On nonlinear rheology of masonries and granular media

We introduce a new rheological nonlinear model for some granular media such as masonries. The latter may demonstrate a rather complex behaviour. In fact, considering a masonry one can see that relative rotations of bricks are most important in comparison with deformation of bricks themselves. As a result, one gets stresses and couple stresses as static characteristics of such a medium. Using the Cosserat point approach for modelling of orientational interactions between masonry elements we provide a deformation energy for such a medium which takes into account both material and geometrical nonlinearity.


Introduction
Granular materials constitute a rather challenging object for continuum mechanics.In fact, they may demonstrate both fluidand solid-like behaviour, see e.g.Capriz et al. (2008), Castellanos (2005), de Gennes (1998de Gennes ( , 1999)), Hutter and Rajagopal (1994), Kachanov and Sevostianov (2005), Nedderman (1992) and Suiker and de Borst (2005).Moreover, their properties may essentially depend on the interactions between granules.Even in the case of solid-like granular media such as a masonry adhesion/friction/sliding of bricks may be more crucial than their mechanical properties.Considering kinematics of granular media and granules themselves one can see that relative rotations of granules are also essential.So one can expect appearance of couple stresses as a response to rotational degrees of freedom.This observation makes rather natural to exploit the Cosserat continuum model for such a medium.
The aim of this paper is to introduce a nonlinear rheological model for a class of granular media which includes masonries.As a masonry represents a highly inhomogeneous material it is quite natural to use some homogenization techniques for introducing an effective medium, see e.g.Addessi et al. (2013), Anthoine (1995), Bacigalupo and Gambarotta (2012), Cecchi and Sab (2002), Diana et al. (2023), Gatta and Addessi (2023), Lourenço (2010), Luciano and Sacco (1997) and Sacco (2009).Such an approach is particularly suitable for masonry-like materials whose mechanical behaviour is strongly influenced by the internal microstructure.Since the arrangement of the bricks plays a crucial role at the macroscopic level, discrete models -where the constitutive elements of the masonry are modelled separately, see for example Lemos (2007) -may be preferable to properly account for this.At the same time, however, the computational requirements limit their applicability to small parts of buildings, requiring the adoption of more efficient models.Macro-element models -where masonry structures are represented by simplified equivalent structures, such as a system of non-linear beams, see for example Magenes and Calvi (1997) -are usually adopted by practitioners thanks to the possibility of analysing real buildings, but the description of the masonry material is too coarse.In this context, continuum models -where the masonry is modelled as an equivalent continuum able to take into account the internal microstructure, i.e. obtained through homogenization techniques, see the review by Lourenço et al. (1997) -can be an effective solution.
The remainder of the paper is organized as follows.First, in Section 2 we consider experimentally observed behaviour of a such medium.We discuss responses to tension-compression, shear, and relative rotations.For simplicity we assume that our medium consists of rigid bricks with highly deformable (soft) thin interfacial layers.We provide three types of response called ''real'', ''simplified'', and ''ideal''.Then, in Section 3 we use these dependencies to introduce a continuum model.First, we briefly recall basic equations of polar media paying most attention to geometrical sense of strain measures as discussed in Eremeyev et al. (2013), Eringen (1999) and Pietraszkiewicz and Eremeyev (2009).Finally, we present the deformation energy as a sum of energies related to tension-compression, shear, torsion, and bending.Its dependence on masonry ordering, i.e. lattice type, is also discussed.

Rheology observations
Masonry is a composite heterogeneous material characterized by an internal microstructure at the mesoscale, consisting of a system of rigid elements -blocks (natural stones, square or rough) or bricks (artificial regular units) -interacting with each other through deformable or non-deformable unilateral interfaces with no or very low tensile strength and resistant to sliding by friction.Masonry makes up a large proportion of the world's existing buildings: much of the historical architectural heritage consists of monumental masonry structures, and in many countries ordinary residential buildings are typically constructed of masonry.Its mechanical behaviour is interesting because it exhibits a highly non-linear behaviour, even for loads far from collapse, and very different responses to compression and tension, with a usually high compressive strength coupled with a very low or uncertain tensile strength, so that it is often considered to be a non-tensile material (Angelillo, 1993;Del Piero, 1989, 1998;Heyman, 1966).Moreover, this complex behaviour is strongly influenced by the constructive characteristics of the masonry: the quality of the constituent materials, the size of the bricks, the thickness of the joints, the arrangement of the bricks, the bonds, etc.The typological diversity, composite nature, heterogeneity and anisotropy of masonry make it difficult to describe its mechanical behaviour.
Here we focus on a simple example of masonry consisting of two bricks connected by an intermediate joint, as shown in Fig. 1.The bricks are assumed to be rigid, while the joint is highly deformable.The following describes the physics of the interaction in terms of compressive-tensile, shear and moment stresses.In particular, the different responses to compression and tension, sliding in relation to shear and relative rotations between the bricks due to moment are highlighted.Both ''real'', ''simplified'' and ''ideal'' behaviour are considered.

Tension-compression
A typical loading curve for tension-compression test is given in Fig. 2 (curve I).Here  and  are engineering stress and engineering strain, respectively.One can see that the mortar material response is not symmetric with respect to tension and compression.Moreover, the material has very low strength under tension.As a result, a simplified linear version can be used as presented in Fig. 2 (curve II).Here  * and  * are maximal tensile stress and strain, respectively.The corresponding strain energy can be defined through the standard formula (1) E. Reccia and V.A. Eremeyev  In particular, for dependence II it could be calculated as follows where  is the Young modulus.Let us note that in the case of simplified version of constitutive equations we may face an issue related to nonexistence of derivative at  =  * .As an essential simplification of such a behaviour the non-tension model can be also used, see curve III in Fig. 2.Here material has no tensile strength and become rigid under compression (has unlimited strength under compression).Formally, this case corresponds to  = ∞ and  * = 0.

Shear and sliding
Similar but symmetric behaviour can be observed for shear (sliding), see Fig. 3, where again we present three models (curves I, II, III).Here  and  are tangent stress and shear strain, respectively, with ultimate values  * and  * .The strain energy is defined by the formula that results in the case of simplified model (curve II) in the equation where  is a shear modulus.
The ideal model III corresponds to the Coulomb friction law.

Bending
Considering relative rotations of two bricks we can also introduce three models, see Fig. 4. As in the case of tension-compression and shear here we face also with limiting behaviour.Here  and  are the moment and relative rotation, respectively.Due to geometrical constraints  has also a limit value  * whereas  * corresponds to maximal admissible moment in an elastic regime.
In the case of linear model (curve II) the general formula for the strain energy results in where  is a bending stiffness.The ideal model (curve III) is similar to rigid-plastic solids but with respect to rotations.It is interesting to note that this constitutive relation is similar to strain-limiting elasticity by Rajagopal (2011).Indeed, unlike the previous cases where we face stress-limiting constitutive relations, in the latter case one see limiting rotations.

Torsion
The case of torsion is similar to shear presented above.Here we again may provide three models with limiting torsional moment  *  and torsion  *  .Related to curve II the linear model corresponds to the following strain energy with where   is a torsional stiffness (see Fig. 5).

Continuum model
In the following, using the previous observations, we discuss the relevant continuum model.

Micropolar finite elasticity
First, let us briefly recall the basic equations of the Cosserat continuum in the case of finite deformations.The kinematics of Cosserat continuum can be described using two descriptors, i.e. translations and rotations (Eremeyev et al., 2013;Eringen, 1999;Pietraszkiewicz & Eremeyev, 2009) where  and  are the position vectors defined in current and reference placements, respectively,  is time,  is the vector of translations, and  is the orthogonal tensor of microrotations.
For hyperelastic materials there exists a deformation energy  introduced as a function of two Lagrangian strain measures where In ( 11)  =   is the deformation gradient,  is the Lagrangian gradient operator,  = − ×  is the permutation tensor,  is the 3D unit tensor, ⋅, ∶, and × stand for the dot, double dot, and cross products, respectively.
Corresponding stress tensors are given by Here  and  are the first kind Piola-Kirchhoff stress and couple stress tensors, respectively.Determination of  constitute a rather complex problem, so in the following we specify it using experimental observations of the previous sections.

Two bricks and mortar: micropolar point of view
Let us consider first interactions of two rigid bricks of dimensions , , , connected via a thin interfacial layer of thickness 2ℎ, see Fig. 6.An energy stored in the layer entirely depends on the relative motion of bricks.The latter consist of relative translations and relative rotations.As in the case of rigid body dynamics, see e.g.Eremeyev et al. (2013), we can introduce the vector of translation as a displacement of their centres of mass.In addition, rotations can be described using a rotation tensor.So for two bricks we introduce two vectors of translation, i.e.  1 and  2 , and two rotation tensors  1 and  2 , respectively.As a result, the total energy of the interfacial layer can be considered as a function of relative translations and relative rotations Fig. 6.Deformation of two bricks with a deformable interface.
Here  1 and  2 are the position vectors of the bricks in the current placement, see Fig. 6.Considering our two bricks as material point of a continuum we can assume that there exist two smooth field  = (, ) and  = (, ) such that   = (  , ) and   = (  , ),  = 1, 2. As a result,  and  can be represented as follows Here  is the thickness of the brick, 2ℎ is the mean thickness of the interfacial layer, and  is the unit normal in the reference placement, see Fig. 6.As a result, we can see that  12 is a function of  and  taken at a point that is the centre of mass of the total structure,  12 =  12 (, ).Considering the system of two bricks and the interface as a representative volume element (RVE), we can see that the deformation energy density  could be introduced as follows:  =  12 ∕ , where  = 2 + 2ℎ is total volume of the system.One can see similarities with the Cosserat point approach by Jabareen and Pestes (2020), Jabareen and Rubin (2008) and Rubin (1985Rubin ( , 2000)).Indeed, here we replaced the 3D interfacial layer by a material point which possesses both strains as in the case of 3D Cosserat continuum.The approach described above can be extended to a system of connected bricks.Depending on the chosen RVE we can obtain similar expression for the deformation energy density.Before we specify the form of  12 using three models based on experimental observations.
First, let us discuss tension-compression deformations only of two bricks as shown in Fig. 6.So here we have stretching/elongation in  2 direction.Note that in this case  =  is constant.So the deformation energy stored in this system depends on  22 only:  =  ( 22 ).Considering the experimental data for tension/compression tests we can use Eq. ( 2) as a deformation energy.
Similarly, we can treat the shear test.From geometrical sense of  it follows that now  depends only on  12 :  =  ( 12 ).Again, we can use Eq. ( 4) as a form of  .
Taking into account the geometrical sense of , see Eremeyev et al. (2013) and Pietraszkiewicz and Eremeyev (2009), we can see that the stored energy related to relative rotations will have of form  =  ( 12 ).As a function we can use Eq. ( 6).
Summarizing our analysis of deformations of two bricks we can came to the following representation of the deformation energy Here  1 ,  2 , and  3 are normalizing coefficients related to the ratio of the volume of the mortar to the whole volume  .In the following for simplicity we assume that they are equal and we omit them just keeping the same notations for the deformation energies.Note that this equation relates to the following deformations: tension-compression in  2 direction, shear in  2 −  1 plane, and relative rotations about  1 -axis.If we consider other shear in  3 -direction, bending about  3 -axis, and torsion about  2 -axis we came to more symmetric case  Obviously, such a deformation energy can describe only a particular oriented material.Indeed, it has preferable direction related to  2 -axis.For a masonry consisting of a system of bricks connected in various ways this approach should be modified accordingly.

Constitutive equations of masonry-like media: micropolar point of view
Let us considering a periodic masonry, see Fig. 7, where block dimensions and joint thickness are the same reported in the previous section, see Fig. 6.The regularity of the brickwork is represented by the so-called ''running bond'' pattern, where each brick is surrounded by six neighbours through six interfaces.In the pattern considered, the width of the horizontal interfaces is equal to half the block width ∕2 and the height of the vertical interfaces is equal to the block thickness .By varying the ratio between the thickness and width ∕ of the bricks, it is also possible to describe the so-called ''head-bond'' pattern and, more generally, to consider the influence of the local size effect due to such a periodic arrangement of bricks on the global behaviour of the masonry.In the case of periodic masonry, it is possible to identify a representative volume element (RVE) that represents the entire masonry panel through its repetition.The RVE provides all the mechanical and geometrical characteristics of the masonry at the micro level required to fully describe the entire masonry at the macro level, thus taking into account the internal microstructure.Two centre-symmetric RVEs are usually chosen for such masonry patterns (Salerno & De Felice, 2009): a classical one with a brick  , in the centre of the cell, surrounded by four horizontal and two vertical joints connecting it to the six adjacent bricks; a second one in which, instead, a vertical joint  , -modelled by means of Cosserat point, a zero-dimension Cosserat continuum -is in the centre of the cell, surrounded by four horizontal joints connecting it to four adjacent bricks.We refer to the latter RVE, reported in Fig. 8, which has been shown to be more suitable for deriving the micropolar continuum in the case of masonry, see Baraldi et al. (2015).

Fig. 2 .
Fig. 2. Tension-compression loading curves.Curves I, II, and III correspond to real, linear simplified, and ideal models.

Fig. 3 .
Fig. 3. Loading curves for shear.Curves I, II, and III correspond to real, linear simplified, and ideal models.

Fig. 4 .
Fig. 4. Moment -relative rotation.Curves I, II, and III correspond to real, linear simplified, and ideal models.

Fig. 5 .
Fig. 5. Torsional moment -relative rotation.Curves I, II, and III correspond to real, linear simplified, and ideal models.