In-plane stresses in polygonal plates due to thermal inclusions

RESEARCH COMMUNICATIONS 0093-6413/82/050301-10503.00/0

MECHANICS

IN-PLANE STRESSES INCLUSIONS

Vol. 9(5),301-310,1982. Printed in the USA Copyright (c) Pergamon Press Ltd

IN P O L Y G O N A L

PLATES

DUE TO T H E R M A L

T h e i n Wah Texas A&I U n i v e r s i t y K i n g s v i l l e , Texas 78363 (Received 18 January 1982; accepted for print ii June 1982)

Introduction

A small t h e r m a l i n c l u s i o n in a thin plate p r o d u c e s s t r e s s e s in the plate. The e v a l u a t i o n of these s t r e s s e s d i v i d e s into two i n d e p e n d e n t problems, one i n v o l v i n g plate f l e x u r e and the o t h e r i n v o l v i n g e x t e n s i o n s and d i s t o r t i o n s in the p l a n e of the]plate. The p r o b l e m of f l e x u r e has b e e n c o n s i d e r e d p r e v i o u s l y [i . In this p a p e r we p r o p o s e a m e t h o d for e v a l u a t i n g the in-plane s t r e s s e s and d i s p l a c e m e n t s . The plate is a s s u m e d to be t r a c t i o n free on the b o u n d a r i e s . The b o u n d a r y c o n d i t i o n s are s a t i s f i e d e i g e n f u n c t i o n s of a n g u l a r regions.

exactly

by the use of

In the area c o n t i g u o u s to the h e a t e d r e g i o n b i h a r m o n i c f u n c t i o n s for c i r c u l a r r e g i o n s are used, and a least square p r o c e d u r e is e m p l o y e d to e s t a b l i s h continuity. The m e t h o d is e x p l a i n e d in d e t a i l w i t h r e f e r e n c e to a t r i a n g u l a r plate. N u m e r i c a l r e s u l t s are g i v e n for r e c t a n g u l a r plates.

Differential

Equations

The s t r e s s e s

in a free p l a t e

function V4¢

¢, w h i c h

is g o v e r n e d

are o b t a i n e d

by the d i f f e r e n c i a l

= -V2N T

with

from A i r y ' s

stress

equation

[2] (i)

h/2

NT = aTE

~Tdz

(2)

. J

-h/2'

301

302

THEIN WAH

T=T(r,@, dinate

Z) is the t e m p e r a t u r e

distribution,

in the d i r e c t i o n of the p l a t e

f r o m the m i d d l e

surface,

z being

the coor-

thickness measured

s T is the c o e f f i c i e n t of

downward

thermal

expan-

sion. V4=(V2) 2 is the b i h a r m o n i c is the p l a t e We

thickness

shall assume

coordinates The

operator,

E is the e l a s t i c m o d u l u s ,

a n d v is P o i s s o n ' s

t h a t all the o p e r a t o r s

h

ratio.

are e x p r e s s e d

in p o l a r

(r,8).

s t r e s s r e s u l a t a n t s are g i v e n by _ 1 D~ + 1 D2~ r r Dr r2 DQ2 D2~ NO - Dr2 N

a ar

NrO =

(3)

1 a~ (r a-@)

The r e q u i r e m e n t

t h a t the b o u n d a r i e s

be t r a c t i o n

free

leads

to the

boundary conditions a~ = a--~ = 0 in w h i c h

(4)

n is the n o r m a l

ing the d i s p l a c e m e n t s by a p p l i c a t i o n Zu

u

a-Y

+

~(r

to the b o u n d a r y .

u, v in the p l a n e

of H o o k e ' s

1 Dv + -r ~-8) -

Law.

Method

v r

2(l+v) Eh

without

involving

(i) a n d

its e d g e s

plays

T h e y are (5)

NT + iL--~)

(6) (7)

Nr@

function

~ e n a b l e s us to solve

the d i p l a c e m e n t s .

(4) that the s t r e s s

lateral deflection all

of the p l a t e are o b t a i n e d

of S o l u t i o n

The use of the s t r e s s

tion

govern-

l-v 2 NT Eh (Nr + ~ )

u + 1 Dv + Du _ l-v 2 ? ? D--O v~ Eh (No 1 ~u + Dv r D0 ~r

The e q u a t i o n s

clamped.

It m a y be n o t e d

function

w of a l a t e r a l l y

loaded

lateral

from equa-

# is a n a l o g o u s isothermal

The r i g h t h a n d m e m b e r

the role of a d i s t r i b u t e d

the p r o b l e m

to the

plate with

of e q u a t i o n

load on the plate.

(i)

THERMAL INCLUSIONS IN POLYGONAL PLATES

Thus equations

(i) and

lem.

Once

(3).

Substitution

with

the necessary

v.

The

ments

¢ has been found,

latter,

means

would

but these

on the plate

(5) to

of evaluating

to be sure,

statement

the stresses

into equation

and rotations,

straints

(4) are a complete

303

of the prob-

are given by equations (7) then provides

the displacements

involve

us u and

rigid body displace-

can be determined

if the con-

are specified.

Ei~enfunctions

Consider plate

the triangular

is heated

Except

plate

over an area

in the heated

area

shown

in Fig.

i.

6A with centroid

¢ is governed

Assume

that the

at 0.

by the biharmonic

equation

v4¢

=

0

(8)

We connect the lines A'B'C'

the point to meet

0 to the vertices

the opposite

of the triangle

sides and complete

and extend

the triangle

as shown in the figure.

It may be shown that,]for eigenfunctions

any given angle

e, an infinite

set of

of the form (9)

¢c = rl+l F(0,1) may be determined values are

of the eigenroots

2 real

single

functions

the boundary

I are generally

satisfying

triangular

function

regions

¢ may be completely

of the type mentioned

plicative

constants.

boundary

region

(4).

complex ~]. Thus

the boundary

designated

functions

edges

conditions

conditions

The there

for a

eigenroot.

For the the

satisfying

conditions

(BC',

BA')

These

3, and of course,

The triangle the thermal

A'B'C'

by three

(AB', AC')

at 0.

region

multi-

depend only on the for region

2 and on the edges

on the angles

sets of e i g e n

with undetermined

eigenfunctions

designated

load centered

described above,

on the edges

for region

i, 2 and 3 in the figure,

i, on the

(CA', CB')

for

el' e2' and ~3" 4 in the figure

contains

304

THEIN WAH

The

general =

#*

solution

Ai,

0 and

known

plates. this

and

1

C.

are

l

We

been

designate

for

this cos

is

n0 c (i0)

constants.

in F i g u r e

used

region

s i n nO c

unknown

0 c is d e f i n e d

and have

(8)

(An Rn + A n + 1R n + ~

(B n R n + B n + 1Rn+2)

B.

the p o i n t well

+ n =El

C 1 + C2R2

+ n =I where

of e q u a t i o n

for

the

them circle

R is the

I.

These

for

from

functions

calculation

functions

radius

are

of c i r c u l a r

the p u r p o s e s

of

paper.

The

Particular

The

function

Integral

}

for

the e f f e c t

of

the h e a t i n g

of an e l e m e n t a r y

O0

area

~A m a y

be

f o u n d by

its a n a l o g y

to the d e f l e c t i o n

w

of a O0

plate

due

~o0

-

to a c o n c e n t r a t e d -NT r [ ~ log ~ 8A .

(ii)

r is the

distance

of

a is a n y

suitable

nondimensionalizing

If we

now consider

effect

of

the

the

load:

thermal

N T as b e i n g

inclusion

load N T from

constant.

constant

at the p o i n t

the p o i n t

over

(x,y)

(x,y)

(Fig.

2)

the a r e a

A,

is g i v e n

by

and

the

the

integral NT ~ o = - 2--~ J A The

integral

of g i v e n circle

at

-

If the

NTa2 2

shape.

"a ~

NTa2 4~

a

c2 R a--g log -a

and y directions

to be p e r f o r m e d

We p r e s e n t

-

the

inclusion

1 c2

1 R2

2

2

--Z a +

final

for a r e g i o n results

is a c i r c l e

for a

of r a d i u s

and

-a Y )

R_< c

c,

(13)

R > c

(14)

is a r e c t a n g l e centered

of d i m e n s i o n s

at 0,

the

{ ~ ( ~ l ' n 1 ) + ~(~ 2' n2)

-qb ( ~ l , r 1 2 ) in w h i c h

has

If the

(°2r_~ log c

inclusion

#o = -

(12)

0 then 2

~o =

(12)

in

a rectangle.

NTa2 ~0

6A

indicated

geometrical

and

centered

r log

-

qb(~j2,r]l)

}

u a n d v in the

function

is g i v e n

x

by (15)

THERMAL

INCLUSIONS

IN POLYGONAL

PLATES

305

nj

(~ k 2 + q 32 ) + ~k 2 tan -I

#((j'qk ) = %k r,j log

(16) ~k

with = - ( ~ + x)/a,

~

=

(17)

(~ u - x)/a

1

= -(~ Equations

+ y)/a, (13),

v2~0

and

(13)

(15)

satisfy

satisfies

the

following

conditions:

the e q u a t i o n

(18)

= _ NT

and t h e r e f o r e fies

(14)

in e q u a t i o n

0

gl = (2 - y ) / a

the basic

equation

in e q u a t i o n

(i).

(14)

satis-

the e q u a t i o n

V2~

= 0 0

and t h e r e f o r e

O

the b i h a r m o n i c

in e q u a t i o n

gle and

(15)

satisfies

The e x p r e s s i o n s the c i r c u l a r IA and

also

equation.

satisfies

(18)

within

(19) outside the h e a t e d

for the stress

and r e c t a n g u l a r

rectan-

rectangle.

resultants

inclusions

the h e a t e d

and d i s p l a c e m e n t s

are

summarized

for

in Table

2A of the Appendix.

It may be v e r i f i e d tinuous

across

from

the Tables

the b o u n d a r y

that

of r e g i o n

the d i s p l a c e m e n t s

are

A for both

the circle

a finite

discontinuity

con

and

rectangle. The

stress

as we pass

resultants, from

however,

the i n t e r i o r

suffer

of the h e a t e d

in the case of the r e c t a n g u l a r suffer These

an infinite results

tangular

agree

equation

for R e g i o n

4 in Fig.

constitute

boundary Since,

discontinuity with

those

(13),

(14)

The

at the corners given

by Nowacki

to the exterior

shearing

stresses

of the rectangle. [51

for the rec-

inclusion.

In brief,

they

inclusion.

region

(15)

Together

with

a complete

conditions

however,

i.

and

integral

are

"particular

integrals"

%* given by e q u a t i o n

of e q u a t i o n

(i).

Only

(i0) the

remain to be satisfied.

the b o u n d a r y

conditions

in regions

i, 2, and

3

306

THEIN WAH

are h o m o g e n e o u s mains 2,

and s a t i s f i e d

to e s t a b l i s h

by the e i g e n f u n c t i o n s ,

continuity

between

region

4 and

it only

re-

3 regions

i,

3.

The

stress

function

of the form C'

for r e g i o n

1 is the series

(9), while t h e s t r e s s

function

is ~' = ~0 + ~* as g i v e n by e q u a t i o n

These C'B'

two f u n c t i o n s which

~= ~

follow

from

four

the plate

(i0)

and

(.14) or

continuity

flexure

A'B'

(15).

conditions

across

analogy

__~' ~n'

~2~

(20)

~2~,

__~ V2q5 = ~n

~ ~n'

n and n' d e f i n e The c o n t i n u i t y lines

C'A'

V2~, normals

of s o l v i n g

[33.

Extension

of M e t h o d

be n o t e d

in no way

also be s a t i s f i e d

these e q u a t i o n s

that w h i l e

restricted

analysis

here

triangle

A'B'C'

gonal

(20) m u s t

adopted

plates,

the

I).

functions

in

only

in m i n d

that each of the r e g i o n s

the p l a t e m u s t be t r i a n g u l a r

fact,

have as m a n y

depend

"circle

interior

of lines

region

sides

apply

also

functions"

in d e f i n i n g

in o r d e r

and

continuity

of symmetry.

to the c h o i c e

One has

being

of

may be

It will, analyzed.

is e s t a b l i s h e d

Such c o n s i d e r a t i o n s

(i0).

There

at the c o r n e r s

plate

the

and poly-

the regions.

of a s u i t a b l e

g i v e n by e q u a t i o n

of

fall w i t h i n

that e i g e n f u n c t i o n s

as the p o l y g o n a l

(15)

the m e t h o d

c o u l d be of any shape.

along which

on c o n s i d e r a t i o n s

symmetry

the

is given

is not so restricted.

flexibility

but the

(14)

In the case of q u a d r i l a t e r a l

the size of i n c l u s i o n

used,

(13)

that the i n c l u s i o n

greater

to k e e p

Squares

as to size of i n c l u s i o n

requires

(Fig.

by L e a s t

is o b v i o u s l y

The n u m b e r

across

and A'B'

details

It w i l l

to the line C'B'

condition

elsewhere

are

satisfy

for the t r i a n g l e

~'

~n

The

must

of e i g e n f u n c t i o n s

will of

s u b s e t of the

in

THERMAL INCLUSIONS IN POLYGONAL PLATES Numerical

We

give

paper. The

Examples

two

numerical

The

first

results

for

a rectangular to be The

at

a circular

center

sumed

a circular

3 for

0=30 ° .

It w i l l the

of be

The

lack

the

of

shear

Of

course

contained

aspect

given

Both

in

ratio

in T a b l e

inclusions

this

b/a=l.2. 1 and

are

for

assumed

rectangle. plate

only.

the

with

The

e=60 °.

results

is a s s u m e d

that

did

while

displacements is

symmetry

we

We

are

to be

have

intersection

diagonals

This

of

of

are

2.

displacements

at

the

symmetrical.

the

inclusion

noticed

center,

plate

in T a b l e

is a r h o m b i c

the

constrained

direction

of

analysis

have

given

located

as-

in T a b l e

at

the inter-

diagonals.

In c a l c u l a t i n g were

the

inclusion

inclusion

The

the

of

is a r e c t a n g u l a r

example

of

examples

inclusion

the

second

section

307

to be in

the

not the in

of

assumed the

change

the

expected

diagonals

at

stresses

that

that

are

the

u displacements

plates

and

t h a t the

point.

symmetrical

rectangular for

the

plate

are

about anti-

v displacements. is d u e

to

the

effect

distortion. no

conclusions

can

displacements

as

dependent

constrained.

But

they

are

they

give

be

some

drawn

from

upon

insight

the

magnitude

the w a y into

the

TABLE 1 Rectangular Plate (axb) =.3, b/a=l.2 Circular Inclusion Uo=N T a/Eh, Radius of inclusion Rc/a=.l

Nxy=O

DISPLACEMENT

& STRESSES ALONG y/b = .5

x/a

u/u °

v/u °

0 .i .2 .3 .4 .5 .6 .8

-.0087 -.0110 -.0178 -.0298 -.0636 0 .0636 .0298 .0178

.9

.0110

1

.0087

.005 .002 .00151 .001 .0005 0 -.0005 -. 001 -.00151 -.002 -.006

.7

Nx/N T

Ny/N T

0 -.0526 -.113 -.283 -.0963 -.0968 -.0963 -.283 -. 113 -.0526

.0513 .1045 .152 .316 -.1044 -.1047 -.1044 .316 .152

.1045 .0513

the

plate

general

of

the

is pattern

308

THEIN WAH

TABLE 2 Rectangular P l a t e (axb) ~=.3, b / a = l . 2 Rectangular Inclusion U o = N T a/Eh,

Nxy=O

Size

of

-a = .I, -a = .i & S T R E S S E S A T y / b = .5

DISPLACEMENTS x/a 0 .I .2 .3 .4 .5 .6 .7 .8 .9 1

G

Inclusion:

u/u

v/u

o

-.124 -.200 -.150 -.i00 -.050 0 .050 .i00 .150 .200 .249

-.0045 -.0030 -.00529 -.00926 -.0198 0 .0198 .00926 .00529 .00302 .00405

Ny/N T

Nx/N T

o

.0125 .0322 .0486 .101 .363 -.iii .363 .101 .0486 .0322 .0125

-.0171 -.0362 ~.0898 -.352 -.109 -.352 -.0898 -.0362 -.0171 0.0

TABLE 3 Rhombic Plate ~=.3, ~ = 6 0 Circular Inclusion (Rc/a = u O = N T a/Eh, N r @ = 0

~axa)

R° =

length

of

.i)

diagonal

DISPLACEMENTS & STRESSES At 0 = 3 0 ° r/R o

.i .2 .3 .4 .5 .6 .7 .8 .9 i

Ur/U o 0 .624x10 -4 0.0 -.00715 -.0316 0.0 .0319 .00726 - 624x10-4 0

v /u

0 0 0 0 0 0

o

Nr/N T 0.0 .00102 -.00103 -.0867 -.386 -.0985 -.392 -.0874 -.00103 .00102 0

No/N T

-.0009 .0151 .115 .426 -.0898 .432 .116 .0151 -.0009 0.0

THERMAL INCLUSIONS IN POLYGONAL PLATES

309

References

i.

T h e i n Wah, "Thermal due to L o c a l i z e d (to appear).

S t r e s s e s in P o l y g o n a l P l a t e s Heating", Aeronautical Quarterly,

2.

B r u n o A. Boley and J e r o m e H. Weiner, "Theory of T h e r m a l Stresses", J o h n W i l e y and Sons, N e w York, 1960.

.

T h e i n Wah, E l a s t i c Q u a d r i l a t e r a l Plates, J o u r n a l of C o m p u t e r s and S t r u c t u r e s , Vol. i0, pp. 456-466.

.

T h e i n Wah, Roots of T r a n s c e n d e n t a l E q u a t i o n s , N a t i o n a l T e c h n i c a l I n f o r m a t i o n Service, A c c e s s i o n No. P B 2 7 2 9 2 9 / A S (1977) U.S. D e p a r t m e n t of Commerce, S p r i n g f i e l d , V A 22161.

5.

W. Nowacki,

"Thermoelasticity",

A d d i s o n Wesley,

Inc.,

1962.

Appendix

TABLE Circular

IA

Inclusion INS IDE

2 @/NTa

-

1

~

1 2

r / NTa

0

D0 Nr/N T

N

1 2

_ _

N@/NT

1 2

Nr0

0

2u v0

r Eh/NT(I+~)

r 0

(

~ a r a

2 log

c

~

-

1 c

~

2

--~

a

+

2

i~

)

a

-

OUTS IDE 2 1 c r ~ --~ log -a a 1 c2 2 ar 0

2 1 c 2 2 r 1 c2 2 2 r 0 2 c m r 0

310

THEIN

TABLE Rectangular

~I

=

-(U/2

+

ql = - ( ~ / 2 X(~<,qj)

NTa

~

~

-

~

v =

(~l,nl)

{-~(q1,<1)

+ <<2

x)/a

(v/2

- y)/a

tan -I ~

+ qj 2 t a ~ 1

~< ]

3<
'nl)+×({2'n2)

- 2qj

+ ~(~1,n2)

-

+ 2~<

} qj tan -I ~ <

+ ~(~2,nl)

~(~2,n2)}

+

~(q1,{2)

+

n~

{tan -I

<-T-

-1

tan

n~

~2-

-1

tan

= NT {in(
~(n2,<1)

-

~(q2,<2)

]

~i ~] ~2 ~2} {tan -I - - - tan -I - - - tan -I - - + tan -I - ~l ~2 ~i n2

NT

Ny = ~

u

(
{X(
{-~

NT N = - -x 2~

xy

log

(u/2

NTa

_

N

=

n2 =

= nj in(~< 2 + qj2)

ax = ~--'~" -

<2

+ y)/a,

NTa2 = 4~D(l-v)

~

2A Inclusion

x)/a,

= {
~(~<,nj)

WAH

~

~

67-1 + tan -I ~2} - in(622+ql 2) + in(£22+n22)

}

~ ~x

(l+v) Eh

~ ~y

A

%

FIG. FIG.

1

DEFINITION

SKETCH

2

E F F E C T OF THERMAL I N C L U S I O N