Semi-quadratic variational problems for multiphase equilibria
Authors:
Eliot Fried and Morton E. Gurtin
Journal:
Quart. Appl. Math. 54 (1996), 73-84
MSC:
Primary 73V25; Secondary 49J45, 73B30, 73C50, 73G05, 80A22
DOI:
https://doi.org/10.1090/qam/1373839
MathSciNet review:
MR1373839
Full-text PDF Free Access
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Additional Information
A. G. Khachaturyan, Some questions concerning the theory of phase transformations in solids, Soviet Phys. Solid State 8, 2163–2168 (1967)
A. G. Khachaturyan and G. A. Shatalov, Theory of macroscopic periodicity for a phase transition in the solid state, Soviet Phys. JETP 29, 557–561 (1969)
A. L. Roitburd, The domain structure of crystals formed in the solid phase, Soviet Phys. Solid State 10, 2870–2876 (1969)
A. L. Roitburd, Domain structure caused by internal stresses in heterophase solids, Phys. Status Solidi 16, 329–339 (1973)
A. L. Roitburd and N. S. Kosenko, Orientational dependence of the elastic energy of a plane interlayer in a system of coherent phases, Phys. Status Solidi 35, 735–746 (1976)
J. A. Wert, The strain energy of a disc-shaped GP zone. Acta Metall. Mater. 24, 65–71 (1976)
A. L. Roitburd, Martensitic transformation as a typical phase transformation in solids, in Solid State Physics, vol. 33 Academic Press, 1979
S. H. Wen, E. Kostlan, M. Hong, A. G. Khachaturyan, and J. W. Morris, The preferred habit of a tetragonal inclusion in a cubic matrix, Acta Metall. Mater. 29, 1247–1254 (1981)
M. Hong, D. E. Wedge, and J. W. Morris, The state and habit of the $F{e_{16}}{N_2}$ precipiatae in b.c.c. iron: elastic theory, Acta Metall. Mater. 32, 279–288 (1984)
E. Kostlan and J. W. Morris, The preferred habit of a coherent thin-plate inclusion in an anisotropic elastic solid, Acta Metall. Mater. 35, 745–777 (1987)
R. V. Kohn, The relaxation of a double-well energy, Cont. Mech. Thermodyn. 3, 193–236 (1991)
J. L. Ericksen, Some phase transitions in crystals, Arch. Rational Mech. Anal. 73, 99–124 (1980)
R. Abeyaratne, Discontinuous deformation gradients away from the tip of a crack in anti-plane shear, J. Elast. 11, 373–393 (1981)
J. L. Ericksen, Continuous martensitic transitions in thermoelastic solids, J. Thermal Stresses 4, 107–119 (1981)
M. E. Gurtin and R. Temam, On the anti-plane shear problem in finite elasticity, J. Elast. 11, 197–206 (1981)
R. D. James, Finite deformation by mechanical twinning, Arch. Rational Mech. Anal. 77, 143–176 (1981)
R. Fosdick and G. MacSithigh, Helical shear of an elastic, circular tube with a non-convex stored energy, Arch. Rational Mech. Anal. 84, 31–53 (1983)
M. E. Gurtin, Two-phase deformations of elastic solids, Arch. Rational Mech. Anal. 84, 1–29 (1983)
R. Fosdick and G. MacSithigh, Minimization in incompressible nonlinear elasticity theory, J. Elast. 16, 267–301 (1986)
R. D. James, Displacive phase transformations in solids, J. Mech. Phys. Solids 34, 359–394 (1986)
J. M. Ball and R. D. James, Fine phase mixtures as minimizers of energy, Arch. Rational Mech. Anal. 100, 13–52 (1987)
I. Fonseca, Variational methods for elastic crystals, Arch. Rational Mech. Anal. 97, 189–220 (1987)
R. D. James, The stability and metastability of quartz, in Metastablity and Incompletely Posed Problems (S. Antman, J. L. Ericksen, D. Kinderlehrer, and I. Müller, eds.), Springer-Verlag, 1987, pp. 147–175
I. Fonseca, The lower quasiconvex envelope of the stored energy function for an elastic crystal, J. Math. Pures Appl. 67, 179–195 (1988)
S. A. Silling, Consequences of the Maxwell relation for anti-plane shear deformations of an elastic solid, J. Elast. 19, 213–239 (1988)
R. D. James and D. Kinderlehrer, Theory of diffusionless phase transformations, in Partial Differential Equations and Continuum Models of Phase Transitions (M. Rascle, D. Serre, and M. Slemrod, eds.), Lecture Notes in Phys., vol. 344, Springer-Verlag, Berlin, 1989, pp. 51–84
K. Bhattacharya, Wedge-like microstructure in martensites, Acta Metall. Mater. 39, 2431–2444 (1990)
P. Rosakis, Compact zones of shear transformation in an anisotropic solid, J. Mech. Phys. Solids 40, 1163–1195 (1992)
E. Fried, Construction of two-phase equilibria in a non-elliptic hyperelastic material, J. Elast. 31, 71–123 (1993)
P. Rosakis and H. Tsai, On the role of shear instability in the modelling of crystal twinning, Mech. Materials 17, 245–259 (1994)
R. V. Kohn, The relationship between linear and nonlinear variational models of coherent phase transitions, in Transactions of the 7th Army Conference on Applied Mathematics and Computing (F. Dressel, ed.), 1989, pp. 279–304
J. M. Ball and R. D. James, Proposed experimental tests of a theory of fine microstructure and the two-well problem, Philos. Trans. Roy. Soc. London A 338, 389–450 (1992)
M. A. Grinfeld, Construction of a physically linear theory of coherent phase transformations, Mechanics of Solids 21, 84–96 (1986)
M. E. Gurtin, An Introduction to Continuum Mechanics, Academic Press, New York, 1981
E. Fried and M. E. Gurtin, Dynamic solid-solid transitions with phase characterized by an order parameter, Physica D 72, 287–308 (1994)
A. G. Khachaturyan, Some questions concerning the theory of phase transformations in solids, Soviet Phys. Solid State 8, 2163–2168 (1967)
A. G. Khachaturyan and G. A. Shatalov, Theory of macroscopic periodicity for a phase transition in the solid state, Soviet Phys. JETP 29, 557–561 (1969)
A. L. Roitburd, The domain structure of crystals formed in the solid phase, Soviet Phys. Solid State 10, 2870–2876 (1969)
A. L. Roitburd, Domain structure caused by internal stresses in heterophase solids, Phys. Status Solidi 16, 329–339 (1973)
A. L. Roitburd and N. S. Kosenko, Orientational dependence of the elastic energy of a plane interlayer in a system of coherent phases, Phys. Status Solidi 35, 735–746 (1976)
J. A. Wert, The strain energy of a disc-shaped GP zone. Acta Metall. Mater. 24, 65–71 (1976)
A. L. Roitburd, Martensitic transformation as a typical phase transformation in solids, in Solid State Physics, vol. 33 Academic Press, 1979
S. H. Wen, E. Kostlan, M. Hong, A. G. Khachaturyan, and J. W. Morris, The preferred habit of a tetragonal inclusion in a cubic matrix, Acta Metall. Mater. 29, 1247–1254 (1981)
M. Hong, D. E. Wedge, and J. W. Morris, The state and habit of the $F{e_{16}}{N_2}$ precipiatae in b.c.c. iron: elastic theory, Acta Metall. Mater. 32, 279–288 (1984)
E. Kostlan and J. W. Morris, The preferred habit of a coherent thin-plate inclusion in an anisotropic elastic solid, Acta Metall. Mater. 35, 745–777 (1987)
R. V. Kohn, The relaxation of a double-well energy, Cont. Mech. Thermodyn. 3, 193–236 (1991)
J. L. Ericksen, Some phase transitions in crystals, Arch. Rational Mech. Anal. 73, 99–124 (1980)
R. Abeyaratne, Discontinuous deformation gradients away from the tip of a crack in anti-plane shear, J. Elast. 11, 373–393 (1981)
J. L. Ericksen, Continuous martensitic transitions in thermoelastic solids, J. Thermal Stresses 4, 107–119 (1981)
M. E. Gurtin and R. Temam, On the anti-plane shear problem in finite elasticity, J. Elast. 11, 197–206 (1981)
R. D. James, Finite deformation by mechanical twinning, Arch. Rational Mech. Anal. 77, 143–176 (1981)
R. Fosdick and G. MacSithigh, Helical shear of an elastic, circular tube with a non-convex stored energy, Arch. Rational Mech. Anal. 84, 31–53 (1983)
M. E. Gurtin, Two-phase deformations of elastic solids, Arch. Rational Mech. Anal. 84, 1–29 (1983)
R. Fosdick and G. MacSithigh, Minimization in incompressible nonlinear elasticity theory, J. Elast. 16, 267–301 (1986)
R. D. James, Displacive phase transformations in solids, J. Mech. Phys. Solids 34, 359–394 (1986)
J. M. Ball and R. D. James, Fine phase mixtures as minimizers of energy, Arch. Rational Mech. Anal. 100, 13–52 (1987)
I. Fonseca, Variational methods for elastic crystals, Arch. Rational Mech. Anal. 97, 189–220 (1987)
R. D. James, The stability and metastability of quartz, in Metastablity and Incompletely Posed Problems (S. Antman, J. L. Ericksen, D. Kinderlehrer, and I. Müller, eds.), Springer-Verlag, 1987, pp. 147–175
I. Fonseca, The lower quasiconvex envelope of the stored energy function for an elastic crystal, J. Math. Pures Appl. 67, 179–195 (1988)
S. A. Silling, Consequences of the Maxwell relation for anti-plane shear deformations of an elastic solid, J. Elast. 19, 213–239 (1988)
R. D. James and D. Kinderlehrer, Theory of diffusionless phase transformations, in Partial Differential Equations and Continuum Models of Phase Transitions (M. Rascle, D. Serre, and M. Slemrod, eds.), Lecture Notes in Phys., vol. 344, Springer-Verlag, Berlin, 1989, pp. 51–84
K. Bhattacharya, Wedge-like microstructure in martensites, Acta Metall. Mater. 39, 2431–2444 (1990)
P. Rosakis, Compact zones of shear transformation in an anisotropic solid, J. Mech. Phys. Solids 40, 1163–1195 (1992)
E. Fried, Construction of two-phase equilibria in a non-elliptic hyperelastic material, J. Elast. 31, 71–123 (1993)
P. Rosakis and H. Tsai, On the role of shear instability in the modelling of crystal twinning, Mech. Materials 17, 245–259 (1994)
R. V. Kohn, The relationship between linear and nonlinear variational models of coherent phase transitions, in Transactions of the 7th Army Conference on Applied Mathematics and Computing (F. Dressel, ed.), 1989, pp. 279–304
J. M. Ball and R. D. James, Proposed experimental tests of a theory of fine microstructure and the two-well problem, Philos. Trans. Roy. Soc. London A 338, 389–450 (1992)
M. A. Grinfeld, Construction of a physically linear theory of coherent phase transformations, Mechanics of Solids 21, 84–96 (1986)
M. E. Gurtin, An Introduction to Continuum Mechanics, Academic Press, New York, 1981
E. Fried and M. E. Gurtin, Dynamic solid-solid transitions with phase characterized by an order parameter, Physica D 72, 287–308 (1994)
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Article copyright:
© Copyright 1996
American Mathematical Society
