Keywords: gradient elasticity, elasticity with microstructure, couple stress, higher-order continua, finite elements, penalty methods.

Conventional constitutive models usually ignore the fact that materials have microstructure, assuming that the dimensions of the modelled structure are much larger than any characteristic microstructural length (e.g. soil grain size), and hence that the effect of the latter is insignificant. Nevertheless there are certain conditions where the above assumption breaks down, and microstructure dominates the mechanical behaviour of a material. In such cases conventional constitutive models, like elastoplasticity, are not appropriate, and one should resort to continuum and constitutive descriptions that take microstructure into account.

A general theoretical framework for elastic materials with microstructure was developed by Mindlin. His Elasticity with Microstructure postulates two distinct levels of deformation, one at the micro- and one at the macro-scale, introducing twelve independent degrees of freedom: three macro-displacements and nine components of micro-deformation. However, it is the special case of vanishing relative deformation, i.e. co-incidence of the micro-deformation with the macro-deformation gradient, that has mostly attracted attention. In that special case, known as Gradient Elasticity, the strain energy density becomes a function of the strains and their gradient.

In this paper we present a finite element discretisation of Mindlin's Elasticity with Microstructure, interpolating the displacement and micro-deformation fields. Conditions of plane strain are assumed, however the formulation is general and the construction of three-dimensional elements equivalent to the ones presented is straight-forward. Appropriate numerical integration schemes are established for each element, so that no zero energy modes exist and all elements pass the single-element and patch tests. As an example the developed elements are used to model the one-dimensional compression of a material layer, and they are shown to accurately capture the resulting boundary layer.

Subsequently, we show how the developed numerical approach can also be used, in a rather simple manner, to produce approximate solutions of boundary value problems of Gradient Elasticity. In particular, appropriate choices of material parameters are shown that essentially convert the presented numerical discretisation of Elasticity with Microstructure to a penalty method approach for Gradient Elasticity. The advantage of the proposed approach is that it is much simpler compared to the C¹ continuous shape functions, mixed formulations, Lagrange multipliers or meshless methods that are normally employed for Gradient Elasticity. As an example, shearing of an infinite material layer of a gradient elastic solid is modelled, showing how the developed elements are able to capture the resulting non-uniformities of the strain profile with accuracy of a fraction of a percent.

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