Acta Polytechnica Vol. 52 No. 6/2012

A Micromechanics-Based Model for Stiffness and Strength Estimation
of Cocciopesto Mortars

Václav Nežerka, Jan Zeman

Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Thákurova 7, 166 29
Praha 6, Czech Republic

Corresponding author: vaclav.nezerka@fsv.cvut.cz, zemanj@cml.fsv.cvut.cz

Abstract
The purpose of this paper is to propose an inexpensive micromechanics-based scheme for stiffness homogenization and
strength estimation of mortars containing crushed bricks, known as cocciopesto. The model utilizes the Mori-Tanaka
method for determining the effective stiffness, combined with estimates of quadratic invariants of the deviatoric stresses
inside phases to predict the compressive strength. Special attention is paid to the representation of the C-S-H gel layer
around bricks and the interfacial transition zone around sand aggregates, which renders the predictions sensitive to particle
sizes. Several parametric studies are performed to demonstrate that the method correctly reproduces the data and trends
reported in the available literature. Moreover, the model is based exclusively on parameters with a clear physical or
geometrical meaning, and as such it provides a convenient framework for its further experimental validation.

Keywords: micromechanics, homogenization, strength estimation, cocciopesto, C-S-H gel coating, interfacial transition
zone.

1 Introduction
The use of lime as a binder in mortars is associated
with well-known inconveniences, such as slow setting
and carbonation, high drying shrinkage and porosity,
and low mechanical strength [1]. Although these limi-
tations have been overcome with the use of Portland
cement in the last 50 years, lime mortars still find use
in the restoration of historic structures. This is mainly
due to their superior compatibility with the original
materials, in contrast to many modern renovation
render systems, e.g. [16, 17, 29].
The mechanical properties of lime mortars can

be improved by the suitable design of the mixture.
The Phoenicians were probably the first ones to add
crushed clay products, such as burnt bricks, tiles or
pieces of pottery, to lime mortars to increase their
durability and strength. The Romans called this
material cocciopesto and utilized this mortar in ar-
eas where other natural pozzolans were not avail-
able. Cocciopesto-based structures exhibit increased
ductility, leading to their remarkable resistance to
earthquakes [3, 21].

Much later, it was found that the mortars contain-
ing crushed clay bricks, burnt at 600–900°C, exhibit
a hydraulic character, manifested by the formation
of a thin layer of Calcium-Silicate-Hydrate (C-S-H)
gel at the lime-brick interface [20]. Since C-S-H gel is
the key component responsible for the favorable me-
chanical performance of Portland cement pastes [23],
it is generally conjectured that the enhanced perfor-
mance of cocciopesto mortars can be attributed to

the high strength and stiffness of the C-S-H gel coat-
ing [3, 20, 21, 29]. This mechanism competes with
the formation of the Interfacial Transition Zone (ITZ)
at the matrix-aggregate interface, which is known to
possess higher porosity and thus lower stiffness in
cement-based mortars, e.g. [24, 28, 36].
The purpose of this work is to interpret these

experimental findings by a micromechanical model
based on the Mori-Tanaka method [19], motivated
by its recent applications to related material sys-
tems. These include estimates of the effective ther-
mal conductivity of rubber-reinforced cement compos-
ites [31], elasticity predictions for early-age cement [5]
or alkali-activated [35] pastes, upscaling the compres-
sive strength of cement mortars [26], and multi-scale
simulations of three-point bending tests of concrete
specimens [34]. Here, we exploit these developments
to propose a simple analytical model for a stiffness
and strength estimation of cocciopesto mortars in
Section 2. In particular, the elasticity predictions uti-
lize Benveniste’s reformulation [4] of the Mori-Tanaka
method [19], whereas the strength predictions build
on recent results by Pichler and Hellmich [26], who
demonstrated that compressive strength is closely re-
lated to the quadratic average of the deviatoric stress
in the weakest phase. Particular attention is paid
to representation of the coatings by C-S-H gel and
ITZ, which renders the predictions sensitive to the
size of the brick particles and aggregates. In Section 3,
we verify predictions of the proposed scheme against
data available in the open literature. These finding
are summarized in Section 4, mainly as a support for

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Acta Polytechnica Vol. 52 No. 6/2012

matrix (0)

brick (2)

void (1)

sand (4)
ITZ (5)C-S-H (3)

Figure 1: Scheme of the micromechanics-based
model. The numbers in parentheses refer to the in-
dexes of the individual phases.

future validation of the model against experimental
results. Finally, in Appendix A we gather technical
details needed to account for coated inclusions in order
to make the paper self-contained.

In what follows, the Mandel representation of sym-
metric tensorial quantities is systematically employed,
e.g. [18, p. 23]. In particular, italic letters, e.g. a
or A, refer to scalar quantities, and boldface letters,
e.g. a or A, denote vectors or matrix representations
of second- or fourth-order tensors. AT and (A)−1
standardly denote the matrix transpose and the in-
verse matrix. Other symbols and abbreviations are
introduced in the text when needed.

2 Model

We consider a composite sample occupying domain Ω,
composed of n distinct phases indexed by r. The value
r = 0 is reserved for the matrix phase and r = 1, . . . ,n
refer to heterogeneities having the shape of a sphere
or spherical shell, see Fig. 1. The volume fraction of
the r-th phase is defined as c(r) = |Ω(r)|/|Ω|, where
|Ω(r)| denotes the volume occupied by the r-th phase,
and the geometry of the coated particles is specified
by their radii R(r) for r = 2, . . . , 5, Fig. 2.

Several comments are in now order concerning sim-
plifications adopted in the model. First, brick particles
and voids are considered spherical in shape, instead
of more realistic ellipsoids, as in e.g. [27, 26]. This
step is known to introduce only minor errors in the
prediction of overall transport [31] or elastic [27] prop-
erties. As demonstrated by Pichler et al. [27], the
up-scaled strength is more sensitive to the shape of
inhomogeneities, but the model is still capable of pre-
dicting the correct trends. Second, the ITZ is taken as
homogeneous and is not resolved down to the level of
microheterogeneities. This arises as a result of the fact
that, in contrast to cement-based materials [24, 28],
we are currently not aware of any work studying the
structure of ITZ in lime-based mortars, thus input
data for a more detailed representation is not available.
Third, only monodisperse distribution of particles in
assumed. Polydispersivity can be incorporated by sim-

ple averaging arguments and results only in a moderate
increase in accuracy [31, and references therein].
The elastic properties of the individual phases are

specified by the material stiffness matrix L(r). As each
phase is assumed to be homogeneous and isotropic,
we have

L(r) = 3K(r)IV + 2G(r)ID for r = 0, . . . ,n, (1)

where K(r) and G(r) are the bulk and shear moduli of
the r-th phase, and IV and ID denote the orthogonal
projections to the volumetric and deviatoric compo-
nents, e.g. [13], so that

ε(x) = (IV + ID)ε(x) = εV(x)1 + εD(x), (2a)
σ(x) = (IV + ID)σ(x) = σV(x)1 + σD(x), (2b)

for x ∈ Ω. In Eq. (2), ε and σ refer to local stresses
and strains, εV and εD are the volumetric and devia-
toric strain components, σV and σD refer to the stress
components, and 1 is the second-order unit tensor (in
the matrix representation).

The development of the model follows the standard
routine of the continuum micromechanics, e.g. [37].
The sample Ω is subjected to the overall strain loading
E. Neglecting the interaction among the phases, the
mean strains inside the heterogeneities are obtained
as

E(r) = A(r)dil E for r = 1, . . . ,n,

where A(r)dil is the dilute concentration factor of the r-th
phase, see Section 2.1. In Section 2.2, after accounting
for the phase interaction, these are combined to the
full concentration factors satisfying

E(r) = A(r)E for r = 0, . . . ,n,

utilized next to estimate the overall stiffness of the
composite material, Leff. Moreover, as outlined in Sec-
tion 2.3, the expression for the overall stiffness also
encodes the mean value of the quadratic invariant of
the local stress deviator σD, defined as

J
(r)
2 =

√
1

2|Ω(r)|

∫
Ω(r)

σD(x)TσD(x) dx, (3)

which can be directly used to estimate the overall
strength of a material.

2.1 Dilute concentration factors

Due to the geometrical and material isotropy of the
individual phases, the dilute concentration factors
attain a form analogous to (1):

A
(r)
dil = A

(r)
dil,VIV + A

(r)
dil,DID for r = 1, . . . ,n. (4)

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Acta Polytechnica Vol. 52 No. 6/2012

(i3)

(i1)

(i2)

R(i1 )

R(i2 )

Figure 2: Scheme of a single-layer inclusion

The expressions for the components are given sep-
arately for the uncoated (r = 1) and coated (r =
2, . . . , 5) particles. Namely, in the first case it holds

A
(1)
dil,V =

K(0)

K(0) + α(0)(K(1) −K(0))
,

A
(1)
dil,D =

G(0)

G(0) + β(0)(G(1) −G(0))
,

with the auxiliary factors following from the Eshelby
solution [9] in the form

α(0) =
1 + ν(0)

3(1 + ν(0))
, β(0) =

2(4 − 5ν(0))
15(1 −ν(0))

,

where ν(0) is the Poisson ratio of the matrix phase.
The coated case is more involved, and was first

solved in its full generality by Herve and Zaoui [10]
for a multi-layered spherical inclusion. To apply their
results in the current setting, we locally number the
phases by the index i = [i1, i2, i3]T, see Fig. 2, where
i = [2, 3, 0]T for the brick–C-S-H conglomerate and
i = [4, 5, 0]T refers to a sand particle coated by ITZ.
Now, we have

A
(i1)
dil,V =

1
Q211

, A
(i2)
dil,V =

Q111
Q211

, (5)

and

A
(i1)
dil,D = A1 −

21
5

R(i1)2

1 − 2ν(i1)
B1, (6)

A
(i2)
dil,D = A2 −

21
5

R(i2)5 −R(i2−1)5

(1 − 2ν(i2))(R(i2)3 −R(i2−1)3)
B2,

where the auxiliary factors are provided in Ap-
pendix A.

2.2 Stiffness estimates

In Benveniste’s [4] interpretation of the original Mori-
Tanaka method [19], the mutual interaction among
the heterogeneities is modeled by loading each particle
by the average strain in the matrix phase E(0) instead
of E. For this purpose, we relate E(0) to E by a

strain compatibility condition, valid under the dilute
approximation,

E =
(
c(0)I +

n∑
r=1

c(r)A
(r)
dil
)
E(0),

from which we express the full concentration factors
as

A(0) =
(
c(0)I +

n∑
r=1

c(r)A
(r)
dil
)−1

,

A(r) = A(r)dil A
(0) for r = 1, . . . ,n.

Utilizing a universal relation

Leff =
n∑
r=0

c(r)L(r)A(r),

we can see that the effective stiffness inherits the
symmetry of individual phases (1) with

Keff =
c(0)K(0) +

∑n
r=1 c

(r)K(r)A
(r)
dil,V

c(0) +
∑n
r=1 c

(r)A
(r)
dil,V

, (7a)

Geff =
c(0)G(0) +

∑n
r=1 c

(r)G(r)A
(r)
dil,D

c(0) +
∑n
r=1 c

(r)A
(r)
dil,D

. (7b)

2.3 Strength estimates
As was first recognized by Kreher [14], the fluctuations
of the stresses and strains in individual phases can be
estimated from the energy conservation condition due
to Hill [11]:

ETLeffE =
1
|Ω|

n∑
r=0

(
9K(r)

∫
Ω(r)

ε2V(x) dx

+
∫

Ω(r)
2G(r)εTD(x)εD(x) dx

)
,

(8)

expressing the conservation of energy on the
macroscale due to E and the average local values
due to εV and εD. Differentiating (8) with respect to
G(r), we obtain

ET
∂Leff
∂G(r)

E =
2
|Ω|

∫
Ω(r)

εTD(x)εD(x) dx,

for r = 0, . . . ,n. Next, we recognize that σ(r)D =
(1/2G(r))ε(r)D and recall the definition of the quadratic
invariant (3) to arrive at

J
(r)
2 = G

(r)
√

1
c(r)

ET
∂Leff
∂G(r)

E. (9)

As was thoroughly demonstrated by Pichler et
al. [27] and Pichler and Hellmich [26], this quantity
is closely related to the compressive strength fc of

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Acta Polytechnica Vol. 52 No. 6/2012

.
Table 1: Reference properties of individual phases; ρ denotes mass density, ft is tensile
strength, m is the mass fraction, and radii R are defined according to Fig. 2

r Phase ρ E ν ft m R Note
[kgm−3] [MPa] [-] [MPa] [-] [µm]

0 Pure lime matrix 1200a 2000[8] 0.25[8] 0.4[8] 3 ×

1 Voids × 10−9 0.25 × × 0.1–100[22] c(1) = 35%a

2 Clay brick 2300a 5000a 0.17a 3.2a 1 500

3 C-S-H gel 2000[12] 22000[7] 0.2[7] × × 510[6]

4 Siliceous sand 2700a 60000[34] 0.17[34] 48[23] 1 500

5 ITZ 1200b 500c 0.25b × × 520c

a Our own (unpublished) data. Densities and porosity were measured by a pycnometer,
elastic constants were determined from strain-gauge data in a compression test, and the
tensile strength follows from a unidirectional tensile test.

b Same value as for the lime matrix.
c Set as in [36] for cement-based concretes, i.e., Young’s modulus to 20–40% of the value
for the matrix phase and thickness to 20 µm.

cement pastes at various degrees of hydration. Here,
we postulate that

fc(p1)
fc(p2)

≈
J

(w)
2 (p2)
J

(w)
2 (p1)

, (10)

where w = 0, . . . ,n is the index of the weakest phase
and p refers to a parameter characterizing the mix-
ture composition, see the next section for concrete
examples.

3 Results and discussion

The purpose of this section is to examine the trends
in mechanical properties as predicted by the proposed
scheme. The default data for individual phases, sum-
marized in Table 1, were assembled from open litera-
ture and complemented with our own, yet unpublished,
measurements. Note that the matrix–brick–sand frac-
tions correspond to a typical composition of historic
lime mortars [3, 2], and that the engineering constants
E and ν are connected to the bulk and shear moduli
through well-known relations, e.g. [18, p. 23],

K =
E

3(1 − 2ν)
, G =

E

2(1 + ν)
.

Given the data in Table 1, the volume fractions of
individual phases are determined from six independent
conditions. The first two relate the volume fractions
of brick and sand particles and their coatings by

c(3) =
((R(3)
R(2)

)3 −1)c(2), c(5) = ((R(5)
R(4)

)3 −1)c(4).

Next, we enforce the value of mass fractions via

c(0) =
m(0)ρ(2)

m(2)ρ(0)
c(2), c(0) =

m(0)ρ(4)

m(4)ρ(0)
c(4),

where m(r) and ρ(r) denote the mass fraction and the
mass density of the r-th phase, respectively. Since
c(1) is given, the remaining condition is provided by∑5
r=0 c

(r) = 1, and the phase volume fractions follow
as the solution of the system of 6 × 6 linear equations.
In the sensitivity analyses, motivated by experi-

mental findings in e.g. [3, 33, 32], we assume that an
increase in the C-S-H gel volume fraction (∆c(3)) is
compensated by the corresponding changes for ma-
trix (∆c(0)), voids (∆c(1)), and clay bricks (∆c(2)), so
that

∆c(0) + ∆c(1) + ∆c(2) + ∆c(3) = 0,
where we set for simplicity ∆c(1) = ∆c(2) = ∆c(3). By
analogy, the increase in the volume fraction of ITZ
corresponds to the decrease in volume of the matrix
phase:

∆c(0) + ∆c(5) = 0.
In the strength estimates, the imposed loading

simulates the uniaxial compression test, for which
Σ = [−1, 0, 0, 0, 0, 0]T and the average strain follows
from

E = (Leff )−1Σ.
We assume that ITZ is the weakest phase, i.e. w = 5
in Eq. (10), and, similarly to [27], we estimate the
derivative in Eq. (9) by the forward difference with a
step size of ∆G(5) = 1 Pa.1

1Our results are reproducible with a MATLAB code Ho-
mogenizator MT, freely available at http://mech.fsv.cvut.
cz/~nezerka/software.

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Acta Polytechnica Vol. 52 No. 6/2012

0 10 20 30 40 50 60 70 80
0.75

0.8

0.85

0.9

0.95

1

C-S-H coating thickness t [µm]

A
(
2
)

d
il
(t

)/
A

(
2
)

d
il
(0

)

 

 

volumetric part

deviatoric part

0 10 20 30 40 50 60 70 80
0.75

0.8

0.85

0.9

0.95

1

ITZ coating thickness t [µm]

A
(
4
)

d
il
(t

)/
A

(
4
)

d
il
(0

)

 

 

volumetric part

deviatoric part

(a) (b)
Figure 3: Influence of the coating thickness on the dilute concentration factors for (a) brick and (b) sand
particles. The vertical lines refer to the default thicknesses that are kept constant in the remaining sensitivity
analyses.

0 10 20 30 40 50
1000

2000

3000

4000

5000

porosity c(1) [%]

E
e
ff

[M
P
a
]

0 10 20 30 40 50
0.4

0.5

0.6

0.7

0.8

0.9

1

porosity c(1) [%]

f
c
(c

(
1
)
)/

f
c
(0

)

(a) (b)
Figure 4: (a) Stiffness-porosity and (b) strength-porosity relations.

3.1 Effect of coatings
The first aspect that we would like to discuss is the
effect of the coating on the dilute concentration factors
of the brick and sand particles. Fig. 3 demonstrates
that, in terms of the volumetric phase strains, the
effects of C-S-H and ITZ are comparable, despite the
fact that the C-S-H is stiffer and ITZ is more com-
pliant than the matrix phase. The differences in the
deviatoric part, which drive the strength estimates
according to Eq. (10), become more pronounced with
the increasing thickness. This indicates that the con-
tribution of brick and sand particles to the overall
properties might still be different, after accounting
phase properties and their interaction, see Section 3.3
for a further discussion.

3.2 Influence of porosity
By analogy with the cement pastes, porosity has a
major influence on the overall properties of lime-based
mortars. This is confirmed by the results of the pro-
posed model, shown in Fig. 4. As for the overall
stiffness, for the realistic range of porosities of 25–
40% [8], the estimates (7) predict Young modulus
values between ≈ 2, 000 and 1, 000 MPa. This is con-
sistent with the values reported in [3] historic lime
mortars (without pozzolan admixtures).
As for the strength estimates, it follows from

Fig. 4 (b) that they reproduce a Power-law relation [23,
p. 280]

fc(c(1)) = fc(0)(1 − c(1))n, (11)

with n .= 1.04, yielding practically the linear strength-
porosity scaling. Unfortunately, we are currently un-
able to validate this prediction against experiments;
the only available work we are aware of by Papayianni
and Stefanidou [25] does not contain enough data.
Still, Eq. (11) complies with the fact that the influ-
ence of porosity is much smaller in lime mortars than
in cement-based materials [15, Section 2.6], for which
n ≈ 3 is typically used, see [25, and references therein].

3.3 Size effects
Now we proceed to clarify the impact of brick and sand
particles on the overall mechanical properties.2 In par-
ticular, when the size of the brick particles increases,
the material becomes more compliant since the stiffen-
ing effect of the C-S-H layer decreases, Fig. 5(a). This
also increases the deviatoric stresses in ITZ, as mani-
fested by the strength reduction visible in Fig. 5(b).
These effects practically stabilize for particles larger
than 0.5 mm, and their magnitude is rather limited:
the stiffness decreases by about 10 % and the strength
only by 4 %. Such trends are qualitatively consistent
with the results presented e.g. in [3, 21, 29].

Larger sand particles, on the other hand, tend to
make the composite material stiffer, Fig. 6(a), by

2In the sensitivity analysis, the mass ratio is kept constant
according to Table 1.

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Acta Polytechnica Vol. 52 No. 6/2012

0.3 0.6 0.9 1.2 1.5
900

1000

1100

1200

1300

1400

radius R(2) [mm]

E
e
ff

[M
P
a
]

0.3 0.6 0.9 1.2 1.5
0.95

0.96

0.97

0.98

0.99

1

radius R(2) [mm]

f
c
(R

(
2
)
)/

f
c
(0

)

(a) (b)
Figure 5: Influence of the brick particle size on the overall (a) stiffness and (b) strength.

0 0.5 1 1.5
800

1000

1200

1400

1600

1800

radius R(4) [mm]

E
e
ff

[M
P
a
]

0 0.5 1 1.5
0.75

0.8

0.85

0.9

0.95

1

radius R(4) [mm]

f
c
(R

(
4
)
)/

f
c
(0

)

(a) (b)
Figure 6: Influence of the sand particle size on the overall (a) stiffness and (b) strength.

compensating for the inferior mechanical properties
of the ITZ. Since the relative thickness of ITZ layer
decreases, the stresses inside this phase increase and
the material becomes weaker in overall, Fig. 6(b).
When compared to brick particles, these effects are
much more pronounced: in the considered range of
radii, the Young modulus increases by about 100 %
and the strength decreases by 25 % with no tendency
to stabilize. This agrees well with the experimental
outcomes reported in [30].

4 Conclusions
In the present work, following the recent developments
presented in [27, 26, 35, 34], a simple micromechanics-
based scheme for strength and stiffness estimates of
cocciopesto mortars has been presented. The model
directly utilizes measurable material and geometrical
properties of individual phases and is free of adjustable
parameters. On the basis of the presented results, we
conclude that the model

1. predicts realistic values of the overall Young mod-
ulus and strength-porosity scaling,

2. captures the “smaller is stiffer” and “smaller is
stronger” trends for crushed brick particles,

3. captures the “larger is stiffer” and “larger is
weaker” trends for sand aggregates,

4. explains the positive role of crushed bricks in
comparison with sand aggregates.

Of course, in order for this model to be accepted
for practical use, it needs to be validated against
comprehensive experimental data at micro- and macro-
scales, and the role of ITZ in lime-based mortars
needs to be clarified. This topic is currently under
investigation and will be reported separately.

Acknowledgements
We wish to thank an anonymous referee for a number
of valuable suggestions on the previous version of
the manuscript. This work was supported by the
Ministry of Culture of the Czech Republic, project
No. DF11P01OVV008.

References appear after the appendix on page 36.

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Acta Polytechnica Vol. 52 No. 6/2012

The effect of coating on the mechanical properties enters the solution through the auxiliary factors Qk in Eq. (5),
and Ak and Bk in Eq. (6). Here, these are provided in the closed form optimized for coding, utilizing the results
and nomenclature by Herve and Zaoui [10]. Note that in order to keep the notation consistent, a(k) corresponds to
a property of the k-th phase, whereas ak denotes a quantity utilized in the Herve-Zaoui solution (independent of
a(k)). Also recall that we employ the local numbering of phases by the index i = [i1, i2, i3]T introduced by Fig. 2.
In particular, the volumetric part is expressed in terms of matrices

Q1 = N1, Q2 = N2Q1,

with

Nk =
1

3K(ik+1) + G(ik+1)


 3K(ik) + 4G(ik+1) 4R(ik)3 (G(ik+1) −G(ik))

3R(ik)3(K(ik+1) −K(ik)) 3K(ik+1) + 4G(ik)


 for k = 1, 2.

The matrices needed to evaluate the deviatoric part follow from

A1 =
P 222

P 211P
2
22 −P 212P 221

, A2 = W 21 , B
1 =

−P 221
P 211P

2
22 −P 212P 221

, B2 = W 22 ,

where

Wk =
1

P 222P
2
11 −P 212P 221

Pk−1
[
P 222 −P 221 0 0

]T
(k = 1, 2), P 1 = M1, P 2 = M2P 1.

The auxiliary matrix Mk admits the expression:

Mk =
1

5(1 −ν(ik+1))




ck

3
R(ik)2(3bk − 7ck)

5(1 − 2ν(ik))
−12αk
R(ik)5

4(fk − 27αk)
15R(ik)3(1 − 2ν(ik))

0
bk(1 − 2ν(ik+1))
5(1 − 2ν(ik))

Mk23
−12αk(1 − 2ν(ik+1))
7R(ik)7(1 − 2ν(ik))

R(ik)5αk

2
−R(ik)7(2ak + 147αk)

70(1 − 2ν(ik))
dk

7
Mk34

Mk41
7αkR(ik)5(1 − 2ν(ik+1))

2(1 − 2ν(ik))
0

ek(1 − 2ν(ik+1))
3(1 − 2ν(ik))




with

Mk23 =
−20αk(1 − 2ν(ik+1))

7R(ik)7
, Mk41 =

−5αkR(ik)3(1 − 2ν(ik+1))
6

,

Mk34 =
R(ik)2(105(1 −ν(ik+1)) + 12αk(7 − 10ν(ik+1)) − 7ek)

35(1 − 2ν(ik))
,

and

ak =
G(ik)

G(ik+1)
(7 + 5G(ik))(7 − 10G(ik+1)) − (7 − 10G(ik))(7 + 5G(ik+1)),

bk =
G(ik)

G(ik+1)
(7 + 5G(ik)) + 4(7 − 10G(ik)),

ck = (7 − 5G(ik+1)) + 2(4 − 5G(ik+1))
G(ik)

G(ik+1)
,

dk = (7 + 5G(ik+1)) + 4(7 − 10G(ik+1))
G(ik)

G(ik+1)
,

ek = 2(4 − 5G(ik)) +
G(ik)

G(ik+1)
(7 − 5G(ik)),

fk = (4 − 5G(ik))(7 − 5G(ik+1)) −
G(ik)

G(ik+1)
(4 − 5G(ik+1))(7 − 5G(ik)),

αk =
G(ik)

G(ik+1)
− 1.

Appendix A: Herve-Zaoui solution

35



Acta Polytechnica Vol. 52 No. 6/2012

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