Crack simulation for the cover of the landfill – A seismic design

A BSTRACT . The stability of the landfill is an environmental issue. The collapse of the landfill causes environmental pollution and influences human life. In the present study, the crack on the cover of the landfill was simulated. Rankine’s theory and the Phantom Node Method were used for the simulation length of the crack and the mechanism of the crack propagation in the nonlinear extended finite element method (NXFEM). Artificial Neural Networks (ANNs) based on Levenberg-Marquardt Algorithm and Abalone Rings Data Set mode were used to predict displacement in critical points of the model. The vibration mechanism of the landfill was changed in each model. During applying seismic load on the model, the optimized thickness of the clay cover on the landfill was discussed . The thickness of the landfill cover controls the seismic response of the landfill. The numerical simulation shows differential displacement of the landfill impacts on the crack propagation and the need for the appropriate design of the cover thickness of the landfill.


INTRODUCTION
growing amount of municipal solid waste (MSW) is being produced due to the industrialization of many urban areas [1], resulting in a massive accumulation of MSW and new landfill construction. Due to changes in the local climate, MSW's physical, chemical, and biological processes can also be affected. As a result of MSW accumulation, high temperatures are generated, which are associated with climate change [2]. During the cover process for landfills with clay, topsoil tends to be more influenced by temperature, and this happens more often in tropical regions, where the soil is exposed to higher temperatures [3]. As temperatures increase in the landfill, cracks form in the clayey cover [2][3][4][5], which extend to the body of the landfill [4][5], causing a change in the moisture content of the landfill. Also, during the rainy season, the amount of leachate increases. In the event of seismic loading, landfill cover cracks extend more rapidly, resulting in a reduction in landfill seismic stability. The landfill's seismic stability is related to local site conditions [6]. Due to the seismic load on the landfill, the damage is divided into major, significant, moderate, minor, slight, or no damage. The crack initiates and occurs in the landfill cover if the damage is moderate. Once the type of damage is recognized, geosynthetics can be utilized to improve landfill seismic stability [7]. Seismic loading causes landfill displacement, and it has been suggested that this phenomenon be quantified through Newmark's one-dimensional analytical sliding-block method [8]. Many researchers have applied the nonlinear finite element method to calculate the nonlinear displacement of the embankment model [9]. Using the appropriate method to simulate a landfill's seismic response accurately is essential. Further, it must be noted that the material's mechanical properties and the model's geometry properties have an impact on the displacement mechanism, and these properties need to be understood correctly by evaluating the input data accurately [10][11][12]. It is well known that the calculation of seismic deformation caused by an earthquake is prone to some error levels [13][14][15]. The impact of seismic loading on the displacement of the landfill has been systematically investigated to identify the seismic stability of the landfill [16][17][18]. Landfill displacement could be minimized using geomembrane [19][20], lattice drainage geocomposite, and flexible polyester [20]. Aside from these methods, clay soil is also used to cover landfills [21,22]. The compacted soil liner (CSL) is required to control a hydraulic conductivity of (≤ 1 × 10−7 cm/s); for this purpose, clay with a thickness in the range of 0.6 to 1.5 m is required [24,25]. Displacement prediction is needed before applying the seismic mitigation method to improve landfill seismic stability. Nonlinear numerical simulations can identify displacement at any point in the modeled landfill. Due to the nonlinearity of the seismic load, numerical simulation results need to be validated using statistical analysis. The current study used artificial neural networks to predict embankment displacement [11][12]26]. The prediction is made from the validation and optimization results of the numerical simulation [27][28][29]. The main objective of the present study is to consider the impact of cracked landfill covers with different thicknesses on the displacement of two critical points in the model. In order to predict displacement at selected points in the model, the Levenberg-Marquardt algorithm was used, which considers Rankine's theory and the phantom node method in crack simulation and propagation. A MODELING oil-like materials (SLM) such as MSW are highly deformable under seismic loading. The crack initiation on the landfill cover was simulated and shown in Fig. 1, which illustrates the entire numerical simulation procedure. Numerical modeling was conducted to design the clay landfill cover under seismic loading. Fig. 2 Subsoil cross section Initial crack path

CRACK SIMULATION
he initial crack in the landfill clay cover is designed based on Rankine's theory (1857) [30]. The crack and solid zones are two significant parts of the model. The ratio of the crack height to the total thickness of the landfill cover for each part changes with respect to the model's mechanical properties. In fact, when the two models have the same geometry but are constructed from two different MSW materials, the crack, and solid zones are changed according to the types of materials used. It is assumed that landfill leachate is confined under undrained conditions. S T According to Rankine's theory, k p and k a are the passive and active earth pressure coefficients and are presented in Eqns. 1 and 2. The pressure and force in the active and passive states have been explained in Eqns. 1-12 [31].
where the Rankine active state in the slip plane is, where the Rankine passive state in the slip plane is, The lateral earth pressure for the Rankine active state is, The lateral earth pressure for Rankine passive state is, The lateral earth force in Rankine's active state is, The lateral earth force in Rankine's passive state is, For the undrained condition in a fully saturated clay, the active and passive pressures are calculated using the parameter c u , (ϕ u is equal to zero) and the total unit weight γ sat . When enough deformation occurs the state of plastic equilibrium occurs. Eqns. 11 and 12 have been used to identify the length of the crack, the solid zone, and the crack initiation zone [32].    2 / u Z Cracked zone c (11)    0 z S Solidzone H Z (12) Based on the theoretical concepts, the crack initiates in association with the mechanical properties of the clay. The crack propagation caused by applying an external load on the model was simulated. Fig. 3 illustrates the crack and solid zone in the landfill cover.

SEISMIC DATA
ccording to the literature report [33], to achieve acceptable results multidirectional seismic acceleration needs to be applied to the model in the numerical simulation. By applying one direction of seismic acceleration to the model, the underestimated strain, stress, and displacement would be obtained in the numerical simulation. Fig. 4 illustrates the multidirectional acceleration history (m/s 2 ) used in NFEM. The seismic acceleration at 0°, 90°, and 360° of the Abra Pampa Earthquake, which occurred on 22 Feb 2022, with 6.0 MWW, in coordinates -22.6625 and -66.2673 in a depth of 242.3 km [34], has been used as the input data in the nonlinear numerical simulation. The seismic accelerations applied to models 1 and 2 are equal. The seismic acceleration in each direction has minimum and maximum domains. The combination of these three accelerations needs to be applied to the model. The critical duration of seismic acceleration at 0°, 90°, and 360° is shown in Fig. 4. The shaking over 0.05 (g) in positive and negative directions in all directions is the most unpleasant and needs to be considered in the nonlinear numerical simulation. Tab. 1 shows peak ground acceleration, displacement, and velocity at 0°, 90°, and 360°.   Table 2: Mechanical properties of the MSW [35][36][37].
The angle of internal friction for clay under undrained conditions is assumed to be equal to zero (ϕ u = 0) [38]. The unit weight of undrained clay (γ c ) is 18 (kN/m 3 ). The undrained modulus of elasticity (E u ) is 12.5 MPa. Poisson's ratio ν c of 0.3 and undrained shear strength equal to 25 kPa have been reported in the literature [39]. The model has three parts, the subsoil, the landfill, and the cover of the landfill. The landfill subsoil and cover were designed using clay. The Landfill has been simulated using MSW.

NONLINEAR EXTENDED FINITE ELEMENT METHOD AND CRACK PROPAGATION
he XFEM provides a situation in that discontinuities act independently, and different types of discontinuities are generating [40]. The crack simulation using XFEM was reported in the literature [41][42]. Eqn. 14 presents the standard form of XFEM [43][44]. .
The phantom node method for modeling fracture is introduced. This technique involves intersecting elements by a crack, erasing the crack, and replacing it with two new elements without modifying the original location of the element. This process does not alter the physical characteristics of the material [45], and later it was upgraded by using a more advanced technique [46]. In the present work, using ABAQUS, the phantom node method has been adopted to simulate crack propagation in the landfill cover. Figure 5: The phantom node method in mesh generation [47].
Fig. 5 explains nodes, elements, and meshes generation by the phantom node method [47], in this process the elements have independent displacement without sharing nodes [48]. Considering Fig. 5, according to the phantom node method, to explain the discontinuous displacement for overlapping elements can be written as [47], where Γ s presents the crack path, , Ω A and Ω B refer to the shaded area the portion of a new element A and B respectively. The N is the standard for finite element method shape functions and U A and U B are the displacements in nodes A and B. Fig. 2 illustrates the boundary condition adopted and applied in the numerical simulation. Two models of the landfill with the cracked cover are simulated. The smallest mesh size is selected for landfill cover. The mesh sizes of 2000 (mm), 4000 (mm), and 6000 (mm) have been selected for landfill cover, landfill, and subsoil of the model respectively. The geometry of the mesh is shown in Fig. 6. The MSW has been covered with clay. In model 1 the cover of the landfill is 3 (m), and in the second model, the cover of the landfill is increased to 6 (m). The selected nodes in numerical simulation for models 1 and 2 are shown in Fig. 6. In these two critical points the displacement, stress, and strain will be analyzed in the next part of the present study.

ARTIFICIAL NEURAL NETWORKS
rtificial neural networks (ANNs) are frequently employed for geotechnical engineering problem prediction, assessment, and solution [12,26]. ANNs are employed in data prediction, categorization, association, and filtering [49]. The Levenberg-Marquardt algorithm was used in this study to perform ANNs for classification and prediction. The four key criteria for predicting displacement are seismic acceleration, stress, strain, and fracture length. The thickness of the landfill cover was varied in the numerical simulation, but the seismic acceleration and model geometries were fixed for models 1 and 2. In MATLAB, Artificial Neural Networks (ANNs) were utilized to forecast displacement in the Y direction of the chosen object. The ANNs in different layers were done, and the numerical simulation results were checked, validated, and predicted based on the training to reach the best outcomes for displacement occurrence.
Eqns. 17-18 introduce the basic concept of the Levenberg-Marquardt Algorithm [50]. If    , , are unknown parameters at , , , ; , , , , , , A Eqn. 17, can be minimized to The Hessian matrix is presented in Eqns. 19-20. If f: R n →R presents a function for input X  R n and output f (X)  R. If partial derivation f is available, subsequently the Hessian matrix H is a square matrix [51]. (20) where J presents the Jacobian matrix the Hessian matrix H is  T H J J (21) If "e" is selected as a vector of network errors, the gradient will be calculated from Eqn. 22, The Updated weights in the Levenberg-Marquardt algorithm can be as To approach second-order training without employing the Hessian matrix, the Levenberg-Marquardt technique was presented [52]. In light of the statistical idea offered in the literature [53], Eqns. 24-25 are proposed. d is the acquired nonlinear displacement in the numerical simulation, d p is the projected displacement using ANNs, and o D is the mean value of obtained nonlinear displacement in these two equations. The accuracy of displacement prediction was assessed using Eqns. 24

RESULTS AND DISCUSSIONS
nonlinear numerical simulation was performed to predict the nonlinear displacement at two critical points in the landfill. The results of the nonlinear numerical simulation showed that the speed and shape of the deformation in each model have distinct behaviors. MSW is a highly deformable material. Because of this characteristic, when MSW is subjected to seismic acceleration, it has a different seismic response compared to clay. Fig. 7 illustrates the nonlinear shear stress-strain for a particular node. These points have been selected for monitoring displacement in the subsoil, the clayey cover of the landfill, and the main body of the landfill. Fig. 8 illustrates the multidirectional nonlinear accelerationdisplacement in the Y direction for a selected node in models 1 and 2. Figs. 7 and 8 reveal the MSW-clay interface's impact on the model's seismic response. In addition, landfill covers of different thicknesses exhibit higher seismic resistance. In addition, models 1 and 2 have different deformability. The unique deformation pattern for each model was observed. According to the nonlinear shear stress-strain curve, model 2 collapses with lower deformation. Increasing the clay landfill cover by more than 3 meters reduces seismic stability. The displacement at the crest and toe of the landfill is dissimilar. The clay landfill cover acts as a surcharge on the landfill and increases its displacement. The seismic frequency causes fill excitation to impact landfill stability considerably [16]. The Abra-Pampa earthquake was simulated and applied to the landfill in the present work. This earthquake has a high frequency. The high frequency of the earthquake accelerates the model collapse before a high displacement level occurs. Due to the seismic load applied to the landfill, a crack develops on the landfill cover during the moderate damage stage [7].  A By comparing the stress-strain curve, it was possible to determine the reduction of stiffness in model 2. The shear and compressive strain in model 1 at node 4 are larger than in model 2 at the same point. In the loading and reloading stages of models 1 and 2, at embankment and subsoil, stress-strain characteristics follow two different patterns. In both models, the stress-strain rate is higher in the subsoil than in the embankment. The nature of MSW-clay landfill cover interaction causes the development of distinct seismic responses and seismic differential displacement for each model. In addition, the numerical simulation results revealed that the model's vibration mechanism is associated with the mechanical properties and material design of landfill construction. Due to the characteristics of MSW, two surfaces of cracks interact with each other, which affects displacement and deformation in the landfill.   Fig. 9 illustrates the displacement in the whole model after 1 second of applying seismic acceleration (g) to the model. The crack shows different morphology at each stage of the numerical simulation. The nature of the seismic acceleration (g) determines the failure mechanism in both models. In addition, the landfill clay cover thickness controls the crack propagation mechanism. With attention to the difference between lower and higher displacement levels in models 1 and 2, it seems that model 1 has a higher collapsing speed. Fig. 9 shows that the model's failure pattern occurs based on the model's lateral sliding. These similarities in the type of failure in models 1 and 2 occur due to the nature of the seismic acceleration. The compacted soil liner (CSL) needs to maintain a hydraulic conductivity of (≤ 1 × 10−7 cm/s). For this purpose, clay with a thickness in the range of 0.6-1.5 m is required [24][25], but from the seismic design perspective, the appropriate clay thickness needs to be recommended. Based on this analysis, it can be concluded that the landfill cover thickness needs to be designed to restrict landfill collapse and control leachate as well.  Tabs. 3, 5, 7, and 9 present the peak values of acceleration (g), stress (MPa), strain, and initial length of the crack (mm) obtained from numerical simulation and theoretical concepts. ANNs based on Levenberg-Marquardt algorithms have been used to predict and validate displacement in the Y direction at the critical point. Due to the huge number of outputs produced in the nonlinear numerical simulation, the peak value of the data has been selected and presented. The Levenberg-Marquardt algorithm in the Abalone Rings Data Set mode was used to validate and predict displacement accuracy in the ANNs process. The test data and results are validated based on the number of training observations in the numerical simulation. In the ANNs, three layers have been created, and mean squared error (MSE) and R have been obtained. Displacement prediction accuracy has been assessed using Eqns. 24

THE CRACK MORPHOLOGY OF LANDFILL
he MSW in each region possesses different mechanical properties. The present simulation work has been done numerically for designing the landfill clay cover. Figs. 18 and 19 illustrate the tension crack in the landfill clayey cover. In Fig. 18, a single initial tension crack in the clayey cover of the landfill is created. This crack we have simulated in the present work, while in Fig. 19, the initial multidirectional tension crack in the clayey cover of the landfill has been developed. Considering the clayey landfill cover for controlling leachate, as presented in the literature [24][25], the seismic stability design of the landfill plays an important role by considering the thickness of the clayey landfill cover when constructing the landfill. Figure 18: The single initial tension crack in the landfill [4].
In each region, MSW has different mechanical properties depending on how it is produced, how it is compacted during landfill construction, how temperature affects the mechanical properties of MSW, and how much moisture it contains due to climate and leachate drainage, etc. Predictions of landfill seismic stability are supported by simulations of landfill design. Landfill sliding and shear failures can be predicted based on landfill design. Fig. 19 shows how the initial tension crack of the landfill can interact during the seismic loading on the landfill. The crack interaction causes landfill failure acceleration. Figure 19: The initial multidirectional tension crack in the cracked zone of the landfill [5]. T CONCLUSION A simulation was conducted to simulate the displacement of the landfill covered by cracked clay under seismic loading. Rankine's theory, the Phantom Node Method, and the Levenberg-Marquardt Algorithm were used to simulate, predict, and validate the numerical simulation results. Stress, strain, and length of the crack were selected to predict displacement at critical points. ANNs were used to predict displacement in the Y direction for the selected nodes. The following findings were achieved in the present study:  In each model, the crack initiation, crack shape modification, deformation speed, and shape are distinct. These phenomena are associated with clay landfill cover. The crack propagation in the landfill model is related to the damage magnitude.  In response to nonlinear seismic acceleration, crack morphology changes and impacts seismic acceleration transmission and landfill seismic stability. The nature of seismic acceleration (g) in both studied models determines the failure mechanism. Furthermore, the landfill clay cover thickness influences crack propagation.  Models 1 and 2 show different patterns of seismic acceleration transfer and dissipation. Based on the nonlinearity in shear stress-strain for models 1 and 2, it can be concluded that landfill deformation is associated with landfill model design.  The nature of the seismic acceleration is an important factor in the collapse of the landfill model.  The clay with a thickness of 0.6-1.5 m controls the leachate, but it is not good enough to ensure landfill stability during earthquakes. A suitable thickness of clay needs to be chosen based on seismic stability analysis.