The aim of this work is, hence, to adopt the computational homogenization techniques to obtain the global response of masonry structures. Since the experimental global response curves, obtained in typical shear tests on masonry panels, show stiffness and resistance degradation, damage is the fundamental ingredients which must be taken into account in such problems.
Moreover, as it is well known, due to the aforementioned softening behavior, regularization techniques are required in order to avoid spurious mesh dependencies when a numerical solution is sought in the framework of finite element method. The first step of this work is the adoption of the standard first order computational homogenization, where Cauchy continuum is used both at the macro and micro-level. This approach is well known in literature and several authors applied it to different engineering problems. An example of the adoption of regularization techniques in the context of multi-scale approaches is found in Massart (2003). Hence a regularization based on the imposition of the macroscopical length scale at the micro-level, in the framework of the fracture energy regularization, is proposed.
However, as previously stated, many authors have pointed out the inner limits of first order computational homogenization. Such a formulation, in fact, may be adopted only if 1)the microstructure is very small with respect to the characteristic size at the macro-scale; 2)the absolute size of the constituents does not affect the mechanical properties of the homogenized medium and in presence of low macroscopic gradients of stresses and strains. As a consequence no localization phenomena typically exhibited by masonry can be analyzed. For masonry structures, instead, microstructural typical sizes are comparable with the macro-structural sizes; shape, size and arrangement of the constituents strongly affect the mechanical global response and high deformation gradients typically appear.
An enriched formulation is then proposed in order to overcome these problems, based on the adoption of a Cosserat medium at the macro-level and a Cauchy medium at the micro-level. The theoretical and computational schemes remain the same as before but for the fact that the two media present different variables. In particular in the Cosserat medium additional strain and stress variables appear, with respect to the Cauchy continuum, as a consequence of the independent rotational degree of freedom assigned to every material point. Thus, a more sophisticated kinematic map, containing higher order polynomial expansions, is needed to state proper bridging conditions between the two levels.
The innovative contribution of this work concerns the adoption of an enhanced multi-scale computational homogenization technique for studying the masonry response, together with the employment of damage models for the constituents description. Thus, by exploiting the inner regularization properties of the Cosserat continuum at the macro-level and by adopting a classical fracture energy regularization at the micro-level, localization phenomena, typically exhibited by masonry structures, are analyzed. Since this material shows a typical strain softening behavior, an ad hoc regularization technique has been developed at both levels in order to obtain objective numerical responses. To the knowledge of the author, no previous examples of Cosserat-Cauchy computational homogenization techniques, taking into account localization effects, have been presented.
A possible objection to the use of a fully-coupled multi-scale technique could be related to the high computational efforts required, but here the use of parallel computing brings them down. In this context, these procedures strike a good balance between the achievement of detailed information at the scale of the constituents and the requirement of holding the computational costs down.
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