Linear grade-consistent micropolar theory

lnt ). l:,ngngSci ~ol. 19, No. 12, pp. 1731-1738, 1981 Prinl'ed in Great Britai0

0020-7225/811121Tq4)8502(N~/O © 1981 Pergamon Press [ d

LINEAR GRADE-CONSISTENT MICROPOLAR THEORY O. BRULIN and S. HJALMARS Royal Institute of Technology,S-10044 Stockholm.Sweden Abstract--A linearization is performed of a previously presented, so-called grade-consistent micropolar

theory, where the free energy is assumed to contain also second order derivatives of the displacement, an order which is argued to match the conventionalfirst order derivatives of the micropolarangle. Dimensionless constitutive constants are introduced, which facilitates the discussion of orders of magnitude. The theory is applied to harmonic plane waves. By means of the dispersion relations the possible magnitudesof the different, conventionaland new micropolareffects are preliminarily discussed. 1. INTRODUCTION:THE NON-LINEAR GRADE-CONSISTENT THEORY Previously, the present authors[l,2] have advocated the proposal that a consistent grade level for a micropolar theory, containing a macro-translational field uk and a micro-rotational field Ck, should contain field derivatives of one order higher in Uk than in Ck. The arguments f~r this are drawn from general dimensional considerations as well as from the analysis of special micro-model cases. The consequence of the proposal is that if we admit the free energy t~ = e - 770 to be a function of q~k up to t~rst order derivatives, as we do in conventional micropolar theory, we have also to admit 4J to be a function of up to s e c o n d order derivatives in uk. The set of independent constitutive variables for ~b on this grade level should thus be u, j, uk.#,, ~k, Ck.~, or equivalently Xk,K, Xk,KL, V/~kK,XkK,L,

( [- [ )

the )¢kK being the orthogonal transformation coefficients of the rigid micro-rotation, as defined by Kafadar and Eringen[3], the notations of whom will be preferably used in the sequel. Now, consider the balance of momentum, angular momentum, energy and entropy, i.e. Pxk = Pfk + tlkj,

(l.~li

pd'k = plk + Eklmllm + mtkj,

(l.,i

pVkVk + p~'kgrk + p~ + Wk,k + qk,* = pW + pq,

(1.~

p(l+(O-lqr),r=pqO-I

{1.5)

+pTr,

rr>_O,

where Wk is the contact work flow and w is the work from volume forces fk and couples lk. It was shown in [1,2] that the constitutive assumption (1.1) requires a generalized form of the contact work, wk = - tk~Vt -- r~ltol -- SktV~ -- Pkjmd~,,,

(1.6)

in which the stress couple is split in two parts, mk! = rkl + Ski,

(1.7~

one part performing the work with the micro-gyration Uk = (I/2)ek,sXrKXsK, and the other with the Cauchy spin tok = (l/2)ekr~Vls,rI, and which also contains a contribution, working with the rate tensor dk~ = V~k,~. A corresponding split of the volume work w = [kVk + rkwk + Skgk,

( 1.8~

lk = rk + Sk,

(I.9,

1731

1732

O. BRULIN and S. HJALMARS

makes it difficult to eliminate lk from the entropy inequality, as required in the conventional termodynamic analysis of constitutive properties. However, since in most cases, where we can conceive a body couple, i.e. mainly in materials with magnetized particles in external magnetic fields[4], only an sk-contribution in (1.8), (i.9) is manifest. An rk-contribution is thus not considered at present. Furthermore, under the assumption (1.1) the free energy $ is seen to be a function only of the scalar integrity basis ~KL = XkKXkL,

~KL = Xk,KXkL,

(1.10)

@RIK"= xkRXK,k,

l

CKML = CKLM = Xk,KXk,LM'

FKL ----2 ~'KMNXkM, LXkN

(l.11)

for the four local vectors (1.1). Under the assumption that ~ is a function of the quantities (1.10), (1.11), the following equations of state for the non-dissipative parts (denoted by') of the constitive variables were derived: t lkl _-

(1.12)

tkt0 + ~k,.t.m, (~k~t = -- (Omkt,

(1.13)

m'kt = mot + r'kt,

(1.14)

S'kl = mot, I P'kt~ = pOkt~ +~ ((Ok,,t + (Okt,,),

(1.15) (1.16)

r'kt = (-Irs~krs, _

1

(1.17)

~krs = ~ (Etrsr'kt + etskr, -- Etkrr'~t), tot = p ~ --

Xk,KXtL + 2p ~ [9 ~

mot = p ~

X(~KXt~.LU

(XI,KXk,LX,.,M + Xk,KXm,LXLM -- Xm,KXI,LXk,M)

Xk,rXtL,

36 pOlm = [9 ~

( -- Xk,KXI,LXm, M + XI,KXm,LXk, M + Xm.KXk,LXI, M ).

,

(1.18) (1.19)

(1.20)

Here the r~,t and ~Dkrs,which uniquely determine each other according to (1.16) and (1.17), are not determined by the bulk ~, and thus, if existent, must originate from surface effects, i.e. must be determined from a possible surface free energy. The non-dissipative parts of the equations (1.2)-(1.5), however, are fully determined by @, i.e. by t°t, mot and pot,., since the ~kt,. and r~,t terms vanish, when t~,t, m~,t, s~t and PI,t,. are inserted in (1.2)-(1.7). The non-dissipative parts of the balance of momentum (1.2) and of angular momentum (1.3), will thus, irrespective of the choice of r~t, ffkt,., take the form PiCk = Pfk + tOk,I,

(1.21)

pd. k = ~l k + #.kl,.tl,. o + m oIkJ,

(1.22)

t°t and m°t being determined uniquely by the free energy ~ through (1.18), (1.19). Consequently the choice of r~a, ffkt,, will only influence the boundary conditions. It will also be noted that t°t generally contains derivatives of one order higher than those originally assumed in $, i.e. in the stress also XkK.LMand Uk,KLM will appear. The purpose of the present paper is to formulate the linear approximation of the above grade-consistent theory for the isotropic case, and to make some preliminary estimates of orders of magnitude.

Linear grade-consistent micropolar theory

17~3

2. THE LINEAR THEORY IN THE ISOTROP1C ELASTIC CASE

For the purpose of linearization we assume ~b to be a function of the following tensors, all small at small deformations:

@KL =- ~KL -- (~KL,

FKLM= eRMLFRK,

CKLM.

(2. }

The general + for the linearized isotropic case can then be obtained as the second order approximation of the general invariant quadric in the tensors (2.1), i.e. pl~ = OI@KL@KL + O2@KL@LK + a3@KK@LL + bIFKLMFKLM + b2FKLMFLKM + bSI'LLKFMMK

+ c, CKLMCKL~+ c2CKLMCLKM -~- C3CKLLCKMM + c4CKLLCMM K @ C5CLLKCMMK + dIFKLMCL~K + d2FLLKC~MK + d3FLLKCKMM,

(2.2}

where a l . . . d3 are constants. The stress and the couple stress tensors are then given by (1.12)-(1.20) up to the arbitrary, 0-independent tensor ffm.m or its equivalent rb. The linearized form of ffKcm can be obtained as the linear approximation of the general tensor, linear in the tensors (2.1), i.e. for the isotropic case

'~KLM = f, (C~LM - GKM) +/'2(CKRRaLM -- C~.R6~ ) + A(CRKr6LM -- CRLRaKM ) + f4FMKL + fS(FKLM -- FLKM ) + f6 (FRRK6LM -- FRRLI~KM),

(2.3'

where f j . . . f6 are constants. Evidently, this approximation gives derivatives in tb up to the same order as those, assumed in 0. The stress, stress couple and work flow are then given b} (1.12)-(1.20) applied on (2.2) and (2.3). If we restrict ourselves to small deformations uk,i and micro-rotations q~k we have xk,i = 8kl + Ug,t,

(2.4)

Xkt = 6k~ - Ekt,~r,

(2.5 t

~kl ~" ltl,k -- ~:klr~gr,

(2.6)

F u =- ~Ok,t F kt,,, = ~:rml~r,k,

(2.7) (2.8)

Ckl~ ~ Uk.l~,

~rk = jvk -~ j~bk.

(2.9)

We obtain from (1.18) to (1.20):

t°' = P ~ -

[ o~,

P ~

a,l,

o4, ]

+ eCk,m aC,.k, ,,.'

(2.10) (2.11)

p%__p[_

a¢ +

o~,

a¢

(2.12)

1734

O. BRULIN and S. HJALMARS

Introducing (2.1)-(2.12) into (1.12)-(1.20) we obtain the general form of the linear approximation of stress, stress couple and work factor: t~,l = (al + a2)(Ut,k + Ilk,l)+ 2a3Ur, r6kl + ( a l - a2)(Ul, k -- Uk,l --2e.klrq~r)

- 2(cl + Cs)(Ukj + Ul,k),mm -1"2(cl - c2 - C4)Um,mkl + 2(cs - c s ) u . . . . . (~1¢1 + (ds - d~)(E..~Or.k., + e~mk~,,.) + (f~ + f2) Uk,,,,,. + (.f3 -- f~) Um.,.k~

-- (f2 + fs) U..... 8kl + (f4 --/6) ek..~O~,,.+ fS(e..rq~r.k., -- etkrq~.m,,,),

(2.13)

s~a = (2b¿ + b2)(q~t,k + ¢Pk,t)+ (2b~ + b2 + 2bs)(~Ot,k - ~Ok.t) -- 2b2~o~,~6kl (2.14)

+ dl ¢lrsUs,rk + d2~lskUr, rs + ds~lsk u ..... r '~t = -- fl~.lrsUr, ks -- Ektr(f2Ur,ss + f3Us.rs) + (f4 -- f5 --/6) q~l,k-- (/4 +/5)q~r,~6kl + f6~Pk,I,

(2.15)

p '~.. = 2Cl(tll,mk + Um,lk ) -Jr 2 ( C 2 -- ¢l)Uk,lm -F 2C3(lll,rr~km q- llm,rr~kl )

+ (C4- 2C3)Uk, rrt~Im + C4(Ur, rl~km + Ur,rm~kl)-~

(2C5-

C4)llr, rk~lm

+ (d~ - d3)eskrCPs,rt]lr n + d3(EslAOs,rt~km + ~smr~s,rt~kl) + dl(erkl~r,m + ~rkrn~r,I) 1 + ~ fl (2 Ukj., -- It l,km -- 2 U,.,kl ) + 2 f2 (2 Uk,~r6t,. -- U~,rr6k,. -- U,.,rr~kl) 1

+ ~ f3(2Ur, kr~lm -- ltr, lr6km -- Ur.mr6kl) + ~ f4(~-rlk~r,m + ~rmk~r,I) 1

+ ~ fs(~.krnrq~r,I q- eklr~r, m ) "}- ~ f6(2erkp~r, p61m -- ~.rlp~r,p6km -- ~rmp~r, p6kl ).

(2.16)

The equations of motion will then be 2alUt, kk + 2(az + a3)Uk,kt -- 2(a1 - aZ)Ektr~Pr.k -- 2(Cl + C3)Ul, kkm m

- 2(c2 + C4 "}" Cs)Uk,lkmm ~- (ds - dl)ElrmCPr, mkk + oft =piil,

(2.17)

2(at + a2)(~.tkrUr, k -- 2q~/) + 2(2bl + b2 + bs)~Ot, kk -- 2(b3 + bz)¢k,kl + (d3 - dl)~.lrmlir, rakk + pit = Pj~,

(2.18)

fully determined by the constants ak, bk, ck and dk of the free energy 6. The possibility, already existing in the non-linear theory, see [1,2], of choosing the Sindependent function ffklmin such a way that the stress couple ink1 vanishes, is here realized by choosing the constants f l . . . f6 so that r[j of (2.15) equals -s~a of (2.14). Introducing the following notations for the occurring independent combinations of the constitutive constants, al + a2 =/z,

al - az = a, 2as = A,

2bm+b2=7, 2bl + b2 + 2 b s = E, - 2b2 = fl, 2(ct + c3) = ~, 2(c2+ C4 + C5) = r/, d3 - d l = ~',

(2.19)

we obtain the following form of the equations of motion (t¢ + Ot)Uk.t! + (1¢ + A -- a)Ul.lk + 2a~klmCPm,! -- ~llk,llrnm -- ~Ul,lkrnm -1- ~krm~r, mll = Ditk,

(2.20)

20t(ekl,,Um,t -- 2~k) + (e + ]/)~OkdI q- ([3 q- "y -- ¢)~Ol.lk + ~EkrrnUr,mtl : pj(Ok.

(2.21)

We see that the conventional micropolar theory is obtained by putting the constitutive constants ~ = 77= ~ = 0. 3. APPLICATION TO HARMONIC ELASTIC WAVES

For the purpose of discussing the possible orders of magnitude of the effects, arising from the different terms in the equations of motion (2.20), (2.21) we will investigate the harmonic plane wave solutions. In order to facilitate a rational discussion of the orders of magnitude,

Linear grade-consistent micropolar theor}

1'73'~

applicable to micro-structures of any level of characteristic micro-lengths l, we introduce 1he following dimensionless constitutive constants, denoted by a bar: (3.1) (3.2)

Evidently this means that we use one of the non-polar Lam6-constants, dominating in all practically known materials, as unit for all material constants, having the same dimension as the Lam6-constants. i.e. for A, c~ and (y, e, ~, ~, ~, ()/l". The equations of motion will now take the form: (1 + d)uk.~/+ (1 + A-- ff)uuk + 267Ektmq~m.t --/'~Uk.//m,, -- I'~Uuk,,,, + l'(Ekr,,,~r,,,,tt -----(p/#)t~k, (3 3) "y- g)¢uk + I~(Ek,,,Ur.,,,H = (p/#)12JCk •

2ff(ek/,,,lt,,,.l -- 2g~k)+ 12(g + 'Y)¢k.U + / z ( / ~ +

(3.4)

Inserting in (3.3), (3.4) an harmonic wave, travelling along the x~-axis, uk = Uk exp [i(qx3 - o9t)],

(35)

Ck = ~k exp [i(qx~ - ogt)],

(3.6)

we obtain 2 equations containing only the amplitudes U~, (I02, 2 equations containing only L'~, q~, each pair defining a transversal, translation-microrotational mode. We also get I equation containing only U3, defining a longitudinal, purely translational mode, and I equation containing only q~, defining a longitudinal, purely micro-rotational mode. The transversal modes give both the following secular determinant: - q2(l + o 7 ) - q412~ + (p/#)o92

iq2d + iq312 ff

- iq2d

-

iq312~

- 467 - (g + ~)q212 + (p~ # )12o92 = O.

(3,7)

In calculating the o92-roots and the corresponding amplitude ratios we expand the solutions in powers of q212, which should be effective, since if, for a sensible continuum interpretation, we set the commonly accepted limit 1 ~
o92=

q2(#/P){[267(l +- 1)/f](q212) '+ 2 [(1 w 1)+ 67(1 + l)+ (¢ + ~)(1 + 1)/[] 1

+~[ = ( g + ~)+ 2~(I T 1)_+4(_+ [](q212)+O(q414)l.

(3. ~

and the corresponding amplitude ratios as i

{[

267(

+~[(1 _+ 1)+67(1 7- l ) - ( g + y)(I + l ) / f + 2[(1 +_ I)/[] 1

+~ [-+ (g+ y)+ 2~(1 _+ 1)- ((1 _+3)7- j - ((1 _+ 1)/6 + (g+ ~){(1 _+ 1)/(67f)- 2~2(1 _+ I)/(67Dl(q2F)+ 0(q414)/.

(3.9t

1736

O. BRULIN and S. HJALMARS

The transversal displacement-rotation modes The lower signs in (3.8), (3.9) give two transversal (t) modes with coupled displacement (u) and microrotation (¢) and with the following properties: (~Ol..~,

-I"'~ [~"[" "y -'1"47-- 4~-- [1(~212) Jr "" "1 ,

(3.10)

[f~2/ (~ qUl) ]t,u = - [(~i/ (~ qU2) ]t,u. (3.11)

= 1 +(46)-1[ - ~ - ~ + 2 ~ + / ] ( q 2 1 2 ) + " ' .

For long waves (ql-~O) the microrotation ¢2 = ¢2exp [" '] tends to be constrained to the Cauchy-rotation rE = (l/2)(rot u)2 = (i/2)qUi exp [...], which turns (3.3) into the ordinary elastic Navier-equations for the displacement u itself. This turns these modes for long waves effectively into u-modes, which we exhibit by using the order u¢ in the index t, u¢, and by referring to them as transversal "displacement-rotation or u¢ modes". The dispersion of these modes is seen to vanish for long waves, ql--) O, for which the phase velocity c = co/q--)const. = V(iz/p), i.e. the ordinary transversal velocity of sound. The group velocity, V = dMdq, given by

}

V,.,,,p=

1 + ~ [~ + '~ + 4~- 4~F- ]] (q212)+... ,

(3.12)

is of course also seen to have the long wave limit c = V'(l~/p).

The transversal rotation-displacement modes Using the upper signs in (3.8), (3.9), we also obtain two transversal modes with coupled displacement (u) and microrotation (¢), but with the following properties:

~ot2.,,= qZ#--4-~ff~(q212)-'{l+l [~+ p /

]'](q212)+ f [-'-'r/+4~ F+ ]'](q212)2+.. "},

4

16d (3.13) t,,u

1

16,~

_--:1 (,:,,,, {,, '-4-a[-E-~ + 2~ + j](q212) )

[]'(,~ + '~ + 4~- 4~F)- 2~'j-/d + 2(,~+ "~)~ - 4~/d - ~] (q~P) 2 + .... J.

(3.14)

These modes behave for long waves (ql--)O) as pure c-modes, since cD/U~, a property, which we exhibit by using the order ~u in the index t, ~u and by referring to them as "rotation-displacement or ~u modes". The dispersion of these modes is such that the phase velocity c = oj/q~oo in the long wave limit ql--)O. Moreover they show a kind of resonance property of admitting in the long wave limit only one unique frequency lim ~ot.,, =

-.

ql.-.~O

l

(3.15)

The group velocity, given by

+~ (-,vanishes in the long wave limit.

~ +4~'+/) (, +'+'Y/-' [ , ](q2l') + • • .},

(3.16)

Lineargrade-consislentmicropolartheory

7~7

The longitudinal displacement mode The equation, containing only U~, arising from (3.3)-(3.6), gives a longitudinal (I), pare displacement (u) mode with the dispersion

~°~,.= q21X(2p+ ~) { 1+ ~~ + ~ (q 2l') } .

(3,17)

This "longitudinal displacement mode" behaves for long waves as an ordinary, non-polar, longitudinal, elastic wave, but has for short waves a dispersion, depending only on the nea polar constants { "0. The group velocity, given by V,.. = ~ / ( - ~ p - - - ) + 3((+ ~-' (q2/?, +. . . . }. #(2+,g) {1 2(2+,~)

(3.!8~

has as long wave limit the normal longitudinal elastic velocity ~[(2# + ,~)/p].

The longitudinal rotation mode The equation, containing only qb3, obtained from (3.3) to (3.6), gives a longitudinal (I), pt re microrotation (¢) mode with the dispersion:

wT,¢ = q 2 ~ - - (q212)-] l + - ~ - - ( q 1

212) ,

(3. !9)

a dispersion which is seen to depend only on the conventional polar constants "2, ft. This "longitudinal rotation mode" shows the same properties as the t, cu-mode, the long wave frequency limit being the same as (3.15) and the group velocity being

V, = ~ ( ~ ) ~ ( , •

Vt4o:j)

ql){l -[(2~+ ~)[(4d)](q212)+. . . }.

(3.20t

4. ORDER OF MAGNITUDE OF POLAR EFFECTS The orders of magnitude of the polar effects, as given by the dispersion, amplitude ratio, and group-velocity formulae, deduced in the preceding section, can be determined both relative to each other and to the dominating non-polar effects, given by the leading unit term and tile ~-( = 1)-terms, by estimating the corresponding dimensionless polar constants 07. g, y,, fi. ~ fl [~ and j~ The most easily estimated of the polar constants is j. Its maximum value should be obtained in making the micropoles as big as possible, as e.g. in a tightly packed composite. A simple calculation shows that for spherical inclusions in tight cubic packing j - 0 . 1 . A slightly higher value (~ 1/6) can be obtained by increased packing, e.g. with cubes, but usually, of course the value will become smaller, e.g. for the crystal KNO3, where ] ~ 0.05 as estimated by Askar[5] and Fischer-Hjalmars[6]. We conclude that the order of magnitude of the maximum value As for the polar constant & its order of magnitude can be estimated by deducing the continuum field equations from realistic micro-structural models. This has been made ky Berglund[7] for a composite, constituted by rigid spheres (radius = a) spaced in cubic lattice (unit cube edge -- d) in an elastic matrix with the Lame-constants #~'~ and ~.~'L The result is that for tight packing (a/d = 0.5) A ~ # ~ 7# (m~,and d = a/# ~ 0.5. For vanishing a/d, of course # and ,~ have as limit values the p,~"~and ,~'~ of the matrix, whereas a. and thus d goes to zero For example, for a/d ~ 0.3 (about one radius length between the spheres), we have ,u, :~ A 2,u~''~ and 07 = (d# =0.2. For less dense micropolar structures d will be much smaller. F~r example for KNO~, Askar[5] and Fischer-Hjalmars[6] have made the estimate d ~ 0.02.

1738

O. BRULIN and S. HJALMARS

Since the close-packed composite, considered by Berglund, seems to be about the most favourable one to produce polar effects, it seems reasonable at present to assume the order of magnitude of the maximum value of ff to be ffmax~ 0.5, a maximum value which may be approached in very favourable cases. As for the stress couple constants/if, ~/and ~ and the contact work constants ~ ~ and ~ no reliable estimates from micro-models have yet been made. A preliminary inspection of the way they will emerge from a model calculation, based on a rational expansion after magnitude levels, indicates that they may be of the same order of magnitude. However, again since from empirical reasons the dispersion effects in the t, u~- and l, u modes as given by (3.10), (3.17) cannot exceed a few per cent at ql = 1, we may conclude that the order of magnitude of the maximum values are (/3, ~2, ~, ~ ~, ~'-)max= 0.1. From the above estimates we may infer that it would be possible to determine the combinations ~ + "~+ 4 ~ - 4 ~ - ./and ( + ~ in cases of values close to the maximum by means of dispersion measurements on the t, u~ and l, u modes near the continuum limit ql ~ 1, i.e. at the short wave limit of the first third of the Brillouin zone. It may be noted that the magnitude of the combination ~+ ~ of the new contact work constants can be determined independently of all other micro-polar constants by investigation only of the l, u mode. Moreover, as shown by Fischer-Hjalmars [6], the continuum equations are likely to be valid as pseudo-continuum field equations for still smaller wave lengths, probably over the whole middle third of the Brillouin zone, i.e. up to ql-~ 2. The dispersion measurement could thus possibly be extendeded also over this region, where the effects, according to (3.10) and (3.17) are much bigger and thus more easy to detect experimentally. Acknowledgements--Thanks are due to Prof. K. Kondo[8], Prof. 1. Fischer-Hjalmars and Tekn. Dr. K. Berglund for illuminatingdiscussions on grade consistency and on the relation between continuum and micro-structure. Note added in proof. Recently all constants have been estimated from a simple cubic lattice model by Fischer-Hjalmarsand Hialmars[9]. REFERENCES [1] O. BRULIN and S. HJALMARS, TRITA-MEK-79-15(Report to be obtained freely from the authors' Department). J. Tech. Phys. To be published. [2] O. BRULIN and S. HJALMARS, POST-RAAG Reports (Edited by K. Kondo), No. 110,June 1980. [3] C. B. KAFADAR and A. C. ERINGEN, Int. J. Engng Sci. 9, 271 (1971). [4] O. BRULIN and S. HJALMARS, J. Tech. Phys. 16, 435 (1975);and R. HSIEH and G. VOROS, Int. J. Engng Sci. (1981). [5] A. ASKAR, Int. J. Engng Sci. 10, 293 (1972). [6] I. FISCHER-I-IJALMARS,Int. J. Engng Sci. 19, 1765(1981). [7] K. BERGLUND, Fibre Sci. Tech. 12, 455 (1979). [8] K. KONDO, POST-RAAGReports (Edited by K. Kondo), No. 113, Sept. 1980. [9] I. FISCHER-HJALMARS and S. HJALMARS, Proc. Continuum models o[ discrete systems, CMDS 4. Holland, Amsterdam (1981). (Recdoed 24 October 1980)