Non-Classical Continuum Theories for Solid and Fluent Continua Aaron Joy

191 Pages·2017·0.71 MB·English

Non-Classical Continuum Theories for Solid and Fluent
Continua
By
Aaron Joy
SubmittedtothegraduatedegreeprograminMechanicalEngineeringandthe
GraduateFacultyoftheUniversityofKansasinpartialfulfillmentofthe
requirementsforthedegreeofDoctorofPhilosophy
Prof. KaranS.Surana,Chairperson
Prof. PeterTenPas
Committeemembers
Prof. RobertSorem
Prof. RayTaghavi
Prof. AlfredParr
Datedefended:
TheDissertationCommitteeforAaronJoycertifies
thatthisistheapprovedversionofthefollowingdissertation:
Non-ClassicalContinuumTheoriesforSolidandFluentContinua
Prof. KaranS.Surana,Chairperson
Dateapproved:
ii
Abstract
This dissertation presents non-classical continuum theories for solid and fluent con-
tinua. In these theories additional physics due to internal rotations and rotation rates
arising from the Jacobian of deformation and the velocity gradient tensor as well as
Cosserat rotations and rotation rates are considered. While the internal rotations and
rotation rates are completely defined by the deformation physics, the Cosserat rota-
tionsandCosseratrotationratesareadditionaldegreesoffreedomatamaterialpoint.
Thenon-classicaltheoriesthatonlyconsidertheinternalrotationsandtheinternalro-
tationratesarereferredtoasinternalpolartheories,whilethosethatconsiderbothare
calledpolarornon-classicaltheories.
The conservation and balance laws and the constitutive theories are derived for non-
classicalcontinuumtheories. Itisshownthatthesenon-classicaltheoriesrequiremod-
ifications of the balance laws used in classical continuum theories. In the presence of
additional rotation and rotation rate physics in non-classical theories, the modifica-
tions of the balance laws used in classical continuum theories are not sufficient to
ensure equilibrium of the deforming matter. It is shown that these theories require
the balance of moments of moments as an additional balance law. The constitutive
theories for solid and fluent continua are derived using the conditions resulting from
theentropyinequalityandtherepresentationtheorem. Useofintegrityintheirderiva-
tions ensures completeness of the resulting constitutive theories. Specific derivations
and details of the constitutive theories for thermoelastic and thermoviscoelastic solids
with and without memory are presented for small deformation, small strain physics.
Detailed derivations of the constitutive theories for compressible as well as incom-
iii
pressible thermoviscous and thermoviscoelastic fluent continua are also presented.
Retardationand/ormemorymoduliarederivedforpolymericsolidsandfluids.
Thepresenttheoriesarecomparedwithpublishedworks,particularlywiththemicrop-
olar theories of Eringen, to highlight the significance and the thermodynamic consis-
tencyofthepresentwork,aswellastocontrastthedifferences.
iv
Acknowledgements
I would like to take this opportunity to acknowledge those who helped me reach this
point. First and foremost I want to thank my advisor, Dr. Karan Surana. His knowl-
edge and enthusiasm have been a constant inspiration during my time here, and all of
the work done in this thesis would have been impossible without his guidance. The
time he has taken to educate and improve me as a researcher and a scientist has had a
profoundimpactonmeandmycareer. IalsowouldliketothankDr. PeterW.TenPas,
Dr. RobertSorem,Dr. DaveParr,andDr. RayTaghaviforservingonmycommittee.
There are others who also deserve to be recognized for their contributions to my suc-
cess. The other students in the Computational Mechanics program, both past and
present, are thanked for the work they have done that has led to this. Dr. Ephraim
Washburn, of the Naval Air Warfare Center, is also thanked for supporting my desire
tocontinuemyeducation,eventhoughhepersonallygainednothingfromthateffort.
Finally, the one person who deserves the most credit for all of this is my wife, An-
drea. Without her endless support, this would not have even been a possibility. Her
dedicationtomysuccessmeansmoretomethanIcandescribe.
v
Contents
1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 LiteratureReview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 ScopeofWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 NotationsandPreliminaryConsiderations 8
2.1 InternalPolarNon-ClassicalContinuumTheories . . . . . . . . . . . . . . . . . . 8
2.2 NotationsandMeasures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 SolidContinua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1.1 ConsiderationsofJ,Stress,andMomentTensors . . . . . . . . 15
2.2.2 FluidContinua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.2.1 CovariantandContravariantBases . . . . . . . . . . . . . . . . 22
2.2.2.2 ConsiderationsofL¯,Stress,Moment,andStrainRateTensors . . 22
2.3 BalanceofMomentsofMomentsBalanceLaw . . . . . . . . . . . . . . . . . . . 26
3 MathematicalModelsforSolidContinua 28
3.1 ConservationandBalanceLaws . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1.1 ConservationofMass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.2 BalanceofLinearMomenta . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.3 BalanceofAngularMomenta . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1.4 BalanceofMomentsofMoments . . . . . . . . . . . . . . . . . . . . . . 34
vi
3.1.5 FirstLawofThermodynamics . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.6 SecondLawofThermodynamics . . . . . . . . . . . . . . . . . . . . . . 39
3.2 RateofWorkConjugatePairsintheEntropyInequality . . . . . . . . . . . . . . . 43
3.2.1 SummaryoftheConservationandBalanceLaws . . . . . . . . . . . . . . 45
3.3 ConstitutiveTheoriesforThermoelasticSolids . . . . . . . . . . . . . . . . . . . 45
3.3.1 DependentVariablesintheConstitutiveTheories . . . . . . . . . . . . . . 46
3.3.2 EntropyInequality: FurtherConsiderations . . . . . . . . . . . . . . . . . 47
3.3.3 ConstitutiveTheoryfor σσσ . . . . . . . . . . . . . . . . . . . . . . . . . . 48
s
3.3.3.1 Constitutive Theory for σσσ Using Φ as a Function of the Invari-
s
antsofεεεandθ . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.3.2 ConstitutiveTheoryfor σσσ UsingtheRepresentationTheorem . 49
s
3.3.3.3 MaterialCoefficientsintheConstitutiveTheoryfor σσσ . . . . . 49
s
3.3.4 ConstitutiveTheoryformmm . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3.4.1 Constitutive Theory formmm Using Φ as a Function of the Invari-
antsoftΘJJJ andθ . . . . . . . . . . . . . . . . . . . . . . . . . . 52
s
3.3.4.2 ConstitutiveTheoryformmmUsingtheRepresentationTheorem . 53
3.3.4.3 MaterialCoefficientsintheConstitutiveTheoryformmm . . . . . . 53
3.3.5 ConstitutiveTheoryfor σσσ . . . . . . . . . . . . . . . . . . . . . . . . . . 54
a
3.3.5.1 Constitutive Theory for σσσ Using Φ as a Function of the Invari-
a
antsof rrr andθ . . . . . . . . . . . . . . . . . . . . . . . . . . 55
a
3.3.5.2 ConstitutiveTheoryfor σσσ UsingtheRepresentationTheorem . 56
a
3.3.5.3 MaterialCoefficientsintheConstitutiveTheoryfor σσσ . . . . . 57
a
3.3.6 LinearConstitutiveTheoriesfor σσσ,mmm,and σσσ . . . . . . . . . . . . . . . 57
s a
3.3.7 MathematicalModelofEringen . . . . . . . . . . . . . . . . . . . . . . . 59
3.3.7.1 ConservationandBalanceLaws . . . . . . . . . . . . . . . . . . 60
3.3.7.2 ConstitutiveTheories . . . . . . . . . . . . . . . . . . . . . . . 61
3.4 ConstitutiveTheoriesforThermoviscoelasticSolidswithoutMemory . . . . . . . 64
vii
3.4.1 DependentVariablesintheConstitutiveTheories . . . . . . . . . . . . . . 65
3.4.2 EntropyInequality: FurtherConsiderations . . . . . . . . . . . . . . . . . 66
3.4.3 ConstitutiveTheoryfor ( σσσ) . . . . . . . . . . . . . . . . . . . . . . . . 71
d s
3.4.3.1 MaterialCoefficientsintheConstitutiveTheoryfor ( σσσ) . . . . 71
d s
3.4.4 ConstitutiveTheoryformmm . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.4.4.1 MaterialCoefficientsintheConstitutiveTheoryformmm . . . . . . 72
3.4.5 ConstitutiveTheoryfor σσσ . . . . . . . . . . . . . . . . . . . . . . . . . . 73
a
3.4.5.1 MaterialCoefficientsintheConstitutiveTheoryfor σσσ . . . . . 74
a
3.4.6 LinearConstitutiveTheoriesfor ( σσσ),mmm,and σσσ . . . . . . . . . . . . . 74
d s a
3.4.6.1 SimplifiedConstitutiveTheoryfor ( σσσ) . . . . . . . . . . . . . 74
d s
3.4.6.2 SimplifiedConstitutiveTheoryformmm . . . . . . . . . . . . . . . 75
3.4.6.3 SimplifiedConstitutiveTheoryfor σσσ . . . . . . . . . . . . . . 75
a
3.5 ConstitutiveTheoriesforThermoviscoelasticSolidswithMemory . . . . . . . . . 77
3.5.1 DependentVariablesintheConstitutiveTheories . . . . . . . . . . . . . . 77
3.5.2 EntropyInequality: FurtherConsiderations . . . . . . . . . . . . . . . . . 79
3.5.3 ConstitutiveTheoryford(sσσσ[msσ]) . . . . . . . . . . . . . . . . . . . . . . 84
3.5.3.1 MaterialCoefficientsintheConstitutiveTheoryford(sσσσ[msσ]) . 85
3.5.4 ConstitutiveTheoryformmm[mm] . . . . . . . . . . . . . . . . . . . . . . . . 86
3.5.4.1 MaterialCoefficientsintheConstitutiveTheoryformmm[mm] . . . 86
3.5.5 ConstitutiveTheoryforaσσσ[maσ] . . . . . . . . . . . . . . . . . . . . . . . 87
3.5.5.1 MaterialCoefficientsintheConstitutiveTheoryforaσσσ[maσ] . . . 87
3.5.6 SimplifiedConstitutiveTheoriesford(sσσσ[msσ]),mmmmm,andaσσσmaσ . . . . . 88
3.5.6.1 SimplifiedConstitutiveTheoryford(sσσσ[msσ]) . . . . . . . . . . . 90
3.5.6.2 SimplifiedConstitutiveTheoryformmm[mm] . . . . . . . . . . . . 92
3.5.6.3 SimplifiedConstitutiveTheoryforaσσσ[maσ] . . . . . . . . . . . . 93
3.5.6.4 RetardationandMemoryModuli . . . . . . . . . . . . . . . . . 95
3.6 ConstitutiveTheoriesforHeatVectorqqq . . . . . . . . . . . . . . . . . . . . . . . 98
viii
4 MathematicalModelsforFluidContinua 100
4.1 ConservationandBalanceLaws . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.1.1 ConservationofMass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.1.2 BalanceofLinearMomenta . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.1.3 BalanceofAngularMomenta . . . . . . . . . . . . . . . . . . . . . . . . 103
4.1.4 BalanceofMomentsofMoments . . . . . . . . . . . . . . . . . . . . . . 106
4.1.5 FirstLawofThermodynamics . . . . . . . . . . . . . . . . . . . . . . . . 108
4.1.6 SecondLawofThermodynamics . . . . . . . . . . . . . . . . . . . . . . 110
4.2 RateofWorkConjugatePairsintheEntropyInequality . . . . . . . . . . . . . . . 114
4.2.1 SummaryoftheConservationandBalanceLaws . . . . . . . . . . . . . . 116
4.3 ConstitutiveTheoriesforThermoviscousFluids . . . . . . . . . . . . . . . . . . . 117
4.3.1 DependentVariablesintheConstitutiveTheories . . . . . . . . . . . . . . 117
4.3.2 EntropyInequality: FurtherConsiderations . . . . . . . . . . . . . . . . . 118
4.3.2.1 DecompositionoftheSymmetricCauchyStressTensor(0)σσσ¯ . . . 120
s
4.3.2.2 Constitutive Theory for Equilibrium Stress ((0)σσσ¯): Compress-
e s
ibleThermoviscousFluids . . . . . . . . . . . . . . . . . . . . 121
4.3.2.3 ConstitutiveTheoryforEquilibriumStress ( σσσ¯(0)): Incompress-
e s
ibleThermoviscousFluids . . . . . . . . . . . . . . . . . . . . 121
4.3.3 Final Choice of the Dependent Variables and Their Argument Tensors in
theConstitutiveTheories . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.3.4 ConstitutiveTheoryfor ((0)σσσ¯) . . . . . . . . . . . . . . . . . . . . . . . . 125
d s
4.3.4.1 MaterialCoefficients . . . . . . . . . . . . . . . . . . . . . . . 125
4.3.4.2 RateConstitutiveTheoryofOrderOne(n = 1)for ( σσσ¯(0)) . . . 127
d s
4.3.4.3 LinearRateConstitutiveTheoryofOrderOne(n = 1)for ((0)σσσ¯)128
d s
4.3.5 ConstitutiveTheoryfor(0)mmm¯ . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.3.5.1 MaterialCoefficients . . . . . . . . . . . . . . . . . . . . . . . 130
4.3.5.2 LinearConstitutiveTheoryfor(0)mmm¯ . . . . . . . . . . . . . . . 131
ix
4.3.6 ConstitutiveTheoryfor(0)σσσ¯ . . . . . . . . . . . . . . . . . . . . . . . . . 131
a
4.3.6.1 MaterialCoefficients . . . . . . . . . . . . . . . . . . . . . . . 132
4.3.6.2 LinearConstitutiveTheoryfor(0)σσσ¯ . . . . . . . . . . . . . . . . 133
a
4.3.7 MathematicalModelofEringen . . . . . . . . . . . . . . . . . . . . . . . 134
4.3.7.1 ConservationandBalanceLaws . . . . . . . . . . . . . . . . . . 135
4.3.7.2 ConstitutiveTheories . . . . . . . . . . . . . . . . . . . . . . . 136
4.3.8 IncompressibleMatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.4 ConstitutiveTheoriesforThermoviscoelasticFluidswithMemory . . . . . . . . . 140
4.4.1 DependentVariablesintheConstitutiveTheories . . . . . . . . . . . . . . 140
4.4.2 EntropyInequality: FurtherConsiderations . . . . . . . . . . . . . . . . . 142
4.4.2.1 DecompositionoftheSymmetricCauchyStressTensor(0)σσσ¯ . . . 144
s
4.4.2.2 Constitutive Theory for Equilibrium Stress ((0)σσσ¯): Compress-
e s
ibleThermoviscoelasticFluids . . . . . . . . . . . . . . . . . . 145
4.4.2.3 ConstitutiveTheoryforEquilibriumStress ((0)σσσ¯): Incompress-
e s
ibleThermoviscoelasticFluids . . . . . . . . . . . . . . . . . . 145
4.4.3 Final Choice of the Dependent Variables and Their Argument Tensors in
theConstitutiveTheories . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.4.4 ConstitutiveTheoryfor ((msσ)σσσ¯) . . . . . . . . . . . . . . . . . . . . . . 148
d s
4.4.4.1 MaterialCoefficients . . . . . . . . . . . . . . . . . . . . . . . 148
4.4.5 ConstitutiveTheoryfor(mm)mmm¯ . . . . . . . . . . . . . . . . . . . . . . . . 149
4.4.5.1 MaterialCoefficients . . . . . . . . . . . . . . . . . . . . . . . 150
4.4.6 ConstitutiveTheoryfor(maσ)σσσ¯ . . . . . . . . . . . . . . . . . . . . . . . . 151
a
4.4.6.1 MaterialCoefficients . . . . . . . . . . . . . . . . . . . . . . . 151
4.4.7 SimplifiedConstitutiveTheoriesford((msσs)σσσ¯),(mm)mmm¯ ,and(maσa)σσσ¯ . . . . . 152
4.4.7.1 SimplifiedConstitutiveTheoryfor ((msσ)σσσ¯) . . . . . . . . . . . 154
d s
4.4.7.2 SimplifiedConstitutiveTheoryfor(mm)mmm¯ . . . . . . . . . . . . 156
4.4.7.3 SimplifiedConstitutiveTheoryfor(maσ)σσσ¯ . . . . . . . . . . . . 158
a
x