#### Abstract

A micromechanical procedure is used in order to evaluate the initiation of damage and failure of masonry with in-plane loads. Masonry material is viewed as a composite with periodic microstructure and, therefore, a unit cell with suitable boundary conditions is assumed as a representative volume element of the masonry. The finite element method is used to determine the average stress on the unit cell corresponding to a given average strain prescribed on the unit cell. Finally, critical curves representing the initiation of damage and failure in both clay brick masonry and adobe masonry are provided.

#### 1. Introduction

Materials used in civil engineering such as cement concrete [1–4] and masonry are subject to deterioration; moreover, constructions built in the past need to carry increasing seismic horizontal as well as vertical loads, leading to the need for strengthening interventions. Fibre reinforced plastics (FRP), polymeric nets embedded in the plaster, and other more classical materials and techniques are effectively used to strengthen existing structures [5–8]. More complex techniques require the insertion of seismic isolation bearings made of rubber, usually modelled as a hyperelastic material [9–16]. New materials are being tested for civil engineering applications; recently, carbon nanotubes (CNTs) are being used in various research works in order to improve the mechanical properties of cement mortar (usable also in strengthening existing structures); therefore many authors are investigating the behaviour of structures at nanoscale with nonlocal material models [17–21].

Numerical methods for determining the structural response of unreinforced and reinforced masonry constructions, such as arches, wall, and dome, are available in the literature [22–27]. It is important to have available methods for estimating not only the structural response but also the material behaviour. To this end, micromechanics and homogenization methods have been applied to the masonry material. In this work, a micromechanical procedure is used in order to evaluate the initiation of damage and failure of masonry panels with in-plane loads following the approach developed in [28–30].

#### 2. Micromechanical Analysis and Material Model

The masonry material is composed of two constituents: the brick and the mortar. These constituents have different mechanical properties and the resulting masonry material can be viewed as a composite so that classical techniques based on micromechanics and homogenization are used in order to determine the local and overall response of the masonry material. The overall mechanical properties of the composite are derived from the geometric and mechanical properties of the constituents, the microstructure of the composite, and so forth. Such information is contained in a reference volume element (RVE), which is statistically representative of the masonry under consideration. The following analyses are performed on the representative volume element. Instead of modelling the masonry as a random composite material (see, e.g., [31–36]), it is assumed that the constituents are arranged in a periodic way [37–45] and a unit cell is adopted as the representative volume element. The masonry considered in this work is shown in Figure 1, where a wall is depicted. The aim is to determine the strength of the masonry when it is subjected to in-plane loads.

The wall in Figure 1 is a three-dimensional structure whose three-dimensional unit cell is shown in Figure 2(a), where , , and are the dimensions of the unit cell along the -, -, and -axes of the coordinate system with origin in the centre of the unit cell.

**(a)**

**(b)**

In micromechanical and homogenization analyses of periodic microstructure, the three-dimensional unit cell is subjected to the following periodic boundary conditions:where is the displacement along the -axis and are the components of the average strain prescribed to the unit cell. Together with (1), the following constraint must be imposed in order to avoid the rigid-body translations:where is the position vector of the centre of the unit cell. Since the in-plane behaviour must be investigated, the plane unit cell shown in Figure 2(b) can be used as representative volume element for a two-dimensional (2D) analysis. In the homogenization analysis, the boundary conditions prescribed to the plane unit cell are

The average stress in the unit cell is denoted by , whereas the local stress in a point of the constituent of the unit cell is the microstress and, for simplicity, will be called stress in the following. The constituents (brick and mortar) are considered linear elastic before their damage and failure. The bond between the constituents is considered perfect. It is assumed that the plane unit cell and the constituents are subject to a plane stress state. Therefore, the principal stress perpendicular to the plane unit cell is equal to zero whereas the principal stresses parallel to the plane of the unit cell can be different from zero. The failure in a point of a constituent occurs when one of the following three failure criteria is satisfied.

*First Criterion*. If the two principal stresses parallel to the plane of the unit cell are nonnegative the failure occurs when the maximum principal stress in a point is equal to the tensile strength of the constituent.

*Second Criterion*. If the two principal stresses parallel to the plane of the unit cell are nonpositive the failure occurs when the following equation is satisfied:where , where and are the two nonpositive principal stresses parallel to the plane of the unit cell and is the compressive strength of the material of the constituent.

*Third Criterion*. When a principal stress parallel to the plane of the unit cell is nonnegative and the other principal stress parallel to the plane of the unit cell is nonpositive, the failure occurs when the following equation is satisfied: where is the tensile strength of the material of the constituent.

Next, the average strain (which is a tensor with components ) is imposed on the plane unit cell by means of the conditions (2) and (3). Then, the average stress corresponding to the prescribed average strain is evaluated and the critical curves of the masonry are determined with a finite element homogenization technique. The critical curves may be plotted in the plane defined by - and -axes or in the plane defined by - and -axes. There are two kinds of critical curves: a generic critical curve may be related to the brick or to the mortar. The generic point of the critical curve related to a constituent represents the average stresses corresponding to the initiation of failure of that constituent. Specifically, the initiation of failure of a constituent is evaluated by prescribing a linear load path , where is a constant tensor, which does not vary in the loading process, and the scalar is an increasing loading positive parameter. Considering both constituents linear elastic, there exists a value such that the microstresses in a constituent ( = brick or mortar) of the unit cell subject to the average strain satisfy some of the above-mentioned criteria in the same constituent , whereas failure criteria are not satisfied in the same constituent when the unit cell is subject to for . The average stress of the unit cell subject to the average strain is denoted by . In the plane defined by - and -axes, the components of define a closed curve which represents the masonry critical curve related to the initiation of failure of the constituent . An analogous critical curve can be plotted in the plane defined by - and -axes.

#### 3. Numerical Examples

If the constituents have the same mechanical properties and the bond between the brick and the mortar is perfect, the masonry behaves like a homogeneous material and the homogenization procedure is not required. Critical cases occur when the mechanical properties of one constituent are significantly different from the mechanical properties of the other constituents. For example, in adobe masonry (AM), bricks and mortar have similar elastic properties, whereas in some kinds of clay brick masonry (CBM) Young’s modulus of the brick may be significantly different from Young’s modulus of the mortar. The masonry critical curves are very sensitive to the ratio /, where and are Young’s moduli of the brick and the mortar, respectively. This is shown in Figure 3, where the masonry critical curve related to the initiation of failure of the constituent is denoted by “” for = brick and by “” for = mortar in the legend of the figure.

The critical curves in Figure 3 are obtained by prescribing the following average strain:

In this section, the mechanical properties of the constituents of CBM are = 0.23, = 0.15, = 10 = 10000 MPa, = 10 = 1 MPa, = , and = ; the mechanical properties of the constituents of AM are = = 0.15, = = 100 MPa, = 10 = 1 MPa, = , and = . In this work, and are Poisson’s ratios of the brick and the mortar, respectively, and are the tensile strengths of the brick and the mortar, and and are the compressive strengths of the brick and the mortar. Both CBM and AM have the following geometric properties: mm (width of the bricks), mm (height of the bricks), mm (thickness of mortar joints), and (each brick corner was approximated as arc of circle with radius ).

In Figure 3, two different values of the ratio / are considered: for the CBM critical curves and for the AM critical curves. Greater values of determine greater concentration of the local stress and, as a consequence, smaller critical areas (the critical area is the area bounded by the critical curve related to the initiation of failure of a constituent). A similar behaviour is observed in Figure 4, where the critical curves are obtained by prescribing the following average strain:

The contrast between the masonry constituents also influences the deformed shape of the unit cell and the distribution of local stress in the unit cell, as shown in Figures 5 and 6, where the local stress on the deformed unit cell subject to the average strain (8) with and is plotted for CBM () and AM (), respectively.

#### 4. Conclusions

In this work, masonry is viewed as a composite constituted of two components (namely, brick and mortar) so that classical techniques based on micromechanics and homogenization are used in order to determine the local and overall response of the masonry material. The critical curves of the masonry are determined with a finite element homogenization technique. The generic point of the critical curve related to a constituent represents the average stresses corresponding to the initiation of failure of that constituent. The numerical analyses show that the critical curves are sensitive to the contrast between brick and mortar: greater contrast involves earlier initiation of damage in the constituents.

#### Competing Interests

There are no competing interests related to this paper.