## 1 Introduction  
Built masonry structures constitute an invaluable heritage worldwide, including in Europe, where many historical centres and UNESCO heritage sites made with masonry can be found. To this extent, the study of their behaviour is one of the main concerns of the international civil engineering community. Existing masonry buildings are often modelled with an assemblage of masonry units with dry interfaces [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Firstly, the mortar used at the construction time was often of low quality, and was even missing in some cases. Secondly, the initial mortar may have been degraded through time because of erosion and lixiviation. Finally, this assumption is conservative and enables experimental tests in the laboratories to validate the developed models to be performed more easily [3, 4, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Indeed, dry masonry allows both faster tests (no mortar hardening time) and scaled tests without needing to scale the mortar size while keeping adequate mechanical properties. Modelling strategies for masonry assemblies depend on the type of structure and the type of loading. While continuum Finite Element Modelling (FEM) strategies have been found useful for modelling large buildings with a relatively good quality mortar [23, 24, 25], Discrete Element Modelling (DEM) and similar discrete approaches often perform better for dry masonry structures, allowing large displacement to occur at dry interfaces [2, 4, 5, 8, 23]. When referring to the type of loading, one can find settlement [3, 4, 13, 17, 21], vertical loading [26] and seismic loading which can then be classified as either pseudo-static (tilting) [6, 10, 14, 20, 27, 28] or dynamic loading [2, 7, 9, 12]. The seismic vulnerability of masonry structures is particularly high because of their heavy weight, brittleness, low tensile strength and poor connection between structural components [29; 30]. As far as the seismic behaviour of masonry structures is concerned, numerical FEM and DEM schemes can be used to perform both types of analysis, pseudo-static (so-called pushover analysis) and dynamic time history [7; 12]. On the other hand, analytical approaches have also been derived with the Limit Analysis framework for pseudo-static loading [6; 10; 31] and rocking theory for dynamic loading [2; 9]. Discrete Element Methods adopt an assemblage of highly rigid blocks in contact at interfaces. The deformability of the whole structure is mainly driven by the joints, while the block deformability is usually taken as either null or very limited [7]. In fact, at interfaces, most DEM strategies assume a stiffness that relates the force transmitted through the interface with the elastic displacement at the same interface. The latter is characterised by a relative displacement between the two blocks in contact. Common values used in the literature lead to a global behaviour (i.e., elastic stiffness) driven by the joint stiffness [4; 7; 12]. In general, the calibration of joint elastic parameters remains complicated, often conducted by inverse fitting, which prevents reliable predictions of structural behaviour. Instead, only post-diction can be carried out. For instance, in dynamic tests, the stiffness parameters can be calibrated through the recorded vibration modes of the structure [7]. Values can also be assumed on a default basis, either taken from the literature or building codes [4; 12], but this hardly applies to tests with assembled dry joints blocks and low vertical precompression. More recently, the joint behaviour of dry masonry assemblages has been more deeply investigated experimentally [32; 33; 34; 35]. It has been shown that the joint behaviour was strongly influenced by the contact stress and geometrical imperfections of blocks in contact (non-planarity of faces, height differences between blocks). In particular, joint stiffness strongly diminishes with decreasing vertical stress, with an approximately linear relationship [33; 34], while block imperfections reduce the joint stiffness even more [32; 35].