International Journal of Rock Mechanics & Mining Sciences 42 (2005) 388–401
Characterizing the deformation behavior of Tertiary sandstones
M.C. Weng, F.S. Jeng
, 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.
r2005 Elsevier Ltd. All rights reserved.
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,
www.elsevier.com/locate/ijrmms
1365-1609/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2004.12.004
Corresponding author. Tel./fax: +886 2 2364 5734. E-mail address: [email protected] (F.S. Jeng).
and found that these Tertiary sandstones can be classified in terms of Grain area ratio (GAR) and porosity (n), as illustrated inFig. 1. Two groups of sandstones, termed as Type A and Type B (with R40:5 and Rp0:5; respec-tively), have been identified. Comparing 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 inFig. 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-mentioned research results, the work focuses on the following aspects:
1. To characterize the deformational behavior of Tertiary sandstones, including elastic and plastic Nomenclature
a1 hardening rule parameter for Cap model af slope of F1
ad state variable of proposed model b1 elastic constant of proposed model
b2 elastic constant of proposed model
b3 elastic constant of proposed model
b1 hardening rule parameter for proposed model b2 hardening rule parameter for proposed model b3 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, dp¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidp ijd
p ij p F ðI1; J2Þ yield surface
GðI1; J2Þ plastic potential surface G shear modulus
G0 initial shear modulus
g shear strain, g ¼ 2 ffiffiffiffiffiJ0 2 p
¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 eijeji p ge elastic shear strain
gp plastic shear strain
gt
total shear strain
gd 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, J02¼
1 2eijeji
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 ¼1 3I1¼
1
3skk (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 CL Lithic graywacke Pliocene Littoral facies Rich calcite content
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 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 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(¼ 1
2eijeij; 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.
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 hydrostatic 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. InFig. 3p ¼1 3I1¼13skk and ffiffiffiffiffiJ2 p (J2¼12SijSij; Sij¼ second deviatoric stress tensor) are measures of hydro-static stress and shear stress, respectively. 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 OACin
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 controlled stress path, could be decomposed, as shown
in Fig. 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 2¼ 1 2eijeij; eij¼ second deviatoric strain tensor) induced by pure 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.
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:
(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 Table 2
Physical properties of the studied sandstones Formation gd(g/cm3) G
s 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 (GPa) Ewet (GPa)
Ewet/Edry Shear 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.
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 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
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) (comp-ression) (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 ACinFig. 3) (c) Volumetric strain induced by shear stress (Path ACinFig. 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.
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 deformation, the total strain tensor can thus be expressed in terms of these two components as t ij¼ e ijþ p ij, (1) where t ij; e ij 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 sij as 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 b1I 3=2
1 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 ð3 2b1I 1=2 1 þb2J2Þdijþ ðb2I1þb3Þsij. (4) According to Eq. (4), the volumetric strain can be obtained as e v¼ e 11þ e 22þ e 33¼ 9 2b1I 1=2 1 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;p¼ 9 2b1I 1=2 1 , (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 ge¼ 2 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, b2 and b3 can
be determined by a curve fitting derived from the experimental curves, by PS test under a constant hydrostatic stress, as shown in Fig. 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
in Table 4. The ranges of b1, b2 and b3 are
100–250 10 6(MPa) 1/2, 0.7 0.1 10 6(MPa) 2 and 50–170 10 6(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 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(10 6(MPa) 1/2) 104.9 167.0 118.3 206.3 246.7 b2(10 6(MPa) 2) 0.13 0.17 0.13 0.26 0.68 b3(10 6(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
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-stone is accordingly proposed to have the following simplified form: F Ið 1; J2Þ ¼J2 adðI1þTÞ m þgdðI1þT Þ 2 ¼0, (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 in Fig. 7. That is, the plastic potential surface propagates as the state variable ad accumulates.
Meanwhile, since the plastic potential surface is capable of propagating when loading is continuously applied, a hardening rule is necessary and proposed as dp¼ b1ab2d þ ad b4 1=b3 , (10) 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 γd 0.5
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
where b1; b2; b3and b4are 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 dp ¼l qF qsij qF qsij 1=2 ¼b1a b2 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 v and 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 b1a b2 d þ ad b4 1=b3 ( ) qF qI1 qF qsij qF qsij 1=2 , ð12Þ dp s ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dp ð Þ2 dp v 2 q . (13)
Based on Eqs. (9)–(13), the proposed plastic model has seven material parameters, with gd; 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 parameters, 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 10 5–9.37 10 3, 2.11–2.40, 0.52–1.23, 7.3 10 11–4.02 10 7, 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).
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 in Fig. 4c. Based on the determined b1 and b2, b3 can then be
obtained from Fig. 4b.
For the parameters determining plastic surfaces, gd can be obtained from the current stress state as gd¼ J2 I1þT ð Þ2. (14) 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.
In turn, m can be determined from gd and the current stress state as m ¼ 2gd gd ½J2=ðI1þTÞ 2 . (15)
For the hardening parameters, b1 and b2 can be determined from the increments of plastic strain during hydrostatic loading as
lnðdp
Þ ¼b1þb2lnðadÞ. (16)
Finally, b3 and b4 can be determined from the following relationship during pure shearing:
lnðdpÞ ¼1=b
3 lnðb4Þ 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 5for comparison.
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 inFig. 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.
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 in Fig. 11b(Table 6).
Table 5
Summary of the analyzed constitutive models
Proposed model Drucker–Prager model Extended Drucker–Prager model Elastic deformation ij¼ ð32b1I
1=2
1 þb2J2Þdijþ ðb2I1þb3Þsij ij¼1þnEsij Eudij ij¼1þnE sij Eudij Plastic deformation
Yield surface F1ðI1; J2Þ ¼J2 F ¼ 0 F1ðI1; J2Þ ¼ ffiffiffiffiffi J2 p afI1 kf¼0 F1ðI1; J2Þ ¼ ffiffiffiffiffi J2 p afI1 kf¼0 F ¼ ½ adðI1þTÞmþgdðI1þTÞ2 F2¼R2cJ2þ ðI1 CÞ2¼R2cb
2 Plastic potential surface GðI1;
ffiffiffiffiffi J2 p Þ F I1; ffiffiffiffiffi J2 p GðI1; ffiffiffiffiffi J2 p Þ F I1; ffiffiffiffiffi J2 p GðI1; ffiffiffiffiffi J2 p Þ F I1; ffiffiffiffiffi J2 p Hardening rule dp¼b 1ab2d þ adb4 1=b3 Not available dp¼a 1expðZ1X 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 εv (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.
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 condition.
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 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 signi-ficant elastic and plastic shear dilation. In addition to 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.
the dilation induced by hydrostatic stress un-loading during tunnel excavation, this dilation contri-butes 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 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 applic-ability 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 in Fig. 5b, this model can be applied.
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 10 6 (MPa) 1/2 b2 0.17 10 6 (MPa) 2 b3 65.5 10 6 (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 gd 0.12 T 40.70 MPa m 3.47 b1 1.30E 5 b2 2.52 b3 0.64 b4 7.34E 8 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.
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. [25] ISRM Ltd. 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. 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.
[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.
