Critical Assessment 18: elastic and thermal properties of porous materials – rigorous bounds and cross-property relations

References

• L. J. Gibson and M. F. Ashby: ‘The mechanics of foams: basic results’, 2nd edn, Chap. 5, ‘Cellular solids – structure and properties’, 175–234; 1997, Cambridge, Cambridge University Press.
   Google Scholar
• L. J. Gibson and M. F. Ashby: ‘Thermal, electrical and acoustic properties of foams’, 2nd edn, Chap. 7, ‘Cellular solids – structure and properties’, 283–308; 1997, Cambridge, Cambridge University Press.
   Google Scholar
• ‘Cellular ceramics – structure, manufacturing, properties and applications’, (ed. M. Scheffler and P. Colombo), 1–645; 2005, Weinheim, Wiley-VCH.
   Google Scholar
• ‘Cellular and porous materials – thermal properties simulation and prediction’, (ed. A. Öchsner et al.), 1–422; 2008, Weinheim, Wiley-VCH.
   Google Scholar
• W. Pabst: ‘The linear theory of thermoelasticity from the viewpoint of rational thermomechanics’, Ceram. Silik., 2005, 49, 254–263.
   Web of Science ®Google Scholar
• S. Torquato: ‘Random heterogeneous materials – microstructure and macroscopic properties’, 403–646; 2002, New York, Springer.
   Google Scholar
• K. Markov: ‘Micromechanics of heterogeneous media’, in ‘Heterogeneous media – micromechanics modeling and simulations’, (ed. K. Markov and L. Preziosi), 1–162; 2000, Boston, Birkhäuser.
   Google Scholar
• G. W. Milton: ‘The theory of composites’, 75–498; 2002, Cambridge, Cambridge University Press.
   Google Scholar
• W. Pabst and E. Gregorová: ‘Effective thermal and thermoelastic moduli of alumina, zirconia and alumina-zirconia composite ceramics’ in ‘New developments in materials science research’ (ed. B. M. Caruta), 77–137; 2007, New York, Nova Science Publishers.
   Google Scholar
• R. W. Rice: ‘Porosity and microcrack dependence of elastic properties at low to moderate temperatures’, Chap. 3, ‘Porosity of ceramics’, 100–167; 1998, New York, Marcel Dekker.
   Google Scholar
• R. W. Rice: ‘The porosity and microcrack dependence of thermal and electrical conductivities and other nonmechanical properties’, Chap. 7, ‘Porosity of ceramics’, 315–374; 1998, New York, Marcel Dekker.
   Google Scholar
• M. Kaviany: ‘Conduction heat transfer’ in ‘Principles of heat transfer in porous media’, 119–156; 1995, New York, Springer.
   Google Scholar
• J. B. Wachtman: ‘Mechanical properties of porous ceramics’ in ‘Mechanical properties of ceramics’, 409–412; 1996, New York, Wiley-Interscience.
   Google Scholar
• L. Gong, Y. Wang, X. Cheng, R. Zhang and H. Zhang: ‘Thermal conductivity of highly porous mullite materials’, Int. J. Heat Mass Transfer, 2013, 67, 253–259. doi: 10.1016/j.ijheatmasstransfer.2013.08.008
   Web of Science ®Google Scholar
• J. A. Choren, S. M. Heinrich and M. B. Silver-Thorn: ‘Young's modulus and volume porosity relationships for additive manufacturing applications’, J. Mater. Sci., 2013, 48, 5103–5112. doi: 10.1007/s10853-013-7237-5
   Web of Science ®Google Scholar
• W. Pabst, E. Gregorová and G. Tichá: ‘Effective properties of suspensions, composites and porous materials’, J. Eur. Ceram. Soc., 2007, 27, 479–482. doi: 10.1016/j.jeurceramsoc.2006.04.169
   Web of Science ®Google Scholar
• R. W. Rice: ‘Evaluation and extension of physical property–porosity models based on minimum solid area’, J. Mater. Sci., 1996, 31, 102–118. doi: 10.1007/BF00355133
   Web of Science ®Google Scholar
• R. W. Rice: ‘Comparison of physical property–porosity behaviour with minimum solid area models’, J. Mater. Sci., 1996, 31, 1509–1528. doi: 10.1007/BF00357860
   Web of Science ®Google Scholar
• A. K. Mukhopadhyay and K. K. Phani: ‘Young's modulus–porosity relations: an analysis based on a minimum contact area model’, J. Mater. Sci., 1998, 33, 69–72. doi: 10.1023/A:1004385327370
   Web of Science ®Google Scholar
• C. Reynaud and F. Thevenot: ‘Porosity dependence of mechanical properties of porous sintered SiC. Verification of the minimum solid area model’, J. Mater. Sci. Lett., 2000, 19, 871–874. doi: 10.1023/A:1006741616088
   Google Scholar
• M. L. Hentschel and N. W. Page: ‘Elastic properties of powders during compaction. Part 3: evaluation of models’, J. Mater. Sci., 2006, 41, 7902–7925. doi: 10.1007/s10853-006-0875-0
   Web of Science ®Google Scholar
• K. U. O'Kelly, A. J. Carr and B. A. O. McCormack: ‘Minimum solid area models applied to the prediction of Young's modulus for cancellous bone’, J. Mater. Sci.: Mater. Med., 2003, 14, 379–384.
   PubMed Web of Science ®Google Scholar
• H. N. Yoshimura, A. L. Molisani, N. E. Narita, P. F. Cesar and H. Goldenstein: ‘Porosity dependence of elastic constants in aluminum nitride ceramics’, Mater. Res., 2007, 10, 127–133. doi: 10.1590/S1516-14392007000200006
   Google Scholar
• M. Lubrick, R. G. Maev, F. Severin and V. Leshchynsky: ‘Young's modulus of low-pressure cold sprayed composites: an analysis based on a minimum contact area model’, J. Mater. Sci., 2008, 43, 4953–4961. doi: 10.1007/s10853-008-2729-4
   Web of Science ®Google Scholar
• L.-H. He, O. C. Standard, T. T. Y. Huang, B. A. Latella and M. V. Swain: ‘Mechanical behaviour of porous hydroxyapatite’, Acta Biomater., 2008, 4, 577–586. doi: 10.1016/j.actbio.2007.11.002
   PubMed Web of Science ®Google Scholar
• O. Wiener: ‘Die Theorie des Mischkörpers für das Feld der stationären Strömung’, Abh. Math. Phys. Klasse Königl. Sächs. Ges. Wiss., 1912, 32, 509–604.
   Google Scholar
• W. Voigt: ‘Über die Beziehung zwischen den beiden Elastizitätskonstanten isotroper Körper’, Wiedemann Ann., 1889, 274, 573–587. doi: 10.1002/andp.18892741206
   Google Scholar
• A. Reuss: ‘Berechnung der Fliessgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle’, Z. Angew. Math. Mech., 1929, 9, 49–58. doi: 10.1002/zamm.19290090104
   Google Scholar
• B. Paul: ‘Prediction of the elastic constants of multiphase materials’, Trans. AIME, 1960, 218, 36–41.
   Google Scholar
• Z. Hashin and S. Shtrikman: ‘A variational approach to the theory of the effective magnetic permeability of multiphase materials’, J. Appl. Phys., 1962, 33, 3125–3131. doi: 10.1063/1.1728579
   Web of Science ®Google Scholar
• Z. Hashin and S. Shtrikman: ‘A variational approach to the theory of the elastic behaviour of multiphase materials’, J. Mech. Phys. Solids, 1963, 11, 127–140. doi: 10.1016/0022-5096(63)90060-7
   Web of Science ®Google Scholar
• W. Pabst and E. Gregorová: ‘Effective elastic properties of alumina–zirconia composite ceramics – Part 2: micromechanical modeling’, Ceram. Silik., 2004, 48, 14–23.
   Web of Science ®Google Scholar
• W. Pabst and E. Gregorová: ‘Effective elastic properties of alumina–zirconia composite ceramics – Part 1: rational continuum theory of linear elasticity’, Ceram. Silik., 2003, 47, 1–7.
   Web of Science ®Google Scholar
• W. Pabst and J. Hostaša: ‘Thermal conductivity of ceramics – from monolithic to multiphase, from dense to porous, from micro to nano’ in ‘Advances in materials science research’, (ed. M. C. Wythers), Vol. 7, 1–112; 2011, New York, Nova Science Publishers.
   Google Scholar
• J. K. Carson, S. J. Lovatt, D. J. Tanner and A. C. Cleland: ‘Predicting the effective conductivity of unfrozen, porous food’, J. Food. Eng., 2006, 75, 297–307. doi: 10.1016/j.jfoodeng.2005.04.021
   Web of Science ®Google Scholar
• W. Pabst and E. Gregorová: ‘The sigmoidal average – a powerful tool for predicting the thermal conductivity of composite ceramics’, J. Phys.: Conf. Ser., 2012, 395, 012021.
   Google Scholar
• W. Pabst, E. Gregorová, I. Sedlárˇová and M. Cˇerný: ‘Preparation and characterization of porous alumina-zirconia composite ceramics’, J. Eur. Ceram. Soc., 2011, 31, 2721–2731. doi: 10.1016/j.jeurceramsoc.2011.01.011
   Web of Science ®Google Scholar
• R. Hill: ‘The elastic behaviour of a crystalline aggregate’, Proc. Phys. Soc. Lond. A, 1952, 65, 349–354. doi: 10.1088/0370-1298/65/5/307
   Google Scholar
• E. Kröner: ‘Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls’, Z. Phys., 1958, 151, 504–518. doi: 10.1007/BF01337948
   Google Scholar
• W. Pabst, G. Tichá and E. Gregorová: ‘Effective elastic properties of alumina–zirconia composite ceramics – Part 3: calculation of elastic moduli of polycrystalline alumina and zirconia from monocrystal data’, Ceram. Silik., 2004, 48, 41–48.
   Web of Science ®Google Scholar
• W. Pabst, E. Gregorová, T. Uhlírˇová and A. Musilová: ‘Elastic properties of mullite and mullite-containing ceramics – Part 1: theoretical aspects and review of monocrystal data’, Ceram. Silik., 2013, 57, 265–274.
   Web of Science ®Google Scholar
• J. C. Maxwell: ‘A treatise on electricity and magnetism, Vol. 1’, 435–449; 1873, Oxford, Clarendon Press ( reprint 1954, New York, Dover).
   Google Scholar
• J. M. Dewey: ‘The elastic constants of materials loaded with non-rigid fillers’, J. Appl. Phys., 1947, 18, 578–581. doi: 10.1063/1.1697691
   Web of Science ®Google Scholar
• J. K. Mackenzie: ‘Elastic constants of a solid containing spherical holes’, Proc. Phys. Soc. Lond. B, 1950, 63, 2–11. doi: 10.1088/0370-1301/63/1/302
   Google Scholar
• J. D. Eshelby: ‘The determination of the elastic field of an elliptical inclusion, and related problems’, Proc. R. Soc. Lond. A, 1957, 241, 376–396. doi: 10.1098/rspa.1957.0133
   Web of Science ®Google Scholar
• W. Pabst and E. Gregorová: ‘Conductivity of porous materials with spheroidal pores’, J. Eur. Ceram. Soc., 2014, 34, 2757–2766. doi: 10.1016/j.jeurceramsoc.2013.12.040
   Web of Science ®Google Scholar
• W. Pabst and E. Gregorová: ‘Young's modulus of isotropic porous materials with spheroidal pores’, J. Eur. Ceram. Soc., 2014, 34, 3195–3207. doi: 10.1016/j.jeurceramsoc.2014.04.009
   Web of Science ®Google Scholar
• D. Bruggeman: ‘Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen’, Ann. Phys. (Leipzig), 1935, 416, 636–664. doi: 10.1002/andp.19354160705
   Google Scholar
• R. Hill: ‘A self-consistent mechanics of composite materials’, J. Mech. Phys. Solids, 1965, 13, 213–222. doi: 10.1016/0022-5096(65)90010-4
   Web of Science ®Google Scholar
• B. Budiansky: ‘On the elastic moduli of some heterogeneous materials’, J. Mech. Phys. Solids, 1965, 13, 223–227. doi: 10.1016/0022-5096(65)90011-6
   Web of Science ®Google Scholar
• D. P. H. Hasselman: ‘On the porosity dependence of the elastic moduli of polycrystalline refractory materials’, J. Am. Ceram. Soc., 1962, 45, 452–453. doi: 10.1111/j.1151-2916.1962.tb11191.x
   Web of Science ®Google Scholar
• R. L. Coble and W. D. Kingery: ‘Effect of porosity on physical properties of sintered alumina’, J. Am. Ceram. Soc., 1956, 39, 377–385. doi: 10.1111/j.1151-2916.1956.tb15608.x
   Web of Science ®Google Scholar
• W. Pabst and E. Gregorová: ‘Note on the so-called Coble–Kingery formula for the effective tensile modulus of porous ceramics’, J. Mater. Sci. Lett., 2003, 22, 959–962. doi: 10.1023/A:1024600627210
   Google Scholar
• W. Pabst: ‘Simple second-order expression for the porosity dependence of thermal conductivity’, J. Mater. Sci., 2005, 40, 2667–2669. doi: 10.1007/s10853-005-2101-x
   Web of Science ®Google Scholar
• W. Pabst and E. Gregorová: ‘Derivation of the simplest exponential and power-law relations for the effective tensile modulus of porous ceramics via functional equations’, J. Mater. Sci. Lett., 2003, 22, 1673–1675. doi: 10.1023/B:JMSL.0000004645.77295.b5
   Google Scholar
• S. Boucher: ‘On the effective moduli of isotropic two-phase elastic composites’, J. Compos. Mater., 1974, 8, 82–89. doi: 10.1177/002199837400800108
   Web of Science ®Google Scholar
• R. McLaughlin: ‘A study of the differential scheme for composite materials’, Int. J. Eng. Sci., 1977, 15, 237–244. doi: 10.1016/0020-7225(77)90058-1
   Web of Science ®Google Scholar
• A. N. Norris: ‘A differential scheme for the effective moduli of composites’, Mech. Mater., 1985, 4, 1–16. doi: 10.1016/0167-6636(85)90002-X
   Web of Science ®Google Scholar
• R. W. Zimmerman: ‘Elastic moduli of a solid containing spherical inclusions’, Mech. Mater., 1991, 12, 17–24. doi: 10.1016/0167-6636(91)90049-6
   Web of Science ®Google Scholar
• L. J. Gibson and M. F. Ashby: ‘The mechanics of three-dimensional cellular materials’, Proc. R. Soc. Lond. A, 1982, 382, 43–59. doi: 10.1098/rspa.1982.0088
   Web of Science ®Google Scholar
• W. Pabst and E. Gregorová: ‘Mooney-type relation for the porosity dependence of the effective tensile modulus of ceramics’, J. Mater. Sci., 2004, 39, 3213–3215. doi: 10.1023/B:JMSC.0000025863.55408.c9
   Web of Science ®Google Scholar
• G. Tichá, W. Pabst and D. S. Smith: ‘Predictive model for the thermal conductivity of porous materials with matrix-inclusion type microstructure’, J. Mater. Sci., 2005, 40, 5045–5047. doi: 10.1007/s10853-005-1818-x
   Web of Science ®Google Scholar
• K. K. Phani and S. K. Niyogi: ‘Young's modulus of porous brittle solids’, J. Mater. Sci., 1987, 22, 257–263. doi: 10.1007/BF01160581
   Web of Science ®Google Scholar
• D. S. McLachlan: ‘Equation for the conductivity of metal-insulator mixtures’, J. Phys. C: Solid State, 1985, 18, 1891–1897. doi: 10.1088/0022-3719/18/9/022
   Web of Science ®Google Scholar
• W. Pabst and E. Gregorová: ‘New relation for the porosity dependence of the effective tensile modulus of brittle materials’, J. Mater. Sci., 2004, 39, 3501–3503. doi: 10.1023/B:JMSC.0000026961.12735.2a
   Web of Science ®Google Scholar
• W. Pabst and E. Gregorová: ‘A new percolation-threshold relation for the porosity dependence of thermal conductivity’, Ceram. Int., 2006, 32, 89–91. doi: 10.1016/j.ceramint.2004.12.007
   Web of Science ®Google Scholar
• R. M. Spriggs: ‘Expression for effect of porosity on elastic modulus of polycrystalline refractory materials, particularly aluminum oxide’, J. Am. Ceram. Soc., 1961, 44, 628–629. doi: 10.1111/j.1151-2916.1961.tb11671.x
   Web of Science ®Google Scholar
• N. Ramakrishnan and V. S. Arunachalam: ‘Effective elastic moduli of porous ceramic materials’, J. Am. Ceram. Soc., 1993, 76, 2745–2752. doi: 10.1111/j.1151-2916.1993.tb04011.x
   Web of Science ®Google Scholar
• A. R. Boccaccini: ‘Comment on “Effective elastic moduli of porous ceramic materials”’, J. Am. Ceram. Soc., 1994, 77, 2779–2781. doi: 10.1111/j.1151-2916.1994.tb04679.x
   Web of Science ®Google Scholar
• W. Pabst and E. Gregorová: ‘The Poisson ratio of porous materials’ in ‘Characterisation of porous solids VIII’, (ed. S. Kaskel et al.), 424–431; 2009, Cambridge, RSC Publishing.
   Google Scholar
• T. Mori and K. Tanaka: ‘Average stress in matrix and average elastic energy of materials with misfitting inclusions’, Acta Metal., 1973, 21, 571–574. doi: 10.1016/0001-6160(73)90064-3
   Google Scholar
• V. M. Levin: ‘Thermal expansion coefficients of heterogeneous materials’, Mech. Solids, 1967, 21, 9–17.
   Google Scholar
• W. Pabst and E. Gregorová: ‘Cross-property relations between elastic and thermal properties of porous ceramics’, Adv. Sci. Technol., 2006, 45, 107–112. doi: 10.4028/www.scientific.net/AST.45.107
   Google Scholar
• J. G. Berryman and G. W. Milton: ‘Microgeometry of random composites and porous media’, J. Phys. D: Appl. Phys., 1988, 21, 87–94. doi: 10.1088/0022-3727/21/1/013
   Web of Science ®Google Scholar
• L. V. Gibiansky and S. Torquato: ‘Connection between the conductivity and bulk modulus of isotropic composite materials’, Proc. R. Soc. Lond. A, 1996, 452, 253–283. doi: 10.1098/rspa.1996.0015
   Web of Science ®Google Scholar
• I. Sevostianov, J. Kovácˇik and F. Simancˇík: ‘Correlation between elastic and electric properties for metal foams: theory and experiment’, Int. J. Fract., 2002, 114, 23–28. doi: 10.1023/A:1022674130262
   Web of Science ®Google Scholar
• I. Sevostianov, J. Kovácˇik and F. Simancˇík: ‘Elastic and electric properties of closed-cell aluminum foams – cross-property connection’, Mater. Sci. Eng. A, 2006, 420, 87–99. doi: 10.1016/j.msea.2006.01.064
   Web of Science ®Google Scholar
• W. Pabst and E. Gregorová: ‘A cross-property relation between the tensile modulus and the thermal conductivity of porous ceramics’, Ceram. Int., 2007, 33, 9–12. doi: 10.1016/j.ceramint.2005.07.009
   Web of Science ®Google Scholar