In this paper we derive necessary and sufficient conditions for strong ellipticity in several classes of anisotropic linearly elastic materials. Our results cover all classes in the rhombic system (nine elasticities), four classes of the tetragonal system (six elasticities) and all classes in the cubic system (three elasticities). As a special case we recover necessary and sufficient conditions for strong ellipticity in transversely isotropic materials. The central result shows that for the rhombic system strong ellipticity restricts some appropriate combinations of elasticities to take values inside a domain whose boundary is the third order algebraic surface defined by x^2 + y^2 + z^2 − 2xyz − 1 = 0 situated in the cube |x| < 1, |y| < 1, |z| < 1. For more symmetric situations, the general analysis restricts combinations of elasticities to
range inside either a plane domain (for four classes in the tetragonal system) or in an one-dimensional interval (for the hexagonal systems, transverse isotropy and cubic system). The proof involves only the basic statement of the strong ellipticity condition
We investigate the spatial behaviour of the steady state and transient elastic processes in an anisotropic elastic body subject to nonzero boundary conditions only on a plane end. For the transient elastic processes, it is shown that at distance x3 > ct from the loaded end, (c is a positive computable constant and t is the time), all the activity in the body vanishes. For x3 < ct, an appropriate measure of the elastic process decays with the distance from the loaded end, the decay rate of end effects being controlled by the factor (1 -~). Next, it is shown that for isotropic materials, in the case of a steady state vibration, an analogue of the Phragm6n-LindelOf principle holds for an appropriate cross-sectional measure. One immediate consequence is that in the class of steady state vibrations for which a quasi-energy volume measure is bounded, this measure decays at least algebraically with the distance from the loaded end. (1991): 73C02, 73CI0, 35B30, 35B40.
Mathematics Subject Classifications
The present paper gives explicit solutions for surface waves propagation in a homogeneous half space filled with an isotropic Cosserat elastic material. Such solutions are important in the study of seismic waves in an earthquake, supposing that the bottom land is modeled as having a microstructure. To construct explicit expressions for the possible surface waves under consideration, we use the Stroh formalism. These solutions are further used to study the Rayleigh waves and to give the explicit equation for the Rayleigh surface wave speed (secular equation). Numerical calculations and graphics corresponding to the analytical solution are given for aluminium-epoxy composite.
This paper deals with the time differential dual-phase-lag heat transfer models aiming, at first, to identify the eventually restrictions that make them thermodynamically consistent. At a first glance it can be observed that the capability of a time differential dual-phase-lag model of heat conduction to describe real phenomena depends on the properties of the differential operators involved in the related constitutive equation. In fact, the constitutive equation is viewed as an ordinary differential equation in terms of the heat flux components (or in terms of the temperature gradient) and it results that, for approximation orders greater than or equal to five, the corresponding characteristic equation has at least a complex root having a positive real part. That leads to a heat flux component (or temperature gradient) that grows to infinity when the time tends to infinity and so there occur some instabilities. Instead, when the approximation orders are lower than or equal to four, this is not the case and there is the need to study the compatibility with the Second Law of Thermodynamics. To this aim the related constitutive equation is reformulated within the system of the fading memory theory, and thus the heat flux vector is written in terms of the history of the temperature gradient and on this basis the compatibility of the model with the thermodynamical principles is analyzed
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