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

[9404484]本项目有以下两个目的：(A)研究用钻孔应变计法测量薄板残余应力的边值问题的适定性。这个边值问题将在具有初始应力的线弹性的一般理论的背景下表述。将特别注意检查钻孔的大小是否会对井位有任何影响。(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。在利用SHO波评价各向异性薄片的应力、织构和弹性常数时，需要准确测定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能得到相速度。本文的研究为根据实测数据确定相速度奠定了理论基础。考虑到复合层压板的应用，将作出规定，以允许板材的强各向异性。本课题包括两个方面的研究：(A)钻孔应变计法评估残余应力的数学基础。具有初始应力的一般线弹性理论研究预应力结构的小变形，为钻孔试验的分析提供了一个自然的背景。本分析将在此背景下进行。根据这一数学研究的结果，将重新审查进行钻孔试验的现行标准程序。(B)。本文的研究为根据实测数据确定相速度奠定了理论基础。考虑到复合层压板的应用，将作出规定，以允许板材的强各向异性。

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.