## Abstract

Models for prediction of the elastic characteristics of natural and synthetic porous materials are re-examined and new models are introduced. First, the Vavakin–Salganik (VS) model for materials with isolated spherical pores is extended in order to take into account various statistical distributions of pore sizes. It is shown that the predictions of the extended VS model are in good agreement with experimental data for porous materials with isolated pores such as foamed titanium, porous glass and sandstone. However, the model is in a considerable disagreement with the experimental data for materials sintered from metal powders. The disagreement is explained by the presence of merged and open pores whose shapes cannot be well approximated as spheres. Using the theory of geometrical probabilities, the amount of pores that are close enough to overlap is estimated, and a model is introduced where merging pores are modelled as corresponding ellipsoids. Another modification is proposed to take into account open pores. This modification is based on the classical Rabotnov–Kachanov approach to damage accumulation in the loaded material. Finally, predictions given by the above models, and their combination is compared with experiments. A good agreement is observed between the combined model and the available experimental data for a variety of sintered materials.

## 1. Introduction

Porous materials are heterogeneous media that contain a number of voids. The main material that is used in the manufacturing process is called the host material or matrix. For example, the host material can be a metal, ceramic or polymer. The voids or pores may have rather different nature: in sintered materials, they are the spaces between the grains of host material that occurred during the manufacturing process, whereas in foamed materials, such as porous glass or titanium foam, they are gas bubbles.

Assuming that the porous material is a linear elastic isotropic solid, the elastic behaviour can be defined by any two constants selected from such material characteristics as elastic modulus (*E*), Poisson's ratio (*ν*), bulk modulus (*K*), shear modulus (*μ*) and Lame parameter (*λ*). These elastic characteristics are connected by the following relations:
1.1

If the pores are filled with gases, their elastic stiffness may be assumed negligible. Therefore, the effective elastic properties of the porous material are different to those of the host material. The effective elastic properties of porous materials are the object of interest in the present work. In a wide class of approaches to estimation of the elastic characteristics of a porous material, it is assumed that the porous structure of the medium repeats a certain periodic pattern. The elastic fields are calculated over the representative unit cell (RUC), and homogenization theory is applied to evaluate the global stress–strain behaviour [1]. The periodic structures representing the porous medium can be chosen in various ways. For example, Chapman & Higdon [2] considered the porous medium as a packed set of spheres; Poutet *et al.* [3] considered three types of unit cells: deterministic, fractal and random; Garboczi & Berryman [4] and Roberts & Garboczi [5] used the finite-element method to calculate the elastic properties of periodic porous media with quasi-random RUCs. The methods based on RUCs are easy to implement; however, real materials have rather random microstructure, and it is difficult to justify the periodic structure assumption.

A number of empirical relations of elastic characteristics of a porous medium with its porosity have also been published, such as the classical Ryshkewitch–Duckworth and Sprigg's equations [6,7],
1.2where *p* is the value of porosity, i.e. the volume fraction of pores per unit volume of the material, *E* and *S* are the elastic modulus and the strength of the porous material, respectively, and *α*,*β* are empiric parameters related to the host material. Here and henceforth, the subscript ‘0’ means that the elastic characteristic is attributed to the host material. These and other similar methods require a set of experimental measurements and are not based on the micromechanical behaviour of pores in the material.

Many approaches to calculate the effective elastic properties are based on physics of composite materials. These approaches consider a material with inclusions subjected to a uniform field, which can have an electrical, elastic or other nature. A statistically homogeneous material is considered, and the individual disturbance of the field caused by a particular inclusion is assumed to vanish far from the disturbance. This leads to a system of equations for the effective properties of the material. The classical models of physics of composites include the effective medium approximation (EMA) introduced by Bruggeman [8] for various physical properties of a composite, and average field approximation (AFA), presented by Polder & Vansanten [9]. These models were intensively studied and further developed by e.g. Avellaneda [10], Milton [11,12] and others. The homogenization methods developed can be applied to calculate both the effective dielectric or elastic properties of the composite; however, the interest of the current paper is in its mechanical properties.

The EMA and AFA methods are limited to cases of inclusions of the same shapes and sizes. Therefore, a class of differential schemes for calculating the effective properties of composite materials was developed [13,14]. First a solution for a single inclusion is considered. If the inclusion has a shape of elliptical origin (sphere, disc, cylinder, prolate or oblate spheroid, arbitrary ellipsoid), then the general procedure for calculating the elastic field outside an ellipsoidal inclusion obtained by Eshelby [15,16] can be applied. The differential methods assume constructing a composite by adding a group of inclusions incrementally to the material. The effective properties are calculated at each step, and are used as properties of the host matrix for the next iteration. The increment of the inclusion phase is then considered to be continuous, which leads to ordinary differential equations for the effective characteristics of the material. Salganik [17] and Vavakin & Salganik [18,19] considered media with pores and rigid inclusions for both two- and three-dimensional cases. Pores were either randomly oriented cracks or isolated spheres of different radii. Based on these works, the effective characteristics of layered media with flattened pores were obtained by Borodich [20,21]. Norris [22] and Norris *et al.* [23] generalized the differential scheme and assumed a fictitious media where inclusions of two or more phases were added incrementally until the volume fraction of the matrix vanishes. A good review of methods based both on periodic and homogenized boundaries is given by Pindera *et al.* [24].

Sevostianov & Kachanov [25] stated that a common drawback of any differential scheme is that a dilute concentration of a newly added group of inclusions in the matrix is considered. The differential schemes therefore neglect the interaction between the individual inclusions within the same group. The interactions between inclusions from different groups is taken into account only in the homogenized sense, i.e. the homogenized contribution of the latest group on the effective properties of the material with inclusions of all the previous groups.

The purpose of the present paper is to give a general procedure for evaluation of the effective elastic properties of porous materials with particular interest in properties of sintered materials. These properties can then be used in engineering applications. The approach developed may be considered as a combination of a differential scheme similar to the Vavakin–Salganik (VS) method and a self-consistent embedding scheme [26,27]. Because the classic form of the VS approach is restricted to spherical pores and does not consider the specific statistical distributions of pore radii, the VS approach has been modified to take into account various distributions of pore radii. In an attempt to account for the interaction of the spherical inclusions of the same or different groups, the probability that the spherical pores overlap and form doublets was calculated. The effect of overlapped pores is then treated as one of ellipsoidal inclusions of the same volume. This modification helps to overcome the dilute concentration limitation of differential schemes mentioned above. It is shown that introducing the elliptical pores into the model improves the predictions of the elastic properties of porous materials with merged pores. However, the micrographic analysis of sintered materials shows a much more complex structure of pores, which cannot be approximated only by isolated and overlapped spheres. An additional modification has been developed that treats complex agglomerates of pores, which cannot be approximated as spherical or elliptical ones, as damaged material. It is assumed that the damaged material does not transfer any load. This approach is similar to the one used by Rabotnov [28] and Kachanov [29] for materials with a large number of cracks. The comparison of the predictions given by models presented to the available experimental data shows that a good fit can be achieved with appropriate choice of the model, which depends on the complexity of the porous microstructure. The predictions of the model are limited to the homogenized material properties, which, however, is consistent with the objective of the current paper.

## 2. Vavakin–Salganik model

Using results obtained by Eshelby [15,16] and assuming a dilute concentration of spherical pores such that their interaction can be neglected, the following formulae were derived by Krivoglaz & Cherevko [30] for the elastic modulus and Poisson ratio of a material with pores of equal size:
2.1where *E* and *ν* are the effective elastic modulus and Poisson ratio of the medium with pores, *E*_{0} and *ν*_{0} are the characteristics of the host matrix and *p* the volume fraction of pores in the unit volume of the host material.

Vavakin & Salganik [18,19] considered the porous medium with spherical pores of *n* different radii *r*_{1},…,*r*_{n} arranged in increasing order, so that
where *r*_{min},*r*_{max} are the minimal and maximal radii, respectively. Using the above assumptions, equations (2.1) are written in terms of the concentration parameter *Ω*,
2.2where *Ω* is the concentration parameter. For the case of single-size spherical pores, *Ω* is defined as
where *N* is the amount of pores of radius *r* per unit volume of the material.

In order to calculate the effective elastic moduli for the material with pores of different sizes, the differential scheme was used, where formulae (2.2) were applied at each step to the medium where only pores of smaller radii are present, starting from the host material, as shown schematically on the flowchart of figure 1. For this purpose, the concentration term *Ω* in (2.2) was replaced by d*Ω*_{i}, which is the concentration of pores of radius *r*_{i} per unit volume of the space between pores of greater radii *r*>*r*_{i},
2.3In other words, d*Ω*_{i} is the increment of the concentration parameter *Ω* due to adding a set of pores of radius *r*=*r*_{i}. The increment d*Ω*_{i} is then assumed to be continuous. The change in porosity d*p* corresponding to taking into consideration pores of radius *r*_{i} is connected to d*Ω*_{i} by the following relation:
where the loss of volume owing to the presence of smaller pores is accounted for by the multiplier (1−*p*).

To calculate the elastic properties *E*_{i} and *ν*_{i} of the medium with pores of radii *r*_{1},…,*r*_{i}, the formulae (2.2) were modified using d*Ω*_{i},
2.4where *E*=*E*_{i−1} and *ν*=*ν*_{i−1} are the elastic moduli of the material with pores of radius *r*<*r*_{i} obtained from the previous step. The moduli of the host material *E*_{0} and *ν*_{0} are used as initial values.

Similar to the definition (2.3) of the change in the concentration value d*Ω*_{i} due to adding pores of the next group, the increments d*E*_{i} and d*ν*_{i} were introduced as corresponding changes in the *E*_{i} and *ν*_{i} values,
2.5This leads to the following system of ordinary differential equations:
2.6with initial conditions *E*=*E*_{0} , *ν*=*ν*_{0} when *Ω*=0. Vavakin & Salganik [18,19] showed that Poisson's ratio *ν*→0.2 when porosity *p*→1. The influence of *ν* on the value of *E* is neglected in the VS model for simplicity; therefore, *ν*(*Ω*) is assumed to be constant. For *ν*_{0}=0.2 and *ν*(*Ω*)=const., the former equations (2.6) can be integrated as
2.7Substituting (2.4) into (2.7), one obtains
2.8this solution is referred to as the VS model.

In the VS approach, the general case of distribution of pores by size is approximated by an asymptotic solution. No specific distribution of pores by size is considered; therefore, it does not allow the influence of pores of a particular radius *r*_{i} on the elastic properties of the material to be considered. In addition, the change in the Poisson ratio due to increasing porosity is neglected. For the two-dimensional case, Vavakin & Salganik [18,19] showed an excellent agreement of their model with experiments, carried out by stretching a plate with a number of circular holes of different sizes drilled in it. For the three-dimensional case, the comparison is presented in §4. Current work is based on the above-described VS model.

## 3. The extended Vavakin–Salganik approach

In this paper, the VS approach is extended to take particular distributions of pores by size into account. Let the distributions of pores by size be defined as a discrete probability density function *f*(*r*_{i}),

The extension of the VS model is implemented by step-by-step application of formulae (2.4) to the porous material with pores of radius *r*_{i}, where the elastic moduli *E*_{i−1} and *ν*_{i−1} are the effective properties of the material where pores of radii *r*_{1},…,*r*_{i−1} only are present. The use of (2.4) is not reduced to the differential equation (2.6) as in the VS model, and therefore, more sophisticated numerical experiments can be implemented, including various statistical distributions of pore sizes. This extended model is referred to as the *extended VS model*.

Various distributions of pores by size with the same value of total porosity have been examined. A group of continuous distribution functions *f*(*r*) with *r*∈[*r*_{min},*r*_{max}] based on the Gaussian distribution is given below. These distributions

— normal: ,

*μ*=(*r*_{max}+*r*_{min})/2,*σ*=(*r*_{max}−*r*_{min})/6— left normal: ,

*μ*_{1}=(*r*_{max}+2*r*_{min})/3,*σ*=(*r*_{max}−*r*_{min})/6— right normal: ,

*μ*_{3}=2(*r*_{max}+*r*_{min})/3,*σ*=(*r*_{max}−*r*_{min})/6— double normal: ,

*μ*_{2}=(*r*_{max}+2*r*_{min})/3,*σ*=(*r*_{max}−*r*_{min})/12

are shown in figure 2*a*, together with the uniform distribution, which provides the same number of pores of each radius.

Skewed distributions such as

— linear descending:

*f*(*r*)=*A*((*r*_{max}−*r*_{min})/2−*r*)+1/*k*,*A*=8/3(*r*_{max}−*r*_{min})^{2}— uniform:

*f*(*r*)=1/(*r*_{max}−*r*_{min})— linear ascending:

*f*(*r*)=*A*(*r*−(*r*_{max}−*r*_{min})/2)+1/(*r*_{max}−*r*_{min}),*A*=8/(3(*r*_{max}−*r*_{min})^{2})— 1/

*R*^{3}distribution:*f*(*r*)=1/*r*^{3}— gamma left:

*f*(*r*)=*r*^{κ−1}e^{−r/θ}/*θ*^{κ}*Γ*(*κ*),*κ*=5,*θ*=(3(*r*_{max}−*r*_{min}))/20— gamma right: ,

*κ*=5,*θ*=3(*r*_{max}−*r*_{min})/20

are illustrated in figure 2*b*. The 1/*R*^{3} distribution provides the same volume fraction of groups of pores of each radius.

Note that for modelling purposes, the discrete distribution functions *f*(*r*_{i}) are used where the set of radii *r*_{i} is discretized as

The effect of different distributions of pores by size on the elastic properties of the material was examined. The effect is summarized in figure 3, which shows the variation in elastic modulus with volume fraction of pores for all the distributions considered. It is clear that the maximum difference does not exceed 1 per cent for porosity of *p*≤0.7. Hence, the difference between the various statistical distributions of pore sizes is considered as negligible, and the extended VS model with various statistical distributions of pore sizes has an excellent agreement with the classical VS model. Thus, it can be concluded that the governing parameter of the models for materials with isolated spherical pores is the total porosity *p*, and the VS model can be applied to describe the elastic characteristics of the materials. The uniform distribution of pores by size is thus considered in this paper.

## 4. Comparison of the extended Vavakin–Salganik model with experiments

A literature survey was carried out to find published experimental data on the influence of the porosity on the elastic moduli of porous materials. The data used in the present paper include experimental values of elastic characteristics for natural materials, foams and sintered materials. If the bulk modulus *K* values are examined in the paper, then substituting *E* and *ν* obtained from the extended VS model into (1.1) allows the predictions of bulk modulus *K* to be calculated and compared with given experimental values.

The experimental data are available for: (I) natural materials and foams with isolated pores such as glass foams [31], synthetic sandstone [32], sandstone and clay [33], sandstone and shales [33,34] and titanium foam [35]; as well as for (II) sintered materials such as silicon carbide [36], and titanium and alumina compacts [7,37,38]; and (III) other sintered materials, including alumina oxides [39,40], magnesium oxides [41], and magnesium and alumina oxide aggregates [42]. For comparison purposes, the experimental data are divided into three groups as specified in tables 1–3.

The natural porous materials considered are clean and clay-bearing sandstones and shales. The laboratory data consist of normalized bulk moduli *K*/*K*_{0} for dry sandstones under 40 MPa confining pressure [33].

Experimental data for both clean and clay-bearing sandstones (with up to 10% of clay) were obtained from Mukerji *et al.* [33]. For the clean sandstones, the elastic moduli of the host matrix (*E*_{0},*ν*_{0}) or (*K*_{0},*μ*_{0}) were those of quartz; while for the clay-bearing ones, they were obtained from the zero porosity interception of linear regression for shaly sandstones (*K*_{0}=31 GPa and *μ*_{0}=34 GPa).

The group of materials also includes foamed titanium, porous glass and synthetic sandstone. The properties of a porous titanium foam have been investigated by Shen *et al.* [35]. Results of a number of experiments on the bulk moduli *K* of glass foams of different porosities are given by Walsh *et al.* [31] and Ji *et al*. [34]. The pores in the samples were nearly spherical and non-interconnecting. The elasticity of synthetic sandstone was obtained by Berge *et al.* [32]. The samples were prepared using sintered glass beads and were in essence porous glass specimens with porosities ranging from 0 to 0.43. The glass characteristics are *E*_{0}=72.3 GPa and *μ*_{0}=29.2 GPa.

The experimental data of these aforementioned natural material and foams are summarized and compared with the predictions given by the extended VS model in figure 4, where the influence of the porosity value on the normalized bulk modulus *K*/*K*_{0} is shown.

The remaining experimental data consider the materials prepared from different powders by compressing and sintering. The powders used include alumina, titanium, silicon carbide, alumina, magnesium oxides and others.

A comparison of experimental data on the elastic modulus of porous materials with the predictions given by the extended VS model is given in figure 5, where the non-dimensional elastic modulus *E*/*E*_{0} is plotted against the value of porosity.

Two main observations can be made from data plotted in figures 4 and 5. First, the data on the bulk modulus of natural materials and foams are in good agreement with the predictions of the extended VS model. This can be explained by the fact that pores are roughly spherical and isolated, as is observed in the optical micrograph of metallographic cross section for a titanium foam with porosity of 0.15 (figure 6*a*) [35]. In §3, the extended VS model was shown to be equal to the classical VS model; therefore, all the statements about the extended VS model are assumed to be true for the classical VS model.

Second, the data on the elastic modulus of materials prepared by sintering do not fit the predicted values. This can be explained by the different types of porous structure present in the sintered materials. Indeed, the metallographic picture of the sintered bearing (figure 6*b*) shows that pores merge to form complex agglomerates and can interconnect to form an open-pore structure.

It can be concluded that for materials whose pores can be approximated as isolated spheres (figure 6*a*), the extended VS model gives good predictions, while for materials with merged pores (figure 6*b*), further development of the physical model is required.

## 5. Merged-pore model

Both the analysis of the microstructure of the porous materials (figure 6*a*,*b*) and comparison of the existing differential scheme with experimental results (figures 4 and 5) suggest that the assumption of dilute concentration is applicable for a class of porous materials given by table 1, but not for materials given by tables 2 and 3. To improve the physical model of the materials with merged pores, a new differential scheme is presented in this section. In this scheme, isolated spherical pores are treated according to the extended VS model, while merged pores are approximated by ellipsoids of the same volume. The effect of merged pores on the elastic properties of the porous material is more significant than the effect of isolated pores of the same volume; consequently, the model has been extended to account for the interaction between individual pores. In order to estimate the amount of merged pores, the theory of geometrical probabilities has been exploited. Geometrical probabilities have been intensively studied [43,44]. In particular, Armitage [45] studied a problem of overlapping particles in application to the counting of cells in biological experiments. In the paper, a number of particles was considered randomly distributed on a plate, and then the probability of the circular particles to be isolated, to form doublets and triplets, was calculated. In the present paper, the approach of Armitage to overlapping particles was generalized to a three-dimensional case of spherical pores that may overlap. For this problem, the probability density function for the distance *d* between two points placed randomly inside the volume *V* is
5.1

The spatial distribution of the point in the volume is assumed to be homogeneous. The merging distance *d* for the two spherical pores of radii *r*_{i} and *r*_{j} is defined as the distance between their centres that allows the pores to overlap, i.e. *r*_{i}<*d*<*r*_{i}+*r*_{j}. The probability *P*_{merge}(*r*_{i},*r*_{j}) for a pore of radius *r*_{i} to overlap a pore of radius *r*_{j} is then calculated as the volume of points that are within merging distance from the centre of the *r*_{i} pore per total volume *V* . This leads to the following formula for *P*_{merge}(*r*_{i},*r*_{j}):
5.2Substituting (5.1) into (5.2) gives
5.3If the distribution of pores by size is given by the probability density function *f*(*r*) for the pore of radius *r*, and the total amount of pores is *N*, then the number of pores of a particular radius *r*_{i} is *Nf*(*r*_{i}), and therefore the number *N*_{ii} of pairs of pores of the same radius *r*_{i} is
5.4Similarly,
5.5is the total number of pairs of pores of different radii *r*_{i} and *r*_{j}, *r*_{i}>*r*_{j}. The concentration of merged pores in the representative volume is then
for merged pores of the same radius *r*_{i}, and, similarly,
for the merged spherical pores of different radii *r*_{i}>*r*_{j}.

The contribution of the merged pores of radii *r*_{i} and *r*_{j} to the mechanical properties of the porous material has been calculated using the results obtained by Luo & Stevens [46] for the material with randomly oriented elliptic inclusions. Merged spherical pores are approximated by ellipsoids (prolate spheroids, see sketch of figure 7) of the same volume.

In order to account for the merged pores of different sizes, the iterative method similar to the extended VS model described in §3 was developed as follows.

(i) Introduce spherical pores of radius

*r*_{1}into the host material with elastic modulus*E*_{0}.(ii) At each

*i*th step, calculate the probabilities*P*_{merge}(*r*_{i},*r*_{j}) of spheres with radii*r*_{i}to merge with spheres of radii*r*_{j}for any*r*_{j}≥*r*_{i}.(iii) Obtain the number of merged pores for each

*j*≥*i*.(iv) Update the number of isolated spherical pores of radius

*r*_{i}subtracting the number of pores involved in merged pairs.(v) Calculate the effective modulus

*E*_{i,0}of the host material with isolated spherical pores as in the extended VS model (§3).(vi) Calculate the effective modulus

*E*_{i,i}of the material with merged pores of the same radius*r*_{i}. Use*E*_{i,0}as the property of the host material.(vii) Calculate the effective modulus

*E*_{i,j}of the material with merged pores of radii*r*_{i}and*r*_{j}for all*j*=*i*+1,*i*+2,…,*n*. Use*E*_{i,j−1}as the property of the host material.(viii) The procedure is repeated for all

*i*=1,…,*n*.

The algorithm is schematically illustrated on the flowchart of figure 8.

In this paper, the above model is referred to as the MPM.

## 6. Comparison of the merged-pore model with experiments

In this section, the predictions given by the MPM are compared with the experimental data, which were described in detail in §4.

Since the elastic field outside the ellipsoidal inclusion is different to that of two isolated spherical pores of the same total volume, the current extension allows the interaction between pores to be taken into account. The difference can be summarized by the graphs of figure 9, where the predictions given by both the extended VS model and the MPM are compared with the experimental data. The results obtained by approximation of merged pores as ellipsoids give an improved fit to the data on some sintered materials (figure 9*a*). However, sintered materials contain a number of complex structures of merged pores that cannot be approximated by either spheres or ellipsoids. Such agglomerates are called open pores. The presence of open pores reduces the elastic moduli more significantly than predicted by the MPM. Hence, the influence of open pores has to be incorporated into the model.

## 7. Open-pore model

The model presented in this section treats the material with open pores as the damaged material with reduced load-carrying capacity. In the 1960s, Rabotnov [28] and Kachanov [29] investigated the creeping behaviour of a material with a large number of microcracks, and introduced two parameters: a damage parameter of the material, *ω* [28], and an integrity parameter *ϕ*=1−*ω* [29]. The damage parameter is a scalar associated with the area fraction of cracks or the area fraction of undamaged material in an arbitrary section of a sample.

In an axially loaded bar with a large number of cracks, the real stress *σ*_{real} is calculated as load *W* over the effective cross-sectional area,
7.1where *A* is the nominal cross-sectional area of the loaded sample, *ω* the damage parameter of the material and *A*_{eff} the effective undamaged area that carries all the load. This makes the nominal uniaxial stress *σ*_{nom} equal to
7.2The effective elastic modulus *E*_{eff} is defined from Hooke's law for the nominal stress,
7.3where *ϵ* is the elastic strain and *E*_{c} the elastic modulus of the host undamaged material. It follows from (7.3) that the effective elastic modulus *E*_{eff} of the damaged material can be obtained as
7.4

The eligibility of the use of the Rabotnov–Kachanov approach in fracture mechanics was discussed by, for example, Salganik & Gotlib [47]. Recently, Kusoglu *et al.* [48,49] applied the Rabotnov–Kachanov idea to porous polymer membranes. They considered materials with open porosity, and assumed that the load is transferred only through the non-porous volume as in the Rabotnov–Kachanov approach for damaged bars.

In the present paper, this idea is generalized and applied to porous materials with both open and isolated pores. The proposed approach can be interpreted as assuming that the entire load is being carried by the parts of the material with isolated pores only; the mechanical model is illustrated schematically in figure 10. Hence, the value of the elastic modulus obtained by the MPM for the material with isolated and merged spherical pores is used to define *E*_{c}, while the damage parameter *ω* can be connected to the amount of open pores,
7.5where *A*_{open} is the area of the cross section of the material that is occupied by open pores. Then, for the case of open pores, one obtains
7.6

To use this approach, the effective area *A*_{eff} that carries all the load has to be established for the known value of open porosity. If the value of open porosity of the sample is *p*_{open}, then the volume of open pores can be calculated as the following:
7.7where *V* _{total} is the total volume of the sample.

Let the representative volume of the sample be a cube of side *h* and let the Cartesian coordinate system be defined so that its origin is on the bottom side of the cube. The *Z*-axis is defined perpendicular to the bottom side of the cube. It follows from the approach of stereology [43] that the volume of a three-dimensional body can be approximated using the sequence of cross sections with measured area. The same approach was applied to approximate the volume of open pores as an integration of the cross-sectional area occupied by pores over the height of the sample,
7.8where *A*_{open}(*z*) is the area occupied by open pores in the cross section of the representative volume parallel to the bottom side of the cubic representative volume, and *z*∈[0,*h*] is the position of the cross section on the *Z*-axis. Since the uniform spatial distribution of pores is considered, *A*_{open}(*z*) is assumed to be constant. Let *A*_{open}(*z*)=*A*_{open}. Then, (7.8) can be presented as *V* _{open}=*hA*_{open}, which together with equation (7.7), leads to
7.9Therefore, the damage parameter *ω* in equation (7.10) is equal to the value of open porosity, *p*_{open},
7.10Thus, the effective elastic modulus for the material with developed open porosity can be derived from equation (7.4) as
7.11where *E*_{c} is calculated according to the MPM with isolated spherical and elliptical pores, and *p*_{open} is the volume of the open pores within the representative volume of the sample. In this paper, the above model is referred to as the open-pore model (OPM).

For the OPM, the volume fraction of open pores needs to be established. Laboratory data on open porosity of porous material were obtained from the literature. In the work of Altman *et al.* [50], the relationship between the total and open porosity of sintered Cu–Sn–C materials was investigated. Other experimental data for open porosity of the sintered titanium compacts were taken from results obtained by Oh *et al.* [38]. Data on total and open porosity of porous bearings sintered from Fe–Cu and Cu–Sn powders have been provided by the manufacturer of porous bearings (GKN Sinter Metals Bruneck, Italy).

In the present study, the relationship between the total and open porosity of the materials has been approximated by an analytical function,
7.12where the coefficients *A* and *α* were selected to fit the available experimental data. Values *A*=10^{−6} and *α*=5 were used in further calculations as they fit pretty well to the experimental data on sintered materials obtained by Oh *et al.* [38]. Values *A*=2.5×10^{−5} and *α*=5 were used to fit the experimental data on sintered Cu–Sn–C materials presented by Altman *et al.* [50]; however, no data on elastic characteristics was available for these materials.

The relation is summarized in the figure 11, where the fraction of open pores within the total porous structure is plotted against the total porosity.

The OPM provides a better agreement with experimental data than the previous model. A comparison of the predictions given for the sintered materials by both MPM and OPM with the experimental data is plotted in figure 12.

## 8. Combined model

The combined model (OPMPM) of the present paper is a combination of the MPM and OPM and incorporates all the extensions described in §§6 and 7.

— Initially, the calculation is performed according to the extended VS model for isolated spherical pores, as described in §3.

— The influence of merged pores is calculated as by Luo & Stevens [46] and incorporated in the model in a step-by-step manner, as described in §5.

— The volume occupied by open pores has been evaluated by the relation (7.12) with appropriate coefficients

*A*and*α*selected for a particular material. Then, the elastic modulus of the porous medium with developed open porosity was calculated as given by equation (7.11) in §7.

A comparison of the predictions given by the combined model with experimental data is shown in figure 13. A good agreement is apparent between the proposed combined model and the materials given by table 3. This can be explained by the choice of the function (7.12) approximating the amount of open pores: it is selected to fit the data for sintered titanium and sintered porous bearings (figure 11). To allow a better agreement with properties of other materials, their open porosity to total porosity ratio has to be evaluated.

## 9. Conclusions

The analysis presented has shown that the elastic characteristics of a porous material are highly dependent on the porous microstructure of the samples. Hence, the manufacturing process used to prepare the material is an important factor for choosing a proper model for prediction of the elastic characteristics of the material. Both the classic VS model and its extended modification with an arbitrary statistical distribution of pore radii (e.g. the uniform distribution function) show good results for materials with isolated spherical pores.

It is proposed that the theory of geometric probabilities is used to estimate the number of overlapping pores. Overlapping spherical pores are treated as merged and approximated by ellipsoids, which allows a better fit to the experimental results. This approach has been shown to be effective, and it can be used in various applications related to the properties of multi-phase materials. A further modification of the geometrical probabilities approach, not implemented in the present paper, considers calculating the probability that three or more spherical pores merge.

For materials with developed porous microstructure (open porosity), the volume fraction of open pores to the total volume of pores has to be established. The elastic properties of sintered materials may be affected not only by pores, but also by cracks that could appear at the manufacturing process (see a discussion by Salganik & Fedotov [51]). However, the consideration of isolated cracks is out of the scope of the present paper. It is assumed, that the load is transferred by the material containing closed pores that are either spherical or elliptical. The parts of the material containing open pores have very low stiffnesses and, therefore, do not carry any load. The function used to approximate the amount of open pores was selected to fit the experimental data for sintered titanium and sintered porous bearings. This results in a good agreement between the proposed combined elastic properties model for both merged and open pores with experimental results for these materials and other similar materials (table 3 and figure 13*b*).

## Acknowledgements

The authors are grateful to Prof. R. L. Salganik (Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow) for valuable discussions and comments. The authors are grateful to Dr W. Pahl (GKN Sintermetals Bruneck, Italy) for valuable discussions and for providing data on porosity of sintered bearings, and to Ms I. A. Neacsu (AC2T Research GmbH) for the metallographic pictures of sintered iron bearings. The authors acknowledge the financial support of MINILUBES (FP7 Marie Curie ITN network 216011-2) by the European Commission. They are also grateful to the Austrian Research Promotion Agency (FFG) for funding the current development of the methods discussed in the paper within the framework of the COMET K2 Excellence Center of Tribology (X-Tribology).

- Received November 20, 2012.
- Accepted March 11, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.