Discretization and hysteresis

Rogers, Robert C.; Truskinovsky, Lev

doi:10.1016/s0921-4526(97)00323-2

Cited by 64 publications

(41 citation statements)

References 10 publications

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“…Moreover one is forced to introduce new possibly non-physical boundary conditions. Motivated by mathematical models for liquid-vapour phase transitions in fluid mechanics still firstorder but non-local approaches have been been suggested in [22] and also in [37]. Following these authors we define for ε > 0 the non-local stored energy…”

Section: The Energy Functional For the Equilibriummentioning

confidence: 99%

“…In this way a unique weak solution of (1.1) is singled out and the modelling of the capillarity mechanism with the particular ε-scaling is justified. However the choice (1.7) to model capillarity effects is not the only possible and considerations from statistical mechanics suggest different non-local models ( [5,22,37]). In this paper we shall focus therefore on these non-local alternatives for (1.7) which typically lead to a capillarity term of the form…”

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confidence: 99%

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Phase transitions and sharp-interface limits for the 1d-elasticity system with non-local energy

Rohde

2005

Interfaces Free Bound.

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Abstract. The one-dimensional system of elasticity with a non-monotone or convex-concave stress-strain relation provides a model to describe the longitudinal dynamics of solid-solid phase transitions in a bar. If dissipative effects are neglected it takes the form of a system of first-order nonlinear conservation laws and dynamical phase boundaries appear as shock wave solutions. In the physically most relevant cases these shocks are of the non-classical undercompressive type and therefore entropy solutions of the associated Cauchy problem are not uniquely determined. Important dissipative effects that lead to unique regular solutions are viscosity and capillarity where the latter effect is usually modelled by at least third-order spatial derivatives. Differently from these models we consider a novel type of non-local regularization that models both effects but avoids high-order derivatives. We suggest a particular scaling for the dissipative terms and conjecture that with this scaling the regular solutions single out unique physically relevant weak solutions of the first-order conservation law in the limit of vanishing dissipation parameter. We verify the conjecture first by proving that the non-local system admits special solutions of traveling-wave type that correspond to dynamical phase boundaries. Moreover it is proven that regular solutions of a general Cauchy problem converge to weak solutions of the system of first-order conservation laws. The proof is achieved by the method of compensated compactness.

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Section: The Energy Functional For the Equilibriummentioning

confidence: 99%

mentioning

confidence: 99%

Phase transitions and sharp-interface limits for the 1d-elasticity system with non-local energy

Rohde

2005

Interfaces Free Bound.

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“…The aim of this paper is to analyze the energy 8) which introduces both the unknown λ and the coupling parameter α > 0. The functional E ε,α has been introduced in [26,27]; we refer to [3,20,21] for related functionals. Note that no derivatives of u appear in (1.8), differently from (1.7).…”

Section: Introductionmentioning

confidence: 99%

Singular limits for a parabolic–elliptic regularization of scalar conservation laws

Corli

Rohde

2012

Journal of Differential Equations

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We consider scalar hyperbolic conservation laws with a nonconvex flux, in one space dimension. Then, weak solutions of the associated initial-value problems can contain undercompressive shock waves. We regularize the hyperbolic equation by a parabolic-elliptic system that produces undercompressive waves in the hyperbolic limit regime. Moreover we show that in another limit regime, called capillarity limit, we recover solutions of a diffusive-dispersive regularization, which is the standard regularization used to approximate undercompressive waves. In fact the new parabolic-elliptic system can be understood as a low-order approximation of the third-order diffusive-dispersive regularization, thus sharing some similarities with the relaxation approximations. A study of the traveling waves for the parabolic-elliptic system completes the paper.

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“…In the last twenty years, a number of researchers have attempted to describe formation of microstructure and hysteresis in crystalline solids within the framework of elasticity theory [1], [2], [6], [7], [14], [21], [22], [28], [29]. A common approach involves minimization of a nonconvex elastic energy for the material, following the pioneering analysis of Ericksen [10].…”

Section: Introductionmentioning

confidence: 99%

“…A common approach involves minimization of a nonconvex elastic energy for the material, following the pioneering analysis of Ericksen [10]. Although the absolute minimization of the total energy captures basic features of the microstructure [17], it cannot account for hysteresis, which arises when the material gets locked in metastable states, as suggested by calculations in [12], [24]; see also [1], [5], [7], [20], [28].…”

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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Discretization and hysteresis

Cited by 64 publications

References 10 publications

Phase transitions and sharp-interface limits for the 1d-elasticity system with non-local energy

Phase transitions and sharp-interface limits for the 1d-elasticity system with non-local energy

Singular limits for a parabolic–elliptic regularization of scalar conservation laws

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

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