Solution of planar elastic stress problems using stress basis functions

• Published in: arXiv.org 2023-04
• Main Authors: Tiwari, Sankalp, Chatterjee, Anindya
• Format: Article
• Language: English
• Subjects: Anisotropy; Basis functions; Boundary value problems; Elasticity; Inhomogeneity; Mathematical analysis; Principles; Strain energy; Stress distribution; Tensors; Thermal expansion
• ISSN: 2331-8422

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.