# Research on Damage Characteristics of Brick Masonry under Explosion Load

Authors: Jifeng Wei; Zhixin Du; Yonghui Zheng; Oundavong Ounhueane
Journal: Shock and Vibration
 (2021)

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## Abstract

As the main structural component of partition wall or load-bearing wall, brick masonry has been widely used in construction engineering. However, brick and mortar are all brittle materials prone to crack. Nowadays, fireworks, gas stoves, high-pressure vessels, and other military explosives may explode to damage nearby structures. Many explosion casualties had shown that the load-bearing capacity of brick masonry decreased dramatically and cracks or fragments appeared. Previous studies mainly focused on noncontact explosion in which shock wave is the main damage element. In fact, the response and damage effect of brick masonry wall under contact explosion are more complex, which attracts more attention now. In order to explore the damage characteristics of brick masonry under explosion load, a series of simulations and verification experiments are conducted. RHT and MO granular material models are introduced to describe the behaviour of brick and masonry, respectively, in simulation. The combination effect of front compressive wave and back tensile wave are main factors influencing the breakage of masonry wall. The experimental results are well in accordance with the simulation results. The front cross section dimension of crater is closely related to the radius of spherical explosive charge. A power function predictive model is developed to express the relationship between the radius of hole and the radius of explosive. Furthermore, with increasing the quantity of explosive charge, the number and ejection velocity of fragments are all increased. The relationship between maximum ejection velocity and the quantity of explosive also can be expressed as a power function model.


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## Body

## 1. Introduction

Brick masonry is a composite structure which consists of brick, mortar, and brick-mortar interfaces. With lots of advantages of easy availability, good durability, low-cost building material, and sound and insulation properties, it has been widely used in civil constructions. However, masonry materials (sintered bricks and mortar) are brittle materials with low ductility which is easy to crack under severe impact or explosion [1–3]. Failure of a masonry wall is likely to be sudden and severe that poses significant debris hazard to building occupants when subjected to blast loads. According to the analysis of human casualties and property losses in explosion accidents, it had been found that the fragmentation of brick masonry also can cause damage to whether human or surrounding devices except high overpressure [4–6].Previous research studies had paid more attention to experiments. Varma et al. [7] tested on several masonry panels with different thickness subjected to near-field and far-field explosions. Gabrielsen et al. [8] investigated experimentally the response of building walls to blast loading from large nuclear weapons. Davidson et al. [9] conducted three explosive tests that involved a sprayed-on polymer to increase blast resistance of unreinforced concrete masonry walls. Dennis et al. [10] conducted both five dynamic experiments to analyse the response and failure of one-way 1/4-scale reinforced masonry walls to blast loads. Ahmad et al. [11] carried out six tests to study the resistance of brick masonry wall exposed to varying blast load at different scaled distances. Keys et al. [12] carried out two experimental trials to obtain a baseline set of results for the breakage mechanisms and debris distribution and developed a new method to predict the spatial debris distribution. Norén-Cosgriff et al. [13] observed the onset of cracking of building and measured air blast overpressure and vibrations on the ground surface.By far many simulations on damage behaviours and phenomena of brick masonry under dynamic loads had been done. Zhou et al. [14] simulated the damage and fragmentation of a typical masonry wall under blast loading at different scaled distances. Keys et al. [15] used applied element method to calculate the breakage and debris distribution of masonry subjected to long duration blast loading. Shadlou et al. [16] presented a brief discussion about the main challenges on modelling the reinforced concrete frames, masonry infills, and interaction between them for structures located in seismic zones. Some scholars [17–20] used brick masonry discrete model to explore the details of damage development process of the masonry and the damage phenomenon at the interface between the brick and the mortar. However, this method would consume lots of central processing unit (CPU) and memory resources of computer during calculations [21]. Other scholars [22–26] adopted equivalent homogenized material model to characterize the overall properties of masonry walls.By now, few researches had been done to study the response of brick masonry under contact explosion loads. In the present study, a series of numerical simulations and experiments are carried out to study the structural response and damage characteristics of brick masonry wall under contact explosion loads. Different explosive charges are deployed in the same contact position to compare the difference among them. The relationship between the dimension of crater and the radius of explosive is obtained. The damage mechanism of brick masonry under explosion load is analysed. Furthermore, the fragmentation and maximum ejection velocities of brick masonry under different conditions are studied.


## 2. Computational Model

### 2.1. Physical Model and Algorithm

The brick masonry wall is composed of mortar and regularly arranged bricks. A separate model is used to establish the brick wall; that is, the brick and mortar are used as independent components. This type of modelling method is intuitive and can accurately simulate the bond between the mortar and the brick. Here, the size of brick is 240 mm (length) × 115 mm (width) × 53 mm (thick). 6 pieces are placed in the length direction of the wall (X-direction), 22 pieces are placed in the height direction (Y-direction), 2 pieces are placed in the thickness direction (Z-direction), and the thickness of mortar is 10 mm. So, the total size of brick wall is 1250 mm × 1490 mm × 240 mm.Both brick and mortar are all divided with Lagrange cells. In Lagrange algorithm, the material is attached to the grid to trace the flow of each mass point. The grid and the material are deformed together and the coordinates move with the material, which is one of the most widely used methods to solve the explosion and impact problems. The boundaries of brick wall are all imposed six degrees of freedom constraints, as shown in Figure1. The spherical composition 4 (C4) explosive is established by Smoothed Particle Hydrodynamics (SPH). As a meshless discretization method, the main advantage of SPH method has advantage to avoid severe problems associated with mesh tangling and distortion which usually occur in Lagrangian analysis involving large deformation impact and explosive loading events. The explosive is attached to the geometric centre of front surface of brick masonry wall and is detonated from the centre of itself. In order to obtain the pressure on the wall and trace the damage zone, a series of gauge points are set along the horizontal and vertical directions of front and back surfaces. The established simulation model is shown in Figure 2.Figure 1
Finite element model of brick masonry.Figure 2
The layout of gauge points.


### 2.2. Material Model and Parameters

There are two materials in this study, C4 explosives (main charge) and brick masonry (including brick and mortar). The material models and parameters used in these materials are described, respectively.


#### 2.2.1. C4 Explosive

As a plastic explosive, C4 is composed of RDX (hexogen), plastic binder, and plasticizer. RDX is the main component of it, about 91%. It is easy to be kneaded into various shapes at will which is often used in breaching obstacles or demolition of large structures. Although this kind of explosive is powerful, it is very safe to use. Even if it is shot directly, it is hard to explode. It can only be detonated with a detonator.The equation of state (EOS) of detonation products is essential for describing the detonation properties of explosives, and it is the core parameter in simulations of detonation process of explosives. In AUTODYN numerical simulation program, there are three EOS suitable for explosives: Becker–Kistiakowsky–Wilson (BKW) EOS, Jones–Wilkins–Lee (JWL) EOS, and Lee–Tarver EOS. Among them, JWL EOS can accurately describe the characteristics of the detonation product volume, pressure, temperature, and energy during the detonation process [27–29]. The main parameters of JWL EOS for C4 explosive are shown in Table 1. Besides, its loading density ρe, detonation velocity De, and CJ pressure PCJ are 1.60 g·cm−3, 8193 m·s−1, and 28 GPa, respectively.Table 1
Parameters of JWL EOS for C4 explosive.
ParameterA (GPa)B (GPa)R1R2ωE0 (J·m−3)ρe (g·cm−3)De (m·s−1)PCJ (GPa)Value609.7712.954.51.40.259.0 × 1091.6819328The expression of JWL EOS is as follows:(1)P=A1−ωR1Ve−R1V+B1−ωR2Ve−R2V+ωE0V,where P is the pressure, E0 is the initial specific energy, V is the relative volume of the detonation product, and A, B, R1, R2, and ω are constants. The parameters A and B have dimensions of pressure, while R1, R2, and ω are dimensionless.


#### 2.2.2. Brick Masonry

Brick masonry is a solid entity composed of many blocks that are bonded by mortar as a whole. RHT strength and failure model is chosen to describe the behaviour of brick material, while Mo granular strength model is used to describe the behaviour of mortar material.RHT model, first proposed by Riedel, Hiemaier, and Thoma, has developed and been widely used in the numerical simulation of explosion shock and other fields. This model is expressed in terms of three stress limit surfaces, namely, initial elastic yield surface, failure surface, and residual strength surface [30, 31]. While the surfaces account for reduction in strength along different meridians as well as strain rate effects, the static compressive meridian surfaces are depicted in Figure 3. The failure surface is formed from material parameters including compressive, tensile, and shear strength of the brick. The model is elastic until the stress reaches the initial yield surface, beyond which plastic strains evolve. When the stress reaches the failure surface, a parameterized damage model governs the evolution of damage, driven by plastic strain, which in turn represents the post-failure stress limit surface by interpolating between failure surface and residual friction surface. There are three stages of linear strengthening and damage softening, as shown in Figure 3.Figure 3
The reference spatial distribution and meridian of RHT constitutive elastic limit surface, failure surface, and residual strength surface.The failure surfaceYfail∗ is expressed as a function of the normalized pressure p, the Lode angle θ, and strain rate ε˙:(2)Yfail∗=YTXC∗p⋅R3θ⋅FRATEε˙,where YTXC∗p represents the compression meridian intensity, R3 (θ) is introduced to describe the reduced strength on shear and tensile meridian, and FRATEε˙ represents the dynamic increase factor (DIF) as a function of the strain rate.The moving elastic limit surfaceYela∗ can be described as follows:(3)Yela∗=Yfail∗FCAPp,where FCAPp is a function that limits the elastic deviatoric stresses under hydrostatic compression.In order to describe the strength of the completely crushed material, a residual failure surfaceYres∗ is defined as(4)Yres∗=B×p∗M,where B is the residual failure surface constant, p∗ is the normalized hydrostatic pressure, and M represents the residual failure surface exponent.The main parameters of brick material are shown in Table2.Table 2
Main parameters of brick material.
Density (g·cm−3)Compressive strength (MPa)Volume modulus (GPa)Shear modulus (GPa)Damage constantD1Damage constantD2Tensile strength (MPa)2.252.611.04.40.0141.06.0MO granular continuum model, proposed by Moxnes et al. in 1999, is an extension of the Drucker–Prager model that considers the effects associated with granular materials [32, 33]. The model uses a hydrostatic compaction curve as a function of the density, a model for the yield stress as a function of pressure and elastic modulus as a function of density. The model does not include any strain rate dependency of yield stress. The parameters are constructed by using a quasistatic unilateral compression test and validated by using a high-speed piston (up to 300 m/s) impacting a granular pyrotechnic bed. The yield stress can be described as(5)σy=σp+σρ⋅F,where σy,σp, and σρ denote the total yield stress, the pressure yield stress, and the density yield stress, respectively, and F is initially supplied by the user as the initial fraction of yield. Failure occurs if the von Mises stress is greater than both the total yield stress σy and FSP, where FS is the user defined variable slope and P is pressure. The material parameters of mortar are shown in Table 3.Table 3
Main parameters of mortar material.
Density (g·cm−3)Poisson’s coefficientVolume modulus (GPa)Shear strength residual coefficient (GPa)Failure principal tensile stress (MPa)1.70.105.92.21.0


## 2.1. Physical Model and Algorithm

The brick masonry wall is composed of mortar and regularly arranged bricks. A separate model is used to establish the brick wall; that is, the brick and mortar are used as independent components. This type of modelling method is intuitive and can accurately simulate the bond between the mortar and the brick. Here, the size of brick is 240 mm (length) × 115 mm (width) × 53 mm (thick). 6 pieces are placed in the length direction of the wall (X-direction), 22 pieces are placed in the height direction (Y-direction), 2 pieces are placed in the thickness direction (Z-direction), and the thickness of mortar is 10 mm. So, the total size of brick wall is 1250 mm × 1490 mm × 240 mm.Both brick and mortar are all divided with Lagrange cells. In Lagrange algorithm, the material is attached to the grid to trace the flow of each mass point. The grid and the material are deformed together and the coordinates move with the material, which is one of the most widely used methods to solve the explosion and impact problems. The boundaries of brick wall are all imposed six degrees of freedom constraints, as shown in Figure1. The spherical composition 4 (C4) explosive is established by Smoothed Particle Hydrodynamics (SPH). As a meshless discretization method, the main advantage of SPH method has advantage to avoid severe problems associated with mesh tangling and distortion which usually occur in Lagrangian analysis involving large deformation impact and explosive loading events. The explosive is attached to the geometric centre of front surface of brick masonry wall and is detonated from the centre of itself. In order to obtain the pressure on the wall and trace the damage zone, a series of gauge points are set along the horizontal and vertical directions of front and back surfaces. The established simulation model is shown in Figure 2.Figure 1
Finite element model of brick masonry.Figure 2
The layout of gauge points.


## 2.2. Material Model and Parameters

There are two materials in this study, C4 explosives (main charge) and brick masonry (including brick and mortar). The material models and parameters used in these materials are described, respectively.


### 2.2.1. C4 Explosive

As a plastic explosive, C4 is composed of RDX (hexogen), plastic binder, and plasticizer. RDX is the main component of it, about 91%. It is easy to be kneaded into various shapes at will which is often used in breaching obstacles or demolition of large structures. Although this kind of explosive is powerful, it is very safe to use. Even if it is shot directly, it is hard to explode. It can only be detonated with a detonator.The equation of state (EOS) of detonation products is essential for describing the detonation properties of explosives, and it is the core parameter in simulations of detonation process of explosives. In AUTODYN numerical simulation program, there are three EOS suitable for explosives: Becker–Kistiakowsky–Wilson (BKW) EOS, Jones–Wilkins–Lee (JWL) EOS, and Lee–Tarver EOS. Among them, JWL EOS can accurately describe the characteristics of the detonation product volume, pressure, temperature, and energy during the detonation process [27–29]. The main parameters of JWL EOS for C4 explosive are shown in Table 1. Besides, its loading density ρe, detonation velocity De, and CJ pressure PCJ are 1.60 g·cm−3, 8193 m·s−1, and 28 GPa, respectively.Table 1
Parameters of JWL EOS for C4 explosive.
ParameterA (GPa)B (GPa)R1R2ωE0 (J·m−3)ρe (g·cm−3)De (m·s−1)PCJ (GPa)Value609.7712.954.51.40.259.0 × 1091.6819328The expression of JWL EOS is as follows:(1)P=A1−ωR1Ve−R1V+B1−ωR2Ve−R2V+ωE0V,where P is the pressure, E0 is the initial specific energy, V is the relative volume of the detonation product, and A, B, R1, R2, and ω are constants. The parameters A and B have dimensions of pressure, while R1, R2, and ω are dimensionless.


### 2.2.2. Brick Masonry

Brick masonry is a solid entity composed of many blocks that are bonded by mortar as a whole. RHT strength and failure model is chosen to describe the behaviour of brick material, while Mo granular strength model is used to describe the behaviour of mortar material.RHT model, first proposed by Riedel, Hiemaier, and Thoma, has developed and been widely used in the numerical simulation of explosion shock and other fields. This model is expressed in terms of three stress limit surfaces, namely, initial elastic yield surface, failure surface, and residual strength surface [30, 31]. While the surfaces account for reduction in strength along different meridians as well as strain rate effects, the static compressive meridian surfaces are depicted in Figure 3. The failure surface is formed from material parameters including compressive, tensile, and shear strength of the brick. The model is elastic until the stress reaches the initial yield surface, beyond which plastic strains evolve. When the stress reaches the failure surface, a parameterized damage model governs the evolution of damage, driven by plastic strain, which in turn represents the post-failure stress limit surface by interpolating between failure surface and residual friction surface. There are three stages of linear strengthening and damage softening, as shown in Figure 3.Figure 3
The reference spatial distribution and meridian of RHT constitutive elastic limit surface, failure surface, and residual strength surface.The failure surfaceYfail∗ is expressed as a function of the normalized pressure p, the Lode angle θ, and strain rate ε˙:(2)Yfail∗=YTXC∗p⋅R3θ⋅FRATEε˙,where YTXC∗p represents the compression meridian intensity, R3 (θ) is introduced to describe the reduced strength on shear and tensile meridian, and FRATEε˙ represents the dynamic increase factor (DIF) as a function of the strain rate.The moving elastic limit surfaceYela∗ can be described as follows:(3)Yela∗=Yfail∗FCAPp,where FCAPp is a function that limits the elastic deviatoric stresses under hydrostatic compression.In order to describe the strength of the completely crushed material, a residual failure surfaceYres∗ is defined as(4)Yres∗=B×p∗M,where B is the residual failure surface constant, p∗ is the normalized hydrostatic pressure, and M represents the residual failure surface exponent.The main parameters of brick material are shown in Table2.Table 2
Main parameters of brick material.
Density (g·cm−3)Compressive strength (MPa)Volume modulus (GPa)Shear modulus (GPa)Damage constantD1Damage constantD2Tensile strength (MPa)2.252.611.04.40.0141.06.0MO granular continuum model, proposed by Moxnes et al. in 1999, is an extension of the Drucker–Prager model that considers the effects associated with granular materials [32, 33]. The model uses a hydrostatic compaction curve as a function of the density, a model for the yield stress as a function of pressure and elastic modulus as a function of density. The model does not include any strain rate dependency of yield stress. The parameters are constructed by using a quasistatic unilateral compression test and validated by using a high-speed piston (up to 300 m/s) impacting a granular pyrotechnic bed. The yield stress can be described as(5)σy=σp+σρ⋅F,where σy,σp, and σρ denote the total yield stress, the pressure yield stress, and the density yield stress, respectively, and F is initially supplied by the user as the initial fraction of yield. Failure occurs if the von Mises stress is greater than both the total yield stress σy and FSP, where FS is the user defined variable slope and P is pressure. The material parameters of mortar are shown in Table 3.Table 3
Main parameters of mortar material.
Density (g·cm−3)Poisson’s coefficientVolume modulus (GPa)Shear strength residual coefficient (GPa)Failure principal tensile stress (MPa)1.70.105.92.21.0


## 2.2.1. C4 Explosive

As a plastic explosive, C4 is composed of RDX (hexogen), plastic binder, and plasticizer. RDX is the main component of it, about 91%. It is easy to be kneaded into various shapes at will which is often used in breaching obstacles or demolition of large structures. Although this kind of explosive is powerful, it is very safe to use. Even if it is shot directly, it is hard to explode. It can only be detonated with a detonator.The equation of state (EOS) of detonation products is essential for describing the detonation properties of explosives, and it is the core parameter in simulations of detonation process of explosives. In AUTODYN numerical simulation program, there are three EOS suitable for explosives: Becker–Kistiakowsky–Wilson (BKW) EOS, Jones–Wilkins–Lee (JWL) EOS, and Lee–Tarver EOS. Among them, JWL EOS can accurately describe the characteristics of the detonation product volume, pressure, temperature, and energy during the detonation process [27–29]. The main parameters of JWL EOS for C4 explosive are shown in Table 1. Besides, its loading density ρe, detonation velocity De, and CJ pressure PCJ are 1.60 g·cm−3, 8193 m·s−1, and 28 GPa, respectively.Table 1
Parameters of JWL EOS for C4 explosive.
ParameterA (GPa)B (GPa)R1R2ωE0 (J·m−3)ρe (g·cm−3)De (m·s−1)PCJ (GPa)Value609.7712.954.51.40.259.0 × 1091.6819328The expression of JWL EOS is as follows:(1)P=A1−ωR1Ve−R1V+B1−ωR2Ve−R2V+ωE0V,where P is the pressure, E0 is the initial specific energy, V is the relative volume of the detonation product, and A, B, R1, R2, and ω are constants. The parameters A and B have dimensions of pressure, while R1, R2, and ω are dimensionless.


## 2.2.2. Brick Masonry

Brick masonry is a solid entity composed of many blocks that are bonded by mortar as a whole. RHT strength and failure model is chosen to describe the behaviour of brick material, while Mo granular strength model is used to describe the behaviour of mortar material.RHT model, first proposed by Riedel, Hiemaier, and Thoma, has developed and been widely used in the numerical simulation of explosion shock and other fields. This model is expressed in terms of three stress limit surfaces, namely, initial elastic yield surface, failure surface, and residual strength surface [30, 31]. While the surfaces account for reduction in strength along different meridians as well as strain rate effects, the static compressive meridian surfaces are depicted in Figure 3. The failure surface is formed from material parameters including compressive, tensile, and shear strength of the brick. The model is elastic until the stress reaches the initial yield surface, beyond which plastic strains evolve. When the stress reaches the failure surface, a parameterized damage model governs the evolution of damage, driven by plastic strain, which in turn represents the post-failure stress limit surface by interpolating between failure surface and residual friction surface. There are three stages of linear strengthening and damage softening, as shown in Figure 3.Figure 3
The reference spatial distribution and meridian of RHT constitutive elastic limit surface, failure surface, and residual strength surface.The failure surfaceYfail∗ is expressed as a function of the normalized pressure p, the Lode angle θ, and strain rate ε˙:(2)Yfail∗=YTXC∗p⋅R3θ⋅FRATEε˙,where YTXC∗p represents the compression meridian intensity, R3 (θ) is introduced to describe the reduced strength on shear and tensile meridian, and FRATEε˙ represents the dynamic increase factor (DIF) as a function of the strain rate.The moving elastic limit surfaceYela∗ can be described as follows:(3)Yela∗=Yfail∗FCAPp,where FCAPp is a function that limits the elastic deviatoric stresses under hydrostatic compression.In order to describe the strength of the completely crushed material, a residual failure surfaceYres∗ is defined as(4)Yres∗=B×p∗M,where B is the residual failure surface constant, p∗ is the normalized hydrostatic pressure, and M represents the residual failure surface exponent.The main parameters of brick material are shown in Table2.Table 2
Main parameters of brick material.
Density (g·cm−3)Compressive strength (MPa)Volume modulus (GPa)Shear modulus (GPa)Damage constantD1Damage constantD2Tensile strength (MPa)2.252.611.04.40.0141.06.0MO granular continuum model, proposed by Moxnes et al. in 1999, is an extension of the Drucker–Prager model that considers the effects associated with granular materials [32, 33]. The model uses a hydrostatic compaction curve as a function of the density, a model for the yield stress as a function of pressure and elastic modulus as a function of density. The model does not include any strain rate dependency of yield stress. The parameters are constructed by using a quasistatic unilateral compression test and validated by using a high-speed piston (up to 300 m/s) impacting a granular pyrotechnic bed. The yield stress can be described as(5)σy=σp+σρ⋅F,where σy,σp, and σρ denote the total yield stress, the pressure yield stress, and the density yield stress, respectively, and F is initially supplied by the user as the initial fraction of yield. Failure occurs if the von Mises stress is greater than both the total yield stress σy and FSP, where FS is the user defined variable slope and P is pressure. The material parameters of mortar are shown in Table 3.Table 3
Main parameters of mortar material.
Density (g·cm−3)Poisson’s coefficientVolume modulus (GPa)Shear strength residual coefficient (GPa)Failure principal tensile stress (MPa)1.70.105.92.21.0


## 3. Results and Discussion

### 3.1. Analysis of the Fracture Process of Brick Masonry

When an explosive is detonated, the expansion of the hot gases at extremely high pressures causes a shock wave to form, moving outward at high velocity. Nearby structures will suffer some degree of damage from it. If the explosive is put on the surface of the structure to explode, higher-pressure, shorter load durations will be obtained. At this time, a part of the shock wave will penetrate into contacted brick masonry. This part of shock wave is so called transmission shock wave. It will also be affected by a reverse incoming sparse wave. This reflected wave will form a reflected shock wave load. The propagation of shock waves not only causes dynamic stress to the medium, but also causes the movement, deformation, or destruction of the medium. Shock wave cannot be supplemented with energy during its propagation process, so it will gradually weaken and finally disappear.When the brick masonry is impacted by the explosion of 2.0 kg C4 spherical charge, the zone suffered from severe impact is damaged to form a crater. Figure4 shows successive frames of fracture development after explosion on the front surface of brick masonry. It can be seen that the dimensions of crater increase gradually; finally, a big hole is formed.Figure 4
The fracture development of brick masonry by 2.0 kg C4 explosive. (a) 0.1 ms. (b) 0.5 ms. (c) 1.0 ms. (d) 2.0 ms. (e) 4.0 ms. (f) 6.0 ms.
(a)(b)(c)(d)(e)(f)The stress contours of brick masonry under explosion loads are shown in Figure5(a). The closer to the centre the element is, the larger the stress is. Accordingly, the larger the strain gets. When the strains of elements are up to failure strain, the elements will be eroded, such as gauge #8, gauge #9, and gauge #23 on the front surface close to the centre of brick masonry (in Figure 5(b)). The strains of other gauges are lower than failure strain, so they do not fail. The zone with a large number of continuous eroded elements is called a crater. The more the effective strains of elements reach failure strain, the larger the carter is. Figure 5(c) shows that the compression behaviour of different gauges. When the value of compression is below zero, the gauges suffer from the effect of tensile stress. When the crater breaks, a few fragments will be extruded from it. The maximum ejection velocity is about 16.2 m·s−1, as shown in Figure 5(d).Figure 5
Simulation results of 2.0 kg C4 explosion load. (a) Contour of stress distribution. (b) Effective strains of gauges. (c) Compression behaviour of gauges. (d) Ejection velocity of fragments.
(a)(b)(c)(d)Figure6 presents the dimensions change of damage zone on the front surface when brick masonry wall is damaged by 2.0 kg C4 spherical charge. With the increase of time, three dimensions become larger and larger. The 240 mm thickness wall cracks at about 2.0 ms while the crater reaches its maximum cross section at about 5 ms. Before the crater breaks, it has reached the maximum cross section surface, beyond 400 mm. The growth rates of X and Y dimensions are almost the same, which are larger than that of Z dimension. So, the shape of cross section of crater is close to be circular.Figure 6
The dimensions in three directions of crater for 2.0 kg C4 explosion load.


### 3.2. Damage Zone of Brick Masonry under Different Charges

In order to study the damage characteristics of the brick masonry under different explosion loads, five C4 spherical charges, 0.5 kg, 1.0 kg, 2.0 kg, and 4.0 kg, are designed. The radii of these charges are 42.10 mm, 53.04 mm, 66.82 mm, and 84.19 mm, respectively. These charges are all placed on the centre of front surface of brick masonry wall to explode.Figure7 gives damaged pattern of the back of brick masonry wall subjected to different explosion loads. As we know, shock wave will be produced when explosive is detonated and then turns to compression wave in brick masonry. When stress wave transmits to the back free surface of brick masonry, a reverse tensile wave is inevitably formed to stretch the back to go farther. When the quantity of charge is smaller, such as 0.5 kg, a small crater is found on the front face just at the contact position. Seen from the back surface of the wall, there is a small hole and few fragments are extruded. When the quantity of charge increases to 1.0 kg, a larger crater is formed on the front. Viewing from the back, the mortar in the centre is extended to form a larger hole and a few fragments are ejected. At the same time, it can be found that left and right sides of masonry wall are slightly affected by such a high explosion. That may be due to the bond strength of transverse mortar layer being lower than that of longitudinal brick-mortar interaction. If 2.0 kg explosive charge is used, deeper crater appears with more distinct bulge on the back. The crater breaks to form more fragments and two sides of masonry wall are heavily affected along transverse mortar layers. When the charge reaches 4.0 kg, an even larger crater appears. The sizes of back fracture zone and hole are all larger than above results.Figure 7
Destruction diagram of brick masonry under different explosive loads. (a) 0.5 kg. (b) 1.0 kg. (c) 2.0 kg. (d) 4.0 kg.
(a)(b)(c)(d)Next, two experiments were conducted to investigate the damage characteristics of masonry wall. 1.0 kg and 2.0 kg spherical explosive charge were chosen to be detonated on the surface of masonry wall. Crater holes are formed under such two equivalent explosives, as shown in Figure8. Compared with the simulation results in Figure 9, the experimental results are in good agreement with them.Figure 8
Experimental results under two different explosion loads. (a) Front view of 1.0 kg. (b) Back view of 1.0 kg. (c) Front view of 2.0 kg. (d) Back view of 2.0 kg.
(a)(b)(c)(d)Figure 9
Simulation results under two different explosion loads. (a) Front view of 1.0 kg. (b) Back view of 1.0 kg. (c) Front view of 2.0 kg. (d) Back view of 2.0 kg.
(a)(b)(c)(d)Table4 shows the dimensions of damage zones of brick masonry wall under different explosion loads. With the increase of the quantity of explosive, the lengths in X-direction and Y-direction increase. Moreover, the size of crater on front surface is smaller than that of fracture zone on back surface. When explosive is attached to brick masonry, released energy would be absorbed by masonry wall and thus compressive stress wave would spread in it. Once compressive stress wave reaches the back free surface, an enhanced tensile stress wave will appear due to impedance mismatch between wall and air. The reflected tensile wave from free surface encounters the inbound compressive stress resulting in total stress that is subtractive. When the total stress is tensile (tensile stress greater than inbound compressive stress) and exceeds the tensile strength of any point in the masonry matrix, cracks or fractures will occur. For brittle brick masonry material, the damage area on the back free surface can be larger than that on the front incident surface.Table 4
The damaged zones of brick masonry under different conditions.
Case no.Charge (kg)Crater in front surfaceFracture in back surfaceX-direction (mm)Y-direction (mm)X-direction (mm)Y-direction (mm)I0.5266252366338II1.0364347524447III2.0473468604618IV4.0637630755751


### 3.3. A Simple Predictive Model of the Dimensions of Hole

When the spherical explosive is detonated on the surface of masonry wall, according to the simulation and experiments, the crater hole is basically determined by the radius of the explosive. The simplified model can be described in Figure10.Figure 10
The relationship between the radius of crater hole and the radius of explosive.Since the explosion energy is released on the front surface of contact point and evenly in all directions, the radius of crater hole is merely controlled by the radius of explosive charge. However, the relationship between these two radii is not given clearly.(6)rd∝r0.According to Table4, the dimensions of front surface are drawn in Figure 11(a).Figure 11
Crater on front surface under different charges. (a) Dimensions of holes. (b) Fitting curve.
(a)(b)The ratios of radius of holes to the radius of explosive charges for four cases increase monotonously from 6.32 to 7.57. When fitting these data in Figure11(b), it can be expressed as(7)rd=1.2859r01.24306.Thus, a power function predicting model of the radius of hole is found.


### 3.4. A Simple Fitting Model of Maximum Fragment Velocity

With the increase in the quantity of the explosive, more and more cracks along the longitudinal direction are noticed. Once the hole is formed, the damaged bricks will be blown out to produce a number of fragments. Seen from the viewpoint of energy, the more quantity of explosive is used, the higher explosive energy loading on the surface of masonry wall is produced, and the higher the velocity of fragment gets. That is to say, the ejection velocities of fragments are determined by the radius of explosive too for spherical explosive charge. The fragment with maximum ejection velocity is found at the centre of crater or the apex of crater before breakup.The maximum velocity of fragmentVmax is proportional to the radius of spherical explosive:(8)Vmax∝r0.Figure12 presents the maximum ejection velocity of fragments under different conditions. Adding coordinate point (0, 0), there are five groups of data. It is found that a power function curve fits these data well.Figure 12
The maximum ejection velocities of fragments under different charges and fitting curve.Now, the maximum ejection velocity is power function to the radius of explosive.(9)Vmax=0.00139r02.21747.When the explosive is 4.0 kg, the maximum ejection velocity can reach about 26 m/s, which may cause severe damage to surrounding equipment or personnel.


## 3.1. Analysis of the Fracture Process of Brick Masonry

When an explosive is detonated, the expansion of the hot gases at extremely high pressures causes a shock wave to form, moving outward at high velocity. Nearby structures will suffer some degree of damage from it. If the explosive is put on the surface of the structure to explode, higher-pressure, shorter load durations will be obtained. At this time, a part of the shock wave will penetrate into contacted brick masonry. This part of shock wave is so called transmission shock wave. It will also be affected by a reverse incoming sparse wave. This reflected wave will form a reflected shock wave load. The propagation of shock waves not only causes dynamic stress to the medium, but also causes the movement, deformation, or destruction of the medium. Shock wave cannot be supplemented with energy during its propagation process, so it will gradually weaken and finally disappear.When the brick masonry is impacted by the explosion of 2.0 kg C4 spherical charge, the zone suffered from severe impact is damaged to form a crater. Figure4 shows successive frames of fracture development after explosion on the front surface of brick masonry. It can be seen that the dimensions of crater increase gradually; finally, a big hole is formed.Figure 4
The fracture development of brick masonry by 2.0 kg C4 explosive. (a) 0.1 ms. (b) 0.5 ms. (c) 1.0 ms. (d) 2.0 ms. (e) 4.0 ms. (f) 6.0 ms.
(a)(b)(c)(d)(e)(f)The stress contours of brick masonry under explosion loads are shown in Figure5(a). The closer to the centre the element is, the larger the stress is. Accordingly, the larger the strain gets. When the strains of elements are up to failure strain, the elements will be eroded, such as gauge #8, gauge #9, and gauge #23 on the front surface close to the centre of brick masonry (in Figure 5(b)). The strains of other gauges are lower than failure strain, so they do not fail. The zone with a large number of continuous eroded elements is called a crater. The more the effective strains of elements reach failure strain, the larger the carter is. Figure 5(c) shows that the compression behaviour of different gauges. When the value of compression is below zero, the gauges suffer from the effect of tensile stress. When the crater breaks, a few fragments will be extruded from it. The maximum ejection velocity is about 16.2 m·s−1, as shown in Figure 5(d).Figure 5
Simulation results of 2.0 kg C4 explosion load. (a) Contour of stress distribution. (b) Effective strains of gauges. (c) Compression behaviour of gauges. (d) Ejection velocity of fragments.
(a)(b)(c)(d)Figure6 presents the dimensions change of damage zone on the front surface when brick masonry wall is damaged by 2.0 kg C4 spherical charge. With the increase of time, three dimensions become larger and larger. The 240 mm thickness wall cracks at about 2.0 ms while the crater reaches its maximum cross section at about 5 ms. Before the crater breaks, it has reached the maximum cross section surface, beyond 400 mm. The growth rates of X and Y dimensions are almost the same, which are larger than that of Z dimension. So, the shape of cross section of crater is close to be circular.Figure 6
The dimensions in three directions of crater for 2.0 kg C4 explosion load.


## 3.2. Damage Zone of Brick Masonry under Different Charges

In order to study the damage characteristics of the brick masonry under different explosion loads, five C4 spherical charges, 0.5 kg, 1.0 kg, 2.0 kg, and 4.0 kg, are designed. The radii of these charges are 42.10 mm, 53.04 mm, 66.82 mm, and 84.19 mm, respectively. These charges are all placed on the centre of front surface of brick masonry wall to explode.Figure7 gives damaged pattern of the back of brick masonry wall subjected to different explosion loads. As we know, shock wave will be produced when explosive is detonated and then turns to compression wave in brick masonry. When stress wave transmits to the back free surface of brick masonry, a reverse tensile wave is inevitably formed to stretch the back to go farther. When the quantity of charge is smaller, such as 0.5 kg, a small crater is found on the front face just at the contact position. Seen from the back surface of the wall, there is a small hole and few fragments are extruded. When the quantity of charge increases to 1.0 kg, a larger crater is formed on the front. Viewing from the back, the mortar in the centre is extended to form a larger hole and a few fragments are ejected. At the same time, it can be found that left and right sides of masonry wall are slightly affected by such a high explosion. That may be due to the bond strength of transverse mortar layer being lower than that of longitudinal brick-mortar interaction. If 2.0 kg explosive charge is used, deeper crater appears with more distinct bulge on the back. The crater breaks to form more fragments and two sides of masonry wall are heavily affected along transverse mortar layers. When the charge reaches 4.0 kg, an even larger crater appears. The sizes of back fracture zone and hole are all larger than above results.Figure 7
Destruction diagram of brick masonry under different explosive loads. (a) 0.5 kg. (b) 1.0 kg. (c) 2.0 kg. (d) 4.0 kg.
(a)(b)(c)(d)Next, two experiments were conducted to investigate the damage characteristics of masonry wall. 1.0 kg and 2.0 kg spherical explosive charge were chosen to be detonated on the surface of masonry wall. Crater holes are formed under such two equivalent explosives, as shown in Figure8. Compared with the simulation results in Figure 9, the experimental results are in good agreement with them.Figure 8
Experimental results under two different explosion loads. (a) Front view of 1.0 kg. (b) Back view of 1.0 kg. (c) Front view of 2.0 kg. (d) Back view of 2.0 kg.
(a)(b)(c)(d)Figure 9
Simulation results under two different explosion loads. (a) Front view of 1.0 kg. (b) Back view of 1.0 kg. (c) Front view of 2.0 kg. (d) Back view of 2.0 kg.
(a)(b)(c)(d)Table4 shows the dimensions of damage zones of brick masonry wall under different explosion loads. With the increase of the quantity of explosive, the lengths in X-direction and Y-direction increase. Moreover, the size of crater on front surface is smaller than that of fracture zone on back surface. When explosive is attached to brick masonry, released energy would be absorbed by masonry wall and thus compressive stress wave would spread in it. Once compressive stress wave reaches the back free surface, an enhanced tensile stress wave will appear due to impedance mismatch between wall and air. The reflected tensile wave from free surface encounters the inbound compressive stress resulting in total stress that is subtractive. When the total stress is tensile (tensile stress greater than inbound compressive stress) and exceeds the tensile strength of any point in the masonry matrix, cracks or fractures will occur. For brittle brick masonry material, the damage area on the back free surface can be larger than that on the front incident surface.Table 4
The damaged zones of brick masonry under different conditions.
Case no.Charge (kg)Crater in front surfaceFracture in back surfaceX-direction (mm)Y-direction (mm)X-direction (mm)Y-direction (mm)I0.5266252366338II1.0364347524447III2.0473468604618IV4.0637630755751


## 3.3. A Simple Predictive Model of the Dimensions of Hole

When the spherical explosive is detonated on the surface of masonry wall, according to the simulation and experiments, the crater hole is basically determined by the radius of the explosive. The simplified model can be described in Figure10.Figure 10
The relationship between the radius of crater hole and the radius of explosive.Since the explosion energy is released on the front surface of contact point and evenly in all directions, the radius of crater hole is merely controlled by the radius of explosive charge. However, the relationship between these two radii is not given clearly.(6)rd∝r0.According to Table4, the dimensions of front surface are drawn in Figure 11(a).Figure 11
Crater on front surface under different charges. (a) Dimensions of holes. (b) Fitting curve.
(a)(b)The ratios of radius of holes to the radius of explosive charges for four cases increase monotonously from 6.32 to 7.57. When fitting these data in Figure11(b), it can be expressed as(7)rd=1.2859r01.24306.Thus, a power function predicting model of the radius of hole is found.


## 3.4. A Simple Fitting Model of Maximum Fragment Velocity

With the increase in the quantity of the explosive, more and more cracks along the longitudinal direction are noticed. Once the hole is formed, the damaged bricks will be blown out to produce a number of fragments. Seen from the viewpoint of energy, the more quantity of explosive is used, the higher explosive energy loading on the surface of masonry wall is produced, and the higher the velocity of fragment gets. That is to say, the ejection velocities of fragments are determined by the radius of explosive too for spherical explosive charge. The fragment with maximum ejection velocity is found at the centre of crater or the apex of crater before breakup.The maximum velocity of fragmentVmax is proportional to the radius of spherical explosive:(8)Vmax∝r0.Figure12 presents the maximum ejection velocity of fragments under different conditions. Adding coordinate point (0, 0), there are five groups of data. It is found that a power function curve fits these data well.Figure 12
The maximum ejection velocities of fragments under different charges and fitting curve.Now, the maximum ejection velocity is power function to the radius of explosive.(9)Vmax=0.00139r02.21747.When the explosive is 4.0 kg, the maximum ejection velocity can reach about 26 m/s, which may cause severe damage to surrounding equipment or personnel.


## 4. Conclusions

In this paper, numerical simulation and verification experiments of damage response of brick masonry under different amounts of C4 spherical charges are carried out. The following conclusions are obtained:(1)
When constrained brick masonry is exposed to contact explosion by spherical charge, a crater is formed on the front. The shape of cross section of crater is almost circular. The cross section dimension of crater increases monotonously with the radius of spherical explosive charge. For four cases in this study, the ratios of radial lengths to the radius of explosive charge increase from 6.32 to 7.57.(2)
Explosion energy releases compressive wave directly on the front surface, and then stress wave (transmission wave) will travel to the back to make brickwork cracked. The combination effect of front compressive wave and back tensile wave are the main factors influencing the crack of brick masonry wall. When the back tensile wave is beyond the tensile strength of brick masonry, fracture zone will be pulled out to form a hole. The radius of hole is directly determined by the radius of the explosive. A power function predictive model is obtained to estimate the size of hole for C4 explosive detonated on the surface of 240 mm thickness masonry wall.(3)
The fragments can get higher velocities under larger explosion load, which may cause lethal damage to surrounding equipment or personnel. In the region of research, there is a power function relationship between maximum ejection velocity and the radius of explosive.



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*Source: 5519231-2021-08-26.xml*