- Email: [email protected]
Mechanical modeling of MX-80–Development of constitutive laws
Physics and Chemistry of the Earth 33 (2008) S504–S507
Contents lists available at ScienceDirect
Physics and Chemistry of the Earth journal homepage: www.elsevier.com/locate/pce
Mechanical modeling of MX-80 – Development of constitutive laws Mattias Åkesson *, Ola Kristensson Clay Technology AB, Ideon, 223 70 Lund, Sweden
a r t i c l e
i n f o
Article history: Available online 11 October 2008 Keywords: Elastoplastic model Swelling pressure Contact stress Unsaturated clay Bentonite Dual porosity
a b s t r a c t In a previous paper [Åkesson, M., Hökmark, H., 2007. Mechanical model for unsaturated MX-80. In: Schanz, T. (Ed.), Theoretical and Numerical Unsaturated Soil Mechanics, Springer Proceeding in Physics, vol. 113, pp. 3–10] a new elastoplastic model for unsaturated swelling clay was presented. The model was formulated as two differential equations, for elastic and plastic strains, respectively. Each equation described a relation between the (air-filled) macro void ratio and the two independent variables stress and (water-filled) micro void ratio. The established concept of swelling pressure and its void ratio dependence were incorporated in the model. To do this, the grain-to-grain contact stress and the contact area had to be considered. In its original form the model only handled one-dimensional problems, e.g. compression tests with uniaxial strain and water uptake test at constant volume with assumed isotropic conditions. This paper describes a first attempt to generalize the model to address multidimensional problems with the intention to integrate it into the framework of Code_Bright. This was made through definition of an expression for the retention properties and through variable substitution. In this way, four functions for the kappa moduli were identified: two elastic and two plastic. These functions provide unique values for each given state. In the used framework of the Barcelona Basic Model (BBM), all strains are treated as elastic, and there is thus no explicit treatment of any deviatoric yield condition. The only remaining parameter outside the new model is the Poisson’s ratio. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Thermo-Hydro-Mechanical modeling of buffer and backfill material is an important subject in the Swedish nuclear waste disposal program and Code_Bright is one of the codes employed in that program. Thermoelastoplastic constitutive laws, based on the BBM (Alonso et al., 1990), which in turn is an extension of the modified Cam–Clay model (Roscoe and Burland, 1968), are used in the code for the description of the mechanical behaviour of compacted bentonite. BBM has, however, certain limitations for expansive soils, and has therefore been further developed into the Barcelona Expansive Model (BExM) (Alonso et al., 1999). An alternative approach to describe the mechanical behaviour of swelling clays, in particular MX-80 bentonite, has recently been presented (Åkesson and Hökmark, 2007). This paper describes a first attempt to generalize this model, from hereon denoted the Bentonite Elastoplastic Swelling pressure and Contact stress based (BESC) model, to address multidimensional problems with the intention to integrate it into the framework of Code_Bright. The basic assumptions, the mathematical formulation and the parameter settings of the BESC model are first described briefly. The method of embedding this model in the BBM framework is thereafter pre* Corresponding author. Tel.: +46 46 286 25 74; fax: +46 46 13 42 30. E-mail address: [email protected] (M. Åkesson). 1474-7065/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.pce.2008.10.014
sented. Application examples for compression and swelling tests with uniaxial strain (oedometers), as well as water uptake tests at constant volume are shown in the subsequent section. Finally, the current limitations of the BESC model are discussed. 2. Short description of the BESC model The model is based on the following assumptions: i. The porosity is two-parted in micro- and macro-porosity. The void ratio e is thus the sum of a micro (em) and a macro (eM) void ratio. The same approach is used in BExM (Alonso et al., 1999). ii. The micro-porosity is water-filled, whereas the macroporosity is air-filled. This is basically a definition, although it can be interpreted as if all water is adsorbed in interlayer pore space. This is supported by the high capacity of water uptake observed for sodium bentonite (e.g. Norrish, 1954). The micro void ratio is thus defined as em = Vw/Vs = wqs/qw, where w is the water content and qs/qw is the particle-water density ratio (=2.78). iii. The two void ratios (em and eM) and the stress (r), are the state variables of the model (r is positive for compressive stresses). The stress is defined as a scalar, denoting the axial
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M. Åkesson, O. Kristensson / Physics and Chemistry of the Earth 33 (2008) S504–S507
iv.
v. vi.
vii.
stress in 1D oedometer tests, and the hydrostatic pressure for isotropic problems. The model is extended through identification of r with the net mean stress and introduction of a Poisson’s ratio (see Section 3). The contact stress (rc) between grains is defined by a ratio, a = r/rc, corresponding to the ratio between the contact area and the section area. a is assumed to be a function of em and eM. The swelling pressure (ps) is a central quantity of the model, and is determined by the micro void ratio only. A condition for elastic strains is that rc < ps, while rc = ps for plastic strains. A transition from elastic to plastic strains is made when the latter condition is fulfilled. No deviatoric condition for plastic strains has yet been defined. The elastic domain is governed by a modulus of compression (K), determined by the micro void ratio only.
The elastic relation can be expressed as:
r @a aK r @a deM ¼ dr de a @eM 1 þ em þ eM a @em m
ð1Þ
The plastic relation can be expressed as:
r @a dp r @a dem deM ¼ dr a s þ a @eM dem a @em
ð2Þ
The functions used in the model for the modulus of compression, swelling pressure and the contact area, respectively, are given as follows:
Kðem Þ ¼ K 1 expð5 ð1 em ÞÞ ðMPaÞ 1 0:24
ps ðem Þ ¼ 2 em ðMPaÞ c 1 þ em aðem ; eM Þ ¼ ðÞ 1 þ em þ eM
ð3Þ
ð5Þ
3. Embedment of the BESC model in the BBM framework The elastic part of the BBM model is defined as: 0
de ¼ ji ded
ð6Þ
where ji ¼ ji ðp0 ; s; eÞ and js ¼ js ðp0 ; s; eÞ are the elastic modulus functions through which the BESC model is embedded in the BBM framework. In the equation above e is the void ratio, p0 is the net mean stress, s the suction, ed the deviatoric strain, rd the deviatoric stress, m Poisson’s ratio and G the shear-modulus. The translation of the BESC model into a state-specific kappa modulus requires a consistent description of retention properties since this model uses em as a state variable rather than suction. The actual suction, s, is calculated using: s ¼ Wðem Þ p0 (Dueck, 2007), where w is suction for free swelling conditions, here identified with the defined swelling pressure function, Eq. (4). With this relation, the state variables e, p0 and s can be substituted for em, eM and r:
8 1 0 > < em ¼ ps ðs þ p Þ 1 eM ¼ e ps ðs þ p0 Þ > : r ¼ p0
0
ds ¼ dps dp ¼
dps 0 dem dp dem
ð8Þ
With the condition that either s or p is constant; the kappa moduli can be identified from the Eqs. (1), (2), and (8) and the following expressions:
8 0 dp > > ðds ¼ 0Þ < dem þ deM ¼ ji 0 p > > 0 : dem þ deM ¼ js ds ðdp ¼ 0Þ s þ 0:1
ð9Þ
The resulting four functions are shown in Table 1. In order to implement the embedded model in the source code of Code_Bright, it is essential that this is made concurrently with the employment of a retention curve consistent with the mechanical model. This can be achieved through expressing the retention properties given by Eq. (7) in terms of degree of saturation (Sl):
Sl ðp0 ; s; eÞ ¼
0 p1 s ðs þ p Þ e
ð10Þ
4. Test of embedded model
ð4Þ
The values of the parameters K1 and c were in this study the same as for isotropic 3D conditions used in the original paper: K1 = 50 MPa and c = 15.
dp ds þ js s þ 0:1 p0 1 3ð1 2mÞ ð1 þ eÞp0 ¼ drd ; G ¼ 2G 2ð1 þ mÞ ji
for elastic, ðji ; js Þe , and plastic conditions, ðji ; js Þp , and provide unique values for each given state. In the used framework of BBM, all strains are treated as elastic, and there is thus no explicit treatment of any deviatoric yield condition. The only remaining parameter outside the BESC model is the Poisson’s ratio, m. In order to derive the kappa modulus functions the suction is differentiated as:
The embedded model has been tested for compression tests with uniaxial strain and constant relative humidity (Fig. 1), swelling tests with uniaxial strain and constant axial load (Fig. 2) and isotropic swelling pressure build-up (Fig. 3). The compression tests and the swelling tests were made in oedometers, consisting of a steel ring around the sample with filters on both sides (Dueck, 2007). The samples were exposed to humidified air, with specified relative humidity, through grooves above and below the filters. Pistons and force transducers were placed axially above the sample, as well as radially through the steel ring. The axial as well as the radial stresses could thereby be measured, although no radial strains were allowed. Calculations were made with simple routines written in the MathCadÒ environment, following the equation system presented in an accompanying paper (Kristensson and Åkesson, 2008). The models of the compression and the swelling tests were chosen to simulate experimental results presented by Dueck and Nilsson (2008). In these calculations, m were fitted to match the experimental results. The model of the swelling pressure build-up at constant volume was generic. The compression tests shown in Fig. 1, were performed at 24 and 28 MPa suction, respectively. With the employed description of the retention properties, Eq. (7), this corresponds to water content of approx. 19%, which was also found in the analysis after the tests. The slopes during compression as well as the apparent pre-
Table 1 Kappa modulus functions.
ji
ð7Þ
The state, elastic or plastic, is determined by the conditions rc < ps and rc = ps, respectively. The kappa modulus functions are defined
js
Elastic (Condition: p’ < aps) 2 3 0 @a 1 pa @e dp1s m 61 dem 7 p0 4 dp þ p0 @ a aK 5 s a @eM 1þem þeM dem 2 3 0 @a 1 pa @e dp1s m 61 dem 7 ðs þ 0:1Þ 4 dp p0 @ a aK 5 s a @eM 1þem þeM de m
Plastic (Condition: p’ = aps) 2 3 0 @a 1 a pa @e dp1s m 61 dem 7 p0 4 dp þ 5 p0 @ a s a @eM de m
" ðs þ 0:1Þ
1
dps dem
0
dps @a a de þ pa @e m m p0
dps @a a @eM dem
#
25
0.8
25
0.7
20
0.7
20
0.6
15
0.6
15
0.5
10
0.5
10
0.4
5
0.4
5
0.3 0.1
1
Void ratio (-)
Radial stress (MPa)
0.8
0.3 0.1
0 100
10
1
Axial stress (MPa)
Radial stress (MPa)
M. Åkesson, O. Kristensson / Physics and Chemistry of the Earth 33 (2008) S504–S507
Void ratio (-)
S506
0 100
10
Axial stress (MPa)
O4_0905; ν = 0.29
O4_1105; ν = 0.25
0.65
15
0.6
10
0.55
5
0.5
0
20
40
60
80
100
2
1
0
0.8
1.5
0.6
1
0.4
0.5
0.2
0
20
Suction (MPa)
40
60
80
100
Radial stress (MPa)
20
Void ratio (-)
0.7
Radial stress (MPa)
Void ratio (-)
Fig. 1. Compression tests with uniaxial strain and constant suction (left: loading at 24 MPa suction; right: loading and unloading at 28 MPa suction). Test results for void ratio (s) and radial stress (). Model results for void ratio (solid lines) and radial stress (dashed lines).
0
Suction (MPa)
O4_1005 & O3_1105; ν = 0.30
O1_1105; ν = 0.467
Fig. 2. Swelling tests with uniaxial strain and constant axial load (left: two swelling tests at 9 MPa; right: swelling at 1 MPa). Test results for void ratio (s) and radial stress () (O4_1005 (large symbols) and O3_1105 (small symbols)). Model results for void ratio (solid lines) and radial stress (dashed lines).
Suction (MPa)
60
40 0.2 Kappa_i
20
Kappa_s 0.15 0
5
10
Kappa
0
Net mean stress (MPa)
Net mean stress (MPa)
0.1
10 0.05
5
0
10
20
30
40
50
Suction (MPa) 0 16
18
20
22
24
26
Water content (%) Fig. 3. Swelling pressure build-up at constant volume (isotropic case, void ratio 0.712, initial water content 17.5%, corresponding to suction 40.5 MPa). Stress path in s–p plane (upper left) and evolution of net mean stress with water content (lower left). The evolution of apparent kappa moduli with decreasing suction (right).
M. Åkesson, O. Kristensson / Physics and Chemistry of the Earth 33 (2008) S504–S507
consolidation stresses are fairly well captured with the model. The slope for unloading is however steeper in the model than in the experiment. This is partly due to the linear elastic basis for the BESC model. The elastic slope of a volume vs. log-stress diagram therefore tends to increase with the stress-level. This is amplified by the constant suction condition which tends to decrease the water content during loading. The experimental results also displayed a stiff behaviour with a ji-value of 0.01 for the last steps, whereas it was 0.06 for the first two steps taken together. The swelling tests shown in Fig. 2, were performed at 9 and 1 MPa axial load, respectively. Results from two tests with 9 MPa axial stress are shown for comparison. Both tests had approximately the same initial condition, but were terminated at two different suction values: at 38 and 13 MPa, respectively. The initial suction value was in all tests approx. 100 MPa. With the employed retention properties, this corresponds to water content of approx. 14%, which was significantly higher than the measured value of approx. 9.5%. The evolution of the void ratio as well as the radial stress is fairly well captured with the model. In the model of the 9 MPa case, plastic conditions were reached at approx. 40 MPa suction, whereas the 1 MPa case never reached plastic conditions. The isotropic swelling pressure buildup at constant volume shown in Fig. 3 had an initial water content of 17.5%, corresponding to 40.5 MPa suction with the employed retention properties, and a void ratio of 0.712. The evolution of the apparent kappa moduli reveals that these parameters display a quite complicated behaviour. The transition from elastic to plastic conditions, in this case at a water content of 20%, have a marked influence of the apparent kappa values. The ji-value starts from zero and increase with decreasing suction values up to the point of plastic conditions, in this case at a suction of 17 MPa and a ji at 0.08. After the transition, the apparent ji-value is significantly higher, 0.12, and continues to increase up to 0.17 at saturated conditions. The js-value starts from approx. 0.12 and remains fairly constant during the elastic conditions. At the transition to plastic conditions, the js-value drops to 0.03. From thereon it decreases down to zero at saturation.
5. Discussion The examples shown above illustrate the robustness of the BESC model and shows that there is generally no need to specify different parameter values for different experimental setups and initial void ratios and water contents. The apparent kappa values shown in Fig. 3 vary, on the other hand, significantly. The plastic relation, Eq. (2), implies that the swelling pressure buildup at confined condition always ends up with value specified by the ps-function, Eq. (4). The used ps-function appears nevertheless to overestimate the true swelling pressure at water contents below 17% (i.e. below approx. 0.5 in void ratio). This is reflected by the high initial water content found for the modeled swelling
S507
tests in Fig. 2. More work is therefore needed to adopt a more general swelling pressure function. The specification of a single m-value is unsatisfactory. The fitted values in the examples shown in Figs. 1 and 2 exhibited a large variation. Some experimental data also indicate that m tends to increase with load during compression. For example, the first two loading steps in the compression test shown in Fig. 1 (left graph) correspond to m-value of 0.1 and 0.25, respectively. The third step corresponds to 0.34, although this deformation is apparently plastic and therefore is this value not relevant. The m-value for the first step in the right graph of Fig. 1 is virtually zero, since the radial stress actually decreased. This behaviour constitutes a challenge for hydromechanical modeling of these types of materials. Deviatoric yield conditions are not represented and the current formulation is therefore inadequate for modeling triaxial compression tests. The inclusion of a deviatoric plastic strain, specified by some type of flow rule, may be a path forward to overcome this limitation. The current formulation is also inadequate for describing shrinkage behaviour. At low constant loads, the elastic relation implies that there is almost no change in the macro void ratio. This appears to be realistic for swelling conditions, but not for shrinkage. A third state, separate from the defined elastic and plastic relations, in which shrinkage is limited, should therefore be developed. Apart from these limitations, the presented results are promising and future work will be devoted to investigate if the approach is applicable for introduction in the source code of Code_Bright as well. Acknowledgement The authors wish to acknowledge that this paper is a result of work funded by the Swedish Nuclear Fuel and Waste Management Company (SKB). References Åkesson, M., Hökmark, H., 2007. Mechanical model for unsaturated MX-80. In: Schanz, T. (Ed.), Theoretical and Numerical Unsaturated Soil Mechanics, Springer Proceeding in Physics, vol. 113, pp. 3–10. Alonso, E.E., Gens, A., Josa, A., 1990. A constitutive model for partially saturated soils. Géotechnique 40, 405–430. Alonso, E.E., Vaunat, J., Gens, A., 1999. Modelling the mechanical behaviour of expansive clays. Engineering Geology 54, 173–183. Dueck, A., Nilsson, U., 2008. Hydro-mechanical properties of unsaturated MX-80, laboratory study 2005–2007. SKB Report in Progress. Dueck, A., 2007. Results from suction controlled laboratory tests on unsaturated bentonite – Verification of a model. In: Schanz, T. (Ed.), Experimental Unsaturated Soil Mechanics, Springer Proceeding in Physics, vol. 112, pp. 329–335. Kristensson, O., Åkesson, M., 2008. Mechanical modeling of MX-80 – Quick tools for BBM parameter analysis. Physics and Chemistry of the Earth 33S1, S508–S515. Norrish, K., 1954. The swelling of montmorillonite. Discussions of the Faraday Society 18, 120–134. Roscoe, K.H., Burland, J.B., 1968. On the generalized stress–strain behaviour of ‘wet’ clay. In: Heyman, J., Leckie, F.A. (Eds.), Engineering Plasticity. Cambridge Univ. Press, pp. 535–609.
Contents lists available at ScienceDirect
Physics and Chemistry of the Earth journal homepage: www.elsevier.com/locate/pce
Mechanical modeling of MX-80 – Development of constitutive laws Mattias Åkesson *, Ola Kristensson Clay Technology AB, Ideon, 223 70 Lund, Sweden
a r t i c l e
i n f o
Article history: Available online 11 October 2008 Keywords: Elastoplastic model Swelling pressure Contact stress Unsaturated clay Bentonite Dual porosity
a b s t r a c t In a previous paper [Åkesson, M., Hökmark, H., 2007. Mechanical model for unsaturated MX-80. In: Schanz, T. (Ed.), Theoretical and Numerical Unsaturated Soil Mechanics, Springer Proceeding in Physics, vol. 113, pp. 3–10] a new elastoplastic model for unsaturated swelling clay was presented. The model was formulated as two differential equations, for elastic and plastic strains, respectively. Each equation described a relation between the (air-filled) macro void ratio and the two independent variables stress and (water-filled) micro void ratio. The established concept of swelling pressure and its void ratio dependence were incorporated in the model. To do this, the grain-to-grain contact stress and the contact area had to be considered. In its original form the model only handled one-dimensional problems, e.g. compression tests with uniaxial strain and water uptake test at constant volume with assumed isotropic conditions. This paper describes a first attempt to generalize the model to address multidimensional problems with the intention to integrate it into the framework of Code_Bright. This was made through definition of an expression for the retention properties and through variable substitution. In this way, four functions for the kappa moduli were identified: two elastic and two plastic. These functions provide unique values for each given state. In the used framework of the Barcelona Basic Model (BBM), all strains are treated as elastic, and there is thus no explicit treatment of any deviatoric yield condition. The only remaining parameter outside the new model is the Poisson’s ratio. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Thermo-Hydro-Mechanical modeling of buffer and backfill material is an important subject in the Swedish nuclear waste disposal program and Code_Bright is one of the codes employed in that program. Thermoelastoplastic constitutive laws, based on the BBM (Alonso et al., 1990), which in turn is an extension of the modified Cam–Clay model (Roscoe and Burland, 1968), are used in the code for the description of the mechanical behaviour of compacted bentonite. BBM has, however, certain limitations for expansive soils, and has therefore been further developed into the Barcelona Expansive Model (BExM) (Alonso et al., 1999). An alternative approach to describe the mechanical behaviour of swelling clays, in particular MX-80 bentonite, has recently been presented (Åkesson and Hökmark, 2007). This paper describes a first attempt to generalize this model, from hereon denoted the Bentonite Elastoplastic Swelling pressure and Contact stress based (BESC) model, to address multidimensional problems with the intention to integrate it into the framework of Code_Bright. The basic assumptions, the mathematical formulation and the parameter settings of the BESC model are first described briefly. The method of embedding this model in the BBM framework is thereafter pre* Corresponding author. Tel.: +46 46 286 25 74; fax: +46 46 13 42 30. E-mail address: [email protected] (M. Åkesson). 1474-7065/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.pce.2008.10.014
sented. Application examples for compression and swelling tests with uniaxial strain (oedometers), as well as water uptake tests at constant volume are shown in the subsequent section. Finally, the current limitations of the BESC model are discussed. 2. Short description of the BESC model The model is based on the following assumptions: i. The porosity is two-parted in micro- and macro-porosity. The void ratio e is thus the sum of a micro (em) and a macro (eM) void ratio. The same approach is used in BExM (Alonso et al., 1999). ii. The micro-porosity is water-filled, whereas the macroporosity is air-filled. This is basically a definition, although it can be interpreted as if all water is adsorbed in interlayer pore space. This is supported by the high capacity of water uptake observed for sodium bentonite (e.g. Norrish, 1954). The micro void ratio is thus defined as em = Vw/Vs = wqs/qw, where w is the water content and qs/qw is the particle-water density ratio (=2.78). iii. The two void ratios (em and eM) and the stress (r), are the state variables of the model (r is positive for compressive stresses). The stress is defined as a scalar, denoting the axial
S505
M. Åkesson, O. Kristensson / Physics and Chemistry of the Earth 33 (2008) S504–S507
iv.
v. vi.
vii.
stress in 1D oedometer tests, and the hydrostatic pressure for isotropic problems. The model is extended through identification of r with the net mean stress and introduction of a Poisson’s ratio (see Section 3). The contact stress (rc) between grains is defined by a ratio, a = r/rc, corresponding to the ratio between the contact area and the section area. a is assumed to be a function of em and eM. The swelling pressure (ps) is a central quantity of the model, and is determined by the micro void ratio only. A condition for elastic strains is that rc < ps, while rc = ps for plastic strains. A transition from elastic to plastic strains is made when the latter condition is fulfilled. No deviatoric condition for plastic strains has yet been defined. The elastic domain is governed by a modulus of compression (K), determined by the micro void ratio only.
The elastic relation can be expressed as:
r @a aK r @a deM ¼ dr de a @eM 1 þ em þ eM a @em m
ð1Þ
The plastic relation can be expressed as:
r @a dp r @a dem deM ¼ dr a s þ a @eM dem a @em
ð2Þ
The functions used in the model for the modulus of compression, swelling pressure and the contact area, respectively, are given as follows:
Kðem Þ ¼ K 1 expð5 ð1 em ÞÞ ðMPaÞ 1 0:24
ps ðem Þ ¼ 2 em ðMPaÞ c 1 þ em aðem ; eM Þ ¼ ðÞ 1 þ em þ eM
ð3Þ
ð5Þ
3. Embedment of the BESC model in the BBM framework The elastic part of the BBM model is defined as: 0
de ¼ ji ded
ð6Þ
where ji ¼ ji ðp0 ; s; eÞ and js ¼ js ðp0 ; s; eÞ are the elastic modulus functions through which the BESC model is embedded in the BBM framework. In the equation above e is the void ratio, p0 is the net mean stress, s the suction, ed the deviatoric strain, rd the deviatoric stress, m Poisson’s ratio and G the shear-modulus. The translation of the BESC model into a state-specific kappa modulus requires a consistent description of retention properties since this model uses em as a state variable rather than suction. The actual suction, s, is calculated using: s ¼ Wðem Þ p0 (Dueck, 2007), where w is suction for free swelling conditions, here identified with the defined swelling pressure function, Eq. (4). With this relation, the state variables e, p0 and s can be substituted for em, eM and r:
8 1 0 > < em ¼ ps ðs þ p Þ 1 eM ¼ e ps ðs þ p0 Þ > : r ¼ p0
0
ds ¼ dps dp ¼
dps 0 dem dp dem
ð8Þ
With the condition that either s or p is constant; the kappa moduli can be identified from the Eqs. (1), (2), and (8) and the following expressions:
8 0 dp > > ðds ¼ 0Þ < dem þ deM ¼ ji 0 p > > 0 : dem þ deM ¼ js ds ðdp ¼ 0Þ s þ 0:1
ð9Þ
The resulting four functions are shown in Table 1. In order to implement the embedded model in the source code of Code_Bright, it is essential that this is made concurrently with the employment of a retention curve consistent with the mechanical model. This can be achieved through expressing the retention properties given by Eq. (7) in terms of degree of saturation (Sl):
Sl ðp0 ; s; eÞ ¼
0 p1 s ðs þ p Þ e
ð10Þ
4. Test of embedded model
ð4Þ
The values of the parameters K1 and c were in this study the same as for isotropic 3D conditions used in the original paper: K1 = 50 MPa and c = 15.
dp ds þ js s þ 0:1 p0 1 3ð1 2mÞ ð1 þ eÞp0 ¼ drd ; G ¼ 2G 2ð1 þ mÞ ji
for elastic, ðji ; js Þe , and plastic conditions, ðji ; js Þp , and provide unique values for each given state. In the used framework of BBM, all strains are treated as elastic, and there is thus no explicit treatment of any deviatoric yield condition. The only remaining parameter outside the BESC model is the Poisson’s ratio, m. In order to derive the kappa modulus functions the suction is differentiated as:
The embedded model has been tested for compression tests with uniaxial strain and constant relative humidity (Fig. 1), swelling tests with uniaxial strain and constant axial load (Fig. 2) and isotropic swelling pressure build-up (Fig. 3). The compression tests and the swelling tests were made in oedometers, consisting of a steel ring around the sample with filters on both sides (Dueck, 2007). The samples were exposed to humidified air, with specified relative humidity, through grooves above and below the filters. Pistons and force transducers were placed axially above the sample, as well as radially through the steel ring. The axial as well as the radial stresses could thereby be measured, although no radial strains were allowed. Calculations were made with simple routines written in the MathCadÒ environment, following the equation system presented in an accompanying paper (Kristensson and Åkesson, 2008). The models of the compression and the swelling tests were chosen to simulate experimental results presented by Dueck and Nilsson (2008). In these calculations, m were fitted to match the experimental results. The model of the swelling pressure build-up at constant volume was generic. The compression tests shown in Fig. 1, were performed at 24 and 28 MPa suction, respectively. With the employed description of the retention properties, Eq. (7), this corresponds to water content of approx. 19%, which was also found in the analysis after the tests. The slopes during compression as well as the apparent pre-
Table 1 Kappa modulus functions.
ji
ð7Þ
The state, elastic or plastic, is determined by the conditions rc < ps and rc = ps, respectively. The kappa modulus functions are defined
js
Elastic (Condition: p’ < aps) 2 3 0 @a 1 pa @e dp1s m 61 dem 7 p0 4 dp þ p0 @ a aK 5 s a @eM 1þem þeM dem 2 3 0 @a 1 pa @e dp1s m 61 dem 7 ðs þ 0:1Þ 4 dp p0 @ a aK 5 s a @eM 1þem þeM de m
Plastic (Condition: p’ = aps) 2 3 0 @a 1 a pa @e dp1s m 61 dem 7 p0 4 dp þ 5 p0 @ a s a @eM de m
" ðs þ 0:1Þ
1
dps dem
0
dps @a a de þ pa @e m m p0
dps @a a @eM dem
#
25
0.8
25
0.7
20
0.7
20
0.6
15
0.6
15
0.5
10
0.5
10
0.4
5
0.4
5
0.3 0.1
1
Void ratio (-)
Radial stress (MPa)
0.8
0.3 0.1
0 100
10
1
Axial stress (MPa)
Radial stress (MPa)
M. Åkesson, O. Kristensson / Physics and Chemistry of the Earth 33 (2008) S504–S507
Void ratio (-)
S506
0 100
10
Axial stress (MPa)
O4_0905; ν = 0.29
O4_1105; ν = 0.25
0.65
15
0.6
10
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Fig. 1. Compression tests with uniaxial strain and constant suction (left: loading at 24 MPa suction; right: loading and unloading at 28 MPa suction). Test results for void ratio (s) and radial stress (). Model results for void ratio (solid lines) and radial stress (dashed lines).
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O4_1005 & O3_1105; ν = 0.30
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Fig. 2. Swelling tests with uniaxial strain and constant axial load (left: two swelling tests at 9 MPa; right: swelling at 1 MPa). Test results for void ratio (s) and radial stress () (O4_1005 (large symbols) and O3_1105 (small symbols)). Model results for void ratio (solid lines) and radial stress (dashed lines).
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Water content (%) Fig. 3. Swelling pressure build-up at constant volume (isotropic case, void ratio 0.712, initial water content 17.5%, corresponding to suction 40.5 MPa). Stress path in s–p plane (upper left) and evolution of net mean stress with water content (lower left). The evolution of apparent kappa moduli with decreasing suction (right).
M. Åkesson, O. Kristensson / Physics and Chemistry of the Earth 33 (2008) S504–S507
consolidation stresses are fairly well captured with the model. The slope for unloading is however steeper in the model than in the experiment. This is partly due to the linear elastic basis for the BESC model. The elastic slope of a volume vs. log-stress diagram therefore tends to increase with the stress-level. This is amplified by the constant suction condition which tends to decrease the water content during loading. The experimental results also displayed a stiff behaviour with a ji-value of 0.01 for the last steps, whereas it was 0.06 for the first two steps taken together. The swelling tests shown in Fig. 2, were performed at 9 and 1 MPa axial load, respectively. Results from two tests with 9 MPa axial stress are shown for comparison. Both tests had approximately the same initial condition, but were terminated at two different suction values: at 38 and 13 MPa, respectively. The initial suction value was in all tests approx. 100 MPa. With the employed retention properties, this corresponds to water content of approx. 14%, which was significantly higher than the measured value of approx. 9.5%. The evolution of the void ratio as well as the radial stress is fairly well captured with the model. In the model of the 9 MPa case, plastic conditions were reached at approx. 40 MPa suction, whereas the 1 MPa case never reached plastic conditions. The isotropic swelling pressure buildup at constant volume shown in Fig. 3 had an initial water content of 17.5%, corresponding to 40.5 MPa suction with the employed retention properties, and a void ratio of 0.712. The evolution of the apparent kappa moduli reveals that these parameters display a quite complicated behaviour. The transition from elastic to plastic conditions, in this case at a water content of 20%, have a marked influence of the apparent kappa values. The ji-value starts from zero and increase with decreasing suction values up to the point of plastic conditions, in this case at a suction of 17 MPa and a ji at 0.08. After the transition, the apparent ji-value is significantly higher, 0.12, and continues to increase up to 0.17 at saturated conditions. The js-value starts from approx. 0.12 and remains fairly constant during the elastic conditions. At the transition to plastic conditions, the js-value drops to 0.03. From thereon it decreases down to zero at saturation.
5. Discussion The examples shown above illustrate the robustness of the BESC model and shows that there is generally no need to specify different parameter values for different experimental setups and initial void ratios and water contents. The apparent kappa values shown in Fig. 3 vary, on the other hand, significantly. The plastic relation, Eq. (2), implies that the swelling pressure buildup at confined condition always ends up with value specified by the ps-function, Eq. (4). The used ps-function appears nevertheless to overestimate the true swelling pressure at water contents below 17% (i.e. below approx. 0.5 in void ratio). This is reflected by the high initial water content found for the modeled swelling
S507
tests in Fig. 2. More work is therefore needed to adopt a more general swelling pressure function. The specification of a single m-value is unsatisfactory. The fitted values in the examples shown in Figs. 1 and 2 exhibited a large variation. Some experimental data also indicate that m tends to increase with load during compression. For example, the first two loading steps in the compression test shown in Fig. 1 (left graph) correspond to m-value of 0.1 and 0.25, respectively. The third step corresponds to 0.34, although this deformation is apparently plastic and therefore is this value not relevant. The m-value for the first step in the right graph of Fig. 1 is virtually zero, since the radial stress actually decreased. This behaviour constitutes a challenge for hydromechanical modeling of these types of materials. Deviatoric yield conditions are not represented and the current formulation is therefore inadequate for modeling triaxial compression tests. The inclusion of a deviatoric plastic strain, specified by some type of flow rule, may be a path forward to overcome this limitation. The current formulation is also inadequate for describing shrinkage behaviour. At low constant loads, the elastic relation implies that there is almost no change in the macro void ratio. This appears to be realistic for swelling conditions, but not for shrinkage. A third state, separate from the defined elastic and plastic relations, in which shrinkage is limited, should therefore be developed. Apart from these limitations, the presented results are promising and future work will be devoted to investigate if the approach is applicable for introduction in the source code of Code_Bright as well. Acknowledgement The authors wish to acknowledge that this paper is a result of work funded by the Swedish Nuclear Fuel and Waste Management Company (SKB). References Åkesson, M., Hökmark, H., 2007. Mechanical model for unsaturated MX-80. In: Schanz, T. (Ed.), Theoretical and Numerical Unsaturated Soil Mechanics, Springer Proceeding in Physics, vol. 113, pp. 3–10. Alonso, E.E., Gens, A., Josa, A., 1990. A constitutive model for partially saturated soils. Géotechnique 40, 405–430. Alonso, E.E., Vaunat, J., Gens, A., 1999. Modelling the mechanical behaviour of expansive clays. Engineering Geology 54, 173–183. Dueck, A., Nilsson, U., 2008. Hydro-mechanical properties of unsaturated MX-80, laboratory study 2005–2007. SKB Report in Progress. Dueck, A., 2007. Results from suction controlled laboratory tests on unsaturated bentonite – Verification of a model. In: Schanz, T. (Ed.), Experimental Unsaturated Soil Mechanics, Springer Proceeding in Physics, vol. 112, pp. 329–335. Kristensson, O., Åkesson, M., 2008. Mechanical modeling of MX-80 – Quick tools for BBM parameter analysis. Physics and Chemistry of the Earth 33S1, S508–S515. Norrish, K., 1954. The swelling of montmorillonite. Discussions of the Faraday Society 18, 120–134. Roscoe, K.H., Burland, J.B., 1968. On the generalized stress–strain behaviour of ‘wet’ clay. In: Heyman, J., Leckie, F.A. (Eds.), Engineering Plasticity. Cambridge Univ. Press, pp. 535–609.