53,722 research outputs found

    Some remarks on materials with memory: heat conduction and viscoelasticity

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    Materials with memory are here considered. The introduction of the dependence on time not only via the present, but also, via the past time represents a way, alternative to the introduction of possible non linearities, when the physical problem under investigation cannot be suitably described by any linear model. Specifically, the two different models of a rigid heat conductor, on one side, and of a viscoelastic body, on the other one, are analyzed. In them both, to evaluate the quantities of physical interest a key role is played by the past history of the material and, accordingly, the behaviour of such materials is characterized by suitable constitutive equations where Volterra type kernels appear. Specifically, in the heat conduction problem, the heat flux is related to the history of the temperature-gradient while, in isothermal viscoelasticity, the stress tensor is related to the strain history. Then, the notion of equivalence is considered to single out and associate together all those different thermal histories, or, in turn, strain histories, which produce the same work. The corresponding explicit expressions of the minimum free energy are compared

    Energy decay rate of a transmission system governed by degenerate wave equation with drift and under heat conduction with memory effect

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    In this paper, we investigate the stabilization of transmission problem of degenerate wave equation and heat equation under Coleman-Gurtin heat conduction law or Gurtin-Pipkin law with memory effect. We investigate the polynomial stability of this system when employing the Coleman-Gurtin heat conduction, establishing a decay rate of type t−4t^{-4}. Next, we demonstrate exponential stability in the case when Gurtin-Pipkin heat conduction is applied

    A two-stage approach to relaxation in billiard systems of locally confined hard spheres

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    We consider the three-dimensional dynamics of systems of many interacting hard spheres, each individually confined to a dispersive environment, and show that the macroscopic limit of such systems is characterized by a coefficient of heat conduction whose value reduces to a dimensional formula in the limit of vanishingly small rate of interaction. It is argued that this limit arises from an effective loss of memory. Similarities with the diffusion of a tagged particle in binary mixtures are emphasized.Comment: Submitted to Chaos, special issue "Statistical Mechanics and Billiard-Type Dynamical Systems

    Asymptotic behavior of a semilinear problem in heat conduction with memory

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    3This paper is devoted to existence, uniqueness and asymptotic behavior, as time tends to infinity, of the solutions of an integro-partial differential equation arising from the theory of heat conduction with memory, in presence of a temperature-dependent heat supply. A linearized heat flux law involving positive instantaneous conductivity is matched with the energy balance, to generate an autonomous semilinear system subject to initial history and Dirichlet boundary conditions. Existence and uniqueness of solution is provided. Moreover, under proper assumptions on the heat flux memory kernel, the existence of absorbing sets in suitable function spaces is achieved.openopenC. GIORGI; MARZOCCHI A.; PATA V.Giorgi, Claudio; Marzocchi, Alfredo; Pata, Vittorin

    The Zoo of Non-Fourier Heat Conduction Models

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    The Fourier heat conduction model is valid for most macroscopic problems. However, it fails when the wave nature of the heat propagation or time lags become dominant and the memory or/and spatial non-local effects significant -- in ultrafast heating (pulsed laser heating and melting), rapid solidification of liquid metals, processes in glassy polymers near the glass transition temperature, in heat transfer at nanoscale, in heat transfer in a solid state laser medium at the high pump density or under the ultra-short pulse duration, in granular and porous materials including polysilicon, at extremely high values of the heat flux, in heat transfer in biological tissues. In common materials the relaxation time ranges from 10−810^{-8} to 10−1410^{-14} sec, however, it could be as high as 1 sec in the degenerate cores of aged stars and its reported values in granular and biological objects varies up to 30 sec. The paper considers numerous non-Fourier heat conduction models that incorporate time non-locality for materials with memory (hereditary materials, including fractional hereditary materials) or/and spatial non-locality, i.e. materials with non-homogeneous inner structure

    Asymptotics of the Coleman-Gurtin model

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    This paper is concerned with the integrodifferential equation \partial_t u-\Delta u -\int_0^\infty \kappa(s)\Delta u(t-s)\,\d s + \varphi(u)=f arising in the Coleman-Gurtin's theory of heat conduction with hereditary memory, in presence of a nonlinearity φ\varphi of critical growth. Rephrasing the equation within the history space framework, we prove the existence of global and exponential attractors of optimal regularity and finite fractal dimension for the related solution semigroup, acting both on the basic weak-energy space and on a more regular phase space.Comment: Accepted in Discrete and Continuous Dynamical Systems, Serie
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