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<article article-type="research-article" dtd-version="1.3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xml:lang="en"><front><journal-meta><journal-id journal-id-type="publisher-id">najo</journal-id><journal-title-group><journal-title xml:lang="en">Nanosystems: Physics, Chemistry, Mathematics</journal-title><trans-title-group xml:lang="ru"><trans-title>Наносистемы: физика, химия, математика</trans-title></trans-title-group></journal-title-group><issn pub-type="ppub">2220-8054</issn><issn pub-type="epub">2305-7971</issn><publisher><publisher-name>Университет ИТМО</publisher-name></publisher></journal-meta><article-meta><article-id custom-type="elpub" pub-id-type="custom">najo-1112</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research Article</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="ru"><subject>Статьи</subject></subj-group></article-categories><title-group><article-title>On new phase intermediate layers in nano composites as a source of increasing the elastic moduli</article-title><trans-title-group xml:lang="ru"><trans-title>О переходных слоях новой фазы в нанокомпозитах как источниках увеличения модулей упругости</trans-title></trans-title-group></title-group><contrib-group><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Филиппов</surname><given-names>Р. А.</given-names></name><name name-style="western" xml:lang="en"><surname>Filippov</surname><given-names>R. A.</given-names></name></name-alternatives><bio xml:lang="en"><p>Saint-Petersburg</p></bio><email xlink:type="simple">rmnfilippov@gmail.com</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Фрейдин</surname><given-names>А. Б.</given-names></name><name name-style="western" xml:lang="en"><surname>Freidin</surname><given-names>A. B.</given-names></name></name-alternatives><bio xml:lang="en"><p>Saint-Petersburg</p></bio><email xlink:type="simple">alexander.freidin@gmail.com</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Вильчевская</surname><given-names>Е. Н.</given-names></name><name name-style="western" xml:lang="en"><surname>Vilchevskaya</surname><given-names>E. N.</given-names></name></name-alternatives><bio xml:lang="en"><p>Saint-Petersburg</p></bio><email xlink:type="simple">vilchevska@gmail.com</email><xref ref-type="aff" rid="aff-1"/></contrib></contrib-group><aff xml:lang="en" id="aff-1"><institution>Institute for Problems in Mechanical Engineering, RAS</institution><country>Russian Federation</country></aff><pub-date pub-type="collection"><year>2011</year></pub-date><pub-date pub-type="epub"><day>17</day><month>08</month><year>2025</year></pub-date><volume>2</volume><issue>2</issue><fpage>107</fpage><lpage>118</lpage><permissions><copyright-statement>Copyright &amp;#x00A9; Filippov R.A., Freidin A.B., Vilchevskaya E.N., 2025</copyright-statement><copyright-year>2025</copyright-year><copyright-holder xml:lang="ru">Филиппов Р.А., Фрейдин А.Б., Вильчевская Е.Н.</copyright-holder><copyright-holder xml:lang="en">Filippov R.A., Freidin A.B., Vilchevskaya E.N.</copyright-holder><license xml:lang="ru" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>Данная работа распространяется под лицензией Creative Commons Attribution 4.0.</license-p></license><license xml:lang="en" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>This work is licensed under a Creative Commons Attribution 4.0 License.</license-p></license></permissions><self-uri xlink:href="https://nanojournal.ifmo.ru/jour/article/view/1112">https://nanojournal.ifmo.ru/jour/article/view/1112</self-uri><abstract><p>We study an opportunity to increase elastic moduli of a nanocomposite due to stress-induced phase transformations which lead to the formation of intermediate new phase layers around nanoparticles. These layers enlarge the effective size of the particles which from now become inclusions made up of kernels (the initial nanoparticles) enclosed by new phase layers shells. Increasing the volume fraction of the inclusions can change the effective elastic moduli of the composite much more than one could expect in a case of the composite with a small volume fraction of initial nanoparticles. As an example we consider an isotropic composite with spherical particles under hydrostatic loading. We begin with considering the new phase formation around an isolated inclusion including the interface stability analysis. We show that stable two-phase states are impossible if both elastic moduli of the matrix increase due to phase transition and possible if the bulk modulus increases and the shear module decreases. Then, basing on a self-consistent approach, we describe the new phase formation around spatially distributed particles and study how external strains effects the new phase areas growth. Finally we demonstrate that the new phase layers formation can lead to increasing the effective bulk modulus of the composite.</p></abstract><trans-abstract xml:lang="ru"><p>.</p></trans-abstract><kwd-group xml:lang="en"><kwd>nanocomposites</kwd><kwd>intermediate layers</kwd><kwd>effective elastic moduli</kwd><kwd>phase transformations</kwd><kwd>stability</kwd></kwd-group><funding-group><funding-statement xml:lang="en">The financial support of RFBR (Grant No 10-01-00670-a), Programs for Fundamental Research No 22,13, Grant from DFG (AL 341/35-1), the Russian Ministry of Education and Science (Contract 14.740.11.0353) is gratefully acknowledged.</funding-statement></funding-group></article-meta></front><back><ref-list><title>References</title><ref id="cit1"><label>1</label><citation-alternatives><mixed-citation xml:lang="ru">Cadek M., Coleman J. N., Barron V., Hedicke K., Blau W. J. Morphological and mechanical properties of carbon-nanotube-reinforced semicrystalline and amorphous polymer composites // Applied Physics Letters, 2002. V. 81. 27. P. 5123–5125.</mixed-citation><mixed-citation xml:lang="en">Cadek M., Coleman J. N., Barron V., Hedicke K., Blau W. J. Morphological and mechanical properties of carbon-nanotube-reinforced semicrystalline and amorphous polymer composites // Applied Physics Letters, 2002. V. 81. 27. P. 5123–5125.</mixed-citation></citation-alternatives></ref><ref id="cit2"><label>2</label><citation-alternatives><mixed-citation xml:lang="ru">Freidin A. B. On new phase inclusions in elastic solids // ZAMM, 2007. V. 87. 2. P. 102–116.</mixed-citation><mixed-citation xml:lang="en">Freidin A. B. On new phase inclusions in elastic solids // ZAMM, 2007. V. 87. 2. P. 102–116.</mixed-citation></citation-alternatives></ref><ref id="cit3"><label>3</label><citation-alternatives><mixed-citation xml:lang="ru">Freidin A.B., Y.B. Fu, Sharipova L.L., Vilchevskaya E.N. Spherically symmetric two-phase deformations ans phase transition zones // Int. J. Solids and Struct, 2006. V. 43. P. 4484–4508.</mixed-citation><mixed-citation xml:lang="en">Freidin A.B., Y.B. Fu, Sharipova L.L., Vilchevskaya E.N. Spherically symmetric two-phase deformations ans phase transition zones // Int. J. Solids and Struct, 2006. V. 43. P. 4484–4508.</mixed-citation></citation-alternatives></ref><ref id="cit4"><label>4</label><citation-alternatives><mixed-citation xml:lang="ru">George R., Kashyap K.T., Rahul R., Yamdagni S. Strengthening in carbon nanotube/aluminum (CNT/Al) composites // Scripta Materialia, 2005. V. 53. P. 1159–1163.</mixed-citation><mixed-citation xml:lang="en">George R., Kashyap K.T., Rahul R., Yamdagni S. Strengthening in carbon nanotube/aluminum (CNT/Al) composites // Scripta Materialia, 2005. V. 53. P. 1159–1163.</mixed-citation></citation-alternatives></ref><ref id="cit5"><label>5</label><citation-alternatives><mixed-citation xml:lang="ru">Freidin A.B., Vilchevskaya E.N. Multiple development of new phase inclusions in elastic solids // International Journal of Engineering Science, 2009. V. 47. P. 240–260.</mixed-citation><mixed-citation xml:lang="en">Freidin A.B., Vilchevskaya E.N. Multiple development of new phase inclusions in elastic solids // International Journal of Engineering Science, 2009. V. 47. P. 240–260.</mixed-citation></citation-alternatives></ref><ref id="cit6"><label>6</label><citation-alternatives><mixed-citation xml:lang="ru">Kanaun S.K., Levin V.M. Self-consistent methods for composites, Vol. 1: Static Problems. — Springer, 2007.</mixed-citation><mixed-citation xml:lang="en">Kanaun S.K., Levin V.M. Self-consistent methods for composites, Vol. 1: Static Problems. — Springer, 2007.</mixed-citation></citation-alternatives></ref><ref id="cit7"><label>7</label><citation-alternatives><mixed-citation xml:lang="ru">Kublanov L. B., Freidin A. B. Solid phase seeds in a deformable material // J. Appl. Math. Mech., 1988. V. 52. P. 382–389.</mixed-citation><mixed-citation xml:lang="en">Kublanov L. B., Freidin A. B. Solid phase seeds in a deformable material // J. Appl. Math. Mech., 1988. V. 52. P. 382–389.</mixed-citation></citation-alternatives></ref><ref id="cit8"><label>8</label><citation-alternatives><mixed-citation xml:lang="ru">Kunin L. A. Elastic media with Microstructure. II. Three Dimensional Models. — Springer Series in Solid State Sciences, Berlin, New York, etc. Springer-Verlag, 1983, 272 p.</mixed-citation><mixed-citation xml:lang="en">Kunin L. A. Elastic media with Microstructure. II. Three Dimensional Models. — Springer Series in Solid State Sciences, Berlin, New York, etc. Springer-Verlag, 1983, 272 p.</mixed-citation></citation-alternatives></ref><ref id="cit9"><label>9</label><citation-alternatives><mixed-citation xml:lang="ru">Morozov N. F., Freidin A. B. Phase transition zones and phase transformations of elastic solids under different stress states // Proc. Steklov Math. Inst., 1998. V. 223. P. 220–232.</mixed-citation><mixed-citation xml:lang="en">Morozov N. F., Freidin A. B. Phase transition zones and phase transformations of elastic solids under different stress states // Proc. Steklov Math. Inst., 1998. V. 223. P. 220–232.</mixed-citation></citation-alternatives></ref><ref id="cit10"><label>10</label><citation-alternatives><mixed-citation xml:lang="ru">Yeremeyev V. A., Freidin A. B., Sharipova L. L. The stability of the equilibrium of two-phase elastic solids // J. Appl. Math. Mech., 2007. V. 71. P. 61-84.</mixed-citation><mixed-citation xml:lang="en">Yeremeyev V. A., Freidin A. B., Sharipova L. L. The stability of the equilibrium of two-phase elastic solids // J. Appl. Math. Mech., 2007. V. 71. P. 61-84.</mixed-citation></citation-alternatives></ref><ref id="cit11"><label>11</label><citation-alternatives><mixed-citation xml:lang="ru">Freidin A. B., Chiskis A. M. Regions of phase transitions in nonlinear-elastic isotropic materials. Part 1: Basic relations // Mechanics of Solids, 1994. V. 29(4). P. 91–109; Mechanics of Solids, 1994. V. 29(5). P. 46-58. Translation from Izv. RAN, Mekhanika Tverdogo Tela, 1994. V. 4. P. 91-109; V. 5. P. 49-61.</mixed-citation><mixed-citation xml:lang="en">Freidin A. B., Chiskis A. M. Regions of phase transitions in nonlinear-elastic isotropic materials. Part 1: Basic relations // Mechanics of Solids, 1994. V. 29(4). P. 91–109; Mechanics of Solids, 1994. V. 29(5). P. 46-58. Translation from Izv. RAN, Mekhanika Tverdogo Tela, 1994. V. 4. P. 91-109; V. 5. P. 49-61.</mixed-citation></citation-alternatives></ref><ref id="cit12"><label>12</label><citation-alternatives><mixed-citation xml:lang="ru">Freidin A. B. Small strains approach to the theory on solid-solid phase transformations under the process of deformation // Studies on Elasticity and Plasticity (St. Petersburg State University), 1999. V. 18. P. 266-290.</mixed-citation><mixed-citation xml:lang="en">Freidin A. B. Small strains approach to the theory on solid-solid phase transformations under the process of deformation // Studies on Elasticity and Plasticity (St. Petersburg State University), 1999. V. 18. P. 266-290.</mixed-citation></citation-alternatives></ref><ref id="cit13"><label>13</label><citation-alternatives><mixed-citation xml:lang="ru">Eremeev V.A., Freidin A.B., Sharipova L.L. Nonuniqueness and Stability in Problems of Equilibrium of Elastic Two-Phase Bodies // Doklady Physics, 2003. V. 48(7). P. 359363. Translated from Doklady Akademii Nauk, 2003. V. 391. 2. P. 189193.</mixed-citation><mixed-citation xml:lang="en">Eremeev V.A., Freidin A.B., Sharipova L.L. Nonuniqueness and Stability in Problems of Equilibrium of Elastic Two-Phase Bodies // Doklady Physics, 2003. V. 48(7). P. 359363. Translated from Doklady Akademii Nauk, 2003. V. 391. 2. P. 189193.</mixed-citation></citation-alternatives></ref><ref id="cit14"><label>14</label><citation-alternatives><mixed-citation xml:lang="ru">Morozov N.F., Nazyrov I.R., Freidin A.B. One-dimensional problem on phase transformation of an elastic ball // Dokl. Akad. Nauk, 1996. V. 346. P. 188-191.</mixed-citation><mixed-citation xml:lang="en">Morozov N.F., Nazyrov I.R., Freidin A.B. One-dimensional problem on phase transformation of an elastic ball // Dokl. Akad. Nauk, 1996. V. 346. P. 188-191.</mixed-citation></citation-alternatives></ref><ref id="cit15"><label>15</label><citation-alternatives><mixed-citation xml:lang="ru">Thostenson E.T., Ren Z., Chou T.-W. Advance in the science and technology of carbon nanotubes and their composites: A review // Composites Sciece and Technology, 2001. V. 61. P. 1899-1912.</mixed-citation><mixed-citation xml:lang="en">Thostenson E.T., Ren Z., Chou T.-W. Advance in the science and technology of carbon nanotubes and their composites: A review // Composites Sciece and Technology, 2001. V. 61. P. 1899-1912.</mixed-citation></citation-alternatives></ref><ref id="cit16"><label>16</label><citation-alternatives><mixed-citation xml:lang="ru">Xie X.-L., Mai Y.-W., Zhou X.-P. Dispersion and alignment of carbon nanotubes in polymer matrix: A review // Materials Science and Engineering, 2005. V. 49. P. 89-112.</mixed-citation><mixed-citation xml:lang="en">Xie X.-L., Mai Y.-W., Zhou X.-P. Dispersion and alignment of carbon nanotubes in polymer matrix: A review // Materials Science and Engineering, 2005. V. 49. P. 89-112.</mixed-citation></citation-alternatives></ref><ref id="cit17"><label>17</label><citation-alternatives><mixed-citation xml:lang="ru">Grabovsky Y., Truskinovsky L. Roughening instability of broken extremals // Arch. Rat. Mech. Anal., 2011. V. 200(1). P. 183202.</mixed-citation><mixed-citation xml:lang="en">Grabovsky Y., Truskinovsky L. Roughening instability of broken extremals // Arch. Rat. Mech. Anal., 2011. V. 200(1). P. 183202.</mixed-citation></citation-alternatives></ref><ref id="cit18"><label>18</label><citation-alternatives><mixed-citation xml:lang="ru">Filippov R.A., Freidin A.B., Vilchevskaya E.N. On transition layers in nanocomposites as new phase areas // In: Proc. XXXIX Summer School-Conference Advanced problems in mechanics (APM 2011), 2011, St. Petersburg (Repino), P. 158–167.</mixed-citation><mixed-citation xml:lang="en">Filippov R.A., Freidin A.B., Vilchevskaya E.N. On transition layers in nanocomposites as new phase areas // In: Proc. XXXIX Summer School-Conference Advanced problems in mechanics (APM 2011), 2011, St. Petersburg (Repino), P. 158–167.</mixed-citation></citation-alternatives></ref></ref-list><fn-group><fn fn-type="conflict"><p>The authors declare that there are no conflicts of interest present.</p></fn></fn-group></back></article>
