Elements of Classical Plasticity Theory

116 Pages·2022·1.974 MB·English

by  Andreas Öchsner

Andreas Öchsner
Elements
of Classical
Plasticity
Theory
Elements of Classical Plasticity Theory
Andreas Öchsner
Elements of Classical
Plasticity Theory
AndreasÖchsner
FacultyofMechanicalandSystems
Engineering
EsslingenUniversityofAppliedSciences
EsslingenamNeckar
Baden-Württemberg,Germany
ISBN 978-3-031-14200-0 ISBN 978-3-031-14201-7 (eBook)
https://doi.org/10.1007/978-3-031-14201-7
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Preface
This monograph provides a compact introduction into the classical, i.e., rate-
independent,plasticitytheory.Startingfromtheengineeringstress–straindiagram,
theconceptofelasticandelasto-plasticmaterialbehaviorisintroduced,aswellasthe
conceptofuniaxialandmultiaxialstressstates.Continuummechanicalmodelingin
theelasto-plasticrangerequires,inregardtotheconstitutiveequation,inadditionto
theelasticlaw(e.g.,Hooke’slaw),ayieldcondition,aflowruleandahardeningrule.
Thesebasicequationsarethoroughlyintroducedandexplainedforone-dimensional
stressstates.Consideringthree-dimensionalplasticity,differentsetsofstressinvari-
antstocharacterizethestressmatrixandthedecompositionofthestressmatrixinits
hydrostaticanddeviatoricpartareintroduced.Furthermore,theconceptoftheyield
condition, flow rule and hardening rule is generalized for multiaxial stress states.
Sometypicalyieldconditionsareintroduced,andtheirgraphicalrepresentationin
differentstressspacesisdiscussedindetail.Thebookconcludeswithanintroduction
intheelasto-plasticfiniteelementsimulationofmechanicalstructures.Inthecontext
ofnumericalapproximationmethods,theso-calledpredictor–correctormethodsare
usedtointegratetheconstitutiveequations.Thisisagainintroducedindetailbased
onone-dimensionalstressstatesandafterwardgeneralizedtothethree-dimensional
case.
Esslingen,Germany AndreasÖchsner
June2022
v
Contents
1 Introduction ................................................... 1
1.1 UniaxialTensileTesting ..................................... 1
1.2 ContinuumMechanicalModelling ............................ 9
References ..................................................... 11
2 TheoryofOne-DimensionalPlasticity ............................ 13
2.1 InitialRemarks ............................................. 13
2.2 YieldCondition ............................................ 14
2.3 FlowRule ................................................. 16
2.4 HardeningRule ............................................ 17
2.4.1 IsotropicHardening .................................. 17
2.4.2 KinematicHardening ................................. 19
2.4.3 CombinedHardening ................................. 20
2.5 Elasto-plasticModulus ...................................... 21
2.6 ConsiderationofUnloading,ReversedLoadingandCyclic
Loading ................................................... 24
References ..................................................... 25
3 TheoryofThree-DimensionalPlasticity .......................... 27
3.1 CommentsontheStressMatrix ............................... 27
3.2 GraphicalRepresentationofYieldConditions .................. 35
3.3 YieldConditions ........................................... 41
3.3.1 MisesYieldCondition ................................ 41
3.3.2 TrescaYieldCondition ............................... 46
3.3.3 Drucker-PragerYieldCondition ........................ 51
3.3.4 SayirYieldCondition ................................. 55
3.4 FlowRule ................................................. 58
3.5 HardeningRule ............................................ 58
3.5.1 IsotropicHardening .................................. 58
3.5.2 KinematicHardening ................................. 59
References ..................................................... 59
vii
viii Contents
4 Elasto-plasticFiniteElementSimulations ......................... 61
4.1 ApproachforOne-DimensionalProblems ...................... 61
4.1.1 IntegrationoftheMaterialEquations ................... 61
4.1.2 Derivation of the Fully Implicit Backward-Euler
AlgorithmforIsotropicHardening ...................... 67
4.1.3 Derivation of the Fully Implicit Backward-Euler
AlgorithmforKinematicHardening .................... 77
4.1.4 Derivation of the Fully Implicit Backward-Euler
AlgorithmforCombinedHardening .................... 78
4.1.5 Derivation of the Semi-implicit Backward-Euler
AlgorithmforIsotropicHardening ...................... 80
4.1.6 SampleProblemsandSupplementaryProblems .......... 81
4.2 ApproachforThree-DimensinalProblems ..................... 94
4.2.1 DifferentiationoftheYieldConditions .................. 94
4.2.2 Derivation of the Fully Implicit Backward Euler
AlgorithmforIsotropicHardening ...................... 97
References ..................................................... 101
Index ............................................................. 103
Symbols and Abbreviations
LatinSymbols(Capital Letters)
A Area
C Elasticitymatrix
Dpl Generalizedplasticmodulus
D Compliancematrix
E Modulusofelasticity
˜
E Modulus
Eelpl Elastic–plasticmodulus
Epl Plasticmodulus(isotropichardening)
F Force,
Yieldcondition
G Shearmodulus
H Kinematichardeningmodulus
I Principalinvariant,
Secondmomentofarea
I◦ Principalinvariantofhydrostaticstressmatrix
I(cid:3) Principalinvariantofdeviatoricstressmatrix
I Identitymatrix(diagonalmatrix),I=(cid:4)111...(cid:5)
J Basicinvariant
J◦ Basicinvariantofhydrostaticstressmatrix
J(cid:3) Basicinvariantofdeviatoricstressmatrix
K Bulkmodulus
K Stiffnessmatrix
L Length
L Scalingmatrix(diagonalmatrix),L=(cid:4)111222(cid:5)
M Massmatrix
O Zeromatrix(diagonalmatrix),O=(cid:4)000000(cid:5)
Q Plasticpotentialfunction(plasticpotential)
ix
x SymbolsandAbbreviations
R Radiusofcurvature,
Stressratio
T Meltingtemperature
mt
LatinSymbols(SmallLetters)
a Factor(kinematichardening)
d Diameter
f Stressfraction
f Columnmatrixofexternalforces
h Evolutionfunctionofhardeningparameter
h Evolutionfunctionmatrixofhardeningparameters
j Cycleindex
k Yieldstress
k Compressiveyieldstress
c
k Shearyieldstress
s
k Tensileyieldstress
t
kinit Initialcompressiveyieldstress
c
kinit Initialtensileyieldstress
t
m Vectorfunction
m˜ ElementoftheJacobianmatrix
n Incrementnumber
p Pressure
q Internalvariable(hardening)
q Columnmatrixofhardeningvariables
r Plasticflowdirection,
Radius,
Residual
r Matrixofplasticflowdirection,
Residuummatrix
s Deviatoricstressmatrix
ij
s Deviatoricstressmatrix
t Time
u Displacement
u Columnmatrixofdisplacements
v Argumentmatrix
w Volume-specificwork
wpl Volume-specificplasticwork
ws Volume-specificdistortionaldeformationenergy
x Cartesiancoordinate
x Columnmatrixofarguments
SymbolsandAbbreviations xi
y Cartesiancoordinate
z Cartesiancoordinate
GreekSymbols(Capital Letters)
Φ Modalmatrix
GreekSymbols(SmallLetters)
α Backstress/kinematichardeningparameter,
Factor
α Columnmatrixofkinematichardeningparameter
γ Shearstrain(engineeringdefinition)
ε Normalstrain
εel Elasticstrain
εpl Plasticstrain
εpl Effectiveplasticstrain
eff
εf Tensilestrainatfracture
t
εinit Initialtensileyieldstrain
t
ε Truestrain
tr
εel Trueelasticstrain
tr
εpl Trueplasticstrain
tr
ε Strainmatrix
ij
ε Strainmatrix
η Parameter
θ Lodeangle
κ Isotropichardeningparameter
λ Lamé’sconstant,
Consistencyparameter
μ Lamé’sconstant,
Proportionalityfactor(kinematichardening)
ν Poisson’sratio
ξ Haigh–Westergaardcoordinate
π Volume-specificenergy
π¯ Volume-specificcomplementaryenergy
ρ Haigh–Westergaardcoordinate
σ Normalstress
σ Stressamplitude
a
σ Effectivestress
eff
σ Hydrostaticstress
m