Analytical Solutions for a Three-Invariant Original Cam Clay Model   Dunja Perić Assistant Professor, Department of Civil Engineering, Kansas State University, Manhattan, Kansas, U.S.A. peric@ksu.edu Bhavik R. Shah Graduate Student, Department of Civil Engineering, Kansas State University, Manhattan, Kansas, U.S.A. and Mbakisya A. Onyango Graduate Student, Department of Civil Engineering, Kansas State University, Manhattan, Kansas, U.S.A.

ABSTRACT

Analytical solutions are derived for a three-invariant original Cam clay model subjected to proportional and circular drained and undrained loading histories. The solutions contain analytical expressions for a generalized shear strain, a volumetric strain and/or an excess pore pressure. Only straight proportional total stress paths are considered, while in the case of circular paths the maximum change in Lode’s angle is limited either by the failure or by the plastic isotropy. A comparison among analytical solutions of original and modified Cam clay models and experimental data confirms that the increased accuracy of the later model is due to its increased physical and mathematical complexity.

Keywords: analytical solutions; circular and proportional paths; three-invariant Cam clay models.

INTRODUCTION

Original and modified Cam clay models were developed by Roscoe et al. (1963) and Roscoe and Burland (1968) respectively. Advantages of these models lie in their apparent simplicity and their capability to model the stress-strain behavior of soft isotropically consolidated clays realistically. Cam clay models incorporate compression and shear behaviors simultaneously. Their simplicity is also reflected in a small number of material parameters, which all have a readily understood physical meaning. Gens and Potts (1987) considered the modified Cam clay model to be the most widely used effective stress plasticity model in the non-linear finite element analysis of practice related problems in geotechnical engineering.

While both Cam clay models belong to the family of cap models, the basic difference between them stems from the assumption about the mode of energy dissipation. The original Cam clay model assumes that energy is entirely dissipated in friction, while according to the modified Cam clay only a portion of the total dissipated energy is dissipated in friction. Two assumptions about the energy dissipation led to two different shapes of yield surfaces, a bullet for the original Cam-clay and an ellipse for the modified Cam clay. Consequently, the former generates a shear strain in isotropic compression. A need to eliminate this shear strain led to development of the modified Cam clay model. It is noted that the models developed by Roscoe et al. (1963) and Roscoe and Burland (1968) were formulated within the two-invariant plasticity framework. Alawaji et al. (1992) presented a three-invariant formulation of the Cam clay model, while Borja et al. (2003) addressed a numerical integration of the three-invariant elastic-plastic models in general.

Perić and Ayari (2002b), and Perić (2006) presented analytical solutions for the three-invariant modified Cam clay model for undrained and drained loading histories respectively. Herein, the analytical solutions are derived for the three-invariant original Cam clay model subjected to two classes of drained and undrained stress histories: proportional and circular. The availability of analytical solutions is useful in confirming that: 1) the simpler underlying physics of the original Cam clay model in comparison to the modified Cam clay model leads to a simpler mathematical form of the analytical solutions, and 2) that the increased mathematical complexity of the modified Cam clay model leads to an increase in the accuracy over the original Cam clay model. The solutions are also useful for an assessment of accuracy of numerical integration schemes and for development of verification and validation methods for non-linear problems thus contributing to further development of simulation based engineering.

PRELIMINARIES

The primary objective of this work is to find analytical solutions for a three-invariant original Cam clay model along proportional and circular stress paths. In addition, these solutions are subsequently compared with corresponding solutions for the modified Cam clay model and with experimental data. The models adopted herein are entirely isotropic, thus placing no restriction on the orientation of principal stresses. Elasticity is not energy conserving due to the assumption that both bulk and shear moduli are mean stress dependent while Poisson’s ratio is constant. The solutions presented herein are theoretically valid for the entire range of stress histories. However, difficulties arise in assessing a performance of the model in the overconsolidated regime because of an increased susceptibility to strain localization, which affects the interpretation of experimental results, and numerical simulations. The later require a properly regularized finite element model to capture the post-peak response successfully. Thus, the analytical predictions presented herein are limited to the normally consolidated and lightly overconsolidated regimes.

A bullet shaped yield surface for the three-invariant model is obtained by a simple extension of Roscoe et al. (1963) as

(1)

and modified stress ratio is defined as

(2)

p', q and q are invariants of the effective stress and stress deviator tensors, which are given by

(3)

(4)

(5)

Total and effective stress tensors are denoted by s_ij and s'_ij, respectively, while the stress deviator tensor is denoted by s_ij. Normal components of stress and strain tensors are negative in compression, while mean effective stress and pore pressure are positive in compression (equation 3). In equation (1) the size of a current yield surface is denoted by p'_c, which represents the past maximum mean effective stress in isotropic compression. The slope of the critical state line in the q, p'-space is described by the parameter M in the case of conventional triaxial compression (). Herein the function g(q) proposed by Willam and Warnke (1974) is adopted for description of dependence of plastic yielding on Lode’s angle. The function is given by

(6)

where,

(7)

(8)

and is the material parameter named ellipticity. It follows from equation (6) that g(5p/3) = 1 and . Thus, Lode’s angles corresponding to conventional triaxial compression (CTC) and conventional triaxial extension (CTE:) are q = 5p/3 and 2p respectively.

Analytical solutions are derived next starting from the consistency condition, which can be expressed as

(9)

where superimposed dots denote the time rates of the corresponding variables. The consistency condition is further rearranged by performing partial differentiation of equation (1) and substituting the obtained derivatives into equation (9). The following is obtained:

(10)

VOLUMETRIC RESPONSE

A volumetric plastic strain rate is linked to the rate of change of the size of the yield surface by the volumetric hardening rule as follows:

(11)

Substituting equation (10) into equation (11) gives the following:

(12)

The general solution for a plastic volumetric strain is obtained by integrating equation (12), thus resulting in

(13)

where the subscripts i denotes initial values. A volumetric elastic strain rate depends on the slope of the swelling line (k) and it is given in accordance with the theory of elasticity as

(14)

An elastic volumetric strain is obtained by integration of equation (14) as

(15)

According to the additive strain decomposition, which is valid for the small strain theory, combining equations (13) and (15) gives the overall rate of volumetric strain as:

(16)

In the case of an undrained loading the condition of zero net volume change is directly reflected in the rate of volumetric strain being equal to zero. Thus, by combining equations (12) and (14) the following is obtained:

(17)

Consequently the effective stress path is obtained by integrating equation (17) as

(18)

where l and k are the slopes of isotropic compression and swelling lines respectively, and L = (l - k )/l represents a ratio of the plastic and overall compressibilities. It is noted that both solutions, for the volumetric strain (equation 16), and for an effective stress path (18) are valid for any loading history.

DEVIATORIC RESPONSE: PROPORTIONAL PATHS

A generalized shear strain e_q is the strain that is energy conjugate to deviatoric stress (q) in the sense that the total incremental work rate can be decomposed into volumetric and deviatoric parts as follows:

(19)

Based on the theory of elasticity a deviatoric elastic strain rate is given by

(20)

whereby the rate of a deviatoric stress was obtained from equation (2). The relationship between shear and bulk elastic moduli is given by the theory of elasticity as

(21)

and according to equation (14) a bulk modulus is mean stress dependent, thus resulting in

(22)

Substituting equations (21) and (22) into equation (20) gives the following:

(23)

Next, a class of straight proportional effective stress paths for drained loading is described by the following equation

(24)

and equation (24) implies that

(25)

After substituting equation (25) into equation (23) the later is integrated to give the following solution for the elastic deviatoric strain:

(26)

A plastic deviatoric strain rate can be obtained from the flow rule as

(27)

In order to find the partial derivates required by equation (27) the differentiation of equation (1) is performed in accordance with the following equation

(28)

which results in the following rearrangement of equation (27)

(29)

It is important to note that equation (29) gives a deviatoric plastic strain rate for any loading history. In the special case of proportional paths (g = g_i = constant), which were described by equations (24) and (25), equation (29) further simplifies into the following form:

(30)

Finally, plastic deviatoric strain is obtained by integration of equation (30) as

(31)

where C_i = (l - k) g_i. The rate of an overall deviatoric strain is obtained by combining equations (26) and (31) as

(32)

Next, a deviatoric response is obtained for the case of undrained loading. First, equation (17) is rearranged to give

(33)

Equation (33) is substituted into equation (23), thus resulting in the following

(34)

and the solution for a deviatoric elastic strain during an undrained proportional loading is obtained by integrating equation (34) as

(35)

A plastic deviatoric strain rate is obtained by substituting equation (33) into equation (29) and setting
. After further rearranging the following is obtained

(36)

After integration of equation (36) the following analytical expression is obtained

(37)

Finally, by combining equations (35) and (37) the analytical solution for deviatoric strain in the case of a proportional undrained loading is obtained as:

(38)

and this completes solutions for proportional stress paths.

DEVIATORIC RESPONSE: CIRCULAR PATHS

A circular loading may be produced for example in a true triaxial apparatus by subjecting a soil sample, which is initially in the state of CTC (q = 5p/3), to a stress change such that Lode’s angle increases towards CTE (2p). During this loading principal stresses are changed in such a way that deviatoric and total mean stresses remain constant. A circular loading may be drained, thus implying that the mean effective stress remains constant, or undrained. While a volumetric strain occurs in a drained circular loading, in undrained loading the excess pore pressure is generated due to the incompressibility constraint. Consequently, in normally consolidated and lightly overconsolidated soils an undrained circular loading will produce a decrease in the initial mean effective stress, thus brining such soils to failure faster than in the case of a drained circular loading. This type of loading is likely to be generated and combined with other loadings during an earthquake or in a plane strain state whereby the later is common in geotechnical prototype situations. Thus, it is of interest to study the response of the original Cam clay model subjected to circular stress paths.

Mathematically a circular loading is defined by the following:

(39)

First, analytical solutions are derived for a drained circular loading, for which mean effective stress remains constant ( p' = p'_i ). Thus, the following holds:

(40)

It follows from equation (15) that no elastic volumetric strain is generated during drained circular loading. Thus, the entire volumetric strain is of plastic (irrecoverable) nature. It is obtained from equation (13) as:

(41)

It is noted that samples subjected to circular stress paths are always loaded from non-zero deviatoric stresses. In the case of cap models this implies that that the samples are overconsolidated. Thus, in calculating the initial specific volume the amount of unloading from the past maximum effective stress must be taken into account as follows:

(42)

and the size of the initial yield surface is described by p'_ci, while the initial mean effective stress is denoted by p'_i. Thus, n_p is the overconsolidation ratio. Since the initial stress state is located on the initial yield surface the initial modified stress ratio can be obtained from equation (1) as

(43)

Similarly to volumetric strain, the entire deviatoric strain rate is of a plastic nature due to a constant deviatoric stress as shown by equation (20). Furthermore, equation (29) is rearranged to reflect a constant mean effective stress and constant stress ratio ( h ). Thus, it takes the following form

(44)

After integration of equation (44) the following solution for a deviatoric strain is obtained

(45)

Next, analytical solutions are found for undrained circular loading, for which a mean effective stress is not constant. Thus, the following holds:

(46)

Due to the incompressibility constraint imposed by an undrained loading the overall volumetric strain is equal to zero. The excess pore pressure can be calculated from the effective stress principle as

(47)

whereby the later expression was obtained from equation (18).

A deviatoric elastic strain rate is again equal to zero due to a constant deviatoric stress. Substituting the expression for from equation (33) into equation (29) and further rearranging gives the plastic deviatoric strain rate as

(48)

Integration of equation (48) gives the solution for a deviatoric strain during a circular undrained loading as

(49)

where e is the natural logarithm base, and Ei is the exponential integral function, which can be evaluated by using a software for symbolic mathematics such as for example Maple or Mathematica. Constant is defined as:

(50)

and this completes all analytical solutions.

APPLICATIONS AND VALIDATIONS

In this section the response curves obtained based on the solutions derived herein are compared with the experimental data from conventional triaxial compression tests on remolded isotropically consolidated Weald clay samples. The experimental data were provided by Henkel (1960) and Wood (1980). In addition, the response curves are also compared with predictions of the modified Cam clay model. The set of model parameters, whose values are listed in Table 1, is used to obtain predictions of both Cam clay models, original and modified. These parameters were taken from Carter (1982) with the exception of Poisson’s ratio and ellipticity, which were determined by Perić and Ayari (2002b). Parameter N is equal to a value of the specific volume at the unit mean effective stress in the state of isotropic compression. Experimental data for circular stress paths are very scarce in the literature. To the best knowledge of the authors no experimental data on circular paths are available for Weald clay.

Table 1. Weald clay parameters for Cam clay models

Parameter	Value
l	0.088
k	0.031
n	0.410
M	0.882
N	2.101
	0.772

Fig. 1 through Fig. 4 depict predictions of the Cam clay models along proportional stress paths. Prior to the application of a deviator stress the samples were isotropically consolidated to the mean effective stress of 207 kPa. Fig. 1 and Fig. 2 depict predictions for the conventional drained triaxial compression. Fig. 3 and Fig. 4 depict predictions for the conventional undrained triaxial compression. The prediction of the original Cam clay model shown in Fig. 1 was obtained from equation (32), and the prediction shown in Fig. 2 was obtained from equation (16). All predictions of the modified Cam clay model shown in this paper were obtained from the solutions presented in previous publications (Perić and Ayari, 2002b, and Perić, 2006). The prediction of the original Cam clay model depicted in Fig. 3 was obtained from equation (32), while the excess pore pressure shown in Fig. 4 was obtained based on the effective stress path given by equation (18).

Fig. 1 through Fig. 4 clearly show that the modified model is superior to the original model. Moreover, the performance of the original Cam clay model is better in drained loading (Fig. 1 and Fig. 2) than in undrained loading (Fig. 3 and Fig. 4). In addition to an under-predicted initial stiffness the original Cam clay model over predicts the volumetric strain and the pore pressure. A larger discrepancy between experimental results and predictions occurs in undrained loading due to the incompressibility constraint. Based on Fig. 4 it can be concluded that the modified model predicts the effective stress path in undrained loading better than the original model thus resulting in under prediction of the deviator stress at failure and over prediction of the pore pressure at failure by the later model.

Figure 1. Deviatoric stress versus deviatoric strain for CIDC-207 on Weald clay

Figure 2. Volumetric strain versus deviatoric strain for CIDC-207 on Weald clay

Figure 3. Deviatoric stress versus deviatoric strain for CIUC-207 on Weald clay

Figure 4. Excess pore pressure versus deviatoric strain for CIUC-207 on Weald clay

Fig. 5 through Fig. 9 show predictions for the response of Weald clay samples subjected to circular stress histories. Since there are no experimental data available for this loading scenario only a direct comparison between the predictions of the two models is possible. Prior to loading along circular paths all samples are anisotropically consolidated having the stress states located at the initial yield surface. Fig. 5 depicts a change in Lode’s angle q measured from initial value of 5p/3 (CTC) versus deviatoric strain for drained circular loadings corresponding to different overconsolidation ratios. Fig. 6 shows a volumetric strain versus deviatoric strain response. The predictions depicted in Fig. 5 were obtained from equation (45), while the predictions shown in Fig. 6 were obtained from equation (41).

Fig. 7 shows a change in Lode’s angle versus deviatoric strain for the undrained circular loading, while Fig. 8 shows the generated excess pore pressure. The response curve in Fig. 7 was obtained from equation (48). The excess pore pressure was obtained by subtracting the effective stress obtained from equation (18) from the initial effective stress.

Qualitatively a unique response trend is observed in drained and undrained loadings albeit with some quantitative differences. Namely, the larger the initial overconsolidation ratio the smaller the change in Lode’s angle is required to fail the sample. This is so because the samples with larger initial overconsolidation ratios are initially closer to the failure state. The difference between the drained and undrained responses is in that the undrained condition facilitates the failure through a generation of the positive excess pore pressure. This is due to a coupling between the mean effective stress and Lode’s angle in three-invariant models (Perić and Ayari, 2002a). Thus, the sample, which does not fail in a drained test, might fail in an undrained test. For example Fig. 5 indicates that among all predictions shown for the original model only the one with overconsolidation ratio of 2.3 fails in drained loading. However, Fig. 7 shows that among all samples shown those with overconsolidation ratios 1.9, 2.1 and 2.3 fail because undrained loading facilitates the failure.

Both models, the original and modified produce a family of response curves over the range of overconsolidation ratios. Within this family there is a gradual transition from non-failing towards the failing samples that occurs with an increase in overconsolidation ratio. The difference between the threshold values of overconsolidation ratios beyond which failure occurs as predicted by original and modified models is a consequence of the underlying assumption about the energy dissipation. The assumption leads to different shapes of yield surfaces, thus ultimately producing a quantitative difference between the thresholds values of overconsolidation ratios required for failure. For example, while the original model predicts that samples with overconsolidation ratios 1.7 and 1.9 do not fail, the modified model predicts that such samples fail in case of drained loading (Fig. 5).

Figure 5. Lode’s angle change versus deviatoric strain for drained circular paths

Figure 6. Volumetric strain versus deviatoric strain for drained circular paths

Figure 7. Lode’s angle change versus deviatoric strain for undrained circular paths

Figure 8. Excess pore pressure versus deviatoric strain for undrained circular paths

Fig. 9 summarizes failure conditions observed along circular stress paths. It shows the change in Lodes’ angles required to bring samples to failure. The plots for modified Cam clay model are taken from previous publications (Perić and Ayari, 2002b; Perić 2006) while the response of the original model is obtained based on the solutions presented herein. These response curves are obtained by simply calculating the value of function g(q) at failure   () and limiting its maximum value to (). Thus, for drained loading the following holds

(51)

Consequently, the failure, which corresponds to the critical state, can be reached only by samples that have the initial overconsolidation ratios larger than 2.16. For these samples the value of function g(q) at failure can be calculated from equation (51), and corresponding value of Lode’s angle can be determined from equation (6). Fig. 9 shows that with an increasing initial overconsolidation ratio the change in Lode’s angle required to bring the sample to failure is decreasing.

For undrained circular loading equation (51) is modified to account for the generation of an excess pore pressure as follows:

(52)

And equations (51) and (52) are plotted in Fig. 9 along with the predictions of the modified model. Both pairs of response curves indicate that the change in Lode’s angle required for failure is larger in drained than in undrained loading.

Figure 9. Lode’s angle at failure versus overconsolidation ratio for circular paths

CONCLUSIONS

The analytical solutions have been found for the three-invariant original Cam clay model for the cases of drained and undrained loadings. While the solutions for the volumetric strain and the excess pore pressure are valid for any loading history, the solutions for deviatoric strain have been derived for straight proportional total stress paths and for circular stress paths. Although both the original and modified Cam clay models require the same number of parameters the modified Cam clay model is more accurate in capturing the experimentally observed response along proportional stress paths, especially in an undrained loading. Thus, the gain in the accuracy can be traced back to the change in the basic underlying assumption about the energy dissipation. The change is in turn is responsible for the change in the shape of the yield surface, from the bullet to the ellipse. Now that all the analytical solutions are readily available, their mathematical form clearly indicates that slightly more mathematical complexity brings a marked improvement in the accuracy especially in regard to the initial stiffness and the undrained response. For example, while deviatoric strain is dominantly the natural logarithm function of the modified stress ratio in the case of the original model an additional inverse tangent term appears in the case of the modified model in the solutions for the proportional paths. In the case of circular undrained paths, the deviatoric strain is the exponential integral function in the case of the original model and the hyper-geometric function of the modified stress ratio in the case of the modified model. The absence of the experimental data along circular paths hampers further evaluation of the models. In addition, the solutions are also useful for validation of numerical integration schemes.

REFERENCES

1. Alawaji, H., K. Runesson, and S. Sture (1992) “Implicit integration in soil plasticity under mixed control for drained and undrained response,” Int. J. Numer. Anal. Meth., Vol. 16, pp 737-756.

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3. Carter, J. P. (1982) “Prediction of the non-homogenous behaviour of clay in the triaxial test,” Geotéotechnique, Vol. 32, No. 1, pp 55-58.

4. Gens, A., and D. M. Potts (1987) “Critical State Models in Computational Geomechanics,” Eng. Comput., Vol. 5, pp 178-197.

5. Henkel, D. J. (1960) “The shear strength of saturated remoulded clays,” In Proc. ASCE Research Conf. Shear Strength of Cohesive Soils, Boulder, CO, pp 533-554.

6. Perić, D. (2006) “Analytical solutions for a three-invariant Cam Clay model subjected to drained loading histories,” Int. J. for Num. and Anal. Methods in Geomechanics, Vol. 30, pp 363-387.

7. Perić, D., M. A. Ayari (2002a) “Influence of Lode’s angle on the pore pressure generation in soils,” Int. J. Plasticity, Vol. 18, pp 1039-1059.

8. Perić, D., M. A. Ayari (2002b) “On the analytical solutions for the three-invariant Cam clay model,” Int. J. Plasticity, Vol. 18, pp 1061-1082.

9. Roscoe, K. H., A. N. Schofield, and A. Thurairajah (1963) “Yielding of clays in states wetter than critical,” Géotechnique, Vol. 13, pp 211-240.

10. Roscoe, K. H, J. B. Burland (1968) “On the generalized stress-strain behavior of ‘wet clay’,” In: Heyman and Leckie (Eds.) Engineering Plasticity. Cambridge University Press, pp 535-609.

11. Willam, K. J., E. P. Warnke (1974) “Constitutive model for triaxial behaviour of concrete” In: Colloquium on Concrete Structures Subjected to Triaxial Stresses, ISMES, Bergamo, IABSE Report, Vol. 19.

12. Wood, D.M. (1990). Soil Behaviour and Critical State Soil Mechanics. Cambridge University Press 1990.

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