Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity

Andrews, Graham; Ball, J. M.

doi:10.1016/0022-0396(82)90019-5

Cited by 147 publications

(130 citation statements)

References 14 publications

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“…Existence and asymptotic behaviour of solutions to (1.11) with λ ≥ 0 has been studied in [3,8]. Furthermore, for the non-stationary local model, global existence and time-asymptotic convergence to a stationary solution for bounded spatial domains in higher dimension were studied (see [8] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Global existence and uniqueness of solutions for a viscoelastic two-phase model with nonlocal capillarity

Dressel¹,

Rohde²

2008

Indiana Univ. Math. J.

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Abstract. The aim of this paper is to study the existence and uniqueness of solutions of an initial-boundary value problem for a viscoelastic two-phase material with capillarity in one space dimension. Therein, the capillarity is modelled via a nonlocal interaction potential. The proof relies on uniform energy estimates for a family of difference approximations: with these estimates at hand we show the existence of a global weak solution. Then, by means of a nontrivial variant of the arguments in [2], uniqueness and optimal regularity are proven. The results of this paper also apply to interaction potentials with non-vanishing negative part and constitute a base for an analysis of the time-asymptotic behaviour.

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Section: Introductionmentioning

confidence: 99%

Global existence and uniqueness of solutions for a viscoelastic two-phase model with nonlocal capillarity

Dressel¹,

Rohde²

2008

Indiana Univ. Math. J.

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“…Viscosity provides energy dissipation, while time-dependent displacement-controlled loading at the ends of the bar supplies energy into the system. Similar dynamical models have been investigated before [3], [4], [6], [13], [15], [19], [27]; they all consider time-independent loads. This forces the energy to decrease with time.…”

Section: Introductionmentioning

confidence: 99%

Hysteresis and Stick-Slip Motion of Phase Boundaries in Dynamic Models of Phase Transitions

Vainchtein

Rosakis

1999

J. Nonlinear Sci.

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Summary. We investigate hysteretic behavior in two dynamic models for solid-solid phase transitions. An elastic bar with a nonconvex double-well elastic energy density is subjected to time-dependent displacement boundary conditions. Both models include inertia and a viscous stress term that provides energy dissipation. The first model involves a strain-gradient term that models interfacial energy. In the second model this term is omitted. Numerical simulations combined with analytical results predict hysteretic behavior in the overall end-load versus end-displacement diagram for both models. The hysteresis is largely due to metastability and nucleation; it persists even for very slow loading when viscous dissipation is quite small. In the model with interfacial energy, phase interfaces move smoothly. When this term is omitted, hysteresis is much more pronounced. In addition, phase boundaries move in an irregular, stick-slip fashion. The corresponding load-elongation curve exhibits serrations, in qualitative agreement with certain experimental observations in shape-memory alloys.

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“…The static problems concerning the soft loading case and the hard loading case have been discussed in [3] and [4], respectively. The dynamical aspects of these loading cases are discussed in Hattori and Mischaikow [10] and Andrews and Ball [1].…”

Section: Introductionmentioning

confidence: 99%

The Korteweg Theory of Cappilarity and the Phase Transition Problems

Hattori¹

1993

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Publc reporting burden for this collenion of information it estimated to a.erage I hour Per resr•one. including the time for reviewing Instructions. searching extsng data sources gathering and maintaining the daita needed and completing and reviewing the collection of informatOn Send comments regaroing fri$ burden ettilae or an! other atpect Korteweg theory of cappilarity in the one-dimensional case and show some numerical results. In the multidimensional case we discuss the existence of local solutions to the system of equations for compressible fluids of Korteweg type.

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Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity

Cited by 147 publications

References 14 publications

Global existence and uniqueness of solutions for a viscoelastic two-phase model with nonlocal capillarity

Global existence and uniqueness of solutions for a viscoelastic two-phase model with nonlocal capillarity

Hysteresis and Stick-Slip Motion of Phase Boundaries in Dynamic Models of Phase Transitions

The Korteweg Theory of Cappilarity and the Phase Transition Problems

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