A single-domain spectral solver for spatially nonsmooth differential equations of quasistatic solid mechanics in polar coordinates

Abstract

In the present work, a spectral solver is developed for integration of certain differential equations of solid mechanics, namely static stress equilibrium in composite materials, described in cylindrical or spherical polar coordinates. The spectral approach is encompassed in approximating the displacement field using expansion into a series of Chebyshev polynomials in the radial coordinate and complex exponents in the angular direction. Consequently, differential operators in real space become algebraic operators in spectral space. The spatial heterogeneity and metric non-flatness pertinent to polar geometry are addressed by an iterative strategy, employing both second-order and first-order iterative solvers. The essence of the new contribution is in addressing the difficulty posed by the inherent nonsmoothness present in composite materials and the polar singularity. The interplay of the two produces instability, which is resolved in the proposed approach, specifically by using a new efficient linesearch algorithm, appropriate for the studied class of problems. The method is illustrated by analysis of 1D and 2D linear-elastic and linear-elastic–perfectly plastic response of composites to prescribed radial surface displacement. The developed method allows performing stress homogenization on polar representative volume elements, which has its conceptual advantages, while allowing similar runtime (for sufficient computing resources and an iterative strategy) to the one exhibited by spectral analysis in Cartesian coordinates.