One-dimensional stress-driven nonlocal integral model with bi-Helmholtz kernel: Close form solution and consistent size effect

Abstract

In this paper, stress-driven nonlocal integral model with bi-Helmholtz kernel is applied to investigate the elastostatic tensile and free vibration analysis of microbar. The relation between nonlocal stress and strain is expressed as first type of Fredholm integral equation which is transformed to first type of Volterra integral equation. The general solution to the axial displacement of nonlocal microbar is obtained through the Laplace transformation with four unknown constants. Taking advantage of boundary and constitutive constraint equations, one can obtain the exact tensile displacements of microbar under different boundary and loading conditions, and the nonlinear characteristic equations about vibration frequency of clamped-free and clamped-clamped nonlocal microbars. Numerical results show that the nonlocal microbar model can be degraded to local bar model when the nonlocal parameters approach to 0, and a consistent toughening response for elastostatic tension and free vibration can be obtained for different boundary and loading conditions.