Math & Engineering

A General Solution in Bipolar Coordinates to Problem Involving Elastic Dislocations

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A General Solution in Bipolar Coordinates to Problem Involving Elastic Dislocations Author(s): T. T. Wu and J. L. Nowinski Reviewed work(s): Source: SIAM Journal on Applied Mathematics, Vol. 19, No. 1 (Jul., 1970), pp. 1-19 Published by: Society for Industrial and Applied Mathematics Stable URL: http://www.jstor.org/stable/2099326 . Accessed: 14/05/2012 05:03 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/ab
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  A General Solution in Bipolar Coordinates to Problem Involving Elastic DislocationsAuthor(s): T. T. Wu and J. L. NowinskiReviewed work(s):Source: SIAM Journal on Applied Mathematics, Vol. 19, No. 1 (Jul., 1970), pp. 1-19Published by: Society for Industrial and Applied Mathematics Stable URL: http://www.jstor.org/stable/2099326. Accessed: 14/05/2012 05:03 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jspJSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org. Society for Industrial and Applied Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to SIAM Journal on Applied Mathematics. http://www.jstor.org  SIAM J. APPL. MATH. Vol.19, No. 1, July1970 AGENERALSOLUTIONINBIPOLARCOORDINATESTOPROBLEMS INVOLVING ELASTICDISLOCATIONS* T. T. WU AND J. L. NOWINSKItSummary. tress anddisplacementields nbipolarcoordinatesrederived ndexpressed nterms fa stressunction,heompatibilityquationbecoming hegoverningquationoftheproblem.Underthe assumption fsingle-valuednessfstress, generalform f stressunctionsgivenom-posedofthe fundamentalartusedbyKoehlerand anauxiliarypartwhichhelpstosatisfyheboundary conditions. The total stress functionyields the desireddiscontinuityfdisplacementcorresponding o theedgedislocation.Ageneralexpressionorthe stressfield husresultswhichcanbeappliedtoparticularroblems eccentricislocationnacircularylinder,islocationnahalf-space, wounlikedislocationsn aninfinitepace) by merelydjusting hevalues ofcoordinatescorrespondingo the boundaries.Numericalxamplesre solved forll threeproblemsndgraphsaregivenllustratinghe stress ields. Introduction.nrecentyearsaprofusionfexperimentalndtheoreticalworkhasbeenpublishedexplainingvariousphenomena,nparticularplasticbehaviorofcrystallineubstances, ymeans of thetheoryfdislocations.Thefundamentaldea is that slip or glide nacrystal akes place as aresultofthemovementfdislocations,heattereing ertainypefinesingularitiesn theotherwiseerfectrystal.ince in theneighborhood fthedislocation inesthedeformationsreso severehatheelasticconstitutivequationscease tobe wellgrounded, tis a commonpracticetoignore orconsidercutaway thincoresaroundthe inesofingularities.hisprocedurencreasesherankofconnectivityofthebody, andordinarily,ollowingVolterra, ntroduces rathertringentconditionhatno stress stransmittedcrossthecylindrical urfacexposedbytheremovalof thecore.Such a modelis alsoanalyzedin thepresentpaperconfinedto two-dimensionaldislocationproblemsinbodies boundedby cylindricalurfacesperpendicularo theplanesofdeformation.nalimitaseoneof thecylindricalboundariesmay degeneratento aplaneorrecede oinfinity.unifiedpproachtothisclass ofproblemss securedby usingasystemfbipolarcoordinates spropounded,.g., byJeffery1]orCoker and Filon[2].Ageneralsolutionsobtained thatcan beadjustedto solution ofparticularroblemsbymerelyn-sertingheconstant oordinate aluesassociatedwithpecificoundaries.Threetypesofproblemsresolvedindetail: adislocation n acylinderfeccentric ircularcrosssection,dislocation nasemi-infiniteedium,wounlikedislocationsnaninfinite edium. nall threeasesthemediums treatedaselastic,omogeneousndisotropicndthedeformationsinfinitesimal.Thefirstroblem ustlistedwasearliernalyzed byKoehler[3]alsouponusinghebipolarcoordinates.However,hissolutiongivesnearlynfinitetresses *Receivedbythe editorsMay 13,1969.This work sanexcerptfthe first uthor'sDoctoraldissertationubmittedo theUniversityf Delaware.The researchwassupportedytheNationalScienceFoundation.tResearchCenter,Uniroyal ncorporated,Wayne,NewJersey, t DepartmentofMechanical andAerospaceEngineering,niversityfDelaware,Newark,Delaware.  2 T. T. WU AND J. L. NOWINSKI on the nneroundaryfthecylinder,hichs physically nsound; t also contra-dicts heusual assumptionhat hetress ield f dislocation s associatedwithself-equilibratedystemfnternal orces o that he boundaries re traction ree.Shivakumar 4] solvedthe sameproblemby means ofconformalmapping;while his solution ould properly atisfy he boundary onditions,t s valid onlyforsingle roblem nd itsgeneralizationo other roblems s not traightforward.his is ust whatpermitsmmediatelyhegeneral orm f olution erived n thispaper. Adislocationn asemi-infiniteediumwas treated by an approximatemethodusinganimagedislocationbyKoehler[3]whendetermininghe forceattractinghedislocation othe freeboundary.Unfortunately,eitherhe outernor the nnerboundary onditions re satisfied y solutionobtained n thisway since the stress ieldf themagedislocationailso cancel theshear stressom-ponent.On theotherhand,bothnormal andshear stressesxistontheinnerboundary and become unbounded whenthe innerradius decreases. Further-more, the validity fthesuperposition rinciplewhen applied to the case ofdislocationssquestionablesexplainedater.Head[5]also solved theproblembyaspecial procedure,ut his solution tillyieldstress nthennerboundary.Two unlike olineardislocationsnan infiniteediumwerenalyzed mongothersby Koehler [3], assuminghat the stressieldf twodislocations xistingsimultaneouslysequivalento the sumoftwo ndividual tressields. fcourse,dislocationsdiffer enerally rom oncentrated orces nd from he theoreticalpoint of view cannot be superposedwithout ualification lthough uch a pro-ceduremaysometimesiveagood approximation.sregardsheelastic modelconsiderednthepresentpaperthisargumentecomeseven moreclear.Itisobviousthat ninfiniteody with wo ingularitiess not equivalentoa combina-tionoftwo infinite odies each witha single singularity,ince the formersadoubly-connectedegion nd the atter re simply onnected.Asa consequence,introducingwosingular urfaces,.e., ntroducingwo multivalued isplacementfieldsassociatedwith wo singularities)nan infiniteodydoes not yield he ameresult s introducing ne singular urface nto each of the component nfinitebodies associatedwith ne singularity). fter ompletion fthepresent nvestiga- tionthework fDean and Wilson 6] was brought o our attention. sing bipolarcoordinateshey ctually tudied wo unlikedislocations avinghediscontinuityofdisplacementsnthe egment etween hedislocations.However, hese uthorsconsideredheproblemsacaseofasingledislocationine nstead ftwo,whichis amistakenotion sincetheproblemsactually problemoftwo unlikedis-locationsn aninfinitemedium. tiseasyto seethat their olutionretainstsvaliditylsoin thecasewhen thediscontinuityfdisplacementsccurs outsidethesegmentbetweenthe dislocations whichis theproblemconsideredhere.Althoughhepresentolutionforhe stress ield akes aformifferingrom hatin[6],thenumericalvalues arethe sameiftheBurgersvector nthepresentsolution ssetequalto K + 1)2T/2.evertheless,t shouldbenotedgainthat hevalidityfthe solutionn[6]is limited o asingleproblem,while thepresentsolution sfairly eneral.1.Generalequations. For futurereference ndto makethepaperself-contained wefirstrieflyeview themain aspectsof thebipolar coordinate  ELASTICDISLOCATIONS 3 system nd the equationsfor hedisplacementnd stress ields xpressed n thesecoordinates.As is well known,the bipolarcoordinate system s definedby the trans-formation (1.1) z = iacoth-2'or equivalently y (1.2)=log - ia' z-lawhere(1.3)z=x+iy,=tc+ifland ax and fi rebipolarcoordinates a= realconst.) see Fig. 1).The curves a= const. represent familyfcoaxial circles with singularities0,?a)asy 0 a-0~~~~~~ FIG. 1.Bipolaroordinates limiting oints; the curves /B const.are a set of circles passing throughhepoints 0,?a).Thesystemfcoordinatessorthogonal.Thevaluesof the co-ordinateBvaryfrom to-7roncrossinghesegmentfthey-axis oiningthesingularpointsin thecounterclockwiseirection. t follows thatthere s a
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