Abstract
The study of the mixed initial-boundary value problem, corresponding to the thermoelasticity of porous micromorphic materials under the influence of microtemperatures, represents the main objective of this article. Achieving qualitative results on the existence, uniqueness and continuous dependence on the initial data and loads, of the solution of the mixed problem, implies a new perspective of approaching these topics, imposed by the large number of unknowns, which increases the complexity of equations and conditions that characterize the thermoelastic porous micromorphic materials with microtemperatures. The use of the semigroup theory of operators represents the optimal solution for deducing these results, the theory being adaptable to the requirements of the demonstrations, the mixed problem turning into a problem of Cauchy type, with regards to an equation of evolution on a Hilbert space, chosen appropriately.
Highlights
The equations describing the behavior of a porous micromorphic thermoelastic material with microtemperatures are expressed by means of the variables, ui being the displacement vector components, φij the microdeformation tensor components, ν the volume fraction directly related to the voids and α, ζ i, the two new variables, introduced by the following relations α( x, t) =
In the theory of the thermoelastic porous micromorphic materials having microtemperatures, the ordered system is a solution of the mixed initial-boundary value problem, denoted by P, if it satisfies the system of Equation (9), together with the initial conditions (11) and the boundary conditions (10), for all ( x, t) ∈ D × (0, ∞)
The variables that describe the deformation of a thermoelastic porous micromorphic material with microtemperatures verify the following equality tij ε ij + sij μij + mkij γijk + ρη α + ξν + λi υi + ρηi ζi + ri α,i + Mij ζ i,j =
Summary
Aiming to remove the inadequacies between the theory and the results of its implementations, occurred, for example, in the case of granular bodies with large particles, where the effect of materials microstructure makes its presence felt, the theory of microstructure is optimally appropriate for applications in solid mechanics. First studies dedicated to the theory of bodies having microtemperatures were published by Grot, which created the foundation of this theory see, for instance [25], considering that the microelements of a continuous body are endowed with microtemperatures, along with microdeformations, extending existing theories of materials with inner structure In this context, the second law of thermodynamics, see [26], is modified in order to include the microtemperatures, and the equations corresponding to the first moment of energy are added to the classical law of equilibrium, related to a material with microstructure, these equations leading to the equations of microtemperatures thermal conductivity. Our present article approaches the effects of microtemperatures on the fundamental features of the mixed problem with initial and boundary values corresponding to the thermoelasticity theory of the micromorphic porous materials In this context, our problem will be transformed into a problem of Cauchy type, attached to an evolutionary equation on a particular Hilbert space. The use of the contractions semigroup theory, see [38,39], to get the existence of the mixed problem solution and the uniqueness of this solution and, a result regarding the continuous dependence of the solution in relation to the initial data and loads, proves to be the most appropriate method
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