A Novel Efficient Prediction Method for Microscopic Stresses of Periodic Beam-like Structures

Abstract

This paper presents a novel superposition method for effectively predicting the microscopic stresses of heterogeneous periodic beam-like structures. The efficiency is attributed to using the microscopic stresses of the unit cell problem under six generalized strain states to construct the structural microscopic stresses. The six generalized strain states include one unit tension strain, two unit bending strains, one unit torsion strain, and two linear curvature strains of a Timoshenko beam. The six microscopic stress solutions of the unit cell problem under these six strain states have previously been used for the homogenization of composite beams to equivalent Timoshenko beams (Acta. Mech. Sin. 2022, 38, 421520), and they are employed in this work. In the first step of achieving structural stresses, two stress solutions concerning linear curvatures are transformed into two stress solutions concerning unit shear strains by linearly combining the stresses under two unit bending strains. Then, the six stress solutions corresponding to six generalized unit beam strains are combined together to predict the structural microscopic stresses, in which the six stress solutions serve as basic stresses. The last step is to determine the coefficients of these six basic stress solutions by the principle of the internal work equivalence. It is found that the six coefficients, in terms of the product of the inverse of the effective stiffness matrix and the macroscopic internal force column vector, are the actual generalized strains of the equivalent beam under real loads. The obtained coefficients are physically reasonable because the basic stress solutions are produced by the generalized unit strains. Several numerical examples show that the present method, combining the solutions of the microscopic unit cell problem with the solutions of the macroscopic equivalent beam problem, can accurately and effectively predict the microscopic stresses of whole composite beams. The present method is applicable to composite beams with arbitrary periodic microstructures and load conditions.