Design of Functionally Graded Composite Structures for Control of Stress
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Design of Functionally Graded Composite Structures for Control of Stress Robert Lipton and Ani Velo1 Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A. 1 Mathematical Sciences Department, United States Military Academy, West Point, NY 10996, U.S.A.
ABSTRACT
In this paper a methodology is introduced for the design of structural components made from composite materials for the control of stress. A numerical method is developed for designing functionally graded materials with minimum stress in prescribed sub-domains inside the structure.
INTRODUCTION
Several technologically important applications have benefited from the use of functionally graded composite materials (FGMs) [1,2]. In many applications the discrete entities forming up the microstructure lie on scales significantly smaller than the characteristic length scale of the loading. Under these conditions, FGMs are modeled using effective thermo-physical properties that depend upon features of the underlying micro-geometry. The effective thermo-physical properties are given by effective constitutive laws relating average flux to average gradient [3]. The primary problem of design of a FGM is the determination of the optimal spatial dependence for the composition. This problem has generated much interest in the engineering community and is the topic of a rapidly developing literature [3-5]. Motivated by the applications, this work treats the problem of controlling the stress in a functionally graded material. In many applications it is of central importance to control stress concentrations inside composite structural components. Regions of large stresses are most likely the first to exhibit failure during service. For this type of problem the effect of microscopic stresses needs to be accounted for. The concept of an effective constitutive law is by its self not sufficient to capture the effect of micro-stresses. This requires new modeling beyond the notion of effective thermo-physical properties. The objective of this paper is to point out a new method that captures the effect of microscopic stresses and to present a methodology for the numerical design of FGMs for stress control.
U10.4.1
THEORY
This article treats the problem of controlling stress in functionally graded reinforced shafts subject to torsion loading. However the approach presented here applies equally well to the control of field gradients for heat conduction problems and problems of electro-statics. The concept of a functionally graded material is introduced by considering a reinforced shaft. Here the microstructure within the composite shaft consists of layers of two materials. Along the length of the shaft the material properties inside each layer are taken to be constant. While material properties change from layer to layer across the shaft cross section. This type of reinforcement geometry is often found in nature. One realization of this type of geometry is seen in tree rings. More generally the layer geometry may resemble that of a parquet floor wit
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