Thermal and thermoelastic behaviour of multiply coated inclusion-reinforced composites

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Thermal and thermoelastic behaviour of multiply coated inclusion-reinforced composites

E. Hervé

To cite this version:

E. Hervé. Thermal and thermoelastic behaviour of multiply coated inclusion-reinforced composites.

International Journal of Solids and Structures, Elsevier, 2002, 39, pp.1041-1058. �10.1016/S0020-

7683(01)00257-8�. �hal-00111351�

Thermal and thermoelastic behaviour of multiply coated inclusion-reinforced composites

E. Herve

aLaboratoire de Meecanique des Solides, Ecole Polytechnique, CNRS, UMR 7649, F-91128 Palaiseau Cedex, France

bUniversiteede Versailles Saint-Quentin-en-Yvelines, 45 Avenue des Etats Unis, 78035 Versailles Cedex, France

Abstract

This work is devoted first to the derivation of the temperature field in an infinite medium constituted of ann-layered isotropic spherical inclusion, embedded in a matrix subjected to a uniform temperature gradient at infinity, under assumptions of no coupling between mechanical and thermal effects and of steady state conditions. These results lead to an estimate of the effective thermal conductivity coefficient of ann-layered inclusion-reinforced material. The second objective of this work is the derivation of the thermoelastic stress and strain fields in the same basic configuration but now, a coupling between stress and thermal properties is considered and the infinite matrix is stress free and subjected to a uniform change of temperature. These results and the solution of the same problem with a hydrostatic loading are used to estimate the effective thermal expansion coefficient and the specific heats for heterogeneous inclusion-reinforced materials.Ó2002 Elsevier Science Ltd. All rights reserved.

Keywords:Thermal behaviour; Thermoelasticity; Micromechanical modelling;n-Layered inclusion-reinforced composites

1. Introduction

The purpose of the present paper is to use an appropriate theory of thermoelasticity to deduce an es- timate for the effective thermal conductivity, the effective thermal expansion coefficient and the specific heats in terms of the constituent properties for composite materials with multiphase inclusions. Rosen (1970) has shown that suitable thermoelastic energy functions can be expressed in terms of macroscopic (or average) state variables and effective composite properties. The problem of relating effective mechanical properties of linear elastic multiply coated inclusion-reinforced materials to constituent properties has al- ready been solved (Hervee and Zaoui, 1993). In contrast, little can be found on the effective thermal be- haviour of this type of composite materials. Nevertheless, for materials with other microstructures, bounds on effective thermal expansion coefficients (Schapery, 1968; Rosen and Hashin, 1970) and on specific heats (Rosen and Hashin, 1970) of anisotropic composites having any number of anisotropic phases (Rosen and

*Address for Correspondence: Laboratoire de Meecanique des Solides, Ecole Polytechnique, CNRS, UMR 7649, F-91128 Palaiseau Cedex, France. Fax: +33-1-39-25-30-15.

E-mail address:[email protected] (E. Herve).

Hashin, 1970) or having two or three isotropic phases (Schapery, 1968) have been derived by employing extremum principles of thermoelasticity. However, in these papers, the connectedness of the matrix sur- rounding the multiply layered inclusion cannot be taken into account. The exception is the work of Stolz (1999) where the Hashin–Shtrikman bounding method is extended to the case of thermoelastic heteroge- neous medium defined by representative morphological patterns. Note that Christensen (1979) takes also into account the connectedness of the matrix but determines only the thermal conductivity and the thermal expansion coefficient of two-phase materials.

The present paper extends the ‘‘ðnþ1Þ-phase’’ model, already used in elasticity (Hervee and Zaoui, 1993, 1995a,b; Hervee et al., 1993; Hervee and Pelleegrini, 1995), in viscoelasticity (Beurthey, 2000; Fen-Chong et al., 1999; Alberola and Mele, 1994, 1997; Alberola and Benzarti, 1997, 1998), in viscoplasticity (Hervee et al., 1995) and in elastoplasticity (Theebaud et al., 1992; Bornert et al., 1993, 1994), to thermal and thermoelastic behaviours. The main results are estimates of the thermal conductivity, of the thermal expansion coefficient and of the specific heats of multiply coated inclusion-reinforced composites.

In the following the studied configuration is composed of an n-layered isotropic spherical inclusion surrounded by an infinite matrix (nis arbitrary). An explicit solution of the temperature field (Section 2) is first proposed if a uniform temperature gradient is applied at infinity and no coupling is assumed between mechanical and thermal effects. This solution is used for the definition of aðnþ1Þ-phase model providing an estimate of the effective thermal conductivity. Then, the coupling between stress and thermal properties is taken into account. We derive in Section 3 the thermoelastic stress and strain fields in such a configu- ration, stress free and subjected to a uniform change of temperature. These results, completed by the stress and strain fields for the same configuration subjected to a hydrostatic pressure (Hervee and Zaoui, 1993) are used to construct a ðnþ1Þ-phase model to predict the effective thermal expansion coefficient and the ef- fective specific heats of multiply coated inclusion-reinforced composites. The effective thermal conductivity, thermal expansion coefficient and specific heats are proved to be the coincident bounds for an isotropic Hashinn-layered composite sphere assemblage (n-CSA) and can thus be derived by a recursive algorithm (Stolz, 1999).

2. Thermal conductivity

2.1. n-Layered spherical inclusion problem

This section is concerned with the derivation, under assumptions of no coupling between mechanical and thermal effects and of steady state conditions, of the temperature field in an infinite medium constituted of ann-layered spherical inclusion, embedded in a matrix subjected to a uniform temperature rise at infinity.

For that purpose we define:

h¼T T₀; ð1Þ

whereT₀is the base temperature andkgrad!

hk 1. Each phase is assumed homogeneous and isotropic.

As in Hervee and Zaoui (1993) let us consider the following infinite medium. Phase 1 constitutes the central core and phase (i) lies within the shell limited by the two concentric spheres with the radiiR_i1and R_i. Let (k¼kiI) be the thermal conductivity tensor of phase (i) (herei2 ½1;nþ1,R₀¼0 andR_nþ1! 1) (Fig. 1), in whichIis the second order unit tensor.

Spherical coordinate systemðr;h;/Þwith origin at the common centre of the above-mentioned spheres is introduced with axial symmetry aroundx₃-axis (x3¼rcosh).

At infinity, the imposed condition is (Christensen, 1979):

hj_r!1¼bx3; ð2Þ

wherex_k are Cartesian coordinates andb is the temperature gradientðb1Þ.

The assumptionkgrad!

hk 1 permits the of use Fourier’s lawof heat conduction

~qq¼ kgrad!

h; ð3Þ

where~qq represents the heat flux vector. With the additional assumption that there is no heat supply, the heat conduction equation writes

div~qq¼0: ð4Þ

From relations (3) and (4) we get the following governing equations forh^ðkÞin each phase k

Dh^ðkÞ¼0; k2 ½1;nþ1; ð5Þ

where theDoperator in the spherical coordinate system reads:

D¼ 1 r²

o or r² o

or

þ 1 r²sinh

o

oh sinh o oh

: ð6Þ

Thus, the general solution of Eq. (5), for the non-zero temperature rise h^ðiÞ, which respects the remote condition (2), is given in phaseðiÞ(Christensen, 1979) by

h^ðiÞ¼ A_ir

þB_i r²

cosh; ð7Þ

whereA_iandB_iare constants. The corresponding heat flux vector has for components:

q^ðiÞ_r ¼ ki A_i

2B_i r³

cosh; q^ðiÞ_h ¼ki A_i

þB_i r³

sinh; q^ðiÞ_/ ¼0; ð8Þ

Fig. 1. Then-layered spherical inclusion embedded in an infinite matrix.

where the coefficientB₁must vanish to avoid a singularity at the origin and the constantA_nþ1is determined by the imposed temperature rise at infinity:

h^ðnþ1Þ!A_nþ1x₃ asr! 1: ð9Þ

Using the condition (2) and relation (9) the following value forA_nþ1is obtained

A_nþ1¼b: ð10Þ

It is worth noticing that the imposed condition at infinity can also be of a flux-type

~qq¼ c~ee3 ð11Þ

and, in that case, we find A_nþ1¼ c

knþ1

: ð12Þ

The two conditions that result from the continuity ofq_randhat the interfacer¼R_kbetween phasesðkÞand ðkþ1Þare written in the form

H_kðRkÞ~VV_k¼H_kþ1ðRkÞVV~_kþ1 ð13Þ

wherek2 ½1;n,~VV_k¼ ðAk;B_kÞandH_k is the following matrix H_kðrÞ ¼ r _r¹2

kk 2^k_r^k3

: ð14Þ

The system of equation (13) is solved for~VV_kþ1 by means of the ‘‘transfer matrices’’I^ðkÞ

~V

V_kþ1¼I^ðkÞ~VV_k; ð15Þ

with

I^ðkÞ¼H¹_kþ1ðRkÞHkðRkÞ: ð16Þ

From Eqs. (14) and (16),I^ðkÞreads I^ðkÞ¼ 1

3k_kþ1

2k_kþ1þkk 2

R³_kðkkþ1kkÞ R³_kðkkþ1kkÞ 2kkþkkþ1

!

: ð17Þ

Therefore one gets

~V

V_kþ1¼Y¹

j¼k

I^ðjÞ~VV1¼O^ðkÞ~VV1; ð18Þ

and, from Eq. (18) withB1¼0 we find A1¼ 1

O^ðnÞ₁₁ A_nþ1: ð19Þ

Thus all the coefficients A_k andB_k can be expressed as:

A_k ¼^O

ðk1Þ 11

O^ðnÞ₁₁ A_nþ1 k2 ½1;n;

B_k ¼^O

ðk1Þ 21 O^ðnÞ₁₁ A_nþ1: 8>

<

>: ð20Þ

2.2. Average temperature gradient and heat flux vector

From the solution derived in the previous section, the third component of the average temperature gradient in each phase is easily calculated with the relation

hh^ðkÞ_;3i ¼ 1 V_k

Z

Vk

h^ðkÞ_;3 dV ¼ 1 V_k

Z

Sk

h^ðkÞn₃dS

Z

Sk1

h^ðkÞn₃dS

: ð21Þ

In Eq. (21)V_k is the volume of the shell occupied by phaseðkÞ, limited byS_k1 andS_k andn3 (n3¼cosh) denotes the third component of the unit outward normal toS_k1orS_kin a cartesian coordinate system (S0is the pointr¼0 and S_nþ1! 1).

The average valuehh^ðkÞ_;3iis given by

hh^ðkÞ_;3i ¼A_k; ð22Þ

if condition (2) or condition (11) is prescribed at infinity. This average is also calculated in the wholen- layered inclusion

hh;3i ¼ A_nþ1

þB_nþ1 R³_n

: ð23Þ

Correspondingly, still with condition (2) or condition (11), the third component of the average of the heat flux vector reads

hq^ðkÞ₃ i ¼ kkA_k; ð24Þ

and is given in the wholen-layered inclusion by hq3i ¼ knþ1 A_nþ1

þB_nþ1 R³_n

: ð25Þ

These solutions can be used if a truen-phase inclusion is considered in a well defined matrix whose thermal conductivity coefficient k_nþ1 is known. It can also be used if k_nþ1 (named then k^eff) denotes the effective thermal conductivity coefficient of a composite material according to the (nþ1)-phase model presented in the next section.

2.3. A (nþ1)-phase model

To determine the effective thermal conductivity coefficient of an isotropic composite material the

‘‘ðnþ1Þ-phase model’’ is defined as follows: if condition (2) is imposed, the third component of the average temperature gradient in the wholen-layered inclusion is the same as the third component of the macro- scopic temperature gradient imposed to the composite medium at infinite ðhh;3i ¼bÞ. From Eq. (23) this condition reduces to

B_nþ1¼0: ð26Þ

Note that a similar self-consistent condition could be proposed from Eq. (25) and boundary condition (11) for the heat flux vector.

Let us introduce the virtual workQ(Hashin, 1972):

Q¼ Z Z Z

V

~qqgrad!

hdv: ð27Þ

Q_compandQ₀are the virtual works calculated over the composite configuration shown in Fig. 1 and over a second configuration. This second configuration is a body identical to the first one except that the inclusion

is replaced by the equivalent homogeneous material characterized byk^eff. Following the same treatment as the one used in (Christensen, 1979) for the elastic energy we can show that:

Q_comp¼Q₀ Z Z

Sn

ðq!⁰

h~qqh⁰Þ ~nndS ð28Þ

where!q⁰

andh⁰designate the field variables calculated in the homogeneous configuration and~qq,hthe ones calculated in the composite configuration.S_ndenotes the surface of the composite inclusion and the sign in front of the integral in Eq. (28) depends on the conditions specified on the outer boundary (imposed heat flux vector or imposed temperature). Combining Eqs. (7) and (8) in the self-consistent conditionQ_comp¼Q₀ yields also relation (26).

The effective thermal conductivity,k^eff, contributes only to the matrixI^ðnÞso that relation (18) is modified to read

~V

V_nþ1¼I^ðnÞ Y¹

j¼n1

I^ðjÞVV~1¼I^ðnÞO^ðn1Þ~VV1: ð29Þ

Relation (17) is nowsubstituted in Eq. (29) forI^ðnÞwithk_nþ1¼k^eff so thatB_nþ1reads B_nþ1¼ 1

3khR³_nðk^eff

k_nÞO^ðn1Þ₁₁ þ ð2knþk^effÞO^ðn1Þ₂₁ i

ð30Þ Setting B_nþ1 to zero provides the effective thermal conductivity coefficient

k^eff¼

knhO^ðn1Þ₁₁ R³_n2O^ðn1Þ₂₁ i R³_nO^ðn1Þ₁₁ þO^ðn1Þ₂₁

: ð31Þ

A recursive algorithm could be invoked to obtain the same result. k^eff_ðiÞ denotes the effective thermal con- ductivity coefficient associated with aðiþ1Þ-phase model. Then the following relation relatesk^eff_ðiÞ andk^eff_ði1Þ

k^eff_ðiÞ ¼k_iþ

ki R³_i1

R³_i ki

k^eff_ði1Þkiþ¹₃ ^R3iR_R3³ⁱ¹ i

: ð32Þ

If i¼2 this equation (with k^eff_ð1Þ¼k1) yields the effective conductivity coefficient which was obtained by Hashin and Shtrikman (1962). It is the coincident bounds of a CSA with the volume fraction R³₁=R³₂. The recursive algorithm can be summarized as follows: any ‘‘two-phase inner spheres’’ in then-layered CSA, is replaced by a homogeneous one with the coefficientk^eff_ð2Þ. Second, the three-layered CSA is considered as a two-layered one with the coefficient k^eff_ð2Þ in the central core with the volume fraction R³₂=R³₃ and k3 in the external layer withinR₂andR₃. These two steps are repeated until the incrementireaches the numbernof phases considered in the CSA. The same results are applicable to electrical conductivity, magnetic per- meability, dielectric constant and diffusivity values.

3. Thermal expansion coefficient and specific heats

3.1. n-Layered spherical inclusion problem 3.1.1. Determination of the strain and stress fields

Let us consider the problem of an infinite medium constituted of a n-layered isotropic spherical inclu- sion, embedded in a stress-free matrix subjected to a prescribed temperatureh¼h0as shown in Fig. 1, with

coupling between stress and thermal properties. This section is concerned with the derivation of the tem- perature and of the displacement fields for all point of this configuration. Under steady state conditions, assuming that there is no heat supply, the heat conduction lawtakes the simple form

Dh^ðiÞ¼0; i2 ½1;nþ1; ð33Þ

whereh^ðiÞ denotes the temperature rise in phase ðiÞ. Letðl_i;k_i;aiÞbe respectively the shear modulus, the bulk modulus and thermal expansion coefficient of phase (i). The equilibrium equation in phase ðiÞreads

3kiþl_i 3 grad!

ðdiv~uuÞ þl_irr~~uu3kiaigrad!

h^ðiÞ¼0: ð34Þ

The problem has spherical symmetry and is thus one-dimensional. The solution, for the non-zero tem- perature fieldh^ðiÞof Eq. (34) is given in phaseðiÞby

h^ðiÞðrÞ ¼ C_i

þD_i r

ð35Þ where the coefficientD1must vanish to avoid the singularity at the origin. The constantC_nþ1is determined by the prescribed condition at infinity:

C_nþ1¼h0: ð36Þ

The interface conditions require the continuity of the temperatureh and of the normal heat flux~qq~nn,~nn denotes the normal vector at the interface. It follows that

Di¼0 and Ci¼C_nþ1¼h0 8 i2 ½1;nþ1 ð37Þ and therefore

hðrÞ ¼h0 8r: ð38Þ

Now, combining Eqs. (34) and (38) and using the spherical symmetry, one obtains

div~uu¼0 ð39Þ

which admits for solution for the non-zero displacement componentu^ðiÞ_r u^ðiÞ_r ¼F_irþG_i

r²: ð40Þ

Then, the corresponding stresses are found to be r^ðiÞ_rr ¼3kiFi4l_i

r³ Gi3kiaih0; r^ðiÞ_hh ¼r^ðiÞ_//¼3kiF_iþ2l_i

r³ G_i3kiaih0;

ð41Þ

while all other stress components vanish. Let us set~VV_k ¼ ðFk;G_kÞ,k2 ð1;nÞ; the continuity conditions forrrr

andu_r at the interfacer¼R_k are written in the form

J_kðRkÞ~VV_kþh0~LL_k ¼J_kþ1ðRkÞ~VV_kþ1þh0~LL_kþ1; ð42Þ where~LL_k ¼ ð0;3kkakÞ ðk2 ½1;nÞandJ_iis the following matrix (Hervee and Zaoui, 1993)

JiðrÞ ¼ r _r¹₂ 3ki 4_r^l3ⁱ

: ð43Þ

The solution of the system of equation (42) is

~V

V_kþ1¼N^ðkÞ~VVkþh0J¹_kþ1ðRkÞð~LLk~LL_kþ1Þ; ð44Þ whereN^ðkÞ is given by

N^ðkÞ¼J¹_kþ1ðRkÞJkðRkÞ: ð45Þ

Substituting forJ_iðrÞfrom Eq. (43) into Eq. (45) yields N^ðkÞ¼ 1

3k_kþ1þ4l_kþ1

3k_kþ4l_{kþ1 R}⁴3 k

ðl_kþ1l_kÞ 3ðk_kþ1k_kÞR³_k 3k_kþ1þ4l_k

!

: ð46Þ

Therefore one gets

~V

V_kþ1¼Q^ðkÞ~VV1þh0M^ðkþ1Þ¼Y¹

j¼k

N^ðjÞ~VV1þh0MM~^ðkþ1Þ; ð47Þ

with M~

M^ðkþ1Þ¼3ðkkþ1a_kþ1k_ka_kÞ 3kkþ1þ4l_kþ1

1 R³_k

þX^k1

i¼1

3ðkiþ1a_iþ1k_ia_iÞ 3kiþ1þ4l_iþ1

Y^iþ1

j¼k

N^ðjÞ

! 1 R³_i

; ð48Þ

withk2 ð1;nÞ,MM~^ð1Þ¼~00 andQ^ð0Þ¼I.

Making use of relation (47), withG₁¼0, we find F1¼F_nþ1h0M₁^ðnþ1Þ

Q^ðnÞ₁₁ : ð49Þ

Thus all the coefficients F_k,G_k are F_k ¼^Q

ðk1Þ 11

Q^ðnÞ₁₁ F_nþ1þ ^h0

Q^ðnÞ₁₁hM₁^ðkÞQ^ðnÞ₁₁ M₁^ðnþ1ÞQ^ðk1Þ₁₁ i

k2 ½1;n;

Gk¼^Q

ðk1Þ 21

Q^ðnÞ₁₁ F_nþ1þ ^h0

Q^ðnÞ₁₁ hM₂^ðkÞQ^ðnÞ₁₁ M₁^ðnþ1ÞQ^ðk1Þ₂₁ i : 8>

<

>: ð50Þ

The infinite matrix being stress free (rrr¼0 at infinity), it results from Eq. (41) that

F_nþ1¼a_nþ1h0: ð51Þ

Then, from Eqs. (50) and (51) F_k ¼ ^h0

Q^ðnÞ₁₁ ½anþ1Q^ðk1Þ₁₁ þM₁^ðkÞQ^ðnÞ₁₁ M₁^ðnþ1ÞQ^ðk1Þ₁₁ k2 ½1;n;

G_k¼ ^h0

Q^ðnÞ₁₁ ½a_nþ1Q^ðk1Þ₂₁ þM₂^ðkÞQ^ðnÞ₁₁ M₁^ðnþ1ÞQ^ðk1Þ₂₁ : 8<

: ð52Þ

Consequently, from Eqs. (40), (41) and (52) the displacement field~uu^ðiÞ and the corresponding stresses are calculated for a prescribed temperature rise at infinity.

3.1.2. Average strains and stresses

From these solutions, the average strain and stress tensor in each phase are nowcalculated following (Hervee and Zaoui, 1993). The average strain tensor in phaseðiÞis given by

h^ðiÞi ¼F_i½~ee₁~ee₁þ~ee₂~ee₂þ~ee₃~ee₃ ð53Þ where~ee₁,~ee₂ and~ee₃ denote the unit vectors if the configuration under study is referred to a cartesian- coordinate system. We obtain therefrom the average strain in the whole n-layered inclusion

hi ¼ F_nþ1

þG_nþ1 R³_n

½~ee₁~ee₁þ~ee₂~ee₂þ~ee₃~ee₃ ¼ a_nþ1h0

þG_nþ1 R³_n

½~ee₁~ee₁þ~ee₂~ee₂þ~ee₃~ee₃: ð54Þ

Similarly, the average stress tensor in phaseðiÞis given by

hr^ðiÞi ¼3kiðF_iaih0Þ½~ee₁~ee₁þ~ee₂~ee₂þ~ee₃~ee₃; ð55Þ and it follows that the average stress in the wholen-layered inclusion, according to Eq. (51), takes the final form

hri ¼ 4l_nþ1

R³_n G_nþ1: ð56Þ

3.2. Thermal expansion coefficient

In order to determine the thermal expansion coefficient of a multiply coated inclusion-reinforced composite, let us consider the problem described in Section 2.1 but nowthe infinite matrix is both subjected to uniform stress conditions at infinity (T0

!¼R~nnÞand to a prescribed temperature (h¼h0) (Fig. 2). We can split this problem (P) into two elementary problems (P⁰) and (P^r) which have been studied in Hervee and Zaoui (1993) and in Section 3.1 respectively.

In this section,ðl_nþ1;k_nþ1;anþ1Þare called ðl^eff;k^eff;a^effÞ. These three scalars denote the effective shear modulus, bulk modulus and thermal expansion tensor according to the ðnþ1Þ-phase model defined in Hervee and Zaoui (1993) forl^eff andk^eff and hereafter fora^eff.

Fig. 2. Decomposition of the problemðPÞinto two elementary problemsðP⁰ÞandðP^rÞ.

The stress fieldr⁰ðxÞis related to the prescribed stressRby the stress-concentration tensorBðxÞwhich can be determined by means of the results of a previous paper (Hervee and Zaoui, 1993)

r⁰ðxÞ ¼BðxÞ:R; ð57Þ

where (:) stands for the second-order inner product (r:¼rij_ij). We definea^eff(according to theðnþ1Þ- phase model) as the following link between the average strainE^F in the wholen-layered inclusion and the prescribed temperature

E^F ¼ h^rðxÞi_VI¼a^effh⁰: ð58Þ

This means that the composite inclusion behaves as the surrounding equivalent homogeneous medium.V_I denotes the volume of the n-layered inclusion. It follows froma_nþ1¼a^eff and from Eq. (54) that

G_nþ1¼0: ð59Þ

Application of result (38) leads to the following expression

^rðxÞ ¼aðxÞh⁰þs:r^rðxÞ; ð60Þ

wheresis the elastic compliance.

In Hervee and Zaoui (1993) it had been shown that

hr⁰i_VI ¼R: ð61Þ

Therefore, accounting for Eq. (57), it can be inferred that

hBi_VI¼I; ð62Þ

whereIdenotes the fourth order unit tensor.

The result given by Levin (1968) leads to the following relation:

a^eff¼ ha:Bi: ð63Þ

Combininga_nþ1¼a^eff and Eq. (62) yields a^eff ¼limV!1 VI

Vha:Bi_VIþ^VV_V ^Ia^eff :hBi_VVI

a^eff ¼limV!1 VI

Vha:Bi_VIþ^VV_V ^Ia^eff

; 8<

: ð64Þ

anda^eff finally reduces to:

a^eff¼ ha:Bi_VI: ð65Þ

Each phase is isotropic, thereforea^eff¼a^effI,aðxÞ ¼aðxÞIand a^eff¼1

3haðxÞB_iikki_VI: ð66Þ

It is worth noting that when R¼ ðr0=3ÞI,B_iikkðxÞmay be derived by means of TrðrðxÞÞ ¼BiikkðxÞr0

3 ð67Þ

and then, by using Eqs. (15) and (34) and results in Hervee and Zaoui (1993) B_iikk is given in phase ðlÞby B_iikk ¼3Q^ðl1Þ₁₁

Q^ðnÞ₁₁ k_l

k^eff: ð68Þ

Consequently, the effective thermal expansion coefficient is provided by the relation

a^eff ¼ 1 Q^ðnÞ₁₁k^eff

Xⁿ

i¼1

c_iaik_iQ^ði1Þ₁₁ ð69Þ

wherec_i denotes the volume fraction of phaseðiÞ.

As in the case of the thermal conductivity, it can be noticed that ifa^eff_ðiÞ denotes the effective thermal ex- pansion coefficient associated with aðiþ1Þ-phase model we get the following relation betweena^eff_ðnÞanda^eff_ðn1Þ

a^eff_ðnÞ¼R³_n1

R³_n a^eff_ðn1Þþ 1

R³_n1 R³_n

a_nþ þ

4l_n^R_R³ⁿ¹3

n 1^R_R³ⁿ¹3 n

k^eff_ðn1Þk_n

2

3k_nþ4l_n

ð Þk_ðn1Þ^eff þ4l_n 1^R_R³ⁿ¹3 n

k_nk_ðn1Þ^eff

a^eff_ðn1Þ a_n

:

ð70Þ

3.3. Illustrative examples

The above results allowus to better understand the problem of debonding along the two interfaces (interface inclusion/coated-phase here represented byr¼R₁ and coated-phase/matrix here represented by r¼R₂) of the interphase of a coated inclusion-reinforced composites. For that purpose Eqs. (41), (52) and (69) and the results fork^eff andl^eff known from Hervee and Zaoui (1993) are used to determine the nor- malized radial stressðrrr=h0k₃a3Þ, along the two above-mentioned interfaces, due to an uniform temperature riseh0. Fig. 3 gives this normalized stress in terms of the normalized bulk modulusðk2=k₃Þof the interface.

The analytical results showthat these radial stresses depend linearly on the thermal expansion coefficient of the interphase ðrrrðR1Þ=h0k₃a3¼a₁a2=a3þb₁ and rrrðR2Þ=h0k₃a3¼a₂a2=a3þb₂Þ. These four parameters are reported in Fig. 4a and b in terms of the normalized bulk modulus ðk2=k₃Þ of the interphase. The illustrative examples (Figs. 3 and 4) prove that these models are easily tractable not only to predict the global behaviour of multiply coated inclusion-reinforced composites but also to determine some local major values, for instance here the sign ofrrr and so the debonding along the interfaces.

3.4. Specific heats

We nowdetermine the effective specific heats for a multiply coated inclusion-reinforced composite. For that purpose, let us consider the configurationðP^rÞdescribed in Fig. 2 whereC_vi denotes the specific heat, per unit volume at constant volume in phase ðiÞandC_v^eff the effective specific heat of the infinite matrix according to theðnþ1Þ-phase model defined hereafter.

For small strain and small temperature changes, the isotropic Helmholtz free energy densityFper unit volume is defined as (Rosen, 1970):

F ¼1

2:C:þC:h1

2C_vh²; ð71Þ

whereCis the tensor of elastic moduli,the strain tensor,Cthe thermal stress tensor,hthe temperature which are related to the stress tensorr and to the thermal expansion tensora by

¼s:rþah ð72Þ

and

C¼ C:a: ð73Þ

Note that in the definition ofFthe term involvingC_vhas a sign reversal from those in Rosen (1970), to be consistent with the classical thermodynamical derivations.T₀C_v=qdenotes here the specific heat at constant deformation.

Consequently, Fmay be represented analogously to expression (71) in the form F ¼1

2r:þ1

2C:h1

2C_vh²: ð74Þ

From Rosen (1970) we know that the average free energy density may be expressed in the following fashion F ¼1

2r:þ1

2C^eff:h⁰1

2C_v^effh²: ð75Þ

wheref ¼_V¹ R R R

VfdV and whereC^eff_v is the unknown effective specific heat.

LetFcompbe the free energy calculated over the composite configuration described in Fig. 5 and letF0be the free energy defined in a body identical with the one just specified, except that the inclusion is replaced by the homogeneous material characterized byðl^eff;k^eff;a^eff;C^eff_v Þ.

Accounting for the boundary conditions~TT₀¼~00 at infinity we find:

Vlim!1

1 2

Z Z Z

V

r:¼ lim

V!1

Z Z

oV

~uu~TT0dS ¼0; ð76Þ

andFcomp andF0 are then expressed as F_comp¼1

2 Z Z Z

V

ðC:h0C_vh²₀ÞdV ð77Þ

and

Fig. 3. Normalized radial stressrrr=h0k3a3along the interface inclusion/coated-phase (r¼R1) and along the interface coated-phase/

matrix here (r¼R2) of the coated phase (phase 2) of a composite material made of coated inclusions versus the normalized bulk modulus (k2=k3) of the coated phase withk1=k3¼2. The Young moduli in the different phases arem1¼0:17,m2¼m3¼0:3, the nor- malized thermal expansion coefficients are given bya1=a3¼1:5 anda2=a3¼3. The volume fractions are given byC1þC2¼0:35 and C1¼0:88ðC1þC2Þ.

Fig. 4. Normalized radial stressrrr=h0k3a3along the interface inclusion/coated-phase (r¼R1) and along the interface coated-phase/

matrix here (r¼R2) of the coated phase (phase 2) of a composite material made of coated inclusions versus the normalized bulk modulus (k2=k3) of the coated phase and subjected to an uniform temperature rise h0. (rrrðR1Þ=h0k3a3¼a1a2=a3þb1 and rrrðR2Þ=h0k3a3¼a2a2=a3þb2). The slopes (a1anda2) are reported in Fig. 4 (a) and the values ofrrr=h0k3a3fora2=a3¼0 are reported in Fig. 4(b).k1=k3¼2. The Young moduli in the different phases arem1¼0:17,m2¼m3¼0:3, the normalized thermal expansion coefficient in phase 1 isa1=a3¼1:5 and the volume fractions are given byC1þC2¼0:35 andC1¼0:88ðC1þC2Þ.

Fig. 5. Self-consistent energy condition.

F₀¼1 2

Z Z Z

V

ðC^eff :⁰h0C_v^effh²₀ÞdV; ð78Þ with⁰¼a^effh0.

It is worth noticing thatF_compF₀can be split into two parts F_compF₀¼1

2 Z Z Z

V

ðC:C^eff:⁰Þh0dV þ1 2

Z Z Z

V

ðC_v^effC_vÞh²₀dV: ð79Þ The isotropy of all materials implies

C:¼Cij_ij ¼ Cijklakl_ij ¼ 3kaTr: ð80Þ Thanks to Eq. (40), Tris given in phase ðiÞby

Tr¼3Fi ð81Þ

and consequently 1

2 Z Z Z

V

ðC:C^eff:⁰Þh0dV ¼9 2

X^nþ1

i¼1

Z Z Z

Vi

ðk^effa^effF_nþ1K_ia_iF_iÞh0dV: ð82Þ Given that k_nþ1¼k^eff,a_nþ1¼a^eff and C_v¼C_v^eff in the matrix around the composite inclusion, FcompF0

finally becomes F_compF₀¼1

2 Xⁿ

i¼1

Z Z Z

Vi

9k^effa^effF_nþ1

K_iaiF_i

h0þ ðC ^eff_v

C_viÞh²₀

dV: ð83Þ

Our criterion for determining the effective specific heat C^eff_v is to set

FcompF0¼0: ð84Þ

It turns out thatC_v^eff is then completely determined and the combination of Eqs. (83) and (84) yields C_v^eff¼hC_vi_VI 9

h0

k^effa^effF_nþ1

"

Xⁿ

i¼1

K_iaiF_i

#

: ð85Þ

Substituting Eqs. (52) and (51) into Eq. (85) and accounting foranþ1¼a^eff provides C_v^eff¼hC_vi_VI9 k^effða^effÞ²

(

Xⁿ

i¼1

cikiai

Q^ðnÞ₁₁ ha^eff

M₁^ðnþ1Þ

Q^ði1Þ₁₁ þM₁^ðiÞQ^ðnÞ₁₁i)

; ð86Þ

and it follows from Eq. (69) that C^eff_v can be put finally in the form C_v^eff¼hC_vi_VI9 k^effa^effM₁^ðnþ1Þ

(

Xⁿ

i¼1

c_ik_iaiM₁^ðiÞ )

: ð87Þ

Note that, as for the thermal conductivity and for the thermal expansion coefficient, if C^eff_v

ðiÞ denotes the effective specific heat associated with aðiþ1Þ-phase model, we get the following relation betweenC_v^eff

ðnÞ and C_v^eff

ðn1Þ

C_v^eff

ðnÞ¼R³_n1 R³_n C^eff_v

ðn1Þþ 1

R³_n1 R³_n

C_vn

27^R_R³ⁿ¹3 n

R³_n1 R³_n 1

k_ðn1Þ^eff a^eff_ðn1Þk_nan

2

3knþ4l_nþ3 1^R_R³ⁿ¹3 n

k_ðn1Þ^eff k_n

: ð88Þ

Ifn¼2, this equation withðC^eff_vð1Þ¼C_v1Þprovides the effective specific heat which is given by (Stolz, 1999) for a two-phase CSA (with a different sign in the definition of C_v).

LetT0C_p=q be the specific heat per unit volume at constant stress. The effective specific heats (C_v^eff and C^eff_p ) are related (Rosen, 1970; Christensen, 1979) by the following expression

C^eff_p C^eff_v ¼a^eff :C^eff:a^eff; ð89Þ

and, since we consider isotropic behaviours, the above-mentioned relation becomes

C^eff_p C^eff_v ¼9k^effða^effÞ²: ð90Þ

4. Results and conclusion

The aim of this paper was to extend the field of possible application of the ðnþ1Þ-phase model to multiply coated inclusion-reinforced composite having a thermal or a thermoelastic behaviour. In order to prove the tractability of these models, illustrative examples of the influence of an interphase lying between the particles and the matrix of a particle-reinforced composite material are given in Figs. 6–8. In the first and third case the volume fraction of the coating phase is varying from 0 up to the total initial volume fraction of the particles (for different values of the interphase conductivity coefficient in the first case and for different values of the thermal expansion coefficient in the third case). In the second case the volume fraction of the coating and particle phases are fixed but both the thermal expansion coefficient and the bulk modulus of the coating phase are varying.

Fig. 6. Normalized effective conductivity coefficientk^eff=k3of a composite material, made of coated inclusions versus the volume fraction (C2) of the coated phase (phase 2) for different values ofa¼k2=k3(1 denotes the inclusion and 3 the matrix).k1=k3¼6, C1þC2¼0:2.

These results can also be used to determine the overall behaviour of material with gradient of properties by discretization as already done in Hervee and Zaoui (1995a) in order to predict the effective behaviour of fibre-reinforced composite having an accommodating interphase.

Fig. 8. Normalized difference of the effective specific heat with the average of the specific heats of each phase of a composite material, made of coated inclusions versus the volume fraction (C2) of the coated phase (phase 2) for different values ofa2=a3(3 denotes the matrix).k1=k3¼2,k2=k3¼0:1,m1¼0:17,m2¼m3¼0:3,a1=a3¼1:5,C1þC2¼0:2.

Fig. 7. Normalized effective thermal expansion coefficienta=a1of a composite material, made of coated inclusions. The volume fraction of the inclusion (phase 1) and of the coated phase (phase 2) are respectively 0.1 and 0.01 and the elastic behaviours of the different phases are characterized byk1=l₃¼15,l₁=l₃¼7,k3=l₃¼2,l₂=l₃¼4,a3=a1¼10.

It is worth noting that the form of these results allows to use a recursive algorithm to determine the effective thermal conductivity, the effective thermal expansion coefficient and the effective specific heats of a composite material according to theðnþ1Þ-phase model.

Moreover, Figs. 3 and 4 prove that all these results are very useful to determine some major local pa- rameters such as the radial stress along the interfaces of a coated phase around the inclusion. The problem of debonding along the interfaces of the coated phase is often meet in the case of fibre-reinforced com- posites. The extension of such a study to multiply coated fibre-reinforced composites should bring inter- esting answers to a lot of practical problems.

Acknowledgement

I am indebted to Claude Stolz for fruitful discussions.

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