a b s t r a c tIn this paper a unified integro-differential nonlocal elasticity model is presented and its use in the bending analysis of Euler-Bernoulli beams is illustrated. A general (for an elastic continuum) finite element formulation for the two-phase integro-differential form of Eringen nonlocal model is provided. The equations are specialized for the case of the Euler-Bernoulli beam theory. Several numerical examples, including the paradoxical cantilever beam problem that eluded other researchers, are provided to show how the present nonlocal model affects the transverse displacement of beams. The examples show that Eringen nonlocal constitutive relation has a softening effect on the beam, except for the case of the simply supported beam. A brief discussion on the applicability of the integro-differential model to other problems is also presented. Finally, the transition from the stiffened nonlocal simply supported beam to the softened nonlocal clamped beam is also investigated.

In this paper, the governing equations and finite element formulations for a microstructure‐dependent unified beam theory with the von Kármán nonlinearity are developed. The unified beam theory includes the three familiar beam theories (namely, Euler‐Bernoulli beam theory, Timoshenko beam theory, and third‐order Reddy beam theory) as special cases. The unified beam formulation can be used to facilitate the development of general finite element codes for different beam theories. Nonlocal size‐dependent properties are introduced through classical strain gradient theories. The von Kármán nonlinearity which accounts for the coupling between extensional and bending responses in beams with moderately large rotations but small strains is included. Equations for each beam theory can be deduced by setting the values of certain parameters. Newton's iterative scheme is used to solve the resulting nonlinear set of finite element equations. The numerical results show that both the strain gradient theory and the von Kármán nonlinearity have a stiffening effect, and therefore, reduce the displacements. The influence is more prominent in thin beams when compared to thick beams.

No abstract

No abstract

In this paper two different nonlinear elasticity theories that account for (a) geometric nonlinearityand (b) microstructure-dependent size effects are revisited to establish the connection betweenthe two theories. The first theory is based on modified couple stress theory of Yang et al. [1]and the second one is based on Srinivasa–Reddy gradient elasticity theory [2]. The modified couplestress theory includes a material length scale parameter that can capture the size effect in a material.The gradient elasticity theory was developed for finitely deforming hyperelastic cosserat continuum,and it is a generalization of small deformation couple stress theories. The Srinivasa–Reddy theorycontains, as a special case, the first one. These two theories are used to derive the governing equationsof beams and plates. In addition, a discrete peridynamics idea as an alternative to the conventionalperidynamics is also presented.