A constitutive model for cemented clays capturing cementation degradation

International Journal of Plasticity 56 (2014) 1–18 Contents lists available at ScienceDirect International Journal of Plasticity journal homepage: www.elsevier .com/locate / i jp las A constitutive model for cemented clays capturing cementation degradation http://dx.doi.org/10.1016/j.ijplas.2014.01.007 0749-6419/Crown Copyright � 2014 Published by Elsevier Ltd. All rights reserved. ⇑ Corresponding author. Address: Centre for Built Infrastructure Research (CBIR), School of Civil and Environmental Engineering, Faculty of Eng and Information Technology, University of Technology, Sydney (UTS), City Campus, PO Box 123, Broadway, NSW 2007, Australia. Tel.: +61 (2) 95 mobile: +61 0413573481; fax: +61 (2) 9514 2633. E-mail address: [email protected] (B. Fatahi). Lam Dinh Nguyen, Behzad Fatahi ⇑, Hadi Khabbaz Centre for Built Infrastructure Research (CBIR), School of Civil and Environmental Engineering, University of Technology Sydney (UTS), Sydney, Australia a r t i c l e i n f o a b s t r a c t Article history: Received 3 September 2013 Received in final revised form 27 December 2013 Available online 5 February 2014 Keywords: A. Yield condition B. Elastic–plastic material B. Constitutive behaviour C. Analytic functions Laboratory experiments show that the effect of cementation on clays gradually diminishes as the confining pressure increases (particularly at high confining pressures) due to the degradation of cementation bonds. The main aim of this paper is to propose a constitutive model for cemented clays, referred to as the Cemented Cam Clay model (CCC), to simulate the cementation degradation during loading. The failure envelope of the proposed model is formulated to describe the behaviour of the cemented clay at a low pressure range similar to over-consolidated soils, while it merges with the Critical State Line of reconstituted sam- ple gradually as the confining pressure continues to increase. In order to examine the stress–strain behaviour of cemented clays, an energy dissipation equation is developed inspired by the Modified Cam Clay model. The characteristics of the proposed model, including a non-associated plastic potential function and elasto-plastic stress–strain rela- tionship, are presented in light of the Critical State concept. Validity of the proposed con- stitutive model derived from the modified energy equation is evaluated against triaxial test results for cemented clays available in literature. Crown Copyright � 2014 Published by Elsevier Ltd. All rights reserved. 1. Introduction With the growth of cities and industries, suitable sites, which can be used without some ground modification, are becom- ing increasingly scarce. Moreover, the cost of replacing soft soils with high quality material has dramatically increased. The design engineers have various options in dealing with problematic soils such as bypassing the poor soil, replacing it with superior soil, redesigning the structure for the poor condition or improving the soil properties by mixing soil with material such as cement, lime, gypsum and fly ash amongst other ground modification techniques. The latter option can be used for surface improvement, such as road and rail subgrade improvement, or in deep soil mixing or jet grouting technologies, which are soil improvement approaches, mixing in situ soil with strengthening agents. A number of laboratory experiments on the effect of cementation have resulted in several constitutive models for cemen- ted clays (Horpibulsuk et al., 2010; Kasama et al., 2000; Liu and Carter, 2002; Yasufuku et al., 1997). Kasama et al. (2000) proposed a constitutive model for cemented clays extending the critical state concept by introducing the cementation effect into the energy dissipation equation. They also modified the Critical State concept to include the increase in the strength of the clay, which results in extending the stress domain. Horpibulsuk et al. (2010) simulated the behaviour of cemented clays ineering 14 7883, http://crossmark.crossref.org/dialog/?doi=10.1016/j.ijplas.2014.01.007&domain=pdf http://dx.doi.org/10.1016/j.ijplas.2014.01.007 mailto:[email protected] http://dx.doi.org/10.1016/j.ijplas.2014.01.007 http://www.sciencedirect.com/science/journal/07496419 http://www.elsevier.com/locate/ijplas Notations A derivative of p0⁄ with respect to p0 a non dimensional anisotropic parameter b cementation degradation parameter CSL critical state line C shear strength contributed by cementation when p0 = 0 dv total volumetric strain increment dev elastic volumetric strain increment dpv plastic volumetric strain increment de total plastic deviatoric strain increment dee elastic deviatoric strain increment dpe plastic deviatoric strain increments dWin internal plastic energy per unit volume e void ratio of the soil e1 axial strain e3 radial strain f yield function g plastic potential function j swelling or recompression index k compression index M slope of failure envelope of reconstituted soil g stress ratio g⁄ modified stress ratio m Poisson’s ratio p0 mean effective stress p0 � modified mean effective stress p0d mean effective stress inducing cementation degradation p0�0 modified mean effective stress on the yield surface when q = 0 p0 � 0;i initial size of the yield surface p00 hardening parameter – mean effective stress on the yield surface when q = 0 p0i mean effective stress at the end of consolidation stage p0y;i initial mean effective yield stress p0X p 0(tension) when q = 0, describing the effect of cementation w⁄ proposed flow rule q deviatoric stress qu unconfined compressive strength r01 axial effective stress r03 radial effect stress 2 L.D. Nguyen et al. / International Journal of Plasticity 56 (2014) 1–18 via the framework of the Structured Cam Clay (SCC) model developed by Liu and Carter (2002). Their constitutive model is an extension of SCC model for cemented clays by modifying the mean effective stress, while the effect of cementation is con- sidered to reinforce the mean effective stress (Horpibulsuk et al., 2010). The failure envelope of the extended SCC model has been assumed to be parallel to that of untreated clay and shifted by a certain intercept, which characterises the effect of cementation similar to the model proposed by Kasama et al. (2000). Furthermore, the constitutive models by Kasama et al. (2000) and Horpibulsuk et al. (2010) assumed an associated plastic potential function while (Shen et al., 2012) ex- plained that the volumetric strains and the transition process from compressibility to dilatancy are correctly described by a non-associated model for clayey material as also suggested by Yuanming et al. (2010). Although these existing models pro- vide a conceptual framework for the development of an appropriate constitutive model capturing the behaviour of cemented clays, the effect of cementation degradation due to the increase in the confining pressure has not been captured in these models. As suggested by Moses et al. (2003), Panda and Rao (1998) and Lo and Wardani (1999), when the confining pressure increases, the beneficial effects of cementation may diminish as a result of cementation degradation. Therefore, the failure envelope of the cemented clay gradually merges with that of reconstituted clay-cement mixture. The aim of this paper is to propose an enhanced model to simulate the behaviour of cemented clays under various con- fining pressures. The degradation of cementation bonds due to increasing confining pressure is presented by a non-linear failure envelope of cemented clay, merging to that of reconstituted soil in high confining pressure. The development of this model is mainly based on the Modified Cam Clay (MCC) model, and when there is no effect of cementation, the model returns to its original form. The paper introduces a new plastic potential function developed through modifying the energy dissipa- tion equation. In addition, a modified stress–strain relationship for cemented clays is presented. L.D. Nguyen et al. / International Journal of Plasticity 56 (2014) 1–18 3 2. Factors affecting the strength of cemented clay As proposed by Diamond and Kinter (1965) and confirmed by several other researchers such as Kamruzzaman et al. (2006), Porbaha (1998) and Chew et al. (2004), the three major categories of reactions, expected in the process of mixing a stabiliser with clay, are: (i) dehydration process, (ii) ion exchange or flocculation, and (iii) pozzolanic reaction. Lorenzo and Bergado (2006) explained that the hydration process occurs rapidly when cement is mixed with pore water of the soil to form primary cementitious products such as hydrated calcium aluminates and hydrated lime. These cementing products result in the dissociation of calcium ions which then react with soil silica and alumina to form pozzolanic products. This sec- ondary cementing product, stabilising the soil and increasing the strength of improved soil with time, is also reported by Porbaha et al. (2000). The effects of cementation on the pre and post failure behaviour of cemented clays can be observed at microstructural level through X-ray diffraction (XRD) and scanning electron microscopy (SEM) images. Kamruzzaman et al. (2009) explained the structuration (formation of structure due to cementation) and destructuration (due to cementation degradation) phe- nomena by analysing SEM images of Singapore marine clay treated with different cement contents. They mentioned that while the untreated clay exhibits an open type of microstructure, with thin and flat clay particles spreading over a large area, the cemented clay shows evidence of flocculated structure with large treated particle clusters interspersed by large open- ings. At the same time, while the flocculated structure becomes more evident as the cement content increases, the platiness of the structure becomes less evident as also observed by Yang et al. (2013) and Locat et al. (1990). According to observations on naturally structured soils, Yin et al. (2009) explained that the surface forces such as elec- trostatic and chemical forces attract clay particles and pull them together to form particle-clusters. The clusters continue to grow in size until their weight, due to gravitation is larger than the inter-particle surface forces. At this time, the clusters stop growing in size and interact with each other mainly through mechanical forces. The effect of cementation degradation can be observed when the sample undergoes isotropic consolidation in the tri- axial tests. When the cemented sample is stressed at a high consolidation (effective confining) pressure, the large treated particle clusters collapse with gradual reduction in the void space until there is a significant reduction of inter-aggregate pore spaces, resulting in the reduced void ratio. Kamruzzaman et al. (2009) observed that when the treated sample is loaded or deformed, the cementation bonds gradually break under the applied pressure. Furthermore, when the treated sample is loaded beyond the peak strength, crushing of cementation bonds occurs. The SEM image of the sheared sample in the shear plane shows negligible clusters remaining. However, as reported by Kamruzzaman et al. (2009), some small clusters still appear outside the failure plane. Therefore, it can be concluded that during isotropic consolidation, the deg- radation of cementation bonds may not be completed as some small clusters will still be present requiring further load to be completely crushed. Laboratory experiments have been conducted broadly by various researchers to report different factors affecting the strength of cemented clays (Lorenzo and Bergado, 2006; Porbaha et al., 2000; Tan et al., 2002; Uddin et al., 1997). According to Tan et al. (2002), there is a linear relationship between the amount of stabilising agents, such as lime and cement, and the strength of the improved clay. The unconfined compressive strength (qu) generally increases with increasing the cement con- tent and the curing time for a fixed amount of water content (Lorenzo and Bergado, 2006), especially for cement contents greater than 5% as observed by Sariosseiri and Muhunthan (2009) and Uddin et al. (1997). The effect of increasing cement content on the strength of improved soils is mainly due to the pozzolanic reactions (Lorenzo and Bergado, 2006). However, expectedly, the water content has an inverse effect on the compressive strength of the soil at any particular cement content (Porbaha et al., 2000). Similar to concrete, the strength of the cemented clay is commonly accepted to develop with curing time (e.g., Porbaha et al., 2000; Uddin et al., 1997). Thus, the effects of increasing cement content and ageing at particular water content play significant roles in the strength development of cemented clays. The confining pressure is also an important parameter, influencing the behaviour of cemented clay (Uddin et al., 1997). As reported by Lorenzo and Bergado (2006), the experimental results from triaxial tests show that increasing the confining pressure leads to an increase in the deviatoric stress at failure; this is even more significant at higher cement contents. Unlike the ductile behaviour of untreated clays, the cemented clays generally exhibit more brittle behaviour as the stress increases to the peak strength state and then drop to the residual strength at the post-peak state (Lorenzo and Bergado, 2006; Porbaha et al., 2000; Yin, 2001). This indicates that the behaviour of cemented clay is similar to over-consolidated soil and the cemen- ted clay is more structured due to chemically induced cementation (Panda and Rao, 1998; Yin, 2001). Lade and Overton (1989) proposed a non-linear failure envelope for cemented clays, which is positioned significantly higher than that of un- treated soil in p0–q graph. The cementation increases the friction angle and the cohesion of the soil and the cemented clays behave like over-consolidated soil at low confining pressures. However, the high stiffness of cemented clays restricts their compressibility during consolidation compared to untreated clays. Thus, as the confining pressure continues to increase, the effect of cementation on the strength of the cemented clay keeps reducing due to cracking and degradation of cementation bonds (Lade and Overton, 1989). Furthermore Zhu et al. (2010) explained that, for cement based materials, the shear strain becomes larger as it approaches the peak strength state and the accumulation of plastic shear strain leads to progressive destruction of the microstructure. Uddin et al. (1997) also observed that the apparent over-consolidation ratio of the soil is reduced as the beneficial effect of cementation is diminished under sufficiently high confining pressures. Due to increasing microcracks in cementation as the confining stress increases, the undrained strength is reduced significantly as the soil 4 L.D. Nguyen et al. / International Journal of Plasticity 56 (2014) 1–18 structure is progressively changed. As a result, the failure envelope of cemented clays gradually approaches to the remoulded clay-cement mixture as observed by Moses et al. (2003), Panda and Rao (1998) and Lo and Wardani (1999). Fig. 1 displays triaxial test laboratory data on Indian marine clay mixed with 3% hydrated lime. Results clearly indicate that a collection of failure points of cemented clay forms a non-linear failure envelope, which is not parallel to the Modified Cam Clay failure envelope (q =Mp0). For the sake of simplicity, Structured Cam Clay (SCC) model (Horpibulsuk et al., 2010; Liu and Carter, 2002) assumes a linear failure line parallel to that of reconstituted soil. In other words, SCC model ignores the reduction in the strength of the cemented clay due to degradation of cementation bonds as a result of confining pressure increase. 3. Development of cemented cam clay model 3.1. Modified mean effective stress In this paper, the definition of stress and strain quantities similar to Modified Cam Clay model for conventional triaxial test is adopted (Roscoe and Burland, 1968) as explained below: � The mean effective stress (p0), the deviatoric stress (q), and the stress ratio (g) are given by: p0 ¼ r 0 1 þ 2r03 3 ð1Þ q ¼ r01 � r03 ð2Þ g ¼ q p0 ð3Þ � The volumetric strain increment (dv) and the deviatoric strain increment (de) are defined as follows: dv ¼ de1 þ 2de3 ð4Þ de ¼ 2ðde1 � de3Þ 3 ð5Þ where, r01 and de1 are axial effective stress and the axial strain increment, respectively, and, r03 and de3 are radial effective stress and the radial strain increment, respectively. According to Huang and Airey (1998) and Rotta et al. (2003), the structure of naturally cemented soils possesses aniso- tropic properties due to various geological processes creating cementation bonds. However, the degree of anisotropy in ce- mented soils is reduced as a result of cementation degradation as suggested by Suebsuk et al. (2010). Naturally cemented soils often have variable densities and degrees of cementation presenting difficulties in studying the fundamental behaviour without disturbing the cementation. To overcome such difficulties, many researchers such as Rotta et al. (2003), Kamruzz- aman et al. (2009) and Suebsuk et al. (2010) used artificially cemented soils in their studies. It should be noted that Huang and Airey (1998) reported that the properties of artificially cemented soils are comparable to naturally cemented soils. 0 50 100 150 200 250 0 50 100 150 200 250 300 350 D ev ia to ri c st re ss (q ) (k Pa ) Mean effective stress (p') (kPa) Shear failure points (data obtained from Panda and Rao, 1998) Structured Cam Clay model failure envelope (Liu and Carter, 2002) Critical State Line ( ) = 0.64 Strength reduction due to cementation degradation Indian marine clay treated with 3% lime Fig. 1. Failure envelope for artificially cemented Indian marine clay. L.D. Nguyen et al. / International Journal of Plasticity 56 (2014) 1–18 5 Furthermore, the variation of mechanical properties of artificially cemented soils can be reasonably considered to be isotro- pic (Suebsuk et al., 2010). Therefore, in order to include the effect of the structure and cementation degradation and to over- come the complexity of anisotropic effects, in this study, the material is assumed to be isotropic, while developing the Cemented Cam Clay model. Suebsuk et al. (2010), in the formulation of the Modified Structure Cam Clay model, assumed that for stiff structured clays, the destructuring process depends on the plastic strains which occur mainly after yielding due to volumetric and shear deformations. However, Horpibulsuk et al. (2004) suggested that the shear resistance induced by the cementation bonds is still present, even when the breaking of cementation bonds has occurred. Suebsuk et al. (2010) further explained that the damage to the cemented bonds caused by the shear deformation before failure is insignificant because the change in the plastic deviatoric strain associated with shearing is very small prior to the shear failure. However, for post failure state where the cementation bonds begin to be crushed, large plastic deformations occurs while the shear resistance reduces. It should be noted that several researchers (e.g., Horpibulsuk et al. (2010), Suebsuk et al. (2010), Yin et al. (2009), and Liu and Carter (2002)) have already investigated the effects of shear stresses on the cementation degradation, particularly when plastic shear deformations occur. However, authors are not aware of any constitutive models capturing cementation degradation due to the effective mean stresses. Thus, in this study authors have tried to develop a new constitutive model capturing the cementation degradation due to the mean effective stress. Based on the framework of the Critical State Soil Mechanics and the basis of MCC model, the proposed model has mod- ified the mean effective stress so that the effect of cementation is clearly obtained and analysed while incorporating destruc- turation as a function of mean effective stress. In other words, the mean effective stress is strengthened by cementation, similar to cohesionless soil (Horpibulsuk et al., 2010; Kasama et al., 2000), while incorporating cementation degradation with increasing mean effective stress. The modified mean effective stress(p0⁄)is proposed to introduce the effect of cemen- tation and its degradation as follows: p0� ¼ p0 þ p0X ð6Þ p0X ¼ C 1þ p 0 d Cþb � � expð �p0 d CþbÞ M ð7Þ where, p0X is a function of p 0 describing the effect of cementation, M is the slope of failure envelope of reconstituted clay-ce- ment mixture and C represents the contribution of the cementation to the shear strength when the mean effective stress is zero. The model fitting parameter (b) influences the degradation rate of cementation due to the mean effective stress. Dp0 P 0 p0d is the mean effective stress inducing cementation degradation which will be the maximum mean effective stress that soil has ever experienced at any point of time. In drained triaxial test (CD), where, p0d ¼ p0, while in undrained triaxial test (CU), when Dp0 < 0, p0d ¼ p0i, where p0i is the mean effective stress at the end of consolidation stage. Due to the effect of cementation, clay particles interact together to form clusters and the creation of the soil structure becomes evident. As discussed in Section 2, the behaviour of the cemented clay is greatly influenced by the mean effective stress. According to Kolovos et al. (2013), during loading, the contact forces between particles increase resulting in develop- ment of micro-cracks. It should be noted that, when micro-cracks appear, the strength of cementation bonds is gradually reduced. The effect of cementation bonds is captured in Eqs. (6) and (7) using parameter p0X. As the mean effective stress (p’) increases, p0X describing the effect of cementation decreases. Furthermore, the cracks continue to extend until some stage where the edges of micro-cracks collide and larger cracks (macro-cracks) are formed. As the applied load increases, macro- cracks reach the edge of the sample leading to a failure as explained by Kolovos et al. (2013). The micro-cracking process leading up to failure is also captured in the formulation of Eq. (7) which p0X e diminishes as p 0 approaches a sufficiently high effective pressure. Furthermore, in the Critical State framework, it is widely accepted that the void ratio (referred to as state variable (e) is a function of applied mean effective stress (p0) (Kamruzzaman et al., 2009; Suebsuk et al., 2010; Vatsala et al., 2001; Yin et al., 2009). In comparison to the natural clay, the cement treated clay needs higher mean effective stress (p’) to reach the same state variable (e), due to the formation of the structure. During the loading, degradation of cementation bonds gradually oc- curs with the change in e. Yin et al. (2009) further explained that clay is regarded as an aggregate of clusters and a representative volume of clay is deformed by compressing all clusters. Hence, the overall deformation can be described as the average deformation of local contact planes in all orientations. Moreover, void ratio is observed to change with the value of the contact number per unit volume as confirmed by SEM images from Kamruzzaman et al. (2009). Furthermore, the micro-variable (the inter-cluster parameter in cementation bonds) is reasonably considered as a function of the macro-state (state variable e) because the behaviour of clusters is influenced by the density or void ratio of the specimen as suggested by Yin et al. (2009). Therefore, it is necessary to introduce the effect of cementation and cementation degradation into the mean effective stress when the Critical State framework is adopted. When the modified mean effective stress is proposed, the stress ratio (g⁄) needs to be modified as follows: g� ¼ q p0� ð8Þ 6 L.D. Nguyen et al. / International Journal of Plasticity 56 (2014) 1–18 3.2. Proposed failure envelope The development of the constitutive model aims to describe the effect of cementation degradation at high effective con- fining pressures as it merges to the Critical State Line of reconstituted soil, while at the low effective confining pressure range, the model shows a beneficial contribution of cementation to the shear strength. Thus, it is critical to model the tran- sition, where the effect of cementation degradation takes place. According to Kamruzzaman et al. (2009), the breaking of cementation bonds can be associated with yielding. Moreover, as confirmed by Rotta et al. (2003), yielding represents the starting point of cementation degradation. Furthermore, Shen et al. (2012) suggested that the clayey materials display inelastic behaviour due to plastic deformation beyond the yield stress as well as the damage caused by micro-cracks growth. The behaviour of clayey materials at macroscopic scale depends on the mineralogical composition and microstructural changes caused by loading (Shen et al., 2012). Therefore, at a high effective confining pressure, the effect of cementation on the sample is apparently destructed during the consolidation process inducing yielding. While at the low effective con- fining pressure, the effect of cementation degradation is insignificant as the effective confining pressure is lower than the yield stress. Suebsuk et al. (2010) and Kamruzzaman et al. (2009) suggested that destructuration occurs with stress state on the yield surface, thus the effect of cementation degradation in the low mean effective stress range can only occur during shearing. The empirical formulation of the proposed model takes into account the effect of cementation and cementation degrada- tion by modifying the mean effective stress (p0⁄). By introducing the model parameter b in the modified mean effective stress, the proposed model can capture the degradation rate, while the failure envelope merges with the Critical State Line. Param- eter b influences the deformation and the peak shear strength of cemented clays. Considering the modified mean effective stress proposed in Eq. (6), the failure envelope of the Cemented Cam Clay model is also modified to capture the degradation effect of cementation ½q ¼ Mðp0 þ p0XÞ�. Thus, the failure envelope is taking the form of: q ¼ Mp0 þ C 1þ p 0 d C þ b � � expð �p0 d CþbÞ ð9Þ Moses et al. (2003) has indicated that the behaviour of cemented clay depends on the strength of cementation, and the bond strength depends upon the stress level within the soil, and other factors such as strain parameter and type of loading. In this study, p0X is proposed in a way that as the mean effective stress (p’) increases to a sufficiently high pressure, p 0 X ap- proaches zero since the effect of cementation is diminished. As a result, the modified mean effective stress can replace the mean effective stress to simulate the behaviour of cemented clays. In Eq. (7), when C = 0 (no binding agent), the effect of cementation prevails ðp0X ¼ 0Þ and Eq. (9) converts to MCC model ðq ¼ Mp 0Þ. Fig. 2 illustrates the proposed failure envelope for cemented clay compared to that of reconstituted clay-cement mixture and Structured Cam Clay model proposed by Hor- pibulsuk et al. (2010). The amount of cementation is significantly reduced to residual strength and diminishes as the failure envelope approaches the Critical State Line (q =Mp0). It can be noted that the Structured Cam Clay model (Horpibulsuk et al., 2010) considers a constant increase in the shear strength (or deviatoric stress at failure) due to cementation ðq ¼ Mp0 þ CÞ and cementation degradation under high mean effective stresses has been ignored. 0 50 100 150 200 250 300 350 D ev ia to ri c st es s (q ) (k Pa ) Mean effective stress (p') (kPa) C Critical State Line (CSL) Cemented Cam Clay model (CCC) Structured Cam Clay model (SCC) -50 0 50 100 150 200 250 300 350 400 450 Fig. 2. Proposed failure envelope compared with Structured Cam Clay model and the Critical State Line. 0 50 100 150 200 250 0 50 100 150 200 250 300 D ev ia to ri c st re ss ( q) ( kP a) Mean effective stress (p') (kPa) Shear failure points (data obtained from Panda and Rao, 1998) Cemented Cam Clay model predicted failure envelope Indian marine clay mixed with 3% hydrated lime LL = 79% PL = 32% 16% Sand, 29% Silt & 55% Clay = 59 kPa Critical State Line ( ) =0.64 C = 30 kPa β = 34.28 kPa Cemented Cam Clay model failure envelope Fig. 3. Predicted failure envelope for 3% cemented Indian marine clay. L.D. Nguyen et al. / International Journal of Plasticity 56 (2014) 1–18 7 Fig. 3 displays a predicted failure envelope Eq. (9) for a set of triaxial tests on Indian marine clay with unconfined com- pressive strength of 59 kPa performed by Panda and Rao (1998). It should be noted that according to Panda and Rao (1998), the unconfined compressive strength of the soil increased from 12 kPa to 59 kPa by adding 3% hydrated lime. As presented in Fig. 3, with an increase in the confining pressure, the contribution of cementation to the shear strength is gradually reduced, and the failure envelope finally merges with the failure envelope of the reconstituted clay-cement mixture with C = 0 and M = 0.64. The failure envelope proposed in this study captures the merging effect of cemented clay with the reconstituted clay-cement mixture, as a result of cementation degradation, reasonably well. Although, there are some disparities while comparing the predicted failure envelope and experimental data for the 3% cemented Indian marine clay particularly at the low mean effective stress range, the coefficient of determination is found to be 0.96, indicating a well fitted failure enve- lope. At the low mean effective stresses, the cemented clays behaved like over-consolidated soils and a reduction in strength was observed when the sample was loaded for a long duration of shear process (Moses et al., 2003; Panda and Rao, 1998). Furthermore, according to Moses et al. (2003), the bond strength depends not only on the stress level within the soil, but also on other factors such as the strain rate, the duration of shear process and the type of loading. Moreover, Liu and Carter (2002) also explained that the disparities between the predicted and measured stress–strain curve and the peak shear strength in low effective pressure ranges may be due to the differences in the size of the initial structural yield surface. However, the main aim of this paper is to develop a constitutive model to capture the cementation degradation which is more pronounced in higher mean effective stresses. The effects of increasing model parameters, C and b on the predicted failure envelope of the proposed model are shown in Figs. 4 and 5, respectively. Referring to Fig. 4, by keeping the parameter b constant at 34.28 kPa, when there is no cementation effect (C = 0), the fail- ure envelope of proposed model converts to Critical State Line (q =Mp0). As the effect of cementation increases (increasing C), higher values of q are obtained at a particular mean effective stress on the proposed failure envelope. Moreover, the tensile strength also increases with the increase in cementation. However, the value of q gradually approaches the Critical State Line at higher confining pressures due to cementation degradation. In contrast to the effect of increasing C on the failure envelope of CCC model, the degradation rate of cementation is re- duced as b increases as shown in Fig. 5. The effect of increasing b is insignificant on the shear strength of cemented clays in low mean effective stress range. However, the effect of b is more notable in high mean effective stresses. It should be noted that, when b decreases, the proposed failure envelope approaches the Critical State Line at a higher mean effective stress. 3.3. Yield function In the proposed model, cemented clay is assumed to be an isotropic material possessing elastic and virgin yielding behav- iour. Following Modified Cam Clay (MCC) model, the yield function in p0–q plane is described as: f ¼ q2 �M2p0�ðp0�0 � p0�Þ ¼ 0 ð10Þ where, p0�0 ¼ p00 þ p0X; p0�0 is the size of the yield surface where it meets with the horizontal axis (q = 0) in p0–q plane, and p00 is a hardening parameter which corresponds to the mean effective stress (p 0 ) on the yield surface when q = 0. It should be noted that the virgin yielding occurs when stress state of the cemented clays is on the yield surface and with positive incremental change (dp0�0>0). The size of the initial yield surface, p 0� 0;i, is assumed to be equal to the initial mean effective yield stress, p 0� y;i. D ev ia to ri c St re ss ( q) ( kP a) Mean Effective Stress (p') (kPa) C = 50 kPa C = 40 kPa C = 30 kPa C = 0 kPa Critical State Line (CSL) = M = 0.64 Cemented Cam Clay model failure envelope Fig. 4. Effect of increasing cementation parameter (C) on the failure envelope of the proposed model. D ev ia to ri c St re ss ( q) ( kP a) Mean Effective Stress (p') (kPa) β = 34.28 kPa β = 60 kPa β = 90 kPa CSL (reconstituted sample) Critical State Line (CSL) = M = 0.64 Cemented Cam Clay model failure envelope Fig. 5. Effect of increasing b on the failure envelope of proposed model. 8 L.D. Nguyen et al. / International Journal of Plasticity 56 (2014) 1–18 The yield function with cementation is adopted to satisfy the critical state condition in which g� ¼ 0 when p0�0 ¼ p0�. The yield curve of the proposed model is taking the shape shown in Fig. 6. The proposed yield function is different than that of MCC model by introducing the new function describing the effect of cementation ðp0XÞ which is influenced by the parameter C. Fig. 7 displays the change in the shape of the proposed yield surface with variation of C. Increasing the value of parameter C results in expansion of the yield surface since p0X increases. As depicted in Fig. 7, when C = 0, the yield surface possesses elliptical shape similar to MCC model. Fig. 8 displays the expansion in the proposed yield surface with an increase in the hardening parameter (p00). In addition, Fig. 9 shows a shrinkage of the yield surface as the cementation degradation increases (reduction in b simulates degradation increase). 3.4. The energy dissipation equation The internal plastic energy per unit volume (dWin) available for dissipation of a soil sample in the triaxial test under the applied mean effective stress (p0) and the shear stress (q) is expressed as (Wood and Graham, 1990): dWin ¼ p0dpv þ qd p e ð11Þ where, dpv is the plastic volumetric strain increment and d p e is the plastic deviatoric strain increments. MCC model assumes an expression for the dissipation of internal energy as follows (Roscoe and Burland, 1968): p0dpv þ qd p e ¼ p0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdpvÞ 2 þ ðMdpeÞ 2 q ð12Þ D ev ia to ri c st re ss ( q) ( kP a) Mean effective stress (p') (kPa) Critical State Line = Cemented Cam Clay yield surface Modified Cam Clay yield surface Cemented Cam Clay model failure envelope Fig. 6. Cemented Cam Clay yield surface presented in this study. D ev ia to ri c st re ss (q ) (k Pa ) Mean effective stress (p') (kPa) C = 50 kPa C = 40 kPa C = 30 kPa C = 0 kPa Ω C = 0, MCC yield surface Critical State Line ( = ) M = 0.64 = 34.28 kPa Fig. 7. Proposed yield surface with increasing effect of cementation. D ev ia to ri c st re ss (q ) (k Pa ) Mean effective stress (p') (kPa) = 400 kPa = 200 kPa = 100 kPa Critical State Line ( = ) M = 0.64 = 34.28 kPa Fig. 8. Proposed yield surface with increasing hardening parameter (p00). L.D. Nguyen et al. / International Journal of Plasticity 56 (2014) 1–18 9 D ev ia to ri c st re ss ( q) ( kP a) Mean effective stress (p') (kPa) β = 90 kPa β = 60 kPa β = 34.28 kPa β = 0 kPa Ω Critical State Line ( = ) M = 0.64 C = 30 kPa Fig. 9. Proposed yield surface with variation of cementation degradation parameter (b). 10 L.D. Nguyen et al. / International Journal of Plasticity 56 (2014) 1–18 A more generalised equation of energy dissipation proposed by Kasama et al. (2000) is described by: p0dpv þ qd p e ¼ p0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdpvÞ 2 þ ðMdpeÞ 2 � Xdpvd p e q ð13Þ The term Xdpvd p e is described as the soil dilatancy dependent coupling term (Kasama et al., 2000). In this study, the authors attempt to make an appropriate selection for parameter X, leading to an appropriate energy dissipation equation for cemen- ted clays capturing the effects of cementation and its degradation. The energy equation for cemented clay proposed in this study is presented in line with Eq. (13) as: p0�dpv þ qd p e ¼ p0� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdpvÞ 2 þ ðMdpe Þ 2 þ 2 q p0� dpvd p e 1� aþ 1 A � �s ð14Þ A ¼ dðp 0�Þ dðp0Þ ¼ 1� p0dC Mðb þ CÞ2 expð �p0 d CþbÞ ð15Þ X ¼ �2 q p0� 1� aþ 1 A � � ð16Þ The expression for the term X is defined by the authors in Eq. (16). The term A is introduced as the derivative of p0⁄ with respect to p0. There is a critical point on the failure curve of cemented clay after which the failure curve approaches the fail- ure line of reconstituted clay-cement mixture (CSL) and by setting A = 0, this critical point can be defined. It should be noted that a is a non-dimensional anisotropic parameter accounting for the coupling of deviatoric and volumetric plastic strain parameters. For an isotropic compression case, where q = 0 and dpe ¼ 0, Eq. (14) will reduce to: dWin ¼ p0�dpv ð17Þ Moreover, on the Critical State Line, where q =Mp0⁄ and dpv ¼ 0, Eq. (14) can be simplified as: dWin ¼ p0�Mdpe ð18Þ It can be noted that when C = 0, and a = 0, Eq. (14) reduces to Eq. (12) of MCC model. Moreover, due to the complexity of the proposed model and based on the semi-empirical method, the choice of X also depends on the simplicity of derivation of flow rules and the plastic potential function. It is also important to choose the energy dissipation equation and the X value in a way that Eqs. (17) and (18) are satisfied. 3.5. Flow rule and plastic potential function One of the important features of the proposed model is the development of the non-associated plastic potential function based on an energy dissipation equation. The non-associated plastic potential function is capable of capturing the plastic deformations more accurately than the associated Cam Clay families. The authors have selected an appropriate expression for the energy equation for cemented clays to capture the effects of cementation and its degradation. In order to derive the L.D. Nguyen et al. / International Journal of Plasticity 56 (2014) 1–18 11 equation for plastic potential function, numerous mathematical procedures have been implemented which can be found in Appendix A. The MCCmodel assumes an associated flow rule which is in the form d p v dpe ¼ M 2�g2 2g � � so that the yield surface is also the plas- tic potential function. However, Bousshine et al. (2001) suggested that, in general, soil materials exhibit non-associated plas- tic flow rule as the strain parameter vectors are not normal to the yield locus. Due to the modified energy equation, the proposed model in this study assumes a non-associated flow rule which takes into account the effect of cementation and the curvature change of the failure envelope. By solving and rearranging the energy equation proposed in Eq. (14) (see Appendix A), the flow rule of Cemented Cam Clay model is derived as follows: dpv dpe ¼ A½M 2 � g�2� 2g�ðaþ 1Þ ð19Þ The partial derivative of the modified stress ratio (g⁄) (with respect to p0 and q) can be presented in the following form: dg� ¼ �q 1� p 0 d C MðbþCÞ2 exp �p0 d Cþb � �( ) p0 þ CM 1þ p0 d Cþb � � exp �p0 d Cþb � �" #2 dp0 þ 1 p0 þ CM 1þ p0 d Cþb � � exp �p0 d Cþb � � dq ð20Þ Combining the proposed flow rule in Eq. (19) with Eq. (20) and taking integration to incorporate the boundary condition ðp0 ¼ p00 when q ¼ 0Þ, the plastic potential function (g) is derived as: g ¼ q2ð1þ 2aÞ þM2p0�2 1� p 0� 0 p0� � � 2aþ1 aþ1ð Þ " # ð21Þ A detailed derivation of the plastic potential function is included in Appendix A. In order to evaluate the effect of a, Fig. 10 displays the shape of the plastic potential function (g) together with variations of a. When a = 0, the plastic potential function coincides with the yield surface and consequently an associated flow rule is obtained. Although, the proposed model con- tains complex mathematical expressions, the developed equations can be readily used by practising geotechnical engineers to calculate deformations and the peak shear strength accurately. 3.6. Elastic deformation When the stress state is within the yield surface, only elastic deformation occurs. Following the conventional MCC model, elastic deformation for soils is formulated following Hooke’s Law widely adopted by many researchers such as Horpibulsuk et al. (2010), Perić and Ayari (2002) and Wood and Graham (1990). With the modified mean effective stress ðp0�Þ, the pro- posed model also assumes an elastic deformation which is also dependent on the effect of cementation: 0 10 20 30 40 50 60 70 80 90 -40 -20 0 20 40 60 80 100 120 140 D ev ia to ri c st re ss (q ) (k Pa ) Mean Effective stress (p') (kPa) α = -0.8 α = -0.6 α = -0.4 α = 0 Critical State Line ( = ) M = 1.8 C = 30 kPa = 34.28 kPa = 0 (Plastic potential function coincides with yield surface) Fig. 10. Plastic potential surface and the effect of increasing a. 12 L.D. Nguyen et al. / International Journal of Plasticity 56 (2014) 1–18 dev ¼ j 1þ e dðp0�Þ p0� � � ð22Þ dee ¼ 2jð1þ mÞ 9ð1� 2mÞð1þ eÞ dq p0� � � ð23Þ where, is elastic volumetric strain increment, dee is elastic deviatoric strain increment, m is Poisson’s ratio, j is the elastic swelling/recompression index and e is the void ratio of the cemented clay. The adoption of this elastic deformation equation with relation to the modified mean effective stress is due to the experimental observations that the elastic deformation in- creases with the bond strength of cementation (Horpibulsuk et al., 2010). 3.7. Virgin yielding behaviour Following the tradition of MCC model, the volumetric hardening law of the proposed model is expressed as: dp0�0 p0�0 ¼ dp 0� p0� þ dg � ðg� þ w�Þ ð24Þ where, w� ¼ d p v dpe ¼ A½M 2�g�2 � 2g�ðaþ1Þ , which is the slope of the current yield locus in p 0–q plane which is also called the flow rule as pre- sented in Eq. (19). When the stress state of the cemented clays is on the yield surface and with dp0�0 > 0, virgin yielding occurs resulting to plastic deformations and it can be calculated adopting virgin isotropic consolidation line similar to MCC model. To predict the plastic deformation of the cemented clay, the following equations are proposed for plastic volumetric strain increment ðdpv Þ and plastic deviatoric strain increment ðd p s Þ, respectively. dpv ¼ ðk � jÞ 1þ e dðp0�Þ p0� þ 2g �ðaþ 1Þdg� 2g�2aþM2 þ g�2 " # ð25Þ dpe ¼ ðk � jÞ 1þ e dðp0�Þ p0� þ 2g �ðaþ 1Þdg� 2g�2aþM2 þ g�2 " # 2g�ðaþ 1Þ AðM2 � g�2Þ ð26Þ where, k is the slope of the normal compression line in e–lnp0⁄ space. When the level of cohesion is diminished C = 0 and a is set to be 0, the proposed plastic deformation is reduced to MCC model which is in the form of: dpv ¼ ðk � jÞ 1þ e dp p þ 2gdg ðM2 þ g2Þ ! ð27Þ dpe ¼ ðk � jÞ 1þ e dp p þ 2gdg ðM2 þ g2Þ ! 2g M2 � g2 ! ð28Þ 3.8. General stress strain relationship Combining Eqs. (22) and (25) yield the following equation to find the total volumetric strain increment (dv): dv ¼ dev þ d p v ¼ 1 1þ e k dðp0�Þ p0� � � þ ðk � jÞ 2g �ðaþ 1Þdg� 2g�2aþM2 þ g�2 !" # ð29Þ . Furthermore, combining Eqs. (23) and (26) gives an equation to determine the total plastic deviatoric strain increment (de), as: de ¼ dee þ d p e ¼ 2jð1þ mÞ 9ð1� 2mÞð1þ eÞ dq p0� � � þ ðk � jÞ 1þ e dðp0�Þ p0� þ 2g �ðaþ 1Þdg� 2g�2aþM2 þ g�2 " # 2g�ðaþ 1Þ AðM2 � g�2Þ ð30Þ When the stress state is increased to sufficiently high pressure, the contribution of cementation is diminished and p0⁄ = p0 (i.e., p0X ¼ 0Þ, and the derivative of p0 ⁄ presented by Eq. (15) becomes 1. Thus, the proposed stress–strain relationship is re- duced to that of MCC model. 4. Estimation of model parameters The proposed model involves nine major parameters. The following steps can be taken to predict the model parameters: L.D. Nguyen et al. / International Journal of Plasticity 56 (2014) 1–18 13 The first five parameters including M, k, j, m and e are deemed to be reasonably independent of the soil structure induced by cementation and consequently can be determined same as the model parameter prediction procedure for MCC model, using a set of conventional isotropic compression tests. However, Yang et al. (2005) emphasised that the selection of Pois- son’s ratio (m) should be below the critical Poisson’ ratio which corresponds to the maximum void ratio of the soil for a solu- tion existence in non-associated model. Moreover, it should be noted that all these model parameters should be estimated for reconstituted clay–cement mixture. However, as reported by Horpibulsuk et al. (2010) and Liu and Carter (2002), these parameters can be reasonably estimated by reconstituted soil properties (excluding cement), particularly when small amount of chemical additives are used. Parameters C and b can be estimated using curve fitting to data obtained by plotting the peak shear stress of cemented clays in p0–q space. In other words, Eq. (9) should be fitted to the peak shear stress data (e.g., Fig. 1) to obtain C and b. Param- eter C indicates the shear strength when p0 = 0, and b is the fitting parameter capturing the cementation degradation with the mean effective stress. Although data fitting is the most accurate method to determine these parameters, C can be estimated by the following equation as suggested by Horpibulsuk et al. (2010) and Liu and Carter (2002): Table 1 Values k j qu (k e m M C (kP b (kP a a Lab b Kam c Hor C ¼ 1 2 qu ð31Þ where, qu is the unconfined compressive strength of the cemented clays. The mean effective stress when virgin yielding occurs ðp0y;iÞ is measured by linking it to the unconfined compressive strength (qu) (Horpibulsuk et al., 2010; Liu and Carter, 2002). It can be estimated by p0y;i ¼ qu Parametera, which is a non-dimensional anisotropic variable, accounting for the coupling of deviatoric and volumetric plastic strain parameters, depends on the selection of the flow rule influencing the plastic potential function (g). The most appropriate a value results in the best fitted stress–strain curves, particularly in the post yielding stages. As explained earlier, assuming a = 0 results in associated flow rule, meaning that the yield and the plastic potential surfaces overlap each other. 5. Performance of cemented Cam Clay model In this section, the performance of the Cemented Cam Clay model is evaluated by comparing the model prediction with available experimental data in literature. The three groups of test data including consolidated-undrained triaxial test results on cemented Aberdeen soil (reported by Sariosseiri (2008)), Singapore marine clay (reported by Kamruzzaman et al. (2009)) and Ariake clay (reported by Horpibulsuk et al. (2004)) are adopted in this study for verification exercise. The amount of ce- ment content is calculated in percentage by a ratio of dry weight of cement and dry weight of clays (Horpibulsuk et al., 2004; Kamruzzaman et al., 2009; Sariosseiri, 2008). The model parameters have been obtained based on the procedure explained in Section 4. The adopted model parameters for the three selected cemented clays are summarised in Table 1. Figs. 11–16 display a comparison amongst the CCC model predictions (the proposed model in this study), SCC model pre- dictions (proposed by Horpibulsuk et al. (2010)) and available experimental data. It is observed that the CCC model predic- tions are in a good agreement with the experimental results, particularly in higher confining pressure ranges. Detailed discussions are presented below. Predicted and measured stress paths up to the peak shear strength for treated Aberdeen soil at 400 and 600 kPa confining pressures are displayed in Fig. 11. The stress path is initially within the yield surface and deformations are only elastic. How- ever, when stress path reaches the yield surface (q = 336.25 kPa), the plastic deformation occurs. At the confining pressure of 600 kPa, it can be seen that the peak strength lies almost on the critical state line of the reconstituted soil as the cementation bonds were destroyed. It should be noted that SCC model is unable to capture this behaviour. Although, there are some dis- parities between the predicted stress–strain relationship at the initial stages of loading and laboratory measurements, the predictions begin to match up with the experimental data when the axial strain exceeds 1% as shown in Fig. 12. of model parameters for cemented clays. Aberdeen soil treated with 5% Portland cementa Singapore clay treated with 10% Portland cementb Ariake clay treated with 6% Portland cementc 0.162 0.73 0.446 0.048 0.067 0.044 Pa) 534.3 300 78 1.97 2.85 4.37 0.25 0.25 0.25 1.4 0.9 1.85 a) 267.15 150 39 a) 84 298 49 �0.6 �0.89 0.15 oratory data obtained from Sariosseiri (2008). ruzzaman et al. (2009). pibulsuk et al. (2004). 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 D ev ia to ri c Sr es s (q ) (k P a) Mean effective stress (p') (kPa) 400 kPa 600 kPa CCC model prediction (current study) SCC model prediction (Horpibulsuk et al., 2010) Aberdeen soil admixed with 5% Cement SCC model failure envelope Strength reduction due to cementation degradation Critical State Line M = 1.4 Data from Sariosseiri, 2008 CCC model failure envelope Aberdeen soil admixed with 5% Cement SCC model failure envelope Strength reduction due to cementation degradation Critical State Line M = 1.4 Data from Sariosseiri, 2008 CCC model failure envelope Fig. 11. Undrained stress paths for Aberdeen Soil treated with 5% cement. 0 100 200 300 400 500 600 700 800 900 0 1 2 3 4 D ev ia to ri c st re ss ( q) ( kP a) Deviatoric strains ( ) (%) 400 Kpa 600 kPa CCC model prediction (current study) SCC model prediction (Horpibulsuk et al., 2010) Aberdeen soil admixed with 5% Cement. Data from Sariosseiri, 2008 Fig. 12. Stress and strain relationship of cemented Aberdeen soil with 5% cement content. 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 D ev ia to ri c st re ss ( q) ( kP a) Mean effective stress (p') (kPa) 500 kPa 1000 kPa CCC model prediction (current study) SCC model prediction (Horpibulsuk et al., 2010) Critical State Line M = 0.9 Strength reduction due to cementation degradation SCC model failure envelope Data from Kamruzzaman et al., 2009 CCC model failure envelope Singapore clay treated with 10% cement Fig. 13. Undrained stress path for Singapore clay treated with 10% cement. 14 L.D. Nguyen et al. / International Journal of Plasticity 56 (2014) 1–18 Kamruzzaman et al. (2009) performed a series of effective undrained triaxial tests on highly plastic Singapore marine clay admixed with 10% cement at different confining pressures ranging from 300 to 1000 kPa. The undrained stress paths and the stress–strain relationship of Singapore clay are simulated in Figs. 13 and 14, respectively. The proposed CCC model predic- tions for the cemented Singapore clay (at confining pressures of 500 and 1000 kPa) are in very good agreement with mea- surements as noticed in Figs. 13 and 14. The stress path at 1000 kPa is observed to be higher than the critical state line indicating that part of cementation is still present in the soil structure at the shearing stage. 0 100 200 300 400 500 600 700 800 900 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 D ev ia to ri c st re ss ( q) ( kP a) Deviatoric strain ( ) (%) 500 kPa 1000 kPa CCC model prediction (current study) SCC model prediction (Horpibulsuk et al., 2010) Singapore soft clay treated with 10% cement. Data from Kamruzzaman et al., 2009 Fig. 14. Stress strain relationship of Singapore clay treated with 10% cement. 0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500 D ev ia to ri c st re ss (k Pa ) Mean effective stress (kPa) 100 kPa 200 kPa 400 kPa CCC model prediction (current study) SCC model prediction (Horpibulsuk et al., 2010) Ariake clay treated with 6% cement SCC model failure envelope CCC model failure envelope Strength reduction due to cementation degradation Critical State Line M = 1.85 Data from Horpibulsuk et al., 2004 Fig. 15. Undrained stress path of Ariake clay treated with 6% cement. L.D. Nguyen et al. / International Journal of Plasticity 56 (2014) 1–18 15 Figs. 15 and 16 display the comparison of CCC and SCC models for the consolidated-undrained triaxial tests data per- formed on Ariake clay admixed with 6% cement content reported by Horpibulsuk et al. (2004). The initial confining pressures of 100, 200, and 400 kPa were adopted for testing. The virgin yielding occurs at 78 kPa (qu = 78 kPa), so the stress state of all samples is outside the yield surface. As observed in Figs. 15 and 16, the results from CCCmodel provide a better prediction on the behaviour of treated Ariake clay than the predictions by SCC model particularly in higher confining pressure ranges. 6. Discussion When the effective confining pressure is on or outside the initial yield surface, the undrained effective stress path of ce- mented clays in p0–q plane initially rises upwards and bends towards to left and approaches the Critical State Line (CSL) of reconstituted cement-clay mixture indicating that plastic deformations occur as shown in the undrained stress paths of Aberdeen Soil (Fig. 11), Singapore clay (Fig. 13) and Ariake clay (Fig. 15). This behaviour of cemented clay is similar to the normally consolidated soils, as observed by Horpibulsuk et al. (2010), Kasama et al. (2000), and Uddin et al. (1997). The undrained stress paths continue to pass the CSL and reach a peak strength state as the effect of cementation is still pres- ent. However, the beneficial contribution of cementation to the peak shear strength (qu) of cemented clay is reduced as the effective confining pressure increases, due to the breaking of cementation bonds during shearing as illustrated in Figs. 11, 13 and 15. For high effective confining stress (p00 ¼ 600 kPa) in case of Aberdeen soil as shown in Fig. 11 with low cement con- tent (5%), the stress path reaches a peak strength state which lies on the CSL as the effect of cementation is completely de- stroyed. In addition, the failure envelope of cemented clay is clearly non-linear and gradually approaches the CSL of reconstituted cement–clay mixture as the effective confining pressure increases. 0 50 100 150 200 250 300 350 400 450 0 1 2 3 4 5 6 D ev ia to ri c st re ss ( q) ( kP a) Deviatoric Strain ( ) (%) 100 kPa 200 kPa 400 kPa SCC model prediction (Horpibulsuk et al., 2010) CCC model prediction (current study) Data from Horpibulsuk et al., 2004 Ariake clay treated with 6% cement ε Fig. 16. Stress and strain relationship of Ariake clay treated with 6% cement. 16 L.D. Nguyen et al. / International Journal of Plasticity 56 (2014) 1–18 The existing constitutive and numerical models do not capture the cementation degradation due to the applied mean effective stresses, resulting in over estimating the peak shear strength of cemented soils and consequently bearing capacity of foundations (i.e., aggressive design). In this study by introducing a nonlinear failure envelope capturing cementation deg- radation induced by the mean effective stress, more realistic behaviour of the cemented soils is presented. By introducing the effect of cementation degradation due to the applied mean effective stress, deformations are captured not only at the shear- ing stage, but also during the consolidation stage where other researchers have chosen to ignore this effect. In this study, the proposed model includes a non-associated plastic potential function to simulate the plastic deformations resulting in a real- istic prediction of the peak shear strength and the associated deformation. Moreover, the plastic potential function is derived from energy dissipation equation which can be extended to simulate different types of soils or loadings. The Structured Cam Clay (SCC) model ignores the reduction in the cementation contribution due to cementation degra- dation by adopting a linear failure envelope while the Cemented Cam Clay (CCC) model proposed in this paper captures this behaviour of cemented clays, thus provides a better agreement with experimental data. 7. Conclusions The effective confining pressure plays a dominant role in the behaviour of cemented clays. Numerous laboratory exper- iments have indicated that the effect of cementation is diminished as the effective confining pressure is increased, due to degradation of cement-soil particle bonding. Various constitutive models were developed to simulate the behaviour of ce- mented clays, however the diminishing effect of cementation, particularly at high effective confining pressures is not cap- tured in these models. In this paper, a predictive constitutive model has been presented to simulate the behaviour of cemented clays referred to as Cemented Cam Clay model (CCC). The model failure envelope has been proposed in a way to merge with the Critical State Line of reconstituted clay-cement mixture, showing the diminishing effect of cementation due to degradation of cementa- tion bonds when the confining pressure increases. The special characteristic of the proposed model includes a modifiedmean effective stress capturing cementation degradation. The main concepts and the formulations of CCC model, including a non- associated plastic potential function and elasto-plastic stress–strain relationship, have been presented within the framework of the critical state concept and inspired by Modified Cam Clay (MCC) model. It can be noted that when the effect of cemen- tation is zero, the Cemented Cam Clay model is reduced to MCC model. The performance of the proposed model has been evaluated by comparing experimental data available in literature with model predictions. Moreover, the predictions from Structured Cam Clay (SCC) model were also included as a comparison tool. The experimental data obtained from undrained triaxial tests performed on Aberdeen clay (admixed with 5% cement), Singapore clay (with 10% cement) and Ariake clay (with 6% cement) have been used in the validation of the proposed model. The results displayed a satisfactory performance of the proposed model and provided a good agreement with experimental data. The main features of the behaviour of cemented clay in undrained stress paths and stress–strain relationship, including the reduction in the cementation contribution in the peak shear strength and cementation degradation have been well cap- tured in this model. Appendix A. Derivation of plastic potential function Expanding and simplifying Eq. (14) results in: L.D. Nguyen et al. / International Journal of Plasticity 56 (2014) 1–18 17 ðqdpe Þ 2 ¼ p0�2ðMdpe Þ 2 � 2p0�qdpvd p e aþ 1 A � � ðA-1Þ Dividing both sides by dpe : Aðq2 � p0�2M2Þdpe ¼ �2p0�qðaþ 1Þd p v ðA-2Þ Rearranging Eq. (A-2) to obtain the flow rule: dpv dpe ¼ Aðq 2 � p0 � 2M2Þ �2p0�qðaþ 1Þ ðA-3Þ Substitutes the modified stress ratio ðg� ¼ qp0�Þ into Eq. (A-3), the flow rule is taking the form of: dpv dpe ¼ A½M 2 � g�2� 2g�ðaþ 1Þ ðA-4Þ Using the normality condition as follows: dpv dpe ¼ � dq dp0 ðA-5Þ Adopting Eq. (15) to obtain A in terms of p0 and then combining Eqs. (A-4) and (A-5) results in: � dq dp0 ¼ 1� pd 0C MðbþCÞ2 exp �p0 d Cþb � �( ) M2 � q2 p0þCM 1þ p0 d Cþb � � exp �pd 0 Cþbð Þ � �2 2 64 3 75 2q aþ1ð Þ p0þCM 1þ p0 d Cþb � � exp �p0 d Cþb � � ! : ðA-6Þ It can be noted that Eq. (A-6) is expressed in terms of p0. Substituting Eqs. (6) and (7) in Eq. (8) results in: g� ¼ q p0 þ CM ð1þ p0 d CþbÞexp �p0 d Cþb � � ðA-7Þ The equation for the partial derivatives of Eq. (A-7) with respect to q and p0 is expressed as follows: dg� ¼ @g � @p0 dp0 þ @g � @q dq ðA-8Þ Evaluating the derivative of Eq. (A-7) with respect top’ results in: @g� @p0 ¼ �q 1� p 0 d C MðbþCÞ2 exp �p0 d Cþb � �( ) p0 þ CM ð1þ p0 d CþbÞexp �p0 d Cþb � �" #2 ðA-9Þ Evaluating the derivative of Eq. (A-7) with respect to q results in: @g� @q ¼ 1 p0 þ CM ð1þ p0 d CþbÞexp �p0 d Cþb � �" # ðA-10Þ Substituting Eqs. (A-9) and (A-10) in Eq. (A-8) leads to: dq dp0 ¼ p0 þ C M ð1þ p 0 d C þ bÞexp �p0 d Cþb � �" # dg� dp0 þ g� 1� p 0 dC Mðb þ CÞ2 exp �p0 d Cþb � �( ) ðA-11Þ Substituting the flow rule presented in Eq. (A-6) into Eq. (A-11) and then integrating the equation using the boundary conditions (p0 ¼ p00 when q ¼ 0Þ results in the plastic potential function as follows: g ¼ q2ð1þ 2aÞ þM2p0�2 1� p 0� 0 p0� � �ð2aþ1aþ1 Þ" # ðA-12Þ 18 L.D. Nguyen et al. / International Journal of Plasticity 56 (2014) 1–18 References Bousshine, L., Chaaba, A., De Saxcé, G., 2001. Softening in stress–strain curve for Drucker–Prager non-associated plasticity. Int. J. Plast. 17, 21–46. Chew, S.H., Kamruzzaman, A.H., Lee, F.H., 2004. Physicochemical and engineering behavior of cement treated clays. J. Geotech. Geoenviron. Eng. 130, 696– 706. Diamond, S., Kinter, E.B., 1965. Mechanisms of soil-lime stabilization. Highway Res. Rec., 83–102. Horpibulsuk, S., Miura, N., Bergado, D.T., 2004. Undrained shear behavior of cement admixed clay at high water content. J. Geotech. Geoenviron. Eng. 130, 1096–1105. Horpibulsuk, S., Liu, M.D., Liyanapathirana, D.S., Suebsuk, J., 2010. Behaviour of cemented clay simulated via the theoretical framework of the structured cam clay model. Comput. Geotech. 37, 1–9. Huang, J., Airey, D., 1998. Properties of artificially cemented carbonate sand. J. Geotech. Geoenviron. Eng. 124, 492–499. Kamruzzaman, A.H., Chew, S.H., Lee, F.H., 2006. Microstructure of cement-treated Singapore marine clay. Proc. ICE Ground Improv. 10, 113–123. Kamruzzaman, A.H., Chew, S.H., Lee, F.H., 2009. Structuration and destructuration behavior of cement-treated Singapore marine clay. J. Geotech. Geoenviron. Eng. 135, 573–589. Kasama, K., Ochiai, H., Yasufuku, N., 2000. On the stress–strain behaviour of lightly cemented clay based on an extended critical state concept. Soils Found. 40, 37–47. Kolovos, K.G., Asteris, P.G., Cotsovos, D.M., Badogiannis, E., Tsivilis, S., 2013. Mechanical properties of soilcrete mixtures modified with metakaolin. Constr. Build. Mater. 47, 1026–1036. Lade, P.V., Overton, D.D., 1989. Cementation effects in frictional materials. J. Geotech. Eng. 115, 1373–1387. Liu, M.D., Carter, J.P., 2002. A structured cam clay model. Can. Geotech. J. 39, 1313–1332. Lo, S., Wardani, S., 1999. Deformation behaviour of cement-flyash stabilised silt, In: Proceedings of the 8th Australia New Zealand Conference on Geomechanics: Consolidating Knowledge. Australian Geomechanics Society, p. 761. Locat, J., Bérubé, M.-A., Choquette, M., 1990. Laboratory investigations on the lime stabilization of sensitive clays: shear strength development. Can. Geotech. J. 27, 294–304. Lorenzo, G.A., Bergado, D.T., 2006. Fundamental characteristics of cement-admixed clay in deep mixing. J. Mater. Civ. Eng. 18, 161–174. Moses, G.G., Rao, S.N., Rao, P.N., 2003. Undrained strength behaviour of a cemented marine clay under monotonic and cyclic loading. Ocean Eng. 30, 1765– 1789. Panda, A.P., Rao, S.N., 1998. Undrained strength characteristics of an artificially cemented marine clay. Mar. Georesour. Geotechnol. 16, 335–353. Perić, D., Ayari, M.A., 2002. On the analytical solutions for the three-invariant Cam Clay model. Int. J. Plast. 18, 1061–1082. Porbaha, A., 1998. State of the art in deep mixing technology. Part I: Basic concepts and overview. Ground Improv. 2, 81–92. Porbaha, A., Shibuya, S., Kishida, T., 2000. State of the art in deep mixing technology. Part III: Geomaterial characterization. Proc. ICE Ground Improv. 4, 91– 110. Roscoe, K.H., Burland, J.B., 1968. On the generalised stress–strain behaviour of ‘‘wet’’ clay. Eng. Plast., 535–609. Rotta, G., Consoli, N., Prietto, P., Coop, M., Graham, J., 2003. Isotropic yielding in an artificially cemented soil cured under stress. Geotechnique 53, 493–501. Sariosseiri, F., 2008. Critical State Framework for Interpretation of Geotechnical Properties of Cement Treated Soils. Washington State University, Pullman. Sariosseiri, F., Muhunthan, B., 2009. Effect of cement treatment on geotechnical properties of some Washington state soils. Eng. Geol. 104, 119–125. Shen, W.Q., Shao, J.F., Kondo, D., Gatmiri, B., 2012. A micro–macro model for clayey rocks with a plastic compressible porous matrix. Int. J. Plast. 36, 64–85. Suebsuk, J., Horpibulsuk, S., Liu, M.D., 2010. Modified structured cam clay: a generalised critical state model for destructured, naturally structured and artificially structured clays. Comput. Geotech. 37, 956–968. Tan, T.S., Goh, T.L., Yong, K.Y., 2002. Properties of Singapore marine clays improved by cement mixing. ASTM Geotech. Test. J. 25, 422–433. Uddin, K., Balasubramaniam, A.S., Bergado, D.T., 1997. Engineering behavior of cement-treated Bangkok soft clay. Geotech. Eng. 28, 89–119. Vatsala, A., Nova, R., Murthy, B.S., 2001. Elastoplastic model for cemented soils. J. Geotech. Geoenviron. Eng. 127, 679–687. Wood, D., Graham, J., 1990. Anisotropic elasticity and yielding of a natural plastic clay. Int. J. Plast. 6, 377–388. Yang, Y., Yu, H.S., Muraleetharan, K.K., 2005. Solution existence conditions for elastoplastic constitutive models of granular materials. Int. J. Plast. 21, 2406– 2425. Yang, Y., Wang, G., Xie, S., Tu, X., Huang, X., 2013. Effect of mechanical property of cemented soil under the different pH value. Appl. Clay Sci. 79, 19–24. Yasufuku, N., Ochiai, H., Kasama, K., 1997. The dissipated energy equation of lightly cemented clay in relation to the critical state model, In: Procceedings of the 9th International Conference on Computer Methods & Advances in Geomechanics, pp. 917–922. Yin, J.-H., 2001. Stress–strain–strength characteristics of soft Hong Kong marine deposits without or with cement treatment. Lowland Technol. Int. 3, 1–13. Yin, Z.-Y., Chang, C.S., Hicher, P.-Y., Karstunen, M., 2009. Micromechanical analysis of kinematic hardening in natural clay. Int. J. Plast. 25, 1413–1435. Yuanming, L., Yugui, Y., Xiaoxiao, C., Shuangyang, L., 2010. Strength criterion and elastoplastic constitutive model of frozen silt in generalized plastic mechanics. Int. J. Plast. 26, 1461–1484. Zhu, Q.Z., Shao, J.F., Mainguy, M., 2010. A micromechanics-based elastoplastic damage model for granular materials at low confining pressure. Int. J. Plast. 26, 586–602. http://refhub.elsevier.com/S0749-6419(14)00016-3/h0005 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0010 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0010 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0015 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0025 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0025 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0020 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0020 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0030 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0035 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0040 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0040 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0045 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0045 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0050 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0050 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0055 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0060 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0070 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0070 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0075 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0080 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0080 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0085 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0090 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0095 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0100 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0100 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0105 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0110 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0115 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0120 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0125 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0130 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0130 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0135 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0140 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0145 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0150 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0160 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0160 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0155 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0170 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0175 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0180 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0180 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0185 http://refhub.elsevier.com/S0749-6419(14)00016-3/h0185 A constitutive model for cemented clays capturing cementation degradation 1 Introduction 2 Factors affecting the strength of cemented clay 3 Development of cemented cam clay model 3.1 Modified mean effective stress 3.2 Proposed failure envelope 3.3 Yield function 3.4 The energy dissipation equation 3.5 Flow rule and plastic potential function 3.6 Elastic deformation 3.7 Virgin yielding behaviour 3.8 General stress strain relationship 4 Estimation of model parameters 5 Performance of cemented Cam Clay model 6 Discussion 7 Conclusions Appendix A Derivation of plastic potential function References