Engineering Elasticity: Elasticity with less Stress and Strain

275 Pages·2022·29.11 MB·English

by  Humphrey Hardy

Humphrey Hardy
Engineering
Elasticity
Elasticity with less Stress and Strain
Engineering Elasticity
Humphrey Hardy
Engineering Elasticity
Elasticity with less Stress and Strain
HumphreyHardy
Pelzer,SC,USA
The book has solutions to the exercises that are available to Professors only. These can be
viewedat:https://link.springer.com/book/10.1007/978-3-031-09157-5.
ISBN978-3-031-09156-8 ISBN978-3-031-09157-5 (eBook)
https://doi.org/10.1007/978-3-031-09157-5
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Preface for Students
Historically,thematerialsavailabletoengineersforconstructionconsistedofmate-
rials that fail when deformed by relatively small amounts (e.g., stone, wood, and
concrete).Tobuildcomputersimulationsforapplicationsofthesematerialsrequired
simulating only small deformations. This has been done quite successfully using
finiteelementsandmatrices.Thisapproachiscalledlinearelasticity.
Theadventofmaterialswhichcanbedeformedbylargeamountswithoutfailing
or fracturing (e.g., rubber, plastics, and biological materials) requires a different
approach. A new approach is needed because large bending, compression, or
extension (finite deformations) results in equations of motion that are nonlinear.
To address the problem of finite deformations, classical elasticity has introduced
many measures of deformation (strain) and the forces to produce the deformations
(stress).Stressandstraincanbedefinedintermsofmatrices(second-ordertensors),
andclassical elasticity textsrelate stress andstrain usinghigher-ordertensormath-
ematics. This makes the mathematics of large deformations difficult, requiring
graduatecoursesinstress,strain,andtensoranalysis.
A different approach is taken in this text. Only one measure of stress and one
measureofstrainisdefined.Therelationshipbetweenthesetwomeasuresisdefined
in terms of energy instead of a direct relationship between stress and strain. The
result is that many finite deformations can be simulated with only matrices
(no higher-order tensors needed). Thus, all that is required is a familiarity with
calculus-basedphysicsandlinearalgebra.
The mathematics needed, example computer simulations, and experiments to
measure material properties for large deformations of materials are all included
here.Themathematicsbeginswithabriefreviewofthephysicsandlinearalgebra.
Deformationsaredescribedintermsofmappingbetweentheinitialandfinalstates
of the material. Forces and the relationship between force and energy follow. The
constraints required on the energy function for isotropic materials are described.
Next,simulationsareshownusingMathematica.Theseareexampleswhichtestand
v
vi PrefaceforStudents
extendthetheory.Averyinexpensiveexperimentalsetupisdescribedtoshowhow
materialpropertiescanbemeasured.Time-dependentsimulationsaredescribed,as
well as the application of this approach to anisotropic materials. A number of
differenttypesofoutputplotsarepresented.
A more eloquent derivation of the equations is made by using the techniques
of Euler and Lagrange, but this requires higher-level physics and mathematics.
A chapter is included to show how the equations of linear elasticity follow from
thefiniteelasticityequations.Finally,achapterisincludedforthosestudentsalready
familiarwithclassicalelasticitytocomparetheapproachofusingasinglestressand
strainmeasurewithclassicalelasticitydescriptions.
Pelzer,SC,USA HumphreyHardy
Preface for Instructors
Thisbookisanengineeringapproachtofinitedeformations.Itisnotintendedtobe
an introduction to classical finite elasticity. Classical finite elasticity introduces a
dozenormorestressandstrainmeasures,makesuseofthetensorpropertyofthese
measures, and expresses the equation of motion in terms of both the reference and
bodycoordinates.Thisapproachdoesnoneofthesethings.Toseetheconnectionof
thisapproachwithclassicalfiniteelasticity,seeChap.16.
Theapproachhereistointroducetheengineertoonlywhatisneededtosolvea
broadrangeofdeformations.Theapproachisbasedonthesamemodelthatisused
in teaching engineers linear elastic deformations. In linear elasticity, engineers are
introduced to Cauchy stress and strain and none of the many other measures of
classical finite elasticity. Cauchy-Green tensors, Kirchhoff stress, and Lagrangian
strain are not introduced in linear elasticity because they are not needed by the
engineertosimulateapplicationsofsmalldeformationsandsmallrotations.Inthis
text,theengineerisintroducedtoonlyonestressandonestrainmeasure,whichisall
thatisneeded tosolvemanyfinitedeformationproblems.Becauseofthissimplifi-
cation, the target audience for this approach can be second-year or third-year
undergraduates, but as discussed below, with a slightly different order of chapters,
thistextcouldalsobeusedforseniorsorgraduatestudents.
Therelationshipbetweenanyfinitestressandstrainmeasureisexpressedinterms
ofnon-lineardifferentialequationsandthereforerequiresnumericalsimulationsfor
applications.ThebodyofthetextusesaminimizationtechniqueusingMathematica
to numerically solve a number of examples. There are many other techniques and
softwareavailablewhichcouldbeusedtosolvetheseexamples,butthisisnotatext
onwritingcomputercode.Instead,theexamplessimplyshowsomeofthebreathof
what is possible with a single stress and strain measure. The codes used are found
onlinewithcommentssothattheenterprisingstudentcanreadandtranslatethiscode
into any language they wish. No homework problems require Mathematica; how-
ever,analgebraicsoftwareorcomputergraphicsprogramwouldbeusefulforsome
of the homework problems. When these are needed, the individual problems
statethis.
vii
viii PrefaceforInstructors
Engineers also require experimental measurements of materials to provide the
material property parameters that occur in the differential equations of elasticity.
Chapter10isincludedtoillustrateonemethodofmakingphysicalmeasurementson
amaterialandturningthesemeasurementsintotheneededmaterialparameters.The
experimentissimpleandcosteffectivesothatitcanbeincludedinanycourse.The
enterprisingstudentshouldhavenoproblemeitherreproducingtheexperimentsor
designingmoreaccurateexperimentsusingmoresophisticated(andexpensive)tools
if this is of interest to the instructor and student. In short, this text provides all an
engineerneedstomodelmanymaterialsinfinitedeformations.
Anyengineerthatrequiresthedescriptionoflargedeformationscanbenefitfrom
thiscourse.Materialslikerubber,softplastics,andbiologicalmaterialsoftenundergo
largedeformationswhenplacedinservice.Materialslikespaghetti,cakeicing,andcar
bumpers are often extruded and also undergo large deformations. Even engineers
producing movies can benefit from a correct description of large deformations of
materials.Thisbook providestheory, experimental measurements of properties, and
numericalsimulationsofthesetypesoflargedeformations.Ineachchapterofthebook,
8–12 problems have been provided. Solutions have been provided in the solutions
manual.Mathematicasoftwarehasbeenprovided(online)fortheexamplesandfigures
inthetext.Thesoftwareisnotdesignedtobecommercialcodebutisprovidedsothat
no steps in the simulation are omitted and the code can be used as a model for the
studenttoeitherextendthemodelsortranslatethemintoothercomputerlanguages.The
Mathematica code can be read using the free Mathematica player found at https://
wolfram.com/player/,buttochangeorexecutethecode,aMathematicalicense,found
athttps:\\wolfram.com\mathematica\,isrequired.
AbasicundergraduatecoursewouldneedtocoverChaps.1,2,3,4,5,and6.After
that,thecoursecangoinmanydirectionsdependingupontheinterestofthestudentand
professor.Forapplicationsofanisotropicmaterials,Chaps.9and12shouldbeadded.For
thestudentwishingtodevelopsoftwaretools,Chaps.7,8,11,and13wouldbehelpful,
butfamiliaritywithprogrammingwouldbeneededforthesechapters.Chapter10covers
experimentalmeasurementsofmaterialproperties.Chapter14wouldbeappropriatefor
moreadvancedseniororgraduatestudentsfamiliarwithLagrangians.Formoreadvanced
students,Chaps.1,2,3,4,5,and6 couldbeomittedandtheapplicationsinChap.7and
onwardapproachedimmediately.Chapters15and16areincludedtomaketheconnec-
tionofthisapproachtolinearelasticityandclassicalfiniteelasticityandwouldalsobe
appropriateforseniorundergraduateorgraduatestudents.
Appendix A derives the mapping between the fixed coordinate system of the
simulationandtheexperimental coordinate system usedtodefine anisotropy prop-
erties.AppendixBprovidesadescriptionofGram-SchmidtQRDwhichisthebasis
oftheanisotropicinvariantsfoundinChap.12.AppendixCprovidesaquickreview
of Euler-Lagrange equations in multiple dimensions for those students with some
familiaritywithLagrangians.AppendixDprovidesthecodeofanexampleshowing
thatfiniteelementtechniquesalsosolvetheequationsoffiniteelasticity.AppendixE
provides a list 50 “projects”, each of which would be appropriate for a class
assignment or an undergraduate or master’s thesis depending upon the project.
Thesemaybeused“asis”orasastartingpointforastudent’sownprojectideas.
Pelzer,SC,USA HumphreyHardy
Contents
1 GettingReady(MostlyReview). . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
LinearAlgebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Scalars,Vector,andMatrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
AdditionandSubtractionofVectors. . . . . . . . . . . . . . . . . . . . . . . 3
MultiplicationofaScalarTimesaVector. . . . . . . . . . . . . . . . . . . . 3
MultiplicationofVectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
AdditionandSubtractionofMatrices. . . . . . . . . . . . . . . . . . . . . . . 5
AScalarTimesaMatrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
MultiplicationofMatrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
MultiplicationofaMatrixTimesaVector. . . . . . . . . . . . . . . . . . . 6
TransposeofaMatrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
InverseofaMatrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
LengthofaVector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
AVectorFieldandMatrixField. . . . . . . . . . . . . . . . . . . . . . . . . . 8
AMatrixTimesaVectorField. . . . . . . . . . . . . . . . . . . . . . . . . . . 8
AMatrixFieldTimesaVectorField. . . . . . . . . . . . . . . . . . . . . . . 9
Calculus-BasedPhysics1Review. . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Newton’sLawsinEquationForm. . . . . . . . . . . . . . . . . . . . . . . . . 9
Force,Work,andEnergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Miscellaneous. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
TheDifferenceBetweenPoint,PointVector,
andaGeneralVector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
MaterialProperties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
IsotropicandAnisotropicMaterials. . . . . . . . . . . . . . . . . . . . . . . . 12
HomogeneousandNon-homogeneousMaterials. . . . . . . . . . . . . . . 12
TaylorExpansionandPhysicalInterpretation. . . . . . . . . . . . . . . . . 12
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
ix
x Contents
2 Deformations. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . . .. . . . . . . .. 17
AMap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
MathematicsofaMapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
TheApplicationofaMapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4 Force-EnergyRelationships. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 43
Springs. . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 43
Young’sModulus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
InternalForces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
EnergyfromSurfaceForces. . . .. . . . . . . . . . . . .. . . . . . . . . . . . . .. 47
EquationofMotioninTermsofEnergy. . . . . . . . . . . . . . . . . . . . . . . 53
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5 IsotropicMaterials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
RotationsandTranslations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
RotationalInvariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6 MinimizingEnergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
SpringModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Discrete3DModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Continuous3DModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7 Simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
UserInput. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
DefineNodesandTheirConnectivity. . . . . . . . . . . . . . . . . . . . . . . . . 77
PositionBoundaryConditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
ForceBoundaryConditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
CalculateInitialVolumes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
CalculateDeformationEnergyperUnitInitialVolume. . . . . . . . . . . . 81
DefineVariableList. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Gravity. . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 83
Go. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
SimulatingIncompressibleMaterials. . . . . . . . . . . . . . . . . . . . . . . 84
ScalinginSimulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
8 Quasi-staticSimulationExamples. . . . . . . . . . . . . . . . . . . . . . . . . . 89
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
SingleGridStudy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90