Thermal Conductivity and Elasticity of Reinforced Plates with Applications to T-ribbed, Π-ribbed and I-ribbed plates

Abstract

Currently, the preponderance of uses of composites is in the form of laminated or reinforced beams, plates and shells. It is clear that the elaboration of micromechanical models that accurately predict the properties of these structures at the design stage, preferably by means of closed-form design-oriented equations, offers significant advantages to the designer and is more likely to promote the use of a new composite/nanocomposite material system in new applications. Because laminated and reinforced plates and shells are often characterized by a periodic or nearly periodic configuration the method of asymptotic homogenization has been at the forefront of the pertinent analytical and semi-analytical modeling efforts. One of the main advantages of asymptotic homogenization is that it successfully decouples the microscopic (fast) and macroscopic (slow) scales characterizing the aforementioned structures; the microscopic scale "zooms in" on the periodicity unit of the composite and deals only with the geometrical and structural features therein, whereas the macroscopic scale "zooms out" on the overall structure and handles the global formulation of the problem. The specific mathematical details of asymptotic homogenization may be found in Bensoussan et al. [1] and others. This paper determines accurate expressions for the effective thermal conduction properties of T-Ribbed, Π-Ribbed and I-ribbed reinforced plates as well as the effective elastic properties of I-ribbed plates. All structures considered are made up of arbitrary orthotropic constituents. To evaluate these properties the authors invoke general, previously developed asymptotic homogenization models pertaining to a thin magnetoelectric composite plate with rapidly varying thickness, see Hadjiloizi et al [2], [3]. In general, the geometrical and material complexity involving reinforced plates, particularly in the case that some or all of the constituents exhibit piezoelectric, piezomagnetic or some other sensor/actuator behavior (which brings coupling of different fields into the foray), renders analytical modeling rather troublesome. In this work, the authors show how to overcome associated problems and effect accurate closed-form expressions for the elasticity and thermal conduction properties of reinforced plates. It is shown in this work that the results obtained can be successfully employed to tailor the effective thermal conduction and elastic properties of composite T-, Π-or I-ribbed plates to the requirements of a particular engineering application by changing certain geometric or material parameters of interest. The validity of the results is illustrated via comparison with other models. Finally, it is shown that in the limiting case of a thin elastic plate of uniform thickness the derived model converges to the familiar classical plate model. The recent emergence of additive manufacturing and 3-D and 4-D printing technologies, see Ge et al., [4], render fabrication of structures with arbitrary complexity more viable than ever before; thus micromechanical models that can design and analyze such structures as the T-ribbed, Π-ribbed and I-ribbed orthotropic plates considered herein are very important.