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Energy minimization and the formation of microstructure in dynamic anti- plane shear. (English) Zbl 0786.73066
The behaviour of a continuum model within the framework of nonlinear viscoelasticity, which provides an insight into the dynamical development of microstructures observed during displacement phase transformations in certain materials, is investigated. The problem of dynamical two- dimensional anti-plane shear with linear viscoelastic damping is considered. Using a transformation due to P. Rybka [Proc. R. Soc. Edinb., Sect. A 121, No. 1/2, 101-138 (1992; Zbl 0758.73004)], the equivalence of this problem to a semilinear degenerate parabolic system can be shown, and the latter system can be treated by the method of semigroup theory to establish existence, uniqueness and regularity of solutions in \(L^ p\) spaces.
The issues of energy minimization and propagation of strain discontinuities are also discussed, but unfortunately, the authors are unable to obtain analogues of the results on non-minimization of the energy and non-propagation of strain discontinuities in one-dimensional models.
Finally, a finite difference approach to the numerical solution of the dynamical anti-plane shear problem is developed, and some numerical simulations of the dynamical evolution of patterns and microstructure in the above problem are presented.

MSC:
74A60 Micromechanical theories
74M25 Micromechanics of solids
74A15 Thermodynamics in solid mechanics
74E10 Anisotropy in solid mechanics
74S20 Finite difference methods applied to problems in solid mechanics
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[1] R. Abeyaratne & J. K. Knowles [1990] On the driving traction acting on a surface of discontinuity in a continuum. J. Mech. Phys. Solids 38, 345-360. · Zbl 0713.73030 · doi:10.1016/0022-5096(90)90003-M
[2] R. Abeyaratne & J. K. Knowles [1991] Kinetic relations and the propagation of phase boundaries in solids. Arch. Rational Mech. Anal. 114, 119-154. · Zbl 0745.73001 · doi:10.1007/BF00375400
[3] R. A. Adams [1975] Sobolev Spaces. Academic Press, New York. · Zbl 0314.46030
[4] G. Andrews [1980] On the existence of solutions to the equation u tt =u xxt +?(ux)x. J. Diff. Eqs. 35, 200-231. · Zbl 0415.35018 · doi:10.1016/0022-0396(80)90040-6
[5] G. Andrews & J. M. Ball [1982] Asymptotic behavior and changes in phase in onedimensional nonlinear viscoelasticity. J. Diff. Eqs. 44, 306-341. · Zbl 0501.35011 · doi:10.1016/0022-0396(82)90019-5
[6] S. S. Antman [1983] Coercivity conditions in nonlinear elasticity, in Systems of Nonlinear Partial Differential Equations (ed. J. M. Ball), D. Reidel, Dordrecht. · Zbl 0542.73047
[7] J. M. Ball [1977] Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63, 337-403. · Zbl 0368.73040 · doi:10.1007/BF00279992
[8] J. M. Ball [1990] Dynamics and minimizing sequences, in Problems Involving Change of Type (ed. K. Kirchgässner) Springer Lecture Notes in Physics 359, 3-16, Springer-Verlag, New York, Heidelberg, Berlin. · Zbl 0705.49023
[9] J. M. Ball, P. J. Holmes, R. D. James, R. L. Pego & P. J. Swart [1991] On the dynamics of fine structure. J. Nonlinear Science 1, 17-70. · Zbl 0791.35030 · doi:10.1007/BF01209147
[10] J. M. Ball & R. D. James [1987] Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100, 13-52. · Zbl 0629.49020 · doi:10.1007/BF00281246
[11] J. M. Ball & R. D. James [1992] Proposed experimental tests of a theory of fine microstructure and the two-well problem. Phil. Trans. Roy. Soc. Lond. A 338, 389-450. · Zbl 0758.73009 · doi:10.1098/rsta.1992.0013
[12] Z. S. Basinski & J. W. Christian [1954] Experiments on the martensitic transformation in single crystals of indium-thallium alloys. Acta Metall. 2, 148-166. · doi:10.1016/0001-6160(54)90104-2
[13] P. Bauman & D. Phillips [1990] A nonconvex variational problem related to change of phase. Appl. Math. Optim. 21, 113-138. · Zbl 0686.73018 · doi:10.1007/BF01445160
[14] C. Bennet & R. Sharpley [1988] Interpolation of Operators. Academic Press, Boston. · Zbl 0647.46057
[15] K. Bhattacharya [1991] Wedge-like microstructures in martensites. Acta Metall. Mater. 39, 2431-2444. · doi:10.1016/0956-7151(91)90023-T
[16] C. Canuto, M. Y. Hussaini, A. Quarteroni & T. A. Zang [1988] Spectral Methods in Fluid Dynamics. Springer-Verlag, Heidelberg, Berlin. · Zbl 0658.76001
[17] M. Chipot [1991] Numerical analysis of oscillations in nonconvex problems. Numer. Math. 59, 747-767. · Zbl 0737.65054 · doi:10.1007/BF01385808
[18] P. G. Ciarlet [1988] Mathematical Elasticity I: Three dimensional Elasticity. North-Holland. · Zbl 0648.73014
[19] C. Collins & M. Luskin [1989] The computation of the austenitic-martensitic phase transition, in Partial Differential Equations and Continuum Models of Phase Transitions (eds. M. Rascle, D. Serre & M. Slemrod). Springer Lecture Notes in Physics 344, 34-50, Springer-Verlag. · Zbl 0991.80502
[20] C. M. Dafermos [1973] The entropy rate admissibility criterion for solutions of hyperbolic conservations laws. J. Diff. Eqs. 14, 202-212. · Zbl 0262.35038 · doi:10.1016/0022-0396(73)90043-0
[21] C. M. Dafermos [1983] Hyperbolic systems of conservation laws, in Systems of Nonlinear Partial Differential Equations (ed. J. M. Ball), D. Reidel, Dordrecht. · Zbl 0536.35048
[22] H. Engler [1989] Global regular solutions for the dynamic antiplane shear problem in nonlinear viscoelasticity. Math. Z. 202, 251-259. · Zbl 0697.73033 · doi:10.1007/BF01215257
[23] J. L. Ericksen [1975] Equilibrium of bars. J. Elast. 5, 191-201. · Zbl 0324.73067 · doi:10.1007/BF00126984
[24] J. L. Ericksen [1980] Some phase transitions in crystals. Arch. Rational Mech. Anal. 73, 99-124. · Zbl 0429.73007 · doi:10.1007/BF00258233
[25] L. C. Evans [1990] Weak convergence methods for nonlinear partial differential equations. CBMS Regional Conference Series in Mathematics No. 74, Amer. Math. Soc., Providence. · Zbl 0698.35004
[26] D. A. French & L. B. Wahlbin [1991] On the numerical approximation of an evolution problem in nonlinear viscoelasticity. Mathematical Sciences Institute. Technical Report 91-49, Cornell University. · Zbl 0783.73069
[27] E. Fried [1991] On the construction of two-phase equilibria in a non-elliptic hyperelastic material (preprint).
[28] D. Fujiwara & H. Morimoto [1977] An Lr-theorem of the Helmholtz decomposition of vector fields. J. Fac. Sci. Univ. Tokyo, Sec. I. 24, 685-700. · Zbl 0386.35038
[29] J. W. Gibes [1876] On the equilibrium of heterogenous substances. Trans. Conn. Acad. Vol III, 108-248, in The Scientific Papers of J. W. Gibbs, Vol I: Thermodynamics. Dover, New York, 1961.
[30] P. Grisvard [1985] Elliptic Problems in Nonsmooth Domains. Pitman, Boston. · Zbl 0695.35060
[31] M. E. Gurtin & A. Struthers [1990] Multiphase thermomechanics with interfacial structure. 3. Evolving phase boundaries in the presence of bulk deformation. Arch. Rational Mech. Anal. 112, 97-160. · Zbl 0723.73018 · doi:10.1007/BF00375667
[32] M. E. Gurtin & R. Temam [1981] On the anti-plane shear problem in finite elasticity. J. Elast. 2, 197-206. · Zbl 0496.73036 · doi:10.1007/BF00043860
[33] J. K. Hale [1988] Asymptotic Behavior of Dissipative Systems. Amer. Math. Soc., Providence. · Zbl 0642.58013
[34] D. Henry [1981] Geometric Theory of Semilinear Parabolic Equations. Springer Lecture Notes in Mathematics 840, Springer-Verlag, New York. · Zbl 0456.35001
[35] P. J. Holmes & P. J. Swart [1991] A mathematical cartoon for the dynamics of fine structure. Transactions of the Eighth Army Conference on Applied Mathematics and Computing, ARO Report 91-1, 11-21, Ithaca. · Zbl 0737.35006
[36] T. J. R. Hughes, T. Kato & J. E. Marsden [1977] Well-posed quasi-linear hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Rational Mech. Anal. 64, 273-304. · Zbl 0361.35046
[37] R. D. James [1979] Co-existent phases in the one-dimensional static theory of elastic bars. Arch. Rational Mech. Anal. 72, 99-139. · Zbl 0429.73001 · doi:10.1007/BF00249360
[38] R. D. James [1980] The propagation of phase boundaries in elastic bars. Arch. Rational Mech. Anal. 73, 125-158. · Zbl 0443.73010 · doi:10.1007/BF00258234
[39] R. D. James [1981] Finite deformations by mechanical twinning. Arch. Rational Mech. Anal. 77, 143-176. · Zbl 0537.73031 · doi:10.1007/BF00250621
[40] T. Kato [1966] Pertubation Theory for Linear Operators. Springer-Verlag, New York. · Zbl 0148.12601
[41] J. K. Knowles [1976] On finite anti-plane shear for incompressible elastic materials. J. Australian Math. Soc. 19B, 400-415. · Zbl 0363.73045
[42] J. K. Knowles [1977] The finite anti-plane shear field near the tip of a crack for a class of incompressible elastic solids. Int. J. Fracture 13, 611-639. · doi:10.1007/BF00017296
[43] J. K. Knowles & E. Sternberg [1975] On the ellipticity of the equations of nonlinear elastostatics for a special material. J. Elast. 5, 341-361. · Zbl 0323.73010 · doi:10.1007/BF00126996
[44] J. K. Knowles & E. Sternberg [1977] On the failure of ellipticity of the equations for finite elastostatic plane strain. Arch. Rational Mech. Anal. 63, 221-236. · Zbl 0351.73061
[45] J. K. Knowles & E. Sternberg [1978] On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics. J. Elast. 8, 329-379. · Zbl 0422.73038 · doi:10.1007/BF00049187
[46] J. K. Knowles & E. Sternberg [1980] Discontinuous deformation gradients near the tip of a crack in finite anti-plane shear: an example. J. Elast. 10, 81-110. · Zbl 0432.73088 · doi:10.1007/BF00043136
[47] R. V. Kohn & S. Müller [1992] Branching of twins near an austenite/twinned-martensite interface, (to appear in) Phil. Mag. A.
[48] S. Larsson, V. Thomeé & L. B. Wahlbin [1991] Finite-element, methods for a strongly damped wave equation. IMA J. of Numer. Anal. 11, 115-142. · Zbl 0718.65070 · doi:10.1093/imanum/11.1.115
[49] J. P. LaSalle & S. Lefschetz [1961] Stability by Liapunov’s Direct Method with Applications. Academic Press, New York. · Zbl 0098.06102
[50] L. J. Leitman & G. M. C. Fisher [1973] The Linear Theory of Viscoelasticity. Handbuch der Physik (ed. S. Flügge) VI a/3, 1-123, Springer-Verlag, Berlin and New York.
[51] T. Meis & U. Marcowitz [1981] Numerical solution of partial differential equations. Springer-Verlag, New York. · Zbl 0446.65045
[52] M. Miklav?i? [1985] Stability for semilinear parabolic equations with non-invertible linear operator. Pacific J. Math. 118, 199-214. · Zbl 0559.35037
[53] R. L. Pego [1987] Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability. Arch. Rational Mech. Anal. 97, 353-394. · Zbl 0648.73017 · doi:10.1007/BF00280411
[54] P. Rosakis [1992] Compact zones of shear transformation in an anisotropic solid. J. Mech. Phys. Solids. 40, 1163-1195. · Zbl 0763.73005 · doi:10.1016/0022-5096(92)90011-P
[55] P. Rybka [1992] Dynamical modeling of phase transitions by means of viscoelasticity in many dimensions, (to appear in) Proc. Roy. Soc. Edinburgh. · Zbl 0758.73004
[56] P. L. Sachdev [1987] Nonlinear diffusive waves. Cambridge University Press, Cambridge. · Zbl 0624.35002
[57] S. A. Silling [1988a] Numerical studies of loss of ellipticity near singularities in an elastic material. J. Elast. 19, 213-239. · Zbl 0634.73040 · doi:10.1007/BF00045617
[58] S. A. Silling [1988b] Consequences of the Maxwell relation for anti-plane shear deformations of an elastic solid. J. Elast. 19, 241-284. · Zbl 0634.73041 · doi:10.1007/BF00045618
[59] M. Slemrod [1989] A limiting ?viscosity? approach to the Riemann problem for materials exhibiting change of phase. Arch. Rational Mech. Anal. 105, 327-365. · Zbl 0701.35101 · doi:10.1007/BF00281495
[60] P. J. Swart & P. J. Holmes [1991] Dynamics of phase transitions in nonlinear viscoelasticity (video animation). Cornell National Supercomputer Facility.
[61] L. Tartar [1983] The compensated compactness method applied to systems of conservation laws, in Material Instabilities in Continuum Mechanics and Related Mathematical Problems (ed. J. M. Ball). Oxford University Press, 263-285. · Zbl 0536.35003
[62] H. Triebel [1978] Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam, New York, Oxford. · Zbl 0387.46032
[63] C. Truesdell & W. Noll [1965] The Non-Linear Field Theories of Mechanics. Handbuch der Physik (ed. S. Flügge) III/3, Springer-Verlag, Berlin. · Zbl 0779.73004
[64] G. Van Tendeloo, J. Van Landuyt & S. Amelinckx [1976] The ?-? phase transitions in quartz and AlPO4 as studied by electron microscopy and diffraction. Phys. Stat. Sol. a33, 723-735.
[65] L. B. Wahlbin [1991] Private communication
[66] W. L. Wood [1990] Practical Time-stepping Schemes. Clarendon Press, Oxford. · Zbl 0694.65043
[67] W. P. Ziemer [1989] Weakly Differentiable Functions. Springer-Verlag, New York. · Zbl 0692.46022
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