# Rheological Analysis of Soft Rock Unloading Combined with Finite Element Analysis Based on H-K Constitutive Model

Authors: Tengge Sun; Houli Fu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The rheological and mechanical features of soft rock in the unloading environment differ in terms of the timeliness under the conditions of loaded rock. For the purpose of analyzing the stability of soft rock in the rheological process accurately, the finite element analysis algorithm based on the H-K constitutive model is used to analyze its nonlinear rheological features and adjust the changes in the surrounding rock stress in a timely manner during the excavation of soft rock. In accordance with the rheological acceleration features of soft rock at the rheological stage, the damage produced at the rheological acceleration stage of soft rock is modified by using the finite element analysis (FEA) method. Through fitting the algorithm to the analysis, it can be concluded that the application of the finite element analysis method based on the H-K constitutive model can present the image of the rheological nonlinearity of soft rock accurately. Finally, the results of the experimental research indicate that through the application of the finite element analysis algorithm based on the H-K constitutive model in the process of rheological analysis of soft rock, the advantageous performance of the rheological and mechanical properties of soft rock can be presented more evidently. However, with the decrease in the surrounding air pressure, the deviatoric stress of the soft rock will become increasingly larger. In the case of an axial change, the corresponding rate of deformation will increase in the range of 12%–35%. The rate of deformation in the lateral soft rock will fluctuate in the range of 9%–28% in accordance with the corresponding growth rate. This suggests that soft rock unloading can reduce the constraint of particles sliding inside the soft rock, which will further increase the rate of deformation change.


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## Body

## 1. Introduction

The rheological properties of soft rocks are an essential feature in rock engineering that cannot be ignored. There are also abundant and complete rheological data and results of rock materials at present [1, 2], and a large number of rheological analyses have been carried out. However, the majority of the viscoelastic and viscoelastic models cannot be used to describe the rheological acceleration phases and limit the application of the models described above [3, 4]. Thus, many scholars have been focused on resolving this problem. Through the introduction of an integrated rheological model constructed using nonlinear viscous damping and viscous damping, three types of rheological deformations can be described at the same time [5, 6]. In general, coal formations contain a large number of soft rocks, and the damage to soft rock engineering is actually the process of the destruction in the internal structure of the rock under the action of external forces. Sudden landslides and rock bursts occur on soft rock slopes, which is often a phenomenon of unstable conditions in soft rock rheological works. In the traditional mineral area mining, the rheological force problems arising in soft rock engineering exposed are often more serious with the increasing mining depth and the expanding mining area. In foreign countries, Grgiggs began to analyze the rheology of different types of soft rocks such as limestone rock, foliage rock, and sandstone in 1930; the results of their experimental analysis indicated that sandstone and siltstone could destroy about 12.6%–81% of the soft rock load under external load, and evident rheological deformation would occur at this point. Li Yong-sheng carried out a compression rheological test for different types of soft rocks such as red sandstone, siltstone, and mudstone to explore the features of the changes in rheological velocity and strain force of the rock materials. In China, Chen Qu carried out uniaxial and triaxial compression-in-formation tests on soft rocks, analyzed the relationship between the deformation that occurred in soft rocks, deformation rate, and time variation, and then put forward the variation of elastic simulated volume and long-term damage volume based on long-term damage amount. Liu Guangting carried out a test of the soft rock rheology under the action of multiaxial strain force, in which the effect of the soft rock rheological state on the rock formation under the action of dryness, full water, and lateral pressure was taken into full consideration. In the process of in-depth analysis of soft rock mechanics, the phenomena and concepts of the rheological process in soft rock engineering mechanics comply with the emergence [7, 8]. Previous studies have indicated that in accordance with the different dimensions of rock testing, such as the change of parameters, the optimization of the constructed model is conducive to an in-depth discussion of the problems that arise throughout the process of soft rock rheology to further investigate the relationship between the rheology of soft rock and stress levels. Thus, in combination with the problem between total strain force and distortion in soft rock, the stability of the soft rock rheology process can be obtained through the detailed analysis of load reduction rheology. In addition, the analytical model parameters and corresponding soft rock stability that occur in the soft rock rheology process can be further explored. Rheological analysis of soft rock unloading is a relatively complex process due to the constant changes in the production of soft rock plant. As a result, it is necessary to consider the overall equilibrium, continuity, uncertainty, and stochasticity of different aspects to ensure effective identification and optimization. The finite element analysis based on the H-K constitutive model is extensively used in various fields. Through the rheological analysis of soft rock unloading, the variation of engineering in soft rock pits can be described [9, 10].The rheological state of soft rocks can be analyzed using damage variables. In particular, where the rheological velocities are constantly changing, the following observations can be made based on different methods in the rheological analysis of soft rocks. According to one method, in the case that soft rock damage threshold in the strain force is constantly changing, the damage force of the rock will be generated immediately as well. In another method, under the premise that the damage threshold is not taken into consideration in the common view, when the soft rock goes through the first and second stages, it can lead to the damage variable in a state of constant acceleration. In other words, under the condition of the existence of the rheological threshold value of soft rock, the rheological mechanics are analyzed in detail by exploring the rheological curve of soft rock and using the damage and hard change of the rock in this paper.


## 2. Finite Element Analysis Algorithm Based on the H-K Constitutive Model

This paper uses the H-K constitutive model finite element analysis method, to set the parameters of the function mainly by using the step length, and follows a certain process finite element gray value set in any direction of the soft rock unloading rheology. The main component of dispersion is the finite element analysis value of this process. Using this step, the corresponding step function value can be obtained, and the finite element analysis curve of the H-K constitutive model can be obtained through continuous calculation. The finite element analysis and calculation process of the H-K constitutive model are shown in Figure1, where the solid line and the dashed line are the numerical sets corresponding to the finite element in step h at 1 and 2.Figure 1
Finite element analysis process of H-K constitutive model.With regard to the computational object, the model is mainly represented by using the square, while at the same time the lattice model in the finite element surface based on the H-K intrinsic model is set toS=T1,T2,…,Tk, which Ti indicates that N random points can be generated on the surface based on the H-K intrinsic model. The process of assigning the corresponding values to the random points can be roughly divided into two steps as follows: the finite elements Tj and Tj are randomly selected, and then, the random points on the surface are taken accordingly. Based on the H-K principal structure model, if the probability of the distribution of random points in this area is proportional to the area in this region, then the following process is implemented.First of all, a finite elementT=P1,P2,P3 is defined, the area of which can be obtained according to the equation as follows:(1)AT=P2−P1×P3−P12.In this way, the surface area of the H-K intrinsic model can be expressed as follows:(2)AS=∑i=1kATi.Thus, the probability of the randomly selected finite elementT can be expressed as follows:(3)pTj=ATjAS.From Figure2, it can be observed that the interval [0, 1] is divided into k segments, in which the j-th segment coincides with pTj in the numerical value. If a random number is generated between [0, 1], then the index number j corresponding to the finite element can be obtained, as shown in the following.Figure 2
Schematic diagram of the randomly selected finite elements.After the finite element is selected, it can be generated randomly on the finite element surface, as shown in Figure3, and the numbers r1 and r2 can be generated in the interval of [0, 1] accordingly. Thus, the location of the random point P can be expressed as follows:(4)P=1−r1P1+r11−r2P2+r1r2P3.Figure 3
Schematic diagram of the randomly generated points on the finite element.The normal vector of each point should be recorded while it is sampled, and the normal vector of each point should be the same as the normal vector of the finite element where the point is located. However, it should be noted that in three cases, that is, whenr1 = 0, r1 = 1, r2 = 0, r1 = 1, r2 = 1, the sampled point is exactly the vertex of the finite element shape, and when r1 = 1, r2 ∈ (0, 1), r2 = 0, r1 (variable 0, 1), r1, r1, r1, r1, r1 ∈ 4 (0, 1) are exactly sampled on the finite element surface. At this point, the normal vector of the sampled point can be calculated in accordance with the weighted mean of the normal vectors with the finite element shapes connected to the point in the area. It is assumed that the point P has n finite element shapes. The normal vector of the finite element shape is denoted as nj, and the area is denoted as Ajj=1,2,⋯,n, and then, the normal vector of the sampled point P can be obtained as follows:(5)n=∑j=1nAjnj∑j=1nAj.According to the actual situation of the specific soft rock, a simulation model of the in-pit visual production system of soft rock is constructed, and the simulation operation is carried out after the model is verified and confirmed, and the production process of the soft rock in the pit is dynamically seen. In addition, the operation status of each stage of the soft rock bioflow system can be analyzed in real time [11, 12]. The development logic program of the dynamic optimization policy decision system for 3D dynamic visualization of engineering under the soft rock is shown below. Information on the soft rock production system is collected for specific soft rock, including parameters related to the tunnel formation system [13, 14]. The influencing factors of soft rock production are analyzed, the boundary of the soft rock logistics system is defined, and subsystems are divided according to the function of the soft rock logistics process. On the basis of a large number of field investigations and statistical analyses, the statistical rules and parameters of each part of the soft rock system are determined. Through the mine layout system and production system of soft rock, the connection and conversion relationship between the various subsystems of soft rock production can be found out, the integration ability of soft rock production system and logistics system equipment can be established, and cost control system model for mining personnel under a fully visualized soft rock can be set up.


### 2.1. Rheological Analysis of Soft Rock Unloading

The rheological analysis of soft rock unloading is mainly to analyze the different mining areas of mining soft rock. The difference in rheological analysis data depends on the different physical properties of soft rock, mainly the color, gloss, and fracture characteristics of soft rock. Among them, there is a big difference in the unloading corresponding to the coal used and the floor rock [15]. The current mining mainly includes bituminous coal and anthracite. The floor rock mainly includes sandstone and mudstone (Figure 4).Figure 4
Characteristic unloading rheology of some mining samples. (a) Bituminous coal. (b) Anthracite. (c) Sandstone. (d) Mudstone.
(a)(b)(c)(d)For the irregularity and randomness of soft rock characteristics, according to the rheological analysis of soft rock, the target feature dimension to be extracted is moderate. The text that comprehensively considers these factors is based on the concept of simulated annealing according to statistical methods and proposes a soft rock unloading rheological analysis method based on the H-K constitutive model finite element analysis.In the selection process of soft rock unloading rheological analysis, finite element analysis is used to complete the soft rock unloading rheological analysis. According to the combination of finite element analysis, the characteristics of unloading rheology of soft rock are expressed by vector as follows:(6)lεg=1−ρlεg−1+γfχig,where f represents the adaptive function corresponding to the finite element eigenvector χi of soft rock unloading rheological analysis. γχig represents the data optimization of the media engineering product corresponding to the εth unloading rheological analysis in the actual application process.The expression for the finite element analysisπp in soft rock unloading rheological analysis II is(7)Acuπp=NMIπp,π∗,where πp and πq represent the finite element analysis of soft rock unloading rheology. If less information is shared with the base finite element for soft rock unloading rheological analysis, the base finite element is less accurate, and otherwise, vice versa.First, the labeled training set is used to train each type of H-K constitutive model. Let the type of soft rock unloading rheological analysis bek=1,2,…,K, and the model parameters of each type are λk. The maximum likelihood criterion is used to maximize the posterior probability, while considering the Bayesian formula, and then(8)k¯=argmax1≤k≤KPλk|O=argmax1≤k≤KPO|λkPλkPO.If different kinds of prior probabilitiesPλk are consistent, the main reason is that there is no relationship between P(O) and k, so the following expression can be used:(9)k¯=argmax1≤k≤KPO|λk.Because the value ofPO|λk is small, different values will change during the calculation process, so it is generally necessary to take its logarithmic value.Based on the accuracy and diversity of soft rock unloading rheological analysis, a comprehensive evaluation standard based on soft rock unloading rheological analysis is defined, including(10)Evalπp=λAcuπp+1−λDivπp.In the formula,λ∈0,1, the correctness and diversity of soft rock unloading rheological analysis are important degrees in the comprehensive evaluation criteria.Formula (10) is based on the diversity of the basic finite element analysis of soft rock unloading rheological analysis Divπp, and each soft rock unloading rheological analysis basic finite element analysis algorithm is selected as the optimization basis to calculate the probability proπp for FEA. The calculation formula is as follows:(11)proπp=Divπp∑p=1BDivπp.Theproπp calculation results of the benchmark equation (11) are used for random selection using roulette, based on the soft rock unloading rheological analysis, the soft rock unloading rheological analysis fusion finite element is obtained.If there areN data in soft rock unloading rheological analysis, and the attribute dimension of the data is represented by w, the position and velocity matrix of soft rock data are represented by N.(12)δ11δ12⋯δ1wδ11δ12⋯δ2w⋯δτ1δτ2⋯δτwδ11⋯δτwf.Distributed overall fitness for soft rock unloading rheological analysis is as follows:(13)ψ2=−∑η=1vfi−favgf,where v represents the number of soft rocks unloading rheological finite element analysis. The optimization of position and velocity for soft rock unloading is updated according to expressions equations (14) and (15).(14)vt=ωvt−1+κ1×rand1pbest−xt−1+κ2×rand2gbest−xt−1,(15)xt=xt−1+vt.The results obtained according to the soft rock unloading rheological analysis method are strongly related to the initial parameters. The transition matrix needs to be initialized to determine whether 0 is in the transition matrix or 0 is obtained after the iterative operation. The observation sequence is defined asO=o1o2⋯oT.(16)PO|λ≥Pλ|O.The method of computingPO|λ is the forward-backward method. For the parameter λ and state i of the H-K constitutive model technique, the forward probability αtiis defined as(17)αti=Po1o2⋯oT,qt=i|λ.That is,αti is the probability that the parameter A produces the sequence o1o2⋯ot and the probability ot of the state at time t.


## 2.1. Rheological Analysis of Soft Rock Unloading

The rheological analysis of soft rock unloading is mainly to analyze the different mining areas of mining soft rock. The difference in rheological analysis data depends on the different physical properties of soft rock, mainly the color, gloss, and fracture characteristics of soft rock. Among them, there is a big difference in the unloading corresponding to the coal used and the floor rock [15]. The current mining mainly includes bituminous coal and anthracite. The floor rock mainly includes sandstone and mudstone (Figure 4).Figure 4
Characteristic unloading rheology of some mining samples. (a) Bituminous coal. (b) Anthracite. (c) Sandstone. (d) Mudstone.
(a)(b)(c)(d)For the irregularity and randomness of soft rock characteristics, according to the rheological analysis of soft rock, the target feature dimension to be extracted is moderate. The text that comprehensively considers these factors is based on the concept of simulated annealing according to statistical methods and proposes a soft rock unloading rheological analysis method based on the H-K constitutive model finite element analysis.In the selection process of soft rock unloading rheological analysis, finite element analysis is used to complete the soft rock unloading rheological analysis. According to the combination of finite element analysis, the characteristics of unloading rheology of soft rock are expressed by vector as follows:(6)lεg=1−ρlεg−1+γfχig,where f represents the adaptive function corresponding to the finite element eigenvector χi of soft rock unloading rheological analysis. γχig represents the data optimization of the media engineering product corresponding to the εth unloading rheological analysis in the actual application process.The expression for the finite element analysisπp in soft rock unloading rheological analysis II is(7)Acuπp=NMIπp,π∗,where πp and πq represent the finite element analysis of soft rock unloading rheology. If less information is shared with the base finite element for soft rock unloading rheological analysis, the base finite element is less accurate, and otherwise, vice versa.First, the labeled training set is used to train each type of H-K constitutive model. Let the type of soft rock unloading rheological analysis bek=1,2,…,K, and the model parameters of each type are λk. The maximum likelihood criterion is used to maximize the posterior probability, while considering the Bayesian formula, and then(8)k¯=argmax1≤k≤KPλk|O=argmax1≤k≤KPO|λkPλkPO.If different kinds of prior probabilitiesPλk are consistent, the main reason is that there is no relationship between P(O) and k, so the following expression can be used:(9)k¯=argmax1≤k≤KPO|λk.Because the value ofPO|λk is small, different values will change during the calculation process, so it is generally necessary to take its logarithmic value.Based on the accuracy and diversity of soft rock unloading rheological analysis, a comprehensive evaluation standard based on soft rock unloading rheological analysis is defined, including(10)Evalπp=λAcuπp+1−λDivπp.In the formula,λ∈0,1, the correctness and diversity of soft rock unloading rheological analysis are important degrees in the comprehensive evaluation criteria.Formula (10) is based on the diversity of the basic finite element analysis of soft rock unloading rheological analysis Divπp, and each soft rock unloading rheological analysis basic finite element analysis algorithm is selected as the optimization basis to calculate the probability proπp for FEA. The calculation formula is as follows:(11)proπp=Divπp∑p=1BDivπp.Theproπp calculation results of the benchmark equation (11) are used for random selection using roulette, based on the soft rock unloading rheological analysis, the soft rock unloading rheological analysis fusion finite element is obtained.If there areN data in soft rock unloading rheological analysis, and the attribute dimension of the data is represented by w, the position and velocity matrix of soft rock data are represented by N.(12)δ11δ12⋯δ1wδ11δ12⋯δ2w⋯δτ1δτ2⋯δτwδ11⋯δτwf.Distributed overall fitness for soft rock unloading rheological analysis is as follows:(13)ψ2=−∑η=1vfi−favgf,where v represents the number of soft rocks unloading rheological finite element analysis. The optimization of position and velocity for soft rock unloading is updated according to expressions equations (14) and (15).(14)vt=ωvt−1+κ1×rand1pbest−xt−1+κ2×rand2gbest−xt−1,(15)xt=xt−1+vt.The results obtained according to the soft rock unloading rheological analysis method are strongly related to the initial parameters. The transition matrix needs to be initialized to determine whether 0 is in the transition matrix or 0 is obtained after the iterative operation. The observation sequence is defined asO=o1o2⋯oT.(16)PO|λ≥Pλ|O.The method of computingPO|λ is the forward-backward method. For the parameter λ and state i of the H-K constitutive model technique, the forward probability αtiis defined as(17)αti=Po1o2⋯oT,qt=i|λ.That is,αti is the probability that the parameter A produces the sequence o1o2⋯ot and the probability ot of the state at time t.


## 3. Experiment and Result Analysis

Mudstone and oil-containing mudstone from a mine are selected for this experiment as experimental materials in this paper. MTS servo rigid machine is used to test the pressure of the rock shape. At this time, the pressure strength is set to 7–15 Mpa. Soft rock rheological tests were carried out in this condition. The soft rock rheological test conducted in this paper was carried out in gravity-loaded rheological mechanics. According to the needs of the experimental test, the rheological curve is tested, and three typical strain force levels are selected as the research targets. In the initial test, the strain force level is set to 8.6 MPa. The carried soft rock test rheology performed in this stage is still in a decaying state, the pressure of the second strain level is set to 9.8 MPa, the third strain force is set to 12.4 MPa, and the third level of stress is set to 12.4 MPa. When the rheological deformation of the sample under this load lasts about 6 h, the fracture curve contains three rheological stages (Figure5).Figure 5
C-3 rheological curve of samples.The rheological experiments of soft rock under different levels of unloading under constant axial pressure and different levels of unloading were carried out for mudstone and mudstone containing oil. According to the experimental results, the lateral rheology of soft rock under different unloading environments will change greatly, and there are also significant changes in different directions. The rheological analysis of soft rock is also quite different at high constant unloading, but the lateral rheology of soft rock will vary greatly depending on the continuous drop in unloading pressure. The samples taken make the soft rock unloading also different in the case of soft rock fracture unloading. According to the experimental test results obtained from the detailed analysis of the soft rock, the HK constitutive model can be used to accurately reflect the decaying rheological and steady-state changes of the rheological analysis curve of the soft rock under different unloading environments. But the acceleration of soft rock under nonconstant unloading cannot be accounted for. Using the finite element analysis of the H-K constitutive model proposed in this paper, and fitting the rheological results of the soft rock, the nonlinear rheological parameters of the soft rock can be obtained. By comparing the fitting curves of soft rock, it can be concluded that the obtained fitting curve is in good agreement with the test curve, indicating that the rheological model can accurately reflect the characteristics of soft rock unloading rheology at different stages. Among them, the relationship between soft rock unloading and lateral rheological rate and time is shown in Figure6.Figure 6
The corresponding relationships between unloading and lateral rheological rate and time.The unloading rheological features of the sample soft rock can be tested under different strains in accordance with the designed rheological test of soft rock unloading. The axial strains thus obtained are 66 MPa and 80 MPa, respectively. The rheological curves of the soft rock under the initial surrounding pressure of 12 MPa and 18 MPa, respectively, are shown in Figures7 and 8 as the following. It can be concluded from the figures that the blue curve marked with 10 MPa indicates the constant pressure at the rheological stage with different levels of unloading, and the curve with 48 h indicated as follows stands for the duration with the continuous use at different levels of unloading rheological conditions. The lateral strain in the figure is positive when it points to the inside of the specimen horizontally and negative when it points to the outside of the specimen horizontally.Figure 7
Axial stressσ = 66 MPa progressive unloading confining pressure rheological curve.Figure 8
Axial stressσ = 80 MPa progressive unloading confining pressure rheological curve.Figure9 is a graph showing the variation curve of mining analysis rate for different unloading quantities under the same test conditions. Here, parameter characteristic error = 0.035, characteristic similarity = 0.28, K = 9. It can be seen from Figure 9 that with the gradual increase in the number of unloading, the analysis rate of the soft rock industry tends to decrease. 70 is taken as the branch point for the number of unloading, and the analysis rate before 70 is relatively stable. The simulated finite element determines that the larger the dimensional unloading quantity of the soft rock unloading rheological vector in the mining process, the higher the time and space cost of the execution with the method, which can be known from the data in the last column of Table 1. As the number of unloading increases, so does the mining analysis time. If the simulated finite element is too small, the dimension of the eigenvector is too small, and it is easy to be buried by the unloading rheological noise.Figure 9
Relationship between unloading quantity and analysis rate.Table 1
Analysis rate under different parameter selection.
Unloading quantityFeature error (g)Feature similarity (a)KAnalysis rate of test sampleNumber of wrong samplesParsing time (s)420.0380.390.9615422.909132400.0380.390.9423132.796078600.0270.19110.8653873.854129800.0420.4170.76923125.741192


## 4. Conclusions

After the detailed analysis of the rheological process of soft rock, it is found that the use of triaxial unloading to test the graded unloading pressure of soft rock can not only identify the features of axial change and lateral strain of the rock at the unloading stage but also take the factors affecting the damage into comprehensive account for different damage acceleration periods. In accordance with the degree of damage in the soft rock and the finite element analysis function, the analysis process of rheological features of soft rock under reduced load can be established accordingly. Based on the total stress and strain relationship of soft rock samples, different parameters can be optimized for various strain force levels. Finally, the stability analysis of soft rock in the rheological process can be implemented. The experiment has verified that the parameters of the H-K constitutive model can be optimized reasonably through processing by the finite element analysis method based on the H-K constitutive model so that the model can be used to describe the features of soft rock at the decay creep stage and the steady-state creep stage properly. The lateral rheological features of soft rock in the unloading rheological environment are more evident than those in the axial direction, and the corresponding lateral expansion effect is significant, which is apparently different from the results of the traditional triaxial rheological test. Thus, it can be concluded that the lateral rheological deformation of soft rock under the unloading rheological conditions is faster than that in the axial direction.



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