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The Masing conditions establish a criterion to relate the loading curve of a hysteretic system (e.g., systems with friction or plasticity) to its complete hysteresis loop. For the field of joint mechanics, where hysteretic models are often used to describe the dissipative, tangential behavior within an interface, the Masing conditions allow for significant computational savings when the normal load is constant. In practice, though, jointed systems experience time varying normal forces that modify the tangential behavior of the system. Consequently, the hysteretic behavior of jointed structures do not adhere to the Masing conditions. In this work, this discrepancy between the Masing conditions and behavior exhibited by jointed structures is explored, and it is hypothesized that if the Masing conditions accounted for variations in normal force, then they would more accurately represent jointed structures. A new set of conditions is introduced to the original set of Masing conditions, yielding a « Masing manifold » that spans the tangential displacement-tangential force-normal force space. Both a simple harmonic oscillator and a built-up structure are investigated for the case of elastic dry friction, and the results show that the hysteresis of both of these systems conforms to the three dimensional Masing manifold exactly, provided that a set of constraints are satisfied, even though the hysteresis does not conform with the original Masing conditions.
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Curve Lengthening via Regularized Motion Against Curvature from the Strong FCH Flow
We present a rigorous analysis of the transient evolution of nearly circular bilayer interfaces evolving under the thin interface limit, ε ≪ 1, of the mass preserving L2-gradient flow of the strong scaling of the functionalized Cahn–Hilliard equation. For a domain Ω ⊂ R2 we construct a bilayer manifold with boundary comprised of quasi-equilibria of the flow and a projection onto the manifold that associates functions u in an H2 tubular neighborhood of the manifold with an interface Γ embedded in Ω. The linearization of the flow about the manifold does not present a clear spectral separation of modes normal and tangential to the manifold. The dimension of the parameterization of the interfaces and the bilayer manifold controls both the normal coercivity of the manifold and the coupling between normal and tangential modes, both of which increase with this dimension. The key step in the analysis is the identification of a range of dimensions in which coercivity dominates the coupling, permitting the closure of the nonlinear estimates that establish the asymptotic stability of the manifold. Orbits originating in a thin, forward invariant, tubular neighborhood ultimately converge to an equilibrium associated to a circular interface. Projections of these orbits yield interfacial evolution equivalent at leading order to the regularized curve-lengthening motion characterized by normal motion against mean curvature, regularized by a higher order Willmore expression. The curve lengthening is driven by absorption of excess mass from the regions of Ω away from the interface, leading to high dimensional dynamics that are ill-posed in the ε → 0+ limit.
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- Award ID(s):
- 1813203
- PAR ID:
- 10359255
- Date Published:
- Journal Name:
- Journal of Dynamics and Differential Equations
- ISSN:
- 1040-7294
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s10884-022-10178-7
