## 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 $\u2113$, $\u2113>>d$, $d$ standing for the characteristic length of inhomogeneities within the RVE, and $\u2113\u226aL$, $L$ 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.

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.

$Es$ | $\nu s$ | ||
---|---|---|---|

(GPa) | (1) | Source^{a} | |

Hydroxyapatite | 114 | 0.27 | [41] |

Glass-ceramics (CEL2) | 85.3 | 0.25 | [42] |

Gypsum | 40 | 0.34 | [13] |

Piezoelectric ceramics | 667 | — | [34] |

Alumina | 400 | 0.23 | [43] |

Zirconia | 210 | 0.31 | [43] |

Silicon carbide | 420 | 0.17 | [43] |

Silicon nitride | 320 | 0.27 | [43] |

Gadolinium(III)oxide ($Gd2O3$) | 410 | — | [44] |

### Averaging Homogenization Upscaling.

$Chom$ 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).

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 $1$.

Both the poroelastic and the porobrittle relations Eqs. (11), (14), and (18) depend on knowledge on the concentration and influence tensors $Apor$, $As(\u03d1,\varphi )$, $dpor$, and $ds(\u03d1,\varphi )$. 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).

In Eqs. (19)–(22), $I$ is the fourth-order unity tensor with
components $Iijkl=1/2(\delta ik\delta jl+\delta il\delta kj)$, $\delta ij$ (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.

$As(\u03d1,\varphi )$ 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 $Chom$.

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

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

Power functions . | |||
---|---|---|---|

Quantity . | Morphology . | $BE$ . | $CE$ . |

$Ehom/Es$ | Needles | 1.040 | 2.596 |

$Ehom/Es$ | Disks | 0.9867 | 2.053 |

Power functions . | |||
---|---|---|---|

Quantity . | Morphology . | $BE$ . | $CE$ . |

$Ehom/Es$ | Needles | 1.040 | 2.596 |

$Ehom/Es$ | Disks | 0.9867 | 2.053 |

Fourth-order polynomial functions . | ||||||
---|---|---|---|---|---|---|

Quantity . | Morphology . | $A\xafE$ . | $B\xafE$ . | $C\xafE$ . | $D\xafE$ . | $E\xafE$ . |

$Ehom/Es$ | Needles | –2.189 | 4.633 | –1.745 | 0.302 | 0 |

$Ehom/Es$ | Disks | –0.511 | 1.5594 | –0.370 | 0.330 | 0 |

Fourth-order polynomial functions . | ||||||
---|---|---|---|---|---|---|

Quantity . | Morphology . | $A\xafE$ . | $B\xafE$ . | $C\xafE$ . | $D\xafE$ . | $E\xafE$ . |

$Ehom/Es$ | Needles | –2.189 | 4.633 | –1.745 | 0.302 | 0 |

$Ehom/Es$ | Disks | –0.511 | 1.5594 | –0.370 | 0.330 | 0 |

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.

Quantity . | Morphology . | $max|e|$ . | $mean|e|$ . | $R2$ . |
---|---|---|---|---|

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 |

Quantity . | Morphology . | $max|e|$ . | $mean|e|$ . | $R2$ . |
---|---|---|---|---|

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 |

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

. | Needles . | Disks . | ||
---|---|---|---|---|

. | $a*$ . | $b*$ . | $a*$ . | $b*$ . |

$A\xaf\nu $ | –2.9425 | 0.5113 | –1.0521 | 0.2197 |

$B\xaf\nu $ | 5.0536 | –0.7564 | 2.2684 | –0.4645 |

$C\xaf\nu $ | –1.4556 | 0.0744 | –0.8121 | 0.1662 |

$D\xaf\nu $ | 0.3410 | –0.0586 | 0.3602 | –0.0718 |

$E\xaf\nu $ | –0.0033 | 0.2313 | 0.2394 | 0.1496 |

. | Needles . | Disks . | ||
---|---|---|---|---|

. | $a*$ . | $b*$ . | $a*$ . | $b*$ . |

$A\xaf\nu $ | –2.9425 | 0.5113 | –1.0521 | 0.2197 |

$B\xaf\nu $ | 5.0536 | –0.7564 | 2.2684 | –0.4645 |

$C\xaf\nu $ | –1.4556 | 0.0744 | –0.8121 | 0.1662 |

$D\xaf\nu $ | 0.3410 | –0.0586 | 0.3602 | –0.0718 |

$E\xaf\nu $ | –0.0033 | 0.2313 | 0.2394 | 0.1496 |

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

Quantity . | Morphology . | $max|e|$ . | $mean|e|$ . | $R2$ . |
---|---|---|---|---|

Power functions | ||||

$\nu hom(\nu s)$ | Needles | 0.035 | 0.009 | 0.922 |

$\nu hom(\nu s)$ | Disks | 0.021 | 0.007 | 0.944 |

Fourth-order polynomial functions | ||||

$\nu hom(\nu s)$ | Needles | 0.005 | 0.002 | 0.996 |

$\nu hom(\nu s)$ | Disks | 0.004 | 0.001 | 0.987 |

Quantity . | Morphology . | $max|e|$ . | $mean|e|$ . | $R2$ . |
---|---|---|---|---|

Power functions | ||||

$\nu hom(\nu s)$ | Needles | 0.035 | 0.009 | 0.922 |

$\nu hom(\nu s)$ | Disks | 0.021 | 0.007 | 0.944 |

Fourth-order polynomial functions | ||||

$\nu hom(\nu s)$ | Needles | 0.005 | 0.002 | 0.996 |

$\nu hom(\nu s)$ | Disks | 0.004 | 0.001 | 0.987 |

### Biot Coefficient and Biot Modulus.

with $q1=A\xafb,B\xafb,C\xafb,D\xafb$, and $E\xafb$; see Table 7 for values of $a\u2227$, $b\u2227$, $c\u2227$, $d\u2227$, and $e\u2227$. 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.

Needles . | |||||
---|---|---|---|---|---|

. | $a\u2227$ . | $b\u2227$ . | $c\u2227$ . | $d\u2227$ . | $e\u2227$ . |

$A\xafb$ | 1067.3 | –962.3 | 226.3 | –15.26 | 2.694 |

$B\xafb$ | –1396.2 | 1326.7 | –317.5 | 18.01 | –4.881 |

$C\xafb$ | 592.9 | –590.7 | 145.8 | –6.149 | 1.584 |

$D\xafb$ | –87.55 | 91.41 | –23.22 | 1.161 | –0.387 |

$E\xafb$ | 3.000 | –3.283 | 0.859 | –0.026 | 1.007 |

Needles . | |||||
---|---|---|---|---|---|

. | $a\u2227$ . | $b\u2227$ . | $c\u2227$ . | $d\u2227$ . | $e\u2227$ . |

$A\xafb$ | 1067.3 | –962.3 | 226.3 | –15.26 | 2.694 |

$B\xafb$ | –1396.2 | 1326.7 | –317.5 | 18.01 | –4.881 |

$C\xafb$ | 592.9 | –590.7 | 145.8 | –6.149 | 1.584 |

$D\xafb$ | –87.55 | 91.41 | –23.22 | 1.161 | –0.387 |

$E\xafb$ | 3.000 | –3.283 | 0.859 | –0.026 | 1.007 |

Disks . | |||||
---|---|---|---|---|---|

. | $a\u2227$ . | $b\u2227$ . | $c\u2227$ . | $d\u2227$ . | $e\u2227$ . |

$A\xafb$ | 1037.2 | –916.4 | 218.5 | –15.48 | 0.917 |

$B\xafb$ | –1391.1 | 1275.6 | –309.6 | 20.08 | –1.818 |

$C\xafb$ | 623.7 | –585.3 | 145.4 | –8.732 | 0.378 |

$D\xafb$ | –95.82 | 93.11 | –23.24 | 1.962 | –0.479 |

$E\xafb$ | 3.497 | –3.454 | 0.882 | –0.052 | 1.004 |

Disks . | |||||
---|---|---|---|---|---|

. | $a\u2227$ . | $b\u2227$ . | $c\u2227$ . | $d\u2227$ . | $e\u2227$ . |

$A\xafb$ | 1037.2 | –916.4 | 218.5 | –15.48 | 0.917 |

$B\xafb$ | –1391.1 | 1275.6 | –309.6 | 20.08 | –1.818 |

$C\xafb$ | 623.7 | –585.3 | 145.4 | –8.732 | 0.378 |

$D\xafb$ | –95.82 | 93.11 | –23.24 | 1.962 | –0.479 |

$E\xafb$ | 3.497 | –3.454 | 0.882 | –0.052 | 1.004 |

Quantity . | Morphology . | $max|e|$ . | $mean|e|$ . | $R2$ . |
---|---|---|---|---|

$b11hom$ | 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 |

Quantity . | Morphology . | $max|e|$ . | $mean|e|$ . | $R2$ . |
---|---|---|---|---|

$b11hom$ | 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 |

with $q2=A\xafN,B\xafN,C\xafN,D\xafN$, and $E\xafN$; 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.

Needles | |||||
---|---|---|---|---|---|

$a$ | $b$ | $c$ | $d$ | $e$ | |

$A\xafN$ | 741.4 | –527.4 | 85.56 | –19.59 | 8.012 |

$B\xafN$ | –1342.7 | 987.9 | –154.0 | 26.93 | –14.55 |

$C\xafN$ | 717.7 | –550.9 | 84.35 | –3.556 | 4.707 |

$D\xafN$ | –128.6 | 101.5 | –18.27 | –3.868 | 1.845 |

$E\xafN$ | 5.221 | –4.219 | 0.705 | 0.010 | 0.020 |

Needles | |||||
---|---|---|---|---|---|

$a$ | $b$ | $c$ | $d$ | $e$ | |

$A\xafN$ | 741.4 | –527.4 | 85.56 | –19.59 | 8.012 |

$B\xafN$ | –1342.7 | 987.9 | –154.0 | 26.93 | –14.55 |

$C\xafN$ | 717.7 | –550.9 | 84.35 | –3.556 | 4.707 |

$D\xafN$ | –128.6 | 101.5 | –18.27 | –3.868 | 1.845 |

$E\xafN$ | 5.221 | –4.219 | 0.705 | 0.010 | 0.020 |

Disks | |||||
---|---|---|---|---|---|

$a$ | $b$ | $c$ | $d$ | $e$ | |

$A\xafN$ | 612.9 | –452.5 | 84.23 | –10.92 | 2.683 |

$B\xafN$ | –1074.2 | 819.5 | –150.6 | 15.38 | –5.359 |

$C\xafN$ | 552.5 | –437.6 | 82.74 | –3.189 | 1.089 |

$D\xafN$ | –102.1 | 80.33 | –18.56 | –1.173 | 1.569 |

$E\xafN$ | 4.013 | –3.250 | 0.645 | –0.035 | 0.012 |

Disks | |||||
---|---|---|---|---|---|

$a$ | $b$ | $c$ | $d$ | $e$ | |

$A\xafN$ | 612.9 | –452.5 | 84.23 | –10.92 | 2.683 |

$B\xafN$ | –1074.2 | 819.5 | –150.6 | 15.38 | –5.359 |

$C\xafN$ | 552.5 | –437.6 | 82.74 | –3.189 | 1.089 |

$D\xafN$ | –102.1 | 80.33 | –18.56 | –1.173 | 1.569 |

$E\xafN$ | 4.013 | –3.250 | 0.645 | –0.035 | 0.012 |

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

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

Power functions | |||
---|---|---|---|

Quantity | Morphology | $B\Sigma $ | $C\Sigma $ |

$\Sigma ult,t/\sigma sult,t$ | Needles | 1.016 | 2.381 |

$\Sigma ult,t/\sigma sult,t$ | Disks | 0.977 | 1.675 |

Power functions | |||
---|---|---|---|

Quantity | Morphology | $B\Sigma $ | $C\Sigma $ |

$\Sigma ult,t/\sigma sult,t$ | Needles | 1.016 | 2.381 |

$\Sigma ult,t/\sigma sult,t$ | Disks | 0.977 | 1.675 |

Fourth-order polynomial functions | ||||||
---|---|---|---|---|---|---|

Quantity | Morphology | $A\xaf\Sigma $ | $B\xaf\Sigma $ | $C\xaf\Sigma $ | $D\xaf\Sigma $ | $E\xaf\Sigma $ |

$\Sigma ult,t/\sigma sult,t$ | Needles | –1.7705 | 3.8896 | –1.4703 | 0.3524 | 0 |

$\Sigma ult,t/\sigma sult,t$ | Disks | 0.0220 | 0.6213 | –0.1855 | 0.5440 | 0 |

Fourth-order polynomial functions | ||||||
---|---|---|---|---|---|---|

Quantity | Morphology | $A\xaf\Sigma $ | $B\xaf\Sigma $ | $C\xaf\Sigma $ | $D\xaf\Sigma $ | $E\xaf\Sigma $ |

$\Sigma ult,t/\sigma sult,t$ | Needles | –1.7705 | 3.8896 | –1.4703 | 0.3524 | 0 |

$\Sigma ult,t/\sigma sult,t$ | Disks | 0.0220 | 0.6213 | –0.1855 | 0.5440 | 0 |

Quantity | Morphology | $max|e|$ | $mean|e|$ | $R2$ |
---|---|---|---|---|

Power functions | ||||

$\Sigma ult,t/\sigma sult,t$ | Needles | 0.018 | 0.010 | 0.9988 |

$\Sigma ult,t/\sigma sult,t$ | Disks | 0.039 | 0.027 | 0.9963 |

Fourth-order polynomial functions | ||||

$\Sigma ult,t/\sigma sult,t$ | Needles | 0.006 | 0.003 | 0.9999 |

$\Sigma ult,t/\sigma sult,t$ | Disks | 0.002 | 0.001 | 0.9999 |

Quantity | Morphology | $max|e|$ | $mean|e|$ | $R2$ |
---|---|---|---|---|

Power functions | ||||

$\Sigma ult,t/\sigma sult,t$ | Needles | 0.018 | 0.010 | 0.9988 |

$\Sigma ult,t/\sigma sult,t$ | Disks | 0.039 | 0.027 | 0.9963 |

Fourth-order polynomial functions | ||||

$\Sigma ult,t/\sigma sult,t$ | Needles | 0.006 | 0.003 | 0.9999 |

$\Sigma ult,t/\sigma sult,t$ | Disks | 0.002 | 0.001 | 0.9999 |

## 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 $P$

Following standard tensor calculus [53], the tensor components of $Pcylhom(\u03d1,\varphi )=Scylesh(\u03d1,\varphi ):Chom,-1$ and $Pdiskhom(\u03d1,\varphi )=Sdiskesh(\u03d1,\varphi ):Chom,-1$, being related to differently oriented inclusions, are transformed into one, single base frame ($e\xaf1$, $e\xaf2$, $e\xaf3$), in order to evaluate the integrals in Eqs. (23)–(30).