- Email: [email protected]
A unified thermo-elasto-viscoplastic model for soft rock
International Journal of Rock Mechanics & Mining Sciences 93 (2017) 1–12
Contents lists available at ScienceDirect
International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
A unified thermo-elasto-viscoplastic model for soft rock a
b,⁎
c
d
MARK e
Yong-lin Xiong , Guan-lin Ye , He-hua Zhu , Sheng Zhang , Feng Zhang a
Faculty of Architectural, Civil Engineering and Environment, Ningbo University, Fenghua Road 818, Ningbo 315211, China Department of Civil Engineering and State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China Department of Geotechnical Engineering, Tongji University, Siping Road 1239, Shanghai 200092, China d Department of Civil Engineering, Central South University, Shaoshan South Road 22, Changsha 410075, China e Department of Civil Engineering, Nagoya Institute of Technology, Showa-ku, Gokiso-cho, Nagoya 466-8555, Japan b c
A R T I C L E I N F O
A BS T RAC T
Keywords: Thermo-elasto-viscoplastic Soft rock Shear strength Confining stress dependency Temperature
A unified advanced thermo-elasto-viscoplastic constitutive model for soft rock is proposed in the critical state framework. The model is able to describe the fundamental mechanical behavior of soft rock such as elastoplastic, strain hardening and softening, time dependency, confining stress dependency, intermediate principal stress dependency and temperature dependency with one set of parameters. The thermal-induced equivalent stress tensor σ∼ij and the transform stress tensor tij are adopted to consider the influence of temperature and intermediate principal stress. Two evolution equations for the shear strength and the overconsolidation are newly introduced to take into consideration the influences of the confining stress and time dependent behavior, respectively. The capability of the model is carefully validated through a series of element tests of different soft rocks. The material parameters involved in the model have clear physical meanings and can be easily determined by the triaxial compression tests and creep tests.
1. Introduction As is well known, soft rock often gives rise to geotechnical engineering problems. For instance, progressive failure of soft rock slope1,2, long-term stability of tunnels in strong weathered soft rock ground3,4 are directly linked to the mechanical behavior of soft rock under general stress condition. The establishment of a rational constitutive model of soft rock, which can describe properly the mechanical behavior of soft rock, has a great significance for predicting the deformation of soft rock. Physically, soft rock has an unconfined compressive strength of 1～30 MPa and its strength lies between that of soil and hard rock. In the past decades, many experimental researches have been carried out to investigate the mechanical behavior of soft rock5–12. In general, the mechanical behavior of soft rock is elastoplastic, strain-hardening and strain-softening, and depends on strain rate, time, confining stress, intermediate principal stress and temperature. Until now, many constitutive models have been proposed to describe the above-mentioned mechanical behavior of soft rock in the framework of continuum mechanics. Based on endochronic theory13, Oka and Adachi14 proposed an elastoplastic model for soft rock that can not only describe the strain hardening-softening of soft rock, but also exhibits less mesh size dependency in finite element analysis
⁎
compared to the other models available at that time. By adopting a transform stress tensor called as tij15 and Matsuoka-Nakai failure criteria17, Zhang et al.18 proposed an elasto-viscoplastic model for soft rocks to consider the influence of the intermediate principal stress that may greatly affect the strength and stress-dilatancy relation of soft rock under different loading paths. In the framework of critical state soil mechanics, Nova19 firstly proposed a constitutive model for soft anisotropic rocks. Subsequently, many constitutive models were established based on the Cam-Clay model20–24. For instance, Amorosi et al.21 dealt with a critical statebased constitutive model for soft rocks, which was successfully applied to the analysis of the response of a pyroclastic rock during in situ plate load tests. Regarding soft rock as a heavily overconsolidated soil, Zhang et al.21 proposed a modified elasto-viscoplastic model for soft rock, which can not only describe the strain hardening-softening, the strain rate dependency and the time dependency, but also the intermediate principal stress dependency, in which the tij stress tensor and the subloading concept25 were adopted. The evolution of the state variable ρ related to overconsolidation ratio, proposed by Nakai and Hinokio26, was redefined to consider the time dependency in the model. The model, however, has to take different value for the shear stress ratio Rf under different confining stresses, in other words, the model could not consider the confining stress dependency.
Corresponding author. E-mail address: [email protected] (G.-l. Ye).
http://dx.doi.org/10.1016/j.ijrmms.2017.01.006 Received 9 December 2015; Received in revised form 6 January 2017; Accepted 9 January 2017 1365-1609/ © 2017 Elsevier Ltd. All rights reserved.
International Journal of Rock Mechanics & Mining Sciences 93 (2017) 1–12
Y.-l. Xiong et al.
In addition, the thermal effect of soft rock is also a very important factor for geotechnical engineering and geophysical science. Various experimental researches27–33 showed that in most cases, the shear strength and the creep failure time of soft rocks show a decreasing trend as temperature increases. At the same time, some constitutive models were proposed to describe the influence of temperature on the deformation and the strength of soft rock34–37. For instance, Gao et al.34 proposed a thermo-viscoplastic model of rock by using thermal expansion coefficient, viscosity attenuation coefficient and damage variable, which can consider the influence of temperature on elasticity, viscosity and damage. Zhang & Zhang35 proposed a thermo-elastoviscoplastic model for soft sedimentary rock in the framework of critical state soil mechanics. Xiong et al.36 modified the model proposed by Zhang & Zhang35 using tij transformed stress to consider the influence of the intermediate principal stress. At present, however, there is still no constitutive model that can take into consideration all the above mentioned aspects of mechanical behavior of soft rock in a unified way with one set of parameters. In this paper, regarding the soft rock as a heavily consolidated soil, a unified thermo-elasto-viscoplastic constitutive model is proposed in the framework of critical state soil mechanics, which has the following features: (1) the equivalent stress36 is adopted to consider the influence of temperature; (2) an evolution equation for the mobilized shear strength under different confining stress is applied to consider the influence of confining stress; (3) an evolution equation for the void ratio difference related to overconsolidation is used to consider the strain hardeningsoftening and the time dependency based on the subloading concept; (4) the transform stress tensor tij is used to consider the influence of intermediate principal stress. Moreover, the performance of the proposed model is carefully investigated through comparison with the test results on different types of soft rocks, such as triaxial compression tests, triaxial creep tests and plane strain tests under different temperatures.
Fig. 1. Similarity of volumetric strain caused by real stress and equivalent stress due to change of temperature.
Γ=
2.2. Thermo-induced equivalent stress It is well known that a change in temperature not only produce elastic volumetric strain, but also cause plastic volumetric strain in a soft rock sample. This effect is similar to the way in which a real stress acts on a soft rock sample. Accordingly, it is assumed in this study that the thermo-elastic volumetric strain is induced by an imaginary stress increment Δσ∼m , namely, the equivalent stress increment. The similarity of elastic volumetric strains caused by the real incremental mean stress and the incremental equivalent stress due to a change in temperature is shown in Fig. 1. Considering the limitation of the variation range for temperature and the fact that temperature T should be greater than or equal to 0 °C, a linear relationship between the change in temperature (T-T0) and the thermo-elastic volumetric strain increment is assumed as:
Matsuoka and Nakai proposed a spatially mobilized plane17 (SMP), in which the shear-to-normal stress ratio is maximized between two principal stresses in three-principal-stress space. Based on the SMP, Nakai and Hinokio26 proposed a simple elastoplastic model using the stress tensor tij to take into consideration the intermediate principal stress on the deformation and strength of soil. A brief derivation of the transformed stress tensor tij is given in Appendix A. Due to the fact that SMP criterion can predict the failure of soft rocks more precisely than the extended Mises criterion, as reported by Ye et al.7, the tij stress tensor is used instead of the normal stress tensor in the derivation of the newly proposed constitutive model for soft rock.
ΔεveT = 3αT (T − T0 )
The total strain rate can be written as,
εij̇ =
+
=
εij̇ eσ
+
εij̇ eT
+
εij̇ p
t∼N = tN + Δt∼N = tN + 3KαT (T − T0 ) (1)
(2)
Eijkl = Γ δij δkl + G ( δik δjl + δil δjk )
(3)
(6)
where K is the bulk modulus of the soft rock, tN is the actual real mean stress, which is the same as p in ordinary stress space.
where ε̇ijeσ is the elastic strain rate induced by the stress rate and ε̇ijeT is the elastic strain rate induced by temperature change. The plastic strain rate ε̇ijp in Eq. (1) can be calculated by Eq. (17), the thermo-elastic strain ε̇ijeT can be easy to obtain using Eq. (5), and the strain rate ε̇ijeσ induced by real stress change is given by Hook's law, −1 εkleσ ̇ = Eijkl σij̇
(5)
where αT is a linear thermal expansion coefficient, and takes a negative value because a compressive volumetric strain is assumed to be positive in geomechanics, T is the actual temperature, and T0 is a reference temperature and is usually taken as 15 °C, an average global temperature of the earth. Based on Hooke's law, it is straightforward to give the expression of equivalent stress:
2.1. Thermo-elasto-viscoplastic strain rate
εij̇ p
(4)
E is Young's modulus, and v is Poisson's ratio.
2. Thermo-elasto-viscoplastic stress-strain relationship for soft rock
εij̇ e
νE E , G= (1 + ν )(1 − 2ν ) 2(1 + ν )
2.3. Yield function The yield function of the proposed model is the same as the model proposed by Xiong et al.36 and its expression is given as:
f = ln
where,
where 2
tN t + ζ (X ) − ln N1 = 0 tN 0 tN 0
(7)
International Journal of Rock Mechanics & Mining Sciences 93 (2017) 1–12
Y.-l. Xiong et al.
ζ (X ) =
1 β
β
( ) X M*
calculated as, (8)
ρ = (λ − κ )ln
tN and X=tS/tN are the mean stress and the stress ratio based on tij stress, and tN1 determines the size of the yield surface (the value of tN at tS=0). tN0, a reference mean stress, is an arbitrary value and is taken as 98 kPa in this paper. β is a parameter that determines the shape of the yield surface in tij stress space. If β takes a value equal to 1, the yield function has a form similar to that of the Cam-clay model. The value of M* is expressed by principal stress ratio XCS=(tS/tN)CS and plastic strain increment ratio YCS=(dεSMP*p/dγSMP*p)CS at critical state: 1/ β
β M * = (XCS + XCS β −1YCS )
2⎛ ⎜⎜ RCS − 3 ⎝
f = ln
YCS =
1−
RCS
2 ( RCS + 0.5)
(10)
RCS = σ1/ σ3 |at critical state is the ratio of principal stress σ1 and σ3 at critical state in conventional triaxial compression tests, dεSMP*p and dγSMP*p are a normal component and a tangential component of the principal plastic strain increment vector on the SMP respectively. In order to describe the behaviors of overconsolidated geomaterial, Hashiguchi & Ueno25 proposed the concept of subloading yield surface, in which the subloading yield surface is similar with the normally consolidated yield surface in geometry and the stress state is always at subloading yield surface. Fig. 2 shows the relation between the subloading yield surface and normally consolidated yield surface and their corresponding e-lntN relations in tij stress space. It is easy to obtain the subloading yield function in the following equation according to Eq. (7), f = ln
⎛ t tN t ⎞ + ζ (X ) − ⎜ln N1e − ln N1e ⎟ = 0 ⎝ tN 0 tN 0 tN1 ⎠
tN1e , tN 0
Cp =
λ−κ 1 + e0
(14)
2.4. Evolution law for shear strength at critical state It is well known that the quantity M* at critical state is constant for a soil, as shown in Eqs. (9) and (10). However, previous experimental results11,37,38 have shown that the shear strength (stress ratio) for soft rock at critical state owned different values under different confining stresses. Therefore, a concept of mobilized shear strength, with the strength varying with the confining stress level, is introduced in this new model. An evolution equation for M* proposed by Iwata11 is given as:
(11)
where tN1 and tN1e represent the size of the subloading yield surface and normal yield surface respectively. Similar to the Cam-clay model, the plastic volumetric strain in tij stress space can be expressed as
εvp = Cp ln
ρ ⎞ tN 1 ⎛ p ⎜εv − ⎟=0 + ζ (X ) − tN 0 Cp ⎝ 1 + e0 ⎠
It should be pointed out that the state variable ρ similarly as the plastic volume strain εvp , has no influence on the critical state. By using the state variable ρ , the model can produce viscoplastic strain in the overconsolidation status and has a smooth transition from the overconsolidation state to the residual state (the critical state for soft rock). It also can be seen from Eq. (13) that the value of tN1e may varies with ρ (whose value will be discussed in Session 2.7), while the tN1 is determined by current stress state. In other words, the expansion or shrinkage of the normal consolidated yield surface is controlled by ρ , however, no matter how it evolves, the subloading yield surface moves toward it and finally the two surfaces overlap.
(9)
1 ⎞ ⎟⎟ , RCS ⎠
(13)
Substituting Eqs. (12) and (13) into Eq. (11), finally the subloading yield surface of thermo-elasto-viscoplastic model is given as,
where
XCS =
tN1e tN1
M * = M0* +
∫ dM *,
dM * = A ln
* p MCS dεd M*
* OCR n M0* = MCS
(15)
(16)
where,M0* is the initial value of M* at the beginning of shear loading, OCR is the overconsolidation ratio of soft rock at initial condition, and * is the value of M0* when OCR=1.0. dεdp is an increment of MCS equivalent plastic shear strain during shear loading, and will be described in detail later. Two new parameters n and A are introduced: n is the exponent in the relationship between OCR and M0*, and A is the parameter that determines the evolution rate of M* during the shearing process. It can be seen from Eqs. (15) and (16) that the shear strength at critical state is mainly determined by two factors, one is the initial
(12)
where e0 is the void ratio at the reference state tN0=98 kPa under isotropic normal consolidated condition. λ is the compression index and κ is the swelling index. tN1e is the pre-consolidation stress, which has the same physical meaning as the pc in the Cam-clay model. At the same time, the density difference ρ in Fig. 2 can be easily
Fig. 2. Shape of subloading yield surface and normally yield surface, and definition of ρ (After Nakai and Hinokio26).
3
International Journal of Rock Mechanics & Mining Sciences 93 (2017) 1–12
Y.-l. Xiong et al.
OCR, the other is the history of plastic shear strain.
2.8. Plastic operator Λ
2.5. Flow rule
Substituting the Eqs. (15)–(17) and (20)–(21) into Eq. (18), the plastic operator Λ can be obtained as
As the same as the Cam-clay model, the associated flow rule is adopted in this model. The thermo-elasto-viscoplastic strain rate is defined as,
Λ=
⎛ ∂f ∂f δij ⎞ ⎛ ∂f ∂f δij ⎞ ∂f ∂f ⎟⎜ ⎟ εij̇ p = Λ , εv̇ p = Λ , dεḋp = Λ ⎜ − − ∂tij ∂tii ∂tkk 3 ⎠ ⎝ ∂tij ∂tkk 3 ⎠ ⎝ ∂tij
where
hTp / Cp − A ln
(17)
where Λ is a positive scalar called as plastic operator. 2.6. Consistency equation
(18)
(23)
(24)
fσ̇ =
∂f σij̇ ∂σij
(25)
(26)
h (tcreep )/ Cp * ∂f MCS M * ∂M *
⎛ ∂f ⎜ ∂t − ⎝ ij
∂f δij ⎞ ⎛ ∂f ⎟⎜ ∂tkk 3 ⎠ ⎝ ∂tij
−
∂f δij ⎞ ⎟ ∂tkk 3 ⎠
(27)
From the above equation, the reason why using the inhomogeneous function in Eq. (21) because Λ will be zero under a pure creep state if homogeneous function is used and the viscoplastic strain will never occur, which will not achieve the purpose of the proposed model. 2.9. Loading criterion In the proposed model, the loading criterion is given as,
εij̇ p > 0 when Λ > 0 (19)
≤0
viscoplastic (loading) εij̇ p = 0 when Λ
elastic (unloading)
(28)
In the case of loading, the plastic operator Λ is positive. From Eq. (23), there exist two different situations listed as follows:
⎧Λ = ⎪ ⎨ ⎪Λ = ⎩
(20)
numerator > 0 deno min ator > 0 numerator < 0 deno min ator < 0
> 0 (i ) > 0 (ii)
(29)
The second part of the numerator of Eq. (23), which contributes to the viscous strain, can be omitted for simplicity. Then the two situations can be rewritten as follow:
⎧ ḟ = ⎪ σ ⎨ ⎪ fσ̇ = ⎩
∂f σ̇ ∂σij ij
> 0 strain − hardening (i )
∂f σ̇ ∂σij ij
< 0 strain − softening (ii)
(30)
From Eq. (30), situation (i) represents the strain hardening because the subloading yield surface is expanding, while situation (ii) represents the strain softening because the subloading yield surface is shrinking. Fig. 3 demonstrates how the model predicts the strain-hardening and strain-softening behaviors in a drained triaxial test of soft rock with a large OCR. The change of deviatoric stress and volumetric strain can be divided into three stages: (1) Before the stress path reaches the critical state line (CSL), the denominator > 0 (situation i) and the current stress point locates to the right side of CSL, leading to a strainhardening (subloading yield surface expanding) and a volumetric contraction. (2) After the stress path goes beyond the CSL, the denominator is still larger than zero, which forces the strain-hardening
(21)
where
t∼N = tN + 3KαT (T − T0 )
∂f δij ⎞ ⎟ ∂tkk 3 ⎠
G (ρ , tcreep ) ∂f + ∂tkk tN + 3KαT (T − T0 )
hTp / Cp − A ln
ε̇v 0 is an initial volumetric strain rate at the time tcreep=0 which represents the time when shearing begins. tcreep1 is a unit time used to standardized the time and always take the value of 1.0. α is a parameter that controls the gradient of strain rate vs. time in logarithmic axes during a creep test. Cn controls the strain rate dependency of soft rocks. a is a parameter controlling the losing rate of overconsolidation. It should be pointed out that the values of the parameters Cn and α are not objective and are dependent on the unit of time. In its application to boundary value problem, the unit of time used in numerical analysis should be the same as the one used in determining the parameters from the laboratory tests. In this model, in order to consider the influence of temperature on the deformation and strength of soft rock, the equivalent stress t∼N induced by temperature is adopted instead of normal mean stress tN in the above evolution equation of state variable ρ, and the corresponding expression is written as, G (ρ , tcreep ) ρ̇ = −Λ + h (tcreep ) 1 + e0 t∼N
−
hTp =
Λ=
⎪ ⎪
∂f δij ⎞ ⎛ ∂f ⎟⎜ ∂tkk 3 ⎠ ⎝ ∂tij
Therefore, Eq.(23) can also be written as
where
⎧ h (tcreep ) = εv̇ 0 (1 + tcreep / tcreep1)−α ⎨ 1+ C ln(1+ tcreep / tcreep1) ⎩G (ρ , tcreep ) = aρ n
⎛ ∂f ⎜ ∂t − ⎝ ij
fσ̇ = (∂f /∂σij ) σij̇ = 0
2.7. Evolution equation for state variable ρ In the consistency equation, it is necessary to given evolution for the development of the state variable ρ . Based on the works by Nakai and Hinokio26, Zhang et al.22 proposed an elasto-viscoplastic model for soft rock, in which the time dependent evolution equation for the state variable ρ is given as an inhomogeneous function,
G (ρ , tcreep ) ρ̇ = −Λ + h (tcreep ) 1 + e0 σm
* ∂f MCS M * ∂M *
When the soil enters into a pure creep state, the following relation holds:
It is known that the plastic potential determines the direction of the development rate of plastic strain, and the yield function gives the yield criterion of the geomaterials, while the consistency equation determines the value of the plastic strain, in other word, the plastic operator Λ can be determined. Because the quantity M* at the critical state is not constant, the change of M* should be taken into consideration. Finally the consistency equation can be obtained according to Eq. (14),
∂f ∂f ρ̇ ⎤ 1 ⎡ p ⎢εv̇ − ⎥=0 σij̇ + Ṁ * − ḟ = 0 ⇒ ∂σij Cp ⎣ ∂M * 1 + e0 ⎦
fσ̇ + h (tcreep )/ Cp
(22)
4
International Journal of Rock Mechanics & Mining Sciences 93 (2017) 1–12
Y.-l. Xiong et al.
Fig. 3. Illustration of how the model predicts the strain-hardening and strain-softening behaviors in a drained triaxial test of soft rock with a large OCR.
density difference ρ is considered by using the thermo-induced equivalent stress.
to continue so that the stress path keeps moving upward to the normal consolidated yield surface. Since the current stress point locates to the left side of CSL, the volume begins to expand. (3) After the stress path reaches the normal consolidated yield surface, the denominator becomes less than zero (situation ii), the subloading yield surface turns to shrink with the normal consolidated yield surface, that is the strainsoftening. The deviatoric stress keeps decreasing and the volume keeps expanding until the stress path reaches the CSL finally. Accordingly, the loading criterion is rewritten as follows:
εij̇ p
> 0 when Λ > 0
3. Performance of the proposed model 3.1. Determination of the parameters involved in the proposed model Thirteen parameters are involved in the proposed model, among which five parameters are the same as the Cam-Clay model. The determination of these five parameters can be found out in the literatures and do not discussed in this paper. The parameter Cn is determined by the difference of peak strengths of two rock samples under drained triaxial compression tests with different constant shear strain rates. Parameter α is the gradient of strain rate vs. time in logarithmic axes and can be uniquely determined with drained triaxial creep test as shown in Fig. 4. The parameter β controls the shape of the yield function and the stress-dilatancy relation (associated flow rule) shown in Fig. 2. β can be determined based on the test data of stress-dilatancy relation. The overconsolidation parameter a, which controls the losing rate of overconsolidation, can be determined from the peak strengths of soft rock specimens under triaxial compression test by curve fitting, in which at least two samples, with the same physical conditions but different initial overconsolidation, should be tested under the same loading condition. The parameter n can be determined by triaxial compression tests with different initial OCR and using Eq. (15). The parameter A, which determines the evolution rate of M* during the shear loading process, can be determined based on the test results of drained triaxial compression tests under different confining stresses. The parameter αT is the coefficient of linear thermal expansion of soft rock, a well-known physical quantity that can be obtained from text books or databases. The calibration methods of material parameters involved in the proposed model is summarized in Table 1.
⎧ f ̇ > 0 hardening ⎪ σ and ⎨ fσ̇ < 0 softening εij̇ p = 0 when Λ ⎪ ̇ ⎩ fσ = 0 pure creep
≤ 0elastic (unloading)
(31)
It is worth noting that different from other constitutive models, this loading criterion not only can predict strain hardening (expansion of failure surface) and strain softening (shrinking of failure surface), but also the creep state where even if the stress tensor rate is equal to zero, it is still regarded as a loading process and viscoplastic strain may happen. Meanwhile, the thermal plastic strain can be taken into
3.2. Parametric study of the proposed model In this section, the influence of parameters on the performance of the new model is investigated by simulating triaxial compression tests and creep tests. The parameters used in the simulations are listed in Table 2. In the simulations, the strain rate is taken as 1.0×10−4 1/min. Fig. 5(a), (b), (c), (d) show the calculated nonlinear stress-strain
Fig. 4. Creep behavior of soft rock under different creep stresses (drained condition).
account automatically because the influence of temperature on the 5
International Journal of Rock Mechanics & Mining Sciences 93 (2017) 1–12
Y.-l. Xiong et al.
Table 1 Calibration methods of material parameters. Parameter
Physical meaning
Calibration method
E Ep(=λ−κ): α a Cn v β M*CS αΤ n A
young's modulus plastic index creep index controls the loss rate of overconsolidation controls the evolution of strain rate Possion's ratio controls the shape of yield surface critical stress ratio when OCR = 1.0 linear thermal expansion coefficient exponential parameter determines the relationship between OCR and critical stress ratio M*CS. determines the evolution rate of M* during shearing
triaxial test consolidation test triaxial creep test triaxial test triaxial test with different strain rates triaxial test stress-dilatancy relation of triaxial test triaxial test conventional thermal test triaxial test with different confining stresses triaxial test with different confining stresses
Table 2 Material parameters for rocks used in parametric study. E (MPa)
Ep (=λ−κ)
a
α
Cn
ν
β
M*CS
αT (K−1)
n
A
900
0.04
500
0.7
0.025
0.0804
1.0
0.40
-8.0×10-6
0.15
1.5
3.4. Plane strain tests under different confining stresses at room temperature
relationship and volume change during triaxial compression tests with different parameters at 15 °C. It can be seen that for larger values of the parameters β and A, the shear strength is low. And for larger values of the parameters a and n, the shear strength is higher. Fig. 5(e) shows the calculated results of triaxial compression tests with different parameter αT and different temperature T. It is shown clearly that the shear strength decreases as the temperature increases, and the larger the linear thermal expansion is, the larger the decrease of shear strength will be. Fig. 5(f) shows the results of creep tests with different parameter α and different temperature T. In the simulation, the creep stress is chosen as 90% of the shear strength with parameter α=0.7 at 15 °C. It can be seen that the creep failure time decreases as the temperature increases, and the smaller the parameter α is, the earlier the creep failure will happen.
In order to validate the ability of the model to consider the influence of the intermediate principal stress on the deformation and strength of soft rock, the test results of manmade soft rock under the plane-strain condition at room temperature conducted by Nishikami and Horii39 are used. The manmade soft rock is made from standard Toyoura sand, gesso and water with the proportion of 69.6%, 11.6% and 18.8% respectively. The plane-strain condition of the samples was realized by clasping the sample with two vertical steel plates that are very stiff and supported by several fixed rollers in the test. The silicon grease is used to reduce the friction between the plate and manmade soft rock. The loading strain rate is given as 0.083% in the tests. Three test results with the confining stress of 0.196 MPa, 0.294 MPa and 0.392 MPa are employed. The physical properties and the material parameters of manmade soft rock involved in the model are listed in Tables 3 and 4. Fig. 7 illustrates the measured and computed stress-strain relationship of manmade soft rock under different confining stress. The comparisons between the test results and the model calculations show that the model is able to reproduce both the test results of the maximum and the intermediate principal stresses. Moreover, the model also can describe the test results of the volumetric strain, which changed from the positive dilatancy to negative dilatancy as the confining stress increased.
3.3. Triaxial shear tests under different confining stresses at room temperature In order to verify the performance of the proposed model, drained triaxial compression tests conducted by Iwata11 are firstly simulated to examine the mechanical behavior of the model. The Ohya stone, a soft sedimentary rock known as green tuff, widely distributed in northeast Japan, is use for testing. The specimen is saturated through repeated process of evacuation in desiccator before testing. The specimen, 100 mm in height and 50 mm in diameter, is isotropically consolidated to a prescribed confining stress and then a strain-rate-controlled vertical load is applied under drained condition. During shearing process, the loading rate is 0.001%/min and the axial strain and volumetric strain are measured at the same time. The physical properties and the material parameters of Ohya stone involved in the model are listed in Tables 3 and 4. Fig. 6 shows the comparison of the stress-strain-dilatancy relation between the simulated results and experimental results. It can be seen that the model can describe well the observed mechanical behaviors, such as strain-hardening and strain-softening, under confining stress from 0.1 to 4.0 MPa at room temperature. It is worth noting that all the parameters used in the simulation are the same except the initial OCR, which took different value under different confining stress.
3.5. Triaxial tests with different loading rates at room temperature The proposed model is used to simulate the mechanical behavior of triaxial compression tests at different strain loading rate on Tomuro stone40, which also is a green tuff. The tests were conducted at two different loading rate of 0.28% and 0.0012%/min respectively with 0.1 MPa confining stress. The initial values of the state parameters and the material parameters of the Tomuro stone are listed in Tables 3 and 4. Fig. 8 shows the comparisons of the calculated and measured deviatoric stress ~ axial strain relationship at different loading rate in
6
International Journal of Rock Mechanics & Mining Sciences 93 (2017) 1–12
Y.-l. Xiong et al.
Fig. 5. Parametric study of the proposed model.
the triaxial compression test. It is found that the rate dependency behavior, namely, the peak strength increases with the loading rate, is described well by the proposed model.
Table 3 Physical properties of soft rocks.
Reference void ratio e0 Specific gravity of soil particles Gs Pre- consolidation pressure p’c (MPa)
Ohya stone
Manmade soft rock
Tomuro stone
Tage stone
0.57 2.51
0.62 2.53
0.63 2.53
0.50 2.52
21.0
8.8
10.0
20.0
3.6. Triaxial tests with different confining stresses and different temperatures In order to investigate the influence of temperature on the deformation and strength of soft rock, Nishimura41 conducted a serial of triaxial compression tests under different confining stress and different temperature. The soft rock used in the tests is Tage stone,
7
International Journal of Rock Mechanics & Mining Sciences 93 (2017) 1–12
Y.-l. Xiong et al.
Table 4 Material parameters for soft rocks.
Ohya stone Manmade soft rock Tomuro stone Tage stone
E (MPa)
Ep (=λ−κ)
a
α
Cn
ν
β
M*CS
αT (K−1)
n
A
600. 600. 350. 1000.
0.005 0.005 0.023 0.015
850. 500. 800. 3000.
0.6 0.85 0.80 0.050
0.02 0.00 0.05 0.25
0.02 0.08 0.0864 0.12
1.1 1.5 1.5 1.1
0.32 0.32 0.58 0.46
-8.0×10-6 -9.0×10-6 -1.0×10-5 -2.5×10-5
0.19 0.11 0.15 0.12
4.0 1.0 1.5 1.0
Fig. 6. Comparisons of plane strain test and calculated results for manmade soft rock under different confining stresses at room temperature (test data from Iwata11).
3.7. Triaxial creep tests under different confining stresses and different temperatures
another kind of Japanese green tuff. Some basic physical properties of Tage stone and a description of it can be found in41. In the tests, the confining stresses were 0.49 MPa and 0.98 MPa and the temperature was 20 ℃ and 80 ℃ respectively. In addition, a back pressure of 0.49 MPa was applied in all the tests to guarantee a full saturation of the specimen. Table 4 lists the material parameters of Tage stone. Fig. 9 presents the calculated and measured stress-strain-dilatancy relations of Tage stone under different confining stresses and different temperatures in the triaxial compression test. The test results indicate that the peak strength of Tage stone decreased and the volumetric strain became contractive as the temperature increased. From the comparisons, it can be known that the model can describe well the influence of temperature and confining stress on the deformation and strength of soft rock with one set of parameters.
The ability of the model to describe the influence of temperature on the creep failure under different temperature is validated through the creep test results on Tage stone41. In the test, instead of applying the axial load abruptly, the load was applied under constant loading rate until the prescribed creep stress was reached. The creep stresses used in the creep tests are 95% of the peak stresses of the specimens under the same confining stress and the temperature in triaxial compression tests. The parameters used in the model, of course, are the same as the triaxial compression test in Table 4. Fig. 10 show the comparisons between the calculated and measured results of Tage stone in drained triaxial creep tests under different constant temperatures and confining stresses. The confining stresses
8
International Journal of Rock Mechanics & Mining Sciences 93 (2017) 1–12
Y.-l. Xiong et al.
Fig. 8. Comparisons of triaxial test and calculated results for Tage stone under different confining stresses and temperatures (test data from Koike40).
approximately one-order of magnitude larger than the one measured in tests, implying that some modification should be made in future study. 4. Conclusions A unified thermo-elasto-viscoplastic model has been proposed within the critical state framework from the viewpoint that the soft rock can be regarded as a heavily overconsolidated geomaterial. The subloading yield surface is utilized to simulate the typical mechanical behavior of soft rock. By introducing a new evolution equation for the critical state parameters M* under different confining stresses, the model is able to describe the confining-stress dependency of soft rock with a set of fixed material parameters whose values are identical under all loading conditions. This is the main difference compared with previous model (Xiong et al.36), in which different parameters were used to describe the mechanical behavior of soft rock under different confining stresses. Based on the fact that a change in temperature produce not only elastic volumetric strain, but also plastic volumetric strain in a soft rock sample, a thermo-induced equivalent stress is introduced to evaluate the influence of temperature on the deformation and the strength of soft rock. The transform stress tensor tij is adopted in the model to consider the influence of intermediate principal stress, which is very importance to describe the stress-strain relation of soft rock under general loading conditions. Thirteen parameters are involved in the proposed model, among which five parameters are the same as Cam-clay model. The other parameters can be definitely calibrated via conventional laboratory tests. The validity of the proposed model is confirmed carefully by serials of element tests under different loading and thermal conditions with one set of parameters. However, further improvement is necessary to increase the accuracy of predictions, such as the onset of creep failure and the volumetric strain during drained triaxial tests.
Fig. 7. Comparisons of triaxial test and calculated results for Tomuro stone under different loading rates at room temperature (test data from Nishikami and Horii39).
Acknowledgement and temperatures are also the same as the triaxial compression tests. It can be known from the comparisons that the model is able to describe the creep behavior of Tage stone under different confining stresses and temperatures quantitatively, with the same parameters as the triaxial compression tests. However, the calculated creep failure time is
The support of the National Nature Science Foundation of China (Grant Nos. 51478345, 41372284, 51608385 and 51478228), National Basic Research Program of China (2014CB047001) and K.C. Wong Magna Fund in Ningbo University are greatly appreciated.
9
International Journal of Rock Mechanics & Mining Sciences 93 (2017) 1–12
Y.-l. Xiong et al.
Fig. 9. Comparisons of triaxial creep test and calculated results for Tage stone under different confining stresses and temperatures (test data from Nishimura41).
Fig. 10. Comparisons of triaxial creep tests and calculated results for Tage stone under different confining stresses and temperatures (test data from Nishimura
41
).
Appendix A As shown in Fig. A1, the Spatial Mobilized Plane (SMP), defined by Nakai and Matsuoka16, can be expressed in a principal stress space (σI, σII, σIII) as
σI σ σ + II + III = 1. σ1 σ2 σ3
(A-1)
Therefore, the normal of the plane (a1, a2, a3) can be evaluated by the following equation as,
ai =
I3 I2 σi
(i = 1, 2, 3)
(A-2)
where I1, I2 and I3 are the first, second and third invariants of effective stress tensor and can be expressed by the following forms using three principal stresses:
I1 = σ1 + σ2 + σ3 I2 = σ1 σ2 + σ2 σ3 + σ3 σ1 I3 = σ1 σ2 σ3
(A-3)
In tij clay and sand models, a symmetric tensor tij is expressed by a product of stress tensor σik and a tensor akj as:
tij = σik akj
(A-4)
10
International Journal of Rock Mechanics & Mining Sciences 93 (2017) 1–12
O
O
O
O
Y.-l. Xiong et al.
O
O
+
Fig. A 1. Explanation of SMP (Nakai and Mihara15).
where the tensor akj can be evaluated by Cayley-Hamilton's theory as
aij =
I3 −1 rij , rik rkj = σij , rij = (σik + I2 δik )(I1 σkj + I3 δkj )−1 I2
(A-5)
Because rij is a one-half power function of σij, rij is a symmetric tensor and its principal directions are the same as σij and have the principal σ2 , σ3 ). Therefore aij is also a symmetric tensor and has the same principal directions as σij and has the principal value of (a1, values of ( σ1 , a2, a3). As a result, the tensor tij is also a symmetric tensor and has the same principal directions as aij and σij. A normal component tn and a tangential component ts of the principal-value vector of tij can be given as (see Fig. A2a),
tN = t1 a1 + t2 a2 + t3 a3 = tij aij ≡ σSMP = σ1 a12 + σ2 a 22 + σ3 a32 = 3 tS =
(t12 + t22 + t32 ) − tN2 =
≡ τSMP =
I3 I2
(A-6)
tij tij − (tij aij )2
(σ1 − σ2 )2a12 a 22 + (σ2 − σ3)2a 22 a32 + (σ3 − σ1)2a32 a12 =
I1I2 I3 − 9I32
(A-7)
I2
It is assumed that the directions of plastic principal strain increments coincide with those of principal axes of tij, and the plastic strain increment *p ) of the principal plastic strain increment vector on the SMP (see *P ) and a tangential component (dγSMP can also be given by a normal component (dεSMP Fig. A2b),
*P = dε1P a1 + dε2P a2 + dε3P a3 = dεijP aij dεSMP
dεijp dεijp − (dεijp aij )2
Fig. A 2. Explanation of stress and strain in principal-value space of tensor tij (Nakai and Mihara15).
11
O
O
O
O
(A-9)
O
*p2 = (dε1p2 + dε2p2 + dε3p2 ) − dεSMP
O
*p = dγSMP
(A-8)
International Journal of Rock Mechanics & Mining Sciences 93 (2017) 1–12
Y.-l. Xiong et al.
The failure criteria of tij models is defined by the SMP failure criteria (Plane ABC in Fig. A1b),
⎛ I I I − 9I 2 ⎛t ⎞ ⎛τ ⎞ 1 2 3 3 Xf ≡ ⎜ S ⎟ ≡ ⎜ SMP ⎟ = ⎜⎜ ⎝ tN ⎠ f ⎝ σSMP ⎠ f ⎝ 9I3
⎞ ⎟ = const . . ⎟ ⎠f
(A-10)
References 22 1 Zhang F, Yashima A, Osaki H, Adachi T, Oka F. Numerical simulation of progressive failure in cut slope of soft rock using a soil-water coupled finite element analysis. Soils Found. 2003;43(5):119–131. 2 Ye GL, Zhang F, Yashima A, Sumi T, Ikemura T. Numerical analysis on progressive failure of slope due to heavy rain with 2D and 3D FEM. Soils Found. 2005;45(2):1–15. 3 Ferrero AM, Migliazza M, Roncella R, Tebaldi G. Analysis of the failure mechanisms of a weak rock through photogrammetrical measurements by 2D and 3D visions. Eng Fract Mech. 2008;75(3):652–663. 4 Yashima A, Matsumoto A, Tanabe K. Slope failure at tunnel entrance due to excavation and its countermeasure. in: Proceedings of the International Conference Modern Tunneling Science and Technology. Kyoto, p.557-562; 2001. 5 Agustawijaya DS. The uniaxial compressive strength of soft rock. Civ Eng Dimens. 2007;9(1):9–14. 6 Liao HJ, Su LJ, Pu WC, Yin JH. Test and numerical analysis of the constitutive relation of a diatomaceous soft rock. Mar Georesources Geotechnol. 2003;21(1):183–200. 7 Ye GL, Zhang F, Yashima A, Aung H, Naito K. Influence of membrane and filter paper on plane-strain testing of soft sedimentary rock. Geotech Test J. 2007;30(6):442–453. 8 Ye GL, Zhang F, Naito K, Aung H, Yashima A. Test on soft sedimentary rock under different loading paths and its interpretation. Soils Found. 2007;47(5):897–909. 9 Ye GL, Ye B, Zhang F. Strength and Dilatancy of Overconsolidated Clays in Drained True Triaxial Tests. J Geotech Geoenviron Eng. 2014;140(4):80–90. 10 You MQ. True-triaxial strength criteria for rock. Int J Rock Mech Min Sci. 2009;46(1):115–127. 11 Iwata M. Modification of Constitutive Model for Soft Rock for Slope Failure Prediction [PhD Thesis], Gifu, Japan: Gifu University; 2011 [in Japanese]. 12 Chiu HK, Johnston IW, Donald IB. Appropriate techniques for triaxial testing of saturated soft rock. Int J Rock Mech Min Sci. 1983;20(3):107–120. 13 Valanis KC. A theory of viscoplasticity without a yield surface. Arch Mech. 1971;23(4):517–533. 14 Oka F, Adachi T. A constitutive equation of geologic materials with memory. In: Proceedings of the 5th International Conference on Numerical Method in Geomechanics; No. 1: 293–300; 1985. 15 Nakai T, Mihara Y. A new mechanical quantity for soils and its application to elastoplastic constitutive models. Soils Found. 1984;24(2):82–94. 16 Nakai T, Matsuoka H. A generalized elastoplastic constitutive model for clay in a three-dimensional stress. Soils Found. 1986;26(3):81–89. 17 Mastuoka H, Nakai T. Stress deformation and strength characteristics of soil under three different principal stresses. Proc JSCE. 1974;232:59–70. 18 Zhang F, Yashima A, Nakai T, Ye GL, Adachi T, Oka F. An elastoplastic strainhardening and strain-softening constitutive model for soft rock considering the influence of intermediate stress. Soils Found. 2003;43(5):107–117. 19 Nova R. An extended Cam-clay model for soft anisotropic rocks. Comput Geotech. 1986;2:69–88. 20 Nova R, Castellanza R, Tamagnini C. A constitutive model for bonded geomaterials subject to mechanical and/or chemical degradation. Int J Numer Anal Meth Geomech. 2003;27(9):705–732. 21 Amorosi A, Aversa S, Boldini D, Laera A, Nicotera MV. Application of a new
23
24
25
26 27 28
29
30 31 32
33 34 35 36
37 38 39
40 41
12
constitutive model to the analysis of plate load tests in a pyroclastic rock. Int J Rock Mech Min Sci. 2015;78:271–282. Zhang F, Yashima A, Nakai T, Ye GL, Aung H. An elasto-viscoplastic model for soft sedimentary rock based on tij concept and subloading yield surface. Soils Found. 2005;45(1):65–73. Zhu HH, Ye B, Cai YC, Zhang F. An elasto-viscoplastic model for soft rock around tunnels considering overconsolidation and structure effects. Comput Geotech. 2013;50:6–16. Fu YK, Iwata M, Ding WQ, Zhang F, Yashima A. An elastoplastic model for soft sedimentary rock considering inherent anisotropy and confining-stress dependency. Soils Found. 2012;52(4):575–589. Hashiguchi K, Ueno M. Elastoplastic constitutive laws of granular materials, Constitutive equations of soils, In: Proceedings 9th ICFSME, Tokyo, JSSMFE, 73– 82; 1977. Nakai T, Hinokio M. A simple elastoplastic model for normally and over consolidated soils with unified material parameters. Soils Found. 2004;44(2):53–70. Nasseri MHB, Goodfellow SD, Wanne T, Young RP. Thermo-hydro-mechanical properties of cobourg limestone. Int J Rock Mech Min Sci. 2013;61:212–222. Okada T. Mechanical properties of sedimentary soft rock at high temperatures (part 1) – Evaluation of temperature dependency based on triaxial compression test. Civ Eng Res Lab Rep. 2005 [No. N04026 (in Japanese]. Okada T. Mechanical properties of sedimentary soft rock at high temperatures (part 2) – Evaluation of temperature dependency of creep behavior based on unconfined compression test. Civ Eng Res Lab Rep. 2006 [No. N05057 (in Japanese]. Ye GL, Nishimura T, Zhang F. Experimental study on shear and creep behaviour of green tuff at high temperatures. Int J Rock Mech Min Sci. 2015;79:19–28. Yang C, Daemen JJK. Temperature effects on creep of tuff and its time—dependent damage analysis. Int J Rock Mech Min Sci. 1997;34(3–4) [345.e1-345.e12]. Zhao Y, Wan Z, Feng Z, Yang D, Zhang Y, Qu F. Triaxial compression system for rock testing under high temperature and high pressure. Int J Rock Mech Min Sci. 2012;52:132–138. Gao F, Xu XL, Yang XJ, Ju Y. Research on thermo-visco-elastoplastic model of rock. Chin J Rock Mech Eng. 2009;28(1):74–80 [in Chinese]. Zhang S, Zhang F. A thermo-elasto-viscoplastic model for soft sedimentary rock. Soils Found. 2009;49(4):583–595. Zhang S, Leng WM, Zhang F, Xiong YL. A simple thermo-elastoplastic model for geomaterials. Int J Plast. 2012;34:93–113. Xiong YL, Zhang S, Ye GL, Zhang F. Modification of thermo-elasto-viscoplastic model for soft rock and its application to THM analysis of heating tests. Soils Found. 2014;54(2):176–196. Adachi T, Ogawa T. Mechanical Properties and failure criteria of soft rock, Proceedings JSCE; 295: 51–62 (in Japanese); 1980. Zhang F. Constitutive models for geologic materials and their application to excavation problems [PhD dissertation], Kyoto, Japan: Kyoto University; 1994. Nishikami H, Horii H. Plane strain compression test of cemented sand and measurement of strain localization, Pro. In: Proceedings of the 28th Annual Conference of Japan Geotech. Soc; 548-548 (in Japanese); 1993. Koike M. Elasto-viscoplastic constitutive model with strain softening for soft rock, Master thesis. Kyoto University, Kyoto, Japan. (in Japanese); 1997. Nishimura T. Experimental research and modeling of thermo-creep behavior of sedimentary soft rock and its application to BVP, Master thesis, Nagoya Institute of Technology, Nagoya, Japan (in Japanese); 2013.
Contents lists available at ScienceDirect
International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
A unified thermo-elasto-viscoplastic model for soft rock a
b,⁎
c
d
MARK e
Yong-lin Xiong , Guan-lin Ye , He-hua Zhu , Sheng Zhang , Feng Zhang a
Faculty of Architectural, Civil Engineering and Environment, Ningbo University, Fenghua Road 818, Ningbo 315211, China Department of Civil Engineering and State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China Department of Geotechnical Engineering, Tongji University, Siping Road 1239, Shanghai 200092, China d Department of Civil Engineering, Central South University, Shaoshan South Road 22, Changsha 410075, China e Department of Civil Engineering, Nagoya Institute of Technology, Showa-ku, Gokiso-cho, Nagoya 466-8555, Japan b c
A R T I C L E I N F O
A BS T RAC T
Keywords: Thermo-elasto-viscoplastic Soft rock Shear strength Confining stress dependency Temperature
A unified advanced thermo-elasto-viscoplastic constitutive model for soft rock is proposed in the critical state framework. The model is able to describe the fundamental mechanical behavior of soft rock such as elastoplastic, strain hardening and softening, time dependency, confining stress dependency, intermediate principal stress dependency and temperature dependency with one set of parameters. The thermal-induced equivalent stress tensor σ∼ij and the transform stress tensor tij are adopted to consider the influence of temperature and intermediate principal stress. Two evolution equations for the shear strength and the overconsolidation are newly introduced to take into consideration the influences of the confining stress and time dependent behavior, respectively. The capability of the model is carefully validated through a series of element tests of different soft rocks. The material parameters involved in the model have clear physical meanings and can be easily determined by the triaxial compression tests and creep tests.
1. Introduction As is well known, soft rock often gives rise to geotechnical engineering problems. For instance, progressive failure of soft rock slope1,2, long-term stability of tunnels in strong weathered soft rock ground3,4 are directly linked to the mechanical behavior of soft rock under general stress condition. The establishment of a rational constitutive model of soft rock, which can describe properly the mechanical behavior of soft rock, has a great significance for predicting the deformation of soft rock. Physically, soft rock has an unconfined compressive strength of 1～30 MPa and its strength lies between that of soil and hard rock. In the past decades, many experimental researches have been carried out to investigate the mechanical behavior of soft rock5–12. In general, the mechanical behavior of soft rock is elastoplastic, strain-hardening and strain-softening, and depends on strain rate, time, confining stress, intermediate principal stress and temperature. Until now, many constitutive models have been proposed to describe the above-mentioned mechanical behavior of soft rock in the framework of continuum mechanics. Based on endochronic theory13, Oka and Adachi14 proposed an elastoplastic model for soft rock that can not only describe the strain hardening-softening of soft rock, but also exhibits less mesh size dependency in finite element analysis
⁎
compared to the other models available at that time. By adopting a transform stress tensor called as tij15 and Matsuoka-Nakai failure criteria17, Zhang et al.18 proposed an elasto-viscoplastic model for soft rocks to consider the influence of the intermediate principal stress that may greatly affect the strength and stress-dilatancy relation of soft rock under different loading paths. In the framework of critical state soil mechanics, Nova19 firstly proposed a constitutive model for soft anisotropic rocks. Subsequently, many constitutive models were established based on the Cam-Clay model20–24. For instance, Amorosi et al.21 dealt with a critical statebased constitutive model for soft rocks, which was successfully applied to the analysis of the response of a pyroclastic rock during in situ plate load tests. Regarding soft rock as a heavily overconsolidated soil, Zhang et al.21 proposed a modified elasto-viscoplastic model for soft rock, which can not only describe the strain hardening-softening, the strain rate dependency and the time dependency, but also the intermediate principal stress dependency, in which the tij stress tensor and the subloading concept25 were adopted. The evolution of the state variable ρ related to overconsolidation ratio, proposed by Nakai and Hinokio26, was redefined to consider the time dependency in the model. The model, however, has to take different value for the shear stress ratio Rf under different confining stresses, in other words, the model could not consider the confining stress dependency.
Corresponding author. E-mail address: [email protected] (G.-l. Ye).
http://dx.doi.org/10.1016/j.ijrmms.2017.01.006 Received 9 December 2015; Received in revised form 6 January 2017; Accepted 9 January 2017 1365-1609/ © 2017 Elsevier Ltd. All rights reserved.
International Journal of Rock Mechanics & Mining Sciences 93 (2017) 1–12
Y.-l. Xiong et al.
In addition, the thermal effect of soft rock is also a very important factor for geotechnical engineering and geophysical science. Various experimental researches27–33 showed that in most cases, the shear strength and the creep failure time of soft rocks show a decreasing trend as temperature increases. At the same time, some constitutive models were proposed to describe the influence of temperature on the deformation and the strength of soft rock34–37. For instance, Gao et al.34 proposed a thermo-viscoplastic model of rock by using thermal expansion coefficient, viscosity attenuation coefficient and damage variable, which can consider the influence of temperature on elasticity, viscosity and damage. Zhang & Zhang35 proposed a thermo-elastoviscoplastic model for soft sedimentary rock in the framework of critical state soil mechanics. Xiong et al.36 modified the model proposed by Zhang & Zhang35 using tij transformed stress to consider the influence of the intermediate principal stress. At present, however, there is still no constitutive model that can take into consideration all the above mentioned aspects of mechanical behavior of soft rock in a unified way with one set of parameters. In this paper, regarding the soft rock as a heavily consolidated soil, a unified thermo-elasto-viscoplastic constitutive model is proposed in the framework of critical state soil mechanics, which has the following features: (1) the equivalent stress36 is adopted to consider the influence of temperature; (2) an evolution equation for the mobilized shear strength under different confining stress is applied to consider the influence of confining stress; (3) an evolution equation for the void ratio difference related to overconsolidation is used to consider the strain hardeningsoftening and the time dependency based on the subloading concept; (4) the transform stress tensor tij is used to consider the influence of intermediate principal stress. Moreover, the performance of the proposed model is carefully investigated through comparison with the test results on different types of soft rocks, such as triaxial compression tests, triaxial creep tests and plane strain tests under different temperatures.
Fig. 1. Similarity of volumetric strain caused by real stress and equivalent stress due to change of temperature.
Γ=
2.2. Thermo-induced equivalent stress It is well known that a change in temperature not only produce elastic volumetric strain, but also cause plastic volumetric strain in a soft rock sample. This effect is similar to the way in which a real stress acts on a soft rock sample. Accordingly, it is assumed in this study that the thermo-elastic volumetric strain is induced by an imaginary stress increment Δσ∼m , namely, the equivalent stress increment. The similarity of elastic volumetric strains caused by the real incremental mean stress and the incremental equivalent stress due to a change in temperature is shown in Fig. 1. Considering the limitation of the variation range for temperature and the fact that temperature T should be greater than or equal to 0 °C, a linear relationship between the change in temperature (T-T0) and the thermo-elastic volumetric strain increment is assumed as:
Matsuoka and Nakai proposed a spatially mobilized plane17 (SMP), in which the shear-to-normal stress ratio is maximized between two principal stresses in three-principal-stress space. Based on the SMP, Nakai and Hinokio26 proposed a simple elastoplastic model using the stress tensor tij to take into consideration the intermediate principal stress on the deformation and strength of soil. A brief derivation of the transformed stress tensor tij is given in Appendix A. Due to the fact that SMP criterion can predict the failure of soft rocks more precisely than the extended Mises criterion, as reported by Ye et al.7, the tij stress tensor is used instead of the normal stress tensor in the derivation of the newly proposed constitutive model for soft rock.
ΔεveT = 3αT (T − T0 )
The total strain rate can be written as,
εij̇ =
+
=
εij̇ eσ
+
εij̇ eT
+
εij̇ p
t∼N = tN + Δt∼N = tN + 3KαT (T − T0 ) (1)
(2)
Eijkl = Γ δij δkl + G ( δik δjl + δil δjk )
(3)
(6)
where K is the bulk modulus of the soft rock, tN is the actual real mean stress, which is the same as p in ordinary stress space.
where ε̇ijeσ is the elastic strain rate induced by the stress rate and ε̇ijeT is the elastic strain rate induced by temperature change. The plastic strain rate ε̇ijp in Eq. (1) can be calculated by Eq. (17), the thermo-elastic strain ε̇ijeT can be easy to obtain using Eq. (5), and the strain rate ε̇ijeσ induced by real stress change is given by Hook's law, −1 εkleσ ̇ = Eijkl σij̇
(5)
where αT is a linear thermal expansion coefficient, and takes a negative value because a compressive volumetric strain is assumed to be positive in geomechanics, T is the actual temperature, and T0 is a reference temperature and is usually taken as 15 °C, an average global temperature of the earth. Based on Hooke's law, it is straightforward to give the expression of equivalent stress:
2.1. Thermo-elasto-viscoplastic strain rate
εij̇ p
(4)
E is Young's modulus, and v is Poisson's ratio.
2. Thermo-elasto-viscoplastic stress-strain relationship for soft rock
εij̇ e
νE E , G= (1 + ν )(1 − 2ν ) 2(1 + ν )
2.3. Yield function The yield function of the proposed model is the same as the model proposed by Xiong et al.36 and its expression is given as:
f = ln
where,
where 2
tN t + ζ (X ) − ln N1 = 0 tN 0 tN 0
(7)
International Journal of Rock Mechanics & Mining Sciences 93 (2017) 1–12
Y.-l. Xiong et al.
ζ (X ) =
1 β
β
( ) X M*
calculated as, (8)
ρ = (λ − κ )ln
tN and X=tS/tN are the mean stress and the stress ratio based on tij stress, and tN1 determines the size of the yield surface (the value of tN at tS=0). tN0, a reference mean stress, is an arbitrary value and is taken as 98 kPa in this paper. β is a parameter that determines the shape of the yield surface in tij stress space. If β takes a value equal to 1, the yield function has a form similar to that of the Cam-clay model. The value of M* is expressed by principal stress ratio XCS=(tS/tN)CS and plastic strain increment ratio YCS=(dεSMP*p/dγSMP*p)CS at critical state: 1/ β
β M * = (XCS + XCS β −1YCS )
2⎛ ⎜⎜ RCS − 3 ⎝
f = ln
YCS =
1−
RCS
2 ( RCS + 0.5)
(10)
RCS = σ1/ σ3 |at critical state is the ratio of principal stress σ1 and σ3 at critical state in conventional triaxial compression tests, dεSMP*p and dγSMP*p are a normal component and a tangential component of the principal plastic strain increment vector on the SMP respectively. In order to describe the behaviors of overconsolidated geomaterial, Hashiguchi & Ueno25 proposed the concept of subloading yield surface, in which the subloading yield surface is similar with the normally consolidated yield surface in geometry and the stress state is always at subloading yield surface. Fig. 2 shows the relation between the subloading yield surface and normally consolidated yield surface and their corresponding e-lntN relations in tij stress space. It is easy to obtain the subloading yield function in the following equation according to Eq. (7), f = ln
⎛ t tN t ⎞ + ζ (X ) − ⎜ln N1e − ln N1e ⎟ = 0 ⎝ tN 0 tN 0 tN1 ⎠
tN1e , tN 0
Cp =
λ−κ 1 + e0
(14)
2.4. Evolution law for shear strength at critical state It is well known that the quantity M* at critical state is constant for a soil, as shown in Eqs. (9) and (10). However, previous experimental results11,37,38 have shown that the shear strength (stress ratio) for soft rock at critical state owned different values under different confining stresses. Therefore, a concept of mobilized shear strength, with the strength varying with the confining stress level, is introduced in this new model. An evolution equation for M* proposed by Iwata11 is given as:
(11)
where tN1 and tN1e represent the size of the subloading yield surface and normal yield surface respectively. Similar to the Cam-clay model, the plastic volumetric strain in tij stress space can be expressed as
εvp = Cp ln
ρ ⎞ tN 1 ⎛ p ⎜εv − ⎟=0 + ζ (X ) − tN 0 Cp ⎝ 1 + e0 ⎠
It should be pointed out that the state variable ρ similarly as the plastic volume strain εvp , has no influence on the critical state. By using the state variable ρ , the model can produce viscoplastic strain in the overconsolidation status and has a smooth transition from the overconsolidation state to the residual state (the critical state for soft rock). It also can be seen from Eq. (13) that the value of tN1e may varies with ρ (whose value will be discussed in Session 2.7), while the tN1 is determined by current stress state. In other words, the expansion or shrinkage of the normal consolidated yield surface is controlled by ρ , however, no matter how it evolves, the subloading yield surface moves toward it and finally the two surfaces overlap.
(9)
1 ⎞ ⎟⎟ , RCS ⎠
(13)
Substituting Eqs. (12) and (13) into Eq. (11), finally the subloading yield surface of thermo-elasto-viscoplastic model is given as,
where
XCS =
tN1e tN1
M * = M0* +
∫ dM *,
dM * = A ln
* p MCS dεd M*
* OCR n M0* = MCS
(15)
(16)
where,M0* is the initial value of M* at the beginning of shear loading, OCR is the overconsolidation ratio of soft rock at initial condition, and * is the value of M0* when OCR=1.0. dεdp is an increment of MCS equivalent plastic shear strain during shear loading, and will be described in detail later. Two new parameters n and A are introduced: n is the exponent in the relationship between OCR and M0*, and A is the parameter that determines the evolution rate of M* during the shearing process. It can be seen from Eqs. (15) and (16) that the shear strength at critical state is mainly determined by two factors, one is the initial
(12)
where e0 is the void ratio at the reference state tN0=98 kPa under isotropic normal consolidated condition. λ is the compression index and κ is the swelling index. tN1e is the pre-consolidation stress, which has the same physical meaning as the pc in the Cam-clay model. At the same time, the density difference ρ in Fig. 2 can be easily
Fig. 2. Shape of subloading yield surface and normally yield surface, and definition of ρ (After Nakai and Hinokio26).
3
International Journal of Rock Mechanics & Mining Sciences 93 (2017) 1–12
Y.-l. Xiong et al.
OCR, the other is the history of plastic shear strain.
2.8. Plastic operator Λ
2.5. Flow rule
Substituting the Eqs. (15)–(17) and (20)–(21) into Eq. (18), the plastic operator Λ can be obtained as
As the same as the Cam-clay model, the associated flow rule is adopted in this model. The thermo-elasto-viscoplastic strain rate is defined as,
Λ=
⎛ ∂f ∂f δij ⎞ ⎛ ∂f ∂f δij ⎞ ∂f ∂f ⎟⎜ ⎟ εij̇ p = Λ , εv̇ p = Λ , dεḋp = Λ ⎜ − − ∂tij ∂tii ∂tkk 3 ⎠ ⎝ ∂tij ∂tkk 3 ⎠ ⎝ ∂tij
where
hTp / Cp − A ln
(17)
where Λ is a positive scalar called as plastic operator. 2.6. Consistency equation
(18)
(23)
(24)
fσ̇ =
∂f σij̇ ∂σij
(25)
(26)
h (tcreep )/ Cp * ∂f MCS M * ∂M *
⎛ ∂f ⎜ ∂t − ⎝ ij
∂f δij ⎞ ⎛ ∂f ⎟⎜ ∂tkk 3 ⎠ ⎝ ∂tij
−
∂f δij ⎞ ⎟ ∂tkk 3 ⎠
(27)
From the above equation, the reason why using the inhomogeneous function in Eq. (21) because Λ will be zero under a pure creep state if homogeneous function is used and the viscoplastic strain will never occur, which will not achieve the purpose of the proposed model. 2.9. Loading criterion In the proposed model, the loading criterion is given as,
εij̇ p > 0 when Λ > 0 (19)
≤0
viscoplastic (loading) εij̇ p = 0 when Λ
elastic (unloading)
(28)
In the case of loading, the plastic operator Λ is positive. From Eq. (23), there exist two different situations listed as follows:
⎧Λ = ⎪ ⎨ ⎪Λ = ⎩
(20)
numerator > 0 deno min ator > 0 numerator < 0 deno min ator < 0
> 0 (i ) > 0 (ii)
(29)
The second part of the numerator of Eq. (23), which contributes to the viscous strain, can be omitted for simplicity. Then the two situations can be rewritten as follow:
⎧ ḟ = ⎪ σ ⎨ ⎪ fσ̇ = ⎩
∂f σ̇ ∂σij ij
> 0 strain − hardening (i )
∂f σ̇ ∂σij ij
< 0 strain − softening (ii)
(30)
From Eq. (30), situation (i) represents the strain hardening because the subloading yield surface is expanding, while situation (ii) represents the strain softening because the subloading yield surface is shrinking. Fig. 3 demonstrates how the model predicts the strain-hardening and strain-softening behaviors in a drained triaxial test of soft rock with a large OCR. The change of deviatoric stress and volumetric strain can be divided into three stages: (1) Before the stress path reaches the critical state line (CSL), the denominator > 0 (situation i) and the current stress point locates to the right side of CSL, leading to a strainhardening (subloading yield surface expanding) and a volumetric contraction. (2) After the stress path goes beyond the CSL, the denominator is still larger than zero, which forces the strain-hardening
(21)
where
t∼N = tN + 3KαT (T − T0 )
∂f δij ⎞ ⎟ ∂tkk 3 ⎠
G (ρ , tcreep ) ∂f + ∂tkk tN + 3KαT (T − T0 )
hTp / Cp − A ln
ε̇v 0 is an initial volumetric strain rate at the time tcreep=0 which represents the time when shearing begins. tcreep1 is a unit time used to standardized the time and always take the value of 1.0. α is a parameter that controls the gradient of strain rate vs. time in logarithmic axes during a creep test. Cn controls the strain rate dependency of soft rocks. a is a parameter controlling the losing rate of overconsolidation. It should be pointed out that the values of the parameters Cn and α are not objective and are dependent on the unit of time. In its application to boundary value problem, the unit of time used in numerical analysis should be the same as the one used in determining the parameters from the laboratory tests. In this model, in order to consider the influence of temperature on the deformation and strength of soft rock, the equivalent stress t∼N induced by temperature is adopted instead of normal mean stress tN in the above evolution equation of state variable ρ, and the corresponding expression is written as, G (ρ , tcreep ) ρ̇ = −Λ + h (tcreep ) 1 + e0 t∼N
−
hTp =
Λ=
⎪ ⎪
∂f δij ⎞ ⎛ ∂f ⎟⎜ ∂tkk 3 ⎠ ⎝ ∂tij
Therefore, Eq.(23) can also be written as
where
⎧ h (tcreep ) = εv̇ 0 (1 + tcreep / tcreep1)−α ⎨ 1+ C ln(1+ tcreep / tcreep1) ⎩G (ρ , tcreep ) = aρ n
⎛ ∂f ⎜ ∂t − ⎝ ij
fσ̇ = (∂f /∂σij ) σij̇ = 0
2.7. Evolution equation for state variable ρ In the consistency equation, it is necessary to given evolution for the development of the state variable ρ . Based on the works by Nakai and Hinokio26, Zhang et al.22 proposed an elasto-viscoplastic model for soft rock, in which the time dependent evolution equation for the state variable ρ is given as an inhomogeneous function,
G (ρ , tcreep ) ρ̇ = −Λ + h (tcreep ) 1 + e0 σm
* ∂f MCS M * ∂M *
When the soil enters into a pure creep state, the following relation holds:
It is known that the plastic potential determines the direction of the development rate of plastic strain, and the yield function gives the yield criterion of the geomaterials, while the consistency equation determines the value of the plastic strain, in other word, the plastic operator Λ can be determined. Because the quantity M* at the critical state is not constant, the change of M* should be taken into consideration. Finally the consistency equation can be obtained according to Eq. (14),
∂f ∂f ρ̇ ⎤ 1 ⎡ p ⎢εv̇ − ⎥=0 σij̇ + Ṁ * − ḟ = 0 ⇒ ∂σij Cp ⎣ ∂M * 1 + e0 ⎦
fσ̇ + h (tcreep )/ Cp
(22)
4
International Journal of Rock Mechanics & Mining Sciences 93 (2017) 1–12
Y.-l. Xiong et al.
Fig. 3. Illustration of how the model predicts the strain-hardening and strain-softening behaviors in a drained triaxial test of soft rock with a large OCR.
density difference ρ is considered by using the thermo-induced equivalent stress.
to continue so that the stress path keeps moving upward to the normal consolidated yield surface. Since the current stress point locates to the left side of CSL, the volume begins to expand. (3) After the stress path reaches the normal consolidated yield surface, the denominator becomes less than zero (situation ii), the subloading yield surface turns to shrink with the normal consolidated yield surface, that is the strainsoftening. The deviatoric stress keeps decreasing and the volume keeps expanding until the stress path reaches the CSL finally. Accordingly, the loading criterion is rewritten as follows:
εij̇ p
> 0 when Λ > 0
3. Performance of the proposed model 3.1. Determination of the parameters involved in the proposed model Thirteen parameters are involved in the proposed model, among which five parameters are the same as the Cam-Clay model. The determination of these five parameters can be found out in the literatures and do not discussed in this paper. The parameter Cn is determined by the difference of peak strengths of two rock samples under drained triaxial compression tests with different constant shear strain rates. Parameter α is the gradient of strain rate vs. time in logarithmic axes and can be uniquely determined with drained triaxial creep test as shown in Fig. 4. The parameter β controls the shape of the yield function and the stress-dilatancy relation (associated flow rule) shown in Fig. 2. β can be determined based on the test data of stress-dilatancy relation. The overconsolidation parameter a, which controls the losing rate of overconsolidation, can be determined from the peak strengths of soft rock specimens under triaxial compression test by curve fitting, in which at least two samples, with the same physical conditions but different initial overconsolidation, should be tested under the same loading condition. The parameter n can be determined by triaxial compression tests with different initial OCR and using Eq. (15). The parameter A, which determines the evolution rate of M* during the shear loading process, can be determined based on the test results of drained triaxial compression tests under different confining stresses. The parameter αT is the coefficient of linear thermal expansion of soft rock, a well-known physical quantity that can be obtained from text books or databases. The calibration methods of material parameters involved in the proposed model is summarized in Table 1.
⎧ f ̇ > 0 hardening ⎪ σ and ⎨ fσ̇ < 0 softening εij̇ p = 0 when Λ ⎪ ̇ ⎩ fσ = 0 pure creep
≤ 0elastic (unloading)
(31)
It is worth noting that different from other constitutive models, this loading criterion not only can predict strain hardening (expansion of failure surface) and strain softening (shrinking of failure surface), but also the creep state where even if the stress tensor rate is equal to zero, it is still regarded as a loading process and viscoplastic strain may happen. Meanwhile, the thermal plastic strain can be taken into
3.2. Parametric study of the proposed model In this section, the influence of parameters on the performance of the new model is investigated by simulating triaxial compression tests and creep tests. The parameters used in the simulations are listed in Table 2. In the simulations, the strain rate is taken as 1.0×10−4 1/min. Fig. 5(a), (b), (c), (d) show the calculated nonlinear stress-strain
Fig. 4. Creep behavior of soft rock under different creep stresses (drained condition).
account automatically because the influence of temperature on the 5
International Journal of Rock Mechanics & Mining Sciences 93 (2017) 1–12
Y.-l. Xiong et al.
Table 1 Calibration methods of material parameters. Parameter
Physical meaning
Calibration method
E Ep(=λ−κ): α a Cn v β M*CS αΤ n A
young's modulus plastic index creep index controls the loss rate of overconsolidation controls the evolution of strain rate Possion's ratio controls the shape of yield surface critical stress ratio when OCR = 1.0 linear thermal expansion coefficient exponential parameter determines the relationship between OCR and critical stress ratio M*CS. determines the evolution rate of M* during shearing
triaxial test consolidation test triaxial creep test triaxial test triaxial test with different strain rates triaxial test stress-dilatancy relation of triaxial test triaxial test conventional thermal test triaxial test with different confining stresses triaxial test with different confining stresses
Table 2 Material parameters for rocks used in parametric study. E (MPa)
Ep (=λ−κ)
a
α
Cn
ν
β
M*CS
αT (K−1)
n
A
900
0.04
500
0.7
0.025
0.0804
1.0
0.40
-8.0×10-6
0.15
1.5
3.4. Plane strain tests under different confining stresses at room temperature
relationship and volume change during triaxial compression tests with different parameters at 15 °C. It can be seen that for larger values of the parameters β and A, the shear strength is low. And for larger values of the parameters a and n, the shear strength is higher. Fig. 5(e) shows the calculated results of triaxial compression tests with different parameter αT and different temperature T. It is shown clearly that the shear strength decreases as the temperature increases, and the larger the linear thermal expansion is, the larger the decrease of shear strength will be. Fig. 5(f) shows the results of creep tests with different parameter α and different temperature T. In the simulation, the creep stress is chosen as 90% of the shear strength with parameter α=0.7 at 15 °C. It can be seen that the creep failure time decreases as the temperature increases, and the smaller the parameter α is, the earlier the creep failure will happen.
In order to validate the ability of the model to consider the influence of the intermediate principal stress on the deformation and strength of soft rock, the test results of manmade soft rock under the plane-strain condition at room temperature conducted by Nishikami and Horii39 are used. The manmade soft rock is made from standard Toyoura sand, gesso and water with the proportion of 69.6%, 11.6% and 18.8% respectively. The plane-strain condition of the samples was realized by clasping the sample with two vertical steel plates that are very stiff and supported by several fixed rollers in the test. The silicon grease is used to reduce the friction between the plate and manmade soft rock. The loading strain rate is given as 0.083% in the tests. Three test results with the confining stress of 0.196 MPa, 0.294 MPa and 0.392 MPa are employed. The physical properties and the material parameters of manmade soft rock involved in the model are listed in Tables 3 and 4. Fig. 7 illustrates the measured and computed stress-strain relationship of manmade soft rock under different confining stress. The comparisons between the test results and the model calculations show that the model is able to reproduce both the test results of the maximum and the intermediate principal stresses. Moreover, the model also can describe the test results of the volumetric strain, which changed from the positive dilatancy to negative dilatancy as the confining stress increased.
3.3. Triaxial shear tests under different confining stresses at room temperature In order to verify the performance of the proposed model, drained triaxial compression tests conducted by Iwata11 are firstly simulated to examine the mechanical behavior of the model. The Ohya stone, a soft sedimentary rock known as green tuff, widely distributed in northeast Japan, is use for testing. The specimen is saturated through repeated process of evacuation in desiccator before testing. The specimen, 100 mm in height and 50 mm in diameter, is isotropically consolidated to a prescribed confining stress and then a strain-rate-controlled vertical load is applied under drained condition. During shearing process, the loading rate is 0.001%/min and the axial strain and volumetric strain are measured at the same time. The physical properties and the material parameters of Ohya stone involved in the model are listed in Tables 3 and 4. Fig. 6 shows the comparison of the stress-strain-dilatancy relation between the simulated results and experimental results. It can be seen that the model can describe well the observed mechanical behaviors, such as strain-hardening and strain-softening, under confining stress from 0.1 to 4.0 MPa at room temperature. It is worth noting that all the parameters used in the simulation are the same except the initial OCR, which took different value under different confining stress.
3.5. Triaxial tests with different loading rates at room temperature The proposed model is used to simulate the mechanical behavior of triaxial compression tests at different strain loading rate on Tomuro stone40, which also is a green tuff. The tests were conducted at two different loading rate of 0.28% and 0.0012%/min respectively with 0.1 MPa confining stress. The initial values of the state parameters and the material parameters of the Tomuro stone are listed in Tables 3 and 4. Fig. 8 shows the comparisons of the calculated and measured deviatoric stress ~ axial strain relationship at different loading rate in
6
International Journal of Rock Mechanics & Mining Sciences 93 (2017) 1–12
Y.-l. Xiong et al.
Fig. 5. Parametric study of the proposed model.
the triaxial compression test. It is found that the rate dependency behavior, namely, the peak strength increases with the loading rate, is described well by the proposed model.
Table 3 Physical properties of soft rocks.
Reference void ratio e0 Specific gravity of soil particles Gs Pre- consolidation pressure p’c (MPa)
Ohya stone
Manmade soft rock
Tomuro stone
Tage stone
0.57 2.51
0.62 2.53
0.63 2.53
0.50 2.52
21.0
8.8
10.0
20.0
3.6. Triaxial tests with different confining stresses and different temperatures In order to investigate the influence of temperature on the deformation and strength of soft rock, Nishimura41 conducted a serial of triaxial compression tests under different confining stress and different temperature. The soft rock used in the tests is Tage stone,
7
International Journal of Rock Mechanics & Mining Sciences 93 (2017) 1–12
Y.-l. Xiong et al.
Table 4 Material parameters for soft rocks.
Ohya stone Manmade soft rock Tomuro stone Tage stone
E (MPa)
Ep (=λ−κ)
a
α
Cn
ν
β
M*CS
αT (K−1)
n
A
600. 600. 350. 1000.
0.005 0.005 0.023 0.015
850. 500. 800. 3000.
0.6 0.85 0.80 0.050
0.02 0.00 0.05 0.25
0.02 0.08 0.0864 0.12
1.1 1.5 1.5 1.1
0.32 0.32 0.58 0.46
-8.0×10-6 -9.0×10-6 -1.0×10-5 -2.5×10-5
0.19 0.11 0.15 0.12
4.0 1.0 1.5 1.0
Fig. 6. Comparisons of plane strain test and calculated results for manmade soft rock under different confining stresses at room temperature (test data from Iwata11).
3.7. Triaxial creep tests under different confining stresses and different temperatures
another kind of Japanese green tuff. Some basic physical properties of Tage stone and a description of it can be found in41. In the tests, the confining stresses were 0.49 MPa and 0.98 MPa and the temperature was 20 ℃ and 80 ℃ respectively. In addition, a back pressure of 0.49 MPa was applied in all the tests to guarantee a full saturation of the specimen. Table 4 lists the material parameters of Tage stone. Fig. 9 presents the calculated and measured stress-strain-dilatancy relations of Tage stone under different confining stresses and different temperatures in the triaxial compression test. The test results indicate that the peak strength of Tage stone decreased and the volumetric strain became contractive as the temperature increased. From the comparisons, it can be known that the model can describe well the influence of temperature and confining stress on the deformation and strength of soft rock with one set of parameters.
The ability of the model to describe the influence of temperature on the creep failure under different temperature is validated through the creep test results on Tage stone41. In the test, instead of applying the axial load abruptly, the load was applied under constant loading rate until the prescribed creep stress was reached. The creep stresses used in the creep tests are 95% of the peak stresses of the specimens under the same confining stress and the temperature in triaxial compression tests. The parameters used in the model, of course, are the same as the triaxial compression test in Table 4. Fig. 10 show the comparisons between the calculated and measured results of Tage stone in drained triaxial creep tests under different constant temperatures and confining stresses. The confining stresses
8
International Journal of Rock Mechanics & Mining Sciences 93 (2017) 1–12
Y.-l. Xiong et al.
Fig. 8. Comparisons of triaxial test and calculated results for Tage stone under different confining stresses and temperatures (test data from Koike40).
approximately one-order of magnitude larger than the one measured in tests, implying that some modification should be made in future study. 4. Conclusions A unified thermo-elasto-viscoplastic model has been proposed within the critical state framework from the viewpoint that the soft rock can be regarded as a heavily overconsolidated geomaterial. The subloading yield surface is utilized to simulate the typical mechanical behavior of soft rock. By introducing a new evolution equation for the critical state parameters M* under different confining stresses, the model is able to describe the confining-stress dependency of soft rock with a set of fixed material parameters whose values are identical under all loading conditions. This is the main difference compared with previous model (Xiong et al.36), in which different parameters were used to describe the mechanical behavior of soft rock under different confining stresses. Based on the fact that a change in temperature produce not only elastic volumetric strain, but also plastic volumetric strain in a soft rock sample, a thermo-induced equivalent stress is introduced to evaluate the influence of temperature on the deformation and the strength of soft rock. The transform stress tensor tij is adopted in the model to consider the influence of intermediate principal stress, which is very importance to describe the stress-strain relation of soft rock under general loading conditions. Thirteen parameters are involved in the proposed model, among which five parameters are the same as Cam-clay model. The other parameters can be definitely calibrated via conventional laboratory tests. The validity of the proposed model is confirmed carefully by serials of element tests under different loading and thermal conditions with one set of parameters. However, further improvement is necessary to increase the accuracy of predictions, such as the onset of creep failure and the volumetric strain during drained triaxial tests.
Fig. 7. Comparisons of triaxial test and calculated results for Tomuro stone under different loading rates at room temperature (test data from Nishikami and Horii39).
Acknowledgement and temperatures are also the same as the triaxial compression tests. It can be known from the comparisons that the model is able to describe the creep behavior of Tage stone under different confining stresses and temperatures quantitatively, with the same parameters as the triaxial compression tests. However, the calculated creep failure time is
The support of the National Nature Science Foundation of China (Grant Nos. 51478345, 41372284, 51608385 and 51478228), National Basic Research Program of China (2014CB047001) and K.C. Wong Magna Fund in Ningbo University are greatly appreciated.
9
International Journal of Rock Mechanics & Mining Sciences 93 (2017) 1–12
Y.-l. Xiong et al.
Fig. 9. Comparisons of triaxial creep test and calculated results for Tage stone under different confining stresses and temperatures (test data from Nishimura41).
Fig. 10. Comparisons of triaxial creep tests and calculated results for Tage stone under different confining stresses and temperatures (test data from Nishimura
41
).
Appendix A As shown in Fig. A1, the Spatial Mobilized Plane (SMP), defined by Nakai and Matsuoka16, can be expressed in a principal stress space (σI, σII, σIII) as
σI σ σ + II + III = 1. σ1 σ2 σ3
(A-1)
Therefore, the normal of the plane (a1, a2, a3) can be evaluated by the following equation as,
ai =
I3 I2 σi
(i = 1, 2, 3)
(A-2)
where I1, I2 and I3 are the first, second and third invariants of effective stress tensor and can be expressed by the following forms using three principal stresses:
I1 = σ1 + σ2 + σ3 I2 = σ1 σ2 + σ2 σ3 + σ3 σ1 I3 = σ1 σ2 σ3
(A-3)
In tij clay and sand models, a symmetric tensor tij is expressed by a product of stress tensor σik and a tensor akj as:
tij = σik akj
(A-4)
10
International Journal of Rock Mechanics & Mining Sciences 93 (2017) 1–12
O
O
O
O
Y.-l. Xiong et al.
O
O
+
Fig. A 1. Explanation of SMP (Nakai and Mihara15).
where the tensor akj can be evaluated by Cayley-Hamilton's theory as
aij =
I3 −1 rij , rik rkj = σij , rij = (σik + I2 δik )(I1 σkj + I3 δkj )−1 I2
(A-5)
Because rij is a one-half power function of σij, rij is a symmetric tensor and its principal directions are the same as σij and have the principal σ2 , σ3 ). Therefore aij is also a symmetric tensor and has the same principal directions as σij and has the principal value of (a1, values of ( σ1 , a2, a3). As a result, the tensor tij is also a symmetric tensor and has the same principal directions as aij and σij. A normal component tn and a tangential component ts of the principal-value vector of tij can be given as (see Fig. A2a),
tN = t1 a1 + t2 a2 + t3 a3 = tij aij ≡ σSMP = σ1 a12 + σ2 a 22 + σ3 a32 = 3 tS =
(t12 + t22 + t32 ) − tN2 =
≡ τSMP =
I3 I2
(A-6)
tij tij − (tij aij )2
(σ1 − σ2 )2a12 a 22 + (σ2 − σ3)2a 22 a32 + (σ3 − σ1)2a32 a12 =
I1I2 I3 − 9I32
(A-7)
I2
It is assumed that the directions of plastic principal strain increments coincide with those of principal axes of tij, and the plastic strain increment *p ) of the principal plastic strain increment vector on the SMP (see *P ) and a tangential component (dγSMP can also be given by a normal component (dεSMP Fig. A2b),
*P = dε1P a1 + dε2P a2 + dε3P a3 = dεijP aij dεSMP
dεijp dεijp − (dεijp aij )2
Fig. A 2. Explanation of stress and strain in principal-value space of tensor tij (Nakai and Mihara15).
11
O
O
O
O
(A-9)
O
*p2 = (dε1p2 + dε2p2 + dε3p2 ) − dεSMP
O
*p = dγSMP
(A-8)
International Journal of Rock Mechanics & Mining Sciences 93 (2017) 1–12
Y.-l. Xiong et al.
The failure criteria of tij models is defined by the SMP failure criteria (Plane ABC in Fig. A1b),
⎛ I I I − 9I 2 ⎛t ⎞ ⎛τ ⎞ 1 2 3 3 Xf ≡ ⎜ S ⎟ ≡ ⎜ SMP ⎟ = ⎜⎜ ⎝ tN ⎠ f ⎝ σSMP ⎠ f ⎝ 9I3
⎞ ⎟ = const . . ⎟ ⎠f
(A-10)
References 22 1 Zhang F, Yashima A, Osaki H, Adachi T, Oka F. Numerical simulation of progressive failure in cut slope of soft rock using a soil-water coupled finite element analysis. Soils Found. 2003;43(5):119–131. 2 Ye GL, Zhang F, Yashima A, Sumi T, Ikemura T. Numerical analysis on progressive failure of slope due to heavy rain with 2D and 3D FEM. Soils Found. 2005;45(2):1–15. 3 Ferrero AM, Migliazza M, Roncella R, Tebaldi G. Analysis of the failure mechanisms of a weak rock through photogrammetrical measurements by 2D and 3D visions. Eng Fract Mech. 2008;75(3):652–663. 4 Yashima A, Matsumoto A, Tanabe K. Slope failure at tunnel entrance due to excavation and its countermeasure. in: Proceedings of the International Conference Modern Tunneling Science and Technology. Kyoto, p.557-562; 2001. 5 Agustawijaya DS. The uniaxial compressive strength of soft rock. Civ Eng Dimens. 2007;9(1):9–14. 6 Liao HJ, Su LJ, Pu WC, Yin JH. Test and numerical analysis of the constitutive relation of a diatomaceous soft rock. Mar Georesources Geotechnol. 2003;21(1):183–200. 7 Ye GL, Zhang F, Yashima A, Aung H, Naito K. Influence of membrane and filter paper on plane-strain testing of soft sedimentary rock. Geotech Test J. 2007;30(6):442–453. 8 Ye GL, Zhang F, Naito K, Aung H, Yashima A. Test on soft sedimentary rock under different loading paths and its interpretation. Soils Found. 2007;47(5):897–909. 9 Ye GL, Ye B, Zhang F. Strength and Dilatancy of Overconsolidated Clays in Drained True Triaxial Tests. J Geotech Geoenviron Eng. 2014;140(4):80–90. 10 You MQ. True-triaxial strength criteria for rock. Int J Rock Mech Min Sci. 2009;46(1):115–127. 11 Iwata M. Modification of Constitutive Model for Soft Rock for Slope Failure Prediction [PhD Thesis], Gifu, Japan: Gifu University; 2011 [in Japanese]. 12 Chiu HK, Johnston IW, Donald IB. Appropriate techniques for triaxial testing of saturated soft rock. Int J Rock Mech Min Sci. 1983;20(3):107–120. 13 Valanis KC. A theory of viscoplasticity without a yield surface. Arch Mech. 1971;23(4):517–533. 14 Oka F, Adachi T. A constitutive equation of geologic materials with memory. In: Proceedings of the 5th International Conference on Numerical Method in Geomechanics; No. 1: 293–300; 1985. 15 Nakai T, Mihara Y. A new mechanical quantity for soils and its application to elastoplastic constitutive models. Soils Found. 1984;24(2):82–94. 16 Nakai T, Matsuoka H. A generalized elastoplastic constitutive model for clay in a three-dimensional stress. Soils Found. 1986;26(3):81–89. 17 Mastuoka H, Nakai T. Stress deformation and strength characteristics of soil under three different principal stresses. Proc JSCE. 1974;232:59–70. 18 Zhang F, Yashima A, Nakai T, Ye GL, Adachi T, Oka F. An elastoplastic strainhardening and strain-softening constitutive model for soft rock considering the influence of intermediate stress. Soils Found. 2003;43(5):107–117. 19 Nova R. An extended Cam-clay model for soft anisotropic rocks. Comput Geotech. 1986;2:69–88. 20 Nova R, Castellanza R, Tamagnini C. A constitutive model for bonded geomaterials subject to mechanical and/or chemical degradation. Int J Numer Anal Meth Geomech. 2003;27(9):705–732. 21 Amorosi A, Aversa S, Boldini D, Laera A, Nicotera MV. Application of a new
23
24
25
26 27 28
29
30 31 32
33 34 35 36
37 38 39
40 41
12
constitutive model to the analysis of plate load tests in a pyroclastic rock. Int J Rock Mech Min Sci. 2015;78:271–282. Zhang F, Yashima A, Nakai T, Ye GL, Aung H. An elasto-viscoplastic model for soft sedimentary rock based on tij concept and subloading yield surface. Soils Found. 2005;45(1):65–73. Zhu HH, Ye B, Cai YC, Zhang F. An elasto-viscoplastic model for soft rock around tunnels considering overconsolidation and structure effects. Comput Geotech. 2013;50:6–16. Fu YK, Iwata M, Ding WQ, Zhang F, Yashima A. An elastoplastic model for soft sedimentary rock considering inherent anisotropy and confining-stress dependency. Soils Found. 2012;52(4):575–589. Hashiguchi K, Ueno M. Elastoplastic constitutive laws of granular materials, Constitutive equations of soils, In: Proceedings 9th ICFSME, Tokyo, JSSMFE, 73– 82; 1977. Nakai T, Hinokio M. A simple elastoplastic model for normally and over consolidated soils with unified material parameters. Soils Found. 2004;44(2):53–70. Nasseri MHB, Goodfellow SD, Wanne T, Young RP. Thermo-hydro-mechanical properties of cobourg limestone. Int J Rock Mech Min Sci. 2013;61:212–222. Okada T. Mechanical properties of sedimentary soft rock at high temperatures (part 1) – Evaluation of temperature dependency based on triaxial compression test. Civ Eng Res Lab Rep. 2005 [No. N04026 (in Japanese]. Okada T. Mechanical properties of sedimentary soft rock at high temperatures (part 2) – Evaluation of temperature dependency of creep behavior based on unconfined compression test. Civ Eng Res Lab Rep. 2006 [No. N05057 (in Japanese]. Ye GL, Nishimura T, Zhang F. Experimental study on shear and creep behaviour of green tuff at high temperatures. Int J Rock Mech Min Sci. 2015;79:19–28. Yang C, Daemen JJK. Temperature effects on creep of tuff and its time—dependent damage analysis. Int J Rock Mech Min Sci. 1997;34(3–4) [345.e1-345.e12]. Zhao Y, Wan Z, Feng Z, Yang D, Zhang Y, Qu F. Triaxial compression system for rock testing under high temperature and high pressure. Int J Rock Mech Min Sci. 2012;52:132–138. Gao F, Xu XL, Yang XJ, Ju Y. Research on thermo-visco-elastoplastic model of rock. Chin J Rock Mech Eng. 2009;28(1):74–80 [in Chinese]. Zhang S, Zhang F. A thermo-elasto-viscoplastic model for soft sedimentary rock. Soils Found. 2009;49(4):583–595. Zhang S, Leng WM, Zhang F, Xiong YL. A simple thermo-elastoplastic model for geomaterials. Int J Plast. 2012;34:93–113. Xiong YL, Zhang S, Ye GL, Zhang F. Modification of thermo-elasto-viscoplastic model for soft rock and its application to THM analysis of heating tests. Soils Found. 2014;54(2):176–196. Adachi T, Ogawa T. Mechanical Properties and failure criteria of soft rock, Proceedings JSCE; 295: 51–62 (in Japanese); 1980. Zhang F. Constitutive models for geologic materials and their application to excavation problems [PhD dissertation], Kyoto, Japan: Kyoto University; 1994. Nishikami H, Horii H. Plane strain compression test of cemented sand and measurement of strain localization, Pro. In: Proceedings of the 28th Annual Conference of Japan Geotech. Soc; 548-548 (in Japanese); 1993. Koike M. Elasto-viscoplastic constitutive model with strain softening for soft rock, Master thesis. Kyoto University, Kyoto, Japan. (in Japanese); 1997. Nishimura T. Experimental research and modeling of thermo-creep behavior of sedimentary soft rock and its application to BVP, Master thesis, Nagoya Institute of Technology, Nagoya, Japan (in Japanese); 2013.