{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/42478"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/42478","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Elastodynamics and wave propagation in fractal media","abstract":"The elastodynamics and wave propagation in three-dimensional fractal media is explored through the application of analytic and computational methods. In particular, two different mechanical models are introduced; with each one applied to characterize an elastodynamic problem pertaining to some fractal media of distinctive properties. The first model considers media whose fractality is uniform in all the directions, thus denoted ``isotropic''. The formulation which governs the propagation of waves in this model is first developed from fractional hydrodynamic laws, and then, boundary value problems are solved analytically and numerically on spherical domains. In the second model, the fractality is direction dependent, thus the designation ``anisotropic''. This model, which implements the concept of product measures to regularize fractional integrals in deriving the balance laws, is assigned to treat the elastodynamics of fractal solid materials. Here, the application of Hooke’s relation (classical elasticity) in the constitutive law is limited to dilatational wave motion. In order to treat general problems, a non-classical (Cosserat-type) constitutive model is incorporated, featuring the introduction of microrotation and couple-stress variables into the micropolar element and, subsequently, the balance laws. Various eigenvalue-type problems of different kinematic configurations are solved analytically, while a transient analysis based on modal excitation is simulated numerically, resulting in validated computational tools capable of solving complex elastodynamic problems of arbitrary settings. The development and verification of these two fractal models promotes the consideration of the more challenging acoustic-solid interaction problems in the fractal paradigm. Indeed an idealized problem is first handled in the continuum framework, where the mathematical steps of the solution are analysed, and then, its demonstration in the fractal domain is performed, illustrating the effectiveness of the fractal models discussed before. In conclusion, our analytical and computational investigation advances the mechanics of fractal media for applications which cannot be studied with classical continuum mechanics.","abstract_html":"The elastodynamics and wave propagation in three-dimensional fractal media is explored through the application of analytic and computational methods. In particular, two different mechanical models are introduced; with each one applied to characterize an elastodynamic problem pertaining to some fractal media of distinctive properties. The first model considers media whose fractality is uniform in all the directions, thus denoted ``isotropic''. The formulation which governs the propagation of waves in this model is first developed from fractional hydrodynamic laws, and then, boundary value problems are solved analytically and numerically on spherical domains. In the second model, the fractality is direction dependent, thus the designation ``anisotropic''. This model, which implements the concept of product measures to regularize fractional integrals in deriving the balance laws, is assigned to treat the elastodynamics of fractal solid materials. Here, the application of Hooke’s relation (classical elasticity) in the constitutive law is limited to dilatational wave motion. In order to treat general problems, a non-classical (Cosserat-type) constitutive model is incorporated, featuring the introduction of microrotation and couple-stress variables into the micropolar element and, subsequently, the balance laws. Various eigenvalue-type problems of different kinematic configurations are solved analytically, while a transient analysis based on modal excitation is simulated numerically, resulting in validated computational tools capable of solving complex elastodynamic problems of arbitrary settings. The development and verification of these two fractal models promotes the consideration of the more challenging acoustic-solid interaction problems in the fractal paradigm. Indeed an idealized problem is first handled in the continuum framework, where the mathematical steps of the solution are analysed, and then, its demonstration in the fractal domain is performed, illustrating the effectiveness of the fractal models discussed before. In conclusion, our analytical and computational investigation advances the mechanics of fractal media for applications which cannot be studied with classical continuum mechanics.","abstract_has_math":false,"creators":["Joumaa, Hady"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mechanical Engineering","degree_department":null,"school":null,"contributors":["Ostoja-Starzewski, Martin","Vakakis, Alexander F.","Masud, Arif","DeVille, Robert E."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013-02-03T19:47:12Z","date_published":"2013-02-03T19:47:12Z","updated_at":"2026-07-22T22:25:33Z","subjects":["Elastodynamics","Fractal Media","Wave Propagation","Micropolar Elasticity","Acoustic-Solid Interaction"],"languages":["en"],"rights":["Copyright 2012 Hady Joumaa"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/42478","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Ostoja-Starzewski, Martin","Vakakis, Alexander F.","Masud, Arif","DeVille, Robert E."]},{"key":"dc:creator","label":"Author","values":["Joumaa, Hady"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2013-02-03T19:47:12Z","2015-02-03T11:00:50Z","2012-12"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mechanical Engineering"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Elastodynamics","Fractal Media","Wave Propagation","Micropolar Elasticity","Acoustic-Solid Interaction"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2012 Hady Joumaa"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/42478"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The elastodynamics and wave propagation in three-dimensional fractal media is explored through the application of analytic and computational methods. In particular, two different mechanical models are introduced; with each one applied to characterize an elastodynamic problem pertaining to some fractal media of distinctive properties. The first model considers media whose fractality is uniform in all the directions, thus denoted ``isotropic''. The formulation which governs the propagation of waves in this model is first developed from fractional hydrodynamic laws, and then, boundary value problems are solved analytically and numerically on spherical domains. In the second model, the fractality is direction dependent, thus the designation ``anisotropic''. This model, which implements the concept of product measures to regularize fractional integrals in deriving the balance laws, is assigned to treat the elastodynamics of fractal solid materials. Here, the application of Hooke’s relation (classical elasticity) in the constitutive law is limited to dilatational wave motion. In order to treat general problems, a non-classical (Cosserat-type) constitutive model is incorporated, featuring the introduction of microrotation and couple-stress variables into the micropolar element and, subsequently, the balance laws. Various eigenvalue-type problems of different kinematic configurations are solved analytically, while a transient analysis based on modal excitation is simulated numerically, resulting in validated computational tools capable of solving complex elastodynamic problems of arbitrary settings. The development and verification of these two fractal models promotes the consideration of the more challenging acoustic-solid interaction problems in the fractal paradigm. Indeed an idealized problem is first handled in the continuum framework, where the mathematical steps of the solution are analysed, and then, its demonstration in the fractal domain is performed, illustrating the effectiveness of the fractal models discussed before. In conclusion, our analytical and computational investigation advances the mechanics of fractal media for applications which cannot be studied with classical continuum mechanics.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2012-10-05T19:56:51Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Joumaa_Hady.pdf: 8194261 bytes, checksum: 12747452e516195f66c49a7bd51c0d22 (MD5)","Made available in DSpace on 2013-02-03T19:47:12Z (GMT). No. of bitstreams: 2 Hady_Joumaa.pdf: 8194261 bytes, checksum: 12747452e516195f66c49a7bd51c0d22 (MD5) license.txt: 4061 bytes, checksum: 87544073fe6fcbda4fc216d8a439b704 (MD5)","Item marked as restricted to the 'Administrator' Group (id=1) by Seth Robbins (srobbins@illinois.edu) on 2013-02-03T19:48:01Z Item is restricted until 2015-02-03T19:47:48Z","Restriction data tranferred 2014-07-01T11:36:03-05:00 Original Data Group with Access Administrator Release Date: 2015-02-03 13:47:48 UTC Reason: Author requested closed access (OA after 2yrs) in Vireo ETD system","Limited Restriction Lifted for Item 42426 on 2015-02-03T11:00:50Z."]},{"key":"dc:title","label":"Title","values":["Elastodynamics and wave propagation in fractal media"]}]}],"canonical_facts":{"dc:contributor":["Ostoja-Starzewski, Martin","Vakakis, Alexander F.","Masud, Arif","DeVille, Robert E."],"dc:creator":["Joumaa, Hady"],"dc:date":["2013-02-03T19:47:12Z","2015-02-03T11:00:50Z","2012-12"],"dc:description":["The elastodynamics and wave propagation in three-dimensional fractal media is explored through the application of analytic and computational methods. In particular, two different mechanical models are introduced; with each one applied to characterize an elastodynamic problem pertaining to some fractal media of distinctive properties. The first model considers media whose fractality is uniform in all the directions, thus denoted ``isotropic''. The formulation which governs the propagation of waves in this model is first developed from fractional hydrodynamic laws, and then, boundary value problems are solved analytically and numerically on spherical domains. In the second model, the fractality is direction dependent, thus the designation ``anisotropic''. This model, which implements the concept of product measures to regularize fractional integrals in deriving the balance laws, is assigned to treat the elastodynamics of fractal solid materials. Here, the application of Hooke’s relation (classical elasticity) in the constitutive law is limited to dilatational wave motion. In order to treat general problems, a non-classical (Cosserat-type) constitutive model is incorporated, featuring the introduction of microrotation and couple-stress variables into the micropolar element and, subsequently, the balance laws. Various eigenvalue-type problems of different kinematic configurations are solved analytically, while a transient analysis based on modal excitation is simulated numerically, resulting in validated computational tools capable of solving complex elastodynamic problems of arbitrary settings. The development and verification of these two fractal models promotes the consideration of the more challenging acoustic-solid interaction problems in the fractal paradigm. Indeed an idealized problem is first handled in the continuum framework, where the mathematical steps of the solution are analysed, and then, its demonstration in the fractal domain is performed, illustrating the effectiveness of the fractal models discussed before. In conclusion, our analytical and computational investigation advances the mechanics of fractal media for applications which cannot be studied with classical continuum mechanics.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2012-10-05T19:56:51Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Joumaa_Hady.pdf: 8194261 bytes, checksum: 12747452e516195f66c49a7bd51c0d22 (MD5)","Made available in DSpace on 2013-02-03T19:47:12Z (GMT). No. of bitstreams: 2 Hady_Joumaa.pdf: 8194261 bytes, checksum: 12747452e516195f66c49a7bd51c0d22 (MD5) license.txt: 4061 bytes, checksum: 87544073fe6fcbda4fc216d8a439b704 (MD5)","Item marked as restricted to the 'Administrator' Group (id=1) by Seth Robbins (srobbins@illinois.edu) on 2013-02-03T19:48:01Z Item is restricted until 2015-02-03T19:47:48Z","Restriction data tranferred 2014-07-01T11:36:03-05:00 Original Data Group with Access Administrator Release Date: 2015-02-03 13:47:48 UTC Reason: Author requested closed access (OA after 2yrs) in Vireo ETD system","Limited Restriction Lifted for Item 42426 on 2015-02-03T11:00:50Z."],"dc:identifier":["http://hdl.handle.net/2142/42478"],"dc:language":["en"],"dc:rights":["Copyright 2012 Hady Joumaa"],"dc:subject":["Elastodynamics","Fractal Media","Wave Propagation","Micropolar Elasticity","Acoustic-Solid Interaction"],"dc:title":["Elastodynamics and wave propagation in fractal media"],"dc:type":["text"],"thesis:degree_discipline":["Mechanical Engineering"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:33Z"}