Modified Timoshenko beam model for bending behaviors of layered materials and structures

Abstract

Layered materials and structures (LMS), such as van der Waals two-dimensional (2D) layered materials and nacre-like layered structures, often exhibit highly anisotropic mechanical properties, i.e., strong in in-plane directions but weak in out-of-plane direction. Despite the strong anisotropy in their mechanical properties, Timoshenko beam model (TBM) is usually used to describe the bending deformation of LMS. We note, however, that there are two fundamental issues in using TBM to describe LMS: First, the stiffness of LMS approaches zero when the interlayer shear modulus G approaches zero; and second, the first derivative of deflection becomes discontinuous at the point of concentrated force. Clearly, both are not true for LMS. In this work, by introducing the bending energy of monolayer into the potential energy of TBM, we develop a modified Timoshenko beam model (MTBM), which is able to not only address these two issues, but also correctly predict the bending stiffness of LMS without any fitting parameters. Our analysis shows that the bending behaviors of LMS are determined by a dimensionless parameter λL, where L is the length of the beam and λ=kGA∕D0+kGA∕(nDbend), where, kGA and D0 are, respectively, the shear and bending rigidity of the beam cross-section, Dbend is the bending rigidity of monolayer, and n is the number of layer. When λL→0, the MTBM degenerates to the multi-beam model with bending stiffness of nDbend; while it degenerates to the TBM whenλL→∞. Furthermore, if kGA is much larger than D0, both MTBM and TBM degenerate to the classical Euler–Bernoulli beam model. We further perform molecular dynamics simulations, finite element simulations and experiments to validate the MTBM. Based on the MTBM, a couple of interesting applications of LMS are also demonstrated. Hence, the MTBM presented here captures the necessary intrinsic deformation modes of LMS and provides an accurate tool for the prediction and optimization of the mechanical properties of LMS.