## Abstract

There are lots of ceramic geological and biological materials whose microscopic load carrying behavior is not dominated by bending of structural units, but by the three-dimensional interaction of disorderedly arranged single crystals. A particularly interesting solution to capture this so-called polycrystalline behavior has emerged in the form of self-consistent homogenization methods based on an infinite amount of nonspherical (needle or disk-shaped) solid crystal phases and one spherical pore phase. Based on eigenstressed matrix-inclusion problems, together with the concentration and influence tensor concept, we arrive at the following results: Young’s modulus and the poroelastic Biot modulus of the porous polycrystal scale linearly with the Young’s modulus of the single crystals, the former independently of the Poisson’s ratio of the single crystals. Biot coefficients are independent of the single crystals’ Young’s modulus. The uniaxial strength of a pore pressure-free porous polycrystal, as well as the blasting pore pressure of a macroscopic stress-free polycrystal, scale linearly with the tensile strength of the single crystals, independently of all other elastic and strength properties of the single crystals. This is confirmed by experiments on a wide range of bio- and geomaterials, and it is of great interest for numerical simulations of structures built up by such polycrystals.

## Introduction

There is general agreement that a material’s porosity is one of the key parameters governing its stiffness and strength. This is reflected by a large number of investigations focusing on how corresponding relationships between porosity and the mechanical properties can be cast in a mathematical form [1–4]. Most famous are the scaling relations for (isotropic) cellular solids (“foams”), proposed by Gibson and Ashby [5–7] on the basis of strut/plate bending arguments: Plate bending in so-called “closed cubic cell models” is related to cubic dependence of Young’s modulus on the relative density, while material failure exhibits a cubic or quadratic dependence—the cubic one being related to local buckling failure and the quadratic one to plastic failure. On the other hand, strut bending in so-called “open cubic cell models” is related to quadratic dependence of Young’s modulus on the relative density, while material failure exhibits a quadratic or power-1.5 dependence (related again to buckling and plastic collapse). This opens a power interval from 2 to 3 for elasticity and from 1.5 to 3 for strength (the power exponent ranges are even wider for anisotropic materials, but this is beyond the scope of the present paper). The aforementioned range of power exponents gives ample room for the discussion of a wide range of experimental data and for use of these foam models in various engineering applications [8].

On the other hand, there are lots of ceramic geological and biological materials whose microscopic load carrying behavior is not dominated by the bending of structural units, but by the three-dimensional interaction of disorderedly arranged single crystals. One obvious possibility of capturing this so-called polycrystalline behavior is through finite element discretization of the 3D network of crystals [9]. In general, this option implies relatively high efforts in terms of microstructure representation and CPU time (especially when investigating a larger number of microstructures) so that more efficient alternatives have been sought: Some success is recorded for differential “mean-field” homogenization methods based on two phases [10], and a particularly interesting solution has emerged in the form of self-consistent homogenization methods based on an infinite amount of nonspherical (needle or disk-shaped) solid crystal phases and one spherical pore phase [11–14]. The “price” for the computational efficiency is the careful use of the semi-analytical and analytical solutions of Eshelby [15] and Laws [16], which, from a practical viewpoint, sometimes appear as demanding. Hence, the question arises whether micromechanics-based porosity-property relationships may be approximated again through power-functions (which may also find a straight-forward use in commercial software packages [17], be it for structural analysis based on nonhomogeneous computer tomographic (CT) data information, or for further “computational” homogenization procedures). Motivated by Salje et al. [18], who showed first power functions for the bulk modulus of mesoporous minerals, we here investigate the approximate existence of power-law type dimensionless structure-property relations of porous polycrystals, based on the concept of influence tensors [19–21] within the framework of continuum poromicromechanics [22] including eigenstressed matrix-inclusion problems [23].

Accordingly, the paper is organized as follows: First, the continuum poromicromechanics of porous polycrystals is introduced. Within representative volume elements (RVEs), material phases are defined: (i) elastobrittle crystals of disk- or needle-shape, oriented in all space directions, and (ii) a pressurized pore space in between (Sec. 2.1). Prescription of homogeneous strains at the boundaries of the RVEs, and of a pore pressure inside, allows for upscaling poroelastic and strength properties (relating to both pressure-free and pressurized pore spaces) from the single crystal scale to that of the porous polycrystal (Sec. 2.2)—provided concentration and influence tensors are known. The latter are derived from matrix-inclusion problems with eigenstresses (Sec. 2.3). Thereafter, the poromicromechanics predictions are discussed from a dimensional analysis view, seeking for the simplest possible structure-property relations between dimensionless quantities, which we approximate by power-law and polynomial functions (Secs. 3 and 4). These relations are compared to a large number of experimental data from different material classes, such as hydroxyapatite [24–27], bioactive glass-ceramics (CEL2) [28], gypsum [29–33], piezoelectric ceramics [34], alumina [1,35], zirconia [35], silicon carbide [36], and silicon nitride [37].

## Continuum Micromechanics of Ceramic Bone Biomaterials

### Representative Volume Element and Phase Properties.

In continuum micromechanics [22,23,38–40], a material is understood as a macrohomogeneous but microheterogeneous body filling a representative volume element with characteristic length $ℓ$, $ℓ>>d$, $d$ standing for the characteristic length of inhomogeneities within the RVE, and $ℓ≪L$, $L$ standing for the characteristic lengths of geometry or loading of a structure built up by the material defined on the RVE. In general, the microstructure within one RVE is so complicated that it cannot be described in complete detail. Therefore, quasi-homogeneous subdomains with known physical quantities (such as volume fractions, elastic properties, strength, and eigenstresses) are reasonably chosen. They are called material phases. The “homogenized” mechanical behavior of the overall material, i.e., the relation between homogeneous deformations acting on the boundary of the RVE and resulting (average) stresses, or the relation between an internal pore pressure and stresses acting on the boundary of the RVE, including the ultimate stresses sustainable by the RVE, can then be estimated from the mechanical behavior of the aforementioned homogeneous phases (representing the inhomogeneities within the RVE), their dosages within the RVE, their characteristic shapes, and their interactions.

We here represent porous ceramics through an infinite amount of needle-shaped or disk-shaped solid phases, which are oriented isotropically in all space directions (quantified through spherical coordinates $ϑ$ and $ϕ$) and one porous phase occupying the remaining space with volume fraction $φ$ (porosity); see Fig. 3. As regards phase elasticity, the fourth-order stiffness tensor $Cs$ relates the (average microscopic) second-order strain tensor in the solid phase $s$, $ɛs(ϑ,ϕ)$, to the (average microscopic) second-order stress tensor in the solid phase $s$, $σs(ϑ,ϕ)$,
$σs(ϑ,ϕ)=Cs:ɛs(ϑ,ϕ)$
(1)
while the pores have zero stiffness but exhibit eigenstresses $-p1$,
$σpor=-p1$
(2)

where $p$ is the pore pressure, and $1$ is the second-order unity tensor. Crystalline solid phases typically exhibit a certain amount of anisotropy, but the latter has only negligible influence on the overall elastic behavior of the porous polycrystal, as shown in Ref. [11].

Isotropic elastic properties of different solid crystalline material phases are given in Table 1.

Table 1

Elastic properties (Young's modulus $Es$ and Poisson's ratio $νs$) of different solid crystal phases

$Es$$νs$
(GPa)(1)Sourcea
Hydroxyapatite1140.27[41]
Glass-ceramics (CEL2)85.30.25[42]
Gypsum400.34[13]
Piezoelectric ceramics667[34]
Alumina4000.23[43]
Zirconia2100.31[43]
Silicon carbide4200.17[43]
Silicon nitride3200.27[43]
Gadolinium(III)oxide ($Gd2O3$)410[44]
$Es$$νs$
(GPa)(1)Sourcea
Hydroxyapatite1140.27[41]
Glass-ceramics (CEL2)85.30.25[42]
Gypsum400.34[13]
Piezoelectric ceramics667[34]
Alumina4000.23[43]
Zirconia2100.31[43]
Silicon carbide4200.17[43]
Silicon nitride3200.27[43]
Gadolinium(III)oxide ($Gd2O3$)410[44]
a

References cited in the table are [13,34,41–44].

As regards solid phase strength, brittle failure is associated to the boundary of an elastic domain $fs[σ(ϑ,ϕ)]<0$,
$ϑ=0,…,π;ϕ=0,…,2π;ψ=0,…,2π:fs[σs(ϑ,ϕ)]=maxϑ,ϕ(βmaxψ|σs,Nn|+σs,NN)-σsult,t=0$
(3)
in case of needle-shaped phases, and
$ϑ=0,…,π;ϕ=0,…,2π;ψ=0,…,2π;ω=0,…,2π:fs[σs(ϑ,ϕ)]=maxϑ,ϕ[maxψ(βmaxω|σs,Nn|+σs,NN)]-σsult,t=0$
(4)
in case of disk-shaped phases. In Eq. (3), $σs,NN(ϕ,ϑ)$ is the normal stress in needle direction, and $σs,Nn(ϕ,ϑ;ψ)$ is the shear stress in planes orthogonal to the needle direction. $N¯$ is the needle orientation vector and $n¯$ is the direction orthogonal to $N¯$, as function of angle $ψ$ (see Fig. 1). $β=σsult,t/σsult,s$ is the ratio between the uniaxial tensile strength $σHAult,t$ and the shear strength $σsult,s$ of pure solid. In Eq. (4), $σs,NN(ϕ,ϑ;ψ)$ is the normal stress acting in the plane of the disk, and $σs,Nn(ϕ,ϑ;ψ;ω)$ is the shear stress acting on planes orthogonal to the disk plane. The orientation vector $N¯$ is defined through angle $ψ$ in the plane of the disk, and direction $n¯$ is orthogonal to $N¯$ and specified through angle $ω$ (see Fig. 2). Strength properties of hydroxyapatite and of gypsum are given in Table 2.
Fig. 1

Needlelike representation of crystals with orientation vector $N¯r=e¯r$, inclined by angles $ϑ$ and $ϕ$ with respect to the reference frame ($e¯1$, $e¯2$, $e¯3$). The local base frame ($e¯ϑ$, $e¯ϕ$, $e¯r$) is attached to the needle.

Fig. 1

Needlelike representation of crystals with orientation vector $N¯r=e¯r$, inclined by angles $ϑ$ and $ϕ$ with respect to the reference frame ($e¯1$, $e¯2$, $e¯3$). The local base frame ($e¯ϑ$, $e¯ϕ$, $e¯r$) is attached to the needle.

Close modal
Fig. 2

Disk-like representation of crystals with normals oriented along vector $e¯r$, inclined by angles $ϑ$ and $ϕ$ with respect to the reference frame ($e¯1$, $e¯2$, $e¯3$). The local base frame ($e¯ϑ$, $e¯ϕ$, $e¯r$) is attached to the disk, and another local frame ($e¯1'$$e¯2'$, $e¯3'$) is introduced for definition of the shear stress direction.

Fig. 2

Disk-like representation of crystals with normals oriented along vector $e¯r$, inclined by angles $ϑ$ and $ϕ$ with respect to the reference frame ($e¯1$, $e¯2$, $e¯3$). The local base frame ($e¯ϑ$, $e¯ϕ$, $e¯r$) is attached to the disk, and another local frame ($e¯1'$$e¯2'$, $e¯3'$) is introduced for definition of the shear stress direction.

Close modal
Fig. 3

(a) RVE of porous ceramic polycrystal and (b) and (c) generalized Eshelby matrix-inclusion problems with eigenstresses: (b) for each solid phase, with phase-specific orientations ($ϑ,ϕ$) (see Fig. 1) and (c) for the pore phase

Fig. 3

(a) RVE of porous ceramic polycrystal and (b) and (c) generalized Eshelby matrix-inclusion problems with eigenstresses: (b) for each solid phase, with phase-specific orientations ($ϑ,ϕ$) (see Fig. 1) and (c) for the pore phase

Close modal
Table 2

Strength properties (uniaxial tensile strength $σHAult,t$ and shear strength $σHAult,s$) of hydroxyapatite and gypsum

$σsult,t$$σsult,s$
(MPa)(MPa)Sourcea
Hydroxyapatite 52.2 80.3 [45,46
Gypsum 17.0 22.1 [13
$σsult,t$$σsult,s$
(MPa)(MPa)Sourcea
Hydroxyapatite 52.2 80.3 [45,46
Gypsum 17.0 22.1 [13
a

References cited in the table are [13,45,46].

### Averaging Homogenization Upscaling.

The central goal of continuum microporomechanics is to estimate the mechanical properties (such as elasticity or strength) of the material defined on the RVE (the macrohomogeneous but microheterogeneous medium) from the aforementioned phase properties (inclusive of eigenstresses). This procedure is referred to as homogenization or one homogenization step. Therefore, homogeneous (macroscopic) strains $E$ are imposed onto the RVE in terms of displacements at its boundary $∂V$:
$∀x¯∈∂V:ξ¯(x¯)=E·x¯$
(5)
with $x¯$ being the position vector for locations within or at the boundary of the RVE. As a consequence, the resulting kinematically compatible microstrains $ɛ(x¯)$ throughout the RVE with volume $VRVE$ fulfill the average condition [39],
$E=〈ɛ〉=1VRVE∫VRVEɛ(x¯)dV=(1-φ)∫ϕ=02π∫ϑ=0πɛs(ϑ,ϕ)sinϑdϑdϕ4π+φɛpor$
(6)
with $ɛpor$ as the average microstrains in the pore space. Equation (6) provides a link between micro and macro strains. Analogously, homogenized (macroscopic) stresses $Σ$ are defined as the spatial average over the RVE of the microstresses $σ(x¯)$,
$Σ=〈σ〉=1VRVE∫VRVEσ(x¯)dV=(1-φ)∫ϕ=02π∫ϑ=0πσs(ϑ,ϕ)sinϑdϑdϕ4π-φ1p$
(7)
For material failure and strength, we need to calculate local strains and stresses. The superposition principle (following from linear elasticity and linearized strain [38]) implies that the solid phase strains $ɛs(ϑ,ϕ)$ are linearly related to both the macroscopic strains $E$ and to the pore pressure $p$ (which can be considered as independent loading parameters),
$ɛs(ϑ,ϕ)=As(ϑ,ϕ):E+ds(ϑ,ϕ)p$
(8)
$ɛpor=Apor:E+dporp$
(9)
with $As(ϑ,ϕ)$ and $Apor$ as the fourth-order concentration tensors of the solid phase oriented in the ($ϑ,ϕ$)-direction and of the pore phase, respectively, and with $ds(ϑ,ϕ)$ and $dpor$ as the second-order influence tensors [19–21] representing the effect of the pore pressure on the strains in the solid phase oriented in ($ϑ,ϕ$)-direction and on the pore strains themselves, respectively. Insertion of phase elasticity law (Eq. (1)) into the concentration-influence relation (Eq. (8)) yields
$σs(ϑ,ϕ)=Cs:As(ϑ,ϕ):E+Cs:ds(ϑ,ϕ)p$
(10)
Combining Eq. (10) with the stress average rule (Eq. (7)) leads to the first macroscopic state equation of linear poroelasticity [22,47]
$Σ=Chom:E-bhomp$
(11)
where $Chom$ and $bhom$ are given by
$Chom=(1-φ)∫ϕ=02π∫ϑ=0πCs:As(ϑ,ϕ)sinϑdϑdϕ4π$
(12)
$bhom=-(1-φ)∫ϕ=02π∫ϑ=0πCs:ds(ϑ,ϕ)sinϑdϑdϕ4π+φ1$
(13)

$Chom$ is the homogenized (macroscopic) stiffness tensor, and $bhom$ is the second-order tensor of Biot coefficients. In the absence of macroscopic strains, $E=0$, this pore pressure provokes macroscopic stresses, which are quantified through $bhom$, see Eq. (11).

For thermodynamic reasons (see, e.g., Coussy [47], p. 72), the same tensor quantifies the change in porosity $(φ-φ0)$ resulting from macroscopic strains in the absence of pore pressure, see Eq. (14), and in case of overall isodeformation conditions ($E=0$), the Biot modulus $Nhom$ quantifies the porosity change provoked by pressure $p$. Accordingly, the second state equation of poroelasticity reads as [22,47]
$φ-φ0=bhom:E+pNhom$
(14)
Combination of the change in porosity,
$φ-φ0=φ01:ɛpor$
(15)
with Eq. (9) allows one to identify $bhom$ and $1Nhom$ as
$bhom=φ01:Apor$
(16)
$1Nhom=φ01:dpor$
(17)

Indeed, it can be shown numerically for all the computations reported in this paper that the expressions for $bhom$ given in Eqs. (13) and (16) are identical. In the present case of isotropy, we have $bhom=bhom1$, with the homogenized Biot coefficient $bhom$ and the second-order unity tensor $1$.

As long as the average phase strains $ɛs$ are relevant for brittle phase failure, resulting in overall failure of the RVE, the concentration-influence relation (Eq. (8)) allows for translation of the brittle microfailure criteria, such as Eqs. (3) and (4), into a macroscopic (homogenized) brittle failure criterion, according to Eqs. (3), (4), (10), and (11),
$fs[σs(ϑ,ϕ)]=0=fs[Cs:(As(ϑ,ϕ):Chom,-1:(Σ+bhomp)+dsp)]=fs[Cs:As(ϑ,ϕ):Chom,-1:Σ+(Cs:As(ϑ,ϕ): Chom,-1:bhom+Cs:ds)p]=F(Σ,p)$
(18)

Both the poroelastic and the porobrittle relations Eqs. (11), (14), and (18) depend on knowledge on the concentration and influence tensors $Apor$, $As(ϑ,ϕ)$, $dpor$, and $ds(ϑ,ϕ)$. They are estimated from generalized Eshelby problems with eigenstrains, as described next.

### Matrix-Inclusion Problems With Eigenstresses.

When considering a polycrystalline morphology where all phases are disordered and in mutual contact, the concentration tensors and the influence tensors can be suitably estimated [21,48] from a generalized Eshelby problem with eigenstresses [23], which is formulated separately for each phase (all solid phases and the pore phase). Thereby, each phase is represented by a single ellipsoidal inclusion embedded in an infinite matrix, which exhibits the elastic stiffness $Chom$ of the overall polycrystal and eigenstresses $-bhomp$ (according to Eq. (11)) and which is subjected at the remote (infinite) boundary to homogeneous auxiliary strains $E0$ (Fig. 3).

Under these conditions, the solid phase strains read as [23,49]
$ɛs(ϑ,ϕ)=As∞(ϑ,ϕ):[E0-Pcylhom(ϑ,ϕ):bhomp]with As∞(ϑ,ϕ)=[I+Pcylhom(ϑ,ϕ):$
(19)
$(Cs-Chom)]-1$
(20)
and the pore strains read as [23,49]
$ɛpor=Apor∞:[E0-Psphhom:(-1+bhom)p]$
(21)
$with Apor∞=[I+Psphhom(ϑ,ϕ):(-Chom)]-1$
(22)

In Eqs. (19)(22), $I$ is the fourth-order unity tensor with components $Iijkl=1/2(δikδjl+δilδkj)$, $δij$ (Kronecker δ) are the components of second-order identity tensor 1, and the fourth-order Hill tensors $Pcylhom$ and $Psphhom$ account for the cylindrical and spherical shapes of phases as an ellipsoidal inclusion embedded in a matrix of stiffness $Chom$. The components of the Hill tensors are based on Eshelby’s 1957 solution [15], and they are given in the Appendix.

The auxiliary strains $E0$ are determined such that the strain average rule (Eq. (6)) is satisfied [23,50]. Accordingly, insertion of Eqs. (19) and (21) into Eq. (6) yields the sought expression for $E0$,
$E0=[(1-φ)∫ϕ=02π∫ϑ=0πAs∞(ϑ,ϕ)sinϑdϑdϕ4π+φApor∞]-1: (E+E0π)$
(23)
with
$E0π= (1-φ)∫ϕ=02π∫ϑ=0πAs∞(ϑ,ϕ):Pcylhom(ϑ,ϕ):bhomp)×sinϑdϑdϕ4π+φApor∞:Psphhom:(-1+bhom)p$
(24)
Insertion of Eq. (23) into Eqs. (19) and (21) delivers the average phase strains as
$ɛs(ϑ,ϕ)=As(ϑ,ϕ):(E+E0π)-As∞(ϑ,ϕ): Pcylhom(ϑ,ϕ):bhomp with As(ϑ,ϕ)=As∞(ϑ,ϕ):$
(25)
$[(1-φ)∫ϕ=02π∫ϑ=0πAs∞(ϑ,ϕ)sinϑdϑdϕ4π+φApor∞]-1$
(26)
and as
$ɛpor=Apor:(E+E0π)-Apor∞:Psphhom:(-1+bhom)pwith Apor=Apor∞:$
(27)
$[(1-φ)∫ϕ=02π∫ϑ=0πAs∞(ϑ,ϕ)sinϑdϑdϕ4π+φApor∞]-1$
(28)

$As(ϑ,ϕ)$ and $Apor$ are the strain concentration tensors of the solid phases and of the pore phase, respectively. Back-substitution of Eq. (26) into Eq. (12) delivers the sought estimate for the homogenized (macroscopic) stiffness tensor $Chom$.

Finally, comparing Eqs. (25) and (27) to Eqs. (8) and (9) yields the eigenstress influence tensors $ds$ and $dpor$. They read as
$ds(ϑ,ϕ)=As(ϑ,ϕ):[(1-φ)∫ϕ=02π∫ϑ=0πAs∞(ϑ,ϕ):Pcylhom(ϑ,ϕ):bhomsinϑdϑdϕ4π+φApor∞:Psphhom:(-1+bhom)]-As∞(ϑ,ϕ):Pcylhom(ϑ,ϕ):bhom$
(29)
and
$dpor=Apor(ϑ,ϕ):[(1-φ)∫ϕ=02π∫ϑ=0πAs∞(ϑ,ϕ):Pcylhom(ϑ,ϕ):bhomsinϑdϑdϕ4π+φApor∞:Psphhom:(-1+bhom)]-Apor∞(ϑ,ϕ):Psphhom:(-1+bhom)$
(30)

For disk-like solid inclusions, the problem is formulated in an analogous way, whereby $Pcylhom(ϑ,ϕ)$ is replaced by $Pdiskhom(ϑ,ϕ)$ in Eqs. (19)(30). The components of the Hill tensor $Pdiskhom(ϑ,ϕ)$ are given in the Appendix.

## Porosity-Dependent Poroelasticity—Micromechanical Results, and Their Power Law and Polynomial Representations

### Young’s Modulus and Poisson’s Ratio.

Equations (12), (20), (22), (26), and (A1)(A6) imply that Young’s modulus $Ehom$ of a polycrystal made up by disk or needle-shaped elements is governed by functional relations of the form
$Ediskhom=FEdisk(Es,νs,φ)$
(31)
$Ecylhom=FEcyl(Es,νs,φ)$
(32)
with $Es$ and $νs$ being Young’s modulus and Poisson’s ratio of the single crystal with isotropic stiffness $Cs$. From dimensional analysis [51,52], it follows that
$EdiskhomEs=ΠEdisk(φ,νs)$
(33)
$EcylhomEs=ΠEcyl(φ,νs)$
(34)
Hence, it turns out that $Ehom$ scales linearly with $Es$ as is confirmed from an evaluation of micromechanics predictions for $Es=11.4…1140$ GPa, see Figs. 4 and 5. Even more remarkably, the dependence on $νs$ in Eqs. (33) and (34) turns out as negligible for $νs=0…0.49$, so that
Fig. 4

Needle-based micromechanics prediction for normalized homogenized Young’s modulus $E/Es$, as function of porosity $φ$, for a wide range of Young’s moduli and Poisson’s ratios of crystal solid phase, and approximated by a power function and a fourth-order polynomial function. Experimental data: hydroxyapatite [24–27], collected in Ref. [12]; gypsum [29–33], collected in Ref. [13]; piezoelectric ceramics [34]; alumina, zirconia [35,43]; silicon carbide [36,43]; and silicon nitride [37,43].

Fig. 4

Needle-based micromechanics prediction for normalized homogenized Young’s modulus $E/Es$, as function of porosity $φ$, for a wide range of Young’s moduli and Poisson’s ratios of crystal solid phase, and approximated by a power function and a fourth-order polynomial function. Experimental data: hydroxyapatite [24–27], collected in Ref. [12]; gypsum [29–33], collected in Ref. [13]; piezoelectric ceramics [34]; alumina, zirconia [35,43]; silicon carbide [36,43]; and silicon nitride [37,43].

Close modal
Fig. 5

Disk-based micromechanics prediction for normalized homogenized Young’s modulus $E/Es$, as function of porosity $φ$, for a wide range of Young’s moduli and Poisson’s ratios of the crystal solid phase and approximated by a power function and a fourth-order polynomial function. Experimental data: CEL2 glass-ceramics [28]; alumina [1]; $Gd2O3$ [44].

Fig. 5

Disk-based micromechanics prediction for normalized homogenized Young’s modulus $E/Es$, as function of porosity $φ$, for a wide range of Young’s moduli and Poisson’s ratios of the crystal solid phase and approximated by a power function and a fourth-order polynomial function. Experimental data: CEL2 glass-ceramics [28]; alumina [1]; $Gd2O3$ [44].

Close modal
$EdiskhomEs=ΠEdisk(φ)$
(35)
$EcylhomEs=ΠEcyl(φ)$
(36)
and this holds for a very wide range of polycrystals tested experimentally [1,24–37]; see experimental data points in Figs. 4 and 5. The latter functions can be reasonably well approximated through power functions
$EhomEs≃BE(1-φ)CE=EappEs$
(37)
where $Eapp$ denotes the power function-approximated Young’s modulus of the porous polycrystal. Alternatively, one may employ polynomial functions
$EhomEs≃ A¯E(1-φ)4+B¯E(1-φ)3+C¯E(1-φ)2+D¯E(1-φ)+E¯E=EappEs$
(38)
Numerical values for $BE$ and $CE$ as well as for $A¯E,B¯E,C¯E,D¯E$, and $E¯E$, corresponding to needle- and disk-type morphologies, are given in Table 3. We assess the quality of these approximations by means of the following error measure,
Table 3

Power function constants and coefficients of polynomial functions for dimensionless Young's modulus ($BE$ and $CE$, given in Eq. (37); $A¯E,B¯E,C¯E,D¯E$, and $E¯E$, given in Eq. (38)), for needle- and disk-type morphologies

Power functions
QuantityMorphology$BE$$CE$
$Ehom/Es$ Needles 1.040 2.596
$Ehom/Es$ Disks 0.9867 2.053
Power functions
QuantityMorphology$BE$$CE$
$Ehom/Es$ Needles 1.040 2.596
$Ehom/Es$ Disks 0.9867 2.053
Fourth-order polynomial functions
QuantityMorphology$A¯E$$B¯E$$C¯E$$D¯E$$E¯E$
$Ehom/Es$ Needles –2.189 4.633 –1.745 0.302
$Ehom/Es$ Disks –0.511 1.5594 –0.370 0.330
Fourth-order polynomial functions
QuantityMorphology$A¯E$$B¯E$$C¯E$$D¯E$$E¯E$
$Ehom/Es$ Needles –2.189 4.633 –1.745 0.302
$Ehom/Es$ Disks –0.511 1.5594 –0.370 0.330
$e(φ)=|Eapp(φ)-Ehom(φ)Es|$
(39)
The mean value of these errors is calculated as
$mean|e|=∫φ=01|e(φ)|dφ$
(40)

Corresponding numbers for the errors are given in Table 4. In terms of the correlation coefficient, the polynomials outperform the already very precise power functions ($R2>99%$) by another two orders of precision.

Table 4

Dimensionless Young's modulus: maximum and mean of errors as well as correlation coefficient, between micromechanical predictions and approximations with power functions and with polynomial functions, for needle- and disk-type morphologies

QuantityMorphology$max|e|$$mean|e|$$R2$
Power functions
$Ehom/Es$ Needles 0.040 0.011 0.9981
$Ehom/Es$ Disks 0.026 0.013 0.9981
Fourth-order polynomial functions
$Ehom/Es$ Needles 0.008 0.004 0.9998
$Ehom/Es$ Disks 0.008 0.002 0.9999
QuantityMorphology$max|e|$$mean|e|$$R2$
Power functions
$Ehom/Es$ Needles 0.040 0.011 0.9981
$Ehom/Es$ Disks 0.026 0.013 0.9981
Fourth-order polynomial functions
$Ehom/Es$ Needles 0.008 0.004 0.9998
$Ehom/Es$ Disks 0.008 0.002 0.9999

For any crystal stiffness $Es$, Eq. (37) can be easily translated into a dimensional relationship
$Ehom=bE(1-φ)cE$
(41)
with
$bE=EsBE and cE=CE$
(42)

being typical input values in commercial image-to-mesh conversion softwares [17].

Equations (12), (20), (22), (26), and (A1)(A6), imply that the polycrystals’ Poisson’s ratios $νhom$ are governed by functional relationships of the form
$νdiskhom=Πνdisk(Es,νs,φ)$
(43)
$νcylhom=Πνcyl(Es,νs,φ)$
(44)
However, since such relationships need to be independent of the chosen units of measurements (here those for stress, e.g., GPa versus MPa), the dimensionless quantity needs to be independent of the dimensional quantity $Es$, hence we have
$νdiskhom=Πνdisk(νs,φ)$
(45)
$νcylhom=Πνcyl(νs,φ)$
(46)
The independence of $Es$ is obviously confirmed by a variation of $Es$ in the micromechanical predictions of Figs. 6 and 7. However, in contrast to the situation encountered with the Young’s modulus, the single crystal’s Poisson’s ratio $νs$ significantly governs the overall polycrystalline Poisson’s ratio (see Figs. 6 and 7).
Fig. 6

Needle-based micromechanics prediction for Poisson’s ratio $ν$, as function of porosity $φ$, for a wide range of Young’s moduli and Poisson’s ratios of the crystal solid phase

Fig. 6

Needle-based micromechanics prediction for Poisson’s ratio $ν$, as function of porosity $φ$, for a wide range of Young’s moduli and Poisson’s ratios of the crystal solid phase

Close modal
Fig. 7

Disk-based micromechanics prediction for Poisson’s ratio $ν$, as function of porosity $φ$, for a wide range of Young’s moduli and Poisson’s ratios of the crystal solid phase

Fig. 7

Disk-based micromechanics prediction for Poisson’s ratio $ν$, as function of porosity $φ$, for a wide range of Young’s moduli and Poisson’s ratios of the crystal solid phase

Close modal
The functions in Eqs. (45) and (46) can be reasonably well approximated in $φ$ through power functions
$νhom(νs)≃Aν(νs)+Bν(νs)×(1-φ)Cν(νs)=νapp(νs)$
(47)
whereby $νapp$ denotes the power function-approximated Poisson’s ratio, and the coefficients $Aν$, $Bν$, and $Cν$ depend on the Poisson’s ratio of the solid crystals, $νs$: The latter dependencies can be approximated through polynomials reading as
$Aν=0.2326Bν={0.1110νs-0.0397 for 0≤νs≤0.20(1.7247νs4-1.8319νs3+0.7221νs2-0.1249νs+0.0080)103 for 0.20<νs<0.300.1020νs-0.0286 for 0.30≤νs≤0.50Cν={-14.9294νs3-30.5060νs2+1.6621νs+1.6764for 0≤νs≤0.25(-0.8145νs3+1.0023νs2-0.4073νs+0.0572)103for 0.25<νs≤0.50$
(48)
in case of needle-type solid inclusions, and
$Aν=0.2466νs+0.1521Bν={0.0720νs-0.0150 for 0≤νs≤0.20-742.9νs4+705.5νs3-248.0νs2+38.31νs-2.199 for 0.20<νs<0.300.1110νs-0.0268 for 0.30≤νs≤0.50Cν={426.7νs3-211.5νs2+17.75νs+2.062for 0≤νs≤0.2555.08νs3-80.71νs2+36.62νs-3.143for 0.25<νs≤0.50$
(49)
in case of disk-type solid inclusions. Alternatively, polynomials can be employed for the relationship between porosity and the polycrystals’ Poisson’s ratios,
$νhom(νs)≃A¯ν(1-φ)4+B¯ν(1-φ)3+C¯ν(1-φ)2 +D¯ν(1-φ)+E¯ν=νapp(νs)$
(50)
Coefficients $A¯ν,B¯ν,C¯ν,D¯ν$, and $E¯ν$ of fourth-order polynomial approximation of $νhom$ depend linearly on $νs$,
$q=a*νs+b*$
(51)
with $q=A¯ν,B¯ν,C¯ν,D¯ν,E¯ν$; see Table 5 for values of $a*$ and $b*$. The precision of the power and polynomial approximations for $νhom$ is documented in terms of an error measure analogous to Eq. (39),
Table 5

Coefficients $a*$ and $b*$ of linear approximation of relationship between coefficients $A¯ν,B¯ν,C¯ν,D¯ν,E¯ν$ and Poisson's ratio of single crystal $νs$, see Eq. (51)

Needles

Disks
$a*$$b*$$a*$$b*$
$A¯ν$ –2.9425 0.5113 –1.0521 0.2197
$B¯ν$ 5.0536 –0.7564 2.2684 –0.4645
$C¯ν$ –1.4556 0.0744 –0.8121 0.1662
$D¯ν$ 0.3410 –0.0586 0.3602 –0.0718
$E¯ν$ –0.0033 0.2313 0.2394 0.1496

Needles

Disks
$a*$$b*$$a*$$b*$
$A¯ν$ –2.9425 0.5113 –1.0521 0.2197
$B¯ν$ 5.0536 –0.7564 2.2684 –0.4645
$C¯ν$ –1.4556 0.0744 –0.8121 0.1662
$D¯ν$ 0.3410 –0.0586 0.3602 –0.0718
$E¯ν$ –0.0033 0.2313 0.2394 0.1496
$e(φ)=|νapp(φ)-νhom(φ)|$
(52)

with the mean calculated in analogy to Eq. (40), see Table 6.

Table 6

Poisson's ratio: maximum and mean of errors as well as correlation coefficient, between micromechanical predictions and approximations with power functions and with polynomial functions, for needle- and disk-type morphologies

QuantityMorphology$max|e|$$mean|e|$$R2$
Power functions
$νhom(νs)$ Needles 0.035 0.009 0.922
$νhom(νs)$ Disks 0.021 0.007 0.944
Fourth-order polynomial functions
$νhom(νs)$ Needles 0.005 0.002 0.996
$νhom(νs)$ Disks 0.004 0.001 0.987
QuantityMorphology$max|e|$$mean|e|$$R2$
Power functions
$νhom(νs)$ Needles 0.035 0.009 0.922
$νhom(νs)$ Disks 0.021 0.007 0.944
Fourth-order polynomial functions
$νhom(νs)$ Needles 0.005 0.002 0.996
$νhom(νs)$ Disks 0.004 0.001 0.987

### Biot Coefficient and Biot Modulus.

Equations (16), (20), (22), (28), and (A1)(A6) imply that the (dimensionless) Biot coefficients of polycrystals $bhom$ are governed by functional relationships of the form
$bdiskhom=Πbdisk(Es,νs,φ)$
(53)
$bcylhom=Πbcyl(Es,νs,φ)$
(54)
As before, dimensional analysis excludes the dependence on $Es$ so that we have
$bdiskhom=Πbdisck(νs,φ)$
(55)
$bcylhom=Πbcyl(νs,φ)$
(56)
The independence of $bhom$ of $Es$ is expectedly confirmed by the micromechanical predictions of Figs. 8 and 9, as is the strong dependence of the Biot coefficient on the Poisson’s ratio $νs$ of the single crystal (see Figs. 8 and 9). The functions in Eq. (55) and (56) can be reasonably well approximated through polynomial functions,
Fig. 8

Needle-based micromechanics prediction for Biot coefficient $b11hom$, as function of porosity $φ$, for a wide range of Young’s moduli and Poisson’s ratios of the crystal solid phase

Fig. 8

Needle-based micromechanics prediction for Biot coefficient $b11hom$, as function of porosity $φ$, for a wide range of Young’s moduli and Poisson’s ratios of the crystal solid phase

Close modal
Fig. 9

Disk-based micromechanics prediction for Biot coefficient $b11hom$, as function of porosity $φ$, for a wide range of Young’s moduli and Poisson’s ratios of the crystal solid phase

Fig. 9

Disk-based micromechanics prediction for Biot coefficient $b11hom$, as function of porosity $φ$, for a wide range of Young’s moduli and Poisson’s ratios of the crystal solid phase

Close modal
$bhom(νs)≃A¯b(1-φ)4+B¯b(1-φ)3+C¯b(1-φ)2 +D¯b(1-φ)+E¯b=bapp(νs)$
(57)
Coefficients $A¯b,B¯b,C¯b,D¯b$, and $E¯b$ of fourth-order polynomial approximation of $bhom$ can be again approximated through fourth-order polynomials reading as
$q1=a∧νs4+b∧νs3+c∧νs2+d∧νs+e∧$
(58)

with $q1=A¯b,B¯b,C¯b,D¯b$, and $E¯b$; see Table 7 for values of $a∧$, $b∧$, $c∧$, $d∧$, and $e∧$. The precision of the polynomial approximations for $bhom$ is documented in terms of an error measure analogous to Eq. (52), with the mean calculated in analogy to Eq. (40), see Table 8.

Table 7

Coefficients $a∧$, $b∧$, $c∧$, $d∧$, and $e∧$ of approximation of relationship between coefficients $A¯b,B¯b,C¯b,D¯b,E¯b$ and Poisson's ratio of single crystal $νs$, see Eq. (58)

Needles
$a∧$$b∧$$c∧$$d∧$$e∧$
$A¯b$ 1067.3 –962.3 226.3 –15.26 2.694
$B¯b$ –1396.2 1326.7 –317.5 18.01 –4.881
$C¯b$ 592.9 –590.7 145.8 –6.149 1.584
$D¯b$ –87.55 91.41 –23.22 1.161 –0.387
$E¯b$ 3.000 –3.283 0.859 –0.026 1.007
Needles
$a∧$$b∧$$c∧$$d∧$$e∧$
$A¯b$ 1067.3 –962.3 226.3 –15.26 2.694
$B¯b$ –1396.2 1326.7 –317.5 18.01 –4.881
$C¯b$ 592.9 –590.7 145.8 –6.149 1.584
$D¯b$ –87.55 91.41 –23.22 1.161 –0.387
$E¯b$ 3.000 –3.283 0.859 –0.026 1.007
Disks
$a∧$$b∧$$c∧$$d∧$$e∧$
$A¯b$ 1037.2 –916.4 218.5 –15.48 0.917
$B¯b$ –1391.1 1275.6 –309.6 20.08 –1.818
$C¯b$ 623.7 –585.3 145.4 –8.732 0.378
$D¯b$ –95.82 93.11 –23.24 1.962 –0.479
$E¯b$ 3.497 –3.454 0.882 –0.052 1.004
Disks
$a∧$$b∧$$c∧$$d∧$$e∧$
$A¯b$ 1037.2 –916.4 218.5 –15.48 0.917
$B¯b$ –1391.1 1275.6 –309.6 20.08 –1.818
$C¯b$ 623.7 –585.3 145.4 –8.732 0.378
$D¯b$ –95.82 93.11 –23.24 1.962 –0.479
$E¯b$ 3.497 –3.454 0.882 –0.052 1.004
Table 8

Normalized poroelastic properties $b11hom$ and $Es/Nhom$: maximum and mean of errors as well as correlation coefficient, between micromechanical predictions and approximations with fourth-order polynomial functions, for needle- and disk-type morphologies

QuantityMorphology$max|e|$$mean|e|$$R2$
$b11hom$ Needles 0.071 0.007 0.9921
$b11hom$ Disks 0.073 0.007 0.9919
$Es/Nhom$ Needles 0.016 0.005 0.9908
$Es/Nhom$ Disks 0.011 0.003 0.9910
QuantityMorphology$max|e|$$mean|e|$$R2$
$b11hom$ Needles 0.071 0.007 0.9921
$b11hom$ Disks 0.073 0.007 0.9919
$Es/Nhom$ Needles 0.016 0.005 0.9908
$Es/Nhom$ Disks 0.011 0.003 0.9910

Equations (17), (22), (28), (30), and (A1)(A6) imply that the Biot moduli $Nhom$ of the porous polycrystals are governed by functional relationships of the form
$Ndiskhom=FNdisk(Es,νs,φ)$
(59)
$Ncylhom=FNcyl(Es,νs,φ)$
(60)
From dimensional analysis, it follows
$NdiskhomEs=ΠNdisk(νs,φ)$
(61)
$NcylhomEs=ΠNcyl(νs,φ)$
(62)
and micromechanical predictions (Figs. 10 and 11) confirm both the linear scaling of the Biot modulus with the Young’s modulus of the single crystal and its significant dependence on the Poisson’s ratio of the single crystal. The functions in Eqs. (61) and (62) can be reasonably well approximated through polynomial functions,
Fig. 10

Needle-based micromechanics prediction for normalized Biot modulus $Es/Nhom$, as function of porosity $φ$, for a wide range of Young’s moduli and Poisson’s ratios of the crystal solid phase

Fig. 10

Needle-based micromechanics prediction for normalized Biot modulus $Es/Nhom$, as function of porosity $φ$, for a wide range of Young’s moduli and Poisson’s ratios of the crystal solid phase

Close modal
Fig. 11

Disk-based micromechanics prediction for normalized Biot modulus $Es/Nhom$, as function of porosity $φ$, for a wide range of Young’s moduli and Poisson’s ratios of the crystal solid phase

Fig. 11

Disk-based micromechanics prediction for normalized Biot modulus $Es/Nhom$, as function of porosity $φ$, for a wide range of Young’s moduli and Poisson’s ratios of the crystal solid phase

Close modal
$NhomEs(νs)≃ A¯N(1-φ)4+B¯N(1-φ)3+C¯N(1-φ)2+D¯N(1-φ)+E¯N=(NhomEs)app(νs)$
(63)
Coefficients $A¯N,B¯N,C¯N,D¯N$, and $E¯N$ of fourth-order polynomial approximation of $Nhom/Es$ can be again approximated through fourth-order polynomials reading as
$q2=aνs4+bνs3+cνs2+dνs+e$
(64)

with $q2=A¯N,B¯N,C¯N,D¯N$, and $E¯N$; see Table 9 for values of $a$, $b$, $c$, $d$, and $e$. The precision of the polynomial approximations for $Nhom/Es$ is documented in terms of an error measure analogous to Eq. (52), with the mean calculated in analogy to Eq. (40), see Table 8.

Table 9

Coefficients $a$, $b$, $c$, $d$, and $e$ of approximation of relationship between coefficients $A¯N,B¯N,C¯N,D¯N,E¯N$ and Poisson's ratio of single crystal $νs$, see Eq. (64)

Needles
$a$$b$$c$$d$$e$
$A¯N$741.4–527.485.56–19.598.012
$B¯N$–1342.7987.9–154.026.93–14.55
$C¯N$717.7–550.984.35–3.5564.707
$D¯N$–128.6101.5–18.27–3.8681.845
$E¯N$5.221–4.2190.7050.0100.020

Needles
$a$$b$$c$$d$$e$
$A¯N$741.4–527.485.56–19.598.012
$B¯N$–1342.7987.9–154.026.93–14.55
$C¯N$717.7–550.984.35–3.5564.707
$D¯N$–128.6101.5–18.27–3.8681.845
$E¯N$5.221–4.2190.7050.0100.020
Disks
$a$$b$$c$$d$$e$
$A¯N$612.9–452.584.23–10.922.683
$B¯N$–1074.2819.5–150.615.38–5.359
$C¯N$552.5–437.682.74–3.1891.089
$D¯N$–102.180.33–18.56–1.1731.569
$E¯N$4.013–3.2500.645–0.0350.012
Disks
$a$$b$$c$$d$$e$
$A¯N$612.9–452.584.23–10.922.683
$B¯N$–1074.2819.5–150.615.38–5.359
$C¯N$552.5–437.682.74–3.1891.089
$D¯N$–102.180.33–18.56–1.1731.569
$E¯N$4.013–3.2500.645–0.0350.012

## Porosity and Pore-Pressure-Dependent Brittle Strength—Micromechanical Results and Their Power Law and Polynomial Representations

Equations (18), (12), (20), (22), (26), and (A1)(A6), when evaluated for nonpressurized pores ($p=0$) and for failure in uniaxial tension $Σ=Σult,te¯3⊗e¯3$, imply that the uniaxial tensile strength $Σult,t$ of a nonpressurized porous polycrystal made up by disk or needle-shaped elements is governed by functional relations of the form
$Σult,t=FΣult,t(Es,νs,σsult,t,σsult,s,φ)$
(65)
From dimensional analysis, it follows that
$Σult,tσsult,t=ΠΣult,t(Esσsult,t,νs,σsult,sσsult,t,φ)$
(66)
Numerical evaluation of the micromechanical model shows that the normalized tensile strength of needle or disk-based porous polycrystals decreases over-linearly with the porosity and that $Σult,t/σsult,t$ is independent of dimensionless quantities $Es/σsult,t$, $νs$, and $σsult,s/σsult,t$, for a wide range of crystal properties ($Es=11.4…1140$ GPa, $νs=0…0.499$, $σsult,t=5.21…521.5$ MPa, $σsult,s=0.802…802$), see Figs. 1213. Hence, the uniaxial tensile strength of porous polycrystals scales linearly with the tensile strength of the solid crystals. The functional relationship in Eq. (66) can be approximated through power functions of the form
Fig. 12

Needle-based micromechanics prediction for normalized uniaxial tensile strength $Σult,t/σsult,t$, as function of porosity $φ$, for different crystal elastic properties ($Es = 11.4…1140$ GPa, $νs = 0…0.499$) as well as for different crystal strength properties ($σsult,t = 5.215…521.5$ MPa, $σsult,s = 0.802…802$ MPa). Experimental data: hydroxyapatite [24,46] and gypsum [13].

Fig. 12

Needle-based micromechanics prediction for normalized uniaxial tensile strength $Σult,t/σsult,t$, as function of porosity $φ$, for different crystal elastic properties ($Es = 11.4…1140$ GPa, $νs = 0…0.499$) as well as for different crystal strength properties ($σsult,t = 5.215…521.5$ MPa, $σsult,s = 0.802…802$ MPa). Experimental data: hydroxyapatite [24,46] and gypsum [13].

Close modal
Fig. 13

Disk-based micromechanics prediction for normalized uniaxial tensile strength $Σult,t/σsult,t$, as function of porosity $φ$, for different crystal elastic properties ($Es = 11.4…1140$ GPa, $νs = 0…0.499$) as well as for different crystal strength properties ($σsult,t = 5.215…521.5$ MPa, $σsult,s = 0.802…802$ MPa)

Fig. 13

Disk-based micromechanics prediction for normalized uniaxial tensile strength $Σult,t/σsult,t$, as function of porosity $φ$, for different crystal elastic properties ($Es = 11.4…1140$ GPa, $νs = 0…0.499$) as well as for different crystal strength properties ($σsult,t = 5.215…521.5$ MPa, $σsult,s = 0.802…802$ MPa)

Close modal
$Σult,tσsult,t≃BΣ(1-φ)CΣ=Σult,t,appσsult,t$
(67)
with $Σult,t,app$ as the power function-approximated tensile strength of the porous polycrystal. Alternatively, one may employ polynomial functions,
$Σult,tσsult,t≃A¯Σ(1-φ)4+B¯Σ(1-φ)3+C¯Σ(1-φ)2 +D¯Σ(1-φ)+E¯Σ=Σult,t,appσsult,t$
(68)

Numerical values for $BΣ$ and $CΣ$ as well as for $A¯Σ$, $B¯Σ$, $C¯Σ$, $D¯Σ$, and $E¯Σ$, corresponding to needle and disk-type morphologies, respectively, are given in Table 10, and respective errors are given in Table 11.

Table 10

Power function constants and coefficients of polynomial functions, for dimensionless uniaxial tensile strength ($BΣ$ and $CΣ$, given in Eq. (67); $A¯Σ$, $B¯Σ$, $C¯Σ$, $D¯Σ$, and $E¯Σ$, given in Eq. (68)) for needle- and disk-type morphologies

Power functions
QuantityMorphology$BΣ$$CΣ$
$Σult,t/σsult,t$Needles1.0162.381
$Σult,t/σsult,t$Disks0.9771.675
Power functions
QuantityMorphology$BΣ$$CΣ$
$Σult,t/σsult,t$Needles1.0162.381
$Σult,t/σsult,t$Disks0.9771.675
Fourth-order polynomial functions
QuantityMorphology$A¯Σ$$B¯Σ$$C¯Σ$$D¯Σ$$E¯Σ$
$Σult,t/σsult,t$Needles–1.77053.8896–1.47030.35240
$Σult,t/σsult,t$Disks0.02200.6213–0.18550.54400
Fourth-order polynomial functions
QuantityMorphology$A¯Σ$$B¯Σ$$C¯Σ$$D¯Σ$$E¯Σ$
$Σult,t/σsult,t$Needles–1.77053.8896–1.47030.35240
$Σult,t/σsult,t$Disks0.02200.6213–0.18550.54400
Table 11

Dimensionless uniaxial tensile strength: maximum and mean of errors as well as correlation coefficient, between micromechanical predictions and approximations with power functions and with polynomial functions, for needle- and disk-type morphologies

QuantityMorphology$max|e|$$mean|e|$$R2$
Power functions
$Σult,t/σsult,t$Needles0.0180.0100.9988
$Σult,t/σsult,t$Disks0.0390.0270.9963
Fourth-order polynomial functions
$Σult,t/σsult,t$Needles0.0060.0030.9999
$Σult,t/σsult,t$Disks0.0020.0010.9999
QuantityMorphology$max|e|$$mean|e|$$R2$
Power functions
$Σult,t/σsult,t$Needles0.0180.0100.9988
$Σult,t/σsult,t$Disks0.0390.0270.9963
Fourth-order polynomial functions
$Σult,t/σsult,t$Needles0.0060.0030.9999
$Σult,t/σsult,t$Disks0.0020.0010.9999
However, compressive uniaxial strengths of a polycrystal made up by disk or needle-shaped elements cannot be represented in an analogous dimensionless way, as is seen in Figs. 14 and 15.
Fig. 14

Needle-based micromechanics prediction for dimensionless uniaxial compressive strength $Σult,c/σsult,t$, as function of porosity $φ$. Influence of dimensionless governing parameters $σsult,s/σsult,t$ and $Es/σsult,t$, $νs=0.27$.

Fig. 14

Needle-based micromechanics prediction for dimensionless uniaxial compressive strength $Σult,c/σsult,t$, as function of porosity $φ$. Influence of dimensionless governing parameters $σsult,s/σsult,t$ and $Es/σsult,t$, $νs=0.27$.

Close modal
Fig. 15

Disk-based micromechanics prediction for dimensionless uniaxial compressive strength $Σult,c/σsult,t$, as function of porosity $φ$. Influence of dimensionless governing parameters $σsult,s/σsult,t$ and $Es/σsult,t$, $νs = 0.27$.

Fig. 15

Disk-based micromechanics prediction for dimensionless uniaxial compressive strength $Σult,c/σsult,t$, as function of porosity $φ$. Influence of dimensionless governing parameters $σsult,s/σsult,t$ and $Es/σsult,t$, $νs = 0.27$.

Close modal
In the absence of macroscopic strains ($E=0$), a pore pressure induces compressive normal strains in all solid phases (see Fig. 16 in combination with Eq. (8)), so that, according to Eq. (3), the polycrystal never fails.
Fig. 16

Hydroxyapatite needle-based micromechanics prediction for components of influence tensor $ds$ in base frame ($e¯ϑ$, $e¯ϕ$, $e¯r$) (1 = $ϑ$, 2 = $ϕ$, 3 = $r$), as function of porosity $φ$

Fig. 16

Hydroxyapatite needle-based micromechanics prediction for components of influence tensor $ds$ in base frame ($e¯ϑ$, $e¯ϕ$, $e¯r$) (1 = $ϑ$, 2 = $ϕ$, 3 = $r$), as function of porosity $φ$

Close modal
In the absence of macroscopic stresses ($Σ=0$, free surfaces), only the pore pressure can induce failure. Equations (12), (18), (16), (20), (22), (26), (28), (29), and (A1)(A6) imply that the pressure at failure of the porous polycrystal made up by disk or needle-shaped elements is governed by functional relations of the form
$pult=Fpult(Es,νs,σsult,t,σsult,s,φ)$
(69)
From dimensional analysis, it follows that
$pultσsult,t=Πpult(Esσsult,t,νs,σsult,sσsult,t,φ)$
(70)
The pore pressure induces hydrostatic tensile stresses in all crystal phases. Brittle failure occurs for all phases at the same pore pressure, when failure criterion (Eq. (3)) is fulfilled. This corresponds to a blasting of the sample through the pore pressure. Numerical evaluation of the micromechanics model shows that $pult/σsult,t$ is independent of dimensionless quantities $Es/σsult,t$, $νs$, and $σsult,t/σsult,t$, for a wide range of crystal properties ($Es=11.4…1140$ GPa, $νs=0…0.49$, $σsult,t=5.21…521.5$ MPa, $σsult,s=0.802…802$ MPa), see Fig. 17.
Fig. 17

Porous sample with free surfaces under the effect of pore pressure only: needle and disk-based micromechanics prediction of dimensionless failure pore pressure $pult/σsult,t$, as function of porosity $φ$, for a wide range of elastic and strength parameters ($Es = 11.4…1140$ GPa, $νs = 0…0.49$, $σsult,t = 5.21…521.5$ MPa, $σsult,s = 0.802…802$ MPa)

Fig. 17

Porous sample with free surfaces under the effect of pore pressure only: needle and disk-based micromechanics prediction of dimensionless failure pore pressure $pult/σsult,t$, as function of porosity $φ$, for a wide range of elastic and strength parameters ($Es = 11.4…1140$ GPa, $νs = 0…0.49$, $σsult,t = 5.21…521.5$ MPa, $σsult,s = 0.802…802$ MPa)

Close modal

## Conclusions

A poromicromechanical analysis of brittle porous polycrystals with needle- or disk-shaped solid phases reveals the following important characteristics of this wide material class:

• Young’s modulus of a polycrystal with disk or needle-shaped elements scales linearly with Young’s modulus of the single crystal phases, and this relation holds independently of the Poisson’s ratio of the solid crystal phases.

• Poisson’s ratio of a porous polycrystal with disk or needle-shaped solid elements is independent of the Young’s modulus of the solid crystal phases, but it depends on the Poisson’s ratio of the solid crystal phases and on the porosity of the porous polycrystal.

• Both dimensionless Young’s modulus and Poisson’s ratio of the porous polycrystal can be very well approximated through power or polynomial functions with the porosity as argument.

• Biot coefficients do not depend on the Young’s modulus of the solid crystal phases, but on their Poisson’s ratio and the porosity of the overall porous polycrystal. Biot modulus scales linearly with Young’s modulus of the single crystals and also depends on their Poisson’s ratio and the overall material’s porosity. Both quantities can be very well approximated through fourth-order polynomial functions with the porosity as argument.

• The uniaxial tensile strength of a pore pressure-free polycrystal with disk or needle-shaped solid elements scales linearly with the tensile strength of the solid crystal phases, independently of all other elastic and strength properties of the solid crystal phases. The dimensionless uniaxial tensile strength can be very well approximated through power or polynomial functions. This does not hold for the uniaxial compressive strength.

• In the absence of macroscopic stresses, the critical pore pressure “blasting” the porous polycrystal from inside scales linearly with the tensile strength of the solid crystal phases, independently of all other elastic and strength properties of the solid crystal phases.

These characteristics are in agreement with independent experimental data found in the open literature.

## Acknowledgment

The authors gratefully acknowledge the financial support of the European Commission, under the theme FP7-2008-SME-1 of the 7th Framework Programme (Grant No. 232164, BIO-CT-EXPLOIT).

### Appendix: Hill Tensors $P$

The fourth-order Hill tensors $Prhom$ account for the shapes of phases $r$ as an ellipsoidal inclusion embedded in a matrix of stiffness $Chom$. They are related to Eshelby tensors $Sresh$ via
$Sresh=Prhom:Chom$
(A1)
where we here discuss shapes being spheres ($r$ = $sph$), cylinders ($r$ = $cyl$), or disks ($r$ = $disk$). The Eshelby tensor $Ssphesh$ corresponding to spherical inclusions reads as [15,23]
$Ssphesh=3ks3ks+4μsJ+6(ks+2μs)5(3ks+4μs)K$
(A2)
with $ks$ and $μs$ as the bulk and the shear modulus of the solid phase being related to Young’s modulus $Es$ and Poisson’s ratio $νs$ via
$Es=9ksμs3ks+μs$
(A3)
$νs=3ks-2μs6ks+2μs$
(A4)
In the base frame ($e¯ϑ$, $e¯ϕ$, $e¯r$)(1 = $ϑ$, 2 = $ϕ$, 3 = $r$; see Fig. 1 for spherical coordinates $ϕ$ and $ϑ$), attached to individual solid needles, the nonzero components of the Eshelby tensor $Scylesh$ corresponding to cylindrical inclusions read as [15]
$Scyl,1111esh=Scyl,2222esh=5-4νs8(1-νs)Scyl,1122esh=Scyl,2211esh=-1+4νs8(1-νs)Scyl,1133esh=Scyl,2233esh=νs2(1-νs)Scyl,2323esh=Scyl,3232esh=Scyl,3223esh=Scyl,2332esh=Scyl,3131esh =Scyl,1313esh=Scyl,1331esh=Scyl,3113esh=14Scyl,1212esh=Scyl,2121esh=Scyl,2112esh=Scyl,1221esh=3-4νs8(1-νs)$
(A5)
In the base frame ($e¯ϑ$, $e¯ϕ$, $e¯r$)(1 = $ϑ$, 2 = $ϕ$, 3 = $r$; see Fig. 2 for spherical coordinates $ϕ$ and $ϑ$), attached to individual solid disks, the nonzero components of the Eshelby tensor $Sdiskesh$ corresponding to oblate inclusions read as [15]
$Sdisk,3333esh=1Sdisk,3311esh=Sdisk,3322esh=νpoly1-νpolySdisk,2323esh=Sdisk,3232esh=Sdisk,3223esh=Sdisk,2332esh =Sdisk,3131esh=Sdisk,1313esh=Sdisk,1331esh=Sdisk,3113esh=12$
(A6)

Following standard tensor calculus [53], the tensor components of $Pcylhom(ϑ,ϕ)=Scylesh(ϑ,ϕ):Chom,-1$ and $Pdiskhom(ϑ,ϕ)=Sdiskesh(ϑ,ϕ):Chom,-1$, being related to differently oriented inclusions, are transformed into one, single base frame ($e¯1$, $e¯2$, $e¯3$), in order to evaluate the integrals in Eqs. (23)(30).

## References

1.
Coble
,
R.
, and
Kingery
,
W.
,
1956
, “
Effect of Porosity on Physical Properties of Alumina
,”
J. Am. Ceram. Soc.
,
39
, pp.
377
385
.
2.
McElhaney
,
J.
,
1966
, “
Dynamic Response of Bone and Muscle Tissue
,”
J. Appl. Physiol.
,
21
, pp.
1231
1236
.
3.
Carter
,
D.
, and
Hayes
,
W.
,
1977
, “
The Compressive Behavior of Bone as a Two-Phase Porous Structure
,”
J. Bone Jt. Surg.
,
59-A
(
7
), pp.
954
962
.
4.
Rice
,
J.
,
Cowin
,
S.
, and
Bowman
,
J.
,
1988
, “
On the Dependence of the Elasticity and Strength of Cancellous Bone on Apparent Density
,”
J. Biomech.
,
21
, pp.
155
168
.
5.
Gibson
,
L.
, and
Ashby
,
M.
,
1982
, “
The Mechanics of Three-Dimensional Cellular Solids
,”
Proc. R. Soc. London, Ser. A
,
382
, pp.
43
59
.
6.
Gibson
,
L.
,
1985
, “
The Mechanical Behavior of Cancellous Bone
,”
J. Biomech.
,
18
, pp.
317
328
.
7.
Gibson
,
L.
, and
Ashby
,
M.
,
1997
,
Cellular Solids: Structure and Properties
, 2nd ed.,
Cambridge University Press
,
Cambridge, UK
.
8.
O’Brien
,
F.
,
Harley
,
B.
,
Yannas
,
I.
, and
Gibson
,
L.
,
2005
, “
The Effect of Pore Size on Cell Adhesion in Collagen-GAG Scaffolds
,”
Biomaterials
,
26
, pp.
433
441
.
9.
Meille
,
S.
, and
Garboczi
,
E.
,
2001
, “
Linear Elastic Properties of 2D and 3D Models of Porous Materials Made From Elongated Objects
,”
Modell. Simul. Mater. Sci. Eng.
,
9
, pp.
371
390
.
10.
Jakobsen
,
M.
,
Hudson
,
J.
,
Minshull
,
T.
, and
Singh
,
S.
,
2000
, “
Elastic Properties of Hydrate-Bearing Sediments Using Effective Medium Theory
,”
J. Geophys. Res.
,
105
, pp.
561
577
.
11.
Fritsch
,
A.
,
Dormieux
,
L.
, and
Hellmich
,
C.
,
2006
, “
Porous Polycrystals Built Up by Uniformly and Axisymmetrically Oriented Needles: Homogenization of Elastic Properties
,”
C. R. Mec.
,
334
(
3
), pp.
151
157
.
12.
Fritsch
,
A.
,
Dormieux
,
L.
,
Hellmich
,
C.
, and
Sanahuja
,
J.
,
2009
, “
Mechanical Behaviour of Hydroxyapatite Biomaterials: An Experimentally Validated Micromechanical Model for Elasticity and Strength
,”
J. Biomed. Mater. Res Part A
,
88A
, pp.
149
161
.
13.
Sanahuja
,
J.
,
Dormieux
,
L.
,
Meille
,
S.
,
Hellmich
,
C.
, and
Fritsch
,
A.
,
2010
, “
Micromechanical Explanation of Elasticity and Strength of Gypsum: From Elongated Anisotropic Crystals to Isotropic Porous Polycrystals
,”
J. Eng. Mech.
,
136
, pp.
239
253
.
14.
Fritsch
,
A.
,
Hellmich
,
C.
, and
Dormieux
,
L.
,
2010
, “
The Role of Disc-Type Crystal Shape for Micromechanical Predictions of Elasticity and Strength of Hydroxyapatite Biomaterials
,”
Philos. Trans. R. Soc. London, Ser. A
,
368
, pp.
1913
1935
.
15.
Eshelby
,
J.
,
1957
, “
The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems
,”
Proc. R. Soc. London, Ser. A
,
241
, pp.
376
396
.
16.
Laws
,
N.
,
1977
, “
The Determination of Stress and Strain Concentrations at an Ellipsoidal Inclusion in an Anisotropicmaterial
,”
J. Elast.
,
7
(
1
), pp.
91
97
.
17.
Simpleware
,
2008
,
“ScanIP, + ScanFE and + ScanCAD Tutorial Guide,” Exeter, UK, http://www.simpleware.com
18.
Salje
,
E.
,
Koppensteiner
,
J.
,
Schranz
,
W.
, and
Fritsch
,
E.
,
2010
, “
Elastic Instabilities in Dry, Mesoporous Minerals and Their Relevance to Geological Applications
,”
Miner. Mag.
,
74
, pp.
341
350
.
19.
Dvorak
,
G.
,
1992
, “
Transformation Field Analysis of Inelastic Composite Materials
,”
Proc. R. Soc. London, Ser. A
,
437
, pp.
311
327
.
20.
Dvorak
,
G.
, and
Benveniste
,
Y.
,
1992
, “
On the Transformation Strains and Uniform Fields in Multiphase Elastic Media
,”
Proc. R. Soc. London, Ser. A
,
437
, pp.
291
310
.
21.
Pichler
,
B.
, and
Hellmich
,
C.
,
2010
, “
Estimation of Influence Tensors for Eigenstressed Multiphase Elastic Media With Nonaligned Inclusion Phases of Arbitrary Ellipsoidal Shape
,”
J. Eng. Mech.
,
136
, pp.
1043
1053
.
22.
Dormieux
,
L.
,
Kondo
,
D.
, and
Ulm
,
F.-J.
,
2006
,
Microporomechanics
,
Wiley
,
New York
.
23.
Zaoui
,
A.
,
2002
, “
Continuum Micromechanics: Survey
,”
J. Eng. Mech.
,
128
(
8
), pp.
808
816
.
24.
De With
,
G.
,
van Dijk
,
H.
,
Hattu
,
N.
, and
Prijs
,
K.
,
1981
, “
Preparation, Microstructure and Mechanical Properties of Dense Polycrystalline Hydroxy Apatite
,”
J. Mater. Sci.
,
16
, pp.
1592
1598
.
25.
Gilmore
,
R.
, and
Katz
,
J.
,
1982
, “
Elastic Properties of Apatites
,”
J Mater. Sci.
,
17
, pp.
1131
1141
.
26.
Liu
,
D.-M.
,
1998
, “
Preparation and Characterization of Porous Hydroxyapatite Bioceramic via a Slip-Casting Route
,”
Ceram. Int.
,
24
, pp.
441
446
.
27.
Charrière
,
E.
,
Terrazzoni
,
S.
,
Pittet
,
C.
,
Mordasini
,
P.
,
Dutoit
,
M.
,
Lemaître
,
J.
, and
Zysset
,
P.
,
2001
, “
Mechanical Characterization of Brushite and Hydroxyapatite Cements
,”
Biomaterials
,
22
, pp.
2937
2945
.
28.
Malasoma
,
A.
,
Fritsch
,
A.
,
Kohlhauser
,
C.
,
Brynk
,
T.
,
Vitale-Brovarone
,
C.
,
Pakiela
,
Z.
,
Eberhardsteiner
,
J.
, and
Hellmich
,
C.
,
2008
, “
Micromechanics of Bioresorbable Porous CEL2 Glass-Ceramic Scaffolds for Bone Tissue Engineering
,”
,
107
, pp.
277
286
.
29.
Ali
,
M.
, and
Singh
,
B.
,
1975
, “
The Effect of Porosity on the Properties of Glass Fibre-Reinforced Gypsum Plaster
,”
J. Mater. Sci.
,
10
, pp.
1920
1928
.
30.
Phani
,
K.
,
1986
, “
Young’s Modulus-Porosity Relation in Gypsum Systems
,”
Am. Ceram. Soc. Bull.
,
65
, pp.
1584
1586
.
31.
Tazawa
,
E.
,
1998
, “
Effect of Self Stress on Flexural Strength of Gypsum-Polymer Composites
,”
,
7
, pp.
1
7
.
32.
Meille
,
S.
,
2001
, “
Etude du comportement mécanique du plâtre pris en relation avec sa microstructure (Study of the Mechanical Behaviour of Gypsum With Regard to its Microstructure)
,” Ph.D. thesis, INSA Lyon, Lyon, France (in French).
33.
Çolak
,
A.
,
2006
, “
Physical and Mechanical Properties of Polymer-Plaster Composites
,”
Mater. Lett.
,
60
(
16
), pp.
1977
1982
.
34.
Craciun
,
F.
,
Galassi
,
C.
,
Roncari
,
E.
,
Filippi
,
A.
, and
Guidarelli
,
G.
,
1998
, “
Electro-Elastic Properties of Porous Piezoelectric Ceramics Obtained by Tape Casting
,”
Ferroelectrics
,
205
, pp.
49
67
.
35.
Pabst
,
W.
,
Gregorová
,
E.
,
Tichá
,
G.
, and
Týnová
,
E.
,
2004
, “
Effective Elastic Properties of Alumina-Zirconia Composite Ceramics–Part 4. Tensile Modulus of Porous Alumina and Zirconia
,”
Ceramics-Silikáty
,
48
(
4
), pp.
165
174
.
36.
Reynaud
,
C.
,
Thévenot
,
F.
,
Chartier
,
T.
, and
Besson
,
J.-L.
,
2005
, “
Mechanical Properties and Mechanical Behaviour of SiC Dense-Porous Laminates
,”
J. Eur. Ceram. Soc.
,
25
, pp.
589
597
.
37.
Díaz
,
A.
, and
Hampshire
,
S.
,
2004
, “
Characterisation of Porous Silicon Nitride Materials Produced With Starch
,”
J. Eur. Ceram. Soc.
,
24
, pp.
413
419
.
38.
Hill
,
R.
,
1963
, “
Elastic Properties of Reinforced Solids: Some Theoretical Principles
,”
J. Mech. Phys. Solids
,
11
, pp.
357
362
.
39.
Hashin
,
Z.
,
1983
, “
Analysis of Composite Materials: A Survey
,”
J. Appl. Mech.
,
50
, pp.
481
505
.
40.
Suquet
,
P.
, ed.,
1997
,
Continuum Micromechanics
,
Springer
,
New York
.
41.
Katz
,
J.
, and
Ukraincik
,
K.
,
1971
, “
On the Anisotropic Elastic Properties of Hydroxyapatite
,”
J. Biomech.
,
4
, pp.
221
227
.
42.
Kohlhauser
,
C.
,
Hellmich
,
C.
,
Vitale-Brovarone
,
C.
,
Boccaccini
,
A.
,
Rota
,
A.
, and
Eberhardsteiner
,
J.
,
2009
, “
Ultrasonic Characterisation of Porous Biomaterials Across Different Frequencies
,”
Strain
,
45
, pp.
34
44
.
43.
Pabst
,
W.
,
Gregorová
,
E.
, and
Tichá
,
G.
,
2006
, “
Elasticity of Porous Ceramics—A Critical Study of Modulusporosity Relations
,”
J. Eur. Ceram. Soc.
,
26
, pp.
1085
1097
.
44.
Haglund
,
J.
, and
Hunter
,
O.
,
1973
, “
Elastic Properties of Polycrystalline Monoclinic Gd2O3
,”
J. Am. Ceram. Soc.
,
56
, pp.
327
330
.
45.
Akao
,
M.
,
Aoki
,
H.
, and
Kato
,
K.
,
1981
, “
Mechanical Properties of Sintered Hydroxyapatite for Prosthetic Applications
,”
J. Mater. Sci.
,
16
, pp.
809
812
.
46.
Shareef
,
M.
,
Messer
,
P.
, and
van Noort
,
R.
,
1993
, “
Fabrication, Characterization and Fracture Study of a Machinable Hydroxyapatite Ceramic
,”
Biomaterials
,
14
(
1
), pp.
69
75
.
47.
Coussy
,
O.
,
2004
,
Poromechanics
,
Wiley
,
Chichester, NJ
.
48.
,
T.
,
Hofstetter
,
K.
,
Hellmich
,
C.
, and
Eberhardsteiner
,
J.
,
2011
, “
The Poroelastic Role of Water in Cell Walls of the Hierarchical Composite ‘Softwood’
,”
Acta Mech.
,
217
, pp.
75
100
.
49.
Zaoui
,
A.
,
1997
, “
Structural Morphology and Constitutive Behavior of Microheterogeneous Materials
,”
Continuum Micromechanics
,
P.
Suquet
, ed.,
Springer
,
New York
, pp.
291
347
.
50.
Benveniste
,
Y.
,
1987
, “
A New Approach to the Application of Mori-Tanaka’s Theory in Composite Materials
,”
Mech. Mater.
,
6
, pp.
147
157
.
51.
Barenblatt
,
G.
,
1996
,
Scaling, Self-Similarity, and Intermediate Asymptotics
, 1st ed.,
Cambridge University Press
,
Cambridge, UK
.
52.
Buckingham
,
E.
,
1914
, “
On Physically Similar Systems. Illustrations of the Use of Dimensional Analysis
,”
Phys. Rev.
,
4
, pp.
345
376
.
53.
Salencon
,
J.
,
2001
,
Handbook of Continuum Mechanics—General Concepts. Thermoelasticity
,
Springer
,
Berlin
.