Thermomechanical evolution of a microstructure
Authors:
Karl-Heinz Hoffmann and Tomáš Roubíček
Journal:
Quart. Appl. Math. 52 (1994), 721-737
MSC:
Primary 73B30; Secondary 35Q72, 73F15, 73S10, 73V25
DOI:
https://doi.org/10.1090/qam/1306046
MathSciNet review:
MR1306046
Full-text PDF Free Access
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Abstract: A nonisothermal microstructure evolution model, governed by a Helmholtz free energy which need not be convex as a function of deformations, is formulated by using a convexified geometry proposed already in [13]. A multidimensional but scalar case is treated. It is shown that, as a special case, this model includes the usual nonlinear thermo-visco-elasticity. In the case of an actual appearance of a microstructure, the existence of a weak solution to a partial linearized model is shown by a semi-implicit time discretization.
H. W. Alt, K.-H. Hoffman, M. Niezgódka, and J. Sprekels, A numerical study of structural phase transitions in shape memory alloys, Preprint No. 90, Institut für Mathematik, Universität Augsburg, 1985
P. Colli, M. Frémond, and A. Visintin, Thermo-mechanical evolution of shape memory alloys, Quart. Appl. Math. 48, 31–47 (1990)
C. M. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal. 29, 241–271 (1968)
C. M. Dafermos, Global smooth solutions to the initial boundary value problem for the equations of one-dimensional thermoviscoelasticity, SIAM J. Math. Anal. 13, 397–408 (1982)
F. Falk, Landau theory and martensitic phase transitions (L. Delaey and M. Chandrasekaran, eds.), Proc. Internat. Conf. on Martensitic Transformations, Les Editions de Physique, Les Ulis, 1982
M. Frémond, Matériaux à mémoire de forme, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers. Sci. Terre 304, 239–244 (1987)
K.-H. Hoffmann, M. Niezgódka, and Zheng Songmu, Existence and uniqueness of global solutions to an extended model of the dynamical development in shape memory alloys, Nonlinear Anal. 15, 977–990 (1990)
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Graylock Press, Baltimore, 1961
J. Nečas, Dynamic in the nonlinear thermo-visco-elasticity, Symposium “Partial Differential Equations” Holzhau 1988 (B.-W. Schulze and H. Triebel, eds.), Teubner-Texte Math. 112, Teubner, Leipzig, 1989, pp. 197–203
J. Nečas, A. Novotný, and V. Šverák, On the uniqueness of solution to the nonlinear thermo-viscoelasticity, Math. Nachr. 149, 319–324 (1990)
R. E. Nickell and J. L. Sackman, Variational principles for linear coupled thermoelasticity, Quart. Appl. Math. 26, 11–26 (1968)
M. Niezgódka and J. Sprekels, Existence of solutions for a mathematical model of structural phase transitions in shape memory alloys, Math. Methods Appl. Sci. 10, 197–223 (1988)
T. Roubíček, Evolution of a microstructure: a convexified model, Math. Methods Appl. Sci. 16, 625–642 (1993)
T. Roubíček, Optimality conditions for nonconvex variational problems relaxed in terms of Young measures (submitted)
T. Roubíček, Finite element approximation of a microstructure evolution, Math. Methods Appl. Sci. 17, 377–393 (1994)
M. Schatzman, A hyperbolic problem of second order with unilateral constraints: the vibrating string with a concave obstacle, J. Math. Anal. Appl. 73, 138–191 (1980)
J. Sprekels, Global existence for thermomechanical processes with nonconvex free energies of Ginzburg-Landau form, J. Math. Anal. Appl. 141, 333–348 (1989)
H. W. Alt, K.-H. Hoffman, M. Niezgódka, and J. Sprekels, A numerical study of structural phase transitions in shape memory alloys, Preprint No. 90, Institut für Mathematik, Universität Augsburg, 1985
P. Colli, M. Frémond, and A. Visintin, Thermo-mechanical evolution of shape memory alloys, Quart. Appl. Math. 48, 31–47 (1990)
C. M. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal. 29, 241–271 (1968)
C. M. Dafermos, Global smooth solutions to the initial boundary value problem for the equations of one-dimensional thermoviscoelasticity, SIAM J. Math. Anal. 13, 397–408 (1982)
F. Falk, Landau theory and martensitic phase transitions (L. Delaey and M. Chandrasekaran, eds.), Proc. Internat. Conf. on Martensitic Transformations, Les Editions de Physique, Les Ulis, 1982
M. Frémond, Matériaux à mémoire de forme, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers. Sci. Terre 304, 239–244 (1987)
K.-H. Hoffmann, M. Niezgódka, and Zheng Songmu, Existence and uniqueness of global solutions to an extended model of the dynamical development in shape memory alloys, Nonlinear Anal. 15, 977–990 (1990)
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Graylock Press, Baltimore, 1961
J. Nečas, Dynamic in the nonlinear thermo-visco-elasticity, Symposium “Partial Differential Equations” Holzhau 1988 (B.-W. Schulze and H. Triebel, eds.), Teubner-Texte Math. 112, Teubner, Leipzig, 1989, pp. 197–203
J. Nečas, A. Novotný, and V. Šverák, On the uniqueness of solution to the nonlinear thermo-viscoelasticity, Math. Nachr. 149, 319–324 (1990)
R. E. Nickell and J. L. Sackman, Variational principles for linear coupled thermoelasticity, Quart. Appl. Math. 26, 11–26 (1968)
M. Niezgódka and J. Sprekels, Existence of solutions for a mathematical model of structural phase transitions in shape memory alloys, Math. Methods Appl. Sci. 10, 197–223 (1988)
T. Roubíček, Evolution of a microstructure: a convexified model, Math. Methods Appl. Sci. 16, 625–642 (1993)
T. Roubíček, Optimality conditions for nonconvex variational problems relaxed in terms of Young measures (submitted)
T. Roubíček, Finite element approximation of a microstructure evolution, Math. Methods Appl. Sci. 17, 377–393 (1994)
M. Schatzman, A hyperbolic problem of second order with unilateral constraints: the vibrating string with a concave obstacle, J. Math. Anal. Appl. 73, 138–191 (1980)
J. Sprekels, Global existence for thermomechanical processes with nonconvex free energies of Ginzburg-Landau form, J. Math. Anal. Appl. 141, 333–348 (1989)
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Article copyright:
© Copyright 1994
American Mathematical Society
