Archive for Rational Mechanics and Analysis

, Volume 133, Issue 3, pp 199–247 | Cite as

Dynamics as a mechanism preventing the formation of finer and finer microstructure

  • G. Friesecke
  • J. B. McLeod


We study the dynamics of pattern formation in the one-dimensional partial differential equation
$$u_u - (W'(u_x ))_x - u_{xxt} + u = 0{\text{ (}}u = u(x,t),{\text{ }}x \in (0,1),{\text{ }}t > 0)$$
proposed recently by Ball, Holmes, James, Pego & Swart [BHJPS] as a mathematical “cartoon” for the dynamic formation of microstructures observed in various crystalline solids. Here W is a double-well potential like 1/4((u x )2 −1)2. What makes this equation interesting and unusual is that it possesses as a Lyapunov function a free energy (consisting of kinetic energy plus a nonconvex “elastic” energy, but no interfacial energy contribution) which does not attain a minimum but favours the formation of finer and finer phase mixtures:
$$E[u,u_t ] = \int\limits_0^1 {(\frac{{u_t^2 }}{2} + W(u_x ) + \frac{{u^2 }}{2})dx.}$$
Our analysis of the dynamics confirms the following surprising and striking difference between statics and dynamics, conjectured in [BHJPS] on the basis of numerical simulations of Swart & Holmes [SH]:
  • •While minimizing the above energy predicts infinitely fine patterns (mathematically: weak but not strong convergence of all minimizing sequences (unvn) of E[u,v] in the Sobolev space W1p(0, 1)×L2(0,1)), solutions to the evolution equation of ball et al. typically develop patterns of small but finite length scale (mathematically: strong convergence in W1p(0,1)×L2(0,1) of all solutions (u(t),ut(t)) with low initial energy as time t → ∞).

Moreover, in order to understand the finer details of why the dynamics fails to mimic the behaviour of minimizing sequences and how solutions select their limiting pattern, we present a detailed analysis of the evolution of a restricted class of initial data — those where the strain field ux has a transition layer structure; our analysis includes proofs that
  • •at low energy, the number of phases is in fact exactly preserved, that is, there is no nucleation or coarsening

  • •transition layers lock in and steepen exponentially fast, converging to discontinuous stationary sharp interfaces as time t → ∞

  • •the limiting patterns — while not minimizing energy globally — are ‘relative minimizers’ in the weak sense of the calculus of variations, that is, minimizers among all patterns which share the same strain interface positions.


Lyapunov Function Strong Convergence Transition Layer Weak Sense Sharp Interface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [AB]
    G. Andrews & J. M. Ball, Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity. J. Diff. Eqns. 44, 306–341, 1982.Google Scholar
  2. [ABF]
    N. Alikakos, P. W. Bates & G. Fusco, Slow motion for the Cahn-Hilliard equation in one space dimension. J. Diff. Eqns. 90, 81–135, 1991.Google Scholar
  3. [AK]
    R. Abeyaratne & J. Knowles, Nucleation kinetics and admissibility criteria for propagating phase boundaries. IMA preprint No. 1044, University of Minnesota, 1992.Google Scholar
  4. [An]
    G. Andrews, On the existence of solutions to the equation u tt=u xxt+σ(u x)x. J. Diff. Eqns. 35, 200–231, 1980.Google Scholar
  5. [AS]
    S. S. Antman & T. I. Seidman, Quasilinear hyperbolic-parabolic equations of one-dimensional viscoelasticity. J. Diff. Eqns. 124, 132–185, 1996.Google Scholar
  6. [Ba]
    J. M. Ball, Dynamics and minimizing sequences. In: Problems involving change of type, ed. K. Kirchgässner, Lecture Notes in Physics 359, 3–16, Springer-Verlag, 1990.Google Scholar
  7. [BCJ]
    J. M. Ball, C. Chu & R. D. James, Metastability and hysteresis in elastic crystals. To appear.Google Scholar
  8. [BF]
    P. W. Bates & P. C. Fife, The dynamics of nucleation for the Cahn-Hilliard equation. SIAM J. Appl. Math. To appear.Google Scholar
  9. [BF1]
    P. Brunovský & B. Fiedler, Connecting orbits in scalar reaction diffusion equations. Dynamics reported Vol. 1., U. Kirchgraber & H. O. Walther (eds.), Wiley and Teubner, 1988.Google Scholar
  10. [BF2]
    P. Brunovský & B. Fiedler, Connecting orbits in scalar reaction diffusion equations II. The complete solution. J. Diff. Eqns. 81, 106–135, 1989.Google Scholar
  11. [BFJK]
    K. Bhattacharya, N. B. Firoozye, R. D. James & R. V. Kohn, Restrictions on microstructure. Proc. R. Soc. Edinb. 124A, 843–878, 1994.Google Scholar
  12. [BFS]
    D. Brandon, I. Fonseca & P. J. Swart, The creation and propagation of oscillations in a dynamical model of displacive phase transformations. To appear.Google Scholar
  13. [BG]
    G. S. Bales & R. J. Gooding, Interfacial dynamics at a first-order phase transition involving strain: dynamical twin formation. Phys. Rev. Lett. 67–24, 3411–3415, 1991.Google Scholar
  14. [Bh]
    K. Bhattacharya. Self-accomodation in martensite. Arch. Rational. Mech. Anal. 120, 201–244, 1992.Google Scholar
  15. [BHJPS]
    J. M. Ball, P. J. Holmes, R. D. James, R. L. Pego & P. J. Swart, On the dynamics of fine structure. J. Nonlinear Sci. 1, 17–70, 1991.Google Scholar
  16. [BJ1]
    J. M. Ball & R. D. James, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100, 13–52, 1987.Google Scholar
  17. [BJ2]
    J. M. Ball & R. D. James, Proposed experimental tests of a theory of fine microstructure and the two-well problem. Phil. Trans. R. Soc. London A, 338, 389–450, 1992.Google Scholar
  18. [BX1]
    P. W. Bates & P. J. Xun, Metastable patterns for the Cahn-Hilliard Equation: Part I, J. Diff. Eqns. To appear.Google Scholar
  19. [BX2]
    P. W. Bates & P. J. Xun, Metastable patterns for the Cahn-Hilliard Equation: Part II. Layer dynamics and slow invariant manifold. J. Diff. Eqns. To appear.Google Scholar
  20. [Ce]
    L. Cesari, Optimization-theory and applications. Springer-Verlag, 1983.Google Scholar
  21. [CFNT]
    P. Constantin, C. Foias, B. Nicolaenko & R. Temam, Integral manifolds and inertial manifolds for dissipative partial differential equations. Springer-Verlag, 1989.Google Scholar
  22. [CK]
    M. Chipot & D. Kinderlehrer, Equilibrium configurations of crystals. Arch. Rational Mech. Anal. 103, 237–277, 1988.Google Scholar
  23. [CP]
    J. Carr & R. L. Pego, Metastable patterns in solutions of u t=ε 2 u xxf(u). Comm. Pure Appl. Math. XLII, 523–576, 1989.Google Scholar
  24. [Da]
    C. M. Dafermos, The mixed initial-boundary value problem for the equations of nonlinear one dimensional viscoelasticity. J. Diff. Eqns. 6, 71–86, 1969.Google Scholar
  25. [DM]
    G. Dolzmann & S. Müller, Microstructures with finite surface energy: the two-well problem. Arch. Rational Mech. Anal. 132, 101–141, 1995.Google Scholar
  26. [FH]
    G. Fusco & J. K. Hale, Slow-motion manifolds, dormant instability, and singular perturbations. J. Dyn. Diff. Eqns. 1, 75–94, 1989.Google Scholar
  27. [FM1]
    P. C. Fife & J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling wave solutions. Bull Amer. Math. Soc. 81, 1076–1078, 1975.Google Scholar
  28. [FM2]
    P. C. Fife & J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Rational Mech. Anal. 65, 335–361, 1977.Google Scholar
  29. [FM3]
    P. C. Fife & J. B. McLeod, A phase plane discussion of convergence to travelling fronts for nonlinear diffusion. Arch. Rational Mech. Anal. 75, 281–314, 1981.Google Scholar
  30. [Fo]
    I. Fonseca, Phase transitions for elastic solid materials. Arch. Rational Mech. Anal. 107, 195–223, 1989.Google Scholar
  31. [Fr1]
    G. Friesecke, Static and dynamic problems in nonlinear mechanics. PhD thesis, Heriot-Watt University, Edinburgh, 1993.Google Scholar
  32. [Fr2]
    G. Friesecke, A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems. Proc. R. Soc. Edinb. 124A, 437–471, 1994.Google Scholar
  33. [Fr3]
    G. Friesecke, A dynamical approach to hysteresis in a bar undergoing a martensitic transformation. In: Mathematics and control in smart structures 2129, H. T. Banks (ed.), 88–93, 1994.Google Scholar
  34. [GB]
    R. J. Gooding & G. S. Bales, Boosted kink-type solitary waves in nonlinear elastic media. Physica D, 55, 251–258, 1992.Google Scholar
  35. [Gu]
    M. E. Gurtin, The nature of configurational forces. Arch. Rational Mech. Anal. 131, 67–100, 1995.Google Scholar
  36. [Ha]
    J. K. Hale, Asymptotic behavior of dissipative systems. Mathemaical Surveys and Monographs 25, Amer. Math. Soc., 1988.Google Scholar
  37. [HaM]
    J. K. Hale & P. Massat, Asymptotic behavior of gradient-like systems. In: Dynamical Systems II, eds. A. R. Bednarek & L. Cesari, Academic Press, 1982.Google Scholar
  38. [HaR]
    J. K. Hale & G. Raugel, Convergence in gradient-like systems with applications to PDE. Z. Angew. Math. Phys. 43, 63–124, 1992.Google Scholar
  39. [He1]
    D. Henry, Geometric theory of semilinear parabolic equations. Springer Lecture Notes in Mathematics 840, 1981.Google Scholar
  40. [He2]
    D. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations. J. Diff. Eqns. 59, 165–205, 1985.Google Scholar
  41. [HM]
    H. Hattori & K. Mischaikov, A dynamical systems approach to a phase transition problem. J. Diff. Eqns. 94, 340–378, 1991.Google Scholar
  42. [HS]
    K. H. Hoffmann & Z. Songmu, Uniqueness for structural phase transitions in shape memory alloys. Math. Methods in the Applied Sci. 10, 145–151, 1988.Google Scholar
  43. [JK]
    R. D. James & D. Kinderlehrer, Theory of magnetostriction with applications to Tb x Dy 1−x Fe 2. Phil. Mag. B 68, 237–274, 1993.Google Scholar
  44. [KH]
    W. D. Kalies & P. J. Holmes, On a dynamical model for phase transformation in nonlinear elasticity. Preprint. 1993.Google Scholar
  45. [KM]
    R. V. Kohn & S. Müller, Branching of twins near an austenite/twinnedmartensite interface. Phil. Mag. A 66, 697–715, 1992.Google Scholar
  46. [KP]
    D. Kinderlehrer & P. Pedregal, Characterizations of Young measures generated by gradients. Arch. Rational Mech. Anal. 115, 329–365, 1991.Google Scholar
  47. [LP]
    J. Lin & T. J. Pence, On the dissipation due to wave ringing in nonelliptic elastic materials. J. Nonlinear Sci. 3, 269–305, 1993.Google Scholar
  48. [Ma]
    J. P. Matos, Young measures and the absence of fine microstructures in a class of phase transitions. Euro. J. Applied Math. 3, 31–54, 1992.Google Scholar
  49. [MS]
    S. Muller & V. Šverák, Attainment results for the two-well problem by convex integration. To appear.Google Scholar
  50. [MW]
    L. Ma & N. Walkington, On algorithms for non-convex optimization. Research Report No. 93-NA-010, Carnegie Mellon University, 1993.Google Scholar
  51. [NP]
    A. Novick-Cohen & R. L. Pego, Stable patterns in a viscous diffusion equation. Trans. Amer. Math. Soc. 324, 331–351, 1991.Google Scholar
  52. [NS]
    M. Niezgodka & J. Sprekels, Existence of solutions of a mathematical model of structural phase transitions in shape memory alloys. Math. Methods in the Applied Sci. 10, 197–223, 1988.Google Scholar
  53. [Pe1]
    R. L. Pego, Phase transitions in one-dimensional nonlinear viscoelasticity: Admissibility and stability. Arch. Rational Mech. Anal. 97, 353–394, 1987.Google Scholar
  54. [Pe2]
    R. L. Pego, Stabilization in a gradient system with a conservation law. Proc. Amer. Math. Soc. 114, 1017–1024, 1992.Google Scholar
  55. [RG]
    A. C. E. Reid & R. J. Gooding, Elastic hydrodynamics and dynamical nucleation in first-order strain transitions. Physica D 66, 180–186, 1993.Google Scholar
  56. [Ry]
    P. Rybka, Dynamical modeling of phase transitions by means of viscoelasticity in many dimensions. Proc. R. Soc. Edinb. 121A, 101–138, 1992.Google Scholar
  57. [SH]
    P. J. Swart & P. J. Holmes, Energy minimization and the formation of microstructure in dynamic anti-plane shear. Arch. Rational. Mech. Anal. 121, 37–85, 1992.Google Scholar
  58. [Sv1]
    V. Šverak, On Tartar's conjecture. Inst. H. PoincaréAnal. Nonl. To appear.Google Scholar
  59. [Sv2]
    V. Šverák, On the problem of two wells. In: Microstructure and phase transitions, D. Kinderlehrer et al. (eds.). Springer-Verlag, 1992.Google Scholar
  60. [Te]
    R. Temam, Infinite-dimensional dynamical systems in mechanics and physics. Springer-Verlag, 1988.Google Scholar
  61. [Yo]
    L. C. Young, Lectures on the calculus of variations and optimal control theory. Chelsea, 1980.Google Scholar
  62. [Zh]
    K. Zhang, Rank-one connections and the three-well problem. Preprint.Google Scholar

Copyright information

© Springer Verlag 1996

Authors and Affiliations

  • G. Friesecke
    • 1
    • 2
  • J. B. McLeod
    • 1
    • 2
  1. 1.Mathematisches InstitutUniversität FreiburgFreiburg
  2. 2.Department of Mathematics and StatisticsUniversity of PittsburghPittsburgh

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