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1、BasicsingularsolutionsforaporoelasticmediuminthedynamicrangeGuyBonnetInstitutdeM•caniquedeGrenoble,DornaineUniversitaire,38402SaintMartind'H•rexCedex,France(Received15February1987;acceptedforpublication18July1987)Thebasicsingularsolutiontotheproblemsofdynamicporeelasticityisproducedin
2、analogywiththermoelasticity.Thesolution,givenforaharmonicexcitation,isvalidforalldistancesandthewholefrequencyrangebyanalyticalexpressions.PACSnumbers:43.20.Bi,43.20.MvINTRODUCTIONp•jtermsareshowntobefunctionsofthefrequencyandofthegeometryofthepores.Toillustratethis,Fig.1showsTheprobl
3、emoffindingabasicsingularsolutionforaexamplesofthedependenceofthesecoefficientsonthefre-poreelasticmediuminthedynamicrange(i.e.,withinertiaterms)hasbeendealtwithbyseveralauthors.•.2However,quencyfordifferentassumptionsconcerningthe"shape"ofthepores.4aswillbeshownbelow,thefieMofapplica
4、tionofthesesolu-tionsislimited.Areviewofthebasicequationsforaporeelasticmedi-B.Time-dependentequationsum,whichareslightextensionsofBlotequations,follows.Theseequationsaregiveninthefrequencydomain.Consequently,thetime-dependentequationsforpore-Thesolutionstotheproblem,asgivenbyBurridge
5、andelasticitymustbewrittenasVargas•andNorris,2areanalyzedandthefieldofapplica-Nu•.k•+[(N+,4)uf.•+Qu•.•,].j--•*(•--•{)tionofthesesolutionsisstrictlydefined,showingthenecessityforacompletesolution.Itwillthereafterbeshownthattheporeelasticityequationscanbereducedtoequationssimilar(2)toth
6、osepertainingtothermoelasticity,whichopenstheway(Qu•.&+Ru•.&).•+D*(fit--•)foracompletesolutiontotheproblem.=-I.EQUATIONSFORDYNAMICPOROELASTICITYwhere•and/50aretheinverseFouriertransformsofb(0•)andpa(•o),andtheasteriskdenotestheconvolutionprod-uct.A.EquationsinthefrequencydomainItispos
7、sible,however,toseethatthebandp•termsareTheequationsforporoelasticityasgivenbyBiot3canbeindependentofthefrequencyinthelow-frequencyrangewrittenasfollowsforaharmonicmotion:(seeFig.1);inthiscaseandonlyinthiscase,theconvolu-tionproductcanberemoved,leadingtoBiot'sequations•inthetimedomain
8、.----
