{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,12]],"date-time":"2026-03-12T05:09:06Z","timestamp":1773292146087,"version":"3.50.1"},"reference-count":25,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2011,4,15]],"date-time":"2011-04-15T00:00:00Z","timestamp":1302825600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":["www.mdpi.com"],"crossmark-restriction":true},"short-container-title":["Micromachines"],"abstract":"<jats:p>The purpose of this paper is to present a simple micromechanics-based model to estimate the effective thermal conductivity of macroscopically isotropic materials of matrix-inclusion type. The methodology is based on the well-established Mori-Tanaka method for composite media reinforced with ellipsoidal inclusions, extended to account for imperfect thermal contact at the matrix-inclusion interface, random orientation of particles and particle size distribution. Using simple ensemble averaging arguments, we show that the Mori-Tanaka relations are still applicable for these complex systems, provided that the inclusion conductivity is appropriately modified. Such conclusion is supported by the verification of the model against a detailed finite-element study as well as its validation against experimental data for a wide range of engineering material systems.<\/jats:p>","DOI":"10.3390\/mi2020129","type":"journal-article","created":{"date-parts":[[2011,4,15]],"date-time":"2011-04-15T14:26:03Z","timestamp":1302877563000},"page":"129-149","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":48,"title":["Mori-Tanaka Based Estimates of Effective Thermal Conductivity of Various Engineering Materials"],"prefix":"10.3390","volume":"2","author":[{"given":"Jan","family":"Str\u00e1nsk\u00fd","sequence":"first","affiliation":[{"name":"Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Zikova 4, 166 36, Prague 6, Czech Republic"}]},{"given":"Jan","family":"Vorel","sequence":"additional","affiliation":[{"name":"Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Zikova 4, 166 36, Prague 6, Czech Republic"}]},{"given":"Jan","family":"Zeman","sequence":"additional","affiliation":[{"name":"Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Zikova 4, 166 36, Prague 6, Czech Republic"}]},{"given":"Michal","family":"\u0160ejnoha","sequence":"additional","affiliation":[{"name":"Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Zikova 4, 166 36, Prague 6, Czech Republic"},{"name":"Centre for Integrated Design of Advanced Structures, Faculty of Civil Engineering, Czech Technical University in Prague, Zikova 4, 166 36, Prague 6, Czech Republic"}]}],"member":"1968","published-online":{"date-parts":[[2011,4,15]]},"reference":[{"key":"ref_1","unstructured":"Torquato, S. (2002). Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Springer-Verlag."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1016\/j.conbuildmat.2006.11.011","article-title":"Thermal conductivity of cement composites containing rubber waste particles: Experimental study and modelling","volume":"22","author":"Benazzouk","year":"2008","journal-title":"Construct. Build. Mater."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"508","DOI":"10.1177\/002199838702100602","article-title":"Effective thermal conductivity of composites with interfacial thermal barrier resistance","volume":"21","author":"Hasselman","year":"1987","journal-title":"J. Compos. Mater."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"582","DOI":"10.1016\/j.scriptamat.2008.12.015","article-title":"The effect of porosity on the thermal conductivity of Al-12 wt.% Si\/SiC composites","volume":"60","author":"Molina","year":"2008","journal-title":"Scripta Mater."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1537","DOI":"10.1016\/0020-7225(86)90162-X","article-title":"The effective conductivity of composites with imperfect thermal contact at constituents interfaces","volume":"24","author":"Benveniste","year":"1986","journal-title":"Int. J. Eng. Sci."},{"key":"ref_6","first-page":"696","article-title":"On the effective thermal conductivity of multiphase composites","volume":"37","author":"Benveniste","year":"1986","journal-title":"J. Appl. Math. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1181","DOI":"10.1016\/j.compscitech.2007.06.009","article-title":"Mori-tanaka models for the thermal conductivity of composites with interfacial resistance and particle size distribution","volume":"68","author":"Nogales","year":"2008","journal-title":"Compos. Sci. Tech."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4067","DOI":"10.1103\/PhysRevLett.75.4067","article-title":"Effect of the interface on the properties of composite media","volume":"75","author":"Torquato","year":"1995","journal-title":"Phys. Rev. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2261","DOI":"10.1063\/1.1337936","article-title":"Thin interphase\/imperfect interface in conduction","volume":"89","author":"Hashin","year":"2001","journal-title":"J. Appl. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1711","DOI":"10.1063\/1.353206","article-title":"The effective thermal conductivity of composites with coated reinforcement and the application to imperfect interfaces","volume":"73","author":"Dunn","year":"1993","journal-title":"J. Appl. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.1016\/0020-7225(86)90011-X","article-title":"Equivalent inclusion method for steady state heat conduction in composites","volume":"24","author":"Hatta","year":"1986","journal-title":"Int. J. Eng. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/0167-6636(87)90005-6","article-title":"A new approach to the application of Mori-Tanaka theory in composite materials","volume":"6","author":"Benveniste","year":"1987","journal-title":"Mech. Mater."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"571","DOI":"10.1016\/0001-6160(73)90064-3","article-title":"Average stress in matrix and average elastic energy of materials with misfitting inclusions","volume":"21","author":"Mori","year":"1973","journal-title":"Acta Metall."},{"key":"ref_14","unstructured":"Str\u00e1nsk\u00fd, J. Micromechnical Models for Thermal Conductivity of Composite Materials with Imperfect Interface. Bachelor Thesis, Available online: http:\/\/mech.fsv.cvut.cz\/zemanj\/teaching\/theses\/09stransky.pdf (accessed on 8 April 2011)."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2878","DOI":"10.1063\/1.345463","article-title":"The effective thermal conductivity of composites reinforced by coated cylindrically orthotropic fibers","volume":"67","author":"Benveniste","year":"1990","journal-title":"J. Appl. Phys."},{"key":"ref_16","unstructured":"Lienhard IV, J.H., and Lienhard V, J.H. (2008). A Heat Transfer Textbook, Phlogiston Press. [3rd ed]."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1016\/j.cossms.2006.02.005","article-title":"Liquid metal infiltration into ceramic particle preforms with bimodal size distributions","volume":"9","author":"Molina","year":"2005","journal-title":"Curr. Opin. Solid State Mater. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1088\/0022-3727\/9\/4\/017","article-title":"The thermal conductivity of epoxy-resin\/metal-powder composite materials from 1.7 to 300 K","volume":"9","author":"Rosenberg","year":"1976","journal-title":"J. Phys. D: Appl. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1016\/0011-2275(95)90896-N","article-title":"Thermal properties of polymer\/particle composites at low temperatures","volume":"35","year":"1995","journal-title":"Cryogenics"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/0956-7151(92)90205-S","article-title":"The effect of particle size on the thermal conductivity of ZnS\/diamond composites","volume":"40","author":"Every","year":"1992","journal-title":"Acta Metall. Mater."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"5486","DOI":"10.1063\/1.347007","article-title":"Effective conductivity of hard-sphere dispersions","volume":"68","author":"Miller","year":"1990","journal-title":"J. Appl. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/S0045-7825(98)00227-8","article-title":"Effective properties of composite materials with periodic microstructure: A computational approach","volume":"172","author":"Michel","year":"1999","journal-title":"Comput. Meth. Appl. Mech. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/S0022-5096(00)00027-2","article-title":"Numerical evaluation of effective properties of graphite fiber tow impregnated by polymer matrix","volume":"49","author":"Zeman","year":"2001","journal-title":"J. Mech. Phys. Solid."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1080\/15376490490492034","article-title":"Micromechanical modeling of porous carbon\/carbon composites","volume":"12","author":"Tsukrov","year":"2005","journal-title":"Mech. Adv. Mater. Struct."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1007\/BF01187726","article-title":"The problem of thermal conduction for two ellipsoidal inhomogeneities in an anisotropic medium and its relevance to composite materials","volume":"111","author":"Chen","year":"1995","journal-title":"Acta Mech."}],"updated-by":[{"DOI":"10.3390\/mi2030344","type":"correction","label":"Correction","source":"publisher","updated":{"date-parts":[[2011,4,15]],"date-time":"2011-04-15T00:00:00Z","timestamp":1302825600000}}],"container-title":["Micromachines"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-666X\/2\/2\/129\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,8,4]],"date-time":"2025-08-04T07:36:51Z","timestamp":1754293011000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-666X\/2\/2\/129"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,4,15]]},"references-count":25,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2011,6]]}},"alternative-id":["mi2020129"],"URL":"https:\/\/doi.org\/10.3390\/mi2020129","relation":{"correction":[{"id-type":"doi","id":"10.3390\/mi2030344","asserted-by":"object"}]},"ISSN":["2072-666X"],"issn-type":[{"value":"2072-666X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,4,15]]}}}