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 & Continuous Dynamical Systems - S, 2017, 10 (6) : 1257-1280. doi: 10.3934/dcdss.2017068
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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[40]

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

[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.  Google Scholar

[42]

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

[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.  Google Scholar

[44]

Ch. Lexcellent, Shape-Memory Alloys Handbook, Wiley, 2013. doi: 10.1002/9781118577776.  Google Scholar

[45]

A. Mainik and A. Mielke, Global existence for rate-independent gradient plasticity at finite strain, J.Nonlinear Sci., 19 (2009), 221-248.  doi: 10.1007/s00332-008-9033-y.  Google Scholar

[46]

A. Mielke, Existence of minimizers in incremental elasto-plasticity with finite strains, SIAM J. Math. Anal., 36 (2004), 384-404.  doi: 10.1137/S0036141003429906.  Google Scholar

[47]

A. Mielke, Evolution of rate-independent systems (ch. 6), in Handbook of Differential Equations, Evolutionary Equations, (eds. C. Dafermos, E. Feireisl), Elsevier B. V., 2 (2004), 461–559.  Google Scholar

[48]

A. Mielke, Differential, energetic and metric formulations for rate-independent processes, in Nonlinear PDE's and Applications (eds. L. Ambrosio and G. Savaré), C. I. M. E. Summer School, Cetraro, Italy 2008, Springer, (2011), 87-170. doi: 10.1007/978-3-642-21861-3_3.  Google Scholar

[49]

A. Mielke, Finite elastoplasticity, Lie groups and geodesics on SL(d), in Geometry, Dynamics, and Mechanics (eds. P. Newton, A. Weinstein, and P. J. Holmes), Springer-Verlag, New York, (2002), 61-90. doi: 10.1007/0-387-21791-6_2.  Google Scholar

[50]

A. Mielke and A. Petrov, Thermally driven phase transformation in shape-memory alloys, Adv. Math. Sci. Appl., 17 (2007), 667-685.   Google Scholar

[51]

A. Mielke and T. Roubíček, Rate-Independent Systems -Theory and Application, Appl. Math. Sci. Series, Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7.  Google Scholar

[52]

A. MielkeT. Roubíček and U. Stefanelli, Γ-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differential Equations, 31 (2008), 387-416.  doi: 10.1007/s00526-007-0119-4.  Google Scholar

[53]

A. Mielke and U. Stefanelli, Linearized plasticity is the evolutionary Γ-limit of finite plasticity, J. Eur. Math. Soc. (JEMS), 15 (2013), 923-948.  doi: 10.4171/JEMS/381.  Google Scholar

[54]

A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Diff. Equations Applications, 11 (2004), 151-189.  doi: 10.1007/s00030-003-1052-7.  Google Scholar

[55]

H.-B. Mühlhaus and E. Aifantis. A variational principle for gradient plasticity, A variational principle for gradient plasticity, Int. J. Solids Struct., 28 (1991), 845-857.  doi: 10.1016/0020-7683(91)90004-Y.  Google Scholar

[56]

R. Paroni and G. Tomassetti, A variational justification of linear elasticity with residual stress, J. Elast., 97 (2009), 189-206.  doi: 10.1007/s10659-009-9217-1.  Google Scholar

[57]

R. Paroni and G. Tomassetti, From non-linear elasticity to linear elasticity with initial stress via Γ-convergence, Contin. Mech. Thermodyn., 23 (2011), 347-361.  doi: 10.1007/s00161-011-0184-y.  Google Scholar

[58]

P. Plecháč and T. Roubíček, Visco-elasto-plastic model for martensitic phase tratsformation in shape-memory alloys, Math. Methods Appl. Sci., 25 (2002), 1281-1298.  doi: 10.1002/mma.335.  Google Scholar

[59]

P. Popov and D. C. Lagoudas, A 3-D constitutive model for shape memory alloys incorporating pseudoelasticity and detwinning of self-accommodated martensite, Int. J. Plasticity, 23 (2007), 1679-1720.   Google Scholar

[60]

S. Reese and D. Christ, Finite deformation pseudo-elasticity of shape memory alloys -constitutive modelling and finite element implementation, Int. J. Plasticity, 24 (2008), 455-482.  doi: 10.1016/j.ijplas.2007.05.005.  Google Scholar

[61]

T. Roubíček, Models of microstructure evolution in shape memory alloys, in Nonlinear Homogenization and its Application to Composites, Polycrystals and Smart Materials, (eds. P. Ponte Castaneda, J. J. Telega, and B. Gambin), NATO Sci. Series Ⅱ, Kluwer, Dordrecht, 170 (2004), 269-304. doi: 10.1007/1-4020-2623-4_12.  Google Scholar

[62]

T. Roubíček, Modelling of thermodynamics of martensitic transformation in shape-memory alloys, Discrete Contin. Dyn. Syst., Dynamical systems and differential equations. Proceedings of the 6th AIMS International Conference, suppl., (2007), 892-902.   Google Scholar

[63]

T. Roubíček, Approximation in multiscale modelling of microstructure evolution in shape-memory alloys, Contin. Mech. Thermodyn., 23 (2011), 491-507.  doi: 10.1007/s00161-011-0190-0.  Google Scholar

[64]

T. Roubíček and M. Kružík, Mesoscopic model of microstructure evolution in shape memory alloys, its numerical analysis and computer implementation, GAMM Mitt., 29 (2006), 192-214.  doi: 10.1002/gamm.201490030.  Google Scholar

[65]

T. Roubíček and U. Stefanelli, Magnetic shape-memory alloys: Thermomechanical modeling and analysis, Contin. Mech. Thermodyn., 26 (2014), 783-810.  doi: 10.1007/s00161-014-0339-8.  Google Scholar

[66]

T. Roubíček and G. Tomassetti, Phase transformations in electrically conductive ferromagnetic shape-memory alloys, their thermodynamics and analysis, Archive Ration. Mech. Anal., 210 (2013), 1-43.  doi: 10.1007/s00205-013-0648-2.  Google Scholar

[67]

T. Roubíček and G. Tomassetti, Thermodynamics of shape-memory alloys under electric current, Z. Angew. Math. Phys., 61 (2010), 1-20.  doi: 10.1007/s00033-009-0007-1.  Google Scholar

[68]

B. Schmidt, Linear Γ-limits of multiwell energies in nonlinear elasticity theory, Contin. Mech. Thermodyn., 20 (2008), 375-396.  doi: 10.1007/s00161-008-0087-8.  Google Scholar

[69]

A.C. SouzaE.N. Mamiya and N. Zouain, Three-dimensional model for solids undergoing stress-induced tranformations, Eur. J. Mech. A Solids, 17 (1998), 789-806.   Google Scholar

[70]

P. Thamburaja and L. Anand, Polycrystalline shape-memory materials: Effect of crystallographic texture, J. Mech. Phys. Solids, 49 (2001), 709-737.  doi: 10.1016/S0022-5096(00)00061-2.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[40]

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

[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.  Google Scholar

[42]

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

[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.  Google Scholar

[44]

Ch. Lexcellent, Shape-Memory Alloys Handbook, Wiley, 2013. doi: 10.1002/9781118577776.  Google Scholar

[45]

A. Mainik and A. Mielke, Global existence for rate-independent gradient plasticity at finite strain, J.Nonlinear Sci., 19 (2009), 221-248.  doi: 10.1007/s00332-008-9033-y.  Google Scholar

[46]

A. Mielke, Existence of minimizers in incremental elasto-plasticity with finite strains, SIAM J. Math. Anal., 36 (2004), 384-404.  doi: 10.1137/S0036141003429906.  Google Scholar

[47]

A. Mielke, Evolution of rate-independent systems (ch. 6), in Handbook of Differential Equations, Evolutionary Equations, (eds. C. Dafermos, E. Feireisl), Elsevier B. V., 2 (2004), 461–559.  Google Scholar

[48]

A. Mielke, Differential, energetic and metric formulations for rate-independent processes, in Nonlinear PDE's and Applications (eds. L. Ambrosio and G. Savaré), C. I. M. E. Summer School, Cetraro, Italy 2008, Springer, (2011), 87-170. doi: 10.1007/978-3-642-21861-3_3.  Google Scholar

[49]

A. Mielke, Finite elastoplasticity, Lie groups and geodesics on SL(d), in Geometry, Dynamics, and Mechanics (eds. P. Newton, A. Weinstein, and P. J. Holmes), Springer-Verlag, New York, (2002), 61-90. doi: 10.1007/0-387-21791-6_2.  Google Scholar

[50]

A. Mielke and A. Petrov, Thermally driven phase transformation in shape-memory alloys, Adv. Math. Sci. Appl., 17 (2007), 667-685.   Google Scholar

[51]

A. Mielke and T. Roubíček, Rate-Independent Systems -Theory and Application, Appl. Math. Sci. Series, Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7.  Google Scholar

[52]

A. MielkeT. Roubíček and U. Stefanelli, Γ-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differential Equations, 31 (2008), 387-416.  doi: 10.1007/s00526-007-0119-4.  Google Scholar

[53]

A. Mielke and U. Stefanelli, Linearized plasticity is the evolutionary Γ-limit of finite plasticity, J. Eur. Math. Soc. (JEMS), 15 (2013), 923-948.  doi: 10.4171/JEMS/381.  Google Scholar

[54]

A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Diff. Equations Applications, 11 (2004), 151-189.  doi: 10.1007/s00030-003-1052-7.  Google Scholar

[55]

H.-B. Mühlhaus and E. Aifantis. A variational principle for gradient plasticity, A variational principle for gradient plasticity, Int. J. Solids Struct., 28 (1991), 845-857.  doi: 10.1016/0020-7683(91)90004-Y.  Google Scholar

[56]

R. Paroni and G. Tomassetti, A variational justification of linear elasticity with residual stress, J. Elast., 97 (2009), 189-206.  doi: 10.1007/s10659-009-9217-1.  Google Scholar

[57]

R. Paroni and G. Tomassetti, From non-linear elasticity to linear elasticity with initial stress via Γ-convergence, Contin. Mech. Thermodyn., 23 (2011), 347-361.  doi: 10.1007/s00161-011-0184-y.  Google Scholar

[58]

P. Plecháč and T. Roubíček, Visco-elasto-plastic model for martensitic phase tratsformation in shape-memory alloys, Math. Methods Appl. Sci., 25 (2002), 1281-1298.  doi: 10.1002/mma.335.  Google Scholar

[59]

P. Popov and D. C. Lagoudas, A 3-D constitutive model for shape memory alloys incorporating pseudoelasticity and detwinning of self-accommodated martensite, Int. J. Plasticity, 23 (2007), 1679-1720.   Google Scholar

[60]

S. Reese and D. Christ, Finite deformation pseudo-elasticity of shape memory alloys -constitutive modelling and finite element implementation, Int. J. Plasticity, 24 (2008), 455-482.  doi: 10.1016/j.ijplas.2007.05.005.  Google Scholar

[61]

T. Roubíček, Models of microstructure evolution in shape memory alloys, in Nonlinear Homogenization and its Application to Composites, Polycrystals and Smart Materials, (eds. P. Ponte Castaneda, J. J. Telega, and B. Gambin), NATO Sci. Series Ⅱ, Kluwer, Dordrecht, 170 (2004), 269-304. doi: 10.1007/1-4020-2623-4_12.  Google Scholar

[62]

T. Roubíček, Modelling of thermodynamics of martensitic transformation in shape-memory alloys, Discrete Contin. Dyn. Syst., Dynamical systems and differential equations. Proceedings of the 6th AIMS International Conference, suppl., (2007), 892-902.   Google Scholar

[63]

T. Roubíček, Approximation in multiscale modelling of microstructure evolution in shape-memory alloys, Contin. Mech. Thermodyn., 23 (2011), 491-507.  doi: 10.1007/s00161-011-0190-0.  Google Scholar

[64]

T. Roubíček and M. Kružík, Mesoscopic model of microstructure evolution in shape memory alloys, its numerical analysis and computer implementation, GAMM Mitt., 29 (2006), 192-214.  doi: 10.1002/gamm.201490030.  Google Scholar

[65]

T. Roubíček and U. Stefanelli, Magnetic shape-memory alloys: Thermomechanical modeling and analysis, Contin. Mech. Thermodyn., 26 (2014), 783-810.  doi: 10.1007/s00161-014-0339-8.  Google Scholar

[66]

T. Roubíček and G. Tomassetti, Phase transformations in electrically conductive ferromagnetic shape-memory alloys, their thermodynamics and analysis, Archive Ration. Mech. Anal., 210 (2013), 1-43.  doi: 10.1007/s00205-013-0648-2.  Google Scholar

[67]

T. Roubíček and G. Tomassetti, Thermodynamics of shape-memory alloys under electric current, Z. Angew. Math. Phys., 61 (2010), 1-20.  doi: 10.1007/s00033-009-0007-1.  Google Scholar

[68]

B. Schmidt, Linear Γ-limits of multiwell energies in nonlinear elasticity theory, Contin. Mech. Thermodyn., 20 (2008), 375-396.  doi: 10.1007/s00161-008-0087-8.  Google Scholar

[69]

A.C. SouzaE.N. Mamiya and N. Zouain, Three-dimensional model for solids undergoing stress-induced tranformations, Eur. J. Mech. A Solids, 17 (1998), 789-806.   Google Scholar

[70]

P. Thamburaja and L. Anand, Polycrystalline shape-memory materials: Effect of crystallographic texture, J. Mech. Phys. Solids, 49 (2001), 709-737.  doi: 10.1016/S0022-5096(00)00061-2.  Google Scholar

[1]

Diego Grandi, Ulisse Stefanelli. The Souza-Auricchio model for shape-memory alloys. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 723-747. doi: 10.3934/dcdss.2015.8.723

[2]

Tomáš Roubíček. Modelling of thermodynamics of martensitic transformation in shape-memory alloys. Conference Publications, 2007, 2007 (Special) : 892-902. doi: 10.3934/proc.2007.2007.892

[3]

Michel Frémond, Elisabetta Rocca. A model for shape memory alloys with the possibility of voids. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1633-1659. doi: 10.3934/dcds.2010.27.1633

[4]

Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli. Thermal control of the Souza-Auricchio model for shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 369-386. doi: 10.3934/dcdss.2013.6.369

[5]

Linxiang Wang, Roderick Melnik. Dynamics of shape memory alloys patches with mechanically induced transformations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1237-1252. doi: 10.3934/dcds.2006.15.1237

[6]

Shuji Yoshikawa, Irena Pawłow, Wojciech M. Zajączkowski. A quasilinear thermoviscoelastic system for shape memory alloys with temperature dependent specific heat. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1093-1115. doi: 10.3934/cpaa.2009.8.1093

[7]

Alessia Berti, Claudio Giorgi, Elena Vuk. Free energies and pseudo-elastic transitions for shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 293-316. doi: 10.3934/dcdss.2013.6.293

[8]

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

[9]

Toyohiko Aiki, Martijn Anthonissen, Adrian Muntean. On a one-dimensional shape-memory alloy model in its fast-temperature-activation limit. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 15-28. doi: 10.3934/dcdss.2012.5.15

[10]

Takashi Suzuki, Shuji Yoshikawa. Stability of the steady state for multi-dimensional thermoelastic systems of shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 209-217. doi: 10.3934/dcdss.2012.5.209

[11]

Ken Shirakawa. Asymptotic stability for dynamical systems associated with the one-dimensional Frémond model of shape memory alloys. Conference Publications, 2003, 2003 (Special) : 798-808. doi: 10.3934/proc.2003.2003.798

[12]

Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli. A rate-independent model for permanent inelastic effects in shape memory materials. Networks & Heterogeneous Media, 2011, 6 (1) : 145-165. doi: 10.3934/nhm.2011.6.145

[13]

Michela Eleuteri, Luca Lussardi. Thermal control of a rate-independent model for permanent inelastic effects in shape memory materials. Evolution Equations & Control Theory, 2014, 3 (3) : 411-427. doi: 10.3934/eect.2014.3.411

[14]

Andrew D. Lewis, David R. Tyner. Geometric Jacobian linearization and LQR theory. Journal of Geometric Mechanics, 2010, 2 (4) : 397-440. doi: 10.3934/jgm.2010.2.397

[15]

Toyohiko Aiki. On the existence of a weak solution to a free boundary problem for a model of a shape memory alloy spring. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 1-13. doi: 10.3934/dcdss.2012.5.1

[16]

Christian Pötzsche, Evamaria Russ. Topological decoupling and linearization of nonautonomous evolution equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1235-1268. doi: 10.3934/dcdss.2016050

[17]

Shu-Ming Sun. Existence theory of capillary-gravity waves on water of finite depth. Mathematical Control & Related Fields, 2014, 4 (3) : 315-363. doi: 10.3934/mcrf.2014.4.315

[18]

Toyohiko Aiki. The position of the joint of shape memory alloy and bias springs. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 239-246. doi: 10.3934/dcdss.2011.4.239

[19]

Filippo Dell'Oro, Olivier Goubet, Youcef Mammeri, Vittorino Pata. A semidiscrete scheme for evolution equations with memory. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5637-5658. doi: 10.3934/dcds.2019247

[20]

Dorin Ieşan. Strain gradient theory of porous solids with initial stresses and initial heat flux. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2169-2187. doi: 10.3934/dcdsb.2014.19.2169

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (21)
  • HTML views (60)
  • Cited by (0)

Other articles
by authors

[Back to Top]