1,720,979 research outputs found
Spatial behaviour for the harmonic vibrations in plates of Kirchoff type
No full textIn this paper the spatial behaviour of the steady-state solutions for an equation of Kirchhoff type describing the motion of thin plates is investigated. Growth and decay estimates are established associating some appropriate cross-sectional line and area integral measures with the amplitude of the harmonic vibrations, provided the excited frequency is lower than a certain critical value. The method of proof is based on a second–order differential inequality leading to an alternative of Phragmèn–Lindelöf type in terms of an area measure of the amplitude in question. The critical frequency is individuated by using some Wirtinger and Knowles inequalities
On spatial growth or decay of solutions to a non simple heat conduction problem in a semi-infinite strip
No full textThe present paper establishes growth and decay spatial properties for the solutions of a fourth–order initial boundary value problem describing the flow of heat in a non–simple heat conductor along a semi–infinite strip in R2. The method of time–weighted line and area integral measures is used. When the time–weighted line integral measure is used, then an alternative of Phragmén–Lindelof type is established. It is shown that the decay rate of the end effects is controlled by the same factor as in the steady–state case (governed by the biharmonic equation), that is exp (-(√2 π)/h x1), where h is the width of the strip and x1 is the distance to the end of the strip. When an appropriate combination of the time–weighted line and area integrals is used as a measure, then a decay estimate of Saint–Venant type is established and it is shown that the end effects decay more rapidly as do their counterparts in the steady–state case
Some continuous data dependence and uniqueness results for nonlinear elastodynamics on unbounded domain
No full textOn a new method for the study of the spatial behaviour in a homogeneous elastic arch-like region
No full textWe consider a two-dimensional homogeneous elastic state in the arch-like region a ≤ r ≤ b,
0 ≤θ≤α where (r,θ) denotes plane polar coordinates. We assume that three of the edges
are traction-free, while the fourth edge is subjected to a (in plane) self-equilibrated load.
The Airy stress function ‘φ’’ satisfies a fourth-order differential equation in the plane polar
coordinates with appropriate boundary conditions. We develop a method which allows us
to treat in a unitary way the two problems corresponding to the self-equilibrated loads
distributed on the straight and curved edges of the region. In fact, we introduce an appropriate
change for the variable r and for the Airy stress functions to reduce the corresponding boundary
value problem to a simpler one which allows us to indicate an appropriate measure of the
solution valuable for both the types of boundary value problems. In terms of such measures
we are able to establish some spatial estimates describing the spatial behavior of the Airy
stress function. In particular, our spatial decay estimates prove a clear relationship with the
Saint-Venant’s principle on such regions
Effects of Saint-Venant type and uniqueness and continuous dependence results for incremental elastodynamics
No full textOn Saint-Venant's principle in a poroelastic arch-like region
No full textIn this paper we consider the state of plane strain in an elastic material with voids occupying a curvilinear strip as an arch-like region described by R:a<r<b,0<h<x, where r and θ are polar coordinates and a, b, and ɷ (<2π) are prescribed positive constants. Such a curvilinear strip is maintained in equilibrium under self-equilibrated traction and equilibrated force applied on one of the edges, whereas the other three edges are traction free and subjected to zero volumetric fraction or zero equilibrated force. In fact, we study the case when one right or curved edge is loaded. Our aim is to derive some explicit spatial estimates describing how some appropriate measures of a specific Airy stress function and volume fraction evolve with respect to the distance to the loaded edge. The results of the present paper prove how the spatial decay rate varies with the constitutive profile. For the problem corresponding to a loaded right edge, we are able to establish an exponential decay estimate with respect to the angle θ. Whereas for the problem corresponding to a loaded curved edge, we establish an algebraical spatial decay with respect to the polar distance r, provided the angle ɷ is lower than the critical value π√2. The intended applications of these results concern various branches of medicine as for example the bone implants
On the asymptotic behaviour of quasi-static solutions in a semi-infinite viscoelastic cylinder
No full text
Spatial Decay Estimates for the Biharmonic Equation in Plane Polars with Applications to Plane Elasticity
No full textThe present paper considers an isotropic and homogeneous elastic body occupying the arch-like region a <= r <= b, 0 <= theta <= alpha, where (r, theta) denote plane polar coordinates. The arch-like body is in equilibrium under an (in plane) self-equilibrated load on the edge r = a, while the other three edges r = b, theta = 0 and theta = alpha are traction-free and the body forces are absent. An appropriate measure is defined in terms of the Airy stress function phi, provided that the opening angle of the arch-like region is lower than 2 pi/root 3. Then the spatial behavior of the solution is studied and a clear relationship is established with Saint-Venant's principle on such regions. In fact, for a bounded arch-like region it is shown that the measure decays at least algebraically with respect to r, while for an unbounded region our result reveals a relationship with the classical Phragmen-Lindelof theorem
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