Energy consistency in homogenisation-based upscaling scheme for localisation in masonry shells

This paper presents an enhanced multi-scale framework for the failure of quasi-brittle thin shells as an improvement of the one proposed in Mercatoris and Massart (Int J Numer Methods Eng 85:1177–1206, 2011). The computational homogenisation-based multi-scale methodology is an attractive solution for heterogeneous materials when their characterisation becomes difficult because of complex evolving behaviour such as damage-induced anisotropy and localisation of degradation. An enhanced upscaling scheme for damage localisation in shell structures is proposed using a periodic computational homogenisation procedure and an energy equivalence between mesostructural material instabilities and aggregate macroscopic cracks. The structural cracking is treated by using embedded strong discontinuities incorporated in the shell formulation, the behaviour of which is deduced by an energetically consistent upscaling scheme. The effects of this energy equivalence are discussed based on results of multi-scale simulations of out-of-plane loaded masonry walls including flexural stair-case failure and compared to the results of direct numerical simulations. A good agreement is observed in terms of the load-bearing capacity and of associated energy dissipation. Based on the homogenisation procedure, the orientation of the structural-scale cracking is detected by means of an acoustic tensor-based failure detection adapted to shell kinematics. A multi-scale bifurcation analysis on a simple loading case is performed in order to discuss the selection of the cracking orientation based on energetic considerations.