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TECHNICAL PAPERS

Parametric Formulation of the Finite-Volume Theory for Functionally Graded Materials—Part II: Numerical Results

[+] Author and Article Information
Marcio A. Cavalcante, Severino P. Marques

Center of Technology, Federal University of Alagoas, Maceio, Alagoas, Brazil

Marek-Jerzy Pindera

Civil Engineering Department, University of Virginia, Charlottesville, VA 22903

J. Appl. Mech 74(5), 946-957 (Dec 22, 2006) (12 pages) doi:10.1115/1.2722313 History: Received June 06, 2006; Revised December 22, 2006

In Part I of this communication, the finite-volume theory for functionally graded materials was further extended to enable efficient analysis of structural components with curved boundaries, as well as efficient modeling of continuous inclusions with arbitrarily-shaped cross sections of a graded material’s microstructure, previously approximated using discretizations by rectangular subcells. This was accomplished through a parametric formulation based on mapping of a reference square subcell onto a quadrilateral subcell resident in the actual microstructure. In Part II, the parametric formulation is verified through comparison with analytical solutions for homogeneous and graded curved structural components subjected to transient thermal and steady-state thermomechanical loading. 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 the matrix phase of large dimensions are also generated and compared with the exact analytical solution, as well as with the results obtained using the standard version of the finite-volume theory based on rectangular discretization and the finite-element method. It is demonstrated that the parametric finite-volume theory is a very competitive alternative to the finite-element method based on the quality of results and execution time.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 1

Thick-walled cylinder subjected to thermomechanical loading

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Figure 2

Discretization of one quarter of a thick-walled cylinder into 15×45 subcells for steady-state analysis of the radially graded and homogeneous cylinder

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Figure 3

Comparison of the analytical solution results with the parametric finite-volume predictions for steady-state thermomechanical loading of radially graded and homogeneous cylinders. Radial temperature and stress distributions in the three cross sections θ=0deg, 45deg, 90deg. (a) T(r,θ=0deg, 45deg, 90deg) distributions, (b) σrr(r,θ=0deg, 45deg, 90deg) distributions, (c) σθθ(r,θ=0deg, 45deg, 90deg) distributions.

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Figure 4

Comparison of the analytical solution results with the parametric finite-volume predictions for transient thermomechanical loading of a homogeneous cylinder. Radial temperature and stress distributions in the cross section θ=45deg at different times. (a) T(r,θ=45deg) distributions, (b) σrr(r,θ=45deg) distributions, (c) σθθ(r,θ=45deg) distributions.

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Figure 5

A circular fiber of radius 1 embedded in a large matrix of dimensions 20×20 subjected to uniform far-field stress σxxo=100MPa and the stress fields obtained from the exact analytical solution for the corresponding Eshelby problem. (a) A circular fiber in large matrix, (b) σxx(x,y) distribution, (c) σxy(x,y) distribution, (d) σyy(x,y) distribution.

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Figure 6

Discretizations of the 2×2 region containing centered fiber used in the standard (a) 4×625 discretization, and parametric (b) 4×325 discretization finite-volume analyses

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Figure 7

Local σxx(x,y) stress distributions in the 3×3 region containing the centered fiber predicted by the standard and parametric finite-volume theories based on 4×(25×25)=4×625 rectangular subcell and 4×325 quadrilateral subcell discretizations, respectively. (a) Standard finite-volume theory, (b) parametric finite-volume theory.

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Figure 8

Local σxy(x,y) stress distributions in the 3×3 region containing the centered fiber predicted by the standard and parametric finite-volume theories based on 4×(25×25)=4×625 rectangular subcell and 4×325 quadrilateral subcell discretizations, respectively. (a) Standard finite-volume theory, (b) parametric finite-volume theory.

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Figure 9

Very fine discretization of the 2×2 region containing centered fiber taken from the 4×(75×75)=4×5625 rectangular subcell mesh used in the standard finite-volume analysis, and the resulting stress distributions in the 3×3 region. (a) 4×5625 discretization, (b) σxx(x,y) distribution, (c) σxy(x,y) distribution, (d) σyy(x,y) distribution.

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Figure 10

Discretizations of the 2×2 region containing centered fiber used in the finite-element (a) 4×1200 Q8-element mesh and parametric finite-volume (b) 4×2600 subcell discretization analyses

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Figure 11

Local σxx(x,y) stress distributions in the 3×3 region containing centered fiber. Comparison of the parametric finite-volume results based on the 4×2600 subcell discretization with the finite-element results based on the 4×1200 Q8-element mesh. (a) Parametric finite-volume theory, (b) finite-element method.

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Figure 12

Local σxy(x,y) stress distributions in the 3×3 region containing centered fiber. Comparison of the parametric finite-volume results based on the 4×2600 subcell discretization with the finite-element results based on the 4×1200 Q8-element mesh. (a) Parametric finite-volume theory, (b) finite-element method.

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Figure 13

Local σyy(x,y) stress distributions in the 3×3 region containing the centered fiber. Comparison of the parametric finite-volume results based on the 4×2600 subcell discretization with the finite-element results based on the 4×1200 Q8-element mesh. (a) Parametric finite-volume theory, (b) finite-element method.

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