Shape Memory Alloy Cellular Solids

128 Pages·2009·10.11 MB·English

Shape Memory Alloy Cellular Solids
by
PetrosA.Michailidis
Adissertationsubmittedinpartialfulfillment
oftherequirementsforthedegreeof
DoctorofPhilosophy
(AerospaceEngineering)
inTheUniversityofMichigan
2009
DoctoralCommittee:
AssociateProfessorJohnA.Shaw,Co-Chair
ProfessorNicolasTriantafyllidis,Co-Chair
AssistantProfessorSamanthaDaly
ProfessorDavidS.Grummon
PetrosMichailidis 2009
(cid:13)c
AllRightsReserved
Acknowledgements
Iwishtothankallwhomadethisworkpossible.... ProfessorsTriantafyllidisandShaw,
my colleagues Jesse Thomas, Chris Churchill, Ben Reedlunn, the staff at the Department
ofAerospaceEngineering,myfriendsandrelatives,andpeoplewhofundedtheproject.
ii
Table of Contents
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
ListofFigures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
ListofTables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
Chapter
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
II. SuperelasticHoneycombs . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.2 ConstitutiveModel . . . . . . . . . . . . . . . . . . . . 7
2.2.3 StabilityoftheSMAHoneycombStructure . . . . . . . 11
2.3 NumericalApproach . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 ConstitutiveParameters . . . . . . . . . . . . . . . . . 17
2.3.2 FiniteElementDiscretizationandNumericalAlgorithm 18
2.4 ResultsandDiscussion . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.1 PrincipalBranchoftheInfinite,PerfectHoneycomb . . 20
2.4.2 StabilityoftheInfinite,PerfectHoneycomb . . . . . . . 31
2.4.3 ResponseoftheFinite,PerfectHoneycomb . . . . . . . 38
2.4.4 ResponseofFinite,ImperfectHoneycombs . . . . . . . 42
iii
2.4.5 ResponseoftheFabricatedNitinolHoneycomb . . . . . 46
III. DesignforEnergyAbsorption . . . . . . . . . . . . . . . . . . . . . . . 51
3.1 DesignObjectives . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2 FamilyofShapes . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 EnergyAbsorptionResults . . . . . . . . . . . . . . . . . . . . . 59
IV. ShapeMemoryHoneycombs . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 ConstitutiveModel . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2.1 Phasefractionspace . . . . . . . . . . . . . . . . . . . 71
4.2.2 Freeenergy . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.3 Kineticlaw . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2.4 Illustrativeexamplesofmaterialmodelbehavior . . . . 76
4.3 Implementationofmodel . . . . . . . . . . . . . . . . . . . . . . 79
4.3.1 Newton-RaphsonScheme . . . . . . . . . . . . . . . . 80
4.4 ShapeMemorySimulations . . . . . . . . . . . . . . . . . . . . . 83
4.4.1 1DElementSimulations . . . . . . . . . . . . . . . . . 83
4.4.2 HoneycombStructuresSimulations . . . . . . . . . . . 89
V. SummaryandConclusions . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.1 Superelastic Behavior of SMA Honeycombs and Design Consid-
erations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.2 ShapeMemoryEffectinSMAHoneycombs . . . . . . . . . . . . 98
Appendix
A. Performanceofcellsusingdifferentmetrics . . . . . . . . . . . . . . . 100
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
iv
List of Figures
Figure
2.1 (a)Referenceconfigurationgeometryandglobalcoordinates(X ,X )of
1 2
the perfect SMA honeycomb. The honeycomb is compressed along the
X direction. The magnified view is the periodic unit cell used in sta-
2
bility calculations for the infinite-perfect honeycomb. (b) Kinematics of
cell-wall deformations with respect to local coordinates (x,y), showing
displacements (v,w) between reference configuration (AB) and current
configuration(A(cid:48)B(cid:48)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Uniaxial superelastic response of SMA material and associated constitu-
tiveparameters. SubscriptsAorM refertoAusteniteorMartensitewhile
superscripts (·)+ or (·)− indicate association with tensile or compressive
response,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 (a) Principal branches of the dimensionless macroscopic compressive
stress (F/EA) vs. compressive strain (δ/H) response of the infinite-
perfect honeycomb, based on constitutive Model 1 (right inset), showing
theinfluenceofnucleationstrain(ε ). . . . . . . . . . . . . . . . . . . . 24
n
2.3 (b)Contoursofphasefraction(ξ),localstrain(ε),andlocalstress(σ/E)
intheslantedcellwall(Model1,ε = 0.006). . . . . . . . . . . . . . . 24
n
2.3 (c) Maximum local tensile strain (ε ) as a function of macroscopic
max
compression(δ/H)forthesameconstitutivelawsof2.3(a). . . . . . . . 25
2.4 Principal branches of the dimensionless macroscopic compressive stress
(F/EA) vs. compressive strain (δ/H) response of the infinite-perfect
honeycomb, based on constitutive Model 1 (right inset), showing the in-
fluenceoftransformationtangentmodulus(E ). . . . . . . . . . . . . . . 26
t
v
2.5 Principal branches of the dimensionless macroscopic compressive stress
(F/EA) vs. compressive strain (δ/H) response of the infinite-perfect
honeycomb,basedonasymmetricconstitutiveModel2(rightinset),show-
ingtheinfluenceoftensiletransformationtangentmodulus(E+). . . . . 27
t
2.6 Principal branches of the dimensionless macroscopic compressive stress
(F/EA) vs. compressive strain (δ/H) load-unload responses of the
infinite-perfecthoneycomb,basedonhystereticconstitutiveModel1(right
inset), showing the influence of stress hysteresis (∆σ/E). Unloading
pathsshownstartatmacroscopicstrainsδ/H = 0.3,0.5,0.7. . . . . . . . 28
2.7 Principal branches of the dimensionless macroscopic compressive stress
(F/EA) vs. compressive strain (δ/H) load-unload responses of the
infinite-perfect honeycomb, based on asymmetric, hysteretic constitu-
tiveModel2(rightinset),showingtheinfluenceoftensiletransformation
modulus(E+). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
t
2.8 Principal branches of the dimensionless macroscopic compressive stress
(F/EA) vs. compressive strain (δ/H) response of the infinite-perfect
honeycomb, based on constitutive Model 1 (right inset), comparing the
unloadingresponseofthehystereticmodelwithan“elastic”one. . . . . 30
2.9 Stability of macroscopic compressive stress-strain, (F/EA) vs. (δ/H),
responses for the perfect, infinite honeycomb under δ/H control, show-
ing the influence of nucleation strain (ε ). Stable and unstable segments
n
are indicated by thick and thin lines, respectively, and stability changes
areshownwithcirclesonthepaths. . . . . . . . . . . . . . . . . . . . . 32
2.10 Loweststrain(δ/H)asafunctionofdimensionlesswavenumbers(ω L ,ω L )
1 1 2 2
on the onset of instability of the infinite-perfect honeycomb calculated
alongtheprincipalpathwithasymmetric,“elastic”constitutivelaw(Model1).
The minimum (δ/H = 0.092) occurs as (ω L ,ω L ) → (0,0), indicat-
1 1 2 2
ing a long wavelength critical mode, which is confirmed from a loss of
ellipticitycalculationofthehomogenizedincrementalmoduli. . . . . . . 35
2.11 Stability of macroscopic compressive stress-strain, (F/EA) vs. (δ/H),
responses for the perfect, infinite honeycomb under δ/H control, show-
ing the influence of transformation modulus (E ) using Model 1 as the
t
base case. Note, multiple changes in stability occur along the response
forE = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
t
vi
2.12 Stability of macroscopic compressive stress-strain, (F/EA) vs. (δ/H),
responses for the perfect, infinite honeycomb under δ/H control based
on asymmetric Model 2, showing the influence of tensile transformation
modulus(E+). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
t
2.13 Stability of macroscopic compressive stress-strain, (F/EA) vs. (δ/H),
load-unload responses for the perfect, infinite honeycomb under δ/H
control,showingtheinfluenceofstresshysteresis(∆σ/E)forsymmetric,
hystereticmaterial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.14 Stability of macroscopic compressive stress-strain, (F/EA) vs. (δ/H),
load-unload responses for the perfect, infinite honeycomb under δ/H
control, showing the influence of tensile transformation modulus (E+)
t
forasymmetric,hystereticmaterial(basedonModel2). . . . . . . . . . 37
2.15 (a) Comparison of compressive loading-unloading responses of the fi-
nite (upper left inset) and infinite (upper right inset) perfect honeycombs
with a symmetric, hysteretic constitutive response (Model 1). (b) Rela-
tive deviation (cid:107)u − u (cid:107)/(cid:107)u (cid:107) of the finite honeycomb’s deformation
# #
(u) from the infinite honeycomb’s principal path deformation (u ) vs.
#
macroscopic compressive strain (δ/H). The finite and periodic configu-
rations are shown in insets at strains 0.2, 0.3 and 0.5 (loading path con-
figurations A1, A2, A3 and the unloading path configurations B1 and
B2.) (c) Magnified view of configuration A2 for the finite and infinite
(periodic)configurations. . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.16 Comparison of compressive loading-unloading responses for the finite
(upperleftinset)andinfinite(upperrightinset)perfecthoneycombswith
anasymmetric,hystereticconstitutiveresponse(Model2). . . . . . . . . 41
2.17 (a) Comparison of compressive loading-unloading responses for finite
perfect and imperfect honeycombs (ζ = 0,0.01,0.1) with a symmet-
ric, hysteretic constitutive response (Model 1). (b) Relative deviation
(cid:107)u−u (cid:107)/(cid:107)u (cid:107)ofthefiniteperfectandimperfecthoneycombdeforma-
# #
tionsvs. macroscopiccompressivestrain(δ/H). . . . . . . . . . . . . . 44
2.18 Comparison of compressive loading-unloading responses for finite per-
fect and imperfect honeycombs (ζ = 0,0.01,0.1) with an asymmetric,
hystereticconstitutiveresponse(Model2). . . . . . . . . . . . . . . . . 45
vii
2.19 (a)PhotographofNitinolhoneycombspecimen. (b)Experimentalisother-
mal compressive, displacement-controlled response subjected to load-
unload cycles in progressively larger 5 % strain increments (data taken
fromFigure5of[26].) . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.20 (a) Macroscopic load-unload stress-strain responses: experimental mea-
surementfromcycle6ofFigure2.19(b)(boldline)andFEMsimulations
(actual geometry using Model 3) with two values of friction coefficient,
µ = 0 (dotted line), µ = 0.3 (thin solid line). The infinite-perfect honey-
comb (upper left inset) principal path (thin line) is also shown for com-
parison. (b) Simulated configuration (µ = 0.3) and experimental image
near30%strain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.1 Principal branch of the maximum local strain ε seen anywhere in the
max
structurevs. compressivestrain(δ/H). . . . . . . . . . . . . . . . . . . 55
3.2 Calculation of the absorbed energy by compressing honeycomb up to
(δ/H)| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
max
3.3 Sampleofunitcellusedinparameterstudyforenergyabsorption,show-
ing the definition of some useful parameters. Here a case with θ > 0 is
depicted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4 Sampleofunitcellusedinparameterstudyforenergyabsorption,show-
ing the definition of some useful parameters. Here a case with θ < 0 is
depicted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.5 Design chart of dimensionless energy absorption per unit cell material
volume for cells with θ = 15◦. A limiting local strain value of ε =
max
0.025isassumed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.6 Design chart of dimensionless energyabsorption per unit cell material
volume for cells with θ = 15◦. A limiting local strain value of ε =
max
0.050isassumed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.7 Design chart of dimensionless energy absorption per unit cell material
volume for cells with θ = −15◦. A limiting local strain value of ε =
max
0.025isassumed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
viii
3.8 Design chart of dimensionless energy absorption per unit cell material
volume for cells with θ = −15◦. A limiting local strain value of ε =
max
0.050isassumed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.9 Design chart of dimensionless energy absorption per unit cell material
volume for cells with θ = 30◦. A limiting local strain value of ε =
max
0.025isassumed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.10 Design chart of dimensionless energy absorption per unit cell material
volume for cells with θ = 30◦. A limiting local strain value of ε =
max
0.050isassumed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.11 Design chart of dimensionless energy absorption per unit cell material
volume for cells with θ = −30◦. A limiting local strain value of ε =
max
0.025isassumed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.12 Design chart of dimensionless energy absorption per unit cell material
volume for cells with θ = −30◦. A limiting local strain value of ε =
max
0.050isassumed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.1 Phasefractionspacenotationusedinthetext. . . . . . . . . . . . . . . . 72
4.2 Transformationpathexhibitingtheshapememoryeffect. . . . . . . . . . 73
4.3 Shape memory effect and pseudoelastic response exhibited by tensile
loadingofonefiniteelement. . . . . . . . . . . . . . . . . . . . . . . . 85
4.4 Phasefractionshistorypresentedinphasefractionspace,duringthermo-
mechanicalprocessdescribedinFigure4.3. . . . . . . . . . . . . . . . . 87
4.5 PhasefractionshistoryduringprocesspresentedinFigure4.3. . . . . . . 88
4.6 Stress-strain response of 1 element in tension for two different tempera-
tures. Curve with letters A,B,C corresponds to detwinning of martensite,
whileD,E,F,Gisatahighertemperature(pseudoelasticbehavior). . . . . 88
4.7 Shape memory effect and pseudoelastic response for the principal solu-
tionofahexagonalhoneycomb. Onequarteroftheunitcellismodeled. . 90
ix