Mathematical Sciences: Some Mathematical Problems Arising from Studies on Measurement of Stress and Texture in Materials

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Man 9404484 This project has the following two objectives: (A) To study the well-posedness of the boundary value problem pertaining to using the hole-drilling strain-gage method for the measurement of residual stress in a thin plate. This boundary value problem will be formulated in the context of the general theory of linear elasticity with initial stress. Particular attention will be paid to examining whether the size of the drilled hole would have any effect on the well- posedness. (B) To develop a mathematical theory on the radiation, focusing, and multiple arrival of SHo waves in anisotropic thin sheets. Accurate determination of phase velocities at different propagation directions is required when we use SHO waves for the evaluation of stress, texture, and elastic constants of anisotropic sheets. Measurement of time delay of signals by cross correlation does not deliver phase velocity. The present study will lay the theoretical groundwork for the determination of phase velocities from measured data. With applications to composite laminate plates in mind, provisions will be made to allow for strong anisotropy of the sheets. This project comprises two studies: (A) Mathematical foundations of the hole-drilling strain-gage method for the evaluation of residual stress. The general theory of linear elasticity with initial stress, which studies small deformations from a prestressed configuration, provides a natural setting for the analysis of the hole-drilling test. The present analysis will be conducted in its context. The current standard procedure for conducting the hole-drilling test will be reexamined in light of findings in this mathematical study. (B) . The present study will lay the theoretical groundwork for the determination of phase velocities from measured data. With applications to composite laminate plates in mind, provisions will be made to allow for strong anisotropy of the sheets.