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

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3.4 Constitutive Equations - Hyperelastic MaterialStresses result from the deformation of the material, and it is necessary to expressthem in terms of some measure of this deformation such as, for instance, the strain.These relationship, known as constitutive equations, obviously depend on the type ofmaterial under consideration and may be dependant or independent on time. Forexample, the behavior of viscous materials is clearly entirely dependent on strainrate. Generally, constitutive equations must satisfy certain physical principles. Theequations must obviously be objective, that is, frame invariant. The constitutiveequations herein described are established for hyperelastic material, whereby stressesare derived from a stored elastic energy function. Although there are a number ofalternative material descriptions that could be introduced, hyperelasticity is aparticularly convenient constitutive equation because of its simplicity and, as aconsequence, it constitutes the basis for more complex material models such aselastoplasticity.3.4.1 HyperelasticityMaterials for which the constitutive behavior is only a function of the current state ofdeformation are generally known as elastic. A so-called perfectly elastic material isby definition a material which produces locally no entropy (Trusdell and Noll, 1992).In other words, the term ‘perfectly’ is used for a certain class of materials which hasthe special merits that for every admissible process the internal dissipationDintiszero (naturally, damage, viscous mechanisms and plastic deformations are excluded).Using as strain measure the deformation gradient F associated with a particle X inthe current configuration and its conjugate first Piola-Kirchhoff stress measure P thebasic material elastic relationship is given byP=P( F( X), X )(3.89)52
where the direct dependency upon X allows the possible inhomogeneity of thematerial.In the special case when the work done by the stresses during a deformation processis dependent only on the initial state at time t 0and the final configuration at time t ,the behavior of the material is said to be path-independent and the material is termedhyperelastic or Green-elastic. As a consequence of the path-independent behavior aHelmholtz stored energy function or elastic potential W per unit undeformed volumecan be established from which the first Piola-Kirchhoff stress is computed using∂W ( FX ( ), X)PFX ( ( ), X)=∂F(3.90)Because of the restrictions imposed by the objectivity, the stored energy functionremain invariant when the current configuration undergoes a rigid body rotation.This implies that W depends on F only via the stretch component U and it isindependent of the rotation component R . For convenience, however, W is oftenexpressed as a function of2 TC= U = F F asW( FX ( ), X) = W ( CX ( ), X)(3.91)Observing that 1 C= E is work conjugate to the second Piola-Kirchhoff stress S2enables a totally Lagrangian constitutive equation to be constructed in the samemanner as equation (3.90) to give( ( ), ) 2 ∂W= =∂WSCX X∂C∂E(3.92)53
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University of CassinoFaculty of Eng
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Contents1. INTRODUCTION ...........
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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 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: ∂∂t=−∂F∂t−1 −1 −1F
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
Page 89 and 90: ⎡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
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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
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7 3D-1D Constitutive Model with 1D
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*∂f6 3σ9 3 2 2 T Ml− hϑ+mz β
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⎡⎤⎢1 0 0 ⎥⎢⎥1ε t= ϑ
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(7.21)σ3 1.5R=− βc( T − Mf)+2
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It is interesting to evaluate how m
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It can be noted that equations (7.3
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The inelastic step is solved by the
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For brevity, only the saturated pha
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Fig. 7.2. Timoshenko’s beam theor
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NNNv1v2v3N θ1= ξξ ( − 1)21= ξ
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to analyze the two-dimensional elem
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300 N . A regular mesh characterize
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8.1.1.3 Superelastic and Shape-Memo
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200180T=285 K160140120Force [N]1008
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8.1.2 Comparison between 3D and 3D-
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PhLFig. 8.7: Geometry and boundary
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250200150100Axial force [N]500-50-1
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908070Bending moment [Nmm]605040302
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The arch is subjected to tension-co
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10864Axial force [N]20-2-4-6-8beam
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Some parameters characterizing the
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9876Applied load [N]54321Numerical
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The proposed constitutive models se
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REFERENCESAdler P.H., Yu W., Pelton
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Auricchio F., Taylor R., A return-m
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Ciarlet P.G., The Finite Element Me
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Huang M.S., Brinson L.C., A multiva
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Raniecki B., Lexcellent Ch., Thermo
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Taylor G.I., Plastic strain in meta
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