Finite Strain Shape Memory Alloys Modeling - Scuola di Dottorato in ...

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As shape-memory alloys show some surprising features not present in materialstraditionally used in engineering, they are the basis for innovative applications.Shape-memory technologies are nowadays exploited in a variety of differentapplicative context ranging from sensors and actuators, to robotics, to dampingdevices. Recent experimental and numerical investigations have also shown thepossibility of using such materials in the area of seismic resistant design. Inparticular, they seem to be effective in improving the response of buildings andbridges under earthquake-induced vibrations. However, the largest commercialsuccess of shape-memory materials is related to some applications in the biomedicalfield. In particular, they are successfully employed in orthodontics (archwires),orthopedics, medical instruments, minimal invasive surgery technology (grippers,cutters), drug delivery systems and vascular scaffolding (cardiovascular stents, aorticaneurysm).A review of the available literature and personal contacts in the industry shows adiffusion of computational tools to support the design process of shape-memoryalloydevices. This is also due to the fact that conventional inelastic models do notprovide an adequate framework able to represent the unusual macrobehavior ofshape-memory materials.The present dissertation focuses on a new family of inelastic models, based on aninternal-variable formalism and known as generalized plasticity (Lubliner et al.,1993; Lubliner and Auricchio, 1996). This theory can be used to describe solid-solidphase transformations and, therefore, it is adopted herein as the framework for thedevelopment of one- and three-dimensional thermomechanical constitutive modelsfor shape-memory materials.The proposed constitutive models reproduce the basic features of shape-memoryalloys, such as superelasticity, the shape-memory effect together with a differentmaterial behavior in tension and compression.2
The implementations of the model in a finite-element scheme and the form of thealgorithmically consistent tangent are discussed in detail. The cases of large- andsmall-deformation regimes are considered.Based on the overall results, it appears that the proposed approach is a viable basisfor the development of an effective computational tool to be used in the simulation ofshape-memory alloys devices behavior.1.1 Motivations of the Research and Outline of the ThesisThe first reported steps towards the discovery of the shape memory effect were takenin the 1930s. According to Otsuka and Wayman (1998), A. Olander discovered thepseudoelastic behavior of the Au-Cd alloy in 1932. Greninger and Mooradian in1938 observed the formation and disappearance of a martensitic phase by decreasingand increasing the temperature of a Cu-Zn alloy. The basic phenomena of the shapememoryeffect governed by the thermoelastic behavior of the martensite phase waswidely reported a decade later by Kurdjumov and Khandros in 1949 and also byChang and Read in 1951. In the early 1960s Buehler and Wiley at the U.S. NavalOrdinance Laboratory discovered the shape memory effect in an equiatomic alloy ofnichel and titanium, which can be considered a breakthrough in the field of shapememorymaterials. This alloy with a composition of 53 to 57 % nickel by weight wasnamed Nitinol (Nichel-Titanium Naval Ordnance Laboratory). It exhibited an usualeffect: severely deformed, specimens of the alloy, with residual strains of 8-15 %,regained their original shape after a thermal cycle. This effect became known as theshape-memory effect, and the alloys exhibiting it were named shape-memory alloys.It was later found that not only other materials show the shape-memory property, butthat at sufficiently high temperatures such materials also possess the property ofsuperelasticity, that is the recovery of large deformations during mechanical loadingunloadingcycles performed at constant temperature.3
Page 1 and 2: University of CassinoFaculty of Eng
Page 3 and 4: Contents1. INTRODUCTION ...........
Page 5 and 6: 5.2.1.2. Finite Element Approximati
Page 7: 1 INTRODUCTIONScience and technolog
Page 11 and 12: applications are presented in order
Page 13 and 14: 2 SHAPE MEMORY ALLOYS2.1 Introducti
Page 16 and 17: transformation (M→A), indicated r
Page 18: Fig. 2.2: SuperelasticityFig. 2.3:
Page 21 and 22: The Differential Scanning Calorimet
Page 23 and 24: 2.3 Biocompatibility of Shape-Memor
Page 25 and 26: only in the angioplasty procedure,
Page 27 and 28: micropumps used to unblock blood ve
Page 29 and 30: introduction of a multi-well free-e
Page 31 and 32: spectral decomposition allowing the
Page 33 and 34: Fig. 3.1: Reference and deformed co
Page 35 and 36: δ δ = δ and δ δ = δ(3.6)iI iJ
Page 37 and 38: and it is a two-point tensor since
Page 39 and 40: Furthermore,( ) 2 2Jdet b= det F =
Page 41 and 42: 3.2.3 Polar DecompositionThe deform
Page 43 and 44: which can now be interpreted as fir
Page 45 and 46: Fig. 3.2: Representation of the pol
Page 47 and 48: ecause, as time progresses, the spe
Page 49 and 50: It can be introduced the rotated ra
Page 51 and 52: traction vector. For the Cauchy’s
Page 53 and 54: Nt per unit area acting on the boun
Page 55 and 56: transforms according to a= Qa. Velo
Page 57 and 58: ∂∂t=−∂F∂t−1 −1 −1F
Page 59 and 60:
where the direct dependency upon X
Page 61 and 62:
3.4.2 Isotropic Hyperelasticity - M
Page 63 and 64:
such phenomena as the Bauschinger e
Page 65 and 66:
superposed to intermediate configur
Page 67 and 68:
whereΨerepresents the elastic stra
Page 69 and 70:
( )( ) E 1 ( )Ψ = h ω β T − M
Page 71 and 72:
4.2.2 Clausius-Duhem InequalityIn o
Page 73 and 74:
−1∂Ψe−TS− 2FtFt= 0∂Ce(4.
Page 75 and 76:
qφ = T : dt− ⋅gradT≥0(4.30)T
Page 77 and 78:
fact the values of ϕ andenergy usu
Page 79 and 80:
4.2.5 Final Format of the Constitut
Page 81 and 82:
1 ( C )*1t−1α = ( β T − M ( )
Page 83 and 84:
5 FINITE STRAIN CONSTITUTIVE MODEL:
Page 85 and 86:
2 3ζ 2 ζ 3exp( ζ g) = 1+ ζ g+ g
Page 87 and 88:
where d is a user-defined parameter
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⎡R R Rt⎢⎢RR Rt⎢⎣R R RtC C
Page 91 and 92:
γ• R,Ct=12EEtt(5.28)• R γ ,
Page 93 and 94:
The spatial virtual work equation s
Page 95 and 96:
[ E E E 2 E 2 E 2 E ]δE = δ δ δ
Page 97 and 98:
ˆNLB = B + B (5.51)a a ain whichBa
Page 99 and 100:
Kirchhoff stress and converting int
Page 101 and 102:
DF( φ)[ u]d=dεε = 0( φ εu)∂
Page 103 and 104:
In terms of the linear strain tenso
Page 105 and 106:
6.2 3D Constitutive Model for Stres
Page 107 and 108:
T − Mf Tη = η0− ⎡⎣h( ω)
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assumed to be ruled by the limit fu
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♦The limit function is assumed fu
Page 113 and 114:
6.3.1.1 Time Integration of the 3D
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⎧∂f∂ ε⎪ C ⎡⎤⎣ ⎦⎪
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RRRRRRRR*( f )2 2 2Xd, X= I+ hΔ ζ
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If equation (6.42) is considered as
Page 121 and 122:
7 3D-1D Constitutive Model with 1D
Page 123 and 124:
*∂f6 3σ9 3 2 2 T Ml− hϑ+mz β
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⎡⎤⎢1 0 0 ⎥⎢⎥1ε t= ϑ
Page 127 and 128:
(7.21)σ3 1.5R=− βc( T − Mf)+2
Page 129 and 130:
It is interesting to evaluate how m
Page 131 and 132:
It can be noted that equations (7.3
Page 133 and 134:
The inelastic step is solved by the
Page 135 and 136:
For brevity, only the saturated pha
Page 137 and 138:
Fig. 7.2. Timoshenko’s beam theor
Page 139 and 140:
NNNv1v2v3N θ1= ξξ ( − 1)21= ξ
Page 141 and 142:
to analyze the two-dimensional elem
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300 N . A regular mesh characterize
Page 145 and 146:
8.1.1.3 Superelastic and Shape-Memo
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200180T=285 K160140120Force [N]1008
Page 149 and 150:
8.1.2 Comparison between 3D and 3D-
Page 151 and 152:
PhLFig. 8.7: Geometry and boundary
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250200150100Axial force [N]500-50-1
Page 155 and 156:
908070Bending moment [Nmm]605040302
Page 157 and 158:
The arch is subjected to tension-co
Page 159 and 160:
10864Axial force [N]20-2-4-6-8beam
Page 161 and 162:
Some parameters characterizing the
Page 163 and 164:
9876Applied load [N]54321Numerical
Page 165 and 166:
The proposed constitutive models se
Page 167 and 168:
REFERENCESAdler P.H., Yu W., Pelton
Page 169 and 170:
Auricchio F., Taylor R., A return-m
Page 171 and 172:
Ciarlet P.G., The Finite Element Me
Page 173 and 174:
Huang M.S., Brinson L.C., A multiva
Page 175 and 176:
Raniecki B., Lexcellent Ch., Thermo
Page 177:
Taylor G.I., Plastic strain in meta
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