September  2014, 3(3): 411-427. doi: 10.3934/eect.2014.3.411

Thermal control of a rate-independent model for permanent inelastic effects in shape memory materials

1. 

Dipartimento di Matematica e Informatica, Università di Firenze, viale Morgagni 67/a, I-50134 Firenze, Italy

2. 

Dipartimento di Matematica e Fisica "N.Tartaglia", Università Cattolica del Sacro Cuore, Via dei Musei 41, I-25121 Brescia, Italy

Received  April 2013 Revised  January 2014 Published  August 2014

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 an extension of the phenomenological Souza-Auricchio model [6,7,8,57] accounting also for permanent inelastic effects [9,11,27]. By assuming to be able to control the temperature of the body in time we determine the corresponding quasi-static evolution in the energetic sense. In a similar way as in [28], using results by Rindler [49,50] we prove the existence of optimal controls for a suitably large class of cost functionals.
Citation: 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
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show all references

References:
[1]

Mathematical aspects of modelling structure formation phenomena, GAKUTO Internat. Ser. Math. Sci. Appl., Tokio, 17 (2002), 144-162.  Google Scholar

[2]

Contin. Mech. Thermodyn., 15 (2003), 463-485. doi: 10.1007/s00161-003-0127-3.  Google Scholar

[3]

Preprint IMATI-CNR 3PV13/3/0, 2013. Google Scholar

[4]

GAMM-Mitt., 34 (2011), 90-96. doi: 10.1002/gamm.201110014.  Google Scholar

[5]

Math. Models Meth. Appl. Sci., 18 (2008), 125-164. doi: 10.1142/S0218202508002632.  Google Scholar

[6]

Internat. J. Numer. Methods Engrg., 55 (2002), 1255-1284. doi: 10.1002/nme.619.  Google Scholar

[7]

Internat. J. Numer. Meth. Engrg., 61 (2004), 807-836. doi: 10.1002/nme.1086.  Google Scholar

[8]

Internat. J. Numer. Meth. Engrg., 61 (2004), 716-737. doi: 10.1002/nme.1087.  Google Scholar

[9]

Int. J. Plasticity, 23 (2007), 207-226. doi: 10.1016/j.ijplas.2006.02.012.  Google Scholar

[10]

Comput. Methods Appl. Mech. Engrg., 198 (2009), 1631-1637. doi: 10.1016/j.cma.2009.01.019.  Google Scholar

[11]

N. Barrera, P. Biscari and M. F. Urbano, Macroscopic modeling of functional fatigue in shape memory alloys,, Eur. J. Mech. A/Solids, ().   Google Scholar

[12]

Math. Modelling Numer. Anal., 45 (2011), 477-504. doi: 10.1051/m2an/2010063.  Google Scholar

[13]

B. Benesova, M. Frost and P. Sedlak, A microscopically motivated constitutive model for shape memory alloys: formulation, analysis and computations,, Preprint NCMM/2013/17, ().   Google Scholar

[14]

Math. Models Meth. Appl. Sci., 21 (2011), 1043-1069. doi: 10.1142/S0218202511005246.  Google Scholar

[15]

Preprint IMATI-CNR, 23PV10/21/0, 2010. Google Scholar

[16]

Discrete Cont. Dyn. S. - Series S, 6 (2013), 293-316.  Google Scholar

[17]

Physica D, 239 (2010), 95-102. doi: 10.1016/j.physd.2009.10.005.  Google Scholar

[18]

Math Studies, 5, North-Holland, Amsterdam/New York, 1973.  Google Scholar

[19]

Applied Mathematical Sciences, 121, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[20]

in Emerging applications in free boundary problems (Montreal, 1990), Pitman Res. Notes Math. Ser., 280, Longman Sci. Tech., Harlow, 1993, 208-214.  Google Scholar

[21]

Numer. Funct. Anal. Optim., 19 (1998), 489-498. doi: 10.1080/01630569808816840.  Google Scholar

[22]

Nonlinear Anal., 18 (1992), 873-888. doi: 10.1016/0362-546X(92)90228-7.  Google Scholar

[23]

ASM International, 2003. Google Scholar

[24]

Physica B: Condensed Matter, 407 (2012), 1415-1416. doi: 10.1016/j.physb.2011.10.017.  Google Scholar

[25]

Comm. Pure Appl. Anal., 12 (2013), 2973-2996. doi: 10.3934/cpaa.2013.12.2973.  Google Scholar

[26]

Discrete Cont. Dyn. S. Series B, 19 (2014). Google Scholar

[27]

Netw. Heterog. Media, 6 (2011), 145-165. doi: 10.3934/nhm.2011.6.145.  Google Scholar

[28]

Discrete Cont. Dyn. S. - Series S, 6 (2013), 369-386.  Google Scholar

[29]

Comput. Mech., 44 (2009), 405-421. doi: 10.1007/s00466-009-0381-8.  Google Scholar

[30]

Internat. J. Numer. Methods Engrg., 81 (2010), 761-785. Google Scholar

[31]

J. Phys. C4 Suppl., 12 (1982), 3-15. Google Scholar

[32]

J. Phys. Condens. Matter, 2 (1990), 61-77. doi: 10.1088/0953-8984/2/1/005.  Google Scholar

[33]

C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre, 304 (1987), 239-244. Google Scholar

[34]

Contin. Mech. Thermodyn., 24 (2012), 63-77. doi: 10.1007/s00161-011-0221-x.  Google Scholar

[35]

Nonlinear Anal., 15 (1990), 977-990. doi: 10.1016/0362-546X(90)90079-V.  Google Scholar

[36]

Adv. Math. Sci. Appl., 7 (1997), 427-436.  Google Scholar

[37]

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.  Google Scholar

[38]

Adv. in Math. Sci. and Appl., 5 (1995), 91-116.  Google Scholar

[39]

M2AN Math. Model. Numer. Anal., 44 (2010), 1239-1253. doi: 10.1051/m2an/2010024.  Google Scholar

[40]

Math. Mech. Solids, 16 (2011), 349-365. doi: 10.1177/1081286510386935.  Google Scholar

[41]

Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9781139172400.  Google Scholar

[42]

in Handbook of Differential Equations, Evolutionary Equations, (editors, C. Dafermos and E. Feireisl), 2 (2005), 461-559.  Google Scholar

[43]

SIAM J. Math. Anal., 41 (2009), 1388-1414. doi: 10.1137/080726215.  Google Scholar

[44]

SIAM J. Numer. Anal., 48 (2010), 1625-1646. doi: 10.1137/090750238.  Google Scholar

[45]

Adv. Math. Sci. Appl., 17 (2007), 667-685.  Google Scholar

[46]

NoDEA, Nonlinear Diff. Equations Applications, 11 (2004), 151-189. doi: 10.1007/s00030-003-1052-7.  Google Scholar

[47]

Preprint WIAS, 1608, (2011). Google Scholar

[48]

Control Cybernet., 29 (2000), 341-365.  Google Scholar

[49]

SIAM J. Control Optim., 47 (2008), 2773-2794. doi: 10.1137/080718711.  Google Scholar

[50]

SIAM J. Numer. Anal., 47 (2009), 3884-3909. doi: 10.1137/080744050.  Google Scholar

[51]

Quaderno 05/2012 del Seminario Matematico di Brescia, (2012), 1-50. Google Scholar

[52]

SIAM J. Math. Anal., 42 (2010), 256-297. doi: 10.1137/080729992.  Google Scholar

[53]

Zeit. angew. Math. Phys., 61 (2010), 1-20. doi: 10.1007/s00033-009-0007-1.  Google Scholar

[54]

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

[55]

Smart Mater. Struct., 16 (2007), 1751-1765. doi: 10.1088/0964-1726/16/5/030.  Google Scholar

[56]

Math. Methods Appl. Sci., 17 (1994), 943-952. doi: 10.1002/mma.1670171204.  Google Scholar

[57]

Eur. J. Mech. A/Solids, 17 (1998), 789-806. doi: 10.1016/S0997-7538(98)80005-3.  Google Scholar

[58]

Phys. B, 407 (2012), 1316-1321. doi: 10.1016/j.physb.2011.06.043.  Google Scholar

[59]

SIAM Journal on Control and Optimization (SICON), 50 (2012), 2836-2861. doi: 10.1137/110839187.  Google Scholar

[60]

Preprint SPP1253-119, 2011. Google Scholar

[61]

Preprint SPP1253-119, 2011. Google Scholar

[62]

SIAM J. Math. Anal., 38 (2007), 1733-1759. doi: 10.1137/060653159.  Google Scholar

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