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Stress trajectories element method for stress determination from discrete data on principal directions

机译：从主方向上的离散数据确定应力的应力轨迹元素方法

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摘要

This paper presents a Trefftz-element numerical method for the reconstruction of stress trajectories and the determination of full stress tensors in two-dimensional elastic bodies from discrete data on principal directions. The conventional techniques cannot be used because neither displacements nor tractions are specified on the boundary. The proposed approach involves the subdivision of the domain into smaller subdomains and the introduction of the Cauchy integrals with unknown densities on element boundaries in order to approximate complex potentials within the elements. For polynomial approximations of the densities, this leads to piecewise polynomial approximations for the complex potentials within the entire domain and, therefore, all elasticity equations are automatically satisfied as in the Trefftz method. Continuity of the complex potentials is forced at the collocation points, which forms the first group of equations. The second group is formed by satisfying the data on principal directions known in some locations. All these equations are homogeneous; therefore, it is assumed that the average value of the maximum shear stresses at data points is unity. This guarantees the existence of a non-trivial solution of the system; however it addresses the non-uniqueness of the reconstruction of the full stress tensor. The technique is validated by reconstructing stress trajectories and determining maximum shear stresses from synthetic and photoelasticity data. It has been applied to reconstruction of tectonic stresses in the Australian region and the results were compared with previous approaches.
机译：本文提出了一种Trefftz元素数值方法，用于从主方向上的离散数据重建应力轨迹并确定二维弹性体中的全应力张量。由于在边界上既未指定位移也未指定牵引力，因此无法使用常规技术。拟议的方法包括将域细分为较小的子域，并引入元素边界上密度未知的柯西积分，以逼近元素内的复杂电势。对于密度的多项式逼近，这导致整个域内复数电位的分段多项式逼近，因此，像Trefftz方法一样，自动满足所有弹性方程。复势的连续性在配置点处被强制，这形成了第一组方程。通过满足某些位置已知的主要方向上的数据来形成第二组。所有这些方程都是齐次的。因此，假设数据点处的最大剪应力的平均值为1。这保证了系统的平凡解决方案的存在；然而，它解决了全应力张量重建的非唯一性。通过重建应力轨迹并从合成和光弹性数据确定最大剪切应力来验证该技术。它已被用于重建澳大利亚地区的构造应力，并将结果与​​以前的方法进行了比较。

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