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Characterization of the thermoelastic behavior of syntactic foams
S1359-8368(98)00002-X
ELSEVIER
Composites Part B 29B (1998) 351-361 © 1998 Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-8368/98/$19.00
Characterization of the thermoelastic behavior of syntactic foams
Gildas L'Hostis a and Fran(~ois Devries b aLaboratoire de M~canique et Mat#riaux, Ecole Centrale de Nantes, 44321 Nantes, Cede)( 3, France bLaboratoire de Mod#lisation et M#canique des Structures, URA 1776 du CNRS, UPMC/ENSAM/ENS Cachan, 4 Place Jussieu, 75252 Paris, Cedex 05, France (Received 22 October 1997; accepted 26 December 1997)
The objective of this article is to highlight the existence of a specific geometrical configuration of syntactic foams (solid or hollow microballs soaked in a resin), which enables the design of a composite material possessing good thermal insulation properties while ensuring optimal mechanical behavior in compression and a significant weight gain. To carry this out, an economical approximated homogenization technique, adapted to the composites being studied herein, is first of all proposed. The results yielded turn out to be in good agreement with those obtained by the application of classical methods which have already demonstrated their effectiveness. In order to highlight the desired geometrical configuration, a minimization problem is then developed and solved. The value of this approach lies in the fact that all the computations necessary can be performed in a quasi-analytical manner, thereby avoiding reliance upon heavy computing resources. The resultant design tool proves to be most costeffective and simple to use. © 1998 Elsevier Science Limited. All rights reserved INTRODUCTION The use of composite materials to develop highperformance structures was constantly increasing over the past few years. This was the case within a wide array of industrial fields. The main particularity of a composite media resides in its strong level of heterogeneity, quite often giving rise to numerical simulations that are costly and difficult to implement. Consequently, when the optimal design for a composite is being sought, it would appear judicious to proceed by substituting a homogeneous media exhibiting an equivalent mechanical behavior. Much research work has in fact been devoted towards achieving this objective of computing the equivalent homogenized behavior of a variety of composites, and has served to derive several homogenization methodsl-4. These cited works all rely upon the existence of an elementary volume representative of the internal structure of the composite under study which can be differentiated in terms of the way the link between the composite's scale and the microstructure's scale is being handled. From such an approach, it thereby becomes possible to compute the homogeneous behavior that is mechanically equivalent to the behavior being exhibited by the composite material studied, on the basis of an understanding of both the specific geometry and the thermomechanical characteristics of the composite's constituents. Our orientation throughout this article stems from the general framework for the optimal design of composite materials 5. It is thus being proposed herein to devise an
economical and reliable computational strategy for the best geometrical configuration of those composites known under the designation of 'syntactic foams '6, which consist of either solid or hollow microballs soaked in a resin. These composites, which have become more commonly used, especially as surfacing elements for spatial and aeronautical structures, must perform certain thermal insulation functions while providing good compressive strength. In order to account for both the relative complexity of the phenomena involved (elasticity/thermal conductance) and the fact that from a general standpoint, the properties of insulation and mechanical strength counteract one another, a minimization problem with constraints is constructed on the basis of the equivalent characteristics of conductance and volumetric compressibility. The solution to this problem will enable determining the composite's specific geometrical configuration that induces the optimal set of thermal insulation and compressibility properties. In order both to avoid being penalized by the high numerical costs associated with seeking such a configuration and to develop an appropriate tool for performing analyses and producing reliable and efficient designs, we will rely herein upon a homogenization technique inspired by the work conducted by 1. A derivative of the homogenization method for fine periodic structural media 7-12, this technique consists of building fields that are both kinematically and statically admissible for the problems to be solved on the representative microstructure of the composite under study. As a result, a framework for the real homogenized behavior is obtained from the perspective
351
Thermoelastic behavior of syntactic foams: G. L'Hostis and F. Devries of the energy. The boundaries of this framework can then be defined in a quasi-analytical fashion, a feature which is of obvious interest when the anticipated minimization problem with constraints is subsequently solved. As a means of measuring the reliability of this homogenization technique, the homogenized characteristics it produces are then compared with those obtained both by utilizing another homogenization method (three-phase media) and by the finite element solution to the classical problems of the periodic-media homogenization method. Lastly, the influence of various geometrical and thermomechanical parameters of the constituents on the optimal configuration will also be assessed.
PRESENTATION OF THE PROBLEM AND THE TARGETED OBJECTIVE It is assumed that the composites being considered herein accept as their representative elementary volume (REV) a tetra-kaidecahedra with a characteristic dimension of L (Figure 1), thereby leading to a cubic pattern centered around the inclusions. Its center consists of a spherical inclusion which is assumed hollow whose exterior diameter is denoted D; e represents the thickness of the layer of material forming this inclusion (0 < e --< D2). The geometry of the composite under study is therefore fully established once values can be determined for the three parameters: L; 7"i and 7, where 7 represents the volumetric rate of inclusion ri, which have been defined respectively inl3: 7 = D' 7 E [0, ~] ri = 4 8 ~ ( I r £)3D ( r i g
[0, 1--~J 7rf13] ~/
where 0 --
Dmax= V/~ L 3 fl = X/11 -t- 3cos20
and 0 = (arcsin ( ~ 3 ) )
isotropic and that the thermal conductance complies with the Fourier law, in such a way that the parameters able to fully describe their behavior are: the volumetric mass, p, Young's modulus, E, Poisson's ratio, v, the dilatation coefficient, or, and the conductance coefficient, X (the indices m and r will be assigned subsequently to the quantities relative both to the matrix and to the reinforcement or inclusion, respectively). In the perspective of designing an optimal composite that combines good properties with respect to both thermal insulation and volumetric compressibility, the existence of specific values for the parameter 7 has already been demonstrated14; this finding has led to stiffening configurations characterized, for example, on the basis of volumetric compressibility modules, by kh/kr, > 1, neutral (kh/km = 1) as well as to softening (kh/km < 1); such configurations can also be determined for the thermal behavior as a function of the ratio xh/Xm (the exponent h is based on the modules that define the homogenized behavior). It is being proposed herein to emphasize whether a geometrical configuration exists for the fixed values of the other parameters, both geometrical and mechanical, that serve to characterize the composite's REV, as defined by a specific value of the parameter 7, which is softening with respect to the thermal and stiffening characteristics from a mechanical standpoint. In the case of the composites being considered, these two properties serve to counteract one another. In fact, on the one hand, insulation characteristics are obtained in part thanks to the existence of spherical, thermally-insulating voids, while on the other hand, the presence of these voids within the composite exert a negative impact on its resistance to a hydrostatic compressive loading. Consequently, the desired optimal composite will be defined by a geometry, if one does exist, close to the configuration that is mechanically and thermally neutral. In order to identify the value of the parameter 7 that results in a thermal as well as mechanical neutrality of the composite's equivalent homogenized behavior, the following two quantities are introduced:
°
Afterwards, we will be considering that the composite exhibits no defects whatsoever, that the presence of voids is caused solely by the inclusions, that the constituents adhere perfectly to one another and that their behavior is governed by the classical linear thermoelastic operator. In addition, it is assumed that the constituents are both homogeneous and
Xh kh C1 = 1 - Xmmand C2 = kmm- 1
(1)
in such a way that the value searched for the design parameter 7 must be a solution to the following minimization problem: min F(7) = min {C12(7) + C~(7) }
(2)
under the constraints
Q Figure 1 Representative elementary volume
352
C1 ~ 0 , C 2 : > 0 ,
C3 ~ 0 , C4-----0
with: 1
C3 = 7 and C 4 = ~ - 7 The problem so defined is in fact a classical minimization problem subjected to positivity constraints 15 which only depends on the single design parameter, 7, and which corresponds to a given volumetric fraction of inclusion, ri. In
Thermoelastic behavior of syntactic foams: G. L'Hostis and F. Devries the aim of simplifying to a standard minimization problem, the Lagrangian associated with this problem is introduced, as defined by:
L=2-2
~mm-t-~m q-~km J
"+- km
- ~1C1 - ~2C2 - ~3C3 - ~4C4
where ~ 1, ~2, ~3 and 44 are the Lagrangian parameters associated with the constraints Cl, C2, C3 and C4. The existence of an extreme point is given by the KuhnTucker conditions which can be reduced in the present case
to:
3£
0 3£,
3L
OL
00L 0 ag4 =
(3)
with these conditions comprising the set of preconditions necessary Ibr the existence of the optimum being searched. The search for this specific configuration thus leads to coupling the solution to a minimization problem with the determination of the homogenized behavior, as
structural media 7'8. This technique relies upon the hypotheses of the composite media's mechanical and geometrical periodicity, as well as upon the periodicity of the magnitudes that characterize the media's behavior (deformations, stresses). These periodicity properties are then complemented by relational data on averages that reflect the link between the studied composite and its microstructure. These relations may be of two types, each of which serves to characterize two distinct approaches that do nonetheless present the same level of precision 16. The imposed deformation and temperature gradient averages give rise to the primal approach, with the dual approach 17 being associated with the conditions of imposed stress and heat flow averages (see eqn (4)). Moreover, for each approach, the macroscopic loading is complemented by the data of a temperature increment, r, assumed constant on the REV ~8. In denoting the microscopic fields of stresses, heat flow, displacement and temperature by m q, u and O, respectively, the problems to be solved for both approaches can be written in the following form:
e(u), VyO Y periodic divyo
o(n), q.n Y antiperiodic
= 0 in Y*
and for the primal method a= ~
e(u) + 3v
tr[e(u)lId - 3k~Idr in Y* <>r = E
divyq = 0 in Y*q = - XVyO in Y*
[ << Vy0 >>V = F
o(p) = O, q.l' = 0 on 3 Y T
and for the dual method
(O-)y= ~, (q)r = q (4) characterized by the modules k h and Xh. A numerical solution procedure can be envisioned5, yet this would require, at each step of the minimum searching algorithm, solving cellular problems which are vital to obtaining the homogenized behavior. In the case of the three-dimensional composites studied herein, such a procedure could only be carried out through the use of numerical methods (for example, finite element methods) which comprises the disadvantage of generating high computation times and costs, thus representing an incompatibility with the objective of developing a simple, economical and efficient tool. In order to circumvent this disadvantage, one obvious solution would consist of determining a literal expression not only for the homogenized behavior but, at the same time, for the value of the ~/ parameter, the solution to the minimization problem, as well. The following section is specifically aimed at finding such a solution.
where u and n represent the unitary normal lines inside to the inclusion and outside to the REV, respectively, where Y* is the solid part of the outside normal, k the volumetric compressibility modulus of the media, and where the divergence and gradient operators have been denoted by divy and Vy in comparison with the space variables (Yl,Y2,Y3) related to the REV. Once these problems were solved, the homogenized behavior is then obtained by specifying the relationships linking the macroscopic quantities r. (stresses), E (deformations), F (temperature gradient) and Q (heat flow) to their corresponding microscopic quantities, such that:r
= ~ = Q h E - Q hethr and
< q > r = Q = - XhF for the primal method << e(u) >>r = E = she + othr and << VyO >>r = - LhQ for the dual method
HOMOGENIZATION AND APPROXIMATED BEHAVIOR The desired homogenized behavior may be derived by utilizing the homogenization technique for fine periodic
where the various averages (caused by the presence of the void) are defined by:
,f
= ~
,f o/n~dr
rfidy and ( <>r)i = ~
353
Thermoelastic behavior of syntactic foams: G. L'Hostis and F. Devries and where Qh (Qh : ( s h ) - l ) , ~h (hh = (Lh)-l) and ~xh represent the homogenized tensors of stiffness, conductance and dilatation, respectively. By using the linearity of these problems with respect to the macroscopic data, this approach leads to solving in the general case 10 cellular problems (seven elasticity problems and three thermal problemsl°'11). For the type of network foreseen (centered cubic), in making use of the fact that the three coordinate planes constitute material symmetry planes for the various cellular problems, it has been demonstrated in 19 that the desired homogenized behavior is necessarily quasi-isotropic in the reference (O,yl,Y2,Y3) relative to the base period Y. The tensor S h of the homogenized softnesses thus only accepts three non-zero components: 1
S~lll I : ~ ,
h
ph
__
h
1 h
Sl122 : Eh, S1212 : ~ G , with [Qh] = [Sh]
-1
and the homogenized conductance tensor h h is spherical, such that:
Figure 2 Decomposition of the representative elementary volume
In order to implement this method, the first step entails decomposing the REV Y into two parts: the sphere V included in Y and its complementary element W (Figure 2), defined by:
Y = W"V The coefficient of volumetric compressibility is also introduced and defined as: 3kh = Q~Illl -~- 2Q~1122=
1 Shll 1 Jr- 2sh122
(5)
As already specified, solving the problems encountered on the REV and then deriving the homogenized behavior requires, in the case of three-dimensional geometries, utilizing adapted numerical techniques (for example, finite elements) which thereby makes it impossible to develop an analytical expression for the characteristics of this behavior. Nonetheless, in order to achieve this objective, in relying on the combined use of both homogenization approaches (primal and dual), we will develop in the remainder of this section a method which was initially introduced to study unidirectional composites 2°-22. This technique consists of building fields that are both kinematically and statically admissible for the problems encountered on the REV from these two approaches. Once these fields have been constructed, it is shown that this technique yields, from an energy standpoint TM, a framework for the desired homogenized behavior, in such a way that: h s h
qijkhgijEkh<- QjkhEijgkh VEij :Eft
and
h ~ i j = ~ji Sijkh~ij~kh--
354
where V is defined in the spherical reference (r,O,dp) as: V = {(r, 0, th); 0 --< r --< R; 0 --< 0 --< r ; 0 --< ~b --< 27r } The total volume of matter within V is also introduced and defined by:
V*= {(r,O, dp);p < r <- R;O -< 0-< 7r;0--< ~b < 2r} along with the two surfaces being defined by: r-ext = {(r,O,z);r=R;O < 0 <- 7r;0 < 4~ -< 2a-} external surface of V r r = {(r,O, dp);r=p;O <-- 0 <-- a';0 <-- 4~ --< 2a'} internal side of the inclusion
Computation of the upper boundaries The computation of the upper boundaries for the homogenized compressibility and thermal conductance moduli, meaning those of the tensors SQh and s~h, requires the construction of fields that are both kinematically admissible for the primal approach's cellular problems. Thus, on Y* (Y* = V*uw), the fields u s and ~bs are defined in the following manner: U S
~-.
U*
inV*
Ey
in W
~)S
6"
inV*
in W
where the fields u* and 4,* are the unique solutions to the following problems: divya* = 0 in V* = ~
E
3vk e(u) + ~ tr[e(u)*]Id in V* u* = Eijyj on rYxt a*(r) = 0 on r r
Thermoelastic behavior of syntactic foams: G. L'Hostis and F. Devries
divyq* = 0 in V* q* = - ),lry4~* in V*
(7)
r* = F y j on ~ext q'r=0
E=
on r T
Classically associated with these fields, by applying behavioral laws, the tensors of deformation e s and of stresses a s, as well as the temperature gradient V4~~ and the heat flow vector q~, are defined by: e(u*) in V* es =
as = E in W
thermal conductance X +, the following macroscopic loadings (macroscopic deformations E, heat flow F) have been chosen:
o* in V* Em 3~mkm l--~vmE+ l~vmtr[E]Id
in W
U*r= A r + B
(
r 2 and ~ * =
1
and F = (0, 0, 1)
0 This choice yields, in the case of problem eqn (7), sphericaltype limit conditions. The solution fields are then sought in the following forms23: After performing a set of cumbersome and fastidious computations which have not been reported herein because of space limitations, the desired homogenized characteristics are derived in the forms:
Cr + ~2) cos0 A, B, C and D are constant
Uo : U4~ : O
k + = 4kral { 7"i(3km - x(3km + 4/Zm)) -- 3kmX } + kmbl {7"i(X(4gm -4- 3km) - 4/ZmX} 12kr(r i - x)a 1 - b 1(3kmT"i+ 4#mX) h+= b2)km(7"i(2
2)~n(b2~km(7"i - X) - a2)tr('ri + X) ) 3X) - 2X) - 2 a 2 ~ k r ( 7 " i ( 2 - 3X) + X)
with: V0 s =
V4~* inV* q S = F in W
q* inV*
It is then verified both that these fields do indeed satisfy the cellular problems' periodicity conditions eqn (4) and that: << eij(uS) >>Y = 2 - ~
a 1 = 1 - ( 1 -21,) 3
- ~,mF in W
oY{u~nj+u}ni} dI'=Eij'
<< [Vy(~S)]i >>r = ~-~ 1 foy~Snid r = Fi in such a way that u s and 4's are kinematically admissible for the set of problems to be solved. In taking account of the R E V ' s geometrical decomposition, and with the help of those admissible fields which have been introduced above, computation can finally be performed for the stiffness and the approximated homogenized conductance, SQh and s h h, as defined by:
IWI m 1 f SQ~okhEkh= ~Y-(aijkhEkh + ~1 .p v* ffqdv sXijFj= h IWIxmF.+ 1 [" IYI 'J J ~Y] J v *q~dv
b1=4+3kr(1 /d'r
a2 = al
-2~/) 3
71.~3 and X =
192
b2 = 2 + (1 - 2~/)3
Computation of the lower boundaries The next step consists of determining the lower boundaries of the homogenized behavior. The procedure adopted to carry this step out, which requires computing the tensor components sSh and SLh, calls for constructing both a stress tensor and a heat flow vector which are statically admissible for the dual approach's cellular problems. This requires defining the fields a s and qS on Y* as follows:
as = a* E
inV in W
qS = o*
inV* in W
where o* and q* are the unique solutions to the following problems:
divyo* = 0 in V*
(divyq* = 0 in V*
v . ]Id in V*~Vy~b*l _-- _ Lq* in V* e(u . ) = ~1-+o v , - ~tr[o
where aijmkh = )km~ij~kh --[-~m(~ik~jh + ~ih~jk)
In order to compute the upper boundaries of the homogenized coefficients of both volumetric compressibility k + and
o*(v) = 0 on I~T
/q*(v) = 0 on F r
o*(n) = Z(n) on ~ext
[,q*(n) = Q(n) on ~ext
(8)
355
Thermoelastic behavior of syntactic foams: G. L'Hostis and F. Devries These tensors are then associated with the deformation tensor e s, the temperature gradient V~bs as well as the displacement field u s and the temperature field th ~, defined as follows:
The solution fields are then sought in the following forlill4: ,
aij
B,
=
,
0 except a*~r = A + -fi
B 3
aoo = a~. = A - ~r
e(u*) in V* vm 1E m+vm~ _ ~ t r [ ~ ] I d
eS = e(u~) =
q* = ( C - 2 ~ ) cosO er - ( C + -~ ) sinO eo
in W
A, B, C are constants Vy$* in V* - LmQ in W
Solving the problems in eqn (8) enables deriving the desired expressions which take on the forms:
u* in V*
k-=
V,# ~ =
uS=
(l+Vm
\
2km {Em(ri - X)bl - Eral (2(1 - 2Vm)ri -F X(1 -t- Vm))}
Det
m
- ~ t r [ r ~ ] I d y in W
Em
/
L'-
~,- =
~b* in V* ~bs =
2Xm{)~m(7"i- x)b 2 - ~kr(2ri + x)a2 } kmb2(%(2 - 3X) - 2X) - 2~ra2(%(2 -- 3X) + X)
with
- LmQy in W The fields constructed in this fashion once again satisfy the periodicity conditions and the following average relationships:
Det = a 1 {2Er(%(3X(1 + bl {7i(2Em -
Vm) -
2(1 - 2 p m ) ) - - X(1 + Vm)}
9kmX(1 - Vm)) - 2EmX}
and
< aS >}" = ~11 ~r, aSdy = r. and
a 1 = 1 --
r
a2= a 1
= ] - ~1 ~r, qSdy = Q
after extension by zero of a* and q* within the void, so that they are statically admissible for the cellular problems associated with the dual homogenization approach. The lower boundaries of softness and homogenized conductance can then be derived in compliance with the following relationships: IwI m 1 f ss~ijkh~kh = ~ aijkh~kh Jv ~-~ ~ v*eij(u)*)dV 1
(1 - 2~/) 3
b 1 = (1 + Vr)(1 - 2*/) 3 +
2(1 - 2Vr)
b 2 = 2 + (1 - 2,/) 3
7r/33 X - 192 Moreover, it is worth highlighting that all of these results can be extended to the case of a cubic network (with the REV thereby being a cube containing an inclusion at its center), provided the following relationships are taken into account:
ri=~
-£
X = ~ andriE
0,~
I (u;nj+ u;ni)dS
21YI or~
sLijQj h = -~Lij 'W' mQj - ~-~ 1 I v* [VYd~*]idV + [-~ 1 f a r q~*nidS or Pm
Aijkh = l ~EVn~(~Si~Sjh+tSih~jk) -- -~m ~ij~kh The presence of a term to be calculated at the level of the void's edge deserves attention; this void edge was introduced by the extension of the various quantities within the void 19. The lower boundaries of both the volumetric compressibility and homogenized conductance moduli are obtained once both problems eqn (8) were solved in the case of a load defined by:
E=
1 0
/i ° 0
356
1
Q=
SOLUTION OF THE MINIMIZATION PROBLEM Knowledge of the approximated homogenized behavior has ultimately enabled us to obtain for both approaches the analytical expressions of the two constraints C1 and C2 of the objective function eqn (2) as well as of the associated Lagrangian. Once this latter expression has been developed, all that is left is to find the solution to the minimization problem which, in the case where the four constraints are inactive (~1 = ~2 = ~3 = ~4 = 0), is obtained by solving the first of the Kuhn-Tucker equations eqn (3), namely:
OL --=0
This solution then results in a 12-degree incomplete polynomial in ~/, which is presented as an appendix to this article, whose solution can be derived through the use of a numerical method for searching the root (for example, bisection method).
Thermoelastic behavior of syntactic foams: G. L'Hostis and F. Devries C 1 and C2, the necessary condition for the existence of a solution to the minimization problem that relates the mechanical and thermal properties of both the inclusion and the matrix in the form of the following inequality eqn (9):
Table 1 Definition of the design field 1
kr < 1
II
kkm r _-- 1
constraints
C2 not valid
km
~~kr > 1
C 1 not valid
X-Lr--< 1 km
C2 and C4 activate;
7"12-< 71 =¢" -[~m -<
~kr -- 1, C1 ' k~
kr > k,,
III
1
#r
Xr X~ -< 1
C2 and Co activate C1 and C2 valid
kr>
C1 and C2 valid if ~/1 -< r/°pt ~< r/2
1
×,,
(1-- 2Vm) ( 3 kr - 2 ~kr ) km )km - 1
1
r/=~
Er = 72 GPA
Em = 3.5 GPA km = 0 . 2 W m j o f - 1
~kr: 1Wm
Pm : 1500 kg m - 3
(9)
-1)
It must be specified that this inequality is identical regardless of the approach employed and that it incorporates the fact that the design field cannot be empty. In Figure 3, the magnitudes indexed by -t- and - are in fact those derived by applying the primal and dual approaches, respectively, yet they don't necessarily constitute the limits of the magnitudes involved herein. For these two approaches and as a function of the parameter 7, the objective function of the minimization problem was represented in the case of a specific configuration for the composite (glass-epoxy, centered cubic network), with the geometrical and mechanical characteristics of the constituents being given by:
At this point, let us specify the conditions related to the design field, as defined by the constraints C~, C2, C3 and C4, as a function of the mechanical and thermal characteristics of the constituents, matrix and reinforcement (Table 1). We will next focus on the case where the characteristics, mechanical as well as thermal, of the inclusion are greater than those of the matrix (part 2 of case III, see Table 1). For this configuration and to assure that the design field does not
ri = 0.60
kr ~m(l+um)(~
p~
Vr = 0.22
Pm -~- 0.4
-I oC-t
= 2500 kg m - 3
The first observation to be made is the small deviation between the values (7 +°p' and 7 -°0') of the obtained optimum by the two approaches; however, it is not possible to construct a framework for this optimal value. The same does apply to the objective function, F. Nonetheless, for an entire range of the design parameter 7 near the optimal value, it can be noticed that the deviation between the extreme values obtained is small in comparison with the deviation resulting from other values of this design parameter, especially from the parameter's extreme values. Let us now turn our attention to the validity of the results
have any contents, it is necessary for the two constraints C~ and C2 not to be violated. This condition thus enables us to obtain the limits of the design interval, 72 and 71 associated with C2 = 0 and C1 = 0, respectively (Figure 3). It should be pointed out that both approaches yield the same values for 7 ~ and 72. The special case where two constraints are active (C1 = C2 = 0; 71 = 72) initially allows us to derive a limit value for the design parameter, denoted by 17lim, for which the design field has been reduced to this single value, and then to generate, on the basis of the analytical expression for the
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Constraints and objective function of the minimization problem
357
Thermoelastic behavior of syntactic foams: G. L'Hostis and F. Devries 3,5 ;-i-;
v"r-Wr"~v-r'T~-r--r;--~
~
12,
.... t
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Elastic and thermal homogenized characteristics
obtained, which requires validating the characteristics of the approximated homogenized behavior. To do so, the results collected are compared both with those yielded by the threephase media homogenization method 24'25'2 and with the finite element solution to the cellular problems associated with the primal approach which had been taken as the reference solution. With respect to the elastic characteristics (Figure 4), in which the determination of the shearing modulus requires solving problems eqns (7) and (8) for another loading scenario (for problem eqn (7), the macroscopic loading defined by: E u = - 1, E22 = - 1 , E33 = 2 is chosen, with the other components of the macroscopic deformation tensor being zero; and for problem eqn (8), ~ 11 = - 1, Z 22 = - - 1 , ~33 = 2 is chosen, with the other components of the macroscopic stress tensor being zero), verification is provided that both approaches (primal and dual) serve to frame the characteristics generated from the other two methods. It can also be observed that for the optimal value, the deviation between the two limits is relatively small, particularly in terms of the volumetric compressibility modulus k h, for which both approximated methods could be considered as leading to its exact value (with reference to the finite elements). For the shearing modulus, it should be noted that despite the hypothesis of isotropy introduced by the specific method being used, the deviation from the optimum in comparison with the finite element reference solution remains small. The thermal characteristics tend to display similar results.
358
0,1
This feature is critical to the results obtained in that the technique used (namely, extending by means of constant fields the derived solution on the included sphere to the base cell) involves to a certain measure neglecting 'the effects of the inclusion', which, in classical terms, for this particular value of the inclusion rate (~'i = 0.6), means a significant deviation between the two resultant limits• (This specific feature is encountered with the extreme values of the parameter 7, which contain the optimal solution; it is thus possible to consider that the various quantities, even though developed as approximations, are indeed exact solutions•) Moreover, it should be remarked that all of the conclusions drawn above could also be obtained for the case of materials with behavior moduli which are different from those selected to present these results• The previous results have enabled us both to observe those properties related to the value of the parameter 7, the solution to the minimization problem, and to verify the validity of the simplified method which has been developed• Since the desired optimal configuration depends on the volumetric fraction of inclusion ri, it would appear necessary to study the sensitivity of the optimal solution with respect to this second geometrical parameter. Figure 5 displays the variations in the optimal value of the parameter ~ as a function of ri w i t h all other parameters being held constant and taken as equal to those found from the previously-defined composite; it also displays the maximum error obtained between the values of the parameter 7/+ (derived for the extreme values of the
Thermoelastic behavior of syntactic foams: G. L'Hostis and F. Devries 0,04765
1,6 ph
4
0,0476
IA
0,04755 0,0475
1
0,04745
0,8
0,0474
0,6
0,04735
0,4 00I
0,2 e
0,3
0,4 5-a
0~
0,6
--....-
0 vl°ot 0,1
02
0,3
0,4
5-b
"gi
0~
'1
(%)
, _1,1o+ . . .
•-----11---20.4
0,7
5 I.~A,
3o.6
tllI, "11~ \"
4-
i
" i~
e=l~
4-
Ino.rs- n o.osl rain( TI;.68- TI;.05)
!
~,k.. ,',L_. u.~',,._2~ . . . . . . . 0 0
5
10
15
20
~r 25 i. m
5-c Figure 5 Influence of the rate of inclusion on the optimal value of the parameter
variation interval of r/) as a function of the ratio "Tr/Xm and for various values of the ratio kr/km. Figure 5 also reveals that the rate of inclusion exerts only a marginal influence on the optimal value of 7, so that it may be possible to consider this value as basically independent of the geometric parameter. In addition, it can be pointed out that this particular feature of the optimal configuration is once again apparent in Figure 5c, provided that the ratio of the conductance (Xfl~,m) resembles that of the compressibility (kr/km). Lastly, it is worth noting that the maximum value of the conductance ratio has been attained by the inequality eqn (9) and that a similar result could be generated for an error defined on the basis of the parameter ~7-, as yielded by the dual approach. This finding constitutes a result of major importance since the rate of inclusion actually does exert a highly significant influence on the homogenized volumetric mass, as Figure 5b attests. This parameter can therefore be chosen such that the composite's mass is minimal; moreover, this particular point is indeed one of the primary objectives set within the scope of optimizing structures.
CONCLUSION During the course of this research work, we have built an optimization problem which serves to highlight, under certain conditions, the existence of a geometrical configuration that best combines the properties of thermal insulation and volumetric compressibility for composites with spherical inclusions. In the aim of devising a simple
and economical analytical tool, we have been led to developing a simplified homogenization technique that provides, first of all, the possibility to compute in a straightforward and cost-effective manner the limits of the exact homogenized behavior for this type of composite. Further, approximations of the desired optimal configuration were generated; results derived have shown that for the optimal configuration, these approximations can be considered as exact (in comparison with the finite element results), given that they turn out to be extremely close to one another. It was also found that this configuration, as defined by an optimal value of the parameter 7, is only very marginally influenced by the value of the rate of inclusion ri, a finding which proves to be of great significance since, as a consequence, the design of a composite, such as those prescribed in this paper, can be carried out economically in guaranteeing a behavior that best combines the properties of thermal insulation and compressibility while making it possible to minimize the composite's weight. Lastly, it should be mentioned that this research into the optimal configuration merely represents a first stage in the optimal design of this type of composite, since taking damage phenomena into account normally proves necessary during the preliminary sizing stage. Even though the simplified homogenization techniques developed herein may be applied whenever damage phenomena (such as viscoelastic slipping at the interfaces) are taken into account, provided that solutions to the cellular problems eqns (7) and (8) can be constructed analytically, these techniques must be excluded when modeling one of these composites' primary damage modes, namely the implosion
359
Thermoelastic behavior of syntactic foams: G. L'Hostis and F. Devries of the inclusions. In order both to develop an appropriate failure criterion and to simulate the evolution of the d a m a g e taking place within the c o m p o s i t e structure, it does b e c o m e critical to compute the stress field accurately at the level of the representative elementary v o l u m e as well as to incorporate a R E V that is more full. Consequently, the i m p l e m e n t a t i o n of adapted numerical methods, such as the finite element method, wind up being unavoidable 26 to the extent that, in this case, the application limits of these simplified techniques have been reached. 27
3"4 - a312aZlbl + 48al b2 -
64b~
1`1 = (a2 -k- b2) 3 F2 = 6a~ +
9a~b2 - 3b32
F 3 = 12a 3 -
9a2 b2 + 3b~
F4 = 8a 3 - 12a2b2 +
6a2b~ - b 3
APPENDIX where A. O p t i m i z i n g the objective function
al = 12kr(7"i- X)
Pi are defined
7r/33
a 2 = ~km(7"i -~- 2×)
b2 = 2~'r(ri - X)
192
Dual approach:
P(~/) =Pit/12 -t-P2r/9 -t-P3~76 -~-P4~4 + P 5 where the constants Primal approach:
b 1 = 3kmr i + 4/~mX with X =
P o l y n o m i a l expression used in the p r o b l e m of optimizing the objective function:
as follows.
Pi =
~'r)(3kmal q- 2Erbl)(1 - ~m)2]Oi
[(1 -
- [2Xra2 -
Pi = [162;krXmlcPi - [4kr(41Zm-k- 3km) 2 (4 + 3 ~r )
~ma2]~i
(i = 1,5)
with
× (3+4~-mm)]o ~ (i=1,5)
~bl = - (2Xr + )km)3" 1
with
q~5 = 2(Xr - ~km)r4
q~l = (2Xr q- ~km)3"l 45 = -- 2(~kr -- ~km)3"4
q~j = -- (2~k r -t- )~n)3"j -1- 2(~kr --
q~j = (2Xr q- ~km)3'j -- 2(3'r -- 3'm)3"q- 1) (J = 2, 4)
01 = (3km(1 + ~r) + 2Er)1`1
o,--
05 = (6km(1 - 2t'r) - 2 E r ) r 4
Oj = 05 = 4 ( 1
(3kin(1
Xm)3"q- 1)
"k-Ur) -t- 2Er)rj -t- (6km(1
(]=2,4)
- 2Pr) --
2Er)r(j_
-- k ~ ) 1 ` 4
( j = 2,4)
Oj=(3~r-l-4k~rm)1"j+4(1-k~mrm)rq_i)
and
(j = 2, 4)
3'1 = (bl(1 +
and
3'2 = 3(al + 2(1 --
3"1=-- [a~-t-9a2bl~r+27alb21(~r)2-t-27b~(~r)31
3"3 = 3(bl(1 + u~) -
"r - 108b3 ( ~ )
3"3 =
r'r '
al) 3
2ur)bl)(b](1 + Ur) al)(al + 2bl(1
3'4 = (al '[- 2b1(1 - 2t~r)) 3
2
--3a31+3a2bl(8--3~)+6alb2(12~-8)k~r
Itr) --
P 1 = (a 2 q-- b2) 3
1`2 = 3(2a2 - b2)(a2 + b2) 2 r 1"3 = 3(a 2 + b2)(2a2 -- b2) 2 1"4 = (2a2 - b2) 3
360
al) 2
- 2Ur)) 2
1)
Thermoelastic behavior of syntactic foams: G. L'Hostis and F. Devries where
a l = 2 E r ( ' r i ( 3 X - 2(1 - 2Vm)) - X(1 + Vm))
rt3 3
b 1 = z i ( 2 E m - 9km(1 -- Vm)X) - 2 E m X with X =
a2 = ~km(7"i(2 -- 3X) _ 2X)
b2 ----2~kr(7i(2 -- 3X) + X)
REFERENCES
1. 2. 3.
4. 5.
6.
7.
8. 9.
10.
I I. 12.
13. 14.
Hashin, Z., Analysis of composite materials a survey Transactions of the ASME. Journal of Applied Mechanics, 1983, 50, 481-505. Christensen, R.M. (1979), Mechanics of Composite Materials, John Wiley, New York. Willis, J.R., Bounds and self-consistent estimate for the overall properties of anisotropic composites. J. Mech. Solids, 1977, 25, 185-202. Aboudi. J. (1991) Mechanics of Composite Materials, Elsevier Science, Amsterdam. Sigmund, O., Materials with prescribed constitutive parameters: an inverse homogenization problem. Int. J. Solids Structures, 1994, 31(17), 2313-2329. Deruntz, J. (1969), Micromechanics and macromechanics of syntactic foams, In Proceedings of international conference on structure, solid mechanics and engineering in civil engineering materials, Held University of Southampton, UK, 1969. Duvaut, G. (1976), Analyse fonctionnelle et m~canique des milieux continus, application ~tl"&ude des matrriaux composites ~lastiques structure pEriodique-homog6nEisation, In Theoretical andApplied Mechanics, ed. W.T. Koiter, North-Holland, New York. Bensoussan, A., Lions, J.-L., Papanicolaou, G. (1978) Asymptotic analysis for periodic structures, North-Holland. Sanchez-Palencia, E. (1980) Non-Homogeneous Media and Vibration Theory,, Lecture Notes in Physics, 127, Springer-Verlag, Berlin, Heidelberg, New-York. Francfort, G.A. (1982), Two variational problems in thermoelasticity: I. homogenization, 2. material momenta, Dissertation for the degree of Doctor of Philosophy of Stanford university. Francfort, G.A., Homogenisation and thermoelasticity. SIAM J. Math. Anal, 1983, 14, 696-708. Bakhalov, N., Panasenko, G. (1989) Homogenization: Averaging Processes in Periodic media. Kluwer Academic publishers, Dordrecht. Gibson, L.J., Ashby, M.F. (1988) Cellular solids, structures and properties, Pergamon Press, UK. L'Hostis, G. (1996) Contribution ~ la conception et /t l'rtude de structures composites thermorlastiques, Thbse de Doctorat de l'Universit6 Pierre et Marie Curie, Paris.
15. 16.
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18.
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20.
21.
22.
23. 24.
25. 26.
27.
192
Haftka, R.T., Gurdal, Z. (1993), Elements of structural Optimization, Kluwer Academic Publishers, Dordrecht. Suquet, P. (1985) Elements of homogenization for inelastic solid mechanics, In Homogenization techniques for composite media, ed. Sanchez-Palencia, E., Zaoui, A., Lecture Notes in Physics, Vol. 272, Springer verlag, Berlin, heidelberg, pp. 194-278. Devri~s, F., Lrnr, F. (1987) Homog~nEisation ~tcontraintes macroscopiques imposEes: implementation numrrique et application, La Recherche Adrospatiale, pp 33-51. Rosen, W. and Hashin, Z., Effective thermal expansion coefficients and specific heats of composite materials. Int. Engng Sci., 1970, 8(1970), 157-173. Lrn6 F. (1984) Contribution h l'~tude des matrriaux composites et de lear endommagement, Th~se de Doctorat d"Etat, Universit~ Pierre et Marie Curie, Paris. Pidrri, C. (1987), Matrriaux composites 61astiques: comparaison de quelques mEthodes d'homogrnrisation., Th~se de Doctorat de troisibme cycle, Universit6 Pierre et Marie Curie, Paris. Devribs, F., Constitutive equations for unidirectional composites with imperfect bonding. The case of elastic, viscous or viscoelastic slipping. Composites Engineering, 1991, 1(5), 261-276. Devribs, F,, Bounds on elastic moduli of unidirectional composites with imperfect bonding. Composites Engineering, 1993, 3(4), 349382. Hashin, Z., The Sperical inclusion with imperfect Interface. Journal of Applied Mechanics, 1991, 58, 444-449. Eshelby, J.D., The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. London, 1957, 4241, 376. Hashin, Z., The elastic moduli of heterogeneous materials. Journal of Applied Mechanics, 1962, 2, 143-162. Devri~s F., LEn6 F., L'Hostis G., Soulat D., Parallel computing and damage in composite structures, In Communication gt I'ICCMI1 (International Conference on Composite materials), Australie, 14-18 Juillet 1997, 2, 374-386. Christensen, R.M. and Lo, K.H., Solutions for effective shear properties in three phase sphere and cylinder models. J. Mech. Phys. Solids, 1979, 27, 315-330.
361
ELSEVIER
Composites Part B 29B (1998) 351-361 © 1998 Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-8368/98/$19.00
Characterization of the thermoelastic behavior of syntactic foams
Gildas L'Hostis a and Fran(~ois Devries b aLaboratoire de M~canique et Mat#riaux, Ecole Centrale de Nantes, 44321 Nantes, Cede)( 3, France bLaboratoire de Mod#lisation et M#canique des Structures, URA 1776 du CNRS, UPMC/ENSAM/ENS Cachan, 4 Place Jussieu, 75252 Paris, Cedex 05, France (Received 22 October 1997; accepted 26 December 1997)
The objective of this article is to highlight the existence of a specific geometrical configuration of syntactic foams (solid or hollow microballs soaked in a resin), which enables the design of a composite material possessing good thermal insulation properties while ensuring optimal mechanical behavior in compression and a significant weight gain. To carry this out, an economical approximated homogenization technique, adapted to the composites being studied herein, is first of all proposed. The results yielded turn out to be in good agreement with those obtained by the application of classical methods which have already demonstrated their effectiveness. In order to highlight the desired geometrical configuration, a minimization problem is then developed and solved. The value of this approach lies in the fact that all the computations necessary can be performed in a quasi-analytical manner, thereby avoiding reliance upon heavy computing resources. The resultant design tool proves to be most costeffective and simple to use. © 1998 Elsevier Science Limited. All rights reserved INTRODUCTION The use of composite materials to develop highperformance structures was constantly increasing over the past few years. This was the case within a wide array of industrial fields. The main particularity of a composite media resides in its strong level of heterogeneity, quite often giving rise to numerical simulations that are costly and difficult to implement. Consequently, when the optimal design for a composite is being sought, it would appear judicious to proceed by substituting a homogeneous media exhibiting an equivalent mechanical behavior. Much research work has in fact been devoted towards achieving this objective of computing the equivalent homogenized behavior of a variety of composites, and has served to derive several homogenization methodsl-4. These cited works all rely upon the existence of an elementary volume representative of the internal structure of the composite under study which can be differentiated in terms of the way the link between the composite's scale and the microstructure's scale is being handled. From such an approach, it thereby becomes possible to compute the homogeneous behavior that is mechanically equivalent to the behavior being exhibited by the composite material studied, on the basis of an understanding of both the specific geometry and the thermomechanical characteristics of the composite's constituents. Our orientation throughout this article stems from the general framework for the optimal design of composite materials 5. It is thus being proposed herein to devise an
economical and reliable computational strategy for the best geometrical configuration of those composites known under the designation of 'syntactic foams '6, which consist of either solid or hollow microballs soaked in a resin. These composites, which have become more commonly used, especially as surfacing elements for spatial and aeronautical structures, must perform certain thermal insulation functions while providing good compressive strength. In order to account for both the relative complexity of the phenomena involved (elasticity/thermal conductance) and the fact that from a general standpoint, the properties of insulation and mechanical strength counteract one another, a minimization problem with constraints is constructed on the basis of the equivalent characteristics of conductance and volumetric compressibility. The solution to this problem will enable determining the composite's specific geometrical configuration that induces the optimal set of thermal insulation and compressibility properties. In order both to avoid being penalized by the high numerical costs associated with seeking such a configuration and to develop an appropriate tool for performing analyses and producing reliable and efficient designs, we will rely herein upon a homogenization technique inspired by the work conducted by 1. A derivative of the homogenization method for fine periodic structural media 7-12, this technique consists of building fields that are both kinematically and statically admissible for the problems to be solved on the representative microstructure of the composite under study. As a result, a framework for the real homogenized behavior is obtained from the perspective
351
Thermoelastic behavior of syntactic foams: G. L'Hostis and F. Devries of the energy. The boundaries of this framework can then be defined in a quasi-analytical fashion, a feature which is of obvious interest when the anticipated minimization problem with constraints is subsequently solved. As a means of measuring the reliability of this homogenization technique, the homogenized characteristics it produces are then compared with those obtained both by utilizing another homogenization method (three-phase media) and by the finite element solution to the classical problems of the periodic-media homogenization method. Lastly, the influence of various geometrical and thermomechanical parameters of the constituents on the optimal configuration will also be assessed.
PRESENTATION OF THE PROBLEM AND THE TARGETED OBJECTIVE It is assumed that the composites being considered herein accept as their representative elementary volume (REV) a tetra-kaidecahedra with a characteristic dimension of L (Figure 1), thereby leading to a cubic pattern centered around the inclusions. Its center consists of a spherical inclusion which is assumed hollow whose exterior diameter is denoted D; e represents the thickness of the layer of material forming this inclusion (0 < e --< D2). The geometry of the composite under study is therefore fully established once values can be determined for the three parameters: L; 7"i and 7, where 7 represents the volumetric rate of inclusion ri, which have been defined respectively inl3: 7 = D' 7 E [0, ~] ri = 4 8 ~ ( I r £)3D ( r i g
[0, 1--~J 7rf13] ~/
where 0 --
Dmax= V/~ L 3 fl = X/11 -t- 3cos20
and 0 = (arcsin ( ~ 3 ) )
isotropic and that the thermal conductance complies with the Fourier law, in such a way that the parameters able to fully describe their behavior are: the volumetric mass, p, Young's modulus, E, Poisson's ratio, v, the dilatation coefficient, or, and the conductance coefficient, X (the indices m and r will be assigned subsequently to the quantities relative both to the matrix and to the reinforcement or inclusion, respectively). In the perspective of designing an optimal composite that combines good properties with respect to both thermal insulation and volumetric compressibility, the existence of specific values for the parameter 7 has already been demonstrated14; this finding has led to stiffening configurations characterized, for example, on the basis of volumetric compressibility modules, by kh/kr, > 1, neutral (kh/km = 1) as well as to softening (kh/km < 1); such configurations can also be determined for the thermal behavior as a function of the ratio xh/Xm (the exponent h is based on the modules that define the homogenized behavior). It is being proposed herein to emphasize whether a geometrical configuration exists for the fixed values of the other parameters, both geometrical and mechanical, that serve to characterize the composite's REV, as defined by a specific value of the parameter 7, which is softening with respect to the thermal and stiffening characteristics from a mechanical standpoint. In the case of the composites being considered, these two properties serve to counteract one another. In fact, on the one hand, insulation characteristics are obtained in part thanks to the existence of spherical, thermally-insulating voids, while on the other hand, the presence of these voids within the composite exert a negative impact on its resistance to a hydrostatic compressive loading. Consequently, the desired optimal composite will be defined by a geometry, if one does exist, close to the configuration that is mechanically and thermally neutral. In order to identify the value of the parameter 7 that results in a thermal as well as mechanical neutrality of the composite's equivalent homogenized behavior, the following two quantities are introduced:
°
Afterwards, we will be considering that the composite exhibits no defects whatsoever, that the presence of voids is caused solely by the inclusions, that the constituents adhere perfectly to one another and that their behavior is governed by the classical linear thermoelastic operator. In addition, it is assumed that the constituents are both homogeneous and
Xh kh C1 = 1 - Xmmand C2 = kmm- 1
(1)
in such a way that the value searched for the design parameter 7 must be a solution to the following minimization problem: min F(7) = min {C12(7) + C~(7) }
(2)
under the constraints
Q Figure 1 Representative elementary volume
352
C1 ~ 0 , C 2 : > 0 ,
C3 ~ 0 , C4-----0
with: 1
C3 = 7 and C 4 = ~ - 7 The problem so defined is in fact a classical minimization problem subjected to positivity constraints 15 which only depends on the single design parameter, 7, and which corresponds to a given volumetric fraction of inclusion, ri. In
Thermoelastic behavior of syntactic foams: G. L'Hostis and F. Devries the aim of simplifying to a standard minimization problem, the Lagrangian associated with this problem is introduced, as defined by:
L=2-2
~mm-t-~m q-~km J
"+- km
- ~1C1 - ~2C2 - ~3C3 - ~4C4
where ~ 1, ~2, ~3 and 44 are the Lagrangian parameters associated with the constraints Cl, C2, C3 and C4. The existence of an extreme point is given by the KuhnTucker conditions which can be reduced in the present case
to:
3£
0 3£,
3L
OL
00L 0 ag4 =
(3)
with these conditions comprising the set of preconditions necessary Ibr the existence of the optimum being searched. The search for this specific configuration thus leads to coupling the solution to a minimization problem with the determination of the homogenized behavior, as
structural media 7'8. This technique relies upon the hypotheses of the composite media's mechanical and geometrical periodicity, as well as upon the periodicity of the magnitudes that characterize the media's behavior (deformations, stresses). These periodicity properties are then complemented by relational data on averages that reflect the link between the studied composite and its microstructure. These relations may be of two types, each of which serves to characterize two distinct approaches that do nonetheless present the same level of precision 16. The imposed deformation and temperature gradient averages give rise to the primal approach, with the dual approach 17 being associated with the conditions of imposed stress and heat flow averages (see eqn (4)). Moreover, for each approach, the macroscopic loading is complemented by the data of a temperature increment, r, assumed constant on the REV ~8. In denoting the microscopic fields of stresses, heat flow, displacement and temperature by m q, u and O, respectively, the problems to be solved for both approaches can be written in the following form:
e(u), VyO Y periodic divyo
o(n), q.n Y antiperiodic
= 0 in Y*
and for the primal method a= ~
e(u) + 3v
tr[e(u)lId - 3k~Idr in Y* <
divyq = 0 in Y*q = - XVyO in Y*
[ << Vy0 >>V = F
o(p) = O, q.l' = 0 on 3 Y T
and for the dual method
(O-)y= ~, (q)r = q (4) characterized by the modules k h and Xh. A numerical solution procedure can be envisioned5, yet this would require, at each step of the minimum searching algorithm, solving cellular problems which are vital to obtaining the homogenized behavior. In the case of the three-dimensional composites studied herein, such a procedure could only be carried out through the use of numerical methods (for example, finite element methods) which comprises the disadvantage of generating high computation times and costs, thus representing an incompatibility with the objective of developing a simple, economical and efficient tool. In order to circumvent this disadvantage, one obvious solution would consist of determining a literal expression not only for the homogenized behavior but, at the same time, for the value of the ~/ parameter, the solution to the minimization problem, as well. The following section is specifically aimed at finding such a solution.
where u and n represent the unitary normal lines inside to the inclusion and outside to the REV, respectively, where Y* is the solid part of the outside normal, k the volumetric compressibility modulus of the media, and where the divergence and gradient operators have been denoted by divy and Vy in comparison with the space variables (Yl,Y2,Y3) related to the REV. Once these problems were solved, the homogenized behavior is then obtained by specifying the relationships linking the macroscopic quantities r. (stresses), E (deformations), F (temperature gradient) and Q (heat flow) to their corresponding microscopic quantities, such that:
= ~ = Q h E - Q hethr and
< q > r = Q = - XhF for the primal method << e(u) >>r = E = she + othr and << VyO >>r = - LhQ for the dual method
HOMOGENIZATION AND APPROXIMATED BEHAVIOR The desired homogenized behavior may be derived by utilizing the homogenization technique for fine periodic
where the various averages (caused by the presence of the void) are defined by:
,f
,f o/n~dr
rfidy and ( <
353
Thermoelastic behavior of syntactic foams: G. L'Hostis and F. Devries and where Qh (Qh : ( s h ) - l ) , ~h (hh = (Lh)-l) and ~xh represent the homogenized tensors of stiffness, conductance and dilatation, respectively. By using the linearity of these problems with respect to the macroscopic data, this approach leads to solving in the general case 10 cellular problems (seven elasticity problems and three thermal problemsl°'11). For the type of network foreseen (centered cubic), in making use of the fact that the three coordinate planes constitute material symmetry planes for the various cellular problems, it has been demonstrated in 19 that the desired homogenized behavior is necessarily quasi-isotropic in the reference (O,yl,Y2,Y3) relative to the base period Y. The tensor S h of the homogenized softnesses thus only accepts three non-zero components: 1
S~lll I : ~ ,
h
ph
__
h
1 h
Sl122 : Eh, S1212 : ~ G , with [Qh] = [Sh]
-1
and the homogenized conductance tensor h h is spherical, such that:
Figure 2 Decomposition of the representative elementary volume
In order to implement this method, the first step entails decomposing the REV Y into two parts: the sphere V included in Y and its complementary element W (Figure 2), defined by:
Y = W"V The coefficient of volumetric compressibility is also introduced and defined as: 3kh = Q~Illl -~- 2Q~1122=
1 Shll 1 Jr- 2sh122
(5)
As already specified, solving the problems encountered on the REV and then deriving the homogenized behavior requires, in the case of three-dimensional geometries, utilizing adapted numerical techniques (for example, finite elements) which thereby makes it impossible to develop an analytical expression for the characteristics of this behavior. Nonetheless, in order to achieve this objective, in relying on the combined use of both homogenization approaches (primal and dual), we will develop in the remainder of this section a method which was initially introduced to study unidirectional composites 2°-22. This technique consists of building fields that are both kinematically and statically admissible for the problems encountered on the REV from these two approaches. Once these fields have been constructed, it is shown that this technique yields, from an energy standpoint TM, a framework for the desired homogenized behavior, in such a way that: h s h
qijkhgijEkh<- QjkhEijgkh VEij :Eft
and
h ~ i j = ~ji Sijkh~ij~kh--
354
where V is defined in the spherical reference (r,O,dp) as: V = {(r, 0, th); 0 --< r --< R; 0 --< 0 --< r ; 0 --< ~b --< 27r } The total volume of matter within V is also introduced and defined by:
V*= {(r,O, dp);p < r <- R;O -< 0-< 7r;0--< ~b < 2r} along with the two surfaces being defined by: r-ext = {(r,O,z);r=R;O < 0 <- 7r;0 < 4~ -< 2a-} external surface of V r r = {(r,O, dp);r=p;O <-- 0 <-- a';0 <-- 4~ --< 2a'} internal side of the inclusion
Computation of the upper boundaries The computation of the upper boundaries for the homogenized compressibility and thermal conductance moduli, meaning those of the tensors SQh and s~h, requires the construction of fields that are both kinematically admissible for the primal approach's cellular problems. Thus, on Y* (Y* = V*uw), the fields u s and ~bs are defined in the following manner: U S
~-.
U*
inV*
Ey
in W
~)S
6"
inV*
in W
where the fields u* and 4,* are the unique solutions to the following problems: divya* = 0 in V* = ~
E
3vk e(u) + ~ tr[e(u)*]Id in V* u* = Eijyj on rYxt a*(r) = 0 on r r
Thermoelastic behavior of syntactic foams: G. L'Hostis and F. Devries
divyq* = 0 in V* q* = - ),lry4~* in V*
(7)
r* = F y j on ~ext q'r=0
E=
on r T
Classically associated with these fields, by applying behavioral laws, the tensors of deformation e s and of stresses a s, as well as the temperature gradient V4~~ and the heat flow vector q~, are defined by: e(u*) in V* es =
as = E in W
thermal conductance X +, the following macroscopic loadings (macroscopic deformations E, heat flow F) have been chosen:
o* in V* Em 3~mkm l--~vmE+ l~vmtr[E]Id
in W
U*r= A r + B
(
r 2 and ~ * =
1
and F = (0, 0, 1)
0 This choice yields, in the case of problem eqn (7), sphericaltype limit conditions. The solution fields are then sought in the following forms23: After performing a set of cumbersome and fastidious computations which have not been reported herein because of space limitations, the desired homogenized characteristics are derived in the forms:
Cr + ~2) cos0 A, B, C and D are constant
Uo : U4~ : O
k + = 4kral { 7"i(3km - x(3km + 4/Zm)) -- 3kmX } + kmbl {7"i(X(4gm -4- 3km) - 4/ZmX} 12kr(r i - x)a 1 - b 1(3kmT"i+ 4#mX) h+= b2)km(7"i(2
2)~n(b2~km(7"i - X) - a2)tr('ri + X) ) 3X) - 2X) - 2 a 2 ~ k r ( 7 " i ( 2 - 3X) + X)
with: V0 s =
V4~* inV* q S = F in W
q* inV*
It is then verified both that these fields do indeed satisfy the cellular problems' periodicity conditions eqn (4) and that: << eij(uS) >>Y = 2 - ~
a 1 = 1 - ( 1 -21,) 3
- ~,mF in W
oY{u~nj+u}ni} dI'=Eij'
<< [Vy(~S)]i >>r = ~-~ 1 foy~Snid r = Fi in such a way that u s and 4's are kinematically admissible for the set of problems to be solved. In taking account of the R E V ' s geometrical decomposition, and with the help of those admissible fields which have been introduced above, computation can finally be performed for the stiffness and the approximated homogenized conductance, SQh and s h h, as defined by:
IWI m 1 f SQ~okhEkh= ~Y-(aijkhEkh + ~1 .p v* ffqdv sXijFj= h IWIxmF.+ 1 [" IYI 'J J ~Y] J v *q~dv
b1=4+3kr(1 /d'r
a2 = al
-2~/) 3
71.~3 and X =
192
b2 = 2 + (1 - 2~/)3
Computation of the lower boundaries The next step consists of determining the lower boundaries of the homogenized behavior. The procedure adopted to carry this step out, which requires computing the tensor components sSh and SLh, calls for constructing both a stress tensor and a heat flow vector which are statically admissible for the dual approach's cellular problems. This requires defining the fields a s and qS on Y* as follows:
as = a* E
inV in W
qS = o*
inV* in W
where o* and q* are the unique solutions to the following problems:
divyo* = 0 in V*
(divyq* = 0 in V*
v . ]Id in V*~Vy~b*l _-- _ Lq* in V* e(u . ) = ~1-+o v , - ~tr[o
where aijmkh = )km~ij~kh --[-~m(~ik~jh + ~ih~jk)
In order to compute the upper boundaries of the homogenized coefficients of both volumetric compressibility k + and
o*(v) = 0 on I~T
/q*(v) = 0 on F r
o*(n) = Z(n) on ~ext
[,q*(n) = Q(n) on ~ext
(8)
355
Thermoelastic behavior of syntactic foams: G. L'Hostis and F. Devries These tensors are then associated with the deformation tensor e s, the temperature gradient V~bs as well as the displacement field u s and the temperature field th ~, defined as follows:
The solution fields are then sought in the following forlill4: ,
aij
B,
=
,
0 except a*~r = A + -fi
B 3
aoo = a~. = A - ~r
e(u*) in V* vm 1E m+vm~ _ ~ t r [ ~ ] I d
eS = e(u~) =
q* = ( C - 2 ~ ) cosO er - ( C + -~ ) sinO eo
in W
A, B, C are constants Vy$* in V* - LmQ in W
Solving the problems in eqn (8) enables deriving the desired expressions which take on the forms:
u* in V*
k-=
V,# ~ =
uS=
(l+Vm
\
2km {Em(ri - X)bl - Eral (2(1 - 2Vm)ri -F X(1 -t- Vm))}
Det
m
- ~ t r [ r ~ ] I d y in W
Em
/
L'-
~,- =
~b* in V* ~bs =
2Xm{)~m(7"i- x)b 2 - ~kr(2ri + x)a2 } kmb2(%(2 - 3X) - 2X) - 2~ra2(%(2 -- 3X) + X)
with
- LmQy in W The fields constructed in this fashion once again satisfy the periodicity conditions and the following average relationships:
Det = a 1 {2Er(%(3X(1 + bl {7i(2Em -
Vm) -
2(1 - 2 p m ) ) - - X(1 + Vm)}
9kmX(1 - Vm)) - 2EmX}
and
< aS >}" = ~11 ~r, aSdy = r. and
a 1 = 1 --
a2= a 1
= ] - ~1 ~r, qSdy = Q
after extension by zero of a* and q* within the void, so that they are statically admissible for the cellular problems associated with the dual homogenization approach. The lower boundaries of softness and homogenized conductance can then be derived in compliance with the following relationships: IwI m 1 f ss~ijkh~kh = ~ aijkh~kh Jv ~-~ ~ v*eij(u)*)dV 1
(1 - 2~/) 3
b 1 = (1 + Vr)(1 - 2*/) 3 +
2(1 - 2Vr)
b 2 = 2 + (1 - 2,/) 3
7r/33 X - 192 Moreover, it is worth highlighting that all of these results can be extended to the case of a cubic network (with the REV thereby being a cube containing an inclusion at its center), provided the following relationships are taken into account:
ri=~
-£
X = ~ andriE
0,~
I (u;nj+ u;ni)dS
21YI or~
sLijQj h = -~Lij 'W' mQj - ~-~ 1 I v* [VYd~*]idV + [-~ 1 f a r q~*nidS or Pm
Aijkh = l ~EVn~(~Si~Sjh+tSih~jk) -- -~m ~ij~kh The presence of a term to be calculated at the level of the void's edge deserves attention; this void edge was introduced by the extension of the various quantities within the void 19. The lower boundaries of both the volumetric compressibility and homogenized conductance moduli are obtained once both problems eqn (8) were solved in the case of a load defined by:
E=
1 0
/i ° 0
356
1
Q=
SOLUTION OF THE MINIMIZATION PROBLEM Knowledge of the approximated homogenized behavior has ultimately enabled us to obtain for both approaches the analytical expressions of the two constraints C1 and C2 of the objective function eqn (2) as well as of the associated Lagrangian. Once this latter expression has been developed, all that is left is to find the solution to the minimization problem which, in the case where the four constraints are inactive (~1 = ~2 = ~3 = ~4 = 0), is obtained by solving the first of the Kuhn-Tucker equations eqn (3), namely:
OL --=0
This solution then results in a 12-degree incomplete polynomial in ~/, which is presented as an appendix to this article, whose solution can be derived through the use of a numerical method for searching the root (for example, bisection method).
Thermoelastic behavior of syntactic foams: G. L'Hostis and F. Devries C 1 and C2, the necessary condition for the existence of a solution to the minimization problem that relates the mechanical and thermal properties of both the inclusion and the matrix in the form of the following inequality eqn (9):
Table 1 Definition of the design field 1
kr < 1
II
kkm r _-- 1
constraints
C2 not valid
km
~~kr > 1
C 1 not valid
X-Lr--< 1 km
C2 and C4 activate;
7"12-< 71 =¢" -[~m -<
~kr -- 1, C1 ' k~
kr > k,,
III
1
#r
Xr X~ -< 1
C2 and Co activate C1 and C2 valid
kr>
C1 and C2 valid if ~/1 -< r/°pt ~< r/2
1
×,,
(1-- 2Vm) ( 3 kr - 2 ~kr ) km )km - 1
1
r/=~
Er = 72 GPA
Em = 3.5 GPA km = 0 . 2 W m j o f - 1
~kr: 1Wm
Pm : 1500 kg m - 3
(9)
-1)
It must be specified that this inequality is identical regardless of the approach employed and that it incorporates the fact that the design field cannot be empty. In Figure 3, the magnitudes indexed by -t- and - are in fact those derived by applying the primal and dual approaches, respectively, yet they don't necessarily constitute the limits of the magnitudes involved herein. For these two approaches and as a function of the parameter 7, the objective function of the minimization problem was represented in the case of a specific configuration for the composite (glass-epoxy, centered cubic network), with the geometrical and mechanical characteristics of the constituents being given by:
At this point, let us specify the conditions related to the design field, as defined by the constraints C~, C2, C3 and C4, as a function of the mechanical and thermal characteristics of the constituents, matrix and reinforcement (Table 1). We will next focus on the case where the characteristics, mechanical as well as thermal, of the inclusion are greater than those of the matrix (part 2 of case III, see Table 1). For this configuration and to assure that the design field does not
ri = 0.60
kr ~m(l+um)(~
p~
Vr = 0.22
Pm -~- 0.4
-I oC-t
= 2500 kg m - 3
The first observation to be made is the small deviation between the values (7 +°p' and 7 -°0') of the obtained optimum by the two approaches; however, it is not possible to construct a framework for this optimal value. The same does apply to the objective function, F. Nonetheless, for an entire range of the design parameter 7 near the optimal value, it can be noticed that the deviation between the extreme values obtained is small in comparison with the deviation resulting from other values of this design parameter, especially from the parameter's extreme values. Let us now turn our attention to the validity of the results
have any contents, it is necessary for the two constraints C~ and C2 not to be violated. This condition thus enables us to obtain the limits of the design interval, 72 and 71 associated with C2 = 0 and C1 = 0, respectively (Figure 3). It should be pointed out that both approaches yield the same values for 7 ~ and 72. The special case where two constraints are active (C1 = C2 = 0; 71 = 72) initially allows us to derive a limit value for the design parameter, denoted by 17lim, for which the design field has been reduced to this single value, and then to generate, on the basis of the analytical expression for the
f(rl) 0,3
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0,035
Figure
3
0,04
0,045
0,05
0,055
0,06
0,065 !1
.
0
0,1
0,2
0,3
0,4
0,5 13
Constraints and objective function of the minimization problem
357
Thermoelastic behavior of syntactic foams: G. L'Hostis and F. Devries 3,5 ;-i-;
v"r-Wr"~v-r'T~-r--r;--~
~
12,
.... t
•
1oi L :
t,5
.
.
.
8i
°! 2
"-
4
o,5 ~
2
o
0 lqopt 0,I
02.
0,3
0A
-q
0,5
i¢ 0
rl
,4
opt 0,1
0,29
0.4
0,3
0,5
3,5
~h
_
. a
2,5 _I
o
stressapproach(+) strain approach (-) 3 p.ha~s model lxmtc elements
i 2
I
1,5
1
0,5 0
'
0
Figure 4
•
opt
0.2
0,3
0,4
11
0,5
Elastic and thermal homogenized characteristics
obtained, which requires validating the characteristics of the approximated homogenized behavior. To do so, the results collected are compared both with those yielded by the threephase media homogenization method 24'25'2 and with the finite element solution to the cellular problems associated with the primal approach which had been taken as the reference solution. With respect to the elastic characteristics (Figure 4), in which the determination of the shearing modulus requires solving problems eqns (7) and (8) for another loading scenario (for problem eqn (7), the macroscopic loading defined by: E u = - 1, E22 = - 1 , E33 = 2 is chosen, with the other components of the macroscopic deformation tensor being zero; and for problem eqn (8), ~ 11 = - 1, Z 22 = - - 1 , ~33 = 2 is chosen, with the other components of the macroscopic stress tensor being zero), verification is provided that both approaches (primal and dual) serve to frame the characteristics generated from the other two methods. It can also be observed that for the optimal value, the deviation between the two limits is relatively small, particularly in terms of the volumetric compressibility modulus k h, for which both approximated methods could be considered as leading to its exact value (with reference to the finite elements). For the shearing modulus, it should be noted that despite the hypothesis of isotropy introduced by the specific method being used, the deviation from the optimum in comparison with the finite element reference solution remains small. The thermal characteristics tend to display similar results.
358
0,1
This feature is critical to the results obtained in that the technique used (namely, extending by means of constant fields the derived solution on the included sphere to the base cell) involves to a certain measure neglecting 'the effects of the inclusion', which, in classical terms, for this particular value of the inclusion rate (~'i = 0.6), means a significant deviation between the two resultant limits• (This specific feature is encountered with the extreme values of the parameter 7, which contain the optimal solution; it is thus possible to consider that the various quantities, even though developed as approximations, are indeed exact solutions•) Moreover, it should be remarked that all of the conclusions drawn above could also be obtained for the case of materials with behavior moduli which are different from those selected to present these results• The previous results have enabled us both to observe those properties related to the value of the parameter 7, the solution to the minimization problem, and to verify the validity of the simplified method which has been developed• Since the desired optimal configuration depends on the volumetric fraction of inclusion ri, it would appear necessary to study the sensitivity of the optimal solution with respect to this second geometrical parameter. Figure 5 displays the variations in the optimal value of the parameter ~ as a function of ri w i t h all other parameters being held constant and taken as equal to those found from the previously-defined composite; it also displays the maximum error obtained between the values of the parameter 7/+ (derived for the extreme values of the
Thermoelastic behavior of syntactic foams: G. L'Hostis and F. Devries 0,04765
1,6 ph
4
0,0476
IA
0,04755 0,0475
1
0,04745
0,8
0,0474
0,6
0,04735
0,4 00I
0,2 e
0,3
0,4 5-a
0~
0,6
--....-
0 vl°ot 0,1
02
0,3
0,4
5-b
"gi
0~
'1
(%)
, _1,1o+ . . .
•-----11---20.4
0,7
5 I.~A,
3o.6
tllI, "11~ \"
4-
i
" i~
e=l~
4-
Ino.rs- n o.osl rain( TI;.68- TI;.05)
!
~,k.. ,',L_. u.~',,._2~ . . . . . . . 0 0
5
10
15
20
~r 25 i. m
5-c Figure 5 Influence of the rate of inclusion on the optimal value of the parameter
variation interval of r/) as a function of the ratio "Tr/Xm and for various values of the ratio kr/km. Figure 5 also reveals that the rate of inclusion exerts only a marginal influence on the optimal value of 7, so that it may be possible to consider this value as basically independent of the geometric parameter. In addition, it can be pointed out that this particular feature of the optimal configuration is once again apparent in Figure 5c, provided that the ratio of the conductance (Xfl~,m) resembles that of the compressibility (kr/km). Lastly, it is worth noting that the maximum value of the conductance ratio has been attained by the inequality eqn (9) and that a similar result could be generated for an error defined on the basis of the parameter ~7-, as yielded by the dual approach. This finding constitutes a result of major importance since the rate of inclusion actually does exert a highly significant influence on the homogenized volumetric mass, as Figure 5b attests. This parameter can therefore be chosen such that the composite's mass is minimal; moreover, this particular point is indeed one of the primary objectives set within the scope of optimizing structures.
CONCLUSION During the course of this research work, we have built an optimization problem which serves to highlight, under certain conditions, the existence of a geometrical configuration that best combines the properties of thermal insulation and volumetric compressibility for composites with spherical inclusions. In the aim of devising a simple
and economical analytical tool, we have been led to developing a simplified homogenization technique that provides, first of all, the possibility to compute in a straightforward and cost-effective manner the limits of the exact homogenized behavior for this type of composite. Further, approximations of the desired optimal configuration were generated; results derived have shown that for the optimal configuration, these approximations can be considered as exact (in comparison with the finite element results), given that they turn out to be extremely close to one another. It was also found that this configuration, as defined by an optimal value of the parameter 7, is only very marginally influenced by the value of the rate of inclusion ri, a finding which proves to be of great significance since, as a consequence, the design of a composite, such as those prescribed in this paper, can be carried out economically in guaranteeing a behavior that best combines the properties of thermal insulation and compressibility while making it possible to minimize the composite's weight. Lastly, it should be mentioned that this research into the optimal configuration merely represents a first stage in the optimal design of this type of composite, since taking damage phenomena into account normally proves necessary during the preliminary sizing stage. Even though the simplified homogenization techniques developed herein may be applied whenever damage phenomena (such as viscoelastic slipping at the interfaces) are taken into account, provided that solutions to the cellular problems eqns (7) and (8) can be constructed analytically, these techniques must be excluded when modeling one of these composites' primary damage modes, namely the implosion
359
Thermoelastic behavior of syntactic foams: G. L'Hostis and F. Devries of the inclusions. In order both to develop an appropriate failure criterion and to simulate the evolution of the d a m a g e taking place within the c o m p o s i t e structure, it does b e c o m e critical to compute the stress field accurately at the level of the representative elementary v o l u m e as well as to incorporate a R E V that is more full. Consequently, the i m p l e m e n t a t i o n of adapted numerical methods, such as the finite element method, wind up being unavoidable 26 to the extent that, in this case, the application limits of these simplified techniques have been reached. 27
3"4 - a312aZlbl + 48al b2 -
64b~
1`1 = (a2 -k- b2) 3 F2 = 6a~ +
9a~b2 - 3b32
F 3 = 12a 3 -
9a2 b2 + 3b~
F4 = 8a 3 - 12a2b2 +
6a2b~ - b 3
APPENDIX where A. O p t i m i z i n g the objective function
al = 12kr(7"i- X)
Pi are defined
7r/33
a 2 = ~km(7"i -~- 2×)
b2 = 2~'r(ri - X)
192
Dual approach:
P(~/) =Pit/12 -t-P2r/9 -t-P3~76 -~-P4~4 + P 5 where the constants Primal approach:
b 1 = 3kmr i + 4/~mX with X =
P o l y n o m i a l expression used in the p r o b l e m of optimizing the objective function:
as follows.
Pi =
~'r)(3kmal q- 2Erbl)(1 - ~m)2]Oi
[(1 -
- [2Xra2 -
Pi = [162;krXmlcPi - [4kr(41Zm-k- 3km) 2 (4 + 3 ~r )
~ma2]~i
(i = 1,5)
with
× (3+4~-mm)]o ~ (i=1,5)
~bl = - (2Xr + )km)3" 1
with
q~5 = 2(Xr - ~km)r4
q~l = (2Xr q- ~km)3"l 45 = -- 2(~kr -- ~km)3"4
q~j = -- (2~k r -t- )~n)3"j -1- 2(~kr --
q~j = (2Xr q- ~km)3'j -- 2(3'r -- 3'm)3"q- 1) (J = 2, 4)
01 = (3km(1 + ~r) + 2Er)1`1
o,--
05 = (6km(1 - 2t'r) - 2 E r ) r 4
Oj = 05 = 4 ( 1
(3kin(1
Xm)3"q- 1)
"k-Ur) -t- 2Er)rj -t- (6km(1
(]=2,4)
- 2Pr) --
2Er)r(j_
-- k ~ ) 1 ` 4
( j = 2,4)
Oj=(3~r-l-4k~rm)1"j+4(1-k~mrm)rq_i)
and
(j = 2, 4)
3'1 = (bl(1 +
and
3'2 = 3(al + 2(1 --
3"1=-- [a~-t-9a2bl~r+27alb21(~r)2-t-27b~(~r)31
3"3 = 3(bl(1 + u~) -
"r - 108b3 ( ~ )
3"3 =
r'r '
al) 3
2ur)bl)(b](1 + Ur) al)(al + 2bl(1
3'4 = (al '[- 2b1(1 - 2t~r)) 3
2
--3a31+3a2bl(8--3~)+6alb2(12~-8)k~r
Itr) --
P 1 = (a 2 q-- b2) 3
1`2 = 3(2a2 - b2)(a2 + b2) 2 r 1"3 = 3(a 2 + b2)(2a2 -- b2) 2 1"4 = (2a2 - b2) 3
360
al) 2
- 2Ur)) 2
1)
Thermoelastic behavior of syntactic foams: G. L'Hostis and F. Devries where
a l = 2 E r ( ' r i ( 3 X - 2(1 - 2Vm)) - X(1 + Vm))
rt3 3
b 1 = z i ( 2 E m - 9km(1 -- Vm)X) - 2 E m X with X =
a2 = ~km(7"i(2 -- 3X) _ 2X)
b2 ----2~kr(7i(2 -- 3X) + X)
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1. 2. 3.
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361