Luận văn A volume-Mass constitutive model for unsaturated soils

on (kPa 0 0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9 1 1 10000 1000 100 10 1 Log ne t mean stress (kPa)1e+06 100000 10000 1000 100 10 1 L g a o soil suction (kP ) 0 0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9 1 1 10000 1000 100 10 1 Log ne t mean stress (kPa)1e+06 100000 10000 1000 100 10 1 g P Lo soil suction (k a) 0.25 0.25 0.4 0.4 0.55 0.55 0.7 0.7 0.85 0.85 1 1 10000 1000 100 10 1 Log ne t mean stress (kPa)1e+06 100000 10000 1000 100 10 1 P Log soil suction (k a) 0.25 0.25 0.4 0.4 0.55 0.55 0.7 0.7 0.85 0.85 1 1 10000 1000 100 10 1 Log ne t mean stress (kPa)1e+06 100000 10000 1000 100 10 1 P Log soil suction (k a) 0 0 0.05 0.05 0.1 0.1 0.15 0.15 0.2 0.2 0.25 0.25 0.3 0.3 0.35 0.35 10000 1000 100 10 1 Log ne t mean stress (kPa)1e+06 100000 10000 1000 100 10 1 P Log soil suction (k a) 0 0 0.05 0.05 0.1 0.1 0.15 0.15 0.2 0.2 0.25 0.25 0.3 0.3 0.35 0.35 G ra vi m et ric w at er c on te nt , w G ra vi m et ric w at er c on te nt , w a) Water content – stress paths #3 d) Water content – stress paths #4 Vo id ra tio , e Vo id ra tio , e b) Void ratio – stress paths #3 e) Void ratio – stress paths #4 D eg re e of s at ur at io n, S D eg re e of s at ur at io n, S c) Degree of saturation – stress paths #3 f) Degree of saturation – stress paths #4 215 2000 1750 1500 1250 1000 750 500 250 Figure 4-13. Volume-mass constitutive surfaces for the artificial clay at low ranges of soil suction and net mean stress (followed by the series stress paths #1) b) Void ratio 2000 1750 1500 1250 1000 750 500 250 Log ne t mean stress (kPa)2000 1750 1500 1250 1000 750 500 250 kP Log soil suction ( a) 1 1 1.5 1.5 2 2 2.5 2.5 3 3 3.5 3.5 V oi d ra tio , e Log ne t mean stress (kPa)2000 1750 1500 1250 1000 750 500 250 P Log soil suction (k a) 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9 1 1 1.1 1.1 1.2 1.2 G ra vi m et ric w at er c on te nt , w a) Gravimetric water content 216 Figure 4-14. Volume-mass constitutive surfaces for the artificial clay on wide ranges of soil suction and net mean stress (followed by the series stress paths #1 and #2). 10000 1000 100 10 1 Log ne t mean stress (kPa)1e+06 100000 10000 1000 100 10 1 o P Log s il suction (k a) 0 0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9 1 1 10000 1000 100 10 1 Log ne t mean stress (kPa)1e+06 100000 10000 1000 100 10 1 Log soil suction (kPa) 0.5 0.5 1 1 1.5 1.5 2 2 2.5 2.5 3 3 3.5 3.5 10000 1000 100 10 1 Log ne t mean stress (kPa)1e+06 100000 10000 1000 100 10 1 ct P Log soil su ion (k a) 0 0 0.25 0.25 0.5 0.5 0.75 0.75 1 1 1.25 1.25 b) Void ratio – stress paths #1 c) Degree of saturation – stress paths #1 f) Degree of saturation – stress paths #2 a) Water content – stress paths #1 G ra vi m et ric w at er c on te nt , w Vo id ra tio , e D eg re e of s at ur at io n, S 10000 1000 100 10 1 Log ne t mean stress (kPa)1e+06 100000 10000 1000 100 10 1 L g Pa o soil suction (k ) 0 0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9 1 1 10000 1000 100 10 1 Log ne t mean stress (kPa)1e+06 100000 10000 1000 100 10 1 L c P og soil su tion (k a) 0.5 0.5 1 1 1.5 1.5 2 2 2.5 2.5 3 3 3.5 3.5 10000 1000 100 10 1 1e+06 100000 10000 1000 100 10 1 g Log ne t mean stress (kPa) Lo soil suction (kPa) 0 0 0.25 0.25 0.5 0.5 0.75 0.75 1 1 1.25 1.25 G ra vi m et ric w at er c on te nt , w d) Water content – stress paths #2 Vo id ra tio , e e) Void ratio – stress paths #2 D eg re e of s at ur at io n, S 217 10000 1000 100 10 1 Figure 4-15. Volume-mass constitutive surfaces for the artificial clay on wide ranges of soil suction and net mean stress (followed by the series stress paths #3 and #4). Log ne t mean stress (kPa)1e+06 100000 10000 1000 100 10 1 g a) Lo soil suction (kP 0 0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 10000 1000 100 10 1 Log ne t mean stress (kPa)1e+06 100000 10000 1000 100 10 1 L Pa og soil suction (k ) 0.5 0.5 0.72 0.72 0.94 0.94 1.16 1.16 1.38 1.38 1.6 1.6 10000 1000 100 10 1 Log ne t mean stress (kPa)1e+06 100000 10000 1000 100 10 1 g o Lo s il suction (kPa) 0 0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9 1 1 10000 1000 100 10 1 Log ne t mean stress (kPa)1e+06 100000 10000 1000 100 10 1 g Lo soil suction (kPa) 0 0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9 1 1 b) Void ratio – stress paths #3 c) Degree of saturation – stress paths #3 f) Degree of saturation – stress paths #4 e) Void ratio – stress paths #4 a) Water content – stress paths #3 10000 1000 100 10 1 10000 1000 100 10 1 Log ne t mean stress (kPa)1e+06 100000 10000 1000 100 10 1 L n og soil suctio (kPa) 0.5 0.5 0.72 0.72 0.94 0.94 1.16 1.16 1.38 1.38 1.6 1.6 d) Water content – stress paths #4 Log ne t mean stress (kPa)1e+06 100000 10000 1000 100 10 1 o Log s il suction (kPa) 0 0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 G ra vi m et ric w at er c on te nt , w G ra vi m et ric w at er c on te nt , w Vo id ra tio , e Vo id ra tio , e D eg re e of s at ur at io n, S D eg re e of s at ur at io n, S 218 4.4.3 Discussions Similar to the void ratio and water content constitutive surfaces predicted for the artificial silt presented using the closed-form equations, the volume-mass constitutive surfaces predicted for the three artificial soils at low soil suctions and net mean stresses using the EPUS software program (Figures 4-7, 4-10 and 4-13) agree well with all the postulates presented by Fredlund et al., (2000). For the volume-mass constitutive relationships predicted following all four series stress paths (i.e., #1, #2 , #3 and #4) for three artificial soils (Figures 4-8 to 4-15), the compression curves of the soils at saturation and the soil-water characteristic curves of the soils at zero net mean stress are reasonable. The volume-mass constitutive surfaces obtained by following series stress paths #1 (i.e., Figures 4-8, 4-11 and 4-14) show that at a higher soil suction or a higher net mean stress, both water content and void ratio are lower. At soil suctions less than the air entry value (i.e., for drying surface) and less than the water entry value (i.e., for wetting surface) all pores are filled with water and the soil behaves exactly like that at saturation; Therefore, the volume-mass constitute relationships are stress path independent (Figures 4-8 to 4-15). When all pores are filled with water, all stress paths give the same predictions for both void ratio and water content. The volume-mass constitutive surfaces starting from initially slurry seem to have steeper slopes than that for starting from air-dried condition. These prediction results appear to be reasonable. The void ratio constitutive surfaces obtained by following stress paths #3 and #4 (Figures 4-9, 4-12 and 4-15) seem to be strange (i.e., at 106 kPa soil suction and 104 kPa net mean stress, void ratio is higher than that at soil suction of 0.1 kPa and net mean stress of 104). The shape of the surfaces is similar to that measured by Matyas and Radhakrishna (1968). It shows that an unsaturated soil can either swell or collapse along a wetting process depending on the magnitude of net mean stress. Explanations on collapse and swell behaviors of an unsaturated soil have been presented by numerous researchers (Clemence & Finbarr, 1981; Popescu, 1986; Lawton et al., 1991a, 1991b; Gens and Alonso, 1992; Gens, 1995; Pereira, 1996). 219 In conclusion, the volume-mass constitutive surfaces plotted in Figures 4-8, to 4- 15 shows that the proposed volume-mass constitutive model is capable of: 1) Predicting the volume-mass constitutive surfaces exhibited stress path dependence when examining a drying process; 2) Predicting volume-mass constitutive surfaces exhibited stress path independence when the surface examining only wetting processes; and 3) Predicting both swell and collapse behaviors of an unsaturated soil. 4.5 Visualization of the unsaturated soil property surfaces In this section, simple application of the volume-mass constitutive surfaces is presented. The shear strength and hydraulic conductivity constitutive surfaces are presented. The shear strength and hydraulic conductivity constitutive surfaces are predicted using the volumetric water content constitutive surface. In this section, the series stress paths #1 is used to predict the shear strength and hydraulic conductivity constitutive surfaces. Most shear strength and hydraulic conductivity equations require volumetric water content of the soil. The volumetric water content constitutive surfaces for the three artificial soils are plotted in Figure 4-16. 220 10000 1000 100 10 1 Figure 4-16. Volumetric water content constitutive surfaces for the three artificial soils 10000 1000 100 10 1 Log ne t mean stress (kPa)1e+06 100000 10000 1000 100 10 1 Lo Pa g soil suction (k ) 0 0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 Log ne t mean stress (kPa)1e+06 100000 10000 1000 100 10 1 c P Log soil su tion (k a) 0 0 0.05 0.05 0.1 0.1 0.15 0.15 0.2 0.2 0.25 0.25 0.3 0.3 0.35 0.35 0.4 0.4 0.45 0.45 Vo lu m et ric w at er c on te nt , θ 10000 1000 100 10 1 a) Artificial sand Log ne t mean stress (kPa)1e+06 100000 10000 1000 100 10 1 L P og soil suction (k a) 0 0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 Vo lu m et ric w at er c on te nt , θ b) Artificial silt Vo lu m et ric w at er c on te nt , θ c) Artificial clay 221 4.5.1 Shear strength surfaces The Fredlund et al. (1996) shear strength equation is used to predict shear strength of the artificial soils at various stress state. It is assumed that, net normal stress in the Fredlund et al. (1996) equation can be replaced by net mean stress as follows: 'tan'tan' φψφτ npc Θ++= (4.6) where: p = net mean stress, ψ = soil suction, c’ = effective cohesion (at saturation), φ’ = friction angle (at saturation), θ = volumetric water content, θ s = saturated volumetric water content, Θ = normalized volumetric water content, sθ θ , and n = soil parameter. Fredlund et al. (1996) suggested that the soil parameter, n, is equal to 1 for sandy soils and increases with plasticity of the soil. For simplicity, let us assume that the soil parameter n is equal to 1, 1.5 and 2 for the artificial sand, silt and clay, respectively. The shear strength constitutive surfaces for the three artificial soils are plotted in Figure 4-16. For a better visualization, two shear strength constitutive surfaces are plotted for each soil at low and high ranges of net mean stress and soil suction. It can be seen in Figure 4-17 that shear strength of a soil is significantly affected by both soil suction and net mean stress. However, the effect of net mean stress appears to be dominant. However, the range of soil suction (i.e., 1 to 106 kPa) in engineering practice is much wider than that of net mean stress (i.e., 1 to 2000 kPa). It is recommended that, both soil suction and net mean stress be considered when studying the shear strength of an unsaturated soil. 222 0 3 6 9 12 15 18 21 24 27 0 500 1000 1500 2000 2500 3000 3500 4000 4500 Figure 4-17. Shear strength constitutive surfaces of the artificial sand, silt and clay for the series stress paths #1. Net mea n st ress (kP a) 03 69 121 5182 124 27 ilSo suction (kPa) 0 01 12 23 34 45 56 67 78 89 910 1011 1112 12 S he ar s tre ss (k P a) S he ar s tre ss (k P a) 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Net mea n st ress (kP a) 0100 00200 00300 00400 00500 00600 00700 00800 00900 00 Soi suction (kPa l ) 0 0 500 500 1000 1000 1500 1500 2000 2000 2500 2500 3000 3000 3500 3500 4000 4000 S he ar s tre ss (k P a) S he ar s tre ss (k P a) Net mea n st ress (kP a) 0100 00200 00300 00400 00500 00600 00700 00800 00900 00 0 0 500 500 1000 1000 1500 1500 2000 2000 2500 2500 S he ar s tre ss (k Pa ) S he ar s tre ss (k Pa ) Soil suction (kPa) 0 20 40 60 80 100 120 140 160 180 a) Artificial sand – low ranges b) Artificial sand – high ranges Net mea n st ress (kP a) 0204 060 8010 012014 01601 80 0 0 10 10 20 20 30 30 40 40 50 50 60 60 70 70 80 80 S he ar s tre ss (k P a) S he ar s tre ss (k P a) Soil suction (kPa) 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Net mea n st ress (kP a) 0100 00200 00300 00400 00500 00600 00700 00800 00900 00 ) Soil suction (kPa 0 0 1000 1000 2000 2000 3000 3000 4000 4000 5000 5000 6000 6000 7000 7000 8000 8000 S he ar s tre ss (k P a) S he ar s tre ss (k P a) 0 100 200 300 400 500 600 700 800 900 c) Artificial silt – low ranges d) Artificial silt – high ranges Net mea n st ress (kP a) 01002 0030 04005 0060 07008 0090 0 ) 0 0 125 125 250 250 375 375 500 500 625 625 750 750 875 875 1000 1000 S he ar s tre ss (k P a) S he ar s tre ss (k P a) Soil suction (kPa e) Artificial clay – low ranges f) Artificial clay – high ranges 223 4.5.2 Hydraulic conductivity surfaces There are a number of equations for calculating unsaturated hydraulic conductivity of an unsaturated soil. Huang et al. (1996) proposed an equation for the hydraulic conductivity of a soil that deforms significantly with changes of net mean stress and soil suction. The equation presents the hydraulic conductivity of a deformable soil as a function of void ratio and degree of saturation. However, the equation appears to be complex and may be beyond the scope of this thesis. For illustration purposes, a simple equation is chosen to predict the entire hydraulic conductivity constitutive surface for the three artificial soils. The microscopic based equation proposed by Irmay (1971) is used in this section. The equation was proposed for insignificant volume change soils. The Irmay (1971) equation is written as follows: 0.3)( Θ= satkk θ (4.7) where: ksat = saturated hydraulic conductivity, Θ = relative degree of saturation, rs r θθ θθ − − , θ = volumetric water content, rθ = residual volumetric water content, and θ s = saturated volumetric water content. The effect of the net mean stress is not mentioned in Equation (4.7). Qualitatively, the equation for the relative degree of saturation can be modified for the case when net mean stress is taken into account as follows: 1)( )( =− −=Θ prs r θθ θθ (4.8) where: )( rθθ − = difference between the current and residual volumetric water content at a certain stress state, and 224 1)( =− prs θθ = difference between the saturated and residual volumetric water content at a net mean stress of 1 kPa. It is assumed that, ksat is the saturated hydraulic conductivity of the soil at 1 kPa mean effective stress. Equation (4.7) requires calculation of the relative degree of saturation of the soil. The residual gravimetric water content is assumed to be a constant with the net mean stress as proposed in the volume-mass constitutive model. The residual gravimetric water content for the artificial sand, silt and clay are: 3%, 8% and 18%, respectively. The residual volumetric water content can be calculated from the residual gravimetric water content using equation (4.4). At soil suctions higher than the residual soil suction, flow of water in the soil is the vapor flow. Ebrahimi-B et al. (2004) suggested that at soil suctions higher than the residual soil suction, the hydraulic conductivity should take a value of 10-12 (cm/s). At soil suctions higher than the residual soil suction, the unsaturated hydraulic conductivity of the three artificial soils is assumed to be equal to 10-12 (cm/s). The predicted hydraulic conductivity constitutive surfaces for the three artificial soils are plotted in Figure 4-18. As can be seen in Figure 4-18, the hydraulic conductivity of a soil is significantly affected by net mean stress and soil suction. It is well accepted in the literature that soil suction plays a dominant role in the hydraulic conductivity of a soil (Mualem, 1986). It may not appropriate to ignore the effect of net mean stress on the hydraulic conductivity for certain soils (Huang et al., 1996). The hydraulic conductivity depends on the magnitude of net mean stress and type of the soil. As can be seen in Figure 4-18c, change in net mean stress may reduce the hydraulic conductivity of a soil up to 2 orders of magnitude (i.e., for artificial clay soil). In summary, it is recommended that both soil suction and net mean stress need to take into consideration when studying the hydraulic conductivity of a soil. Application of the Irmay (1971) equation in this section may not correctly describe the hydraulic conductivity constitutive surface for an unsaturated soil. The equation proposed by Huang et al. (1996) should be used in the future research. 225 Figure 4-18. Hydraulic conductivity constitutive surfaces of three artificial soils for the series stress paths #1. 10000 1000 100 10 1 Log n et m ean s tress (kPa ) 1e+0 61000 00100 0010 001 00 10 1 Lo soil suctio (kPa g n ) 10-13 10-12 10-9 10-8 10-10 10-11 H yd ra ul ic c on du ct iv ity (c m /s ) c) Artificial clay 10000 1000 100 10 1 10000 1000 100 10 1 Log net me an s tres s (k Pa) 1e+ 06100 000100 00100 0100 10 10.10. 01 ) Log soil suction (kPa b) Artificial silt 10-5 10-13 10-11 10-7 10-9 H yd ra ul ic c on du ct iv ity (c m /s ) Log net me an s tres s (k Pa) 1e+ 06100 00010 00010 0010 010 1 il ) Log so suction (kPa H yd ra ul ic c on du ct iv ity (c m /s ) 10-2 10-4 10-7 10-10 10-13 a) Artificial sand 226 CHAPTER 5 Laboratory Testing Program 5.1 General A number of test programs related to the volume-mass constitutive relationships have been presented in the research literature (see Chapter 2). Each test program has been carried out with specific objectives. There appears to be three primary limitations associated with the soil datasets found in the literature: a. Most datasets present only volume change information of the soil with respect to the changes of soil suction and net mean stress. Quite a few soil datasets have data on the water content constitutive relationship, b. The hysteretic nature of the soil-water characteristic curve is not fully considered, and c. Most experimental programs have focused on the measurement of volume mass constitutive relations of compacted soil specimens which have a complex stress history as described in Chapter 3. It would appear to be better to study the mechanical behavior of unsaturated soils using slurry specimens which have a simple and well defined stress history. 227 In this chapter, the objectives of the testing program are first described, followed by a description of the soils, equipment and testing procedures. 5.2 Objectives of the Testing Program The laboratory program is designed to overcome several limitations of existing soil datasets in the research literature. The objectives of the laboratory testing program include: 1. Study the compression characteristics of several soils at saturation and at completely dry conditions. 2. Study the effect of net mean stress on the air entry value, the water entry value and the residual water content of a soil. 3. Study the hysteretic nature of the soil-water characteristic curve for various types of soils over a wide range of soil suctions. 4. Measure both the void ratio and water content constitutive surfaces. The testing results are used to verify the proposed volume-mass constitutive model. The uniqueness of the volume-mass constitutive surface is also explored. 5.3 Materials and Preparations: Soil specimens and methods of preparation are required for the study of the fundamental volume-mass soil behavior. 5.3.1 Materials Six soils are used in the testing program; namely, i) Beaver Creek sand; ii) Saskatchewan Silty Sand; iii) Processed silt; iv) Botkin silt; v) Indian Head till; and vi) Silty clay. The grain-size distributions for the Saskatchewan Silty Sand and Indian Head Till were measured in the laboratory testing program in general accordance with ASTM D422. The grain-size distributions for the other soils were collected from the research literature. The grain size distributions for the six soils are plotted in Figure 5-1. The liquid limit and plastic limit of the Indian Head Till and Silty Clay were measured in the laboratory testing program in general accordance with ASTM D-4318-00. The liquid 228 limit and plastic limit for the Processed Silt and Botkin Silt were collected from the research literature. The specific gravities for the Saskatchewan Silty Sand, Indian Head Till, and Silty Clay were measured in the laboratory testing program in general accordance with ASTM C 127. The specific gravities for the other soils were collected from the research literature. The classification properties of the soils are presented in Table 5-1. 0 10 20 30 40 50 60 70 80 90 100 0.001 0.01 0.1 1 10 Grain size (mm) P er ce nt fi ne r t ha n (% ) Beaver Creek sand (Bruch, 1993) Processed silt (Bruch, 1993) Sask. silty sand Indian Head till Botkin silt (vanapalli et al., 2000) Silty clay Figure 5-1. Grain-size distributions for the six soils used in the laboratory testing program. Table 5-1. Soil properties of the six soils used in the testing program. Soil Liquid limit (LL%) Plastic limit (PL%) Specific gravity (Gs) Beaver Creek sand N/A N/A 2.65 (Bruch, 1993) Saskatchewan Silty sand N/A N/A 2.65 Processed silt 26.8 (Wilson, 1990) 25.4 (Wilson, 1990) 2.70 (Wilson, 1990) Botkin silt 29.4 (Wang, 2000) 13.8 (Wang, 2000) 2.71 (Vanapalli, 2000) Indian Head till 36.1 16.4 2.73 Silty clay 51.9 20.5 2.79 229 5.3.2 Specimen Preparation Soil specimen preparation is important for determining the mechanical behavior of an unsaturated soil. A soil specimen can be prepared as: i) a slurry specimen; ii) a compacted specimen; iii) a consolidated specimen (with a specific stress history); or iv) an undisturbed specimen. A compacted specimen has a complex unknown stress history and therefore, it is not the most suitable for studying the fundamental behavior of an unsaturated soil. Two types of soil specimens were used in the laboratory testing program; namely, i) slurry specimens and ii) specimens that were air-dried from slurry. For preparing a slurry soil specimen, dry soil was mixed with water to approximately 1 to 2 times the liquid limit of the soil. For specimens air-dried from a slurry, a special technique was required to ensure that there were no cracks in the soil specimen. The following procedures were implemented to prepare air-dried specimens from a slurry: • Mix dry soil with water to approximately 1 to 2 times the liquid limit of the soil. • Prepare a mold with a size that is much bigger than the expected size of the air- dried soil specimen (depending on the soil type). • Spread a thin plastic wrapping paper over the bottom and around the wall of the mold. Pour the slurry into the mold. • Apply a pressure of 1 kPa on top of the soil specimen to help ensure there are no entrapped air bubbles in the soil specimen. • Place the s