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In this paper, the authors propose the application of the Gibbs–Appell equations to obtain the equations of motion in the case of a mechanical system that has elements with a micro-polar structure, containing voids. Voids can appear as a result of the processing or manufacturing of the parts, or can be intentionally introduced. This research involves a model of the considered solid material containing voids. To determine the dynamic behavior of such a system, the Gibbs–Appell (GA) method is used to obtain the evolution equations, as an alternative to Lagrange’s classical description. The proposed method can be applied to any mechanical system consisting of materials with a micro-polar structure and voids. The study of such systems is interesting because the literature shows that even a reduce number of small voids can produce significant variations in physical behavior. The proposed method requires a smaller number of mathematical operations. To apply this method, the acceleration energy is calculated, which is then used to derive the equations. The method comes with advantages in the application to multibody systems having the mentioned properties and, in particular, in the study of robots and manipulators. Using the GA method, it is necessary to do a fewer differentiation operations than applying the Lagrange’s equations. This leads to a reduced amount of computation for obtaining the evolution equations.

In this paper we approach the linear mixed problem with initial and boundary values for a Cosserat body which is elastic and has pores. We coupled the equations characterizing the pores evolution with basic equations that describing the elastic deformations of the Cosserat body. The coupling is done by means of predetermined coefficients. To prove the continuous dependence of the solutions, we introduce an adequate measure which is useful in obtaining of some estimations with respect to the gradient of elastic deformations and regarding the gradient of a specific the application, introduced to characterize the behavior of the pores. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a generalization of the Gronwall’s inequality to cover the study of porous elastic media considering their internal state variables. Based on some estimations obtained in three auxiliary results, we use this form of the Gronwall’s inequality to prove the uniqueness of solution for the mixed initial-boundary value problem considered in this context. Thus, we can conclude that even if we take into account the internal variables, this fact does not affect the uniqueness result regarding the solution of the mixed initial-boundary value problem in this context.

In this paper, the authors propose the application of the Gibbs–Appell equations to obtain the equations of motion in the case of a mechanical system that has elements with a micro-polar structure, containing voids. Voids can appear as a result of the processing or manufacturing of the parts, or can be intentionally introduced. This research involves a model of the considered solid material containing voids. To determine the dynamic behavior of such a system, the Gibbs–Appell (GA) method is used to obtain the evolution equations, as an alternative to Lagrange's classical description. The proposed method can be applied to any mechanical system consisting of materials with a micro-polar structure and voids. The study of such systems is interesting because the literature shows that even a reduce number of small voids can produce significant variations in physical behavior. The proposed method requires a smaller number of mathematical operations. To apply this method, the acceleration energy is calculated, which is then used to derive the equations. The method comes with advantages in the application to multibody systems having the mentioned properties and, in particular, in the study of robots and manipulators. Using the GA method, it is necessary to do a fewer differentiation operations than applying the Lagrange's equations. This leads to a reduced amount of computation for obtaining the evolution equations. [ABSTRACT FROM AUTHOR]

Abstract With a wide applicability in solid state mechanics, the theory of continuous dipolar materials with voids is a distinctive part of microstructure theory, a theory that emerged with the necessity of eliminating the discrepancies between the classical theory and its experimental applications. These inconsistencies are caused by the influence of the microstructure of materials on the general deformations of the bodies, such as ceramics, graphite or polymers. In the present work, we study the mixed boundary value problem with initial data for the thermoelasticity of three-phase-lag dipolar materials with voids, in order to obtain the corresponding constitutive laws, using the same method as in the classical elasticity. By going deeper into the study of these materials, we achieve a uniqueness result and a reciprocal relation, using less common methods, like the dissipative inequality regarding the result of uniqueness. Along with these aspects, we demonstrate a variational principle for anisotropic and non-homogeneous materials, derived from a well-known variational principle, but studied in the context of thermoelasticity of three-phase-lag dipolar materials with voids.

Abstract Our study is dedicated to the problem with initial and boundary conditions in the theory of thermoelasticity for micropolar materials with pores. We obtain some results regarding the existence and uniqueness of a finite energy solution of this mixed problem by generalizing the corresponding results obtained by Dafermos in the context of classical elasticity. In some specific conditions we prove some properties regarding the possibility to control the finite energy solution.

Abstract The goal of this paper is to obtain an extension of the relaxed Saint–Venant principle in order to cover the thermoelasticity of dipolar porous bodies. According to this principle, for a finite time t>0 $t>0$, we identify a bounded domain Dt $\mathcal{D}_{t}$ so that outside of this domain the displacement field ui $u_{i}$, the dipolar displacement field φjk $\varphi_{jk}$, the temperature θ, and the change in volume fraction ϕ do not generate disturbances.

Composite-based polymer materials are manufactured in a wide variety of types with different compositions, structures, geometries, and topological descriptions. Among these, micropolar materials with voids have become increasingly studied in the literature. This paper establishes the equations of motion for such a material for the purpose of dynamic analysis via the finite element method (FEM). The Euler–Lagrangian formalism, based on the expressions of kinetic energy, potential energy, and mechanical work, is used. Hence, it is possible to study the dynamic response of such a system in the most general configuration case. The choice of the shape functions will determine the matrix coefficients for each particular case. An application illustrates the presented results.

In our study we analyse the vibration of a right cylinder which consists of an elastic material with dipolar structure and has pores. One end of this cylinder is subjected to an excitation, harmonically in time. The other end of the cylinder and its lateral surface are free of loads. We prove that the presence of the voids does not affect the spatial decay of effects away from the excited end, if the harmonic excitation level is below a predetermined threshold.

In the present article, the theory of linear thermoelasticity without energy dissipation is addressed from the perspective of the analysis of the spatial evolution of harmonic vibrations in time, in the context of a porous micropolar media. Some preliminary identities are determined that will lead to estimates of the harmonic vibration amplitude, some of these estimates being consequences of the distance influence from the disturbed base, provided that a certain critical value for the vibration frequency is considered. [ABSTRACT FROM AUTHOR]

We consider a porous thermoelastic body, including voidage time derivative among the independent constitutive variables. For the initial boundary value problem of such materials, we analyze the temporal behaviour of the solutions. To this aim we use the Cesaro means for the components of energy and prove the asymptotic equipartition in mean of the kinetic and strain energies.

In our study, we will extend the domain of influence in order to cover the thermoelasticity of initially stressed bodies with voids. In what follows, we prove that, for a finite time t > 0 , the displacement field u i , the dipolar displacement field φ j k , the temperature θ and the change in volume fraction ϕ generate no disturbance outside a bounded domain B.

The present work investigated the effect of Thomson and initial stress in a thermo-porous elastic solid under G-N electromagnetic theory. The Thomson coefficient affects the heat condition equation. A constant Thomson coefficient, instead of traditionally a constant Seebeck coefficient, is assumed. The charge density of the induced electric current is taken as a function of time. A normal mode method is proposed to analyze the problem and to obtain numerical solutions. The results that were obtained for all physical sizes are graphically illustrated and we offer a comparison between the type II G-N theory and the G-N theory of type III, both in the present case and in the absence of specific parameters, as initial stress, pores and the Thomson effect. Some particular cases are also discussed in the context of the problem. The results indicate that the effect of initial stress, Thomson coefficient effect, and magnetic field are very pronounced.

In our article, we deduce a reciprocal relation in the context of dipolar porous materials. As an application, we analyze the effect of the dipolar structure on bodies charges which are similar to a dislocation of seismic type. Our results appear as generalizations of equivalent statements from the simpler theory of classical elastic bodies and they cover the more complicated theory of thermoelastic bodies with a dipolar structure. [ABSTRACT FROM AUTHOR]

In our paper we formulate a theory for thermoelastic porous dipolar bodies in which we consider a new independent variable, namely the heat-flux vector. Furthermore, we add, to the differential equations that describe the behavior of the body, a new differential equation which is an equation of evolution which is satisfied by the components of the heat-flux vector. The basic system of the mixed initial-boundary value problem in this context consists of equations of the hyperbolic type. In order to ensure the consistency of the constructed theory, we formulate and prove an uniqueness result, with regards to the solution of the mixed problem. [ABSTRACT FROM AUTHOR]

The goal of this article is to combine the theory based on fractional order of strain with the theory of thermoelasticity of dipolar bodies with voids, in order to obtain the basic thermoelasticity equations with fractional order strain for dipolar materials with voids. [ABSTRACT FROM AUTHOR]

Composite-based polymer materials are manufactured in a wide variety of types with different compositions, structures, geometries, and topological descriptions. Among these, micropolar materials with voids have become increasingly studied in the literature. This paper establishes the equations of motion for such a material for the purpose of dynamic analysis via the finite element method (FEM). The Euler–Lagrangian formalism, based on the expressions of kinetic energy, potential energy, and mechanical work, is used. Hence, it is possible to study the dynamic response of such a system in the most general configuration case. The choice of the shape functions will determine the matrix coefficients for each particular case. An application illustrates the presented results. [ABSTRACT FROM AUTHOR]

The present work investigated the effect of Thomson and initial stress in a thermo-porous elastic solid under G-N electromagnetic theory. The Thomson coefficient affects the heat condition equation. A constant Thomson coefficient, instead of traditionally a constant Seebeck coefficient, is assumed. The charge density of the induced electric current is taken as a function of time. A normal mode method is proposed to analyze the problem and to obtain numerical solutions. The results that were obtained for all physical sizes are graphically illustrated and we offer a comparison between the type II G-N theory and the G-N theory of type III, both in the present case and in the absence of specific parameters, as initial stress, pores and the Thomson effect. Some particular cases are also discussed in the context of the problem. The results indicate that the effect of initial stress, Thomson coefficient effect, and magnetic field are very pronounced. [ABSTRACT FROM AUTHOR]