Time-dependent deformation behaviors of weak sandstone

Time-Dependent Deformation Behaviors of Weak Sandstones

L.S. Tsai1 _{Y.M. Hsieh}2 _{M.C. Weng}3 _{T.H. Huang}4 _{F.S. Jeng}4, *

1 _{Engineer, China Engineering Consultants, Inc., Taipei, Taiwan.}

2 _{Assistant professor, Department of Construction Engineering, National Taiwan University of Science and Technology,}

Taipei, Taiwan.

3 _{Assistant professor, Department of Civil and Environmental Engineering, National University of Kaohsiung,}

Kaohsiung, Taiwan..

4 _{Professor, Department of Civil Engineering, National Taiwan University, Taipei, Taiwan.} * _{Corresponding author, [email protected]; Tel/Fax: +886 2 2364 5734}

Abstract

Time-dependent deformation behaviors of rocks have significant impact on the stability of rock slopes or underground constructions. This paper presents systematic experimental data regarding time-dependent deformations of a typical weak sandstone, known as the Mushan sandstone. The observed deformations are further separated to distinguish elastic and visco-plastic material behaviors of the weak sandstone through the use of multi-staged loading-unloading-reloading-creep tests. Meanwhile, the stress path is designated to be a purely hydrostatic loading followed by a pure shearing so that the deformations induced by these two types of stress can be distinguished.

For elastic behaviors, although the nonlinear stress-strain relations vary according to the applied hydrostatic stress, these stress-strain relations can be normalized by the applied hydrostatic stress or the bulk modulus and converted into a single consistent stress-strain curve. Inelastic behaviors are then obtained by subtracting elastic deformations from total deformations. As a result, the characteristics of the visco-plastic deformation are found to be: 1) the direction of the visco-plastic flow is time independent, and 2) it has a similar direction as the conventionally defined plastic flow. As such, the visco-plastic potential has a similar shape as the plastic potential, but the size of the former one changes with time, while the later has a size which is time independent. Meanwhile, through the calculation of irreversible works, direct evidence of orthogonality between * Manuscript

the yield surface and the plastic flow, as well as the visco-plastic flow, is observed in experiments. As a result, it is reasonable to assert that the yield surface, the plastic potential, and the visco-plastic potential all have the same geometry. Consequently, the associated flow rules are applicable to modeling the time-dependent deformational behaviors of weak sandstones.

Keywords: weak sandstone, plastic potential surface, visco-plastic potential surface,

Nomenclature

a0 Elastic constant of proposed regression _curve

a1 Elastic constant of proposed regression _curve

a2 Elastic constant of proposed regression

curve

CTC Conventional triaxial compression test

ij

e Deviatoric strain tensor

' 1

I

First strain invariant J2

Second deviatoric stress invariant,

ji ijs s J 2 1 2=

J2f Second deviatoric stress invariant at _failure

' 2

J

Second deviatoric strain invariant,

ji ije e J 2 1 ' 2 = H Yield function, 2 2 ( , ) _H( ) _D( , ) H p J ≡H p +H p J H

H Yield function under hydrostatic _compression

D

H Yield function of under pure shearing _condition p Hydrostatic stress p I σ_kk 3 1 = = 1 3 1 _(MPa) ij

S Deviatoric stress tensor

H

T The time at the end of hydrostatic _compression T The time corresponding to a certain time _{during pure shearing}

I

W Irreversible stress work,

( ) ( ) ( ) I I I H D W t ≡W t +W t I H

W irreversible stress work under hydrostatic _compression

I D

W irreversible stress work under pure _{shearing condition} β1

Plastic flow angle,

1 1 tan / p p v d d

β

₌ − _

γ

ε

_   β2

Visco-plastic flow angle,

( ) 0 0 1 2 tan / vp vp t t v t t β − γ ε − −   = _{ } v ε Volume strain, ' 1 v

I

kk

ε

= =

ε

p d_ε

Increment of plastic strain,

( ) ( )

2 2 p p p v dε = dε + dγ p v

d

ε

Plastic volumetric strain increment

t ij

ε

Second strain tensor of total strain

e ij

ε

Second strain tensor of elastic component

e p v,

ε

Elastic volume strain induced by hydrostatic stress

e s v,

ε Elastic volume strain induced by shear _stress

vp ij

ε

Second strain tensor of visco-plastic _component vp

v

ε Creep volumetric strain on the stabilizing _boundary

( 0)

vp v t t

ε ₋ Creep volumetric strain increment between time t₀ and t

I v

ε Rate of Irreversible volumetric strain _{during deviatoric shearing}

'

I v

ε Rate of Irreversible volumetric strain during pure shearing

'I

ε Rate of Irreversible strain during deviatoric shearing

p

dγ Plastic shear strain increment

vp

γ Creep shear strain on the stabilizing _boundary

( 0)

vp t t

γ ₋ Creep shear strain increment between _time

0

t and t

I

γ Rate of shearing strain under deviatoric _shearing; η Shear stress ratio, η≡ J2 / J2f

'

σ Rate of deviatoric stress Superscripts e _{Elastic deformation} p _{Plastic deformation} t Total deformation vp _{Visco-plastic deformation}

1. Introduction

The time-dependent deformation behavior of rocks has a significant impact on the stability of rock slopes and underground excavations such as power plants, nuclear waste storage facilities and tunnels. These engineering projects are susceptible to hazards induced by the time-dependent material behaviors during or after the constructions. In tunneling, squeezing ground is one of the typical engineering challenges associated with time-dependent deformations of rocks. After a tunnel excavation and a support installation, the ground may continue to deform as time progresses, and may cause severe damages to tunnel linings and supports if they were not designed to account for time-dependent ground deformations [1-3]. The squeezing ground phenomenon is known to have caused problems in many tunnels built in the sandstone of Western Foothill Range in Taiwan, and significant remediation costs were required to repair the damage induced by time-dependent deformations.

Most rocks exhibit various degrees of time-dependent deformations. In order to understand and to characterize the time-dependent deformation behaviors, tests, within either a very short term (several minutes) or a short term (several hours), were conducted by the rock mechanics laboratory at National Taiwan University [4-6]. These tests and prior researches [7-10] regarding the creep behavior of rocks show that many different kinds of rocks such as rock salt, limestone, granite, chalk and sandstone do exhibit time-dependent visco-plastic deformations. Additionally, irrecoverable and pronounced time-dependent deformations have been observed, even under relatively low stress, for weak rocks such as sandstone, rock salt, and chalk.

It is necessary to understand the general deformation characteristics of rocks before looking into its time-dependent behaviors. The deformation of rock is conventionally decomposed into elastic and plastic components (which ignore time-dependent behaviors in the very short term of loading), or has elastic and visco-plastic components (with considerations of time-dependent behaviors). However, Perzyna[11] comments that since the separation of plastic and viscous deformations is difficult to be done practically, the separation of deformation into elastic, plastic, and viscous

components can be even more difficult to achieve. Furthermore, a common practice of describing elastic deformations may not be ideal. Elastic deformations are commonly described using the isotropic linear elasticity. Maranini[10] and Cristescu[12] proposed empirical expressions of the linear elastic modulus, in which the influence of confining pressures on the deformational behavior of granite and rock salt, is considered. However, experimental results [13-15] suggest that rocks can exhibit nonlinear elastic deformation behaviors, and coupling between shear stress and elastic volumetric strains, e.g. the shear dilation or the shear contraction, is obvious. Microscopically, the nonlinear elastic deformation characteristics of rocks are possibly related to the opening and the closing of the void space and the micro-cracks in rock during the loading-unloading processes [12]. Most constitutive models for rocks assume that shear stress will not induce elastic volumetric strain, but this assumption does not conform to the experimental results, especially for weak rocks.

Furthermore, the non-linear characteristics of the elastic deformation should be of concern when separating the elastic and the visco-plastic deformations from the total deformation. Otherwise, the obtained visco-plastic deformation is likely to involve the elastic deformation and not the precise visco-plastic deformation. The incorrectly deduced visco-plastic deformation may inherently lead to the increase of complexity in a constitutive modeling, rendering that the use of a non-associated flow rule becomes necessary to reach agreements between measured and modeled behaviors.

In this paper, the authors concernthe nonlinear elastic–visco-plastic deformation characteristics, in conjunction with the shear dilation/contraction for weak sandstones. The detailed description of deformation behaviors was made possible due to the experiment to unambiguously separate the nonlinear elastic and the visco-plastic deformations. The experiment consists of triaxial tests with pure shear stress paths (to distinguish shear dilation/contraction) and multi-staged loading-unloading-creep procedures (to separate the visco-plastic deformations from elastic ones). Based on this effort, details of plastic flow, visco-plastic flow, plastic dilation threshold, visco-plastic creep dilation threshold, yield surface of rocks under various stress conditions are

found and presented.

2. Methodology

The Mushan sandstone, which has caused squeezing ground hazards of tunnels in Taiwan, was studied. This sandstone has void ratio of 14.1 %, dry density of 2.3 g/cm3, saturated water content of 5.8 %, and an average uniaxial compressive strength of 37.1 MPa. Based on micro-structure analyses, the percentages of particles, matrix, and voids are 67.5 %, 18.5 %, and 14.1 %, with an average particle diameter of 0.72 mm. Mineralogically, the Mushan sandstone consists of 90.7 % of quartz, 9.0 % of rock debris, and thus is classified as lithic greywacke. The oven-dried specimen used for experiments has a diameter of 55 mm and a height of 129 mm.

During the course of conventional triaxial compression (CTC) tests, both confining pressures and shear stresses vary with each increment of loading. As a result, it is difficult to distinguish the effect of shear and volumetric stresses on the volumetric deformations, which includes elastic, plastic and creep components. In order to separate the effects of volumetric and shear stresses on the volumetric deformation, the pure shear stress path is used in triaxial tests. Triaxial tests with pure shear stress paths make it possible to separate the effect of shear stresses and volumetric stresses on deformations, and have been adopted in recent years to study rock mechanics in the three-dimensional stress space [13-18]. In triaxial tests with pure shear stress paths, confining pressure is held constant by decreasing cell pressure increments to one half of the increment of axial stresses (∆σ2 ====∆σ3= −= −= −= −0.5∆σ1). Such stress path makes it possible to study the effect of shear

stresses on the shear deformations and the volumetric deformations.

The experiments and test conditions used in this study are summarized in Table 1. These experiments are designed with engineering applicability in mind, thus confining pressure between 20 ~ 60 MPa has been chosen. For the Mushan sandstone, the pure shear triaxial tests under confining pressure of 60 MPa have shear strength close to the one obtained from CTC with 20 MPa

confining pressure or CTC with 20 MPa of confining pressure is equivalent to the shear strength with roughly 800 m of overburden. Therefore, the experimental data should be applicable to a wide rage of engineering projects.

To measure the deformation of the specimen, strain gauges of 20 mm in length, 120 Ω in electrical resistance, maximum measurable strain of 2 %, working temperature between -20 ~ 80 ℃ , and a measurement accuracy of 0.85 [(mm/m)/ ] were directly attached to the surface of the specimen inside the membrane. In order to connect these strain gauges to the outside of the membrane, authors made small openings on the membrane, connected wires through these openings, and then sealed the opening with silicon (as illustrated in Fig. 1). This method has been validated by checking the existence of cell fluid inside the membrane after the completion of experiments, and no cell fluid was found inside the membrane, thus there is no oil leakage into the specimen. Two full bridges are formed by the attached strain gauges to measure axial (ε1) and circumferential

strains (ε2 = ε3) of the specimen. Finally, volumetric strain εv= =I1' εkk and deviatoric shear

strain ' 2

2 J (where I is the first strain invariant; 1' ' 2

J is the second deviatoric strain invariant) of the specimen can be measured and recorded. The procedure was further validated by comparing the measured deformation to the one measured by LVDT, and satisfactory consistency between these two different measurements was obtained.

For the hydrostatic compression stage of loading, experimental results showed that the creep volumetric strain monotonically decreases while confining pressure increases, suggesting that the specimen approaches a stable condition upon a greater compression. Meanwhile, the creep volume strain induced by hydrostatic stress is much smaller than the one induced by shear stress. As such, the creep deformations induced by hydrostatic pressure during the shearing stage of loading were ignored so as to focus on the creep deformational behavior under pure shearing. The same set of loading rates for triaxial tests and creep tests (shown in Table 1) was applied to all experiments under a constant room temperature of 25 ℃. Multi-staged loading – unloading -

reloading tests, as shown as Fig. 2, were performed on selected experiments in order to obtain elastic deformations. Authors recognized that creep could potentially contribute to the “elastic” strain _εe _{and “visco-plastic” strain}

0

0

vp t

ε _→ , as shown in the Fig.2, during the unloading-reloading cycle. However, it was found that the influence of this creep is insignificant, and the framework of the traditional plasticity theory is accordingly adopted in this study so that the separated deformation obtained in this research and others can be compared on the same basis. As such, _εe

and

0

0

vp t

ε _→ are regarded as elastic and plastic strain, respectively. For creep tests, this work applied multi-staged loadings shown in Fig. 2d, and the stress applied in each stage has a shear stress ratio

η

= 0.2 ~ 0.95 (η≡ J₂ / J_2f , J₂ is the second deviatoric stress invariant, and

2 f

J is the second deviatoric stress invariant at failure).

The selection of duration for creep tests is based on Cristescu[18], which suggests the use of the stabilization boundary, defined by data points at the end of primary. This concept was adopted in determining the time interval of creep for each stage of creep. As a result, the time to the end of primary creep was found to be proportional to the shear stress during the shearing phase of triaxial tests. Therefore, authors have used creep time of 6 hours for η = 0.2 ~ 0.6, 12 hours for η = 0.6 ~ 0.8, and 24 hours for η = 0.8 ~ 0.95. After creep tests have reached the designated time period, the next stage of loading/creep is subsequently conducted. Figs. 3 and 4 show the creep tests for different η under confining pressure of 20 MPa. It was kept in the authors' mind that the test procedure, the multi-staged creep test, should be easily conducted in laboratories to increase its applicability when the creep behavior is investigated. A too long creep test cannot fulfill the needs of engineering projects, as only limited time is available to wait for the experimental results. As such, the total test time for one specimen is about five days, while each single step of the creep test can be as long as 24 hours.

Based on the small strain assumption and the work done by Peryna [11], the total strain is the sum of the elastic and the visco-plastic strains and can be presented as:

t e vp

ij ij ij

ε =ε +ε (1)

3.1 Volumetric stress – elastic volume strain relationships

In order to separate the total strain into elastic and inelastic components, the unloading-reloading cycle is used to find elastic strains. Based on the unloading-reloading test, confining pressure – volumetric strain curves obtained from several experiments are shown in Fig. 5, and a non-linear, regression curve of the following form was obtained as:

, 0 e v p o p a p ε = × (2)

In Eq. 2, a is a material constant, and ₀ p = 1 MPa. For the Mushan sandstone, o a0 =1120

(the bold curve in Fig. 5) is obtained by fitting the experiment data. Based on Eq. 2, it can be seen the elastic bulk modulus increases nonlinearly with the confining pressure.

3.2 Elastic volumetric strain induced by shearing

From triaxial tests with pure shear stress paths, shear stress – volumetric strain relationships were obtained. In this work, three different confining pressures p of 20, 40, and 60 MPa are used to conduct unloading – reloading pure shearing triaxial tests with four different shear stress ratios η of 0.22, 0.44, 0.66, and 0.88. Significant amount of shear dilation was observed when

η

= 0.88, suggesting that the stress state should be close to the yield condition. As a result, unloading – reloading cycles with

η

= 0.22, 0.44, 0.66 and p= 20, 40, and 60 MPa (nine different curves) are used for regression to obtain the shear stress – volumetric strain relationships.

can be expressed as: 2 2 , 1 e v s J a p ε = −       (3)

In Fig. 6, it can be seen that Eq. 3 with a₁ = 488 can closely match nine sets of different experiment data observed under different conditions. From these results, it is seen that pure shearing induces elastic volumetric dilation ( e_,

v s

ε ) on the unloading – reloading path. Thus, the tested weak sandstone indeed exhibits a nonlinear coupling between shear stresses and elastic volumetric strains.

3.3 Shear stress – elastic shear strain relationships

By using the sets of data obtained from unloading – reloading during pure shearing triaxial tests, the relationship between c shear stress and elastic shear strain can be obtained. Fig. 7a shows the shear stress – elastic shear strain relationships obtained from these nine sets of data. A linear relationship obtained by fitting these data has the following form:

'

2 2 2

2 J =a J (4)

From these data, it seems a₂ is a function of p or K, and regressions shown in Fig. 7a suggest a larger confinement result with a larger shear stiffness. If this data is further analyzed by normalizing the shear stress J with respect to the bulk modulus K (derived from ₂ Fig. 5), all these data converge to a single linear relationship, as shown in Fig. 7b.

4. Visco-plastic deformation behaviors

As described in Eq. 1, visco-plastic strains can be computed by subtracting elastic strains from total strains. In section 3, we have obtained regression Eqs. 2, 3 and 4 to describe elastic volumetric stress – volumetric strain, shear stress – volume strain, and shear stress – shear strain

relationships based on unloading – reloading tests. In this section, visco-plastic deformation is deduced and its characteristics are then discussed.

The deformations obtained from the pure shearing state of loading were used to calculate plastic strains (plastic flow) based on methods suggested by Jeng et al. [1, 15]; the deformations obtained from creep tests can be presented as normalized resultant vectors, which are composed of creep volumetric strain and creep shear strain, to indicate the visco-plastic strain increment (visco-plastic flow). Based on the plasticity theory [16], we can define β₁ as the angle between plastic flow and the strain axis (horizontal), and β₂ as the angle between visco-plastic flow and the strain axis as: 1 1 tan p p v d d γ β ε −   = _{ }   (5) ( ) 0 0 1 2 tan vp t t vp v t t γ β ε − − −   =       (6)

where β₁ is the plastic flow angle; β₂ is the visco-plastic flow angle; _dγp _{is the plastic shear}

strain increment; p v

dε is the plastic volumetric strain increment; _{( )}

0 vp

t t

γ ₋ is the creep shear strain increment between time t and t ; _{0 ( )}

0 vp v t t

ε ₋ is the creep volumetric strain increment between time

0

t and t; to is the time at the beginning of creep; t is an arbitrary time after creep starts.

Figure 8 shows the variation of β₂ with respect to time for the specimen under 20 MPa of confining pressure. Very small variations of the visco-plastic angle are observed during experiments. Figure 9 presents the variation of plastic flow angle β₁ and visco-plastic flow angle

2

β with respect to shear stress ratio η, and a nearly linear relationship between β₁, β₂ and η is observed. It should be noted that plastic flow angle β = °90 marks the transition between compressive (β1< °90 ) to dilative (β1> °90 ) volumetric plastic behaviors. It is apparent from Fig.

2

β are very close.

As shown in Figs. 8 and 9, it was found that the directions of plastic flow and visco-plastic flows coincided with each other, suggesting similarities between plastic strain (from pure shearing tests) and visco-plastic strain (from creep tests). The plastic strain is often obtained from a very short term (a few minutes) loading, in which the time-dependent deformation can still exist and contribute to the deduced "plastic" deformation. The embedment of the visco-plastic deformation in the plastic deformation most likely accounts for the similarity between the directions of the plastic flow and the visco-plastic flow. For the sake of convenience, the very short term visco-plastic deformation is referred to as the plastic deformation hereinafter.

4.1 Plastic dilation threshold and visco-plastic creep dilation

threshold

As shown in Fig. 10, both the plastic deformation and the visco-plastic deformation exhibit shear contraction followed by shear dilation when subjected to pure shearing. Therefore, a dilation threshold for both the plastic and the visco-plastic deformations exists. The plastic dilation threshold is a transitional stress state from the volume contraction behaviors to the volume dilatation behaviors of materials in plastic states under shearing, and is often considered to be associated with the initiation of unstable crackings in sandstones.

Figure 10 illustrates the relationship between shear stresses and total strains, plastic strains, and visco-plastic strains. Points A, B and C mark corresponding transition points between the compressive volumetric behavior and the dilative volumetric behavior under one specific confining pressure. Among the thresholds shown in Fig. 10, the dilation thresholds of plastic and visco-plastic deformation (Points B and C) are similar and correspond to a greater shear stress level than the threshold of total deformation (Point A) does. It has been observed that specimen under sustained creep compression, in which the specimen is shear contracted, is less likely to yield [19]. If the initiation of unstable cracks within the rock is related to the shear dilation, then the thresholds

of plastic or visco-plastic (Points B or C in Fig. 10; rather than Point A) should represent the initiation of unstable crack growth.

Figure 11 further summarized the experiment data for these transitional stress states under different confining pressures and shear stress ratios. It is seen that these dilation thresholds are not significantly affected or slightly increased by the increase of confining pressure, and this result is consistent with the references for sandstones [20-22]. It is also seen that the plastic dilation and the visco-plastic dilation thresholds are still very close to each other for varying hydrostatic stresses.

4.2 Plastic strain increment and visco-plastic strain

The plastic strain increment _dεp _{and visco-plastic strain ε}vp_{are defined as:}

( ) ( )

2 2 p p p v d

ε

= d

ε

+ d

γ

(7)

( ) ( )

2 2 vp vp vp v

ε

=

ε

+

γ

(8) where p v

d

ε

is the plastic volumetric strain increment; _d

_γ

p _{is the plastic shear strain increment;} vp

v

ε

and

_γ

vp _{are the visco-plastic volumetric and shear strain increment on the stabilizing}

boundary. The plastic strain increment used in this work corresponds to stress increment of 1 MPa in J₂ , and Eq. 8 is used to calculate the visco-plastic strain increment.

Figure 12 shows the

_ε

vp _{under various shear stress ratio η under different confining}

pressures. It is seen that

_ε

vp _{increases with}

_η

_{under all confining pressures p, and the}

relationship between

_ε

vp _and

_η

_{appears to be independent of p. It is also seen from Fig. 12}

that plastic strains can occur even when η is very small, suggesting that yielding of weak sandstones should occur at very low shear stress levels, which is very different from the behavior of hard rocks. Figure 12 also shows significant decrease of stiffness (inverse of the slope of curves) of factors 2 ~ 3 under high shear stress ratio

η

for both plastic and visco-plastic behaviors.

5.

Yield, plastic potential and visco-plastic potential surfaces

Based on the method described in Section 4, plastic strains throughout the course of experiment were accordingly obtained. Based on the experiments, yielding at small strains was observed, suggesting that the stress state at the beginning of the experiment should be close to the plastic potential surface, which evolves with the increase of shearing. By applying techniques described in Reference [15], plastic potential surface can be deduced from plotting the plastic flow, as shown in Fig. 13. It is seen in Fig. 13 that the plastic potential surface prior to reaching yielding in shear is an elliptical cap. The cap of the plastic potential surface increases its size with increasing stress levels, and coincides with the yield envelope when the material reaches yielding in shear. Similar results are obtained for visco-plastic potential surfaces, suggesting that the weak sandstone has the same surface for both plastic potential and visco-plastic potential by separating appropriate elastic deformations from total deformations.

Under the assumption of homogeneous, isotropic, and small strain, the yield function

(

, 2

)

H p J can be written as [7, 18]:

2 2

( , ) H( ) D( , )

H p J ≡ H p +H p J (9)

In Eq. 9, H_H is controlled by hydrostatic compression; H_D is controlled by pure shearing. On the stabilization boundary, the value of the yield function equals the irreversible stress work,

( )

I

W t . This is because the irreversible strain increment and stress increment are both zero on the stabilization boundary, and the material is in elastic condition inside and on this boundary. Thus, the yield surface can be defined using _{W t}I

( )

_{. Cristescu [7, 18] suggested the irreversible stress}

work _{W t}I

( )

_{can be expressed as:}

' ' 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) H H I I I H D T _I T _I v _T W t W t W t p t

ε

t dt

σ

t

ε

t dt ≡ + =

_∫

 +

_∫

⋅ (10)

where I( )

H

W t is the irreversible stress work under hydrostatic compression; I( )

D

W t is the irreversible stress work under pure shearing condition; p is the hydrostatic compression;

σ

' is the deviatoric stress; I

v

ε

 is the irreversible volumetric strain increment during the hydrostatic compression;

_ε

_{ is the strain increment during deviatoric shearing;} ' I

H

T is the time at the end of hydrostatic compression, and T is the time corresponding to a certain time during pure shearing.

Cristescu assumed no coupling between shear stress and elastic volumetric strain, and treated volumetric strain during pure shearing as irrecoverable deformations. Based on the experiment evidence shown in Section 4.1.2, however, the coupling between shearing and volumetric deformations is evident. The elastic bulk modulus is related not only to the confining pressure, but also to the shear stress state. Therefore, for weak sandstones and materials alike, we recommend the irreversible stress work to be modified as:

0 ( ) _I( ) TH ( ) ( )_I H H v H p =W t =

∫

p t

ε

 t dt (11) ' ' 2 ' 2 ( , ) ( ) ( ) ( ) ( ) ( ) ( ) H H H T I I D D _T T _I T _I v T T H p J W t t t dt J t t dt p t dt

σ

ε

γ

ε

= = ⋅ = +

∫

∫

∫

   (12)

In Eq. 12,

_γ

_I _{is the irreversible visco-plastic shearing strain increment under deviatoric}

shearing; and I'

v

ε

 is the irreversible volumetric strain increment in shear. Remarkably, these two increments do not involve elastic volumetric strain; while in Cristescu's approach [7, 18] elastic volumetric strain was not subtracted. Figure 14 shows the variation of I

H

W versus confining pressure p during hydrostatic load for the weak sandstone with four different final confining pressures. It is seen that I

H

W increases with increasing p , suggesting that sandstones exhibit the work-hardening behavior.

Figure 15 shows I D

W versus shear stress J under confining pressures of 20, 30, 40, 50, ₂ and 60 MPa. An exponential increase of I

D

pressures, suggesting sandstones also exhibit work-hardening during pure shearing. This is because each unit increase of the shear stress results in exponentially increasing strain increments due to plasticity, thus larger increment of I

D

W is rendered.

By using Eq. 9, along with data on Figs. 14 and 15, yield locus can be obtained by plotting contours of _WI _{in the stress space as shown in Fig. 16, which shows: 1) plastic flow direction, 2)}

visco-plastic flow direction, 3) hypothesized yield surface (contours of irreversible stress work), and 4) failure envelope. It is seen from Fig. 16 that the both plastic flow and visco-plastic flow are orthogonal to yield surfaces, suggesting both plastic potential and visco-plastic potential take the same form of yield surface, thus associated flow rules can be used when formulating constitutive models for weak sandstones. However, the plastic potential surface is not necessarily the same as the visco-plastic potential surface, as the plastic potential is time-independent, while visco-plastic is time dependent. The size of the visco-plastic potential surface will increase as time elapses.

6. Conclusion

Details and corresponding insights of the time-dependent deformational behaviors of weak sandstones are presented and analyzed based on a series of pure shear stress path triaxial tests and multi-staged creep tests. It is found that the elastic behavior of weak sandstones, which were used to deduce plastic or visco-plastic strains, is nonlinear and differs from the linear one. Strong coupling between pure shear stress and elastic volumetric behaviors is also found, and can be described by a regression function normalized by the confining pressure (Eq. 3).

It is found that the plastic dilation threshold of weak sandstones is, in fact, quite close to the visco-plastic dilation threshold. This is quite different from results based on the assumptions that there is no shear dilatancy and no shear contraction. Such assumptions lead to a significantly lower plastic dilation threshold than creep dilation threshold. By introducing the shear dilation/contraction, it is found that the plastic dilation threshold is close to visco-plastic dilation threshold, and this threshold can be regarded as the initiation of unstable cracks inside the rock

material, suggesting the initiation of micro-crack opening inside the specimen. It is also concluded that the direction of plastic flow is similar to the visco-plastic flow, suggesting that plastic potentials coincide with visco-plastic potentials. Based on experimental data, the plastic strain and visco-plastic strain increments are orthogonal to the yield surface, which implies that the associated flow rule in plasticity theory is applicable. In other words, the yield surface, the plastic potential, and the visco-plastic potential may take the same form.

Consequently, separations of the coupling between pure shearing and volumetric deformations could potentially simplify the formulation of the constitutive behaviors of sandstone materials. Conventionally constitutive models ignore this coupling or assume linear relationships between pure shearing and volumetric behaviors prevent themselves from describing the plastic and visco-plastic behaviors in a more precise manner, and require complex formulations, such as non-associated flow rule, to achieve reasonable modeling capabilities. Base on the findings presented in this work, the coupling between pure shearing and volumetric deformations should be taken into account when formulating a constitutive model for weak sandstones. This coupling enables the use of associated flow rules for describing time-dependent behaviors, and thus greatly simplifies formulations of constitutive models

7. References

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[2] Jeng FS, Huang TH. Time dependent behavior of weak sandstone on Mushan Formation. In: Proceedings of the 13th Southeast Asian Geotechnical Conference, Taipei, 1998. p. 75-80. [3] Jeng F.S. Lin M.L. and Huang T.H. Study of the Geological Barriers of the Tunnels in

Northern Taiwan (in Chinese). Taipei: Ministry of Transportation, Research Report, 1996. [4] Jeng FS, Ju GT, Huang TH. Properties of some weak rock in Taiwan (in Chinese). In:

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2000;37:143-160.

8. Acknowledgements

The research is supported by the National Science Council of Taiwan, Grant no. NSC-92-2211-E-002-047and NSC 93-2211-E002-047.

List of figure captions

Figure 1 Schematic illustration of procedures for measuring deformations of the

specimen.

Figure 2 Schematic illustration of multi-step loading creep tests performed. (a) shear

stress versus time; (b) shear stress versus axial strain; (c) axial strain versus tie;

(d) measured shear and volumetric strain versus time with shear stress ratio of

0.22 – 0.91; thin lines presents shearing stage, and bold lines presents creep

stage.

Figure 3 Measured volumetric creep strain versus time (in log scale) under varying

shear stress ratio

η

. It is seen volumetric creep shows compressive behavior at

low shear stress ratio

η

, and the creep rate decreases with increasing

η

.

η

of

0.55 serves as the threshold of the transition from compressive to dilative

volumetric creep. Further increase of

η

raises the rate of dilative volumetric

creep. It should be noted that

η

larger than 0.55 suggests a continuous increase

of volumetric creep, and the material could fail.

Figure 4 Measured shear creep strain under varying shear stress ratio

η

. Increasing

shear creep rate is observed with increasing

η

.

Figure 5 Measured elastic volumetric strains during hydrostatic loading. Elastic

volumetric strain versus confining pressure relationship is a curve concaved

towards horizontal axis, suggesting an exponential increase of elastic bulk

modulus with increasing confining pressure.

Figure 6 Normalized elastic volumetric strain versus normalized stress. Pronounce

coupling between shear stress and elastic volumetric strain can be observed, and

this coupling is related not only to shear stress, but also to the confining

pressure.

Figure 7 Measured elastic shear stress – strain during unloading-reloading. (a)

measured elastic shear strain versus shear stress; (b) elastic shear strain versus

shear stress normalized by bulk modulus.

figure correspond to calculated

β

₂

from measurements done every 30 minutes

under different shear stress ratio.

Figure 9 Measured plastic flow angle

β

₁

and visco-plastic flow angle

β

₂

with

varying shear stress ratio. It is seen from the figure that

β

₁

and

β

₂

are

similar under the same shear stress ratio. Therefore, in the stress space of p

versus

J₂

, it can be assumed that the plastic flow and visco-plastic flow have

the same flow direction under the same stress condition.

Figure 10 Illustration of dilations thresholds. Three points, A, B, and C marks

thresholds of three different deformation behaviors. A: dilation threshold of

total strain during triaxial shearing; B: dilation threshold of plastic strain during

triaxial shearing; C: dilation threshold of time-dependent visco-plastic strain

during shear-creep tests.

Figure 11 Measured dilation thresholds under various stress conditions. Similar

visco-plastic and plastic dilation thresholds are under the same confining

pressure.

Figure 12 Plastic strain increments (hollow data points) and visco-plastic strains (solid

data points) versus increasing shear stress ratios under various confinements.

Figure 13 Illustration of plastic and visco-plastic potentials of sandstones. Nearly

identical plastic flow ad visco-plastic flow are seen in p-

J₂

stress space,

suggesting identical plastic potential and visco-plastic potential can be assumed

after reasonable separation of elastic deformations from total deformations for

weak sandstones.

Figure 14 Measured irreversible volumetric stress work during hydrostatic loading.

Figure 15 Measured pure shearing irreversible stress work during pure shearing

Figure 16 Derived yield Surface, plastic and visco-plastic potentials. Plastic flow and

visco-plastic flow are reasonably orthogonal to yield surface under varying

hydrostatic stress, suggesting associated flow rule can be applied to model weak

sandstones.

Table 1 List of triaxial tests and testing conditions

Test

Confining

Pressure

(MPa)

Number of

Tests

Testing Conditions

20 3+2*

30 1

40 1+2*

50 1

Pure shearing

stress path

triaxial tests

60 3+2*

20 2+1*

30 1

40 1

50 1

Pure shearing

stress path creep

tests

60 2+1*

¾ Loading rate

Hydrostatic stage: 5 MPa/min Shearing stage: Axial: 16/3 MPa/min Radial: -8/3 MPa/min ¾ Loading/unloading tests at η = 0.22, 0.44, 0.66, 0.88 ¾ 8 creep stresses at η = 0.2~0.95 ¾ Creep time, tc: η = 0.2~0.60, tc=6 hours η = 0.6~0.80, tc=12 hours η = 0.8~0.95, tc=24 hours Note:

1. η is shear stress ratio 2. dry specimen

3. * indicates loading / unloading tests Table 01

Strain gauge

Data

logger

Specimen

Terminal Silicon

Exterior of membrane

Interior of

membrane

Step 1: attach strain gauge onto the surface of specimen Step 2: Wrap the specimen with membrane, and make

openings on the membrane.

Step 3: Connect lead wires of the strain gauge through the opening on the membrane to the terminal.

Step 4: Attach the terminal to the exterior of the membrane Step 5: Connect wires from the data logger to the terminal.

Figure 1

t

t

₀

σ

_c

σ

a b a c d

t

t

₀

σ

_c

σ

a b a c d

ε

σ

_c

σ

0 0 vp t

ε

→ εe 0 vp t t ε _→ b a c d

ε

σ

_c

σ

0 0 vp t ε _{→ ε}e 0 vp t t ε _→ b a c d

(a) Stress versus time (b) Stress versus strain

ε

t

a b a c d

t

₀

ε

t

a b a c d

t

₀

-4000 -2000 0 2000 4000 6000 8000 0 20 40 60 80 100 120 time (hour) St ra in ( 10 -6) η=0.91 0.86 0.76 0.65 0.55 0.33 0.22 0.47 Shearing stage Creep stage Shear strain Volumetric strain p=20 MPa Compression Dilation

(c) Strain versus time (d) measured shear and volumetric strain

versus time with shear stress ratio of

0.22 – 0.91

Figure 2

-1600 -1400 -1200 -1000 -800 -600 -400 -200 0 200 400 0.0001 0.001 0.01 0.1 1 10 100

log t (hour)

V

ol

um

etr

ic

cr

eep

s

tra

in

(

10

-6

)

p=20 MPa η=0.22 η=0.33 η=0.55 η=0.65 η=0.86 η=0.91 η=0.76

Figure 3

Figure 03

0 200 400 600 800 1000 1200 1400 0.0001 0.001 0.01 0.1 1 10 100

log t (hour)

Sh

ea

r c

ree

p s

tr

ai

n

2

*(J

'

2

)

1/ 2

(1

0

-6

)

p=20 MPa η=0.33 η=0.55η=0.65 η=0.86 η=0.91 η=0.76

Figure 4

Figure 04

0 10 20 30 40 50 60 70 0 2000 4000 6000 8000 10000

Elastic volumetric strain (10

-6

)

H

ydr

os

ta

ti

c s

tr

es

s

(MP

a)

p=15 MPa p=30 MPa p=45 MPa p=60 MPa , 0 * e v p

p

a

p

ε

=

a0= 1120, R2=0.97

Figure 5

Figure 05

0 0.2 0.4 0.6 0.8 1 1.2 1.4 -700 -600 -500 -400 -300 -200 -100 0

Elastic volumetric strain (10

-6

)

Sh

ea

r s

tres

s

(J

2

)

1/ 2

/ H

ydr

os

ta

ti

c

st

re

ss

p=20 MPa p=40 MPa p=60 MPa Regrssional curve a1=488, R2=0.86 2 2 1 v

J

a

p

ε

= −





_





_





Figure 6

Figure 06

0 10 20 30 40 50 60 0 1000 2000 3000 4000 5000

Elastic shear strain 2*(J'₂)1/2 ₍₁₀-6₎

She ar s tr ess (J2 ) 1/ 2 (M P a) p=20 MPa p=40 MPa p=60 MPa Regressional curve ' 2 2 2 J =74.3 J R2_=0.987 ' 2 2 2 J =86.6 J ' 2 2 2 J =107.17 J R2_=0.974 R2_=0.959 ' 2 2 2 2 J = a J

(a) Elastic shear strain versus shear stress

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0 1000 2000 3000 4000 5000

Elastic shear strain 2*(J'₂)1/2 (10-6)

She ar st re ss ( J2 ) 1/ 2 / K p=20 MPa p=40 MPa p= 60 MPa 2 ' 2 3 2 J a J K = a₃=6.769×105_{, R}2_=0.98

(b) Elastic shear strain versus shear stress normalized by bulk modulus

Figure 7

p (MPa) K (MPa) 20 5234 40 7404 60 9068 Figure 07

0 30 60 90 120 150 0 5 10 15 20 25 30 35

Time (hour)

ββββ

2222

(

D

eg

ree)

η=0.22 η=0.33 η=0.44 η=0.55 η=0.65 η=0.76 η=0.86 η=0.91 p =20 MPa vp t t v(−₀) ε vp t t ) (−0 γ Dilation Compressio β 2

Figure 8

Figure 08

0 30 60 90 120 150 0 0.2 0.4 0.6 0.8 1

D

eg

re

e)

TC test ; p=20 MPa TC test ; p=20 MPa TC creep test ; p=20 MPa TC creep test ; p=20 MPa

Dilation Compression vp t t ) (−0

γ

_δγ

p p v

δε

β 2 vp β 1 t t v(−0) ε

Figure 9

Figure 09

Volumetric strain she ar st re ss ( J2 ) 1/ 2 B C

A : Total dilation threshold

B : Platic dilation threshold

C : Viscoplatic dilation threshold

A 0 Compression Dilation , vp v s

ε

, p v s

ε

, t v s

ε

Figure 10

Figure 10

0 0.2 0.4 0.6 0.8 1 0 20 40 60 80

Hydrostatic stress (MPa)

ηηηη

Platic dilation threshold Viscoplatic dilation threshold Total dilation threshold

Figure 11

1 10 100 1000 10000 0 0.2 0.4 0.6 0.8 1

ηηηη

d

ε

p

,

ε

vp

(

10

-6

)

TC test p=20 MPa TC test p=40 MPa TC test p=60 MPa TC creep test p=20 MPa TC creep test p=40 MPa TC creep test p=60 MPa

Figure 12

Figure 13

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0 10 20 30 40 50 60 70

Hydrostatic stress (MPa)

(M

Pa

)

(M

Pa

)

(M

Pa

)

(M

Pa

)

p=30 MPa p=40 MPa p=50 MPa p=60 MPa

Figure 14

Figure 14

0 0.05 0.1 0.15 0.2 0.25 0.3 0 10 20 30 40 50 60 70 80 90

Shear stress (J

₂

)

1/2

(MPa)

(M

P

a)

p=20 MPa p=30 MPa p=40 MPa p=50 MPa p=60 MPa p =20 MPa p =30 MPa p =40 MPa p =50 MPa p =60 MPa

Figure 15

Figure 15

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Hydrostatic stress (MPa)

Shea

r s

tr

es

s (J

2

)

1/2

(M

P

a)

Plastic flow Viscoplastic flow Irreversible stress work Hypothesize yield surface

WI_{=0.04 W}I_=0.05 WI_=0.06 WI_=0.08 WI_=0.1 WI_=0.16 WI_=0.035 Failure envelope

Figure 16

Figure 16