\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Thermal control of the Souza-Auricchio model for shape memory alloys

Abstract Related Papers Cited by
  • We address the thermal control of the quasi-static evolution of a polycrystalline shape memory alloy specimen. The thermomechanical evolution of the body is described by means of the phenomenological SOUZA$-$AURICCHIO model [6,53]. By assuming to be able to control the temperature of the body in time we determine the corresponding quasi-static evolution in the energeticsense. By recovering in this context a result by RINDLER [49,50] we prove the existence of optimal controls for a suitably large class of cost functionals and comment on their possible approximation.
    Mathematics Subject Classification: 74C05, 49J20.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    F. Auricchio, A. Reali and U. Stefanelli, A macroscopic 1D model for shape memory alloys including asymmetric behaviors and transformation-dependent elastic properties, Comput. Methods Appl. Mech. Engrg., 198 (2009), 1631-1637.

    [2]

    F. Auricchio, A.-L. Bessoud, A. Reali and U. Stefanelli, A phenomenological model for ferromagnetism in shape-memory materials, In preparation, (2011).

    [3]

    F. Auricchio, A.-L. Bessoud, A. Reali and U. Stefanelli, A three-dimensional phenomenological models for magnetic shape memory alloys, GAMM-Mitt., 34 (2011), 90-96.doi: 10.1002/gamm.201110014.

    [4]

    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.

    [5]

    F. Auricchio, A. 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.

    [6]

    F. Auricchio and L. Petrini, Improvements and algorithmical considerations on a recent three-dimensional model describing stress-induced solid phase transformations, Internat. J. Numer. Methods Engrg., 55 (2002), 1255-1284.

    [7]

    F. Auricchio and L. Petrini, A three-dimensional model describing stress-temperature induced solid phase transformations. Part I: Solution algorithm and boundary value problems, Internat. J. Numer. Meth. Engrg., 61 (2004), 807-836.doi: 10.1002/nme.1086.

    [8]

    F. Auricchio and L. Petrini, A three-dimensional model describing stress-temperature induced solid phase transformations. Part II: Thermomechanical coupling and hybrid composite applications, Internat. J. Numer. Meth. Engrg., 61 (2004), 716-737.doi: 10.1002/nme.1087.

    [9]

    F. Auricchio, A. Reali and U. Stefanelli, A phenomenological 3D model describing stress-induced solid phase transformations with permanent inelasticity, in "Topics on Mathematics for Smart Systems," 1-14. World Sci. Publ., Hackensack, NJ, 2007.

    [10]

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

    [11]

    A.-L. Bessoud and U. Stefanelli, Magnetic shape memory alloys: Three-dimensional modeling and analysis, Math. Models Meth. Appl. Sci., 21 (2011), 1043-1069.

    [12]

    A.-L. Bessoud, M. Kružík and U. StefanelliA macroscopic model for magnetic shape memory alloys, Z. Angew. Math. Phys., to appear.

    [13]

    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.

    [14]

    M. Brokate and J. Sprekels, Optimal control of shape memory alloys with solid-solid phase transitions, in "Emerging Applications in Free Boundary Problems" (Montreal, 1990), 280 of Pitman Res. Notes Math. Ser., 208-214. Longman Sci. Tech., Harlow, 1993.

    [15]

    N. Bubner, J. Sokołowski and J. Sprekels, Optimal boundary control problems for shape memory alloys under state constraints for stress and temperature, Numer. Funct. Anal. Optim., 19 (1998), 489-498.doi: 10.1080/01630569808816840.

    [16]

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

    [17]

    V. Evangelista, S. 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.

    [18]

    V. Evangelista, S. Marfia and E. Sacco, A 3D SMA constitutive model in the framework of finite strain, Internat. J. Numer. Methods Engrg., 81 (2010), 761-785.

    [19]

    F. Falk, Model free energy, mechanics and thermodynamics of shape memory alloys, Acta Metallurgica, 28 (1990), 1773-1780.doi: 10.1016/0001-6160(80)90030-9.

    [20]

    G. 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.

    [21]

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

    [22]

    M. Frémond, "Non-Smooth Thermomechanics," Springer-Verlag, Berlin, 2002.

    [23]

    M. Frémond and S. Miyazaki, "Shape Memory Alloys," CISM Courses and Lectures, vol. 351, Springer-Verlag, 1996.

    [24]

    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), 63-77.

    [25]

    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.doi: 10.1016/S0045-7825(01)00271-7.

    [26]

    D. Helm and P. Haupt, Shape memory behaviour: modelling within continuum thermomechanics, Intern. J. Solids Struct., 40 (2003), 827-849.doi: 10.1016/S0020-7683(02)00621-2.

    [27]

    K.-H. Hoffmann and D. Tiba, Control of a plate with nonlinear shape memory alloy reinforcements, Adv. Math. Sci. Appl., 7 (1997), 427-436.

    [28]

    K.-H. Hoffmann and A. żochowski, Control of the thermoelastic model of a plate activated by shape memory alloy reinforcements, Math. Methods Appl. Sci., 21 (1998), 589-603.doi: 10.1002/(SICI)1099-1476(19980510)21:7<589::AID-MMA904>3.0.CO;2-D.

    [29]

    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.

    [30]

    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.

    [31]

    D. C. Lagoudas, P. B. Entchev, P. Popov, E. Patoor, L. C. Brinson and X. Gao, Shape memory alloys, Part II: Modeling of polycrystals, Mech. Materials, 38 (2006), 391-429.doi: 10.1016/j.mechmat.2005.05.027.

    [32]

    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.doi: 10.1016/j.ijplas.2007.03.011.

    [33]

    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.

    [34]

    M. Eleuteri, L. Lussardi and U. Stefanelli, A rate-independent model for permanent inelastic effects in shape memory materials, Netw. Heterog. Media, 6 (2011), 145-165.

    [35]

    A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99.

    [36]

    A. Mielke, Evolution of rate-independent systems, in "Handbook of Differential Equations, Evolutionary Equations" (eds., C. Dafermos and E. Feireisl), Elsevier, 2 (2005), 461-559.

    [37]

    A. Mielke, L. Paoli and A. Petrov, On existence and approximation for a 3D model of thermally induced phase transformations in shape-memory alloys, SIAM J. Math. Anal., 41 (2009), 1388-1414.doi: 10.1137/080726215.

    [38]

    A. Mielke, L. Paoli, A. Petrov and U. Stefanelli, Error estimates for space-time discretizations of a rate-independent variational inequality, SIAM J. Numer. Anal., 48 (2010), 1625-1646.doi: 10.1137/090750238.

    [39]

    A. Mielke, L. Paoli, A. Petrov and U. Stefanelli, Error bounds for space-time discretizations of a 3d model for shape-memory materials, in "IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials" (editor, K. Hackl), 185-197. Springer, 2010. Proceedings of the IUTAM Symposium on Variational Concepts, Bochum, Germany, Sept.\ 22-26, 2008.

    [40]

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

    [41]

    A. Mielke and F. Rindler, Reverse approximation of energetic solutions to rate-independent processes, NoDEA Nonlinear Differential Equations Appl., 16 (2009), 17-40.

    [42]

    A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA, Nonlinear Diff. Equations Applications, 11 (2004), 151-189.

    [43]

    B. Peultier, T. Ben Zineb and E. Patoor, Macroscopic constitutive law for SMA: Application to structure analysis by FEM, Materials Sci. Engrg. A, 438-440 (2006), 454-458.

    [44]

    L. Paoli and A. Petrov, Global existence result for phase transformations with heat transfer in shape memory alloys, Preprint WIAS, 1608 (2011).

    [45]

    I. Pawłow and A. Żochowski, A control problem for a thermoelastic system in shape memory materials, Sūrikaisekikenkyūsho Kōkyūroku, No. 1210 (2011), 8-23, Free boundary problems (Japanese) (Kyoto, 2000).

    [46]

    {R. Peyroux, A. Chrysochoos, Ch. Licht and M. Löbel}, Phenomenological constitutive equations for numerical simulations of SMA's structures. Effect of thermomechanical couplings, J. Phys. C4 Suppl., 6 (1996), 347-356.

    [47]

    B. Raniecki and Ch. Lexcellent, $R_L$ models of pseudoelasticity and their specification for some shape-memory solids, Eur. J. Mech. A Solids, 13 (1994), 21-50.

    [48]

    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.

    [49]

    F. Rindler, Optimal control for nonconvex rate-independent evolution processes, SIAM J. Control Optim., 47 (2008), 2773-2794.doi: 10.1137/080718711.

    [50]

    F. Rindler, Approximation of rate-independent optimal control problems, SIAM J. Numer. Anal., 47 (2009), 3884-3909.doi: 10.1137/080744050.

    [51]

    T.Roubíček, Models of microstructure evolution in shape memory alloys, in "Nonlinear Homogenization and its Appl.to Composites, Polycrystals and Smart Materials" (eds. P. Ponte Castaneda, J. J. Telega and B. Gambin), NATO Sci. Series II/170, Kluwer, Dordrecht, 2004, 269-304.

    [52]

    J. Sokołowski and J. Sprekels, Control problems with state constraints for shape memory alloys, Math. Methods Appl. Sci., 17 (1994), 943-952.

    [53]

    A. C. Souza, E. N. Mamiya and N. Zouain, Three-dimensional model for solids undergoing stress-induced tranformations, Eur. J. Mech. A Solids, 17 (1998), 789-806.doi: 10.1016/S0997-7538(98)80005-3.

    [54]

    U. Stefanelli, Magnetic control of magnetic shape-memory single crystals, Phys. B, 407 (2012), 1316-1321.

    [55]

    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.

    [56]

    G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, Part I: Existence and discretization in time, SIAM Journal on Control and Optimization (SICON), to appear (2012).

    [57]

    G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, part II: Regularization and differentiability, Preprint SPP1253-119, 2011.

    [58]

    G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening, part III: Optimality conditions, Preprint SPP1253-119, 2011.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(111) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return