| Summary: | Optical component performance increasingly demands materials with tailored properties. Optical systems see applications where ambient conditions can drastically reduce performance. Optics used in space, for example, may undergo severe changes in temperature, which results in large thermally induced stresses and distortions. To minimize these thermal effects, it was desired to manipulate the coefficient of thermal expansion (CTE) within the material. However, wholesale reductions in CTE may not be optimum since synthetic manipulation of CTE often leads to undesirable effects on other material properties, such as strength or ductility. Consequently, there was interest in distributing the CTE/material property trades over the optical structure to achieve optimum results for competing requirements. Performing such studies relied heavily on Finite Element Analysis (FEA) programs coupled with closed form solutions that give basic understanding of CTE design. To begin fulfilling this need, it is presented here first a revisit of the problem of two infinitely long nested cylinders. From this study, design equations were developed that related the dependent variables of stress, strain, and displacement to the elastic modulus, CTE, and temperature change of the system. The equations provide insight into how the dependent variables were affected by the layer CTEs. The analysis was then extended to three nested cylinders. Extrapolation of this approach to N nested cylinders is shown. This work provided a starting point for analysis of optical cylindrical troughs. To strengthen the analytical techniques for the cylindrical troughs, the problem of the bi-metallic strip was also revisited. Design equations were likewise developed that related the radius of curvature and surface distortion to the elastic modulus, CTE, and temperature change of the system. The analysis was extended to a tri-metallic strip and eventually a strip of N layers. Each layer in the model represents a material differing in CTE of neighboring layers. Finally, a system was observed that was traceable to an optical backing structure, namely a system of several nested cylindrical troughs. Design equations and observations of CTE distribution as well as quantifying data for this system are provided.
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