an observer would see a series of

isochromatic bands, corresponding to regions of equal principal stress (or

strain) difference upon which is superimposed the black isoclinies. In most

cases it is desirable to remove these isoclinics as they obscure the stress patterns.

If the polarizer and analyzer with their axes of polarization

per-pendicular to each other. were to rotate, the isochromatics would remain

stationary while the isoclinics would move with every different orientation of

the lenses. If the rotation were fast enough the isoclinics would no longer be

visible to the eye, leaving only the isochromatics. A device which achieves

this effect by purely optie al means is the circular polariscope (see Fig. 7)

This type of polariscope consists of a plane polariscope with

two quarter wave plates ~A and Qp inserted immediately before and af ter

the analyzer and polarizer respectively. A quarter wave plate is a crystal plate which has two mutually perpendicular polarizing axes which affect the light in the same way as a permanently stressed birefringent plastic. The

thickness is chosen such that the phase difference

pCt, -

t2.) introduced

be-tween the two wave components passing through it is

1T'

J

2. One axis of the

quarter wave plate is called the fast axis (i. e. the plane which advances the

incident light faster than its orthogonal counterpart plane) and the other is

called the slow axis.

A ray of light passing through a circular polariscope is

indi-cated in Fig. 7.

8

..

(21)

Let:

the polarized vibration vector OA be given by:

(17 )

The quarter wave plate

Qp

is aligned such that its fast axis makes an angle of 450 _{with the polarizer's axis.} _{OB is the component of OA} corresponding to the fast axis of

C;;>p ,

oe

being the component of OA correspand-ing to the slow axis. Upon emergcorrespand-ing from

cOl"

OB and OA are given by:

0{3

= : ;

~f-t

oe=

~ ~(pt- 7(/2)::

fi

(18) (19)

As can be seen from these equations, a point moving with these displacement components traces out a circular helix (i. e. a helical path of constant amplitude

=

.9r ).

Thus the light is said to be circularly polarized.

(see Fig. 6).

t/2

The principal stress

CJi

is inclined at an angle ~ to the fast axis of Qp. The components OB and

Oe..

(of equal constant amplitude) in the direction 1 and 2 are given by:

SI

-=

OC ~(->

-

0 '13

StM..

~ (vectorial addition)

S~

=

oe

.s.v1t

~

of-

oB

~

f>

which up on substitution of equations (18) and (19), reduce to

SI::::'

Sz ':

If the times for these two components to pass through the plastic, and emerge, are tI and t2 then S 1

ir

S2 become

5,':

fi

~(pt--pt,-(S)

5

2.1-:

~ ~

(pt-

ftL -

~

)

(20) (21) ,(22) (23) (24) (25)

is the relative phase change caused by the principal strain difference(e 1-!2~ The quarter wave plate

QA

has its fast axis aligned at 900 to that of

Qp.

The components of 5~

!,

~1. in the directions 3 and 4 are:

5,3-==

~ [s~~ ~(pt-

pt2.

-~)

+

Ccg~~(pt-~i:\-rS)J

(26)

54

=

a-

[~~ ~

C

pt-

pt2-~) ~ ~(3

M{

pt-

f-t

,

,-~)J

(27)

(22)

where

I

'S, Co-;)

rS

(28)

I

'a..

SI S01-? (29)

After passing through QA, a relative phase change of 1j' /2 is introduced, whence 8 3 and S4 become:

s/

~ [s';"'~=(pt

-p

t

_n

5)

+Cff.>~Sv>JPt-ptl-f'»J

(30)

54'=.ii

[c...r.>

(!>

c..r.l

(ft -

ph -

(S - Ir/?. )

-S';"1->51;',J;i

-ptl

1;lfz)]

-

~

[w:lf->

s.n.

(pt-

ft>.

-f-.)

+-

SUtf>CtYJ(

pt: -

pb

-;.3)]

Since the analyzer axis is at an angle of 900 with respect to the polarizer axis, the components of S3' and 84 ' transmitted by the analyzer also given by

,,:=

'5>4'

53'

{ i

Q

=~ ~~(r>t-Pt2

-2(3)) -ScM.(Pt-lrtl-2(3)]

=

0.[

~

l

pt - p(

10,

+;~)+ 4~

H

s"".

P

(~-!:7.)}]

( 32)

) pt -

PI

(tl"tk2.)

+-

2.«J

The factor ~

t

2. \'

J

represents the simple harmonic variation with time

The amplitude is

Thus for a circular polariscope, the amplitude of the re-sultant vibration is independent of the orienta'f:ion of the principal stresses with respect to the polarizer axis.

However, the conditions for iSlJchromatics are the same as for the plane polariscope.

(23)

White Light Analysis:

-From Eqs. 4 and 12 it can be shown that using a monochromatic crossed polariscope, an isochromatic fringe will be observed only when

én

=

Vl À. That is to say, each time the principal strain difference at some

point in the plastic is of such a magnitude that it causes a relative retardation cfn between the two wave components passing through it equal to some multiple

of the wave length of the light used, the two wave forms interfere, and no light reaches the viewer' s eye.

Suppose white light, consisting of wave lengthsÀ l' À 2 ... etc is used. Then, when

dn

= n À.) Yl/\2 . . . etc. the wave lengths )\ 1,

A

2'

À 3' .. would be extinguished and a coloured band, characteristic of the wave length extiriguished, would appear. The first colour to be extinguished is violet, leaving the complémentary colour yell ow . As

én

increases, the successive extinction of the spectral colours from violet to red takes place. The complementary colours observed through the polariscope are, in the

order of increasing strain (or stress); yellow, deep red, deep blue, and green. The line of demarcation from one colour to the next is, for most colours vague and poorly defined. There is, however, one colour known as the "tint of passage", which is suitable as a reference fringe. It is a dull purplish shade which sharply marks the transition from red to blue, or green. The tint of passage corresponds to the extinction of the yellow light, of wave-length 2.27 x 10- 5 inches. Hence, for the tint of passage of first order (n ::: 1),

dn :::

2.27 x 10- 5 inches.

A"s

J",

is increased beyond the first order, the colour sequence repeats. However, it is found that the shades of yellow, green etc. observed are slightly different in the higher orders due to overlapping of the comple-mentary colours. Fortunately, the tint of passage is still quite distinctive and can be used as a reference colour for calibration and stress measurements.

Despite the fact that these tints of passage are more difficult to observe than the d-ark isochromatics of monochromatic analysis, there are

several definite advantages for using white light. The fringe orders are more easily determined because of the progressive coloured bands between tints of passage, and isotropic points (where

cr

1 - \j' 2 ::: 0) show up unmistakably as black regions. lntermediate stresses between tints can be estimated by the colour of the bands.

Oblique lncidence

Formulae:-Figure 8 represents an element of the metal structure under analysis, coated with birefringent plastic of thickness tp. The axes 1 and 2 co- incide with the directions of the principal stresses at the point

"0".

Since the plastic is bonded to the metal, the strains in the plastic and the metal are identical.

(24)

..

To find the separate values of the principal stresses in the metal,

it is necessary to take two polariscope readings - one in normal incidence and the other in oblique incidence.

One normal incidence reading (light incident alongothe axis 3) yields the difference of the principal strains

(33) An additional measurement in oblique incidence provides another relationship between 2.., ,fz. and

do

where

th

is the relative retardation in °

oblique incidence. These two values, ~o and

eI;,)

permit the calculation of the separate values of ~I and !2')hence the value of the principal stresses.

For light propagating through the plastic in oblique incidence , the relative retardation is proportional to the secondary principal strain

difference in the plane perpendicular to the direction of propagation (see Fig. 9). Assume the light is propagating along the axis 3/ at an angle 9 to the axis 3 and in the plane 2-3. The secondary principal strains in the plane

l' -2' are given by Mohr's Circle relation (see Fig. 10),

Therefore Let

,

z.

Z. :: (34) (35)

Since (J3

=

0 (because stresses are applied in the 1-2 plane only).

_ -Yr

I

-Y\,

'E,(I-.2.Yp)

~.)Jp I-)lr

f,

~

(.

)JE

\-Yp

Now

do

=

2.t

o

K CE / -

'fr{)

where

-l

Cl ' :

-t~

/

Co:)

e

(36)

(37 )

(38)

do:::

2.Kt.

p [

fo.

('2.-\.>"

-VrU02~)

-

~7..(I-2.Yp

-c.a:J2eJ]

~e

\

.

-vp

(39)

[

~I

(1-)iE

c:.co'e ) - al.

(C<fol"'e

-!f

)d

1-

y

P

1- Yp

B

(25)

From Hookets

Law:

!ol

=

~W\

-

VW\

<fi",

E'WI

~\IY\

E2.:

Gi~

-

yW\

~

Y'Y\

EW\

EM

Substituting these into equation for:. .

J'~

m = metal p = plastic

we get

10 EWI

Cif.j

e

=

OlM (Ai

BYm) -

<J

_2M

(Ay~

+

8 )

~I(

tp

~

J

n relation becomes:

cf"

EjIY1

:l )(

-fp

(HVrn )

(40) (41)

lf we again let(A

+

B~)

=

M

,(A~

+B)=N,then a solution ofdn and

J'

0 equations

yields:-or where and

v,t)\=

EM

f

d'oCa:;e

cm

IV

2 :<

Jä:p( M-N)

2

I

+

YJ)?

J

V2fr1

=

Ewt

f

_{ioQJJG -}

cFn

M

{

2.

J(t/> (

/Yf-IV)

1+

YW1

J

\Tl"",

-=

E~

À

• ~ ho~t)

\(3 - hl"lGd:)'tS t(2

~

.2

l<

-lp (\typ)

sv>\?e

L

J

'"

E"",

À Z

)'

Y'lo

CJ:1

e

1<.3 -

Y)~ K',

I

'l1.V\'\ -.:=. {

.2

K

t

p (

lot

Yp)

SlM 'f,)

ho) Y)V\ are fringe orders. given by

~

0 -::.

Jo /

X

,

V)'"

=

dY1/

À

Kl.

=

I-)Jr

'YM

Cy

r -

y'W\ \

c.n1..

e

( \-Vr'r'\')

~

-Yp

YVY\ -

()lp

-YW\

J/

GdO"t.,;

(\ -YW\')

\-)Jp

t-y~ (42) (43) (44) (45)

Note the angle 9 must include the corrected angle of incidence due to index of 13

(26)

refraction of birefringent material viz.

I I

A correction factor must now be employed to account for the reinforcing effect of the plastic. However, if a photoelastic model were em-ployed, then no correction term would be necessary.

Reinforcing Effect of Birefringent Coatings

--When mechanicalor structural parts are coated with a photo-elastic plastic and subjected to their working loads, the coating follows the surface deformations of the part. The deformed coating then behaves as a photoelastic specimen and reveals the distribution of surface strains through-out the part. Since the birefringent coatings do carry a portion of the lo'ad that would otherwise be carried by the structure, strains at the structure

-coating interface are somewhat less than those in an uncoated part. In addition,

strain gradients through the coating thickness must be taken into account to determine the surface strains that would be developed in an uncoated part. Correction factors are derived in Reference 5 for plane stress, flexure of plates, torsion of shafts and cylindrical pressure vessels. For plane and flexural loading, the correction factors are equally applicable to regions surrounding common" geometrical discontinuities.

Plane Stress Problems: (Fig. 11)

Consider a structural part with a birefringent coating in a state of plane stress, and let figure 11 represent an infinitesimal element taken from the member. Since birefringence is proportional to the difference of principal strains in the coating, the obj ective is to relate the principal strain difference in a coating to that in an uncoated part. Thus the element of figure 11 is chosen such that x and y are principal axes and

q-x, ~y,

'Lx, ~y

are principal stresses and strains.

The influence of load carried by the coating can be determined by equating the forces acting on the composite element to the forces acting on the same element in the uncoated structure. Thus:

Subscripts

o structure with no coating

s composite structure - coating member c coating in composite member

Therefore ( 46)

(27)

(47)

Strains are independent of z for plane stress problems, and are equal at the interface between the structure and the coating.

or

Accordingly;

) ~~

Cc)

=

~~

(s)

₍₄₈₎

Using Hooke' s law of elasticity for 2-D plane stress: requires;

o

(49)

(49)- which applies for both the structure and the coating.

Substitution of (48) and (49) into equation (47) yields:

2.~

(0) -

t'j(o)

=[h(G) -

L'j

Cc)]

[I

+

::~!: ~::':l)J

(50)

[ 'Lx

Cc.) -

~'j

(C)]

where Cl represents the correction factor required to convert measured strains fx(c:.) ,

ey(c)

into strains that would be developed on the uncoated structural part.

a) b)

c)

It is assumed in this analysis that:

structure and coating are isotropic elastic ma~erials identical deformation of structure and coating exists at the interface

plane stress prevails in structure and coating.

Experimental results of Ref. 5 along with computed data indicate that even for very thick coatings, the correction factor is only a few percent.

(28)

Flexure

Problems:-An analysis of birefringenet coating behaviour is very important for fiexure problems, since large correction terms/factors are possible in

this case. Besides the reinforcing effect resulting from the bending

moments carried by the coating, two additional factors must be considered, namely, the gradient of strain through the coating thickness and the dis-placement of the neutral plane in the member if the coating is applied on one side of the plate or bar.

For the case of thin and medium -thick plates i. e. for plates in which the bending stresses play a predominant role in the deformation of the plate, while vertical shear stresses have negligible influence, the· re-inforcing term /factor is

, (51) where

(52) where

d:::

distance between neutral surface and interface

0.(

-=

Ec (,-

V.s

2

)

(-fix

+

~) ~

Fe:.

Es

(I _ Vel)

Ry

~

( -'-

_T2~

+

_R.y

.v~

)

The above resultCanalysis)does not apply to the general case of very thick plates where vertical shear stresses may become the primary cause of deformation. This case is of no importance because the coating thickness can be made small in relation to plate thickness and the corrections associated with reinforcement become negligible.

Correction factors for other types of loading are derived in Ref. 5 and will not be presented here.

In conc1usion, in order to minimize the effects of dissimilar Poissonts ratios and local reinforcement thin coatings are preferabie.

(29)

Goniometric Compensation:

-Using an initially crossed circular polariscope. let ~'" be the angle by which the analyzer is rotated in a clockwise direction from its crossed position (see Fig. 12).

The resultant vibration th us emerging fr om the analyzer may be written as (c. f. Eq. 32)

V '::

S~' ~

(45"+oln)

5 _{3 / ~}

(45°

+cocI\,\ ')

(53)

Now from Eqs. (3:» and (31). we get:

54'-

$.3'=

~

[

~

(p-!: -

pt ...

-2~)

-

s-<M.

(j.t-

pi,-~)]

(54)

54'

+

503'

~ ~

[

AA(

pt - ptJ.)

-I-

SvYt- (

»t -

p-t.)

1

(55)

Suppose that the crossed system had previously been rotated such that the polarizer axis was parallel to the principal stress ~,. This can be achieved by using a crossed plane polariscope and by observing the isoclinics. The angle

(3

must then be 450.

Therefore Eqs. (54) and (55) become

(56)

(57 )

and Eq. (53) becomes:

\j -

-%

~

Cif.I "'''

l-

~

(pt-

pt~)

+

C<r.l(

rt -

~tl

)]

~~Yl [~(p-t-pt2)

+slÀ-\.(r

t -

p-lI)J}

-

';" { -

~

(pt-

ph -.(,,)

4~

(\*-

pob

+"",,)

1

(58) - <À

sV.t.

[pt-

f(-tt

+h) ]

$~

[Js.(tt-t

l ) -

"".,J

17

(30)

The amplitude is then

[?

(t,-tl.)

-O<I"1J

2

For a minimum of intensity, P(t1 - t2)

=

2(n1]\-

+

cm) But p(t 1 - t 2) = 211"'

cf

nl

À

where n = 0, 1, 2, an integer

d

O.

fractional fringe order.

where n' is the ( 59)

The application of goniometrie compensation-to estimating fractional fringe orders can best be illustrated by an example (see Fig. 13).

Suppose the point under observation lies somewhere between a fringe order n and order (n

+

1). A rotation of the analyzer in the clock-wise direction will cause the fringe of lower order to move towards the \>oint of observation. After a rotation of cm degrees, the nth order fringe (tint of passage) coincides with the point of observation. The fractional frïnge order at that point is then given by

'()/==

n+

cXy)

\80

Hence, fractional fringe orders in both normal and oblique incidence can be determined using goniometrie compensation.

Calibration of Plastic:

-In order to determine stresses and strains it is necessary to know the fringe value 'f' or sensitivity constant K of the plastic - that is, the increment of strain corresponding to a change in the fringe order of one.

Therefore for n = 1, the fringe value f is given by

f

=

(~\

-

'i:

2) •

n=

1

( 60)

( 61)

Using the technique of Goniometrie compensation, f is then that value of the strain corresponding to 1800 ₍₌ _cm) _{rotation of the analyzer.}

Suggested Experimental Technique

:-Using an aluminum bar having the dimensions say 1/2" x 1" x 12" with a strip of photoelastic plastic bonded to one side, either calibrate the plastic by a cantilever load curve or by a tension test.

Take all readings at a given point. Using the tint of passage as a reference curve (in which case À = 2.27 x 10- 5 in. ) plot positive angular analyzer rotations (an, clockwise) versus applied load until Q'll ~ 1800 .

(31)

As each increment of load js applied, the analyzer is rotated until the tint of passage coi_. ,ncides. with the point of measurement.

_.

In a tensile test

cr

₂= <J""'3

=

0, and the fringe value f may be obtained from the following equation:

J

- l .

e

_{p •}

t.p

\ -rYp

t

w\ ( (2)

wherecr1 is the applied stress at an

=

1800 _i._{e. the necessary stress for the}

given plastic and specimen to produce the first fringe (n

=

1).

The above equation takes into account the reinforciqg effect of the plastic coating to the metal part under plane stress conditions.

,

A Summary of Important

Points:-(a) The foregoing analysis and techniques are valid for 2-D plane stress problems only.

(b) The index of refraction must be taken into account especially in the oblique incident formulas. s~

i.

~ 1.59 for c1.lrrent plastics in use.

S~

y-(c) An inherent error in oblique readings is obvious. The light incident on the plastic is at some angle 8 (corrected for refraction effects) . However the separation of the incident and emergent light rayon the surface of the plastic may then be of the order of 1/2 ". Thus the net retardation observed thru the analyzer is actually an integrated value over this distance. In regions of high stress gradients, the errors could be very large.

(d) The strain sensitivity constant K or f varies for temperatures outside the range of -440 to + 850_{F. Corrections for K can be made for other}

temperatures (see Fig. 14).

(e) An exceptionaUy wide temperature variation during a test may result in a parasitic birefringence caused by the difference in expansion be-tween the work piece and the attached plastic coating. Therefore it is neces-sary to take a measurement of the birefringence before and af ter loading, so as to cancel the birefringence present under no load.

(f) Thickness effect of plastic: if the thickness of the coating is comparable to the thickness of the part under study, especially in the case of thin plates subj ected to bending:

(1) Error due to the fact that the photoelastic indication corresponds to a measurement which is not made on the surface of the part, but somewhere inside the plastic layer, if the part is subjected to bending.

(32)

'"

(2) . Error due to the reinforcement of the part by coating it with a plastic ( E. plastic varies anywhere from 4. 5 x 10 5 psi to 4. 0 x 104 psi) (g) These errors can be corrected by the curve shown in Fig. 15 for bending reinforcement on Aluminum and Steel.

(h) The. thinner the plastic, the less sensitive it is to strain (&ee formula 12). Therefore relatively thick sheets of plastic may be used to give larger strain sensitivity, but corrections must be made for the above

enumerated effects.

20

(33)

Colour Stress Conversion Table (Ref.

3)

Colour

Ll

Strain* Relative Retardation*

-6

IN/tN.

tI'

in O.

00001

mrn XIO

In crossed In parallel

system system ( ( I - El.)

black white

0

gray yellow-white

170

10

white light red

430

26

pale yellow dar k red brown

460

27.5

tint of passage (ni)

473

28.7

light yellow indigo

500

30

brown yellow gray-blue

720

43

red-orange blue-green

840

50.5

red pale green

900

54 ..

( Ol) tint of passage

945

57.5

indigo gold yellow

980

59

blue orange

1100

66

green red

1250

75

tint of pas sage{

n,)

1418

86.2

green-yellow violet

1450

87

pure yellow indigo

1520

91

orange green-blue

1670

100

dark red green

1830

110

"11 ( 01.) tint of pas sage

1890

115

indigo gray yellow

1910

116

green brown-red

2200

133

tint of passage(n3)

2363

143.7

green-yellow gray-indigo

2380

145

carmine red green

2550

153 "'

C

n

3) tint of passage

2835

172

green pale pink

2900

174

* This chart is valid for t p

=

O. 120"

and

K

=

O. 1

(plastics type S or

A).

If

thickness of plastic used in test is different from O.

120",

then (el -f2) have to be muIt. by a factor

0.120

where

-Lu..

=

thick.of plastic used.

i: u.

If

1<

factor is different from O. 1 we must muIt. readings by O. 1 where

K

\4 = constant for plastic used. K~

Note (ZI-i1.)= (5:r,-Ûl.)(~).- Hooke's Law is assumed.

E.

(34)

(35)

1. Frocht, M. M. 2. Hetenyi, M. 3. Timoshenko, S. Goodier, J. N. 4. Zandman, F. Riegner, E. I. Redner,

S.

S. 5. Zandman, F. 6. Redner, S. S.

7.

REFERENCES

Photoelasticity (New York, J. Wiley &

Sons) Vol. 1 (1941), Vol. 2 (1948).

Handbook of Experimental Stress Analysi~

(New York, J. Wiley & Sons) 1950.

Theory of Elasticity, (New York, McGraw-Hill) 2nd edit ion, 1951.

Reinforcing Effect of Birefringent Coatings, S. E. S. A. #588, 1959.

Photostress --- Principles and Applica-tions (Phoenixville, Pennsylvania:

Instruments DivisiQJl, The Budd Co. Oblique Incidence Formulae and Data Reduction, (The Budd Co., Bulletin PS - 5052, 1960).

Instructions for the Use of the Large Field Universal Photostress Meter

(The Budd Co., 1960, Bulletin PS-3043).

(36)

(37)

APPENDIX I (Ref. 1) Derivation of Principal Stresses in a Plane Positive Sign Convention:

-"v,

'iJ"...,

~~~

-~H,I--""'~?t-)

cr

')t.

~ ~

' - -_ _ - ' I ~

cr,

Assume a thin flat model loaded in its plane (2-D stress system).

Let an element or smal! block of the body being considered. in-cluding the desired point of measurem ent, be severed from the whole. by planes. on which the stresses at the point are assumed to be given.

Assume the force on each face of the block to be uniformly distributed, since each face is very smal!. (force

=

stress x area of face). Neglect the weight of the block element. Consider point

0:-y'

Y •

<Or""-

y'

t<l~d"

-

cr~do.'.Wu.

~----r----~--.~

~~

D

~

~--~---~---..

r

ch

('

~"'&

- Cfy

do-Square Element. sides of area cl~ Area of face BOE is

0.0...:::

cJ.o!

~G>

x./

By applying the conditions for static equilibrium we find:

Z

t"x'-O

<fw

~I

'=-

cr}( da..'GJ.)e

C(])

è

+

cr'

da..l

SII\'\.G CtJ:) 0

or

<f"x'

(J"y

da'

StM..G)

s~e

-+

+

L'dcL'

S\~

.

e ~

g

23

(38)

or

<f')(

I::.

_

~

(o-x

+Cl~)

+

i (

0-

x-

cr\j)

C-if.) 2.e

~

(64)

q:

SWt.

:z.e

Z

'Fy' ::.

0

_~/-,~I

""""-l._'

-

....,.,-1 \ . , (65)

L.- c..AU.- - l uu. ~ ê ~

G -

'-

'--tC\.. S Vh..9 5\.1);\.9

cr

)(.do..'4öe

'SvY\..e

+

\j"y

<::la

I 5

lIh.e

CU)B

'1'

~

."

(<î"

;<J

Y )

s';"'2e -

'T

C<r:> W (66)

By definition. the principal stresses are the maximum and minimum stresses at a point.

Therefore.

d

(~X/)

': -

S~2..t> (<J)(-<Ju)

+

n:

CJ:J.) l.G ow: 0 (67)

.

..

oe

I

which is· a maximum (or minimum) for

(68)

(<:j)( -c:J~)

Whence the maximum point are defined by equation (68).

(and minimum) normal stresses at a

I

If we substitute (68) into (66) we find ~

=

o.

Whence the principal stresses occur on planes for which the shearing stress es vanish.

Equation (68) defines principal stresses and it can be seen that there are two possible values for the angle 26' (less than 3600 ) which differ by 1800_. _{Therefore the two principal stresses !ie on principal planes at 90}0 to each other.

By definition>we let

\li

~ maximum principal stress and

<fl..::

minimum principal stress.

Note:

cr,

is algebraically ~

<r2.

~where we accept tension as positive and compression as negative.

If Eq. (68) is substituted into Eq. (64) we get:

~.'

=

~(<r"

...

IJ~) ~

l

[Uî>-<J':j)' ..

4'1'1. ] ,/...

(69) whence:

<;J,

=:

~

(0-)(

4-

<r'-j)

+

i

l (

q'l-<J~

')"l. -+

4"1"2

J

lIL

-

~

L]

,~

t

(6

~

ot

cr

~ ~

-

1L

[cr~ -

<:r

'1 )

4 4

~

24 (70) ('10

(39)

\'

-And the maximum shearing stress obtained from Eq. (66) and Mohr' s circle is

(72)

Principal stresses and strains may be easily related by Rooke's Law ( :L-l) plane stress):

For pure stresses and pure strains

(73)

However for a two-dimensional plane stress system (<::tof; = 0)

~)( ::-

cr-)(

-

y

<J

_j

E

F-:

(74) ~~ ~

~~

-

y

<ft(.

E

F- (75)

!.-e ':

-

~

C

~

4-

Cf''1 )

(76)

In terms of principal stresses, the principal strains ~I) I I I

E3

may be obtained from Eqs. (74) (7S) (76) by setting x, y, z, equal to 1, 2, 3 and noting that <\)3' =

o.

(40)

(41)

, :

"

APPENDIX II

Isostatics - To Determine Principal Stress Diredions from Isoclinics from nsm-axial loading

ISOSTA,JC

\.

'Iï , q-"z. ARe' s~ow~

I~ P R l't\olC , pA,. L s. Ta.

e

sS

PI' t'te-Cï I ON S • (srlZ.Ess TRA.,,reCTö/ZY

--~~~~---~----AB

Be

Note Example REF'E.1ZENCe U ~E'

is obtained by drawing a straight line of slope(91

+

92)/2 etc. Actual stress trajectory, by fairing in a curve.

;;:; angular rotation of polarizer-·analyzer assembly from zero position, for a constant load (viewed thru a plane polariscope set-up)

( cr

1 ~ \r

2 algebraically)

see Figures 16 and 17, 18.

(42)

Summary of Important Properties of Isoclinics:

-a) Isoclinie lines do not interseet each other, except at an isotropic point (Ref. 1).

b) An isotropie point

=-

point at which the two principal stresses are equal and are inclined at every conceivable angle. Therefore all

isoc1inics pass thru such a póint.

c) A straight free foundary is also an isoclinie line.

d) Isoclinie lines only interseet a free boundary when it has the inclination indicated from the isoclinie (except at a point of zero stress where all isoclinics may run int 0 the boundary).

e) , An axis which is symmetrical with respect to both the loads and with the geometry of the model, coincides with one isoclinic.

f) To determine whether a region is an isoclinie or an isotropic point, insert quarter wave plates into the plane polariscope set-up. lf the black region vanishes, then we have an isoclinic.

(43)

, ..

PLAHE OF

PoLARI~ED

UNPOLARIi!ED \...\G-HT IS SPL\I IN,O TWO

BEAMS POlAR\:2.ED AT R.\GHT ANGLES

B Y THE BIREFRINGENT MATERlAL

EIG· l 'i'AlO olU~OGON .... L wÄ"'J:S) RE:T .... RDEP (OUT O~ ~I-t~se) ey AMoV NT

'cf'.

"~TARD""'T\ON FIG. 2.

BI RSF I<:INGE NT IVIAT E\'2IAL

PRINCIPAL STRESSE.$

(44)

/' /' / ,/ / / /' / Nore IZEFR.ACTI"N EPFECTS'

p

/' /' ,/ PHOTOELAS TIC a:::)4T!NG A

IZ.E I=LEC T I VI:

.5~IZt=ACE !Nel D~ '" T

RA""

P~AN!! POLAR/ScOPE NORMAL / ,/ /' / ,/ / / ' / REFLECTI"E SUI!FACE REFRACTION EFFECT Sc>URCE ~

s'"

t

$1r'1.r

-

INt>!!X

(45)

,.

t

.

'

,-

_,

POLARI ~ER A'A.\ea

STR.ES'SEt> . PLASilC

COATING

--..:::lOK,.. -~"....---+--~<:jl.) l:,

A)(.IS

OA PLANE OF" POLAQI~A"ION Ot: lNCID1=N"'r LIGHT

\....IG~T "IECTOQS eMER6ING FflDM ANAL'( rE-12.

m

E'r'E. .

RESObUTION Of P\...~Nl::: fOLA R\.:zED Lol G HT ALDNG. PRINCI PAL STRESS A)lES

FIG.S

FIG- <ê

A

'/

iHe. E:~ER.GING \lECTOR. \-\AS A ~'AGN\TuDE

=

C

\~i~

'

+

\)'12 ) '/2 -=:

Q/~2

AT

ANY

ANG Lr:

e.)

+ClJy\

e

=-

~

pt

S_ VAR\ES '\{j I'-M îlME, \.(J HENCE ~A"tH . TRACED 'i!

'I

O/~ ) OF Cö'rJ STÄN'ï Àf'Il1>L I iUDE..) IS A C I

!lev

I..A R HEU

X. '

Cl R<..ULAR P01-A{<.\~ATION

(46)

QU-"IRT'aR. VoIA'iE

PL~Tl!

S

LIC?/IH S.OU RCe

4 QUAR~1'Z.

'<.J

JIo..vE P\..P<ïE. --+~"""Tliftil\r:-=;-+--A:-:-:N7-A~\..~"" Zo ER A ~ \~ oF I

~

_.fi oF" TR.A~::' Mlse,\ oN

CIRCULAR. POLARISCQPE

(47)

· PLA~TIC(~)

CJ

1

aaL'

G\)E; \NCIDENcE

~\G-

e

PLANE Orz.T~oNA L TO INClt)';'NT

I , .0.

L.IG"T AT OSLlQUE. At-lGLE \'7

OBLIQV§ PLANS

(48)

e.l.

MOHR~S C.IRCLE FelS ST&AIN

F IG· \0

----.,.~_+tJ- q-~ (c )

_ _ ~_ 'l"~(s.)

..J---".,...-I- '(

ELEMENT TAKEN F~o"l\ COATED F'LA.NE. SrRSSSËp BODy

(49)

---~~~~~---~~--~,

ANALY%ER A)I.\S

GONIOMETRIe ÇOMPE hI.sATloN

EIG. 12.

--+-.:..:-... DI R.Ee. T , 0 N O r

F1'2.INGE OF

ORD~R ((\+1)

ROTA"'-'ON

I .... ~_- NEw pos ITION O~

FfZ.IWGE 'V» A~EQ

~

ROTA"t'O'tJ 0<",.

PoINT O~

OBSIrRV Ai ION

(50)

C

0.10

~

0.08

0.02.

o

....

"\

~

,..."

K

=

SENS' ST'" IT~

FAc.TOR

-$0

-25

0 .2S".50 7.> /00 /25 I.S() J7S FIG.14 FIG .. 15 4-0 3-.5 ,310 2.11$

z·o

l,S .... ~ ~

"0

ALUM\N\JM a_oS"

o

0'2 0.4 0. E>

o·e

1.0 \.2 I,~ I,b

,·S

Z·O REF. 3

~

PlAST Ic

/4:

"'1 ET

AI...

CQIV~ECTIONS Tl>

K EOJl

SPEC/ME-NS S Ue..zECTlfP 70

(51)

FAMILY OF ISOCLINICS AT ROOT

OF' CRACK

ISOCLINI~5

WERE PHOTOGRAPHED ThROUGH A PLANE POLArtI5COPE

_ _ _ 00

10°

CRACK

'10

(52)

PHOTOGRAPHS OF STRESS PATTLRNS WHICH SHOW

ISOCLINICS, USING A PLANE-POLARISCOPE.

THIN SHEi:T WITH

~TERNAL

CRACK

LOADED 'IN TENSION

PIG. 17

(53)

Isoclinics of Buckled Circular Cylindrical Shells Under Axial Compression

(54)

(55)

UTIA REVIEW NO. 23

Institute of Aerophysics, University of Toronto

A Review of the Theory of Photoelasticity

R. C. Tennyson December, 1962

1. Research Techniques and Equipment

3. Experimental Stress Analysis

1. Tennyson, R. C.

• ~

W

27 pages 18 figures 2. Photoelasticity 4. Reference Manual

II. UTIA Review No. 23

The theory of both the plane and circular polariscope employing the reflected light

technique is developed. The solution for the principal stresses using goniometrie compensation in conjunction with normal and oblique incidence readings for a two dimensional analysis is outlined in detail, with a suggested technique for plotting

principal stress trajectories. Also discussed, are correction factors for the

pre-ceding analysis, such as the index of refraction and the reinforcing effect of bire-fringent coatings.

Available copies of this report are limited. Return this card to UTlA, if you re'luire a copy.

UTIA REVIEW NO. 23

R. C. Tennyson December 1962

1. Tennyson, R. C.

•

27 pages 18 figures

2. Photoelasticity 4. Reference Manual

Il. UTIA Review No. 23

technique is developed. The solution for the principal stresses using goniometrie

compensation in conjunction with norm al and oblique incidence readings for a two

dimensional analysis is outlined in detail, with a suggested technique for plotting

principal stress trajectories. Also discussed, are correction factors far the

pre-ceding analysis, such as the index of refraction and the reinfol·cing effect of bire-fringent coatings.

Available copies of this report are limited. Return this card to UTIA, if yau re'luire a copy.

UTIA REVIEW NO. 23

1. Research Techniques and Equipment 3. Experimental Stress Analysis

1. Tennyson, R. C.

• ~

W

The theory of bath the plane and circular polariscope emplaying the reflected light technique is develaped. The solution far the principal stresses using goniometrie

compensatian in conjunction with normal and oblique incidence readings far a two

dimensional analysis is outlined in detail, with a suggested technique for plotting principal stress trajectories. Also discussed, are correction factors for the

pre-ceding analysis, such as the index of refraction and the reinforcing effect of bire -fringent coatings.

Available copies of th is report are limited. Return th is card ta UTlA, jf you require a copy .

UTIA REVIEW NO. 23

I. Tennyson, R. C.

~

m

w

27 pages 18 figures

The theory of both thc plane and circular polariscope employing the reflected light

technique is developed. The salution for the principal stresses using goniam etric

compensation in conjunction with normal and oblique incidence readings for a two dimensianal analysis is outlined in detail, with a suggested technique for plotting

principal stress trajectories. Also discussed, are correction factors for the

pre-ceding analysis, such as the index of refraction and the reinforcing effect of

bire-fringent coatings.

(56)

(57)

UTIA REVIEW NO. 23

Institute of Aerophysics, University of Toronto A Review of the Theory of Photoelasticity

1. Tennyson, R. C.

• ~

W

11. UTIA Review No. 23

technique is developed. The solution for the principal stresses using goniometric compensation in conjunction with norm al and oblique incidence readings for a two

dimensional analysis is outlined in detail, with a suggested technique for plotting

principal stress trajectories. Also discussed, are correction factors for the pre-ceding analysis, such as the index of refraction and the reinforcing effect of bire-fringent coatings.

Available copies of this report are limited. Return this card to UTlA, if you re9uire a copy. UTIA REVIEW NO. 23

Institute of Aerophysics, University of Toronto A Review of the Theory of Photoelasticity R. C. Tennyson December 1962

1. Tennyson, R. C.

27 pages 18 figures

• ~

_~

The theory of both the plane and circular polariscope employing the reflected light technique is developed. The solution for the princ·ipal stresses using goniometric compensation in conjunction with normal and oblique incidence readings for a two dimensional analysis is outlined in detail, with a suggested technique for plotting

principal stress trajectories. Also discussed, are correction factors for the pre-ceding analysis, such as the index of refraction and the reinforcing effect of bire-fringent coatings.

Available copies of this report are limited. Return this card to UTIA, if yoll require a copy.

UTIA REVIEW NO. 23

Institule of Aerophysics, University of Toronto

1. Tennyson, R. C.

,.

•

technique is developed. The solution for the principal stresses using goniometric compensation in conjunction with normal and oblique incidence readings for a two dimensional analysis is outlined in detail, with a suggested technique for plotting principal stress trajectories. Also discussed, are correction factors for the pre

-ceding analysis, such as the index of refraction and the reinforcing effect of bire

-fringent coatings.

Available copies of this report are I imited. Return th is card to UTlA, if yau require a copy.

UTIA REVIEW NO. 23

Institute of Aerophysics, University of Toronto A Review of the Theory of Photoelasticity

R. C. Tennyson December, 1962 1. Research Techniques and Equipment 3. Experimental Stress Analysis

I. Tennyson, R. C.

..

~

W

27 pages 18 figures 2. Photoelasticity 4. Refèrence Manual

Il. UTIA Review No . . 23

The theory of both th" plane and circular polariscope employing the reflected light technique is developed. The solution for the principal stresses using goniometric compensation in conjunction with normal and oblique incidence readings for a two dimensional analysis is outlined in detail, with a suggested technique for plotting principal stress trajectories. Also discussed, are correction factors for the pre-ceding analysis, such as the index of refraction and the reinforcing effect of

bire-fringent coatings.