Home Numerical simulation of damage mechanism in rock with cracks impacted by self-excited pulsed jet based on SPH-FEM coupling method: The perspective of nonlinear engineering and materials science
Article Open Access

Numerical simulation of damage mechanism in rock with cracks impacted by self-excited pulsed jet based on SPH-FEM coupling method: The perspective of nonlinear engineering and materials science

  • Shixian Chen , Yizhou Zhang , Junren Jia , Xinlong Sun , Aibaibu Abulimiti and Hang Dong EMAIL logo
Published/Copyright: September 12, 2025
Download Article (PDF)
Become an author with De Gruyter Brill
Submit Manuscript Author Information
Nonlinear Engineering
From the journal Nonlinear Engineering Volume 14 Issue 1

Abstract

The pulsed water jet (PWJ) rock-breaking technology is vital in developing oil and gas resources. The presence of fractures in the rocks in the reservoir is one of the prominent features in the renovation project of the old well. However, the dynamic damage process of cracked rocks under the action of PWJ has not yet been revealed. Therefore, from the perspective of nonlinear engineering, in this study, a new PWJ coupled rock-breaking model was established by using the smooth particle fluid dynamics-finite element method (SPH-FEM) coupling algorithm to clarify the macroscopic damage mechanism of rocks with crack characteristics. From the perspective of materials science, this study verified the reliability of the model through comparative experiments and investigated the failure mechanism induced by the impact of pulsed jets on rocks containing natural fractures. The experimental and simulation results illustrate that the rock-breaking effect of the PWJ is significantly superior to that of continuous water jet, combined with the theoretical framework of nonlinear engineering and materials science. The presence of cracks hinders the propagation of the stress waves. At a crack inclination angle of 75°, the broken pit depth increased by approximately 48.5% compared to 30°. The significant improvement in rock-breaking efficiency is closely related to crack spacing, especially when the spacing is less than 2 cm.

Keywords: rock-breaking; SPH-FEM method; damage characteristic

1 Introduction

The petroleum industry widely employs high-pressure water jet technology in multiple facets of resource extraction processes, such as hydraulic cutting, rock breaking, hydraulic slit penetration. Pulsed water jet (PWJ) technology has gained widespread application owing to its highly efficient rock-breaking capabilities [1]. However, its complex rock-breaking mechanisms are not yet fully understood and have become a significant factor limiting further development and application of PWJ technology [2]. When using PWJ technology to enhance production in older wells, cracks in the reservoir rocks significantly affect the rock-breaking efficiency and mechanisms, making them markedly different from those observed in intact rocks. Therefore, the study of the rupture characteristics and damage mechanisms of rocks with varying crack characteristics under PWJ impact is of significant scientific and engineering value for advancing the application of PWJ technology in oil drilling and extraction. Moreover, understanding the failure mechanisms of rocks with different crack characteristics can enhance rock-breaking efficiency during subsurface drilling operations, which is critical for efficient underground energy extraction.

Many specialists and scholars have extensively studied the rock fracturing mechanisms and rock-breaking patterns of PWJ technology, achieving notable progress in this field. A systematic evaluation of PWJ oscillation characteristics and their impact on rock fragmentation efficiency was conducted by Zhang et al. [3]. The analysis revealed a positive correlation between proximity to resonance conditions (frequency ratio near 1) and optimized rock fragmentation performance. Through an experimental analysis, Li et al. investigated the effect of stress waves on the erosion damage in rocks subjected to pulsating fracturing erosion processes. Their research found that the power of the stress wave weakened more significantly as the distance from the impact point of the jet increased [4]. Xiao et al. studied the dynamic fracture patterns of PWJ eroded concretes. The results indicated that the PWJ could repeatedly utilize high water hammer pressure, reduce the water cushion effect, and deliver enhanced concrete fragmentation efficiency compared to the continuous water jet (CWJ) [5]. Ling et al. designed a novel pressurized PWJ, and field test results demonstrated that this tool significantly improves the drilling speed compared to the conventional CWJ [6]. Wei et al. investigated the PWJ flow and pulse frequency characteristics. Their study found that the “peak” and “trough” of the PWJ were not perfectly symmetrical [7]. Owing to the opacity of rocks, it is difficult to capture the damage characteristics inside the rock at the instant of jet erosion using only experimental methods. This limitation has hindered the exploration of the mechanisms by which water jets erode rocks. Consequently, scholars have turned to numerical models to simulate and analyze microscopic damage mechanisms within rocks.

The coupling of smoothed particle hydrodynamics (SPH) and the finite element method (FEM) in water jet rock breaking primarily addresses the mesh distortion issues inherent in traditional FEM when handling large fluid deformations. This approach combines the mesh-free advantages of SPH for fluid simulation with the high-precision characteristics of FEM in solid mechanics analysis, enabling more accurate simulation of the dynamic failure process during water jet impact on rock. Liu et al. constructed numerical models to examine the erosion characteristics of various jets with numerical magnitudes on concrete with different properties. Their studies found that as the jet diameter increased, the response of the erosion hole aperture size to the flow velocity increased. In the non-optimal domain of action of a water jet injection system, stress reduction within the concrete causes cracks to propagate along the aggregates [8,9,10]. Zhao et al. numerically modeled the erosional dynamics of sedimentary formations under abrasive jet impingement. This study found that compressive and shear stresses primarily govern rock failure [11]. Li et al. conducted a simulation study on the effect of the jet inclination angle on rock damage. They found that rock breakage was the most efficient when the inclination angle was between 15° and 20° [12].

In summary, existing research on the rock fragmentation mechanisms of PWJ primarily focuses on the jet characteristic parameters, the type of rock, and applied stress conditions. However, the fracture characteristics and damage evolution of rocks with different crack features under pulsed jet impact remain unclear. This study presents the first quantitative investigation into the influence of natural crack characteristics (dip angle/spacing/length) on PWJ damage mechanisms using an SPH-FEM coupling approach. It reveals the synergistic damage mechanism arising from crack-induced stress wave reflection coupled with dynamic PWJ loading. Furthermore, this work provides the first quantification of the quantitative relationship between crack dip angle and the threshold of crack spacing governing the damage mechanism. The study uncovered the failure mechanisms of cracked rocks under PWJ erosion, offering theoretical insights for the application of PWJs in rock breaking.

2 Simulation methods and models

2.1 SPH-FEM method

SPH particle technology can accurately capture the pronounced deformation effects generated by water jets during impact, which are marked by substantial reflection and distortion. This method also circumvents the computational disruptions arising from mesh deformation [13]. Although the rock experienced small fracture pits in the impact area after water jet impingement, its overall deformation remained relatively minor. Furthermore, existing studies have shown that the SPH-FEM approach has the advantage of being more reliable than the Lagrangian/Eulerian approach when dealing with problems in which fluids and solids interact with each other [14,15]. This combination offers unique advantages in solving impact problems and is widely applied to fluid-structure coupling, high strain rates, and large deformation scenarios [16,17]. The coupling principle is illustrated in Figure 1, where a node-based algorithm defines the interaction between the PWJ and rock, facilitating the transfer of force and velocity. A penalty function approach enables bidirectional data exchange between SPH discrete entities and FEM continuum domains, ensuring interfacial compatibility. As shown in Figure 2, a virtual spring is introduced between SPH discrete entities and FEM continuum domains to prevent overpenetration when the two come into contact. The stiffness of the spring K, is proportional to the penetration depth L, of the SPH particles, and the force exerted at this point is referred to as the penalty force. During the computational solution, the kernel function w ( x − x ′ , h ) is first approximated using an integral expression given by

(1) f ( x ) = ∫ Ω f ( x ′ ) W ( x − x ′ , h ) d x ′ ,

where f ( x ) is an expression of x in the spatial coordinate system and the distance between the particles is denoted by ( x − x ′ ) . The length of the smooth particles is h.

Figure 1 Principle of coupling.
Figure 1

Principle of coupling.

Figure 2 Principles of the contact process.
Figure 2

Principles of the contact process.

Then, the above expression is transformed as follows:

(2) ∇ ⋅ f ( x i ) = ∫ Ω f ( x ′ ) ⋅ ∇ i W ( x − x ′ , h ) d x ′ .

Discretizing the above expression using the particle approximation method yields

(3) ∇ f ( x i ) = ∑ j = 1 N m j ρ j f ( x i ) ⋅ ∇ i W i j ,

where the mass and density of SPH particle j are denoted by m j and ρ j , respectively, and N stands for the number of particles.

2.2 Geometric model conditions assumptions

This study established an infinite water jet rock-breaking numerical model using the LS-DYNA software. Figure 3(a) shows the geometrical modeling of the established model. The geometric model of the rock has dimensions of a cube with a width of 8 cm and a height of 15 cm. The jet diameter was 6 mm, and the velocity of the CWJ was 206 m/s. The velocity of the PWJ is defined using a velocity function curve, with the idealized velocity curve of the pulsed jet approximating a sine wave (Figure 3(b)) ( v = v 0 + A sin 2 π ft ) . The pulse frequency is set to 1,000 Hz, and the amplitude is set to 50 m/s. Both jets impact for the same duration. The remaining five surfaces of the rock model except for the impact surface were uniformly set as non-reflective wall boundaries to prevent the boundary conditions from influencing rock fragmentation. The rocks were meshed using uniformly distributed FEM square elements with an element size of 1 mm.

Figure 3 Numerical modeling and velocity discretization methods. (a) Geometric model and (b) velocity discrete curve.
Figure 3

Numerical modeling and velocity discretization methods. (a) Geometric model and (b) velocity discrete curve.

2.3 Material constitutive

The Riedel Hiermaier Thoma (RHT) constitutive model, proposed by Johnson and Holmquist [18], is based on the Holmquist Johnson Cook model and combines the stress–strain relationship of the material with the damage criterion to consider the mechanical response of the rock material under different collision, blasting, and dynamic loading conditions [19]. The model introduces three damaged surfaces that describe the initial changes in the rock [20,21] as shown in Figure 4.

Figure 4 Three limit surfaces in the RHT model.
Figure 4

Three limit surfaces in the RHT model.

Figure 5 illustrates that the rock experiences elastic deformation as long as its pore compression pressure surpasses the external pressure P. Beyond this threshold, the rock initiates fracturing, diminishing its overall stiffness and reducing the effective bulk modulus. With further pressure increments, the volumetric strain exhibits a nonlinear response.

Figure 5 P–α equation of state.
Figure 5

P–α equation of state.

The relevant rock parameters were determined experimentally based on the strength of the rocks, and the remaining parameters were calculated by referring to Borrvall and Riedel and Wang et al. [20,22]. Table 1 shows the determined rock parameters.

Table 1

Material parameters of granite

Parameters Value Parameters Value Parameters Value
G 2.76 GPa p el 1.67 × 10−4 β t 0.0443
ρ 0 2.39 g/cm3 p co 0.06 ε ̇ 0 c 3.0 × 10−11
f c 25.15 MPa g c ⁎ 0.53 ε ̇ 0 t 3.0 × 10−12
f t ⁎ 0.1352 g t ⁎ 0.7 ε ̇ c 3.0 × 1019
f s ⁎ 0.18 XI 0.5 ε ̇ t 3.0 × 1019
α 0 1.1 A 2.44 D 1 0.04
n p 3 N 0.76 D 2 1
A 1 25.7 GPa Q 0 0.68 ε p m 0.01
A 2 31.35 GPa B 0.05 A f 1.62
A 3 6.59 GPa β c 0.0419 n f 0.6

In LS-DYNA, the jet’s equation of state is characterized by the Gruneisen equation, with the constitutive equation for water formulated as follows:

(4) P = ρ ω C 2 μ 1 + 1 − γ 0 2 μ − a 2 μ 2 1 − ( S 1 − 1 ) μ − S 2 μ 2 μ + 1 − S 3 μ 3 ( μ + 1 ) 2 2 + ( γ 0 + α μ ) E ,

where P denotes the pressure. ρ ω signifies the water density. C represents the intersection of the V s − V p curve in the shock wave velocity–particle velocity relationship. S 1 , S 2 and S 3 are the slopes of the V s − V p curve. γ 0 is Gruneisen’s constant. a is the adjustment factor relating Gruneisen’s coefficient to volume. μ is the volumetric parameter.

The parameters of water are referenced from the study by Li et al. [2], as shown in Table 2.

Table 2

Parameters related to water

ρ ( g cm − 3 ) C ( cm μs − 1 ) S 1 S 2 S 3 γ 0 a
1 0.148 2.56 −1.986 0.2886 0.49 1.397

2.4 Model validation

Mechanisms of rock erosion by water jet is challenging to quantify experimentally owing to its instantaneous and dynamic processes [23,24,25]. However, experiments using qualitative comparison methods can be used to verify the accuracy of numerical simulations. This study carried out indoor physical simulation experiments of the CWJ and PWJ, and the experimental setup and the schematic are shown in Figure 6. The PWJ device in the figure was independently developed and designed by an oilfield company in Xinjiang [26]. The experiments were conducted by our research group and numerical simulations were performed using LS-dyna software. CWJ and PWJ had the same working conditions and lasted for the same impact duration. The 40 cm cubic rock specimens were eroded by continuous water jetting through a 6 mm diameter nozzle, with the pressure not exceeding 60 MPa and an inlet flow rate maintained at 350 L/min for 2 min.

Figure 6 Experimental equipment.
Figure 6

Experimental equipment.

Figure 7 compares the experimental results of CWJ rock-breaking with numerical simulation results. Figure 7(a) shows a relatively regular circular hole formed by the CWJ on an eroded granite surface. After the erosion experiment, the granite was symmetrically cut along the erosion hole, the cut surface is shown in Figure 7(b). A circular hole was formed in the area of the impact surface of the granite, with no noticeable cracks observed, and the erosion hole exhibited a narrow and deep characteristic. The measured diameter and depth of the erosion hole were 39.22 and 77.44 mm, respectively. The simulation results indicated that the hole diameter and depth were 37.85 and 76.0 mm, respectively, demonstrating good agreement with the experimental measurements, with a discrepancy of less than 5%. The fracture patterns from both the CWJ impact test and simulation are primarily consistent, which verifies that the parameters and models in this study are usable and accurate.

Figure 7 Experimental results of CWJ with numerical simulation (locally enlarged). (a) Rock surface and (b) cutting surface.
Figure 7

Experimental results of CWJ with numerical simulation (locally enlarged). (a) Rock surface and (b) cutting surface.

The PWJ formed a relatively regular circular hole on the granite erosion surface, with a diameter of 48.32 mm. In comparison, the diameter of the hole formed by the CWJ impact was only 39.22 mm. In addition, significant cracks appeared on the surface of the rock. These cracks considerably weaken the integrity of the rock, thereby enhancing the effectiveness of reservoir stimulation. Comparing and analyzing the results of the experiments in Figure 8, the research reveals that the rock surface under the PWJ exhibits an approximately “cross-shaped” crack pattern, similar to the experimental findings. The numerically simulated aperture size of 45.94 mm shows a deviation of no more than 5% from the experimental result. The damage value in Figure 8 is a dimensionless quantity that represents the degree of damage accumulation, with values typically ranging from 0 to 1, indicating the material’s state from being fully intact and undamaged to completely failed.

Figure 8 Experimental results of PWJ with numerical simulation (locally enlarged).
Figure 8

Experimental results of PWJ with numerical simulation (locally enlarged).

To summarize the comparison results, the effect of PWJ rock-breaking compared with the CWJ is significantly larger. The PWJ impact broken pit diameter and depth are considerably more significant than the CWJ’s, and PWJ, under the action of the rock’s surface, produces approximate “cross-shaped” cracks. The rock fragmentation state and the size of erosion holes under the action of PWJ significantly exceed the impact effect of CWJ. Meanwhile, the reliability of the numerical model is further confirmed.

2.5 Design of the numerical modeling tests

The numerical model used in this section is consistent with Section 2.4, using the same material parameters and removing fracture elements using Boolean difference operations. The model is applicable under specific working conditions and rock quality conditions: The jet diameter parameter is 6 mm, covering the typical working conditions of the oilfield. The experimental rock samples were taken from the granite buried 300 m underground in the Karamay area of Xinjiang. The model can reveal the synergistic damage process and damage evolution law of rocks with different fracture characteristics in PWJ dynamic impact, provide new theoretical support for optimizing key parameters of water jet rock breaking, and effectively promote efficient rock breaking in geologically complex reservoirs. This study independently examined the effects of the crack length, spacing, and inclination angle on rock damage. Figure 9 shows a close-up view of the rock profile showing the distribution of cracks in the rock. The model shown in Figure 9(a) is an intact rock in an ideal state; Figure 9(b) shows a crack 1 cm from the rock surface perpendicular to the direction of the jet; Figure 9(c) illustrates a rock with two horizontal cracks; and Figure 9(d) shows a rock featuring a crack at an inclined angle. The angle between the projection of the fissure on the rock erosion surface and the fissure contour line is defined as the angle of inclination, φ .

Figure 9 Positioning of fractures within rocks (locally enlarged). (a) Intact rock, (b) rock containing horizontal cracks, (c) rock containing two horizontal cracks and (d) rock containing inclined cracks.
Figure 9

Positioning of fractures within rocks (locally enlarged). (a) Intact rock, (b) rock containing horizontal cracks, (c) rock containing two horizontal cracks and (d) rock containing inclined cracks.

3 Results and analyses

3.1 Analysis of the effect of the presence of a single crack on the damage mechanism of the rock

Figure 10(a) shows the damage process of the rock at different moments for a crack length of 40 mm. The PWJ was ejected from the outlet and contacted the rock surface 100 μs later. Comparing Figure 10(a) and (b), the damaging cloud at 10 µs after the contact of the PWJ with the rock, it can be seen that the breakage district of the rock without cracks shows a “V” shape, while the lower part of the breakage district of the rock with cracks shows a “Λ” shape. This indicates that cracks interfere with the expansion of the damaged area of the rock. Before the PWJ penetrates the crack, the diffusion process of the stress wave is impeded and repeatedly superimposed above the crack, resulting in the concentration of the damage region above the crack. By the 100th μs, microcracks are formed and extended to the ends of the horizontal cracks. Shear fracture surfaces gradually formed around rock elements in the central erosion area. A collapsed area was found above the crack, and by 180 μs, the rock elements in the collapsed area were entirely removed. Stress wave propagation resumes only once the PWJ traverses the crack.

Figure 10 Damage cloud diagram of rock (locally enlarged). (a) Cracked rock and (b) rock without cracks.
Figure 10

Damage cloud diagram of rock (locally enlarged). (a) Cracked rock and (b) rock without cracks.

3.1.1 Analysis of the effect of horizontal cracking on rock damage

The diffusion characteristics of the stress waves around the cracks can be determined by selecting the rock elements at the corresponding locations in the uncracked and cracked rocks and plotting the damage values of the corresponding rock elements against the stress curves. Based on the shape and extent of the damage to the broken hole shown in Figure 10, several representative rock elements were selected for further analysis. Figure 11 shows a schematic of the location of the corresponding rock element. A2, B2, and C2 are horizontally distributed rock elements above the crack, and D2 is located directly below the crack. The corresponding rock elements without cracks are A1, B1, C1, and D1.

Figure 11 Schematic distribution of rock elements (locally enlarged). (a) Rock without cracks and (b) rock containing cracks.
Figure 11

Schematic distribution of rock elements (locally enlarged). (a) Rock without cracks and (b) rock containing cracks.

After the rock is impacted, a strong impact force is generated at the moment of contact with both different rock types owing to the large water hammer pressure generated by the PWJ. Figure 12 shows the diffusion of stress waves in all directions within the rock in a hemispherical shape centered on the jet erosion zone. The standard stress wave had a regular “hemispherical” shape. In contrast, the stress wave in the cracked rock is affected by reflections from the horizontal cracks underneath and therefore, has an irregular shape at the bottom.

Figure 12 Stress waves in rocks: (a) Rocks without cracks and (b) rocks with cracks.
Figure 12

Stress waves in rocks: (a) Rocks without cracks and (b) rocks with cracks.

This can be summarized from the damage history variables and the stress change curves for the uncracked rock elements A1, B1, and C1 in the horizontal direction in Figure 13.

Figure 13 Damage history variables and stress curves for rock elements.
Figure 13

Damage history variables and stress curves for rock elements.

Since A1 is located in the region directly underneath the jet, the element A1 is subjected to higher compressive and shear stresses after jet erosion, which gradually leads to element failure and deletion, with the maximum compressive and shear stresses reaching 53.8 and 18.65 MPa, respectively. Elements B1 and A1 are separated by 10 mm, and the cumulative damage time of element B1 is a little longer than that of element A1. The trend of the stress curve shows that element B1 is mainly subjected to compressive shear damage, and there is also a little bit of tensile stress, but tensile damage is not the main damage factor. The C1 element is far away from the center of the jet. From the stress diagram, it can be concluded that the initial contact of the PWJ with the rock mainly dominates the tensile and compressive stresses. Then, it is also affected by the shear stresses, but the element has not been damaged and failed in this time range.

The damage history variable curves for elements A2, B2, and C2, which contain cracked rock, show that the two rocks do not have the same damage time, and the cracked rock can be damaged more quickly. Branching damage zones (Figure 10(a) 100 μs) were developed on both sides of the center of the jet impact area, forming a “collapse zone.” During this process, the damage value of element A2 reaches one instantaneously. Element A2 is dominated by transient damage, with significant effects of tensile and shear stresses. Element B2 is relatively close to the shear fracture surface, where the effect of shear stresses is more pronounced. The instantaneous accumulation of damage values up to 1, followed by rapid destruction, indicates that the “collapse zone” is fractured. Since a single crack is subjected to a stress wave, a stress concentration is generated at both ends of the crack, resulting in the C2 element being instantaneously subjected to extremely high compressive and shear stresses, with compressive stresses of up to 28.5 MPa. After the horizontal crack is penetrated, the PWJ reaches the location of the C2 rock element instantaneously along the inside of the crack. The C2 rock element can be directly affected by the PWJ, and the damage value increases first gradually to 0.39 and then increases to 1 instantaneously. Compared with the damage value of 0.17 for the rock element C1 without cracks, the element C2 is much larger than that of C1, because horizontal cracks accelerate the spalling of parts of rock. The distance of the unfractured rock element C1 from the scope of erosion of the jet center is farther away, resulting in a gradual weakening of the energy of the stress wave diffusion.

Comparison of rock elements D1 and D2 at exact locations. The initial perturbation time of rock element D2 by the stress wave is slightly slower than that of rock element D1 but the damage failure time of rock element D2 is a little faster than that of element D1. This was because the presence of a horizontal crack prevented the propagation of the stress wave. After the horizontal crack penetrated, the rock element is destroyed owing to the “water wedge effect” of the PWJ and the vast water hammer pressure that instantly impacts element D2 resulting in an instantaneous increase in its stress and damage values. In conclusion, even though both types of rock elements are located in the same position, horizontal cracks lead to very different damage mechanisms.

3.1.2 Characterization of rock damage by cracks of different lengths

The different crack lengths were set as 2, 3, 4, and 6 cm. Figure 14 shows the profile damage cloud of the rock at 110 μs, and Figure 15 shows the surface damage cloud of the rock. It is evident that shorter crack lengths result in smaller rock breakage holes; conversely, longer crack lengths not only enlarged the diameter of the breakage holes, but also made the collapse zone increasingly visible.

Figure 14 Damage cloud for different crack sizes at the same moment in time (locally enlarged). Crack size: (a) 2 cm, (b) 3 cm, (c) 4 cm, and (d) 6 cm.
Figure 14

Damage cloud for different crack sizes at the same moment in time (locally enlarged). Crack size: (a) 2 cm, (b) 3 cm, (c) 4 cm, and (d) 6 cm.

Figure 15 Damage clouds diagram of rock surfaces. Crack size: (a) 2 cm, (b) 3 cm, (c) 4 cm, and (d) 6 cm.
Figure 15

Damage clouds diagram of rock surfaces. Crack size: (a) 2 cm, (b) 3 cm, (c) 4 cm, and (d) 6 cm.

The reason for this phenomenon can be explained by applying the principle of bending deformation to the middle part of a solid support beam under the action of a concentrated force in engineering mechanics, which simplifies the rock in the upper part of the crack to a solid support beam [27,28]. Based on the simplified mechanical model shown in Figure 16, an expression for the plastic bending displacement can be derived as follows:

(5) w = m 2 v 2 24 M M 0 1 + b ( 1 + a ) 2 − 1 + 2 b 1 + b + 2 b 1 + a + 2 ln 1 + a 1 + b ,

(6) M 0 = σ 0 D h 2 / 4 ,

(7) a = M ξ / m ,

(8) b = M x / m ,

(9) v = v 0 + A sin 2 π ft,

where A is the amplitude, and f is the frequency. m and M are the masses of PWJ and fixed beam unit mass, respectively. Where D and h are the width and height of the fixed beam, respectively. ξ is the moving hinge that varies over time, σ 0 is the yield strength, x is the center point of the theoretical model, and L is the length of the fixed beam.

Figure 16 Schematic diagram of force bending of a solid fixed beam. (a) Support beam impacted by load and (b) bending deformation of the support beam.
Figure 16

Schematic diagram of force bending of a solid fixed beam. (a) Support beam impacted by load and (b) bending deformation of the support beam.

3.2 Analysis of the effect of the inclination angle of cracks on rock damage

In the modeling process, the inclination of the cracks was set to four different angles such that the center of symmetry of the crack intersected the axial extension of the jet without changing the length between the erosion surface of the rock and the nearest endpoint of the crack. Figure 17 shows the damage characteristics of the PWJ impact at 1.3 ms.

Figure 17 Damage cloud for crack inclination change at the same instant of time (locally enlarged).
Figure 17

Damage cloud for crack inclination change at the same instant of time (locally enlarged).

The presence of inclined cracks significantly affected the shape of the broken pit, as shown in Figure 17. Cracks containing smaller inclination angles have a more significant effect on the PWJ because the overall bending angle of the crushing and broken pits is curved significantly toward the diagonal of the crack. In the radial direction of the jet, the middle part of the broken hole profile gradually narrowed as the inclination angle of the cracks increased, indicating that the cracks had less influence on the damaged area in the radial direction of the jet. In the axial direction of the jet, the damage to the rock gradually increased as the angle of inclination of the crack increased, but the increase in depth was not significant beyond 60°.

Figure 18 plots the rock broken pit volume and depth vs the crack’s inclination angle. The depth of the broken hole increases with the change in inclination. However, the broken hole volume decreases gradually and nonlinearly with increasing tilt angle. This is due to the fact that when the tilt angle of the crack is slight, the extent of the reflected stress wave is larger, thus affecting the rock elements above the crack, and the rock elements are deleted in large numbers. On the contrary, when the crack’s inclination angle is larger, fewer rock elements are removed. A nonlinear fit regression ( R 2 = 0.9744 ) to the data for the broken hole volume V and inclination φ in Figure 18 yields the following expression:

(10) V = 0.1805 φ − 0.0035 φ 2 + 15.274 .

Figure 18 Damage volume and depth of rock as a function of the angle of inclination of the crack.
Figure 18

Damage volume and depth of rock as a function of the angle of inclination of the crack.

It should be noted that the purpose of the fitting formulas used here and later in this study is to clearly show the changing trends of the numerical simulation results. They can be regarded as empirical relationships within the range of numerical simulation parameters and do not represent other physical meanings.

3.3 Effect of the distance between two cracks on rock damage

Figure 19 shows the numerical simulation results of the rock model containing different crack spacings at the moment of 1.1 ms. The rock element between the cracks corresponded to a “solid square plate.” When the PWJ penetrated the crack above, the stress wave in the impact damage region spread rapidly through the “solid square plate.” This causes the square plate, fixed at both ends, to undergo plastic bending deformation until element damage is removed. The upper part of the “solid square plate” is mainly subjected to compressive stresses resulting from PWJ impacts. In contrast, the underside of the “solid square plate” was mainly damaged by tensile stresses, as the stress waves were affected by reflections. As seen from the damage cloud in Figure 19 with a crack spacing of 3 cm, microcracks were created underneath the “solid square plate” connecting the tensile and compressive zones in the middle of the two cracks at an angle of approximately 45°. When the “solid square plate” is thinner, it is easier for the jet to penetrate, thus causing the plane to act in a deeper direction. As the distance between the two cracks increased, the thickness of the “solid square plate” increased, increasing the impact energy required to bend the square plate, thus the extent of damage underneath the “solid square plate” was reduced. The deceleration of the jet’s erosion velocity along the axial direction extended the interaction time in the radial direction, leading to an enlargement of the eroded pit diameter.

Figure 19 Damage clouds diagram of rocks at crack spacing (locally enlarged).
Figure 19

Damage clouds diagram of rocks at crack spacing (locally enlarged).

Figure 20 shows the volume of the damaged crater, depth and caliber of the damaged crater in relation to the distance between different cracks. As the distance between cracks increases, the size and depth of the resulting damaged crater initially expand and subsequently contract. The results show an optimal spacing of cracks for better crushing of rock. This is because the stress wave between the two cracks interferes with each other, making the “solid square plate” more susceptible to breakage. When the spacing is greater than 2 cm, the interaction between the cracks is significantly weakened, thus reducing the broken pit volume. A nonlinear fit regression ( R 2 = 0.9951 ) to the data for the volume V of the damaged crater and the crack distance x in Figure 20 yields the following expression:

(11) V = 13.7244 + 14.6624 x − 4.4288 x 2 .

Figure 20 Volume and caliber size of damage pits in relation to crack spacing.
Figure 20

Volume and caliber size of damage pits in relation to crack spacing.

4 Conclusion

This study considered the damage effect of rocks with cracks in the old well renovation project under the impact of pulsed jets. Based on the SPH-FEM method, a new numerical model for the coupling of pulsed jets and fractured rocks was established. The laws that influence different crack characteristics on the rock breakage evolution under PWJ erosion were analyzed. The results indicate that

  1. Compared with the round hole-shaped erosion crater formed by the continuous jet, the erosion crater formed by the pulsed jet is a composite structure of “center crater + cross crack.”

  2. The stress wave reflection and concentration above the crack create a zone of reduced stress, subjecting the rock to the combined effect of compressive and tensile shear forces, whereas results in damage. The inclination angle of the cracks significantly affected the impact effect of the PWJ, resulting in a curved internal shape of the broken pit with significant differences. As the crack inclination increased, the volume of the broken pit decreased “non-linearly,” and the broken pit depth increased by approximately 48.5%. When the crack inclination angle was slight, the effect of the reflected stress wave is significant; when the inclination angle was large, the impact of the reflected stress wave was insignificant.

  3. The volume and depth of the broken pit tended to increase and decrease, respectively with the crack spacing. The volume and depth of the broken pit were maximized when the crack spacing was 2 cm. This indicates that optimal crack spacing exists for better rock breaking.

This study provides new insights and data support for understanding the failure mechanisms of cracked rocks under pulse jet impact, accelerating rock fragmentation, optimizing construction schemes, and enhancing oil and gas permeability.

  1. Funding information: This work was sponsored by the Key Research and Development Projects of Xinjiang Uygur Autonomous Region (2023B01016), Natural Science Foundation of Xinjiang Uygur Autonomous Region (Grant No. 2022D01C672), National Natural Science Foundation of China (Grant Nos 52105486, 52465054, 52265061), and Special Funds for Local Science and Technology Development under the Guidance of the Central Government (Grant No. ZYYD2023A07).

  2. Author contributions: Shixian Chen: conceptualization, data curation, formal analysis, investigation, methodology, resources, validation, visualization, writing – original draft, and writing – review and editing. Yizhou Zhang: Investigation, supervision, and writing – review and editing. Junren Jia: formal analysis, investigation, methodology, and resources. Xinlong Sun: validation, visualization, and writing – review and editing. Aibaibu Abulimiti: investigation. Hang Dong: investigation. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: Authors state no conflict of interest.

  4. Data availability statement: The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

References

[1] Szada-Borzyszkowska M, Kacalak W, Banaszek K, Pude F, Perec A, Wegener K, et al. Assessment of the effectiveness of high-pressure water jet machining generated using self-excited pulsating heads. Int J Adv Manuf Technol. 2024;133(9):5029–51.10.1007/s00170-024-14040-6Search in Google Scholar

[2] Li H, Liu S, Jia J, Wang F, Guo C. Numerical simulation of rock-breaking under the impact load of self-excited oscillating pulsed waterjet. Tunn Undergr Space Technol. 2020;96:103179.10.1016/j.tust.2019.103179Search in Google Scholar

[3] Zhang J, Zhang B, Liu B, Li B. Investigation on the influence of the frequency of pulsed water jet on the rock-breaking effect. Powder Technol. 2024;431:119054.10.1016/j.powtec.2023.119054Search in Google Scholar

[4] Li Y, Zhao Y, Tang J, Zhang L, Zhou Y, Zhu X, et al. Rock damage evolution model of pulsating fracturing based on energy evolution theory. Energy Sci Eng. 2020;8(4):1050–67.10.1002/ese3.567Search in Google Scholar

[5] Xiao S, Qin H, Zhang W, Ren Q, Xiao J, Li W, et al. On the concrete breakage by pulsed water jet impact: Fracture characteristic, stress and damage evolution laws. Case Stud Constr Mater. 2023;19:e02634.10.1016/j.cscm.2023.e02634Search in Google Scholar

[6] Ling Y, Wang X, Tang J, Zhang Y. Experimental investigation on rock fragmentation charactersitics of pressurized pulsed water jet. Sci Rep. 2025;15(1):232.10.1038/s41598-024-84194-6Search in Google Scholar PubMed PubMed Central

[7] Wei J, Du Y, Liu Y, Wang M, Zhao L. The influence of pulse frequency on the energy evolution law and rock-breaking effect of pulsed abrasive water jet. Phys Fluids. 2024;36(4):12.10.1063/5.0183531Search in Google Scholar

[8] Liu J, Sun H, Zhu Y. Fracturing mechanism and crack expansion rule of concrete impacted by high pressure water jet. Mater Struct. 2021;54(5):178.10.1617/s11527-021-01774-ySearch in Google Scholar

[9] Yu R, Dong X, Li Z, Du M, Zhang Q. SPH-FEM simulation of concrete breaking process due to impact of high-speed water jet. AIP Adv. 2021;11(4):045226.10.1063/5.0049213Search in Google Scholar

[10] Xiao S, Xiao J, Ren Q, Cheng Y, Li W, Zhang W, et al. Damage evolution and fracture characteristics of heterogeneous concrete with coarse aggregate impacted by high-velocity water jet. Constr Build Mater. 2024;416:135128.10.1016/j.conbuildmat.2024.135128Search in Google Scholar

[11] Zhao H, Jiang H, Li H, Zhang X, Zhao M. Numerical research on rock cutting by abrasive jet under confining pressure based on SPH-FEM method. Powder Technol. 2024;433:119196.10.1016/j.powtec.2023.119196Search in Google Scholar

[12] Li L, Wang F, Li T, Dai X, Xing X, Yang X. The effects of inclined particle water jet on rock failure mechanism: experimental and numerical study. J Pet Sci Eng. 2020;185:106639.10.1016/j.petrol.2019.106639Search in Google Scholar

[13] Ju A, Zhang R, Cai Y, Ling J, Yang J, Su C. Study on characteristics and prediction model of jet impact concrete crushing based on SPH modeling. Structures. 2022;44:1523–31.10.1016/j.istruc.2022.08.065Search in Google Scholar

[14] Sayehvand HO, Haftlang PB. A novel numerical investigation of particle waterjet parameters on the cutting characteristics of rock samples. J Manuf Process. 2025;135:161–78.10.1016/j.jmapro.2025.01.027Search in Google Scholar

[15] Zhao H, Jiang H, Warisawa S, Li H. Numerical study of abrasive water jet rotational slits in hard rock using a coupled SPH-FEM method. Powder Technol. 2023;426:118622.10.1016/j.powtec.2023.118622Search in Google Scholar

[16] Yang T, Zhao W, Zhu X. Morphological study of cortical bone damage caused by high-pressure water jet erosion based on SPH-FEM coupling algorithm. J Manuf Process. 2024;118:195–205.10.1016/j.jmapro.2024.03.044Search in Google Scholar

[17] Yao C, Xu C, Zhou X, Liu Q, Qiang B. Study on the destruction process of piers by debris flow impact using SPH-FEM adaptive coupling method. KSCE J Civ Eng. 2024;28(8):3162–75.10.1007/s12205-024-1233-ySearch in Google Scholar

[18] Johnson GR, Holmquist TJ. An improved computational constitutive model for brittle materials. In AIP Conference Proceedings. Vol. 309, Woodbury (NY): American Institute of Physics; 1994. p. 981–4.10.1063/1.46199Search in Google Scholar

[19] Wang Z, Ni Y, Wang J, Li S. Improvement and performance analysis of constitutive model for rock blasting damage simulation. Simul Model Pract Theory. 2025;138:103043.10.1016/j.simpat.2024.103043Search in Google Scholar

[20] Borrvall T, Riedel W. The RHT concrete model in LS-DYNA. In Proceedings of The 8th European LS-DYNA User Conference; 2011 May 23-24. Strasbourg, France: 2011. p. 23–4.Search in Google Scholar

[21] Fang S, Wang J, Zhang Y, Li H, Liu J, Xue Q. Numerical simulation and experimental investigation of the propagation law of plasma pulse shock waves. Phys Scr. 2024;100(1):015608.10.1088/1402-4896/ad9ae4Search in Google Scholar

[22] Wang H, Wang Z, Wang J, Wang S, Wang H, Yin Y, et al. Effect of confining pressure on damage accumulation of rock under repeated blast loading. Int J Impact Eng. 2021;156:103961.10.1016/j.ijimpeng.2021.103961Search in Google Scholar

[23] Yu R, Dong X, Li Z, Fan M. A coupled SPH–DEM model for erosion process of solid surface by abrasive water-jet impact. Comput Part Mech. 2023;10(5):1093–112.10.1007/s40571-023-00555-4Search in Google Scholar

[24] Tripathi R, Hloch S, Chattopadhyaya S, Klichová D, Ščučka J, Das AK. Application of the pulsating and continous water jet for granite erosion. Int J Rock Mech Min Sci. 2020;126:104209.10.1016/j.ijrmms.2020.104209Search in Google Scholar

[25] Zhang M, Li D, Kang Y, Huang M, Yuan M. Experimental study on the rock erosion performance of a pulsed abrasive supercritical CO2 jet. J Pet Sci Eng. 2021;201:108489.10.1016/j.petrol.2021.108489Search in Google Scholar

[26] Abulimiti A, Zheng C, Liu Y, Pang H, Pang D, Anwaier M, et al. Study on the impacting performance of a self-excited oscillation pulsed jet nozzle. J Pet Sci Eng. 2021;207:109120.10.1016/j.petrol.2021.109120Search in Google Scholar

[27] Wang Z, Lei X, Zhou W, Wang Y, Cao J, Li L, et al. Numerical simulation of the damage process of rock containing cracks by impacts of steel-particle water jet. Powder Technol. 2023;422:118465.10.1016/j.powtec.2023.118465Search in Google Scholar

[28] Jones N. Several phenomena in structural impact and structural crashworthiness. Eur J Mech A Solids. 2003;22(5):693–707.10.1016/S0997-7538(03)00077-9Search in Google Scholar
Received: 2025-05-14
Revised: 2025-06-16
Accepted: 2025-07-02
Published Online: 2025-09-12

© 2025 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Research Articles
  2. Generalized (ψ,φ)-contraction to investigate Volterra integral inclusions and fractal fractional PDEs in super-metric space with numerical experiments
  3. Solitons in ultrasound imaging: Exploring applications and enhancements via the Westervelt equation
  4. Stochastic improved Simpson for solving nonlinear fractional-order systems using product integration rules
  5. Exploring dynamical features like bifurcation assessment, sensitivity visualization, and solitary wave solutions of the integrable Akbota equation
  6. Research on surface defect detection method and optimization of paper-plastic composite bag based on improved combined segmentation algorithm
  7. Impact the sulphur content in Iraqi crude oil on the mechanical properties and corrosion behaviour of carbon steel in various types of API 5L pipelines and ASTM 106 grade B
  8. Unravelling quiescent optical solitons: An exploration of the complex Ginzburg–Landau equation with nonlinear chromatic dispersion and self-phase modulation
  9. Perturbation-iteration approach for fractional-order logistic differential equations
  10. Variational formulations for the Euler and Navier–Stokes systems in fluid mechanics and related models
  11. Rotor response to unbalanced load and system performance considering variable bearing profile
  12. DeepFowl: Disease prediction from chicken excreta images using deep learning
  13. Channel flow of Ellis fluid due to cilia motion
  14. A case study of fractional-order varicella virus model to nonlinear dynamics strategy for control and prevalence
  15. Multi-point estimation weldment recognition and estimation of pose with data-driven robotics design
  16. Analysis of Hall current and nonuniform heating effects on magneto-convection between vertically aligned plates under the influence of electric and magnetic fields
  17. A comparative study on residual power series method and differential transform method through the time-fractional telegraph equation
  18. Insights from the nonlinear Schrödinger–Hirota equation with chromatic dispersion: Dynamics in fiber–optic communication
  19. Mathematical analysis of Jeffrey ferrofluid on stretching surface with the Darcy–Forchheimer model
  20. Exploring the interaction between lump, stripe and double-stripe, and periodic wave solutions of the Konopelchenko–Dubrovsky–Kaup–Kupershmidt system
  21. Computational investigation of tuberculosis and HIV/AIDS co-infection in fuzzy environment
  22. Signature verification by geometry and image processing
  23. Theoretical and numerical approach for quantifying sensitivity to system parameters of nonlinear systems
  24. Chaotic behaviors, stability, and solitary wave propagations of M-fractional LWE equation in magneto-electro-elastic circular rod
  25. Dynamic analysis and optimization of syphilis spread: Simulations, integrating treatment and public health interventions
  26. Visco-thermoelastic rectangular plate under uniform loading: A study of deflection
  27. Threshold dynamics and optimal control of an epidemiological smoking model
  28. Numerical computational model for an unsteady hybrid nanofluid flow in a porous medium past an MHD rotating sheet
  29. Regression prediction model of fabric brightness based on light and shadow reconstruction of layered images
  30. Dynamics and prevention of gemini virus infection in red chili crops studied with generalized fractional operator: Analysis and modeling
  31. Qualitative analysis on existence and stability of nonlinear fractional dynamic equations on time scales
  32. Fractional-order super-twisting sliding mode active disturbance rejection control for electro-hydraulic position servo systems
  33. Analytical exploration and parametric insights into optical solitons in magneto-optic waveguides: Advances in nonlinear dynamics for applied sciences
  34. Bifurcation dynamics and optical soliton structures in the nonlinear Schrödinger–Bopp–Podolsky system
  35. Review Article
  36. Haar wavelet collocation method for existence and numerical solutions of fourth-order integro-differential equations with bounded coefficients
  37. Special Issue: Nonlinear Analysis and Design of Communication Networks for IoT Applications - Part II
  38. Silicon-based all-optical wavelength converter for on-chip optical interconnection
  39. Research on a path-tracking control system of unmanned rollers based on an optimization algorithm and real-time feedback
  40. Analysis of the sports action recognition model based on the LSTM recurrent neural network
  41. Industrial robot trajectory error compensation based on enhanced transfer convolutional neural networks
  42. Research on IoT network performance prediction model of power grid warehouse based on nonlinear GA-BP neural network
  43. Interactive recommendation of social network communication between cities based on GNN and user preferences
  44. Application of improved P-BEM in time varying channel prediction in 5G high-speed mobile communication system
  45. Construction of a BIM smart building collaborative design model combining the Internet of Things
  46. Optimizing malicious website prediction: An advanced XGBoost-based machine learning model
  47. Economic operation analysis of the power grid combining communication network and distributed optimization algorithm
  48. Sports video temporal action detection technology based on an improved MSST algorithm
  49. Internet of things data security and privacy protection based on improved federated learning
  50. Enterprise power emission reduction technology based on the LSTM–SVM model
  51. Construction of multi-style face models based on artistic image generation algorithms
  52. Research and application of interactive digital twin monitoring system for photovoltaic power station based on global perception
  53. Special Issue: Decision and Control in Nonlinear Systems - Part II
  54. Animation video frame prediction based on ConvGRU fine-grained synthesis flow
  55. Application of GGNN inference propagation model for martial art intensity evaluation
  56. Benefit evaluation of building energy-saving renovation projects based on BWM weighting method
  57. Deep neural network application in real-time economic dispatch and frequency control of microgrids
  58. Real-time force/position control of soft growing robots: A data-driven model predictive approach
  59. Mechanical product design and manufacturing system based on CNN and server optimization algorithm
  60. Application of finite element analysis in the formal analysis of ancient architectural plaque section
  61. Research on territorial spatial planning based on data mining and geographic information visualization
  62. Fault diagnosis of agricultural sprinkler irrigation machinery equipment based on machine vision
  63. Closure technology of large span steel truss arch bridge with temporarily fixed edge supports
  64. Intelligent accounting question-answering robot based on a large language model and knowledge graph
  65. Analysis of manufacturing and retailer blockchain decision based on resource recyclability
  66. Flexible manufacturing workshop mechanical processing and product scheduling algorithm based on MES
  67. Exploration of indoor environment perception and design model based on virtual reality technology
  68. Tennis automatic ball-picking robot based on image object detection and positioning technology
  69. A new CNN deep learning model for computer-intelligent color matching
  70. Design of AR-based general computer technology experiment demonstration platform
  71. Indoor environment monitoring method based on the fusion of audio recognition and video patrol features
  72. Health condition prediction method of the computer numerical control machine tool parts by ensembling digital twins and improved LSTM networks
  73. Establishment of a green degree evaluation model for wall materials based on lifecycle
  74. Quantitative evaluation of college music teaching pronunciation based on nonlinear feature extraction
  75. Multi-index nonlinear robust virtual synchronous generator control method for microgrid inverters
  76. Manufacturing engineering production line scheduling management technology integrating availability constraints and heuristic rules
  77. Analysis of digital intelligent financial audit system based on improved BiLSTM neural network
  78. Attention community discovery model applied to complex network information analysis
  79. A neural collaborative filtering recommendation algorithm based on attention mechanism and contrastive learning
  80. Rehabilitation training method for motor dysfunction based on video stream matching
  81. Research on façade design for cold-region buildings based on artificial neural networks and parametric modeling techniques
  82. Intelligent implementation of muscle strain identification algorithm in Mi health exercise induced waist muscle strain
  83. Optimization design of urban rainwater and flood drainage system based on SWMM
  84. Improved GA for construction progress and cost management in construction projects
  85. Evaluation and prediction of SVM parameters in engineering cost based on random forest hybrid optimization
  86. Museum intelligent warning system based on wireless data module
  87. Optimization design and research of mechatronics based on torque motor control algorithm
  88. Special Issue: Nonlinear Engineering’s significance in Materials Science
  89. Experimental research on the degradation of chemical industrial wastewater by combined hydrodynamic cavitation based on nonlinear dynamic model
  90. Study on low-cycle fatigue life of nickel-based superalloy GH4586 at various temperatures
  91. Some results of solutions to neutral stochastic functional operator-differential equations
  92. Ultrasonic cavitation did not occur in high-pressure CO2 liquid
  93. Research on the performance of a novel type of cemented filler material for coal mine opening and filling
  94. Testing of recycled fine aggregate concrete’s mechanical properties using recycled fine aggregate concrete and research on technology for highway construction
  95. A modified fuzzy TOPSIS approach for the condition assessment of existing bridges
  96. Nonlinear structural and vibration analysis of straddle monorail pantograph under random excitations
  97. Achieving high efficiency and stability in blue OLEDs: Role of wide-gap hosts and emitter interactions
  98. Construction of teaching quality evaluation model of online dance teaching course based on improved PSO-BPNN
  99. Enhanced electrical conductivity and electromagnetic shielding properties of multi-component polymer/graphite nanocomposites prepared by solid-state shear milling
  100. Optimization of thermal characteristics of buried composite phase-change energy storage walls based on nonlinear engineering methods
  101. A higher-performance big data-based movie recommendation system
  102. Nonlinear impact of minimum wage on labor employment in China
  103. Nonlinear comprehensive evaluation method based on information entropy and discrimination optimization
  104. Application of numerical calculation methods in stability analysis of pile foundation under complex foundation conditions
  105. Research on the contribution of shale gas development and utilization in Sichuan Province to carbon peak based on the PSA process
  106. Characteristics of tight oil reservoirs and their impact on seepage flow from a nonlinear engineering perspective
  107. Nonlinear deformation decomposition and mode identification of plane structures via orthogonal theory
  108. Numerical simulation of damage mechanism in rock with cracks impacted by self-excited pulsed jet based on SPH-FEM coupling method: The perspective of nonlinear engineering and materials science
  109. Cross-scale modeling and collaborative optimization of ethanol-catalyzed coupling to produce C4 olefins: Nonlinear modeling and collaborative optimization strategies
  110. Unequal width T-node stress concentration factor analysis of stiffened rectangular steel pipe concrete
  111. Special Issue: Advances in Nonlinear Dynamics and Control
  112. Development of a cognitive blood glucose–insulin control strategy design for a nonlinear diabetic patient model
  113. Big data-based optimized model of building design in the context of rural revitalization
  114. Multi-UAV assisted air-to-ground data collection for ground sensors with unknown positions
  115. Design of urban and rural elderly care public areas integrating person-environment fit theory
  116. Application of lossless signal transmission technology in piano timbre recognition
  117. Application of improved GA in optimizing rural tourism routes
  118. Architectural animation generation system based on AL-GAN algorithm
  119. Advanced sentiment analysis in online shopping: Implementing LSTM models analyzing E-commerce user sentiments
  120. Intelligent recommendation algorithm for piano tracks based on the CNN model
  121. Visualization of large-scale user association feature data based on a nonlinear dimensionality reduction method
  122. Low-carbon economic optimization of microgrid clusters based on an energy interaction operation strategy
  123. Optimization effect of video data extraction and search based on Faster-RCNN hybrid model on intelligent information systems
  124. Construction of image segmentation system combining TC and swarm intelligence algorithm
  125. Particle swarm optimization and fuzzy C-means clustering algorithm for the adhesive layer defect detection
  126. Optimization of student learning status by instructional intervention decision-making techniques incorporating reinforcement learning
  127. Fuzzy model-based stabilization control and state estimation of nonlinear systems
  128. Optimization of distribution network scheduling based on BA and photovoltaic uncertainty
  129. Tai Chi movement segmentation and recognition on the grounds of multi-sensor data fusion and the DBSCAN algorithm
  130. Special Issue: Dynamic Engineering and Control Methods for the Nonlinear Systems - Part III
  131. Generalized numerical RKM method for solving sixth-order fractional partial differential equations
https://doi.org/10.1515/nleng-2025-0169
Keywords for this article
rock-breaking; SPH-FEM method; damage characteristic
Creative Commons
BY 4.0

Articles in the same Issue

  1. Research Articles
  2. Generalized (ψ,φ)-contraction to investigate Volterra integral inclusions and fractal fractional PDEs in super-metric space with numerical experiments
  3. Solitons in ultrasound imaging: Exploring applications and enhancements via the Westervelt equation
  4. Stochastic improved Simpson for solving nonlinear fractional-order systems using product integration rules
  5. Exploring dynamical features like bifurcation assessment, sensitivity visualization, and solitary wave solutions of the integrable Akbota equation
  6. Research on surface defect detection method and optimization of paper-plastic composite bag based on improved combined segmentation algorithm
  7. Impact the sulphur content in Iraqi crude oil on the mechanical properties and corrosion behaviour of carbon steel in various types of API 5L pipelines and ASTM 106 grade B
  8. Unravelling quiescent optical solitons: An exploration of the complex Ginzburg–Landau equation with nonlinear chromatic dispersion and self-phase modulation
  9. Perturbation-iteration approach for fractional-order logistic differential equations
  10. Variational formulations for the Euler and Navier–Stokes systems in fluid mechanics and related models
  11. Rotor response to unbalanced load and system performance considering variable bearing profile
  12. DeepFowl: Disease prediction from chicken excreta images using deep learning
  13. Channel flow of Ellis fluid due to cilia motion
  14. A case study of fractional-order varicella virus model to nonlinear dynamics strategy for control and prevalence
  15. Multi-point estimation weldment recognition and estimation of pose with data-driven robotics design
  16. Analysis of Hall current and nonuniform heating effects on magneto-convection between vertically aligned plates under the influence of electric and magnetic fields
  17. A comparative study on residual power series method and differential transform method through the time-fractional telegraph equation
  18. Insights from the nonlinear Schrödinger–Hirota equation with chromatic dispersion: Dynamics in fiber–optic communication
  19. Mathematical analysis of Jeffrey ferrofluid on stretching surface with the Darcy–Forchheimer model
  20. Exploring the interaction between lump, stripe and double-stripe, and periodic wave solutions of the Konopelchenko–Dubrovsky–Kaup–Kupershmidt system
  21. Computational investigation of tuberculosis and HIV/AIDS co-infection in fuzzy environment
  22. Signature verification by geometry and image processing
  23. Theoretical and numerical approach for quantifying sensitivity to system parameters of nonlinear systems
  24. Chaotic behaviors, stability, and solitary wave propagations of M-fractional LWE equation in magneto-electro-elastic circular rod
  25. Dynamic analysis and optimization of syphilis spread: Simulations, integrating treatment and public health interventions
  26. Visco-thermoelastic rectangular plate under uniform loading: A study of deflection
  27. Threshold dynamics and optimal control of an epidemiological smoking model
  28. Numerical computational model for an unsteady hybrid nanofluid flow in a porous medium past an MHD rotating sheet
  29. Regression prediction model of fabric brightness based on light and shadow reconstruction of layered images
  30. Dynamics and prevention of gemini virus infection in red chili crops studied with generalized fractional operator: Analysis and modeling
  31. Qualitative analysis on existence and stability of nonlinear fractional dynamic equations on time scales
  32. Fractional-order super-twisting sliding mode active disturbance rejection control for electro-hydraulic position servo systems
  33. Analytical exploration and parametric insights into optical solitons in magneto-optic waveguides: Advances in nonlinear dynamics for applied sciences
  34. Bifurcation dynamics and optical soliton structures in the nonlinear Schrödinger–Bopp–Podolsky system
  35. Review Article
  36. Haar wavelet collocation method for existence and numerical solutions of fourth-order integro-differential equations with bounded coefficients
  37. Special Issue: Nonlinear Analysis and Design of Communication Networks for IoT Applications - Part II
  38. Silicon-based all-optical wavelength converter for on-chip optical interconnection
  39. Research on a path-tracking control system of unmanned rollers based on an optimization algorithm and real-time feedback
  40. Analysis of the sports action recognition model based on the LSTM recurrent neural network
  41. Industrial robot trajectory error compensation based on enhanced transfer convolutional neural networks
  42. Research on IoT network performance prediction model of power grid warehouse based on nonlinear GA-BP neural network
  43. Interactive recommendation of social network communication between cities based on GNN and user preferences
  44. Application of improved P-BEM in time varying channel prediction in 5G high-speed mobile communication system
  45. Construction of a BIM smart building collaborative design model combining the Internet of Things
  46. Optimizing malicious website prediction: An advanced XGBoost-based machine learning model
  47. Economic operation analysis of the power grid combining communication network and distributed optimization algorithm
  48. Sports video temporal action detection technology based on an improved MSST algorithm
  49. Internet of things data security and privacy protection based on improved federated learning
  50. Enterprise power emission reduction technology based on the LSTM–SVM model
  51. Construction of multi-style face models based on artistic image generation algorithms
  52. Research and application of interactive digital twin monitoring system for photovoltaic power station based on global perception
  53. Special Issue: Decision and Control in Nonlinear Systems - Part II
  54. Animation video frame prediction based on ConvGRU fine-grained synthesis flow
  55. Application of GGNN inference propagation model for martial art intensity evaluation
  56. Benefit evaluation of building energy-saving renovation projects based on BWM weighting method
  57. Deep neural network application in real-time economic dispatch and frequency control of microgrids
  58. Real-time force/position control of soft growing robots: A data-driven model predictive approach
  59. Mechanical product design and manufacturing system based on CNN and server optimization algorithm
  60. Application of finite element analysis in the formal analysis of ancient architectural plaque section
  61. Research on territorial spatial planning based on data mining and geographic information visualization
  62. Fault diagnosis of agricultural sprinkler irrigation machinery equipment based on machine vision
  63. Closure technology of large span steel truss arch bridge with temporarily fixed edge supports
  64. Intelligent accounting question-answering robot based on a large language model and knowledge graph
  65. Analysis of manufacturing and retailer blockchain decision based on resource recyclability
  66. Flexible manufacturing workshop mechanical processing and product scheduling algorithm based on MES
  67. Exploration of indoor environment perception and design model based on virtual reality technology
  68. Tennis automatic ball-picking robot based on image object detection and positioning technology
  69. A new CNN deep learning model for computer-intelligent color matching
  70. Design of AR-based general computer technology experiment demonstration platform
  71. Indoor environment monitoring method based on the fusion of audio recognition and video patrol features
  72. Health condition prediction method of the computer numerical control machine tool parts by ensembling digital twins and improved LSTM networks
  73. Establishment of a green degree evaluation model for wall materials based on lifecycle
  74. Quantitative evaluation of college music teaching pronunciation based on nonlinear feature extraction
  75. Multi-index nonlinear robust virtual synchronous generator control method for microgrid inverters
  76. Manufacturing engineering production line scheduling management technology integrating availability constraints and heuristic rules
  77. Analysis of digital intelligent financial audit system based on improved BiLSTM neural network
  78. Attention community discovery model applied to complex network information analysis
  79. A neural collaborative filtering recommendation algorithm based on attention mechanism and contrastive learning
  80. Rehabilitation training method for motor dysfunction based on video stream matching
  81. Research on façade design for cold-region buildings based on artificial neural networks and parametric modeling techniques
  82. Intelligent implementation of muscle strain identification algorithm in Mi health exercise induced waist muscle strain
  83. Optimization design of urban rainwater and flood drainage system based on SWMM
  84. Improved GA for construction progress and cost management in construction projects
  85. Evaluation and prediction of SVM parameters in engineering cost based on random forest hybrid optimization
  86. Museum intelligent warning system based on wireless data module
  87. Optimization design and research of mechatronics based on torque motor control algorithm
  88. Special Issue: Nonlinear Engineering’s significance in Materials Science
  89. Experimental research on the degradation of chemical industrial wastewater by combined hydrodynamic cavitation based on nonlinear dynamic model
  90. Study on low-cycle fatigue life of nickel-based superalloy GH4586 at various temperatures
  91. Some results of solutions to neutral stochastic functional operator-differential equations
  92. Ultrasonic cavitation did not occur in high-pressure CO2 liquid
  93. Research on the performance of a novel type of cemented filler material for coal mine opening and filling
  94. Testing of recycled fine aggregate concrete’s mechanical properties using recycled fine aggregate concrete and research on technology for highway construction
  95. A modified fuzzy TOPSIS approach for the condition assessment of existing bridges
  96. Nonlinear structural and vibration analysis of straddle monorail pantograph under random excitations
  97. Achieving high efficiency and stability in blue OLEDs: Role of wide-gap hosts and emitter interactions
  98. Construction of teaching quality evaluation model of online dance teaching course based on improved PSO-BPNN
  99. Enhanced electrical conductivity and electromagnetic shielding properties of multi-component polymer/graphite nanocomposites prepared by solid-state shear milling
  100. Optimization of thermal characteristics of buried composite phase-change energy storage walls based on nonlinear engineering methods
  101. A higher-performance big data-based movie recommendation system
  102. Nonlinear impact of minimum wage on labor employment in China
  103. Nonlinear comprehensive evaluation method based on information entropy and discrimination optimization
  104. Application of numerical calculation methods in stability analysis of pile foundation under complex foundation conditions
  105. Research on the contribution of shale gas development and utilization in Sichuan Province to carbon peak based on the PSA process
  106. Characteristics of tight oil reservoirs and their impact on seepage flow from a nonlinear engineering perspective
  107. Nonlinear deformation decomposition and mode identification of plane structures via orthogonal theory
  108. Numerical simulation of damage mechanism in rock with cracks impacted by self-excited pulsed jet based on SPH-FEM coupling method: The perspective of nonlinear engineering and materials science
  109. Cross-scale modeling and collaborative optimization of ethanol-catalyzed coupling to produce C4 olefins: Nonlinear modeling and collaborative optimization strategies
  110. Unequal width T-node stress concentration factor analysis of stiffened rectangular steel pipe concrete
  111. Special Issue: Advances in Nonlinear Dynamics and Control
  112. Development of a cognitive blood glucose–insulin control strategy design for a nonlinear diabetic patient model
  113. Big data-based optimized model of building design in the context of rural revitalization
  114. Multi-UAV assisted air-to-ground data collection for ground sensors with unknown positions
  115. Design of urban and rural elderly care public areas integrating person-environment fit theory
  116. Application of lossless signal transmission technology in piano timbre recognition
  117. Application of improved GA in optimizing rural tourism routes
  118. Architectural animation generation system based on AL-GAN algorithm
  119. Advanced sentiment analysis in online shopping: Implementing LSTM models analyzing E-commerce user sentiments
  120. Intelligent recommendation algorithm for piano tracks based on the CNN model
  121. Visualization of large-scale user association feature data based on a nonlinear dimensionality reduction method
  122. Low-carbon economic optimization of microgrid clusters based on an energy interaction operation strategy
  123. Optimization effect of video data extraction and search based on Faster-RCNN hybrid model on intelligent information systems
  124. Construction of image segmentation system combining TC and swarm intelligence algorithm
  125. Particle swarm optimization and fuzzy C-means clustering algorithm for the adhesive layer defect detection
  126. Optimization of student learning status by instructional intervention decision-making techniques incorporating reinforcement learning
  127. Fuzzy model-based stabilization control and state estimation of nonlinear systems
  128. Optimization of distribution network scheduling based on BA and photovoltaic uncertainty
  129. Tai Chi movement segmentation and recognition on the grounds of multi-sensor data fusion and the DBSCAN algorithm
  130. Special Issue: Dynamic Engineering and Control Methods for the Nonlinear Systems - Part III
  131. Generalized numerical RKM method for solving sixth-order fractional partial differential equations
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 14.11.2025 from https://www.degruyterbrill.com/document/doi/10.1515/nleng-2025-0169/html
Scroll to top button