Elastic strain energy and interactions of thin square plates which have undergone a simple shear

Scripta M E T A L L U R G I C A

Vol. II, pp. 477-484, Printed ~n the United

1977 States

Pergamon

Press,

Inc.

ELASTIC STRAIN ENERGY AND INTERACTIONS OF THIN SQUARE PLATES WHICH HAVE UNDERGONE A SIMPLE SHEAR Jong K. Lee and William C. Johnson Department of M e t a l l u r g i c a l E n g i n e e r i n g Michigan Technological University Houghton, Michigan 49931 (Received March 28, 1977) (Revised April 22, 1977)

The elastic strain energy of and the elastic strain interaction energies between thin precipitate plates which have undergone a simple shear parallel to their broad faces is of great interest in connection with respect to twinning and martensitic phase transformations. Although the elastic strain energy associated with an equivalently sheared ellipsoidal precipitate has been exactly formulated by Eshelby (I), the interaction energy between two such precipitates has not been computed due to the considerable amount of numerical calculation involved. On the other hand, it has been shown that the calculation of harmonic and blharmonic functions and their derivatives can be performed without difficulty for any point of interest inside or outside a right rectangular parallelepiped (2). Utilizing this mathematical characteristic, we investigate in this Letter, the elastic strain energy and the interaction energies of thin square plates which have undergone a simple shear under the simplifying assumption that the matrix and precipitates are elastically isotropic and have the same elastic constants. Let an infinitely v, be subjected domain

extended matrix,

to a uniform

of a precipitate,

the constrained

V.

with shear modulus

stress-free

displacement

of u ci(~)

pTkj is the stress derived

Green's

functions

Gik(~ , ~'),

is given by (1,3)

u (~) = P j where

transformation

In terms of the elastic

~ and Polsson's ratio T strain e.. Ij within the

j(~, r')dV(r')

from the stress-free

(i)

transformation

T strain eij

using Hooke's law. Here the usual tensor suffix notation is employed. A repeated suffix is to be summed over the values i, 2 and 3. Suffixes preceded by a comma denote d i f f e r e n t i a t i o n with respect to Cartesian coordinates. For the isotropic form (3):

case,

the elastic

÷ Gik(r' where

Green's

function

i ~ik ~') - ~-~U I~-~' I

6ik is the Kronecker

delta

function.

i 16~(l-v)U

the r e s u l t i n g

for the constrained

strain eCij" " = (u~,jl + u~j,l.)/2: T

~)

relationship

,klij

Eq.

with respect

- 2ve~k¢(~)

T + ejk~(r),ki}]

22 ÷ + ~xi~x k r-r'l

Substituting

and d i f f e r e n t i a t i n g

e~ij (~) : I • ~[ekl~(

Gik(~ , ~') is given in the

,ij

(2) into Eq.

(2) (i)

to xi, one obtains

- 2(l-v)

T (~) {eik~ ,kj (3)

477

478

STRAIN ENERGY OF SHEARED THIN PLATES

Vol, ii, No, 6

where

(4)

¢(~) = f f f dV(~') v

Ir-r'l

and

dV(~').

~(~) = f f f l ~ - ~ ' J V

¢(~) is the ordinary harmonic harmonic potential function.

potential

function

while

(5) ¢(~)

is

called

the

bi-

Now we c o n s i d e r a right rectangular parallelepiped whose center is taken as the origin and whose edges are parallel to the Cartesian coordinate axes. Let the edges of the p a r a l l e l e p i p e d be 2a, 2b and 2c. For a d i l a t a t i o n a l stressfree t r a n s f o r m a t i o n Faivre

strain e T T ~ ~ 0 if i = J, otherwise ei~ T = 0), (i.e., ei~ ij c (~) by i n t e g r a t i n g Eqo (i) and (4) obtained the c o n s t r a i n e d strain ei~

d i f f e r e n t i a t i n g the e q u a t i o n thus obtained with respect to x i.

In a different

approach, Sass, Mura and Cohen (5) expressed the d i s p l a c e m e n t a s s o c i a t e d with a cube p r e c i p i t a t e in the Fourier series. In this study, we assume that the T stress-free t r a n s f o r m a t i o n strain ei~J is a simple shear in the Xl-X 3 plane T T (= x - z plane), i.e., el3 = e31 ~ 0 and all other components are zero. The elastic strain energy per unit volume of a precipitate, is given by (i): w

IH ( Tj -

=

V

T ~e13

= a~

{8abc

W, in an infinite matrix

ij

T

el3 - f

c

f

b

a c

+

/ el3(r')dx'dy'dz').

(6)

-c -b -a For the elastic

strain interaction energy between two precipitates,

we consider,

as shown in Figure i, an identical p r e c i p i t a t e whose center is at ~. For s~mpllcity, let the edges of the second p r e c i p i t a t e also be parallel to the Cartesian coordinate axes and I~] be sufficiently large so that there is no contact between the two precipitates. The interaction energy becomes (6):

Eint

l = -fffP j B>e j A

dV,

VA z+c

=

T -4~em3

f

y+b x+a

f

z-cy-bx-a

f

e lc 3 ~. -( ~ ' ) d x ' d y ' d z ' ,

where P icj (B) is the c o n s t r a i n e d stress due to the p r e c i p i t a t e whose center is T at the origin and e .(A) is the stress-free transformation strain of the ij p r e c i p i t a t e whose center is at ~.

(7)

Vol.

II, No.

6

STRAIN ENERGY OF SHEARED THIN PLATES

479

[o01] Z,X3

~ / / /...... /L

F;/

FIG. ]_ Sketch o f the orientation relationships between two square plates. The square plate at ~he orizin is denoted as B while the other one is designated as A.

1/ ~Y,X 2

[010]

/ X,X, [100] The harmonic biharmonic MacMillan.

function ~(~) has been obtained by MacMillan

(2) and the

function ~(~) is derived using techniques similar to those of Employing, with Faivre (4), the fol!owin~ notation: [f(x,y,z)] x2'y2'z2 -Xl,Yl,Z I

one obtains Cor a right rectangular ~(~) = [xy~n(r+z)

(8)

(-l)i+j+kf(xi,Yj ,z k) i,j,k=l parallelepiped:

+ yz%n(r+x)

+ zx~n(r+y)

o ] - 1 {x~tan - . ( y z ) xr

+ y2 tan-].(~_~) x y ~ xs +Jx_a, a , y +y_b, b , z +z_c c zx + z 2 t ~~ . - i t~-~j

(9)

and ~(~) = [~xyzrl ÷ ~l{(r2 x2)yzZn(r+x)

+ (r2_y2)zxgn(r+y)

+ (r2_z2)xy~n(r+z)}_~{x4tan-l(x_~)yz

+ Y4~-an-l'zx~,

]x+a,y+b,z+c + z4tan-l(z~r)}~x_a,y_b,z_c ' where r 2 = x 2 + y2 + z 2. e

e ]e3 ( r ) -~

From Eq.

(lO)

(3), el3( c ~) becomes:

T

]3 [ ~b ( ~) , ] - 1 3 3 = ~TF~-(T2b-]7

._ ( t _ v ) { ~ ( ~ ) 1

] + ~(~) , 33}]

T

f k~ --.

~ - ,I- ( 1 - v ) { t , a . ~ - - ( ~ } ;

+ ta, n - l ( ~ - t ~ ; ] - I

x-'a,?'~4b,z-'-a

'"

480

STRAIN ENERGY

The fo].lowing integration found to give: I(~)

=

OF SHEARED

which

THIN PLATES

is required

Vol.

in ca!culat~.ng

gint(~)

11, No.

is also

z+c y+b x+a ~ ~ ~ [ uw~ + (l_v){tan -I uv _ -l,wv~u+a,v+b,w+c z-c y-b x-a p(v2-p 2) (wP) + ~an ~J;]u-a,v-b,w-c

= [[(l-v)xyz{tan-l(~zY)

+ tan-l(x~)}

+ ~-~{x(y2_z2)£n

6

dudvdw

r-x

+ z(y2-x2)£n r-z _ ¼z(r2_y2)~nr7 y r+z r+y + where

2

£

, 2,~x+a,y+b,z+c]x+a,y+b,z+c 6{(6-3v)y 2 - (3-v)r IJx_a,y_b,z_c~x_a,y_b,z_c

= u 2 + v 2 + w 2.

(12)

We note that T 2 ~(e13) W = ab--~{8abc

I(~=O) } - 4n(l-v)

(13)

and

T

÷ Eint(r) For a squar e plate

2

U(el3) = - ~--~-v~-- I(~).

(14)

for which a = b = c/8,

W becomes:

2 f(B) 3 ) {I - 4 ~ ( l _ v ) g }

W = 2~(e

(15)

where f(B)

= 8(l-v)B{tan-l(

1

_4(2_v)B2tanh-1

) + tan-l(

1 l~+B~

B

)} + 4{v + (2-v)B2}tanh -I

i

4 ~{v + (3-v)B2} 2~+B2

+ ~{v + 2(3-~)B2} l + ~ B2 - ~(3-v)B 3 + v{2£n/~-1 + 4 ' ~ - 1 ) } ~+1

In Figure

2, the constrained

8 : 0.i is plotted different

strain

as a function

directions,

[I00],

C +

el3(r)

(Eq.

(Ii))

of the distance

[010],

and

[001].

whether

it is inside

We note

changes

sign from negative

For comparison, (ellipsoid

to positive

at

is calculated,

in the

that

(6),

of position

[00!]

the results

calcu-

the e~o~r) ~J

with an ellipsoidc!

using

with in three

out by Eshelby

is a function

I~I=I3B

the el3c (~) field associated

of revolution)

plate

center

In all of the numerical As pointed

a precipitate.

"

of a square

from the plate

lations, Poisson's ratio v is taken to be 1/3. c the el3(r) field associated with a square plate or outside

(16)

3"

f~e!d

direction. precioitate

of Eshe!by

(3).

Vol.

II, No.

6

STRAIN ENERGY OF SHEARED THIN PLATES

The aspect ratio,

i.e., the ratio of minor axis to m a j o r axis,

tb be 0.i, and the results are p l o t t e d

in Figure

3.

481

is again taken

c (~) field in the The el3

[001] d i r e c t i o n d i m i n i s h e s more r a p i d l y in the case of a square plate than in that of an e l l i p s o i d while in the other two directlons, [I00] and [010], the situation is reversed.

ioi.... , 1 ~oo~]

I

oo /

-~-

/I

FIG. C

o,

]

06

÷

The el3(r)

2

field a s s o c i a t e d with a

square plate (B=0.1) vs. distance from the center in the three d i r e c t i o n s [i00], [010], and [001].

~o4 03

0.1 0 -0.1 ''

-O2

O

L

02

L

0.4

I

I

I

I

0.6 08 10 12 Distance f r o m Center, r l a

1.4

16

O7r I

0.8 0 7

1001,[010]

II

k

O 5

810.1

FIG. O10]

~o.4 O3 0.2

~'

,~,oo]

j

-o.1 ~

C

02

04

06 0.8 10 1.2 Dlstancg f r o m Center, r f a

3

c (~) field a s s o c i a t e d with an The el3 e l l i p s o i d a l precipitate (B=0.1) vs. distance from-the center in the three d i r e c t i o n s [100], [010] and [001].

14

•~

482

STRAIN ENERGY

OF SHEARED

THIN PLATES

Vol,

II~ No,

6

Figure 4 shows the elastic strain energy a s s o c i a t e d with a square plate (Eq. (15)) as a function of B (= c/a). Again, for comparative purposes, Eshelby's results on the ellipsoidal p r e c i p i t a t e (i) are included in this figure. When B~0.35, the ellipsoidal p r e c i p i t a t e has a smaller elastic shear strain energy than the square plate while the situation is reversed for B~0.35. The ratio of W c u b e / W s p h e r e is found to be 0.94. 0.7

0,6

Sphere

~./~ /

0,5

Square P'°'"

~-- 0 . 4

FIG.

~ 0.3

/~/// 0.2

~):113

111II

0.1

%

o!~ o!4 o!6 o!8

,!o 12~........ 14 ,6

The (self) elastic shear strain energy per unit volume of a precipitate vs. the aspect ratio, B. The solid line shows the strain energy a s s o c i a t e d with a square plate while the dashed line indicates that of an ellipsoidal precipitate.

Figures 5 and 6 show the ratio of the interaction energy

(Ein t) to the

total (self) elastic shear strain energy (2VW) as a function of distance between the centers of the two square plates. In Figure 5, ~ is 0.01 while Figure 6 shows the results when ~=0.i. As one might expect, the m a x i m u m interaction energy occurs when the centers are aligned along the [001] d i r e c t i o n and the two p r e c i p i t a t e s are nearly in contact; the ratio of Ein t to 2VW is 0.71 when B=0.01 and 0.45 for B=0.1. Figure 7 shows the m a x i m u m possible interaction energy, in the same terms, as a continuous function of B. In the other directions, the i n t e r a c t i o n energy diminishes rapidly with i n c r e a s i n g distance relative to the d i m e n s i o n of the plate in that direction. We note that in the i n t e r a c t i o n energy calculation, transformation

the same stress-free

strain e~. Hence, if the centers of ij is assumed for both plates. the two square plates are aligned along the [001] direciton, and their transformation strains are simple shears of opposite sign in the x-z plane, the m a g n i t u d e s of their elastic interaction energies are the same but the signs are reversed. Thus, when B=0.1, the elastic interaction energy can decrease the total (self) shear strain energy of plates aligned in the [001] d i r e c t i o n by a factor of 0.48.

Vol.

Ii, No.

6

STRAIN

ENERGY

OF SHEARED

THIN

PLATES

483

080 070 060 /3:001 050 040

~Tn.

030

The

e4 ~O2O

ratio

center

5

E i n t / 2 V W , vs.

distance,

where

the

inter-

Ein t is the

shear i n t e r a c t i o n e n e r g y and V W is the self shear s t r a i n e n e r g y a s s o c i a t e d w i t h a square p l a t e , 6=0.01.

01

-01 -0.2' -0 30~Inter-center di.~tance, rla

0.8

i

i

1.0

0.7

09

06 08

0.5

/3=01 07

0.4 >0.6 >~0.3

E

e,l

~e5

jO.2

,2

~01]

"~04 w

0.1

03

0 -0.,

[010}-/&~O]

02 0.1

-0.2 -0"30

1

3

012

4

Inter-center distQnce, rio

FZG. The ratio, center

6

E i n t / 2 V W , vs.

distance,

where

0 4J

#

FIG. the

inter-

Ein t is the

s h e a r i n t e r a c t i o n e n e r g y and V W is the self shear s t r a i n e n e r g y a s s o c i a t e d w i t h a s q u a r e plate. 6 = 0.i.

I __ 10 L 0.8

016

--1.2

7

The p o s s i b l e m a x i m u m ratio, E i n t / 2 V W , vs. the a s p e c t r a t i o , 6. These v a l u e s are d e v e l o p e d at the [001] o r i e n t a t i o n w i t h T an el3 s i m p l e shear.

484

STRAIN ENERGY OF SHEARED THIN PLATES

Vol.

II, No.

Acknowledsements This work was supported by the Divsion of Materials Research of the National Science Foundation under Grant DMR76-06855 for which much appreciation is expressed. The authors are indebted to Professor H. I. Aaronson for introducing us to this problem and his invaluable discussions of this work and to Professors D. M. Barnett and K. C. Russell for their helpful comments. References i. 2. 3. 4. 5. 6.

J. D. Eshelby, Proc. Roy. Soc. A241, 376 (1957). W. D. MacMillan, The Theory of the Potential, p. 78, McGraw-Hill, (1930). J. D. Eshelby, Proc. Roy. Soc. A252, 561 (1959). G. Faivre, Phys. Stat. Sol. 35, 2-~-9 (1964). S. L. Sass, T. Mura and J. B__Cohen, Phil. Mag., 16, 679 (1967). J. D. Eshelby, Progress in Solid Mechanics, 2, 89-~1961).

New York

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