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

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martensite has 24 variants, i.e. 24 subtypes with different crystallographicorientations. Instead, if there is a preferred direction for the formation of themartensitic phase, just one (single) variant is formed (see Fig. 2.2 Fig. 2.). When thematerial is in its martensitic form, it is soft and ductile and can be easily deformed(somewhat like soft pewter). Stress-induced martensite NiTi, one of the most usedshape memory alloys, is highly elastic (rubber-like), while austenite NiTi is quitestrong and hard (similar to titanium). The NiTi material has all these properties andtheir specific expression depend on the temperature in which it is used.8
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 and 8: 1 INTRODUCTIONScience and technolog
Page 9 and 10: The implementations of the model in
Page 11 and 12: applications are presented in order
Page 13: 2 SHAPE MEMORY ALLOYS2.1 Introducti
Page 17 and 18: Shape-memory effect or one way shap
Page 20 and 21: Ni-Ti 49/51 at % Ni -50 to 110 30Fe
Page 22 and 23: Fig. 2.5: Transformation diagram ob
Page 24 and 25: design capabilities, which make it
Page 26 and 27: 2.4.3 Orthopedic ApplicationsAnothe
Page 28 and 29: 2.5 Constitutive ModelsDue to the s
Page 30 and 31: Probably the first example along th
Page 32 and 33: 3 Review of Some Basic Results in C
Page 34 and 35: whereEIare unit orthogonal base vec
Page 36 and 37: never zero. Indeed, since det ∇ i
Page 38 and 39: ( ) 2 2Jdet C= det F = > 0(3.16)Alt
Page 40 and 41: 1 ⎡1⎤e= ( 1− F F ) = F ( )2
Page 42 and 43: 3∑C= λ N ⊗N (3.31)α = 12α α
Page 44 and 45: 3∑α = 1( ) ( )V = λ RN ⊗ RN (
Page 46 and 47: ∂φ( X, t)VX ( , t)=∂t(3.42)Obs
Page 48 and 49: d ⎛ ∂φ⎞ ∂ ⎛∂φ⎞F =
Page 50 and 51: is focused on that part of the body
Page 52 and 53: Tτ = PF or τ = PF(3.65)ij iI jIRe
Page 54 and 55: ⎛ρ⎞0 1where ρ= ⎜φ⎜⎝
Page 56 and 57: or, in components:⎡⎢∂σ⎣∂
Page 58 and 59: 3.4 Constitutive Equations - Hypere
Page 60 and 61: The strain-energy function vanishes
Page 62 and 63: 4 A FINITE STRAIN CONSTITUTIVE MODE
Page 64 and 65:
4.2 Constitutive ModelThe micromech
Page 66 and 67:
thermodynamic state of each materia
Page 68 and 69:
last term of equation (4.7) ensures
Page 70 and 71:
it is a function of the heat capaci
Page 72 and 73:
C =− l C + F CF −C l(4.18)T
Page 74 and 75:
The tensor T represents the driving
Page 76 and 77:
∂ϕ∂ϕT= gradT=−∂d∂t( q T
Page 78 and 79:
with R a material parameter definin
Page 80 and 81:
has been introduced in order to def
Page 82 and 83:
( ) 1 1T T T TFt dev T Ft = Ft ( M
Page 84 and 85:
C = ζf( C, C ) = ζg( CC , ) C(
Page 86 and 87:
5.1.2 Time Integration of the Const
Page 88 and 89:
If the above solution is not admiss
Page 90 and 91:
−1 −1( )Q = U fU(5.19)lt t tlt
Page 92 and 93:
∫( ) ( ρ )f η = divσ+ b ⋅ η
Page 94 and 95:
∫ ∫ ∫S:δEdv = ρB⋅ δudV +
Page 96 and 97:
5.2.1.2 Finite Element Approximatio
Page 98 and 99:
ˆ Tˆ T ˆ ˆ ∂B∂fK = ∫B D B
Page 100 and 101:
6 LINEARIZATION OF THE FINITEDEFORM
Page 102 and 103:
DE[ u]= ε(6.5)Similarly, the right
Page 104 and 105:
where*I1and*I2are the principal inv
Page 106 and 107:
∂Ψ ∂Ψ ∂Ψ gradTσ: ε− :
Page 108 and 109:
Accordingly, the tensor α plays a
Page 110 and 111:
Fig. 6.1: Experimental stress-strai
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♦The deviatoric form of the trans
Page 114 and 115:
The trial state is determined solel
Page 116 and 117:
6.3.1.2 Newton-Raphson MethodAs sta
Page 118 and 119:
( )( ) ( )2 dev X X2X2X∂ f X JN
Page 120 and 121:
2∂ fε =Δζ∂ X∂fεt,Δζ=
Page 122 and 123:
in which only one scalar variable
Page 124 and 125:
under the constrain:⎧ 30 ≤ ϑ <
Page 126 and 127:
variant martensite transformation (
Page 128 and 129:
⎛⎞⎜⎟1 1.5RAs = Mf− ⎜ +
Page 130 and 131:
components of the transformation st
Page 132 and 133:
[ ]⎧ σl= E ε −ϑ⎪⎪ ⎛*3
Page 134 and 135:
⎡R⎢⎣ RRΔζΔζ, η , γγ γ
Page 136 and 137:
in which:ϑ, γ= 0(7.47)It follows
Page 138 and 139:
w= N w + N w + N ww w w1 1 2 2 3 3
Page 140 and 141:
8 Numerical Results8.1 Introduction
Page 142 and 143:
700060005000Axial force [N]40003000
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350300h/L=1/10250Shear force [N]200
Page 146 and 147:
In Fig. 8.6 the applied force is pl
Page 148 and 149:
140120T=223K100Force [N]80604020sma
Page 150 and 151:
• Pseudo-Elastic EffectA clamped
Page 152 and 153:
3D-1D simplified, are performed con
Page 154 and 155:
at the value of T = 285Kand after t
Page 156 and 157:
cycle in which the maximum value of
Page 158 and 159:
compression except from the stress
Page 160 and 161:
8.1.3 Experimental ComparisonThe ab
Page 162 and 163:
450400350Axial stress [MPa]30025020
Page 164 and 165:
CONCLUSIONSThe research on shape me
Page 166 and 167:
The overall discussion leads to con
Page 168 and 169:
Auricchio F., Petrini F., Extension
Page 170 and 171:
Bonet J., Wood R.D., Nonlinear cont
Page 172 and 173:
Fremond M., Non-Smooth Thermomechan
Page 174 and 175:
Lubliner J., A simple model of gene
Page 176 and 177:
Simo J.C., Marsden J.E., On the Rot
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