Tuning the load-deflection curve of beams via densely distributed edge cracks: Analytic modeling and solutions

Abstract

Tuning load-deflection curves is of significant importance in advanced structural design. In this paper, by introducing densely distributed edge cracks artificially, the N-shaped load-deflection curve with snap-through instability is realized in the axially pre-compressed, simply supported beam made of tough material under transverse uniform pressure. A simple mechanics model is established based on the Bernoulli-Euler beam theory with large rotations, which enables a satisfactory description of quasi-static opening and closing of the edge cracks. A second-order nonlinear differential equation that governs the quasi-static evolution of the envelope of densely distributed edge cracks is derived, which exceptionally yields an analytic solution expressed in terms of incomplete elliptic integrals. Computational results show that the realization of N-shaped pressure-deflection curves is mainly attributed to the competition between edge-crack opening induced softening and axial force releasing induced stiffening. Parametric analysis indicates that the shape of pressure-deflection curve is sensitive to the edge-crack length, which provides a useful means to tune the load-deflection curve of beams made of tough materials on the structural level.

Keywords:

• Analytical solution
• crack
• load-displacement curve
• snap-through instability
• structural softening

Additional information

Funding

J. Y. acknowledges the support from the National Natural Science Foundation of China (Grant Nos. 11772272 and 11402133) and the support from the Fundamental Research Funds for the Central Universities (Grant Nos. 2682019LK06 and 2682019LXCGKY001). Y. H. acknowledges the support from the National Natural Science Foundation of China (Grant Nos. 11972027 and 11502128). X. L. acknowledges the support from the National Natural Science Foundation of China (Grant No. 11702258).