A quasistatic simulation of highly nonlinear problems under fault movements was carried out using the EXPLICIT module of ABAQUS. Combined with the secondary development program of the software, the application of the strain softening Mohr–Coulomb model in the simulation was realized. Free field-fault systems were simulated with two types of fault types (normal and reverse faults), four fault dip angles (45°, 60°, 75°, and 90°), and two kinds of soil (sand and clay). Moreover, the rupture laws and sensitivities of the sand and clay were studied with different soil thicknesses and different fault dip angles in the free field. The results show that the width of the ground zone with obvious deformation, which represents the point of the fault outcrop, the critical displacement of the fault, and the rupture characteristics of the overlying soil are closely related to the fault type and soil parameters. The critical displacement of the reverse fault is larger than that of the normal fault. The width of the ground zone with obvious deformation varies from 0.65 to 1.3 and does not exhibit a regular relationship with the type of soil. Compared with a normal fault, the rupture of a reverse fault is not prone to exposure at the surface.
Understanding the failure mechanisms of the free field under fault movements can provide a reference to analyse complex structures, indicating that the free field is an important part of engineering design and research. The simulation of the free field is simple, and there are no additional structural interactions with the fault, making it a relatively simple case of fault movement. When a bedrock fault is covered with a certain depth of soil, the dip angle, the displacement of the fault, and the soil parameters can be determined. However, there are still two problems worthy of our attention: (1) whether the fault rupture zone can be exposed at the surface along a fault outcrop and (2) what the deformation of the ground is and what causes significant deformation of the ground caused by a rupture of the overlying soil. These problems in practical applications of engineering and research are particularly important, and thus, many scholars’ attention has been directed toward this issue in recent years.
At present, many scholars have studied the rupture laws of the overlying soil due to fault movements, and most studies adopted numerical simulation methods [
Above all, most previous numerical analyses were concerned with the free field-fault system and were focused on studying the fracture characteristics of the overlying soil. However, few studies have been conducted on obvious ground deformations or the sensitivity of the overlaying soil deformation and rupture with different faults and soils. To further study the response characteristics of the free field under fault movements and understand the influences of fault movements on the overlaying soil, this paper comprehensively considered four key factors through numerical simulations, including the quasistatic analysis, mesh size, strain softening, and material damping. The ABAQUS/EXPLICIT software was used for the simulations and calculations; the simulations used the Mohr–Coulomb yield criterion in consideration of the properties of material strain softening. In addition, this study focused on two questions. The first is whether the fault rupture surface can be exposed at the ground surface and the outcrop position, and the second addresses the ground deformation mode and the zone of obvious ground deformation due to overlying soil fractures. Therefore, the fracture deformation characteristics of the overlying free field under a reverse fault and a normal fault can be obtained.
Combined with the quasistatic method, the numerical simulation of the EXPLICIT module in the finite element software ABAQUS is used for this numerical simulation. The quasistatic method has been widely used in previous studies to solve problems of fault movement and propagation [
The explicit dynamics analysis procedure is based on the implementation of an explicit integration rule together with the use of diagonal or “lumped” element mass matrices. The equations of motion for the body are integrated using the explicit central difference integration rule:
A small amount of damping is introduced to control high frequency oscillations. With damping, the stable time increment is given by
The explicit integration rule is simple but cannot provide the computational efficiency associated with the explicit dynamics procedure. The explicit procedure requires no iterations and no tangent stiffness matrix. A special treatment of the mean velocities, including
The central difference operator is not self-starting because the value of the mean velocity
Substituting (
The selected soil constitutive model is the Mohr–Coulomb model, which can realize the numerical simulation of most geotechnical engineering problems and achieve good simulation results. To consider the strain softening characteristics, this paper utilizes the user subroutine USDFLD (VUSDFLD) to supplement the Mohr–Coulomb model. The main purpose is to change the field variables at the material points to alter the properties of the material. The following interface programs can be customized: SUBROUTINE USDFLD(FIELD, STATEV, PNEWDT, DIRECT, T, CELENT, TIME, DTIME, CMNAME, ORNAME, NFIELD, NSTATV, NOEL, NPT, LAYER, KSPT, KSTEP, KINC, NDI, NSHR, COORD, JMAC, JMATYP, MATLAYO, LACCFLA) C INCLUDE 'ABA_PARAM.INC' C CHARACTER∗80 CMNAME, ORNAME CHARACTER∗3 FLGRAY(15) DIMENSION FIELD(NFIELD), STATEV(NSTATV), DIRECT(3,3), T(3,3), TIME(2) DIMENSION ARRAY(15), JARRAY(15), JMAC(∗), JMATYP(∗), COORD(∗) RETURN END SUBROUTINE VUSDFLD(nblock, nstatev, nfieldv, nprops, ndir, nshr, jElemUid, kIntPt, kLayer, kSecPt, stepTime, totalTime, dt, cmname, coordMp, direct, T, charLength, props, stateOld) C INCLUDE 'VABA_PARAM.INC' C CHARACTER∗80 CMNAME, ORNAME CHARACTER∗3 FLGRAY(15) DIMENSION FIELD(NFIELD), STATEV(NSTATV), DIRECT(3,3), T(3,3), TIME(2) DIMENSION ARRAY(15), JARRAY(15), JMAC(∗), JMATYP(∗), COORD(∗) RETURN END
The vertical movement velocity of the fault is calculated and compared with
The displacement of the ground surface: (a) vertical displacement and (b) tilt displacement.
According to Figure
The ground vertical displacement curves and ground tilt curves with different grid sizes are shown in Figure
The displacement of the ground surface: (a) vertical displacement and (b) tilt displacement.
The overlying soil will produce a relatively large deformation and strain, and the reduction of the soil strength should be considered at this point. If the numerical simulation does not consider the impact of this factor, the results may be different from those in practical situations. In addition, previous scholars have made a good comparison of this result [
Strain softening and nonstrain softening are both carried out in this simulation. When considering strain softening, the residual friction angle and residual dilation angle are
The displacement of the ground surface: (a) vertical displacement and (b) tilt displacement.
It can be concluded from the comparison of the plastic strain zone and ground deformation under the two conditions that considering strain softening for this model is necessary to fully estimate the effects of fault movements on the overlying soil.
For the quasistatic problem, the method can greatly reduce the influence of dynamic waves on the model, but the damping of the material cannot be ignored. In view of this, Rayleigh damping is considered in the model [
The displacement of the ground surface: (a) vertical displacement and (b) tilt displacement.
From Figure
In this model, the overlying soil is a homogeneous single body. The fault plane is flat and penetrates the bedrock to reach the lower part of the soil. Therefore, all of the attention can be focused on the internal response of the overlying soil due to the fault movements. The structure and parameters of the finite element model are shown in Figure
Schematic diagram of the model structure and parameters.
The model grid and boundary conditions with a 20 m deep overlying soil layer are shown in Figure
The model grid and boundary conditions.
The lower boundary is divided into two parts. The left side of the footwall is established with fixed constraints in the horizontal and vertical directions. A displacement is applied to the right side of the hanging wall equivalent to a displacement applied to the fault. The transformation of the fault dip angle is simulated by adjusting the ratio of the horizontal and vertical displacement components. The horizontal displacement on the left side of the hanging wall is equivalent to that of the bedrock. The right side of the footwall is established with a fixed constraint in the horizontal direction, and both sides of the boundary are unconstrained in the vertical direction. The contact between the overlying soil layer and the bedrock is considered to be fully complete. In addition, since this hypothesis is more reasonable than establishing a rough interface between the rock and soil layers, relative slip at the interface can be completely avoided.
In this case,
The contact constraint is enforced with a Lagrange multiplier representing the contact pressure in a mixed formulation. The virtual work contribution is
This simulation considers only normal and reverse faults; therefore, strike-slip faults are not studied. The fault vertical dislocation velocity
The soil parameters.
Soil type | Dry sand | Clay |
---|---|---|
|
1.7 | 1.8 |
|
|
|
|
0.3 | 0.35 |
|
0 | 2 |
|
— |
|
|
35 | 0 |
|
25 | — |
|
15 | 0 |
|
0 | — |
|
10 | — |
If the numerical parameters are in accordance with the established model, the simulation results are in good agreement with the actual investigations of fault movements [
The ground displacement curves with different dip angles for the normal fault in sand: (a) 45°, (b) 60°, and (c) 75°.
The ground displacement curves with different dip angles for the reverse fault in sand: (a) 45°, (b) 60°, (c) 75°, and (d) 90°.
The ground displacement curves with different dip angles for the normal fault in clay: (a) 45°, (b) 60°, and (c) 75°.
The ground displacement curves with different dip angles for the reverse fault in clay: (a) 45°, (b) 60°, (c) 75°, and (d) 90°.
From Figures
The ground incline curves are shown in Figures
The ground incline curves with different dip angles for the normal fault in sand: (a) 45°, (b) 60°, and (c) 75°.
The ground incline curves with different dip angles for the reverse fault in sand: (a) 45°, (b) 60°, (c) 75°, and (d) 90°.
The ground incline curve with different dip angles for the normal fault in clay: (a) 45°, (b) 60°, and (c) 75°.
The ground incline curves with different dip angles for the reverse fault in clay: (a) 45°, (b) 60°, (c) 75°, and (d) 90°.
The relationship curves between the ground maximum absolute inclination angle and the fault dip angle.
As indicated in Figure
With an increase in the fault movement, the maximum inclination of the ground increases gradually; in addition, the increase in the speed is slow at the beginning, for example, when
The relationship curves between
There are two approaches to determine the location of the fault outcrop. One approach is to find the intersection between the main rupture zone and the ground, and the other is the maximum inclination point of the ground. The shortest horizontal distances from the above two points to the fault are expressed as
Fracture zone inclination.
Fault inclination | Sand | Clay | ||
---|---|---|---|---|
|
|
|
|
|
135 | 42.8 | 44.4 | 50.5 | 49.6 |
120 | 56.0 | 53.1 | 60.1 | 59.0 |
105 | 76.0 | 67.0 | 81.5 | 69.4 |
90 | 87.1 | 87.1 | 89.9 | 90.0 |
75 | 80.1 | 73.3 | 80.1 | 65.8 |
60 | 65.8 | 64.6 | 55.1 | 55.0 |
45 | 59.1 | 59.0 | 47.3 | 47.2 |
The relationship between
The relationship curves between
To illustrate the influences of different soil depths
The relationship curves between
From Figure
The finite element software ABAQUS is utilized to simulate the free field-fault system and to study the fracture mode, ground displacement, and deformation characteristics of the overlying soil in the free field. The direction of a fault in the fracture zone of the overlying soil may deflect or bend with respect to the fault dip. When the dip angle of the normal fault is less than or equal to 45°, a second fracture zone may appear in the overlying soil that may lead to a downward movement of a block of the triangular fracture, which is similar to Coulomb’s Earth pressure theory of soil mechanics. However, this phenomenon will be eliminated with an increase in the fault dip angle. Compared with a normal fault, the fracture zone in a reverse fault cannot easily outcrop at the ground surface; that is, a larger critical fault displacement is required before the outcrop is achieved. Moreover, with an increase in the fault dip angle, the critical fault displacement increases gradually, and the width of the obvious surface deformation zone at the ground surface also increases gradually from the normal fault at 45° to the reverse fault at 45° ( Under the conditions with the same fault dip angle and displacement, the dynamic response of sand is greater than that of clay. In addition, sand exhibits a larger deformation at the ground surface in close relation to the soil strength. Sand is a compressive and nontensile material, and the compression of sand is less than that of clay due to its high strength. Therefore, the response of the overlying soil surface to the movement of the fault is greater, and the fracture zone dip angle of the sand is smaller than that of clay. In a normal fault, the cohesion and a portion of the tensile strength of clay can reduce the effects of the extension and shearing coincident with the fault movements. Therefore, the fracture zone dip angle of sand is larger than that of clay. The width of the obvious deformation zone at the ground surface and the critical displacement of the fault are directly proportional to the depth of the overlying soil. However, the proportional relationship between the horizontal distance of the two outcrop points to the fault and the soil depth H is not obvious.
The authors declare that they have no conflicts of interest.
This study was supported by the National Natural Science Foundation of China under Grant no. 41602332 and the Student Innovation and Entrepreneurship Training Program of China under Grant no. 201710615051.