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Bounds of third order for the overall response of nonlinear composites
J. Mech
Pergamon
Phw. Solrds, Vol. 45, No. I. pp. X7- I I I, lYY7 Copyright ‘i: 1996 Elsevier Science Ltd Printed in Great Bntain. All nghts reserved
0022~ 5096/97 $17.00+0.00
PII : SO0225096(96)00069-S
BOUNDS OF THIRD ORDER FOR THE OVERALL RESPONSE OF NONLINEAR COMPOSITES D. R. S. TALBOTt
and J. R. WILLIS::
?School of Mathematical and Information Sciences. Coventry University. Priory Street. Coventry CVI 5FB ; and JDepartment of Applied Mathematics and Theoretical Physics. University of Cambridge, Silver Street. Cambridge CB3 YEW. U.K.
(Rrcrirwi
24 January
1996; in rwisrd
form
I7 Mu! 1996)
ABSTRACT Composites whose response can be described in terms of a convex potential function are discussed. Bounds are constructed for the overall, or effective. potential of the composite. given the individual potentials of its constituents. Steady-state creep is considered explicitly but the results apply equally well to physically nonlinear elasticity, or deformation-theory plasticity, if strain-rate is reinterpreted as infinitesimal strain. Earlier work employed a linear “comparison” medium. This permitted the construction of only one hound -either an upper bound or a lower bound--or even in some cases no bound at all. Use of a nonlinear comparison medium removes this restriction but at the expense of requiring detailed exploration of the properties of the trial fields that are employed. The fields used here-and previously-have the property of “bounded mean oscillation” ; the use of a theorem that applies to such fields permits the construction of the bounds that were previously inaccessible. Illustrative results, which allow for three-point correlations. are presented for an isotropic two-phase composite, each component of which is isotropic, incompressible and conforms to a power-law relation between equivalent stress and equivalent strain-rate. Generalized Hashin-Shtrikman-type bounds follow by allowing the parameter corresponding to the three-point correlations to take its extreme values. Copyright c 1996 Elsevier Science Ltd Keywords: A. creep, B. constitutive ational calculus.
behaviour,
1.
B. inhomogeneous
material.
C. energy methods,
C. vari-
INTRODUCTION
The first bounds for the overall response of composites displaying nonlinear constitutive behaviour, analogous to the Hashin-Shtrikman bounds for linear behaviour. were developed by Talbot and Willis (1985) and Willis (1986), from an idea proposed by Willis (1983). In common with the approach of Hashin and Shtrikman (1963). a “comparison medium” was introduced ; then, some elementary results from convex analysis (Van Tiel, 1984) were exploited to generate, very directly, a generalization to nonlinear material behaviour of the Hashin-Shtrikman variational structure. The formulation starts from a variational principle of “minimum energy” type, and is applicable only to composites whose response permits such a description. Exploitation of the variational structure requires the construction of suitable trial fields from the class of admissible fields. This was accomplished by taking the comparison medium to be linear and homogeneous, and constructing trial fields by employing the Green’s 87
88
D. R. S. TALBOT
and J. R. WILLIS
function for this medium, exactly as in the analysis of composites with linear response. Reasoning like that advanced by Willis (1977, 1981, 1982) then generated a bound for the potential featuring in the variational principle, involving piecewise-constant “polarizations” (defined below), and the two-point geometrical correlation function(s) for the composite. Optimization of the bound required the solution of a set of nonlinear algebraic equations. Subsequently, Ponte Castaiieda (199 1) introduced an alternative formulation, involving a comparison linear composite, in which the bound for the nonlinear problem was expressed in terms of a corresponding bound for the linear composite. For many problems, the approach is very efficient: if a HashinShtrikman bound for the linear composite is employed, it usually provides the same bound for the nonlinear problem as the Talbot-Willis approach, though there are cases in which it fails (Willis, 1991, 1992; Ponte Castafieda and Willis, 1993). When it succeeds, however, there is the possibility of further improvement. by use of a higher-order bound for the linear composite. Talbot and Willis (1992) developed a unified formulation, which contained both the original Talbot-Willis and the Ponte Castaiieda prescriptions as special cases: it permits allowance for statistical correlations involving more than two points and always produces results at least as good as obtained from the original TalbottWillis prescription. There is, however, a limitation inherent to all of the developments described above : restriction to a linear comparison medium (whether or not this is a composite) generally precludes the construction of both upper and lower bounds. The remedy is to employ a nonlinear comparison medium. but this introduces a major hurdle relating to the construction and employment of trial fields. Talbot and Willis (1994) took a first step towards resolving this difficulty, in the context of nonlinear electrostatics, by employing a uniform comparison medium which was linear up to some magnitude of the field and nonlinear thereafter. Then, trial fields were employed which were related just to the linear part of the comparison medium’s response. Such fields have “bounded mean oscillation” (John and Nirenberg, 1961) and a property of such fields was used in an essential way in the construction of the bound. A new bound was obtained but the “penalty” incurred from the introduction of nonlinearity into the comparison medium was sufficiently severe to permit only slight improvement over bounds obtainable by elementary means. Corresponding results for a physicallynonlinear elastic composite were given by Talbot and Willis (1995). The present work employs a “comparison composite” whose behaviour is linear up to some magnitude of the field but nonlinear thereafter, in conjunction with the general formulation of Talbot and Willis (1992). The trial field that is substituted into the resulting functional is one generated from piecewise-constant polarizations in a further underlying medium, which is uniform and linear. In the absence of the nonlinearity, this type of field generates bounds of third order for the response of a linear composite (Milton, 1981 ; Willis, 1982), except that the problem for the linear composite contains a term analogous to thermoelasticity, with arbitrary thermal distortion tensors for each component whose choice is subsequently to be optimized. In the absence of the nonlinear terms, the procedure generates a three-point bound of the type introduced by Beran (1965) ; optimization of the parameters, allowing for the nonlinearity, produces different parameter values but still the bound is sensitive to three-point statistics.
x9
Bounds of third order for nonlinear composites
In the particular case of a two-phase composite with isotropic micro-geometry, the effect of the three-point statistics is felt through one or two parameters (two in the case of compressible elasticity ; one for incompressible behaviour or electrostatics). As shown by Milton (1981), these parameters must lie in certain ranges (they are normalized, in fact, so that they take values between 0 and 1). If the three-point statistics are unknown, bounds valid for all three-point statistics are obtained by allowing the parameters to take their “worst” values. For linear media, this produces two-point bounds that coincide with the HashinShtrikman bounds (always for incompressible elasticity or electrostatics; usually for compressible elasticity). In the nonlinear case, however, the procedure described delivers a significantly better twopoint bound than that obtained through the procedure of Talbot and Willis (1994. 1995). The reason is given in the body of the paper. The plan of the paper is as follows. First, in Section 2. both “primal” and “dual” formulations of the problem are presented, the general approach introduced by Talbot and Willis (1992) is summarized and its relation to earlier formulations is briefly outlined. In conjunction with a nonlinear “comparison composite” and a trial field of the type described above, it generates the new bounds. From Section 3 onwards, to avoid complicating the exposition, the general approach is demonstrated through consideration of a particular problem : a two-phase composite, each component of which is isotropic and incompressible, and subject to power-law creep. The general formulae are given in Section 3, and numerical results are presented in Section 4. Mathematical technicalities, relating to the property of bounded mean oscillation of the trial fields (John and Nirenberg, 1961 ; Stein, 1967) are recorded in the Appendix.
2. The constitutive in the form
response
d
FORMULATION
of the composites
=
F’(e;x),
or
treated
in this work can be represented
0,, = $e:x). i,
(2.1)
Here, c denotes stress, with components (T,,. In the context of steady creep, e denotes strain-rate, with components (u,,+zQ/2 where ui is the i-component of the velocity u. and F(e ; x) is the stress potential ; it is a convex function of e. Equivalently. e = V(a; where U(a ; x) is the strain-rate
x),
potential.
or
c#, = ;;(C:x), r, It is the Legendre
(2.2) transform
(2.3)
U(b ; x) = F*(o : x) = s!p [fr,,~,, - F(e ; x)] and is a convex function
of 6. Since F(e ; x) is a convex function
F(e; x) = U*(e ; x) = stp [o,,e,, - u(a ; x)]
of F(e ; x)
of e, (2.4)
90
D. R. S. TALBOT
and J. R. WILLIS
and the suprema in (2.3) and (2.4) are attained when o and e are related by (2.1) ; then, (2.2) is also satisfied. [The differentiability of the potentials implicitly assumed here is relaxed to subdifferentiability in the proofs given by Van Tie1 (1984).] Physically nonlinear small-strain elasticity, or deformation-theory plasticity, is incorporated in (2.1))(2.4), simply by interpreting u(x) as displacement rather than velocity, so that e becomes infinitesimal strain. For a composite, the potential F(e; x) takes the form F,(e) when x lies in material oftyper(r= 1,2 ,... n).Then,
where xr is the characteristic function of the region occupied by material of type r, taking the value 1 in that region and zero elsewhere. The strain-rate potential is similarly decomposed in terms of potentials U,(a) = F,*(a). Perfect bonding (implying continuity of velocity and traction) across interfaces is assumed. It is envisaged that the composite has a fine microstructure (relative to macroscopic dimensions) so that, on the macroscale, it appears homogeneous, with “overall” or “effective” stress potential &) and strain-rate potential o(5). The detailed strainrate and stress fields fluctuate rapidly about mean values e and 8. It is convenient to consider a unit cube COof material and to construct from it a composite of infinite extent by extending the characteristic functions xI so that they are periodic of period C,,. Then, @) is characterized as the mean value of F(e ; x) over CO,when all field equations are satisfied and e(x) is periodic with mean value E. Thus, Fis characterized by the variational principle &!) = inf
F(E+e; x) dx, I (‘0
(2.6)
where the infimum is taken over strain-rate fields e(x) related to velocity fields a(x) which are periodic with period CO(so that &has zero mean value). Dually, the strainrate potential is characterized by O(a) = inf
U@+B;x)dx, s cl1
(2.7)
where the infimum is taken over self-equilibriated stress fields a(x) which are C,periodic and have mean value zero. The overall potentials are convex (Ponte Castaiieda and Willis, 1988). Exact conditions under which the definitions (2.6) and (2.7) generate a Legendre transform pair were discussed, for a closely-related problem, by Toland and Willis (1989). The definitions (2.6) and (2.7) immediately induce bounds for F”and 0: F(;(e),<
I++G;x), s C‘O
for any admissible Z and 8. Furthermore, bound for F, and conversely
O(a) <
U@+&;x)dx s C‘U
(2.8)
an upper bound Us for 0 induces a lower
Bounds
of third order for nonlinear
u”(6)
<
U,(ii)
F(E)
d
F&5)
=S U,*(E)
91
composites d F(E).
=a F&T?)
d
(2.9)
O(C).
directly from the definition (2.4) or (2.5) of the Legendre transform. The difficulty is the choice of suitable fields @and 6. The simplest choices, C = 0 and 8 = 0, generate the elementary bounds O”(E) < F(:(e, < F(e). P(C)
(2. IO)
< u”(C) < U(b).
where
I+)
=
s
F(E ; x) dx, (‘0
c/(5 ; x) dx.
U(C) =
(2.11)
i (0
The bound P(E) is called the Taylor bound, after Taylor (1938). who proposed e(x) = C as an approximation. In the context of linear elasticity. the same proposal was made by Voigt (1889). The elementary bound U(C) is called the Sachs bound, after Sachs (1928), who proposed a(x) = t7 as an approximation. Reuss (1929) made the same proposal in the context of linear elasticity. 2.1.
Improved
Following define
bounds : primal problem
Talbot
and Willis (1992), introduce
(F-F,)*(r;x)
a comparison
potential
= sup[s,,e,,-(F-F,,)(e;x)]. r
F,,(e; x) and
(2.12)
Then.
F(e;x) >, z,,e,,+F,,(e;x)-(F-F,,)*(r;x)
(2.13)
for every e and r and hence, from (2.6)
09
3
I
inf ~,,~[r:(e+c)+f;(B+C;~)-(F-F,,)*(~:~)ld~
for any chosen field of “polarizations” r(x)_ The infimum is taken over set of admissible fields &. It is sensitive to the choice of polarizations comparison potential FO. One interesting choice of F,, corresponds to with exactly the same microgeometry as the given composite, but with that are linearly viscous (or elastic) : F,)(e;x)
= i
$e:L,
:exr(x).
(2.14)
the relevant z and to the a composite constituents
(2.15)
,=I
say, where CL,..r = 1,. n$ are tensors of viscosities (or elastic moduli). the polarization field t(x) is taken to be piecewise constant. so that
If. further.
92
D. R. S. TALBOT
and J. R. WILLIS
z(x) = i
r=l
rrxr(x),
(2.16)
the problem of evaluating the infimum in (2.14) becomes analogous to that of finding the overall thermoelastic response of a linear composite. Bounds of this type were introduced and discussed by Talbot and Willis (I 992). Two cases are worthy of special mention. First, if the potential F. is independent of x, and so corresponds to a uniform comparison medium rather than a comparison composite, the bound (2.14) coincides with that developed originally by Talbot and Willis (1985). If F,(e) for some constant
=+e:L,:e
tensor L,,, the infimum 0)
is attained
(2.17) by the field (2.18)
= -(r,r)(x),
where the linear operator To is related to the Green’s function for the uniform comparison medium. Calculation of the bound then follows exactly the pattern established for linear composites (Willis, 1977, 198 1, 1982), for any given piecewiseconstant polarization field r. Optimization over the parameters rr then reduces to solving a set of algebraic nonlinear equations. A special property of the operator T,, ensures that the resulting bound depends on the microgeometry through points taken two at a time, for the nonlinear problem just as for the linear one. The bound is one of Hashin-Shtrikman type. The other special case is obtained by choosing z = 0. The problem of evaluating the infimum now reduces to that of finding the overall viscosity tensor (or tensor of elasticity) of a linear composite, and (F-&)*(0; x) = - min {F-F,}(x). Then, if any lower bound L, for the linear composite is known, it follows from (2.14) that
F(E) >fE:L,:E+
i c, min {FJe)-ie:L,.:e} I= I ( e
:
(2.19)
)
where c, =
I
L(X) dx
(2.20)
co
is the volume fraction of material of type Y. Bounds of this type were introduced by Ponte Castafieda (1991). If a Hashin-Shtrikman bound L, is employed, optimizing over the parameters L, usually generates the same bound as is produced by the original “nonlinear Hashin-Shtrikman” approach of Talbot and Willis (1985), but it can leave a gap in unfavourable cases (Willis, 1992 ; Ponte Castafieda and Willis, 1993). An upper bound for I@) can be developed similarly. Choosing perhaps some different F,,(e ; x), define (F-F&(r;x) so that
= i~f[7,ie,,-(F--“)(e;x)l,
(2.21)
Bounds
F(e;x)
of third order for nonlinear
d T,,e,,+F,,(e:x)-(F-F,),(t;x)
for any e and T. Then, from the first of equations
F(ce)<
composites
(2.6),
[T::(e+e)+F,(~+~;x)-(F-F,,),(z;x)ldx.
(2.23)
for any z and any admissible &: since the inequalities all run the same way, it is not obligatory to evaluate the infimum over $. In particular, it is possible to employ a trial field of the form (2.18), for some additional tensor of viscosities L,, and new “polarizations” 2,. say :
@W= - i [r,,(Q,.Xl)](x).
(2.24)
,= I
The “original” polarization z can also be taken to be piecewise constant. The quadratic part of the functional (2.23) then provides an upper bound for the energy of a linear composite, with the polarizations z,. playing a role analogous to thermoelastic coupling. Such bounds (optimized with respect to the parameters e,) are of a type first produced by Beran (1965), and formulated more generally by Milton (1981) and Willis (1982). In practical terms, the number of parameters introduced above is excessive : some can be eliminated by performing fairly simple optimizations. However. the concepts will find use in constructions that follow later. 2.2. Improwd bounds : dual problem Similar arguments (2.7). First, introduce
can be developed, based on the dual variational formulation a comparison strain-rate potential U,,(a ; x). and define
(C’- v,,)*(V
; x) = sy h,,~,,- (CT-r/,,)(a: x)1.
(2.25)
It follows that o(a)
2 inf
[q:(a+8)+U,,(c+&;x)-(U-U,,)*(v;x)]dx. i (‘11
(2.26)
where the infimum is taken over self-equilibriated stress fields 6 which are C,,-periodic and have mean value zero. In the particular case of a uniform linear comparison medium, with strain-rate potential U,(a) the infimum
is attained
=;a:M,+,
(2.27)
when S(x) = -(&V)(X).
(2.28)
The operator A0 is also related to the Green’s function for the comparison medium; it is given explicitly later. A prescription of Ponte Castaiieda type follows by choosing q = 0 and U,, to correspond to a linear composite with
94
D. R. S. TALBOT
U,(d,X)
= f i
and J. R. WILLIS
a:M,:axr(x).
(2.29)
7-I
An upper bound
for o(B) follows by defining
(U-U,),(tj;x) The bound
= i~f[riia,i-(U-U,)(b;x)l.
(2.30)
is (2.31)
where B is any admissible
stress field : for instance,
W = - i Lh(it,xr)lW, where A,, is calculated tensor M,.
for a medium
with uniform
viscosity
(2.32) tensor
L, and inverse
2.3. Limitations If the comparison medium is taken to correspond to linear material response (whether uniform or a composite), and any one of the potentials F, grows less rapidly than quadratically as )Ie 11+ GO,then (F-F,)*(r) = co, at least for some x, and (2.14) gives F” > - CC. [Here and in the sequel, I/e I/ = (e : e)“’ = (ej,e;,)“2.] Also, still with a linear comparison medium, if any I;; grows more rapidly than quadratically as 11e 11+ co, then (F-F,),(z) = - co for some x, and (2.23) gives F”< + co. Similarly, if U, is taken as a quadratic function of G and any one of the potentials U, grows less rapidly than quadratically as I/ TV/I + co, then (2.26) gives u” 3 -cc, and if any (/, grows more rapidly than quadratically as 11 D 11+ co, then (2.31) gives 0< +co. The bounds (2.14) and (2.26) require the specification of a linear comparison medium (composite or uniform), because it is necessary to evaluate (or at least to bound) the infimum, and this is only feasible for a linear comparison medium. It may be noted, also, that, if all of the potentials F, are “superquadratic”, then all of the potentials C: are “subquadratic” ; thus, in general, at most one of (2.14), (2.26) provides a finite bound. The formulae (2.23) and (2.31) display similar limitations if they are employed in conjunction with linear comparison media. They do, however, admit the possibility of use with nonlinear comparison media; this is discussed next. 2.4. New upper bounds The trial fields (2.24) and (2.28) are known to produce not only bounds of Hashin Shtrikman type, but also more accurate bounds, of third order, for linear composite media. The approach to be adopted here will be to employ a comparison composite that is linear up to some level of intensity of the field, and nonlinear thereafter if the
Bounds
of third order for nonlinear
behaviour of the actual nonlinear composite bound (2.23) the potential F,, is taken as F,(e;x)
warrants
it. Thus, considering
~e:L,:ex,(x)+N(e)H(lIell-j.).
= i r=
composites
first the
(2.33)
I
The exact form of N(e) should be tailored to the particular application. It is important that it should grow at least as fast as any of the potentials F,- as I/e I/ --+ xc. The Heaviside function If(*) ensures that it “switches on” only at high field values (the parameter i. is still open to choice), so that the comparison medium behaves “almost” like a linear medium, and therefore trial fields of the form (2.24) should be able to capture its response. Substituting the field (2.24) into the inequality (2.23), with F,, given by (2.33). gives
+
(e- i r=
) ( ,= 1 i!+t! /! IS N(E+C)H(l/
[r,(~,xr)l(x) : i L,L(x): I
-(F-F,,),(z;x)
I=
E- i
I
[r,(?,x,)](x)
I
-i)dx.
dx+
(2.34)
C‘,,
It may be noted that all of the terms except the one involving the function N(e) contain one, two, or three factors xI, xY,etc., and so depend on points in the composite taken one, two. or three at a time. The function N(e) can be taken to be zero, if each of the potentials F,.(e) grows no more than quadratically as I/e 11+ x. In such a case, the bound is explicit (even if in need of simplification) and is the required three-point bound. If any of the functions F,(e) grows more rapidly, however, a non-zero function N(e) is necessary, and the last term in (2.34), which may be regarded as a “penalty” incurred from the use of a nonlinear comparison material, has to be estimated. This can. in fact, be done because fields e of the form e(x) = G+&(x),
(2.35)
with & given by (2.24), have the property of “bounded mean oscillation” (John and Nirenberg, 1961 ; Stein, 1967). The relevant theory is summarized in Appendix B. The result is that. if m(s) denotes the measure of the set of points in C,, where 11e /I > .s m(s) = measjx
E C,, : 1)e(x) 11> .sj.
(2.36)
then m(s) d A exp( -KS). where A and CIdepend that, if
on the parameters
from which & is constructed.
N(e) d fi(II e (with A monotone
increasing),
then
II)
(2.37) It follows then
(2.3X)
D. R. S. TALBOT
96
Ne(x))H(Il s c‘o
e II - 4 dx
d
fi(ll
s
“0
=
e(x)
m(s)
and J. R. WILLIS
^
IIMII
e
^
II-4 dx
g(s)ds < A
I.
exp( - cIs)$$)
ds.
(2.39)
sI
The entire bound (2.34) is now explicit, and requires optimization with respect to the parameters contained therein. Similarly, let U,(a;x)
(2.40)
= f: ~a:M,:a~~(x)+N(a)H()Ia/l-~). r=
I
Substituting the trial stress field (2.32) into (2.31) then gives
+ c-- i r=
[&(hxr)l(x> : i M,x,W:
-(u-U,)&;x)
a- i [&(il,x,)1W
*=I
I
1
dx+
I=
s
N@+6)H(l~
a+6
I
1)-i)
dx.
(2.41)
c,1
Again, the field c = z+B has the property of bounded mean oscillation and the nonlinear “penalty” term can be bounded as in (2.39). The penalty term is not needed if each U, grows less than quadratically as //d jl -+ CG,which is equivalent to growth more rapid than quadratic for each F,. Since, by (2.9), an upper bound for Finduces a lower bound for 0 and conversely, the two prescriptions (2.34) and (2.41) provide upper and lower bounds for both potentials, which are sensitive to the three-point statistics of the composite, without restriction on the potentials Fr or U,.
3.
EXAMPLE : AN ISOTROPIC TWO-PHASE
COMPOSITE
The composite that will be discussed has two phases, Y= 1,2, with strain-rate potentials ~. ,!+I I/,(a) = !?!F? 2 (3.1) nfl 0 or ’ where ge denotes the equivalent stress, (3/2)“21(r~’11,with a:i = c,~-+~~,cJ~~.The potentials are thus isotropic and independent of hydrostatic stress, so that each phase is incompressible. The exponent n is taken to be greater than 1. It follows that the overall potential 0 for the composite is a homogeneous function of degree n+ 1 of the mean stress 6 and is independent of hydrostatic mean stress, corresponding to
Bounds
of third order for nonlinear
incompressibility. The microgeometry isotropic, so that 0 will be an isotropic
97
composites
will, in fact, be assumed to be statistically function of 8. It can therefore be written as (3.1)
but here the characterizing stress cr,)is a homogeneous function of degree 0 in CF.and so depends on the pattern of stress, as discussed previously by Dendievel ct (11.( 199 1). for example. There is no need to discuss lower bounds for such a composite. because they are available from the work of Ponte Castarieda (I 992). Attention is here directed towards finding an upper bound, based on (2.41). 3. I.
General
t\zwphase
composite
It is helpful to note that, because the operator A,, is symmetric and has mean value zero, its effect on a polarization field ij is the same as that on fi - 4. For any two-phase composite, now
m,-5 =(it, -fid(x,(x)-c,)
(3.3)
a(x) = -4,(x,
-~.,)(il, -b).
(3.4)
(2.41) therefore
yields
and so
The upper bound c?(a) 6
formula
[(x,1, +(I -x,)v~):(*--,r(x, s (‘0 +;(+A&
-c,)(ir,
-j/Z))
-~.,)(il, :(m+(M,
-42))
-M:)(x,
--(‘I )):
@-Ao(x, -c,)(il, -ildJ
+
s
N(c+3)N(II
(3.5)
t?+6 1)-2)dx.
(0
Here, C,,(a; x) has been decomposed U,(a;x)
into
= U,,(a)x,(x)+
~~o,(~)Xz(X).
(3.6)
with U,,,(o) =~o:M,.:a+N,(a)H(IIaIl-R) Employing
(3.7)
the notation
s
(x, -c,)&(x,
Cl,
-c,)dx
= c,(.lQ,
(3.X)
D. R. S. TALBOT
and J. R. WILLIS
s
(XI -Wb@Ao(x, -c,)dx
=A
(3.9)
(‘I,
and
s
(x, -c,)AoW,
-Md(x,
-c,)Ao(x, -c,)dx
= B>
(3.10)
CO
optimization
of (3.5) with respect to the parameters
O(C) +I%~+c,(L’,
-U,,),,(a-czQ:(ij,
+c,(Uz-U”~)**(~+c,Q:(il, +;(4, -42) :(A+B) The property
ql, q2 gives -ijz))
-iL)>-c,c@:(M,
:(9, -42)+
-Md:Q:(fi,
-42)
N(i?+B)H(II a+6 I/-1) dx. s c.1,
that A, has mean value zero was also employed
(3.11)
here.
3.2. Isotropic two-phase composite The bound (3.11) applies to any two-phase composite. Now it is specialized to the isotropic, incompressible composite, described above. The tensors M, are taken to be isotropic and to correspond to incompressible behaviour, so that they are characterized by shear viscosities nL,, and the “comparison medium” relative to which A0 is calculated, is taken to be isotropic and incompressible, with shear viscosity ,LL”. The operator A0 generates both shear and hydrostatic stress but, since the latter has no influence on the overall constitutive response, it is necessary only to consider that part of A0 which maps the sub-space of tensor fields with zero trace to itself. As shown in Appendix A, this restricted operator, A6 say, is a convolution operator over C,, : A; = 2~,(16’-
E),
(3.12)
where I has components zi,L, = i L6&6,, + 6,,6,L -: 6’ is a scalar convolution kernel with components
operator
with kernel
6,j6k,13
(3.13)
(6(x) - 1) and the operator
E has
ICI”1
(3.14)
2 - --[i5,&&
exp(-i<*x).
The sum in (3.14) extends over all reciprocal lattice sites except the origin; thus, components of < are integer multiples of 27~. It is also demonstrated in Appendix A that, when the microgeometry of the composite is statistically isotropic, the trace-free part of the tensor Q is
Bounds of third order for nonlinear composites
99
Q'=(2poj(3. Furthermore,
the trace-free
(3.15)
part of A is A,
(because (I&-E) is a projection). B is expressible in the form
mkd
cm (-1
=
Finally,
B’ = ;(2/4’[$~
Q,
the trace-free
-c,
(3.16) part of the three-point
++,]I.
tensor
(3.17)
Here, M, is a parameter introduced by Milton (1982) and shown by him to be restricted to the interval [0,11, to avoid crossover of upper and lower bounds for linear composites. In fact, for the corresponding linear problem, the extreme values for m, produce the Hashin-Shtrikman upper and lower bounds. These are known to be attainable and so microgeometries must exist which correspond to any m, in [0, I]. Finally, since hydrostatic mean stress is of no interest, 5 is taken to be trace-free, and, because of statistical isotropy, the “polarizations” 4, are selected so that it, -ir2 With the above notation,
+
s
the bound
= 85/(2/4,).
(3.18)
(3.1 I) becomes
N(ii + i+)H( (15+
ciI/ -
2) dx.
(3.19)
(‘0
where m2 = I --m,. It is noted in passing that the corresponding bound for a linear composite with viscosities ~1, and pL?is contained in (3.19) and is obtained by disregarding all of the terms associated with the nonlinearity. This gives, upon optimizing with respect to the parameter p.
Setting m, = 0 or m, = 1 generates the bounds of Hashin and Shtrikman this composite ; the range [0, l] for this parameter is thus confirmed.
(1963) for
D. R. S. TALBOT and J. R. WILLIS
100
3.3. Further detail
The remaining optimizations are best addressed in the context of the example, to avoid introducing further notation. The general approach will in any case be easy to recognize. The potentials UO,from which U, is composed are selected to be (3.21) Here, i, = (3/2)“2A, to match the definition of gc. The variation of (r/,- U,,.)(a) with or is illustrated in Fig. 1, in which three cases are distinguished. In Fig. 1(a), 3ne,(A,/c~,)“~’ < or/p,.. (U,- U,,) is a concave function of CJand hence coincides with (U,-U,,),,. In Fig. l(b), 6eo/(n+ l)(&/~~)~- < 0,./p, < 3ne,,(3Ja,)“-‘, and in Fig. l(c), G,/K < W(n + 1Wo,.Y’ . (UT- u,,> IS no longer concave ; the corresponding plots of (U,- U,,),, are shown by the dashed lines. The only complicated feature is the value (TV of ge which corresponds to the point of tangency between (U,- Uo,) and (U,-- UoJ** in Fig. 1(b). This is given as the solution of the equation (3.22) With a view towards optimizing the bound (3.19) with respect to p,, p2, it is helpful to present (U,- U,,,),,(a) as a function of l/~~, with (T, taking a fixed value. First, when ~~ > A, n+ I
(U~-Uo,)**(a)
=z 0 2 -g.
=(u~-u(,,)(o)
(3.23)
I
I
When oe < A,, different ranges for l/~~ have to be distinguished
0-A /.lr 6e, (ZiJGK
he0 m$
3
II-
,r+I
I
- $_
(3.24) I
0
)LC“-’ 0 (U,.- Uo,)**(a) = 2
$ n+’ -&-a,) 0 I
(U,-
The bound (3.19) is expressible in the form
U,,)**(a)
[co ($
= fy+
z]-
2,
(3.25)
(3.26)
Bounds
of third order for nonlinear
composites
101
(4
Fig. 1. Schematic 3nr,(%,/o,)“~ I <
plots
of (r/,-
U,,)
(solid
(~,,‘p,;(b) 6c,/(n+ 1)(&/a,)“--
lines), and (U,- U,,),, i 6,;~~ < 3nq,(i.,io,)“+’
(dashed
lines),
against
cr.
: (c) cr,!pr< 6e,,I(n+ I)(i,,‘a,)”
(a) ‘.
102
o’(a)
D. R. S. TALBOT
< c,
2
+(u1
-
1
Uo,)**(P,)
[
+CZ
and J. R. WILLIS
-A,) Ii N(a)H(a,
~+w(i,,d**(P,)
dx,
+
(3.27)
CO
[
where X, = [~+C2(~C2+~m,)p*-~c,P]aeZ
= [~(I-~c,B)*+~c,m,B2]aez,
x2
=
=
[~+c,(~c,+~m,)~‘+~c,~]a~
p, =(l-;c*jQa,
[ By introducing
dimensionless 1 _=-
eo
PI
Cl
variables -
01
+cu,
-
(3.27) with
Uo,)**(P,>
1
Xl
=&c,
A
=p,,L
(3.30)
PI
)
it is easy to verify that the minimum
min 5 VI’0 [ p,
-
optimizing
so that
,!p I 1 ,>
Pie
(
of p,, p2. Therefore,
2 +(Ul
(3.28) (3.29)
p* =(l+~c,~)a.
The nonlinear penalty term is independent respect to p, requires the evaluation of
min ~, >.
[~(l+~c,B)‘+~c,m,B*]a,2,
is expressible
Uo,)**(P
in the form
, >] = eo(T, p)“”
V(_J?,J,).
(3.31)
Similarly,
min Ir,>o
2
+ (U2 -
(3.32)
Uo2)**(P2)
where 113and J2 are defined so that x* = p:J?*, The function The bound
U”(a)d
eoo2
/I, = p&.
V requires computation. (3.27), optimized with respect to p, and p2, now becomes
(3J+‘[cl (3(1-$,p)il- V(R,J,)-t
(3.33)
103
Bounds of third order for nonlinear composites
3.4. The nonlinrar
penalty
term
Suppose, to be definite, that cr, < o?. Then, chosen as in (3.21), it follows that
with the comparison
potentials
U,,,
(3.35)
N(a) d H(a), where
(3.36)
With the polarizations deviatoric part
chosen
so that (3.18) is satisfied,
b’ = [Id’-P(Id’-E)(x,
the trial stress field has
(3.37)
-~,)]a.
Now let (3.38)
It is demonstrated and hence that
in Appendix
B that t(x) has bounded
mean
(square)
oscillation,
(3.39)
meas{ x E C, : /(t /I > s} < A exp( - KY). The constants A, a depend on the volume fraction c, and on the “pattern” a/(\ ii // ; their calculation is outlined in Appendix B. It follows that meas{xECO:I/5--fiBtli
>s/(511}
drneasjxECO:~lflt~~ = measjxE
>s-lj
C,, : I/ t // > (s- 1)/p]
d Aexp[-r(s-l)ifi]. Elementary
calculation
s( 0
(3.40)
now gives n+l
N(a)H(a,
of loading.
-A,) dx < eOdl
5 0 01
* A exp [ - x(s - 1)/fl].s” ds s I,’ a
where (3.42)
104
D. R. S. TALBOT
and J. R. WILLIS
u = 11B 11s/A. Finally, repeated inte-
having employed the variable transformation gration by parts shows that
s
n(n- l)*.*(n-m)
3c
u”e-‘u
(3.43)
m+l P
I
where m is the integer satisfying m < n < m + 1.
4.
RESULTS AND DISCUSSION
Calculations have been performed for a two-component composite of the type described above. The overall potential u”(a) has the form (3.2). The characterizing stress crOis a homogeneous function of degree zero in a, that is, a function of the stress pattern a/II 5 I). For any fixed choice of stress pattern, a lower (or upper) bound for 0 generates an upper (or lower) bound for go. Figure 2 shows such a result. The stress pattern corresponds to the in-plane shear
bfj = 0 otherwise.
(4.1)
The figure plots bounds for oO/ozas a function of 1/n, for an isotropic composite with c, = Cl = 0.5, and 0, = a,/2. The uppermost line in the figure corresponds to the classical “Taylor” bound, obtained from the first of equations (2.11). The lines labelled (a) and (b) correspond to “improved” lower bounds for 0 (and hence upper bounds for oO), related to (2.26). These do not require the use of a nonlinear
(4
0.72 0.7 -
-
(b)
0.68 0.66 0.64 0.62 0.6 i 0.58 0.56 F 0.54 ' 0.1
Fig. 2. Bounds
I
I
I
I
1
I
I
I
I
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
for a,/rr, plotted
against
l/n, for a composite
for which c, = cz = 0.5 and 0, = ~~12
Bounds
of third order for nonlinear
composites
105
comparison medium, and were already calculated by Ponte Castaiieda [ 1992, equation (4.41)]. These two lines correspond to the extreme values of the Milton parameter : nt, = 0 for (a), and m, = 1 for (b). If no information on three-point statistics is available, the only guaranteed bound is that obtained by assuming the worst, to give line (a) (m, = 0). This is, in fact, the “nonlinear Hashin-Shtrikman bound” for this composite. reducing to the classical linear bound of Hashin and Shtrikman (1963) in the case n = 1. The remaining lines in Fig. 2 are lower bounds for a,,/az (corresponding to upper bounds for 0. The lowest bound is the Sachs bound, obtained from the second 01 equations (2.11). The other two lines, labelled (c) and (d). are “improved” bounds, making use of three-point statistics and obtained. as described in Section 3. by employing a nonlinear comparison composite and allowing for the nonlinear “penalty” thereby incurred. Line (c) corresponds to the value 177,= 0 of the Milton parameter. Note that it is a lower bound but coincides with the upper bound (a) when II = 1 : this corresponds to the fact that the Hashin-Shtrikman upper bound for the linear composite is attained when m, = 0. The lower of the two new lower bounds, labelled (d), corresponds to the value 111,= 1. This coincides with the corresponding upper bound, (b), when y1= 1, and is consistent with the attainment of the linear Hashin-Shtrikman lower bound in this case. When the Milton parameter is unknown. the only guaranteed bounds are those obtained by adopting the “worst-case” values : 177, = 0 for the upper bound, and m, = 1 for the lower bound. These are independent of three-point statistics and can be considered as generalizations to nonlinear material behaviour of the classical linear Hashin--Shtrikman bounds. Calculations have also been performed for other volume fractions and for other ratios (T~!~J,.The results have the same form as shown in Fig. 2. As remarked previously, the “easy” bound, not requiring use of a nonlinear comparison medium, was obtained earlier, by Ponte Castaiieda (1992), and methodology has been available since 1985 for construction of the nonlinear Hashin-Shtrikman bound. Progress in constructing the more difficult bounds is of more recent origin. The first steps were taken by Talbot and Willis (1994) in the context of electrostatics and Talbot and Willis (1995) for incompressible elasticity. Those works focussed directly on the nonlinear Hashin-Shtrikman bound. by taking as a starting point the nonlinear Hashin-Shtrikman variational principle as introduced by Talbot and Willis (I 985). The new contribution here is to take the formulation of Talbot and Willis (1992) as the basis. The direct, and predictable, benefit is that this allows the construction of bounds of higher order, such as the three-point bound given explicitly here. Additional benefits, however, have also emerged. First. the additional flexibility contained in the present formulation, far from introducing more complication, has actually led to simplification. Several of the optimizations could be performed analytically, and the remaining computation was actually simpler than that required previously. The other benefit is still more interesting. Although nothing is gained in the linear problem, in the nonlinear case the polarization has to be chosen so as to provide the best trade-off between the “ordinary” part of the bound expression and the nonlinear penalty. It appears that the “ordinary” part of the bound (3.34) varies more slowly about its minimum value than the corresponding expression employed by Talbot and Willis (1995), even in the extreme case III, = I. The new lower bound
106
D. R. S. TALBOT
and J. R. WILLIS
of Hashin-Shtrikman type produced here is correspondingly larger, when 12> 1, than that produced before. It is of course unlikely that the new lower bound is optimal when n > 1. Improvement over the classical Sachs bound declines as n increases. It should be noted, however, that numerical experiments on a “composite sphere” model, performed for a similar problem in electrostatics by Barrett and Talbot (1996), displayed exactly this trend with increasing n, when a sphere of the “harder” material was surrounded by a shell of the “softer”.
REFERENCES Barrett, K. E. and Talbot, D. R. S. (1996) Bounds for the effective properties of a nonlinear two-phase composite dielectric. Proc. 8th Int. Symp. Continuum Models and Discrete Systems (ed. K. Z. Markov), pp. 92-99. World Scientific, Singapore. Beran, M. J. (1965) Use of the variational approach to determine bounds for the effective permittivity of a random medium. Nuovo Cimento 38, 771-782. Dendievel, R., Bonnet, G. and Willis, J. R. (1991) Bounds for the creep behaviour of polycrystalline materials. Inelastic Deformation of Composite Materials (ed. G. J. Dvorak), pp. 175-192. Springer-Verlag, New York. Hashin, Z. and Shtrikman, S. (1963) A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11, 127-140. John, F. and Nirenberg, L. (1961) On functions of bounded mean oscillation. Communs. Pure Appl. Math. 14,41 S-426. Milton, G. W. (1981) Bounds on the electromagnetic, elastic, and other properties of twocomponent composites. Phys. Rev. Lett. 46, 542-545. Milton, G. W. (1982) Bounds on the elastic and transport properties of two-component composites. 1. Mech. Phys. Solids 30, 177-I 9 1. Ponte Castafieda, P. (1991) The effective mechanical properties of nonlinear isotropic composites. J. Mech. Phys. Solids 39,45-71. Ponte Castafieda, P. (1992) New variational principles in plasticity and their application to composite materials. J. Mech. Phys. Solids 40, 1757-l 788. Ponte Castafieda, P. and Willis, J. R. (1988) On the overall properties of nonlinearly viscous composites. Proc. R. Sot. Lond. A416,217-244. Ponte Castafieda, P. and Willis, J. R. (1993) The effective behavior of nonlinear composites : a comparison between two methods. Materials Science Forum 123-125, 351-360. Reuss, A. (1929) Calculation of the flow limits of mixed crystals on the basis of the plasticity of mono-crystals. Z. Anger. Math. Mech. 9, 49-58. Sachs, G. (1928) Zur Ableitung einer Fleissbedingun. Z. Ver. Dtsch. Ing. 72, 734736. Stein, E. M. (1967) Singular integrals, harmonic functions, and differentiability properties of functions of several variables. Proc. Symp. Pure Math. 10, 316335. Talbot, D. R. S. and Willis, J. R. (1985) Variational principles for inhomogeneous nonlinear media. IMA J. Appl. Math. 35, 39-54. Talbot, D. R. S. and Willis, J. R. (1992) Some simple explicit bounds for the overall behaviour of nonlinear composites. Int. J. Solids Struct. 29, 1981-1987. Talbot, D. R. S. and Willis, J. R. (1994) Upper and lower bounds for the overall properties of a nonlinear composite dielectric. I. Random microgeometry. Proc. R. Sac. Lond. A447, 365384. Talbot, D. R. S. and Willis, J. R. (1995) Upper and lower bounds for the overall properties of a nonlinear elastic composite. Proceedings, IUTAM Symposium on Anisotropy, Inhomogeneity and Nonlinearity in Solid Mechanics (ed. D. F. Parker and A. H. England), pp. 4099414. Kluwer, Dordrecht. Taylor, G. I. (1938) Plastic strain in metals. J. Inst. Metals 62, 307-324.
Bounds of third order for nonlinear
composites
107
Toland, J. F. and Willis, J. R. (1989) Duality for families of natural variational principles in nonlinear electrostatics. SIAM J. Math. Anal. 20, 1283-l 292. Van Tiel, J. (1984) Convex Analysis. Wiley, New York. Voigt, W. (1889) Ueber die Beziehung zwischen den beiden Elasticitatsconstanten isotroper K&per. Ann. Phys. (Leipzig) [3] 38, 573-587. Willis, J. R. (1977) Bounds and self-consistent estimates for the overall properties of anisotropic composites. J. Mech. Phys. Solids 25, 185-202. Willis. J. R. (1981) Variational and related methods for the overall properties of composites. Ad~unc~~.s in .4pplird Mechanics (ed. C.-S. Yih), Vol. 21. pp. I 78. Academic Press. New York. Willis, J. R. (1982) Elasticity theory of composites. Mrchanics of’ Solids. Thr Rodm~~~ Hi// Si.\-ticth Anniwrsrrry Volume (ed. H. G. Hopkins and M. J. Sewell). pp. 653-686. Pergamon. Oxford. Willis, J. R. (1983) The overall elastic response of composite materials. .I. App/. Mcch. SO, 1202 1209. Willis, J. R. (1986) Variational estimates for the overall response of an inhomogeneous nonlinear dielectric. Homogenization und IZfjtictiw Propertics of’MatcTriu1.r and Mediu (ed. J. L. Ericksen. D. Kinderlehrer, R. V. Kohn and J.-L. Lions). pp. 245-263. Springer-Verlag, New York. Willis, J. R. (1991) On methods for bounding the overall properties of nonlinear composites. J. Mrch. Phys. Solids 39, 73-86. Willis, J. R. (1992) On methods for bounding the overall properties of nonlinear composites : Correction and addition. J. Mech. Phys. Solids 40,441 -445.
APPENDIX
A : THE
DELTA
OPERATOR
The operator A,, introduced in Section 2.2 is derived from the Green’s function medium with tensor of elastic moduli L,,. If L,, has components L,,k, the Green’s satisfies the equations I,,&.,,
+ &,,]6(x) -
G for a function
II = 0, x 6 cc,.
(A.1)
where it is assumed that C,) has unit volume and contains the origin in its interior. The Green’s function is also required to be C,,-periodic and to have mean value zero. It follows immediately by Fourier transforms that G(x) =(2rc-’ where e(c)
c c(<)exp(-it-x). <#O
(A?)
is the inverse of the matrix K(c) with components (A.3)
K,, (<) = L,,&,&
The sum in (A.2) ranges over reciprocal vectors < whose components are integer multiples of 2rl. As discussed. for example, by Willis (198 I, 1982), the operator A,, is related to an operator F,, which, in the present case, is of convolution form over C,, and has kernel with components ftiil given by (A.4)
F,,/J,(x) = - G‘?,,,,‘,(X) I,/,,,,‘<,I. Then, A0 is given as A0 =L,6 where 6’ is a scalar components
operator
with kernel
-LO:Fo:L, 6(x)-
I. In suffix notation,
(A.5) A0 has kernel
with
D. R. S. TALBOT and J. R. WILLIS
108
&k/(x) = &~(&x) Equivalently,
in terms of Fourier
- 1) - Li,pyrpy~.s?i(x)L,~,.
c4.6)
transforms,
d”(C) =L”-LD:L,:r”({):L”,
(A.7)
where f,,(r) denotes the Fourier transform of I,,. In the particular case of an isotropic reference medium with Lame moduli lo, p,,, calculation shows that
It may be noted that, in the limit of incompressible
behaviour
(& + co), relation
(A.8) gives (A.9)
where E is defined by (3.14). The transform of A0 follows from (A.7). Taking the incompressible limit gives the expression (3.12) for its “trace-free” part A&, that maps the space of trace-free tensors into itself. Some elementary properties of Ah are required. First, it is noted that (M-E) defines a projection. This follows directly from its Fourier transform and use of the convolution theorem I:I=I
and
E:E=E.
(A. 10)
The tensor Q’ is also required. This is defined by (3.8). When the micro-geometry is “statistically isotropic”, and has fine scale relative to the dimensions of C,,, the tensor Q’ is isotropic and so of the form Q’ = Q’I. The constant Q’ can be found by calculating the (ijtj) invariant of Q’. It follows, since I,,i, = 5 that the (ijij) invariant The result (3.15) follows The tensor B is also calculation is not needed
and
Ei,,, = 2,
(A.11)
of (IS’-E) has transform 3 (independent of 5) and hence equals 36’. immediately. isotropic when the composite is statistically isotropic. Its detailed but the form (3.17) for B’ is a direct consequence of the isotropy.
APPENDIX Talbot and Willis (1994) introduced
B : THE BMO PROPERTY a definition
which, adapted
to present
notation,
reads :
“Let t(x) be defined and let 11 t 11be square-integrable over a cube C,,. The field t(x) has the property of bounded mean (square) oscillation if there is a constant K such that, for every parallel sub-cube C of C,,, and some constant t,-, the inequality 1 (1t-tc ~ m(C) s C”
ll’dx < K2
holds. Here, m(C) denotes the Lebesgue measure of C.” This is a slight variation on the original definition of John and Nirenberg (1961), in using “square-integrable” instead of “integrable”. The difference is not fundamental but it was found to lead, in the present context, to better values of the constants. By tracking the reasoning of John and Nirenberg, Talbot and Willis showed that, for any such field t(x), constants A and c( could be found such that the inequality (3.39) holds. The constant A is given as
IO9
Bounds of third order for nonlinear composites /It(x) II2 dx.
A = B
and B and c( appear in equation B
=
(B.2)
(A 40) of Talbot and Willis (1994):
(t/K-
I)*
2 ln(t/K) ’
t/I-Q’
K’(fi+
(B.3)
’ = K&t/K)
Here, t is a parameter required to satisfy t > K. Its value is to be chosen optimally in relation to minimizing the penalty term (3.41). The inequality (3.39) in fact holds subject to the restriction s > (J?K+ t)/(t/K1). Equation (B.2) can be simplified, as follows dx(X, -c,)(M’-E):(I6’-E)(x,
-1.1):
dx(x,-c,)(I6’-E)(X,-c,):
03.4)
since (Id’- E) is a projection. In view of the overall isotropy, the tensor appearing in the last expression above is a constant times I, and the constant can be evaluated as it was in the calculation of Q’ : it has the value 3c,c2/5. Thus, (B.2) simplifies to A = 3c,c,B/5.
(B.5)
The task that remains is the estimation of the constant K. The construction proposed by Stein (1967). It is observed first that (3.38) can be given in the alternative form
t(x) =(IS-E”)*(x,
,
-c,)
follows
one
(B.6)
where the convolution * is over all space, the operator S has kernel 6(x) and E” is the infinite-body limit of E, obtained by replacing the Fourier sum in (3.14) by a Fourier integral (interpreted in the sense of generalized functions). For a chosen cube C, let B be a ball centred at the centre xc of the cube, and with sufficiently large radius to enclose C. Now define,f,(x) and,f,(x) over all space so that f,(x)
=x,(x)-c,> =O,
xgB
x$B
(B.7)
and
f2c4 =
XI
(x)-c, -f,(x).
(B.8)
Let
03.9)
It follows immediately,
because
(IS-E”)
is a projection,
that
D. R. S. TALBOT and J. R. WILLIS
110 11 t, )I2dx d
(B. 10)
(x, -c,)* dx < max(c:, &m(B),
sc and thus
&
sc
II t, II2 dx < cmax(c?,&,
(B.11)
where
c = W>lm(c)
(B.12)
is a parameter open to choice (c > $&r/2). Next,
s c
IIt, (9 --t,(xc) II2dx =
s11s c
dx
2
dY(E”(x--Y)-E”(xc-~)):(~/Il~ll)f,(~) YdB
2
-EPY,,(xc -Y))(flk,/ll 8 II)1 max(c:, c:) > (B.13)
= m(C)A’* max(c:, c:),
the last equality defining A’. The relations (B. 11) and (B. 13) already show that the field t(x) has the BMO property. The calculations upon which (B.3) is based require t, to be identified with the mean value over C oft. With the notation t ,,-, t2Cfor the mean values oft,, t2 over C, Minkowski’s inequality gives I;2
II t, -t,c
IItz -cc II2dx
/I* dx
(B.14)
Now
s <
ll t, -t,,
II2 dx =
sc
(II t, II2 - /Kc II2 )dx
G
sc
II t, II2 dx
(B.15)
< cm(C) max(c:, c:), from (B.11). Next, 112
(s c
IItz--zc II*dx
> (s <
c
II f2 --t,C+)
112
II* dx
+ >
(s c
II t,(xc)--t,c
l.‘Z
II2 dx
> >
(B. 16) and
s c
II f2 (xc) - f2c II’ dx = II tz (xc> - t,c II‘m(c)
[tz - f2(xc)1 dx
Bounds
of third order for nonlinear
composites
IItz--tzk) <
(S
IItz -t,(xc)
III
IIdx (B.17)
II*dx,
C’
by an application
of Hiilder’s inequality.
s
/ltz-ttzc C’
Finally, relation
It follows,
from (B. 16), (B. 17) and (B. 13) that
l)‘dx < 4m(C)A”max(cT,ci).
(B.18)
(B.1) follows from (B.14), (B.15) and (B.18) with K=(c’
All that remains components
is the calculation
2+2A’)max(c,,cz).
of A’. Inverting
its Fourier
(B.19) transform
shows that E ’ has
(B.20) where r = 1x I. The homogeneity
of E” permits
A” to be calculated
dy I(E;dy -x) -E$,(~)H~ull~
as
II) I .
(B.21)
where Q is a cube centred at the origin, with sides of length 2, and R is a ball centred at the origin, with radius v > & ( so that c = nv’/6). This expression has been evaluated when the stress pattern is given by (4.1). The y-integrals were expressed in polar coordinates r = 1y /, I), 4, and then Q was replaced by u = cos 0. The mid-point rule was used for integration with respect to u and Gauss-Legendre formulae were used for all other integrals. As a result of this calculation, the value K = 16 max(c,, cJ was employed for the BMO estimates, since this clearly overestimated K, allowing for any numerical inaccuracy.
Pergamon
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0022~ 5096/97 $17.00+0.00
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BOUNDS OF THIRD ORDER FOR THE OVERALL RESPONSE OF NONLINEAR COMPOSITES D. R. S. TALBOTt
and J. R. WILLIS::
?School of Mathematical and Information Sciences. Coventry University. Priory Street. Coventry CVI 5FB ; and JDepartment of Applied Mathematics and Theoretical Physics. University of Cambridge, Silver Street. Cambridge CB3 YEW. U.K.
(Rrcrirwi
24 January
1996; in rwisrd
form
I7 Mu! 1996)
ABSTRACT Composites whose response can be described in terms of a convex potential function are discussed. Bounds are constructed for the overall, or effective. potential of the composite. given the individual potentials of its constituents. Steady-state creep is considered explicitly but the results apply equally well to physically nonlinear elasticity, or deformation-theory plasticity, if strain-rate is reinterpreted as infinitesimal strain. Earlier work employed a linear “comparison” medium. This permitted the construction of only one hound -either an upper bound or a lower bound--or even in some cases no bound at all. Use of a nonlinear comparison medium removes this restriction but at the expense of requiring detailed exploration of the properties of the trial fields that are employed. The fields used here-and previously-have the property of “bounded mean oscillation” ; the use of a theorem that applies to such fields permits the construction of the bounds that were previously inaccessible. Illustrative results, which allow for three-point correlations. are presented for an isotropic two-phase composite, each component of which is isotropic, incompressible and conforms to a power-law relation between equivalent stress and equivalent strain-rate. Generalized Hashin-Shtrikman-type bounds follow by allowing the parameter corresponding to the three-point correlations to take its extreme values. Copyright c 1996 Elsevier Science Ltd Keywords: A. creep, B. constitutive ational calculus.
behaviour,
1.
B. inhomogeneous
material.
C. energy methods,
C. vari-
INTRODUCTION
The first bounds for the overall response of composites displaying nonlinear constitutive behaviour, analogous to the Hashin-Shtrikman bounds for linear behaviour. were developed by Talbot and Willis (1985) and Willis (1986), from an idea proposed by Willis (1983). In common with the approach of Hashin and Shtrikman (1963). a “comparison medium” was introduced ; then, some elementary results from convex analysis (Van Tiel, 1984) were exploited to generate, very directly, a generalization to nonlinear material behaviour of the Hashin-Shtrikman variational structure. The formulation starts from a variational principle of “minimum energy” type, and is applicable only to composites whose response permits such a description. Exploitation of the variational structure requires the construction of suitable trial fields from the class of admissible fields. This was accomplished by taking the comparison medium to be linear and homogeneous, and constructing trial fields by employing the Green’s 87
88
D. R. S. TALBOT
and J. R. WILLIS
function for this medium, exactly as in the analysis of composites with linear response. Reasoning like that advanced by Willis (1977, 1981, 1982) then generated a bound for the potential featuring in the variational principle, involving piecewise-constant “polarizations” (defined below), and the two-point geometrical correlation function(s) for the composite. Optimization of the bound required the solution of a set of nonlinear algebraic equations. Subsequently, Ponte Castaiieda (199 1) introduced an alternative formulation, involving a comparison linear composite, in which the bound for the nonlinear problem was expressed in terms of a corresponding bound for the linear composite. For many problems, the approach is very efficient: if a HashinShtrikman bound for the linear composite is employed, it usually provides the same bound for the nonlinear problem as the Talbot-Willis approach, though there are cases in which it fails (Willis, 1991, 1992; Ponte Castafieda and Willis, 1993). When it succeeds, however, there is the possibility of further improvement. by use of a higher-order bound for the linear composite. Talbot and Willis (1992) developed a unified formulation, which contained both the original Talbot-Willis and the Ponte Castaiieda prescriptions as special cases: it permits allowance for statistical correlations involving more than two points and always produces results at least as good as obtained from the original TalbottWillis prescription. There is, however, a limitation inherent to all of the developments described above : restriction to a linear comparison medium (whether or not this is a composite) generally precludes the construction of both upper and lower bounds. The remedy is to employ a nonlinear comparison medium. but this introduces a major hurdle relating to the construction and employment of trial fields. Talbot and Willis (1994) took a first step towards resolving this difficulty, in the context of nonlinear electrostatics, by employing a uniform comparison medium which was linear up to some magnitude of the field and nonlinear thereafter. Then, trial fields were employed which were related just to the linear part of the comparison medium’s response. Such fields have “bounded mean oscillation” (John and Nirenberg, 1961) and a property of such fields was used in an essential way in the construction of the bound. A new bound was obtained but the “penalty” incurred from the introduction of nonlinearity into the comparison medium was sufficiently severe to permit only slight improvement over bounds obtainable by elementary means. Corresponding results for a physicallynonlinear elastic composite were given by Talbot and Willis (1995). The present work employs a “comparison composite” whose behaviour is linear up to some magnitude of the field but nonlinear thereafter, in conjunction with the general formulation of Talbot and Willis (1992). The trial field that is substituted into the resulting functional is one generated from piecewise-constant polarizations in a further underlying medium, which is uniform and linear. In the absence of the nonlinearity, this type of field generates bounds of third order for the response of a linear composite (Milton, 1981 ; Willis, 1982), except that the problem for the linear composite contains a term analogous to thermoelasticity, with arbitrary thermal distortion tensors for each component whose choice is subsequently to be optimized. In the absence of the nonlinear terms, the procedure generates a three-point bound of the type introduced by Beran (1965) ; optimization of the parameters, allowing for the nonlinearity, produces different parameter values but still the bound is sensitive to three-point statistics.
x9
Bounds of third order for nonlinear composites
In the particular case of a two-phase composite with isotropic micro-geometry, the effect of the three-point statistics is felt through one or two parameters (two in the case of compressible elasticity ; one for incompressible behaviour or electrostatics). As shown by Milton (1981), these parameters must lie in certain ranges (they are normalized, in fact, so that they take values between 0 and 1). If the three-point statistics are unknown, bounds valid for all three-point statistics are obtained by allowing the parameters to take their “worst” values. For linear media, this produces two-point bounds that coincide with the HashinShtrikman bounds (always for incompressible elasticity or electrostatics; usually for compressible elasticity). In the nonlinear case, however, the procedure described delivers a significantly better twopoint bound than that obtained through the procedure of Talbot and Willis (1994. 1995). The reason is given in the body of the paper. The plan of the paper is as follows. First, in Section 2. both “primal” and “dual” formulations of the problem are presented, the general approach introduced by Talbot and Willis (1992) is summarized and its relation to earlier formulations is briefly outlined. In conjunction with a nonlinear “comparison composite” and a trial field of the type described above, it generates the new bounds. From Section 3 onwards, to avoid complicating the exposition, the general approach is demonstrated through consideration of a particular problem : a two-phase composite, each component of which is isotropic and incompressible, and subject to power-law creep. The general formulae are given in Section 3, and numerical results are presented in Section 4. Mathematical technicalities, relating to the property of bounded mean oscillation of the trial fields (John and Nirenberg, 1961 ; Stein, 1967) are recorded in the Appendix.
2. The constitutive in the form
response
d
FORMULATION
of the composites
=
F’(e;x),
or
treated
in this work can be represented
0,, = $e:x). i,
(2.1)
Here, c denotes stress, with components (T,,. In the context of steady creep, e denotes strain-rate, with components (u,,+zQ/2 where ui is the i-component of the velocity u. and F(e ; x) is the stress potential ; it is a convex function of e. Equivalently. e = V(a; where U(a ; x) is the strain-rate
x),
potential.
or
c#, = ;;(C:x), r, It is the Legendre
(2.2) transform
(2.3)
U(b ; x) = F*(o : x) = s!p [fr,,~,, - F(e ; x)] and is a convex function
of 6. Since F(e ; x) is a convex function
F(e; x) = U*(e ; x) = stp [o,,e,, - u(a ; x)]
of F(e ; x)
of e, (2.4)
90
D. R. S. TALBOT
and J. R. WILLIS
and the suprema in (2.3) and (2.4) are attained when o and e are related by (2.1) ; then, (2.2) is also satisfied. [The differentiability of the potentials implicitly assumed here is relaxed to subdifferentiability in the proofs given by Van Tie1 (1984).] Physically nonlinear small-strain elasticity, or deformation-theory plasticity, is incorporated in (2.1))(2.4), simply by interpreting u(x) as displacement rather than velocity, so that e becomes infinitesimal strain. For a composite, the potential F(e; x) takes the form F,(e) when x lies in material oftyper(r= 1,2 ,... n).Then,
where xr is the characteristic function of the region occupied by material of type r, taking the value 1 in that region and zero elsewhere. The strain-rate potential is similarly decomposed in terms of potentials U,(a) = F,*(a). Perfect bonding (implying continuity of velocity and traction) across interfaces is assumed. It is envisaged that the composite has a fine microstructure (relative to macroscopic dimensions) so that, on the macroscale, it appears homogeneous, with “overall” or “effective” stress potential &) and strain-rate potential o(5). The detailed strainrate and stress fields fluctuate rapidly about mean values e and 8. It is convenient to consider a unit cube COof material and to construct from it a composite of infinite extent by extending the characteristic functions xI so that they are periodic of period C,,. Then, @) is characterized as the mean value of F(e ; x) over CO,when all field equations are satisfied and e(x) is periodic with mean value E. Thus, Fis characterized by the variational principle &!) = inf
F(E+e; x) dx, I (‘0
(2.6)
where the infimum is taken over strain-rate fields e(x) related to velocity fields a(x) which are periodic with period CO(so that &has zero mean value). Dually, the strainrate potential is characterized by O(a) = inf
U@+B;x)dx, s cl1
(2.7)
where the infimum is taken over self-equilibriated stress fields a(x) which are C,periodic and have mean value zero. The overall potentials are convex (Ponte Castaiieda and Willis, 1988). Exact conditions under which the definitions (2.6) and (2.7) generate a Legendre transform pair were discussed, for a closely-related problem, by Toland and Willis (1989). The definitions (2.6) and (2.7) immediately induce bounds for F”and 0: F(;(e),<
I++G;x), s C‘O
for any admissible Z and 8. Furthermore, bound for F, and conversely
O(a) <
U@+&;x)dx s C‘U
(2.8)
an upper bound Us for 0 induces a lower
Bounds
of third order for nonlinear
u”(6)
<
U,(ii)
F(E)
d
F&5)
=S U,*(E)
91
composites d F(E).
=a F&T?)
d
(2.9)
O(C).
directly from the definition (2.4) or (2.5) of the Legendre transform. The difficulty is the choice of suitable fields @and 6. The simplest choices, C = 0 and 8 = 0, generate the elementary bounds O”(E) < F(:(e, < F(e). P(C)
(2. IO)
< u”(C) < U(b).
where
I+)
=
s
F(E ; x) dx, (‘0
c/(5 ; x) dx.
U(C) =
(2.11)
i (0
The bound P(E) is called the Taylor bound, after Taylor (1938). who proposed e(x) = C as an approximation. In the context of linear elasticity. the same proposal was made by Voigt (1889). The elementary bound U(C) is called the Sachs bound, after Sachs (1928), who proposed a(x) = t7 as an approximation. Reuss (1929) made the same proposal in the context of linear elasticity. 2.1.
Improved
Following define
bounds : primal problem
Talbot
and Willis (1992), introduce
(F-F,)*(r;x)
a comparison
potential
= sup[s,,e,,-(F-F,,)(e;x)]. r
F,,(e; x) and
(2.12)
Then.
F(e;x) >, z,,e,,+F,,(e;x)-(F-F,,)*(r;x)
(2.13)
for every e and r and hence, from (2.6)
09
3
I
inf ~,,~[r:(e+c)+f;(B+C;~)-(F-F,,)*(~:~)ld~
for any chosen field of “polarizations” r(x)_ The infimum is taken over set of admissible fields &. It is sensitive to the choice of polarizations comparison potential FO. One interesting choice of F,, corresponds to with exactly the same microgeometry as the given composite, but with that are linearly viscous (or elastic) : F,)(e;x)
= i
$e:L,
:exr(x).
(2.14)
the relevant z and to the a composite constituents
(2.15)
,=I
say, where CL,..r = 1,. n$ are tensors of viscosities (or elastic moduli). the polarization field t(x) is taken to be piecewise constant. so that
If. further.
92
D. R. S. TALBOT
and J. R. WILLIS
z(x) = i
r=l
rrxr(x),
(2.16)
the problem of evaluating the infimum in (2.14) becomes analogous to that of finding the overall thermoelastic response of a linear composite. Bounds of this type were introduced and discussed by Talbot and Willis (I 992). Two cases are worthy of special mention. First, if the potential F. is independent of x, and so corresponds to a uniform comparison medium rather than a comparison composite, the bound (2.14) coincides with that developed originally by Talbot and Willis (1985). If F,(e) for some constant
=+e:L,:e
tensor L,,, the infimum 0)
is attained
(2.17) by the field (2.18)
= -(r,r)(x),
where the linear operator To is related to the Green’s function for the uniform comparison medium. Calculation of the bound then follows exactly the pattern established for linear composites (Willis, 1977, 198 1, 1982), for any given piecewiseconstant polarization field r. Optimization over the parameters rr then reduces to solving a set of algebraic nonlinear equations. A special property of the operator T,, ensures that the resulting bound depends on the microgeometry through points taken two at a time, for the nonlinear problem just as for the linear one. The bound is one of Hashin-Shtrikman type. The other special case is obtained by choosing z = 0. The problem of evaluating the infimum now reduces to that of finding the overall viscosity tensor (or tensor of elasticity) of a linear composite, and (F-&)*(0; x) = - min {F-F,}(x). Then, if any lower bound L, for the linear composite is known, it follows from (2.14) that
F(E) >fE:L,:E+
i c, min {FJe)-ie:L,.:e} I= I ( e
:
(2.19)
)
where c, =
I
L(X) dx
(2.20)
co
is the volume fraction of material of type Y. Bounds of this type were introduced by Ponte Castafieda (1991). If a Hashin-Shtrikman bound L, is employed, optimizing over the parameters L, usually generates the same bound as is produced by the original “nonlinear Hashin-Shtrikman” approach of Talbot and Willis (1985), but it can leave a gap in unfavourable cases (Willis, 1992 ; Ponte Castafieda and Willis, 1993). An upper bound for I@) can be developed similarly. Choosing perhaps some different F,,(e ; x), define (F-F&(r;x) so that
= i~f[7,ie,,-(F--“)(e;x)l,
(2.21)
Bounds
F(e;x)
of third order for nonlinear
d T,,e,,+F,,(e:x)-(F-F,),(t;x)
for any e and T. Then, from the first of equations
F(ce)<
composites
(2.6),
[T::(e+e)+F,(~+~;x)-(F-F,,),(z;x)ldx.
(2.23)
for any z and any admissible &: since the inequalities all run the same way, it is not obligatory to evaluate the infimum over $. In particular, it is possible to employ a trial field of the form (2.18), for some additional tensor of viscosities L,, and new “polarizations” 2,. say :
@W= - i [r,,(Q,.Xl)](x).
(2.24)
,= I
The “original” polarization z can also be taken to be piecewise constant. The quadratic part of the functional (2.23) then provides an upper bound for the energy of a linear composite, with the polarizations z,. playing a role analogous to thermoelastic coupling. Such bounds (optimized with respect to the parameters e,) are of a type first produced by Beran (1965), and formulated more generally by Milton (1981) and Willis (1982). In practical terms, the number of parameters introduced above is excessive : some can be eliminated by performing fairly simple optimizations. However. the concepts will find use in constructions that follow later. 2.2. Improwd bounds : dual problem Similar arguments (2.7). First, introduce
can be developed, based on the dual variational formulation a comparison strain-rate potential U,,(a ; x). and define
(C’- v,,)*(V
; x) = sy h,,~,,- (CT-r/,,)(a: x)1.
(2.25)
It follows that o(a)
2 inf
[q:(a+8)+U,,(c+&;x)-(U-U,,)*(v;x)]dx. i (‘11
(2.26)
where the infimum is taken over self-equilibriated stress fields 6 which are C,,-periodic and have mean value zero. In the particular case of a uniform linear comparison medium, with strain-rate potential U,(a) the infimum
is attained
=;a:M,+,
(2.27)
when S(x) = -(&V)(X).
(2.28)
The operator A0 is also related to the Green’s function for the comparison medium; it is given explicitly later. A prescription of Ponte Castaiieda type follows by choosing q = 0 and U,, to correspond to a linear composite with
94
D. R. S. TALBOT
U,(d,X)
= f i
and J. R. WILLIS
a:M,:axr(x).
(2.29)
7-I
An upper bound
for o(B) follows by defining
(U-U,),(tj;x) The bound
= i~f[riia,i-(U-U,)(b;x)l.
(2.30)
is (2.31)
where B is any admissible
stress field : for instance,
W = - i Lh(it,xr)lW, where A,, is calculated tensor M,.
for a medium
with uniform
viscosity
(2.32) tensor
L, and inverse
2.3. Limitations If the comparison medium is taken to correspond to linear material response (whether uniform or a composite), and any one of the potentials F, grows less rapidly than quadratically as )Ie 11+ GO,then (F-F,)*(r) = co, at least for some x, and (2.14) gives F” > - CC. [Here and in the sequel, I/e I/ = (e : e)“’ = (ej,e;,)“2.] Also, still with a linear comparison medium, if any I;; grows more rapidly than quadratically as 11e 11+ co, then (F-F,),(z) = - co for some x, and (2.23) gives F”< + co. Similarly, if U, is taken as a quadratic function of G and any one of the potentials U, grows less rapidly than quadratically as I/ TV/I + co, then (2.26) gives u” 3 -cc, and if any (/, grows more rapidly than quadratically as 11 D 11+ co, then (2.31) gives 0< +co. The bounds (2.14) and (2.26) require the specification of a linear comparison medium (composite or uniform), because it is necessary to evaluate (or at least to bound) the infimum, and this is only feasible for a linear comparison medium. It may be noted, also, that, if all of the potentials F, are “superquadratic”, then all of the potentials C: are “subquadratic” ; thus, in general, at most one of (2.14), (2.26) provides a finite bound. The formulae (2.23) and (2.31) display similar limitations if they are employed in conjunction with linear comparison media. They do, however, admit the possibility of use with nonlinear comparison media; this is discussed next. 2.4. New upper bounds The trial fields (2.24) and (2.28) are known to produce not only bounds of Hashin Shtrikman type, but also more accurate bounds, of third order, for linear composite media. The approach to be adopted here will be to employ a comparison composite that is linear up to some level of intensity of the field, and nonlinear thereafter if the
Bounds
of third order for nonlinear
behaviour of the actual nonlinear composite bound (2.23) the potential F,, is taken as F,(e;x)
warrants
it. Thus, considering
~e:L,:ex,(x)+N(e)H(lIell-j.).
= i r=
composites
first the
(2.33)
I
The exact form of N(e) should be tailored to the particular application. It is important that it should grow at least as fast as any of the potentials F,- as I/e I/ --+ xc. The Heaviside function If(*) ensures that it “switches on” only at high field values (the parameter i. is still open to choice), so that the comparison medium behaves “almost” like a linear medium, and therefore trial fields of the form (2.24) should be able to capture its response. Substituting the field (2.24) into the inequality (2.23), with F,, given by (2.33). gives
+
(e- i r=
) ( ,= 1 i!+t! /! IS N(E+C)H(l/
[r,(~,xr)l(x) : i L,L(x): I
-(F-F,,),(z;x)
I=
E- i
I
[r,(?,x,)](x)
I
-i)dx.
dx+
(2.34)
C‘,,
It may be noted that all of the terms except the one involving the function N(e) contain one, two, or three factors xI, xY,etc., and so depend on points in the composite taken one, two. or three at a time. The function N(e) can be taken to be zero, if each of the potentials F,.(e) grows no more than quadratically as I/e 11+ x. In such a case, the bound is explicit (even if in need of simplification) and is the required three-point bound. If any of the functions F,(e) grows more rapidly, however, a non-zero function N(e) is necessary, and the last term in (2.34), which may be regarded as a “penalty” incurred from the use of a nonlinear comparison material, has to be estimated. This can. in fact, be done because fields e of the form e(x) = G+&(x),
(2.35)
with & given by (2.24), have the property of “bounded mean oscillation” (John and Nirenberg, 1961 ; Stein, 1967). The relevant theory is summarized in Appendix B. The result is that. if m(s) denotes the measure of the set of points in C,, where 11e /I > .s m(s) = measjx
E C,, : 1)e(x) 11> .sj.
(2.36)
then m(s) d A exp( -KS). where A and CIdepend that, if
on the parameters
from which & is constructed.
N(e) d fi(II e (with A monotone
increasing),
then
II)
(2.37) It follows then
(2.3X)
D. R. S. TALBOT
96
Ne(x))H(Il s c‘o
e II - 4 dx
d
fi(ll
s
“0
=
e(x)
m(s)
and J. R. WILLIS
^
IIMII
e
^
II-4 dx
g(s)ds < A
I.
exp( - cIs)$$)
ds.
(2.39)
sI
The entire bound (2.34) is now explicit, and requires optimization with respect to the parameters contained therein. Similarly, let U,(a;x)
(2.40)
= f: ~a:M,:a~~(x)+N(a)H()Ia/l-~). r=
I
Substituting the trial stress field (2.32) into (2.31) then gives
+ c-- i r=
[&(hxr)l(x> : i M,x,W:
-(u-U,)&;x)
a- i [&(il,x,)1W
*=I
I
1
dx+
I=
s
N@+6)H(l~
a+6
I
1)-i)
dx.
(2.41)
c,1
Again, the field c = z+B has the property of bounded mean oscillation and the nonlinear “penalty” term can be bounded as in (2.39). The penalty term is not needed if each U, grows less than quadratically as //d jl -+ CG,which is equivalent to growth more rapid than quadratic for each F,. Since, by (2.9), an upper bound for Finduces a lower bound for 0 and conversely, the two prescriptions (2.34) and (2.41) provide upper and lower bounds for both potentials, which are sensitive to the three-point statistics of the composite, without restriction on the potentials Fr or U,.
3.
EXAMPLE : AN ISOTROPIC TWO-PHASE
COMPOSITE
The composite that will be discussed has two phases, Y= 1,2, with strain-rate potentials ~. ,!+I I/,(a) = !?!F? 2 (3.1) nfl 0 or ’ where ge denotes the equivalent stress, (3/2)“21(r~’11,with a:i = c,~-+~~,cJ~~.The potentials are thus isotropic and independent of hydrostatic stress, so that each phase is incompressible. The exponent n is taken to be greater than 1. It follows that the overall potential 0 for the composite is a homogeneous function of degree n+ 1 of the mean stress 6 and is independent of hydrostatic mean stress, corresponding to
Bounds
of third order for nonlinear
incompressibility. The microgeometry isotropic, so that 0 will be an isotropic
97
composites
will, in fact, be assumed to be statistically function of 8. It can therefore be written as (3.1)
but here the characterizing stress cr,)is a homogeneous function of degree 0 in CF.and so depends on the pattern of stress, as discussed previously by Dendievel ct (11.( 199 1). for example. There is no need to discuss lower bounds for such a composite. because they are available from the work of Ponte Castarieda (I 992). Attention is here directed towards finding an upper bound, based on (2.41). 3. I.
General
t\zwphase
composite
It is helpful to note that, because the operator A,, is symmetric and has mean value zero, its effect on a polarization field ij is the same as that on fi - 4. For any two-phase composite, now
m,-5 =(it, -fid(x,(x)-c,)
(3.3)
a(x) = -4,(x,
-~.,)(il, -b).
(3.4)
(2.41) therefore
yields
and so
The upper bound c?(a) 6
formula
[(x,1, +(I -x,)v~):(*--,r(x, s (‘0 +;(+A&
-c,)(ir,
-j/Z))
-~.,)(il, :(m+(M,
-42))
-M:)(x,
--(‘I )):
@-Ao(x, -c,)(il, -ildJ
+
s
N(c+3)N(II
(3.5)
t?+6 1)-2)dx.
(0
Here, C,,(a; x) has been decomposed U,(a;x)
into
= U,,(a)x,(x)+
~~o,(~)Xz(X).
(3.6)
with U,,,(o) =~o:M,.:a+N,(a)H(IIaIl-R) Employing
(3.7)
the notation
s
(x, -c,)&(x,
Cl,
-c,)dx
= c,(.lQ,
(3.X)
D. R. S. TALBOT
and J. R. WILLIS
s
(XI -Wb@Ao(x, -c,)dx
=A
(3.9)
(‘I,
and
s
(x, -c,)AoW,
-Md(x,
-c,)Ao(x, -c,)dx
= B>
(3.10)
CO
optimization
of (3.5) with respect to the parameters
O(C) +I%~+c,(L’,
-U,,),,(a-czQ:(ij,
+c,(Uz-U”~)**(~+c,Q:(il, +;(4, -42) :(A+B) The property
ql, q2 gives -ijz))
-iL)>-c,c@:(M,
:(9, -42)+
-Md:Q:(fi,
-42)
N(i?+B)H(II a+6 I/-1) dx. s c.1,
that A, has mean value zero was also employed
(3.11)
here.
3.2. Isotropic two-phase composite The bound (3.11) applies to any two-phase composite. Now it is specialized to the isotropic, incompressible composite, described above. The tensors M, are taken to be isotropic and to correspond to incompressible behaviour, so that they are characterized by shear viscosities nL,, and the “comparison medium” relative to which A0 is calculated, is taken to be isotropic and incompressible, with shear viscosity ,LL”. The operator A0 generates both shear and hydrostatic stress but, since the latter has no influence on the overall constitutive response, it is necessary only to consider that part of A0 which maps the sub-space of tensor fields with zero trace to itself. As shown in Appendix A, this restricted operator, A6 say, is a convolution operator over C,, : A; = 2~,(16’-
E),
(3.12)
where I has components zi,L, = i L6&6,, + 6,,6,L -: 6’ is a scalar convolution kernel with components
operator
with kernel
6,j6k,13
(3.13)
(6(x) - 1) and the operator
E has
ICI”1
(3.14)
2 - --[i5,&&
exp(-i<*x).
The sum in (3.14) extends over all reciprocal lattice sites except the origin; thus, components of < are integer multiples of 27~. It is also demonstrated in Appendix A that, when the microgeometry of the composite is statistically isotropic, the trace-free part of the tensor Q is
Bounds of third order for nonlinear composites
99
Q'=(2poj(3. Furthermore,
the trace-free
(3.15)
part of A is A,
(because (I&-E) is a projection). B is expressible in the form
mkd
cm (-1
=
Finally,
B’ = ;(2/4’[$~
Q,
the trace-free
-c,
(3.16) part of the three-point
++,]I.
tensor
(3.17)
Here, M, is a parameter introduced by Milton (1982) and shown by him to be restricted to the interval [0,11, to avoid crossover of upper and lower bounds for linear composites. In fact, for the corresponding linear problem, the extreme values for m, produce the Hashin-Shtrikman upper and lower bounds. These are known to be attainable and so microgeometries must exist which correspond to any m, in [0, I]. Finally, since hydrostatic mean stress is of no interest, 5 is taken to be trace-free, and, because of statistical isotropy, the “polarizations” 4, are selected so that it, -ir2 With the above notation,
+
s
the bound
= 85/(2/4,).
(3.18)
(3.1 I) becomes
N(ii + i+)H( (15+
ciI/ -
2) dx.
(3.19)
(‘0
where m2 = I --m,. It is noted in passing that the corresponding bound for a linear composite with viscosities ~1, and pL?is contained in (3.19) and is obtained by disregarding all of the terms associated with the nonlinearity. This gives, upon optimizing with respect to the parameter p.
Setting m, = 0 or m, = 1 generates the bounds of Hashin and Shtrikman this composite ; the range [0, l] for this parameter is thus confirmed.
(1963) for
D. R. S. TALBOT and J. R. WILLIS
100
3.3. Further detail
The remaining optimizations are best addressed in the context of the example, to avoid introducing further notation. The general approach will in any case be easy to recognize. The potentials UO,from which U, is composed are selected to be (3.21) Here, i, = (3/2)“2A, to match the definition of gc. The variation of (r/,- U,,.)(a) with or is illustrated in Fig. 1, in which three cases are distinguished. In Fig. 1(a), 3ne,(A,/c~,)“~’ < or/p,.. (U,- U,,) is a concave function of CJand hence coincides with (U,-U,,),,. In Fig. l(b), 6eo/(n+ l)(&/~~)~- < 0,./p, < 3ne,,(3Ja,)“-‘, and in Fig. l(c), G,/K < W(n + 1Wo,.Y’ . (UT- u,,> IS no longer concave ; the corresponding plots of (U,- U,,),, are shown by the dashed lines. The only complicated feature is the value (TV of ge which corresponds to the point of tangency between (U,- Uo,) and (U,-- UoJ** in Fig. 1(b). This is given as the solution of the equation (3.22) With a view towards optimizing the bound (3.19) with respect to p,, p2, it is helpful to present (U,- U,,,),,(a) as a function of l/~~, with (T, taking a fixed value. First, when ~~ > A, n+ I
(U~-Uo,)**(a)
=z 0 2 -g.
=(u~-u(,,)(o)
(3.23)
I
I
When oe < A,, different ranges for l/~~ have to be distinguished
0-A /.lr 6e, (ZiJGK
he0 m$
3
II-
,r+I
I
- $_
(3.24) I
0
)LC“-’ 0 (U,.- Uo,)**(a) = 2
$ n+’ -&-a,) 0 I
(U,-
The bound (3.19) is expressible in the form
U,,)**(a)
[co ($
= fy+
z]-
2,
(3.25)
(3.26)
Bounds
of third order for nonlinear
composites
101
(4
Fig. 1. Schematic 3nr,(%,/o,)“~ I <
plots
of (r/,-
U,,)
(solid
(~,,‘p,;(b) 6c,/(n+ 1)(&/a,)“--
lines), and (U,- U,,),, i 6,;~~ < 3nq,(i.,io,)“+’
(dashed
lines),
against
cr.
: (c) cr,!pr< 6e,,I(n+ I)(i,,‘a,)”
(a) ‘.
102
o’(a)
D. R. S. TALBOT
< c,
2
+(u1
-
1
Uo,)**(P,)
[
+CZ
and J. R. WILLIS
-A,) Ii N(a)H(a,
~+w(i,,d**(P,)
dx,
+
(3.27)
CO
[
where X, = [~+C2(~C2+~m,)p*-~c,P]aeZ
= [~(I-~c,B)*+~c,m,B2]aez,
x2
=
=
[~+c,(~c,+~m,)~‘+~c,~]a~
p, =(l-;c*jQa,
[ By introducing
dimensionless 1 _=-
eo
PI
Cl
variables -
01
+cu,
-
(3.27) with
Uo,)**(P,>
1
Xl
=&c,
A
=p,,L
(3.30)
PI
)
it is easy to verify that the minimum
min 5 VI’0 [ p,
-
optimizing
so that
,!p I 1 ,>
Pie
(
of p,, p2. Therefore,
2 +(Ul
(3.28) (3.29)
p* =(l+~c,~)a.
The nonlinear penalty term is independent respect to p, requires the evaluation of
min ~, >.
[~(l+~c,B)‘+~c,m,B*]a,2,
is expressible
Uo,)**(P
in the form
, >] = eo(T, p)“”
V(_J?,J,).
(3.31)
Similarly,
min Ir,>o
2
+ (U2 -
(3.32)
Uo2)**(P2)
where 113and J2 are defined so that x* = p:J?*, The function The bound
U”(a)d
eoo2
/I, = p&.
V requires computation. (3.27), optimized with respect to p, and p2, now becomes
(3J+‘[cl (3(1-$,p)il- V(R,J,)-t
(3.33)
103
Bounds of third order for nonlinear composites
3.4. The nonlinrar
penalty
term
Suppose, to be definite, that cr, < o?. Then, chosen as in (3.21), it follows that
with the comparison
potentials
U,,,
(3.35)
N(a) d H(a), where
(3.36)
With the polarizations deviatoric part
chosen
so that (3.18) is satisfied,
b’ = [Id’-P(Id’-E)(x,
the trial stress field has
(3.37)
-~,)]a.
Now let (3.38)
It is demonstrated and hence that
in Appendix
B that t(x) has bounded
mean
(square)
oscillation,
(3.39)
meas{ x E C, : /(t /I > s} < A exp( - KY). The constants A, a depend on the volume fraction c, and on the “pattern” a/(\ ii // ; their calculation is outlined in Appendix B. It follows that meas{xECO:I/5--fiBtli
>s/(511}
drneasjxECO:~lflt~~ = measjxE
>s-lj
C,, : I/ t // > (s- 1)/p]
d Aexp[-r(s-l)ifi]. Elementary
calculation
s( 0
(3.40)
now gives n+l
N(a)H(a,
of loading.
-A,) dx < eOdl
5 0 01
* A exp [ - x(s - 1)/fl].s” ds s I,’ a
where (3.42)
104
D. R. S. TALBOT
and J. R. WILLIS
u = 11B 11s/A. Finally, repeated inte-
having employed the variable transformation gration by parts shows that
s
n(n- l)*.*(n-m)
3c
u”e-‘u
(3.43)
m+l P
I
where m is the integer satisfying m < n < m + 1.
4.
RESULTS AND DISCUSSION
Calculations have been performed for a two-component composite of the type described above. The overall potential u”(a) has the form (3.2). The characterizing stress crOis a homogeneous function of degree zero in a, that is, a function of the stress pattern a/II 5 I). For any fixed choice of stress pattern, a lower (or upper) bound for 0 generates an upper (or lower) bound for go. Figure 2 shows such a result. The stress pattern corresponds to the in-plane shear
bfj = 0 otherwise.
(4.1)
The figure plots bounds for oO/ozas a function of 1/n, for an isotropic composite with c, = Cl = 0.5, and 0, = a,/2. The uppermost line in the figure corresponds to the classical “Taylor” bound, obtained from the first of equations (2.11). The lines labelled (a) and (b) correspond to “improved” lower bounds for 0 (and hence upper bounds for oO), related to (2.26). These do not require the use of a nonlinear
(4
0.72 0.7 -
-
(b)
0.68 0.66 0.64 0.62 0.6 i 0.58 0.56 F 0.54 ' 0.1
Fig. 2. Bounds
I
I
I
I
1
I
I
I
I
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
for a,/rr, plotted
against
l/n, for a composite
for which c, = cz = 0.5 and 0, = ~~12
Bounds
of third order for nonlinear
composites
105
comparison medium, and were already calculated by Ponte Castaiieda [ 1992, equation (4.41)]. These two lines correspond to the extreme values of the Milton parameter : nt, = 0 for (a), and m, = 1 for (b). If no information on three-point statistics is available, the only guaranteed bound is that obtained by assuming the worst, to give line (a) (m, = 0). This is, in fact, the “nonlinear Hashin-Shtrikman bound” for this composite. reducing to the classical linear bound of Hashin and Shtrikman (1963) in the case n = 1. The remaining lines in Fig. 2 are lower bounds for a,,/az (corresponding to upper bounds for 0. The lowest bound is the Sachs bound, obtained from the second 01 equations (2.11). The other two lines, labelled (c) and (d). are “improved” bounds, making use of three-point statistics and obtained. as described in Section 3. by employing a nonlinear comparison composite and allowing for the nonlinear “penalty” thereby incurred. Line (c) corresponds to the value 177,= 0 of the Milton parameter. Note that it is a lower bound but coincides with the upper bound (a) when II = 1 : this corresponds to the fact that the Hashin-Shtrikman upper bound for the linear composite is attained when m, = 0. The lower of the two new lower bounds, labelled (d), corresponds to the value 111,= 1. This coincides with the corresponding upper bound, (b), when y1= 1, and is consistent with the attainment of the linear Hashin-Shtrikman lower bound in this case. When the Milton parameter is unknown. the only guaranteed bounds are those obtained by adopting the “worst-case” values : 177, = 0 for the upper bound, and m, = 1 for the lower bound. These are independent of three-point statistics and can be considered as generalizations to nonlinear material behaviour of the classical linear Hashin--Shtrikman bounds. Calculations have also been performed for other volume fractions and for other ratios (T~!~J,.The results have the same form as shown in Fig. 2. As remarked previously, the “easy” bound, not requiring use of a nonlinear comparison medium, was obtained earlier, by Ponte Castaiieda (1992), and methodology has been available since 1985 for construction of the nonlinear Hashin-Shtrikman bound. Progress in constructing the more difficult bounds is of more recent origin. The first steps were taken by Talbot and Willis (1994) in the context of electrostatics and Talbot and Willis (1995) for incompressible elasticity. Those works focussed directly on the nonlinear Hashin-Shtrikman bound. by taking as a starting point the nonlinear Hashin-Shtrikman variational principle as introduced by Talbot and Willis (I 985). The new contribution here is to take the formulation of Talbot and Willis (1992) as the basis. The direct, and predictable, benefit is that this allows the construction of bounds of higher order, such as the three-point bound given explicitly here. Additional benefits, however, have also emerged. First. the additional flexibility contained in the present formulation, far from introducing more complication, has actually led to simplification. Several of the optimizations could be performed analytically, and the remaining computation was actually simpler than that required previously. The other benefit is still more interesting. Although nothing is gained in the linear problem, in the nonlinear case the polarization has to be chosen so as to provide the best trade-off between the “ordinary” part of the bound expression and the nonlinear penalty. It appears that the “ordinary” part of the bound (3.34) varies more slowly about its minimum value than the corresponding expression employed by Talbot and Willis (1995), even in the extreme case III, = I. The new lower bound
106
D. R. S. TALBOT
and J. R. WILLIS
of Hashin-Shtrikman type produced here is correspondingly larger, when 12> 1, than that produced before. It is of course unlikely that the new lower bound is optimal when n > 1. Improvement over the classical Sachs bound declines as n increases. It should be noted, however, that numerical experiments on a “composite sphere” model, performed for a similar problem in electrostatics by Barrett and Talbot (1996), displayed exactly this trend with increasing n, when a sphere of the “harder” material was surrounded by a shell of the “softer”.
REFERENCES Barrett, K. E. and Talbot, D. R. S. (1996) Bounds for the effective properties of a nonlinear two-phase composite dielectric. Proc. 8th Int. Symp. Continuum Models and Discrete Systems (ed. K. Z. Markov), pp. 92-99. World Scientific, Singapore. Beran, M. J. (1965) Use of the variational approach to determine bounds for the effective permittivity of a random medium. Nuovo Cimento 38, 771-782. Dendievel, R., Bonnet, G. and Willis, J. R. (1991) Bounds for the creep behaviour of polycrystalline materials. Inelastic Deformation of Composite Materials (ed. G. J. Dvorak), pp. 175-192. Springer-Verlag, New York. Hashin, Z. and Shtrikman, S. (1963) A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11, 127-140. John, F. and Nirenberg, L. (1961) On functions of bounded mean oscillation. Communs. Pure Appl. Math. 14,41 S-426. Milton, G. W. (1981) Bounds on the electromagnetic, elastic, and other properties of twocomponent composites. Phys. Rev. Lett. 46, 542-545. Milton, G. W. (1982) Bounds on the elastic and transport properties of two-component composites. 1. Mech. Phys. Solids 30, 177-I 9 1. Ponte Castafieda, P. (1991) The effective mechanical properties of nonlinear isotropic composites. J. Mech. Phys. Solids 39,45-71. Ponte Castafieda, P. (1992) New variational principles in plasticity and their application to composite materials. J. Mech. Phys. Solids 40, 1757-l 788. Ponte Castafieda, P. and Willis, J. R. (1988) On the overall properties of nonlinearly viscous composites. Proc. R. Sot. Lond. A416,217-244. Ponte Castafieda, P. and Willis, J. R. (1993) The effective behavior of nonlinear composites : a comparison between two methods. Materials Science Forum 123-125, 351-360. Reuss, A. (1929) Calculation of the flow limits of mixed crystals on the basis of the plasticity of mono-crystals. Z. Anger. Math. Mech. 9, 49-58. Sachs, G. (1928) Zur Ableitung einer Fleissbedingun. Z. Ver. Dtsch. Ing. 72, 734736. Stein, E. M. (1967) Singular integrals, harmonic functions, and differentiability properties of functions of several variables. Proc. Symp. Pure Math. 10, 316335. Talbot, D. R. S. and Willis, J. R. (1985) Variational principles for inhomogeneous nonlinear media. IMA J. Appl. Math. 35, 39-54. Talbot, D. R. S. and Willis, J. R. (1992) Some simple explicit bounds for the overall behaviour of nonlinear composites. Int. J. Solids Struct. 29, 1981-1987. Talbot, D. R. S. and Willis, J. R. (1994) Upper and lower bounds for the overall properties of a nonlinear composite dielectric. I. Random microgeometry. Proc. R. Sac. Lond. A447, 365384. Talbot, D. R. S. and Willis, J. R. (1995) Upper and lower bounds for the overall properties of a nonlinear elastic composite. Proceedings, IUTAM Symposium on Anisotropy, Inhomogeneity and Nonlinearity in Solid Mechanics (ed. D. F. Parker and A. H. England), pp. 4099414. Kluwer, Dordrecht. Taylor, G. I. (1938) Plastic strain in metals. J. Inst. Metals 62, 307-324.
Bounds of third order for nonlinear
composites
107
Toland, J. F. and Willis, J. R. (1989) Duality for families of natural variational principles in nonlinear electrostatics. SIAM J. Math. Anal. 20, 1283-l 292. Van Tiel, J. (1984) Convex Analysis. Wiley, New York. Voigt, W. (1889) Ueber die Beziehung zwischen den beiden Elasticitatsconstanten isotroper K&per. Ann. Phys. (Leipzig) [3] 38, 573-587. Willis, J. R. (1977) Bounds and self-consistent estimates for the overall properties of anisotropic composites. J. Mech. Phys. Solids 25, 185-202. Willis. J. R. (1981) Variational and related methods for the overall properties of composites. Ad~unc~~.s in .4pplird Mechanics (ed. C.-S. Yih), Vol. 21. pp. I 78. Academic Press. New York. Willis, J. R. (1982) Elasticity theory of composites. Mrchanics of’ Solids. Thr Rodm~~~ Hi// Si.\-ticth Anniwrsrrry Volume (ed. H. G. Hopkins and M. J. Sewell). pp. 653-686. Pergamon. Oxford. Willis, J. R. (1983) The overall elastic response of composite materials. .I. App/. Mcch. SO, 1202 1209. Willis, J. R. (1986) Variational estimates for the overall response of an inhomogeneous nonlinear dielectric. Homogenization und IZfjtictiw Propertics of’MatcTriu1.r and Mediu (ed. J. L. Ericksen. D. Kinderlehrer, R. V. Kohn and J.-L. Lions). pp. 245-263. Springer-Verlag, New York. Willis, J. R. (1991) On methods for bounding the overall properties of nonlinear composites. J. Mrch. Phys. Solids 39, 73-86. Willis, J. R. (1992) On methods for bounding the overall properties of nonlinear composites : Correction and addition. J. Mech. Phys. Solids 40,441 -445.
APPENDIX
A : THE
DELTA
OPERATOR
The operator A,, introduced in Section 2.2 is derived from the Green’s function medium with tensor of elastic moduli L,,. If L,, has components L,,k, the Green’s satisfies the equations I,,&.,,
+ &,,]6(x) -
G for a function
II = 0, x 6 cc,.
(A.1)
where it is assumed that C,) has unit volume and contains the origin in its interior. The Green’s function is also required to be C,,-periodic and to have mean value zero. It follows immediately by Fourier transforms that G(x) =(2rc-’ where e(c)
c c(<)exp(-it-x). <#O
(A?)
is the inverse of the matrix K(c) with components (A.3)
K,, (<) = L,,&,&
The sum in (A.2) ranges over reciprocal vectors < whose components are integer multiples of 2rl. As discussed. for example, by Willis (198 I, 1982), the operator A,, is related to an operator F,, which, in the present case, is of convolution form over C,, and has kernel with components ftiil given by (A.4)
F,,/J,(x) = - G‘?,,,,‘,(X) I,/,,,,‘<,I. Then, A0 is given as A0 =L,6 where 6’ is a scalar components
operator
with kernel
-LO:Fo:L, 6(x)-
I. In suffix notation,
(A.5) A0 has kernel
with
D. R. S. TALBOT and J. R. WILLIS
108
&k/(x) = &~(&x) Equivalently,
in terms of Fourier
- 1) - Li,pyrpy~.s?i(x)L,~,.
c4.6)
transforms,
d”(C) =L”-LD:L,:r”({):L”,
(A.7)
where f,,(r) denotes the Fourier transform of I,,. In the particular case of an isotropic reference medium with Lame moduli lo, p,,, calculation shows that
It may be noted that, in the limit of incompressible
behaviour
(& + co), relation
(A.8) gives (A.9)
where E is defined by (3.14). The transform of A0 follows from (A.7). Taking the incompressible limit gives the expression (3.12) for its “trace-free” part A&, that maps the space of trace-free tensors into itself. Some elementary properties of Ah are required. First, it is noted that (M-E) defines a projection. This follows directly from its Fourier transform and use of the convolution theorem I:I=I
and
E:E=E.
(A. 10)
The tensor Q’ is also required. This is defined by (3.8). When the micro-geometry is “statistically isotropic”, and has fine scale relative to the dimensions of C,,, the tensor Q’ is isotropic and so of the form Q’ = Q’I. The constant Q’ can be found by calculating the (ijtj) invariant of Q’. It follows, since I,,i, = 5 that the (ijij) invariant The result (3.15) follows The tensor B is also calculation is not needed
and
Ei,,, = 2,
(A.11)
of (IS’-E) has transform 3 (independent of 5) and hence equals 36’. immediately. isotropic when the composite is statistically isotropic. Its detailed but the form (3.17) for B’ is a direct consequence of the isotropy.
APPENDIX Talbot and Willis (1994) introduced
B : THE BMO PROPERTY a definition
which, adapted
to present
notation,
reads :
“Let t(x) be defined and let 11 t 11be square-integrable over a cube C,,. The field t(x) has the property of bounded mean (square) oscillation if there is a constant K such that, for every parallel sub-cube C of C,,, and some constant t,-, the inequality 1 (1t-tc ~ m(C) s C”
ll’dx < K2
holds. Here, m(C) denotes the Lebesgue measure of C.” This is a slight variation on the original definition of John and Nirenberg (1961), in using “square-integrable” instead of “integrable”. The difference is not fundamental but it was found to lead, in the present context, to better values of the constants. By tracking the reasoning of John and Nirenberg, Talbot and Willis showed that, for any such field t(x), constants A and c( could be found such that the inequality (3.39) holds. The constant A is given as
IO9
Bounds of third order for nonlinear composites /It(x) II2 dx.
A = B
and B and c( appear in equation B
=
(B.2)
(A 40) of Talbot and Willis (1994):
(t/K-
I)*
2 ln(t/K) ’
t/I-Q’
K’(fi+
(B.3)
’ = K&t/K)
Here, t is a parameter required to satisfy t > K. Its value is to be chosen optimally in relation to minimizing the penalty term (3.41). The inequality (3.39) in fact holds subject to the restriction s > (J?K+ t)/(t/K1). Equation (B.2) can be simplified, as follows dx(X, -c,)(M’-E):(I6’-E)(x,
-1.1):
dx(x,-c,)(I6’-E)(X,-c,):
03.4)
since (Id’- E) is a projection. In view of the overall isotropy, the tensor appearing in the last expression above is a constant times I, and the constant can be evaluated as it was in the calculation of Q’ : it has the value 3c,c2/5. Thus, (B.2) simplifies to A = 3c,c,B/5.
(B.5)
The task that remains is the estimation of the constant K. The construction proposed by Stein (1967). It is observed first that (3.38) can be given in the alternative form
t(x) =(IS-E”)*(x,
,
-c,)
follows
one
(B.6)
where the convolution * is over all space, the operator S has kernel 6(x) and E” is the infinite-body limit of E, obtained by replacing the Fourier sum in (3.14) by a Fourier integral (interpreted in the sense of generalized functions). For a chosen cube C, let B be a ball centred at the centre xc of the cube, and with sufficiently large radius to enclose C. Now define,f,(x) and,f,(x) over all space so that f,(x)
=x,(x)-c,> =O,
xgB
x$B
(B.7)
and
f2c4 =
XI
(x)-c, -f,(x).
(B.8)
Let
03.9)
It follows immediately,
because
(IS-E”)
is a projection,
that
D. R. S. TALBOT and J. R. WILLIS
110 11 t, )I2dx d
(B. 10)
(x, -c,)* dx < max(c:, &m(B),
sc and thus
&
sc
II t, II2 dx < cmax(c?,&,
(B.11)
where
c = W>lm(c)
(B.12)
is a parameter open to choice (c > $&r/2). Next,
s c
IIt, (9 --t,(xc) II2dx =
s11s c
dx
2
dY(E”(x--Y)-E”(xc-~)):(~/Il~ll)f,(~) YdB
2
-EPY,,(xc -Y))(flk,/ll 8 II)1 max(c:, c:) > (B.13)
= m(C)A’* max(c:, c:),
the last equality defining A’. The relations (B. 11) and (B. 13) already show that the field t(x) has the BMO property. The calculations upon which (B.3) is based require t, to be identified with the mean value over C oft. With the notation t ,,-, t2Cfor the mean values oft,, t2 over C, Minkowski’s inequality gives I;2
II t, -t,c
IItz -cc II2dx
/I* dx
(B.14)
Now
s <
ll t, -t,,
II2 dx =
sc
(II t, II2 - /Kc II2 )dx
G
sc
II t, II2 dx
(B.15)
< cm(C) max(c:, c:), from (B.11). Next, 112
(s c
IItz--zc II*dx
> (s <
c
II f2 --t,C+)
112
II* dx
+ >
(s c
II t,(xc)--t,c
l.‘Z
II2 dx
> >
(B. 16) and
s c
II f2 (xc) - f2c II’ dx = II tz (xc> - t,c II‘m(c)
[tz - f2(xc)1 dx
Bounds
of third order for nonlinear
composites
IItz--tzk) <
(S
IItz -t,(xc)
III
IIdx (B.17)
II*dx,
C’
by an application
of Hiilder’s inequality.
s
/ltz-ttzc C’
Finally, relation
It follows,
from (B. 16), (B. 17) and (B. 13) that
l)‘dx < 4m(C)A”max(cT,ci).
(B.18)
(B.1) follows from (B.14), (B.15) and (B.18) with K=(c’
All that remains components
is the calculation
2+2A’)max(c,,cz).
of A’. Inverting
its Fourier
(B.19) transform
shows that E ’ has
(B.20) where r = 1x I. The homogeneity
of E” permits
A” to be calculated
dy I(E;dy -x) -E$,(~)H~ull~
as
II) I .
(B.21)
where Q is a cube centred at the origin, with sides of length 2, and R is a ball centred at the origin, with radius v > & ( so that c = nv’/6). This expression has been evaluated when the stress pattern is given by (4.1). The y-integrals were expressed in polar coordinates r = 1y /, I), 4, and then Q was replaced by u = cos 0. The mid-point rule was used for integration with respect to u and Gauss-Legendre formulae were used for all other integrals. As a result of this calculation, the value K = 16 max(c,, cJ was employed for the BMO estimates, since this clearly overestimated K, allowing for any numerical inaccuracy.