Elastoplastic Analysis of Circular Opening Based on a New Strain-Softening Constitutive Model and Its Engineering Application in Hydraulic Fracturing

Constitutive effect is extremely important for the research of the mechanical behavior of surrounding rock in hydraulic fracturing engineering. In this paper, based on the triaxial test results, a new elastic-peak plastic-softening-fracture constitutive model (EPSFM) is proposed by considering the plastic bearing behavior of the rock mass. ,en, the closed-form solution of a circular opening is deduced with the nonassociated flow rule under the cavity expansion state. Meanwhile, the parameters of the loadbearing coefficient and brittles coefficient are introduced to describe the plastic bearing capacity and strain-softening degrees of rock masses. When the above two parameters take different values, the new solution of EPSFM can be transformed into a series of traditional solutions obtained based on the elastic-perfectly plastic model (EPM), elastic-brittle plastic model (EBM), elasticstrain-softening model (ESM), and elastic-peak plastic-brittle plastic model (EPBM). ,erefore, it can be applied to a wider range of rock masses. In addition, the correctness of the solution is validated by comparing with the traditional solutions. ,e effect of constitutive relation and parameters on the mechanical response of rock mass is also discussed in detail. ,e research results show that the fracture zone radii of circular opening presents the characteristic of EBM > EPBM > ESM > EPSFM; otherwise, it is on the contrast for the critical hydraulic pressure at the softening-fracture zone interface; the postpeak failure radii show a linear decrease with the increase of load-bearing coefficients or a nonlinear increase with the increasing brittleness coefficient.,is study indicates that the rock mass with a certain plastic bearing capacity is more difficult to be cracked by hydraulic fracturing; the higher the strain-softening degree of rock mass is, the easier it is to be cracked. From a practical point of view, it provides very important theoretical values for determining the fracture range of the borehole and providing a design value of the minimum pumping pressure in hydraulic fracturing engineering.


Introduction
e stresses and plastic zone distribution of the circular opening are extremely important for evaluating the tunnel stability and hydraulic fracturing effect in underground engineering.However, the mechanical response of the surrounding rock is closely related to rock mass lithology.In fact, the constitutive relation of different lithology rock masses generally shows obviously diversity and complexity under the effect of internal fissures, joints, components, and external environment.erefore, it is difficult to choose a certain simplified constitutive equation to study this problem [1][2][3][4].In the early stage, the elastoplastic analysis of the circular opening was firstly investigated by Fenner and then corrected by Kastner.However, they regarded the rock mass as the elastic-perfectly plastic material (EPM).It is obviously not reasonable for the brittle plastic and strainsoftening rock masses.In recent years, as shown in Figure 1, many studies have been carried out by using the elasticbrittle plastic model (EBM), elastic-strain-softening model (ESM), and elastic-peak plastic-brittle plastic model (EPBM) with the associated and nonassociated flow rule [5][6][7][8][9].
Nevertheless, each constitutive model has its own application scope.e EBM applies only to the poor-quality rock mass, while ESM is suitable for average-quality rock mass and EPM for high-quality rock mass [10][11][12][13][14][15].In addition, EPBM is suitable for the brittle rock masses with a certain plastic bearing capacity [16][17][18].In fact, many strainsoftening rock masses also show a certain plastic bearing behavior after load peak.
As shown in Figure 2, the silty mudstone and marble were, respectively, taken from the Yangzhuang coal mine, Wushan, and Ya'an area of China.A large number of rock masses rstly showed the strain-softening characteristics after the peak plastic zone and then entered the fracture stage.erefore, according to the total stress-strain curve, the rock mass approximately experienced four stages in the process of the triaxial test.
at is elastic, peak plastic, softening, and fracture stages.en, the elastic-peak plasticsoftening-facture constitutive model (EPSFM) was proposed in this paper and then applied to the engineering practice.
Most of the above investigations focus on the compression problems of the circular opening.However, the problems of cavity expansion have also attracted much attention in geotechnical engineering with the application to wellbore instability, coal-gas exploration, and hydraulic fracturing [3,4,19,20].Actually, the circular opening expansion is mainly applied in the hydraulic fracturing, which has been widely used in the hard roof fracturing, coalbed methane extraction, and in situ stress measurement.
Since the early 1950s, numerous analytical solutions to the circular opening expansion have been studied in materials and geotechnical engineering.For instance, Gibson and Anderson applied the cavity expansion theory to in situ measurement of soil properties with the pressure meter test [19].Li et al. obtained the closed-form solution for the hydraulic fracturing borehole, which was only applied to hard rock, depending on the elastic fracture theory [20].Bishop and Mott derived the quasistatic expansion equations of cylindrical cavities in an in nite medium and applied it to the materials processing [21].Cheng discussed the errors arising from the assumption of small displacement around the cavity with no volume change in the plastic zone and modi ed Kastner's formula for cylindrical cavity contraction and expansion in the Mohr-Coulomb rock masses [22].Li et al. derived the stresses and plastic zone radii of the circular borehole excavated in the strain-softening coal seam by considering contraction and expansion problems [23].
In this paper, based on the triaxial test results, a new elastic-peak plastic-softening-fracture constitutive model (EPSFM) is rstly proposed and then used to study the borehole expansion problems in underground engineering.Furthermore, the validity of this solution is veri ed by comparing with a series of traditional solutions based on EBM, EPM, ESM, and EPBM.Finally, the in uences of the parameters and constitutive models on the mechanic responses of rock mass are discussed in detail.

Establishment of EPSFM.
As shown in Figure 3, a borehole with the inner radius R 0 drilled in an in nite, isotropic, and homogeneous EPSFM rock masses is subjected to an inner hydraulic pressure p in at r R 0 and hydrostatic pressure p 0 at in nite boundary.Originally, the surrounding rock is in the elastic state.As p in gradually increase, the peak plastic rstly occurs around the borehole when p in is more than the initial yield stresses.e stage is not an in nite extension whose range should be restricted by some factors.In this paper, assuming the plastic shear strain increment of the peak plastic zone reaches a certain value, the surrounding rock of the borehole will enter the softening stage in which the strength parameters gradually decrease.Until a residual value is reached, the surrounding rocks start to enter the fracture stage.Finally, it will have four zones around the borehole that is elastic zone, peak plastic zone, softening zone, and fracture zone.Meanwhile, the radius of peak plastic, softening, and fracture zones are, respectively, denoted as R 3 , R 2 , and R 1 .e mechanical model should satisfy the following assumption conditions: (i) e borehole is drilled in an in nite geological body, so the problem can be regarded as a plane strain problem (ii) e total strain of the postpeak failure zone only consists of plastic strain and the e ect of elastic strain is ignored For axisymmetric plane strain problems, when p in > p 0 , the hoop stress σ θ and radial stress σ r are, respectively, the minimum and maximum principal stresses; ε θ and ε r are the minimum and maximum principal strains, respectively  Advances in Civil Engineering [20,21].Supposing that the rock mass satis es the linear Mohr-Coulomb yield criteria, the stress-strain relation at any postpeak stages can be expressed as follows [22,23]: where σ cc and σ RR are, respectively, the initial uniaxial compressive strength and residual compressive strength, σ cc 2c 3 cos φ/(1 − sin φ), σ RR 2c 1 cos φ/(1 − sin φ); c 3 and c 1 are, respectively, initial and residual cohesion of rock mass; and K is a constant which is related to the strength parameter φ, K (1 + sin φ)/(1 − sin φ).

Basic Equations and Boundary Condition.
For the axisymmetric plane strain problems, the equilibrium di erential equation in the "i" zone can be expressed as follows (ignoring the body force) [7,9]: where σ ri and σ θi are the radial and hoop stresses in the "i" zone, respectively.e subscript symbol "i" represents di erent zones of surrounding rock, which can be replaced by the numbers "0, 1, 2, and 3." Based on the supposition of small deformation, the geometric equation for the axisymmetric plane strain problem can be denoted as [12,13] ε ri du ri dr ,  12 and ε A 32 are, respectively, the maximum and minimum principal strains at "A" point; ε B 12 and ε B 32 are, respectively, the maximum and minimum principal strains at "B" point).

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where ε ri and ε θi are the radial and hoop strains in the "i" zone, respectively, and u ri represents the radial displacement.
Supposing that the volume of rock mass is changing, the relationship between hoop strain ε θi and radial strain ε ri can be established by adopting a nonassociated flow rule and small strain theory as follows [22,23]: where Both the radial stress and radial displacement should be continuous at the elastic-peak plastic, peak plastic-softening, and softening-fracture zone interfaces.
erefore, the boundary conditions around the borehole can be summarized as

Stresses and Displacement of Elastic Zone.
Based on the elasticity theory, the solution of a thick-walled cylinder under hydrostatic pressure can be easily obtained.e stresses and displacement for the elastic zone can be expressed as [18,23] where p exp min is the minimum critical inner hydraulic pressure at elastic-peak plastic zone interface; A 0 � (1 + ]) (p 0 − p exp min )/E; and E and ] are Young's modulus and Poisson's ratio.
For the borehole expansion problem, both radial and circumferential stresses satisfy the Mohr-Coulomb yield criteria at the elastic-peak plastic zone interface.Hence, the parameters p exp min can be easily deduced by substituting equations ( 8) and ( 9) into equation (1) as follows: Considering the boundary condition (u r(i−1) ) r�R i � (u ri ) r�R i by equation ( 7), the radial displacement and strains in the postpeak failure zones can be easily deduced based on the small deformation supposition and volume expansion assumption by substituting equation ( 5) into equation ( 6).
e calculation results are shown in Table 1.

Stresses Distribution of Peak Plastic and Fracture Zones.
When the inner hydraulic pressure remains at a certain value, the surrounding rock of the borehole is in the stress equilibrium state in the peak plastic and fracture zones.erefore, the principal stresses should satisfy the equations ( 1) and ( 4) in the peak plastic zone or equations ( 3) and ( 4) in the fracture zone.
In the above two zones, the equilibrium differential equation can be rewritten by substituting equation (1) or equation (3) into equation ( 4) as follows: where σ jj equals to σ cc in the peak plastic zone or equals to σ RR in the fracture zone.Solving equation ( 12), the stresses in the peak plastic zone can be obtained by combining with the boundary condition (σ r2 ) r�R 3 � p exp min : Meanwhile, the stresses in the fracture zone can be also easily deduced by considering (σ r0 ) r�R 0 � p in :

Stresses Distribution of Softening Zone.
By considering the condition σ c (ε r1 ) � σ cc at r � R 2 and dσ c � λd(ε r1 ), the compressive strength in the softening zone can be obtained as where α � λ/E, which can be defined as a brittleness coefficient and represents the strain-softening degree of rock mass and λ may be called the strain-softening modulus.
Introducing equations ( 2) and ( 15) into equation ( 4), the equilibrium differential equation in the softening zone can be deduced as e radial stress at the peak plastic-softening interface must be coincided; thus it can be obtained by solving equation ( 16) and considering the boundary condition σ r1 � σ r2 at r � R 2 : en, by introducing equations ( 15) and ( 17) into equation ( 2), the hoop stress is As the inner hydraulic pressure p in gradually increasing, the surrounding rock of the borehole will experience four stages.at is elastic stage, elastic-peak plastic stage, elastic-peak plastic-softening stage, and elastic-peak plastic-softeningfracture stage.

Elastic-Peak Plastic Stage.
In this stage, the surrounding rock of the borehole only consists of elastic and peak plastic zones.
e range of the peak plastic zone gradually increases with the increase of the inner hydraulic pressure.As shown in Figure 3, when the plastic shear strain increment of the peak plastic zone increases to a particular value, the rock mass will reach the maximum peak plastic state in which the softening zone is just not arisen.Hence, we can define a load-bearing coefficient Δc which can be calculated by the difference of the plastic shear strain in section "AB" of Figure 3 to describe the plastic bearing capacity of rock mass.e parameter Δc can be expressed as follows: where c B and c A represent the plastic shear strain at points "B" and "A," respectively.ey can easily be determined by the experiment.Hence, the radius of the peak plastic zone can be obtained as Presently, the middle critical inner hydraulic pressure p exp mid at the peak plastic-softening zone interface can be solved by introducing equation (20) into equation ( 13):

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3.4.2.Elastic-Peak Plastic-Softening Stage.When R 3 /R 0 > T, the softening zone appears.If assuming that the surrounding rock is in the critical state where the fracture zone is not yet arisen, equation ( 20) can be rewritten as By integrating equation ( 15), according to σ c (ε r1 ) � σ RR at r � R 0 , the relationship between R 3 and R 2 can be obtained as follows: en, by substituting equation ( 22) into equation ( 23), the softening zone radii can be expressed as At this state, introducing equations ( 22) and (24) into equation (17), the maximum critical inner hydraulic pressure p exp max can be calculated as follows: (25) it means that the rock mass has entered into the fracture stage.According to equations ( 22) and ( 24), the relationship of R 3 � TR 2 and R 2 � tR 1 is easily deduced.In addition, the radial stress should be consistent at the softening-fracture zone interface.erefore, we can obtain Integrating equation (26), the fracture zone radius can be obtained as follows: ) . ( en, the radius of peak plastic and softening zones can also be calculated by introducing equation ( 27) into R 3 � TR 2 and R 2 � tR 1 .

Discussion and Transformation with Traditional Model.
e new closed-form solution based on the EPSFM can be degenerated for different traditional solutions based on the EPM, EBM, ESM, and EPBM in a particular situation.For instance, only when Δc � 0, the results of EPSFM can be translated into the results of ESM [23]; when Δc � 0, α ⟶ ∞, the EPSFM converts to the EBM; if assuming that Δc � 0 and α � 0, the EPSFM solution degenerates for EPM solution [22]; only when α ⟶ ∞, the EPSFM solution changes to the EPBM solution.It includes not only the traditional results but also a series of new results compared with the traditional ones.Hence, it can be regarded as a unified analytical solution.In other words, the new closedform solution can generate a broad range of theoretical and practical values in circular opening expansion engineering, especially in the hydraulic fracturing.
When load-bearing coefficient Δc and brittleness coefficient α take special values, the new analytical solution will degenerate for a series of traditional solutions.It mainly includes four different cases.
Case 1.When Δc � 0 and T � lim Δc⟶0 T � 1, the peak plastic zone will disappear, and then the EPSFM degenerates into the elastic-strain-softening model.

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In this state, the softening and fracture zones radius can be obtained by solving equation ( 27): ) . (29) When β 1 � 1, equations ( 28) and ( 29) are the solutions obtained by Li et al. [23] for the circular opening expansion.
en, integrating equation ( 25), the maximum critical inner hydraulic pressure p exp max at r � R 1 can be rewritten as follows: T � 1, and t � 1, the EPSFM converts to the elastic-brittle plastic model.e stress at the elastic-fracture zone interface presents instantaneous dropping characteristics.However, the radius of the fracture zone cannot be given directly.e fracture zone radius can be deduced by considering the boundary condition p exp min � (σ r0 ) r�R 1 as follows: t � 1, the softening zone will disappear.us, the EPSFM degenerates into the elastic-peak plastic-brittle plastic model.Meanwhile, the maximum principal stress between peak plastic and fracture zones shows obvious drop characteristics.In this state, the radius of peak plastic and fracture zones can be deduced by integrating equations ( 22) and ( 27): T � 1, and σ cc � σ RR , the surrounding rock is only composed of the elastic and peak plastic zones.erefore, the EPSFM becomes the elastic-perfectly plastic model.e radius of the peak plastic zone can also be deduced by considering the boundary condition (σ r2 ) r�R 0 � p in : e analytical solution of equation ( 33) is the same with reference results (Cheng [22]).

Case I: Comparative Analysis.
Constitutive effect is extremely important for researching the mechanics and deformation behavior of rock mass.To validate the developed model in this paper and study the influence of constitutive relation on the mechanics response of the rock mass, the geometrical and physical parameters of a circular opening are shown in Table 2.Moreover, the load-bearing coefficient is assumed as 0.004.e circular opening expansion theory is mainly applied to hydraulic fracturing in underground engineering.e stresses distribution law under different constitutive models is shown in Figure 4.In addition, Table 3 presents the maximum inner hydraulic pressure p exp max at the softeningfracture zone interface.It can be seen from Figure 4 and Table 3 that the maximum critical pressure shows the characteristics of EBM < EPBM < ESM < EPSFM.By comparing with the EBM, EPBM, and ESM rock masses, the maximum critical pressure of EPSFM increases by 9.895 MPa, 7.752 MPa, and 1.286 MPa, respectively.It means that the EPSFM rock mass is the hardest to be cracked, whereas the EBM rock mass is the easiest in the process of hydraulic fracturing.
e influence of constitutive relation on the postpeak failure radii is shown in Figure 5.When the inner hydraulic pressure is equal to 40 MPa, the radii of R 1 , R 2 , and R 3 show the characteristics of EBM > EPBM > ESM > EPSFM.erefore, the above results indicate that the rock mass with a certain plastic bearing capacity is more difficult to be cracked in hydraulic fracturing engineering.In other words, Advances in Civil Engineering the design of hydraulic fracturing pressure should take full account of the in uence of lithology to achieve the best crack e ect.

Case II: Parameter Analysis.
A case of hydraulic fracturing in coal seam is used to study the mechanical response of rock masses with the change of hydraulic pressure.e in uence of parameters on the surrounding rock state is also discussed.e hydraulic fracturing case was implemented in No. 7601 coal seam with high gas in Wuyang Coal Mine of China for improved gas extraction.e coalbed was buried at about 480 m underground.e average value of hydrostatic pressure p 0 is 7.16 MPa; the radius of the borehole R 0 is 0.1 m; Young's modulus E and Poisson's ratio ] are 3.0 GPa and 0.28, respectively; the initial cohesion c 3 and the internal friction angle φ are 1.5 MPa and 30 °; and σ cc and σ RR are, respectively, about 5.2 MPa and 1.2 MPa.Moreover, the load-bearing coe cient Δc and brittleness coe cient α are 0.0006 and 1.2, respectively.It should be noted that the in uence of the dilatancy coe cient is ignored (β i 1) in order to avoid the errors arising from the volume change of postpeak rock mass.

Stresses and Postpeak Failure Radii Evolution Law.
Figure 6 shows the stress evolution law with the change of the critical hydraulic pressure.In the present example, it can be seen that there is only elastic zone around the borehole when 7.16 MPa ≤ p in ≤ 12.039 MPa (Figure 6(a)).ere are elastic and peak plastic zones when 12.039 MPa ≤ p in ≤ 12.711 MPa (Figure 6(b)).en, the surrounding rock of the borehole is composed of elastic, peak plastic, and softening zones if 12.711 MPa ≤ p in ≤ 14.917 MPa (Figure 6(c)).Finally, the surrounding rock consists of four zones if p in ≥ 14.917 MPa (Figure 6(d)).In addition, σ r > σ θ is commonly found in Figure 6 for the borehole expansion.
e radius of the postpeak failure zone is also signicantly important for evaluating the hydraulic fracturing e ect and optimizing the layout of the boreholes.e radius of the peak plastic, softening, and fracture zones evolution law under di erent hydraulic pressures are shown in Figure 7.It is clear that there is no postpeak failure zone when p 0 ≤ p in ≤ p exp min .e radius gradually increases with the increasing of the hydraulic pressure in the range p in ≥ p exp min for the circular opening expansion.Figure 7 is of great practical signi cance because the threshold of the critical hydraulic pressure p exp max has an important theoretical value for providing a design value of the minimum pumping pressure compared with the traditional empiricism [23].In this case, the threshold of calculation is 14.917 MPa and is in good accordance with the eld test results (14.54 MPa).

In uence of Load-Bearing Coe cient.
e loadbearing coe cient Δc re ects the plastic bearing capacity of rock mass and is extremely important for determining the fracture range and the critical hydraulic pressure in the process of hydraulic fracturing.e radii of the postpeak failure zone evolution law are shown in Figure 8.It can be seen that the postpeak failure radii obviously decrease with the increase of the load-bearing coe cient.However, the decreasing rate of softening zone radii is the maximum.For instance, when Δc transforms from 2 × 10 −3 to 5 × 10 −4 , the radii R 1 , R 2 , and R 3 , respectively, decrease by 15.8 mm, 25.9 mm, and 1.8 mm.It means that the greater the Δc is, the stronger the plastic bearing capacity of the rock mass and the smaller the fracture range of the drill hole are.Here, the inner hydraulic pressure is set at 20 MPa (>15.986MPa) (Table 4) in order to make the rock mass enter the residual state.
In addition, the load-bearing coe cient also has a very important e ect on the critical hydraulic pressure.As shown in Table 4, p exp mid and p exp max , respectively, decrease by 1.987 MPa and 1.525 MPa with the load-bearing coe cient Δc decreasing from 2 × 10 −3 to 5 × 10 −5 .e conclusion can provide exceedingly important reference for determining the threshold of maximum critical hydraulic pressure in hydraulic fracturing engineering.

In uence of Brittleness Coe cient.
Figure 9 shows the in uence of brittleness coe cients (α) on the postpeak failure radii.With the parameter (α) increasing, the postpeak failure radii show a nonlinear increase characteristic.However, the increase rate is gradually decreasing.For instance, when α changes from 0.6 to 2, the radii R 1 , R 2 , and R 3 , respectively, increase by 35.6 mm, 6.7 mm, and 7.2 mm.In addition, as shown in Table 5, the maximum critical hydraulic pressure p exp max is negatively correlated with the brittleness coe cient (α).e above result shows that the higher the strain-softening degree of rock mass is, the easier it is to be cracked by hydraulic fracturing.

Conclusions
Based on the triaxial test results, a new elastic-peak plasticsoftening-fracture constitutive model (EPSFM) is proposed by considering the plastic bearing behavior of the silty mudstone.
en, the closed-form solution of a circular opening based on the new proposed constitutive model is deduced with the nonassociated ow rule under the cavity expansion state.e correctness of the solution is also veri ed by comparing with the traditional solutions.e e ect of the constitutive relation and parameters on the mechanical response of rock mass is also discussed in detail.e primary conclusions can be summarized as follows: (1) e new closed-form solution based on EPSFM, considering the e ect of plastic bearing capacity of rock masses, can be regarded as a uniform solution compared with the traditional research results.Only when the load-bearing coe cient is equal to zero, the calculated results of the EPSFM can be converted to the ESM's solution; only when the brittleness coe cient is large enough or zero, the EPSFM's solution turned to the result by EPBM or EPM.Meanwhile, when the load-bearing coe cient is zero and the brittleness coe cient is large enough, the calculated results of the EPSFM was found to be in accordance with the closed-form solution of the EBM.(2) In hydraulic fracturing engineering, when the hydraulic pressure remains at a certain values,    Advances in Civil Engineering the fracture zone radii of circular opening present the characteristic of EBM > EPBM > ESM > EPSFM; otherwise, it is on the contrast for the critical hydraulic pressure at the softening-fracture zone interface.erefore, the EPSFM rock mass is hardest to be cracked, whereas the EBM rock mass is easiest in the process of hydraulic fracturing.(3) e postpeak failure radii show obviously a linear decrease with the increase of load-bearing coe cients or a nonlinear increase with the increasing brittleness coe cient.It means that, for the best fracturing e ects, the design of hydraulic fracturing pressure should take full account of the in uence of rock mass lithology, load-bearing coe cient, and brittleness coe cient.

Data Availability
e article data used to support the ndings of this study are included within the article.

Figure 3 :
Figure 3: Computational mechanical model of EPSFM (note: ε A12 and ε A 32 are, respectively, the maximum and minimum principal strains at "A" point; ε B 12 and ε B 32 are, respectively, the maximum and minimum principal strains at "B" point).

Figure 2 :
Figure 2: e total stress-strain curves by di erent lithology rock masses.(a) Silty mudstone from Yangzhuang coal mine.(b) Silty mudstone from Wushan.(c) Marble from Ya'an area of China.

Figure 4 :
Figure4: Stress distribution law for borehole expansion (note: because the maximum critical inner hydraulic pressure p exp max is taken as the calculated inner pressure, the fracture zone does not appear).

Figure 5 :
Figure 5: e radii distribution laws for circular opening expansion.

Figure 6 :
Figure 6: Stresses evolution law for the borehole expansion.

Figure 7 :
Figure 7: Radius of postpeak failure zones evolution law with hydraulic pressure.

Figure 8 :
Figure 8: e radii of the postpeak failure zone under di erent load-bearing coe cients.

Figure 9 :
Figure 9: e radii of the postpeak failure zone under di erent brittleness coe cients.

Table 1 :
Radial displacement and strain of the postpeak failure zones.

Table 3 :
e maximum critical inner hydraulic pressure p exp max (MPa).

Table 4 :
e critical hydraulic pressure under di erent load-bearing coe cients.StateΔc p con max p exp min (MPa) p con mid p exp max (MPa) p con min p exp max (MPa)