Integro-differential nonlocal theory of elasticity

The second-order integro-differential nonlocal theory of elasticity is established as an extension of the Eringen nonlocal integral model. The present research introduces an appropriate thermodynamically consistent model allowing for the higher-order strain gradient effects within the nonlocal theory of elasticity. The thermodynamic framework for third-grade nonlocal elastic materials is developed and employed to establish the Helmholtz free energy and the associated constitutive equations. Establishing the minimum total potential energy principle, the integro-differential conditions of dynamic equilibrium along with the associated classical and higher-order boundary conditions are derived and comprehensively discussed. A rigorous formulation of the third-grade nonlocal elastic Bernoulli-Euler nano-beam is also presented. A novel series solution based on the modified Chebyshev polynomials is introduced to examine the flexural response of the size-dependent beam. The proposed size-dependent beam model is demonstrated to reveal the stiffening or softening flexural behaviors, depending on the competitions of the characteristic length-scale parameters. The higher-order gradients of strain fields are illustrated to have more dominant effects on the beam stiffening.