Nonlocal continuum formulations exist in integral formats and differential formats. The latter, also known as gradient-enriched continua, have successfully been applied in elasticity, plasticity and damage and provide a robust framework to analyse size effects and dispersive waves. Moreover, gradient enhancement can be used to remove singularities from elastic fields as well as guarantee well-posedness in the modelling of post-peak phenomena. In this paper, the focus will be on novel formulations for gradient elasticity.
Aifantis and coworkers suggested a simple format of gradient elasticity in which the usual stress-strain is augmented with an additional term, namely the Laplacian of the strain. This then leads to a fourthorder differential equation in terms of the displacement and, hence, C1 continuity requirements in case of a numerical implementation. However, it is possible to split the fourth-order equations into two sets of second-order equations, whereby the first set coincides with the equations of classical elasticity and the second set comprises a set of diffusion-type equations that introduce the gradient enrichment. Hence, a staggered solution algorithm is obtained, whereby the displacements of classical elasticity are computed first and then used as input for the second set of equations in order to compute the gradient-enriched displacements.
The staggered, displacement-based approach will be scrutinised together with two alternative formulations of gradient elasticity: (i) a staggered, strain-based approach, and (ii) another strain-based approach that can be derived from Pade approximations of the previous method. The three approaches will be presented with their boundary conditions, and it will be verified whether all singularities are removed from the strain field. Also, size effect tests will be reported.
In the accompanying paper, the displacement-based approach will be used in the formulation of nonlinear gradient models.