Summary
Initial-boundary value problems describing the mechanics of nonelliptic elastic materials give rise to solutions that involve phase boundaries, the motion of which can dissipate mechanical energy. We investigate whether this dissipation, acting alone, can drive such a system toward equilibrium. Moving phase boundaries are regarded as a localized dissipative mechanism, and we consider a model which specifically excludes dissipation away from a phase boundary (such as that due to viscoelastic damping). In the problem under consideration, wave packets reverberate between the fixed external boundary and a single internal phase boundary. The phase boundary remains stationary unless it is acted upon by one of these wave packets, and each such interaction dissipates a finite amount of energy while causing the initiating wave packet to split into a reflected wave packet and a transmitted wave packet. Consequently, the number of wave packets increases in a geometric fashion. Each individual interaction of a wave packet with the phase boundary is, in a certain sense, mechanically underdetermined, and we augment the mechanical theory with two alternative energy criteria, each of which determines a different interaction dynamics. These alternative energy criteria are motivated by considerations of maximizing the energy dissipation in the system. We treat a system that is perturbed out of an initial minimum energy equilibrium state by a disturbance at the external boundary. A framework is developed for treating the resulting wave reverberations and calculating the energy dissipation for large time. Numerical computation indicates that the total energy dissipated in both versions of the dynamical problem is that which is necessary to settle into a new energy-minimal equilibrium state. We then establish the same result analytically for a meaningful limit involving a vanishingly small dynamical perturbation.
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References
R. Abeyaratne (1980). Discontinuous deformation gradients in plane finite elastostatics of incompressible materials.J. Elasticity 10, 255–293.
R. Abeyaratne (1983). An admissibility condition for equilibrium shocks in finite elasticity.J. Elasticity 13, 175–184.
R. Abeyaratne and J. K. Knowles (1987). Non-elliptic elastic materials and the modeling of dissipative mechanical behavior: an example.J. Elasticity 18, 227–278.
R. Abeyaratne and J. K. Knowles (1988). On the dissipative response due to discontinuous strains in bars of unstable elastic material.International Journal of Solids and Structures 24, 1021–1044.
R. Abeyaratne and J. K. Knowles (1990). On the driving traction acting on a surface of strain discontinuity in a continuum.J. Mechanics and Physics of Solids 38, 345–360.
R. Abeyaratne and J. K. Knowles (1991a). Kinetic relations and the propagation of phase boundaries in solids.Arch. Ration. Mech. & Anal. 114, 119–154.
R. Abeyaratne and J. K. Knowles (1991b). Implications of viscosity and strain gradient effects for the kinetics of propagating phase boundaries in solids.SIAM J. on Applied Math. 51, 1205–1221.
J. M. Ball (1977). Convexity conditions and existence theorems in nonlinear elasticity.Arch. Ration. Mech. & Anal. 63, 337–403.
J. M. Ball and R. D. James (1987). Fine phase mixtures as minimizers of energy.Arch. Ration. Mech. & Anal. 100, 13–52.
J. M. Ball, P. J. Holmes, R. D. James, R. L. Pego, and P. J. Swart (1991). On the dynamics of fine structure.J. Nonlinear Sci. 1, 17–70.
C. M. Dafermos (1973). The entropy rate admissibility criterion for solutions of hyperbolic conservation laws.J. Differ. Equations 14, 202–212.
J. L. Ericksen (1975). Equilibrium of bars.J. Elasticity 5, 191–201.
H. Fan (1992). Global versus local admissibility criteria for dynamic phase boundaries. To appear inProc. Royal Soc. of Edinburgh.
H. Fan and M. Slemrod (1991). The Riemann problem for systems of conservation laws of mixed type. To appear inShock Induced Transitions and Phase Structures in General Media. Ed. R. Fosdick, E. Dunn, & M. Slemrod, vol. 52. IMA, Springer-Verlag.
R. L. Fosdick and G. Macsithigh (1983). Helical shear of an elastic, circular tube with a nonconvex stored energy.Arch. Ration. Mech. & Anal. 84, 31–53.
E. Fried (1991). Stability of a two-phase process in an elastic solid. To appear inJ. Elasticity.
M. E. Gurtin (1983). Two-phase deformations of elastic solids.Arch. Ration. Mech. & Anal. 84, 1–29.
M. E. Gurtin (1991). On thermomechanical laws for the motion of a phase interface.ZAMP 42, 370–388.
M. E. Gurtin and A. Struthers (1990). Multiphase thermomechanics with interfacial structure 3. Evolving phase boundaries in the presence of bulk deformation.Arch. Ration. Mech. & Anal. 112, 97–160.
H. Hattori (1986). The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion—isothermal case.Arch. Ration. Mech. & Anal. 92, 247–263.
R. D. James (1979). Co-existent phases in the one-dimensional static theory of elastic bars.Arch. Ration. Mech. & Anal. 72, 99–140.
R. D. James (1980). The propagation of phase boundaries in elastic bars.Arch. Ration. Mech. & Anal. 73, 125–158.
R. D. James (1981). Finite deformation by mechanical twinning.Arch. Ration. Mech. & Anal 77, 143–176.
R. D. James (1984). On the stability of phases.International Journal of Engineering Science 22, 1193–1197.
R. D. James (1986). Displacive phase transformations in solids.J. Mechanics and Physics of Solids 34, 359–394.
N. Kikuchi and N. Triantafyllidis (1982). On a certain class of elastic materials with non-elliptic energy densities.Quarterly of Applied Mathematics 40, 241–248.
J. K. Knowles (1979). On the dissipation associated with equilibrium shocks in finite elasticity.J. Elasticity 9, 131–158.
J. K. Knowles and E. Sternberg (1978). On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics.J. Elasticity 8, 329–379.
J. Lin and T. J. Pence (1991). Energy dissipation in an elastic material containing a mobile phase boundary subjected to concurrent dynamic pulses.Transactions of the Ninth Army Conference on Applied Mathematics and Computing. Ed. F. Dressel, 437–450.
R. L. Pego (1987). Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability.Arch. Ration. Mech. & Anal. 97, 353–394.
T. J. Pence (1986). On the emergence and propagation of a phase boundary in an elastic bar with a suddenly applied end load.J. Elasticity 16, 3–42.
T. J. Pence (1987). Formulation and analysis of a functional equation describing a moving one-dimensional elastic phase boundary.Quarterly of Applied Mathematics 45, 293–304.
T. J. Pence (1991a). On the encounter of an acoustic shear pulse with a phase boundary in an elastic material: reflection and transmission behavior.J. Elasticity 25, 31–74.
T. J. Pence (1991b). On the encounter of an acoustic shear pulse with a phase boundary in an elastic material: energy and dissipation.J. Elasticity 26, 95–146.
T. J. Pence (1992). On the mechanical dissipation of solutions to the Riemann problem for impact involving a two-phase elastic material.Arch. Ration. Mech. & Anal. 117, 1–52.
M. Shearer (1982). The Riemann problem for a class of conservation laws of mixed type.J. Differ. Equations 46, 426–443.
M. Shearer (1986). Nonuniqueness of admissible solutions of Riemann initial value problems for a system of conservation laws of mixed type.Arch. Ration. Mech. & Anal 93, 45–59.
M. Shearer (1988). Dynamic phase transitions in a van der Waals gas.Quarterly of Applied Mathematics 46, 631–636.
S. A. Silling (1988). Consequences of the Maxwell relation for anti-plane shear deformations of an elastic solid.J. Elasticity 19, 241–284.
S. A. Silling (1989). Phase changes induced by deformation in isothermal elastic crystals.J. Mechanics and Physics of Solids 37, 293–316.
S. A. Silling (1992). Dynamic growth of martensitic plates in an elastic material.J. Elasticity 28, 143–164.
M. Slemrod (1983). Admissibility criteria for propagating phase boundaries in a van der Waals fluid.Arch. Ration. Mech. & Anal. 81, 301–315.
M. Slemrod (1989). A limiting “viscosity” approach to the Riemann problem for materials exhibiting change of phase.Arch. Ration. Mech. & Anal. 105, 327–365.
L. M. Truskinovsky (1982). Equilibrium phase interfaces.Sov. Phys. Dokl. 27, 551–553.
L. M. Truskinovsky (1985). Structure of an isothermal phase discontinuity.Sov. Phys. Dokl. 30, 945–948.
L. M. Truskinovsky (1987). Dynamics of non-equilibrium phase boundaries in a heat conducting non-linearly elastic medium.J. Applied Mathematics and Mechanics (PMM, USSR) 51, 777–784.
L. M. Truskinovsky (1990). Kinks versus shocks. To appear inShock Induced Transitions and Phase Structures in General Media. Ed. R. Fosdick. E. Dunn, & M. Slemrod, vol. 52. IMA, Springer-Verlag.
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Communicated by Philip Holmes
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Lin, J., Pence, T.J. On the dissipation due to wave ringing in nonelliptic elastic materials. J Nonlinear Sci 3, 269–305 (1993). https://doi.org/10.1007/BF02429867
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DOI: https://doi.org/10.1007/BF02429867