Finite element simulation for elastic dislocation of the North-Tehran fault: The effects of geologic layering and slip distribution for the segment located in Karaj

Abstract

The present study uses the finite element method for simulating the crustal deformation due to the dislocation of a segment of the North-Tehran fault located in the Karaj metropolis region. In this regard, a geological map of Karaj that includes the fault segment is utilized in order to create the geometry of finite element model. First, finite element analysis of homogeneous counterpart of the fault’s domain with two different sections was performed, and the results were compared to those of Okada’s analytical solutions. The fault was modeled with the existing heterogeneity of the domain having been considered. The influences of both uniform and non-uniform slip distributions were investigated. Furthermore, three levels of simplification for geometric creation of geological layers’ boundaries were defined in order to evaluate the effects of the geometric complexity of the geological layering on the displacement responses obtained with the finite element simulations. In addition to the assessment of slip distribution, layering complexity and heterogeneity, the results demonstrate both the capability and usefulness of the proposed models in the dislocation analysis for the Karaj segment of North-Tehran fault.

Graphical abstract

Keywords

finite element method / fault dislocation / slip distribution / the North-Tehran fault / heterogeneity / geological layering

1 Introduction

Earthquakes are one of the most destructive natural hazards resulting often in significant loss of life. Earthquake engineering focuses in part on the estimation and mitigation of an earthquake’s consequences. In this regard, understanding the seismic behavior of faults influencing the various lifelines is inevitable because many oil and gas transmission lines are near to faults [1,2]. The study of earthquake scenarios is performed to acquire a better understanding of fault behavior, which can lead to safer engineered structures. Predicting the surface deformation of ground caused by earthquake fault dislocation is an important topic in computational seismology. This topic is known as kinematic rupture simulation, in which the slip history on a fault is predefined [3]. Nowadays there are many solution methods for the elastic dislocation problem that was initially introduced by Volterra and Somigliana [4]. These solution methods can be classified into analytical and numerical approaches. A collection of analytical solutions for the elastic dislocation problem with simplified assumptions such as homogeneity, isotropy and uniformity of the available domain was proposed by Okada. These analytical solutions give consistent results based on geological observations [5,6]. Okada’s analytical solutions assume that both sides of the fault dislocation are rigid. Although the analytical solutions are promising in both computational cost and accuracy [7], they cannot support every condition of the dislocation problem such as heterogeneity. The crustal properties are much more complex in practice, thus a more comprehensive analysis method should be sought as a solution to address a wide range of issues [7–9]. In this sense, the numerical method does not have the limitations associated with the analytical method. Therefore, many numerical methods have been applied to the analysis of fault dislocation [10–13]. Also, the stochastic spectral element method has recently been developed which considers the dislocation-induced wave propagation in random media [14]. Obviously, the faults are known as discontinuities in the ground. In this regard, many numerical studies have been carried out in the field of discontinuity simulation such as the following examples:

The phase field model for complex patterns of cracks in rocks [15], the cracking element method for quasi-fragility rupture [16,17], the meshfree and particle methods for crack growth [18], the finite cell method and the stochastic finite cell method for complex geometries [19,20], and the dual-horizon peridynamics [21], among others.

Here, we summarize a part of relevant investigations undertaken. A study on the effect of superficial layer overlying elastic half-space on the surface displacements due to a dip-slip dislocation of rectangular fault incorporating a two-dimensional finite element model showed that an underestimated seismic moment release can occur when the effect of an existing low-rigidity layer is not considered [22]. Zhao et al. [23] used three-dimensional finite element models to determine deformation and stress induced by strike-slip, thrust and tensile faults, considering the layering in the crust. Megna et al. [11] investigated the effects of heterogeneities considering different slip distributions for two typical earthquakes in the Central Apennines, and computed their resulting displacement fields. Lavecchia et al. [24] obtained the ground deformation under the 2016 Amatrice earthquake (Central Italy) by the DInSAR measurements; the finite element method (FEM) was applied along with DInSAR measurements for the fault model, and displacement fields of both approaches were reported. Zakian et al. [25] utilized a finite element approach based on the Monte Carlo simulation for the analysis of faults with geometric uncertainty using the kinematic approach. Gómez et al. [26] determined surface displacements subjected to a synthetic megathrust earthquake which took into consideration elastic dislocation models with/without the gravity effects. Hearn [27] employed two types of models including the finite element and GPS-constrained cases to detect the discrepancies between geodetic and geologic fault slip rates in southern California, and to assess the extent of off-fault deformation.

Tehran metropolitan area contains several active faults and has experienced earthquakes with magnitudes greater than M6.5 in its history. The most important faults in the Tehran region are named as follows: the North-Tehran, Parchin, Eyvanakey, Mosha and Garmsar [28] faults. Although no destructive earthquakes have struck Tehran over the last 180 years, there are historical evidences of earthquakes from moderate magnitude M~5.5–6.5 which occurred around 864 and 1665 AD to large magnitude M > 7 earthquakes that occurred around 743, 855, 958, 1177, and 1830 AD [29,30]. Ritz et al. [31] indicated that eight earthquakes with magnitudes greater than M6.5 have happened during the past 30000 years, from studies on a ~3 m fault scarp between Tehran and Karaj located on the North-Tehran fault. Investigations demonstrate that there is paleoseismological evidence of at least two M~7 ruptures on the North-Tehran fault that occurred during the past 8000 years, thereby this fault can be categorized as an active fault which can potentially generate earthquakes of magnitudes up to M7.2–7.4.

The southern faults of the central Alborz region, which are located at north of Tehran, dip northward and pass beneath the mountains. The tectonic deformation of this area is caused by the convergent movements of central Iran and the Caspian blocks, which results in distinct reverse and strike-slip faults. The Mosha and Taleghan faults are two strike-slip faults located near the Tehran province, and the North-Tehran fault is considered to be a reverse fault accommodating a shortening of almost 0.7 mm/year perpendicular to the fault strike. Also, a part of the North-Tehran fault intersects with the Mosha fault [29]. Two eastern and western segments constitute the North-Tehran fault at the southern boundary of the Alborz area [29]. The length of this V-shaped fault is 110 km. The 58 km-long eastern segment of the fault is located beneath the northern area of the Tehran metropolitan and is known as a mostly reverse fault, while the western segment which crosses the northern area of Karaj has a reverse mechanism with a negligible left-lateral component. Since both fault segments can be hazardous for highly populated Tehran and Karaj cities, appropriate investigations of the fault slip simulation are necessary.

Geological observations demonstrate that the crust is elastically inhomogeneous [23], thus the effects of layering and heterogeneities on the crustal deformation need to be considered in order to detect the magnitude and patterns of deformation and stress. In this paper, the FEM is employed for simulating the crustal deformation subjected to the dislocation of the western segment of North-Tehran fault located in Karaj (i.e., the Karaj segment). Accordingly, the geological map of Karaj metropolis including the fault segment is used for geometry creation of the finite element model. In the first stage, the homogeneous counterpart of the fault domain is analyzed using two geologic sections, and results are compared to those of Okada’s analytical solutions. In the second stage, the heterogeneous properties of the fault domain are considered according to the geologic layering. Also, the effects of both uniform and non-uniform slip distributions are studied. Furthermore, three independent simplifying assumptions are considered for defining the boundaries of layers in order to assess the effects of geometric simplification for geological layering in the finite element analysis of surface deformations under fault slip.

2 Elastic dislocation problem

In the fault dislocation problem, the crustal domain is a semi-infinite medium having a fault leading to discontinuity, as shown in Fig.1 where {\Gamma }_{\text{1}} and {\Gamma }_2 show the force and displacement boundary conditions, respectively, such that \mathit{\text{Γ}}={\mathit{\text{Γ}}}_1\cup {\mathit{\text{Γ}}}_2. b is the slip vector over the dislocation plane \mathcal{F} embedded in \text{Ω}. The elastostatic governing equation is stated as follows:

(1)\text{div} \sigma (\mathit{u})+\mathit{f}=\mathbf{0}\text{in} \text{Ω}/\mathcal{F},

for which displacements u in domain \mathrm{\Omega } should be sought under the following boundary conditions:

(2a)\begin{array}{}\text{(2a)}& [{\sigma }_{\text{v}}(\mathit{u})]=\mathbf{0}\text{on}\mathcal{F},\end{array}

(2b)\begin{array}{}\text{(2b)}& [\mathit{u}]=\mathit{b}\text{on}\mathcal{F},\end{array}

(2c)\begin{array}{}\text{(2c)}& {\sigma }_{\text{n}}(\mathit{u})=\mathbf{0}\text{on}{\Gamma }_1,\end{array}

(2d)\begin{array}{}\text{(2d)}& \mathit{u}=\mathbf{0}\text{on}{\Gamma }_2,\end{array}

where \sigma denotes the stress field which is continuous in \text{Ω}/\mathcal{F}; Eq. (2a) represents the normal stress difference on the dislocation plane \mathcal{F}; Eq. (2b) indicates that the displacement created on the dislocation plane is equal to the slip vector b; Eqs. (2c) and (2d) represent the normal stress on the free surface of the earth and zero-displacement at the lateral boundaries, respectively. Also, the jump operator is defined as follows:

(3)[\mathit{u}]={\mathit{u}}^{+}- {\mathit{u}}^{-},

where {\mathit{u}}^{+} and {\mathit{u}}^{-} denote the displacements at two opposite sides of \mathcal{F}[7].

Discretizing the weak form of governing equation with the FEM finally gives the following relation:

(4)\mathit{K}\mathit{U}=\mathit{F},

in which K, U, and F are the stiffness matrix, displacement and force vectors, respectively. In this study, infinite elements are utilized in order to impose the effects of unbounded boundaries [32]. The surface of domain is a traction-free boundary. Fig.2 shows a finite element model of the problem considering the boundary conditions, in which finite and infinite elements are also shown. Infinite elements can efficiently capture the displacement decays at infinite boundaries such that the solution for a truncated domain can be obtained with higher accuracy and with lower number of elements than the case where only finite elements are used for those boundaries. In this study, the contact elements are used for modeling the fault. As shown in Fig.3, the contact problem is considered using the surface-to-surface method. It is worth mentioning that contact modeling in the present problem is based on the classical theory of non-adhesive contact of elastic bodies using the penalty method [33,34].

3 Geological aspects of the Karaj fault

Due to the large population of Karaj metropolis, crisis management is a vital process to be planned for mitigating the casualties, economic losses and structural damages under natural disasters like earthquake [35]. Since some engineering structures are categorized as being underground infrastructures such as oil and natural gas pipelines [36], a better understanding of the displacement distribution caused by earthquake fault is vital for detecting the hazardous zones. In this study, dislocation of a segment of the North-Tehran fault located in the Karaj (so-called, the Karaj fault) is investigated. For this purpose, two-dimensional numerical simulation of the Karaj fault is performed with the FEM employing the geometry of the fault domain given in the geological map shown in Fig.4 [37]. Two geological sections of the domain are used for finite element analyses, as illustrated in Fig.5. To validate the numerical simulation, the surface deformation of the homogeneous counterpart of the domain is also calculated with Okada’s analytical solutions. This validation is carried out without defining the geologic layers, because the analytical solutions are limited to homogeneous domains [5,7]. As depicted in Fig.6, since the fault reaches the domain boundaries, the unbounded boundaries of this domain (i.e., the left, right, and bottom sides) are extended in order to implement the infinite elements in the numerical simulations.

4 Finite element models: the homogeneous domain

In this section, the Karaj fault domain is assumed as a homogeneous domain in order to make use of the analytical solutions [5,6]. As depicted in Fig.5, the surface of domain is not flat, and this curved surface slightly increases the differences between the FEM and analytical solutions apart from the traditional differences seen between the accuracy of numerical and analytical methods. Since the fault in the domain is not a completely straight line, the fault is divided into six parts as shown in Fig.6. Each part of the fault can have its own slip distribution, thus it is necessary to determine the slip distribution of each one as given by [38,39]:

(5)\log u=- 1.38+1.02\log L,

where u denotes the maximum slip of the fault in m, and L is the length of fault rupture in km. The rupture length can be taken as the fault width in a two-dimensional problem.

To use the analytical solutions, the displacements due to all parts of the fault are superimposed according to the principle of superposition. On the other hand, Fig.7 shows a coarse mesh of finite element analysis for both sections where the infinite elements are utilized for the unbounded boundaries. Here, both uniform and non-uniform slip distributions are considered. The average slip over the fault parts is used for the case with the uniform slip distribution, while each part has an independent slip for the case with non-uniform slip distribution which is more similar to a corresponding real-world problem. Also, three finite element meshes are generated for each type of slip distribution to check the effects of mesh density on the results. Tab.1 and Tab.2 summarize the slip values defined for the fault parts. It should be noted that all the sizes in the computer-aided design (CAD) data of geological map [37] have a relative scale of 1:25. For example, the height level of 1400 units in Fig.5(a) is equivalent to 56 units in the CAD data. Therefore, the scaled slip values in Tab.1 and Tab.2 are utilized for the FEM models, and all the displacements resulting from the FEM models as well as the sizes should be multiplied by 25, as already performed in the following Sections of this paper.

4.1 Section A-A

The geometrical characteristics of section A-A are listed in Tab.3. Material properties are taken as follows: Poisson’s ratio 0.3, and elasticity modulus 14.1 GPa. The surface displacements due to uniform and non-uniform distributions are shown in Fig.8 where the FEM solutions provide the desirable accuracy in comparison with those of the analytical solutions. Also, the results of three meshes are compared, which demonstrate the suitability of the spatial discretization, and indicate the small sensitivity of the numerical model to the mesh density. Tab.4 reports details of the generated meshes. Furthermore, Fig.8 shows that the FEM solutions for the case with non-uniform slip distribution are accurate compared to the analytical solutions. Topographical effect of the ground surface leads to some response differences between numerical and analytical solutions at the right side of domain with steeper surface slope.

4.2 Section B-B

Tab.5 includes the geometric characteristics of the fault parts in section B-B whose material properties are identical to those of section A-A. The characteristic lengths of elements are shown in Tab.6. Horizontal and vertical displacements of the surface are drawn in Fig.9, demonstrating that the response sensitivity with respect to mesh density is higher than that of section A-A, because some inaccurate displacements are visible in the peak values of displacements obtained by the coarser meshes (see Fig.9).

5 Finite element models: the heterogeneous domain

In this section, numerical simulation of the Karaj fault in the heterogeneous domain is examined according to the geological map. To capture the effects of inclusions and/or boundaries within the crustal layers, three simplification levels are introduced for each section of the fault domain. Similar to Section 4, the analysis of each model is carried out with three different meshes, and also two slip distributions are imposed to the fault. The slip values mentioned in Tab.1 and Tab.2 are utilized here. Obviously, due to the limitation of the analytical solutions, only the FEM is herein employed for the analysis of the heterogeneous domain.

5.1 Section A-A

Three definitions are provided for the layering properties, as shown in Fig.10. As visible, the first model contains the maximum simplifications with respect to details of the geological sections in Fig.5, whereas the third model contains the minimum simplifications (almost without simplification). Thus, the second model contains a moderate level of simplification. Tab.7 reports the material properties of every layer of the fault domain in section A-A [40–46]. For any layer composed of multiple materials, an equivalent material definition is assumed in Tab.7. To achieve a fair judgment for the mesh assessment, an identical characteristic length for infinite elements at the unbounded boundaries is considered in all the three models. Also, identical meshes are employed for the analysis of the domain under two slip distributions in order to capture the effects of the slip distribution solely on the results. Tab.8 shows the characteristic length of the elements used for those meshes.

5.1.1 The Karaj fault with uniform slip

Ground surface displacements resulting from three FEM models with three meshes are compared in Fig.11. Consequently, these models with different layering properties provide similar displacement curves. Also, the displacement responses determined with the finest mesh of each model are illustrated in Fig.12, demonstrating the effectiveness of the simplified layering properties. As visible, the differences between horizontal displacements are higher than those of vertical displacements, depending on the model employed. Based on Fig.12, the results of model 2 and model 3 are close together overall. Therefore, the inclusions within the geologic layers of model 1, as appeared in model 3 (see Fig.10), are not very effective in the displacement responses due to the uniform dislocation of fault. Fig.13 shows the displacement fields obtained by the FEM solution of each model using the finest mesh.

5.1.2 The Karaj fault with non-uniform slip

In the case where non-uniform slip is imposed to the Karaj fault, Fig.14 illustrates horizontal and vertical surface displacements calculated for three models shown in Fig.10. Each model uses three meshes to evaluate the convergence of the FEM solutions. Fig.15 shows the surface displacements calculated based on the finest meshes used for the three models. In comparison with Fig.12, Fig.15 demonstrates that the results are not very sensitive to the slip distribution here. The distribution of displacement values in the section A-A is indicated in Fig.16, where the slip distribution is more effective for displacements around the fault with non-uniform slip than what is shown in Fig.13 for uniform slip. However, similar to the case with uniform slip, the results of model 2 and model 3 are compatible when the surface displacements are needed, as shown in Fig.15.

5.2 Section B-B

Similar to section A-A, three models with different levels of simplification for the layering properties are implemented, as indicated in Fig.17. These differences can provide various degrees of geological heterogeneity for section B-B whose material properties are presented in Tab.9 [40–46]. Similar to Tab.7, an equivalent material definition is assumed when a layer is composed of multiple materials in Tab.9. In addition, material properties of geological layers are assumed based on available data on these layers. Mesh details are listed in Tab.10, and the other definitions of section B-B are identical to those of section A-A.

5.2.1 The Karaj fault with uniform slip

The average slip of fault parts is used to analyze the problem under uniform slip. Again, the effects of heterogeneity in geological layers are investigated with three different models. In addition to the effects of heterogeneity of layers, the effect of mesh density is also studied for the displacements of the ground surface. Deformations of the ground surface for the employed models are illustrated in Fig.18, where model 3 is more sensitive to mesh density. Also, Fig.19 represents the deformations corresponding to the finest meshes, which illustrates that horizontal displacements are more sensitive to the simplifications of layers. Indeed, model 2 provides sufficient accuracy because its results are close to those of model 3 including more complexity due to heterogeneity, and hence the higher effects of inclusions are negligible for section B-B. The contours of displacement field are illustrated in Fig.20. These contours also demonstrate that the values of displacements are similar for various models of section B-B. The differences between the simplified models are lower than those of section A-A.

5.2.2 The Karaj fault with non-uniform slip

In the last case of this study, horizontal and vertical displacements of the ground surface due to the non-uniform slip distribution of the Karaj fault are considered, as depicted in Fig.21. The results show that model 3 is more sensitive to the meshing than models 1 and 2. To find out the effect of heterogeneity in geological layers, as in the previous section, the responses corresponding to the finest mesh of all three models are illustrated in Fig.22 which shows that the horizontal displacements of three models are not very close to each other, whereas they are close for vertical displacements. Fig.23 shows that the displacement contours corresponding to models 2 and 3 are fairly matched and as such it is not necessary to employ the complicated layering definitions of model 3.

6 Conclusions

Investigating the seismic behavior of faults can enhance the safety of engineering structures against earthquakes. In the present study, two-dimensional simulation of the Karaj fault dislocation in elastic half-space was investigated using FEM. Two sections of the existing geological map were used for the numerical simulations. Also, two different slip distributions were utilized to analyze the problem. Since the analytical methods are only applicable to homogeneous domains, the dislocation problem for homogeneous counterpart of the layered domain was also solved with the analytical methods in order to validate the numerical solutions. Then, the heterogeneous form consisting of various geological layers (as in the geological map of Karaj) was considered with the proposal of three models having various levels of simplifications in their layering definitions. Thus, the effects of these simplifications are also studied.

Clearly, heterogeneity of geological layers affects the coseismic displacement field. Also, when the heterogeneity level (the geological layering irregularity) increases, the results become more sensitive to the mesh density. The results indicate that the complexity level of geological layering in model 2 is sufficient for this dislocation analysis, and therefore it is not necessary to consider model 3 having a higher level of complexity. When non-uniform slip distribution is occurred on fault, the responses are significantly influenced by the choice of mesh. Furthermore, the surface displacement curves show that the heterogeneous domain is more sensitive to the choice of mesh than its homogeneous counterpart. Two geological cross-sections (i.e., sections A-A and B-B) show different behaviors such that the calculated ground displacements depend on the slip distribution, mesh definition and the heterogeneity of domain. Given the importance of identifying the high-risk areas due to their potential for fault dislocation, the illustration of displacement contours can be very useful. These contours confirm that, obviously, the displacements are maximum at the vicinity of the fault regardless of whether or not the slip distribution is uniform. Finally, this study can provide valuable information about the kinematic modeling of the Karaj segment of North-Tehran fault to distinguish its seismicity further.

References

[1] Sabagh M, Ghalandarzadeh A. Centrifuge experiments for shallow tunnels at active reverse fault intersection. Frontiers of Structural and Civil Engineering, 2020, 14( 3): 731–745

[2] Izadi M, Bargi K. Improvement of mechanical behavior of buried pipelines subjected to strike-slip faulting using textured pipeline. Frontiers of Structural and Civil Engineering, 2019, 13( 5): 1105–1119

[3] IgelH. Computational Seismology: A Practical Introduction. Oxford: OUP Oxford, 2016

[4] Fichera G. The Italian contribution to the mathematical theory of elasticity. Meccanica, 1984, 19( 4): 259–268

[5] Okada Y. Surface deformation due to shear and tensile faults in a half-space. Bulletin of the Seismological Society of America, 1985, 75( 4): 1135–1154

[6] Okada Y. Internal deformation due to shear and tensile faults in a half-space. Bulletin of the Seismological Society of America, 1992, 82( 2): 1018–1040

[7] van Zwieten G J, Hanssen R F, Gutiérrez M A. Overview of a range of solution methods for elastic dislocation problems in geophysics. Journal of Geophysical Research. Solid Earth, 2013, 118( 4): 1721–1732

[8] Zakian P. An efficient stochastic dynamic analysis of soil media using radial basis function artificial neural network. Frontiers of Structural and Civil Engineering, 2017, 11( 4): 470–479

[9] Ranjbarnia M, Zaheri M, Dias D. Three-dimensional finite difference analysis of shallow sprayed concrete tunnels crossing a reverse fault or a normal fault: A parametric study. Frontiers of Structural and Civil Engineering, 2020, 14( 4): 998–1011

[10] Megna A, Barba S, Santini S. Normal-fault stress and displacement through finite-element analysis. Annals of Geophysics, 2005, 48( 6): 1009–1016

[11] Megna A, Barba S, Santini S, Dragoni M. Effects of geological complexities on coseismic displacement: Hints from 2D numerical modelling. Terra Nova, 2008, 20( 3): 173–179

[12] van Zwieten G J, van Brummelen E H, van der Zee K G, Gutiérrez M A, Hanssen R F. Discontinuities without discontinuity: The weakly-enforced slip method. Computer Methods in Applied Mechanics and Engineering, 2014, 271 : 144–166

[13] Zakian P, Khaji N. Spectral finite element simulation of seismic wave propagation and fault dislocation in elastic media. Asian Journal of Civil Engineering, 2016, 17( 8): 1189–1213

[14] Zakian P, Khaji N. A stochastic spectral finite element method for solution of faulting-induced wave propagation in materially random continua without explicitly modeled discontinuities. Computational Mechanics, 2019, 64( 4): 1017–1048

[15] Zhou S, Zhuang X, Zhu H, Rabczuk T. Phase field modelling of crack propagation, branching and coalescence in rocks. Theoretical and Applied Fracture Mechanics, 2018, 96 : 174–192

[16] Zhang Y, Zhuang X. Cracking elements: A self-propagating strong discontinuity embedded approach for quasi-brittle fracture. Finite Elements in Analysis and Design, 2018, 144 : 84–100

[17] Zhang Y, Zhuang X. Cracking elements method for dynamic brittle fracture. Theoretical and Applied Fracture Mechanics, 2019, 102 : 1–9

[18] Rabczuk T, Belytschko T. Cracking particles: A simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61( 13): 2316–2343

[19] Giraldo D, Restrepo D. The spectral cell method in nonlinear earthquake modeling. Computational Mechanics, 2017, 60( 6): 883–903

[20] Zakian P. Stochastic finite cell method for structural mechanics. Computational Mechanics, 2021, 68( 1): 185–210

[21] Ren H, Zhuang X, Cai Y, Rabczuk T. Dual-horizon peridynamics. International Journal for Numerical Methods in Engineering, 2016, 108( 12): 1451–1476

[22] Cattin R, Briole P, Lyon-Caen H, Bernard P, Pinettes P. Effects of superficial layers on coseismic displacements for a dip-slip fault and geophysical implications. Geophysical Journal International, 1999, 137( 1): 149–158

[23] Zhao S, Müller R, Takahashi Y, Kaneda Y. 3-D finite-element modelling of deformation and stress associated with faulting: Effect of inhomogeneous crustal structures. Geophysical Journal International, 2004, 157( 2): 629–644

[24] Lavecchia G, Castaldo R, Nardis R, De Novellis V, Ferrarini F, Pepe S, Brozzetti F, Solaro G, Cirillo D, Bonano M, Boncio P, Casu F, De Luca C, Lanari R, Manunta M, Manzo M, Pepe A, Zinno I, Tizzani P. Ground deformation and source geometry of the 24 August 2016 Amatrice earthquake (Central Italy) investigated through analytical and numerical modeling of DInSAR measurements and structural-geological data. Geophysical Research Letters, 2016, 43( 24): 12389–12398

[25] Zakian P, Khaji N, Soltani M. A Monte Carlo adapted finite element method for dislocation simulation of faults with uncertain geometry. Journal of Earth System Science, 2017, 126( 7): 105

[26] Gómez D D, Bevis M, Pan E, Smalley R Jr. The influence of gravity on the displacement field produced by fault slip. Geophysical Research Letters, 2017, 44( 18): 9321–9329

[27] Hearn E H. Kinematics of southern California crustal deformation: Insights from finite-element models. Tectonophysics, 2019, 758 : 12–28

[28] Berberian M, Yeats R S. Patterns of historical earthquake rupture in the Iranian Plateau. Bulletin of the Seismological Society of America, 1999, 89( 1): 120–139

[29] Majidinejad A, Zafarani H, Vahdani S. Dynamic simulation of ground motions from scenario earthquakes on the North-Tehran fault. Geophysical Journal International, 2017, 209( 1): 434–452

[30] Majidinejad A, Zafarani H, Vahdani S. Broad-band simulation of M7. 2 earthquake on the North-Tehran fault, considering non-linear soil effects. Geophysical Journal International, 2018, 213( 2): 1162–1176

[31] Ritz J F, Nazari H, Ghassemi A, Salamati R, Shafei A, Solaymani S, Vernant P. Active transtension inside central Alborz: A new insight into northern Iran–southern Caspian geodynamics. Geology, 2006, 34( 6): 477–480

[32] LiuG RQuek S S. Finite Element Method: A Practical Course. Oxford: Elsevier Science, 2003

[33] KimN H. Introduction to Nonlinear Finite Element Analysis. New York: Springer, 2014

[34] BoulbesR J. Troubleshooting Finite-Element Modeling with Abaqus. Cham: Springer, 2020

[35] HoriM. Introduction to Computational Earthquake Engineering. London: World Scientific, 2011

[36] Zhang J, Xiao Y, Liang Z. Mechanical behaviors and failure mechanisms of buried polyethylene pipes crossing active strike-slip faults. Composites. Part B, Engineering, 2018, 154 : 449–466

[37] KhezriMHeidarzadeh GShahidiAOroujniaPVakil Baghmisheh FGhaemiJHaddadanM. Urban geological map of Karaj. Geological Survey & Mineral Exploration of Iran, 2013

[38] Wells D L, Coppersmith K J. New empirical relationships among magnitude, rupture length, rupture width, rupture area, and surface displacement. Bulletin of the Seismological Society of America, 1994, 84( 4): 974–1002

[39] Kim Y S, Sanderson D J. The relationship between displacement and length of faults: A review. Earth-Science Reviews, 2005, 68( 3−4): 317–334

[40] Chen G, Chenevert M E, Sharma M M, Yu M. A study of wellbore stability in shales including poroelastic, chemical, and thermal effects. Journal of Petroleum Science Engineering, 2003, 38( 3−4): 167–176

[41] OkuHHaimsonB SongS R. True triaxial strength and deformability of the siltstone overlying the Chelungpu fault (Chi-Chi earthquake), Taiwan (China). Geophysical Research Letters, 2007, 34(9): L09306

[42] Sanahuja J, Dormieux L, Meille S, Hellmich C, Fritsch A. Micromechanical explanation of elasticity and strength of gypsum: From elongated anisotropic crystals to isotropic porous polycrystals. Journal of Engineering Mechanics, 2010, 136( 2): 239–253

[43] Hooshmand A, Aminfar M H, Asghari E, Ahmadi H. Mechanical and physical characterization of Tabriz Marls, Iran. Geotechnical and Geological Engineering, 2012, 30( 1): 219–232

[44] Li L, Meng Q, Wang S, Li G, Tang C. A numerical investigation of the hydraulic fracturing behaviour of conglomerate in Glutenite formation. Acta Geotechnica, 2013, 8( 6): 597–618

[45] Liu H, Bu Y, Nazari A, Sanjayan J G, Shen Z. Low elastic modulus and expansive well cement system: The application of gypsum microsphere. Construction & Building Materials, 2016, 106 : 27–34

[46] Turichshev A, Hadjigeorgiou J. Triaxial compression experiments on intact veined andesite. International Journal of Rock Mechanics and Mining Sciences, 2016, 86 : 179–193

RIGHTS & PERMISSIONS

Higher Education Press 2022