Abstract
In order to solve the problem of accumulation of glass waste seriously affecting people’s life safety, it is of great significance to study the crushing mechanism of glass waste. At present, no scholars have studied and analyzed the waste glass head, especially the single-axis compression test. In this study, a three-dimensional waste glass is established by discrete element software, and the single-axis compression characteristics of waste glass at different loading rates are analyzed by using Hertz contact model provided by the software. The results show that the loading rate is influenced by the equilibrium iteration rate and stress loading step length. The study also analyzes the pressure plate extruding waste glass in different positions. Finally, the stress–strain strength of five groups of waste glass under the action of uniaxial compression is analyzed, and the stress strength is between 0.3 and 0.56 MPa. This study provides a theoretical basis for further study of the breakage characteristics of waste glass under composite loading.
1 Introduction
Glass has been widely used in construction, automobile, household packaging, and other industries. China is the world’s largest producer of glass, and glass manufacturing is associated with high carbon emissions and energy consumption. Cheng et al. [1] pointed out that China produces a large amount of waste glass each year and has a low recycling rate. In theory, waste glass can be completely recycled, and its own characteristics make its utilization value pay much attention.
Therefore, it is very important to study the breakage glass, and relevant scholars have also carried out in-depth research. Zeng [2] studied the mechanical properties of waste glass powder (GP) concrete under dynamic load, and concluded that the breakage of the test block is graded from middle to edge. Qiu et al. [3] found that the compression and breaking of ceramics is an instantaneous process that releases tremendous energy and flies out at high speed in fragments. Qaidi et al. [4] found that recycled glass waste is one of the most attractive wastes that can be used to manufacture sustainable concrete compounds. By studying the influence of recycled glass waste on the freshness and mechanical properties of concrete, the environmental impact of using recycled glass waste in concrete is reviewed and discussed. Jiang et al. [5] studied the influence of waste GP as a supplementary material of ordinary Portland cement on paste samples at normal temperature and exposed to high temperature through experiments. Introducing waste GP into ordinary Portland cement reduced the degradation of ordinary Portland cement matrix at high temperature, and may reduce the alkali-aggregate reaction effect. Aslam et al. [6] studied the behavior of concrete with 15% fly ash (FA) instead of cement containing 2% coconut fibers (CFs) and concrete with waste beverage glass as sand at different substitution levels (14, 15, 16, 17, 18, 19, and 20%). Nodehi and Mohamad Taghvaee [7] reviewed the relationship between construction industry and circular economy and recycled glass. Therefore, by partially replacing cement or aggregate with recycled glass, on average, greenhouse gases can be reduced by 19%, energy consumption can be reduced by 17%, and significant cost savings can be achieved.
Wu et al. [8] found that particle fragmentation increased with stress levels and was significantly influenced by corresponding stress pathways. Zhang et al. [9] found that single particles in gravel had the strongest fragmentation characteristics and single particles in mortar had weakest fragmentation properties. Yu and Tong [10] analyzed the mass distribution of broken particles by introducing fractal theory and revised the related model. The load-displacement curve of regenerated aggregate is increasing in multiparticle compression test. Huseien et al. [11] used blast furnace slag in FA-based alkali activated mortar (AAMs) as raw material to prepare glass bottle waste nano-powder (BGWNP). The main purpose is to evaluate the energy consumption, cost effectiveness, and mechanical and chemical properties of the realized BGWNP hybrid AAM. Ali-Boucetta et al. [12] studied the influence of two mineral additives in limestone filler on the strength and durability of self-compacting concrete. These are GP recovered from glass bottles of the same color and granular slag from blast furnace, a by-product of the iron and steel industry of El-Hadjar complex in Annaba region (eastern Algeria).
Song et al. [13] used finite element software to analyze the load variation and strain of glass beads under impact. Jian et al. [14] studied the crushing process of double particles under dynamic impact by analyzing the impact test of Shi Ying glass beads. Fang et al. [15] obtained the distribution of the breakage zone by analyzing the breaking process and damage modes of glass spheres at different velocities. Cai [16] analyzed the relationship between fragmentation and shock energy and concluded that brittle particles are not infinitely crushed with increased shock energy, but have crushing limits. Li et al. [17] indicate that crushing ratio is an indicator of the crushing effect of crusher. Through the experimental study on the crushing ratio of waste concrete by Liu et al. [18], it can be known that the crushing ratio is proportional to the increase in material strength and size. Although this accords with the actual situation, the experimental operation process is complicated and there are many uncontrollable factors. Therefore, a discrete element software that can accurately analyze glass breakage is very important.
At present, the problem of beer bottles in used daily glass is getting more and more attention, and the research on the head of waste beer is very important. Current research has focused on impact breakage, and no one has yet studied compression test on spent beer bottle heads. In order to improve the utilization rate of waste glass resources and reduce the waste of resources and environmental pollution, it is very important to make a scientific analysis of the broken characteristics of spent beer bottle heads. This study will be of great significance for the further analysis of the application of waste glass on the road. At the same time, it also provides theoretical basis for further study of the breakage characteristics of waste glass under composite load.
2 Discrete element model and simulation principle
2.1 Contact model between particles
Discrete element models are created by random stacking and bonding to create objects with specific mechanical properties (Figure 1a) with surface friction force, normal force, and tangential force between them; normal force (Kn) and normal deformation (Xn) between particles are simulated by normal springs between particles and are given according to the following formula:
(a)–(c) Schematic diagram of particle bonding and its linear elastic modeling.
where Kn is the normal stiffness, Xn is the normal relative displacement (Figure 1b), and Xb is the fracture displacement.
At first, the particles are interconnected, and when they come under pressure or spring pressure, there is a normal force between them. When the positive relative displacement of Xn between two particles exceeds the fracture displacement Xb, the tension between the connecting spring and the particle disappears. When two particles are compressed, the repulsion between them increases. By dispersing the solution space into the element matrix of discrete elements, and combining with practical problems, we can choose reasonable connecting elements to connect adjacent elements. The basic variable between elements is relative displacement. According to the relationship between force and relative displacement, the normal force and tangential force between two elements can be obtained by Newton’s law of motion. The interaction forces between discrete elements and other elements in all directions, as well as the resultant force and moment of the resultant force, are caused by the influence of other physical fields on the discrete elements. Then, the acceleration of the discrete element can be obtained according to the relation between force and acceleration of Newton’s law of motion, and then the velocity and displacement of the discrete element can be obtained by integrating it in time. Therefore, the motion characteristics of all units at any time are obtained, and the force and motion are combined.
Using tangential spring force F s, the tangent force between particles is simulated in the tangent direction:
where K
s is the tangential stiffness and X
s is the tangent relative displacement (Figure 1c). In the tangential direction, the spring is subjected to a criterion of failure, where
where
When an external force exceeds the maximum shear force, it breaks.
At this time, only relative sliding occurs between particles.
2.2 Principle of discrete element numerical simulation of uniaxial compression test
2.2.1 Damping force between elements
In discrete element method, viscous damping is used to simulate the gradual attenuation of mechanical energy in order to avoid the accumulation of stress waves. The damping in the software used in this study is global damping. As defined by Liu et al. [19], damping force mode is given by
where x′ is the current velocity of particles, and
2.2.2 Time step iteration
On the basis of obtaining the force on each particle, the discrete element method uses time-step iterative algorithm Cundall and Strack [20] calculate the particle displacement.
In the absence of damping, in order to better reflect the advantages of numerical simulation of elastic processes, it is necessary to consider that the iterative time step should be less than the damped simple harmonic vibration period.
where
2.3 Energy conversion
In discrete element numerical simulation, the total energy is the sum of all mechanical energy and heat, which is gradually converted into thermal energy by damping force, friction force and fracture force. In MatDEM discrete element system, various mechanical energies and corresponding heat can be accurately calculated.
3 Numerical model establishment
3.1 Random stacking
Based on the three-dimensional discrete element software MatDEM developed by Liu et al. [21], the three-dimensional model of the waste glass head was designed independently. According to the characteristics of the software, the particle aggregates with particle sizes ranging from 0.008 to 0.012 are randomly generated, and the cuboid structure diagram is established after the corresponding gravity deposition. Based on the macro–micro mechanical parameters conversion formula of the compact packing discrete element model, the micro-mechanical parameters suitable for our model are obtained. But at present, this transformation formula still has some limitations. This transformation formula is derived from a tetrahedral model composed of four elements based on the assumption of small deformation, and then the number of effective boundary elements will gradually decrease with the increase in the number of elements. The boundary effect will lead to the actual mechanical properties of the multi-element model usually lower than the set value.
For the whole model, the boundary formed above is a positive side pressure plate formed by stacking cluster particles unique to the software. In the whole numerical simulation, according to the stress and particle area to be applied, the vertical force of each particle on the pressure plate is calculated and applied to the particles, thus generating vertical pressure. The lower boundary particles of the specimen are fixed particles. In the numerical simulation, the force on the lower boundary is automatically recorded, and the stress–strain curve is calculated. At the same time, the software will automatically record the energy changes of the system during the simulation process.
The process is as follows.
First the model is initialized, then particle deposition and standard equilibrium are carried out to obtain compaction accumulation. However, due to the specificity of the model, further modifications are required to meet the requirements. The three-dimensional particle stacking model is shown in Figure 2.
Three-dimensional particle accumulation model.
3.2 Geometric modeling and assignment of materials
MatDEM uses its own material training function to endow unit particles with properties so as to simulate real-world objects. According to the macro–micro transformation formula developed by Liu et al., the macro-mechanical parameters of the sample can be transformed into micro-mechanical parameters by using the analytical solution between the element mechanical parameters of the closely packed discrete element model and the overall mechanical properties of the model, but the corresponding random packing model will have various mechanical properties reduced. This software substitutes macroscopic material properties into the transformation formula to obtain its initial microscopic parameters and multiplies them by a ratio. Through automatic uniaxial compression, compressive strength, and tensile strength tests, the corresponding elastic modulus and strength are obtained. The ratio of measured value and set value is used to readjust the corresponding ratio, until the mechanical parameters converge to the set values, the microscopic unit parameters that are most in line with the reality are obtained, so we set the material parameters that are most in line with our own model, and then use the unique numerical simulation ability of the software to model and assign materials. Then, the material parameters after many simulations, screening, and adjustment are given to the particle element in this model, and then the overlying load is applied. Through the corresponding iterative calculation, the sample is gradually compacted and reaches a stable state under the set load.
MatDEM can first build a structure that meets the requirements, and then import it into the model. Here you first build a model of each component, then you piece together a complete model. Because the coordinates of the two will deviate during the import process, it is necessary to adjust the position to reach the correct position and facilitate the processing and analysis. The establishment of the model is determined step by step according to the actual situation, thus reducing the error and ensuring the reliability of the corresponding model. The 3D model of the spent glass head is shown in Figure 3.
Three-dimensional model of waste glass head.
Since MatDEM itself has the function of training materials automatically, the properties of the allocated materials better reflect their actual performance. Based on the relationship between the mechanical formula of sealed packaging and macroscopic mechanical properties, Liu et al. [22] derived the parameters that can be used for the mechanical formula of sealed packaging. However, this stochastic stacking model reduces the performance of various mechanical parameters. On the basis of making full use of the module of automatic training material, the micro-unit parameters are obtained by adjusting the module.
Hertz contact models are used here, with corresponding macro- and micromechanical parameters as shown in Table 1.
Macro- and micromechanical parameters of waste glass heads
| Macroscopic mechanical parameters | Micromechanical parameters | ||
|---|---|---|---|
| Young’s modulus (GPa) | 77 | Tangential stiffness (Kn) | 0.8077 |
| Poisson’s ratio | 0.17 | Normal stiffness (Ks) | 0.2275 |
| Tensile strength (MPa) | 158 | Fracture displacement Xb | 0.0057 |
| Compressive strength (MPa) | 400 | Shear resistance FsO | 0.0057 |
| Angle of internal friction | 0.5 | Friction coefficient | 0.1208 |
After many simulation tests, the optimal micromechanical parameters are determined and the materials are given, as shown in Figure 4.
Model assignment material.
4 Analysis of numerical simulation results and experimental verification
A digital model of particle accumulation was used to build 11,673 particles. It contains a boundary element with an average radius of 2 mm. MatDEM uses novel discrete element GPU matrix algorithms to support dynamic simulation of millions of discrete elements. It mimics the breakdown of objects in the real world. The spent glass head is crushed under the action of the single uniaxial stress.
In the numerical simulation of uniaxial compression of waste glass, the stress of the upper pressure plate is gradually increased, and the stress of the lower boundary will be recorded correspondingly after the equilibrium iterative calculation, until the specimen is destroyed, that is, the corresponding lower boundary stress drops suddenly, so as to obtain the uniaxial compressive strength of the specimen. In this process, when a new stress is applied to the upper surface of each pair of models, a downward stress wave will suddenly be generated on the surface due to extrusion. Because of the damping in the system, the stress wave will gradually decay during the transmission process. After that, when the stress wave propagates completely to the lower boundary, the kinetic energy in the system will be fully attenuated to achieve the balance between stress and kinetic energy.
Simulated data plotting.
Uniaxial compressive strength at different load rates
| Balanced iteration rate | |||||
|---|---|---|---|---|---|
| Stand-up loading step | 0.1 (kPa) | 0.5 (kPa) | 0.9 (kPa) | 1.6 (kPa) | 2.4 (kPa) |
| 30 | 200 | 226 | 253 | 246 | 237 |
| 50 | 220 | 241 | 218 | 234 | 426 |
| 70 | 241 | 208 | 181 | 425 | 425 |
| 90 | 264 | 209 | 424 | 423 | 424 |
| 110 | 268 | 423 | 424 | 422 | 422 |
The loading intervals set by the system were used in this study. The parameter settings at different loading rates and for different stress loading steps are shown in Figure 5 and Table 2. The horizontal coordinate is the equilibrium iteration rate and the vertical coordinate is the stress loading step. When you use a larger Qd, then the incremental rate per step will be smaller and the stress wave after loading will be smaller. After each incremental step is used, an iterative calculation is required to balance the stress and kinetic energy. In this study, a standard balance is defined as 50, according to which the loading rate can be studied.By analyzing 25 sets of data from combinations of different loading rates and different loading steps, it is concluded that the compressive strength increases the compressive strength increasing the uniaxial compression experiment iteration rate, especially when the equilibrium iteration rate is 1.6, and then tends to be stable, indicating that the increase in equilibrium iteration rate has little effect on the larger variation equilibrium iteration compressive strength, which is 0.42 MPa. When equilibrium iteration rate Rb is constant, the turning point occurs at two points: stress loading step length is 70, equilibrium iteration rate is 0.9, mutation occurs, and the compressive strength is 0.18 MPa, close to the crushing point of the glass. Then, we can see that the compressive strength increases linearly as the pressure increases, which is consistent with the reality. At a stress loading step of 90, the change in equilibrium iteration rate largely no longer affects the later change in compressive strength, reaching critical mass, followed by mutations, then linear changes, until the compressive strength stabilized at 0.5 Rb, when 0.43 MPa was achieved. The validity of simulation data is verified by experiment. The accuracy of simulation data is verified. At the same time, numerical simulation can save cost, shortens experimental time, and enables understanding microchanges in detail.
Then, the experimental operation was carried out to verify the relevant data. The test process diagram is shown in Figure 6.
Force–time curve.
4.1 Curves of pressure plate pressing waste glass at different positions
Here we show that only half of the pressure plate are in action, i.e., the displacement is set at 0.015 m in the simulation, as shown in Figure 7.
Stress–strain curve.
The glass breaking under uniaxial pressure has obvious breaking characteristics. The strain strength is 1.3 kPa at the front fracture in the Z direction, and the stress strength is stable at 20 kPa at the Z direction after multiple fractures, which provides a data basis for further research on the fracture under the combined load. When the loading speed is increased, the maximum force of breakage is reduced, which is because a slight impact occurs near the time, making its internal cracks increase and expand. According to the brittle fracture theory, the more cracks and the larger the crack size, the smaller the external force required to break or make it break. The actual loading is through the extrusion test of waste glass in different parts, as shown in Figures 8 and 9.
Uniaxial compression waste glass head.
Uniaxial compression of waste glass sheet.
In the actual test, it is verified that when only half of the pressure plate acts on the object, the obvious breaking phenomenon appears under uniaxial stress.
Through the simulation process of the software itself, the uniform stress wave propagation load is applied to the upper boundary, and the transmission reaches the lower boundary. The compression curve of waste glass head is obtained by setting the lower boundary to receive the stress. As you can see from Figure 10, when the strain is 1.2 z, the maximum stress is 0.118 MPa, and that is when the pressure plate is pressed down, so there is a sudden change here. The stress-strain curve exhibits variations at different pressure positions, wherein when the applied displacement reaches half of the glass bottle's height, the stress-strain curve becomes half of its originalmagnitude. This indicates that different forces are generated at various loading displacements, resulting in variations in the stress-strain curve.
Boundary stress changes with time and stress strain in the Z direction.
4.2 Stress–strain curves of waste glass under different edges and corners
Numerical simulation of the crushing test of waste glass at different angles is carried out and the test charts are drawn as shown in Table 3.
Stress strengths at different angles
| Different edges and corners | Angle 1 | Angle 2 | Angle 3 | Angle 4 | Angle 5 |
|---|---|---|---|---|---|
| Uniaxial compressive strength (kPa) | 543 | 530 | 507 | 405 | 301 |
At present, due to the fact that the glass production process is harmful to the human body and causes great pressure on the environment, it is very important to use resources to dispose the waste glass. At present, waste glass comes from a wide range of sources and is easy to extract. To ensure that the research factors are controllable, some experimental materials are extracted and pretreated. A uniform stress load is applied to the upper boundary, which is transmitted to the lower boundary, and the lower boundary is set as the received stress, so the compression curve of the waste glass head can be obtained. From the figure, it can be seen that the maximum stress is 0.118 MPa when the strain is 1.2 in the Z direction, which is exactly when the pressure plate is pressed down, so a sudden change occurs here. Under the pressure at different positions, it can be seen that the position of the pressure is different, and the stress–strain curve is not the same, but when the displacement is the general radius of the bottle, it is just in line with half of the stress during all the pressure.
By comparing five groups of experiments, the variation in stress between 300 and 560 kPa is obtained, which shows that different edges affect the corresponding stress values and has scientific guiding function. In particular, when the margin is greater than 30°, a sudden change in the breakage value, known as the third test, indicates that this is the breaking point, allowing for better control of broken waste glass in practice and for resource recovery and recycling.
5 Conclusion
The glass particle discrete element model based on MatDEM is established through the powerful secondary development function of the software. The stress–strain curves of waste glass at different loading rates and broken glass at different pressure plate locations are studied.
The faster the one-step iteration, the more stress wave propagation and kinetic energy the attenuation are. When the stress loading step length is constant and the equilibrium iteration rate is greater than 1.6 Rb, the experimental results under quasi-static loading can be obtained with minimum computation. Among them, the compressive strength at stress loading step length is 0.18 MPa, which is close to the practical compression glass threshold. Second, when the stress loading step length was 90, the equilibrium iteration rate was 0.5 and the compressive strength is 0.43 MPa.
The uniaxial compression test of glass is carried out by moving the pressure plate in accordance with certain rules. The pressure plate moves slightly at a time, with the pressure plate working best in the middle, with a compression value of 0.118 MPa.
By adjusting five groups of glass parameters from different angles, the compressive breaking characteristics of waste glass are obtained. The stress ranged from 0.3 to 0.56 MPa compared to actual tests. The experimental results are consistent with the numerical simulation results and conform to the relevant national standards. At the same time, it also provides theoretical basis for further study of the breakage characteristics of waste glass under composite load.
Acknowledgement
This project is funded by Fujian University of Technology and is a scientific and technological project of Fujian University of Technology. Project number is (GY-Z220201).
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Author contributions: Huangyi Wu: data analysis; Peng Sun: experimental operation; Saifei Han: experimental data sorting; and Luojian Yu: simulation analysis.
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Conflict of interest: The authors state no conflict of interest.
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Data availability statement: The data will be provided upon request, and they can be used without any conflict of interest.
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