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TECHNICAL PAPERS

Designing Optimal Volume Fractions For Functionally Graded Materials With Temperature-Dependent Material Properties

[+] Author and Article Information
Florin Bobaru

Department of Engineering Mechanics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0526fbobaru2@unl.edu

J. Appl. Mech 74(5), 861-874 (Oct 10, 2006) (14 pages) doi:10.1115/1.2712231 History: Received March 18, 2006; Revised October 10, 2006

We present a numerical approach for material optimization of metal-ceramic functionally graded materials (FGMs) with temperature-dependent material properties. We solve the non-linear heterogeneous thermoelasticity equations in 2D under plane strain conditions and consider examples in which the material composition varies along the radial direction of a hollow cylinder under thermomechanical loading. A space of shape-preserving splines is used to search for the optimal volume fraction function which minimizes stresses or minimizes mass under stress constraints. The control points (design variables) that define the volume fraction spline function are independent of the grid used in the numerical solution of the thermoelastic problem. We introduce new temperature-dependent objective functions and constraints. The rule of mixture and the modified Mori-Tanaka with the fuzzy inference scheme are used to compute effective properties for the material mixtures. The different micromechanics models lead to optimal solutions that are similar qualitatively. To compute the temperature-dependent critical stresses for the mixture, we use, for lack of experimental data, the rule-of-mixture. When a scalar stress measure is minimized, we obtain optimal volume fraction functions that feature multiple graded regions alternating with non-graded layers, or even non-monotonic profiles. The dominant factor for the existence of such local minimizers is the non-linear dependence of the critical stresses of the ceramic component on temperature. These results show that, in certain cases, using power-law type functions to represent the material gradation in FGMs is too restrictive.

Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 1

Various realizations of material gradations as given by the power law xp for different values of the parameter p. The types of volume fraction variations is rather limited; for instance, non-monotonic variations are excluded.

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Figure 2

Schematic representation for a possible continuous, non-monotonic volume fraction function along the radial direction of the hollow cylinder in Fig. 3, and sample design variables (y1,…,y5) that control its profile. The end points may also serve as design variables.

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Figure 3

Boundary conditions for the hollow cylinder under thermomechanical loading: temperature values imposed on the inner (Tint) and outer (Text) surfaces, inner pressure (t¯), and symmetry conditions for the thermal flux and displacements

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Figure 4

Analytical and numerical results for the temperature (a) and temperature gradient (b) along the radial direction of a hollow FGM cylinder under temperature-imposed boundary conditions. Dimensionless quantities are as in Eq. 17.

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Figure 5

The initial guess (∗) and optimal designs for the metal volume fraction function with five (◻), seven (▵), and nine (엯) design variables selected between the pure metal and ceramic coatings

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Figure 6

The temperature along the radial direction of the hollow cylinder at the initial design (엯) and final metal volume fraction (▵) in the case of seven design variables chosen between the fixed coatings

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Figure 7

The history of the objective for the case of seven design variables. We use dimensionless mass M*=M∕Mc, where Mc is the mass of a purely ceramic hollow cylinder.

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Figure 8

History of constraint value for the case of seven design variables. The constraint in Eqs. 19,20 is slightly violated, but the use of the safety factor gives stresses that are nowhere larger than the critical stresses.

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Figure 9

The stress profiles along the radial direction of the hollow cylinder for two intermediate iterations for the case of seven design variables in the problem defined by Eqs. 19,20. The critical stresses are dependent on the temperature and, implicitly, on the design variables. Shown are the tangential (σθ, ◻) and radial (σr , 엯) stresses, and the critical tensile (σtc, ▿) and compressive (σcc, ▵) stresses.

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Figure 10

Volume fraction profile for the mass minimization with no enforced coatings. The initial guess (∗) and optimal designs for the metal volume fraction function with six (◻), nine (▵), and 13 (엯) design variables.

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Figure 11

Metal volume fraction profile for the stress minimization case without inner pressure. The initial guess (*) and optimal metal volume fraction function with five (◻), six (▵), and nine (엯) design variables selected. The solution with the modified Mori-Tanaka scheme with six design variables case is also shown (▿).

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Figure 12

Temperature profiles along the radial direction for the initial and final material design for the stress minimization problem with six design variables (test B)

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Figure 13

History of the objective function for the stress minimization problem with six design variables (test B)

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Figure 14

The best metal volume fraction profile for the stress minimization problem with no inner pressure. Initial guess (∗) and the optimal solutions with six design variables with the ROM (엯) or the modified Mori-Tanaka method (▿).

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Figure 15

Initial and final profile of the metal volume fraction for the stress minimization problem with inner pressure. The designs of tests G and K provide the best values of the objective function. Convergence in terms of the number of design variables is shown by comparing test K and L final profiles.

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