On Knowles' version of Saint-Venant's Principle in two-dimensional elastostatics

Flavin, J. N.

doi:10.1007/bf00281492

Cited by 45 publications

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“…In [5] an explicit estimate (lower bound) is obtained for the rate of energy decay with distance from a portion of the domain boundary carrying a self-equilibrated load. A modification of the analysis of [5] was given by Flavin [7], yielding an improved estimate of the decay rate. An alternative argument, leading to the same estimated decay rate as that obtained in [7], has been provided by Oleinik and Yosifian [8], [9].1 The quality of the estimate for the decay rate obtained in [7][8][9] may be tested by comparison with the exact decay rate for the semi-infinite strip problem.2 It turns out that the results of [7][8][9] underestimate the exact value by a factor of nearly one-half.…”

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confidence: 99%

“…A modification of the analysis of [5] was given by Flavin [7], yielding an improved estimate of the decay rate. An alternative argument, leading to the same estimated decay rate as that obtained in [7], has been provided by Oleinik and Yosifian [8], [9].1 The quality of the estimate for the decay rate obtained in [7][8][9] may be tested by comparison with the exact decay rate for the semi-infinite strip problem.2 It turns out that the results of [7][8][9] underestimate the exact value by a factor of nearly one-half.In [1], a third type of argument is employed to establish energy decay for the biharmonic equation in a semi-infinite strip. The new feature contained in [1] is the consideration of a "higher-order energy" in addition to the physical energy associated with the problem.…”

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confidence: 99%

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Decay estimates for the biharmonic equation with applications to Saint-Venant principles in plane elasticity and Stokes flows

Horgan¹

1989

Quart. Appl. Math.

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1. Introduction. In a recent paper [1], J. K. Knowles has established new energy decay estimates for solutions of the biharmonic equation in a semi-infinite strip, subject to nonzero boundary conditions on the near end only. Such estimates, which predict an exponential decay of energy with axial distance from the end, have been used in the analysis of Saint-Venant's principle in plane elastostatics. (See [2] for a review of recent work on principles of Saint-Venant type; for a discussion of earlier results in the linear theory of elasticity, see [3].) These results are also relevant to the study of the spatial evolution of stationary Stokes flows in a semi-infinite parallel plate channel to fully-developed Poiseuille flow [4], Energy decay arguments involving differential inequalities have been employed previously by Knowles [5] in the analysis of Saint-Venant's principle in plane isotropic elastostatics for bounded, simply-connected domains of general shape. Similar arguments were used by Toupin [6] in his investigation of the corresponding issue for the three-dimensional elastic cylinder. In [5] an explicit estimate (lower bound) is obtained for the rate of energy decay with distance from a portion of the domain boundary carrying a self-equilibrated load. A modification of the analysis of [5] was given by Flavin [7], yielding an improved estimate of the decay rate. An alternative argument, leading to the same estimated decay rate as that obtained in [7], has been provided by Oleinik and Yosifian [8], [9].1 The quality of the estimate for the decay rate obtained in [7][8][9] may be tested by comparison with the exact decay rate for the semi-infinite strip problem.2 It turns out that the results of [7][8][9] underestimate the exact value by a factor of nearly one-half.In [1], a third type of argument is employed to establish energy decay for the biharmonic equation in a semi-infinite strip. The new feature contained in [1] is the consideration of a "higher-order energy" in addition to the physical energy associated with the problem. This method provides an improved estimated decay rate over that of [7][8][9], although still underestimating the exact value.

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Decay estimates for the biharmonic equation with applications to Saint-Venant principles in plane elasticity and Stokes flows

Horgan¹

1989

Quart. Appl. Math.

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“…It is well known that another interprelation for the biharmonic equation in the plane is that of the stream function in two dimensional Stokes flow, hence the results of the Saint-Venant's principle in plane elastostatics are also relevant to the study of the spatial evolution of stationary stokes flows in a semi-infinite paralled plate channel. Numerous authors have dealt with Saint-Venant type decay estimate for solutions of the biharmonic equations in a semi-infinite channel in R 2 , we mention in particular the papers of Flavin [1], Horgan [5], Knowles [8] and Flavin et al [2]. In a paper [10], Lin has established energy decay estimates for solutions of the plane Stokes flow in a semi-infinite channel, subject to nonzero boundary conditions on the end only.…”

Section: Introductionmentioning

confidence: 99%

Spatial Behavior of Solution for the Stokes Flow Equation

Liu¹,

Liao²,

Cheng³

2011

Bulletin of the Korean Mathematical Society

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Abstract. In this paper, the equation of the transient Stokes flow of an incompressible viscous fluid is studied. Growth and decay estimates are established associating some appropriate cross sectional line and area integral measures. The method of the proof is based on a first-order differential inequality leading to an alternative of Phragmén-Lindelöf type in terms of an area measure of the amplitude in question. In the case of decay, we also indicate how to bound the total energy.

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“…A comprehensive review of research on the spatial behavior of solutions is given by [Toupin 1965;Knowles 1966;Flavin 1974;Horgan and Knowles 1983;Gregory and Wan 1985;Horgan 1989;1996;Mielke 1988].…”

Section: Introductionmentioning

confidence: 99%

Decay properties of solutions of a Mindlin-type plate model for rhombic systems

Passarella

Tibullo

Zampoli

2010

JOMMS

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In the present paper, we investigate the spatial behavior of transient and steady-state solutions for the problem of bending applied to a linear Mindlin-type plate model; the plate is supposed to be made of a material characterized by rhombic isotropy, with the elasticity tensor satisfying the strong ellipticity condition. First, using an appropriate family of measures, we show that the transient solution vanishes at distances greater than cT from the support of the given data on the time interval [0, T ], where c is a characteristic material constant. For distances from the support less than cT , we obtain a spatial decay estimate of Saint-Venant type. Then, for a plate whose middle section is modelled as a (bounded or semiinfinite) strip, a family of measures is used to obtain an estimate describing the spatial behavior of the amplitude of harmonic vibrations, provided that the frequency is lower than a critical value.A list of symbols can be found on page 337.

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On Knowles' version of Saint-Venant's Principle in two-dimensional elastostatics

Cited by 45 publications

References 4 publications

Decay estimates for the biharmonic equation with applications to Saint-Venant principles in plane elasticity and Stokes flows

Decay estimates for the biharmonic equation with applications to Saint-Venant principles in plane elasticity and Stokes flows

Spatial Behavior of Solution for the Stokes Flow Equation

Decay properties of solutions of a Mindlin-type plate model for rhombic systems

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