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Materials displaying microstructures with random porous distribution are found in a myriad of ways in nature and engineering applications: foams, bones, reservoirs, architectured materials with defects, etc. Minimally constrained multiscale models (Reuss-like for solid mechanics) losses the well-posedness when voids reach the boundary in a non-periodic fashion (situation which the convenient periodic boundary conditions cannot be applied). Generally, only linear boundary models (Voigt-like) can be straightforwardly applied in such cases, notwithstanding the well-known drawback of overstimation of effective stiffness. Hence, we investigate a novel multiscale mechanical formulation based on a proper formulation of a generalised minimally constrained multi-scale model that remedies this issue for general combination of solid and void parts on the boundary, as recently proposed in [1]. The Method of Multiscale Virtual Power serves as the foundation for developing the proposed model, with particular focus on reformulating the gradient homogenisation formula [6]. This reformulation leads to the development of not only a novel Minimally Constrained Kinematical Multiscale Model but also inspires a broader family of multiscale models suited for practical scenarios. The key ingredient is the introduction of an averaged boundary normal vector modifying the point-wise normal appearing in the non-local kinematical constraints arising in fibre networks [2]. Although special attention is given to the classical first-order theory of solid mechanics at continua, the present method can be straightforwardly adapted ot other applications, including strain-gradient of architected materials with defects [4], homogenisation of fibres networks [2] and fluid saturated porous media [5]. Implementations aspects of such generalised kinematical constraint will be discussed in the context of FEniCSx-based opensource implementation [3]. Several numerical examples demonstrating the potentialities of such method are also addressed.

[1] Blanco P.J., Sanchez P.J., Rocha F., Toro S., Feijoo R.A., 2023, A consistent multiscale mechanical formulation for media with randomly distributed voids, 2023, International Journal of Solids and Structures, 283, pp. 112494. 2

[2] Rocha, F.F., Blanco, P.J., Sanchez, P.J. and Feijoo, R.A., 2018. Multi-scale modelling of arterial tissue: Linking networks of fibres to continua. Computer Methods in Applied Mechanics and Engineering, 341, pp.740-787.

[3] Rocha F., 2023, micmacsfenics. Zenodo. doi: https://zenodo.org/records/7643777.

[4] dos Santos W. , Lopes I.A.R, Pires F.M.A., Proença S.P.B., 2023, a Second-order multi-scale modelling of natural and architected materials in the presence of voids: Formulation and numerical implementation, Computer Methods in Applied Mechanics and Engineering, 416, pp. 116374

[5] Anonis R.A., Mroginski J.L., Sanchez P.J., 2024, Multiscale formulation for materials composed by a saturated porous matrix and solid inclusions, Computer Methods in Applied Mechanics and Engineering, 429, pp. 117162.

[6] P.J. Blanco, P.J. Sanchez, E.A. de Souza Neto, and R.A. Feijoo, 2016, Variational foundations and generalized unified theory of RVE-based multiscale models. Archives of Computational Methods in Engineering, 23:191–253.