Thermal effects in orthotropic porous elastic beams

Abstract

This paper is concerned with the linear theory of anisotropic porous elastic bodies. The extension and bending of orthotropic porous elastic cylinders subjected to a plane temperature field is investigated. The work is motivated by the recent interest in the using of the orthotropic porous elastic solid as model for bones and various engineering materials. First, the thermoelastic deformation of inhomogeneous beams whose constitutive coefficients are independent of the axial coordinate is studied. Then, the extension and bending effects in orthotropic cylinders reinforced by longitudinal rods are investigated. The three-dimensional problem is reduced to the study of two-dimensional problems. The method is used to solve the problem of an orthotropic porous circular cylinder with a special kind of inhomogeneity. (2000). 74E10, 74E05, 74F05, 74F10, 74G05, 74K10, 74L15. Mathematics Subject Classification

Theory of thin thermoelastic rods made of porous materials

“…In view of relations (50), we remark that the one-dimensional Eq. (66) have exactly the same form as the governing equations in the direct approach (39)- (41). In line with the decompositions (52) 1,2 , we also denote byŵ…”

“…In this section, we show that a set of rod equations similar to (39)- (41) can be obtained from the three-dimensional equations by integration over the cross-section. Consider a three-dimensional rod made of a thermoelastic material with pores that occupies the domain…”

“…(69) should be considered and the equilibrium equations are given by (65) with vanishing right-hand sides. If we impose that the elastic potential is positive definite, then the constitutive coefficients satisfy the following inequalities [41,43] In the particular case of isotropic materials, this problem has been solved in [20] using a semi-inverse method.…”

“…In what follows, we turn our attention to orthotropic and homogeneous rods. The constitutive equations for orthotropic thermoelastic materials with pores are [41] t * 11 = c 11 e * 11 +c 12 e * 22 +c 13 e * 33 +β 1 ϕ * −b 1 T * , t * 22 = c 12 e * 11 +c 22 e * 22 +c 23 e * 33 +β 2 ϕ * −b 2 T * , t * 33 = c 13 e * 11 +c 23 e * 22 +c 33 e * 33 +β 3 ϕ * −b 3 T * , t * 23 = 2c 44 e * 23 , t * 31 = 2c 55 e * 31 , t * 12 = 2c 66 e * 12 ,…”

A beam theory consistent with three-dimensional thermo-elasticity

“…In this paper, we combine such nonstandard uses of the Principle to derive a thermo-elastic beam theory consistent with three-dimensional thermo-elasticity. Several models for beam thermomechanics have been proposed in the literature, some of them direct [6,12,19,21,30,1,28], others 1 Actually, in [3,11], it is argued that non-local and higher gradients continuum mechanics was considered already in Piola's works, by means of a suitable version of the Principle of Virtual Powers; the reader is referred to [35] and other works by the same author cited in [3,11]. Therein it is possible to find other references to papers using the same spirit and methods as Piola to introduce generalized stress tensors, e.g.…”