Structural stability risk threshold of classical garden rockeries based on elastoplastic theory

Abstract

In previous studies, linear elastic fracture mechanics (LEFM) combined with the extended finite element method (XFEM) was used to predict crack propagation paths in rockeries and to calculate the associated risk thresholds. However, when rockeries develop through-cracks—the most structurally critical cracks that propagate linearly and directly contribute to overall instability—LEFM alone cannot fully capture the complete evolution of structural stability. Therefore, this study introduces an elastoplastic constitutive model and conducts numerical simulations in Abaqus to analyze the full process of instability in traditional garden rockeries within stone cultural heritage. The objective is to determine the instability risk threshold from the crack-level stage to the structural-level stage, thereby providing garden managers with quantitative data on the staged evolution of overall instability in rockery structures.

Introduction

The rockeries in classical gardens are currently affected by various forms of deterioration, among which structural damage is of particular concern. Previous studies indicate that such damage mainly includes stress concentration, peak stone inclination, foundation settlement, adhesive detachment, and plant-induced root splitting¹. In current research on the stability of traditional garden rockeries, some researchers have conducted structural risk assessments based on physical vulnerability indicators derived from finite element simulations², while others have identified stress-concentrated areas in rockeries under self-weight and visitor loading through finite element analysis³.

In the field of historical monument studies, many structures exhibit significant deterioration, with cracks being one of the most distinct surface defects. Consequently, crack behavior has attracted considerable attention in the numerical analysis of historical monuments. Many researchers have investigated advanced computational methods for finite element modeling of crack propagation in brittle materials^4,5. Detailed on-site visual inspections combined with finite element modeling are commonly used to evaluate the structural stability of such heritage objects. In our previous work, we investigated the crack expansion risk threshold of the Small Rock Mountain Adobe (SRMA) rockery in He Garden⁶.

It is worth noting that once cracks develop, they progressively accelerate the deterioration of cultural heritage, leading to further structural damage and even severe risks such as overall instability or collapse⁷. However, no existing literature has applied the finite element method to numerically simulate the staged evolution of overall instability in rockery structures or to determine the corresponding risk thresholds. Establishing a numerical simulation methodology for assessing the structural stability of cultural heritage is essential for scientifically predicting the damage process, implementing targeted monitoring and rescue measures, and prolonging the lifespan of heritage assets⁸. This approach also provides practical engineering guidance for heritage managers.

Finite element analysis offers a robust computational framework for evaluating structural risks in cultural heritage. Current research includes finite element modeling of heritage structures⁹, static and dynamic analyses of structural responses under various loading conditions^10,11,12, and comprehensive assessments of structural stability risks¹³. These advancements facilitate sustainable conservation through deeper structural understanding and evidence-based restoration strategies¹⁴. For stone cultural relics such as masonry structures and grottoes, software like Abaqus and elastoplastic constitutive models are commonly used to evaluate structural stability, for example, employing plastic models to diagnose damage in historic masonry within large cultural heritage sites¹⁵. Some researchers also use the elastoplastic damage model and finite element method to analyze the structural safety of the hollow defense gun emplacements of the Great Wall¹⁶. In the finite element analysis of rock deformation, Drucker-Prager and MC constitutive models are often adopted.

In previous studies, the risk levels and corresponding risk thresholds for the expansion of the most structurally critical cracks in the central cave area of Small Rock Mountain Adobe (SRMA) in He Garden were evaluated⁶. Because the rockeries in Jiangnan gardens are predominantly built from limestone—a brittle rock material—crack propagation mainly occurs within the linear elastic stage, and different propagation risk levels can be classified using the key parameter SIF.

However, that study focused solely on the crack scale and did not address the overall instability risk threshold of the rockery when cracks advance into the high-risk stage. Moreover, it did not incorporate the deformation and failure behavior of the plastic stage¹⁷ into the numerical simulation framework for assessing structural stability. Therefore, this study attempts to incorporate an elastoplastic constitutive model to conduct a finite element numerical simulation of the entire process of the instability of the rockery structure, in order to determine the various stages of the instability risk of the rockery structure and the corresponding risk thresholds.

For brittle, hard limestone, a limit strain value exists under cyclic loading; as this limit is approached, the material experiences continuous deformation followed by rupture¹⁸. In current research on rock-structure stability, determining the overall failure behavior of rock materials through the L–D curve is a feasible approach¹⁹. Furthermore, based on the stress-strain curve, the failure process of brittle rocks can be divided into several stages^20,21.

In this study, the assessment of the stability risk threshold of the rockery structure was extended from the crack-level stage to the structural-level stage. The crack-level stage refers to the stage from the initial formation of cracks to the stage of linear crack propagation in the rockery structure. The structural-level stage refers to the stage involving overall structural damage, instability, and eventual collapse of the rockery structure.

In previous studies, linear elastic fracture mechanics (LEFM)-based numerical simulations were used to model the cracking patterns and crack propagation paths of rockeries, and corresponding risk levels and risk thresholds were determined. LEFM theory (adopted to model crack propagation), which assumes a linear relationship between stress and strain until plastic deformation occurs, is effective for modeling brittle rock failure²². The stress-intensity factor solution obtained from the LEFM theory can be used to evaluate crack propagation²³. The maximum principal stress criterion^24,25 was adopted in this study, whereby damage initiation starts when the maximum principal stress at a point exceeds the tensile strength of the material. Crack propagation follows a linear evolution model based on the energy release rate, propagating in the direction that maximizes the total energy release.

The LEFM theory assumes a linear elastic response near crack tips and characterizes the stress field using the stress intensity factor (SIF). For Mode I loading, the SIF is defined as

$${K}_{I}=\sigma \sqrt{\pi a}$$

(1)

where $\sigma$ is the applied stress, and $a$ is the crack length.

The maximum principal stress criterion is used as the criterion for crack initiation, which holds that the crack will occur when the maximum principal stress reaches the ultimate strength of the material. Material failure is assumed to occur when the maximum principal stress ${\sigma }_{\max }$ reaches the ultimate strength of the material ${\sigma }_{c}$, i.e.,

$${\sigma }_{\max }={\sigma }_{c}$$

(2)

The energy release rate ($G$) describes the energy change during crack propagation. This represents the energy released by the system per unit increase in the crack area. For three-dimensional crack propagation, when the energy release rate $G$ reaches the critical energy release rate ${G}_{c}$ of the material, the crack continues to propagate.

$$G={G}_{C}$$

(3)

One of the main drawbacks of the aforementioned linear elastic fracture mechanics is that it neglects plasticity. Even when the overall stress is elastic, plastic deformation can still occur at the crack tip, even in materials that are generally considered brittle.

Therefore, an elastoplastic constitutive model is adopted.

Using the Drucker–Prager yield function, we get

$$F=\sqrt{{J}_{2}}+\alpha {I}_{1}-k\left(\kappa \right)$$

(4)

Where ${I}_{1}$ is the first invariant of the stress tensor, ${J}_{2}$ is the second invariant of the stress deviator, and $\alpha$ is a material constant²⁶. The internal variable $\kappa ={\varepsilon }_{{eq}}^{p}$ represents the equivalent plastic strain (an output variable in Abaqus, PEEQ), and $k\left(\kappa \right)$ denotes the strength parameter that governs the evolution of the yield surface during plastic deformation.

In this section, the ABAQUS numerical simulation adopts the uniaxial compression mode, that is, ${\sigma }_{1}\,=\,{\sigma }_{{\rm{c}}}\,$, and ${\sigma }_{2}\,=\,{\sigma }_{3}=0$. Therefore $k\left(\kappa \right)$ is written as

$$\frac{{\sigma }_{{\rm{c}}}}{\sqrt{3}}+\alpha {\sigma }_{{\rm{c}}}=k\left(\kappa \right)$$

(5)

In this section, the hardening-softening function is adopted in the Drucker–Prager function setting of ABAQUS to observe the deformation behavior of the rockery material before and after the peak, and it is expressed as follows:

In the pre-peak stage, the material exhibits hardening behavior, which is expressed as:

$$\frac{{dk}}{d\kappa } > 0,0\le \kappa \le {\kappa }_{p}$$

(6)

In the post-peak stage, the material exhibits softening behavior, expressed as:

$$\frac{{dk}}{d\kappa } < 0,\kappa > {\kappa }_{p}$$

(7)

Among them, ${\kappa }_{p}={\varepsilon }_{{eq},p}^{p}$ denotes the equivalent plastic strain at the point of reaching the peak stress.

These key nodes include (1) crack initiation, (2) formation of a through-crack, (3) structural instability, and (4) structural collapse. Based on these risk thresholds, the instability risk of the rockery structure can be divided into five stages: Ⅰ Stable structure stage, Ⅱ Stage of linear crack propagation, Ⅲ Stage of structural damage, Ⅳ Stage of structural instability, and Ⅴ Stage of structural collapse.

Using the above formula, the key nodes of (1) crack initiation and (4) structural collapse can be obtained. Based on these results, and in combination with the L-D curve, the stability risk stages of the rockery structure can be further divided in a more detailed manner. The physical meanings and calculation bases of the key nodes are as follows:

The risk threshold during the crack-level stage includes the following two key points:

Crack initiation: The maximum principal stress criterion is used to determine crack initiation. According to this criterion, a crack occurs when the maximum principal stress reaches the ultimate strength of the material, i.e., ${\sigma }_{\max }={\sigma }_{c}$.

Formation of a through-crack: A significant turning point appears on the L-D curve.

The risk threshold during the structural-level stage includes the following two key points:

Structural instability: The trend of the L–D curve undergoes a significant change.

Structural collapse: The bearing strength of the rockery materials reaches its peak, i.e., ${\varepsilon }_{{eq},p}^{p}={\kappa }_{p}$.

The instability risk threshold proposed in this study is more innovative compared to previous work. In our previous work, we used LEFM combined with XFEM to predict the rockery cracking patterns and crack propagation paths.

In this study, it was observed that once the L-D curve reached the critical point corresponding to the “formation of a through crack”, its overall trend no longer changed significantly. This indicates that the overall structural response behavior of the rock formation can no longer be captured using LEFM alone. By incorporating the elastoplastic constitutive model, the key node corresponding to the peak point of stress-strain curve can be quantified to determine the critical node of “structural collapse.” Furthermore, by analyzing the evolution of the L-D curve, the risk stages of the rock formation structural instability can be further quantified. This approach enables the prediction of the entire process of structural stability risk in rock formations through numerical simulations, forming a comprehensive methodological framework with engineering practicality.

This addresses the existing gap in quantifying unstable risk stages in rockery structures and supports garden managers in implementing preventive and emergency protection measures, particularly in areas undergoing plastic deformation.

Methods

Verification of mechanical parameters

The material parameters in the linear elastic stage are directly adopted from verified data in the relevant literature, as shown in the Table 1.

Table 1 Linear elastic fracture mechanics parameters

Among these parameters, the density, Poisson’s ratio, elastic modulus², and tensile strength are directly determined based on the parameters and conclusions of previous finite element analysis of rockery cultural heritage³.

Once the elastic modulus is known, the corresponding fracture toughness can be derived from the fracture energy. As noted above, fracture toughness is a key parameter for assessing the risk level and critical threshold of crack propagation in rockery structures. Therefore, the appropriate determination of the fracture energy is of great importance. The relationship between fracture energy and fracture toughness can be expressed as ${K}_{{IC}}=\sqrt{{G}_{C}E}$. In addition, the fracture toughness of limestone specimens in reported literature (ranging from 0.6 to 1.76 MPa⋅m^1/2) should also be taken into consideration.

The fracture energy of limestone is typically in the range of 20–50 J/ ${{\rm{m}}}^{2}$²⁷. In our previous study on the critical threshold of crack propagation in classical garden rockeries, a fracture energy value of $G=30$J/${{\rm{m}}}^{2}$ was adopted, corresponding to a fracture toughness of ${K}_{{IC}}=1.5$MPa·${{\rm{m}}}^{1/2}$. In this section, a sensitivity analysis of this fracture energy value is performed to further assess the reasonableness of its selection.

Sensitivity analysis of fracture energy: In this study, numerical simulations of the Brazilian splitting test were conducted to examine the force-displacement curves and the differences in crack patterns under different energy release rates, so as to evaluate the effect of fracture energy on the material response.

The mechanical parameters of the limestone used in the simulation were as follows: elastic modulus, 75 GPa; Poisson’s ratio, 0.32; and density, 2700 kg/m³. The specimen geometry was defined with a diameter of D = 50 mm and a thickness of L = 10 mm. In the Abaqus parameter settings, the damage initiation criterion was defined using the maximum principal stress criterion with a tensile strength of 2 MPa, while the damage evolution criterion was based on fracture energy. Numerical tests were conducted using four fracture energy values: 20 J/m², 30 J/m², 40 J/m², and 50 J/m². Table 2 presents the mechanical parameter values of the four materials used in the comparative experiment.

Table 2 The parameter values of the comparative experiment of four types of stones

Figure 1 presents a comparative line graph of the force–displacement curves corresponding to four typical fracture energy values within the displacement range of 0–0.025 mm. The overall trends of the curves remain consistent across the different energy values.

Fig. 1: Force–displacement curves for fracture energy values ranging from 20 to 50 J/m².

a Displacement range: 0–0.01 mm, (b) Displacement range: 0.005–0.025 mm.

As shown in Fig. 1a, the force-displacement curves obtained using the four sets of material parameters exhibited a highly consistent trend when the applied load was less than 1000 N, i.e., prior to the first pronounced inflection in the force-displacement curve.

Figure 2 shows the visualized results of the Brazilian splitting simulations for the four fracture energy values, revealing that the crack propagation paths are nearly identical. Overall, comparison of the force–displacement curves under varying fracture energy levels (20–50 J/m²) indicates that the crack propagation behavior of limestone is largely unaffected. Therefore, a moderate fracture energy value of 30 J/m² was selected as the representative energy release rate for this study.

Fig. 2: Crack propagation results from Brazilian splitting simulations under different fracture energy values.

a ${G}_{f}$ = 20 J/m²; (b) ${G}_{f}$ = 30 J/m²; (c) ${G}_{f}$ = 40 J/m²; (d) ${G}_{f}$ = 50 J/m².

For the plasticity stage calculation, the Drucker–Prager model was implemented in Abaqus. Limestone materials exhibit a tensile-to-compressive strength ratio ranging from 0.01 to 0.067, an internal friction angle of 35–50°, and a cohesion value of 10–50 MPa. Due to the absence of precise experimental data, the following assumptions were adopted: the initial yield stress is typically approximately two-thirds of the peak strength²⁸. In this study, a compressive strength of 100 MPa was used as the peak stress, and the tensile-to-compressive strength ratio was set to 0.02.

The parameters required for the Drucker-Prager model in Abaqus^29,30 are summarized in Table 3. Since the elastic parameters have been given separately above, only the parameters associated with Drucker-Prager plasticity are dilation angle, yield stress ${\sigma }_{y}$, peak stress ${\sigma }_{p}$, and peak equivalent plastic strain ${\varepsilon }_{{eq},p}^{p}$. Among these parameters, the sensitivity analysis of the dilation angle is presented below as a representative case. The simulation results indicate that the parameter set listed in Table 3 provides a stable and moderate plastic evolution.

Table 3 Plasticity mechanical parameters

Effect of dilatancy angle: In Abaqus, the Drucker–Prager model requires a dilatancy angle parameter. For brittle rock materials, the dilatancy angle is typically low, and in practical calculations is usually selected within the range of $0\le \psi \le \varphi$³⁰. Therefore, in this study, dilatancy angle values within the range of 0–35° were compared. The numerical simulation results of the stress–strain curves over this range were used to evaluate the influence of dilatancy angle on the numerical outcomes and to determine the final value.

The Brazilian splitting numerical simulation method was employed. The mechanical parameters of limestone used in the simulation were: elastic modulus 75 GPa, Poisson’s ratio 0.32, and density 2700 kg/m³. The specimen geometry was defined with a diameter D = 50 mm and thickness L = 10 mm. In the Abaqus settings, the damage initiation criterion used the maximum principal stress criterion with a tensile strength of 2 MPa, and the damage evolution criterion was based on fracture energy, set to 30 J/m².

The plastic material parameters employed the Drucker–Prager model with an internal friction angle of 35°. The dilatancy angle was tested across the 0–35° range. In the Drucker–Prager Hardening settings, the two point pairs (0.0000, 100 MPa) and (0.02, 100 MPa) were used.

Figure 3 presents a comparison of the Smises–PEEQ curves for dilatancy angles from 0° to 35°. Except for the two extreme values (0° and 35°), the curve trends are similar. Thus, a typical value of 15° was selected for this study.

Fig. 3

Comparison of Smises–PEEQ curves for typical dilatancy angles ranging from 0° to 35°.

Therefore, the final choice was to use the equivalent plastic strain ${\varepsilon }_{{eq}}^{p}$ (Abaqus: PEEQ) as the internal variable for the hardening-softening evolution ${\sigma }_{c}({\varepsilon }_{{eq}}^{p})$. The yield stress was set to ${\sigma }_{y}(0)=66.7$MPa, and the peak strength was taken as the UCS, ${\sigma }_{p}=100$ MPa, which reached the peak at ${\varepsilon }_{{eq},p}^{p}=0.00179$. After the peak, the strength gradually degrades according to a segmented linear pattern to the residual strength ${\sigma }_{r}$ (corresponding to ${\varepsilon }_{f}$) to characterize the post-peak load attenuation.

As shown in Fig. 4, this section also compares the differences in the mechanical response outputs of the Brazilian splitting numerical simulation experiments under the same loading conditions for the LEFM and elastoplastic constitutive models. The crack propagation path predicted by LEFM occurs before the development of plastic deformation. A plastic zone usually forms around the crack tip area. At the stage when plastic deformation has not significantly developed, the distribution of the plastic zone is consistent with the crack propagation path under LEFM.

Fig. 4: Output results under the same loading conditions for LEFM and elastoplastic constitutive models.

a Crack initiation and the corresponding plastic deformation area. b Crack propagation and the corresponding plastic deformation area. c Formation of a through-crack and the corresponding plastic deformation area.

This comparison clarifies the respective applicability of the two theoretical approaches. LEFM is effective in predicting the crack propagation path, but it is unable to capture the overall plastic deformation region of the structure. In contrast, the elastoplastic model can describe both the development of the plastic zone near the crack tip and the evolution of the global structural response.

Research foundation

This study refines the previously established numerical simulation system for assessing the stability of classical garden rockery structures, enabling the analysis of stability risks to advance from the crack level to the structural-level stage. The complete framework of the numerical simulation system is presented in Fig. 5.

Fig. 5

Framework of the numerical simulation system for evaluating the stability of the rockery structure.

This study builds upon previous research on the SRMA in He Garden, Yangzhou, China. The central cave area is again selected as a representative case to evaluate the feasibility of the proposed methodology. The location of the SRMA within He Garden and the position of the central cave area are shown in Fig. 6.

Fig. 6: Plan view of He Garden and location of the SRMA.

a Illustration showing the positions of five rockeries in He Garden; (b) Photo of the SRMA with the central cave area highlighted.

Following the workflow of ground laser scanning → point cloud preprocessing → topological simplification → generation of a C3D8R voxel finite element model⁶, the resulting model was imported into the ABAQUS finite element analysis software. The mechanical parameter settings for the linear elastic stage are provided in Table 1.

The boundary constraints and interaction relationships were defined according to the primary cause of structural deterioration diagnosed for the central cave area—namely, localized foundation settlement. The settlement zone of the central cave in the SRMA and the corresponding interaction settings in Abaqus are illustrated in Fig. 7.

Fig. 7: The foundation settlement area of the central cave in the SRMA and the corresponding interaction settings in Abaqus.

a Plan of the foundation settlement area; (b) Interaction relationships.

The central cavern was obtained by using a 0.2 × 0.2 × 0.2(m) C3D8R voxel finite element model. Boundary conditions included full constraint of the rockery base, and the vertical displacement was set in the settlement area. Furthermore, the rockery was subject to self-weight load (−9.8 m/s²) as a whole.

Figure 7 illustrates the local subsidence area of the foundation of the central cave area in SRMA and the setting of the interaction relationship in Abaqus⁶.

In our previous study, under the same loading conditions, the numerical simulation results were in good agreement with the location of the most critical crack in the central cave of the SRMA at He Garden, Yangzhou. Accordingly, the same modeling assumptions were adopted in this study.

Results

Risk threshold during the crack-level stage of the central cave area in SRMA

This study builds upon the previous research on the SRMA in He Garden, continuing to use the finite element model developed in the earlier stage along with the damage-diagnosis results of the rockery. In the 2021 phase, predictive simulations of the cracking pattern in the central cave area of the SRMA were conducted under local foundation-settlement loading conditions.

Earlier research verified that linear elastic fracture mechanics (LEFM) is an effective numerical approach for determining crack propagation paths. In that work, the stage at which a through-crack formed was identified based on the characteristic that the crack area no longer changed, though this phenomenon was not examined in depth.

Building on this foundation, the present study validates the feasibility of using the L–D curve to identify the key stages at which overall failure behavior occurs in classical garden rockeries. This chapter examines the relationship between variations in the L–D curve during crack propagation and the cracking pattern of the rockery. Through this analysis, the key intervals governing the transition of instability risk—from the crack level to the structural level—along with their corresponding risk thresholds, are determined.

As shown in Table 4, the relevant data obtained from the LEFM-based prediction of crack patterns in the central cave of the SRMA were categorized according to the changes in crack area and crack volume. The point at which the crack area ceased to change was treated as the key node marking the formation of a through-crack.

Table 4 Division of key nodes in the crack propagation stage

If both crack area and crack volume remain at 0, the rockery structure is considered stable.

If the crack area continues to increase while crack volume and settlement remain slight, the structure is in the stage of linear crack expansion.

If the crack area no longer increases, but crack volume and settlement increase significantly, the structure has entered the stage of overall damage.

The key nodes and corresponding risk thresholds for instability of the classical garden rockery structure under the linear elastic model are presented in Table 4.

As shown in Fig. 8a, the L-D curve exhibits a pronounced turning point at FrameTime 238, after which its overall trend becomes nearly unchanged. Figure 8b illustrates the crack propagation path corresponding to FrameTime 238. After this point, the crack area no longer increases, while the crack volume and settlement increase significantly. This stage corresponds to the key node identified in Table 4 as the formation of a through-crack.

Fig. 8: Numerical simulation results of the key node forming a through-crack.

a L–D curve under LEFM, displacement range 0–3 mm; (b) Visualization of the node forming a through-crack, FrameTime 238.

These two observations show consistent mechanical responses of the rockery structure, thereby supporting the feasibility of using the L-D curve to identify the key node corresponding to the formation of a through-crack.

Risk threshold for the structural-level stage in the central cave area in SRMA

In the previous section, the method for identifying the key nodes of structural stability in the linear elastic stage of classical garden rockeries was established. Through the L–D curve, the key node at which a through-crack forms in the linear elastic stage can be determined, and the corresponding risk threshold can be obtained, as shown in Fig. 8.

However, a limitation of LEFM is its inability to capture the plastic-stage behavior of the rockery’s overall structural stability, as illustrated in Fig. 9, it can only simulate the elastic-stage response of the structure.

Fig. 9: Comparison of the mechanical response of the rockery predicted by the LEFM and elastoplastic constitutive models.

a At through-crack initiation; (b, c) after through-crack formation.

In contrast, the elastoplastic model is capable of capturing the development of the plastic zone at the crack tip and the evolution of the overall structural response.

As illustrated in Fig. 9, LEFM is applicable to the numerical evaluation of crack propagation risk, whereas elastoplastic theory is more appropriate for predicting the extent and evolution of plastic deformation zones. In Fig. 9a, the crack propagation path predicted using LEFM coupled with XFEM develops earlier than the plastic zone, which is generally concentrated around the crack tip. Figure 9b indicates that, before the plastic zone becomes significantly developed, its distribution remains consistent with the crack propagation path obtained from LEFM. As shown in Fig. 9c, the plastic zone in the rockery cave extends from the cave wall to the cave roof and subsequently propagates toward the cave pillar.

In LEFM-XFEM simulations, through-crack formation is a local geometrical event at the crack-level and is controlled by the fracture energy ${G}_{f}$. By contrast, in the Drucker–Prager elastoplastic model, plastic initiation is defined by the yield criterion $F(\sigma )=0$, whereas the peak point corresponds to the attainment of the maximum overall load-carrying capacity of the structure and thus reflects a structural-level response.

The primary difference between the two models lies in the response after the formation of a linear through crack. Beyond this point, LEFM no longer accounts for the expansion of the plastic zone or the subsequent energy dissipation, whereas the elastoplastic model is able to continuously capture response evolution—from the development of the crack-tip plastic zone to the post-peak stage of structural loading.

Based on the elastic mechanical parameters, an elastoplastic model was introduced. For the definition of the plastic response, the Drucker-Prager constitutive model was adopted, with a hardening-softening function used to characterize the constitutive behavior. The positions of the corresponding force–displacement curve and their relationships with the PEEQ–Smises curves are shown in Fig. 10.

Fig. 10: Numerical simulation results of rockery structure stability under the elastoplastic constitutive model.

a Force–displacement curve under the elastoplastic constitutive model; (b) PEEQ–Smises curve under the elastoplastic constitutive model, Node 11280.

The output results indicate that the trends of the L–D curve and the stress–strain curve are consistent. From the curve trends, two key nodes can be identified: the node where the L–D curve exhibits the first significant change, and the node corresponding to the stress peak in the stress–strain curve. Based on these observations, the key nodes for the overall structural instability risk of the classical garden rockery are presented in Table 5.

Table 5 Division of key nodes for the overall structural instability

Figure 11 presents the visualization of the plastic zone during the overall structural instability process of the classical garden rockery. Figure 11a, b illustrate the development of the plastic zone during the stage of damaged structure. Figure 11c shows the mechanical response at the onset of structural instability. Figure 11d illustrates the evolution of the plastic zone during the stage of structural instability. Figure 11e presents the mechanical response during structural collapse, while Fig. 11f shows the post-peak mechanical response.

Fig. 11: Visualization of the plastic zone during the overall structural instability process of the rockery.

a FrameTime 1; (b) FrameTime 6; (c) FrameTime 12; (d) FrameTime 18; (e) FrameTime 23; (f) FrameTime 40.

Risk diagnosis of the structural instability of the central cave area in SRMA

Table 6 shows the risk thresholds corresponding to each stage of structural instability risk of the rockery from the crack level to the whole-structure level. As the dominant cause of the disease in the central cave area in SRMA is local settlement, its risk threshold is characterized by the quantified data of foundation settlement.

Table 6 Risk thresholds for each stage of the structural instability of the rockery

Figure 12 shows the visualization diagrams corresponding to each stage of structural instability risk of the rockery from the crack level to the whole-structure level.

• I. Stage of stable structure: The crack area and volume remain at 0, reaching the critical node of “ Crack initiation”.

• II. Stage of linear crack propagation: The crack area changed significantly, increasing from 0 to 1.921 m². The crack cracked vertically upward from 1.2 m above the ground at the corner of the central cave, expanded to the top of the west entrance of the central cave, and cracked downward along the corner of the chamber. However, the crack volume and settlement remain slight, indicating that the crack is at the linear expansion level and has not yet affected the overall structural stability until it reaches the critical node of “Forming a through-crack”.

• III. Stage of damaged structure: The crack volume and settlement increase significantly, affecting the overall structural stability. That is, the structural stability risk of the rockery has shifted from the crack level to the whole-structure level. The deformation areas include the crack tip areas (cave walls and ceilings) in the linear elastic stage as well as the water-facing cave columns.

• IV. Stage of structural instability: The overall structure of the rockery undergoes significant deformation. From the water-facing cave columns to the cave ceiling and the east side cave wall, the overall structure gradually becomes unstable until it reaches the critical point of “structural collapse”.

• V. Stage of structural collapse: The overall structure of the central cave collapsed.

Fig. 12: Visualization diagram of key nodes for instability risk of rockery structure.

a Crack initiation (b) Forming a through-crack (c) Structural instability (d) Structural collapse.

Discussion

This study employs the finite element method to systematically investigate the stability of classical garden rockery structures. By integrating both linear elastic and elastoplastic constitutive models, it provides a comprehensive visualization of the entire process of structural stability changes in classical garden rockeries under the dominant causes of damage. The analysis clarifies that the instability risk of the rockery structure initiates with the most critical cracks, gradually propagates throughout the entire rockery, and ultimately leads to structural collapse. Furthermore, the study demonstrates the practical significance of evaluating both crack expansion risk and overall structural instability risk for the preventive protection and conservation of classical garden rockeries.

Applying this methodology to the central cavern area of the Small Rock Mountain Adobe confirms the feasibility of determining the risk thresholds for each stage of instability in classical garden rockery structures using the L–D curve and stress–strain curve. These key nodes include (1) Crack initiation, (2) Forming a through-crack, (3) Structural instability, and (4) Structural collapse. Based on these risk thresholds, the instability risk of the rockery structure can be divided into five stages: (1) Stable structure stage, (2) Stage of linear crack propagation, (3) Stage of damaged structure, (4) Stage of structural instability, and (5) Stage of structural collapse.

The LEFM-based crack propagation numerical simulation results are classified through the L–D curve, representing an advancement over previous risk classifications based solely on stress intensity factors. Specifically, the critical-node risk threshold for the formation of through-cracks is determined, allowing for the division of the crack extension and overall structural damage stages. More specifically, in Stage (2), i.e., the stage of linear crack propagation, the most critical crack undergoes linear expansion, with the crack area increasing while the corresponding settlement and crack volume change only slightly. In contrast, upon entering the (3) Stage of damaged structure, the crack area stabilizes, while the corresponding settlement and crack volume continue to grow, significantly compromising the overall structural stability of the rockery.

Using the elastoplastic constitutive model combined with Abaqus finite element analysis, the instability risk of the overall rockery structure is further divided into stages. During the damage stage of the overall structure, the plastic zone distribution is consistent with the crack propagation path obtained under LEFM in the previous analysis. It can also be observed that when cracks progress from the linear expansion stage to the overall damage stage, significant plastic deformation occurs throughout the rockery, not limited to the crack tip area. During the instability stage, which is indicated by the first significant change in the L–D curve, the plastic deformation region expands substantially. In the structural collapse stage, when the stress-strain curve of the initiating unit reaches its peak, the L–D curve exhibits another significant change, and the plastic deformation zone stabilizes, indicating that the collapse process has completed.

The methodology presented in this paper for dividing the risk threshold of the stability of rockeries is superior. In previous studies, the risk levels of crack expansion in rockeries were classified by calculating the stress intensity factor. However, there was no clear indication of the importance and method of obtaining the key node “formation of a through crack,” but in this paper, we proved that through the L-D curve, this key node can be effectively extracted and used as the risk threshold for the overall structural stability risk of the rockery, which transitions from the linear expansion stage of cracks to the overall structural damage stage. Moreover, by introducing the elastoplastic constitutive model, the plastic deformation area in the overall structural stability change of the rockery can be effectively observed. Furthermore, the basis for dividing the risk stages of the overall structural stability of the rockery through stress-strain and L-D curves is provided. Hence, the garden managers can predict the areas that need to be protected in each stage of the rockery (from the crack level to the structural level) through numerical simulation.

By comparing numerical simulation results for different dilatancy angle values, it was determined that the stability simulation results for the rockery structure were largely consistent within the range of $0\le \psi \le \varphi$. Only in the extreme cases of $\psi =0^\circ$ and $\psi =35^\circ$ did the curves show significant differences. Based on these observations, a dilatancy angle of $\psi =15^\circ$ was selected for the study.

According to our monitoring campaigns conducted in 2021 and 2025, the rockery remains in Ⅱ Stage of linear crack propagation. The principal contribution of this study is to address the limitations of previous work that focused solely on individual crack behavior by establishing a comprehensive numerical simulation framework for assessing the structural stability risks of classical garden rockeries. This framework has practical engineering value for the preventive conservation of rockery cultural heritage.

In terms of the study object, the proposed methodology has so far been applied only to the SRMA in He Garden, Yangzhou, as a case study, providing a preliminary validation of its engineering applicability. Future work will further refine the simulation framework and extend the case studies to deterioration mechanisms beyond foundation settlement that may threaten rockery integrity. In addition, regarding the establishment of a comprehensive risk prevention and conservation system for rockery structural stability, the present study focuses primarily on the numerical simulation of structural stability risk thresholds. Future research should also address the development of monitoring and early-warning systems for classical garden rockeries, for example, through the deployment of damage-monitoring instruments across representative rockery sites. At the methodological level, future studies will further examine the influence of modeling assumptions, such as mesh size, boundary conditions, and constitutive relations, on the quantified threshold values for structural stability risk, thereby enhancing the practical applicability of this research. In future work, laboratory-scale experiments could be conducted to further validate the numerical simulation results, thereby enhancing the practical applicability of the proposed method.