Nonlinear theory of micro-elastic solids—II

int. .r. Engng. Sci. Vol. 2, pp. 389-W Pergamon Press 1964. Printed in Great Britain.

NONLINEAR THEORY OF MICRO-ELASTIC SOLIDS-II*

E. S. Square AND A. CEMAL ERINGEN

Purdue XJniversity, Lafayette, Indiana, U.S.A.

Abstract-The present paper is a contusion of our previous work on the same topic. After the introduc- tion of material and strain measures of micro-elasticity we give the specific form of constitutive equations for isotropic micro-elastic materials, various approximate theories, the hnear and the determinate theory of couple stress. Field equations of the linear theory for the latter case are obtained and applied to the study of the Rayleigh surface waves in micro-elasticity.

1. INTRODUCTION

IN OUR previous work [I] we gave the formuhrtion of a nonlinear simple micro-elastic solid. The basic field equations, jump conditions, thermodynamics and constitutive equations of this theory were studied in detail. In the present paper first we establish interrelations of various material and spatial measures of strain and afterward obtain the explicit forms of the constitutive equations in terms of some of these strain measures for a macro-isotropic material. For small micro-deformations we give various special theories and study the linear ~ete~~~~ate couple stress theory in detail. For the tatter case two sets of field equations are derived in terms of uncoupled macro-and micro-displacements. We conclude the present paper by the investigation of Rayleigh surface waves in micro-elastic half space.

2. STRAIN MEASURES

Here we introduce material strain measures E, (3 and spatial strain measures e, E, by

2&L z c,, - GK, ) 2ekl = gki - ckl Qt = Yly~ - G,, , 91= @kt - &cl

CJ.1)

where CKL, YKt and ckr, &r are respectively given in (6.5) and (6.16) of [I). The third order micro-strain measures r,, and yklm were also introduced in the same place.

For the differentials of the material and spatial position vectors dX and dx we have

dX=dX+d==X,kdxk -t-Xkdrk +~,&kdX ,

dx=dx+d~=x,,dX+~KdZK+~K,LZKdXL. (2.2)

From these there follows the squares of the material and spatial arc lengths dS =dX .dX and dY2 =dx edx so that

* A part of the present work aas supported by the OfIice of Naval Research. t Postdoctoral Research Associate, on leave from Technical University of Istanbul, Turkey.

389

390 E.S. SUH~BI~UN~ A. CEMALERENOEN

+2g~~~~~~~~~~d~~dX M = 2e,,dxkdx f- 2(E,xdkdg

+2y,,,,,t;dxdx +(g,, - Gro.~g~~L,)d~kd%- G, &&,,:nSkYdxldx

- 2GKL~KkxLI:m~d

Nonlinear theory of micro-elastic solids-II 391

Strain tensors E, e, 6, E, I and y can now be expressed, by use of (3.1) and (3.2), as

(3.4)

4. CONSTITUTIVE EQUATIONS FOR ISOTROPIC MICRO-ELASTIC SOLIDS

In our previous work [l] constitutive equations for stresses t, s and stress moment h were expressed in terms of the derivatives of a potential I: with respect to strain measures e, E and y. We now derive the specific forms of these equations in the special cases of linear dependence on y, linear de~ndence on 7 and E and finally obtain the linear theory.

(i) Constitutive equations linear in y In this case c may be expressed in the quadratic form in y, Le.,

x = CO f~~ClklmnyijkYi~~ 9

z ijklmn = pnnijk (4.1)

where Z, and ZtiUmn are isotropic tensor functions of the tensor variables c and *. Into the argument of these isotropic functions we must include the transpose of \lr, JI* because of the asymmetry of JI, i.e.,

CO = &(e, $9 $*) ,

c i jklmn =

Pkl(c, q/, Jr*> . (4.2)

In the expansion (4. I) the linear term in y does not appear because of inexistence of odd order isotropic tensors. The constitutive equations now take the form

(4.3)

We thus see that the stress tensors t and s are independent of strain measure y in this theory. We assume that CO as well as Eijkl? are expressible as polynomials in their arguments.

Following the results of [3] any scalar invariant polynomial X,, of three matrices c, \I, and Jr* may be shown to be a polynomial in the following 34 invariants.

I1 =trc,

f3 = treZ ,

I7 = trc3,

IZ=tr\L,

1, = trJi2,

1,=tr$,

I, = trcJl ,

I, = trcJr2,

I, = trJIJI* ,

I 10 = trc2\li,

(4.4)

392 E. S. SUHUBI and A. CEMAL ERINGEN

II1=trcJIJr*,

I,,= trc+ ,

Z18 =trc+*JI ,

Izo = trc+JI* ,

f2,=trc\lr\lr*J12,

Iz6 = trc2+2c$,

I,,=trc\lr*JI, 113=tr\)rJI*2,

l,,=trJr2\lr*2, I,, = trc$*+ , r, , = trc'$JI* ,

1 19 = trc\)$*2,

I,, = trc'JrcJr* , I,, = trcJr2$*2, I,, =trc2rjr*c\lr, I,, =trc\lr'\lr*\lr ,

I,, =trc'$"$*\lr , I,, = trJl\lr*$z$*2, f29 = trc\(r*\(r2+*2 , I30 = trJrc$2c2 , I, I= tr\(r*cqf2c2 ,

I,2 =tr$Jr*c2$*2, I33 =tr$*JIJI*2$2, .13_+ = trc$$*J12 .

The most general sixth order isotropic function satisfying (4.1), has the form

(4.5)

where coefficients cl, g2, . . . and (r, 1 are polynomials in the 34 invariants Z,, Z2, . . . and 1,, listed in (4.4).

We calculate the various partial derivatives appearing in (4.3) by the chain rule of differentiation, e.g.,

&z, XX0 81, -=-- s. 0~~~ ar, ackl cl=1,2, . ..) 34.

After some lengthly manipulations we obtain the following constitutive equations for t, s and h:

t=- P[2a,c+a2~*+4u,c+2u,~*2+a,(~c+c3r*+~*c)+2u,JrJ/*+6a,c3 PO

~3a~\lr*3+ug(JI2C+C~*~+~*~C+~*C~*)+ulO(~C~+C~c+C~~*tC~*C+~*C~)

+2all(JI\CI*c+c\lr\lr*)+2a,,(JI*~c+~cJI*)+u,,(J12JI*+~Jl*2+JI*~~*) +a~~(JI2c~+C2\b*~+C~~C+CJI*~C+~*~C~+~*C~~*)+2u~,(~~\lr*~+~*.J1~\CI*)

+ 24+*+c2 + *c2\lr* + c\lr*+c) + 2al,(c2$+* $ JIJI*c2 + c\lr+*c) -I- al &2c$r* + JIc\lr*2 + \(r*2Jrcf \(r*\(r2c+ +*I/@*) + u19(c\lrZJr* f c\lrJI2 + $$r*%s \lr2**c + +*c\lr$*) + uzo(CZ~2~* + cZ$\lr*2 + *r\lr*c2 + +2Jr*c2 + \cI*c2*+* + c\lr+*% +c\lr\lr*c)+2a2,(~c\lr*c~+c\lrc\lr*c~\Ir*c~~c+C~~c~*)+2u22(~~J/*~C+C\C1~\CI*~ + **c+2\lr*> + 2a2,(+*c\lrc + c**cIJK+ IJK+*c+ c\lrc\lr*) + u2&J*2Jl\lr*c + \Ir+*\lr2c+ c\cI*2\lr$* + Jr%\lr+* + \Ir**c+* ++*c\lr+*) + u2&+*\lrc + \Ir*+\lr*%+ c\lr*+**2 + +*c+*\lr** + \Irc\lr+* f \Ir\lr*%$*) +uz6(+*2c2+*c + \Irc2I/lc + +*c+*V + c\lr%\lrc + c+*c\cI*2c + +2c\lrc + c+*V+* (4.6) + c2**c+*2 + \Ir*c2+*c\lr*> + a2,(\lr*+$*c + c\lr+*\lrc+ cJI*+rJI*%+ \Ir\lr*$c + c2**\lr\lr* + \Ir*c2+*\lrJI* + \Irc2J12\lr* + \Ir$*c\lr*) + 2u2*(+2\lr*2+JI* + \Ir**JIJI* f +*JI\)*Q2+*) + u29(+*J12+*c + +2JI*+JJc+ Jj2JI*2c+* +JlcJI2JI*+~*\)rc~~~*+~*cJI*\)~~*+c~*J1~~*~)+u~~(~*c~JI*~c+~~c2~c + +*2cJI*c2 + cJlcJ12c + c+*2cJr*c+ Jlc\lrc + czJj*2cJI* $ cJI*c2+*2

Nonlineal theory of micro-elastic solids-11 393

s= - ~[~U,c+U2(~+~*)+~U,E+2U,(~+~*)+Ui(c9+~C+~*e+C~*)

+ 2a,($$* + +*+I) + 6a,c3 + 3a,($ + $*> + ag(cG2 + Jlc + $cJI + cJ~*~ +\Ir*2C+~*CJI*)+a,,(~C2+C2~+C~C+JI*C2+C~*C+C2~*)+2u,,(JrJr*c + cJIJI* + +*0/f) +2a&*\lrc + cl/f** + $c$*) + a13(\lr2$* + +*\lrJI* + \lr*2JI -t $\lr*2 + JrJI*JI + lJJr*\lr2) + u14($2c2 + c2qJ2 + ce2c + $c2\lr + c2JI*2 + +*2c2

+ cJr*2c + \Ir*cZJI*) +2u,,(\jr$*2$ + Jr*Jr*JI* + e2+* + JI*2$2) + 2Ul&r*$C2 + c2\lr*\lr +cJl*qK + JIc2+*) + 2u&JfJI*c2 +c2$$* + c*\lr*c + +*c%Jl> + a&Jf*2+c + \jr*\lr2c+ Jr2c\lr* + \Irc**JI + c**qf2 + c\lr*\lr + qmJl*2 + \JI*lJm+*) + ul&lJI*c+ e2Jr*c + $*2c$ + Jr*c*+* + cqf2Jr* + a#* + JI*c* + JrJI*c\lr> + u2&$2\lr*c+ *IJI*c2 + +qJ*c2 + +*c2+1JI* + JI*c%J? + clJftJ#*% + c%JAJ/* +c2Jl\lr*2+qnJI*c2JI+**2 2 c $) + 2u, l($cJr*c2 + **c2JIc + c2+cqf* + c+*cqJ + c\lrc\lr*c> + 2u22(lJf*c~2JI* + Jr2\lr*c + JI$*2cJr f cqAjf*2 + \lr*2c*q + 2U23($*C\JIC2 + $C2\jr*C+ cJI*c+ + c\lrc2$* + c$*c~c) + u24(\lr*2@/f*c + *+*qJ2c + $2c$$* + +*++*c** + **\lr2c* + c***\lr2 + c\lr*2+cl\lr* + \Ir\lr*cJ/*2 + *cw*\lr + $r*c$r*\lr) + u2&Jf2+*+c+ \Ir*+$*2c+ ~*c+*~l\lr* + tjK*2JI* + \Ir*+c\lr2 + c**\lr+* + cQ2$r*\lr + \Ir+*\lrc\lr + lJJ$*2cJ/* + +**c\lr*\lr> + u2&*%2$*c + +*cqf*v + qK2\lr2c + c\lr%\lrc + qf%$c2 + $*c+*c$* + c$c2qJ2 + c2$2c++ c~*%2~* + c\lr*c+*2c+ c2+*c**2 + \Irc\cIcqf) (4.7) + u&*+l\lr*c + c\lr2\lr**c + \lr2+*+c2 + \Ir*c2+*++* + \IrcqJr\lr* + +*l/Ky/2 + c2qf2\lr*\lr + c$*$+*2c + c2\lr*$$* + ~Jr*~c2+ + ql$*cqf* + +*%qt*+>

+ 2a2*042**2N* + +*w*\lr2** + w*\1/2$*2 + w*$b*Q + JI*$2\lr*2+ ++*2$$*+2) +u29(9*~2\lr*2c+ $2$*2qw+ $2+*c\lr* + q/*Jrc\lr2$*

+ **c\lr*e2\lr* + qJ*2c$*+2 + c+2$*2+ + c$*qf2Jr* + \lrc+2\lr*2 + ~+*2c~*JI

+ \IrJr*zJIc$ + $*24fc\lr2> + u3&*cZ+**c + qf2c2$c + 4!*2cJI*c* + cJrcJ12c

+ Jrc$2c2 + JI*c\)*c2\lr* + c$2c$ + cJI*c2JI*2 + c~c\lr* + cJI**ctJ/*c

+ c2Jc*2c$* + Jrc2gwJI) + u3~(~*cJlc + c+*cJI2c + *2c2JI*c + qf*c\lrc2\lr*

+ Jr*oJ12c2 + Jrc2Jr*c + c2qI*cJI2 + cJr*2c+c + ctJlc2ql* + *c2JI*c\lr + cJI*2cg/

+ CJ12C2$*) + U3*(\lrJI*\lr2C2 + C$*JrJI*C + $*2JIJI*C2 + lJ/2C2JIlJ/* + Jr*JIJI*CJr*

+ \lr*c2JI*2\lr + cZ$*\lr+* + c\lr\JI*lJJlc + c2+cI\JI*+* + +ljJ*c\lr* + \Irc2\Jl\lr*+

+ +*+c*\lr) + 2U&*/2j*\lr$* + \Ir*+$*\lr + \Ir+*$$*$ + $*\lr$*\lr

+ w*+JI* + q2JI*q.$*2) -t a34(JI$*2$2C + $*24hJt*c + ~*\lr*cJr$* + JI*2tJ?cJI + qJ*+*+*cJI* + +2c++*2 + cljJr*$\JI* + clJ/~*Jr2 + tJ!+*c**2JI + $*c$*+I

+ wJJI*Jr + \Ir2~*cJI*)l where

a=l, 2, . ..) 34.

From (4.5) and (4.3), we find

(4.8)

394 E.S. SUHIJBI and A. CEMAL ERINGEN

The above expressions are reducible further involving only matrix products up to and including fifth power. We have not attempted to make this reduction since the coefficients will be extremely complicated.

The above expressions are useful in finite deformation theories. For approximate theories these expressions must be transformed to those involving strain measures e and E, For the stress tensor t and s these are given below:

t=~~jb,(I-2e)+b,(I-E*)+2b,(e-2eZ)+2b,(E*-E*2)+blCe+(E)S-2(Ee)S-E*e]

+ 2b,(e - EE) + 3b7(e2 - 2e3) -t 3b,(e* - &*3) + b,(ea* + E*e+ (E2)s- 2(C2e), -E*2e-E*eE*]+b,,[e2+(eE+~e),-2(EeZ +eee),-E*eZ]i-b,,[2ee+~~* -4(EE*e),]+b,,(2Ee+E*&-2&*Ee-2&e&*)+b13[E2 +&*&+&&*-~(&E*)~-~*EE*] + b14[e2E* + e*e2 + (eE2 + E2e)s - 2(E2e2 -t eE2e), - E*2e2 --*e2E*] + 2bl 5(~2~* + E*E2 -E2E*2 -E*&2E)+2h16[(e&*&)s+Ee2 - E*Ee2 -Ee2E*-~E*&e]$2b,,[(eEE*), + e2e - 2(e2EE*), -ees*e] +b,,[~~e+~e~* + ~*ce+(~*~~)~-22(~~eE*)~-E*~~e

-E*&2e-E*EeE*]+b,,[eE2+&*eE+eEE*+(E&*2),-2(e&2E*+e&E*2)~-E*e&E*] + b2,[e2EE* f E*eE+ e2E2 + (eE2E* + e&E*2)S - 2(e2E2E* + e2EE*2 + e&E*2e)S -&e2E&*]+621[2(Ee&*e)S+E*e2E+2e2Ee-4(~e~*e2)s-2e~e~*e-2~*e2~e] + b22[E2E*2 + 2e&E* + 2E*eE2 - 4(E2E*2e), - 2E*eE2E*] +b,,[2(&*e&e), + &e2&* + 2eEe2 - E*2E&*e- 2(EEE2e),- E2eEE* - E&*e&*2 - &*eE2E*] + b2,[eE*2E + E2eE + EE*eE* + E*E&*e + (EE*E2)S - 2(&E*E2e $ E2e&E*)S- E*2EE*e- &*eE2E*] +b2,[eE*EE*+~*eE*E+EeE2+&E*2e+(E2E*E)~-2(E2E*Ee+Ee&2E*)~-E*EE*2e - E*ea*EE*] + h2,[eeh2e2 + eE*eE* -t- (Ee2E2 + es2e& + e2eEe)S- 2(Ee2C2e f eaese + E2eEe2)s- E*e&*2e2 - E*eE+e -E*e2E*e&*] + bz7[e2e*EE* + E*e2E*E + Ee2E2 + EEh2e2 + (eE2&*E + eE*EE*2)g- 2(eE2E*Ee + E2E*&e2 + Ee2E2E*)a- E*EE*2e2 - E*e2E*EE*] + 2b2,[E2E**E+ E&*e2E* + E*EE*E - 2(E2E*2&E*),- E*EE*E2E*] (4.10) + b29[(E2E*2E)s+ E2Ee2e + EeEE* + E*El?E2 +E*eE*E2 +eEE2E* - 2(&e.E2Ee2 +E2E*2Ee)S-&*E2E*2e-E4EeE2E* - E*eE*E2&*] + b,,[(E2e2E+ eseE2 + eea2e)a +e2E**efeE*e2Eh +E*e&*e2-2(E2e2EefeEeE2e+EeE2e2)S-E*e2E*2e-E*2eE*e2 - E*eE*eE*] + b, l[(eE*ett2 + E2e2E + E*eE2e)S + eE2e2 +es*e2c* + s*ee*e2 - 2(ea*eE2e + ec2e2E* + E2e2E*e)S - E*2eEe2 - E*ec2e2 - E*eEe2E*] + bS2[(E8*E2e + eEE*E2)S f E2e2E+ e2Ee2& + E*E&*e2 + EE*e2E* - 2(EE*E2e2 + eEE*E2e + E2e2&E*)s - E*2EE*e2 - &*&E*eE*] + 2h,3(&E*2&2 + E2E*EE* $ &*EE*E - EE*2E2E* - &*EE*&&* - E2E*E&*2) + b34[(E&*2E2)S+ eE*2E2 + E*E2eE + E2eEE* + EE*eE* f E*E2E*e - 2(EE*2&2e + &2e&E*2)s - Ee2E2E*e - EE2eEE* - E*E2E*eE*]},

s=~{b,(I-2e)+2b2(I-s)s+2b,(e-2e2)+4b~(E-El),+b,(2e+E-2e~-2~e)~

+ 2bJ2(~)~- EE* - E*E] + 3b,(e2 - 2e3) + 6b8(~2 - F.~)~ + bg[~2 + 2(Ee+ e&) - 2(eE2 -I- E2e + EeE)ls + b,,[eE + Ee + 2e2 - 2(ce2 + e2E + eae)ls + bl ,[4(eE - e&e*), + E&* -2E*eE]+ b,,[4(Ee-eE*E),+E*E-2EeE*]+2b,,(E2 + E*&+ EE* - E2E* -GLEE* - E*2~)s + b,,[eE f E2e + 2(e2E + Ee) - 2(E2e2 + e2E2 + eE2e+ Ee2E)ls +2b,,[2(&2E*+&*E2-EE*2E)~-E2E*2-E*2&2]+26~~[(eE*&+2Ee2-2E*E~2)~

-e~*Ee-&e~~*]+2b,,[(eEE*+2e~E-2es*e~),-eee*e-E*e~E]+b~~[E*E~ +2(E2e+&eE* + E*&e)-2(&*2Ee+E*E2e+E2eE* $&e&*&)],+b,~[&E*~ +2(e&

Nonlinear theory of micro-elastic solids-II 395

where

+~*~E+~EE*)-~(EE*~~+E~E*~+E*~~E+E*~EE*)]~+~~~[~E~E*+~EE*~~~(~~EE* + E*es + e2E2) - 2(eE2s*e + a*e2 + E2E*e2 + E*e2rx* + E*e2E2)]s + b, 1 [2(seE*e +2e2~e-~es*e2-2~*e2~e)s+~*eZ~-2e~e~*e]+b22[4(e~2~*+~*ea2-~*e~Z~* - E2E*2e)s + E2E*2 -2E*ea] + 6,,[2(E*eEe+ 2e&e2 - 2E*eae2 - 2Ee2E*e), + Ee2&* - 2es*eee] + b2J~~*c2 +2(eE*2& + E2e& + E&*eE* + E*eE*e) - 2(E*2EE*e+ &&*E2e + E2eEE* + E*EE*eE* + E*EeE)],+ b2,[E2E*& + 2(eE*&E* + E*eEE $ EeE2 $ EE*2C!)

- 2(E2&*Ee + &*EE*2e + E*eE*EE* + EeE2&* + E*EeE2)lS + b2,[&e2E2+ eE2eE + E2eEe (4.11) +2(es*2e2+e2E*es*+E*e2E*e)-2(E*2e2E*e+E*ea*2e2+Ee2E2e+esze~e+~2ece2 +&*e2E*eE*)], + b2,[eE2&*E + EE*&e +2(e2E*EE + E*e2E*& + Ee2E2 +&E*e2) - 2(E*E&*2e2 + eE2E*Ee+ E2E*Ee2 f E*eE*EE* + Ee2E2E*)ls + 2b2,[2(E2E*2&

+ EE*E2E* + E*EE*E2 - E2&*2EE* - E*EE*E2E*)S- E*E2E*2E - E*E&*E] + b2,[E2&*2E + 2(EZE*2e + EeE2E* + E*&eE2 + &*e&*e2 + eE*E2E*) - 2(E*&2&*2e+ E2E*2Ee+ E2E*2eE

+ E*EeEE* + E*eE*EE* + E*2eE*E2)]s f b,,[E2e2E + eEeE2 + EeE2e+2(eE2e2 + EeEe + e2EeE) -2(E*eZE*2e + E2e2Ee+ E*2eE*e2 + eEec2e + EeE2e2 + Ee2EeE)ls

+ 63 JeE*eE2 + E2e2E* + eE*2eE + 2(eE2e2 + Ee2Ee+ eces) - 2(E*2ee2e2 + eE*eEe + E2e2E*e+ &*eEeE* + E*eE2e2 + Ee2E*2e)]s + b,,[&&*E2e+ eEE*& +2(E2e2E + e2Ee2& + E*EE*e +EE*e2E*) - 2(EE*E2e2 + t?Ee2EE*C?+ E*E&*e2 + E2e2EE* + E*&E*eE

+ E*e2E*2E)]s++b33[2(EE*2E2 + EE*EE* + &*E2E*E- E*E2E*EE* - E*E2E*& - EE*EE* - E2E*E&*2] + b,,[EE*E2 + 2(eE*2E2 + E*E2eE + EzeEE* f E2E*e&* + E*EE*e) -2(E&*2E2e + E*EE*e + &*EeE&* f E*2E2eE + E*EC*e&* f E2eEE*2)]S}

7:-T;,-- (a=1 3 , -9 .., 34) . (4.12)

The invariants J, are identical in form to those listed in (4.4) except that they are formed by e, E, and E* respectively instead of c, $ and $*.

The parentheses carrying a subscript S indicate the symmetric part of the matrices enclosed within, i.e.,

(A),=t(A+A*).

The constitutive equation for h is given by

+mmkYrlr +6k,YmrJ fGj~klYrm+~,ylmk +Zg(ymkl +yktrn) +rgyIkm +Io?

mk I f~~~~km~-~~(~m~Yk'~+g'"k~r~YPpr)-~~(~m~~rkr+Ek,~,r") (4.13)

- 3&m,yr,k - ZqgmkErlylp - T,(gmkE,ypr, +Ekrymrr) - Z6Eklyrm- Z,&;yrmk

where - T8(ErlYmkr +&lykrm)- ZgE;y,km- T,,E;ymrk- Z1 lEr,ykmr]

G=z,(JI, J,, . . . . 534). (4.14)

(ii) Constitutive Equations Linear in E and y In the ewessims (4.10), (4.11) and (4.13) if we keep only the linear terms in E we get

t=-$po +B;, +@I +PsV2 +(P3 +A#5 +a$ -wJ103~+m3 +/%3 -Po

396 E.S.SUHUBI and A. CEMAL ERINGEN

where the foI~owing expansions are used

B1=P,+PlJz+P3Js+P4J10, b,=P13+P19J2fP2OJ5-/.82,J1*, b16=Pm bz=Po+P5J2+B6J5+P,JID, b=&z, b17=B31, bJ=Ps+PsJz+B,oJs3.BitJlo, &O &=B121

=2(P,,+P24J2f825J$$P26J10), ;=gD,2. &1=P27, 2?-- 33%

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ b12=82~5, k=Pxn b14=&&

and &I> A, 81, 3 833 are functions of the i~variants J1, J,, J, only.

+fZg(gmkYrir -i+y,) +Zg&klYrmr f2glmk +2*(ymki +ykgrn)

+ZgYIkln+Tt*Ymlk+Z*lYkmll (4.17)

where 2, are functions of invariants J,, J3 and J7 only. From these expressions by retaining various powers of e in the polynomial constitutive

coefficients we obtain various order theories involving the elastic strain e. Below we only give the linear theory.

{iii) The Linear ~o~stitut~ve ~quutions The linearization of the foregoing constitutive equations with respect to e resuits in

t=[:-n+(i+~)ekk]I+2(1(~cr)e+qk,lfve*~rc&, (4.18)

s= I- x +(A ik)e$ +2(@ +k)e f(2q-z)EkkI +(v +K-u(r){& +&*) (4.19)

where n, 1, p, r, CJ, y, JJ and K are material constants. From the above expressions it follows that the only possible initial stress in isotropic micro-elastic solid is a hydrostatic pressure x. Thus for a natural stress free state n==O, The stress potential X corresponding to these constitutive equations is

Nonlinear theory of micro-elastic solids-II 397

T9 ijk +zy Yikj +yYijkyjik +++jkykji . (4.20)

The const~tLltive equation for h is identical in form to (4.17) except that the coefficients t, are now constants.

Thus the micro-elastic properties of an isotropic linear elastic solid are completely described by 18 material constants.

It is to be noted that in conformity with the above linearization the strain must also be linearized leading to

(4.21 j

5. FIELD EQUATIONS IN LINEAR THEORY

The partial differential equations of linear micro-elastic solids are the union of the equations of co~lservation of mass, balance of momenta and the constituti~e equations obtained in Lyiii). Were we list them

(5s)

l5.2)

(5.3)

(5.4)

(5.5)

(5.6)

Were ekl, ekl and lj& are given by (4.21) and the spin tensor tin has the general expression

5 *!m=FgfNgmp(SPN +@PM)$N, . (5.7)

In conformity with the foregoing linearization we get for the linear theory

where the synlmetric tensor

398 E. S. SUHUBI and A. CEMAL ERINOEN

s a function of X only since IRM is so, The material tensor PM depends on the position X of a material point for inhomogeneous materials and it is constant for homogeneous materials. If the material possesses micro-isotropy then IRM=I,GRM and Ym = I,g so that (5.8) reduces to

1,20. (5.10)

Combining (5.1) to (5.6) and using (4.21) and (5.10) we obtain the partial differential equations of the displacement fields

If we take z = r~ = v = q = hc = z, = I0 = I= 0 equations (5.12) reduce to an identity and equa- tions (5.11) reduce to the celebrated Naviers equations of the classical theory of elasticity.

For a traction and surface moment boundary value problem the appropriate boundary conditions are:

(5.13)

where ttnj and a,,, are prescribed on the deformed surface 9 of the solid. There are twelve boundary conditions to determine twelve unknowns uk and 4*. In place of (5.13) a displacement boundary value problem may be set up by prescribing uk and +k on 9. Various mixed boundary value problems are possible. The problems that are well set require the proof of existence and uniqueness theorems.

6. LINEAR TKEORY OF COUPLE STRESS*

In the couple stress theory the equation of local balance of moment of momentum has the form [2, Art. 401

mklm ;k + tlm +p( I** - 3) = 0 (6.1)

where nzklnr PI and cirm are skew-symmetric in the indices f and M and f pm3 E 3(p _ p) . (6.2)

Comparing (6.1) with the equations of stress moments (5.3) we see that if in (5.3) we set

th _ $l s ttkl , ;IWM z 0 3 jlkCM z _ ,#m , pm) 2zz& 0 ,

pl+ _ fm 6.3) t p0 _= 0 , (j[hl, _p

* This theory is diierent than that discussed recently by Toupin [4] and MindI~ and Tiersten [5] in the sense that in the theory presented by these authors the skew symmetric parts of the stress tttil and couple stress &flm. are indeterminate. In the present theory no such indeterminacy arises.

Nor&near theory of micro-elastic sdids-II 399

we obtain equation (6.1). Conditions (6.3), and (6.3), impose restrictions on the constitu- tive equations (5.4), (5.5) and (5.6). These conditions are satisfied for all motions if and only I:f

z=fJ=~=o, PC=-v,

zl=z~=z~=o, zq= -z,=r6-@a/2 f (6.4)

r,= -2r*--p/2, z*== -zg= -r,,z6/2 0

The condition (6.3), of skew-symmetry of ri requires that

which reduces the number of independent Cp, from nine to three. For a micro-isotropic solid, (6.5) reduces to

(6.5)

tjfklLof

The constitutive equations (5.4) to (5.6) now become

rk = llem,gkr + 2pek + 21cerki1 ,

Sk1 = Aem,gk + 2,uek ,

(6.6)

(6.7)

(6.8)

(6.9)

where on account of (6.6) we also have

Y ktm= _ tkm

Y *

The stress potential corresponding to the present case is obtained by carrying (6.4) into (4.20)

I

E=$tre)2 +j.&re+~~traa*-tre2)+c((traa*-tral)+(/I-J)traa* +6(&a) (6.10)

where akt = t&ijtYk . (6.12)

For the classical theory of elasticity YE is non-negative if and only if 3,I-t 2~ > 0 and p > 0. For constant u and $ fields e = a = 0 and we see that rc must be non-negative. Since each of the terms containing a are non-negative if we select c( 3 0, /? - 6 > 0 and S 2 0 the contribution of a in E; would be non-negative. Thus a su$icient condition for C to be non-negative is

31+2@0, P>O, Ic>o, a>o, B-S>O, 6>0 s (6.12)

Clearly these conditions are too restricted and the necessary conditions for the non- negativeness of the stress potential needs a separate proof. However, for the present investigation these conditions are found to be adequate.

Carrying (5.10), (6.6) to (6.9) into (5.2) and (5.3) we obtain the field equations for the displacement fields u, and q&, = - 41k.

=0 (6.13)

(~-~)(#~k3.(bks~~)$~#~m~k-22h-~tn3-~(~~; -u;r) +~*~~m-~~)=O . (6.14)

400 E. S. SIEIUBI and A. CEMAL ERINCSEN

The boundary conditions corresponding to (5.13) now read

~~~~?~ = $a)i

mklmrtk = rncnjim 011 Y (6.15)

where m(,, is the prescribed surface couple on the surface of the body. We can obtain uncoupled partial differential equations for II and Q, by an elimination

process. These are

(~+2~+~)~l-a~*vz]vv * u- ( )

p+f (1-;1,2v2)v xv x u+p,f-;/&?oVV * f

+R42Pov x v x f+kPoV * 1 (6.16)

_a6272(4 k m a21 ln

at2 t ; k -+ cbk;kt) - PO*

I

(6.27)

p42 = P +I,@ + G Polo CC-6 u !h2=-, pfj2=-. K

7. PROPAGATION OF SURFACE WAVES IN A MICRO-ELASTIC HALF SPACE

In this article we investigate the propagation of surface waves for a micro-elastic material characterized by the couple stress theory developed above.

We select our rectangular coordinates x1, x2, x3 with x3 =0 plane being the bounda~ of the half space and xg pointing into the medium (Fig. 1).

A micro-surface wave may be characterized by the forms

u,=a,exp[ - [x& + i&+Skl - ct)] , = ,

(birn= b~mex~~-+@k3 + i&kdki - ct>] , bin= - brnf (7.1)

where al, bl,, 5, q and c are constants,

Nonlinear theory of micro-elastic solids--II 401

Fro. 1.

Substituting (7.1) into (6.13) and (6.14) with f=l=O gives

[(~+~)(~~-q2)-(A+~-~)q2+p~q2c2]a,-i~(~+~-~)qa,-rKb,~=O,

-i~(n+~-Ic)qa,+[52(1+~--)+(~++)(52-q2)+p,q2c2]u,-ilcqb3,=O, (7.2)

~h-a,+ik-qu,+[(ar+~-6)(i2-42)-2ti+~~I~q2C2]b~~=0

and

(7.3)

For plane waves travelling in the direction of positive x1 axis a2 =0 and therefore b,, = b =0 so that equations (7.3) are satisfied identically. The necessary condition for the eiktence of a non-trivial solution of (7.2) for a,, a3 and b,, is the vanishing determinant of the coefficient. This gives

[I,ew(l9- -$)+2&]{(1 +4jl-[2+2E- .$ -(l +e)$-J%y

+(I- $)(I+&- $)CP]+e2[+2- $)W+(l- $]=o (7.4)

402 E. S. SUHUBI and A. Ch.w %uNGEN

where

J

A-i-2@ cl= -,

PO

of which c1 and c2 are, of course, res~ct~ve~y the velocities of irrotationat and equi- voluminal waves. We see from the inequalities (6.12) that c3 and c4 are real.

We now proceed to soive (7.4) for I. To this end we assume that 8 are uegti~ib~e. In this case the roots of (7.4) are

On the boundary x 3 = 0 we have the boundary conditions

&1=&=&=0, ??23~2=n-l3~3=Ilt323- -0. (7.7)

Thecondition t32=m312-m 323 =0 are satisfied identically so that the boundary conditions, in terms of the displacement fieids reduce to

0--2k)u,, 1 -f-u,, 3=0 f

U3,1+U1,3~&(U1,3-U3,t-2~31)=0t (7.8)

(f, -0 31,3 - where

krc,2[c,2 . (7.9)

In order for the waves to be surface waves we must only consider positive values of the roots cl, Cz and &. Thus the displacement fields will have the forms

where

(7.10)

(7.11)

Substituting (7.10) into (7.8) we obtain a set of three linear homogeneous equations for A*, A, and A,. The determinant of the coe~cients must vanish. Hence

~3-8~2~~~3-2k)~-16(~-k)-~6s(l-k~)~0, (7.12)

Nonlinear theory of micro-elastic solids-11 403

where

i#j=O (7.13)

8 c&r% (7.14) c2

For s=O, (7.12) is the classical expression of the Rayleigh surface waves. Denoting the value of w for this case by oO we obtain for the root of (7.12), to a first order approxi- mation in E.

w-0*+ 16(1 -km,)

3w&- 16we +8(3-2k)a * (7.15)

For the Poissons material k= l/3 (which corresponds to Poissons ratio l/4) and for the incompressible solid k=O we have respectively the surface wave velocities

e= 0*919(1 sO.932s)cz , (k= l/3)

c=o~955(1+0*783E)c, , (k=O). (7.16)

We note that the terms containing E represent the corrections to the Rayleigh wave velocity in each case.

Equation (7.13) gives rise to the speed propagation of a new wave not encountered in the classical theory of elasticity. This is given by

c==[2&/&,(l +&--~~34-1NN(2Ei10)f4-1 . (7.17) c2

We note that this wave depends on .,,/E and that it is ~~~~er~i~e. By accurate measurements of the surface wave velocities one can deduce the material

constants of the present theory. This must await future developments.

REFERENCES

[lf A. C. ERINGEN and E. S. SUHUI, Inr. .7. &gng. Sci. 2, 189-203 (1964) [2] A. C. ERINGEN, ~onI~ne~r Theory of Cu~~~~~off~ Mediu, McGraw-bill, New York (1962). [3J A. J. M. SPENCER and R. S. RIVLIN, Arch. rat. Mech. Anal. 2,309,435 (1959); 4,214-230 (1960). 141 R. TOUPIN, Arch. vaf. Me& Anal. 11, 385-414 (1962). [S] R. D. MINDLIN and H. F. TIERSTEN, Arch. rat. Mech. Anal. 11,415-448 (1962).

(Received 6 Jmuary 1964)

R&umL-Le present article constitue une suite a notre precedent travail sur Ie mCme sujet. Apr&s avoir introduit les grandeurs mat&elles et la mesure des deformations en micro-elasticite, nous donnons la forme spticifique des equations dttat pour les materiaux isotropes, en macro-clasticite, ainsi que queiques theories approx~atives et la th&orie lineaire d~te~inante des couples de contraintes. Nous ohtenons, pour ce demier cas, les equations de champ de la theorie lineaire et nous les appljquons & letude des ondes de surface de Rayleigh en micro-~lasticjt~.

Zusammenfassung-Die gegenwgrtige Abhandhmg ist eine Fortsetzung unserer vorhergehenden Arbeit tiber dasselbe Thema. Nach der Einftihrung von Materials- und Dehnungsmessungen der Mikro-Elastizitlit geben wir die kennzeichnende Art von Aufstellungsgleichungen ftir isotropische makro-elastische Materialien, verschiedene Annaherungstheorien, die lineare und die massgebliche Theorie der Dehmmgsdralle. Feldgleichungen der linearen Theorie fur den letzteren Fall werden aufgestellt und zur Studie der Rayleigb- Oberfliiehenwellen in der Mikro-Elastizitat angewandt.

Sommario--I1 presente Iavoro costitnisce ~nt~u~ione di quell0 precedente sullo stesso argom~to, Dopo aver fatto una introduzione delie misure sui materiali e sulle soIl~i~zioni toccanti Ia micro-elasticita, si

404 E. S. SUHUBI and A. CEMAL ERINGEN

presentano in forma specifica equazioni fondamentali per i materiali isotropi macro-elastici, varie teorie di approssimazione nonche la teoria lineare e di calcolo della sollecitazione combinata. Equazioni fondamentali della teoria lineare vengono ricavate per lultimo case detto e se ne fa lapplicazione allo studio di onde superficiali Rayleigh nella micro-elasticita.

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