Gradient elasticity solutions of 2D nano-beams

Özer T.

Applications in Engineering Science, vol.15, 2023 (Scopus) identifier

  • Publication Type: Article / Article
  • Volume: 15
  • Publication Date: 2023
  • Doi Number: 10.1016/j.apples.2023.100140
  • Journal Name: Applications in Engineering Science
  • Journal Indexes: Scopus
  • Keywords: Analytical solutions, Bi-harmonic differential equations, Cantilever beams, Micro-size effects, Nonlocal elasticity, One and two-dimensional nano-beams, Stress and strain gradient elasticity
  • Istanbul Technical University Affiliated: Yes


In this study, the exact analytical solutions of a two-dimensional linear homogeneous isotropic nano-beam in gradient elasticity are studied. Four different types of two-dimensional cantilever beams and related boundary conditions are considered. The cases are a cantilever beam under a concentrated force at the end, a cantilever beam under a uniform load, a propped cantilever beam under a uniform load, and a fixed-end beam under a uniform load. The two-dimensional stress gradient fields are investigated and obtained from the analytical solutions of a linear second-order partial differential equation written in terms of the classical and the gradient Airy stress functions. Additionally, the micro-size effects in the displacement components for different loads and support conditions for the two-dimensional cantilever beams by using strain gradient elasticity theory are investigated. Furthermore, for one-dimensional Euler–Bernoulli beam model, the associated stress and strain elasticity solutions are obtained from two-dimensional analytical solutions. The graphical presentations of the exact closed-form solutions are provided and discussed.