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Dynamical modelling of phase transitions by means of viscoelasticity in many dimensions

Published online by Cambridge University Press:  14 November 2011

Piotr Rybka
Piotr Rybka
Affiliation:
Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A
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Synopsis

We study the equations of viscoelasticity in a multidimensional setting for the ‘no-traction’ boundary data. For the sake of modelling phase transitions we do not assume elliptieity of the stored energy function W. We construct dynamics in W1,2(Ωℝn) globally in time. Next, we study the question of stability for a class of equilibria. Moreover, we show a certain kind of decay in time of solutions for arbitrary initial conditions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 121 , Issue 1-2 , 1992 , pp. 101 - 138
Copyright
Copyright © Royal Society of Edinburgh 1992

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References

1Acerbi, E. and Fusco, N.. Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal. 86 (1984), 125–145.Google Scholar
2Agmon, S., Doughs, A. and Nirenberg, L.. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Comm. Pure App. Math. 17 (1964), 35–92.Google Scholar
3Andrews, G.. On the existence of solutions to the equation utt = uxxt + σ(ux)x. J. Differential Equations 35 (1980), 200–232.Google Scholar
4Andrews, G. and Ball, J. M.. Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity. J. of Differential Equations 44 (1982), 316–341.Google Scholar
5Ball, J. M. and James, R.. Fine phase mixture as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987), 13–52.CrossRefGoogle Scholar
6Ball, J. M. and Marsden, J. E., Quasiconvexity at the boundary, positivity of the second variation and elastic stability. Arch. Rational Mech. Anal. 86 (1984), 251–277.Google Scholar
7Barsch, G. R., Horovitz, B. and Krumhansl, J. A.. Dynamics of twin boundaries in martensites. Phys. Rev. Lett. 100 (1987), 1251–1254.Google Scholar
8Clements, J.. Existence theorems for a quasi-linear evolution equation, SIAM J. Appl. Math. 26 (1974), 745–752.Google Scholar
9Dafermos, C. M.. The mixed initial-boundary value problem for the equations of non-linear one-dimensional viscoelasticity. J. Differential Equations 6 (1969), 71–86.CrossRefGoogle Scholar
10DiPerna, R.. Convergence of approximate solutions to conservation laws. Arch. Rational Mech. Anal. 82 (1983), 27–70.CrossRefGoogle Scholar
11Engler, H.. Strong solutions for strongly damped quasilinear wave equations. Contemporary Math. 64 (1987), 219–237.Google Scholar
12Evans, L. C. and Gariepy, R. F.. Some remarks on quasiconvexity and strong convergence. Proc. Roy. Soc. Edinburgh Sect. A 106 (1987), 53–61.CrossRefGoogle Scholar
13Falk, F.. Ginzburg–Landau theory of static domain walls in shape-memory alloys. Z. Phys. B 51 (1983), 30–55.Google Scholar
14Fujiwara, D. and Morimoto, H.. An Lr-theorem of the Helmholtz decomposition of vector fields. F. Fac. Sci. Univ. Tokyo, Sec. I 24 (1977), 685–700.Google Scholar
15Fujiwara, D.. On the asymptotic behavior of the Green operators for elliptic boundary problems and the pure imaginary powers of some second order operators. J. Math. Soc. Japan 21 (1969), 481–522.Google Scholar
16Friedman, A.. Partial Differential Equations (New York: Holt, 1969).Google Scholar
17Friedman, A. and Necas, J.. Systems of nonlinear wave equations with nonlinear viscosity. Pacific J. Math. 135 (1988), 30–55.Google Scholar
18Greenberg, J. M.. On the existence, uniqueness and stability of the equation ρºXtt = E(Xx)Xxx + λXxxt. J. Math. Anal. Appl. 25 (1969), 575–591.CrossRefGoogle Scholar
19Greenberg, J. M., MacCamy, R. C. and Mizel, V.. On the existence, uniqueness and stability of the equation ρ'(ux)uxx + λuxxt = ρºutt. J. Math. Mech. 17 (1968), 707–728.Google Scholar
20Henry, D., Geometric theory of semilinear parabolic equations (Berlin: Springer, 1981).CrossRefGoogle Scholar
20aHenry, D.. Geometricheskaya teoriya polulineinykh parabolicheskikh uravnenii (in Russian; Moscow: Mir, 1985).Google Scholar
21Hörmander, L.. The analysis of linear partial operators, vol 3 (Berlin: Springer, 1985).Google Scholar
22Hughes, T., Kato, T. and Marsden, J.. Well-posed quasi-linear hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Rational Mech. Anal. 63 (1977), 273–304.Google Scholar
23Kato, T.. Perturbation Theory for Linear Operators (New York: Springer, 1966).Google Scholar
24Krasnoselskii, M. A.et al. Integral Operators in Spaces of Summable Functions (Leyden: Noordhoff, 1976).CrossRefGoogle Scholar
25Kuttler, K. and Hicks, D.. Initial-boundary value problems for the equation u tt = (ρ(ux) + α(ux)uxt)x + f, Quart. Appl. Math. 46 (1988), 393–407.Google Scholar
26Lions, J. L. and Magenes, E.. Non-Homogeneous Boundary Value Problems and Applications vol. I (Berlin: Springer, 1972).Google Scholar
27Maddocks, J. H. and Parry, G. P., A model for twinning. J. of Elasticity 16 (1986), 113–135.CrossRefGoogle Scholar
28Miklavčič, M.. Stability for semilinear parabolic equations with noninvertible linear operator. Pacific J. Math. 118 (1985), 199–214.Google Scholar
29Morrey, C. B., Jr. Quasiconvexity and the lower semicontinuity of multiple integrals. Pacific J. Math. 2 (1952), 25–35.Google Scholar
30Niezg, M.ódka and Sprekels, J.. Existence of solutions for a mathematical model of structural phase transitions in shape memory alloys. Math. Methods Appl. Sci. 10 (1988), 197–223.Google Scholar
31Pecher, H.. On global regular solution of third order partial differential equations. J. Math. Anal. Appl. 73 (1980), 278–309.Google Scholar
32Pego, R.. Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability. Arch. Rational Mech. Anal. 97 (1987), 353–394.Google Scholar
33Potier-Ferry, M.. On the Mathematical foundations of elastic stability, I. Arch. Rational Mech. Anal. 78 (1982), 55–72.Google Scholar
34Renardy, M., Hrusa, W. J. and Nohel, J. A.. Mathematical Problems in Viscoelasticity (New York: Longman, 1987).Google Scholar
35Rybka, P.. Dynamical Modeling of Phase Transitions in solids by Means of Viscoelasticity in Many Dimensions (Thesis, New York University, New York, 1990).Google Scholar
36Valent, T.. Boundary Value Problems of Finite Elasticity (New York: Springer, 1988).CrossRefGoogle Scholar
37Ziemer, W. P.. Weakly Differentiable Functions (New York: Springer, 1989).CrossRefGoogle Scholar