The consistency of the nonlocal strain gradient integral model in size-dependent bending analysis of beam structures

Despite the extensive applications of the nonlocal strain gradient theory to predict the size-effect of beam-like structures, the inconsistencies of its differential form have been uncovered recently. For bounded beam structures, there are additional constitutive boundary conditions should be taken into account to ensure the strict equivalence between the differential constitutive relation and the originally integral ones. Such an improvement would lead the problems to have more mandatory boundary conditions than those required, and therefore, there must be a trade-off between these boundary conditions. A strategy by omitting the higher-order (non-standard) boundary conditions has been presented recently and from which consistently size-effects can be obtained. However, it was pointed out that this treatment no longer fully meets the requirements of the energy-variational principle. In this paper, an alternative methodology by directly using the originally integral model is proposed for the first time. The advantages of such a strategy are pertinently evidenced by investigating Euler-Bernoulli curved beams made of functionally graded materials. The Laplace transform technique is applied to solve the integral constitutive equations and differential governing equations, which is an original contribution of this paper. In contrast to the differential forms, the originally integral model can satisfy energy conservation well. Meanwhile, adopting specific higher-order boundary conditions, numerical results show consistently stiffening and softening effects for all boundary edges and loading conditions.