A beam theory consistent with three-dimensional thermo-elasticity

Abstract

We propose a model for thermo-elastic beams, consistent with the theory of linear threedimensional thermo-elasticity and deduced by a suitable version of the Principle of Virtual Powers. Dimensional reduction is achieved by postulating convenient a priori representations for mechanical and thermal displacements, the latter playing the role of an additional kinetic variable. Such representations are regarded as internal constraints, some involving the first, others the second gradient of deformation and thermal displacements; these constraints are maintained by reactive stresses and hyper-stresses of the type occurring in non-simple elastic materials of grade two, and by reactive entropy influxes and hyper-influxes.

A multiphysics and multiscale approach for modeling microcracked thermo-elastic materials

“…Let P be a subbody of C and I ¼ ðt i ; t f Þ a subinterval of T . Following the approach of [54,24], we define as internal virtual power the following functional over V:…”

“…The continuum with a just mechanical structure has been proposed in [72,42,79,78] and applied for particle/fiber reinforced polymer/ceramic matrix and masonry. Here we extend the microstructure to the thermomechanical framework, stemming from the ideas exposed in [24], where a simple and paradigmatic example of continua with (micro-)structure is exposed within the beams theory framework. The effects of cracks on the thermal conductivity have been investigated on various materials such as fiber-reinforced composites [41], optical coatings [58], graphite materials [52], thermal barrier coatings [49,16,65], clay [19] and others.…”

Multiscale Modeling of Thermo-Elastic Properties of Microcracked Material

“…The continuum here described, with just reference to the mechanical structure, has been proposed in [18] and developed in [31,37,30] with applications to particle/fiber reinforced polymer/ceramic matrix and masonry-like materials. Recently, the microstructure has been extended to the thermomechanical framework in [13], stemming from the ideas exposed in [11], where a simple and paradigmatic example of continua with micro-structure is exposed within the beams theory framework. An application of a two-temperatures model to rod theory can be also found in [1].…”

“…As Equations (12) reveal, h and k can be interpreted as the entropy and micro-entropy influxes, respectively; the fields h and k are measures of internal dissipation, as anticipated in the previous section. The extended version of the PVL, whence (11) and (12) come out, is based upon the introduction of: (i) a space-time integral, responsible of the appearance of time derivatives, and (ii) those thermal kinetic variables already introduced in the discrete system. On substituting (8), (9) in (11) and (12), we obtain a system of partial differential equations in the unknowns u, d, α and β.…”