Analytical formulation of three-dimensional dynamic homogenization for periodic elastic systems
Homogenization of the equations of motion for a three-dimensional periodic elastic system is considered. Expressions are obtained for the fully dynamic effective material parameters governing the spatially averaged fields by using the plane wave expansion method. The effective equations are of Willi...
Gespeichert in:
| Veröffentlicht in: | Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Mathematical, physical, and engineering sciences, 2012-06, Vol.468 (2142), p.1629-1651 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| container_end_page | 1651 |
|---|---|
| container_issue | 2142 |
| container_start_page | 1629 |
| container_title | Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences |
| container_volume | 468 |
| creator | Norris, A. N. Shuvalov, A. L. Kutsenko, A. A. |
| description | Homogenization of the equations of motion for a three-dimensional periodic elastic system is considered. Expressions are obtained for the fully dynamic effective material parameters governing the spatially averaged fields by using the plane wave expansion method. The effective equations are of Willis form with coupling between momentum and stress and tensorial inertia. The formulation demonstrates that the Willis equations of elastodynamics are closed under homogenization. The effective material parameters are obtained for arbitrary frequency and wavenumber combinations, including but not restricted to Bloch wave branches for wave propagation in the periodic medium. Numerical examples for a one-dimensional system illustrate the frequency dependence of the parameters on Bloch wave branches and provide a comparison with an alternative dynamic effective medium theory, which also reduces to Willis form but with different effective moduli. |
| doi_str_mv | 10.1098/rspa.2011.0698 |
| format | Article |
| fullrecord | <record><control><sourceid>jstor_royal</sourceid><recordid>TN_cdi_royalsociety_journals_10_1098_rspa_2011_0698</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><jstor_id>41511082</jstor_id><sourcerecordid>41511082</sourcerecordid><originalsourceid>FETCH-LOGICAL-c516t-da26bbf63688cdcf3a8f45d84133ce3551f5d6b8e81abfe7f03bbae97f2312be3</originalsourceid><addsrcrecordid>eNp9kc2L1TAUxYsoOI5u3QlduukzN19Nl49BHeGBMqOzDWl64-RN29SkFTt_vXlWBkR0lZBzzu9yT4riJZAdkEa9iWkyO0oAdkQ26lFxBryGijZcPs53JnklCIWnxbOUjoSQRqj6rND70fTr7K3pSxfisPRm9mEsgyvn24hYdX7AMeWnbOjW0QzelrdhCF9x9PebN-fKCaMPXdawNynjyrSmGYf0vHjiTJ_wxe_zvPjy7u3ni8vq8PH9h4v9obIC5Fx1hsq2dZJJpWxnHTPKcdEpDoxZZEKAE51sFSowrcPaEda2BpvaUQa0RXZevN64UwzfFkyzHnyy2PdmxLAkDVISkuGEZ-tus9oYUoro9BT9YOKqgehTk_rUpD41qU9N5gDbAjGseYdgPc6rPoYl5lLSv1N3_0tdXX_af-dSeQqcaqIYEC5IZt37aUNlUfuUFtS_LH_i_572apt2THOIDxtxEABE0axXm-7zt_x40E2807JmtdA3imvCaHN5fbjRV-wnFwe3OA</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>1660068804</pqid></control><display><type>article</type><title>Analytical formulation of three-dimensional dynamic homogenization for periodic elastic systems</title><source>Jstor Complete Legacy</source><source>Alma/SFX Local Collection</source><source>JSTOR Mathematics & Statistics</source><creator>Norris, A. N. ; Shuvalov, A. L. ; Kutsenko, A. A.</creator><creatorcontrib>Norris, A. N. ; Shuvalov, A. L. ; Kutsenko, A. A.</creatorcontrib><description>Homogenization of the equations of motion for a three-dimensional periodic elastic system is considered. Expressions are obtained for the fully dynamic effective material parameters governing the spatially averaged fields by using the plane wave expansion method. The effective equations are of Willis form with coupling between momentum and stress and tensorial inertia. The formulation demonstrates that the Willis equations of elastodynamics are closed under homogenization. The effective material parameters are obtained for arbitrary frequency and wavenumber combinations, including but not restricted to Bloch wave branches for wave propagation in the periodic medium. Numerical examples for a one-dimensional system illustrate the frequency dependence of the parameters on Bloch wave branches and provide a comparison with an alternative dynamic effective medium theory, which also reduces to Willis form but with different effective moduli.</description><identifier>ISSN: 1364-5021</identifier><identifier>EISSN: 1471-2946</identifier><identifier>DOI: 10.1098/rspa.2011.0698</identifier><language>eng</language><publisher>The Royal Society Publishing</publisher><subject>Bloch Spectrum ; Bloch waves ; Constitutive equations ; Dynamical systems ; Dynamics ; Elastic systems ; Elastodynamics ; Homogenization ; Homogenizing ; Mathematical analysis ; Mathematical functions ; Mathematical independent variables ; Mathematical models ; Mathematical vectors ; Matrices ; Periodic Elastic Medium ; Scalars ; Tensors ; Waves</subject><ispartof>Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences, 2012-06, Vol.468 (2142), p.1629-1651</ispartof><rights>2012 The Royal Society</rights><rights>This journal is © 2012 The Royal Society</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c516t-da26bbf63688cdcf3a8f45d84133ce3551f5d6b8e81abfe7f03bbae97f2312be3</citedby><cites>FETCH-LOGICAL-c516t-da26bbf63688cdcf3a8f45d84133ce3551f5d6b8e81abfe7f03bbae97f2312be3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/41511082$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/41511082$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>314,776,780,799,828,27901,27902,57992,57996,58225,58229</link.rule.ids></links><search><creatorcontrib>Norris, A. N.</creatorcontrib><creatorcontrib>Shuvalov, A. L.</creatorcontrib><creatorcontrib>Kutsenko, A. A.</creatorcontrib><title>Analytical formulation of three-dimensional dynamic homogenization for periodic elastic systems</title><title>Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences</title><addtitle>Proc. R. Soc. A</addtitle><addtitle>Proc. R. Soc. A</addtitle><description>Homogenization of the equations of motion for a three-dimensional periodic elastic system is considered. Expressions are obtained for the fully dynamic effective material parameters governing the spatially averaged fields by using the plane wave expansion method. The effective equations are of Willis form with coupling between momentum and stress and tensorial inertia. The formulation demonstrates that the Willis equations of elastodynamics are closed under homogenization. The effective material parameters are obtained for arbitrary frequency and wavenumber combinations, including but not restricted to Bloch wave branches for wave propagation in the periodic medium. Numerical examples for a one-dimensional system illustrate the frequency dependence of the parameters on Bloch wave branches and provide a comparison with an alternative dynamic effective medium theory, which also reduces to Willis form but with different effective moduli.</description><subject>Bloch Spectrum</subject><subject>Bloch waves</subject><subject>Constitutive equations</subject><subject>Dynamical systems</subject><subject>Dynamics</subject><subject>Elastic systems</subject><subject>Elastodynamics</subject><subject>Homogenization</subject><subject>Homogenizing</subject><subject>Mathematical analysis</subject><subject>Mathematical functions</subject><subject>Mathematical independent variables</subject><subject>Mathematical models</subject><subject>Mathematical vectors</subject><subject>Matrices</subject><subject>Periodic Elastic Medium</subject><subject>Scalars</subject><subject>Tensors</subject><subject>Waves</subject><issn>1364-5021</issn><issn>1471-2946</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2012</creationdate><recordtype>article</recordtype><recordid>eNp9kc2L1TAUxYsoOI5u3QlduukzN19Nl49BHeGBMqOzDWl64-RN29SkFTt_vXlWBkR0lZBzzu9yT4riJZAdkEa9iWkyO0oAdkQ26lFxBryGijZcPs53JnklCIWnxbOUjoSQRqj6rND70fTr7K3pSxfisPRm9mEsgyvn24hYdX7AMeWnbOjW0QzelrdhCF9x9PebN-fKCaMPXdawNynjyrSmGYf0vHjiTJ_wxe_zvPjy7u3ni8vq8PH9h4v9obIC5Fx1hsq2dZJJpWxnHTPKcdEpDoxZZEKAE51sFSowrcPaEda2BpvaUQa0RXZevN64UwzfFkyzHnyy2PdmxLAkDVISkuGEZ-tus9oYUoro9BT9YOKqgehTk_rUpD41qU9N5gDbAjGseYdgPc6rPoYl5lLSv1N3_0tdXX_af-dSeQqcaqIYEC5IZt37aUNlUfuUFtS_LH_i_572apt2THOIDxtxEABE0axXm-7zt_x40E2807JmtdA3imvCaHN5fbjRV-wnFwe3OA</recordid><startdate>20120608</startdate><enddate>20120608</enddate><creator>Norris, A. N.</creator><creator>Shuvalov, A. L.</creator><creator>Kutsenko, A. A.</creator><general>The Royal Society Publishing</general><general>The Royal Society</general><scope>BSCLL</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>H8D</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20120608</creationdate><title>Analytical formulation of three-dimensional dynamic homogenization for periodic elastic systems</title><author>Norris, A. N. ; Shuvalov, A. L. ; Kutsenko, A. A.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c516t-da26bbf63688cdcf3a8f45d84133ce3551f5d6b8e81abfe7f03bbae97f2312be3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2012</creationdate><topic>Bloch Spectrum</topic><topic>Bloch waves</topic><topic>Constitutive equations</topic><topic>Dynamical systems</topic><topic>Dynamics</topic><topic>Elastic systems</topic><topic>Elastodynamics</topic><topic>Homogenization</topic><topic>Homogenizing</topic><topic>Mathematical analysis</topic><topic>Mathematical functions</topic><topic>Mathematical independent variables</topic><topic>Mathematical models</topic><topic>Mathematical vectors</topic><topic>Matrices</topic><topic>Periodic Elastic Medium</topic><topic>Scalars</topic><topic>Tensors</topic><topic>Waves</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Norris, A. N.</creatorcontrib><creatorcontrib>Shuvalov, A. L.</creatorcontrib><creatorcontrib>Kutsenko, A. A.</creatorcontrib><collection>Istex</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>Aerospace Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Norris, A. N.</au><au>Shuvalov, A. L.</au><au>Kutsenko, A. A.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Analytical formulation of three-dimensional dynamic homogenization for periodic elastic systems</atitle><jtitle>Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences</jtitle><stitle>Proc. R. Soc. A</stitle><addtitle>Proc. R. Soc. A</addtitle><date>2012-06-08</date><risdate>2012</risdate><volume>468</volume><issue>2142</issue><spage>1629</spage><epage>1651</epage><pages>1629-1651</pages><issn>1364-5021</issn><eissn>1471-2946</eissn><abstract>Homogenization of the equations of motion for a three-dimensional periodic elastic system is considered. Expressions are obtained for the fully dynamic effective material parameters governing the spatially averaged fields by using the plane wave expansion method. The effective equations are of Willis form with coupling between momentum and stress and tensorial inertia. The formulation demonstrates that the Willis equations of elastodynamics are closed under homogenization. The effective material parameters are obtained for arbitrary frequency and wavenumber combinations, including but not restricted to Bloch wave branches for wave propagation in the periodic medium. Numerical examples for a one-dimensional system illustrate the frequency dependence of the parameters on Bloch wave branches and provide a comparison with an alternative dynamic effective medium theory, which also reduces to Willis form but with different effective moduli.</abstract><pub>The Royal Society Publishing</pub><doi>10.1098/rspa.2011.0698</doi><tpages>23</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 1364-5021 |
| ispartof | Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences, 2012-06, Vol.468 (2142), p.1629-1651 |
| issn | 1364-5021 1471-2946 |
| language | eng |
| recordid | cdi_royalsociety_journals_10_1098_rspa_2011_0698 |
| source | Jstor Complete Legacy; Alma/SFX Local Collection; JSTOR Mathematics & Statistics |
| subjects | Bloch Spectrum Bloch waves Constitutive equations Dynamical systems Dynamics Elastic systems Elastodynamics Homogenization Homogenizing Mathematical analysis Mathematical functions Mathematical independent variables Mathematical models Mathematical vectors Matrices Periodic Elastic Medium Scalars Tensors Waves |
| title | Analytical formulation of three-dimensional dynamic homogenization for periodic elastic systems |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-02-13T17%3A45%3A43IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-jstor_royal&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Analytical%20formulation%20of%20three-dimensional%20dynamic%20homogenization%20for%20periodic%20elastic%20systems&rft.jtitle=Proceedings%20of%20the%20Royal%20Society.%20A,%20Mathematical,%20physical,%20and%20engineering%20sciences&rft.au=Norris,%20A.%20N.&rft.date=2012-06-08&rft.volume=468&rft.issue=2142&rft.spage=1629&rft.epage=1651&rft.pages=1629-1651&rft.issn=1364-5021&rft.eissn=1471-2946&rft_id=info:doi/10.1098/rspa.2011.0698&rft_dat=%3Cjstor_royal%3E41511082%3C/jstor_royal%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=1660068804&rft_id=info:pmid/&rft_jstor_id=41511082&rfr_iscdi=true |