December  2017, 10(6): 1257-1280. doi: 10.3934/dcdss.2017068

Existence and linearization for the Souza-Auricchio model at finite strains

1. 

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz, A-1090 Vienna, Austria

2. 

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz, A-1090 Vienna, Austria

3. 

Istituto di Matematica Applicata e Tecnologie Informatiche E. Magenes, CNR, via Ferrata 1, I-27100 Pavia, Italy

Received  July 2016 Revised  October 2016 Published  June 2017

We address the analysis of the Souza-Auricchio model for shape-memory alloys in the finite-strain setting. The model is formulated in variational terms and the existence of quasistatic evolutions is obtained within the classical frame of energetic solvability. The finite-strain model is proved to converge to its small-strain counterpart for small deformations via a variational convergence argument.

Citation: Diego Grandi, Ulisse Stefanelli. Existence and linearization for the Souza-Auricchio model at finite strains. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1257-1280. doi: 10.3934/dcdss.2017068
References:
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F. AuricchioA. Mielke and U. Stefanelli, A rate-independent model for the isothermal quasi-static evolution of shape-memory materials, Math. Models Meth. Appl. Sci., 18 (2008), 125-164.  doi: 10.1142/S0218202508002632.

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F. Auricchio and L. Petrini, Improvements and algorithmical considerations on a recent three-dimensional model describing stress-induced solid phase transformations, Intern. J. Numer. Methods Engng., 55 (2002), 1255-1284.  doi: 10.1002/nme.619.

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F. Auricchio and L. Petrini, A three-dimensional model describing stress-temperature induced solid phase transformations: solution algorithm and boundary value problems, Intern. J. Numer. Methods Engng., 61 (2004), 807-836.  doi: 10.1002/nme.1086.

[7]

F. Auricchio and L. Petrini, A three-dimensional model describing stress-temperature induced solid phase transformations: thermomechanical coupling and hybrid composite applications, Intern. J. Numer. Methods Engng., 61 (2004), 716-737.  doi: 10.1002/nme.1086.

[8]

F. AuricchioA. Reali and U. Stefanelli, A three-dimensional model describing stress-induced solid phase transformation with permanent inelasticity, Int. J. Plasticity, 23 (2007), 207-226.  doi: 10.1016/j.ijplas.2006.02.012.

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K. Bhattacharya, Microstructures of Martensites, Oxford Series on Materials Modeling, Oxford University Press, Oxford, 2003.

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J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., 63 (1977), 337-403.  doi: 10.1007/BF00279992.

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J. M. Ball, Minimizers and the Euler-Lagrange equations, in Trends and Applications of Pure Mathematics to Mechanics (Palaiseau, 1983), Lecture Notes in Physics, 195, Springer, Berlin, 1984, 1-4. doi: 10.1007/3-540-12916-2_47.

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J. M. Ball, Some open problems in elasticity, in Geometry, Mechanics, and Dynamics. Volume in honor of the 60th birthday of J. E. Marsden (eds. P. Newton et al. ), Springer, New York, NY, 2002, 3-59. doi: 10.1007/0-387-21791-6_1.

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B. Benešová and T. Roubíček, Micro-to-meso scale limit for shape-memory-alloy models with thermal coupling, Multiscale Model. Simul., 10 (2012), 1059-1089.  doi: 10.1137/110852176.

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V. EvangelistaS. Marfia and E. Sacco, A 3D SMA constitutive model in the framework of finite strain, Intern. J. Numer. Methods Engng., 81 (2010), 761-785.  doi: 10.1002/nme.2717.

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A. Giacomini and A. Musesti, Quasi-static evolutions in linear perfect plasticity as a variational limit of finite plasticity: a one-dimensional case, Math. Models Methods Appl. Sci., 23 (2013), 1275-1308.  doi: 10.1142/S0218202513500097.

[32]

S. Govindjee and C. Miehe, A multi-variant martensitic phase transformation model: Formulation and numerical implementation, Comput. Methods Appl. Mech. Engrg., 191 (2001), 215-238. 

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D. Grandi and U. Stefanelli, The Souza-Auricchio model for shape-memory alloys, Discret. Contin. Dyn. Syst. Ser. S, 8 (2015), 723-747.  doi: 10.3934/dcdss.2015.8.723.

[34]

D. Grandi and U. Stefanelli, Finite plasticity in $\mathbf P^\top\mathbf P $. Part Ⅰ: Constitutive model, Contin. Mech. Thermodyn., 29 (2017), 97-116.  doi: 10.1007/s00161-016-0522-1.

[35]

D. Grandi and U. Stefanelli, Finite plasticity in $\mathbf P^\top\mathbf P $. Part Ⅱ: Quasistatic evolution and linearization, SIAM J. Math. Anal., 49 (2017), 1356-1384.  doi: 10.1137/16M1079440.

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D. Helm and P. Haupt, Shape memory behaviour: Modelling within continuum thermomechanics, Intern. J. Solids Struct., 40 (2003), 827-849. 

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P. Krejčí and U. Stefanelli, Existence and nonexistence for the full thermomechanical Souza-Auricchio model of shape memory wires, Math. Mech. Solids, 16 (2011), 349-365.  doi: 10.1177/1081286510386935.

[39]

P. Krejčí and U. Stefanelli, Well-posedness of a thermo-mechanical model for shape memory alloys under tension, M2AN Math. Model. Numer. Anal., 44 (2010), 1239-1253.  doi: 10.1051/m2an/2010024.

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show all references

References:
[1]

V. AgostinianiG. Dal Maso and A. DeSimone, Linear elasticity obtained from finite elasticity by Γ-convergence under weak coerciveness conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 715-735.  doi: 10.1016/j.anihpc.2012.04.001.

[2]

L. Ambrosio, N. Fusco and D. Percivale, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford, 2000.

[3]

F. Auricchio and J. Lubliner, A uniaxial model for shape-memory alloys, Internat. J. Solids Structures, 34 (1997), 3601-3618.  doi: 10.1016/S0020-7683(96)00232-6.

[4]

F. AuricchioA. Mielke and U. Stefanelli, A rate-independent model for the isothermal quasi-static evolution of shape-memory materials, Math. Models Meth. Appl. Sci., 18 (2008), 125-164.  doi: 10.1142/S0218202508002632.

[5]

F. Auricchio and L. Petrini, Improvements and algorithmical considerations on a recent three-dimensional model describing stress-induced solid phase transformations, Intern. J. Numer. Methods Engng., 55 (2002), 1255-1284.  doi: 10.1002/nme.619.

[6]

F. Auricchio and L. Petrini, A three-dimensional model describing stress-temperature induced solid phase transformations: solution algorithm and boundary value problems, Intern. J. Numer. Methods Engng., 61 (2004), 807-836.  doi: 10.1002/nme.1086.

[7]

F. Auricchio and L. Petrini, A three-dimensional model describing stress-temperature induced solid phase transformations: thermomechanical coupling and hybrid composite applications, Intern. J. Numer. Methods Engng., 61 (2004), 716-737.  doi: 10.1002/nme.1086.

[8]

F. AuricchioA. Reali and U. Stefanelli, A three-dimensional model describing stress-induced solid phase transformation with permanent inelasticity, Int. J. Plasticity, 23 (2007), 207-226.  doi: 10.1016/j.ijplas.2006.02.012.

[9]

K. Bhattacharya, Microstructures of Martensites, Oxford Series on Materials Modeling, Oxford University Press, Oxford, 2003.

[10]

J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., 63 (1977), 337-403.  doi: 10.1007/BF00279992.

[11]

J. M. Ball, Minimizers and the Euler-Lagrange equations, in Trends and Applications of Pure Mathematics to Mechanics (Palaiseau, 1983), Lecture Notes in Physics, 195, Springer, Berlin, 1984, 1-4. doi: 10.1007/3-540-12916-2_47.

[12]

J. M. Ball, Some open problems in elasticity, in Geometry, Mechanics, and Dynamics. Volume in honor of the 60th birthday of J. E. Marsden (eds. P. Newton et al. ), Springer, New York, NY, 2002, 3-59. doi: 10.1007/0-387-21791-6_1.

[13]

B. Benešová and T. Roubíček, Micro-to-meso scale limit for shape-memory-alloy models with thermal coupling, Multiscale Model. Simul., 10 (2012), 1059-1089.  doi: 10.1137/110852176.

[14]

H. Brézis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, Math Studies, Vol. 5, North-Holland, Amsterdam/New York. 1973.

[15]

P. G. Ciarlet, Mathematical Elasticity, Volume 1: Three Dimensional Elasticity, Elsevier, 1988.

[16]

G. Dal Maso, An Introduction to Γ-convergence, Progress in Nonlinear Differential Equations and their Applications, Vol. 8, Birkhäuser Boston Inc. , Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8.

[17]

G. Dal MasoM. Negri and D. Percivale, Linearized elasticty as $Γ$-limit of finite elasticity, Set-Valued Anal., 10 (2002), 165-183.  doi: 10.1023/A:1016577431636.

[18]

E. Davoli, Linearized plastic plate models as $Γ$-limits of 3D finite elastoplasticity, ESAIM Control Optim. Calc. Var., 20 (2014), 725-747.  doi: 10.1051/cocv/2013081.

[19]

E. Davoli, Quasistatic evolution models for thin plates arising as low energy Γ-limits of finite plasticity, Math. Models Methods Appl. Sci., 24 (2014), 2085-2153.  doi: 10.1142/S021820251450016X.

[20]

E. De Giorgi and T. Franzoni, On a type of variational convergence, in Proceedings of the Brescia Mathematical Seminar, Univ. Cattolica Sacro Cuore, Milan, 3 (1979), 63-101. 

[21]

T. W. Duerig and A. R. Pelton (Eds. ), SMST-2003 Proceedings of the International Conference on Shape Memory and Superelastic Technology, ASM International, 2003.

[22]

V. EvangelistaS. Marfia and E. Sacco, Phenomenological 3D and 1D consistent models for shape-memory alloy materials, Comput. Mech., 44 (2009), 405-421.  doi: 10.1007/s00466-009-0381-8.

[23]

V. EvangelistaS. Marfia and E. Sacco, A 3D SMA constitutive model in the framework of finite strain, Intern. J. Numer. Methods Engng., 81 (2010), 761-785.  doi: 10.1002/nme.2717.

[24]

N. A. Fleck and J. W. Hutchinson, Strain gradient plasticity, Adv. Appl. Mech., 33 (1997), 295-361.  doi: 10.1016/S0065-2156(08)70388-0.

[25]

N. A. Fleck and J. W. Hutchinson, A reformulation of strain gradient plasticity, J. Mech. Phys. Solids, 49 (2001), 2245-2271.  doi: 10.1016/S0022-5096(01)00049-7.

[26]

G. A. Francfort and A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies, J. Reine Angew. Math., 595 (2006), 55-91.  doi: 10.1515/CRELLE.2006.044.

[27]

M. Frémond, Matériaux á mémoire de forme, C. R. Acad. Sci. Paris Sér. Ⅱ Méc. Phys. Chim. Sci. Univers Sci. Terre, 304 (1987), 239-244. 

[28]

M. Frémond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9.

[29]

G. FrieseckeR. D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., 55 (2002), 1461-1506.  doi: 10.1002/cpa.10048.

[30]

S. Frigeri and U. Stefanelli, Existence and time-discretization for the finite-strain Souza-Auricchio constitutive model for shape-memory alloys, Contin. Mech. Thermodyn., 24 (2012), 63-67.  doi: 10.1007/s00161-011-0221-x.

[31]

A. Giacomini and A. Musesti, Quasi-static evolutions in linear perfect plasticity as a variational limit of finite plasticity: a one-dimensional case, Math. Models Methods Appl. Sci., 23 (2013), 1275-1308.  doi: 10.1142/S0218202513500097.

[32]

S. Govindjee and C. Miehe, A multi-variant martensitic phase transformation model: Formulation and numerical implementation, Comput. Methods Appl. Mech. Engrg., 191 (2001), 215-238. 

[33]

D. Grandi and U. Stefanelli, The Souza-Auricchio model for shape-memory alloys, Discret. Contin. Dyn. Syst. Ser. S, 8 (2015), 723-747.  doi: 10.3934/dcdss.2015.8.723.

[34]

D. Grandi and U. Stefanelli, Finite plasticity in $\mathbf P^\top\mathbf P $. Part Ⅰ: Constitutive model, Contin. Mech. Thermodyn., 29 (2017), 97-116.  doi: 10.1007/s00161-016-0522-1.

[35]

D. Grandi and U. Stefanelli, Finite plasticity in $\mathbf P^\top\mathbf P $. Part Ⅱ: Quasistatic evolution and linearization, SIAM J. Math. Anal., 49 (2017), 1356-1384.  doi: 10.1137/16M1079440.

[36]

W. Han and B. D. Reddy, Plasticity, Mathematical Theory and Numerical Analysis, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4614-5940-8_13.

[37]

D. Helm and P. Haupt, Shape memory behaviour: Modelling within continuum thermomechanics, Intern. J. Solids Struct., 40 (2003), 827-849. 

[38]

P. Krejčí and U. Stefanelli, Existence and nonexistence for the full thermomechanical Souza-Auricchio model of shape memory wires, Math. Mech. Solids, 16 (2011), 349-365.  doi: 10.1177/1081286510386935.

[39]

P. Krejčí and U. Stefanelli, Well-posedness of a thermo-mechanical model for shape memory alloys under tension, M2AN Math. Model. Numer. Anal., 44 (2010), 1239-1253.  doi: 10.1051/m2an/2010024.

[40]

E. Kröner, Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen, Arch. Ration. Mech. Anal., 4 (1960), 273-334.  doi: 10.1007/BF00281393.

[41]

M. KružíkA. Mielke and T. Roubíček, Modelling of microstructure and its evolution in shape-memory-alloy single-crystals, in particular in CuAlNi, Meccanica, 40 (2005), 389-418.  doi: 10.1007/s11012-005-2106-1.

[42]

E. Lee, Elastic-plastic deformation at finite strains, J. Appl. Mech., 36 (1969), 1-6.  doi: 10.1115/1.3564580.

[43]

V. I. Levitas, Thermomechanical theory of martensitic phase transformations in inelastic materials, Intern. J. Solids Struct., 35 (1998), 889-940.  doi: 10.1016/S0020-7683(97)00089-9.

[44]

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