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Research Articles

Micromechanics-Derived Scaling Relations for Poroelasticity and Strength of Brittle Porous Polycrystals

[+] Author and Article Information
Andreas Fritsch

Postdoctoral Associate
e-mail: Andreas.Fritsch@tuwien.ac.at

Christian Hellmich

Professor
e-mail: Christian.Hellmich@tuwien.ac.at
Institute for Mechanics of Materials and Structures,
Vienna University of Technology (TU Wien),
1040 Vienna, Austria

Philippe Young

Associate Professor
School of Engineering,
Computer Sciences and Mathematics,
University of Exeter,
Exeter EX4 4QF, UK;
Simpleware Ltd.,
Exeter EX4 3PL,
UK
e-mail: philippe.g.young@exeter.ac.uk

1Corresponding author.

Manuscript received August 23, 2011; final manuscript received March 14, 2012; accepted manuscript posted October 25, 2012; published online February 4, 2013. Assoc. Editor: Younane Aboulsleiman.

J. Appl. Mech 80(2), 020905 (Feb 04, 2013) (12 pages) Paper No: JAM-11-1300; doi: 10.1115/1.4007922 History: Received August 23, 2011; Revised March 14, 2012

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.

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Figures

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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

Grahic Jump Location
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].

Grahic Jump Location
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].

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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].

Grahic Jump Location
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)

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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

Grahic Jump Location
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)

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