Regular and singular kernel problems in rigid heat conduction with memory

Materials whose mechanical and/or thermodynamical behaviour is determined not only by their present status but also by their past history can be termed materials with memory. Some problems which arise in the study of this kind of materials are considered. Specifically, the model of a rigid heat conductor with memory is introduced. The quantities of interest are related via the constitutive equations connecting the energy, the heat flux, the temperature gradient and the heat flux relaxation function. It is follows to depend only on the time variable since the heat conductor is assumed to be homogeneous and isotropic. Then, the evolution equation which describes the heat conduction phenomenon is a linear integro-differential one. A Dirichlet boundary value problem with homogeneous initial conditions is considered. The attention is focussed on the kernel of the integro-differential evolution equation: it represents the heat flux relaxation function. Both the two different cases of regular as well as of singular kernel problems are considered. The term “regular” denotes a bounded heat flux relaxation function: it is integrable together with its time derivative; conversely, the term “singular” is used when a heat flux relaxation function, unbounded near t = 0, is considered. Note that thermodynamical admissibility requires the integrability of the heat flux relaxation function which corresponds to finite heat flux and energy. The outline of the method devised to prove the existence and uniqueness result, recently proved also in the case of a singular kernel problem, is provided. Some closing remarks are concerned about connections with other related problems and some perspective projects which refer to other materials with memory such as viscoelastic and magneto-viscoelastic ones.

Heat conduction with memory is considered. Specifically, the model of a rigid heat conductor is introduced. The quantities of interest are related via the constitutive relations connecting the energy, the heat flux, the temperature gradient and the heat flux relaxation function. The expression of the free energy is also given. The evolution equation which describes the heat conduction phenomenon is a linear integro-differential one. The kernel of such equation is represented by the heat flux relaxation function. Different cases of regular as well as of singular kernel problems are considered: the corresponding results are, then, compared.