Characterizing the deformation behavior of Tertiary sandstones

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International Journal of Rock Mechanics & Mining Sciences ] (]]]]) ]]]–]]]

Characterizing the deformation behavior of Tertiary sandstones

F.S. Jeng

, M.C. Weng, T.H. Huang, M.L. Lin

Department of Civil Engineering, National Taiwan University, Taipei, 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan Accepted 22 December 2004

Abstract

Tertiary sandstones possess deformational behavior different from hard rocks, especially the relatively larger amount of volumetric dilation during shearing. Such excess dilation contributes to the increase of crown settlement during tunnel excavation and accounts for several cases of tunnel squeezing within Tertiary sandstones. Therefore, the deformation behavior of Tertiary sandstones sampled from more than 13 formations was studied. To distinguish the volumetric deformation induced by hydrostatic stress or by shear stress as well as to decompose the elastic and the plastic components of strains, special experimental techniques, including pure shear tests and cycles of loading–unloading were applied.

The experimental results reveal that the deformation of Tertiary sandstone exhibits the following characteristics: (1) signiﬁcant amount of shear dilation, especially elastic shear dilation; (2) non-linear elastic and plastic deformation; (3) plastic deformation occurs prior to the failure state. Furthermore, features of plastic deformation were inferred from experimental results and, as a result, the geometry of plastic potential surface and the hardening rule were accordingly determined. A constitutive model, involving nonlinear elastic/plastic deformation and volumetric deformation induced by shear stress, is proposed. This proposed model simulates the deformational behavior for the shear-dilation-typed rocks reasonably well. Furthermore, tests on the versatility of the proposed model, including varying hydrostatic stress and stress paths, indicate that the proposed model is capable of predicting deformational behavior for various conditions.

Keywords: Sandstone; Constitutive model; Deformation; Shear dilation

1. Introduction

Tertiary sandstones have a digenetic age of no more than 70 million years and such relatively short rock forming period is insufﬁcient to classify them as hard rocks. For instance, the typical strength of Tertiary

sandstones in Taiwan ranges from 10 to 80 MPa[1].

While tunneling through the Tertiary strata, several

unsuccessful cases were reported[2]. Difﬁculties,

includ-ing severe squeezinclud-ing and ravelinclud-ing, were encountered during construction of these tunnels. For instance, a crown settlement of 180 cm of a 12.4 m wide highway tunnel passing through a faulted zone of Tertiary formations was reported. A crown settlement ranging

from 14 to 30 cm occurred in several sections of the tunnels under construction, in which Tertiary sandstone (Mushan Formation) was encountered. The crown settlement in other sandstones’ strata is often within several centimeters. Therefore, the deformational char-acteristics of Tertiary sandstones should be involved while the deformation of a constructing tunnel is analyzed.

When compared to hard rock, it was found that the deformational behavior of Tertiary sandstones is characterized by large amount of nonlinear deforma-tion, shear dilation and plastic deformation prior to the

failure state [3–5]. Jeng et al. [6] compared the

mechanical properties of sandstone, the uniaxial com-pressive strength (UCS) and the reduction of strength

due to wetting (R ¼ UCSdry=UCSwet) with the

petro-graphic features of the 13 sandstones listed in Table 1,

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 www.elsevier.com/locate/ijrmms

Corresponding author. Tel./fax: +886 2 2364 5734. E-mail address: [email protected] (F.S. Jeng).

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and found that these Tertiary sandstones can be

classiﬁed in terms of Grain area ratio (GAR) and

porosity (n), as illustrated in Fig. 1. Two groups of

sandstones, termed as Type A and Type B (with R40:5

and Rp0:5; respectively), have been identiﬁed.

Compar-ing to Type A, Type B sandstone is characterized by greater degree of deformation (or being ‘‘softer’’) and by

having a more signiﬁcant reduction not only in strength

but also in stiffness, as shown in Fig. 2. This

characteristic highlights that Type B can be the problematic rock type, which is prone to tunnel squeezing.

This paper explores the deformational behavior of Tertiary sandstones in details. In addition to the above-1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 Nomenclature

a1 hardening rule parameter for Cap model

af slope of F1

ad state variable of proposed model

b1 elastic constant of proposed model

b1 hardening rule parameter for proposed model

b₄ hardening rule parameter for proposed model

CTC conventional triaxial compression test

ij second strain tensor

e

v;p elastic volume strain induced by hydrostatic

stress

e

v;s elastic volume strain induced by shear stress

eij second deviatoric strain tensor

v volume strain

dp _{increment of plastic strain, d}p_¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_dp

ijd p ij

q

F1 failure envelope of Drucker–Prager model

F2 plastic potential surface of cap model

G shear modulus

G0 initial shear modulus

g shear strain, g ¼ 2pffiffiffiffiffiJ0₂¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 eijeji

ge _{elastic shear strain}

gp _{plastic shear strain}

gt _{total shear strain}

g_d square of slope of failure envelope F

Z₁ hardening rule parameter for cap model

J2 second deviatoric stress invariant, J2¼1₂sijsji

J20 second deviatoric strain invariant, J02¼12eijeji

K bulk modulus

kf interception of F1

l positive scalar factor of proportionality

m parameter for plastic potential surface

O strain energy density function

p hydrostatic stress p ¼1₃I1¼1₃skk (MPa)

PS pure shear test

R strength reduction ratio ¼ UCSwet/UCSdry

RTE reduced triaxial extension test

Rc axis ratio deﬁned by Cap model

sij second deviatoric stress tensor

sij second stress tensor

T interception of failure envelope with I1axis

UCS uniaxial compressive strength (MPa)

Superscripts e elastic deformation p plastic deformation t total deformation Table 1

Sandstones of this research

Formation Classiﬁcation (Pettijohn et al., 1987) Geological age Sedimentary facies Remark WGS1 Lithic graywacke Oligocene Marine–terrestrial mixed facies

WGS2 Lithic graywacke Oligocene Marine–terrestrial mixed facies Apparent preferred orientation

MS1 Lithic graywacke Miocene Littoral facies

MS2 Lithic graywacke Miocene Littoral facies

MS3 Lithic graywacke Miocene Littoral facies Apparent preferred orientation

TL1 Lithic graywacke Miocene Marine facies

TL2 Lithic graywacke Miocene Marine facies

ST Lithic graywacke Miocene Littoral facies

NK Lithic graywacke Miocene Marine facies

TK Lithic graywacke Miocene Littoral facies Apparent preferred orientation

SFG1 Quartzwacke Miocene Littoral facies

SFG2 Lithic graywacke Miocene Littoral facies Apparent preferred orientation, rich mica content

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mentioned research results, the work focuses on the

following aspects:

1. To characterize the deformational behavior of Tertiary sandstones, including elastic and plastic components of deformation for both volumetric and shear deformation;

2. To distinguish the volumetric deformation induced by hydrostatic stress or shear stress so that the contribution of shear dilation can be clariﬁed. 3. To propose a constitutive model able to describe the

deformational characteristics of Tertiary sandstones, for example, the elastic shear dilation.

4. To validate the proposed model, the predictions for various testing conditions, including stress paths and different hydrostatic pressures, are compared to actual test results.

Furthermore, to further differentiate the proposed constitutive model from other geo-material models, the simulations of other models, including the Drucker–-Prager (DP) model and the cap model are compared with the proposed model, which also reveals the distinct deformational features of the Tertiary sandstones.

2. Set-up of experimental study

The deformation of sandstone, soft rocks or cemented soil has been explored by means of conventional triaxial

compression tests (CTC), simple shear tests (SS) and triaxial extension tests (TE) under hydrostatic pressure

from several kPa’s to 27.6 MPa [7–10]. Based on the

studied behavior, theoretical constitutive models were

accordingly proposed[7–9,11–13]. A review on previous

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 Fig. 1. Geotechnical classiﬁcation of the studied sandstones in terms

of n and GAR. The empirical UCS and R are shown by solid and dashed contour lines. The formation name of each sandstone is marked near each symbol. The sandstone is classified on one hand in terms of n and GAR, which affects its strength (UCS) and in terms of degree of wetting softening (R) on the other hand. Accordingly, the sandstones are classified into two groups: Type A (R40.5) and Type B (Rp0.5). The classification of strength (Types I– IV) is based on the definition of ISRM (1981). Volumetric strain Volumetric stress I1 /3 Type A Type B Shear strain 2*(J'2)0.5 Shear stress ( J2 ) 0.5 Type A Type B Volumetric strain Shear stress ( J2 ) 0.5 Type A Type B dilation compression 0 (a) (b) (c)

Fig. 2. Typical deformational behavior of the two types of Tertiary sandstones. Compared to Type A sandstone (MS2), Type B sandstone (WGS2) is characterized by more volumetric and shear deformation and less shear strength as well. J0

2(¼12eijeij; eij¼second deviatoric strain tensor) is a measure of shear strain. (a) Volumetric strain induced by volumetric stress, (b) shear strain induced by shear stress and (c) volumetric strain induced by shear stress.

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works[14–20] has been provided by Desai and Salami

[8].

Since shear dilation was of concern especially when dealing with tunnel squeezing and was reported as a result of the opening of micro-cracks due to shearing [21–24], this research focused on identifying the volu-metric deformation induced by shearing from hydro-static stress. Stress-path-controlled tests, with the applications of hydrostatic stress and shear stress separated into two stages, were adopted so that the inﬂuence of hydrostatic stress and shearing on volu-metric stress could be clearly distinguished.

Secondly, the components of elastic and plastic strains were decomposed to distinguish the contribution of each component to the total strain, which is essential when establishment of a constitutive model is intended. Cycles of loading unloading were performed to identify the elastic and the plastic components of strains at various stages of loading.

The specimens were loaded in a triaxial cell, to which

the axial pressure (s1) and the conﬁning pressure (s3)

were applied, under room temperature. The triaxial cell was able to sustain the hydrostatic pressure up to 175 MPa. A pressure transducer monitored the hydro-static pressure with an accuracy of 0.1 MPa. The axial load was provided by a high stiffness MTS loading frame, which has a maximum load and stiffness of

4448 kN and 13.1 109N/m, respectively. Both the axial

stress and the hydrostatic stress were servo-controlled so that tests with various stress paths would be feasible. The load was applied at a rate of 5 MPa/min.

The physical and mechanical properties of the studied sandstones were measured according to the methods

suggested by ISRM[25]. The specimen size was 5.5 cm in

diameter and 12.5 cm in height. The specimen was oven-dried (105 1C) to remove its natural water content. The longitudinal and transverse types of deformation were separately measured by a full Wheatstone bridge consisted of four strain gages, which were capable of

measuring strains up to 2% with an accuracy of70.85

(mm/m)/1C. In Fig. 3 p ¼1 3I1¼13skk and ffiffiffiffiffiJ2 p

(J2 ¼1₂SijSij; Sij¼ second deviatoric stress tensor) are

measures of hydrostatic stress and shear stress, respec-tively. The specimen was ﬁrst applied with hydrostatic stress (volumetric stress) along a stress path OA, followed by subsequent shearing. If the hydrostatic stress is not altered during the subsequent shearing, a pure shear (PS; path AC) condition is achieved and the volumetric strain induced in this stage of loading is

surely the shear dilation/contraction [26–30].

Further-more, to verify the applicability of the proposed constitutive model, other stress paths, including CTC (path AB) and reduced triaxial extension tests (RTE;

path AD and path AE), as shown in Fig. 3, were also

conducted[31].

Moreover, petrographic analysis, X-ray diffraction tests (XRD) and SEM observation were also conducted in this research.

3. Deformation characteristics of Tertiary sandstone The physical properties of the tested sandstones are

summarized in Table 2. In general, the studied

sandstones are mainly composed of quartz (greater than 75%) with minor content of rock fragments and very little of feldspar (less than 5%); accordingly, these sandstones are classiﬁed as lithic graywacke or

quartz-wacke based on Pettijohn’s deﬁnition [32]. The

poros-ities range from 11% to 25%.

The mechanical properties of sandstones, including uniaxial strength and Young’s modulus, are listed in Table 3. A fairly wide spectrum of mechanical behavior was obtained, in which the dry and wet uniaxial compressive strengths varied from 7–86 and 3–45 MPa, respectively. The (dry) strength was thus classiﬁed from

low strength to medium strength, based on ISRM [25]

deﬁnition. Remarkably, signiﬁcant wetting softening could be observed, either in strength or in stiffness. The reduction ratio R ranged from 0.85 to as low as 0.14.

Fig. 4illustrates the results of a PS test (path OAC in Fig. 3). During the hydrostatic loading (path OA), the bulk modulus increased as hydrostatic stress increased and it resulted in a nonlinear deformation curve, as

shown inFig. 4a. When cycles of loading, unloading and

reloading were applied, the results showed that plastic volumetric strain existed. Based on the plastic strain revealed by the cycles of loading, the elastic deformation and the plastic deformation, at all stages along a 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 Fig. 3. Schematic illustration of the conducted stress paths. PS ¼ pure shear test, CTC ¼ conventional triaxial compression test, RTE ¼ reduced triaxial extension tests.

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controlled stress path, could be decomposed, as shown

inFig. 4.

In addition to the volumetric strain induced by the

hydrostatic stress shown in Fig. 4a, Fig. 4b illustrates

the shear strain (defined as 2 ffiffiffiffiffiJ0

2

p

(J0

2¼12eijeij; eij¼

second deviatoric strain tensor) induced by pure shearing (path AC). The elastic component is relatively linear when compared to the plastic component. When the stress path approached the failure envelope, namely the failure state, signiﬁcant increases of plastic shear

strain occurred, as shown in Fig. 4b, while the elastic

component was linearly on the increase.

Fig. 4c shows the volumetric strain induced by

shearing. The total strain experienced contraction and was followed by dilation above point A, as marked in Fig. 4c. After decomposing of elastic and plastic strains,

it was found that elastic deformation tended to dilate throughout the shearing process while the plastic deformation went through contraction and then dila-tion. Therefore, if the establishment of a representative constitutive model for such sandstone is intended, shear dilation should be considered in addition to the conventional elasticity, to which pure shearing would induce no volumetric deformation.

As long as the plastic strain, including volumetric

strain shown inFig. 4a and cand shear strain shown in

Fig. 4b, is obtained, the increments of plastic strains along any particular stress path (CTC or PS) can be

presented as vectors shown inFig. 5.Fig. 5reveals the

following characteristics of plastic deformation of sandstone: 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 Table 2

Physical properties of the studied sandstones

Formation g_d(g/cm3₎ _Gs _{n (%)} _{GAR (%)} _{Matrix (%)} _{Mineralogy of grains}

Quartz (%) Feldspar (%) Rock fragment (%)

WGS1 2.19 2.66 17.4 65.0 17.6 90.3 0.0 9.7 WGS2 2.26 2.72 16.7 25.3 58.0 85.8 0.0 7.2 MS1 2.24 2.52 11.5 50.4 38.2 88.0 0.2 10.3 MS2 2.28 2.66 14.1 67.5 18.5 90.7 0.2 9.0 MS3 2.32 2.67 13.1 51.0 35.9 85.0 2.3 12.2 TL1 2.36 2.72 13.1 36.4 50.5 86.5 1.7 9.9 TL2 2.35 2.70 12.8 50.0 37.2 87.3 0.7 9.7 ST 2.18 2.67 18.2 40.4 41.4 77.7 4.4 12.7 NK 2.31 2.71 14.8 28.6 56.6 90.0 2.3 5.6 TK 2.30 2.65 12.8 28.2 59.0 84.5 0.5 13.0 SFG1 2.01 2.66 24.6 52.6 22.8 95.6 0.8 3.0 SFG2 2.21 2.66 16.9 42.8 40.4 78.4 1.6 8.9 CL 2.14 2.70 20.7 39.4 40.0 83.7 1.0 5.5 Table 3

Mechanical properties of sandstones

Formation UCSdry(MPa) UCSwet(MPa) R Strength classiﬁcation ISRM (1981) Edry(MPa) Ewet(MPa) Ewet/EdryShear dilation No. specimen Dry Sat WGS1 34.1 25.4 0.74 Moderate 11.9 4.4 0.37 Yes 8 9 WGS2 47.5 6.7 0.14 Moderate 5.0 1.6 0.32 Yes 10 8 MS1 48.5 28.9 0.60 Moderate 7.6 3.4 0.45 Yes 15 2 MS2 37.1 28.3 0.76 Moderate 12.7 10.0 0.79 Yes 27 23 MS3 82.7 43.3 0.52 Medium 14.0 9.9 0.71 Yes 3 3 TL1 68.7 23.2 0.34 Medium 9.7 4.3 0.44 Yes 11 9 TL2 77.5 44.2 0.57 Medium 12.2 7.3 0.60 Yes 3 3 ST 38.4 7.8 0.20 Moderate 5.6 1.6 0.29 Yes 5 3 NK 86.0 43.2 0.50 Medium 12.1 8.1 0.67 Yes 4 3 TK 69.0 29.4 0.43 Medium 5.2 2.2 0.43 Yes 10 2

SFG1 14.5 12.2 0.84 Low strength 2.6 2.2 0.85 Yes 3 3

SFG2 46.4 19.9 0.43 Moderate 5.2 2.5 0.48 Yes 3 3

CL 19.9 3.1 0.16 Low strength — — — Yes 7 6

Remarks: The strength reduction ratio R due to wetting softening is deﬁned as: R ¼ UCSwet=UCSdry: Shear dilation occurs for all tests on both dry and wet specimens.

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(a) Plastic deformation exists prior to the failure state.

Concerning this, a plastic potential surface (or a cap) in addition to the failure envelope is necessary for describing the deformation behavior of sandstone [6].

(b) When approaching failure state, these plastic defor-mation vectors tend to be perpendicular to the

failure envelope of sandstone, regardless the stress paths. That is, the plastic deformation at failure state appears to meet with the so-called associated ﬂow rule. Therefore, a plastic potential surface with

geometry illustrated by Fig. 5b, in which plastic

deformation vectors meet with the associated ﬂow rule, can be assumed. This plastic potential surface would propagate upon increases of loading and during the loading process. The vector continues to be perpendicular to this plastic potential surface and 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 Volumetric strain Volumetric stress p

Plastic def. Elastic def. Total def.

(a) Shear strain 2*(J2')1/2 Shear stress ( J2 ) 1/2 Plastic def. Total def. Elastic def. (b) Volumetric strain Shear stress ( J2 ) 1/2 Elastic def. Plastic def. Total def. A (dilation) (c)

Fig. 4. Schematic illustration of typical deformation curves. Cycles of loading–unloading–reloading, which are indicated by gray lines, were applied to identify plastic deformation in separate stages of loading so that the total strain could be decomposed into elastic and plastic components, as shown in the ﬁgure. (a) Volumetric strain induced by hydrostatic stress (Path OA inFig. 3) (b) Shear strain induced by shear stress (Paths AC inFig. 3) (c) Volumetric strain induced by shear stress (Path AC inFig. 3). 0 20 40 60 80 100 120 0 20 40 60 80 100 120

Hydrostatic stress p (MPa)

Shear stress ( J2 ) 1/2 (MPa) Failure envelope p v δε p δγ (a) 0 20 40 60 80 100 0 20 40 60 80 100 120 v δε p s δε p Failure Envelop e Plastic Flow Plastic Potential Surface J2 (b)

Fig. 5. Vectors of plastic strain increments during triaxial testing. (a) Vectors obtained from various stress paths. (b) Vector obtained from the pure shear test. The final vector (when the stress path meet the failure envelope) tends to be perpendicular to the failure envelope, which indicates a typical associated rule of plastic flow. A conceptual plastic potential surface is also plotted, provided the associated flow rule is assumed. This conceptual plastic potential surface would propagate upon subsequent increases of loading. It should be noticed that all the plastic potential surfaces have a similar geometry; i.e. not evolving of such surface is necessary.

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UNCORRECTED PROOF

eventually be perpendicular to the failure envelope.

Therefore, the plastic potential surface should have a tangent contact with the failure envelope, as shown inFig. 5b.

(c) Inferred from the comparison of the experimental

results shown in Fig. 5a with the plastic potential

surfaces shown inFig. 5b, it can be asserted that the

plastic potential surfaces tend to have a similar geometry, except that differ in sizes.

4. Proposed constitutive model

Given the deformational behavior of sandstone observed during the experimental study, a constitutive model can accordingly be developed to describe the aforementioned behavior found in this study. As the sandstone possesses both elastic and plastic deforma-tion, the total strain tensor can thus be expressed in terms of these two components as

t_ij¼e_ijþp_ij, (1)

where t

ij; eij and

p

ij are total strain tensor, elastic strain

tensor and plastic strain tensor, respectively. 4.1. Elastic component of deformation

In view of the non-linear elastic behavior shown in Fig. 4a and c, a Green elastic model (or hyper-elastic

model) is adopted[31]. On the basis of such a model, the

strain tensor can be determined using strain energy density function O partially differentiated by the stress

tensor sijas the following expression:

e_ij¼ qO

qsij

. (2)

Fig. 4illustrates that the studied sandstones exhibit non-linear, elastic volumetric deformation, induced either by

hydrostatic stress (Fig 4a) or by shear stress (Fig. 4c)

and linear shear deformation (induced by shear stress; Fig. 4b). Given such deformational behavior on one hand and the nature of Green model (Eq. (2)) on the other hand, a strain energy density function O is accordingly proposed as

O ¼ b1I3=21 þb2I1J2þb3J2, (3)

where b1, b2and b3are material constants.

Substituting Eq. (3) into Eq. (2), the elastic strain

tensor e

ij has the following form:

e_ij¼ qO

qsij

¼ ð3₂b1I1=21 þb2J2Þdijþ ðb2I1þb3Þsij. (4)

According to Eq. (4), the volumetric strain can be obtained as

e_v¼e₁₁þe₂₂þe₃₃¼9₂b1I1=21 þ3b2J2. (5)

Furthermore, the elastic volumetric strain can be

further separated into two components, e_v;p and e_v;s as

the following expression:

e_v;p¼9₂b1I1=2₁ , (6) e_v;s¼3b2 ffiffiffiffiffi J2 p 2 , (7)

where I1is a measure of hydrostatic stress and ffiffiffiffiffiJ2

p is a measure of shear stress.

The shear strain g ¼ 2 ffiffiffiffiffiJ0

2

p

can also be determined based on Eq. (5) as ge_¼₂ ffiffiffiffiffiffi_Je0 2 q ¼2ðb2I1þb3Þ ffiffiffiffiffi J2 p . (8)

Namely, the proposed O enables (1) non-linear

volu-metric deformation induced by hydrostatic stress (Fig.

4a); (2) non-linear volume dilation by shear stress (Fig.

4b); and (3) linear shear dilation (Fig. 4c). Accordingly,

the material constants b1, b2and b3can be determined

by a curve ﬁtting derived from the experimental curves, by PS test under a constant hydrostatic stress, as shown inFig. 4. For the tested Tertiary sandstones, two Type A sandstones (WGS1 and MS2) and three Type B sandstones (WGS2, TL1 and CL) the corresponding

b1, b2and b3are summarized inTable 4. The ranges of

b1, b2 and b3 are 100–250 106(MPa)1/2,

0.70.1 106(MPa)2 and 50–170

106(MPa)1, respectively.

The factor b1alone controls the magnitude of elastic

volumetric deformation induced by hydrostatic stress

(e

v;p), as revealed by Eq. (6). A greater b1increases the

amount of volumetric strain, as shown in Fig. 6a.

Similarly, based on Eq. (7), the factor b2alone controls

the amount of elastic shear dilation. A less b2increases

the amount of shear dilation, as shown inFig. 6b. The

amount of elastic shear strain is jointly controlled by b2

and b3by Eq. (8). A greater b3increases the amount of

shear strain, as illustrated inFig. 6c.

4.2. Plastic component of deformation

If a plastic potential surface (or yield cap) with associated ﬂow rule is assumed for the studied sand-stones, the plastic potential surface will have the

geometry, as illustrated inFig. 6b. Similar yield surfaces

have been proposed [33–36,7,11]. This coincidence

reveals the similarity in the deformational nature of geo-material, including sandstones and soils.

An existing plastic model developed by Desai [37–38,8], which possesses a plastic potential surface similar to the above-mentioned features of the tested sandstone, is adopted and further modiﬁed. Assuming that the material has uniform strength on the deviatoric plane, the plastic potential surface for Tertiary sand-1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111

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UNCORRECTED PROOF

stone is accordingly proposed to have the following

simpliﬁed form:

F Ið 1; J2Þ ¼J2 a dðI1þTÞmþgdðI1þT Þ2¼0,

(9)

where T is the I1stress at which shear strength vanishes

and m and gd are material constants controlling the

geometry of the surface. The square root of gdrepresents

the slope of the failure envelope, as illustrated inFig. 7.

In Eq. (9), adis a state parameter; it increases when the

material is subjected to continuous loading and leads to the propagation of the plastic potential surface. Greater

ad represents a greater plastic potential surface, as

depicted inFig. 7. That is, the plastic potential surface

propagates as the state variable adaccumulates.

Meanwhile, since the plastic potential surface is capable of propagating when loading is continuously applied, a hardening rule is necessary and proposed as

dp¼b1a b2 d þ ad b₃ 1=b4 , (10)

where b1; b2; b3 and b4 are material constants

controlling the magnitude of plastic strain increments. Assuming that the associated plastic ﬂow rule is applicable, the magnitude of plastic strain increment can then be expressed as dp¼l qF qsij qF qsij 1=2 ¼b₁ab2 d þ ad b₃ 1=b4 , (11)

where l is a positive scalar factor of proportionality. Accordingly, the magnitude of plastic strain

incre-ment dp; the plastic volumetric strain increment dp_vand

the plastic shear strain increment dp_s can be obtained as

dp_v ¼3l qF qI1 ¼ 3dp qF qI1 qF qsij qF qsij 1=2 ¼ 3 b₁ab2 d þ ad b3 1=b4 ( ) qF qI1 qF qsij qF qsij 1=2 , ð12Þ dp_s ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dp ð Þ2dpv2 q . (13)

Based on Eqs. (9)–(13), the proposed plastic model

has seven material parameters, with g_d; T, m controlling

the geometry of plastic potential surface and b₁; b₂; b₃

and b₄ controlling the magnitude of plastic strains. All

these parameters can be obtained by a curve ﬁtting derived from the experimental results. As for the detailed process of determining these material

para-meters, the reference can be found in Desai [8,37–38]

and Jeng et al. [6]. These material parameters for the

studied sandstones are also listed inTable 4. The ranges

of gd; T, m, b1; b2; b3 and b4 are 0.10–0.14, 31–52 MPa,

3.3–4.1, 1.3 105–9.37 103, 2.11–2.40, 0.52–1.23,

7.3 1011–4.02 107, respectively. The inﬂuence of

b2 and b3 on the plastic volumetric strain and plastic

shear strain is illustrated inFig. 8. In general, greater b₃

results in greater shear contraction followed by dilation

and a greater b₂ leads to greater volumetric strain.

4.3. Determination of parameters

The proposed model includes 3 elastic parameters (b1,

b2 and b3) and 7 plastic parameters (gd; T and m,

determining the geometry of failure envelope and plastic

yield surface; b₁; b₂; b₃ and b₄; determining the

hardening rule). 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 Table 4

List of material parameters of the studied sandstones for the proposed constitutive model

Dry sandstone WGS1 MS2 WGS2 TL1 CL Elastic deformation b1(106_(MPa)1/2₎ _104.9 _167.0 _118.3 _206.3 _246.7 b2(106_(MPa)2₎ _0.13 _0.17 _0.13 _0.26 _0.68 b3(106(MPa)1) 56.7 65.5 75.5 103.0 165.5 Plastic deformation gD 0.11 0.12 0.14 0.12 0.10 T (MPa) 42.95 40.70 26.81 51.30 31.50 M 3.52 3.47 3.60 3.33 4.04

b1 9.37E3 1.30E5 3.88E4 1.22E4 6.50E5

b2 2.36 2.52 2.40 2.39 2.11

b₃ 0.52 0.64 0.65 0.63 1.23

(9)

UNCORRECTED PROOF

For the elastic parameters, b1is determined from the

elastic volumetric strain–volumetric stress relationship

as shown in Fig. 4a. b2 is determined from the elastic

shear stress–shear strain relationship shown inFig. 4c.

Based on the determined b1 and b2, b3 can then be

obtained fromFig. 4b.

For the parameters determining plastic surfaces, g_d

can be obtained from the current stress state as

g_d¼ J2

I1þT

ð Þ2. (14)

In turn, m can be determined from g_dand the current

stress state as

m ¼ 2gd

g_d ½J2=ðI1þT Þ

. (15)

For the hardening parameters, b₁ and b₂ can be

determined from the increments of plastic strain as

lnðdp_{Þ ¼}_b

1þb2lnðadÞ. (16)

Finally, b3 and b4 can be determined from the

following relationship:

lnðdpÞ ¼1=b₄ lnðb₃Þ lnðadÞ

. (17)

5. Performance of the proposed constitutive model To evaluate the performance of the proposed constitutive model, comparisons between simulated deformation and the actual deformation were con-ducted. In addition, the simulations of other existing models were also carried out and compared. In this way the characteristics of the proposed model could be highlighted. Accordingly, the D–P model and Cap model were included. The expressions determining the geometry of the failure envelope, plastic potential surface and hardening rules of the proposed model, DP model and D–P model with a cap (Extended

Drucker–Prager model, EDP) are summarized in Table

5 for comparison. 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 0 10 20 30 40 50 0 5000 10000 15000 20000 25000 30000

Elastic volumetric strain ε εv (10-6)

Volumetric stress p (MPa) b1=100 b1=250 b1=500 (a) 0 10 20 30 40 50 -8000 -6000 -4000 -2000 0

Elastic volumetric strain εv (10-6)

Shear stress ( J2 ) 1/2 (MPa) b2=-0.1 b2=-0.5 b2=-1 (b) 0 10 20 30 40 50 0 4000 8000 12000

Elastic shear strain εs (10-6)

Shear stress ( J2 ) 1/2 (MPa) b3=10 b3=50 b3=100 (c)

Fig. 6. Inﬂuences of the material parameters, b1, b2 and b3 on the simulated elastic deformation based on the proposed model. The units of these parameters are shown inTable 4. (a) Elastic volumetric strain induced by hydrostatic stress, (b) Elastic volumetric strain induced by shear stress, (c) Elastic shear strain induced by shear stress.

Volumetric stress p Shear stress ( J2 ) 1/2 Failure envelope 0 α1 α5 α4 α3 α2 γd0.5

Fig. 7. Schematic illustration of the propagation of proposed plastic potential surfaces, in which a1; a2; a3;y represent different magnitudes of the state variable ad: ffiffiffig

p

d represents the slope of the failure envelope.

(10)

UNCORRECTED PROOF

Typical deformation curves obtained from

experi-ments are shown inFigs. 9a,10a and 11a, in which the

deduced elastic and plastic components were also plotted. The simulated volumetric deformation induced

by volumetric strain is shown in Fig. 9. Since

conven-tional linear elasticity was incorporated into the other two models, they yielded a linear elastic volumetric deformation curve; however, the proposed model allowed the non-linear behavior and resulted in a reasonable volumetric deformation when compared to the actual one.

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 0 5 10 15 20 25 0 1000 2000 3000 4000

Plastic volumetric strain εv (10-6)

Volumetric stress p (MPa) β2=2.3 β2=2.4 β2=2.5 (a) 0 5 10 15 20 25 30 35 -750 -550 -350 -150 50 250

Plastic shear strain εs (10-6)

Shear stress ( J2 ) 1/2 (MPa) β3=0.6 β3=0.65 β3=0.7 (b) 0 5 10 15 20 25 30 35 0 250 500 750 1000 1250 1500

Plastic shear strain εs (10-6)

Shear stress ( J2 ) 1/2 (MPa) β3=0.6 β3=0.65 β3=0.7 (c)

Fig. 8. Inﬂuences of the material parameters, b2 and b3 on the simulated plastic deformation based on the proposed model. (a) Plastic volumetric strain induced by hydrostatic stress, (b) Plastic volumetric strain induced by shear stress, (c) Plastic shear strain induced by shear stress.

Table 5

Summary of the analyzed constitutive models

Proposed model Drucker–Prager model Extended Drucker–Prager model

Elastic deformation _ij_{¼ ð}3 2b1I

1=2

1 þb2J2Þdijþ ðb2I1þb3Þsij ij¼ 1þn

E sijEudij ij¼1þnE sijEudij Plastic deformation

Yield surface F1ðI1; J2Þ ¼J2F ¼ 0 F1ðI1; J2Þ ¼ ffiffiffiffiffiJ2 p

afI1kf¼0 F1ðI1; J2Þ ¼pJ2ffiffiffiffiffiafI1kf¼0 F ¼ ½adðI1þT ÞmþgdðI1þT Þ2 F2¼R2cJ2þ ðI1CÞ

2 ¼R2

cb 2 Plastic potential surface GðI1; ffiffiffiffiffiJ2

p Þ F I1; ffiffiffiffiffiJ2 p GðI1; ffiffiffiffiffiJ2 p Þ F I1; ffiffiffiffiffiJ2 p GðI1; ffiffiffiffiffiJ2 p Þ F I1; ffiffiffiffiffiJ2 p Hardening rule dp_¼_b 1abd2þ abd3

1=b4 Not available dp¼a₁expðZ₁X Þ

0 10 20 30 40 50 0 2000 4000 6000 8000 10000 12000 14000 Volumetric strain εv (10-6) Volumetric stress p (MPa) Measured def. Deduced elastic def. Deduced plastic def. Simulated total def.

(a) 0 10 20 30 40 50 0 5000 10000 15000 Volumetric strain ε_v (10-6₎ Volumetric stress p (MPa) Measured def. Drucker-Prager model Extended Drucker-Prager model This research

(b)

Fig. 9. Simulation of volumetric strain induced by hydrostatic stress using different models. (a) The total strain and the decomposed elastic and plastic components. (b) Simulation comparison for the proposed model and other two models. The sandstone is MS2.

(11)

UNCORRECTED PROOF

As shear deformation was concerned, the simulations

of the other two models had linear deformation prior to the failure state and were followed by signiﬁcant plastic

deformation, as shown inFig. 10b. The proposed model

enabled plastic deformation before the failure state and thus resulted in gradual transition before and after the

failure state (Fig. 10b). As to the volumetric

deforma-tion induced by shearing, the proposed model was capable of simulating shear contraction and shear dilation, which is compared well to the actual

deforma-tion as illustrated inFig. 11b(Table 6).

6. Discussion—inﬂuence of hydrostatic stress and stress path

Based on the result of comparison shown in previous section, the simulation of the proposed model is made under constant hydrostatic stress and sheared under pure shearing state (PS). It appears to meet the actual deformation reasonably well. It would be interesting to further investigate the performance of the proposed

model under various hydrostatic stresses and stress paths, with material parameters obtained during one single test.

Fig. 12 shows the results of deformation simulation under various hydrostatic stresses, p ¼ 20; 40 and 60 MPa, in which the material parameters were obtained from p ¼ 40 MPa condition. It can be seen that the proposed model can also yield reasonable simulation close to the actual behavior, both in shear deformation (Fig. 12a) and in volumetric deformation (Fig. 12b) induced by shearing of other hydrostatic stress condi-tion.

When triaxial tests, other than PS-typed, are con-cerned, the simulated and the actual types of deforma-tion of CTC test, in which axial compression occurs, are

shown in Fig. 13. The simulated results are still

reasonably close to the actual deformation of CTC tests with some discrepancy exist. This discrepancy possibly arises from the slightly evolution of the actual yield surface or the heterogeneity of sandstone samples; however, since the discrepancy is minor, the assumption of a no-evolving yield surface seems to be acceptable. As 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 0 10 20 30 40 50 60 0 4000 8000 12000 Shear strain 2*(J'2)1/2 (10-6) Shear stress ( J2 ) 1/2 (MPa) Measured def. deduced elastic def. deduced plastic def. Simulated total def.

(a) 0 10 20 30 40 50 60 0 5000 10000 15000 Shear strain 2*(J'2)1/2 (10-6) Shear stress ( J2 ) 1/2 (MPa) Measured def. Drucker-Prager model Extended Drucker-Prager model This research

(b)

Fig. 10. Simulation of shear strain induced by shear stress using different models. (a) The total strain and the decomposed elastic and plastic components. (b) Simulations comparison for the proposed model and other two models. The sandstone is MS2.

0 10 20 30 40 50 60 -8000 -6000 -4000 -2000 0 2000 Volumetric strain εv (10-6) Shear stress ( J2 ) 1/2 (MPa) Measured def. Deduced elastic def. Deduced plastic def. Simulated total def.

(a) 0 10 20 30 40 50 60 -4000 -3000 -2000 -1000 0 1000 Volumetric strain εv (10-6) Shear stress ( J2 ) 1/2 (MPa) Measured def. Drucker-Prager model Extended Drucker-Prager model This research

(b)

Fig. 11. Simulation of volumetric strain induced by shear stress using different models. (a) The total strain and the decomposed elastic and plastic components. (b) Comparison of the simulations obtained from the proposed model and other two models. The sandstone is MS2.

(12)

UNCORRECTED PROOF

the axial-elongation type of triaxial test is concerned,

e.g., the RTE, Fig. 14 shows the comparison of the

simulated and the actual deformation and again reveals that both of them are in a reasonably good agreement. In summary, the tests over varying hydrostatic stress and stress paths indicate that the proposed model is capable of providing reasonable simulation under the above-mentioned situations.

7. Conclusion

Tertiary sandstones are characterized with signiﬁcant elastic and plastic shear dilation. In addition to the dilation induced by hydrostatic stress unloading during tunnel excavation, this dilation contributes to the tunnel deformation, especially the crown settlement. This phenomenon indicates that, while allocating the source of tunnel squeezing for any type of rocks and as long as elastic shear dilation is signiﬁcant, the vast unloaded elastic zone should be considered in addition to the plastic zone adjacent to the tunnel wall.

To simulate more sophisticated deformational beha-vior, the number of parameters increases simulta-neously. In the proposed model, a total of 10 parameters are needed. However, tests on the model 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 Table 6

Material parameters of MS2 sandstone based on different material models

Parameter Magnitude Elastic deformation

Linear elasticity K 3.1 GPa

G 11.2 GPa Non-linear elasticity (proposed model) b1 167.0 106(MPa)1/2 b2 0.17 106(MPa)2 b3 65.5 106(MPa)1 Plastic deformation Drucker–Prager model af 0.34 kf 11.02 MPa Extended Drucker–Prager model Rc 1.67 a1 0.29 Z₁ 0.05 Proposed model g_d 0.12 T 40.70 MPa m 3.47 b1 1.30E5 b2 2.52 b3 0.64 b4 7.34E8 0 20 40 60 80 0 5000 10000 15000 Shear strain 2*(J'2)1/2 (10-6) Shear stress ( J2 ) 1/2 (MPa) Measured –p=20 MPa Measured–p=40 MPa Measured–p=60 MPa Simulated (a) 0 20 40 60 80 -6000 -4000 -2000 0 Volumetric strain εv (10-6) Shear stress ( J2 ) 1/2 (MPa) Measured–p=20 MPa Measured–p=40 MPa Measured–p=60 MPa Simulated (b)

Fig. 12. Simulation of shear and volumetric strains induced by shear stress under different constant hydrostatic stress (PS tests). Figs. 12–14, the material parameters were obtained from PS test under p ¼ 40 MPa: The sandstone is MS2.

0 20 40 60 80 0 5000 10000 15000 Shear strain 2*(J'2)1/2 (10-6) Shear stress ( J2 ) 1/2 (MPa) CTC - p=10 MPa CTC - p=18 MPa Predicted (a) 0 20 40 60 80 -8000 -6000 -4000 -2000 0 2000 4000 Volumetric strain εv (10-6) Shear stress ( J2 ) 1/2 (MPa) CTC -p=10 MPa CTC -p=18 MPa Simulated (b)

Fig. 13. Simulation of shear and volumetric strains induced by shear stress for CTC tests.

(13)

UNCORRECTED PROOF

showed that one set of parameters obtained from one

particular testing condition could be applied to various conditions, stress paths and hydrostatic stresses. This more or less reduces the efforts needed to determine the parameters. Furthermore, the applicability of the proposed model is not limited to Tertiary sandstones only. For any rock, as long as it possesses linear or non-linear plastic and elastic deformation, including shear dilation, and plastic potential surface similar to the ones

shown inFig. 5b, this model can be applied.

Acknowledgements

The research is supported by the National Science Council of Taiwan, Grant no. NSC-89-2211-E-002-152, NSC-90-2211-E-002-094 and NSC-91-2211-E-002-046. The considerable effort and valuable suggestions pro-vided by the anonymous reviewers are gratefully acknowledged.

References

[1] Jeng FS, Ju GT, Huang TH. Properties of some weak rock in Taiwan. In: Proceedings of Taiwan rock engineering symposium, Taipei, 1994. p. 259–67 [in Chinese].

[2] Jeng FS, Lee IT, Huang TH. Deterioration of weak rock induced by wetting process. Proceedings of Taiwan rock engineering symposium, Taipei, 1996. p. 373–82 [in Chinese].

[3] 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.

[4] Jeng FS, Huang TH. Shear dilatational behavior of weak sandstone. In: Proceedings of regional symposium on sedimentary rock engineering, Taipei, 1998. p. 262–7.

[5] Jeng FS, Weng MC, Huang TH, Lin ML. Deformational characteristics of weak sandstone and impact to tunnel deforma-tion. Tunnelling Underground Space Technol 2002;17:263–74. [6] Jeng FS, Weng MC, Lin ML, Huang TH. Inﬂuence of

petrographic parameters on geotechnical properties of Tertiary sandstones from Taiwan. Eng Geol 2004;73:71–91.

[7] Lagioia R, Nova R. An experimental and theoretical study of the behaviour of a calcarenite in triaxial compression. Ge´otechnique 1995;45:633–48.

[8] Desai CS, Salami MR. A constitutive model and associated testing for weak rock. Int J Rock Mech Min Sci Geomech Abstr 1987;24:299–307.

[9] Papamichos E, Tronvoll J, Vardoulakis I, Labuz JF, Skjærstein A, Unander TE, Sulem J. Constitutive testing of red Wildmoor sandstone. Mech Cohes-Frict Mater 2000;5:1–40.

[10] Hawkins AB, McConnell BJ. Sensitivity of sandstone strength and deformability to changes in moisture content. Q J Eng Geol 1992;25:115–30.

[11] Lagioia R, Puzrin AM, Potts DM. A new versatile expression for yield and plastic potential surfaces. Comput Geotech 1996;19(3):171–91.

[12] Sulem J, Vardoulakis I, Papamichos E, Oulahna A, Tronvoll J. Elasto-plastic modeling of red Wildmoor sandstone. Mech Cohes-Frict Mater 1999;4:215–45.

[13] Vatsala A, Nova R, Srinivasa Murthy BR. Elastoplastic model for cemented soils. J Geotech Geoenviron Eng 2001;127(8):679–87.

[14] Lade P. Elasto-plastic stress–strain theory for cohesionless soil with curved yield surfaces. Int J Struct 1977;13:1019–35. [15] Kim MK, Lade PV. Modelling rock strength in three dimensions.

Int J Rock Mech Min Sci Geomech Abstr 1984;21:21–33. [16] Gerogiannopolous N, Brown ET. The critical state concept

applied to rock. Int J Rock Mech Min Sci Geomech Abstr 1978;15:1–10.

[17] Brown ET, Michelis PN. A critical state yield equation for strain softening rock. In: Proceedings of the 19th US symposium on rock mechanics, Lake Tahoe, Nevada, 1978. p. 515–9.

[18] Price AM, Farmer IW. Application of yield models to rock. Int J Rock Mech Min Sci Geomech Abstr 1979;16:157–9.

[19] Dragon A, Mroz Z. A continuum model for plastic-brittle behavior of rock and concrete. Int J Rock Mech Min Sci Geomech Abstr 1979;16:253–9.

[20] Ichikawa Y, Yamabe T, Obara Y, Ito F, Kawamoto T. Brittle-ductile fracture of a tuffaceous rock and plasticity theory. In: Desai CS, Gallagher RH, editors. Proceedings of the international conference on constitutive laws for engineering materials. Tucson: Arizona; 1982. p. 349–55.

[21] Brace WF, Paulding BW, Scholz C. Dilatancy in the fracture of crystalline rocks. J Geophys Res 1966;71:3939–53.

[22] Hardy Jr HR, Kim RY, Stefanko R, Wand YJ. Creep and microseismic activity in geologic materials. In: Proceedings of the 11th symposium on rock mechanics (AIME) 1970. p. 377–414. [23] Goodman RE. Introduction to rock mechanics. New York:

Wiley; 1989.

[24] Martin CD, Chandler NA. The progressive fracture of Lac du Bonnet Granite. Int J Rock Mech Min Sci 1994;31:643–59. 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 0 10 20 30 40 50 0 2000 4000 6000 8000 10000 Shear strain 2*(J'2)1/2 (10-6) Shear stress ( J2 ) 1/2 (MPa) RTE1 - p=40 MPa RTE2 - p=40 MPa Simulated (a) 0 10 20 30 40 50 -10000 -6000 -2000 2000 Volumetric strain εv (10-6) Shear stress ( J2 ) 1/2 (MPa) RTE1 - p=40 MPa RTE2 - p=40 MPa Simulated (b)

Fig. 14. Simulation of shear and volumetric strains induced by shear stress for RTE tests.

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UNCORRECTED PROOF

[25] ISRM. In: Brown ET, editor. Rock characterization testing & monitoring—ISRM suggested methods. New York: Pergamon; 1981.

[26] Aristorenas GV. Time-dependent behavior of tunnels excavated in shale. Doctoral thesis, Department of Civil Engineering, Massachusetts Institute of Technology, MA, USA, 1992. [27] Bernabe Y, Fryer DT, Shively RM. Experimental observations of

the elastic and inelastic behaviour of porous sandstones. Geophys J Int 1994;117:403–18.

[28] Be´sulle P, Desrues J, Raynaud S. Experimental characterization of the localisation phenomenon inside a Vosges sandstone in a triaxial cell. Int J Rock Mech Min Sci Geomech Abstr 2000;37:1223–37.

[29] Dyke CG, Dobereiner L. Evaluating the strength and deform-ability of sandstones. Q J Eng Geol 1991;24:123–34.

[30] Handin J, Hager RV. Experimental deformation of sedimentary rock under a conﬁning pressure tests at room temperature on dry samples. Bull Am Assoc Pet Geol 1957;41:1–50.

[31] Chen WF, Saleeb AF. Constitutive equations for engineering materials, vol. I: elasticity and modeling. New York: Wiley; 1982.

[32] Pettijohn FJ, Potter PE, Siever R. Sand and sandstone. Berlin: Springer; 1987 306pp.

[33] DiMaggio ML, Sandler IS. Material model for granular soils. J Eng Mech Div ASCE 1971;97:935–50.

[34] Mullenger G, Davis RO. An uniﬁed yield criterion for cohesion-less granular materials. Int J Numer Analytical Methods Geomech 1981;5:285–94.

[35] Gens A, Nova R. Conceptual bases for a constitutive model for bonded soils and weak rocks. In: Anagnostopoulos, et al., editors. Geotechnical engineering of hard soils-weak rocks. Rotterdam: Balkema; 1993. p. 485–94.

[36] Zhang J, Wong TF, Davis DM. Micromechanics of pressure-induced grain crushing in porous rocks. J Geophys Res 1990;95(B1):341–52.

[37] Desai CS, Varadarajan A. A constitutive model for quasi-static behavior of rock salt. J Geophys Res 1987;92:11445–56. [38] Desai CS, Somasundram S, Frantziskonis G. A hierarchical

approach for constitutive modeling of geologic materials. Int J Num Anal Methods Geomech 1986;10:225–57.

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