The use of global displacement basis functions to solve boundary-value problems in linear elasticity is well established. No prior work uses a global stress tensor basis for such solutions. We present two such methods for solving stress problems in linear elasticity. In both methods, we split the sought stress $\sigma$ into two parts, where neither part is required to satisfy strain compatibility. The first part, $\sigma_p$, is any stress in equilibrium with the loading. The second part, $\sigma_h$, is a self-equilibrated stress field on the unloaded body. In both methods, $\sigma_h$ is expanded using tensor-valued global stress basis functions developed elsewhere. In the first method, the coefficients in the expansion are found by minimizing the strain energy based on the well-known complementary energy principle. For the second method, which is restricted to planar homogeneous isotropic bodies, we show that we merely need to minimize the squared $L^2$ norm of the trace of stress. For demonstration, we solve eight stress problems involving sharp corners, multiple-connectedness, non-zero net force and/or moment on an internal hole, body force, discontinuous surface traction, material inhomogeneity, and anisotropy. The first method presents a new application of a known principle. The second method presents a hitherto unreported principle, to the best of our knowledge.

翻译：使用全局位移基函数求解线弹性边值问题已有充分研究。然而，目前尚无研究采用全局应力张量基函数进行此类求解。本文提出两种求解线弹性应力问题的方法。在这两种方法中，我们将待求应力 $\sigma$ 分解为两部分，且无需任一部分满足应变协调条件。第一部分 $\sigma_p$ 是与载荷平衡的任意应力场，第二部分 $\sigma_h$ 则是无载荷体上的自平衡应力场。两种方法中，$\sigma_h$ 均通过采用前人发展的张量值全局应力基函数展开。在第一种方法中，根据已知的余能原理通过最小化应变能确定展开系数。第二种方法局限于平面均匀各向同性体，我们证明仅需最小化应力迹线的平方 $L^2$ 范数。为演示效果，我们求解了涉及尖角、多连通、内孔非零净力和/或力矩、体力、非连续表面牵引力、材料非均匀性及各向异性等八类应力问题。第一种方法展示了一个已知原理的新应用，而第二种方法据我们所知提出了此前未见报道的新原理。