Anisotropy in strain gradient elasticity: Simplified models with different forms of internal length and moduli tensors

Anisotropy of centro-symmetric (first) strain gradient elastic materials is addressed and the role there played by the dual gradient directions (i.e. directions of strain gradient and of double stress lever arm) is investigated. Anisotropy manifests itself not only through the classical fourth-rank elasticity tensor C (21 independent constants) in the form of moduli anisotropy, but also through a sixth-rank elasticity tensor B (171 independent constants) in a unified non-separable form as compound internal length/moduli anisotropy. Depending on the microstructure properties, compound anisotropy may also manifest itself in a twofold separable form through a decoupled tensor B=LC, consisting of a moduli anisotropy attached to C, and an internal length anisotropy attached to a symmetric positive definite second-rank internal length tensorL (6 independent constants). Tensor L confers a tensorial character to the concept of internal length and is the basis of a mutual one-to-one relationship between the dual gradient directions. Indeed, at every point of a material with separable compound anisotropy, a characteristic ellipsoid exists whereby the generic radius and the associated normal to the ellipsoidal surface constitute dual gradient directions. Therefore, at every point, there are at least three mutually orthogonal directions in each of which the dual gradient directions are collinear, but infinite in number when the ellipsoid is rotational or even spherical. With restrictions on dual gradient directions, simplified models with a reduced number of independent constants are derived, namely: i) Gradient symmetric materials which obey a reciprocity principle and generally exhibit a non-separable compound anisotropy (21 + 126 independent constants in total); ii) Materials with separable compound anisotropy, a subclass of gradient symmetric materials (21 + 6 independent constants in total). A typical boundary-value problem for materials with separable compound anisotropy is presented and a way to transform the three governing fourth-order PDEs into six second-order ones is suggested. An application to an Euler–Bernoulli beam is analytically worked out. [Display omitted] •Macro-scale and micro-scale anisotropies in strain gradient elasticity.•Influence of dual gradient directions on micro-scale anisotropy.•Notion of internal length as a tensor carrying in collinearity classes.•Internal length anisotropy locally described by a characteristic ellipsoid.•Simplified