Conjugate stress and tensor equation ∑r=1mUm−1XUr−1=C

CONJUGATE STRESS AND TENSOR EQUATION i UP’-‘Xu’- I = C r= I Department

of hlathematlos.

Peking University.

Beijing 100871. China

and CHI-SING MAN Department

of hlathrmntic~.

University

of Kentucky.

Lexington.

KY 30506.

U.S.A.

&bstrnct --in this paper we seek an explicit tcnsorial cxprcssion which. for each integer nr # 0. gives the stress T’-’ conjugate to the strain Lv”’ in the Seth -1 lill class. The preceding prohlom 14s us IO lid an intrinsic (i.e. coordinarc-free) rcprcscnlatinn of the solution of the tensor equation display& m the title. We obtain the requisite rcprcscntation by using tiill’s principal axis method and the rcprc
I. lN~i$Ol~lJCl’fON

The notion of work conju_cncy of stress and stmin in solid mechanics W;IS introduced Hill (1968) and by M:tcvcitn (196X). in the sense may be derived in it naturttl

way.

of conjugrtcy,

by

;t class of stress mc;tsuros

Some of those stress mC;tsurcs have alre;tdy proved useful

[cf. e.g. Hill (196X); Guo (1980)]. According

to Hill, the definition of work conjug:tcy is as follows : For a given Lagran-

gean-type strain E, if there is LLsymmetric second-order tensor T such that the stress power (the rate of specific work) per unit reference volume

,i’=

If[a:D

(=

III tr(aD))

(1)

can be recast as G+=T:l?

(G tr(Tl?)),

(2)

then T may serve LIStt stmss measure and (T, E) is crrlled a conjugtttc pair. Here a is the Cauchy stress tensor, D is the stretching

tensor (i.e. the symmetric

gradient L) nnd Ill = det U is the third principal Let U = x&N,

@ N,. whcrc f2,J and (N,f

pnrt of the velocity

invariant of the right stretch tensor U. are the principal

stretches and the sub-

ordinate orthonorm~ll eigcnvcctors of U, respectively. Hill (1968) proposed ;t class of strain measures given by

E(U)= ~j’(W,

where /(*)

is 3 smooth

strictly-increasing

0 N,.

scalnr function

(3)

that satisfies J’f I) = 0 and

f’(l) = I. Earlier Seth (1964) discussed n subclass of strain measures E(U) parameter m. where the function

indexed by the

I,1, (i.“--

j-c;.,=

I).

if m # 0.

Iln L.

(4)

if 112= 0.

We suggest calling strain tensors from this subclass the Seth-Hill

strain measures, and we

denote them with superscript by

E l,“, __ ‘r(;_-l)S,~u,=~(~-I,.

ifr7lfO.

I!1 4

(5)

E”” = 2 In E.,N, 0 N, = In U.

We mention the following

often used strain measures as special cases of (5) :

(i) Green’s strain : E’” = i(U’-I), (ii)

Almansi strain : EC ‘I = !(I - U

(iii)

r\;ominal strain : E’ ” = U - I,

(iv)

Logarithmic

strain : E”” = In U.

The stress tensors conjugate (19M)j.

to E”‘.

They are the second Piolz

(whcrc I; is the ilcformation

” and E”’ are well known [cf. Hill (1978) ; Guo tensor

E’

KirchhoCstrcss ,I.“’

=

gradient). I

‘I””

1”

E‘

‘gl;

‘,

(6)

the weighted convected stress tensor

I ‘,

[I’rucsdcll and Nell (1905) callctl l;‘aV

Recently tloger

‘).

2) =

I.-‘.aI;

“I

(7)

the convected stress] and the Jaumann stress tensor =

{(T’?‘U+U’,““),

(8)

(19X7) found ;t tensorial expression for the stress ‘I““’ conjugate

to In U.

From

p

It

-u’

_ -

‘*

. i;’

1) =

_ i

2

,i, = T’

:‘:$;I

Z.5i (T’

:)=‘I_,

?‘U

(u

‘U

21:

l+U

i((J

‘+U

IQ

IT.!‘):&’

‘u ‘).

‘1+&y

I’U

I)

I),

we may add the conjugate stress

to the list of conjugate stress tensors for which tensorid expressions have been found. For positive integers ttJ, we obtain by differentiating (5)

Conju_rate stressandtensorequation

U”-‘XIJm

’= C

x65

I- I @ml = t

f

um-riy_y-

I_

r=l

Substituting

the preceding expression into the identity

yields

(,=

T”‘:~=;

f

:u.

um-q-~“qJ’-’

I

The arbitrariness

>

of ir implies that the stress tensor T’m’ conjugate to E’“” satisfies the tensor

equ::tion

i

’ =

U”.-‘Su’-

,,,,-‘1).

(10)

,=I

Analogously.

for negative integers --nr (where 111> 0). we have

:lntl wc drnw the analogous conclusion

that the stress tensor 1” .‘“’ conjugate to I? ‘“’

s:ttistics the tensor cqu;ltion 0,

1

“XU’

U’

-r = ,,,T’~

‘1.

(II)

.--I

I’re- and postmultiplying

(I I) by U’” -‘, WC get an equivalent equation t ,-

I =

“m-‘xur-

I),,”

I’,-‘-

n,u”-

(13

I

which difrcrs from (IO) only by the right-hand side. Thus, explicit formula

- 1,

for the conjugatc stress T”“’ or T’-“‘.

the crucial point in obtaining an

where HI is a positive integer. is to

solve the tensor equation

,g,

U”‘_3u’-

=

(13)

c,

where C is ;I given symmetric tensor. WC shall investigate eqn (13) in the next section.

2. TENSOR

EQUATION

f U”-‘Xu’-’ r-1

= C

We shall investigate the tensor equation

f r=

Iy-‘XIJ-’

=

c;

(14)

I

here U and C are given second-order tensors;

U is symmetric and positive-definite;

rn > 2

Gvo

X66

ZHONG-HENG

and

CHI-SING

MAS

is an integer. We set aside the case m = 1 because it is trivial. Since simple tensorial the case expressions for T”’ and T’ - ” are already available in (6) and (7) respectively. m = 2 is irrelevant for our present purpose. Nevertheless we mention that our analysis below remains valid for m = 1 and m = 2. When m = 2. (14) assumes the form US + XU = C. which has been studied extensively in the literature. In particular the reader is invited to put m = _1 in our analysis below and compare it with the work of Sidoroff (1978) on the equation UX +XU = C; both start by appealing to the same representation formula for the solution X [see (17) and (38) below]. but they depart in taking different paths to arrive at the coefficients that appear in the representation formula. The following assertions follow easily from eqn (14) and from the positive definiteness of u: (i) When C = 0. the (ii) Equation (14) has function of U and (iii) The solution splits

homogeneous equation has only a trivial solution. a unique solution X(U. C). which is linear in C and is an isotropic C. into two parts :

X(U.0 (iv) The right-hand symmetric.

= X(U, gc+c’))+x(u,

side C and the solution

By decomposing

the tensors

X arc simultaneously

X and C under the principal

X=x.r,,N,@N,.

obtain

the unique

symmetric

or skcw-

frame (N,) of U as

C=xc,,N,@N,,

(15)

1.1

‘.I

wc irnmcdiatcly

l(C-CT)).

solution

of (14) in principal

rcprcscntation

“8,

x,, =

:

(16)

1#,

C 2:” ‘2.; ’ r-

I

Our objective in this paper is to lind an expression of this solution in tensorial form. Let us first consider the case of symmetric C. By virtue of assertion (ii) and Rivlin’s representation formula [cf. Rivlin (1955). eqn (IO. 17) ; Spencer (1971). Table V and Section 8; Wang (1970). p. 2151, we may cast the (symmetric) solution (for symmetric C) in the following form : X(U,C)

=

r,I+r~U+r,U~+z,C+c(~(UC~CU)+r,(U~C+CU~),

where z,,, z~, rb are functions

of the three principal

invariants

(17)

of U :

I = 2, +i.2+i.,, II = i2il+i.,j.,

+j.,iz, (18)

III = i.,lzi.,,

and x,,zz,zJ linear in C :

are functions

of 1. II, III and three common

invariants

of U and C that are

trC=c,,+cz2+c,,. tr(UC) tr(U’C) (I 9) can be written

as

= i.,c,,+i.,c~:+i.,~,,, = i.~c,, +E-icz2+j.Tc,,.

(19)

Conjugate stressand tensor equation 1 U’“-‘XL”,- I

2067

’ = C

m-J)

where

is the van der Monde matrix. Because of the linearity of X in C. a,, ctzand a3 must be linear combinations of tr C. tr (UC) and tr (U’C) and can be presented as

(22)

where the matrix A := (a,,).

Comparing the componcntial form of (I 7) with (16). WChave

‘ii a~ci,fa~(A,+R,)c~,,f~,(%,~+j~,~)Ci, = -y-------, z,

L:” -‘E,‘-

i #j.

(25)

’

We divide our further discussion into the case where all the eigenvalues of U are distinct and that which they are not. Henceforth we shall refer to the eigenvalues of U as the principal stretches. Cuse

I. All principal

stretches

are distinct

Since C can be arbitrary and

(25) is equivalent to

Since the determinant of the coefficient matrix

206X

C&o ZHOP.G-HEW and CHI-SISG \f.ts A = (L1-L,)(i,-i.,)(i.,

-i,)

f

0.

(26) has a unique solution :

(7s)

Here the summation (i’..j’.k’)]

C (or C later on) is carried out for all even permutations

of (1. 2.3). iubstitilting

(i.,j,,k)

[or

(7X) back to (24). we have

whcrc I p, = ,,,

i.,’ “I

Owing to (17). solving (29) is cquivaknt ¬ing

--cl,

-

7x,i., --Ir,,L,2.

(30,

to liriding lhc m;ltrix A tlclincti in (23). In fact. by

the malriccs

L = cling ( I, I, I ).

ii’ = tliag (i.,. i,. i.,).

(31)

(29) can bc written in matrix form as follows :

(3’)

Substituting have

(21) into the left-hand

side of (32). in virtue of the arbitrariness

I ]. ol ,(I, I

WC

-

I

(33) whcrc /I’:) := ill

’ L;:‘\l

It can be seen that .A is symmetric. Having

r.

< = I - rlf.0, 1. 2.

(3-J)

Conjugate

stressand tensorequation z U”~‘XU’I- I

’= C

at hand. we can easily use (34) to calculate A”’ and to get the matrix A with the following entries :

a1.l =

$c

B,%,R,(%,-%p)2.

I

i: a:> = ~-

1 B,(%,-I,)(%;-%I). I

(36) where

(37)

Thus. an intrinsic expression for the solution of (14) is

X(U,C) = [CL,,trC+a,ztr(UC)+cr,,tr(UW]I +[a,ltrC+rzztr(UC)+rz,tr(U’C)]U +[a,, tr C+rzj

tr(UC)+a,,

tr(U’C)]U*

+a~C+r,(UC+CU)+a,(U~C+CU~).

(38)

A glance at (24) and (25) reveals that the solution set of the six simultaneous linear equations for a, (i = 1,2.. .., 6) is infinite. Each solution 6-tuple of a,, when substituted into (17). will give a representation of the solution X(U, C) of (14). Nevertheless, as we shall see, the right-hand sides of (28) and (36) may be extended by continuity so that their domains of definition will include the possibility of coalescent stretches. Moreover. when we use the values of ak (k = 4.5.6) and a,, (i,j = I, 2.3) thus determined from (28) and (36). eqn (38) still delivers one representation of the unique solution X(U,C) in tensorial form. First let us revert to the case of distinct principal stretches and examine (25) more closely. By Cramer’s rule,

‘OX

Z~WS'G-HESG ml

Gco

CHI-SISG MA\

(39)

It is clear that when bve expand the numerator

function

of i.,.

f’(i_,, r.2. i.&‘o,

i.,.

E.,. For

wherej’is

definiteness.

a polynomial

determinant

let this

rational.

of xl NT obtain function

a rational

be denoted by

in L,, L,. j.%. and

(40)

hcrc the product is mkxnt to bu carried out for a11even pcrmutntions

if we set i, = i., in the numcr;ltor minant

will

bc the

dctcrminant

hcncc U’C ooncludc

S;IIW;

q ( i.:. i.:. i.,) > 0, WC must have J‘(i.2.j.2.j.I) simply ;I potyrromial in i.,. f3y the rcnxlind~r is

;I

factor of./: Similarly

,/’ = A-g.

wc

c;tii

whcrc g is ;I polynomial

ucll dcliticti

cwn

for the prcsctit

of (39). the first

that j”(L,. i.:. i.,)/n(i.L. i_:, i.,) = 0. Since = 0. If we regard /!? and i., as given, /‘ is

thcorcm,_f’(L~. L,

prove that i., - 2,

and

i. t -

in E.,, L:. L,. It follows c;isc

(i..i. X-) of (I. 2. 3).

two rows of the dctcr-

j.,) = 0 imptics that 2, -i.,

arc also factors of,/: [Icncc that xj = ~/‘a. bvhich is clcarty L ,

oi’coalcsccnt strctchcs. The discussion

for r5 and rh

is similar. Al’tcr the v;ilucs ot’ z_~. x5 and x,, art’ dcturmincd siniult;mcous

linear quations

for the unknowns x ,, xl

iIS ab0Vc. and

xl.

(2-I)

bccomcs

;I

Systcrll

Ot

Again, WCstart by considering

the C;W of distinct principal strc’tchcs. Then, by Cramer’s rule.

t y, = --

(41)

A

Since x4. x5 and xh satisfy (25), if we set i., = i.:. we obtain the equation

(42)

It follows

that the numerator determinant

is on expansion a rational function P(j. ,, il.

of‘ xl vanishes when 2, = i.:. This dctorminant i-J/Q(j. ,, i.:. i.,). where the denominator poly-

nomial

is always positive. Sincc P(h2, i.>, ;.,)iQ(i.:. is a factor of the polynomial P. Similarly

i.z. E.,) = 0 and Q > 0, WCconclude that 2, - j.: WI: can prove that ;.: -i., and i.,--i., are ~SO

Conjugatestress and

tensor

equation 1 L’“~‘SU’~ r-

’ =C

3371

1

factors of P. In other words P = A./r,. where /~,(E.,.i.~.i.,) is a polynomial in i,, E.: and %,. and we conclude that z, = h,/Q. Similarly. a: = h,iQ and xl = h,/Q. where h, and h, are polynomials in i ,. il and i.,. In fact, a closer examination of (41) and of its counterparts for rI, zJ reveals that a, =

.!.. (h,,c,, +h,,c:z -th,s,,). Q

i= 1.7.3,

(44)

where h,, are polynomials in il. I., and i,,. Let &:= h,,/Q. We see from (22) and (44) that the matrix B:= (p,,) =

AMT.

(45)

Since both h,, and Q are defined even for coalescent stretches, so do the entries #Iiiof the matrix B. Indeed we claim that

hi = Btk. if S=E.,.

(46)

Let us prove the preceding assertion for /I,, and /?,:. Proofs for other cases are similar. When the principal stretches are all distinct. pII and PI2 are given by the formulae:

The system of simultaneous equations (25) rcmnins invariant under the exchitngc of J., and %,(i f: j). Since we are discussing the cast of distinct principal strctchcs for the moment. we conch& from the uniqueness of the solution of (25) that rd. as and ah arc symmetric functions of L,, R2and i3_ It follows that

when the principal stretches are distinct. As mentioned earlier, we may extend the domain of definition of /~‘rr and /Jr? to include the possibility of coalescent stretches. In fact, PI, = h,,/Q and (ii2 = ~z~~~Q, where h,, and 11,: are polynomials in I,. ,Jz and lJ, and Q > 0 is defined in (43). Since Q is a symmetric function of the principal stretches. we conclude from (48) that /r,l(j.l,j.,.j.J) = /r,z(L,.i.,,i.,). which implies h,,(i:,i.,.i.,) = /t,?(i.,. j.z. i.,). It follows that /it, = /1,: ifi.,

= 2:.

We obtain the matrix A := (cc,,)by multiplying the matrix B by itf -‘. We may write down the entries of rt by using (35) and (45). For instance,

The numemtor of zll in (49) is a rational function of L,, L_tand i, whose denominator polynomial is always positive. If we set I, = ,?:, we see from (46) that this numerator vanishes. Hence we conclude that i,, -i2 is a factor of the numerator polynomial of the rational function in question. Similarly i.z - i., and i., - J., are also factors. On cancellation of A from its numerator and denominator, Q, , is expressed as a rational function whose denominator polynomial is always positive. Therefore we may extend ai i by continuity so that its domain of definition includes the present case of coalescent stretches. By similar arguments. we can reach the same conclusion for the other glj.

Guo

2072

For

definiteness

functions

let i,,(L,.i.l.i.j)

that result from

zc by continuity coefficients

and

CHI-SI*G

to include

the possibility deliver

of coalescent

OIICJrepresentation

Iet us rewrite

We claim

that

the

of S(U. C) via (38) even if two or

(2-t) and (25) as

F/(2,, , . . .r!!.Tl.li.r,,:L,.I.-,L~)

=O.

G,,,(1~.r,.r,:i.,.i.~.i.,) respectively;

stretches.

be the x1, and

stretches coalesce.

For simplicity

(i. j =

MAX

i.,) (i. j = I. 2.3 ; k = 4,5.6) after we extend the functions

and z&(i.,.j.,,

(36) and (28). respectively.

z?,,and 5, always

all of the principal

ZHO~GHE~G

here,

in rewriting

I, 2.3). The functions

Let us now consider loss of generality,

(24).

=O.

we have

(50)

used (22) to replace

the case where exactly E., = i., # i.,. Let

1.2.3,

nr=4.5.6,

F, and G,,, are continuous

suppose

I=

two principal

by I,,

stretches are equal.

=(I.

be a sequence

Without

lim I.(,“’ = i.:. n--r and 2’;’ # j.:. i-‘,“I # E., for each n. Since i.‘;“. L, and E., are distinct. we know from our previous discussion that Z,,(i.‘,“‘, j.:. i.,) and f,(i.‘;“. L:. i. ,) satisfy the equations

F,(f , , . . . r?).i,.r,.Z,:j.‘;“.i.:.j.,)

(;.‘,“‘l

z,, zz and r,

in all their arguments.

G,,,(Z,.ZS,2,;j.‘;‘,E.2,j.?)

for each II. Since P’,. G ,,,. z,, and ri arc aII continuous conclude that as n -+ %, r,,(i.:. E.,. i.,) and I,(L:.j.:,L,)

F,(ci,, ,....

r‘,,.r,.~,.~,,;i.l.i.:,L,)

Hence js,, and YiLstill dclivcr

such that

=O,

(51)

in their respective arguments. satisfy the equations

(;,,,(r.,.r,,r,,:j.:,%?.%,)

via (3X) one rcprcsentation

=O.

for the unique

=O.

solution

wc

(52)

X(U, C) of

(14). It is clear that when the principal ;I similar

strctchcs

all coaicscc wc can prove

Wc summarize

our finding

For a skew-symmetric

VIII

X(U,C)

and Section

=

in

(17) arc similar.

when the principal

for dctcrmining

IliaI

For

(53)

/jJ. /jii. /I, is the same as (25)

strctchcs arc distinct.

(28) ; morcovcr, these cocfflcicnts can be extcndcd by continuity coalescent strctchcs. Hcncc WC’arrive at the following : Corollary.

of (14) can bc cast in the form

of I, II. 111. The form of (53) and the last three terms of

and the system of equations

for z,, rsq zo. Thcrcforc.

solution

X]:

~~,c+~I,(uc+cu)+~~~,(u~c+cu~),

where pa. [I5 and /I,, arc functions

will

assertion

:

in lhc following

C, the skew-symmetric

[cf. Spencer (1971). Table

exccpl

our

Wily.

(I .skclc.-.s?,,tlttlcrric C. ull t/w cotrc1u.sion.s x,, .rl1orrld hc Itlkc’tl us ndl.

/Id, /II. Ph are given also by to allow

for the possibility

c$ the preccdiy

theorem

of

stand.

lhc c0qtticicwI.s

By assertion (iii) at the beginning of this section, the theorem together provide a tensorial expression for the unique solution

for any given

second-order

tensor

C. Using an algorithm

and corollary above X(U, C) of eqn (14)

(cf. Appendix)

based on the

Conjugate stressand tensor equation s C” -‘XC I_ I

fundamental

theorem of symmetric

polynomials,

2073

’ = C

we can express all the coefficients z, ,,

z’:..... r>>. X4. 35. x, of the solution (38) in terms of the principal invariants I. 11, III of U. We shall illustrate the entire procedure by finding an explicit intrinsic expression for the conjugate stress T’ _“.

3. EXPRESSION By (12) the stressT’-“conjugate

FOR CONJUGATE

STRESS T’-”

to the strain E’-”

= :(I - U-‘)

is the unique solution

of the tensor equation

U?S+CiSU+Sc’ Denoting T’-”

= 3U2T’-“U’.

(54)

= c t,,S, @ N,. we obtain the solution of (53) in principal representation

:

1.1

(55)

An intrinsic expression of X is

S(U.T’

“) = i’,I+~2U+;/JU-‘+;‘JT’ +‘r’o(U:T’.

“+?”

“+y,(UT’-“+T’

-“U)

-“U:)

= [.I’, tr’l”

“-t-:,,2 tr(UT’

+[;.‘ztrT’

“+;~,tr(U’I”

“)+7z,tr(U2’T’

“)]U

+[;“,trT’

“+y2,tr(tJ’l”

“)+;‘,,tr(U”l”

“)JU’

fi’,,T

“+7s(u’I”

The equations for determining

“)+ylr

“+‘I”

tr(lJ’T’

“)]I

“U)+;‘,,(U:‘I”

“+‘I”

“U?).

(56)

the cocliicicnts ‘iI, y2.. . . , y6 arc

yl+~~L,+y~i.,~+~,r,,+2ys%,r,,+~~,,i.’f,,

yJr,,+~s(i.,+L,)r,,+;~,,(i.~+L~)r,,

= &‘I,,,

= -.-; .A,- +

To start with, we proceed as if the principal

i = 1,2,3,

~-~---.-%,i.,

+

E.,’’

i # j.

stretchcs arc distinct.

(57)

(58)

Solving the system

cquivnlcnt to (58). namely

we obtain the solution 7,. ‘is, yh in the form of rational functions of I.,, I.,. 2,. Cancelling the common skew-symmetric factor A in the numerators and denominators, we use the algorithm in the Appendix to get

$1

I.

,5

=

II’-JZJJ

.,,b = $1

111-11‘111).

II 111-11~).

(59)

where

q:=n(j.i+j.,&+i.t)

= J’JJ’-J”IJJ-JJ‘>O.

(60)

Substituting (59) into (57) and following the procedure described in Section 2, we obtain the expression for the matrix :

J-z=(y,,) = -;‘,.P”-2~5,4”‘+(I where A”’ = arc j,,

for < = 0, 1.2 [cf. (34)]. Let fi=A’ur

iCI-‘U’M-’

= 81 I II’ III”+66

111‘111-50

I’ II’ III’+3

- I5 1’ JJJ III+ I I 1’ JJ~JJJ~+4 j,z = II7 I II> JJJ~+ I2 I IJo -54 + I4 JJ II’ III-X 9,) = 541 [I JJJ’+62 -HI II’ III’jz2 = 54 I II III’s26 +J’JJ5-3

I” III’-81

1: II III’--60

I! II III’-54 I JJJ [[I‘--36

I’ II”+27

= (jr,). Theentries

I’ III’-

18 I” JJ [I[’

II’ III’-- I’ II’. 1: II” 111+2x I’ 11: IlIZ-31’

II? III’-8 I’ II’ III’+3

off

II5

II’ III. J’IJ‘--

I4 I’ JJ’JJJ+8

I JJ JJJ’

I2 II”. I JJJJJJ-631’

I’ II? III-4

-j:> = 27 I [I’ III’+4

(61)

_2i’o)#I),

I II’-

Jf’IJJ’-31’

I”IIJ~-27

JJ’+HJ’JJ’JJI-27J’JJf’+26JJ

II’ III?--4

I4 I’ II’ III+27

II [[I’

II”,

I’ III’--

I8 I’ II III’-I’

[I’

+3 JJ II’ III-t-4 I5 III’, j,,

= 361 II’ III+2

I’ II’--8

I’ II’ III-54

II’ lJJ~--8

II’.

(6 2)

Extracting the factor A’ = (j.?-j.,)‘(j.l-j.I)‘(j.I

-i.2)’

= I8 I II Ill + I’ Jl’--4

I’ III -4

II’-27

111~

(63)

from the above expressions for y,,. WCarrive at the intrinsic cxprcssion

T’-“=

6

{[(-3JJJ~JJJ-J~JJJ~+3JJ’)trT’~“+(-3JJI’+2J~JJ1JJ+2JJ~lJl)tr(UT’~“)

+(-211[111+311’)tr(U~T’~“)]1+[(-31[1’+21~11111+211’11[)trT’~” +(-21

JJJJJ+J~II~+J’JJJ+lJ’)

+[(-2111111+311’)

trT’-“-(I

tr(UT’-“)-(I JJ~+J’JJJ)

JJ’+J~IJJ) tr(UT’-I’)+2

tr(U’T’-“)]U JJ’tr(I?T’-“)]U’

+3((JJJ’IJJ+J~JJJ~-JJJ)T’-“+(JJJ’-J2JJJJJ-JJ~JJI)(UT~-“+Tr-”U) +(I11 III-JI”)(U’T’-“+T’-“U’)]).

(64)

Conjugate

stress and tensor equation

f U”-‘XL”,- I

’ = C

2075

which is valid for any given U and T' - I).

4. CONCLUDING

REMARKS

For each integer m with lml > 2. the method proposed above delivers an intrinsic expression for the stress T”“’ conjugate to the strain measure E ‘m’in the Seth-Hill class. The invested labor, however. will increase rapidly with increasing /ml. Symbolic computation could be useful here, because all operations in this method are algebraic. .4cknoHl~~~emenrs-The collaborative research reported here was carried out when Guo held a visiting professorship at the Department of Mathematics. University of Kentucky. Lexington. That professorship was supported in part with funds from the National Science Foundation (Grant No. RI)-8610671) and the Commonwealth of Kentucky through the Kentucky EPSCoR Program. This research was supported in part by the University of Kentucky under Faculty Grant No. 2-04449.

REFERENCES Guo. Z.-h. (1980). The unified theory of variational principles in nonlinear elasticity. Arch. Mech. 32.577-596. Guo. Z.-h. (1984). Basic aspects of Hill’s method in solid mechanics. SM Arch. 9. 353-380. Hill, R. (1968). On constitutive inequalities for simple materials-l. J. Mech. Phys. Solids 16. 229-242. Hill. R. (1978). Aspects of invariance in solid mechanics. In A&ance.r in Applied Mechanics (Edited by C.-S. Yih). Vol. 18. pp. I-75. Academic Press. New York. Hoger. A. (1987). The stress conjugate to logarithmic strain. Inf. 1. So/idv Swucrures 23. 16451656. Macvean. D. 8. (1968). Die Elementararbeit in eincm Kontinuum und die Zuordnung von Spannungs- und Verzerrungstensoren. ZAAfP 19, 157-185. Rivlin. R. S. (1955). Further remarks on the strcsssdeformation relations for isotropic materials. J. Ro(ionttl Mech. And. 4 . 68 I -702. Seth. B. R. (1964). Generalized strain mcasurc with applications to physical problems. In Sccontl-nrdcr Eficfs in fkriciry. Plusticify und F/d Dynumic.~ (Edited by M. Rciner and D. Abir), pp. 162-172. Pcrgamon Press. Oxford. Sidoroff. F. (1978). Sur I’equation trnsoricllc A.U+>J = H. C.R. Accrcl. Sci. Puris 2R6. Sirie A, 71 -73. Spencer, A. J. M. (1971). Theory of invariants. In Conrinuum Physics (Edited by A. C. Eringen). Vol. I. pp. 239353. Academic Press, New York. Truesdcll, C. and Nell. W. (1965). The nonlinear licld thcorics of mechanics. In Encyckoprdiu ofPhysics (Edited by S. Fliigge). Vol. ill/3. Springer. Berlin. Wang, C.-C. (1970). A new representation thcorcm for isotropic functions: an answer to Professor G. F. Smith’s criticism of my papers on reprrscnlations for isotropic functions. Part 2. Arch. Rutionul Mech. And. 36, 19X -

“1 em_. APPENDIX Throughout this paper we make use of an algorithm that recasts a symmetric polynomial of the eigenvalues i.,. i.,, 1, of the symmetric tensor U in terms of its principal invariants I, II, Ill. As an illustration. we demonstrate this algorithm for

which is a homogeneous symmetric polynomial as follows : (i) Write down all the clrmcntary

la.5 29:16-E

homogeneous

of degree six. In this instance the algorithm

polynomials

in question proceeds

of degree six in L,. 1r and ,I,. namely.

GLO

‘076 (ii) Express in terms of the rlementq

ZHONG-HE.G

polynomials III’

and

CHI-SIX

each monumlal

bl\x

of I. II. III that IS ofdrgree

SIX In i.,.

L:.

iI.

= .4.

I II III

= 3,-t -5.

1: III

= 6.-l -38-C.

II’=6.-1+3B+D. I’ll’=

15.4+8B+lC+20+E.

1’11 = 366+2XT+YC+60+1E+F. I” = Y0,4+6OE+?OC+‘OD~ (Iii) Solve the above hnear system wth .aInangular of I. II and 111:

A=

ISE+hF-G

coefficient matron to wrltr

the elementary polqnomtals

111’.

H = I II III -3 C‘ = 1: III -3 n = -3

111:. I II III t?

I II III+Il’+3

E = -‘1’111+1’11~+41

III:. Ill!. II Ill-Zll’-3

Ill’,

f=

I’II-I’Ill-JI’11’+7l

11111+?11’-31112.

G-

1”-6l’I1+61’111+91’11-‘-1~11!II1-21I’~?l11~

A’ ~: -o..I+~lI-~~‘-~I)+I; IS I

II III + I‘ II‘-4

I’ III

--4 II’ -27 III’.

in terms