American Institute of Mathematical Sciences

Advanced
April  2013, 6(2): 277-291. doi: 10.3934/dcdss.2013.6.277

A new "flexible" 3D macroscopic model for shape memory alloys

Ferdinando Auricchio 1, and  Elena Bonetti 2,

1. 

Dipartimento di Meccanica Strutturale, Università di Pavia, via Ferrata 1, 27100 Pavia, Italy
2. 

Dipartimento di Matematica, Università di Pavia, via Ferrata 1, 27100 Pavia

Received  November 2011 Revised  January 2012 Published  November 2012

In this paper we introduce a 3D phenomenological model for shape memory behavior, accounting for: martensite reorientation, asymmetric response of the material to tension/compression, different kinetics between forward and reverse phase transformation. We combine two modeling approaches using scalar and tensorial internal variables. Indeed, we use volume proportions of different configurations of the crystal lattice (austenite and two variants of martensite) as scalar internal variables and the preferred direction of stress-induced martensite as tensorial internal variable. Then, we derive evolution equations by a generalization of the principle of virtual powers, including microforces and micromovements responsible for phase transformations. In addition, we prescribe an evolution law for phase proportions ensuring different kinetics during forward and reverse transformation of the oriented martensite.
Keywords: Shape memory, reorientation, phase transformation, internal variables..
Mathematics Subject Classification: Primary: 35Q74; Secondary: 74C05, 74N3.
Citation: Ferdinando Auricchio, Elena Bonetti. A new "flexible" 3D macroscopic model for shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 277-291. doi: 10.3934/dcdss.2013.6.277
References:
[1]

J. Arghavani, F. Auricchio, R. Naghdabadi, A. Reali and S. Sohrabpour, A 3-D phenomenological constitutive model for shape memory alloys under multiaxial loadings,, International Journal of Plasticity, 26 (2010), 976.  doi: 10.1016/j.ijplas.2009.12.003.  Google Scholar

[2]

F. Auricchio and L. Petrini, A three-dimensional models describing stress-temperature induced solid phase transformations. part I: Solution, algorithm and boundary value problems,, International Journal of Numerical Methods in Engineering, 6 (2004), 807.   Google Scholar

[3]

F. Auricchio, A. Reali and U. Stefanelli, A macroscopic 1D model for shape memory alloys including asymmetric behaviors and transformation-dependent elastic properties,, Computer Methods in Applied Mechanics and Engineering, 198 (2009), 1631.  doi: 10.1016/j.cma.2009.01.019.  Google Scholar

[4]

E. Bonetti, Global solvability of a dissipative Frémond model for shape memory alloys. I. Mathematical formulation and uniqueness,, Quart. Appl. Math., 61 (2003), 759.   Google Scholar

[5]

E. Bonetti, M. Frémond and Ch. Lexcellent, Global existence and uniqueness for a thermomechanical model for shape memory alloys with partition of the strain,, Math. Mech. Solids, 11 (2006), 251.  doi: 10.1177/1081286506040403.  Google Scholar

[6]

H. Brézis, "Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert,", Number 5 in North Holland Math. Studies. North-Holland, (1973).   Google Scholar

[7]

M. Frémond, "Non-Smooth Thermomechanics,", Springer-Verlag, (2002).   Google Scholar

[8]

J. Lubliner, "Plasticity Theory,", Macmillan, (1990).   Google Scholar

[9]

J. J. Moreau, "Fonctionelles Convexes,", Universià di Roma Tor Vergata Pub., (2003).   Google Scholar

[10]

A. C. Souza, E. N. Mamiya and N. Zouain, Three-dimensional model for solids undergoing stress-induced phase transformations,, European Journal of Mechanics, 17 (1998), 789.  doi: 10.1016/S0997-7538(98)80005-3.  Google Scholar

show all references

References:
[1]

J. Arghavani, F. Auricchio, R. Naghdabadi, A. Reali and S. Sohrabpour, A 3-D phenomenological constitutive model for shape memory alloys under multiaxial loadings,, International Journal of Plasticity, 26 (2010), 976.  doi: 10.1016/j.ijplas.2009.12.003.  Google Scholar

[2]

F. Auricchio and L. Petrini, A three-dimensional models describing stress-temperature induced solid phase transformations. part I: Solution, algorithm and boundary value problems,, International Journal of Numerical Methods in Engineering, 6 (2004), 807.   Google Scholar

[3]

F. Auricchio, A. Reali and U. Stefanelli, A macroscopic 1D model for shape memory alloys including asymmetric behaviors and transformation-dependent elastic properties,, Computer Methods in Applied Mechanics and Engineering, 198 (2009), 1631.  doi: 10.1016/j.cma.2009.01.019.  Google Scholar

[4]

E. Bonetti, Global solvability of a dissipative Frémond model for shape memory alloys. I. Mathematical formulation and uniqueness,, Quart. Appl. Math., 61 (2003), 759.   Google Scholar

[5]

E. Bonetti, M. Frémond and Ch. Lexcellent, Global existence and uniqueness for a thermomechanical model for shape memory alloys with partition of the strain,, Math. Mech. Solids, 11 (2006), 251.  doi: 10.1177/1081286506040403.  Google Scholar

[6]

H. Brézis, "Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert,", Number 5 in North Holland Math. Studies. North-Holland, (1973).   Google Scholar

[7]

M. Frémond, "Non-Smooth Thermomechanics,", Springer-Verlag, (2002).   Google Scholar

[8]

J. Lubliner, "Plasticity Theory,", Macmillan, (1990).   Google Scholar

[9]

J. J. Moreau, "Fonctionelles Convexes,", Universià di Roma Tor Vergata Pub., (2003).   Google Scholar

[10]

A. C. Souza, E. N. Mamiya and N. Zouain, Three-dimensional model for solids undergoing stress-induced phase transformations,, European Journal of Mechanics, 17 (1998), 789.  doi: 10.1016/S0997-7538(98)80005-3.  Google Scholar

[1]

Barbora Benešová, Miroslav Frost, Lukáš Kadeřávek, Tomáš Roubíček, Petr Sedlák. An experimentally-fitted thermodynamical constitutive model for polycrystalline shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020459
[2]

Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021006
[3]

Mohamed Dellal, Bachir Bar. Global analysis of a model of competition in the chemostat with internal inhibitor. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1129-1148. doi: 10.3934/dcdsb.2020156
[4]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031
[5]

Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147
[6]

Tomáš Oberhuber, Tomáš Dytrych, Kristina D. Launey, Daniel Langr, Jerry P. Draayer. Transformation of a Nucleon-Nucleon potential operator into its SU(3) tensor form using GPUs. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1111-1122. doi: 10.3934/dcdss.2020383
[7]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467
[8]

Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325
[9]

Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 501-514. doi: 10.3934/dcdsb.2020350
[10]

Lin Shi, Dingshi Li, Kening Lu. Limiting behavior of unstable manifolds for spdes in varying phase spaces. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021020
[11]

Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021012
[12]

Tomáš Smejkal, Jiří Mikyška, Jaromír Kukal. Comparison of modern heuristics on solving the phase stability testing problem. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1161-1180. doi: 10.3934/dcdss.2020227
[13]

Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267
[14]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215
[15]

Xiaorui Wang, Genqi Xu, Hao Chen. Uniform stabilization of 1-D Schrödinger equation with internal difference-type control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021022
[16]

Manil T. Mohan. Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory". Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020105
[17]

Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089
[18]

Cung The Anh, Dang Thi Phuong Thanh, Nguyen Duong Toan. Uniform attractors of 3D Navier-Stokes-Voigt equations with memory and singularly oscillating external forces. Evolution Equations & Control Theory, 2021, 10 (1) : 1-23. doi: 10.3934/eect.2020039
[19]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, 2021, 15 (1) : 159-183. doi: 10.3934/ipi.2020076

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences