A Layered-Shell Model of Anisotropic Composites:
Extension of the Milgrom and Shtrikman Model
ELHASSANE BARHDADI
Department of Mechanical Engineering,
Abdelmalak Asaadi University,
Sidi Bouafif, 32003 Al-Hoceima,
MOROCCO
Abstract: - In this work, the extension of the Milgrom and Shtrikman model to anisotropic composite materials
containing n-layered hollow ellipsoidal inclusions, is presented. The effective properties of such materials are
determined using the Green function techniques and interfacial operators. Here, the basic unit of the
microstructure is a hollow system of contacting concentric ellipsoidal shells, each of which is made of one of
the components. Space is packed with such units of different sizes, but the same proportions; the cavity within
each such shell system is then packed with similar systems and this continues in an infinite nesting sequence. In
the final configuration, the effective properties are inside and outside the basic unit of layered shells (n+1). For
n=2 and in the case of isotropic material, it is shown that the effective compressibility covers all ranges of the
Hashin-Strikman bounds.
Key-Words: - Layered Shells, composite materials, anisotropy, ellipsoidal inclusions, Micromechanics,
Milgrom and Shtrikman Model, Four-Phase Model, Hashin-Shtrikman bounds.
Received: April 5, 2024. Revised: August 14, 2024. Accepted: September 16, 2024. Published: November 13, 2024.
1 Introduction
The multilayered model is widely studied in the
literature. [1] extend the composite sphere model,
[2] to n-layered spherical inclusion and give an
exact solution of the effective elastic properties. [3],
studied the thermo-elastic behavior of such
materials. [4], obtained a general form for the
piezoelectric properties of -layered ellipsoidal
inclusion. [5] and [6], give another form of the
problem of multicoated inclusion.
Historically, [7] was the first to study
theoretically the effect of an interphase on local
stress and strain fields. More recently, [8] developed
a new analytical method based on Green’s function
technics [9] and interfacial operators [10] for the
determination of effective elastic moduli of the so-
called four-phase model.
Here, an extension of [11] is given for a
composite consisting of hollow multilayered
ellipsoidal inclusions. Space is packed with such
units of different sizes, but the same proportions; the
cavity within each such shell system is then packed
with similar systems and this continues in an infinite
nesting sequence. All phases are assumed
homogeneous and anisotropic and perfect bonding is
supposed at the interfaces. It shows that the obtained
effective properties depend on the evaluation of the
localization tensors.
2 Micromechanical Modelling
Following [11] and using the generalized self-
consistent scheme (GSCS) [12] the effective
properties are inside and outside the composite
inclusion (Fig.1). The elementary problem described
in Fig. 1 is constituted by ellipsoidal shells
surrounded by a matrix layer and the whole
composite inclusion is embedded in the effective
medium with unknown elastic properties. In
this composite inclusion, c1 = Ceff denotes the elastic
properties of ellipsoidal inclusion and c2 , c3 ,...., ck
,….cn+1 denote the elastic properties of outer shells.
Fig. 1: The (n+1) phase model
The present model takes a step beyond the
multi-layered ellipsoid model in that the equivalent
Effective medium
k
1
2
k
+
1
n
n+1
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medium is both inside and outside the basic unit of
layered shells (Figure 1).
2.1 Integral Equation
To solve this problem, the kinematical integral
equation [9] linking the elastic local strain with
the global uniform strain, is given by:
(1)
Where:
(2)
The average strain of composite inclusion in
Fig.1 is given by:
(3)
and denote the average strain and volume
fraction of phase k, respectively.
The composite inclusion of volume consists of
all phases, such that:
(4)
On the other hand, is calculated by using
equation (1):
(5)
The tensor is deduced from [13]:
(6)
Combining equations (3) and (5), one can give:
(7)
The concentrations tensor for the composite
inclusion and the concentrations tensors for each
phase k can be introduced [14] so that:
(8)
(9)
with from equation (3):
(10)
where I is the unit fourth order tensor. From
equations (8) and (9), one can also write:
(11)
where: (12)
Using equation (7), it comes:
(13)
Equation (13) expresses a relation between
tensors and . To solve the problem, another
equation should be derived.
2.2 Interfacial Operators
Perfect bonds between all phases are assumed and
then the displacement and traction vectors are
continuous through the interfaces. Using the elastic
properties of two phases (k) and (k+1), the strain
jump through their common interface is written as
follows [10]:
(14)
is the interfacial operator.
For each level (k), denotes the
volume of the composite formed by the phases 1 to
k. Then, in order to solve the problem is
substituted by the averaged value . Thus, by
performing the average strain over the phase (k + 1)
of volume denoted , the following
recurrence relation at each level (k) from eq. (14) is
giving by:
(15)
where:
(16)
and:
(17)
for , is given by:
(18)
Tensor is given by [15]:
(19)
where:
(20)
(21)
From equation (15), the following form is obtained:
(22)
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Where;
(23)
The Hooke’s law implies that:
(24)
By using equations (17), (22) and (24), the
expression of is obtained:
(25)
where:
(26)
Then, by recurrence, equation (25) is
transformed into the following equation:
(27)
with the recurrence relations for :
(28)
Thus, it is sufficient to derive to completely
solve the problem. This is done by applying
equation (10) such that:
(29)
3 Framing of Any Compressibility by
[16]
The composite under consideration is an isotropic
material consisting of isotropic matrix containing
isotropic spherical and coated inclusions. The
resulting four phase model is shown in Fig.2 where
the inclusion 1, the interphase 2 and the matrix layer
3 are characterized by the radii , , and ,
respectively. The interphases 2 and 3 are
characterized by the elastic moduli , and
respectively. The inclusion and the equivalent
homogeneous medium are characterized by the
effective elastic moduli .
Fig. 2: Four-phase model
The volume fraction of the sphere occupied by
the components [11] is given by:
(30)
The special case is the well-known
coated sphere model. Another special case is
(very thin shells).
The volume fractions of phases 1 and 2 are,
respectively:
(31)
(32)
The effective elastic moduli [8] of the
composite of Fig.2 are given by:
(33)
For isotropic elastic bodies, the tensors , ,
, and are written as sum of volumetric
and deviatoric parts and and are given by the
following relations:
(34)
(35)
(36)
(37)
(38)
where and are bulk and shear moduli and
the tensors and result from the decomposition of
the unit tensor such that:
(39)
Basing on the forgoing equations, one can
express the effective compressibility as follow:
(40)
and denote the volumetric parts of
tensors and , respectively, and are given
by: (41)
(42)
Where:
Effective medium
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(43)
(44)
(45)
(46)
For a material having , ,
, and varying p between 0
and 1, an exact model that cover the whole range of
allowed values of the effective compressibility is
given in Fig.3.
Fig. 3: Framing of any compressibility by [16]
4 Conclusion
The expression for the effective properties of a new
model for anisotropic composite materials
containing -layered hollow ellipsoidal inclusions,
is derived. The obtained results present the
extension of the Milgrom and Shtrikman model who
gave an expression for the effective response matrix
of isotropic composites.
Analytical formulations obtained using the
integral equation, interfacial operators and the
obtained effective properties require the estimation
of the strain localization tensors in each phase of the
multi-layered inclusion. It is shown that any
compressibility of isotropic material lies within the
Hashin-Shtrikman bounds.
References:
[1] Herve, E., Zaoui, A., Elastic behaviour of
multiply coated fibre reinforced composites,
Int. J. Eng. Sci., Vol. 33, No. 10, 1995, pp.
1419–1433,
https://doi.org/10.1016/j.ijsolstr.2020.03.013.
[2] Hashin, Z., The elastic moduli of
heterogeneous materials, J. Appl. Mech., Vol.
29, No. 7, 1962, pp.143-150,
https://doi.org/10.1115/1.3636446.
[3] Hervé, E., Thermal and thermoelastic
behaviour of multiply coated inclusion-
reinforced composites, Int. J. Sol. Stru., Vol.
39, No. 4, 2022, pp. 1041-1058,
https://doi.org/10.1016/S0020-
7683(01)00257-8.
[4] Koutsawa, Y., Cherkaoui, M., Daya, D.,
Multi-coating inhomogeneities problem for
effective viscoelastic properties of particulate
composite materials, J. Eng. Mat. Tech., Vol.
131, No. 2, 2009, pp. 1–11,
https://doi.org/10.1115/1.3086336.
[5] Berbenni, S., Cherakoui, M., Homogenization
of multi-coated inclusion-reinforced linear
elastic composites with eigenstrains:
application to the thermo-elastic behavior,
Phil. Maga. & Phil. Mag. L., Vol. 90, No. 22,
2010, pp. 3003-3026,
https://doi.org/10.1080/14786431003767033.
[6] Bonfoh, N, Dizart, F., Sabar, H., New exact
multi-coated ellipsoidal inclusion model for
anisotropic thermal conductivity of composite
materials, App. Math. Mod., Vol. 87, No. 4,
2020, pp. 584-605,
https://doi.org/10.1016/j.apm.2020.06.005.
[7] Walpole, L. J., Elastic behavior of composite
materials: theoretical foundations, Ad. App.
Mech., Vol. 21, No. 8, 1981, pp. 169-242,
https://doi.org/10.1016/S0065-
2156(08)70332-6.
[8] Barhdadi, E. H., Lipinski, P., Cherkaoui, M.,
Four phase model: A new formulation to
predict the elastic moduli of composites, J.
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2024.19.12
Elhassane Barhdadi
E-ISSN: 2224-3429
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Volume 19, 2024
Eng. Mat. Tech., Vol. 129, No. 2, 2007, pp.
313-320, https://doi.org/10.1115/1.2712472.
[9] Dederichs, P., H., Zeller R., Variational
treatment of the elastic constants of disordered
materials, Z. Phy. Vol. 259, No. 2, 1973, pp.
103-113,
https://doi.org/10.1007/BF01392841.
[10] Hill, R., Interfacial operators in the mechanics
of composite media, J. Mech. Phy. Sol., Vol.
31, No. 4, 1983, pp. 347-357,
https://doi.org/10.1016/0022-5096(83)90004-
2.
[11] Milgrom, M., Shtrikman, S., A layered-shell
model of isotropic composites and exact
expressions for effective properties, J. App.
Ph., Vol. 66, No. 8, 1989, pp. 3429–3436,
https://doi.org/10.1063/1.344097.
[12] Christensen, R., M., Lo, K., H., Solutions for
effective shear properties in three Phase
sphere and cylinder models, J. Mech. Phy.
Sol., Vol. 27, No. 4, 1979, pp. 315-330,
https://doi.org/10.1016/0022-5096(79)90032-
2.
[13] Eshelby, J., D., The determination of the
elastic field of an ellipsoidal inclusion and
related problems, Proc. R. Soc. London, Ser.
A. Vol. 241, No. 4, 1957, pp. 376-396,
https://doi.org/10.1098/rspa.1957.0133.
[14] Benveniste, Y., Dvorak, G., J., in The Toshio
Mura Anniversary volume: Micromechanics
and Inhomogeneity, (eds.), Springer, New
York, 1989, pp. 65–81,
https://doi.org/10.1007/978-1-4613-8919-4.
[15] Cherkaoui, M., Sabar, H., Berveiller, M.,
Micromechanical approach of the coated
inclusion problem and applications to
composite materials, ASME J. Eng. Mat.
Tech., Vol. 116, No. 3, 1994, pp. 274-278,
https://doi.org/10.1115/1.2904286.
[16] Hashin, Z., Shtrikman, S., A variational
approach to the theory of the elastic behavior
of multiphase materials, J. Mech. Phy. Sol.,
Vol. 11, No. 2, 1963, pp. 127-140,
https://doi.org/10.1016/0022-5096(63)90060-
7.
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Conflict of Interest
The author has no conflict of interest to declare.
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