Eshelby-1951 The force on an elastic singularity

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1、[87]THEFORCEONANELASTICSINGULARITYBYJ.D.ESHELBYH.H.WillsPhysicalLaboratory,UniversityofBristol(CommunicatedbyN.F.Mott,F.R.S.-Received18April1951)CONTENTSPAGEPAGE1.INTRODUCTION876.THECENTREOFDILATATION:DISLOCA-TIONS942.NOTATIONANDINTEGRALTHEOREMS897.THEIMAGEFORCEANDTHEFORCES3.TYPES

2、OFSINGULARITY91BETWEENSINGULARITIES96BETWEENSINGUARITI~ES964.THESTRESSFIELDOFASINGULARITYIN8.THEFORCEONANINHOMOGENEITY102AFINITEBODY929.THEGENERALCASE1045.THEFORCEEXERTEDONASINGULARITY10.THEDYNAMICCASE109BYSURFACETRACTIONS93REFERENCES111Theparallelbetweentheclassicaltheoryofelasti

3、cityandthemodernphysicaltheoryofthesolidstateisincomplete;theformerhasnothinganalogoustotheconceptoftheforceactingonanimperfection(dislocation,foreignatom,etc.)inastressedcrystallattice.ToremedythisageneraltheoryoftheforcesonsingularitiesinaHookeanelasticcontinuumisdeveloped.Thesi

4、ngularityistakentobeanystateofinternalstresssatisfyingtheequilibriumequationsbutnotthecompatibilityconditions.Theforceonasingularitycanbegivenasanintegraloverasurfaceenclosingit.Theintegralcontainstheelasticfieldquantitieswhichwouldsurroundthesingularityinaninfinitemedium,multipli

5、edbythedifferencebetweenthesequantitiesandthoseactuallypresent.Theexpressionfortheforceisthusofessentiallythesameformwhethertheforceisduetoappliedsurfacetractions,othersingularitiesorthepresenceofthefreesurfaceofthebody('imageforce').Aregionofinhomogeneityintheelasticconstantsmodi

6、fiesthestressfield;ifitismobileonecandefineandcalculatetheforceonit.Thetotalforceonthesingularitiesandinhomogeneitiesinsideasurfacecanbeexpressedintermsoftheintegralofa'Maxwelltensorofelasticity'takenoverthesurface.Possibleextensionstothedynamicalcasearediscussed.1.INTRODUCTIONMod

7、erntheoriesofthesolidstatemakeuseoftheideaoftheforcesactingonimperfectionsinacrystallattice,suchas,forexample,dislocations,foreignatoms,vacantlatticepoints,grainboundaries.Thestressinthematerialarisesfromthepresenceoftheimperfectionsandfromanyexternallyappliedsurfaceandbodyforces.

8、Iftheappliedforcesareheldconstant