Maria Cristina Patria

Saint-Venant's Principle for Antiplane Shear Deformations of Linear Piezoelectric Materials

This paper is concerned with investigation of the effect of mechanical/electrical coupling on the decay of Saint-Venant end effects in linear piezoelectricity. Saint-Venants principle and related results for elasticity theory have received considerable attention in the literature but relatively little is known about analogous issues in piezoelectricity. The current rapidly developing smart structures technology provides motivation for the investigation of such problems. We examine the decay of Saint-Venant end effects in the context of antiplane shear deformations for linear homogeneous piezoelectric solids. For a rather general class of anisotropic piezoelectric materials, the governing partial differential equations of equilibrium are shown to be a coupled system of second-order partial differential equations for the mechanical displacement u and electric potential

Energy bounds in dynamical problems for a semi-infinite magnetoelastic beam

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Saint-Venant's principle for a piezoelectric body

. The traction boundary-value problem with prescribed surface charge is formulated as an oblique derivative boundary-value problem...

Energy bounds for a mixture of two linear elastic solids occupying a semi-infinite cylinder

The aim of this paper is to investigate the behaviour of the total energy of a magnetoelastic conductor occupying a semi-infinite prismatic cylinder in dynamical conditions. Precisely, we deduce some estimates for the energyW(x3,t) of the portion of the medium at distance greater thanx3 from the base in terms of the data. First of all, we prove that the total energyW(0,t) is finite for allt > 0 providedW(0, 0) is finite. Then, using the first Korn inequality, we obtain that the estimate forW(x3,t) depends only on the initial data iftx3/V then the bound forW(x3,t) depends on all the data of the problem.

Uniqueness in the boundary-value problems for the static equilibrium equations of a mixture of two elastic solids occupying an unbounded domain

For a semi-infinite piezoelectric body with uniform or variable cross section we establish the exponential decay of the internal energy relative to the portion of the body beyond a distance x3 from the base: Saint-Venants principle. All possible combinations of physically admissible boundary conditions are examined. The stabilizing effect of the electric field on the decay length is shown. Finally the total internal energy is estimated in terms of the data, making use also of the Korn inequality.

Discontinuity waves through a linear thermo-viscoelastic solid of integral type

SummaryThe aim of this paper is to establish some forms of the Saint-Venant principle for a mixture of two linear elastic solids occupying a semi-infinite prismatic cylinder. We examine the behaviour of the energy for both static and dynamical problems. Under mild assumptions on the asymptotic behaviour of the unknown fields at infinity, we show that in the static case the elastic energy of the portion of the cylinder beyond a distancex3 from the loaded region decays exponentially withx3. For the dynamical problem we estimate through the data the total energy stored in that part of the cylinder whose minimum distance from the loaded end isx3; these estimates, which are based on the assumption that the initial total energy is finite, depend uponx3 but do not depend upon the timet.

MHD orthogonal stagnation-point flow of a micropolar fluid with the magnetic field parallel to the velocity at infinity

Abstract In this paper the Authors study the uniqueness of the solutions to the most important boundary-value problems for the static equilibrium equations of a mixture of two linear elastic solids. Some uniqueness theorems concerning the mixed boundary-value problem and the displacement problem are proved for unbounded domains. If the mixture is anisotropic, mild assumptions are imposed on the displacement fields at infinity. If the mixture is isotropic, uniqueness is proved for exterior domains without artificial restrictions upon the behavior of the unknown fields at infinity.

Decay and other estimates for a semi-infinite magnetoelastic cylinder : Saint-Venant's principle

Abstract In this paper we investigate the propagation of discontinuity waves of order N > 1 through a homogeneous linear anisotropic thermo-viscoelastic solid whose heat flux vector depends upon the past history of the temperature gradient. We show that the normal speeds of propagation are independent of the order of the wave. Our analysis is simplified in the case of generalized longitudinal and transverse waves. For these waves we get also the evolution law of the discontinuities (which is the same for any N >= 1) along the rays associated with the wave front.

Stability questions for a third-grade fluid in exterior domains

An exact solution is obtained for the steady MHD plane orthogonal stagnation-point flow of a homogeneous, incompressible, electrically conducting micropolar fluid over a rigid uncharged dielectric at rest. The space is permeated by a not uniform external magnetic field He and the total magnetic field H in the fluid is parallel to the velocity at infinity. The results obtained reveal many interesting behaviours of the flow and of the total magnetic field in the fluid and in the dielectric. In particular, the thickness of the layer where the viscosity appears depends on the strength of the magnetic field. The effects of the magnetic field on the velocity and on the microrotation profiles are presented graphically and discussed.

Reverse flow in magnetoconvection of two immiscible fluids in a vertical channel

An analogue of the well known Toupins version of Saint-Venants principle is proved for a semi-infinite magnetoelastic cylinder under very mild assumptions on the asymptotic behaviour of the Dirichlet integral of the magnetic field and of the elastic energy. With regard to the elastic fields, we assign on the base either the stress or the displacement vector while we assume that the lateral surface is either traction free or held fixed at zero displacement. We make use of the first Korn inequality and estimate the total energy of the conductor in terms of the data for all the problems considered.