Theory of thin thermoelastic rods made of porous materials

Abstract

In this paper, we consider thin rods modeled by the direct approach, in which the rod-like body is regarded as a one-dimensional continuum (i.e., a deformable curve) with a triad of rigidly rotating orthonormal vectors attached to each material point. In this context, we present a model for porous thermoelastic curved rods, having natural twisting and arbitrary shape of cross-section. To describe the porosity, we employ the theory of elastic materials with voids. The basic laws of thermodynamics are applied directly to the one-dimensional continuum, and the nonlinear governing equations are established. We formulate the constitutive equations and determine the structure of constitutive tensors. We prove the uniqueness of solution to the boundaryinitial-value problem associated with the deformation of porous thermoelastic rods in the framework of linear theory. Then, we show the decoupling of the bending-shear and extension-torsion problems for straight porous rods. Using a comparison with three-dimensional equations, we identify and give interpretations to the relevant fields introduced in the direct approach. Finally, we consider the case of orthotropic materials and determine the constitutive coefficients for deformable curves in terms of three-dimensional constitutive constants by means of comparison between simple solutions obtained in the two approaches for porous thermoelastic rods.

Nonlinear resultant theory of shells accounting for thermodiffusion

“…Postulating some reasonable relations between the three surface temperatures, one can reduce the number of independent 2D temperature fields to two or to one, whichever is appropriate. Review of several such relations available in the literature is given in [7,20,22,42,47,54,58], see also [2,8,55] for thermoelastic beams.…”

On a thermodynamic theory of rods with two temperature fields

“…This is an important advantage of considering two temperature fields for describing thermal effects in rods. In this sense, our approach is an improvement of the theory presented in [41], where the temperature field was coupled only with the extensional deformation in the linear theory of straight rods.…”

“…(38)(39)(40)(41) into Eq. (36) we find the explicit form of the constitutive equations for N · P, M · P, η and F. For instance, the constitutive equations for η and F are expressed by…”

“…-porosity effects in the mechanics of rods [40,41], using the theory of thermoelastic materials with voids [42]; -functionally graded materials in the mechanics of rods [43] or -electrical and magnetic fields, phase transitions, diffusion, and other coupled effects known in 3D continuum mechanics or in 2D shell theories.…”

A beam theory consistent with three-dimensional thermo-elasticity

“…One may ask why a zeroth order interaction has not been taken into account for the mechanical structure; as pointed out in[39], such a term would be cancelled by the requirement of translational invariance of δW (i) 6. For a revised exposition of Green-Naghdi theory, the reader is referred to[5].…”