Designing composite microstructures with targeted properties
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We present a numerical method to find specific composite microstructures with targeted effective properties. The effective properties that may be prescribed are quite diverse and include transport, mechanical, and electromagnetic properties, as well as properties associated with coupled phenomena, such as piezoelectric and thermoelectric coefficients. We formulated the target problem as an optimization problem. To illustrate our general target optimization technique, we have successfully found two-phase composite microstructures having specified effective electrical or thermal conductivities at fixed volume fractions. The method can also be used to design microstructures with multifunctional characteristics.
I. INTRODUCTION
An important goal of materials science is to have exquisite knowledge of structure/property relations in order to design material microstructures with desired properties and performance characteristics. Although this objective has been achieved in certain cases through trial and error, a systematic means of doing so is currently lacking. For certain physical phenomena at specific length scales, the governing equations are known and the only barrier to achieving the aforementioned goal is the development of an appropriate method to attack the problem. The purpose of this article is to introduce a methodology to design at will composite microstructures with targeted effective properties under required constraints. In general terms, this is accomplished by formulating the task as an optimization problem that we call target optimization. Target optimization is an adaptation of traditional structural optimization techniques.1,2 Specifically, an initial microstructure is allowed to evolve to the targeted state by extremizing an appropriately defined objective function. The types of effective properties that we can address are quite general and include transport, mechanical, and electromagnetic properties, as well as properties associated with coupled phenomena, such as piezoelectric and thermoelectric coefficients. To illustrate our general target optimization technique, we find two-phase composite microstructures in two dimensions having specified effective electrical or thermal conductivities at fixed volume fractions. In the first example, we use the geometric-mean formula for the targeted effective conductivity. In the second example, we a)
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II. LOCAL AND HOMOGENIZED EQUATIONS
Consider a two-phase composite material consisting of a phase with a property K1 and volume fraction 1 and another phase with a property K2 and volume fraction 2 (⳱1 − 1). The property Ki is perfectly general: It may represent a transport, mechanical, or electromagnetic property, or properties associated with coupled phenomena, such as piezoelectricity or thermoelectricity. For steady-state situations, the generalized flux F(r) at some local position r in the composite obeys the following conservation law in the phases: ⵜ ⭈ F(r) ⳱ 0
.
(1)
In the case of electrical conduction and elasti
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