R Hill

A SELF-CONSISTENT MECHANICS OF COMPOSITE MATERIALS

THE MACROSCOPIC elastic moduli of two-phase composites arc estimated by a method that takes account of the inhomogeneity of stress and strain in a way similar to the Hershey-Kr<>ncr theory of crystalline aggregates. The phases may be arbitrarily acolotropic and in any concentrations, but arc required to have the eharactcr of a matrix and effectively ellipsoidal inclusions. Detailed results arc given for an isotropic dispersion of spheres.

I.

IxTRODUCTION

PttEDll'TIOKs of macroscopic properties of two-phase solid composites have mostly been restricted to stating universal bounds on various overall elastic moduli (IlAsmN 196-t;Hl65;[START_REF] Hill | a 1965b 1958 1956[END_REF]. Such hounds depend only on the relative volumes and do not reflect any particular geometry, except when one phase consists of continuous aligned fibres (IIASllIN and RosEx 196-�; HILL lfl(}.q. Howenr, when one phase is a dispersion of ellipsoidal inclusions, not necessarily dilute, a much more direct approach is availablet. This is the ' self-consistent method. ' of HERSHEY (1954) and KRONER (rn.58), originally proposed for aggregates of crystals. In that connexion it has recently been reviewed and elaborated by the writer (1965a).

The method draws on the familiar solution to an auxiliary elastic problem, namely a uniformly loaded infi nite mass containing an ellipsoidal inhomogeneity. In applying this solution the properties and orientation of a typical crystal are assigned to the inclusion, and the macroscopic properties of the polycrystal to the matrix. For self-consist.ency the orientation average of the inclusion stress or strain is set equal to the overall stress or strain. The result is an implicit tensor formula for the macroscopic moduli.

The analysis for the composite proceeds in similar spirit but necessarily differs in an important respect : only the particulate phase can reasonably be treated on this footing. However, as is well known (op. cit. 1963; § 2 (iii)), a knowledge of average stress or strain in this one phase suffices to determine the o\•erall properties when the matrix is homogeneous. As a matter of fact, notwithstanding this difference in viewpoint, the entire analysis is found to remain structurally close to that for a crystal aggregate (as given in op. cit. l965a, § § 3 and 4 ).

tNo mention of it in this context has been traced in the literature. But Professor B. Budiansky recently informe<l me that he tried the approach in 1961; his conclusions appenr elsewhere in this issue of the Journal. My own investiga� tion dates from March 1962, when preliminary results were given in a letter to Dr. J. D. Eshelby. 1

SYMBOLIC NOTATION

For brevity cartesian tensors of second order are denoted simply by their kernel letter, u say, set in lower case bold face as if for a vector. Correspondingly, their tensor components are considered to be arranged in some definite sequence as a 9 X I column. Tensors of fourth order are denoted by an ordinary capital, A say, a nd are regarded as 9 X 9 matrices. More precisely, the leading pair of indices is set in correspondenc e with rows, and the terminal pair with columns (both in the chosen sequence), so that the second-order inner product of tensors A and u can be written as the matrix product Au. Similarly, AB can stand for the fourth-order inner product of A and B.

We shall only be concerned with fourth-order tensors that are symmetric with respect to interchange of the leading pair of indices and also of the terminal pair. The representative matrices are consequently singular, with rank � 6. Neverthe less, equations of type u = Av are compatible when u and v are any symmetric second-order tensors and matrix A has rank 6. In this sense we can define a unique inverse A-1 as the solution of

AA-1 = 1 or A-1 A = I
where I is the suitably symmetric ' unit ' tensor formed with the Kronecker delta. One can then verify that

A-1 u = A-1 Av= Iv= v
as required, for any A, u and v with the stated properties.

THE At'XILIARY PROBLEM

A single inclusion, arbitrarily ellipsoidal in shape, is imagined to be embedded in a homogeneous mass of some different material. The tensors of elastic moduli. not necessarily isotropic, are denoted by £ 1 and L, respectively, and their inverse compliances by JI 1 and M. In addition to the symmetries mentioned already in § 2, the representative matrices have full diagonal symmetry so that all cross moduli and compliances are pairwise equal. The displacement at infinity is prescribed to correspond to a uniform overall strain i. Across the phase interface both displacement and traction are required to be continuous. The solution, certainly unique when the tensors of moduli arc positive definite, has the character of a uniform field locally perturbed in the neighbourhood of the inclusion. In particular the overall average, or macroscopic. stress a is equal to Li, since the contribution from the inclusion is vanishingly small; furthermore, a and i are also the local field values at infinity. The principal feature of the solution is that the inclusion is strained uniformly, though not necessarily coaxially (EsHELBY 1957;1961 ).

This property prompts the introduction of an ' overall constraint ' tensor L* for the L phase. with im•erse 111*, in respect of loading over the interface by any distribution of traction-rate compatible with a uniform fi eld of stress, a* say.

That is, if E* is the accompanying uniform strain of the ellipsoid, a* = -L* E*, E* = -M* a*.

(1)

The corresponding matrices naturally have diagonal symmetry, as may be shown by Betti's reciprocal theorem, and are functions of L or M and the aspect ratios of the ellipsoid. Once L* and M* have been determined, the solution of the auxiliary problem follows by superimposing the uniform fi elds a and i, and identifying a* with a1 -a and E* with E 1 -i where a1 and E1 are the actual fields in the inclusion. Then { 2) and so which furnish the required stress and strain in the inclusion in terms of the macro scopic quantities (HERSHEY 1954 ).

In an alternative approach (EsHELBY 1957), seemingly adopted by all later writers, attention is focussed first on a certain transformation problem for an infinite homogeneous elastic continuum with stiffness tensor L. In this, an ellipsoi dal region would undergo a transformation strain e if free, but attains only the strain Se in situ. The components of tensor S, being dimensionless, are functions of the moduli ratios and of the aspect ratios of the ellipsoid and its orientation in the frame of reference. When L is isotropic, explicit formulae for the components on the principal axes have been given by Eshelby (op. cit.). When Li s orthotropic and the transformed region is an elliptic cylinder whose axes coincide with the material axes, explicit formulae ha,•e been given by BHARGAVA and RADHAKRISHNA (1964); when L has cubic symmetry equivalent results have also been given by WILLIS (1964).

The

T = L* (L* + L)-1 = (M* + M)-1 M. } } (5) (6)
The significance of T is that the stress a* in the transformed region can be written as Ts, where s is the stress that would remo\•e the strain e. Separate symbols P and Q ha,•e heen introduced for the products in (5) since these appear frequently hcrcnftcr. \Ve note the further connexions and p-1 = L* TL, Q-1 = JI* + JI .

} (i)
PL+ JIQ �I,

P = JI (I -'l'), <J =c L (I --S),
.

From the latter pair one secs that matrices P and Q ha,•c the diagonal symmetry stipulated for the moduli and compliances (while S and T generally do not). This can of course also he established purely within the context of the transformation problem by means of Betti•s reciprocal theorem. The interpretation of Q is that an ellipsoidal cavity in a medium under stress QE at infinity would deform by amount E: a dual intrrprctation ma�• he gin•n for I'.

•L

SEJ.F-COXSISTE�T TllEOltY

We consider statistically homogeneous dispersions in which the inclusions can he treated, on aYerage, either as variously-sized spheres or as similar ellipsoids with corresponding axes alignedt.

Each phase may he arbitrarily anisotropic but is assumed homogeneous in situ. Consequently, in a common frame of reference, every tensor in the generic auxiliary problem has the same components for all inclusions.

Let the respecti,•e phase properties be distinguished by subscripts 1 and 2, and let c 1 and c 2 be the fractional concentrations by ,•olumc, such that c1 + c 2 = 1. The elementary rdations between the phase an(l on-rail averages of stress and strain arc c1 (0-1 -a) + c2 (a2 -a-) = o.

c1 (e1 -e) + c 2 (e2 -e) = o.

} (8 )

These incidentally imply the qrnishing of the a ,•crages of the ' polarization ' stress or strain :

Cl (cr1 -/,'i.1) + C2 (a2 -li.2) = 0, c1 (i.1 -Jla-1) + c2 (i.2 -JJ0-2) = 0,
since a = Li. and e =-Jiu.

}

Now, according to the basic postulate of the self-eonsistcnt method.

U1cr = L* (e -E1), from the leading equation (2). It follows antomaticall�• from (8) that

i1 2 -a = L* ('i. -€2), ( 9
) (10) (11)
and vice i•ersa. Tims. right at the outset, it is evident that both phases will enter subsequent formulae on the same footing. HoweYer. this does not imply• that the tFibns of elliptic section rnuy be t•nvisaged as n limiting t•ao;; e in whiC'h one principal axis beC'Offi{'S infinite. :\ direct analysis is given elsewlwre (IIJI.L 1H65b).

matrix phase also is treated as particulate in the theory, through a kind of con ceptual fragmentation. It simply means that the same overall moduli are predicted for another composite in which the roles of the phases are reversed : that is, where the first phase forms a coherent matrix and the second phase is distributed as inclusions shaped and oriented as before, both in their original concentrations.

It is also obvious that either of (8) would imply the 3thcr, and then (9), if both (Hl) and ( 11) were postulated. This, indeed. is the standpoint in the polycrystal theory, where an equation corresponding to (2) is assumed for grains of all orienta tions. But, as already remarked, such an a priori standpoint for a dispersion would seem unconYincing.

Equations (10) and ( 11), which may as well now be takm together, can he re-arranged as or dually as } (12) as in (3). Combining these with (8) yields a pair of equivalent formulae for the overall stiffness and compliance tensors L and JJ :

C i (L* + L i )-1 + C2 (L* + L2)-i = (L* + L)-i = P, 7"-Q . } (13) C i (111 * + lll i )-1 + C2 (111 * + 111 2)-i =(JI* + M)-i .
Since the constraint tensor L* and its inverse JI* are themselves functions of L and 111, these formulae are actually quite complex. Variants obtainable with the help of the last pair in (7) are

Ci [ (L 1 -L)-1

+ P]-i + C2 [ (L2 -L)-i + PJ-1 = 0, } (14) C1 [(Mi -M)-1 + Q]-i + C2 [ Oll2 -Jl)-i + Q]-i = 0,
which are essentially in the form ( 9). An inversion immediately produces Ci (L -L2)-i + C2 (L --L 1 )-i = P. Ci (11 1 -JJ2)-1 + C2 (M -JJ i )-i = Q, } (15) which seem to be the simplest obtainable, superficially at least.

Finally, we can read off from ( 12) the phase 'concentration-factor' tensors,

A 1 and A2 for strain, B1 and B2 for stress, which arc defined by Thus:

A i -i = P(L* +L i ) =I+ P(L 1 -L), A2-1 = P (L* + L2) =I + P (L2 -L), B 1 -1 = Q (M* + M 1) =I+ Q (M 1 -M), B2-1 = Q (JI* + M2) =I + Q (M2 -M).
Equations ( 13) are of course an expression of the basic connexions When the dispersion is dilute, with c 1 small, (14) reduces to L -L2 '.::::'. c1 ( L 1 -La) [I + P2 (L1 -L2) ]-1, M -M2 '.'.:::'. c1 (M1 -M2) (I+ Q2 (M1 -M2}]-1, } (16) correct to first order. These can alternatively be obtained (HILL 1962, § 7) by substituting the zeroth order approximation for the concentration factors in which are exact relations.

ISOTROPIC DISPERSION OF SPHERES

Suppose that the inclusions are spheres distributed in any way such that the composite is statistically isotropic overall. The first equation ( 15) then reduces to a pair of scalar formulae for the bulk and shear moduli, K and p. :

where

C1 C2 � ----+--=-, /L -1'-2 /L -1'-1 /L OC = 3 -5� = K/( K + t p.). ( 17
) (18) (19)
The dimensionless quantities oc and � are those that appear in the specific form of Eshelby's S tensor in the auxiliary problem for a sphere (cf. HrLL 1965a, § 4 (ii)): StJkl =0 t (oc -�) '011 'Ok1 + t� ('Otk '011 + 'Ou '01k).

After substituting for oc, (17) can be solved for K parametrically in terms of p., for instance in the form (20) It is noteworthy that this is identical with the known exact solution for composites with arbitrary geometry, when the phases have equal shear moduli (HILL 1963, § 4;1964, § 6), and also with the solution for a spherical composite element whose shell has rigidity p.. To discuss (18) in general terms one may retain �as a parameter in view of its restricted range, namely i � � < � when K, p. � 0.

Then, clearing fractions,

(1 -�) p. 2 + { � (.u1 + f.L2) -(c1 f.L1 + C2 f.L2)} f.L -� /L1 f.L2 = O .
The left side is found to be positive or negative respectively when p. is put equal in turn to the so-called Voigt and Reuss estimates :

Consequently, the required root lies between these limits. It follows that ic is certainly in the interval obtained by substituting µR and µ v in the monotonic relation (20), and hence a fortiori between the rigorous best-possible bounds for arbitrary geometry, which are known to correspond to µ1 and µ2 in (20) (Hn.L 196 3 , § 5). These are further satisfactory features of the theory.

To derive the explicit equation for µ in its most convenient form, however, we express both sides of the first of ( 19) in terms of µ with the help of (18) and (20).

The result is

( Cl IC ! + C 2 IC : -) + 5 ( � + C 2 f-' 1 ) + 2 = 0. (21) /Cl + 3 f-L K2 + 3 f-L f-L -f-' 2 f-L -f-Lt
[ This could be multiplied out as a quartic but is far better left as it stands for iterative or graphical solution, by tabulating c1 or c2 as a function of µ between µ1 and µ2). As µ increases from 0 to oo, the first bracketed function decreases monotonically to zero from

1 if K1 K2 ¥ 0, from c 1 if K2 = 0, from c2 if K1 = O,
and vanishes if both K1 and K2 are O. If µ1 µ2 # 0, with µ1 > µ2 say, the second bracketed function decreases monotonically froml tooo in the range (0, µ2); from + oo tooo in (µ2, µ1), with values O and -l at f-L R and µv; and from + oo to O in (µ1, oo ). It is thereby confirmed again, provided neither phase rigidity vanishes, that there is precisely one positive root and that it lies between the Reuss and Voigt estimates.

This root can be stated explicitly when the dispersion is dilute. Thus, if c 1 � I, we find µ :::: µ2 (1 -+ ,\ c1) where 21), correct to zeroth order. That is, (22) which is a special case of (16). This coincides with formulae of OLDROYD (1956, equation (40)) and EsHELBY (1957, § 5; 1961, equation (6.10)).

I f-' 1 -(2 + OC2 ) � = 1-'1 -1-'2 from (
When one phase is vacuous, say K2 and µ2 --+ 0, equation (21) has a positive root when and only when the concentration of this phase is less than f; for instance the root is µ1 (1 -2c2)/(1 -! c?) when K1 --+ oo. On the other hand, when both phases are incompressible ( K1, K2 --+ w ), equation ( 21) can be reduced to

(3 µ -+ 2µ2 ) -(3 µ -+ 2µ1 ) C1 - Cz µ -1-' 2 1-' 1 -µ
which gives either volume fraction explicitly in terms of µ, or to

3 µ 2 + { (2 -5 c 1) µ 1 + (2 -5c2) J-L 2 } µ -2µ1 µ2 = O
giving µ as a function of concentration. In particular, when µ1/µ2 --+ oo, the re levant root of this quadratic behaves asymptotically like

A study of these examples makes it plain that the theory is unreliable under extreme conditions, except when the dispersed phase is .mfficiently dilute. Some such restriction on the range of validity was already to be expected : the general formulae in § 4 do not distinguish between the phases and yet the actual overall properties arc totally different according to which of two disparate materials is the matrix. With this proviso it appears that, in practice, the theory should be useful when rigorous bounds arc eitlwr not known or are too far apart for empirical interpolation. This conclusion, already indicated for the bulk modulus, is reinforced by consideration of the only non-trivial bounds presently a\•ailable for the rigidity modulus. These have been given by HAsm:s and SHT111KMA1' (1 963). It is con venient to re-arrange their formulae so as to isolate the \'olnme fractions :

µ, ' -/L2 [ l + f3 1 (� -l ) l C p 1 /L 1 -/L2 (23) {! " --f1-1 = [ 1 + (32 (��' -1 ) l C2, J /L 2 -/L 1
where µ' is the upper and µ" the lower bound when the materials are numbered so that µ1 > µ2, and where {31 and {32 denote the phase values of f3 in ( 19). The coefficients of the volume fractions arc thC' shear-strain concentration factors in the respective phases. These bounds WC'rc deri\'ed under the apparently essential restriction K1 > K 2 •

We now reeast (18) similarly in thC' altcrnativC' ways

/L -/L1 - - -�•- -- [1 +f3 (; -1 )r 1 c 1 . [ 1 + {3 ( : 2 - 1 ) i-l c2• } (2.t.)
These, of course, are scalar counterparts of the matrix formulae in § .i.. Our plan is to compare the respective \'alues of c1 and c2 defined by ( 23) and ( 24) when µ, ' and µ " are formally set equal to any chosen value of µ, between µ1 and µ2•

That is, we take a horizontal section of the respective (modulus v. concentration) relations, in preference to a \'ertical section which would here be unprofitable.

In preparation we note the identity together with a similar one in the second subscript. Now ( 25) is a monotonicall�• increasing function of both K and µ. Hence the right-hand bracket in the identity is positive when K1 � K (as is the case when K1 � K2, by what was proved before). It follows that each bracketed factor on the left of the identity exceeds the reciprocal of the other, and thus that the value of c 1 defi ned in ( 24) is greater than that defi ned in (23) whenµ.' = µ.. Similarly, the value of c 2 in ( 24) is greater than that in (23) when µ." = µ..

It may be concluded that, when (µ1 -µ2 ) ( K 1 -K 2) > 0, the theoretical rigidity lies between the Hashin-Sktrikman bounds at any concentration. More especially, for a dilute dispersion, ( 22) coincides with the fi rst-order approximation to one of the bounds (namely µ" when c 1 is small). On the other hand, when (µ1 -µ2) (K1 -K2) < 0, nothing can be concluded from the identity by this line of argument, just as the status of the Hashin-Shtrikman expressions themselves is then also undecided. Indeed, their difference is given by Finally, it should be observed that the modulus (25) has a precise mechanical signifi cance. The overall constraint tensor of an isotropic continuum with a spherical cavity is

(1 -/3) L*11k1 = t µ 811 'bk1 + --13µ ('btk 811 + Sa 81k -t 811 'bk1), from the fi rst of (4) and the components of S given in § 5 (cf. Hill 1965a, equation (20)). That is, by the defi nition (1), 2 (1 -/3) -a*tj = t µ £*kk 811 + --µ--µ (E*tJt E*kk 8tJ)• Consequently, a unit fractional increase in radius calls for an internal pressure 4 µ , while a unit shear of the cavity calls for tractions corresponding to an internal fi eld of shear stress 2 µ ( 1 -/3 ) / /3 .

ACKNOWLEDGMENT

This work is part of a programme of research on mechanics of materials which is supported by a grant from the Department of Scientific and Industrial Research.

  general connexion with L* or M* is most easily obtained by imagining the transformation problem soh•ed from the viewpoint of (1 ). That is, we substitute E* =Se, a* = L (E* -e) in a* = -L* E*. Then, since these hold for all e, L* S = L (I -S), (I -S) 111 * =Sill, (4) where I is the unit tensor defined in § 2. These are equivalent formulae for L* or ill* in terms of S. Or they can be put inversely as S = (L* + L)-1 L = M* (21:1* + M)-1 for S in terms of L* or Jll*. Another dimensionless tensor T, the dual of S, could just as well be admitted on this footing. Set so that and M* T =Sill = P, say, TL= L*S = Q, say, M* T = M (I -T), (I -T) L* = TL,

[

  (1 -/31) ( 1 -/32) ] µ2) 2 = [ µl + /31 C1 (µ2 -µ1)] [ µ2 + /32 C2 (µ1 -• µ2)] ' (26)its sign being controlled by a precisely similar factor in the right-hand numerator.


Géraldine Lucchi
email: geraldine.lucchi@inrae.fr

Jean-Luc Le Quéré

Karine Gourrat

Marine Crépin

Comparison of 3 ionization methods -electron ionization, chemical ionization and atmospheric pressure photoionization -for the characterization of volatile organic compounds (VOCs)

oligosaccharide chain GD3 ceramide choline phosphate glycerol PC

Gas chromatography -Mass Spectrometry (GC-MS) is the method of choice to identify and quantify VOCs in food. The main ionization method is the Electron Ionization (EI): high energy exchanges occur, causing reproducible molecular fragmentations. Chemical Ionisation (CI) is another ionization method where a reactive gas (i.e. methane or ammonia) is ionized to form reactant ions. These ions react with the analytes by proton transfer or charge transfer to produce quasi-molecular ions, sometimes accompanied by characteristic adduct ions. Atmospheric Pressure PhotoIonization (APPI) is the most recent source [1]. Emitted photons give rise to an odd-electron radical cation; hydrogen atom abstraction frequently occurs during in-source collisions, and produces a large quantity of protonated molecules, MH + .

These preliminary results allowed to set up optimized parameters to better characterize VOCs by GC-APPI-HRMS. LOD were calculated for different chemical classes and compared to other ionization methods. EI remains the most appropriate one to identify chemical compounds in databases, while CI (CH 4 ) could provide supplementary information for molecular characterization. APPI, a promising technique to resolve coelution problems, has to be improved, especially to reduce in-source fragmentation. The use of a dopant gas (acetone) should be tested in the near future.

LOD comparison according to the ionization method

In our research platform, we recently coupled a GC Trace 1310 to a High Resolution Mass Spectrometer (HRMS) Orbitrap Fusion (ThermoScientific) with the APPI source developed by Mascom (Bremen, Germany). In this work, first, we present a general overview of the technical developments carried out on 13 VOCs with the GC-APPI-HRMS hyphenated technique. Secondly, we compare the ionization methods listed above. For this purpose, we used 6 VOCs of different chemical classes to determine the Limit Of Detection (LOD) for each source.

Methodological development in GC-APPI-HRMS

Methodological development in GC-APPI-HRMS

Figure 1 -MS profile in GC-APPI-HRMS for 3 VOCs.

 Fragmentation occurs in MS mode for several VOCs (Figure 1)

 MS parameters have been optimized (Table I) to limit these fragmentations and to enhance the sensitivity  In-source fragmentation remains important for several chemical classes: in fact, the radical cation of the linear oxygenated aroma compounds is subject to rearrangements that require very little activation energy. Ionization energy of oxygenated molecules is relatively low, then, fragmentations occur.

PTR-MS LOD comparison

[1] Raffaelli A. et al. (2003). Mass Spec. Reviews, 22, 318.

[2] Revel'skii, I. In blue, optimized parameters  Great disparity in sensitivity according to the chemical classes and the ionization method (Table II)  EI and CI (CH 4 ) are the most sensitive ionization methods for the studied acid, ketone and ester  For the alcohol, EI is the best one while CI (CH 4 ) is most appropriate for the aldehyde  Relatively high background noise in CI (NH 3 ) spectra compared to CH 4 is observed, which makes data processing very complex, and consequently, a higher LOD  The studied pyrazine is the better-detected molecular species for the 4 considered methods  LODs in APPI are better than those described in the literature, where the limit of detection in GC-APPI-MS is between 1 and 100 ng/µL [2] Table II -LOD (ng/µL) of different VOCs according to the ionization method.



  Standard solutions in triplicate (0.1 to 200 ng/µL)  4 to 7 points in the compound linearity domain  Selection of 1 to 3 ions for each ionization method  ∑ abundances of these ions for each triplicate and each solution  Average and standard deviation for each solution  LOD calculation  Calculation of minimal detectable concentration

LOD (ng/µL) Compound names Chemical classes EI CI (CH 4 ) CI (NH 3 ) APPI

	2,3-dimethylpyrazine	[M + H] +	Heptan-2-ol	[M + H] +
					2-Methylbutanoic acid	Acids	0.029	0.038	2.706	0.331
					Heptan-2-ol	Alcohols	0.006	0.028	0.719	0.165
					Heptanal	Aldehydes	0.023	0.008	6.872	0.052
					2,3-dimethylpyrazine	Pyrazines	0.002	0.002	0.005	0.004
	Butanoic acid		[M + H] +		Heptan-2-one Isoamyl acetate	Ketones Esters	0.003 0.003	0.004 0.009	0.052 0.064	0.020 0.223
	MS	Optimum
	Detector	Pos
	AGC	2 e 5
	Resolution	15000 (0.006 Da à 89 m/z)
	% RF Lens	20 -40
	MS2
	HCD	20 % (best fragmentation)
	CID	22 %
	AGC	5 e 4
	Resolution	15000
	Sheath gas	2
	Source T°C	150°C

Table I -

 I Optimized parameters for VOCs analysis.