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

男士9404484 本计画有以下两个目的：（一）研究钻孔应变片法量测薄板残余应力之边值问题适定性。 这个边值问题将在具有初应力的线性弹性力学的一般理论的范围内进行表述。 将特别注意检查钻孔的大小是否对适定性有任何影响。 (B)建立各向异性薄板中SHo波的辐射、聚焦和多次到达的数学理论。当我们使用SHO波来评估各向异性片材的应力、织构和弹性常数时，需要准确地确定在不同传播方向上的相速度。通过互相关测量信号的时间延迟不提供相速度。本研究为从实测数据中确定相速度奠定了理论基础。 考虑到复合材料层压板的应用，将做出规定以允许片材的强各向异性。 该项目包括两项研究：（一）钻孔应变片法评估残余应力的数学基础。线弹性力学的一般理论 初始应力， 研究 来自预应力配置的小变形为钻孔测试的分析提供了自然设置。 本分析将在这一背景下进行。 目前进行钻孔试验的标准程序将根据这项数学研究的结果进行重新审查。(B) .本研究将为确定 相速度 从 测量数据。 与应用程序 考虑到复合材料层压板， 以允许片材的强各向异性。

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.