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Effect of voids in a heat-flux dependent theory for thermoelastic bodies with dipolar structure
(2020)

In our present paper, we approach the mixed problem with initial and boundary conditions, in the context of thermoelasticity without energy dissipation of bodies with a dipolar structure. Our first result is a reciprocal relation for the mixed problem which is reformulated by including the initial data into the field equations. Then, we deduce a generalization of Gurtin’s variational principle, which covers our generalized theory for bodies with a dipolar structure. It is important to emphasize that both results are obtained in a very general context, namely that of anisotropic and inhomogeneous environments, having a center of symmetry at each point.

Our study is dedicated to a composite, which, in fact, is a mixture of two thermoelastic micropolar bodies. We formulate the mixed initial boundary value problem in this context and define the domain of influence for given data. For any solution of the mixed problem we associate a measure and prove a second-order differential inequality for it. Based on the maximum principle for the heat equation and on the second-order differential inequality, we establish an estimate which proves that the thermal and the mechanical effects, at large distance from the domain of influence, are dominated by an exponential decay.

This study is concerned with the linear elasticity theory for bodies with a dipolar structure. In this context, we approach transient elastic processes and the steady state in a cylinder consisting of such kind of body which is only subjected to some boundary restrictions at a plane end. We will show that at a certain distance d=d(t), which can be calculated, from the loaded plan, the deformation of the body vanishes. For the points of the cylinder located at a distance less than d, we will use an appropriate measure to assess the decreasing of the deformation relative to the distance from the loaded plane end. The fact that the measure, that assess the deformation, decays with respect to the distance at the loaded end is the essence of the principle of Saint-Venant.

This paper aims to analyze the stress and strain states appearing in the elbow of a tube, such as those commonly used in a city’s water supply network. The stress field is characterized by the fact that there is a significant stress increase when compared to a straight tube. As a result, the strength of such an elbow must be investigated and guaranteed for such a network to be well designed. A practical solution used is to anchor the elbow in a massive concrete block. The paper compares the stress field that occurs in the elbow when it is free, buried in the ground, and when it is anchored in a massive concrete block. Furthermore, we investigate how a crack appears and propagates in the elbow. This happens especially for the elbow buried in the ground where the stress and strain are higher than when the elbow is anchored in concrete. The results obtained can be used in the current practice in the case of water supply networks made by high-density polyethylene pipes.

In our study, we consider the linear mixed initial boundary value problem for a porous elastic body having a dipolar structure. The equations that describe the elastic dipolar deformations are coupled with the equations which describe the evolution of the voids by means of certain coefficients. Our main result proves the continuous dependence of solutions for the mixed problem with regard to the coefficients which perform this coupling. Using an adequate measure, we can evaluate the continuous dependence by means of some estimate regarding the gradient of deformations and the gradient of the function that describes the evolution of the voids.