A gradient elastic homogenisation model for brick masonry

Abstract

The elastic macro-mechanical properties of masonry are investigated herein by taking into account the presence of a weakly non-local heterogeneity within a simple gradient elasticity model. Masonry has a heterogeneous structure composed of masonry units bound by mortar. The homogenisation of masonry walls is a challenging task but also a very appealing method for modelling heterogeneity effects exhibited by masonry elements. In particular, it allows the use of smeared mechanical properties, thus avoiding the need of knowing the exact unit-to-unit and joint-to-joint geometry. Current codes provide very simplified empirical expressions to estimate an isotropic elastic modulus of masonry on the basis of its strength properties. The respective equations which do not take into account the anisotropy of masonry, present high scatter resulting in ambiguous safety. The homogenization argument employed in this work is based on a simple procedure utilising Aifantis’ gradient elasticity (GradEla) model. The GradEla model is a straight-forward extension of Hooke’s law by enhancing it with the addition of the Laplacian of the classical expression of the Hookean stress multiplied by an internal length accounting for the local heterogeneity. It has been successfully used to eliminate singularities from dislocation lines and crack tips, as well as in interpreting size effects. However its use in masonry structures has not yet been explored. A first step in this direction is attempted in this paper with emphasis on obtaining practical easy-to-use results rather than exhausting all other possibilities and complexities encountered in GradEla and its generalisation, as well as in more involved homogenisation procedures. In our analysis uniform vertical, horizontal and shear loads are assumed to act on the boundaries of the representative volume/surface element. The components of masonry are assumed to follow a gradient elastic stress distribution resulting in a gradient elastic homogenised model (GREHM). GREHM comprises a set of closed-form concise equations which estimate the elastic moduli in the longitudinal and transverse directions, the shear modulus and the Poisson’s ratio. The aforementioned orthotropic material properties are verified using experimental results and also, compared to other homogenisation models. The validation shows that the proposed equations can effectively estimate with considerable precision the elastic properties of masonry walls. To illustrate the resulting estimation of the orthotropic elastic properties, normalised graphs are provided.