American Institute of Mathematical Sciences

Advanced
March  2011, 6(1): 145-165. doi: 10.3934/nhm.2011.6.145

A rate-independent model for permanent inelastic effects in shape memory materials

Michela Eleuteri 1, , Luca Lussardi 2, and  Ulisse Stefanelli 3,

1. 

Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38100 Povo (Trento), Italy
2. 

Fakultät für Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, 44227 Dortmund, Germany
3. 

Istituto di Matematica Applicata e Tecnologie Informatiche – CNR, Via Ferrata 1, 27100 Pavia

Received  May 2010 Revised  October 2010 Published  March 2011

This paper addresses a three-dimensional model for isothermal stress-induced transformation in shape memory polycrystalline materials in presence of permanent inelastic effects. The basic features of the model are recalled and the constitutive and the three-dimensional quasi-static evolution problem are proved to be well-posed. Finally, we discuss the convergence of the model to reduced/former ones by means of a rigorous $\Gamma$-convergence analysis.
Keywords: Shape memory materials, $\Gamma$-convergence., well-posedness, permanent inelastic effects.
Mathematics Subject Classification: Primary: 74C05; 35K8.
Citation: 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
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show all references

References:
[1]

J. Math. Soc. Japan, 57 (2005), 903-933. doi: 10.2969/jmsj/1158241940.  Google Scholar

[2]

in the IMA Volumes in Mathematics and its Applications, 3, Springer-Verlag, New York, 1987.  Google Scholar

[3]

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

[4]

Journal de Physique IV, 11 (2001), 577-582. Google Scholar

[5]

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

[6]

Internat. J. Numer. Meth. 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]

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

[9]

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

[10]

Int. J. Non-Linear Mech., 32 (1997), 1101-1114. doi: 10.1016/S0020-7462(96)00130-8.  Google Scholar

[11]

Comput. Mech. Appl. Mech. Engrg., 146 (1997), 281-312. doi: 10.1016/S0045-7825(96)01232-7.  Google Scholar

[12]

Math. Models Meth. Appl. Sci. (2010) to appear. Google Scholar

[13]

Int. J. Eng. Sci., 37 (1999), 1175-1203. doi: 10.1016/S0020-7225(98)00115-3.  Google Scholar

[14]

vol. 121 of Applied Mathematical Sciences, Springer-Verlag, New York, 1996.  Google Scholar

[15]

Math. Methods Appl. Sci., 29 (2006), 209-233. doi: 10.1002/mma.672.  Google Scholar

[16]

Nonlinear Anal., 24 (1995), 1565-1579. doi: 10.1016/0362-546X(94)00097-2.  Google Scholar

[17]

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

[18]

Birkhäuser-Boston, 1993.  Google Scholar

[19]

Springer-Berlin, 1976.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

Springer-Verlag, 2002.  Google Scholar

[24]

Springer-Verlag, 1996. Google Scholar

[25]

Phys. D, 72 (1994), 287-308. doi: 10.1016/0167-2789(94)90234-8.  Google Scholar

[26]

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[27]

J. Intell. Mater. Syst. Struct., 8 (1997), 815-823. doi: 10.1177/1045389X9700801001.  Google Scholar

[28]

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[29]

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

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[31]

Math. Mech. Solids (2010), to appear. Google Scholar

[32]

M2AN Math. Model. Anal. Numer., (2010), to appear. Google Scholar

[33]

Meccanica, 40 (2005), 389-418. doi: 10.1007/s11012-005-2106-1.  Google Scholar

[34]

IMA J. Appl. Math., (2010), to appear. Google Scholar

[35]

Mech. Mater., 38 (2006), 391-429. doi: 10.1016/j.mechmat.2005.08.003.  Google Scholar

[36]

Mech. Mater., 36 (2004), 865-892. Google Scholar

[37]

Calc. Var. Partial Differential Equations, 22 (2005), 73-99.  Google Scholar

[38]

Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1992.  Google Scholar

[39]

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

[40]

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

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Proc. of the IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials (Bochum 2008), IUTAM Bookseries, Springer, 2009. Google Scholar

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Adv. Math. Sci. Appl., 17 (2007), 160-182.  Google Scholar

[44]

Multiscale Model. Simul., 1 (2003), 571-597. doi: 10.1137/S1540345903422860.  Google Scholar

[45]

Calc. Var. Partial Differential Equations, 31 (2008), 387-416. doi: 10.1007/s00526-007-0119-4.  Google Scholar

[46]

Proc. of the Workshop on Models of Continuum Mechanics in Analysis and Engineering, eds. H.-D Alber, R. Balean and R. Farwig (Shaker-Verlag, 1999), 117-129. Google Scholar

[47]

Arch. Rational Mech. Anal., 162 (2002), 137-177. doi: 10.1007/s002050200194.  Google Scholar

[48]

J. Phys. IV, C2-5 (1995), 423-431. Google Scholar

[49]

Internat. J. of Solid. Struct., 42 (2005), 3439-3457. doi: 10.1016/j.ijsolstr.2004.11.006.  Google Scholar

[50]

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

[51]

Materials Sci. Engrg. A, 438-440 (2006), 454-458. doi: 10.1016/j.msea.2006.01.104.  Google Scholar

[52]

Int. J. Plasticity, 23 (2007), 1679-1720. doi: 10.1016/j.ijplas.2007.03.011.  Google Scholar

[53]

European J. Mech. A Solids, 13 (1994), 21-50. Google Scholar

[54]

Int. J. Plasticity, 28 (2008), 455-482. doi: 10.1016/j.ijplas.2007.05.005.  Google Scholar

[55]

Interfaces Free Bound., 4 (2002), 111-136. doi: 10.4171/IFB/55.  Google Scholar

[56]

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, 2004), 269-304.  Google Scholar

[57]

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

[58]

J. Mech. Phys. Solids, 49 (2001), 709-737. doi: 10.1016/S0022-5096(00)00061-2.  Google Scholar

[59]

Comput. Materials Sci., 41 (2007), 208-221. doi: 10.1016/j.commatsci.2007.04.006.  Google Scholar

[60]

Progress in Nonlinear Differential Equations and their Applications, 28. Birkhäuser Boston, MA, 1996.  Google Scholar

[61]

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

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