XFEM Composite Failure Criterion and Slope Failure Simulation Based on ABAQUS

Abstract

Conventional numerical methods face major challenges in simulating the complex failure process of soil slopes effectively and accurately. This paper introduces a tension-shear composite failure criterion that elucidates the compound failure mechanisms of soil slopes, realized in the ABAQUS software through the secondary development of a user subroutine and simulated via the extended finite element method (XFEM) module. This method is utilized to simulate the process of soil slope failure under conditions that include heaping load at the crest and excavation at the toe, accompanied by analyses of the failure patterns. The methodology's validity and accuracy are substantiated through comparison with experimental data. The proposed approach adeptly captures the initiation, propagation, and ultimate penetration of cracks during the slope failure process, offering an effective method for simulating the entire slope failure.

1 Introduction

Landslides represent a prevalent geological hazard, capable of inflicting substantial destruction. They have consistently been a focal point of scholarly inquiry within geotechnical engineering. Conventional analyses of slope stability are predominantly founded on the principles of limit equilibrium, augmented by assorted techniques for identifying potential sliding planes. Yet, these methodologies are contingent upon a rigid-plastic assumption, implying that the shear strength of the sliding surface attains its threshold simultaneously, and they neglect the nonlinear stress-strain relationship within the soil, as well as the integral procession of slope failure, encompassing the failure initiation, evolution, and ultimate rupture [1]. Contrastingly, the finite element method (FEM) is favored for its comprehensive capture of the deformation and the failure mechanics of the slopes, accomplished by integrating the constitutive relationship of the soil mass. FEM refines the assessment of slope stability by incorporating a strength reduction factor to modulate the strength parameters of the slope. As the slope approaches instability, the nonlinear finite element computation ceases to converge, with the reduction coefficient concurrently serving as the slope's safety factor [2]. Slope failures seldom manifest as a sudden incident; rather, the onset of discontinuity originates at the most vulnerable juncture or nature flaw, proceeding through a succession of stages within which the stress at the crack tip transfer to adjacent area and the discontinuity grows, culminating in the slope's eventual total penetrative slip failure [3]. However, neither the limit equilibrium approach nor the strength reduction method can authentically and precisely encapsulate the entire progression from the inception of the slip surface to its ultimate penetration [4, 5].

In the application of the traditional finite element method for simulating discontinuities, such as cracks and shear bands, the mesh boundary must coincide with the discontinuity. As the discontinuity progresses, the mesh requires ongoing refinement to adapt the discontinuity geometry, which costs heavy. The extended finite element method (XFEM) enhances the finite element approach by integrating local enrichment functions, primarily the Heaviside function and the near tip asymptotic functions, into the FE approximation, based on the theory of partition of unity (PU). The interpolation of the enriched degrees of freedom reproduces the local enrichment functions within the element, thus permitting the simulation of a crack's arbitrary growth path without remeshing [6,7,8,9,10,11]. The corpus of research dedicated to the utilization of XFEM in the depiction of soil's discontinuous deformation and failure mechanisms is expanding. Yu et al. [12] devised a nonlinear analysis system that model the discontinuous internal boundaries within nonlinear materials employing the XFEM. A contact algorithm for the XFEM enriched soil discontinuities was established, incorporating a cohesive crack model and Willner's theory to simulate the adhesive, sliding, and separation states at the contact interface [13]. Wang et al. [14] formulated a methodology for assessing soil cracking through elemental stress analysis and stress backtracking, and utilized a sector control domain for the weighted mean stress evaluation and cracking direction determination, enhancing the precision and adaptability in ascertaining the nature, timing, and orientation of soil failure. Yu et al. [15] advanced a hybrid integration strategy within the extended finite element method, amalgamating Gaussian integration's precision with the expediency of single-point integration. Building upon this, they simulated the expansion of an arbitrary three-dimensional failure surface characterized by tensile-shear composite failure [16].

Since the release of version 6.9, ABAQUS, a widely utilized premier commercial CAE software, has integrated the essential functionalities of the extended finite element method. Nonetheless, the incorporation of more sophisticated and recently proposed methodologies necessitates the implementation via user-defined subroutines [17,18,19,20]. Giner et al. [17] initially employed a user subroutine to facilitate the implementation of the extended finite element method within ABAQUS in a two-dimensional framework, subsequently analyzing crack propagation in the context of a fretting fatigue problem. Cheng et al. [21] studied the desiccation shrinkage and cracking of soils by a 3D hydro-mechanical model established with ABAQUS via the subroutine of UMAT and the XFEM tools in ABAQUS. Cruz et al. [19] employed a user subroutine to surmount the constraints of ABAQUS on multiple cracks or crack intersection within a single element, thereby facilitating the simulation of crack intersection in porous rock formations. To replicate the progression of hydraulic fracture propagation, Tawfik et al. [22] notched plain and reinforced concrete beams were investigated numerically to study their flexural response using the contour integral technique (CIT), the extended finite element method (XFEM), and the virtual crack closure technique (VCCT) in ABAQUS. Teimouri [23] amalgamated the virtual crack closure technique (VCCT) with the extended finite element method to model the delamination growth of Type I fatigue fractures in composites, employing the direct cycling method within Abaqus.

From a mechanical perspective, the discontinuities in soils can be bifurcated into two distinct types: tensile cracks induced by tensile forces, and shear slips induced by shear forces. Presently, XFEM-based simulations predominantly address the formation of tension cracks, with simulations about the sliding phenomena under shear in soils being notably sparse. Specifically, the implementation of shear and tension-shear compound failure cracks on the ABAQUS platform remains unobserved. This study integrates an XFEM crack propagation criterion tailored for soil tension-shear composite failure via the ABAQUS user subroutine interface. In conjunction with ABAQUS's robust nonlinear computing capabilities, it presents an innovative approach for simulating the tension-shear composite failure process in soil slopes.

The latter part of this paper arranged as follows: Sect. 2 encapsulates the foundational principles of the extended finite element method (XFEM) and delineates the procedure for simulating crack propagation in ABAQUS utilizing its XFEM module. In Sect. 3, the tension-shear compound failure criterion is introduced and implemented via the UDMGINI subroutine. Concurrently, the weighted average stress is deployed to ascertain the crack propagation direction. The fourth section illustrates the application of this method in simulating centrifugal model experiments of soil slope failure under two disparate operational conditions (heaping load at the crest of the slope and excavation at the toe), corroborating the method’s rationality and accuracy.

2 Numerical Method

2.1 Nodal Enrichment Functions

XFEM introduces the asymptotic crack-tip functions [6] to capture the singularity of the crack tip and the Heaviside step function to characterize the displacement jump at the crack [24]. The displacement vector, inclusive of both displacement enrichment functions, is articulated as follows:

$$ {\bf{u}} = \sum_{I = 1}^n {N_I } (x)\left[ {{\bf{u}}_I + H(x){\bf{a}}_I + \sum_{\alpha = 1}^4 {F_\alpha } (x){\bf{b}}_I^\alpha } \right] $$

(1)

where, x is the coordinates, $N_I$ $(x)$ is the shape function, u_I is the node displacement vector describing continuous deformation, a_I is the enriched degree of freedom scaling the displacement jump, $H(x)$ is the Heaviside step function, ${\bf{b}}_I^\alpha$ is the crack tip enriched degree of freedom scaling the displacement singularity, $F_\alpha$ $(x)$ is the asymptotic crack-tip function.

The displacement description adheres to the partition of unity principle and is capable of reproducing the discontinuity and singularity of the crack displacement field within the element. The Heaviside step function is defined as follows:

$$ H(x) = \left\{ {\begin{array}{*{20}c} {1,} & {{\text{ if }}\left( {{\bf{x}} - {\bf{x}}^{\bf{*}} } \right) \cdot {\bf{n}} \ge 0} \\ { - 1,} & {\text{ otherwise }} \\ \end{array} } \right. $$

(2)

where, ${\bf{x}}^{\bf{*}}$ is the point on the crack closest to ${\bf{x}}$, and ${\bf{n}}$ is the unit normal vector on the crack.

The asymptotic crack-tip functions consist of four functions, which are as follows:

$$ F_\alpha (x) = \left[ {\sqrt {r} \sin \frac{\theta }{2},\sqrt {r} \cos \frac{\theta }{2},\sqrt {r} \sin \theta \sin \frac{\theta }{2},\sqrt {r} \sin \theta \cos \frac{\theta }{2}} \right] $$

(3)

where, ($r$, $\theta$) is the polar coordinate system with the crack tip as the origin, and $\theta$ = 0 is the crack tangent direction.

2.2 Phantom Node Method

To facilitate the enrichment of degrees of freedom and delineate the discontinuities within cracked elements, phantom nodes (overlaid atop the original authentic nodes) are embedded within the elements of the enriched region in the extended finite element module of ABAQUS. For the sake of coherence, each phantom node is entirely synchronized with its respective genuine node when the element remains intact, as illustrated in Fig. 1. As the crack penetrates the element, the element is cleaved into two parts, each associated with actual nodes at one side and phantom nodes at the other side. At this point, the phantom nodes are decoupled from their real counterparts, allowing for independent movement. By generating phantom nodes and managing their independent movement, the displacement description within ABAQUS’s XFEM module behaves good flexibility and accommodates the variations in discontinuities during simulating.

Fig. 1.

Phantom nodes of the cracking element during crack propagation.

2.3 Damage Initiation Criteria

Within the XFEM module of ABAQUS, six damage initiation criteria are built in: the maximum principal stress criterion, the maximum principal strain criterion, the maximum nominal stress criterion, the maximum nominal strain criterion, the quadratic separation interaction criterion and the quadratic traction interaction criterion. These criteria are pertinent to tensile failures and do not precipitate failure under compressive conditions. Yet, there exists no equivalent criterion for the prevalent shear failure in soils. To more accurately replicate the failure of a soil slope, secondary development through a user subroutine is indispensable. This entails the integration of a shear failure criterion within the purview of the XFEM module.

Within the ABAQUS’ XFEM module, the damage initiation criteria are amenable to customization, permitting users to define these thresholds via the subroutine [25]. Considering that soil slopes may concurrently undergo shear and tensile failures under varying stress conditions, any newly formulated failure criterion must encompass both eventualities. This research adopts the maximum tensile stress criterion for tensile failures and the Mohr-Coulomb criterion for shear failures. By the combination of these two criteria, the particular failures modes are determined and composite failure mechanism is revealed, depending on the stress conditions.

2.4 Damage Evolution Mechanism

Upon fulfillment of the pertinent damage initiation criterion, the damage evolution mechanism delineates the degradation rate of the bond stiffness. D is included to indicate the average global damage at the junction of the crack surface and the edge of the crack element. During the damage evolution process, D incrementally increases from 0 to 1. The normal and tangential stresses of the element, as influenced by the damage, are articulated as follows:

$$ t_n = \left\{ {\begin{array}{*{20}c} {(1 - D)T_n ,\,\,\,\,\,T_n \ge 0} \\ {T_n ,\,\,\,\,\,T_n < 0} \\ \end{array} } \right. $$

(4)

$$ t_s = (1 - D)T_s $$

(5)

$$ t_t = (1 - D)T_t $$

(6)

where, $D$ is the damage variable, $t_n$, $t_s$, $t_t$ are the normal stress component and two tangential stress components respectively, $T_n$ are the normal stress component without damage, $T_s$ and $T_t$ are the first and second tangential stress components without damage respectively. For user defined damage initiation criterion, corresponding damage evolution criterion must be defined simultaneously [26]. The cumulative effect of normal and tangential traction displacements is typically quantified through effective traction displacement, which characterizes the ensuing damage evolution. The effective traction displacement is defined as follows:

$$ \delta_m = \sqrt {{\left\langle {\delta_n } \right\rangle^2 + \delta_s^2 + \delta_t^2 }} $$

(7)

where, $\delta_n$ is normal traction displacement, $\delta_s$ and $\delta_t$ is the first and second tangential traction displacement.

3 Initiation and Propagation of Cracks

3.1 Crack Initiation

ABAQUS, by default, employs the stress or strain at the centroid of the element preceding the crack tip as the foundational metric for determining whether the damage initiation criteria are met. This method is both precise and efficient given a sufficiently refined mesh. However, when the precision of the mesh near the crack tip is coarse relative to the stress or strain field, the accuracy of stress or strain measurements based on the centroid's position diminishes. To remedy this discrepancy, ABAQUS offers an option to shift the damage initiation assessment from the centroid of the element to the position of the crack tip itself, thus leveraging the stress or strain at the crack tip for a more accurate evaluation, as illustrated in Fig. 2.

When tensile or shear failure occurs, the element is regarded to have met the damage initiation requirement, and damage evolution occurs, with the damage variable increasing from 0 to 1, resulting in the creation and spread of crack. To identify the failure criteria, the tensile stress level T_l and shear stress level S_l are set in this study for the tensile failure and shear failure, respectively. If T_l > 1, the element is considered to have tensile failure. Shear failure occurs in the judgment element when S_l > 1. The tensile stress level T_l and shear stress level S_l is defined as follows:

$$ T_l = \left| {\frac{\sigma_3 }{{f_{\text{t}} }}} \right|\quad \left( {\sigma_3 < 0} \right) $$

(8)

$$ S_l = \frac{\sigma_1 - \sigma_3 }{{\sin \varphi \left( {\sigma_1 + \sigma_3 + 2c/\tan \varphi } \right)}} $$

(9)

where $\sigma_1$ is the maximum principal stress, $\sigma_3$ is the minimum principal stress, $f_{\text{t}}$ is the tensile strength of the soil, $\varphi$ is the internal friction angle of the soil, and $c$ is the cohesion of the soil. When the stress level at the crack tip reaches 1, the appropriate type of failure is regarded to have occurred.

Fig. 2.

Location of the crack initiation.

3.2 Propagation of Crack

The mode of failure dictates the trajectory of crack expansion. Tensile failure transpires when the tensile stress level, T_l, exceeds unity, aligning the tensile failure plane with the normal direction of the minimum principal stress. Similarly, shear failure ensues when the shear stress level, S_l, surpasses unity, positioning the shear failure plane at an angle of 45° + $\varphi$/2 relative to the direction of the minimum principal stress.

It is essential to recognize that the stress estimation at the centroid or crack tip is a localized computation for an individual element. In cases of inferior mesh quality, a non-local averaging method may be employed to more accurately evaluate the stress or strain field preceding the crack tip, thus refining the precision of the estimated crack propagation direction.

To precisely govern the range of non-local averaging and smoothing for the crack growth direction, a semi-circular local zone is established with the crack tip as the epicenter and a predefined influence radius r_c positioned anterior to the crack tip, designated as the control domain (illustrated in Fig. 3). Within this control domain, non-local averaging and smoothing are exclusively conducted. The average stress within the control domain is determined as the weighted mean stress of all integration points within that domain, and the Gaussian weighting function is employed to assign greater significance to the integral points in proximity to the crack tip. The Gaussian weighting function is delineated as follows:

$$ \omega (r) = \frac{1}{{(2\pi )^{3/2} r_c^3 }}\exp \left( {\frac{ - r^2 }{{2r_c^2 }}} \right) $$

(10)

where, $r$ is the distance between an integral point in the control domain and the crack tip, $r_{\text{c}}$ is the control domain's influence radius, and the default is three times the enrichment element feature length.

Fig. 3.

Scope of control domain.

3.3 The Realization of Compound Failure Criterion

For the management of crack growth, the UDMGINI subroutine in ABAQUS can be configured to report the conditions of crack initiation and the orientation of crack propagation. Crafted in Fortran, the UDMGINI subroutine is capable of real-time extraction of stress, strain, coordinate positions, and temporal data from the element integration points. Utilizing these data, the subroutine computes the magnitude of the index value pertinent to the prevailing damage initiation criterion via a bespoke algorithm, thereby discerning whether, at a given moment, the element is experiencing cracking and the specific direction of such cracking.

The implementation of the tension-shear composite failure criterion within the UDMGINI subroutine involves the following steps:

• 1) The UDMGINI subroutine retrieves the stress tensor from the integration point of the element.

• 2) The principal stress magnitude and its directional orientation at the element's integration point are ascertained utilizing ABAQUS's SPRIND function.

• 3) The principal stress is then input into the formulas for tensile stress level (Eq. 8) and shear stress level (Eq. 9) for evaluation. An element is deemed to have undergone tensile failure if its tensile stress level exceeds the prescribed threshold, while the shear stress level remains below it. Conversely, an element is considered to have experienced shear failure if its shear stress level surpasses the threshold, irrespective of the tensile stress level. Should both tensile and shear stress levels meet or exceed their respective thresholds, a comparison of the two levels is conducted, and the failure mode corresponding to the higher stress level value is selected.

• 4) After ascertaining the mode of failure, the trajectory of crack propagation is established by converting the local coordinates to global coordinates, using the orientation of the minimum principal stress as a benchmark. In the event of tensile failure, the crack propagates in alignment with the normal to the minimum principal stress; conversely, during shear failure, the crack advances at an angle of 45° + $\varphi$/2 relative to the normal to the minimum principal stress.

4 Verification by Numerical Examples

4.1 Simulation of Soil Slope Failure Process Under the Condition of Heaping Load on Top of Slope

Regueiro et al. [27] probed the behavior of pronounced discontinuous fields in pressure-responsive plastic materials by employing the embedded discontinuity approach in conjunction with the augmented Drucker-Prager elastoplastic model. Within their paper, they developed a computational model depicting soil slope instability induced by a heaping load on the top of the slope, as represented in Fig. 4, wherein the slope ultimately succumbed to a comprehensive slide under the influence of the load. It is significant to note that the paper revealed the figuration of the slip surface (Fig. 5) but refrains from detailing the processes of crack initiation and propagation.

Fig. 4.

Instability problem of pushing slope.

Fig. 5.

Crack track diagram after slope failure.

The parameters employed in this analysis are derived from the data presented in the paper [27]. Encompassing an elastic modulus of 10 MPa, Poisson's ratio of 0.4, soil cohesion of 40 kPa, a friction angle of 10°, and a unit weight of 20 kN/m3. To enhance the comparability of the calculated results, these parameters align with those specified in the computational examples [27]. Additionally, these values fall within the typical spectrum of soil properties and do not undermine the objectives or conclusions of this study, thereby affirming the significance of the research findings. The model configuration is as follows: a rigid foundation is emplaced atop the slope, intimately integrated with the subjacent soil; the slope's gravity stress is factored into the calculation and a downward displacement u is exerted upon the center of the foundation.

As the loading displacement reaches u=4 cm, the soil in proximity to the bottom right corner of the robust foundation at the summit of the slope starts to fracture, as shown in Fig. 8; the tensile crack only cuts two layer of element before it turned to shear failure. Upon escalation to u=5 cm, the shearing slip plane proliferates swiftly, with its depth extending to 8.18m, as shown in Fig. 9. Then, the inclination of the slip plane moderates and the pressure induced by the loading amplifies the friction along the fracture surface, decelerating the slip plane’s propagation. Complete penetration of the slope by the cracks transpires at a displacement load of u = 10 cm, as shown in Fig. 10. With the increase of the displacement load, the slope undergoes a process of uniform deformation, nonuniform deformation, formation of the slip surface and finally overall failure. At this time, The vertical and horizontal displacement cloud maps of the slope are shown in Fig. 6 and Fig. 7. The analytical outcomes reveal that the configuration and location of the identified shear zone are congruent with the findings documented in the literature [27]. Plastic deformation is localized within the elements of the shear zone, while the elements beyond this zone exhibit predominantly elastic deformation. The results corroborate the ability of the methodology applied in this investigation to meticulously monitor the initiation, progression, and ultimate penetration of cracks and shear zones in soil slopes subjected to top slope loading. This research elucidates the morphology and progression patterns of the shear zone with greater clarity and precision than the embedded discontinuity approach.

Fig. 6.

Vertical displacement diagram.

Fig. 7.

Horizontal displacement diagram.

Fig. 8.

Mesh deformation diagram when u is 4 cm.

Fig. 9.

Mesh deformation diagram when u is 5 cm.

Fig. 10.

Mesh deformation diagram when u is 10 cm.

4.2 Simulation of Soil Slope Failure Process Under Excavation at the Toe of the Slope

Excavation-induced landslides, as a common mode of soil slope failure, have garnered considerable focus. Li [28] conducted a thorough investigation into the processes and mechanisms underpinning the instability and collapse procession of such landslides through centrifugal model tests. This study employs the C1 experiment as a paradigm to simulate and analyze the inception and propagation of the slip failure of the tested slope.

The experimental model features a slope with a height of 25 cm, and an excavation depth at the toe of the slope measuring 8 cm. Following the attainment of 50g during centrifugation and the sample achieving relative stability, the excavation at the slope's toe is executed in one continuous process.

Figure 11 illustrates the failure surface post-landslide occurrence within the test specimen, while Fig. 12 presents the trajectory of the associated crack patterns. The specimen exhibits a principal slip plane near the slope's surface and two tension cracks at the crest of the slope. For simplicity purpose, this study primarily examines the emergence and progression of a principal crack.

Fig. 11.

A photo of the test sample after it was destroyed.

Fig. 12.

Crack path diagram.

In the course of the centrifuge experiment, Li [28] observed that the primary slip plane on the slope originated at the excavation front and progressively ascended. Plastic deformation is localized within the shear zone elements, whereas elements outside this zone display characteristics of near-elastic deformation. The slip surface, to an extent, serve as demarcations between the zones of elastic and plastic deformation. Table 1 details the specific physical and mechanical properties of the test soil specimen.

Table 1. Physical and mechanical parameters of C1 test soil1.

The specific simulation process is as follows:

• 1) Three steps of the analysis are established. Step 1 applies a self-weight load to the entire model and conducts geo-stress equilibrium, yielding a model that sustains the self-weight load without undergoing any displacement.

• 2) Centrifugal load of 50g is applied to the model gradually with 5g each increment.

• 3) The birth-and-death element technique is utilized to simulate the slope excavation process, with Step 2 initiating the birth-and-death elements prearranged in the excavation zone, effectively rendering the regional elements inactive. The self-weight load is preserved, and calculations for crack propagation are carried out.

Figure 15 illustrates the simulation results from this study. At Step Time 0.3178 in STEP-2, shear failure manifests in the element at the base of the excavation face (Fig. 15a). This shear failure continues, reaching a height of 3.2 cm at Step Time 0.9431 (Fig. 15b). Subsequently, the crack height swiftly escalates to 24.3 cm, transitioning into a tensile failure, which persists until it completely cleaves through the slope's summit, culminating in a breach (Fig. 15c). At this time, The vertical and horizontal displacement cloud maps of the slope are shown in Fig. 12 and Fig. 13. The simulated trajectory and ultimate form of the principal crack correlate with the experimental findings, affirming the effectiveness of the proposed methodology. Specifically, it confirms the precision with which the position of slope fissures can be pinpointed and the accuracy of the predicted direction of their expansion.

The numerical simulation indicates that the shear crack within the slope predominantly propagates within the strain localization zone observed in the centrifuge test, ascending gradually over time. However, these fissures are situated considerably lower than the corresponding areas shown in Fig. 12. Moreover, the crosspoint of shear and tension cracks near the slope's apex is markedly higher than illustrated in Fig. 12. This variance could stem from two potential factors: Initially, the physical model test may be subject to three-dimensional effects, in contrast to the numerical simulation, which is constrained to plane strain conditions, potentially leading to significant discrepancies between the two. Secondly, the excavation face's surface area might have neared a critical state, and the centrifugal test model could have fractured due to a minor, spontaneously occurring local crack on the excavation face (Fig. 14).

Fig. 13.

Vertical displacement diagram.

Fig. 14.

Horizontal displacement diagram.

Fig. 15.

Crack track diagram.

5 Conclusion

To replicate the slope failure, the tension-shear composite failure criterion is encoded into a UDMGINI subroutine and incorporated into the ABAQUS extended finite element module. Employing this methodology, the failure patterns of the slope under two distinct loading conditions are simulated and meticulously analyzed.

The simulation outcomes corroborate that the methodology delineated in this study can vividly portray the inception, development, and terminal failure state of the slope's shear zone. It precisely identifies the type of onset, the timing, and the trajectory of crack propagation, thus bridging the gap left by conventional slope failure simulations, which typically illustrate only the terminal failure state without the evolutionary process. The two cited instances demonstrate the viability of this approach in emulating complex soil slope failures. The simulation results are readily interpretable and instrumental in elucidating the causation and progression of soil slope collapse.