A bi-Helmholtz type of two-phase nonlocal integral model for buckling of Bernoulli-Euler beams under non-uniform temperature

Abstract

It is well-acknowledged by the scientific community that Eringen’s nonlocal integral theory is not applicable to nanostructures of engineering interest due to conflict between equilibrium and constitutive requirements. In this paper, a well-posed two-phase nonlocal integral elasticity with the bi-Helmholtz kernel is developed to study the size-dependent buckling response of Bernoulli-Euler beams under non-uniform temperatures. The governing equation is derived by invoking the variational principle of virtual work, and the temperature effect is equivalent to the thermal load along the axial direction, which is determined by nonlocal heat conduction. The two-phase nonlocal integral constitutive equation is transformed into a differential one equipped with four constitutive boundary conditions, and then exact solutions for the buckling loads of the beam with various boundary edges are obtained. Numerical results are validated by comparing them with those from local elasticity. Moreover, the effects of parameters related to the two-phase nonlocal elastic model and the nonlocal heat conductive model are investigated.

Keywords:

• Bi-Helmholtz kernel
• buckling loads
• exact solution
• nonlocal heat conduction
• two-phase nonlocal elasticity

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No. 11672131) and the Priority Academic Program Development of Jiangsu Higher Education Institutions. Zhang appreciates the support of the China Scholarship Council (CSC).