Tuning the elastic buckling of a soft block with graded material stiffness. (1st March 2023) - Chen, Lingling; Xing, Xinyu; Yang, Shengyou
Abstract: Elastic buckling of slender structures is of high interest in engineering since the buckling could bring many unintended and disastrous consequences. The critical load of buckling in nonlinear elasticity can be highly affected by either the structure's geometry or the material properties. This paper studies the buckling behaviors of an elastic block of soft materials with graded modulus at finite deformation. We give the total potential energy of the conservative system and formulate a boundary-value problem and its incremental forms. We solve the buckling problem analytically for an elastic block of neo-Hookean materials. Though our problem is different from Euler's buckling, our results agree well with the Euler's formula in the case of slender columns. In addition, our analysis can be used to predict the buckling load of short columns. Since our loading device controls the displacement of the block, the critical load is computed by using the critical stretch at which the loaded block begins to buckle. Depending on the varied forms of the moduli, the graded modulus can tune the critical load. The forms of the linear and exponential functions of the Young moduli of the material can only reduce the critical load. However, the quadratic form can either reduce or increase the critical load by about 50%, making a weaker or stronger structure without changing its geometry. This paper contributes to our understanding of the buckling behaviors of elastic structures with graded moduli. Highlights: A closed-form analysis of the buckling of a finite block of soft materials. The energy formulation and the complete linear bifurcation analysis are given. A system of PDEs and ODEs are solved analytically and numerically. The graded modulus can either reduce or increase the critical load by about 50%.