# Tensile Fracture Mechanism of Masonry Wallettes Parallel to Bed Joints: A Stochastic Discontinuum Analysis

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## Abstract

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## 1. Introduction

- To investigate the effect of nonlinear contact properties on the macro behavior of the masonry wallettes subjected to tension parallel to bed joints.
- To gain a better understanding of the tensile fracture mechanism in masonry, considering strong masonry unit–weak mortar joint and weak masonry unit–strong mortar joint systems.

## 2. Computational Framework

#### 2.1. Contact Constitutive Models

#### 2.2. Benchmark Study and Testing Setup

**i**) tensile and shear debonding at the unit–mortar interfaces (failure mode I) and (

**ii**) tensile failure in the mortar joint and cracking at the units (failure mode II). The former was obtained when the weak mortar joint with strong masonry units was used, whereas the latter was observed when weaker masonry units and relatively strong mortar joint were utilized. Typical failure patterns obtained from the experiment are shown in Figure 6a. Furthermore, these two failures modes led to different stress–displacement (or stress–strain) curves, where the failure mode I resulted in an approximately constant residual stress after the peak due to interlocking effect of the units and friction, and the failure mode II demonstrated a sharp descending branch after the maximum tensile stress was obtained (see Figure 6b), as typical in quasi-brittle materials such as concrete. For further details about the testing, readers are referred to [2].

## 3. Statistical Representation of the Material Properties

## 4. Results of the Computational Models

#### 4.1. Failure Mode I: Strong Unit–Weak Bond (SU-WB) Behavior

#### 4.2. Failure Mode II: Weak Unit–Strong Bond (WU-SB) Behavior

## 5. Conclusions

- The proposed modeling strategy allows detailed and realistic (based on comparison with experimental results) fracture patterns of masonry wallettes. This is accomplished by representing the masonry components (masonry units and unit–mortar interfaces) using mechanically interacting discrete polyhedral blocks.
- The inherent uncertainty in the mechanical properties of masonry constituents yields various possible fracture mechanisms. Through this research, all possible fracture patterns of masonry wallettes under tension parallel to bed joints are explored. There is a great economy in performing such a comprehensive study computationally instead of a very broad and costly experimental campaign.
- Stochastic analyses enable us to attain the dispersion of the load-carrying capacity concerning the modeling parameters ($E,{f}_{t,u},{f}_{t,i},c$, and $\varphi $). They highlight both the direct and indirect influence of these parameters on the fracture patterns and stress–displacement behavior. Direct correlation of the parameters such as the bond tensile strength, friction angle, or the unit tensile strength with the capacity reflects the physically expected phenomena. In the future, results of these determined correlations can be used to estimate the capacity of different arrangements of masonry materials without using numerical simulations and may lead to the derivation of new empirical formulas to predict the tensile capacity of masonry parallel to bed joints.
- For the given probability distributions of the material parameters, the fracture mechanisms and their likelihood have been found. Moreover, the influence of the modeling parameters on the tensile strength of masonry has been quantified. However, it should be noted that the results presented within this study may be restricted to the statistical distributions and the parameters used.

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Figure A1.**Influence of block-size on the capacity and stress–displacement response,

**Left**: failure mode I;

**Right**: failure mode 2 (taken from [6]).

**Figure A2.**Influence of morphology on the stress–displacement and fracture mechanism (deterministic discontinuum analysis).

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**Figure 1.**Left: Illustration of a masonry wallette, Right: Discontinuum representation of the masonry.

**Figure 4.**Contact constitutive models. (

**a**) Tension-compression stress–displacement behavior, (

**b**) Shear stress–displacement behavior.

**Figure 6.**Failure modes and macro behavior of masonry wallettes, obtained from the experiment [2]. (

**a**) Left: Failure mode I, Right: Failure mode II, (

**b**) Stress–displacement response of masonry wallettes, corresponding to different failure modes.

**Figure 7.**Histograms and probability distributions of the sampled random variables for WU-SB masonry assemblages.

**Figure 10.**Stress–displacement behavior of SU-WB and WU-SB masonry wallettes under tension parallel to bed joints with different material properties. (

**a**) SU-WB (failure mode I), (

**b**) WU-SB (failure mode II).

**Figure 11.**Fracture mechanism and stress–displacement behavior of SU-WB masonry wallettes (experimental result indicated via a thick red dashed line in part b). (

**a**) Typical fracture mechanism of SU-WB, (

**b**) Macro behavior of SU-WB wallettes (DEM and Experiment).

**Figure 13.**Different fracture mechanisms (FM) of the masonry wallettes subjected to tension parallel to bed joints (WU-SB). (

**a**) Left: FM-1; Right: FM-2, (

**b**) Left: FM-3; Right: FM-4, (

**c**) FM-5.

**Figure 14.**Macro behavior of WU-SB wallettes (DEM results (FM-1) vs. experiment, which is plotted via a thick red dashed line).

**Figure 15.**Effect of contact properties on the capacity for each fracture mechanism. (

**a**) Influence of unit tensile strength (${f}_{t,u}$), (

**b**) Influence of bond tensile strength (${f}_{t,i})$ and cohesion ($c),$ (

**c**) Influence of unit–mortar interface friction angle ($\varphi $).

Random Variable | Prob. Distribution | $\mathbf{Mean}\left(\mathit{\mu}\right)$ | Coeff. of Variation | $\mathbf{Lognormal}\mathbf{Mean}\left({\mathit{\mu}}_{\mathit{l}\mathit{n}}\right)$ | $\mathbf{Lognormal}\mathbf{Std}\mathbf{Dev}\left({\mathit{\sigma}}_{\mathit{l}\mathit{n}}\right)$ |
---|---|---|---|---|---|

$E$ (GPa) | Normal | 4 | 0.30 | N/A | N/A |

${f}_{t,u}\left(\mathrm{MPa}\right)$ | Lognormal | 1.0 | 0.45 | −0.0922 | 0.4294 |

${f}_{t,i}\left(\mathrm{MPa}\right)$ | Lognormal | 0.10 | 0.45 | −2.3948 | 0.4294 |

$\varphi $ (degrees) | Normal | 35 | 0.30 | N/A | N/A |

$c\left(\mathrm{MPa}\right)$ | Lognormal | 0.15 | 0.45 | −1.9893 | 0.4294 |

$\psi $ (degrees) | Lognormal | 2 | 0.30 | 0.6501 | 0.2936 |

Random Variable | Prob. Distribution | $\mathbf{Mean}\left(\mathit{\mu}\right)$ | Coeff. of Variation | $\mathbf{Lognormal}\mathbf{Mean}\left({\mathit{\mu}}_{\mathit{l}\mathit{n}}\right)$ | $\mathbf{Lognormal}\mathbf{Std}\mathbf{Dev}\left({\mathit{\sigma}}_{\mathit{l}\mathit{n}}\right)$ |
---|---|---|---|---|---|

$E$ (GPa) | Normal | 4 | 0.30 | N/A | N/A |

${f}_{t,u}\left(\mathrm{MPa}\right)$ | Lognormal | 0.25 | 0.45 | −1.4785 | 0.4294 |

${f}_{t,i}\left(\mathrm{MPa}\right)$ | Lognormal | 0.20 | 0.45 | −1.7016 | 0.4294 |

ϕ (degrees) | Normal | 35 | 0.30 | N/A | N/A |

$c\left(\mathrm{MPa}\right)$ | Lognormal | 0.30 | 0.45 | −1.2962 | 0.4294 |

$\psi $ (degrees) | Lognormal | 2 | 0.30 | 0.6501 | 0.2936 |

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**MDPI and ACS Style**

Pulatsu, B.; Gonen, S.; Erdogmus, E.; Lourenço, P.B.; Lemos, J.V.; Hazzard, J. Tensile Fracture Mechanism of Masonry Wallettes Parallel to Bed Joints: A Stochastic Discontinuum Analysis. *Modelling* **2020**, *1*, 78-93.
https://doi.org/10.3390/modelling1020006

**AMA Style**

Pulatsu B, Gonen S, Erdogmus E, Lourenço PB, Lemos JV, Hazzard J. Tensile Fracture Mechanism of Masonry Wallettes Parallel to Bed Joints: A Stochastic Discontinuum Analysis. *Modelling*. 2020; 1(2):78-93.
https://doi.org/10.3390/modelling1020006

**Chicago/Turabian Style**

Pulatsu, Bora, Semih Gonen, Ece Erdogmus, Paulo B. Lourenço, Jose V. Lemos, and Jim Hazzard. 2020. "Tensile Fracture Mechanism of Masonry Wallettes Parallel to Bed Joints: A Stochastic Discontinuum Analysis" *Modelling* 1, no. 2: 78-93.
https://doi.org/10.3390/modelling1020006