Elastic Properties of Reinforced Solids_some Theoretical Principles

8/7/2019 Elastic Properties of Reinforced Solids_some Theoretical Principles

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358 R . HILLA detailed review of the older literature can be found in REINER (1958). The

most recent and authoritative survey is by HASHIN (1963).

2. ELE MENTARY CONSIDER ATIONS(i) Basic not&ionIt is found most convenient to take for the fundamental independent pair of

isotropic elastic constants the bulk modulus K and the shear (or rigidity) modulusp. The combination K + %p occurs frequently later : it is the modulus in acompression test with lateral expansion prevented. Poissons ratio v and Youngsmodulus E are regarded as dependent properties, the connexions being

v= (t-&)/(1 +&-), i=$+&; -) (2.1)E = 2~ (1 + v) = 3~ (1 - 2~). J

In certain formulae it makes for conciseness to introduce dimensionless parametersa and fi whereK (1 + v)a= ----=3(1 -v)Kf9CL (2.2)2(K+2P)

=,(K++,)=2(4 - 5v)15(1 -v) - 1

All subsequent results are valid under the usual restrictions ensuring a positivestrain-energy function, namely K and TVpositive with - 1 < v < l/2. Then0 < a < 1 and 315 > / > Z/5 ; /I is very near 0.45 for most metals.

Moduli for the two phases will be distinguished by subscripts 1 and 2. Exceptin formulae explicitly dependent on geometrical detail, it is unnecessary to statewhich subscript refers to the included phase. Moduli without numerical subscriptsrefer to the macroscopic, or average, properties of the mixture.The fractional concentrations by volume of the phases will be denoted byc1 and c,, where c1 + c, = 1. While, in subsequent formulae, one concentrationcan always be eliminated in favour of the other, reasons of symmetry usuallymake it desirable to retain both.For brevity of notation, and to lay bare the structure of formulae and proofs,the nine components of stress and of strain will be regarded as forming vectorsin a O-dimensional space, and accordingly denoted collectively by boldface symbolsu and 6, respectively. Tensor sufhx notation is used only where necessary. Asthey stand, these symbols will denote the actual stress and strain tensors at ageneric point of the mixture. A subscript 1 or 2 will be added when referringexclusively to one or other phase.Average values will be further distinguished by a bar placed above. By the average of any quantity is meant, quite straightforwardly, its integral over aspecified region divided by the volume of the region. We then have the obviousconnexions a = Cl Zl + c, z*, ;e = Cl r1 + ca r2, (2.3)fo r a n y r e g ion c o n t a in in g t h e p h a s e s i n t h e g iv e n o v e r a l l co n c e n t r a t i o n s .


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Elas t i c p r up er t i e s o f r e in f or ced so l id s : Some tbeomt ioa l p r inc ip l es 35 9The relations between stress andstr ain tensors at any point in the phases are

written concisely ascr, = L, el, 02 = L, (8,

with inversesl = M, q, P 2 = Iw , 0,. 1

(2.4)

Here t he capitals sta nd for s ymmetr ic 9 x 9 ma tr ices (replacing fort h-r~kten sors) whose element s ar e th e st iffnesses or complian ces resp ectively. Since th epha ses ar e ass um ed uniform an d isotr opic, identical r elations with the sam eoperat ors also hold between th e corr esponding averaged qua nt ities. On substitutingthese in (2.3)

This phr ase will be used when r eferring to a sam ple tha t (a) is str uctur allyent irely typical of th e whole mixtur e on avera ge, an d (b) cont ain s a su fficientnu mber of inclusions for th e ap par ent overa ll moduli to be effectively indepen dentof th e surface values of tr action an d displacement, so long as th ese values ar e macroscopically uniform. That is, they fluctu at e about a mean with a wavelengthsma ll compa red with t he dimensions of the sample, a nd the effects of such fluctu a-tions become insignifican t within a few wavelength s of th e su rface. The contribu-tion of th is su rface layer to any avera ge can be ma de negligible by ta king t hesample large enough.(iii) Relations between averages

In the circumst an ces described in (ii) th ere is a unique dependence of th e averagestrains in the phases upon the overall strain in the mixture. Let this be written as

r1 = A, Z, s=A,z, with clA,+c,A,=I (2.6)where I is the unit matrix. A, and A, me matrices dependent on theconcentra-tions an d pha se moduli, etc. an d are generally un symmetr ic. Combining(2.6)with the first of (2.5) :

i? = Lz where L ==clLIAl +c,L,A, (2.7)is th e required ma tr ix for the mixture (juxtaposed capita ls indicat e ordina ry ma tr ixproducts ). Equ ivalently, in ter ms of st res ses, if

thenZ1 = B, Cr, Zz = B, ?;, with c, B, + c2 B, = I, (2.8)

-e=MGwhere M=cl Ml B, +c,M,B, (2.9)is th e compliance ma tr ix for the mixtu re.

If, therefore, th e average st ress or stra in in either const ituent can be foun dfor ar bitra ry overall values (so deter minin g one of th e As or Bs), the elasticpropert ies of th e mixtu re ar e complet ely specified th rough L or M by (2.7) or(2.9). Explicitly, in ter ms only of avera ges over t he firs t ph as e (say),

L-L 2 = Cl WI - L2) 4 M -- M, = cl (M, - M,) B,. (2.10)


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360 R. HILLConversely, th ese may be rea d as form ulae for A, an d R, when the overall pr opert iesL and M are known.

B, L = L, A,, A,M=M,B1, (2.11)with similar ones in th e other su bscript. The first equa tion follows by expres singiY1 in t he altern at ive ways B, L z [from (2.7) and (2.8)] and L, A, I [from theaver aged form of (2.4) with (2.6)]. The second follows by dua l opera tions with'zl.For a mixture whose overa ll properties ar e isotr opic th e preceding an alysissimplifies somewh at . Since only two elast ic cons ta nt s h ave now to be deter min ed,na mely th e bulk a nd rigidity moduli K an d I*, it is no longer necessa ry to impose aperfectly genera l overa h distort ion. It is enough to cons ider just two indepen dentstrains, chosen at will.

Take first a pur e dila~t ion, with fra ctiona l volum e increa se B sa y, for whichth e corr esponding overall st res s in an isotropic mixtu re (or even one with onlycubic symm etr y) is an all roun d ten sion, 5 sa y. Ins tea d of (2.5) we need onlythe scalar equations

zt=C,K1@1 +C,Kn_&, (Ly +c%-J!,

Here a ,, a, arc th e hydrostat ic par ts of th e str ess (i.e. ar ithm etic mean of normalcom~n en~) an d 8;, 8, ar e th e volumetr ic par ts of th e str ain (Le. sum of normalcomponen ts ), all aver aged over th e resp ective pha ses (in each of which th e ent irestr ess an d strain t ensors ar e, of cour se, not necessarily purely dilat at iona l, noreven th eir avera ges). In place of (26) writ e

3 z.z b,,i-3 3? -_ b,, with b, c1 + b, e, = 1 ;2rwhere th e as a nd bs are fun ction of th e concent ra tions an d moduli, etc. ThenV = Kg where

K = a , c1 u1 + a, ca K2 orin place of (2.7) and (2.9). In st ead of (2.10),

K - K,d = a, Cl or l/K - liKq: _ b c- 1 '1''%--KS 1/'Q-l/'%

(2.12)

(2.13)

These are equivalent express ions for th e overall bulk modulus (valid also whenth e mixtu re is an isotropic wit h cubic symmet ry). The equivalence can easily beverified by int ercha nging a1 an d b, via K = a, Kl/bl, which is the scalar form of(2.11).Take, secondly, a pur e shear stra in, 7 say, its plane a nd direction being chosenar bitrar ily. The overall st ress is likewise a pur e shear , si = 2~?, when the mixtu reis isotr opic. Form ula e similar to (2.12) an d (2.13) ar e th ereby obta ined for th erigidity modulus f~ n term s of pI an d pa. Th e as and bs now stand for the various


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Elastic properties of reinforced solids : Some theoretical principles 361ratios form ed from t he averaged shear ing str ess an d str ain components having t heplane an d direction of th e overall sh ear.

Equations (2.12) an d (2.13), togeth er with t heir shea r an alogues, follow also,of cour se, by decomp osing th e gener al rela t ions (2.7), (2.9) an d (2.10) into hydro-sta tic an d deviat oric par ts.

Rasic as th e results of th is section ar e. th ey ar e appa rent ly not to be foun d in th eliterature.(iv) Reuss and Voigt estim ates

A crude approximat e treat ment of reinforcement assum es tha t the stra inth roughout th e mixture is un iform . With A, = A, = I, equa tion (2.7) yieldsLv = cl L , + c2 L,. This is an isotropic ten sor since L, and L, are, and so

KV =r Cl K1 + C2 Kz, PV = ClPl + CzP2. (2.14)These a re also obta ina ble from th e firs t of (2.12) an d its sh ear an alogue witha, = a2 = 1. These appr oxima te values ar e distinguished by th e subscript Vbecau se th e un iform st ra in assu mpt ion was firs t proposed long ago by Voigt inconn exion with th e rela ted polycryst al problem. Obviously (2.14) is ta nt am oun t toa simple volum e weight ing of th e pha se st iffnesses.

The dua l as su mpt ion, du e to Reuss a nd also origiually for a polycryst al, isth at t he str ess is uniform . This implies B, = B, = I in (2.9), or b, = b, = 1in the second of (2.12) and its analogue, giving

1 c1 +c, =c1+5,zzz- (2.15)KR 9 K2 PR PI P2

which a re simple weight ings of th e compliances.Nat ura lly, neither a ssum ljtion is corr ect : th e implied Voigt str esses ar e such

th at t he tr actions at phase boun dar ies would n ot be in equilibrium, while th eimplied Reu ss str ains a re such th at inclusions an d ma trix could n ot rem ain bonded.

The difference between the estimates can be put as(2.16)

with us e of c1 + c2 = I, with a similar expression for th e shea r moduli. Thusth e Voigt values a lways exceed th e Reus s ones, the differen ce being only a second-order infinitesima l for sma ll differen ces between th e pha ses. The differen ce becomeslar ge, however, when one pha se is compa ra tively rigid (so th at KV an d PV + cowhile KR an d /LR ar e finit e) or weak (KR an d PR + 0 with KV and pv finite), theoth er pha se having finite moduli. And, in general, both estima tes a re ra th er poorwhen th e pha se moduli differ by more t ha n a factor of two or so.

3. ENERGY APPR OACH

(i) Eurt her rem ark s on av eragesIt h as been shown how the centr al problem is reducible to the calculat ion ofaverage st ress or stra in in one or oth er ph ase. A more versat ile appr oach stem sdirectly from classical t heorems in elasticity an d focusses at tent ion on stra inenergies.


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362 R. HI LLIn th e spirit of th e previous shorth an d th e local energy per unit volume, U say,

will be written as a scalar product 4 u E. Expressed entirely in term s of str ainor stress, the energy density is a quadratic form denoted by

U= &Lc or T&MU. (3.1)That L and M ar e symmetr ic tens ors is proved below. In par ticular, when th estr ain is a pur e dilat at ion 8, th e str ess in an isotropic ma terial is an all-roundtens ion u = Ke an d the energy density is 4 &a or U2/2K. Again, if the strainis a pure shear y (half the engineering definition), th e str ess is a pure sh earr = 2py an d th e energy density is 2~9 or 42~.

Consider an y volume of th e mixture, of unit m agnitu de on an appr opriat escale. Let it be subjected to prescribed sur face displacements of th e kind th atwould produce a uniform str ain T in a homogeneous ma terial. This is also th eaverage st ra in in the inh omogeneous mixture itself, since th e avera ge is uniquelydeterm ined by the sur face displacement s, irrespective of th e ma terial (providedth at it rema ins coheren t)*. Now th e total st ra in energy, which is th e integralof th e dens ity over t he un it volum e, can also be evalua ted as th e integr al of ux/2.For th e integra l of u (E - s) is zero since u is an equilibra ted field of str ess whileth e (virt ua l) str ain field E - B is derived from a cont inuous displacement vanishingon the surface. Hen ce t he total ener gy is just G r/2, by definition of avera ge st res s.

In words : the average strain energy in any region can be calculated from theaverage str ess an d stra in, when th e surface const ra ints ar e of th e specified kind.This can be proved similarly when sur face tra ctions ar e prescribed, of th e kind t ha twould produce uniform intern al str ess in a homogeneous ma terial?. In exactlyth e sam e fash ion one can sh ow th at th e infinitesimal Gds is th e increment inavera ge ener gy per un it volum e an d th erefore a perfect differen tia l. It followsthat tensor L is symmetr ic, an d hence also its inverse M, just like the array ofcoefficients in the local stress/strain relations.

Suppose th at th e considered region is a representative volume. Then the relationsbetween average st ress an d stra in are the same for both types of boun dar y condition.We can then write, without discriminating,

20=Zz=izLr=ZMG. (3.2)Fu rt herm ore, th ere is no difficulty in proving th at t his rema ins valid when theloadin g is mer ely macroscopically uniform, in the limiting sen se defined inSection 2 (ii), Notice,, on the oth er ha nd, t ha t sim ilar rela tions do not invariablyhold for t he average energy densit y in a pha se, in terms of its own average str essand strain.(ii) Elementary bounds on bulk and shear moduli

According to th e familiar p rin ciple of minim um potent ial ener gy for an elast iccont inuum , th e actua l stra in energy in th e mixture does not exceed th e energy

*The ctua lormula is, in suffix notation,s ij dV = t s (n i ~~ + n j p i ) dsby an application of Gausss theorem, where V is the region, S ita surface, t&i he diiplacement,and nl the unitoutward normal.


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Elastic properties of reinforced solids : Some theoretical principles 866of any fictitious (unequilibrated) sta te of distort ion with the sam e sur face displace-ment s. When t hese ar e compa tible with a un iform str ain 2, th is can itself be ta kenas a compa rison fictitious sta te, having a n energy density P L, ~12 throughout onepha se an d IL, ~$2 th roughout th e oth er. Hence, by considering a represent at ivevolume u nder such sur face const ra ints an d using (3.2),

2:L 1 < P (cl L, + cg L,) P for arbitrary r. (3.3)Thus, the matrix c1 L , + c2 L , - L is positive semi-definite an d th e cons equen trest rictions on its componen ts (and on th ose of L alone) can be writ ten down byany of several standard methods. In this way upper bounds ar e obtained on eachmodulus appearing as a coefficient in the normal form of the overall energydensity ; i.e. when convert ed int o a su m of squa res of indepen dent linear combina -tions of th e comp onen ts of avera ge str ain .

When t he mixtur e is isotr opic, one can proceed in a more element ar y ma nn er.The overall stra in is put equal in tur n to a pure dilatation and a pure shear. Th egenera l inequa lity (3.3) redu ces to

K < KY, LLL clF (3.4)in ter ms of th e Voigt est ima tes (2.14). Similar ly, by considering loadin g un dersu rface tr actions compa tible with a fictitious un iform field of st res s, an d applyingth e dua l principle of minimum complement ar y energy,

ii: M G < Cr ci M, + cz iMe) i3 for ar bitr ar y G. (3.5)This fur nishes upper bounds on compliances an d therefore lower boun ds on moduli.When th e mixtu re is isotr opic, th e overa ll str ess is put equal in tu rn t o an all-roundtension and a pure shear, giving

K > KR. p I PRt (3.6)in terms of the Reuss estimates (2.15).

It ha s th ereby been proved rigorously th at th e actua l overall moduli lie some-where in the interval between the Reuss and Voigt values, no matter what thegeometry may be. The result is illuminat ing but not part icularly sha rp in general ;better universal boun ds ar e sta ted later. This th eorem was first proved by HILL(1952) for t he relat ed polycrysta l problem, wher e it is more effective for actu alma terials, a nd after wards in th e present cont ext (1955 un published) ; and forp, but not K, by PAUL (1960). The writer a lso at that time derived impr oved boun dsfor specific geomet ries by cons tr uct ing piecewise-un iform comp ar ison fields ; t heresu lts, extended to fibre-like inclusions with ar bitrar y properties, ar e detailed byCROSSLIST (1963) and summarized by HILL (1962).

From (2.12) and the first of both (3.4) and (3.6) one easily deducesa, >< Kz.

There a re similar inequalities for t he as an d bs associated with the shear moduli.Expressed in words : un der dilat at ion or pur e shea r th e average corr esponding str ainin the weaker phase exceeds that in the stronger phase, while the reverse holdsfor the average st ress. This accords with intuitive expecta tion, but would perha psbe un convincing as an a p r ior i bas is for the conver se derivat ion of (3.4) and (3.6).


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364 R. HILL(iii) Elementary bounds on Youngs modulus

Let ER and EV denote values of Youngs modulu s calcula ted from th e secondequation in (2.1) ith t he Reuss a nd Voigt est ima tes , respectively, of th e bulkshea r moduli given by (2.14) r ,d (2.15).t follows easily from (3.4)nd (3.6)hat

where

But , due to th e linear relat ion between th e reciprocal of Ycungs moduh rs andth e reciprocals of th e bulk an d shea .r moduli, one h as from (2.18) th at

1-=- Cf+$ER E, 2 (3.8

Thu s, the Reuss est ima te (3.7) of Youngs modulu s is wha t would be obta ined bycalculat ing th e average longitudinal str ain in a tension t est on the assum ptionth at th e local st ress in th e mixture is uniform . That th is provides a lower boun d,as in (3.Q can alt ern at ively be proved directly from the complem ent ar y ener gyprinciple (34, by put ting th e overall st ress equal to a uniaxial tension.

By contrast, Ev # cl El + c2 E, in genera l, since Youngs modulu s is not alinear combina tion of th e two prim ar y moduli. In fact one can show, after a littlealgebra, that Ev > cl E, $- c2 E, with equality only when the Poisson ratios areequal.Finally, therefore, we have the chain of comparisons :

Ther e is no un iversal order ing of E and c1 E, + c2 E,. When the Poisson ratiosare equal it can be shown that E is th e smaller (by m eans of th e potent ial energyprinciple) ; whereas, when the shear moduli are equal, E is th e greater (from th eexact s ohit ion in Section 4 below). In any event n ot mu ch significan ce can befoun d for th e simple weighting form ula , since th ere is no un iform tr iaxial str ainsuch tha t th e longitudinal component of str ess in each phas e is proport iona lsimply t o th e resp ective Youn gs modulu s.

It may be mentioned that PAUL (1960) ives a very involved form ula for anupper bound on E (obta ined by th e potent ial ener gy prin ciple a pplied to a ten sionspecimen under prescribed axial extension and no tractions on the cylindricalsur face), Algebraic reduction reveals tha t th is form ula is in reality n one oth er th anEv hea vily disguised. And, in fact, a more circum spect use of th e ener gy principleth an Pa uls yields E < Ev very easily, confirming (3.7). hus, by taking anyfictitious un iform tr iaxial s tr ain with given axial component E, we clearly h ave tha tEG/2 is less th an th e energy density of th e tr iaxial str ain in an isctr opic ma terialwith moduli KV, PV. And t his density is least when the st ra in is such tha t th eactual stress in th e latt er mat erial is uniudn l an d produces extension E ; that is ,when the density is EV G/2. This rema rk follow-s immedia tely by applying aga inth e potent ial energy principle in th e same way, this time t o th e stat ed isotr opicmat,erial.


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Elastic properties of reinforced solids : Some theoretical principles(iv) Comparison of energies

365

The reinforcement problem can also be appr oached profita bly in th e followingway. Compa re th e ener gy in unit representative volume of mixtur e with tha t inunit volume of a single const ituen t (say th e second). By (3.1)nd (3.2) the differencein the elast ic energies is z (L - L,) E/Z if th e overa ll str ains ar e th e sam e, andG (N - M,) G/Z if th e overall str esses ar e th e same. In this second case thedifferen ce in th e total potential energies (i.e. including th at of th e ma inta inedsur face loads ) is precisely minu s t he differen ce in elastic energies?. The expressionsare equal t o

$ c1 z (L, - L,) %I and 4 c1 0 (M, - Ma) G1 (3.10)respectively, by (2.10) ith t he help of (2.6) an d (2.8). An adva nt age of th esescalar expressions over (2.10) (to which th ey ar e in principle ent irely equivalent)is tha t th ey can in practice be approximated more conveniently tha n the separat ecomponent s of th e unk nowns a, an d 2,.

Corr esponding form ulae, in ter ms of integra ls in stea d of avera ges, h ave beender ived for an y volum e of mixtur e by HASHIN [ 1960, 1962, equations (11) and(12)]. The terminology polarization stress and strain is sometimes used for(L, - La) q and (M, - Ma) aI respectively.

There is an interesting connexion between the energy changes in (3.10). Fromthe first of (2.9) and (2.4) they can be written as

4 c1 5 M (L, - L,) q and 4 c1 a 111, (L, - L,) g,, (3.11)remembering that L, ~11, = I = L, M,. Thes e form s differ prim ar ily in signbut seconda rily in th at M appea rs in one and 111, in th e oth er. If th e concent ra tionof th e firs t pha se is sma ll, ill is corr espondin gly close to 111,. In th at event , t owithin a second-order infinitesimal, th e expressions (3.11) gree in magnitude ;th e differen ces in potent ial ener gy are th en appr oximat ely equal, and th e differen cesin elastic ener gy eclual and opposite.

4. EQUAL PIIASF RIGIIIITIES : EXACT SOLUTION(i) Statevnent oj results

Suppose th at t he phases ha ve equal rigidity moduli II, only th eir bulk modulidiffering (or equivalently, in this insta nce, only th eir Poisson ra tios). Then thecomplete solution is known in principle for an y geomet ry wha tever (HILL 1962).The detailed fields of str ess and stra in ar e expressible in ter ms of th e gravitat iona lpotent ial of a certain distribut ion of ma tt er, and ar e derived in (ii) below.

It is foun d, rema rka bly enough, th at t he overa ll bulk modulus of an isotr opicmixtur e, or one with cubic symm etr y, depends solely on th e concentr at ions andth e separ at e moduli, and is una ffected by th e sha pes of th e inclusions. It is givenby a simple symm etr ic form ula :

;=(l+g/(l+E). (4-l)tBy a simple but very general theorem : if the moduli of an elastic continuum under tied loading are changed

in any way (inhomogeneously), the work done by the loads (or potential energy lost) in the accompanying surtsceadjustments is equal to twice the gain in strain energy.


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366 H. HILLdlterna tive ways of writing th is ar e

K-KR =l/(l+?X?),KV - KH

Kv-K =l/(l+E).KV - KR

(4.2)

(4.3)Term s ident ifiable with t he Reuss a nd Voigt est ima tes (2.14) an d (2.15) occur inth e formu lae an d are so denoted. It is th ereupon appa rent from (4.2) an d (4.3)th at th e tr ue value of th e bulk modulus is bracketed by these estima tes, in accor-da nce with t he genera l conclusion in Section 3 (ii). The overall r igidity modulu sis of course just TVtself, th e value common to both pha ses.

An unsymmet ric variant of th e above expressions isK - KS-zzzz .-

Kl - K2 1 + (Kl - K,E;esi(% +$P1 (4-J)

or th e an alogous equat ion with num erical subscripts int erchanged ; or, yet again,with a ll moduli (including 4 CL)repla ced by th eir reciprocals. Ry compa risonwith (2.13) th e concentra tion factors for th e avera ge hydrostatic str ess an d volu-metric strain in each phase can be read off :

a1 = aa 1Ks ++tL K1 +*cl = Cl Kz + cz K1 + 4 r

4 b2 1%(s ++p)= %(Q ++p)= Kl$ ++CL(%Q +%G).

The Poisson ra tio of th e mixtu re is obta ined via (2.1) :

(4.5)

(4.3)

v= Cl v1 -t- ca VB VI v21 - Cl v* - c, YrFor fixed sh ear modulu s, P oissons ra tio increa ses monotonically from - 1 to+ Q with increasing bulk modulus. Consequ ent ly, th e exact valu e lies betweenth ose calculat ed from t he Reuss an d Voigt estima tes of th e bulk modulus. Withsome fur th er a lgebra , th e following chain of inequa lities can be esta blished (stillun der th e restriction of equal shea r moduli) :

-1< VR 6 cl Vl + C2 V2 < V < VV.

Youngs modulu s of th e mixtu re is obta ina ble as E = 2p (1 + v), which canbe arranged as

E = ~1% +cnE z --E ,E,I~cL1 - (~1 Es + ~2 E1)/4t~ (4.7)

In ad dition to th e relat ions (3.9) it can be verified th at , in th is special case of equa lrigidities, E exceeds cl E, + cBE,.(ii) Derivation

The solut ion just out lined depen ds on th at for th e following au xiliar y problem?.*Prompted by the treatment of E~HELBY (1967, 1961) for a single misfltting inclusion in an inI3nite matrix of the

same material, when the misfit is equivalent to 8 uniform expansion. No dilatation is induced in the matrix, whosebulk modulus is consequently without inIl.uence and can be varied at will. It is because of such a propert y that thepresent analysis succeeds.


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Elastic properties of reinforced solids : Some thecretical piinciples 367Ima gine th at one phase (say the first) ha s uniform ma ss-density while th e oth erha s none. Let 4 be th e gravitational potent ial in an ar bitrar y volume of mixtureun der boun dar y conditions to be stipulat ed later ; in general these call for thepresence of additiona l external ma tt er, but t his is of no concern. In suitableun its, an d with Vs denoting th e Laplacian ,

- 1 in the first phasev= $ =

I(4.8)0 in the second phase

while across inter faces th e potent ial an d its vector gradient ar e required to becont inu ous. Ther e ar e, however, discont inu ities in th e second derivat ives :

wher e x1 (i = 1, 2, 3) ar e Cartesian co-ordina tes, 111 s th e local unit normal at th econs idered point of an int erface [for a proof see HILL (1961), Section 21. Thesejumps, un avoidable a ccompa niment s of (4.8), ar e pivota l in th e ensuing ana lysis.

Now let a cont inuous an d irrotational displacement uf be defined in the con-sidered volume by

3&

where 7 is an infinitesimal const an t par am eter. The first right-han d term re-present s a displacement proport iona l to th e gravitational field, while th e lastterm represen ts a un iform expansion. The resulting stra in tens or l1 is giveneverywhere by

(4.11)

where &J is th e usu al Kronecker delta. The str ain is discont inuous across pha seboundaries, by (4.9), and in particular the dilatational part is piecewise constant,and respectively

01 = (3 + + P)7/b 0, = (Q + + /)7/P,in each pha se. The avera ge dilatat ion in th e mixture is th erefore

(4.12)

e = cl 0, +- 4 B, = (cl K2 + c2 Kr + + p) 7/p, (4.13)an d th e concentra tion factors ar e as sta ted in (4.5) and (4.6). Conversely, when theoverall dilata tion is assigned, the par am eter 7 is determ ined by (4.13). In fact,when the normal component of displacement is prescribed a rbitra rily over thesu rface of th e considered volume, hen ce fixing 8, th is relat ion ens ur es th e existenceof a solut ion to th e Neu ma nn problem set by (4.8) with boun dar y values obta inedfrom (4.10) for th e norm al der ivative of potent ial.

The s tr ess tens or a(3 corr esponding to str ain (4.11) is obtained asb2 4- + 8gg in phase 1--

7K2 (K1 +- +p) &, /2 (K1 .._ ,,(* ) _

P 1 yq (4.14)--3xt 3q in phase 2.


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368 R. HILLWith the help of (4.9) the stress jump at an interface is accordingly

27 (K1 - 4 (& - -6 4) (4.15)in crossing from the second phase to the first. Clearly, however, the interfacialtr action is itself cont inu ous , since (603 - nr n f) n j = o or by recognizing thelast bracket in (4.15) as a unit biaxial tens ion in th e local t an gent plane to theinterface. Moreover, the distribution (4.14) is self-equilibrated th oughout th emixture, since ~U&XZ~= 0 by (4.8).

It follows th at we have obtained th e actua l displacement field in an y volumeof mixture under surface values of type (4.10) with arbitrary normal component.This family of sohr tions has th e distinctive property th at t he rat io of averagehydrostat ic str ess to average dilatat ion is always (cl K~8, + cp as 8s)/(c1 8, -+ cs 8,),where 8, an d 8, ar e th e un iform dilat at iona l par ts (4.12) of the phase strains.This is precisely t he modulus sta ted in (-1.1). It remains to establish that it isactua lly th e conventiona l bulk modulus when, in par ticular , th e mixture ha s cubicsymm etr y or is isotr opic. We have to verify th at t he fam ily cont ain s a solut ioncorr esponding to a ma croscopically un iform pure dilat at ion in a represent at ivevolum e of su ch a mixtu re. To th is end it is enough t o pres cribe a norm al com-ponent of su rface displacemen t equa lling t ha t in a un iform expan sion g; orequivalent ly to pres cribe a norm al der ivative of potent ial as in a cent ra l field- c1 43 (like th at in a solid spher e with a ma ss density of c1 un its). For a mixturewith the sta ted properties it is apparent tha t the solution of this Neuman n problemwill fluctu at e about th e centr osymmetr ic potent ial as its mean with a wavelengthof th e order of th e dista nce between inclusions. Corr espondingly, th e local~elasticstrains will fluctuate similarly about a pure un iform expan sion. The rema rks inSection z (ii) complet e th e ar gumen t.

Fina lly, it is noted th at th e potent ial would be #, where 4 + # = - @-22j6,if all the mass were assigned to th e second pha se inst ead of th e first. The endresult s ar e nat ur ally un alter ed, while in th e form ulae for displacement an d str ain4 is repla ced by II,and the numerical subscripts interchanged.

5. BOUNDS ON BULK MODULW FOR ARBITRARY MIXTURE(i) Derivation

Suppose th at t he mixture is isotr opic or has cubic symmet ry, th e geometr ybeing other wise a rbitr ar y. It is intu itively evident, an d is proved in (iii) below,th at th e overall bulk m odulus would be increased (decrea sed) if both pha ses ha da comm on rigidity modulus equal to the lar ger (sma ller) or their actua l values.Consequent ly, th e actua l bulk modulus mu st lie between th ose of two similarlyproportioned mixtures, in one of which th e phases ha ve the sam e rigidity pFL1an d in th e oth er t he sam e rigidity ps (th e separ at e bulk moduli being fixed).The signi~~anee of th is obse~ation is th at th e exact solution ha s just been foun dwhen the phases have equal rigidities. In fact, imm ediat ely from (4.1), we nowhave

4Pz KV + 3% % +& _If_ < 411% V + 3% K24fJzKR +3QQj KR 4P1 KR + 3x1 % (5.1)


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Ela st ic pr operties of reinforced solids : Some t heoretical principleswith the phases num bered so tha t p1 > ps.

The variant obtained from (4.4) is

369

Cl K - K2 Cl1 + (4 - x2)Cd(K2 $ Et)


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370 Ft. HILLFor a mixture consisting entirely of spherical composite element s with th e appr o-priat e moduli a nd concentra tions, an d with sizes ra nging from finite to vanishinglysmall such that the whole space is filled.(iii) An auxiliary theorem

We supply now th e proof th at t he overa ll m oduli a re increas ed when either orboth of th e bulk a nd rigidity moduli ar e increa sed in one or both ph as es. Accord-ing to th e rema rk s in Section 3 (i) th is follows, for a n isotropic mixtu re, if th e tota lelast ic ener gy increas es un der fixed overa ll st ra .in (or decreases u nder fixed overallst res s). We prove, even more gener ally, for any heterogeneous elastic continuum,that the energy increases under fixed surface constraints when the material is strengthened in an y (non-un iform) way. Neith er t he th eorem nor its proofis app ar ent ly t,o be foun d in th e textbooks.

By strengthened is mean t t ha t, in each cha nged element of ma terial, th eenergy density afterwards (U) would exceed that beforehand (U) if the localstr ain were th e sam e. But it is not the same. We th erefore appea l first to th eminimium energy principle applied t o th e original body, where th e actu a.1 str ainis E, ta king for compar ison th e actua l field L in th e cha nged body. Then , in ter msof average densities, the principle yields

0 (C) < 0 (P).By hypoth esis, for a ny st ra in C an d at each point of

U (E) < U (d).Whence

the body,

an d the desired result follows by compa ring th e first and last members.

6. FI NAL REMARKSThere rema ins th e difficult problem of boun ding th e sheur modulus. The idea

behind th e met hod leading to (5.1),hile valid her e also, is not yet pr acticablesince the formula analogous to (4.1)s not known when only the shear modulidiffer. In an y event it would probably depen d on geomet ry an d not on concent ra -tion alone.

Boun ds ha ve, h owever, been pr oposed by H ASHIN an d SH TRIKMAN 1963)via th eir varia tion prin ciple, as follows :

wheii p1 - pz an d K1 - KS ar e both positive, an d with inequa lities revers ed (orsubscripts interchan ged) when th ey a re both negative, th e /Is being defined as in(2.2). When th e differences of th e moduli h ave opposite signs th e met hod fails.Notice th at t he houn ds each involve only cne or other bulk modulu s. However,they coincide for a rbitra ry concentra tions, not when the bulk moduli a re equal,but when p1 (1 - ,&)/rS, = p8 (1 - &)/Bs. This is a complicat ed rela tionsh ipa,nd th e a na logues of (5.1) nd (5.2) are not as simple.


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Elastic properties of reinforced solids : Some theoretical principles 371Xn Section 5 (ii) a conn exion was esta blished between each boun d on th e overall

bulk modulus of a mixture an d th e exact formu la for the appa rent bulk modulusof a sph erical composite element . Ther e is a similar th ough less specific eonn exionin respect of th e shear modulus, which helps towards clarifying th e sta tu s of (6.1).Thus, HASHIN (1960, 1962) obta ined exact s olut ions for th e int ern al fields in asph erical composite element u nder two types of su rfsce loadin g : (a) where thetr action is prescribed an d (b) where the displacement is prescribed, both distribu-tions being of th e kind tha t would be complement ar y an d produce un iform shear ingof a ~o?7logel7eous pher e. In th e composite element , h owever, th ese loadin gs a renot complementary : the respective internal fields are distinct and not macroscopi-ta lly u niform (by cont ra st with what t he situ at icn would be for representa fivevolume of mixture similarly loaded). Consequent ly, th e appa rent shea r moduliof th e element, which m ay h ere be calculat ed from t he appr opriat e componentof average shear strain in (a) and of average shear stress in (b), are themselvesdistinct, with modulus (b) exceeding (a).

Ha shin ma kes the empirical observation th at t he right (left) side of (6.1)falls between the apparent moduli (a) and (b) when the shell of the compositeelement is ma de from th e first (second) const ituent . He indicat es, furt herm ore,how the sta ndar d extremum principles can be used to show that the intervalbetween (a) an d (b) also cont ain s th e (un kn own) shea r modulu s of th e corr espondingspecial distr ibution of composite element s described in Section 5 (ii). It rema insan open question wheth er this shea r modulus is actua lly given by th e appr opriat eside of (6.1).

ACKNOWLEDGMENTThis paper is essentiall y the major part of B.I.S.R.A. Report P/19/62. I am greatly indebtedto the Director of the British Iron and Steel Research Association for a grant and for permissionto publish.

REFEILENCESCROSSLEY, A.E&RELBY,J.D.

HASHIN, 2.

HASEIIN,~. andSHTBIKMAN,S.

HILL, R.

1963 M.Sc.Thesis, University of Nottingham.1957 PTOC. Rq& &X, A 241,376.1961 Progress @a So&id ~ec~a~i~ Vol. 2, . 89 (Noah-Holland1960 Harvard University, Div. of Eng. Tech. Rep. 9.1962 Trans. A.S.M.E. (3. Appl, Me&.) 29, 143.1963 Appt. Mech. Rev., in press.1962196319521961

PAUL,B.REINER,M.

196319601958

Publishing Co.).

J. Me&. Phys. Solids 10, 335.Ibid. 11, 127.Proc. Phys. Sot. A 65, 349.Progress in Solid Mechanics Vol. 2, p. 247 (North-Holland

Publishing Co.).ST& frost Steel Z&X. Ass-n., Rep. P/19/62.W. Prayer 60th Anniversary Volume (MacMillan & Co.).Trans. A.I.M.E. 218, 36.Encyclopedia of Physics Vol. B, p. 522 Springer Verlag).

APPENDIX : SPHERICAL COMPOSITE ELEMENT : BULK MODUL~JSWe are oncerned here with a spherical inclusion of the first constituent cemented in a matching

spherical shell of the second constituent. A uniform pressure P applied to the external surface


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37 2 R . HILLof radius R produces an (in~nitesim~) inward d~pla~ment U. The fractional decrease in totalvolum e is 3U/R, while th e aver age str ain is easily shown to be an a&r ound compr ession of am ountU/R [cf. th e first footnote in Section 3 (i)]. The appa ren t bulk m odulus of th e composite element is

K = PRIBU (1)either directly from the ra tio of press ur e to fractional volum e chan ge, or by equa ting th e workdone by the pressur e to the total strain energy expressed in terms of aveiage stra in[cf. Section S(i)!.

The sotu tion is easily const ru cted as a combina tion of element ar y, spher ically-symmetr ic,elastic fields (which are given in any text on strength of mat erials). The Anal result is tha t t hedisplacement (which is radially inaa rd) has magnitu de u at distan ce r from th e centr e, where

1(us + Q ps) r/R . . . . . . . . . . . . . . . . (inclusion)

ka /U = @I(4 + * k ) r/R - (9 -- K~) l R 2jr z . . . . . . . sheI1)

In the she11 he deforma tion is a uniform compression together with a non-uniform strain involvingno chan ge of volnm e. The inclusion is simply compress ed un iformly an d so its shea r modulusdoes not ent er. Consequen tly, th e an alysis of Section 4 could h ave been applied h ere ; thecorr esponden ces with (4.10), (4.12) and (4.13) ar e appa ren t.

The compressive radial and hoop stresses, magnitu des p and q respectively, ar e given bykpR/3U =

1K~ us + + ps) . . . . . . . . . . . . . . (inclusion)

(3)

~Ri3~ = l9 (K~ +-4 pz) , . . . . . . . . . . . . . (incl~ien)

(4)It nil1 be seen tha t both the displacement and ra dial stress are indeed continu ous across theinterface, in conformity with the stipulated bonding ther e. The jum p in hoop stress is, of cour se,adm issible a nd inevitable.

The relation bet ween surface pressure and displacement is obtained on putt ing r = R in (3) :P R 4 1s C l q + 2 4 + K1 9_^ =_3u Cl Ka + c2 Kl + +r2

vlitb (1) this supplies a n express ion for th e overa ll bulb modulus th at is str uctur ally an alogousto th e bounds (5.1)

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Elastic Properties of Reinforced Solids_some Theoretical Principles