Micromechanical modelling of mortar joints and brick-mortar interfaces in masonry Structures: A review of recent developments

Structures 23 (2020) 831–844

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Micromechanical modelling of mortar joints and brick-mortar interfaces in masonry Structures: A review of recent developments Masoud Shadlou, Ehsan Ahmadi, Mohammad Mehdi Kashani

T

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University of Southampton, Faculty of Engineering and Physical Sciences, UK

A R T I C LE I N FO

A B S T R A C T

Keywords: Micromechanical modelling Constitutive models Mortar joints Brick-mortar interface Masonry structures

Suitable numerical models of masonry structures are very important in their response evaluation under various loading events. Masonry is a heterogeneous material, made of mortar and masonry units, and joined together by interfaces. Constitutive models of mortar joints and masonry-mortar interfaces play a crucial role in achieving high-fidelity numerical models for masonry structures. Hence, this review paper particularly collates the most commonly available constitutive models of mortar joints and brick-mortar interfaces in the literature. The previous experimental studies on mechanical characteristics of mortar joints and brick-mortar interfaces are first discussed in detail. The existing constitutive models developed based on theory of plasticity, fracture mechanics, and damage theory are then mathematically described, and their strengths and shortcomings are fully explained. It is found that the literature lacks reliable experimental calibration of the current constitutive models, and combined loading experiments are required for better understanding of nonlinear behaviour of mortar joints and brick-mortar interfaces. It is also seen that most current constitutive models are two dimensional, use many theoretical assumptions and hypotheses with no experimental verifications, and do not account for three dimensional irregular interface bonding, bonding degradation, and relevant post-yielding deformational pattern. Effects of unloading-reloading, dilatancy, surface asperities, and crack formations also need further investigations.

1. Introduction Masonry structures, particularly masonry infill walls and frames, are still used in seismic regions to withstand lateral earthquake loading events. Assessment of existing and design of new masonry structures require very rigorous numerical models. The nonlinear seismic behaviour and structural evaluation of masonry structures during earthquake events extensively depend on modelling approach, and constitutive models of masonry elements directly implemented in powerful numerical tools. Masonry structures are composite structural systems composed of building blocks/units and mortars, which itself is a composite material. The masonry units can be made of bricks, stones, concrete masonry, etc. Mortars are used between masonry units as adhesives, and are divided into cement-based and mud-based mortars depending on construction circumstances. Mortars join masonry units horizontally and vertically, termed as bed joints and head joints respectively. These joints are the most important parts of any masonry structure as they provide the entire integrity of masonry structures and their dominant failure mode. The largest material variations are seen in mortar joints, in particular at

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masonry unit-mortar interfaces due to the material discontinuity, and thus, their constitutive models play a crucial role in capturing realistic behaviour of masonry structures under various loading conditions. During the last three decades, several micromechanical constitutive models have been developed for masonry joints (e.g. mortar joints and masonry unit-mortar interfaces), and implemented in powerful numerical tools such as nonlinear finite element methods [1–7]. Mortar joints and masonry unit-mortar interfaces are often expressed at mesoscale (10−3 to 10−1 m). In this scale, the brick is usually modelled using elastic continuum solid elements while the mortar (i.e. cementaggregate paste and voids) and brick-mortar interfaces are modelled by means of interface elements. The microscale modelling of masonry unitmortar interface under loading conditions is equivalent to mesoscale, commonly discussed in multiscale methods. In mesoscale modelling of masonry structures, the actual size of the masonry unit and mortar are used (e.g. two masonry units and mortar joints). Fig. 1 shows an overview of the microscale modelling of a masonry panel. Macroscopic characteristics of a masonry panel depend on different components of the multi-component composite system including masonry units and mortar. Response of the mortar under any loading event is dependent

Corresponding author. E-mail address: [email protected] (M.M. Kashani).

https://doi.org/10.1016/j.istruc.2019.12.017 Received 2 August 2019; Received in revised form 7 November 2019; Accepted 16 December 2019 2352-0124/ © 2019 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.

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Nomenclature a c c0 d de df dg dp dup dn dn1 dn2 dt deff dne dnf dnp dn,up dt p dng dnM f fcdil fσdil g h knn m mn mt n q q0 qr r r0 rr s s0 sr w Cu De Dep D10 Dnn Dnn0 Dtt E F

Fc Ft Fs G Gmax G fI GfII H Q Qs T α γ δ ω εp εnp εtp εv η θ κc κm κs κt λ λc λs λt μ μ0 μr υ ξ ρ σ σd σu σ σdil σe σ' σ'’ σu σ¯c σ¯m σ¯r σ¯s σ¯t τ τe φ φ0 ϕ χ

A constant for quantifying direction of the plastic flow Interface cohesion Initial interface cohesion Relative displacement vector Elastic relative displacement vector Fracture displacement vector Geometric relative displacement vector Plastic displacement vector Unrecoverable plastic displacement vector Normal component of relative displacement Additional internal variable in Eq. (13) Additional internal variable in Eq. (13) Tangential component of relative displacement Effective relative displacement Normal component of elastic relative displacement Normal component of fracture relative displacement Normal component of plastic relative displacement Normal component of unrecoverable plastic relative displacement Tangential component of plastic relative displacement Normal component of geometric relative displacement Maximum normal relative displacement Frictional sliding in internal joints A coefficient which controls dilatancy and depends on cohesion A coefficient which controls dilatancy and depends on normal stress A function which indicates opening of mortar joint Closing distance of the interface Normal elastic stiffness in tension Unit direction of the plastic flow Normal component of vector m Tangential component of vector m Unit direction of the yield surface State vector Initial state vector Residual state vector Radius of the yield surface Initial radius of the yield surface Residual radius of the yield surface Tensile strength of the interface Initial tensile strength of the interface Residual tensile strength of the interface Material parameter in Eq. (18c) Coefficient of uniformity of grains Elastic stiffness matrix of the interface Elastoplastic stiffness matrix of the interface Effective grain size Normal elastic stiffness of the interface Initial normal elastic stiffness of the interface Tangential elastic stiffness of the interface Softening modulus Yield surface function

Compression yield surface function Tension yield surface function Shear yield surface function Shear modulus Maximum shear modulus Mode I fracture energy Mode II fracture energy Heaviside function Plastic potential function Shear plastic potential function Transpose of a vector or matrix Dilatancy-shear degradation coefficient Shear strain Increment of a scalar, vector, or matrix Projection vector Plastic strain vector Normal component of plastic strain Tangential component of plastic strain Volumetric strain Material parameter for scaling dilatancy A function which indicates sliding of mortar joint Compression plastic work Material parameter in Eq. (18c) Shear plastic work Tension plastic work Plastic multiplier vector Compression plastic multiplier Shear plastic multiplier Tension plastic multiplier Slope of friction angle of the interface Initial slope of friction angle of the interface Residual slope of friction angle of the interface Dilatancy coefficient Damping ratio Elastic deterioration parameter Stress vector Damaged stress vector Undamaged stress vector Stress component normal to the interface Normal stress above which dilatancy is removed Normal elastic predictor tractions Mean effective stress Deveiatoric stress in triaxial stress condition Compression stress at zero dilatancy Yield compressive stress Material parameter in Eq. (18c) Residual compressive stress Yield shear stress Yield tensile stress Stress component tangential to the interface (shear) Tangential elastic predictor traction Dilatancy angle Dilatancy angle at zero normal stress Friction angle Damage parameter

interfaces are very limited compared to other structural materials. Furthermore, sources of experimental data available for calibration of such constitutive models are very finite [9]. On the other hand, bricks are constantly being used as common masonry units in construction of many new masonry structures. Therefore, a research review is imperative to collate recent studies on microscale modelling of mortar joints and brick-mortar interfaces, identify knowledge gaps to provide recommendations for future research, and accordingly improve

on its micro-pore structure, surface characteristics, thermodynamic states associated with durability, structural mechanics, and damage state. To realistically simulate the behaviour of masonry structures, material science and structural mechanics of masonry units, mortar, mortar joints, and masonry unit-mortar interfaces must be combined with the macroscopic features of the multi-component composite masonry panel [8]. Studies on constitutive models of mortar joints and masonry-mortar 832

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Fig. 1. Overview of the microscale modelling of masonry strutures.

2. Mortar and mortar-brick interface behaviours

modelling technique of masonry structures. The focus of this work is on micromechanical modelling of cement-based mortar joints and brickmortar interfaces in masonry structures. The results of previous experimental tests on mortar joints and brick-mortar interfaces are first discussed in detail. The recent achievements on relevant constitutive models developed based on theories of plasticity, damage, and nonlinear fracture mechanics are then explained. It will be seen that the commonly used constitutive models available in the literature do not account for the effects of some important phenomena such as dilatancy, softening, and crack propagation as well as relevant material properties while these effects are very essential for capturing more realistic behaviour of masonry structures. Hence, more detailed constitutive models should be developed for simulating behaviour of mortar joints at microscale.

Tension and shear modes of failure are often dominant in mortar joints and brick-mortar interfaces compared to compression modes of failure, and thus, tension and shear modes of failure have received much attention in the literature. Combinations of tension, shear, and compression failure modes were discussed by Mann and Mueller [10] and Van der Pluijm [11] where they carried out one of the most complete experimental studies on tension and shear behaviours at the University of Eindhoven. Furthermore, cementation is a technique for soil enhancement and stabilization in the field of geotechnical engineering through adding cement and water to granular materials. Similarly, in the area of structural engineering, inter-particle cementation can elucidate key concepts of dilatancy, stress softening, and yield 833

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surface (i.e. the boundary of elastic behaviour), which are very important in understanding micromechanical behaviour of mortar joints and brick-mortar interfaces. Therefore, in this section, previous tensile tests and their results are first discussed; inter-particle cementation as a key concept in understanding shear behaviour of mortar is then explained; comparisons are made in terms of uncemented and highly cemented granular materials where highly cemented granular materials resemble mortars; afterwards, existing shear experiments are presented, and various possible modes of failure for mortar joints and brick-mortar interfaces are discussed.

section of the brick-mortar interface. This phenomenon accordingly affects the stress-displacement behaviour of the brick-mortar interface under uniaxial tension, compression [13], and shear loadings. van der Pluijm [11] reported 34% and 18% reductions respectively in coefficients of variation of tensile bond strength and fracture energy. It should be noted that tensile bond strength and fracture energy are parameters which control softening behaviour of a material under uniaxial tension.

2.1. Tensile tests

A cement-based mortar is mostly made by mixing water, cement, and sand. Cement is the binding component of mortar, and makes a strong bond between sand grains. Around 130–200 kg cement is required for each 1 m3 mortar depending on cement-sand ratio. Cementation adds extra strength and stiffness to bonded granular materials. This gives higher loading resistance for cemented granular materials compared to uncemented granular materials [14,15]. Further, influence of compressive stress on maximum shear modulus of cemented materials, Gmax, decreases as cement-sand ratio increases. Higher amount of cement allows sufficient inter-particle bonding. As a result, compressive normal stress will have a negligible effect on shear modulus of cemented materials unless it exceeds breaking stress of cementation bonds [16]. Amount of compressive stress has a larger effect on small-strain shear modulus of cemented sands than uncemented sands [16]. Fig. 3, schematically shows effects of cementation on shear modulus, G, and damping of cemented sand, ξ (original work from Saxena et al. [17]). Highly cemented sands (e.g. mortar) are more brittle, and thus begins to degrade at a far lower shear strain, γ, compared to uncemented sand (see Fig. 3a). Increasing cement-sand ratio also results in an increase in damping at lower strain levels (see Fig. 3b). Experiments by Chang and Wood [18] showed significant influence of inter-particle contacts and bonding network on low-strain shear modulus of cemented sands. Effective grain size, D10, coefficient of uniformity, Cu, cementsand ratio, and cement type were reported as the most important factors affecting inter-particle network and small-strain shear modulus of cemented sands [18]. Numerous bond breakages were reported after initial small strains, and bonding between granular materials were gradually removed during plastic distortional or volumetric straining beyond the yield surface [19]. Fig. 4 shows the results of triaxial experiments carried out by Huang and Airey [20]. As seen, artificial cementation (e.g. increasing cohesion between granular materials) increases yield surface and alters its shape compared to uncemented sand. Cracks form and grow parallel to the direction of the largest compressive stress in any material under loading. This crack formation and growth leads to increase of the material’s volume, referred to as dilatancy. Dilatancy plays a significant role in mechanical properties and

2.2. Inter-particle cementation

Tensile failure of mortar joints and brick-mortar interfaces is preceded by micro-cracking. These micro-cracks increase in size and number, are localized as their deformation increases, and finally merge into one large crack. In mortars, cracks are developed at weaker regions i.e. between aggregates and cement paste, where stress concentrations are very high. Fig. 2a schematically shows normal stress-displacement relationship under a uniaxial tension test for a quasi-brittle material such as mortar joint. The normal stress, σ, gradually reduces to zero after the peak stress point, i.e. so-called softening phenomenon. The normal stress-displacement behaviour of the brick-mortar interface under uniaxial tension is different from mortar. This is because micromechanical properties of the brick-mortar interface are significantly affected by curing process. Fig. 2b schematically shows normal stress-displacement relationships of the brick-mortar interface under different loading boundary conditions. As shown in Fig. 2b, different boundary conditions of testing device cause a clear distinct between normal stress-displacement relationships of the same sample. The hinged boundary condition causes the development of one crack while fixed boundary condition gives more plastic behaviour and fracture energy (i.e. the energy required to open unit area of crack surface of a material) by multi-crack growth. van der Pluijm [11] reported that fixed boundary condition is the most suitable for masonry behaviour in tension. Further details on the effects of boundary condition of testing devices on strength and stress-displacement of mortar joints and mortar-brick interfaces and relevant challenges can be found in [12] and [11]. In addition to loading boundary condition, many environmental factors affect material behaviour of mortar joints and brick-mortar interfaces under tensile loadings. These environmental factors mainly influence mortar workability throughout bricklaying as well as shrinkage after bricklaying and during curing. However, knowledge on the influence of these factors on the behaviour of mortar joints and brick-mortar interfaces is very limited and needs further research. Moreover, experimental studies [11] showed an irregular interface bonding with a bonded central area smaller than the nominal cross-

Fig. 2. Schematic normal stress-displacement relationships under uniaxial tension for: (a) quasi-brittle material such as mortar, and (b) brick-mortar interface for various boundary conditions [12]. 834

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Fig. 3. (a) Normalized shear modulus, and (b) damping ratio versus shear strain for highly cemented sand (mortar) and uncemented sand.

concrete [33]. The volumetric strain, εv, increases after the yielding point under shear loading, known as delayed dilatancy of mortar joints and brickmortar interfaces (see Fig. 5b, solid line). This state may be the onset of the in-plane failures for mortar joints or brick-mortar interfaces, or starting point of additional loading mechanism into other mortar joints (e.g. head joints and bed joints) around bricks. Numerous bond breakages are produced at large shear strains, and thus many decemented particles and bonded clusters are already detached from their parent bonding network in large shear strains. Hence, lower shear strength is expected at large shear strains (see Fig. 5a, solid line).

behaviour of porous-cohesive-frictional materials such as sand, cemented sand, mortar joints, and brick-mortar interfaces during distortional loading. Dilatancy was experimentally observed in sand [21,22], concrete [23], mortar joints and brick-mortar [24]. At small strains, dilatancy is prevented by the intact bonding network that generates a web-patterned force chain. After yielding at higher strains, dilatancy accelerates. Effects of cement-sand ratio and cement type on dilatancy and strength of cemented sands were investigated using triaxial tests [25–29]. While Portland cement provides more ductile bonding agents between particles, Gypsum cement gives higher shear strength and brittleness. Porosity-cement ratio is also reported as a crucial parameter which affects initial stiffness and unconfined compressive strength of cemented sands [30]. Bond breakages cause a shear strength reduction while dilatancy increases shear strength [28]. Previous laboratory experiments demonstrate that cemented granular materials are dilative after yielding. However, for uncemented sands, dilatancy increases or decreases depending on initial void ratio and shear strain level [31]. Fig. 5, schematically summarizes triaxial testing results of shear loading on highly cemented sands, i.e. mortars [28,31], uncemented sand [32], and

2.3. Shear tests To understand shear behaviour of mortar joints and brick-mortar interfaces, direct shear tests are often conducted on a masonry element of bricks and mortar bed joint (see [24,34-39]). A schematic view of a generic shear test set-up is shown in Fig. 6. In couplet and triplet tests, a masonry element including brick and mortars are loaded by a combined force normal to the bed joint plane, σ, and shear force tangential to the

Fig. 4. Yield surface for artificially cemented sands in triaxial stress invariant spaces, σ' is the mean effective stress and σ'' is the deviatoric stress [20]. 835

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Fig. 5. (a) Shear stress–strain, and (b) volumetric strain of highly cemented sand (mortar), uncemented sand, and concrete.

bed joint plane, τ. The displacements in normal and tangential directions are respectively denoted by dn and dt. The triplet test is the standard test for evaluation of shear strength of masonry joints [40]. The results of direct shear test under different normal compression stresses, σ, are shown in Fig. 7. The yield surface is illustrated in Fig. 7b. Upon shear-slipping along bed joints, brick units in masonry element also undergo upward translation and dilatancy, causing global volume increase (see Fig. 7c). If this dimensional change is prevented, large compressive stresses may build up, and this increases the resistance against slip by Coulomb friction [36]. As demonstrated in Fig. 7a and reported in [24], shear strength, mode II fracture energy, cohesion softening, and dilatancy are dependent on the amount of normal stress. Softening part of shear stress-displacement (See Fig. 7a), and subsequently mode II fracture energy become less steep as the normal stress increases. Experimental results by Atkinson et al. [35] and van der Pluijm [24] showed that dilatancy is a function of tangential plastic relative displacement, dtp, and normal stress, σ. It was demonstrated that increasing normal stress and total plastic strain, a transition from joint dilation to joint compaction takes place [3,41,42]. The loss of cohesion, in adhesive joints of brick-mortar interface, is generally accompanied

by local phenomena such as crack kinking, voids, and micro-debonding leading to formation of rough fracture surfaces. Effects of the roughness on frictional joint response are well documented in rock mechanics. Therefore, one possible solution to modelling brick-mortar interface after onset of de-bonding is to use asperity (i.e. unevenness of surface or roughness in material science) degradation and discontinuity concepts available in the rock mechanics [43–45]. Use of shear test apparatus of rock mechanics for investigation of shear behaviour of mortar joints was proposed by Atkinson et al [35], and accordingly, a series of shear tests were carried out and reported in [46].

2.4. Failure modes Failure modes of mortar joints and brick-mortar interfaces depend on direction and amplitude of normal and tangential stresses on masonry units. Lourenco and Rots [47] and van der Pluijm [24] listed possible modes of failure in a mesoscale masonry elements including two bricks and a bed joint of mortar: (i) cracking in the joint, (ii) sliding along a bed joint at low values of normal stress, (iii) cracking of the bricks in shear, (iv) splitting masonry units in tension as a result of mortar dilatancy at high values of normal stress, and (v) diagonal

Fig. 6. Schematic view of the shear test setup for mortar joints, and brick-mortar interfaces: (a) couplet test, and (b) triplet test. 836

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Fig. 7. Shematic view of shear tests for mortar joints and brick-mortar interface [24]: (a) shear under constant compression tests, (b) yield surface, and (c) dilatancy.

tension cracking of masonry units at sufficient values of normal stress for development of friction in joints. Fig. 8 illustrates different failure modes along the brick-mortar interface, through the mortar, or in bricks observed in experiments by Fouchal et al. [37]. These failure modes includes all possible modes due to the combination of shear, tension and compression loadings such as those mentioned by [47] and [24] except cracking of the bricks alone. Mann and Mueller [10] suggested a failure envelope curve for a mesoscale masonry element of two masonry units and mortar joints (see Fig. 9). The tensile and friction failures of the bed joints and cracking of the bricks were considered. The basic mechanism of small to large deformations of a masonry element is mainly linked to: (1) shear failure of the bed joints, (2) tensile failure of the brick, and (3) crushing of the masonry element. A comprehensive comparison between experimental results of couplets and triplets tests for full and hollow bricks demonstrates possible reasons for crack and slippage growth in bed joints of a masonry element under combined tension and shear loadings [24,34,35,38,39]:

Fig. 9. Failure envelope for a mesoscale masonry element containing two masonry units and mortar joints [10].

Fig. 8. Different failure modes of fracture of a msaonry unit: (a) interface failure and shear failure of mortar joint, (b) interface failure only, (c) interface failure and tensile failure of bricks, and (d) failure in mortar joints. 837

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(1) Damage on mortar-brick interface due to compression, shear, or tension stresses. (2) Plastic friction/slip and dilatancy on mortar-brick interface (3) Plastic friction/slip in mortar joints (4) Dilatancy on mortar joints (5) Microscopic structure of mortar-brick interface 3. Mortar and mortar-brick interface modelling As previously mentioned, mortar joints consist of more or less rigid particles (e.g. sand) surrounded by cement and voids. The bonding network of mortar joints remains intact at small strains. Thus, all sand particles in the bonding network can withstand loading, and a webbedpattern force-chain distribution is induced. As the strain increases, the force-chain distribution gradually changes, inter-particle cementing starts cracking, and consequently volume changes (dilatation/contraction) occur. This cracking behaviour significantly affects mechanical properties of mortar joints and brick-mortar interfaces. Thus, the first step in achieving an appropriate numerical model for mortar joints and brick-mortar interfaces is to develop an approach for modelling mortar cracking. Afterwards, suitable models need be developed based on mechanics of the chosen modelling approach to link between stress and relative displacement of mortar joints and brick-mortar interfaces, i.e. constitutive models. Correct choice of relative displacement increment and stress variables are very important in development of suitable constitutive models [48]. Many crack modelling approaches and constitutive models for mortar joint and brick-mortar interfaces exist in the literature. However, herein, the focus is on the most common approaches and models that are implemented in finite element programs. Further, constitutive models developed based on thermodynamic laws and coupling of adhesion, friction, and unilateral contacts [37,49,50–52], which account for roughness and micro-cracks, are not discussed here. These constitutive models are beyond the three main fundamental concepts of plasticity theory, fracture mechanics, and damage theory, and may not be general for different loading conditions and material parameters. Therefore, in this section, crack modelling approaches and current micromechanical constitutive models using concepts of plasticity theory, fracture mechanics, and damage theory are discussed in detail.

Fig. 10. Zero-thickness element for modelling mortar joints and brick-mortar interfaces: (a) isoparameteric interface model, and (b) four-node/six-node zerothickness interface element.

(SCFEM) increases number of nodal points, unknown degrees of freedom, and computational time. One technique to reduce computational cost is to model mortar joints and mortar-brick interface by zerothickness element (see Fig. 10a). On the other hand, mortar-brick interface may be modelled by an interface element where brick and mortar are modelled by separated continuum approach (see Fig. 10b). Zero-thickness interface element has some shortcomings: (a) failure of the brick-mortar interface is not distinguished from that of the mortar layer itself, (b) brick-mortar interaction cannot be considered, and (c) tensile splitting of the brick units under compression cannot be modelled [6]. Hence, some material properties for continuum elements representing the brick reflect the properties of masonry element rather than those of brick itself. 3.2. Plasticity-based discrete constitutive models Several plasticity-based constitutive models exist in the literature. Stankowski et al. [61] developed a plasticity-based zero-thickness interface model using concepts of kinematic hardening and softening, in which a single yield surface is described in stress space. Lotfi and Shing [1] extended Stankowski et al.’s model [61], considered different yield surface and plastic potential surface (i.e. the rate of plastic deformation functions) through concepts of isotropic-kinematic hardening and softening, and incorporated softening mechanism due to tensile strength degradation and frictional strength degradation. They thus developed a nonlinear-dilatant interface constitutive model to simulate initiation and propagation of fracture, and capture combined normal and shear stresses as well as the dilatancy seen in experimental tests by Atkinson et al. [35].

3.1. Crack modelling approaches Several approaches are used to model cracking of mortar. Smeared crack approach is used for implementation in a finite element programme where cracking is described within a continuum medium [53]. Two important concepts of decomposed strain and total strain are employed to develop constitutive models using smeared crack approach [53]. Although the smeared crack is a powerful approach for modelling mortar cracking, it has some drawbacks. Numerous problems inherent to the smeared crack models such as mesh size dependency of numerical solution [54], failure to capture diagonal shear cracks [55], directional bias, spurious kinematic modes, and stress locking [56] have been identified in the literature for quasi-brittle materials of mortar, brick, and concrete. Discrete crack approach is more suitable for modelling crack in a quasi-brittle material which is based on the theory of fracture mechanics and fictitious crack model [57]. Unlike classical smeared crack approaches, the discrete crack approach does not exhibit strong mesh sensitivity [54,58]. The only disadvantage of discrete crack approach is to modify finite element mesh at each crack increment. This modification can be avoided in some cases by introducing interface element along all possible paths at the beginning of any analysis (see for example [59;60]). Cracks in masonry structures may be located at brick-mortar interfaces, mortar joints, or mortar-brick together (see Section 2.4). Discretization on brick and mortar in a Single-Scale Finite Element Model

3.2.1. Single-yield surface models In this section, single-yield surface models are mathematically described. Plasticity-based discrete models in [61] and [1] decompose relative displacement between top and bottom faces of interface into elastic and plastic parts: (1)

d = de + d p T

in which d = { dt dn } ; T stands for transpose of a matrix or vector; dt and dn are relative tangential and normal displacements to interface, respectively. The hypoelastic constitutive relationship is given in terms of incremental stresses, δσ, and incremental elastic relative displacements, δd: (2)

δσ = D eδd

τ and σ are shear and normal stresses, respecwhere σ = { τ tively. Assuming isotropic hypoelasticity, the elastic stiffness matrix, De, is given by:

σ }T ;

Dtt 0 ⎤ De = ⎡ ⎣ 0 Dnn ⎦

(3)

Dtt and Dnn are elastic interface parameters, respectively elastic 838

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shear stiffness and normal stiffness, which can be assumed as a function of stress state as well as a number of internal variables. It is also assumed that there is no coupling between normal and tangential components due to zero dilatancy in elastic range. Stankowski et al. [61] introduced a fracture criterion using a curvilinear extension of MohrCoulomb yield criterion. The yield criterion contains a tension-cutoff, termed as the state vector, q = { s c }T , which controls evolution of the yield surface, F:

F (σ , q) = |τ|s0/ μc0 − c0s0/ μc0/ s0 (s − σ )

(4)

in which, μ is the slope of asymptotes of yield surface. s and c are the interface tensile resistance and cohesion, respectively; s0 and c0 are initial interface tensile resistant and cohesion, respectively. μ is assumed constant during evolution of plasticity and crack growth in the interface element. Dilatancy was accounted for through non-associated flow rule (i.e. the vector of the plastic strain rate is not normal to the yield surface) [61], and the plastic potential surface, Q, is given by:

Q (σ , q) = μ/ υ |τ|s0/ μc0 − c0s0/ μc0/ s0 (s − σ )

(5)

in which, υ is dilatancy coefficient. The interface model of Stankowski et al. [61] considers degradation of tensile strength as the only mechanism for fracture and slip of the interface. Material parameters of the model can be experimentally obtained using uniaxial tension and shear tests. However, dilatancy coefficient, υ, friction coefficient, μ, and state vector, q, must be extrapolated from curve fitting of additional combined tension-shear and compression-shear tests. Going a step further, Lotfi and Shing [1] developed a model based on a hyperbolic Mohr-Coulomb yield criterion. The model also includes a tension-cutoff including three internal variables, q = { s r μ }T , and the yield surface is given by:

F (σ , q) = τ 2 − μ2 (σ − s )2 + 2r (σ − s )

Fig. 11. Elastic tension-compression behaviour of interface model developed by Mehrabi and Shing [4].

geometric part, dg:

dg

(6)

Dnn =

(8)

D emnT D e − pT t

nT D em

(11)

in which, h is closing distance of the interface, and dn is the normal component of the elastic relative displacement. When compressive strength of the masonry material is reached, Dnn remains constant. To account for normal contraction of the interface under shear sliding, Mehrabi and Shing [4] used a different plastic potential surface to that adopted by Lotfi and Shing [1]. An elliptical function independent of state vector, q, was defined for the plastic potential surface to reduce compaction of the interface material with higher shear strength:

(7)

and,

D ep = D e −

dne ⩾ 0 ⎧ knn, h ⎨ d e + h knn, dne < 0 ⎩ n e

in which, η is the material parameter which scales dilatancy and controls direction of the plastic flow. The elastoplastic constitutive relationship is given in terms of incremental stresses and incremental relative displacements:

δσ = D epδd

dng }T .

= {0 This geometric vector accounts for rein which, versible shear dilatancy, and is applied after fracture where the dilatation appears. In this constitutive model, the elastic response of the interface is due to compressive stress, and the total shear dilatation is combined effects of normal compaction and geometric dilatation. The geometric dilatancy is expressed as a function of plastic shear displacement and inclination angle of asperities. To address compressive hardening behaviour of the interface, Mehrabi and Shing [4] assumed a constant elastic shear stiffness and a constant elastic normal stiffness, knn, in tension. However, as shown in Fig. 11, the elastic normal stiffness increases exponentially when in compression:

The radius of the yield surface, r, is determined using cohesion, c, slope of the yield surface, μ, and interface tensile resistance, s. The evolution of s is expressed as work-softening variables due to the tensile strength degradation, mode I, and mode II fracture energies. Further, the evolution of r and μ is determined by work-softening variables due to the frictional strength degradation. In this model, initial and residual states of internal variables are respectively stated as q0 = { s0 r0 μ0 }T and qr = { 0 rr μr }T . Like Stankowski et al.’s model, a non-associated flow rule is used, and the plastic potential surface is:

Q (σ , q) = ητ 2 + (r − rr )(σ − s )

(10)

d = de + d p + d g

Q (σ ) = 0.5[ητ 2 + (σ + a)] (9)

(12)

in which, a is a small positive constant which quantifies direction of the plastic flow when τ = σ = 0. In Mehrabi and Shing’s model, the evolution of state vector, q, is governed by the same softening rules for tensile strength degradation and frictional strength degradation as in [61] and [1]. However, a slight modification was made on the compression-shear region where shear capacity and residual shear capacity were increased. The evolution of inclination angle of asperities was also defined as a function of cumulative plastic distortional work. Koutromanos et al. [62] extended Mehrabi and Shing’s model [4] adding crack opening and closing, and shear sliding. This was done by decomposing into normal relative displacement:

in which, n and m are unit vectors; n is normal to the yield surface, ∂F/∂σ, and m is normal to unit direction of the plastic potential surface, ∂Q/∂σ; p = ∂F/∂q and t = E(∂q/∂κ)m where κ is plastic work, and E is the softening modulus determined from plastic work considering tensile strength degradation and frictional strength degradation. The constitutive model of Lotfi and Shing [1] has three main drawbacks: (i) reversible shear dilatancy, (ii) compressive hardening behaviour of interface, and (iii) normal contraction of interface under shear sliding. Mehrabi and Shing [4] put effort into improving these shortcomings. To consider reversible dilatation due to the wedging action of asperities, Mehrabi and Shing [4] added an extra component to the relative displacement of interface (expressed in Eq. (1)) using a

σ = −Dnn 〈dn1 − dn〉 + Dnn 〈dn − dn2〉 839

(13)

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in which, σ¯r is residual compression strength; κm, σ¯m , and w are material parameters, and include hardening/softening law in the cap model (i.e. cap model captures hardening/softening behaviour of material under compression). Associated flow rule (i.e. the vector of plastic strain rate is normal to the yield surface) was used for tension and compression behaviours, and non-associated flow rule was considered for shear behaviour. Plastic potential surface was assigned to shear mode by introducing dilatancy angle, φ:

in which, 〈. 〉 is Macaulay bracket; dn1 and dn2 are additional internal variables which are related to crack opening and closing during loading-unloading-reloading mechanism in zero-thickness crack model. Koutromanos et al. [62] also removed parameter a in the plastic potential surface (see Eq. (12)), and accordingly recommended a new formula for evolution of the inclination angle of asperities. The direction of plastic flow for an interface under tension was given by:

mn = σ e /[Dnn (σ e / Dnn )2 + (τ e / Dtt )2 ]; mt = τ e /[Dtt (σ e / Dnn )2 + (τ e / Dtt )2 ]

Qs (σ ) = |τ| − σ tan(φ) − σ¯s (κs )

(14)

The dilatancy angle is as a function of plastic relative tangential displacement, dtp, and normal stress. The dilatancy angle can be determined as [36]:

where mn and mt are normal and tangential components of unit direction of the plastic potential surface; σe and τe are elastic predictor tractions for normal and tangential directions, respectively. Recently, Zhai et al. [7] modified Koutromanos et al.’s model [62]. They accounted for the degradation of the elastic tensile stiffness where a memory variable named maximum normal displacement, dnM, is reached during loading. This was implemented by modifying Eq. (13):

− Dnn 〈dn1 − dn〉 + Dnn 〈dn − dn2〉, dn < dnM σ=⎧ Dnn (dn − dnp), dn ⩾ dnM ⎨ ⎩

p

tan(φ) = tan(φ0 ) 〈1 − σ / σu 〉 e−αdt

All models discussed above [1,4,62,7] lack experimental calibration their material parameters. Experimental data for calibration of the material parameters is not readily available [6] as material parameters depend on the size of aggregates, composition of mixture, and etc. Hyperbolic yield surfaces were used in zero-thickness interface models, where it has considerable benefits compared to multi-surface yield functions [63]: (1) no discontinuities on the tip of the yield functions, so numerical solution is straight forward and does not need extra computation time, and (2) asymptotically equivalent to Mohr-Coulomb yield criterion. 3.2.2. Multi-yield surface models Despite advantages of hyperbolic yield surfaces, Lourenco and Rots [47] proposed a multi-surface yield function defined by three different yield surfaces for tension, shear, and compression. Using similar approach for composite plasticity model of concrete [64,47], they developed a zero-thickness interface element using following yield functions: (16a)

Fs (σ , q) = |τ| + σμ (κs ) − σ¯s (κs )

(16b)

and,

Fc (σ , q) = (σT Pσ )0.5 − σ¯c (κ c )

(16c)

in which, P is a matrix containing a set of material parameters; σ¯t , σ¯s , and σ¯c are yield tensile, shear, and compressive stresses; μ = tan(ϕ) represents friction coefficient where ϕ is the friction angle; q = { κt κs κ c }T is the state vector, controls evolution of the yield surface and plasticity, and contains components of the plastic works for tension, shear, and compression, respectively. The evolution of isotropic hardening variables is determined linking the state vector parameters, q, plastic multipliers for tension, shear, and compression (λt, λs, and λc), and rate of relative displacements:

κṫ = |Δdṅ | = λṫ ; κṡ = |Δdṫ | = λṡ ; κ ċ = σT ω̇ / σ¯c = λ ċ

(17)

in which ω is projection vector; as per Eqs. (16a)–(16c), σ¯t and σ¯s are analogous to fictitious crack mechanics [57] using an exponential decay law. σ¯c is experimentally determined using stress-displacement curve: I

σ¯t (κt ) = s0 e−s0 κt / Gf σ¯s (κs ) =

II c0 e−c0 κs / Gf

σ¯c (κ c ) = σ¯r + (σ¯m − σ¯r

) e w (κ c − κm)/(σ¯m− σ¯r )

μ (κs ) = μ0 + (μr − μ0 )(c0 − σ¯s )

(20)

where φ0, σu, and α are material parameters; φ0 is dilatancy angle at zero normal stress; σu is the compression stress at which dilatancy becomes zero; and α is dilatancy-shear degradation coefficient. The composite plasticity model of Lourenco and Rots [47] is totally composed of sixteen material parameters: (i) two for elastic stiffness, (ii) two for uniaxial tension, (iii) five for pure shear, (iv) four for hardening/ softening of the cap model, and (v) three for the cap model of the yield surface. Oliviera and Lourenco [65] expanded Lourenco and Rots’s model [47] where they introduced unloading surfaces and back-stress for simulation of unloading-reloading behaviour of the interface. The results of the model were compared with static cyclic experiments on three masonry walls (without frames). It was found that the model can reasonably capture stiffness degradation, energy dissipation, and deformed pattern of masonry walls. The material parameters under monotonic loading condition can be determined by uniaxial cyclic experiments under tension and compression, but number of material parameters need be increased for a better fit with experimental results [9]. Once the crack is initiated in the interface element, the shear stress in the brick unit is set to zero in a single load step, in order to achieve the numerical convergence [66]. Even though bifurcations in the load path could not be resolved at certain load increments, it addresses higher amount of computation time as well as a shortcoming of Lourenco’s model. Simplifying the yield surface used in [47], Sutcliffe et al. [67] adopted a linear approximation of nonlinear cap model to study behaviour of masonry shear walls with lower number of material parameters. However, Sutcliffe et al.’s model [67] ignores the softening behaviour. Cahimoon and Attard [66] used linear yield surfaces similar to those suggested by Sutcliffe et al. [67]. Unlike stress-displacement relations used in interface elements, Cahimoon and Attard applied 2D force-displacement relations to nodal points between two elastic triangular meshes. In this case, the yield surface is defined by normal vectors of yield planes and inelastic failure vector. Associated flow rule is set when yielding is initiated by tension and non-associated flow rule is applied in cases of shear and compression. Even though pushover experiments on masonry shear walls showed a promising results, cyclic behaviour of the walls was not validated due to undeniable effects of evolution of cohesion, friction angle, and dilatancy. Further, Cahimoon and Attard‘s model is not computationally efficient as it is applied to pairs of nodal points. Some concepts of rock mechanics were used in the plasticity formulation, and a classical bilinear Coulomb yield surface was recommended [68]:

(15)

Ft (σ , q) = σ − σ¯t (κt )

(19)

(18a)

F1 (σ , q) = |τ| + μσ − c; F2 (σ , q) = σ − s

(18b)

where, μ = tan(ϕ) is the slope of the q = { s c }T controls evolution of the yield surface:

(18c)

(21) yield

q = q0 − Aλ

(18d)

in which, q0 = { s0 c0 840

surface; (22)

}T is

initial state vector; λ = { λ1 λ2 } where λ1 T

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and λ2 are plastic multipliers; Matrix A controls hardening and softening behaviours of the interface [68]. A non-associated flow rule was used where F1 = 0. However, an associated flow rule was adopted for F2 = 0. As a result, the plastic relative displacement is given by:

δd p = λ1 ∂Q/ ∂σ + λ2 ∂F2/ ∂σ

mortar and mortar-brick interface due to lack of information on the calibration of material parameters. In an attempt to develop a zero-thickness interface element for discrete crack analysis or model one crack plane in smeared crack approach, Carol et al. [63] proposed a constitutive model using nonlinear fracture mechanics and mixed-mode fictitious crack model. Fracture energy was used to control the evolution of tensile strength and cohesion, and two material parameters governed the rate of evolution. The yield surface was given as in [71] and [72] (see Eq. (25)), in which, μ = tan(ϕ) is the slope of asymptotes of the yield surface. Constant slopes were initially assumed. However, Lopez et al. [75] considered μ as a function of fracture energy and a material parameter. Generally, three internal variables, q = { s c μ }T , control the evolution of the yield surface. On crack and plasticity development, Carol et al. [63] assumed tha: (i) all fracture energy dissipated in the crack goes into fracture process when the interface is in tension, and (ii) a part of fracture energy is dissipated by shear work when the interface is under compressive stress. The associated flow rule was used for the interface model in tension, and non-associated flow rule was used for the interface model in compression. The dilatancy was removed when compressive normal stress approaches a threshold value, σdil, as a material parameter. Dilatancy is controlled by following differentiation over the plastic potential surface [63,75]:

(23)

and, the plastic potential surface is given by:

Q (σ , λ ) = |τ| + σ tan(φ) − r

(24)

It is assumed that tan(φ) decreases linearly as cohesion, c, decreases. The wedge asperity model, in which two plates slide relatively along asperities, was applied to simulate rough joint behaviour. The evolution of cohesion and dilatancy are controlled using values of shear and tensile stresses. The hyperbolic wedge asperity model [69], was implemented to simulate rough joint response and describe evolution of frictional interface response. The roughness of the contact surface was considered using additional geometric dilatancy and residual tangential strength. The constitutive model of the interface exhibits a good agreement with experimental results in [35]. However, at the first cycle of loading, the model is unable to simulate irreversible joint thickness decrement due to compression stresses. In a different study, Sheih-Beygi and Pietruszczak (2008) proposed an advanced constitutive model, and addressed pre and post-localization behaviour (the onset of localization is associated with the formation of macrocrack) using pre and postfailure yield criteria [70].

∂Q ∂F dil dil ∂Q ∂F = f f ; = ∂σ ∂σ σ c ∂τ ∂τ

3.3. Fracture mechanics-based joint constitutive models

(28)

fσdil

controls the amount of dilatancy, and is dewhere, coefficient fined as a function of σ/σdil and a material parameter. Coefficient fcdil is defined as a function of c/c0 and a material parameter. Although Carol et al. [63] and Lopez et al. [75] models show good agreement with mixed-mode fracture tests for concrete specimens under monotonic loading conditions [76], the constitutive models of interface have not been formulated for masonry yet and also need be experimentally validated for cyclic tests. Using evolution laws [77,78], Carol et al. [63] developed a 3D constitutive model including a cap model, non-associated flow rule for shear-tension loading mechanism, and associated flow rule for compression behaviour. Compression strength of masonry element (instead of brick-mortar interface or brick only) was used to represent their cap model. This cap model was then employed in the authors’ nonlinear interface element. Due to trigonometric function used for evolution law, post-peak softening of the interface element differs from brick-mortar interface tests.

Cervenka and Saouma [71] and Cervenka et al. [72] developed a fracture mechanics joint model where a hyperbolic function and two internal variables, q = { s c }T , were used to define a yield surface:

F (σ , q) = τ 2 − (c − μσ )2 + (c − μs )2

(25)

The evolution of tensile resistance, s, and cohesion, c, depend on a parameter named effective plastic displacement, deffp. The rate of the effective plastic displacement is defined as a norm of rates of plastic relative displacements. Cervenka et al. [72] assumed a constant frictional coefficient, μ, and divided plastic relative displacement vector, dp, into two components of unrecoverable displacement, dup, and fracture displacement, df, which is recoverable only in tension. An elastic deterioration parameter, ρ, is defined using damage parameter, χ, and the hypoelastic constitutive relation in Eq. (2) is rewritten as:

δσ = ρD eδd = (1 − χ 〈σ 〉/|σ|) D eδd

(26) 0

Damage parameter is defined by initial normal stiffness, Dnn , tensile resistance, s, and effective plastic displacement, deffp: 0 χ = 1 − s /[s + dnf deffp Dnn /(dnp, u + dnf )]

dpn,u

f

3.4. Damage-based constitutive models Gambarotta and Lagomarsino [3] developed a damage model for mortar joints and brick-mortar interfaces. The mortar joint model is characterized by frictional dissipation and stiffness degrading under compressive stresses, and a brittle behaviour under tensile stresses. The constitutive model of the mortar joint and brick-mortar interface is described by a plane stress condition which includes stress-strain relationships. Plastic strains are expressed as:

(27) p

f

where and dn are normal components of du and d , respectively. A non-associative flow rule was assumed which considers effects of initial normal and tangential stiffnesses, and dilatancy angle. The evolution of the dilatancy angle is a function of effective plastic displacement and a threshold effective plastic displacement, in which the dilatancy angle becomes zero. This constitutive model [72] has twelve material parameters including elastic stiffness (normal and tangential), dilatancy (threshold relative displacement and initial dilatancy coefficient), initial state parameters (friction coefficient, tensile strength and cohesion), fracture energies (mode I and mode II), and residual cohesion and tensile resistance as well as their relative displacements. Puntel et al. [73] modified the fracture mechanics-based model of Cervenka et al. [72]. To address reverse cyclic loading and accompanying macroscopic surface degradation, they combined Cervenka et al.’s model [72] with a frictional-based model [41]. An asperity curve characterizing first-order joint roughness [41,74] was also used to link plastic irreversible normal and tangential displacements. None of the fracture mechanics-based models above [71–73] has been evaluated for

p

g(χ ) H(σn ) σn ⎫ εn ε p = ⎧ p⎫ = ⎧ ⎨ ⎬ ⎨ θ ε t ⎩ ⎭ ⎩ (χ ) (τ − f ) ⎬ ⎭

(29)

in which, g and θ are positive functions which represent opening and sliding of mortar joints as a function of the damage variable, χ. H is Heaviside function, and f is the frictional sliding in internal joints. This constitutive model has two internal variables of f and χ. The evolution of these variables depend on a frictional strength limit and a damage condition. The damage evolution is defined based on some simplifications over R-curve approach. The R-curve approach is imposed by limiting the release rate of damage energy (independent of damage variable) to be less than or equal to the mortar joint toughness [3]. The 841

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Gambarotta and Lagomarsino’s model [3] has five material parameters. Even though the constitutive model exhibits lateral stress versus lateral relative displacement curves, which are in good agreement with experimental results [35], it ignores dilatancy and effects of shearing on axial displacement and volumetric strains. In another attempt to use damage theory for simulating behaviour of masonry structures [79,5], a phenomenological interface model was developed by addressing mortar-brick interface as a main part of crack growth in masonry. Combined effects of damage and sliding friction were addressed in a cohesive zone model through a de-cohesion process. In the interface constitutive model, stress-relative displacement relationship is decomposed into undamaged and damaged parts of the interface area, and the damage is quantified by a damage factor, χ:

σ u = D eδd, χ=0 ⎧ d = D e (δd − δd p ), 0 < χ ⩽ 1 ⎨ σ ⎩

(30)

Combining damaged and undamaged stress vectors relations gives:

σ = (1 − χ ) σ u + χσ d = D e (δd − χδd p)

(31)

in which, dp contains unilateral contact displacement, dnp, and plastic sliding displacement, dtp. The evolution of the damage factor, χ, is determined coupling fracture modes I and II as a function of relative displacement history. For the evolution of plastic sliding displacement, Classical Coulomb yield function using the damaged stress, σd, and a constant friction coefficient following a non-associated flow rule were used. This constitutive model has seven material parameters and has not been experimentally validated using testing data such as those reported in [35].

Fig. 12. Dilatancy curves for two stress paths.

interfaces and mortar joints. Mortar joints and brick-mortar interfaces under any loading results in volumetric strains development, overcomes inter-particle friction, and eventually degrades inter-particle cementation. There is a significant paucity of experimental data in the literature for combined tension-shear stress paths [2,11]. For low compressive normal stresses, brittle failure was found with potential instability of the test set-up [2]. This brittle failure is shown schematically in Fig. 12. The relationship between normal relative displacement and tangential relative displacement is well documented for combined compression-shear stress path as the shape of dilatancy curve is mainly dependent on the shape of asperities either saw-tooth or sine-shaped/spiral [74]. The dilatancy curve is different in tension-shear stress path as tangential stress causes bond breakage. Thus, the number of de-cemented particles increases, and this eventually increases normal relative displacement towards brittle failure, as illustrated in Fig. 12. None of current constitutive models available in the literatures account for shear-tension interaction to model the shear-tension stress path in mortar joints and brick-mortar interface. Furthermore, current constitutive models have many material parameters, which may not be easy to calibrate with standard tests. Hence, when relevant experimental data are not available, it is difficult to calibrate these material parameters.

4. Open challenges for future research Current 2D constitutive models of brick-mortar interfaces and mortar joints do not take into account many important effects. The behaviour of mortar joints and brick-mortar interfaces under external loading is influenced by many environmental parameters and in-situ material properties. The water-induced processes due to freezing/ thawing of water, deposition of pollutants, soluble salts and crystallization are among the environmental parameters. Environmental and weather conditions affect mortar workability during bricklaying, mortar shrinkage after bricklaying, and curing of the masonry constituents. The influence of these factors on the behaviour of mortar joints and brick-mortar interfaces has not been fully understood yet and needs detailed experimental testing. Furthermore, results of experimental studies in the literature [11,13,37] show an irregular interface bonding with a bonded central area smaller than the nominal crosssection of the brick-mortar interface. This phenomenon causes irregularities on distributing cracks on the brick, mortar joints, and mortarbrick interface. This will affect stress-strain of the interface under uniaxial tension, compression, and shear. Results from current constitutive models evidence some common features. Mainly, it is imperative to define a consistent conceptual framework for analysis of mortar joints and mortar-brick interface. Among many features, the most phenomena are bonding degradation and related post-yielding deformational pattern. Effects of unloading-reloading, dilatancy, surface asperities, and crack formation and pattern have not been commonly evaluated, and many theoretical assumptions and hypotheses are involved. Geotechnical engineers evaluate behaviour of cemented granular materials using critical state soil mechanics and stress-dilatancy relationships. However, structural engineers model dilatancy using exponentially decaying functions in forms of plastic tangential relative displacement, plastic work, mode II fracture energy, etc. Therefore, combining inter-particle cementation discussed in geomechanics and rough surface discontinuities in rock mechanics, might lead to develop more consistent and accurate 2D constitutive models for brick-mortar

5. Conclusion Advanced numerical models for masonry structures are fed by constitutive models of mortar joints and brick-mortar interfaces. Hence, these constitutive models are a necessary means for assessment of existing masonry structures and design of new masonry infilled panels. In this research review, recent developments in micromechanical constitutive models for mortar joints and brick-mortar interfaces in the context of plasticity, nonlinear fracture mechanics, and damage theory were discussed in mesoscale. The results of numerous experimental tests were described, and knowledge gaps were identified. Brick-mortar interfaces and mortar joints are complex to model because of response nonlinearity, softening, dilatancy, three-dimensional effects of crack 842

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propagation, and lack of experimental calibration of material models. The numerical micro-modelling of mortar joints and brick-mortar interfaces are not straightforward by current developments and may present inconsistent results due to high variability of material constituents and casting process. On the other hand, current solutions present diverse yield functions, plastic potentials, and evolution laws which are the result of inconsistent assumptions on the pre- and postfailure mechanisms of brick-mortar interfaces. None of current constitutive models available in the literatures accounts for shear-tension interaction to model the shear-tension stress path in mortar joints and brick-mortar interface, while this mechanism is very important particularly in head joints. Furthermore, current constitutive models have many material parameters, which may not be easy to calibrate with standard tests. The difficulties significantly increase in cases of weakly to moderately bonded materials (i.e. lower amount of cement-sand ratio and cement-water ratio) and under the effect of environmental factors where current methods do not present a promising solution. This effect is due to the enhanced sensitivity of mortar joints to even small disturbances induced by small in-situ loading or environmental conditions. It should be noted that microscale material models will presumably change depending on different mixture of cement-sand ratio, water absorption into the brick, and age of the construction. Therefore, a reliable constitutive model must be developed by combining continuum mechanics and material science.

[16]

[17]

[18] [19] [20]

[21] [22] [23] [24]

[25]

[26] [27]

[28]

Declaration of Competing Interest [29]

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

[30]

[31]

Acknowledgement

[32]

The authors acknowledge the support received by the UK Engineering and Physical Sciences Research Council (EPSRC) for a Global Challenges Research Fund [grant number EP/P028926/1: Seismic Safety and Resilience of Schools in Nepal].

[33]

[34] [35]

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