Airy Stress Function

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AIRYSTRESSFUNCTIONFORTWODIMENSIONALINCLUSIONPROBLEMS

by

DHARSHINIRAOKAVATI

PresentedtotheFacultyoftheGraduateSchoolofTheUniversityofTexasatArlingtoninPartialFulfillmentoftheRequirementsfortheDegreeof

MASTEROFSCIENCEINMECHANICALENGINEERING

THEUNIVERSITYOFTEXASATARLINGTONDecember2005 ii

ACKNOWLEDGEMENTS

Iwouldliketoexpressmygratitudetoallthosewhogavemethepossibilitytocomplete this thesis. I am deeply indebted to my supervising professor, Dr. SeiichiNomura for his guidance, patience and motivation. I sincerely thank him for his helpnot only in completing my thesis but also for his invaluable suggestions andencouragement.I sincerely thank committee members Dr. Wen Chan and Dr. Dereje Agonaferfor serving on my committee. They have been a continuous source of inspirationthroughoutmystudyatTheUniversityofTexasatArlington.Iwouldliketoendbysayingthatthisthesiscametoasuccessfulendduetothesupportandblessingsfrommyfamily.LastbutnottheleastIthankSameerChandragiriandallmyotherfriendsfortheirhelpandencouragement.November18th,2005

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ABSTRACT

AIRYSTRESSFUNCTIONFORTWODIMENSIONALINCLUSIONPROBLEMS

PublicationNo.______

DharshiniRaoKavati,MS

TheUniversityofTexasatArlington,2005

SupervisingProfessor:Dr.SeiichiNomura This thesis addresses a problem of finding the elastic fields in a two-dimensional body containing an inhomogeneous inclusion using the Airy stressfunction. The Airy stress function is determined so that the prescribed boundarycondition at a far field and the continuity condition of the traction force and thedisplacementfieldattheinterfacearesatisfiedexactly.Allthederivationsandsolvingsimultaneous equations are carried out using symbolic software.

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TABLEOFCONTENTS

ACKNOWLEDGEMENTS...................................................................................... ii

ABSTRACT.............................................................................................................. iii

LISTOFILLUSTRATIONS.................................................................................... vi

Chapter

1. INTRODUCTION 1

1.1Overview. 1

1.2UseofSymbolicSoftware ...................................................................... 3

2.FORMULATIONOFTHEAIRYSTRESSFUNCTION........................... 6

2.1AiryStressFunction .............................................................................. 6

2.1.1PolarCoordinateFormulation.................................................. 8

2.2ComplexVariableMethods.. 9

2.2.1PlaneElasticityProblemusingComplexVariables. 10

2.3InvestigationofComplexPotentials..... 13

2.3.1FiniteSimplyConnectedDomain.. 13

2.3.2FiniteMultiplyConnectedDomain... 14

2.3.3InfiniteDomain.. 15

3.ADVANCEDAPPLICATIONSOFTHEAIRYSTRESSFUNCTION 17 3.1FinitePlatewithaHoleSubjectedtoTensileLoading.. 17

v 3.2InfinitePlatewithaHoleSubjectedtoTensileLoading ................... 19

3.3TwoDimensionalCircularInclusion................................................. 23 3.3.1StressFieldInsidetheTwoDimensional CircularInclusion.. 24

3.3.2InfinitePlateSurroundingtheDisc 26

4.CONCLUSIONSANDSUGGESTIONSFORFUTUREWORK.............. 37

Appendix

A.MATHEMATICACODE.......................................................................... 39

REFERENCES.......................................................................................................... 52

BIOGRAPHICALINFORMATION........................................................................ 53

vi

LISTOFILLUSTRATIONSFigure Page

2.1 TypicalDomainforthePlaneElasticityProblem6

2.2 ComplexPlane..9

2.3 TypicalDomainsofInterest:(a)FiniteSimplyConnected, (b)FiniteMultiplyConnected,(c)InfiniteMultiplyConnected.....................14 3.1 FinitePlatewithaHoleSubjectedtoTensileLoading ...................................17

3.2 InfinitePlatewithaHoleSubjectedtoTensileloading.......20

3.3 InfinitePlatewithaCircularInclusionSubjectedtoTensileLoading23

3.4 Variationofthetensilestressalongthey-axis.....33

3.5 Variationofthetensilestressalongthex-axis.....33

3.6 Variationofthecompressivestressalongthey-axis...34

3.7 Variationofthecompressivestressalongthex-axis.......34

3.8 Variationoftheshearstressovertheplate......35

3.9 VariationoftheVonMisesstressovertheplate.35

3.10 VariationoftheVonMisesstressalongthex-axis.36

3.11 VariationoftheVonMisesstressalongthey-axis..36

1

CHAPTER1INTRODUCTION1.1OverviewElasticityisanelegantandfascinatingsubjectthatdealswiththedeterminationof the stress, strain and distribution in an elastic solid under the influence of externalforces. A particular form of elasticity which applies to a large range of engineeringmaterials, at least over part of their load range produces deformations which areproportionaltotheloadsproducingthem, giving risetotheHookes Law.Thetheoryestablishes mathematical models of a deformation problem, and this requiresmathematical knowledge to understand the formulation and solution procedures. Thevariable theory provides a very powerful tool for the solution of many problems inelasticity. Employing complex variable methods enables many problems to be solvedthat would be intractable by other schemes. The method is based on the reduction ofthe elasticity boundary value problem to a formulation in the complex domain. Thisformulation then allows many powerful mathematical techniques available from thecomplexvariabletheorytobeappliedtotheelasticityproblem.Another problem faced is the complexity of the elastic field equations asanalytical closed-form solutions to fully three-dimensional problems are very difficulttoaccomplish.Thus,mostsolutionsaredevelopedforreducedproblemsthattypicallyinclude axisymmetry or two-dimensionality to simplify particular aspects of the

2formulation and solution. Because all real elastic structures are three-dimensional, thetheoriessetforthherewillbeapproximatemodels.Thenatureandaccuracyoftheapproximation depend on the problem and loading geometry. Two basictheories, plane stress and plane strain represent the fundamental plane problem inelasticity.Thesetwotheoriesapplytosignificantlydifferenttypesoftwo-dimensionalbodiesalthoughtheirformulationsyieldverysimilarfieldequations.Numerous solutions to plane stress and plane strain problems can be obtainedthroughtheuseofaparticularstressfunctionstechnique.ThemethodemploystheAirystress function [1] and will reduce the general formulation to a single governingequation in terms of a single unknown. The resulting governing equation is thensolvablebyseveralmethodsofappliedmathematics,andthusmanyanalyticalsolutionsto problems of interest can be generated. The stress function formulation is based onthe general idea of developing a representation for the stress field that satisfiesequilibrium and yields a single governing equation from the compatibility statement.This thesis is a successful attempt to apply the above-mentioned method to a plate ofinfinite length and width with a central hole and a disc separately. The problem of acircular hole in an infinite plate has been studied for many years with variousapproaches [1] including the Airy stress function approach. This problem has manyapplications in engineering as it can reveal stress singularity around the hole.However, due to the complexity of algebra involved, there has been no work about atwo-dimensional plate with a circular inclusion (disc) using the Airy stress function to

3our best knowledge. This thesis addresses such a problem using symbolic algebrasoftware1.2UseofSymbolicSoftwareThe development of hardware and software of computers has made availablesymbolicalgebrasoftwarepackagessuchasMATLAB,MAPLE,MATHEMATICA[2].Older packages such as Macsyma- one of the very first general-purpose symboliccomputationssystemswerewritteninLISPwhereasnewonessuchasMathematicaarewritten in the C language and its variations and is one of the most widely availablesymbolicsystems.Using symbolic algebra systems, one can evaluate mathematical expressionsanalytically without any approximation. Differentiations, integrations, expansions andsolvingequationsexactlyarethemajorfeaturesofsymbolicalgebrasystems.Mostofthe symbolic algebra systems have been used by mathematicians and theoreticalphysicists[8].Theabilitytodealwithsymbolicformulae,aswellaswithnumbers,isone of its most powerful features. This is what makes it possible to do algebra andcalculus.Ithasbeendemonstratedthatincertaincircumstancesthewidelyheldviewthatone can always dramatically improve on the CPU time required for lengthycomputations by using compiled C or Fortran code instead of advanced quantitativeprogramming environments such as Mathematica, MATLAB etc is wrong. A wellwrittenCprogramcanbeexpectedtooutperformMathematica,R,S-PlusorMATLAB[7] but, if the C program is not efficiently programmed using the best possible

4algorithm then in fact it may take longer than using a symbolic software byte-codecompiler.At a technical level, Mathematica performs both symbolic and numericcalculations of cross-sectional properties such as areas, centroids, and moments ofinertia.SymbolicsoftwaressuchasMathematicacanderiveclosed-formsolutionsforbeamswithcircular,elliptical,equilateral-triangular,andrectangularcrosssections[2].Symbolic software also addresses the finite element method and is useful in findingshape functions, creating different types of meshes and can solve problems for bothisotropicand anisotropicmaterials. They are alsousefulinthekinematicmodelingoffullyconstrainedsystems[2].This thesis will fo