Parametric Formulation of the Finite-Volume Theory for Functionally Graded Materials—Part I: Analysis

Abstract

The recently reconstructed higher-order theory for functionally graded materials is further enhanced by incorporating arbitrary quadrilateral subcell analysis capability through a parametric formulation. This capability significantly improves the efficiency of modeling continuous inclusions with arbitrarily-shaped cross sections of a graded material’s microstructure previously approximated using discretization based on rectangular subcells, as well as modeling of structural components with curved boundaries. Part I of this paper describes the development of the local conductivity and stiffness matrices for a quadrilateral subcell which are then assembled into global matrices in an efficient manner following the finite-element assembly procedure. Part II verifies the parametric formulation through comparison with analytical solutions for homogeneous curved structural components and graded components where grading is modeled using piecewise uniform thermoelastic moduli assigned to each discretized region. Results for a heterogeneous microstructure in the form of a single inclusion embedded in a matrix phase are also generated and compared with the exact analytical solution, as well as with the results obtained using the original reconstructed theory based on rectangular discretization and finite-element analysis.

A Model for Homogenization of Linear Viscoelastic Periodic Composite Materials with Imperfect Interface

“…where denote the vertex coordinates of the subvolume and , are the interpolation function given by (Cavalcante et al, 2007) Figure 3: Mapping of the reference square subvolume onto a quadrilateral subvolume of the actual microstructure (after Cavalcante et al, 2007).…”

“…An attractive alternative to the finite-element method in the solution of periodic repeating unit cell (RUC) problems is the parametric finite-volume theory developedby Cavalcante et al (2007) having as basis the original version constructed by Bansal and Pindera (2003). In that parametric version, the heterogeneous material microstructure is discretized using quadrilateral subvolumes which are mapped into corresponding reference square subvolumes.…”

“…Using the parametric FVDAM theory, the unit cell domain is discretized into quadrilateral subvolumes whose locations inside the unit cell are specified by their vertices referred to a fixed coordinate system (Cavalcante et al, 2007). Here, it is assumed that the periodic multiphase composite material is continuously reinforced along the y1 axis so that the homogenization procedures require only the unit cell discretization in the y2-y3 plane.…”

A model for evaluation of effective thermal conductivity of periodic composites with poorly conducting interfaces

“…In the parametric finite-volume formulation the actual material microstructure is discretized into quadrilateral subvolumes whose geometry and location are identified by their node coordinates. It is based on a mapping of a reference square in the parametric plane h -x onto a quadrilateral subvolume in the Cartesian plane y 2 -y 3 of the actual material microstructure 7 , as shown in Figure 2. The mapping of the point (h, x) in the reference square to the corresponding point (y 2 -y 3 ) in the quadrilateral subvolume of the actual discretized microstructure is expressed in the form…”

“…An attractive alternative to the finite-element method in the solution of periodic repeating unit cell (RUC) problems is the parametric finite-volume theory developed by Cavalcante et al 7 having as basis the original version constructed by Bansal and Pindera 8 . In that parametric version the heterogeneous material microstructure is discretized using quadrilateral subvolumes which are mapped into corresponding reference square subvolumes.…”