Abstract
Barodesy is a constitutive model based on proportional paths and the asymptotic behaviour of soil. It was originally developed for sand in 2009 by Kolymbas, and a version for clay was introduced in 2012. A shortcoming of former barodetic models was that tensile stresses can occur for certain dilative deformations. In this article, an improved version of barodesy for clay and a simplified calibration procedure are proposed. Basic features are shown, and simulations of element tests are compared with experimental data of several clay types.
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1 Introduction
The constitutive model for soil called barodesy, proposed by Kolymbas [9–12], is based on the asymptotic behaviour of granulates expressed by the two rules proposed by Goldscheider [6], which have been experimentally confirmed for sand and clay [3, 6, 26, 27]: (1) starting at the stress-free state, \({\mathbf { T}}=\mathbf {0}\), proportional strain pathsFootnote 1 lead to proportional stress paths (see footnote 1); and (2) starting at \({\mathbf { T}}\ne \mathbf {0}\), proportional strain paths lead asymptotically to the corresponding proportional stress paths starting at \({\mathbf { T}}=\mathbf {0}\). This means that proportional stress paths function as attractors.
Barodesy exhibits similarities to hypoplasticity and was introduced by Kolymbas [9] for sand in 2009. In 2012, Medicus et al. [20] modified the sand version [10] and introduced barodesy for clay. A major component of barodesy is the so-called \({\mathbf { R}}\)-function, which links proportional strain paths to proportional stress paths and thus acts as a stress-dilatancy relation. Former versions of \({\mathbf { R}}\) in barodesy [4, 9–12, 20] allow proportional stress paths to reach the tensile area. Experiments by Bergholz [1] of saturated reconstituted clay show that, also for highly overconsolidated clay, the stress ratio q/p (in triaxial compression) does not exceed three (i.e., the stress paths stay in the compression regime). The \({\mathbf { R}}\)-function according to Medicus et al. [21] explicitly prohibits tensile stresses and is chosen as one of the equations for the improved version of barodesy for clay. However, an article [21] reviews existing experimental evidence on stress-dilatancy relations and discusses it in the framework of barodesy, but does not provide a constitutive model.
The main differences of barodesy for clay [20] and the version presented here are shown in this article. The tensor \({\mathbf { R}}\) and the scalar quantities f and g have been changed, and the calibration procedure is simplified as compared to [20]. The calibration procedure is explained, and simulations of element tests are compared with experimental data.
As in other constitutive models for clay, e.g., the hypoplastic models by Mašín[15, 18] and the Modified Cam Clay model [23], barodesy does not take argotropy (i.e., rate-dependence, relaxation, creep) into account.
2 Notation
We use the symbolic notation for Cauchy stress \({\mathbf { T}}\) and stretching \({\mathbf { D}}\). In some cases, the more familiar symbol \(\sigma _i\) instead of \(T_i\) is used for the principal stresses. Stress and stretching are defined as negative for compression. Tensors are written in bold capital letters (e.g., \({\mathbf { X}}\)). \(|{\mathbf { X}}|:=\sqrt{\text{tr}{\mathbf {X}}^2}\) is the Euclidean norm of \({\mathbf { X}}\) and \(\text{ tr }\,{\mathbf { X}}\) is the sum of the diagonal components of \({\mathbf { X}}\). The superscript 0 marks a normalized tensor, i.e., \({\mathbf { X}}^0 = {\mathbf { X}}/|{\mathbf { X}}|\). \(\mathbf {1}\) denotes the second-order unit tensor. Stresses are considered as effective ones, and the normally used dash is omitted. \(\mathring{{\mathbf { T}}}\) is the objectiveFootnote 2 (co-rotational) stress rate resulting from barodesy, and \(\dot{{\mathbf { T}}}\) is the time derivative according to Zaremba/Jaumann, which is obtained by \(\dot{{\mathbf { T}}}=\mathring{\mathbf { T}}-{\mathbf { W}}{\mathbf { T}}+{\mathbf { W}}{\mathbf { T}}\), with \({\mathbf { W}}\) being the antimetric part of the velocity gradient. For rectilinear extensions, the objective stress rate \(\mathring{{\mathbf { T}}}\) is equal to \(\dot{{\mathbf { T}}}\) and will therefore be used in Sect. 4.
The stretching tensor \({\mathbf { D}}\) is the symmetric part of the velocity gradient.Footnote 3 The mean effective stress is \(p:=-\frac{1}{3}\text{tr}\,{\mathbf {T}}\), and \(\varepsilon _{\text {vol}}=\text{tr} {\varvec{\varepsilon} }\) is the volumetric strain. In this article, \(\delta :=\text{tr }{\mathbf {D}}^0={\dot{\varepsilon}_{\text {vol}}}/{| {\dot{\varvec{\varepsilon} }}}|\) is used as a dilatancy measure. For axisymmetric conditions, e.g., conventional triaxial or oedometric compression, the axial stress is denoted with \(\sigma _1\) and the radial stress is denoted with \(\sigma _2 (=\sigma _3)\). The associated strains are \(\varepsilon _1\) and \(\varepsilon _2=\varepsilon _3\). The void ratio e is the ratio of the volume of the voids \(V_p\) to the volume of the solids \(V_s\). The deviatoric stress is written as \(q=-(\sigma _1-\sigma _3)\), and the deviatoric strain reads \(\varepsilon _q=2/3\cdot (\varepsilon _1-\varepsilon _3)\). The stress ratio \(K=\sigma _2/\sigma _1\) at critical states equals \(K_c\), and for oedometric normal compression K is denoted by \(K_0\).
3 Barodesy for clay
Barodesy is expressed by an evolution equation of the rate type \(\mathring{{\mathbf { T}}}\) \(= {\mathbf { h}}({\mathbf { T}}, {\mathbf { D}}, e)\). The general form of the constitutive relation is [9]:
with
Note that \(c_4\) equals 1 for clay, and therefore all material constants are dimensionlessFootnote 4. \({\mathbf { R}}^0\) and \({\mathbf { T}}^0\) are the normalized tensors of proportional stress paths \({\mathbf { R}}\) and actual stress, respectively. \({\mathbf { R}}\) is a tensorial argumentFootnote 5 of normalized stretching \({\mathbf { D}}^0\) and is chosen according to Medicus et al. [21]:
The scalar functions f and g take into account asymptotic states, critical states, the influence of stress level (barotropy) and density (pyknotropy). They are chosen as follows:
with the critical void ratio \(e_c\)
and the scalar functions \(\beta\) and \(\varLambda\)
where \(\sigma ^*\) is the reference pressure 1 kPa; \(c_1\) − \(c_6\) are material constants which depend on the soil parameters \(\varphi _c\), N, \(\lambda ^*\) and \(\kappa ^*\), cf. Table 1; \(\varphi _c\) is the critical friction angle; N is the ordinate intercept of the isotropic normal compression line (NCL) in the \(\ln p\) versus \(\ln (1+e)\) plot; \(\lambda ^*\) is the slope of the NCL; and \(\kappa ^*\) is the slope of the unloading line under isotropic compression in the \(\ln p\) versus \(\ln (1+e)\) plot. A detailed description of the soil parameters is given in Sect. 5.
Gudehus and Mašín [8] consider the angles \(\psi _{\dot{\varepsilon }}\) and \(\psi _\sigma\) according to Fig. 1a, b, for a graphical representation of proportional strain and stress paths. In Fig. 1c a \(\psi _{\dot{\varepsilon }}\) - \(\psi _\sigma\) plot of barodesy for clay [20] is compared with the improved version presented in this article. The main difference of the models is the choice of the \({\mathbf { R}}\)-function. Note that the \({\mathbf { R}}\)-function according to (3)–(5) prohibits tensile stresses.Footnote 6 For volume decreasing paths, i.e., \(-90^\circ<\psi _{\dot{\varepsilon }}<90^\circ\) the models almost coincide, for certain volume increasing pathsFootnote 7. Moreover, the proportional stress paths obtained with the 2012 version [20] lie in the tensile stress region (\(\psi _\sigma <-35.3^\circ\) or \(\psi _\sigma >54.7^\circ\)). The consequences are shown in Fig. 2. Deforming soil with proportional strain paths, and the stress paths approach the corresponding proportional stress paths. Certain stress paths approach proportional stress paths in the tensile region with the old version [20], cf. Fig. 2a. In Fig. 2b, all stress paths stay in the compressive area. Note that, for simulations with barodesy for sand [4, 9–12], qualitatively the same is observed for dense samples. However, this shortage for sand is not as drastic as it is for overconsolidated clay in Fig. 2a.
4 Calibration
In barodesy, material constants are denoted by \(c_i\), unless other symbols are established, such as \(\varphi _c\), N, \(\lambda ^*\) and \(\kappa ^*\) in the case of barodesy for clay. All constants \(c_1\) – \(c_6\) can be determined on the basis of \(\varphi _c\), N, \(\lambda ^*\), \(\kappa ^*\), see Table 1. In order to calibrate the four parameters \(\varphi _c\), N, \(\lambda ^*\) and \(\kappa ^*\) a consolidated undrained triaxial test (CU) is sufficient. From consolidation, we get the parameters N, \(\lambda ^*\) and \(\kappa ^*\) and from undrained compression the critical friction angle \(\varphi _c\) is obtained. In Sect. 5 the determination of \(\varphi _c\), N, \(\lambda ^*\) and \(\kappa ^*\) is illustrated by element tests. In Table 2, the parameters are shown for several clay types.
Below, the approach for the determination of \(c_1\) − \(c_6\) is explained.
4.1 Constants \(c_1\) and \(c_2\)
The constants \(c_1\) and \(c_2\) can be calculated from \(\varphi _c\), cf. Table 1. The \({\mathbf { R}}\)-function (Eqs. 3–5) includes \(c_1\) and \(c_2\), captures critical states and Jáky’s relation \(K_0=1-\sin \varphi _c\) for oedometric compression and produces similar results to Chu and Lo’s relation [3]. Results are presented in Appendix 1. A detailed explanation and further results are given by Medicus et al. [21].
4.2 Constants \(c_4\) and function \(\beta\)
The NCL
is used for the determination of \(c_4\) and \(\beta\).Footnote 8 \(\sigma ^*\) is a reference pressure equal to 1 kPa. The NCL according to Butterfield [2] as well as the critical state line (CSL) are assumed to be linear in the \(\ln (1+e)\) − \(\ln p\) plot, cf. Mašín [15, 18].
The constant \(c_4\) and the function \(\beta\) are chosen in order to ensure that a simulation of hydrostatic normal compression with barodesy starting from \(e=\exp N-1\) yields the NCL. A detailed derivation of \(c_4\) and \(\beta\) is shown in Appendix 2.
4.3 Constant \(c_3\)
Gudehus and Mašín [8] propose the following graphical representation of admissible states with respect to void ratio and proportional stress paths. Figure 3 shows how proportional stress paths (in terms of \(\psi _\sigma\)) are assumed to be connected with \(p_e/p\). Hvorslev’s equivalent consolidation pressure \(p_e\) is the value of mean stress on the NCL, which refers to the current specific volume \((1+e)\), cf. Fig. 3a:
The distance of a state characterized by e and p from the isotropic normal compression line is therefore indicated by \(p_e/p\). For example, for hydrostatic compression it applies \(p_e/p=1\) and \(\psi _\sigma =0\), and at critical states \(p_e/p_c\) is assumed to be equal to 2. Proportional stretching will eventually lead to constant values of \(p_e/p\) for compressive stretching, as well as for extensive stretching, so-called asymptotic extension states [7, 8, 16]. Asymptotic extension states correspond to so-called normal extension lines in the \(\ln p\) − \(\ln (1+e)\) plot [8, 16]. Gudehus and Mašín [8] propose the \(p_e/p\)−\(\psi _\sigma\) plot (Fig. 3b) for the directions of proportional stretching in the range of \(-d<\psi _{\dot{\varepsilon }}<d\), according to Fig. 1a. The directions \(-d\) and d are theoretical limits of asymptotic behaviour according to Gudehus and Mašín [8]. Discrete element simulations by Mašín [16] demonstrated that asymptotic states could only be obtained in a narrower range of \(\psi _{\dot{\varepsilon }}\). However, in barodesy, asymptotic states are obtained for the whole range of \(-180^\circ<\psi _{\dot{\varepsilon }}<180^\circ\).
The following procedure for the determination of \(c_3\) is proposed: barodesy predicts for sufficiently long proportional compressive stretching \(p_e / p =\) const., e.g., for \(\psi _{\dot{\varepsilon }}=0^\circ\) the NCL is reached, and \(p_e / p_i =1\) (Fig. 3). This also applies for extension paths, which lead to normal extension lines in the \(\ln p\) − \(\ln (1+e)\) plot. In particular for an isotropic extension path, which is denoted with \(-i\) in Fig. 1a, the isotropic normal extension line is reached, see Fig. 3a. It follows that the unloading stiffness in the \(\ln p\) − \(\ln (1+e)\) plot is characterized by the parameter \(\lambda ^*\):
With the general form of the barodetic constitutive relation (1), isotropic compression (i.e. \({\mathbf { T}}= -p\mathbf{1}, {\mathbf { R}}^0={\mathbf { T}}^0=-\frac{1}{\sqrt{3}}{} \mathbf{1}\)) is expressed by:
For a proportional isotropic extension paths, Eq. 15 follows from Eqs. 13 and 14 (with \(\delta =\sqrt{3}\)):
With f, g and \(\beta\) from (6), (7) and (34) and with \((1+e)/(1+e_c)=(2\cdot p_{-i}/p_e)^{\lambda ^*}\), we obtain:Footnote 9
Releasing \(c_3\), yields:
Choosing \(p_{-i}/p_e=1/1000\) in (16) yields:
Note that \(p_{-i}/p_e=1/1000\) is arbitrary and cannot be acquired by experiments. However, the overall performanceFootnote 10 of barodesy for clay is best by choosing \(p_{-i}/p_e=1/1000\) and helps to present a calibration procedure which is simple and applicable also for practitioners without performing any least square optimization. Figure 3b shows how \(\psi _\sigma\) is related to \(p_e/p\) in barodesy.
4.4 Constant \(c_5\)
The constant \(c_5\) has been determined by trial. Setting \(c_5=1/K_c\) gives the best fit concerning overall performance. Setting \(c_5=1\) would highly overestimate radial stress under oedometric compression.
4.5 Constant \(c_6\)
It appears reasonable to require that, under isotropic extension, the stress paths follow the shortest way to the origin regardless of its actual stress state, cf. Fig. 4a. From this requirement, we get for isotropic extension:
and with \({T_1}/{T_2}={T_1^0}/{T_2^0}\) and (1) we obtain:
Equation (20) is valid for a proportional isotropic extension or compression paths or if \(f=0\). Setting \(f=0\) in (6), we obtain with \(\delta =\sqrt{3}\) and \(\beta\) from (34):
In Fig. 4b, simulations of barodesy show that the stress paths for isotropic extension follow the shortest way to the origin. Response envelopes for London clay are added in Fig. 4b.
5 Simulations of element tests
In this section, simulations of element tests with and without rotation of principal axes are shown. Element tests in general are an idealization and, as in all experiments inhomogeneities occur. Especially with shearing, localization takes place and the loss of homogeneity is unavoidable. Thus, the comparison of simulations of element tests with experimental data only serves as an approximate reference.
5.1 Rectilinear extensions
Isotropic compression: In Fig. 5, an isotropic compression test and its simulation with barodesy is shown. In Fig. 5b, the calibration of the parameters N, \(\lambda ^*\) and \(\kappa ^*\) is illustrated with experimental data of Dresden clay [1]. As isotropic normal compression is included in the formulation and calibration of barodesy, normally consolidated isotropic compression test results are therefore in agreement with the simulated NCL, see Fig. 5. The unloading stiffness is described through the parameter \(\kappa ^*\). The term \(({1+e})/({1 + e_c}) = \text{ const. }\) in (22) indicates a straight line (A-B) in the \(\ln p\)-\(\ln (1+e)\) plot in Fig. 5b, cf. [19]
On this line, the tangential unloading stiffness under isotropic extension is \({\dot{p}}/{\dot{e}}=-{p}/(\kappa ^*(1+e))\) with barodesy. Closer to the NCL, the unloading stiffness is slightly higher, and for lower mean stresses p, the stiffness is lower, cf. Fig. 5b and [19].
Triaxial compression: The critical friction angle \(\varphi _c\) is calibrated with a normally consolidated Weald clay sample and can be obtained from the slope of the critical state line in the p-q plot, cf. Fig. 6a. Test results and numerical simulation with barodesy of a normally consolidated and overconsolidated sample are shown. Note that the simulation of the overconsolidated sample does not allow a higher mobilized friction angle \(\varphi _m\) than \(\varphi _c\). Barodesy therefore underestimates the peak friction (i.e., the maximum mobilized friction) angle in CU tests, cf. Fig. 6b. The simulations of the normally consolidated and overconsolidated samples in the q/p-\(\varepsilon _1\) plot coincide for CU tests, cf. Fig. 6b.
In Fig. 7, limit points of normally consolidated samples obtained by true triaxial tests are shown. The data refer to San Francisco Bay Mud from Lade [13] and are compared with predictions by barodesy. Note that the critical state locus of barodesy practically coincides with the locus according to Matsuoka-Nakai, cf. Fellin and Ostermann [5].
In Fig. 8 drained triaxial compression and extension tests of normally consolidated and overconsolidated Weald clay are shown. The simulations with barodesy are realistic. Contractant behaviour for the normally consolidated samples and dilatant behaviour for the overconsolidated clay is observed. Peak strength is well predicted.
In Fig. 9, a more general picture of drained triaxial tests simulated with barodesy is shown. Triaxial tests are shown as p–e and p–q plots as well as plots in the normalized stress plane (i.e., \(p/p_e\)−\(q/p_e\)). The paths approach the critical state line in the p–e and p–q plots. Highly overconsolidated samples dilate to approach the CSL in the p–e plot, and slightly overconsolidated and normally consolidated samples exhibit contractant behaviour to approach the CSL. Highly overconsolidated samples overshoot the CSL in the p–q plot.
Oedometric compression: In Fig. 10, oedometric compression of London clay is shown. The normal compression behaviour gives reasonable results in the e − p plot, as well as in the \(\sigma _1\) − \(\sigma _2\) plot.Footnote 11 For unloading, the radial stress is overestimated.
5.2 Rotation of principal stress and strain axes
Simple shear test: Figure 11 presents a simulation of a simple shear test with a constant vertical stress of \(\sigma _y=-100\) kPa. The evolution of the shear stress \(\tau _{xy}\) is plotted over the shear strain \(\gamma\) (in radian). The angle \(\alpha _\sigma\) denotes the inclination of major principal stress to the horizontal direction x, and \(\alpha _D\) is the inclination of major principal stretching, respectively. In Fig. 11, a Weald clay sample with \(K_0=1-\sin \varphi _c\) is sheared. The major principal stress direction \(\alpha _\sigma\) is \(90^\circ\) at zero shear strain and decreases to \(\approx\) \(45^\circ\) with ongoing shear strain. The difference between the angles \(\alpha _D\) and \(\alpha _\sigma\), i.e., the angle of non-coaxiality \(\alpha _D-\alpha _\sigma\) becomes very smallFootnote 12, i.e., \(\alpha _\sigma \approx \alpha _D\approx 45^\circ\) at the critical state. Similar results with hypoplasticity and an elasto-plastic model are shown in Schranz and Fellin [24]. Experiments on sand according to Roscoe et al. [23] and DEM simulations [25, 29] yield similar results, cf. Yu [28].
In Fig. 12, the evolution of the angle of non-coaxiality with ongoing shear strain is shown for different initial \(K_0\) values. In Fig. 12a, DEM simulations from Thornton and Zhang[25], Zhang [29] show that the angle of non-coaxiality is small for \(K_0=1\). For \(K_0=2\), the angle of non-coaxiality decreases with ongoing shear strain to \(\approx 0^\circ\); and for \(K_0=0.5\) it increases to \(\approx 0^\circ\). It is stated that non-coaxiality is significant before 10% shear strain [29]. The predictions with barodesy in Fig. 12b are in good agreement with the DEM simulations in Fig. 12a.
The results of the DEM simulations and experiments [23] apply for sand. Therefore, only a qualitative comparison of barodesy for clay (Figs. 11b, 12b) is possible. However, the comparison demonstrates that barodesy is applicable for general deformation, i.e., rotation of principal stress and strain axes.
Appendix 3 summarizes all equations of barodesy for clay.
6 Summary and conclusions
Barodesy comprises fundamental characteristics of soil behaviour, such as critical states, asympotic states, barotropy, pyknotropy, and a stress-dilatancy relation. Barodesy can be written symbolically as a single equation of the form \({\mathring{{\mathbf { T}}}}\,=\,{\mathbf { h}}({\mathbf { T}}, {\mathbf { D}}, e)\), i.e., the stress rate is expressed as a function of the stress, stretching and the void ratio. As in basic hypoplastic models, barodesy uses only \({\mathbf { T}}\) and e as memory parameters. This covers many phenomena and is insufficient to capture strong memory effects. Consequently, ratcheting and unrealistic small-strain behaviour are obtained.
In order to calibrate barodesy for clay, four well-known material parameters of soil mechanics, which can be determined from a consolidated undrained compression test, are sufficient. The model provides realistic results, as compared with experimental results of various clay types.
Notes
Proportional strain paths are paths with constant ratios of the principal strains, i.e., \(\varepsilon _1\) : \(\varepsilon _2\) : \(\varepsilon _3\) = const. In the same sense, paths with constant ratios of principal stresses are called proportional stress paths, i.e., \(\sigma _1:\sigma _2:\sigma _3=\)const., cf. Fig. 1a.
The term objectivity points to the fact that material behaviour is frame-indifferent, i.e., the behaviour is independent of the observers’ motion.
In general, stretching \({\mathbf { D}}\) is only approximately equivalent to the strain rate \(\dot{\boldsymbol{\varepsilon }}\). For rectilinear extensions, \({\mathbf { D}}\) equals \(\dot{\boldsymbol{\varepsilon }}\), with \(\dot{\boldsymbol{\varepsilon }}\) being the logarithmic strain tensor.
The exponential of the tensor \({\mathbf { R}}\) can be defined by means of its eigenvalues \(r_i\):
\(\exp \mathbf R = \exp \begin{pmatrix} r_1 &{} 0 &{} 0 \\ 0 &{} r_2 &{} 0 \\ 0 &{} 0 &{} r_3 \end{pmatrix} = \begin{pmatrix} \exp r_1 &{} 0 &{} 0 \\ 0 &{} \exp r_2 &{} 0 \\ 0 &{} 0 &{} \exp r_3 \end{pmatrix}\).
E.g., for \(\varphi _c=22.6^\circ\) (London clay), tensile stresses occur for \(\psi _{\dot{\varepsilon }}<-128.3^\circ\) and \(\psi _{\dot{\varepsilon }}>138.9^\circ\), cf. Fig. 1c.
At isotropic extension, \(\ln ({1+e})\) equals \({N-\lambda ^*\ln (p_e/p_{-i} \cdot p/\sigma ^*)}\), cf. Fig. 3a. We therefore get \(\dfrac{1+e}{1+e_c}=\dfrac{\exp (N-\lambda ^*\ln (p_e/p_{-i} \cdot p/\sigma ^*))}{\exp (N-\lambda ^*\ln (2 \cdot p/\sigma ^*))}=\left( 2\dfrac{p_{-i}}{p_e}\right) ^{\lambda ^*}\).
The parameter \(c_3\) does not only affect extension states, but also shear stiffness.
Note that barodesy for clay predicts \(K_0\)-values according to Jáky’s relation \(K_0=1-\sin \varphi _c\).
At critical states \(\alpha _D-\alpha _\sigma \approx 0.5^\circ\). Neglecting the Zaremba/Jaumann expression \(-{\mathbf { W}}{\mathbf { T}}+{\mathbf { W}}{\mathbf { T}}\) yields \(\dot{{\mathbf { T}}}=\mathring{\mathbf { T}}\). It follows that \(\alpha _D-\alpha _\sigma =0^\circ\) at failure.
The equation \(\frac{\dot{e}}{1+e}=\text{tr}\,{\mathbf {D}}\) holds for incompressible grains.
Note that \(\varLambda\) in (35) is chosen as a function of \(\delta\). \(\varLambda\) equals \(\lambda ^*\) for isotropic compression (\(\delta =-\sqrt{3}\)). For isotropic extension (\(\delta =\sqrt{3}\)), \(\varLambda\) equals \(\kappa ^*\). The consequences are described below. The values in between \(\varLambda =\lambda ^*\) and \(\varLambda =\kappa ^*\) are interpolated linearly.
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Appendices
Appendix 1: Determination of \(c_1\) and \(c_2\)
The parameters \(c_1\) and \(c_2\) are determined according to Eqs. 23 and 24 and are included in the \({\mathbf { R}}\)-function [21] (Eqs. 3–5).
The stress ratios \({\sigma _2}/{\sigma _1}\) under oedometric compression \(K_0\) and at critical states \(K_c=\dfrac{1-\sin \varphi _c}{1+\sin \varphi _c}\) are comprised. If we include Jáky’s relation under oedometric compression, i.e., \(K_0=1-\sin \varphi _c\), Eq. 24 can be simplified as follows:
In Fig. 13, results of the \({\mathbf { R}}\) - function are compared with the relation by Chu and Lo [3].
Appendix 2: Determination of \(c_4\) and \(\beta\)
Derivation of the NCL (Eq. 11) with respect to time t yields:
With \(\frac{\dot{e}}{1+e}=\text{ tr }{\mathbf { D}}=-\sqrt{3}|{\mathbf { D}}|\) and \(|{\mathbf { T}}|=\sqrt{3}p\) the NCL readsFootnote 13:
With the general form of the barodetic constitutive relation (1), isotropic compression (i.e. \({\mathbf { T}}= -p\mathbf{1}, {\mathbf { R}}^0={\mathbf { T}}^0=-\frac{1}{\sqrt{3}}{} \mathbf{1}\)) is expressed by the following form:
Comparing (27) and (28) yields:
and
Now, we write \(f+g\) from (6) and (7) for hydrostatic compression (\(\delta =-\sqrt{3}\)), use (30), and obtain:
Introducing the NCL (Eq. 11) and CSL (Eq. 8) into (31) leads to:
Setting
with
yields the following:Footnote 14 Equation 30 is satisfied under isotropic compression, i.e., for \(\delta =-\sqrt{3}\).
Appendix 3: Equations of barodesy for clay
In this appendix, all equations of barodesy for clay are summarized.
In Table 1 the determination of constants \(c_1\) - \(c_6\) on the basis of \(\varphi _c\), N, \(\lambda ^*\) and \(\kappa ^*\) (Table 2) is shown.
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Medicus, G., Fellin, W. An improved version of barodesy for clay. Acta Geotech. 12, 365–376 (2017). https://doi.org/10.1007/s11440-016-0458-4
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DOI: https://doi.org/10.1007/s11440-016-0458-4