{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T16:29:12Z","timestamp":1774628952982,"version":"3.50.1"},"reference-count":25,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2019,4,19]],"date-time":"2019-04-19T00:00:00Z","timestamp":1555632000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In our study, we will extend the domain of influence in order to cover the thermoelasticity of initially stressed bodies with voids. In what follows, we prove that, for a finite time     t &gt; 0    , the displacement field     u i    , the dipolar displacement field     \u03c6  j k     , the temperature    \u03b8    and the change in volume fraction    \u03d5    generate no disturbance outside a bounded domain B.<\/jats:p>","DOI":"10.3390\/sym11040573","type":"journal-article","created":{"date-parts":[[2019,4,22]],"date-time":"2019-04-22T03:15:53Z","timestamp":1555902953000},"page":"573","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Thermoelasticity of Initially Stressed Bodies with Voids: A Domain of Influence"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1552-3763","authenticated-orcid":false,"given":"Marin","family":"Marin","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Science, Transilvania University of Brasov, 500093 Brasov, Romania"}]},{"given":"Mohamed I. A.","family":"Othman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Zagazig University, P.O. Box 44519 Zagazig, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8679-2579","authenticated-orcid":false,"given":"Sorin","family":"Vlase","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Transilvania University of Brasov, 500096 Brasov, Romania"}]},{"given":"Lavinia","family":"Codarcea-Munteanu","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Transilvania University of Brasov, 500093 Brasov, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2019,4,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1007\/BF00249363","article-title":"A nonlinear theory of materials with voids","volume":"72","author":"Nunziato","year":"1979","journal-title":"Arch. Rat. Mech. Anal."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/BF00041230","article-title":"Linear elastic materials with voids","volume":"13","author":"Cowin","year":"1983","journal-title":"J. Elast."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1007\/BF00284326","article-title":"A continuum theory of granular materia","volume":"44","author":"Goodman","year":"1971","journal-title":"Arch. Rat. Mech. Anal."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/BF01302942","article-title":"A theory of thermoelastic material with voids","volume":"60","author":"Iesan","year":"1986","journal-title":"Acta Mech."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/S0020-7225(97)00019-0","article-title":"Thermoelasticity of initially stressed bodies, asymptotic equipartition of energies","volume":"36","author":"Marin","year":"1998","journal-title":"Int. J. Eng. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/BF01181996","article-title":"Cesaro means in thermoelasticity of dipolar bodies","volume":"122","author":"Marin","year":"1997","journal-title":"Acta Mech."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1016\/0020-7225(72)90089-4","article-title":"Theory of micromorphic materials with memory","volume":"10","author":"Eringen","year":"1972","journal-title":"Int. J. Eng. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1291","DOI":"10.1016\/0020-7225(90)90076-U","article-title":"Theory of thermo-microstretch elastic solids","volume":"28","author":"Eringen","year":"1990","journal-title":"Int. J. Eng. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/BF00248490","article-title":"Micro-structure in linear elasticity","volume":"16","author":"Mindlin","year":"1964","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1007\/BF00253051","article-title":"Multipolar continuum mechanics","volume":"17","author":"Green","year":"1964","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1007\/s00161-007-0053-x","article-title":"Thermomechanics of the interface between a body and its environment","volume":"19","author":"Fried","year":"2007","journal-title":"Continuum Mech. Thermodyn."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1016\/j.amc.2014.07.059","article-title":"A GN model based upon two-temperature generalized thermoelastic theory in an unbounded medium with a spherical cavity","volume":"245","author":"Abbas","year":"2014","journal-title":"Appl. Math Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4193","DOI":"10.1007\/s12206-014-0932-6","article-title":"Eigenvalue approach for an unbounded medium with a spherical cavity based upon two-temperature generalized thermoelastic theory","volume":"28","author":"Abbas","year":"2014","journal-title":"J. Mech. Sci. Technol."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1166\/jctn.2014.3402","article-title":"On the numerical solution of thermal shock problem for generalized magneto-thermoelasticity for an infinitely long annular cylinder with variable thermal conductivity","volume":"11","author":"Abbas","year":"2014","journal-title":"J. Comput. Theor. Nanosci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1403","DOI":"10.1139\/p03-100","article-title":"State Space Approach to Generalized Thermoelasticity Plane Waves with Two Relaxation Times under the Dependence of the Modulus of Elasticity on Reference Temperature","volume":"81","author":"Othman","year":"2003","journal-title":"Can. J. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"4243","DOI":"10.1016\/j.ijsolstr.2006.11.016","article-title":"Effect of Rotation on Generalized Thermo-viscoelastic Rayleigh-Lamb Waves","volume":"44","author":"Sharma","year":"2007","journal-title":"Int. J. Solids Struct."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1139\/cjp-2013-0398","article-title":"Effect of Rotation on Micropolar Generalized Thermoelasticity with Two-Temperatures using a Dual-Phase-Lag Model","volume":"92","author":"Othman","year":"2014","journal-title":"Can. J. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1365","DOI":"10.1007\/s00161-017-0585-7","article-title":"The effect of a dipolar structure on the Holder stability in Green-Naghdi thermoelasticity","volume":"29","author":"Marin","year":"2017","journal-title":"Contin. Mech. Thermodyn."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2749","DOI":"10.1016\/j.physleta.2018.06.026","article-title":"Convective heat transfer flow of nanofluid in a porous medium over wavy surface","volume":"382","author":"Hassan","year":"2018","journal-title":"Phys. Lett. A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1645","DOI":"10.1007\/s00161-016-0503-4","article-title":"Existence and stability results for thermoelastic dipolar bodies with double porosity","volume":"28","author":"Marin","year":"2016","journal-title":"Contin. Mech. Thermodyn."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/j.jmaa.2009.08.045","article-title":"Lagrange identity method for microstretch thermoelastic materials","volume":"363","author":"Marin","year":"2010","journal-title":"J. Math. Anal. Appl."},{"key":"ref_22","first-page":"278","article-title":"The influence of dimensional and structural shifts of the elastic constant values in cylinder fiber composites","volume":"15","author":"Modrea","year":"2013","journal-title":"J. Optoelectron. Adv. Mater."},{"key":"ref_23","first-page":"1233","article-title":"Optimum stacking in a multi-ply laminate used for the skin of adaptive wings","volume":"5","author":"Niculita","year":"2011","journal-title":"J. Optoelectron. Adv. Mater."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1007\/BF00041663","article-title":"Energy inequalities in classical elastodynamics","volume":"14","author":"Carbonaro","year":"1984","journal-title":"J. Elast."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1007\/BF00127556","article-title":"An uniqueness theorem in the theory of elastic","volume":"18","author":"Chandrasekharaiah","year":"1987","journal-title":"J. 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