The gauge theory of dislocations static solutions

arXiv:0802.0670v1[cond-mat.mtrl-sci]5Feb2008Thegaugetheoryofdislocations:staticsolutionsofscrewandedgedislocationsMarkusLazar∗,CharalamposAnastassiadis,EmmyNoetherResearchGroup,DepartmentofPhysics,DarmstadtUniversityofTechnology,Hochschulstr.6,D-64289Darmstadt,GermanyFebruary6,2008AbstractWeinvestigatetheT(3)-gaugetheoryofstaticdislocationsincontinuoussolids.Weusethemostgenerallinearconstitutiverelationsbilinearintheelasticdistor-tiontensoranddislocationdensitytensorfortheforceandpseudomomentstressesofanisotropicsolid.Theconstitutiverelationscontainsixmaterialparameters.Inthistheoryboththeforceandpseudomomentstressesareasymmetric.Thetheorypossessesfourcharacteristiclengthsℓ1,ℓ2,ℓ3andℓ4whicharegivenexplicitely.Wefirstderivethethree-dimensionalGreentensorofthemasterequationfortheforcestressesinthetranslationalgaugetheoryofdislocations.Wetheninvestigatethesituationofgeneralizedplanestrain(anti-planestrainandplanestrain).Usingthestressfunctionmethod,wefindmodifiedstressfunctionsforscrewandedgedislocations.Thesolutionofthescrewdislocationisgivenintermsofoneindepen-dentlengthℓ1=ℓ4.Fortheproblemofanedgedislocation,onlytwocharacteristiclengthsℓ2andℓ3arisewithoneofthembeingthesameℓ2=ℓ1asforthescrewdislocation.Thus,thistheorypossessesonlytwoindependentlengthsforgener-alizedplanestrain.Ifthetwolengthsℓ2andℓ3ofanedgedislocationareequal,weobtainanedgedislocationwhichisthegaugetheoreticalversionofamodifiedVolterraedgedislocation.Inthecaseofsymmetricstresseswerecoverwellknownresultsobtainedearlier.Keywords:Gaugetheory;dislocations;defects;incompatibleelasticity,torsion,Greentensor.∗E-mail:[e-mail](M.Lazar).11Introduction1.1ReviewingremarksGaugetheoriesareverysuccessfulinhighenergyphysicstounderstandandtoexplainfundamentalinteractions(see,e.g.,[1,2]).Importantsolutionsingaugetheoriesaretopologicalstringsliketheso-calledAbrikosov-Nielson-Olesenstring[3,4].ItisalsoknownthatthetheoriesofgravityandgeneralizedgravitycanbeunderstoodasgaugetheoriesofthePoincar´egroupandtheaffinegroup,respectively[5,6].Forexample,theteleparallelformulationofEinstein’sgeneralrelativitycanbeunderstoodasagaugetheoryofthefour-dimensionaltranslationgroup(see,e.g.,Hehl[7]).Thus,thetorsiontensorisnotjustatensor,butratheraveryspecialtensorthatisrelatedtothetranslationgroup[6].Insuchtheoriesmathematicalsolutionsforcosmicstringsandothertopologicaldefectsinspace-timearefound[8].Butitishardtoobservethembyexperiment.Anapplicationofthethree-dimensionaltranslationalgaugetheoryisthegaugetheo-reticalformulationofdislocations.Thetorsiontensorplaystheroleasthetranslationalfieldstrength.Insuchatheorydislocationsappearquitenaturallyastopologicaldefects(strings).Dislocationsplayanimportantroleinsolidstatephysics[9].Itiseasytoinvestigatethemexperimentally.Therefore,thegaugetheoryofdislocationsisthebestexampletoinvestigatethetranslationalgaugetheoryasaphysicalfieldtheory.PioneeringworkinthefieldofthetranslationalgaugetheoryofdefectsinsolidswasmadebyKadi´candEdelen[10,11],EdelenandLagoudas[12].Itiswellknownthatdislocationsarethefundamentalcarrierofplasticity.Thehopeofthegaugetheoryofdislocationsistoun-derstandplasticityasa‘fundamental’interactioninsolids.Materialsdisplaystrongsizeeffectswhenthecharacteristiclengthscalesareoftheorderofmicrometersdowntoafewnanometers.Withtheinterestinminiaturizationofdevices,thelengthscalesassociatedwithnanodevicesareverysmallandclassicaltheoriesofelasticityandplasticityarenotapplicable.Classicalelasticityandplasticitytheoriescannotexplainthesizedependencebecausetheirconstitutivelawspossessnointernalmateriallengths.Inmetals,thephys-icaloriginofsizeeffectscanberelatedtodislocations.Gaugetheoryofdislocationsmaybeusedtoexplainsizeeffectsandtomodelmechanicalpropertiesofminiaturizeddevices.Inthegaugetheoryofdislocationsthedislocationdensitytensorwhichisthecurloftheelasticdistortionorofthenegativeplasticdistortionmultipliedbymaterialparameterswhichdefineinternalmateriallengthsentertheconstitutiverelations.Thegaugetheoryofdislocationsshallbeastepinthedirectionofamicroscopictheoryofplasticitythatincorporatesinteractionofstructuraldefects.ThedislocationshouldplayasimilarroleasagaugebosonlikethephotonasagaugebosoninMaxwell’stheory.Nevertheless,thetheoryofKadi´candEdelen[11],EdelenandLagoudas[12]hassomelacks.Theyusedveryspecialconstitutiverelations.Forexample,theforcestresstensorissymmetric.Butinthecontinuumtheoryofdislocationsitisknownthattheforcestresstensorhastobeasymmetricatleastinthedislocationcoreregion[13,14]andthatmomentstressisthespecificresponsetodislocations[15].Suchamomentstresscannotbecalculatedfromelasticmodulionly.Arealisticconstitutivelawbetweenthemomentstressandthedislocationdensitytensorhastobeusedtobuiltaphysicaldislocationtheory.Therefore,suchatheoryisnotthemostgeneralisotropicdislocationgaugetheory.Theirtheoryisjustabletogiveanacceptablesolutionofascrewdislocationaftersomeadhocassumptions[16,17].Moreover,itisnotpossibletogiveacorrectsolutionofanedge2dislocation.Especially,thestresscomponentσzzisincorrectandtheconditionofplanest