Analytical foundations of the isodyne photoelasticity

MECHANICS RESEARCH COMMUNICATIONS 0093-6413/81/060391-07502.00/0

Vol. 8(6),391-397,1981. Printed in the USA. Copyright (c) Pergamon Press Ltd

ANALYTICAL FOUNDATIONS OF THE ISODYNE PHOTOELASTICITY

Jerzy T. Pindera Department of Civil Engineering, University of Waterloo Waterloo, Ontario, Canada N2L 3GI (Received 10 August 1981; accepted for print 20 August 1981)

Introduction:

Photoelastic Isodynes

It has been reported by Pindera and Mazurkiewicz in 1977, Ill that for plane stress problems - when the signal/noise ratio is optimized,[2] - it is possible to obtain photoelastically particular set of families of lines of constant scattered light intensities, each of which being related to an arbitrarily chosen direction, x or y, Isx = Isx (x,y,msx) = const;

Isy = Isy (x,y,msy ) = const

(la,b)

These isodensitometric lines were labelled "photoelastic isodynes", since they represent lines of constant values of total normal forces acting on cross-sections of a plate of a thickness b collinear with the chosen direction, and which are bounded by two adjacent isodynes, Fig. I; in other words the values of the total normal forces P acting on the characteristic sections As between the isodynes I (m)I and I ( m ) , or their intensities p, are constant, where s K S S £ k a n d Z denoteSthe order of isodyne: P(x,y,ms) = bp(x,y,ms) = const

(2)

Basic relationships between the photoelastic isodynes and the derived quantities are presented in Fig. i. Material of the object satisfies the conditions which follow from the concepts of Hooke's body and Ramachandran-Ramaseshan body. Summarizing, when the stress field in a plane object is two-dimensional, the isodyne fields yield information on related normal stress component, in chosen sections (x,y o) or (Xo,Y): d d 1 d °yy(X'Yo) = dxx py(X'Yo) = dxx [b Py (X'Yo)] = Ss dxx msx (X'Yo)

(3a)

o xx (Xo,Y) = dyy d Px(Xo 'y) = dyy d [i d msy (Xo,Y) b Px (Xo ,y)] = S s dyy

(3b)

It can be shown experimentally, and confirmed by using the known equilibrium relations for plane stress fields [3] that Oy x = Oxy =-S s dxx d msyx (X,Yo) + fxx (y) =-S s dyy d msxy (Xo,Y) + fyy (x) '

391

(3c)

392

JERZY T. PINDERA

where parameter

S

s

is identical with S

s

= S

the stress-optic

coefficient,S

o

= S

(I).

c7

(>,): (3d)

It was shown by this author and his co-workers that various'photo-elastic methods and techniques can be developed on the basis of the concept of isodynes, [3-11]. It is convenient to introduce the term "isodyne photoelasticity" to denote this system of methods. More discussion concerning their physical foundations is given in [12-15]. Certain conditions for optimization of measurements using isodyne photoelasticity, and the classical methods of photoelasticity as well, follow from phenomena presented and analyzed in [1622]. The mathematical model of photoelastic effect presented by Ramachandran and Ramaseshan [23], with some modifications, [6] has been accepted as a sufficiently comprehensive model of the basic photoelastic phenomena. The purpose of this paper is to present the analytical foundations of the isodyne photoelasticity in a more general and aesthetically pleasing fashion than that given originally. The new presentation allows one to draw more comprehensive conclusions from the experimental data and to analyse them more efficiently.

ELASTIC AND PHOTOELASTIC ISODYNES: DEFINITIONS, DERIVED QUANTITIES CONDITIONS: Y

~

=(x 'Y'ms'-I)

;

- HOOKE'S BODY - RAMACHANDRAN-RAMASESHAN - PLANE S T R E S S FIELD

l - O'xy (Xo,Y)

BOOY

SYMBOLS Is= ,Isy : x- OR y - I S O D Y N E S m : ORDER (PARAMETER) OF ISODYNE S x : x-CHARACTERISTIC DIRECTION F FORCE ]

Ss : ELASTIC ISODYNE FACTOR [

%

---~'-~'Sx ~-°'yy(X'Yo) ! XoL

0

LE-T~'T~ J Py :TOTAL NORMAL FORCE ON SECTION s x Py : N O R M A L FORCE INTENSITY (= b ' l p y ) T~y: TOTAL SHEAR FORCEON SECTION Sy txy: SHEAR FORCE INTENSITy (=b-IT~y) o'ij:STRESS COMPONENTS ( i , j = x , y ) f,C:BOUNDARY CONDITIONS FUNCTION

i

I Sx=I x_xoJ ~ . ~ x

L

I

"~--X

xo

MAJOR FEATURE ISODYNES ARE RELATED TC CHOSEN CHARACTERISTIC DIRECTIONS: Isx =Isx (x,y,ms=) = const.; Isy= Isy( x ,y, rnsy) = const. CHARACTERISTIC CROSS-SECTIONS THROUGH ISODYNE FIELDS msu = msxx( x 'Yo ) = Ss'Pyy(X'Yo) = ( bSs )_1 pyy (X,Yo) rnsxy" rnsxy ( Xo,Y ) = Ssitxy(Xo,Y)

RELATIONS FOR STRESS COMPONENTS d I, %y ( x ,yo ) = Ss-~.x m,..( x ,Yo ) d

2. %,1Xo.Y).S, ~ 3 ~.y(~o,y)

= ( bSs 1-1 [Txy (Xo,Y) + C ( y ) ]

msyy= msyy ( Xo,Y ) = Sslpxx ( Xo,Y ) : ( bSs )-I Pxx( xo, Y )

m,yylXo.y)

:-s~ d %.(Xo,yl+f~y(~

)

d 4. O'yx(X,y o ) :-Ss~--msy,lx,y o) + f'x,(Y )

rnsyx=rnsy'(x,yo) :S~'Ityx(X,Yo)=(bSs)-' [Ty, (x,yo) ÷ C(x)J Fig. 2.

1

Concept of isodynes - basic relations

Isodynes

as Functions

Related

to the First Derivatives

of Airy

Stress Function Let us consider

a homogeneous

and isotropic

flat plate of a uniform thickness

b, the surface of which is coplanar with the plane

(x,y).

The plate is of

ANALYTICAL FOUNDATIONS OF ISODYNE PHOTOELASTICITY

393

such a geometry and is loaded in such a manner that the stress state in plate is two-dimensional a condition,

and does not depend on the value of z.

for instance,

restricts

and requires

such

the values of the thickness b and the

maximal radius of curvature of boundary, b/0 m

Obviously,

Pm' such that

<
(4)

that the strain gradient in the planes

(x,y) are negligible,

other

wise the stress components normal to the face of the plate cannot be neglected. Consequently,

it is assumed in the theory of thin plates loaded by the forces

normal to the boundary that the stress components

do not depend on the coordi-

nate z normal to the plate face, regardless of the geometry of the boundary, and regardless of the values of the strain gradients. such a mathematical model of a hypothetical ised by Airy stress function, yields the known expressions tesian coordinate system, O~(x,y) ~x

plate is conveniently character-

~(x,y), which - in the absence of body forces for the stress components with respect to a Car-

(x,y,z):

= ~ x (x,y); ~2~

The stress state in

O~(x,y) ~y

~)

= ~ y (x,y) ~2q~

o xx - ~y2 = - Oy ~y"'

oyy = ix2 = - ~x ~x

~2~ ~ ~ yx =- -~x~y =_ -~y - ~X =_ -~X - ~ Y

Oxy

(5a)

(5b)

(5C)

There exist particular relations between the stress components o.., the intenzj sities Pi of normal forces acting on sections between isodynes, and the functions ~ , ~ : x y f Oyy dx = f __~2~ ~x 2 dx = ~x + fxx(Y) = my (x,y O) = s s msxx (X'Yo)

(6a)

f

(6b)

OXX

dy = f ~2~ dy = ~ + f ~y2 y yy (x) = Px (Xo'Y) = Ss msyy (Xo,Y)

and d d Oyy = ~ x ~x = ~x Py = Ss ~ x d

d

o xx = - -~y ~y = - dy - Px = S s ~ y where py and Px denote intensities collinear with the corresponding cross-sections Followingly,

msxx (X'Yo)

(7a)

m syy (Xo ,y)

(7b)

of normal forces acting on cross-sections

directions x and y, and m

and m denote sxx syy through x- and y- isodyne fields in the x- and y- directions,

the equation of isodynes

can be presented in the form:

394

JERZY T. PINDERA

?

I

= I

sx

sx

(x,y,m

sx

) =

!!--¢--'dx = py (x,y,m ) = S m (x,y) = const ~x 2 sx s sx

(8a)

2 t

Isy

Isy (x,y,msy)

~ 2~ dy ]F 3y

Px (x'y'msy)

The loci of point described by relations isodynes" because they represent acting on corresponding which elastic isodynes tic directions", The functions

Functions

(8a) and

S s m sy (x,v)

const

(8b)

(8b) can be called "elastic

lines of constant values of normal forces

cross-sections. are related,

The arbitrary directions

can be conveniently

and S . x y ~ and ~ could be called "generalized x y

x and y, to

called "characteris-

S

isodynes":

x (x'y'msx) = Isx (x'y'msx)

- fxx (Y) = const

(9a)

Y (x,y,msy)

- fyy (x) = const

(9b)

= Isy (x,y,msy)

(x) can be determined from the boundary conditions. YY The generalized isodynes ~ and ~ are simply related to the corresponding x y isodyne fields characterized by the isodyne orders m and m : sx sy x

= S

s

fxx(Y) and f

msx(X,y)

Relations and o

yx '

- fxx(Y) = const;

~

y

= S

s

m

(10a,b) can be used to determine and their intensities

°xy =- ~--y ~ x =- S s ~$y

t

xy

and t

yx

sy

(x,y) - f

yy

(x) = const

the shear stress components

m sx (x,y) + ~Dy fxx(Y) =- Ss ~dy msxy(X,y)

xy

+ f'xx (Y) (lla)

8---x ~y =- S s ~3x msy

(x,y) + ~~x fyy (x) =- Ss ~dx msyx (x,y) + f,YY (x)

= d__ t (x,y) dx yx

txy = txy(X'y'msxy)

o

.

d (x,y) = d-y txy °yx=-

(10a,b)

(llb)

~x = I °xy dY = - Ij3y

dY =- S s msxy(X,y)

+ fxx(Y) + C 1

(12a)

ty x = ty x (x,y,msy x) = f Oyx dx =- fa~a-~-X dx =- Ss msyx (x ,y) + fyy(X) + C2

(12b)

Obviously,

Txy

=

btxy

At any chosen point S

s

[ d ~ y msxy

Relations

and o

(x,y)

xy

= o

Ty x = bty x yx

(13a,b)

, thus

d (x,y)] = f' (y) - f' (x) - ~ x msyx xx yy

(14)

(14) can be used to check the accuracy of determination of the

ANALYTICAL

functions

FOUNDATIONS

and f' (x). YY to relations (7a•b) and

OF ISODYNE PHOTOELASTICITY

395

f~(y)

According

two different - preferably

(lla•b)

mutually

two families

perpendicular

of isodynes

- directions•

related

to

for instance•

I

(x,y) and I (x,y) or shortly (x,y) - isodynes, yield four independent sx sy pieces of information on the values of the normal and shear stress components, Oxx, Oyy, Obviously•

o

and

xy

°yx'

where o

the elastic and photoelastic (Isi)

In addition•

elastic

= o

xy

yx

isodynes

~ (Isi) photoelastic;

the fields of

(x,y) - isodynes

. are formally

identical•

i = x,y.

(15)

are related by the condition

o xx + o yy = o I + 0 2, thus the equation of isopachics ~

+2 x

= S

y

where m i denotes

may be presented

in the form:

~d d ~-- m + m ) = S m. = const dx sxx ~y syy s l

s

the order of isopachics

and S

s

denotes

(16)

the elastic

coeffici-

ent of isopachics. When the boundary

conditions

are such that the functions

f

(y) XX

C I and C 2 vanish then the elastic and photoelastic with the first derivatives It is convenient

of Airy stress

isostatics,

isoclinics,

such a framework teristic

characterizing

isochromatics,

lines of plane stress

the stress

isopachics,

the elastic isodynes

represent

etc.

fields, [24],

fields"

such as the [25].

Within

a new family of the charac-

lines of plane stress fields.

3. Determination

of Stress

These four independent

Components

pieces

Using Elastic Isodynes

of information

stress

information

and can be used to increase

is redundant

evaluation

of stress

entiation of isodyne fields

components,

given by (x,y) - isodynes

scribe three independent

mental

•

are identical

function.

to use the term "characteristic

to denote a set of functions

isodynes

fyy (x) '

components.

Oxx, Oyy• Oxy

Various

de-

Oyx; one piece of

the accuracy of experi-

known techniques

can be applied or adapted,

of differ-

to determine

the quan-

tities of interest. 4. Evaluation

of Airy Stress Function

When the boundary different

characteristic

ically determine It appears

conditions

are known,

directions

the stress

two isodyne

to two to numer-

function.

data sufficient

approach Airy stress

related

are - in general - sufficient,

that the isodyne photoelasticity

which supplies

fields

is the only experimental

to construct Airy stress

function represents

function.

method

In this

a surface having particular

proper-

396

JERZY T. PINDERA

ties, such as: portional

slopes of this surface in characteristic

directions are pro-

to the total normal forces acting on corresponding

through the object,

and the curvatures

cross-sections

are related to the stress components.

5. Comments on the Efficacy of the Isodyne Photoelasticity The versatility of the isodyne photoelasticity

as an experimental method

based on the application of Airy stress function is limited by two factors: i. the degree of correlation between the predictions

of Airy stress function

and the actual stress state in a plate of a finite thickness, the regions where the ratio thickness/(radius

especially in

of curvature of boundary)

is

not very small, and where the strain gradients are high. 2. the practical limits of the applicability

of the elementary mathematical

models of the interaction between radiation and matter; - the major constraints are given by the limits of the applicability

of the assumption that

in a deformed body the light propagation is rectilinear and that both related rays do not diverge noticeably,

[16-22].

On the other hand the isodyne fields supply information on the magnitude and distribution of the local deviations

from a plane stress state, which is

caused by the applied load, the high boundary curvature,

etc.

Acknowledgement This research was supported by NSERC Canada under Grant No. A2939. References i. Pindera, J.T., Mazurkiewicz, S.B., Mech. Res. Comm., 4, 4 (1977). 2. Pindera, J.T., Straka, P., Journ. Strain Analysis, 8, 1 (1973). 3. Pindera, J.T., Mazurkiewicz, S.B., Theory and Technique of Photoelastic Isodynes. (To be published). 4. Pindera, J.T., Mazurkiewicz, S.B., Kepich, T.Y., Photoelastic Isodynes in Static and Dynamic Stress Analysis. In: Proc. Seventh Hungarian Congr. Testing of Materials, Budapest, (1978). 5. Mazurkiewicz, S.B., Pindera, J.T., Exp. Mech., 19, 7, (1979). 6. Pindera, J.T., "Elements of More Rigorous Theory-and Techniques of the Isodyne Method and their Applications to other Optical Methods". In: Optical Methods in Mechanics of Solids, Sijthoff & Noordhoff, (1981). 7. Pindera, J.T., Mazurkiewicz, S.B., Optimization of Photoelastic Stress Analysis Using Isodyne Method. In: Proc. Eighth All-Union Conf. Photoelasticity, Akademia Nauk Estonskoi SSR, Tallinn, (1979). 8. Pindera, J.T., On Theoretical Foundations of Photoelastic Measurements. In: Proc. Eighth All-Union Conf. Photoelasticity, Akademia Nauk Estonskoi SSR, Tallinn, (1979). 9. Pindera, J.T., Mazurkiewicz, S.B., Studies of Contact Problems Using Photoelastic Isodynes. In: Proc. SESA's Fourth Int. Congr. Experimental Mechanics, Boston, (1980). (To be published). i0. Pindera, J.T., Issa, S.S., Krasnowski, B.R., Isodyne Coatings in Strain Analysis. In: Extended Summaries of the Technical Papers SESA (1981).

ANALYTICAL FOUNDATIONS OF ISODYNE PHOTOELASTICITY

397

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(1981). 23. Ramachandran, G.N. and Ramaseshan, S., "Crystal Optics". In: Handbuch der Physik, Fl~gge, S., ed., Springer-Verlag, Berlin, (1961). 24. Pirard,A., La Photoelastlcmte. Dunod, Paris, (1947). 25. Pindera, J.T., Outline of Photoelasticity (in Polish). PWT., Warszawa, (1953).