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Elastic instabilities in dry, mesoporous minerals and their relevance to geological applications

Published online by Cambridge University Press:  05 July 2018

E. K. H. Salje ,
J. Koppensteiner ,
W. Schranz  and
E. Fritsch
E. K. H. Salje*
Affiliation:
Department of Earth Sciences, Downing Street, Cambridge CB2 3EQ, UK
J. Koppensteiner
Affiliation:
Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Wien, Austria
W. Schranz
Affiliation:
Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Wien, Austria
E. Fritsch
Affiliation:
Institut de Minéralogie et de Physique des Milieux Condensés, Université Pierre et Marie Curie, Université Paris Diderot, CNRS, IRD and IPGP, 140 rue de Lourmel, 75015 Paris, France
*
*E-mail: es10002@esc.cam.ac.uk
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Abstract

The collapse of minerals and mineral assemblies under external stress is modelled using a master curve where the stress failure is related to the relative, effective elastic moduli which are in turn related to the porosity of the sample. While a universal description is known not to be possible, we argue that for most porous materials such as shales, silica, cement phases, hydroxyapatite, zircon and also carbonates in corals and agglomerates we can estimate the critical porosity ϕc at which small stresses will lead to the collapse of the sample. For several samples we find ϕc ~0.5 with an almost linear decay of the bulk moduli with porosity at ϕc <0.5. The second scenario involves the persistence of elasticity for porosities until almost 1 whereby the bulk modulus decreases following a power law κ ~ (1–ϕm, m>2, between ϕ = 0.5 and ϕ = 1.

Keywords

Elasticityporous materialsuniversalitycalibration of elastic moduli
Type
Research Article
Information
Mineralogical Magazine , Volume 74 , Issue 2 , April 2010 , pp. 341 - 350
Copyright
Copyright © The Mineralogical Society of Great Britain and Ireland 2010

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References

Aifantis, K.E. and Willis, J.R. (2005) The role of interfaces in enhancing the yield strength of composites and polycrystals. Journal of Mechanics and Physics of Solids, 53, 1047–1070.CrossRefGoogle Scholar
Akao, M., Aoki, H. and Kato, K. (1981) Mechanical properties of sintered hydroxyapatite for prosthetic applications. Journal of Materials Science, 16, 809–812.CrossRefGoogle Scholar
Arita, I.H., Wilkinson, D.S., Mondragon, M.A. and Castaño, V.M. (1995) Chemistry and sintering behaviour of thin hydroxyapatite ceramics with controlled porosity. Biomaterials, 16, 403–408.CrossRefGoogle ScholarPubMed
Atkinson, A.J., Carpenter, M.A. and Salje, E.K.H. (1999) Hard mode infrared spectroscopy of plagioclase feldspars. European Journal of Mineralogy, 11, 7–21.CrossRefGoogle Scholar
Barton, T.J., Bull, L.M., Klemperer, W.G., Loy, D.A., McEnaney, B., Misono, M., Monson, P.A., Pez, G., Scherer, G.W., Vartuli, J.C. and Yaghi, O.M. (1999) Tailored porous materials. Chemistry of Materials, 11, 2633–2656.CrossRefGoogle Scholar
Bismayer, U. and Salje, E. (1981) Ferroelastic phases in Pb3(PO4)2–Pb3(AsO4)2 – X-ray and optical experiments. Acta Crystallographica A, 37, 145—153.Google Scholar
Bobko, C. and Ulm, F.J. (2007) The nano-mechanical morphology of shale. Mechanics of Materials, 40, 318–337.CrossRefGoogle Scholar
Boker, A., He, J., Emrick, T. and Russell, T.P. (2007) Self-assembly of nanoparticles at interfaces. Soft Matter, 3, 1231–1248.CrossRefGoogle ScholarPubMed
Carpenter, M.A. and Salje, E.K.H. (1998) Elastic anomalies in minerals due to structural phase transitions. European Jounal of Mineralogy, 10, 693–812.CrossRefGoogle Scholar
Charrire, E., Terrazzoni, S. and Pittet, C. (2001) Mechanical characterization of brushite and hydroxyapatite cements. Biomaterials, 21, 2937–2945.CrossRefGoogle Scholar
Constantinides, G. and Ulm, F-J. (2007) The nanogranular nature of C–S–H. Journal of Mechanics and Physics of Solids, 55, 64–90.CrossRefGoogle Scholar
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, 49–67.CrossRefGoogle Scholar
DeWitt, G., van Dijk, H., Hattu, N. and Prijs, K. (1981) Preparation, microstructure and mechanical properties of dense polycrystalline hydroxyapatite. Journal of Material Sciences, 16, 1592–1958.Google Scholar
Dmitriev, V.P., Rochal, S.B. and Gufan, Y.M. (1988) Definition of a transcendental order parameter for reconstructive phase transitions. Physical Review Letters, 60, 1958–1961.CrossRefGoogle ScholarPubMed
Dmitriev, V.P., Toledano, P., Torgashev, V.I. and Salje, E.K.H. (1998) Theory of reconstructive phase transitions between SiO2 polymorphs. Physical Review B, 58, 11911–11921.CrossRefGoogle Scholar
Eschricht, N., Hoinkins, E., Mädler, F. and Schubert-Bischoff, P. (2002) Reconstruction of the mesoporous silica glass Gelsil50. Surface Science and Catalysis, 144, 355–262.CrossRefGoogle Scholar
Elmer, T.H. (1992) ASM Engineered Materials Handbook 4: Ceramics and Glasses. Materials Park, Ohio, US.Google Scholar
Fritsch, A., Dormieux, L., Hellmich, C. and Sanahuja, J. (2007) Micromechanics of crystal interfaces in polycrystalline solid phases of porous media: fundamentals and application to strength of hydroxyapatite biomaterials. Journal of Material Science, 42, 8824–8837.CrossRefGoogle Scholar
Fritsch, A., Dormieux, L. and Hellmich, C. (2009) Mechanical behavior of hydroxyapatite biomaterials: An experimentally validated micromechanical model for elasticity and strength. Journal of Biomedical Materials Research A, 88, 149–161.CrossRefGoogle ScholarPubMed
Gatta, G.D. (2008) Does porous mean soft? On the elastic behaviour and structural evolution of zeolites underpressure. Zeitschrift fur Kristallographie, 223, 160–170.Google Scholar
Gibson, L.J. and Ashby, M.F. (1997) Cellular solids: Structure and and properties, second ed. Cambridge Univ. Press, Cambridge, UK.CrossRefGoogle Scholar
Gilmore, R.S. and Katz, J.L. (1982) Elastic properties of apatites. Journal of Material Sciences, 17, 1131–1141.CrossRefGoogle Scholar
Gross, J., Schlief, T. and Fricke, J. (1993) Ultrasonic evaluation of elastic properties of silica aerogels. Materials Science and Engineering A, 168, 235–238.Google Scholar
Harrison, R.J., Redfern, S.A.T. and Salje, E.K.H. (2004) Dynamical excitation and anelastic relaxation of ferroelastic domain walls in LaAlO3 . Physical Review B, 69, 144101.CrossRefGoogle Scholar
Herrmann, A., Schwarzer, N. and Richter, F. (2006) Determination of Young's modulus and yield stress of porous low-k materials by nanoindentation. Surface Coatings Technology, 201, 4305–4310.CrossRefGoogle Scholar
Hill, R (1972) On constitutive macro-variables for heterogeneous solids at finite strain. Proceedings of the Royal Society A, 326, 131–147.Google Scholar
Hornby, B.E., Schwartz, L.M. and Hudson, J.A. (1994) Anisotropic effective-medium modeling of the elastic properties of shales. Geophysics, 59, 1570–1583.CrossRefGoogle Scholar
Jakobson, M., Hudson, J.A., Minshull, T.A. and Singh, S.C. (2000) Elastic properties of hydrate-bearing sediments using effective medium theory. Journal of Geophysical Research, 105, 561–577.CrossRefGoogle Scholar
Kityk, A.V., Schranz, W., Sondergeld, P. and Salje, E.K.H. (2000) Low-frequency superelasticity and nonlinear elastic behavior of SrTiO3 crystals. Physical Review B, 61, 946–956.CrossRefGoogle Scholar
Knackstedt, M.A., Arns, C.H., Senden, T.J. and Gross, K. (2006) Structure and properties of clinical coralline implants measured via 3D imaging and analysis. Biomaterials, 27, 2776–2786.CrossRefGoogle ScholarPubMed
Koppensteiner, J., Schranz, W. and Puica, M.R. (2008) Confinement effects on glass forming liquids probed by dynamic mechanical analysis. Physical Review B, 78, 054203.CrossRefGoogle Scholar
Lee, W.T., Salje, E.K.H. and Dove, M.T. (1999) Effect of surface relaxations on the equilibrium growth morphology of crystals: platelet formation. Journal of Physics, Condensed Matter, 11, 7385–7410.CrossRefGoogle Scholar
Lee, W.T., Dove, M.T. and Salje, E.K.H. (2000) Surface relaxations in hydroxyapatite. Journal of Physics: Condensed Matter, 12, 9829–9841.Google Scholar
Levitz, P., Ehert, G., Sinha, S.K. and Drake, J.M. (1991) Porous Vycor glass – the microstructure as probed by electron-microscopy, dielectric energy-transfer, small-angle scattering and molecular absorption. Journal of Chemical Physics, 95, 6151–6161.CrossRefGoogle Scholar
Liu, D-M. (1998) Preparation and characterisation of porous hydroxyapatite biocheramic via a slip-casting route. Ceramics International, 24, 441–446.CrossRefGoogle Scholar
Ma, H-S., Prévost, J-H., Jullien, R. and Scherer, G.W. (2002) Elasticity of DLCA model gels with loops. International Journal of Solids and Structures, 39, 4606–4614.CrossRefGoogle Scholar
Michel, J.C., Lopez-Pamies, O., Castaneda, P.P. and Triantafyllidis, N. (2007) Microscopic and macroscopic instabilities in finitely strained porous elastomers. Journal of Mechanics and Physics of Solids, 55, 900–938.CrossRefGoogle Scholar
Migliori, A. and Maynard, J.D. (2005) Implementation of a modern resonant ultrasound spectroscopy system for the measurement of the elastic moduli of small solid specimens. Review of Scientific Instruments, 76, 121301.CrossRefGoogle Scholar
Milton, G.W. (2002) The Theory of Composites. Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
Miyoshi, H., Yamada, K., Kohmura, K., Fujii, N., Matsuo, H., Tanaka, H., Oku, Y., Seino, Y., Hata, N. and Kikkawa, T. (2005) Theoretical investigation of dielectric constant and elastic modulus of three-dimensional isotropic porous silica films with cubic and disordered pore arrangements. Japanese Journal of Applied Physics, 44, 5982–5986.CrossRefGoogle Scholar
Muralidharan, V. and Hui, C-Y. (2004) Stability of nanoporous materials. Macromolecular Rapid Communications, 25, 1487–1490.CrossRefGoogle Scholar
Murray, C., Flannery, C. and Streiter, I. (2002) Comparison of techniques to characterise the density, porosity and elastic modulus of porous low-k SiO2 xerogel films. Microelectronic Engineering, 60, 133–141.CrossRefGoogle Scholar
Nogues, J-L.R. and Moreshead, W. (1990) Porous gel silica, a matrix for optically-active components. Journal of Non-Crystalline Solids, 121, 136–142.CrossRefGoogle Scholar
Nur, A., Mavko, G., Dvorkin, J. and Galmudi, D. (1998) Critical porosity: A key to relating physical properties to porosity in rocks. The Leading Edge, 357–362.CrossRefGoogle Scholar
Pekala, R.W., Hrubesh, L.W., Tillotson, T.M., Alviso, C.T., Poco, J.F. and Lemay, J.D. (1991) A comparison of mechanical properties and scaling law relationships for silica aerogels and their organic counterparts. Materials Research Society Symposium Proceedings, 207, 197–200.CrossRefGoogle Scholar
Pertsev, N.A. and Salje, E.K.H. (2000) Thermodynamics of pseudoproper and improper ferroelastic inclusions and polycrystals: effect of elastic clamping on phase transitions. Physical Review B, 61, 902–908.CrossRefGoogle Scholar
Puchegger, S., Brandhuber, D., Hsing, N. and Peterlik, H. (2006) Changing poisson's ratio of mesoporous silica monoliths with high temperature treatment. Journal of Non-Crystalline Solids, 352, 5251–5256.CrossRefGoogle Scholar
Puchegger, S., Dose, F., Loidl, D., Kromp, K., Janssen, R., Brandhuber, D., Hsing, N. and Peterlik, H. (2007) The dependence of the elastic moduli of reaction bonded alumina on porosity. Journal of the European Ceramic Society, 27, 35–39.CrossRefGoogle Scholar
Roberts, A.P. and Garboczi, E.J. (2000) Elastic properties of model porous ceramics. American Ceramic Society, 83, 3041–3048.CrossRefGoogle Scholar
Roberts, A.P. and Garboczi, E.J. (2002) Computation of the linear elastic properties of random porous materials with a wide variety of microstructure. Proceedings of the Royal Society of London A, 458, 1033–1054.CrossRefGoogle Scholar
Salje, E.K.H. (1992) Application of Landau theory for the analysis of phase-transitions in minerals. Physics Reports, 215, 49–99.CrossRefGoogle Scholar
Salje, E.K.H. (1993) Phase Transitions in Ferroelastic and Co-Elastic Crystals. Cambridge University Press, Cambridge, UK.Google Scholar
Salje, E.K.H. (2006) Elastic softening of zircon by radiation damage. Applied Physics Letters, 89, 131902.CrossRefGoogle Scholar
Salje, E.K.H. (2007) An empirical scaling model for averaging elastic properties including interfacial effects. American Mineralogist, 92, 429–432.CrossRefGoogle Scholar
Salje, E.K.H. (2008 a) A pre-martensitic elastic anomaly in nanomaterials: elasticity of surface and interface layers. Journal of Physics: Condensed Matter, 20, 485003.Google Scholar
Salje, E.K.H. (2008 b) (An)elastic softening from static grain boundaries and possible effects on seismic wave propagation. Physics and Chemistry of Minerals, 35, 321–330.CrossRefGoogle Scholar
Salje, E. and Wruck, B. (1991) Order parameter saturation at low-temperatures – numerical results for displacive and o/d systems. Ferroelectrics, 124, 185–188.CrossRefGoogle Scholar
Salje, E.K.H. and Zhang, H.L. (2009) Domain boundary pinning and elastic softening in KMnF3 and KMn1-xCaxF3 . Journal of Physics, Condensed Matter, 21, 035901.CrossRefGoogle Scholar
Salje, E., Palosz, B. and Wruck, B. (1987) In situ observation of the polytypoic phase-transition 2H-12R in PbI2 – investigation of the thermodynamic, structural and dielectric properties. Journal of Physics C: Solid State Physics, 20, 4077–4096.CrossRefGoogle Scholar
Salje, E.K.H., Wruck, B. and Thomas, H. (1991) Order parameter saturation and low-temperature extension of Landau theory. Zeitschrift fr Physik B – Condensed Matter, 82, 399–404.CrossRefGoogle Scholar
Salje, E.K.H., Ridgwell, A., Guttler, B., Wruck, B., Dove, M.T. and Dolino, G. (1993 a) on the displacive character of the phase transition in quartz – a hard-mode spectroscopy study. Journal of Physics: Condensed Matter, 4, 571–577.Google Scholar
Salje, E.K.H., Schmidt, C. and Bismayer, U. (19936) Structural phase-transition in titanite, CaTiSiO5 – a Raman-spectroscopic study. Physics and Chemistry of Minerals, 19, 502–509.Google Scholar
Salje, E.K.H., Hayward, S.A. and Lee, W.T. (2005) Ferroelastic phase transitions: structure and microstructure. Acta Crystallographica, 61, 3–18.CrossRefGoogle ScholarPubMed
Salje, E.K.H., Zhang, H., Planes, A. and Moya, X. (2008) Martensitic transformation B2-R in Ni-Ti-Fe: experimental determination of the Landau potential and quantum saturation of the order parameter. Journal of Physics, Condensed Matter, 20, 275216.CrossRefGoogle ScholarPubMed
Scherer, G.W. (1998) Crystallization in pores. Cement and Concrete Research, 29, 1347–1358.CrossRefGoogle Scholar
Scherer, G.W., Smith, D.M., Qiu, X. and Anderson, J.M. (1995) Compression of aerosols. Journal of Non-Crystalline Solids, 186, 316–320.CrossRefGoogle Scholar
Schranz, W., Trster, A., Kityk, A.V. and Salje, E.K.H. (2003) Ultralow-frequency elastic response in KMn1-xCaxF3 . Europhysics Letters, 62, 512–518.CrossRefGoogle Scholar
Sonmez, H., Tuncay, E. and Gokceoglu, C. (2004) Models to predict the uniaxial compressive strength and the modulus of elasticity for Ankara Agglomerate. International Journal of Rock Mechanics and Mining Sciences, 41, 717–729.CrossRefGoogle Scholar
Speer, D. and Salje, E. (1986) Phase transitions in langbeinites. 1 crystal chemistry and structures of K-double sulfates of the langbeinite type M2 ++K2(SO4)3, M++ = Mg, Ni, Co, Zn, Ca. Physics and Chemistry of Minerals, 13, 17–24.CrossRefGoogle Scholar
Takada, S., Hata, N., Seino, Y., Fujii, N. and Kikkawa, T. (2006) Dependences of Young's modulus of porous silica low dielectric constant films on skeletal structure and porosity. Journal of Applied Physics, 100, 123–512.CrossRefGoogle Scholar
Toledano, P., Fejer, M.M. and Auld, B.A. (1983) Non-linear elasticity in proper ferroelastics. Physical Review B, 27, 5717–5746.CrossRefGoogle Scholar
Torquato, S., Gibiansky, L.V., Silva, M.J. and Gibson, L.J. (1998) Effective mechanical and transport properties of cellular solids. International Journal of Mechanical Sciences, 40, 71–82.CrossRefGoogle Scholar
Traore, D., Beauvais, A., Chabaux, F., Peiffert, C., Parisot, J.C., Ambrosi, J.P. and Colin, F.D. (2008) Chemical and physical transfers in an ultramatic rock weathering profile: Part 1. Supergene dissolution of Pt-bearing chromite. American Mineralogist, 93, 22–30.CrossRefGoogle Scholar
Ulm, F-J., Vandamme, M., Bobko, C., Ortega, J.A., Tai, K. and Ortiz, C. (2007) Statistical indentation techniques for hydrated nanocomposites: Concrete, bone, and shale. Journal of the American Ceramic Society, 90, 2677–2692.CrossRefGoogle Scholar
Willis, J.R. (1977) Bounds and self-consistent estimates for the overall properties of anisotropic composites. Journal of Mechanics and Physical Solids, 25, 185–202.CrossRefGoogle Scholar
Woignier, T., Phalippou, J. and Vacher, R. (1989) Parameters affecting elastic properties in silica aerogels. Journal of Materials Research, 4, 688–692.CrossRefGoogle Scholar