Consider a composite material consisting of a periodic (with a period d) collection of alternating flat plates (laminates) with two different values of thermal conductivity. Suppose that thermal conductivity is equal to 1 in the first set of plates and it is equal to c in the second set of plates. We focus on finding the effective conductivity of the laminated composite when the parameter c is large and the parameter d is small.
This problem can be solved by determining the partial differential equation governing the homogenized temperature distribution. Rather than go this route, we choose to formulate the problem in a variational form. Then the conductivity of a composite for the given values c and d can be determined by minimizing an appropriate energy functional. We use the theory of Γ-convergence to find the asymptotic limits of both minimizers and their respective minimum energy values as (c–1, d) approaches (0, 0). We show that these limits are independent of the direction in which (c–1, d) approaches (0, 0) in the parameter space. Further, we show that the minimum energy values do not exceed the minimum value of the limiting energy functional.
|School:||The University of Akron|
|School Location:||United States -- Ohio|
|Source:||MAI 57/05M(E), Masters Abstracts International|
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