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An improved statistical damage constitutive model for warm frozen clay based on Mohr–Coulomb criterion
Cold Regions Science and Technology 57 (2009) 154–159
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Cold Regions Science and Technology j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / c o l d r e g i o n s
An improved statistical damage constitutive model for warm frozen clay based on Mohr–Coulomb criterion Shuangyang Li ⁎, Yuanming Lai, Shujuan Zhang, Deren Liu State Key Laboratory of Frozen Soil Engineering, Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences, Lanzhou, 730000, China
a r t i c l e
i n f o
Article history: Received 15 December 2008 Accepted 24 February 2009 Keywords: Warm frozen clay Warm ice-rich frozen clay Statistical damage constitutive model Mohr–Coulomb criterion
a b s t r a c t There are many microdefects distributed randomly in frozen soil, which will lead to great uncertainty and randomness of mechanical properties and behaviors under applied load, therefore, it is more scientific to study stress–strain relationship of frozen soil by stochastic method instead of deterministic way. For the warm frozen clay and warm ice-rich frozen clay, a stochastic damage constitutive model has been proposed on the foundation of a large number of experimental data, in which the axial strain is regarded as random variable. In this paper, according to these experimental data under three temperature conditions (−0.5 °C, −1.5 °C and −2.0 °C) and the above research results, the strength of soil element is selected as random variable and an improved statistical damage constitutive model is deduced, and in this new model, the Mohr–Coulomb failure criterion is also used to judge whether the soil element is damaged. Finally, compared with original model, it is found that the new improved model can better describe the experimental data and reflect deformation characteristics. Especially, when the stress reaches its peak value, the experimental data and new theoretical curves overlap with each other. © 2009 Elsevier B.V. All rights reserved.
1. Introduction From the point of view of the science of materials, frozen soil is a natural particulate composite, which is composed of four different constituents: solid grains (mineral or organic), ice, unfrozen water, and gases. The most important characteristic by which it differs from other similar materials — such as unfrozen soils and the majority of artificial composites — is that under natural conditions its matrix, which is composed mostly of ice and water, changes continuously with varying temperature and applied stress. Despite the presence of unfrozen water, when ice fills most of the pore space, the mechanical behavior of a frozen soil closely reflects that of the ice. And the strength of ice depends on many factors; the most important are temperature, pressure, and strain rate, as well as the size, structure, and orientation of grains. The strength of ice increases with decreasing temperature, and its model of failure is strain-rate dependent. With varying temperature and strain rate, its response to loading is found to vary from viscous to brittle (Ladanyi and Andersland, 2004). Therefore, temperature has a marked effect on all aspects of the mechanical behavior of frozen soil because of its direct influence on the strength of intergranular ice and on the amount of unfrozen water in a frozen soil. In general, a decrease in temperature results in an increase in strength of a frozen soil, but at the same time it increases its brittleness, which
⁎ Corresponding author. Tel.: +86 931 4967296. E-mail address: [email protected] (S. Li). 0165-232X/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.coldregions.2009.02.010
is manifested by a larger drop of strength after the peak and an increase in the ratio of compressive strength to tensile strength (Ladanyi and Andersland, 2004; Sayles and Haines, 1974; Haynes, 1978). For warm frozen clay (its temperature ranges from 0 to − 1.5 °C) and warm ice-rich frozen clay (its temperature ranges from 0 to − 1.5 °C and volumetric ice content is greater than 25%) (Cheng, 2003), they are less stable than clays at colder temperatures. Nevertheless, because of the complexity of the frozen soils and the limitation of the experimental conditions, it is difficult to make the frozen soil specimen, whose temperature is close to 0 °C. So, there are a few published papers concerning this aspect. For example, through in-situ experiment, Ma (2006) found that large deformations in warm ice-rich frozen soil would occur even if the load was small, which may be an important reason for the differential settlement and many cracks along the subgrade of the Qinghai–Tibet Railway (QTR) during the last 5 years. In addition, Ma (2006) researched the uniaxial compression strengths of warm ice-rich frozen clay at different water contents (20%, 40%, 60%, 90% and 120%) and found that the constitutive relationship of warm ice-rich frozen clay was of a strain hardening manner and the form of failure was plastic. Meanwhile, the compressive strength of warm ice-rich frozen clay with water contents of 40%–90% increased linearly with a decrease in temperature. In fact, there are many defects, such as fissures and cavities, in warm frozen clay and warm ice-rich clay. And these defects are distributed randomly, which make the mechanical properties of these clays exhibit great uncertainty and randomness.
S. Li et al. / Cold Regions Science and Technology 57 (2009) 154–159
155
Thus it is somewhat unreasonable to study the stress–strain relationships of them by deterministic method. Fortunately, Lai et al. (2008) paid their attentions on this problem. In their study, a vast amount of uniaxial tests on warm frozen clay and warm ice-rich frozen clay, under different temperatures (− 0.5 °C, − 1.0 °C and − 2.0 °C), were firstly carried out. Following, after comparing study, it was found that the Weibull distribution, among the many potential distribution laws, could well describe strength law of these two kinds of clays. Then, a stochastic damage constitutive model was developed using the continuous damage theory and probability method, as well as statistical theory. And comparisons with the experimental data showed that the whole stress–stain process could be described by this model to some extent. But in their paper, the axial strain is regarded as a random variable and the soil element will be damaged if the axial strain reaches a specific value, which is somewhat unreasonable and whose physical concept is also ambiguous in the classical plastic theory (Zheng et al., 2002). In soil mechanics, some failure criterions, such as Mohr–Coulomb, Drucker–Prager, Hoek–Brown and so on, are often used to judge whether the soil is damaged. Therefore, it will be better that the strength of soil element should be regarded as random variable if it is based on a failure criterion. In this study, a new improved statistical damage constitutive model for warm frozen clay and warm ice-rich frozen clay is proposed on the basis of the experimental data and previous research results (Lai et al., 2008), in which Mohr–Coulomb criterion is applied to judge whether the clay element is damaged. The results show that the improved statistical damage constitutive model is closer to the experimental data than the original model deduced by Lai, et al. (2008). Meanwhile, since the new model is derived from three-dimensional general stress state, its three-
Fig. 2. Comparisons of experimental and theoretical data on warm frozen clay at −0.5 °C.
dimensional form can be obtained by further deduction. In addition, the unknown parameters can be calculated directly or obtained by experimental data regression and the numerical form of the new model could be inserted into some commercial software, such as Ansys, Adina, Flac3D, Marc, Abaqus and so on, so this model is useful in practical engineering. 2. Statistical damage constitutive model based on Mohr–Coulomb criterion Based on the conclusion that the Weibull distribution could well describe the strength distribution law for the warm frozen clay and warm ice-rich frozen clay (Lai et al., 2008), an improved statistical damage constitutive model is deduced by using the Mohr–Coulomb failure criterion and theories of damage mechanics and probability, as well as statistic theory. 2.1. Damage constitutive relationship According to the Lemaitre principle of equivalent stress (Lemaitre and Chaboche, 1970), the strain caused by [σ] applied to a damaged material is equivalent to the strain caused by [σ̃] acting on the undamaged material. Hence, the stress–strain relation for frozen soil can be expressed as the following:
Fig. 1. The experimental specimens.
fσ g feg = h i = E˜
n o ˜ σ ½E
=
fσ g ð1 − DÞ½E
ð1Þ
156
S. Li et al. / Cold Regions Science and Technology 57 (2009) 154–159
The above expression can be rewritten as: fσ g = ð1 − DÞ½Efeg
ð2Þ
where [E˜] is the elastic constant matrix of the damaged material; [E] is the elastic constant matrix of the undamaged material; D denotes the damage variable, and it is defined as: D=
Nt N
ð3Þ
where Nt is the number of damaged elements, and N represents all the elements. 2.2. Statistical damage constitutive model based on Mohr–Coulomb failure criterion Here, the strength of frozen soil element is chosen as random variable, the probability density function of the Weibull distribution can be written as (Li, 2008): β β F β − 1 − FF0 e ; ðF N 0Þ f ðF Þ = F0 F0
ð4Þ
where β and F0 are both unknown parameters of the Weibull distribution.
Fig. 4. Comparisons of experimental and theoretical data on warm frozen clay at −2.0 °C.
When the stress reaches a stress level, the number of damaged elements, Nt, may be expressed as: 2
ZF
Nf ðxÞdx = N41 −e
Nt ðF Þ =
β 3 −
F F0
5
ð5Þ
0
Substituting Eq. (5) into Eq. (3), the damage variable can be written as: D=
− Nt = 1 −e N
β F F0
ð6Þ
Then, substituting Eq. (6) into Eq. (2), we have: fσ g β
feg = e
−
F F0
ð7Þ ½E
As far as triaxial compressive test is concerned, the previous formula can be changed as: β σ 1 = Ee1 e
−
F F0
+ 2μσ 3
ð8Þ
And under uniaxial test condition, there is: β Fig. 3. Comparisons of experimental and theoretical data on warm frozen clay at −1.0 °C.
σ 1 = Ee1 e
−
F F0
ð9Þ
S. Li et al. / Cold Regions Science and Technology 57 (2009) 154–159 Table 1 Statistics on the average values for the warm frozen clay. Temperature
E/MPa
- 0.5 °C - 1.0 °C - 2.0 °C
56.765 65.958 79.189
β
1.406 1.483 1.754
F0
3120 4904 6345
εp
0.031 0.038 0.038
Experiment
σp/MPa
σp/MPa
σp/MPa
In original model
In improved model
0.840 1.297 1.629
0.791 1.246 1.587
0.840 1.297 1.629
The Mohr–Coulomb failure criterion is: F = σ˜ 1 ð1 + sin uÞ − σ˜ 3 ð1 − sin uÞ = 2c cos u
ð10Þ
where σ˜1 and σ˜3 are effective stress; c and φ are cohesion and internal friction n angle. o ˜ = fσ g into Eq. (10) leads to: Substituting σ ð1 − D Þ
σ1 σ3 F= ð1 + sin uÞ − ð1 − sin uÞ 1− D 1− D
ð11Þ
Meanwhile, there exists: e1 =
1˜ 1 ðσ − 2μσ 3 Þ σ 1 − 2μ σ˜ 3 = E ð1 − DÞE 1
ð12Þ
Combining Eqs. (11) and (12) yields the following expression: F=
Ee1 ½σ 1 ð1 + sin uÞ − σ 3 ð1 − sin uÞ σ 1 − 2μσ 3
ð13Þ
157
After the values of β and F0 are calculated by experimental data, the improved statistical damage constitutive model for frozen soil can be determined by Eqs. (9), (14), (18) and (19). As noted in the above deduction process, there isn't any experimental information included in the improved statistical damage constitutive model after the specimen fails. Thus, after yielding and failure of specimen, its stress–strain behavior can't be expressed by this new model, which has been proved by some research results (Xu et al., 2007; Li et al., 2007; Liu and Yang, 2006 and Liu et al., 2007). However, the analytical method in original model could be borrowed to solve this problem, and in this model, the unknown parameters are obtained through experimental data regression. 3. Verification of the improved statistical damage constitutive model based on Mohr–Coulomb failure criterion Two groups of warm frozen clay and warm ice-rich frozen clay specimens (shown Fig. 1(a, b)) were made of the clay brought from the Beiluhe site along the QTR and mixed with the desired water contents of 17.6% and 30.0%, respectively, and the samples had an average dry unit weight of 18.1 kN/m3. Under three negative conditions (− 0.5 °C, −1.0 °C and −2.0 °C), 50 specimen experiments for each group of clay were carried out. The detailed experimental description and results can be found in previous study (Lai et al., 2008). According to the experimental results of the warm frozen clay and warm ice-rich frozen clay at different temperatures, the unknown parameters are determined and then the improved statistical damage constitutive model for each specimen can be obtained. In order to demonstrate rationality and accuracy of the statistical damage
For the uniaxial test, the above expression can be rewritten as: F = Ee1 ð1 + sin uÞ
ð14Þ
Eq. (9), should satisfy the following conditions: ① when ε1 = 0, σ1 = 0; 1 = E; ② when ε1 = 0, dσ de1 ③ when ε1 = εp, σ1 = σp; 1 = 0. ④ when ε1 = εp, dσ de1 where εp is strain at which the stress is equal to a peak stress σp. Differentiating Eq. (9), we can obtain the following expression: − Aσ 1 = Ee Ae1
β β − 1 F F 1 AF F0 1 + e1 −β F0 F0 Ae1
ð15Þ
Obviously, Eqs. (9) and (15) satisfy the conditions of ① and ②. If the conditions of ③ and ④ are substituted into Eqs. (9) and (15), the following formulae can be obtained. β σ p = Eep e
Fp F0
−
" 1 + ep
β − 1 # Fp 1 −β Eð1 + sin uÞ = 0 F0 F0
ð16Þ
ð17Þ
where Fp = Eεp(1 + sin φ). Solving Eqs. (16) and (17), the expressions of β and F0 can be written as: β= −
1 σ
ln Eep
ð18Þ
p
1 β β F0 = βFp
ð19Þ
Fig. 5. Comparisons of experimental and theoretical data on warm ice-rich frozen clay at – 0.5 °C.
158
S. Li et al. / Cold Regions Science and Technology 57 (2009) 154–159
shown that the experimental data is little difference from those of the two theoretical models. But in the improved model, the peak stresses is more close to experimental data, which is indicated in Table 2. In addition, because of the large quantity of ice in the warm ice-rich frozen clay, its stress–strain relationship is of an elastic-perfectly plastic manner and the form of failure is plastic, which is also reflected in Fig. 1(b). 4. Conclusions and discussions Due to many microdefects distributed randomly in warm frozen clay and warm ice-rich clay, the mechanical properties and behaviors of these clays display great uncertainty, so it is more reasonable to research the stress–strain relationship of them by stochastic method. In this study, an improved statistical damage constitutive model for these two types of clays is deduced on the basis of experimental data and original model, and the new model is verified by large numbers of experimental data under different temperatures. Through this study, we can find that, it is more definite to use the strength of soil element as random variable instead of the axial strain. Meanwhile, the improved model includes the Mohr–Coulomb failure criterion, which is very widely applied in the classical plastic theory and soil mechanics. Compared with original model, the new statistical damage constitutive model can better describe the experimental data. In particular, when the stress reaches its peak value, the experimental and theoretical curves overlap with each other. In addition, the new model is derived from three-dimensional general stress state, so the three-dimensional form of the new model could be deduced and obtained.
Fig. 6. Comparisons of experimental and theoretical data on warm ice-rich frozen clay at − 1.0 °C.
constitutive model proposed in this paper, some typical experimental curves are selected and compared with the new theoretical model and original model (Lai et al., 2008). 3.1. Comparisons and analyses of the stress–strain models for warm frozen clay Figs. 2–4 display the comparisons of experimental and theoretical stress–strain curves for warm frozen clay. Obviously, compared with original model, the new statistical damage constitutive model can better describe the experimental data. In particular, when the stress reaches its peak point, the experimental and theoretical curves overlap with each other. But there is large difference between the experimental data and original model at the moment (Listed in Table 1). From the whole stress–strain process, it is evident that the simulation effect of the improved model is better than that reported in previous research. Furthermore, we also find that the Young's modulus and strength of warm frozen clay increase as the temperature decreases, which is described in some documents (Ladanyi andAndersland, 2004; Lai et al., 2008; Haynes, 1978). 3.2. Comparisons and analyses of the stress–strain models for warm icerich frozen clay Similarly, several typical experimental and theoretical stress–strain curves for warm ice-rich frozen clay are shown in Figs. 5–7. It is apparent that the two damage constitutive models could both well express the whole stress–strain process of the warm ice-rich frozen clay under uniaxial condition. However, from these Figs. 6(b) and 7(a, b), it is
Fig. 7. Comparisons of experimental and theoretical data on warm ice-rich frozen clay at − 2.0 °C.
S. Li et al. / Cold Regions Science and Technology 57 (2009) 154–159
References
Table 2 Statistics on the average values for the warm ice-rich frozen clay. Temperature E/MPa
β
F0
εp
Experiment σp/MPa σp/MPa
- 0.5 °C - 1.0 °C - 2.0 °C
849.711 0.199 43.36 0.124 0.686 1042.624 0.208 91.73 0.121 0.974 1371.813 0.221 235.72 0.097 1.493
σp/MPa
In original In improved model model 0.704 0.967 1.503
159
0.687 0.972 1.495
However, when the specimen fails, there is no experimental data used to calculate some unknown parameters, thus, the improved statistical damage model can't accurately simulate stress–strain behavior of the frozen clay after its failure and the approximate analytical method is adopted. And this is also a common problem in experiment mechanics of rock and soil, which deserves further study. Notwithstanding its limitation at this point, this study does suggest that the improved model is more reasonable on the concept of soil mechanics. Acknowledgements This research was supported by the National Natural Science Foundation of China (No. 40730736, No. 40801029 and No. 40601023), the National Hi-Tech Research and Development Plan (2008AA11Z103), the Project for Incubation of Specialists in Glaciology and Geocryology of National Natural Science Foundation of China (J0630966), the grant of the Western Project Program of the Chinese Academy of Sciences, No. KZCX2-XB2-10.
Cheng, G.D., 2003. Construction of Qinghai–Tibetan railway with cooled roadbed. China Railway Science 24 (3), 1–4 (in Chinese). Haynes, F.D., 1978. Strength and deformation of frozen silt. Proc. 3rd Int. Conf. on Permafrost, Edmonton, Alberta, vol. 1. National Research Council of Canada, Ottawa, pp. 656–716. Ladanyi, B., Andersland, O.B., 2004. Frozen ground engineering, 2nd edition. ASCE, pp. 106–120. Lai, Y.M., Li, S.Y., Qi, J.L., et al., 2008. Strength distribution of warm frozen clay and its stochastic damage constitutive model. Cold Regions Science and Technology 53 (2), 200–215. Lemaitre, J., Chaboche, J.L., 1970. Mechanics of solid materials. Cambridge University Press, London. Li, S.C., Xu, J., Wang, H., et al., 2007. Research on statistical damage constitutive model of rock and determination of its parameters. Mining R & D 27 (2), 6–8 (in Chinese). Li, S.Y., 2008. Numerical study on the thermal–mechanical stability of railway subgrade in permafrost regions. Ph D Dissertation for Graduate University of the Chinese Academy of Sciences, Beijing. (in Chinese). Liu, C.X., Yang, L.D., 2006. A new damage softening constitutive model for rocks. Hydro-Science and Engineering 3, 25–28 (in Chinese). Liu, C.X., Yang, L.D., Cao, W.G., et al., 2007. A statistical damage softening constitutive model for rock and back analysis of its parameters. Chinese Journal of Underground Space and Engineering 3 (3), 453–457 (in Chinese). Ma, X.J., 2006. Research on strength and creep of warm and ice-rich frozen soil. A Dissertation Submitted to Graduate School of Chinese Academy of Sciences for the Degree of Master. Lanzhou. (in Chinese). Sayles, F.H., Haines, D., 1974. Creep of frozen silt and clay. U. S. Army Cold Regions Research and Engineering Laboratory Technical Report, p. 252. Xu, J., Li, S.C., Liu, Y.B., et al., 2007. Damage constitutive model of rock based on Drucker– Prager criterion. Journal of Southwest Jiaotong Universtiy 42 (3), 278–282 (in Chinese). Zheng, Y.R., Shen, Z.J., Gong, X.N., 2002. General plastic mechanics—the principles of geotechnical plastic mechanics. China Architecture and Building Press, Beijing.
Contents lists available at ScienceDirect
Cold Regions Science and Technology j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / c o l d r e g i o n s
An improved statistical damage constitutive model for warm frozen clay based on Mohr–Coulomb criterion Shuangyang Li ⁎, Yuanming Lai, Shujuan Zhang, Deren Liu State Key Laboratory of Frozen Soil Engineering, Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences, Lanzhou, 730000, China
a r t i c l e
i n f o
Article history: Received 15 December 2008 Accepted 24 February 2009 Keywords: Warm frozen clay Warm ice-rich frozen clay Statistical damage constitutive model Mohr–Coulomb criterion
a b s t r a c t There are many microdefects distributed randomly in frozen soil, which will lead to great uncertainty and randomness of mechanical properties and behaviors under applied load, therefore, it is more scientific to study stress–strain relationship of frozen soil by stochastic method instead of deterministic way. For the warm frozen clay and warm ice-rich frozen clay, a stochastic damage constitutive model has been proposed on the foundation of a large number of experimental data, in which the axial strain is regarded as random variable. In this paper, according to these experimental data under three temperature conditions (−0.5 °C, −1.5 °C and −2.0 °C) and the above research results, the strength of soil element is selected as random variable and an improved statistical damage constitutive model is deduced, and in this new model, the Mohr–Coulomb failure criterion is also used to judge whether the soil element is damaged. Finally, compared with original model, it is found that the new improved model can better describe the experimental data and reflect deformation characteristics. Especially, when the stress reaches its peak value, the experimental data and new theoretical curves overlap with each other. © 2009 Elsevier B.V. All rights reserved.
1. Introduction From the point of view of the science of materials, frozen soil is a natural particulate composite, which is composed of four different constituents: solid grains (mineral or organic), ice, unfrozen water, and gases. The most important characteristic by which it differs from other similar materials — such as unfrozen soils and the majority of artificial composites — is that under natural conditions its matrix, which is composed mostly of ice and water, changes continuously with varying temperature and applied stress. Despite the presence of unfrozen water, when ice fills most of the pore space, the mechanical behavior of a frozen soil closely reflects that of the ice. And the strength of ice depends on many factors; the most important are temperature, pressure, and strain rate, as well as the size, structure, and orientation of grains. The strength of ice increases with decreasing temperature, and its model of failure is strain-rate dependent. With varying temperature and strain rate, its response to loading is found to vary from viscous to brittle (Ladanyi and Andersland, 2004). Therefore, temperature has a marked effect on all aspects of the mechanical behavior of frozen soil because of its direct influence on the strength of intergranular ice and on the amount of unfrozen water in a frozen soil. In general, a decrease in temperature results in an increase in strength of a frozen soil, but at the same time it increases its brittleness, which
⁎ Corresponding author. Tel.: +86 931 4967296. E-mail address: [email protected] (S. Li). 0165-232X/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.coldregions.2009.02.010
is manifested by a larger drop of strength after the peak and an increase in the ratio of compressive strength to tensile strength (Ladanyi and Andersland, 2004; Sayles and Haines, 1974; Haynes, 1978). For warm frozen clay (its temperature ranges from 0 to − 1.5 °C) and warm ice-rich frozen clay (its temperature ranges from 0 to − 1.5 °C and volumetric ice content is greater than 25%) (Cheng, 2003), they are less stable than clays at colder temperatures. Nevertheless, because of the complexity of the frozen soils and the limitation of the experimental conditions, it is difficult to make the frozen soil specimen, whose temperature is close to 0 °C. So, there are a few published papers concerning this aspect. For example, through in-situ experiment, Ma (2006) found that large deformations in warm ice-rich frozen soil would occur even if the load was small, which may be an important reason for the differential settlement and many cracks along the subgrade of the Qinghai–Tibet Railway (QTR) during the last 5 years. In addition, Ma (2006) researched the uniaxial compression strengths of warm ice-rich frozen clay at different water contents (20%, 40%, 60%, 90% and 120%) and found that the constitutive relationship of warm ice-rich frozen clay was of a strain hardening manner and the form of failure was plastic. Meanwhile, the compressive strength of warm ice-rich frozen clay with water contents of 40%–90% increased linearly with a decrease in temperature. In fact, there are many defects, such as fissures and cavities, in warm frozen clay and warm ice-rich clay. And these defects are distributed randomly, which make the mechanical properties of these clays exhibit great uncertainty and randomness.
S. Li et al. / Cold Regions Science and Technology 57 (2009) 154–159
155
Thus it is somewhat unreasonable to study the stress–strain relationships of them by deterministic method. Fortunately, Lai et al. (2008) paid their attentions on this problem. In their study, a vast amount of uniaxial tests on warm frozen clay and warm ice-rich frozen clay, under different temperatures (− 0.5 °C, − 1.0 °C and − 2.0 °C), were firstly carried out. Following, after comparing study, it was found that the Weibull distribution, among the many potential distribution laws, could well describe strength law of these two kinds of clays. Then, a stochastic damage constitutive model was developed using the continuous damage theory and probability method, as well as statistical theory. And comparisons with the experimental data showed that the whole stress–stain process could be described by this model to some extent. But in their paper, the axial strain is regarded as a random variable and the soil element will be damaged if the axial strain reaches a specific value, which is somewhat unreasonable and whose physical concept is also ambiguous in the classical plastic theory (Zheng et al., 2002). In soil mechanics, some failure criterions, such as Mohr–Coulomb, Drucker–Prager, Hoek–Brown and so on, are often used to judge whether the soil is damaged. Therefore, it will be better that the strength of soil element should be regarded as random variable if it is based on a failure criterion. In this study, a new improved statistical damage constitutive model for warm frozen clay and warm ice-rich frozen clay is proposed on the basis of the experimental data and previous research results (Lai et al., 2008), in which Mohr–Coulomb criterion is applied to judge whether the clay element is damaged. The results show that the improved statistical damage constitutive model is closer to the experimental data than the original model deduced by Lai, et al. (2008). Meanwhile, since the new model is derived from three-dimensional general stress state, its three-
Fig. 2. Comparisons of experimental and theoretical data on warm frozen clay at −0.5 °C.
dimensional form can be obtained by further deduction. In addition, the unknown parameters can be calculated directly or obtained by experimental data regression and the numerical form of the new model could be inserted into some commercial software, such as Ansys, Adina, Flac3D, Marc, Abaqus and so on, so this model is useful in practical engineering. 2. Statistical damage constitutive model based on Mohr–Coulomb criterion Based on the conclusion that the Weibull distribution could well describe the strength distribution law for the warm frozen clay and warm ice-rich frozen clay (Lai et al., 2008), an improved statistical damage constitutive model is deduced by using the Mohr–Coulomb failure criterion and theories of damage mechanics and probability, as well as statistic theory. 2.1. Damage constitutive relationship According to the Lemaitre principle of equivalent stress (Lemaitre and Chaboche, 1970), the strain caused by [σ] applied to a damaged material is equivalent to the strain caused by [σ̃] acting on the undamaged material. Hence, the stress–strain relation for frozen soil can be expressed as the following:
Fig. 1. The experimental specimens.
fσ g feg = h i = E˜
n o ˜ σ ½E
=
fσ g ð1 − DÞ½E
ð1Þ
156
S. Li et al. / Cold Regions Science and Technology 57 (2009) 154–159
The above expression can be rewritten as: fσ g = ð1 − DÞ½Efeg
ð2Þ
where [E˜] is the elastic constant matrix of the damaged material; [E] is the elastic constant matrix of the undamaged material; D denotes the damage variable, and it is defined as: D=
Nt N
ð3Þ
where Nt is the number of damaged elements, and N represents all the elements. 2.2. Statistical damage constitutive model based on Mohr–Coulomb failure criterion Here, the strength of frozen soil element is chosen as random variable, the probability density function of the Weibull distribution can be written as (Li, 2008): β β F β − 1 − FF0 e ; ðF N 0Þ f ðF Þ = F0 F0
ð4Þ
where β and F0 are both unknown parameters of the Weibull distribution.
Fig. 4. Comparisons of experimental and theoretical data on warm frozen clay at −2.0 °C.
When the stress reaches a stress level, the number of damaged elements, Nt, may be expressed as: 2
ZF
Nf ðxÞdx = N41 −e
Nt ðF Þ =
β 3 −
F F0
5
ð5Þ
0
Substituting Eq. (5) into Eq. (3), the damage variable can be written as: D=
− Nt = 1 −e N
β F F0
ð6Þ
Then, substituting Eq. (6) into Eq. (2), we have: fσ g β
feg = e
−
F F0
ð7Þ ½E
As far as triaxial compressive test is concerned, the previous formula can be changed as: β σ 1 = Ee1 e
−
F F0
+ 2μσ 3
ð8Þ
And under uniaxial test condition, there is: β Fig. 3. Comparisons of experimental and theoretical data on warm frozen clay at −1.0 °C.
σ 1 = Ee1 e
−
F F0
ð9Þ
S. Li et al. / Cold Regions Science and Technology 57 (2009) 154–159 Table 1 Statistics on the average values for the warm frozen clay. Temperature
E/MPa
- 0.5 °C - 1.0 °C - 2.0 °C
56.765 65.958 79.189
β
1.406 1.483 1.754
F0
3120 4904 6345
εp
0.031 0.038 0.038
Experiment
σp/MPa
σp/MPa
σp/MPa
In original model
In improved model
0.840 1.297 1.629
0.791 1.246 1.587
0.840 1.297 1.629
The Mohr–Coulomb failure criterion is: F = σ˜ 1 ð1 + sin uÞ − σ˜ 3 ð1 − sin uÞ = 2c cos u
ð10Þ
where σ˜1 and σ˜3 are effective stress; c and φ are cohesion and internal friction n angle. o ˜ = fσ g into Eq. (10) leads to: Substituting σ ð1 − D Þ
σ1 σ3 F= ð1 + sin uÞ − ð1 − sin uÞ 1− D 1− D
ð11Þ
Meanwhile, there exists: e1 =
1˜ 1 ðσ − 2μσ 3 Þ σ 1 − 2μ σ˜ 3 = E ð1 − DÞE 1
ð12Þ
Combining Eqs. (11) and (12) yields the following expression: F=
Ee1 ½σ 1 ð1 + sin uÞ − σ 3 ð1 − sin uÞ σ 1 − 2μσ 3
ð13Þ
157
After the values of β and F0 are calculated by experimental data, the improved statistical damage constitutive model for frozen soil can be determined by Eqs. (9), (14), (18) and (19). As noted in the above deduction process, there isn't any experimental information included in the improved statistical damage constitutive model after the specimen fails. Thus, after yielding and failure of specimen, its stress–strain behavior can't be expressed by this new model, which has been proved by some research results (Xu et al., 2007; Li et al., 2007; Liu and Yang, 2006 and Liu et al., 2007). However, the analytical method in original model could be borrowed to solve this problem, and in this model, the unknown parameters are obtained through experimental data regression. 3. Verification of the improved statistical damage constitutive model based on Mohr–Coulomb failure criterion Two groups of warm frozen clay and warm ice-rich frozen clay specimens (shown Fig. 1(a, b)) were made of the clay brought from the Beiluhe site along the QTR and mixed with the desired water contents of 17.6% and 30.0%, respectively, and the samples had an average dry unit weight of 18.1 kN/m3. Under three negative conditions (− 0.5 °C, −1.0 °C and −2.0 °C), 50 specimen experiments for each group of clay were carried out. The detailed experimental description and results can be found in previous study (Lai et al., 2008). According to the experimental results of the warm frozen clay and warm ice-rich frozen clay at different temperatures, the unknown parameters are determined and then the improved statistical damage constitutive model for each specimen can be obtained. In order to demonstrate rationality and accuracy of the statistical damage
For the uniaxial test, the above expression can be rewritten as: F = Ee1 ð1 + sin uÞ
ð14Þ
Eq. (9), should satisfy the following conditions: ① when ε1 = 0, σ1 = 0; 1 = E; ② when ε1 = 0, dσ de1 ③ when ε1 = εp, σ1 = σp; 1 = 0. ④ when ε1 = εp, dσ de1 where εp is strain at which the stress is equal to a peak stress σp. Differentiating Eq. (9), we can obtain the following expression: − Aσ 1 = Ee Ae1
β β − 1 F F 1 AF F0 1 + e1 −β F0 F0 Ae1
ð15Þ
Obviously, Eqs. (9) and (15) satisfy the conditions of ① and ②. If the conditions of ③ and ④ are substituted into Eqs. (9) and (15), the following formulae can be obtained. β σ p = Eep e
Fp F0
−
" 1 + ep
β − 1 # Fp 1 −β Eð1 + sin uÞ = 0 F0 F0
ð16Þ
ð17Þ
where Fp = Eεp(1 + sin φ). Solving Eqs. (16) and (17), the expressions of β and F0 can be written as: β= −
1 σ
ln Eep
ð18Þ
p
1 β β F0 = βFp
ð19Þ
Fig. 5. Comparisons of experimental and theoretical data on warm ice-rich frozen clay at – 0.5 °C.
158
S. Li et al. / Cold Regions Science and Technology 57 (2009) 154–159
shown that the experimental data is little difference from those of the two theoretical models. But in the improved model, the peak stresses is more close to experimental data, which is indicated in Table 2. In addition, because of the large quantity of ice in the warm ice-rich frozen clay, its stress–strain relationship is of an elastic-perfectly plastic manner and the form of failure is plastic, which is also reflected in Fig. 1(b). 4. Conclusions and discussions Due to many microdefects distributed randomly in warm frozen clay and warm ice-rich clay, the mechanical properties and behaviors of these clays display great uncertainty, so it is more reasonable to research the stress–strain relationship of them by stochastic method. In this study, an improved statistical damage constitutive model for these two types of clays is deduced on the basis of experimental data and original model, and the new model is verified by large numbers of experimental data under different temperatures. Through this study, we can find that, it is more definite to use the strength of soil element as random variable instead of the axial strain. Meanwhile, the improved model includes the Mohr–Coulomb failure criterion, which is very widely applied in the classical plastic theory and soil mechanics. Compared with original model, the new statistical damage constitutive model can better describe the experimental data. In particular, when the stress reaches its peak value, the experimental and theoretical curves overlap with each other. In addition, the new model is derived from three-dimensional general stress state, so the three-dimensional form of the new model could be deduced and obtained.
Fig. 6. Comparisons of experimental and theoretical data on warm ice-rich frozen clay at − 1.0 °C.
constitutive model proposed in this paper, some typical experimental curves are selected and compared with the new theoretical model and original model (Lai et al., 2008). 3.1. Comparisons and analyses of the stress–strain models for warm frozen clay Figs. 2–4 display the comparisons of experimental and theoretical stress–strain curves for warm frozen clay. Obviously, compared with original model, the new statistical damage constitutive model can better describe the experimental data. In particular, when the stress reaches its peak point, the experimental and theoretical curves overlap with each other. But there is large difference between the experimental data and original model at the moment (Listed in Table 1). From the whole stress–strain process, it is evident that the simulation effect of the improved model is better than that reported in previous research. Furthermore, we also find that the Young's modulus and strength of warm frozen clay increase as the temperature decreases, which is described in some documents (Ladanyi andAndersland, 2004; Lai et al., 2008; Haynes, 1978). 3.2. Comparisons and analyses of the stress–strain models for warm icerich frozen clay Similarly, several typical experimental and theoretical stress–strain curves for warm ice-rich frozen clay are shown in Figs. 5–7. It is apparent that the two damage constitutive models could both well express the whole stress–strain process of the warm ice-rich frozen clay under uniaxial condition. However, from these Figs. 6(b) and 7(a, b), it is
Fig. 7. Comparisons of experimental and theoretical data on warm ice-rich frozen clay at − 2.0 °C.
S. Li et al. / Cold Regions Science and Technology 57 (2009) 154–159
References
Table 2 Statistics on the average values for the warm ice-rich frozen clay. Temperature E/MPa
β
F0
εp
Experiment σp/MPa σp/MPa
- 0.5 °C - 1.0 °C - 2.0 °C
849.711 0.199 43.36 0.124 0.686 1042.624 0.208 91.73 0.121 0.974 1371.813 0.221 235.72 0.097 1.493
σp/MPa
In original In improved model model 0.704 0.967 1.503
159
0.687 0.972 1.495
However, when the specimen fails, there is no experimental data used to calculate some unknown parameters, thus, the improved statistical damage model can't accurately simulate stress–strain behavior of the frozen clay after its failure and the approximate analytical method is adopted. And this is also a common problem in experiment mechanics of rock and soil, which deserves further study. Notwithstanding its limitation at this point, this study does suggest that the improved model is more reasonable on the concept of soil mechanics. Acknowledgements This research was supported by the National Natural Science Foundation of China (No. 40730736, No. 40801029 and No. 40601023), the National Hi-Tech Research and Development Plan (2008AA11Z103), the Project for Incubation of Specialists in Glaciology and Geocryology of National Natural Science Foundation of China (J0630966), the grant of the Western Project Program of the Chinese Academy of Sciences, No. KZCX2-XB2-10.
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