Mechanics Research Communications 38 (2011) 131–135
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Effective elastic properties of porous materials: Homogenization schemes vs experimental data K. Miled a,∗ , K. Sab b , R. Le Roy b a b
Université Tunis El Manar, Laboratoire de Génie Civil (LGC), Ecole Nationale d’Ingénieurs de Tunis, BP 37, Le Belvédère, 1002 Tunis, Tunisia Université Paris-Est, Laboratoire Navier (ENPC/LCPC/CNRS 8205), Ecole des Ponts ParisTech, 6 et 8, avenue, Blaise-Pascal, 77455 Marne-la-Vallée Cedex 2, France
a r t i c l e
i n f o
Article history: Received 13 April 2010 Received in revised form 13 November 2010 Available online 21 January 2011 Keywords: Porous materials Elastic moduli Mean-field homogenization EPS concrete
a b s t r a c t In this paper, we focus on the prediction of elastic moduli of isotropic porous materials made of a solid matrix having a Poisson’s ratio vm of 0.2. We derive simple analytical formulae for these effective moduli based on well-known Mean-Field Eshelby-based Homogenization schemes. For each scheme, we find that the normalized bulk, shear and Young’s moduli are given by the same form depending only on the porosity p. The various predictions are then confronted with experimental results for the Young’s modulus of expanded polystyrene (EPS) concrete. The latter can be seen as an idealized porous material since it is made of a bulk cement matrix, with Poisson’s ratio 0.2, containing spherical mono dispersed EPS beads. The Differential method predictions are found to give a very good agreement with experimental results. Thus, we conclude that when vm = 0.2, the normalized effective bulk, shear and Young’s modulus of isotropic porous materials can be well predicted by the simple form (1 − p)2 for a large range of porosity p ranging between 0 and 0.56. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction We focus in this paper on isotropic porous materials made of a solid matrix having a Poisson’s ratio vm of 0.2. Analytical formulae for their overall elasticity moduli will be derived based on five well-known Mean-Field Eshelby-based Homogenization (MFH) schemes. Then, in order to determine the best scheme, these predictions will be confronted with experimental results obtained on an idealized isotropic porous material (polystyrene concrete) made of a matrix with Poisson’s ratio 0.2.
¯ x)dV and where x is the position vector in a local (1/V ) ω f (x, frame attached to the RVE. The later is subjected to linear boundary displacements corresponding to a macro strain ε. The strain averages per phase are related through a still unknown strain con¯ centration tensor Bε as follows: εωi = Bε : εωm and εωi = Aε : −1
with Aε = Bε : [i Bε + (1 − i )ı] , where ı designates the fourthorder symmetric identity tensor. For any homogenization scheme defined by Bε or Aε , the macro stiffness is therefore given by one of the two equivalent expressions: C = [i Ci : Bε + (1 − i )Cm ] : [i Bε + (1 − i )ı]
−1
,
(1)
1.1. MFH of two-phase linear elastic composites Consider a two-phase composite in isothermal linear elasticity. All inclusions have the same shape, aspect ratio, orientation and elastic stiffness tensor Ci . The matrix phase has a stiffness tensor Cm , a volume Vm and a volume fraction (or concentration) m = Vm /V, where V is the volume of the representative volume element (RVE). We define the same quantities for the inclusions phase (Vi and i = Vi /V). It is easy to check that the averages over the entire RVE (ω), the matrix phase (ωm ) and the inclusions phase (ωi ) are related by: f ω = i f ωi + (1 − i )f ωm , with f ω =
∗ Corresponding author. Tel./fax: +216 71 871476. E-mail addresses:
[email protected] (K. Miled),
[email protected] (K. Sab),
[email protected] (R. Le Roy). 0093-6413/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2011.01.009
C = Cm + i Aε : (Ci − Cm )
(2)
1.2. Eshelby’s tensors Mean-Field Homogenization is based on Eshelby’s tensors (Eshelby, 1957). The latter are used to estimate the average strains and stresses inside the micro-inclusions embedded in an elastic matrix and account for their shape and their orientation. Moreover, they can be computed analytically (Mura, 1987; Christensen, 1991; Nemat-Nasser and Hori, 1999) and therefore they provide effective means for estimating the overall elastic moduli of heterogeneous materials. Hereafter, we will recall Eshelby’s tensors expressions for isotropic porous materials made of a solid matrix having a Poisson’s ratio vm of 0.2.
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1.2.1. Eshelby’s single inclusion problem Within this problem, an infinite matrix with uniform stiffness Cm is considered. An ellipsoidal volume (I) is cut out, undergoes a stress-free eigenstrain ε* and is then “welded back” into the hole it occupied in the matrix. Eshelby showed that the strain field inside the ellipsoid (I) is uniform and he gave an explicit expression for the tensor which relates the resulting uniform strain inside (I) to the prescribed eigenstrain: ε(x) = : ε*, ∀ x ∈ (I), where is the fourth-order Eshelby’s tensor depending on Cm and on the shape and orientation of (I). For the particular case of a spherical inclusion and an isotropic stiffness Cm , depends only on the matrix Poisson’s ratio vm . Moreover, if vm = 0.2, = (0.5)ı.
for this model corresponds to that of the Eshelby’s single heterogeneity problem, i.e. Bε = Hε . Based on Eq. (1), the Mori–Tanaka (M–T) macro stiffness is equal to: C¯ M−T = [i Ci : H ε + (1 − i )Cm ] : −1 [i H ε + (1 − i )ı] . In the case of a porous isotropic elastic matrix with vm = 0.2, the overall effective stiffness tensor is given by: C¯ M−T = Cm (1 − p)/(1 + p), and the effective normalized bulk and shear modulus are therefore given by the same following hyperbolic form: h kM−T (p)
km
hM−T (p)
=
m
=
1−p 1+p
(4)
2.3. The Self Consistent model and its generalized form 1.2.2. Eshelby’s single ellipsoidal heterogeneity problem Within this problem, a single ellipsoidal heterogeneity (I) of uniform stiffness Ci is embedded in an infinite matrix having a uniform stiffness Cm and subjected to a uniform strain ε on its boundary (at ∞). Eshelby showed that the strain field inside (I) is uniform and −1
−1 that: ε(x) = H ε : ε¯ , ∀x ∈ (I), where H ε = [ı + : (Cm : Ci − ı)] , and is the Eshelby’s tensor related to the single inclusion problem. Moreover, if (I) is a spherical pore and vm = 0.2, it follows that Hε = 2.
2. MFH of elastic isotropic porous materials with vm = 0.2 Consider an RVE (ω) made of an isotropic linear elastic matrix having a Poisson’s ratio vm of 0.2 and containing stress free spherical cavities having a volume fraction or a macro porosity p. Assume that the overall effective behavior of this porous material is isotropic. This RVE, having a total volume V, is bounded externally by surface ∂ω, and it is subjected to linear boundary displacements corresponding to a macro strain ε. Hereafter, analytical formulae for the overall elasticity moduli of this isotropic porous material will be derived based on five Mean-Field Eshelby-based Homogenization (MFH) schemes: the Dilute method, the Mori–Tanaka model, the Self Consistent model, the Generalized Self Consistent model and the Differential method.
The Self Consistent (SC) model is due to Budiansky (1965) and Hill (1965). It can be defined directly by assuming that each inclusion is embedded in a fictitious homogeneous matrix possessing the ¯ which means that: ε(x) = H ε : ε¯ . It composite unknown stiffness C, follows that: Aε = Hε . According to this model, the effective bulk and shear modulus of a two phase isotropic elastic material are given by the following expressions: h kSC = km + i
ki − km (1 − ˛SC ) + (ki /km )˛SC
hSC = m + i
and
i − m , (1 − ˇSC ) + (i /m )ˇSC
where ˛SC = (1 + SC )/3(1 − SC ) and ˇSC = 2(4 − 5SC )/15(1 − SC ) (SC is the effective Poisson’s ratio given by the S-C model). The Generalized Self Consistent (GSC) model formalized by Christensen and Lo (1979) and Christensen (1990) can be defined directly by assuming that each spherical inclusion is embedded in a concentric spherical annulus of the matrix material of the prescribed volume fraction, which in turn is embedded in an infinite medium possessing the unknown effective properties and subjected to a uniform strain ε. Christensen and Lo (1979) and later Christensen (1990) had derived the analytical exact solution for this three phase medium equilibrium problem and had shown that the effective bulk modulus is given by the following expression:
2.1. The Dilute or Eshelby’s method h kGSC = km +
i (ki − km ) 1 + (1 − i )(ki − km )/(km + (4/3)m )
The Dilute method does not account for the interaction between inclusions. It stipulates that each inclusion (I) behaves like an isolated inclusion in an infinite matrix subjected to a uniform strain ε applied on its boundary, which corresponds to the Eshelby’s ellipsoidal heterogeneity problem. Thus, Aε = Hε . Based on Eq. (2), the overall effective stiffness tensor is equal to: C¯ Dilute = [Cm + i H ε : (Ci − Cm )]. In the case of a porous isotropic elastic matrix with vm = 0.2, we have Ci = 0 and Hε = 2ı. This gives: C¯ Dilute = (1 − 2p)Cm . Thus, the effective normalized bulk and shear modulus are given by the same form depending linearly on the porosity p:
where A, B and C are three constants depending on i , m , i , m and on inclusions volume fraction i . In the case of a porous isotropic elastic matrix with m = 0.2, ki = i = 0 and the effective volumetric modulus is reduced to:
h kDilute (p)
h (p) kGSC
km
=
hDilute (p) m
= 1 − 2p
(3)
However, the Dilute method does not cover the full range of porosity up to p = 1, since it gives non physical predictions for porosities greater than 0.5. 2.2. The Mori–Tanaka model This model is credited to Mori and Tanaka (1973) but the most recent and simplest derivation of it has been proposed by Benveniste (1987). According to this author: “Each inclusion (I) behaves like an isolated inclusion in the matrix seeing εωm as a far-field strain”. Consequently, the strain concentration tensor
They had shown also that the effective shear modulus is the positive solution of the following quadratic equation:
A
hGSC
2 + 2B
m
km
=
hGSC m
1−p 1+p
+ C = 0,
(5)
Thus, the Generalized Self Consistent model gives the same prediction for the effective bulk modulus as the Mori–Tanaka model. Moreover, in this case, constants A, B and C are reduced to: A = 1344p10/3 + 4872p7/3 − 7056p5/3 + 6888p + 3360, B = −4872p7/3 + 7056p5/3 − 1176p − 1008 and C = −1344p10/3 + 4872p7/3 − 7056p5/3 + 4872p − 1344. Consequently, the normalized effective shear modulus is given by the following formula: hGSC m
=
−B +
B2 − AC A
(6)
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Fig. 2. A SEM view showing the EPS concrete microstructure.
Fig. 1. Variation of the normalized effective shear modulus with porosity computed with the Generalized Self Consistent model.
Fig. 1 shows that this form is very close to the one given by the simple form (1 − p)/(1 + p) already obtained for the normalized bulk modulus. Thus, we conclude that for isotropic porous materials made of a solid matrix with m = 0.2, the Generalized Self Consistent scheme predicts a similar decrease on the effective elastic moduli h kGSC and hGSC when porosity p increases from 0 to 1. 2.4. The Differential method The starting point of the Differential method (Bruggeman, 1935; Roscoe, 1952) is the Dilute suspension results for effective bulk and shear modulus of a macroscopically isotropic composite containing non-interacting spherical inclusions. The basic concept of the method is to view the overall composite as a sequence of dilute suspensions. That is, the first inclusions which are added to the matrix are used to calculate the effective properties from the previous equations. Next, that suspension is viewed as a homogeneous medium of those properties, to which a new increment of inclusions is added and the new effective properties are obtained always under assumed dilute conditions. The process is continued up to the condition of full packing of the inclusion phase, i.e. i → 1. Mathematically, the process involves going to the limit where the increments of added inclusions become infinitesimal and a differential form results. The end result of this differential process is the following set of governing equations: h dkdif
di dhdif di
+
+
h −k kdif i h )/(kh + (4/3)h )] (1 − i )[1 + (ki − kdif dif dif h )(h − ) 15(1 − dif i dif h + 2(4 − 5h ) /h ] (1 − i )[7 − 5dif i dif dif
=0
must be proportional to each other over the full range of porosities, h with ˇ = 3/4. Consequently, the effective normalized i.e. hdif = ˇkdif bulk and shear moduli are given by the same following form: h (p) kdif
km
= 0,
h is related to kh and h where the effective Poisson’s ratio dif dif dif h = (3kh − 2h )/2(3kh + h ). through the usual relation: dif dif dif dif dif These equations represent a highly nonlinear coupled set of difh , h and h . ferential equations to be solved for the properties kdif dif dif The solution must satisfy also the following conditions: at i = 0, h = k and h = and at = 1, kh = k and h = . kdif m m i i i dif dif dif In the case of a porous isotropic elastic matrix with m = 0.2, the effective shear and bulk modulus given by the differential scheme
hdif (p)
= (1 − p)2
m
(7)
This homogenization scheme gives the fastest decrease on the h and h when p increases from 0 to 1, effective elastic moduli kdif dif compared to the others schemes presented previously. 2.5. Effective Poisson’s ratio and Young’s modulus For isotropic porous materials made of a solid matrix having m = 0.2, we have shown that the Dilute, the Mori–Tanaka and the differential schemes give the same form for the effective normalized bulk and shear modulus. We have shown also that the Generalized Self Consistent model gives nearly the same form for these moduli. Thus, we consider that for m = 0.2, we have kh (p)/km = h (p)/m , ∀ p. This implies that the normalized effective Young’s modulus has the same form: E h (p) kh (p) h (p) = = , Em m km
∀p
(8)
And the effective Poisson’s ratio is also 0.2 for all p, h
(p) = m = 0.2,
∀p
(9) h (p) =
Moreover, assuming that m = 0.2, the Self Consistent model described above gives the same predictions as the Dilute method: h (p) kSC
km and
=
=
hSC (p) m
= 1 − 2p
(10)
Actually, for all the considered homogenization schemes, the effective normalized bulk, shear and Young’s modulus are given by the same form for m = 0.2. The later can vary from an homogenization scheme to another. Hereafter, we will determine the scheme which best fits with a real porous material. 3. Comparison with EPS concrete experimental data Expanded polystyrene (EPS) lightweight concrete can be seen as an idealized porous material since it is made of a bulk cement matrix containing spherical mono dispersed EPS beads. These artificial aggregates are hydrophobic beads having a closed cell structure consisting essentially of air (Fig. 2). Thus, they have negligible density, rigidity and strength compared to the cement matrix and they
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K. Miled et al. / Mechanics Research Communications 38 (2011) 131–135
Fig. 3. Variation of the EPS concrete Young’s modulus according to porosity (p): homogenization schemes vs experimental data.
can be therefore seen as randomly distributed air voids which total volume represents the concrete macro porosity p. Macroscopically, EPS concrete can be considered as an isotropic material. Moreover, the bulk matrix has a Poisson’s ratio very close to 0.2 which is the average Poisson’s ratio of concrete and cement-based materials (Le Roy, 1996). We had conducted a previous experimental investigation (Miled, 2005; Miled et al., 2007) on EPS concrete with a view to highlighting and explaining the particle size effect observed experimentally on its compressive strength. In fact, it had been observed that EPS concrete compressive strength increases significantly with a decrease in EPS beads size for the same concrete density. Thus, three concretes with three mono-sized EPS beads of diameter 1, 2.5 and 6.3 mm had been used to examine the effect of EPS aggregates size on the lightweight concrete mechanical properties. These concretes are made from the same matrix. The latter is an ultra high strength mortar matrix made from cement, sand, silica fume, water and a superplasticizer and has a compressive strength of 154 MPa and a Young’s modulus of 41000 MPa. Moreover, five densities (1.2, 1.4, 1.6, 1.8 and 2) had been investigated for each EPS concrete by varying the polystyrene volume fraction from 10% to 50%. The latter considered as the concrete macro porosity is determined by the following formula: p = ( m − c )/( m − EPS ), where m , c and EPS are respectively the densities of matrix, EPS concrete and EPS beads. Totally, 15 mix designs had been produced (mix designs proportions, production and curing of EPS concretes had been described earlier; Miled et al., 2007). For each concrete density and EPS beads size, four concrete cylindrical samples of diameter 110 mm and height 220 mm have been casted on which static modulus of elasticity test had been carried out. Note also that this experimental investigation is the continuation of the one made by Parant and Le Roy (Le Roy et al., 2005) who had tested the same three mono-sized EPS concretes but had explored another domain of concrete densities ranging from 0.6 to 1.4 which corresponds to a macro porosity (p) ranging from 0.4 to 0.74.
4. Discussion on the experimental results The EPS concrete considered in the experimental investigation can be seen as a two-phase composite since a scanning electron microscope (SEM) view shows that the porous interfacial transition zone (ITZ) classically observed between cement paste and aggregates is practically non-existent (Fig. 2). This can be explained by the presence of silica fume that improves the bonding between EPS beads and cement paste and also by the very small water to cement ratio (w/c = 0.26) used for the mortar matrix. Experimental results show that the EPS concrete Young’s modulus decreases with increasing the polystyrene volume fraction p (Fig. 3), which is expected since EPS beads have negligible rigidity and behave like pores. Moreover, it is observed that this mechanical property does not depend on the inclusion size since the three EPS concretes have very close moduli for the same concrete density. Thus, it is concluded that EPS concrete does not exhibit a particle size effect on its Young’s modulus and that the normalized EPS concrete Young’s modulus (which is the ratio of the EPS concrete Young’s modulus with respect to that of the mortar matrix) depends only on the concrete macro porosity p. Fig. 3 shows also that for small porosities (0 ≤ p ≤ 0.2), the theoretical averaging schemes give similar results and a good agreement with experimental results. However, they move away from experimental data for greater porosities, except for the differential method predictions which keep a good agreement for a wide range of porosities (0 ≤ p ≤ 0.56). Thus, we conclude that when the matrix Poisson’s ratio of an isotropic porous material is equal to 0.2, its normalized effective Young’s, bulk and shear modulus can be well evaluated by the same simple form (1 − p)2 given by the differential scheme. 5. Conclusions In this paper, we have first derived analytical formulae for the effective moduli of isotropic porous materials made of a solid
K. Miled et al. / Mechanics Research Communications 38 (2011) 131–135
matrix having a poisson ratio of 0.2 based on five well-known Mean-Field Eshelby-based Homogenization schemes: the Dilute method, the Mori–Tanaka model, the Self Consistent model, the Generalized Self Consistent model and the Differential method. For each scheme, we have shown that the normalized bulk, shear and Young’s moduli are given by the same form depending only on the porosity p. Then, the different models predictions have been confronted with experimental Young’s modulus of an idealized isotropic porous material (expanded polystyrene concrete) made of a solid matrix with Poisson’s ratio close to 0.2. This comparison shows that the differential method is the best homogenization scheme. Therefore, we conclude that the normalized effective Young’s, bulk and shear modulus of an isotropic porous material made of a matrix with m = 0.2 can be very well evaluated by the same form (1 − p)2 for a large range of porosity p ranging between 0 and 0.56. References Benveniste, Y., 1987. A new approach to the application of Mori Tanaka’s theory in composite materials. Mech. Mater. 6, 147–157. Bruggeman, D., 1935. Berechnung verschiedener physikalischer konstanten von heterogenen substanzen. Annal en der Physik 24, 636–679.
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