Domain Of Influence Theorem Research Articles

Spatial Behavior of Solutions in Linear Thermoelasticity with Voids and Three Delay Times

This brief contribution aims to complement a study of well-posedness started by the same authors in 2020, showing—for that same mathematical model—the existence of a domain of influence of external data. The object of investigation, we recall, is a linear thermoelastic model with a porous matrix modeled on the basis of the Cowin–Nunziato theory, and for which the heat exchange phenomena are intended to obey a time-differential heat transfer law with three delay times. We therefore consider, without reporting it explicitly, the (suitably adapted) initial-boundary value problem formulated at that time, as well as some analytical techniques employed to handle it in order to address the uniqueness and continuous dependence questions. Here, a domain of influence theorem is proven, showing the spatial behavior of the solution in a cylindrical domain, by activating the hypotheses that make the model thermodynamically consistent. The theorem, in detail, demonstrates that for a finite time t>0, the assigned external (thermomechanical) actions generate no disturbance outside a bounded domain contained within the cylindrical region of interest. The length of the work is deliberately kept to a minimum, having opted where possible for bibliographic citations in favor of greater reading fluency.

Uniqueness, continuous dependence, and spatial behavior of the solution in linear porous thermoelasticity with two relaxation times

This work aims to contribute to the verification of the well-posedness question, as for the uniqueness and continuous dependence issues, for a linear thermoelastic model with two main features: (i) a porous material matrix modeled on the basis of the Cowin–Nunziato theory; (ii) a heat transfer process obeying a time-differential constitutive equation with two relaxation times, derived from the dual-phase lag theory with an appropriate selection of the Taylor series expansion orders. Imagining to deal with very small spatial scales (in the order of the micro or nanometer), we assume it is reasonable to accept that the deformations caused by temperature variations are small enough to be realistically modeled under hypotheses of linearity, thus making the mathematical theory under investigation particularly meaningful, e.g. in the study of miniaturized devices in very fast transients. The work is concluded by proving a domain of influence theorem, and with a reference to the future research activities to carry out starting from it.

A domain of influence theorem for generalized thermoelasticity with memory-dependent derivative

The present article theoretically deals with the domain of influence theorem for generalized thermoelasticity with memory-dependent derivative (MDD). A mixed initial-boundary value problem is considered in an isotropic medium in the context of three-phase-lag model (TPL). The theorem is proved with the help of a domain of dependence inequality. This theorem certifies that the pair of displacement and temperature associated with the thermoelastic problem generates no thermal signals outside a suitably defined bounded domain V ∗ ( t ) for a finite time t > 0 and the set V ∗ ( t ) is the required domain of influence of the associated thermoelastic problem.

A domain of influence theorem for a natural stress–heat-flux disturbance in thermoelasticity of type-II

ABSTRACTThe aim of the present work is to establish the domain of influence theorem for a stress–heat-flux disturbance under Green and Naghdi thermoelasticity theory of type-II. We consider a mixed problem of natural type represented as stress–heat-flux disturbance in the context of Green–Naghdi thermoelasticity theory of general type. We establish a general energy identity for the problem. Then we establish the domain of influence theorem for natural stress–heat-flux disturbance in the context of Green–Naghdi model of thermoelasticity oftype-II. We prove that for a finite time t > 0, the pair of stress and the heat-flux field generates no disturbance outside a bounded domain. The domain of influence is shown to be dependent on the thermoelastic parameters.

A Domain of Influence Theorem for Thermoelasticity with Three-Phase-Lag Model

We establish the theory of domain of influence for a potential-temperature disturbance under the generalized theory of thermoelasticity with three-phase-lag model. For a finite time t > 0, it has been proved that the potential due to the displacement and the temperature fields does not produce any disturbances outside a bounded domain and the domain of influence is dependent on the support of load, the thermoelastic coupling constant and the phase-lag parameters. It has been shown that the domain of influence of the present case reduces to the domain of influence for classical thermoelasticity theory and the thermoelasticity theory with one relaxation parameter.

Domain of influence theorem and uniqueness results in magneto-thermoelasticity

In this paper, some energy inequalities are developed for an unbounded perfect electrically conducting generalized thermoelastic region moving in a variable magnetic field. Sufficient restrictions on the magneto-thermoelastic moduli and solutions are given in order that the perturbations initially generated in a bounded domain of the body cannot invade the whole body. Some uniqueness results are readily proved provided that such restrictions are met.

A Well-Posed Problem for the Dual-Phase-Lag Heat Conduction

The dual-phase lag theory based on the law q(x, t + τ q ) = −K ∇ T(x, t + τ T ) was proposed from an intuitive point of view by Tzou. This equation shows that the temperature gradient established across a material volume at the position x at time t + τ T results in a heat flux to flow at a different instant of time t + τ q . Though it proposes a law which could be compatible with our intuition, when we adjoin it with the energy equation − ∇ q(x, t) = c (x, t), we may obtain an ill posed problem, that is, a problem which has a sequence of eigenvalues such that its real part is positive (and goes to infinity). A consequence of this fact is that the problem may be unstable and we cannot obtain continuous dependence on the initial data.1 In this note we combine this constitutive equation with a two temperatures heat conduction theory and we show in this new context the problem is well posed. We also show that whenever the constitutive constants satisfy suitable relations, the real part of a point spectrum of the problem is negative. We also discuss a 2-temperature thermoelasticity and a higher-order theory corresponding to the higher order Taylor expansions of the fields involved. 1 In Ignaczak (2001), found in Ref. [9], a uniqueness theorem for arbitrary non-negative τ T and τ q was proved; together with a domain of influence theorem when τ q −τ T > 0. Similar results were also obtained in the literature described herein. These results show to what extent the dual-phase lag one-temperature theory makes sense.

Some results on the spatial behaviour for elastic mixtures

This article studies the spatial behaviour of solutions for some classes of homogeneous and isotropic simple elastic mixtures, for which the internal energy density is not always a positive definite quadratic form. By introducing some appropriate measures, we obtain the domain of influence theorem and the spatial decay estimates with time-independent decay rate inside the domain of influence.

A domain of influence theorem in linear thermo-elasticity with thermal relaxation and internal variable

A domain of influence theorem is proved for a linear thermoelastic solid with a Cattaneo's type heat conduction law and a scalar internal variable. The obtained result is applied to prove the hyperbolicity of a semiempirical heat conduction theory, describing the propagation of thermal waves in crystals at low temperatures.

Some Results in the Dynamical Theory of Porous Elastic Bodies

This paper is concerned with the linear dynamic theory of elastic materials with voids. First, a spatial decay estimate of an energetic measure associated with a dynamical process is established. Then, a domain of dependence inequality associated with a boundary-initial-value problem is derived and a domain of influence theorem is established. It is shown that, for a finite time, a solution corresponding to data of bounded support vanishes outside a bounded domain.

Domain of influence theorem in the theory of elastic materials with voids

A domain of influence theorem in the linear theory of elastic materials with voids, proposed by Cowin and Nunziato, has been established. The theorem shows that for a finite time t > 0, the displacement field u i and the change in volume fraction ϑ generate no disturbance outside a bounded domain Ω t .

SOME THEOREMS IN TEMPERATURE-RATE-DEPENDENT THERMOELASTICITY FOR UNBOUNDED DOMAINS

Abstract A domain of influence theorem, work and energy theorem, and reciprocity theorem are proved for a linear anisotropic nonhomogeneous temperature-rate-dependent thermoelastic unbounded body. The theorems cover the case where the constitutive fields may attain unbounded values at infinity, and they are based on a hypothesis from which it follows that a thermoelastic disturbance produced by an external ther-momechanical load of a bounded support cannot invade the whole body in a finite time

DOMAIN OF INFLUENCE THEOREM IN THERMOELASTIC1TY WITH ONE RELAXATION TIME

Abstract A domain of influence theorem is proved for a general mixed problem of linear nonhomogeneous anisotropic coupled dynamic thermoelasticity with one relaxation time. The proof is based on a domain of dependence inequality associated with the problem. The domain of influence is constructed in terms of the bounds for ther-momechanical constitutive fields, the relaxation time, and the data supports.

Energy inequalities and the domain of influence theorem in classical elastodynamics

In this paper we derive a number of a priori estimates for classical solutions of the system of linear elastodynamics, and we prove an analogue of the classical domain of influence theorem for a class of unbounded elastic bodies that may stiffen at infinity.