Abstract
For the motion of a one-dimensional viscoelastic material of rate type with a non-monotonic stress-strain relation, a mixed initial boundary value problem is considered. A simple existence theory is outlined, based on a novel transformation of the equation into the form of a degenerate reaction-diffusion system. This leads to new results characterizing the regularity of weak solutions. It is shown that each solution tends strongly to a stationary state asymptotically in time. Stable stationary states are characterized. Stable states may contain coexistent phases, i.e. they may have discontinuous strain. They need not be minimizers of energy in the strong sense of the calculus of variations; “metastable” and “absolutely stable” phases may coexist in a stable state. Furthermore, such states do arise as long-time limits of smooth solutions.
Beyond the above, “hysteresis” and “creep” phenomena are exhibited in a model of a loaded viscoelastic bar. Also, a viscosity criterion is proposed for the admissibility of propagating waves in the associated purely elastic model. This criterion is then applied to describe the formation of some propagating phase boundaries in a loaded elastic bar.
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References
E. C. Aifantis & J. B. Serrin (1983), The mechanical theory of fluid interfaces and Maxwell's rule, J. Coll. Int. Sci. 96, 517–529.
G. Andrews (1980), On the existence of solutions to the equation u tt = uxxt + σ(ux)x,J. Diff. Eqns. 35, 200–231.
G. Andrews & J. M. Ball (1982), Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity, J. Diff. Eqns. 44, 306–341.
C. M., Dafermos (1969), The mixed initial-boundary value problem for the equations of nonlinear one dimensional viscoelasticity, J. Diff. Eqns. 6, 71–86.
J. L. Ericksen (1975), Equilibrium of bars, J. Elasticity 5, 191–201.
J. W. Gibbs (1906), On the equilibrium of heterogeneous substances, in The Scientific Papers of J. Willard Gibbs, Longmans, London.
J. M. Greenberg, R. C. MacCamy, & V. J. Mizel (1968), On the existence, uniqueness and stability of solutions of the equation σ′(ux) u xx + λuxxt= ρ0utt, J. Math. Mech. 17, 707–728.
J. M. Greenberg (1969), On the existence, uniqueness and stability of the equation ρ0X tt = E(Xx)Xxx + λXxxt,J. Math. Anal. Appl. 25, 575–591.
J. M. Greenberg & R. C. MacCamy (1970), On the exponential stability of solutions of E(ux) uxx + λuxtx= ρu tt,J. Math. Anal. Appl. 31, 406–417.
R. Hagan & M. Slemrod (1983), The viscosity-capillarity admissibility criterion for shocks and phase transitions, Arch. Rational Mech. Anal. 83, 333–361.
D. Henry (1981), Geometric theory of semilinear parabolic equations, Lecture Notes in Math. v. 840, Springer, New York.
D. Hoff & J. Smoller (1985), Solutions in the large for certain nonlinear parabolic systems, Anal. Non Lin. 2, 213–235.
J. Hunter & M. Slemrod (1983), Viscoelastic fluid flow exhibiting hysteretic phase changes, Phys. Fluids 26, 2345–2351.
R. D. James (1980a), Coexistent phases in the one dimensional static theory of elastic bars, Arch. Rational Mech. Anal. 72, 99–140.
R. D. James (1980b), The propagation of phase boundaries in elastic bars, Arch. Rational Mech. Anal. 73, 125–158.
T.-P. Liu (1976), The entropy condition and the admissibility of shocks, J. Math. Anal. Appl. 53, 78–88.
I. Müller & K. Wilmanski (1981), Memory alloys—phenomenology and ersatzmodel, in Continuum Models of Discrete Systems 4, eds. O. Brulin & R. K. T. Hsieh, North-Holland, Amsterdam.
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.
M. Shearer (1982), The Riemann problem for a class of conservation laws of mixed type, J. Diff. Eqns. 46, 426–443.
M. Shearer (1983), Admissibility criteria for shock wave solutions of a system of conservation laws of mixed type, Proc. Roy. Soc. Edinburgh 93 A, 233–244.
M. Shearer (1986), Nonuniqueness of admissible solutions of Riemann initial value problems for a system of conservation laws of mixed type, Arch. Rational Mech. Anal. 93, 45–59.
M. Slemrod (1983), Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Rational Mech. Anal. 81, 301–316.
H. F. Weinberger (1982), A simple system with a continuum of stable inhomogeneous steady states, Lecture Notes in Num. Appl. Anal. 5, 345–359.
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Communicated by J. M. Ball
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Pego, R.L. Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability. Arch. Rational Mech. Anal. 97, 353–394 (1987). https://doi.org/10.1007/BF00280411
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DOI: https://doi.org/10.1007/BF00280411