a b s t r a c tIn this paper, we discuss various formats of gradient elasticity and their performance in static and dynamic applications. Gradient elasticity theories provide extensions of the classical equations of elasticity with additional higher-order spatial derivatives of strains, stresses and/or accelerations. We focus on the versatile class of gradient elasticity theories whereby the higher-order terms are the Laplacian of the corresponding lower-order terms. One of the challenges of formulating gradient elasticity theories is to keep the number of additional constitutive parameters to a minimum. We start with discussing the general Mindlin theory, that in its most general form has 903 constitutive elastic parameters but which were reduced by Mindlin to three independent material length scales. Further simplifications are often possible. In particular, the Aifantis theory has only one additional parameter in statics and opens up a whole new field of analytical and numerical solution procedures. We also address how this can be extended to dynamics. An overview of length scale identification and quantification procedures is given. Finite element implementations of the most commonly used versions of gradient elasticity are discussed together with the variationally consistent boundary conditions. Details are provided for particular formats of gradient elasticity that can be implemented with simple, linear finite element shape functions. New numerical results show the removal of singularities in statics and dynamics, as well as the size-dependent mechanical response predicted by gradient elasticity.

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Summary The use of higher-order strain-gradient models in mechanics is studied. First, existing second-gradient models from the literature are investigated analytically. In general, two classes of second-order strain-gradient models can be distinguished: one class of models has a direct link with the underlying microstructure, but reveals instability for deformation patterns of a relatively short wave length, while the other class of models does not have a direct link with the microstructure, but stability is unconditionally guaranteed. To combine the advantageous properties of the two classes of second-gradient models, a new, fourth-order strain-gradient model, which is unconditionally stable, is derived from a discrete microstructure. The fourthgradient model and the second-gradient models are compared under static and dynamic loading conditions. A numerical approach is followed, whereby the element-free Galerkin method is used. For the second-gradient model that has been derived from the microstructure, it is found that the model becomes unstable for a limited number of wave lengths, while in dynamics, instabilities are encountered for all shorter wave lengths. Contrarily, the secondgradient model without a direct link to the microstructure behaves in a stable manner, although physically unrealistic results are obtained in dynamics. The fourth-gradient model, with a microstructural basis, gives stable and realistic results in statics as well as in dynamics.Keywords Strain-gradient Models, Higher-order Continuum, Microstructure, Wave Propagation, Stability IntroductionClassical continuum theories assume that the stresses in a material point depend only on the first-order derivative of the displacements, i.e. on the strains, and not on higher-order displacement derivatives. As a consequence of this limitation on the kinematic field, a classical continuum is not always capable of adequately describing heterogeneous phenomena. For instance, unrealistic singularities in the stress and/or strain field may occur nearby imperfections. Furthermore, severe problems in the simulation of localisation phenomena with classical continua have been encountered, such as loss of well-posedness in the mathematical description and pathological mesh-dependence in numerical simulations (see [25] for an overview). To avoid these types of deficiencies, it has been proposed to include higher-order strain gradients into the constitutive equations, so that the defects of the classical continuum may be successfully overcome, [4,12,17,19,22,24,25]. The second-order strain gradients that are normally used for these purposes introduce accessory material parameters that reflect the microstructural properties of the material. However, the second-gradient term is often postulated, rather than derived from the microstructure. Hence, this class of models can be denoted as phenomenological.

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