Combining the method of boundary states and the Lindstedt–Poincaré method in geometrically nonlinear elastostatics

Abstract

The study considers a statement of geometrically non-linear problems of isotropic theory of elasticity for a simply connected bounded body. The Lindstedt–Poincaré method reduces the problem to a weakly non-linear variation containing a small parameter. The solution is then effected at each step of the perturbations method through the means of the method of the boundary states with perturbations (MBSP). Each iteration is accompanied by a generation of fictitious volume forces of a polynomial nature. The study proposes a strict particular solution and a solution of an elasticity problem for iteration. The approach is also applicable to thermoelasticity problems. The method is tested on classical axially symmetrical bodies, i.e., cylinders subjected to uniaxial tensioning or twisting. The study confirms the applicability of flat section hypotheses to linear engineering calculations. Nonlinear distortions mostly happen at the first iteration, followed by a weak correction. We also provide a calculation for and an illustration of a stress-strain state (SSS) of a semicylinder under unbalanced axial loads and a concentration of compression stresses in extended bodies’ middle areas that are remote from load-free surfaces.