## Abstract

We investigate the behavior of a continuum model designed to provide insight into the dynamical development of microstructures observed during displacive phase transformations in certain materials. The model is presented within the framework of nonlinear viscoelasticity and is also of interest as an example of a strongly dissipative infinite-dimensional dynamical system whose forward orbits need not lie on a finite-dimensional attracting set, and which can display a subtle dependence on initial conditions quite different from that of classical finite-dimensional “chaos”.

We study the problem of dynamical (two-dimensional) anti-plane shear with linear viscoelastic damping. Within the framework of nonlinear hyperelasticity, we consider both isotropic and anisotropic constitutive laws which can allow different phases and we characterize their ability to deliver minimizers and minimizing sequences of the stored elastic energy (Theorem 2.3). Using a transformation due to Rybka, we recast the problem as a semilinear degenerate parabolic system, thereby allowing the application of semigroup theory to establish existence, uniqueness and regularity of solutions in *L*
^{p}spaces (Theorem 3.1). We also discuss the issues of energy minimization and propagation of strain discontinuities. We comment on the difficulties encountered in trying to exploit the geometrical properties of specific constitutive laws. In particular, we are unable to obtain analogues of the absence of minimizers and of the non-propagation of strain discontinuities found by Ball, Holmes, James, Pego & Swart [1991] for a one-dimensional model problem.

Several numerical experiments are presented, which prompt the following conclusions. It appears that the absence of an absolute minimizer may prevent energy minimization, thereby providing a dynamical mechanism to limit the fineness of observed microstructure, as has been proved in the one-dimensional case. Similarly, viscoelastic damping appears to prevent the propagation of strain discontinuities. During the extremely slow development of fine structure, solutions are observed to display local refinement in an effort to overcome incompatibility with boundary and initial conditions, with the distribution and shape of the resulting finer scales displaying a subtle dependence on initial conditions.

This is a preview of subscription content, log in to check access.

## References

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.R. Abeyaratne & J. K. Knowles [1991] Kinetic relations and the propagation of phase boundaries in solids.

*Arch. Rational Mech. Anal.***114**, 119–154.R. A. Adams [1975]

*Sobolev Spaces*. Academic Press, New York.G. Andrews [1980] On the existence of solutions to the equation

*u*_{ tt }=u_{ xxt }+σ(u_{x})_{x}.*J. Diff. Eqs.***35**, 200–231.G. Andrews & J. M. Ball [1982] Asymptotic behavior and changes in phase in onedimensional nonlinear viscoelasticity.

*J. Diff. Eqs.***44**, 306–341.S. S. Antman [1983] Coercivity conditions in nonlinear elasticity, in

*Systems of Nonlinear Partial Differential Equations*(ed. J. M. Ball), D. Reidel, Dordrecht.J. M. Ball [1977] Convexity conditions and existence theorems in nonlinear elasticity.

*Arch. Rational Mech. Anal.***63**, 337–403.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.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.J. M. Ball & R. D. James [1987] Fine phase mixtures as minimizers of energy.

*Arch. Rational Mech. Anal.***100**, 13–52.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.Z. S. Basinski & J. W. Christian [1954] Experiments on the martensitic transformation in single crystals of indium-thallium alloys.

*Acta Metall.***2**, 148–166.P. Bauman & D. Phillips [1990] A nonconvex variational problem related to change of phase.

*Appl. Math. Optim.***21**, 113–138.C. Bennet & R. Sharpley [1988]

*Interpolation of Operators.*Academic Press, Boston.K. Bhattacharya [1991] Wedge-like microstructures in martensites.

*Acta Metall. Mater.***39**, 2431–2444.C. Canuto, M. Y. Hussaini, A. Quarteroni & T. A. Zang [1988]

*Spectral Methods in Fluid Dynamics*. Springer-Verlag, Heidelberg, Berlin.M. Chipot [1991] Numerical analysis of oscillations in nonconvex problems.

*Numer. Math.***59**, 747–767.P. G. Ciarlet [1988]

*Mathematical Elasticity I: Three dimensional Elasticity*. North-Holland.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.C. M. Dafermos [1973] The entropy rate admissibility criterion for solutions of hyperbolic conservations laws.

*J. Diff. Eqs.***14**, 202–212.C. M. Dafermos [1983] Hyperbolic systems of conservation laws, in

*Systems of Nonlinear Partial Differential Equations*(ed. J. M. Ball), D. Reidel, Dordrecht.H. Engler [1989] Global regular solutions for the dynamic antiplane shear problem in nonlinear viscoelasticity.

*Math. Z.***202**, 251–259.J. L. Ericksen [1975] Equilibrium of bars.

*J. Elast.***5**, 191–201.J. L. Ericksen [1980] Some phase transitions in crystals.

*Arch. Rational Mech. Anal.***73**, 99–124.L. C. Evans [1990]

*Weak convergence methods for nonlinear partial differential equations*. CBMS Regional Conference Series in Mathematics No. 74, Amer. Math. Soc., Providence.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.E. Fried [1991] On the construction of two-phase equilibria in a non-elliptic hyperelastic material (

*preprint).*D. Fujiwara & H. Morimoto [1977] An L

_{r}-theorem of the Helmholtz decomposition of vector fields.*J. Fac. Sci. Univ. Tokyo, Sec. I.***24**, 685–700.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.P. Grisvard [1985]

*Elliptic Problems in Nonsmooth Domains.*Pitman, Boston.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.M. E. Gurtin & R. Temam [1981] On the anti-plane shear problem in finite elasticity.

*J. Elast.***2**, 197–206.J. K. Hale [1988]

*Asymptotic Behavior of Dissipative Systems.*Amer. Math. Soc., Providence.D. Henry [1981]

*Geometric Theory of Semilinear Parabolic Equations.*Springer Lecture Notes in Mathematics**840**, Springer-Verlag, New York.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.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.R. D. James [1979] Co-existent phases in the one-dimensional static theory of elastic bars.

*Arch. Rational Mech. Anal.***72**, 99–139.R. D. James [1980] The propagation of phase boundaries in elastic bars.

*Arch. Rational Mech. Anal.***73**, 125–158.R. D. James [1981] Finite deformations by mechanical twinning.

*Arch. Rational Mech. Anal.***77**, 143–176.T. Kato [1966]

*Pertubation Theory for Linear Operators*. Springer-Verlag, New York.J. K. Knowles [1976] On finite anti-plane shear for incompressible elastic materials.

*J. Australian Math. Soc.***19B**, 400–415.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.J. K. Knowles & E. Sternberg [1975] On the ellipticity of the equations of nonlinear elastostatics for a special material.

*J. Elast.***5**, 341–361.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.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.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.R. V. Kohn & S. Müller [1992] Branching of twins near an austenite/twinned-martensite interface, (to appear in)

*Phil. Mag. A.*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.J. P. LaSalle & S. Lefschetz [1961]

*Stability by Liapunov's Direct Method with Applications*. Academic Press, New York.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.T. Meis & U. Marcowitz [1981]

*Numerical solution of partial differential equations.*Springer-Verlag, New York.M. Miklavčič [1985] Stability for semilinear parabolic equations with non-invertible linear operator.

*Pacific J. Math.***118**, 199–214.R. L. Pego [1987] Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability.

*Arch. Rational Mech. Anal.***97**, 353–394.P. Rosakis [1992] Compact zones of shear transformation in an anisotropic solid.

*J. Mech. Phys. Solids.***40**, 1163–1195.P. Rybka [1992] Dynamical modeling of phase transitions by means of viscoelasticity in many dimensions, (to appear in)

*Proc. Roy. Soc. Edinburgh*.P. L. Sachdev [1987]

*Nonlinear diffusive waves*. Cambridge University Press, Cambridge.S. A. Silling [1988a] Numerical studies of loss of ellipticity near singularities in an elastic material.

*J. Elast.***19**, 213–239.S. A. Silling [1988b] Consequences of the Maxwell relation for anti-plane shear deformations of an elastic solid.

*J. Elast.***19**, 241–284.M. Slemrod [1989] A limiting “viscosity” approach to the Riemann problem for materials exhibiting change of phase.

*Arch. Rational Mech. Anal.***105**, 327–365.P. J. Swart & P. J. Holmes [1991] Dynamics of phase transitions in nonlinear viscoelasticity (

*video animation)*. Cornell National Supercomputer Facility.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.H. Triebel [1978]

*Interpolation Theory, Function Spaces, Differential Operators*. North-Holland, Amsterdam, New York, Oxford.C. Truesdell & W. Noll [1965]

*The Non-Linear Field Theories of Mechanics*. Handbuch der Physik (ed. S. Flügge) III/3, Springer-Verlag, Berlin.G. Van Tendeloo, J. Van Landuyt & S. Amelinckx [1976] The α-β phase transitions in quartz and AlPO

_{4}as studied by electron microscopy and diffraction.*Phys. Stat. Sol.***a33**, 723–735.L. B. Wahlbin [1991]

*Private communication*W. L. Wood [1990]

*Practical Time-stepping Schemes*. Clarendon Press, Oxford.W. P. Ziemer [1989]

*Weakly Differentiable Functions*. Springer-Verlag, New York.

## Author information

### Affiliations

## Rights and permissions

## About this article

### Cite this article

Swart, P.J., Holmes, P.J. Energy minimization and the formation of microstructure in dynamic anti-plane shear.
*Arch. Rational Mech. Anal.* **121, **37–85 (1992). https://doi.org/10.1007/BF00375439

Received:

Issue Date:

### Keywords

- Energy Minimization
- Parabolic System
- Dynamical Mechanism
- Absolute Minimizer
- Semigroup Theory