Permissible range of model parameters for natural finegrained materials
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Abstract
This paper presents a threedimensional constitutive model for natural clay that includes creep, anisotropy and structure, as well as a theoretical means to estimate the range for anisotropy and structurerelated parameters, as needed for parameter optimisation. CreepSCLAY1S is an extension of the CreepSCLAY1 model proposed by Sivasithamparam et al. (Comput Geotech 69:46–57, 2015) which includes the effects of bonding and destructuration. The model needs 14 model parameters, of which five are similar to those used in the modified Cam–Clay model. A method is developed to quantify the range for the three parameters related to structure and anisotropy that cannot be derived directly from experimental data. The theoretically derived range compares favourably with the values found in the literature. As a result, the model now can be used with more confidence, enabling sensitivity analysis and systematic parameter derivation with optimisation techniques.
Keywords
Anisotropy Constitutive behaviour Creep Finegrained material Optimisation1 Introduction
Modelling of saturated finegrained matter such as natural soft soils has always been a challenge in engineering. The strain–stress behaviour of these materials is very complex and highly nonlinear. Numerous of different features of soil behaviour, such as time/rate dependency (sometimes called creep), anisotropy as well as bonding/destructuration influence the relation between strain and stress as a function of strain rate. Advanced models taking into account these different features are required to simulate the responses of these materials accurately.
This paper uses an extension of the CreepSCLAY1 model by Sivasithamparam et al. [19]. The CreepSCLAY1S model adds the effect of structure to CreepSCLAY1 that already takes into account anisotropy and creep. The model accounts for structure in the same way as the SCLAY1S model developed by Karstunen et al. [8] based on the formulation proposed by Gens and Nova [4]. Existing models developed by e.g. Yin et al. [28] and Grimstad et al. [5] already take into account these three different features. The model has some major similarities with Yin et al.[28] and Grimstad et al. [5] models, as well as differences. In order to avoid any confusion in different definitions of some model parameters and key equations, the name of CreepSCLAY1S is used to refer to the model in the format as introduced in Sivasithamparam et al. [19] with addition of bonding and destructuration. The advantage of CreepSCLAY1, and therefore CreepSCLAY1S, similarly to Leoni et al. [11] and Yin et al. [28] models over [5], is the use of the modified creep index parameter, which is directly related to the secondary compression coefficient \(C_\alpha\) commonly used internationally.
The main drawback for the use of these types of advanced constitutive models is the number of parameters required. The CreepSCLAY1S model requires in total 14 parameters, of which most can be directly derived from experimental data. Nevertheless, some are not directly measurable, such as some parameters used to describe the evolution of anisotropy and structure (material degradation). These parameters are estimated through indirect methods, such as calibration of the model response against the soil response measured in nonstandard laboratory tests or optimisation methods [3, 12, 15, 16, 20, 25, 27]. For optimisation methods, however, it is of paramount importance to know the bounds for the values of these parameters prior to calibration. In this paper, a method to estimate these bounds is proposed for the three most important parameters for calibration: two related to structure and one related to anisotropy. In addition, the parameter relating Lode angle dependency is discussed. The current work will not only benefit the CreepSCLAY1S model presented here, but the principles can be applied to a wide range of models that include formulations for structure and anisotropy, such as [2, 5, 6, 8, 11, 13, 19, 22, 24, 28]. The validity of the range proposed will be compared against the parameter values found in studies.
2 Description of the CreepSCLAY1S model

the modified swelling index \(\kappa ^{*}\),

the Poisson’s ratio \(\nu ^{\prime }\),

the modified intrinsic compression index \(\lambda _{i}^{*}\),

the slope of critical state line in compression \(M_{c}\),

the slope of critical state line in extension \(M_{e}\),

the intrinsic modified creep index \(\mu _{i}^{*}\),

the reference time \(\tau\),

the absolute effectiveness of rotational hardening \(\omega\),

the relative effectiveness of rotational hardening \(\omega _{d}\),

the absolute rate of destructuration a,

the relative rate of destructuration b,

the preoverburden pressure POP or the overconsolidation ratio OCR,

the initial inclination of the ICS, CSS and NCS \(\alpha _{0}\),

the initial amount of bonding \(\chi _0\).
3 Bounds for parameters related to structure
In order to take into account the apparent bonding, CreepSCLAY1S uses three parameters: the initial amount of bonding \(\chi _0\), the relative rate of destructuration b and the absolute rate of destructuration a. The initial amount of bonding \(\chi _0\) is generally related to the experimentally obtained soil sensitivity, which is a simple routine test. Parameters a and b, however, cannot be measured directly and hence require an optimisation procedure. Some specialist tests (i.e. drained consolidation at two constant stress paths, one with high stress ratio and one with almost zero/negative stress ratio, see Koskinen et al. [10] and Karstunen et al. [8] for details) may be used to increase the accuracy of the calibration process, but these tests are normally not available. To perform such optimisation procedure, appropriate range for the values for a and b is required. In soft clays a reasonable assumption is that the deviatoric creep strains have less or equal influence as the volumetric creep strains on the destructuration process, giving b bounds \(0<b<1\). Usually, a is unknown before model calibration against experimental results. In the following, a method is proposed to get a range of values for the future calibration of a.
3.1 Upper bound for a
3.2 Lower bound for a
4 Bounds for parameters related to anisotropy
In order to take into account anisotropy of the soil and its evolution, three parameters are needed: the initial inclination of ICS, CSS and NCS represented by \(\alpha _{0}\), the relative effectiveness of creep strains in rotational hardening \(\omega _{d}\) and the absolute effectiveness of rotational hardening \(\omega\).
4.1 \(\omega\) range for model without structure
4.2 \(\omega\) range for models with bonding and bond degradation
5 Comparison of the new bounds with available data
The values of parameters commonly used in the literature will be compared with the range derived. Given the relatively recent formulation of CreepSCLAY1S, data of existing models using comparable model formulations and parameters for structure and anisotropy evolution will be used. Data corresponding to \(\omega\) for models that do not account for structure (see Table 1) are first compared, followed by data corresponding to \(\omega\) for models which account for bonding and degradation of bonds (see Table 2). Finally, data corresponding to parameter a will be presented (see Table 3).
5.1 Bounds for \(\omega\) for models without structure
Comparison between \(\omega\) values used in the literature and the range of values proposed; without structure
References  \(\xi ^{*}\)  \(\omega\) literature  \(\omega\) range (Eq. 42) 

Leoni et al. [11]  0.089  28  17–47 
Leoni et al. [11]  0.060  43  25–70 
Sivasithamparam et al. [19]  0.093  50  16–45 
Sivasithamparam et al. [19]  0.062  45  24–67 
Sivasithamparam et al. [19]  0.168  25  9–25 
Sivasithamparam et al. [19]  0.102  25  15–41 
Grimstad et al. [5]  0.102  25  15–41 
Grimstad et al. [10]  0.166  20  9–25 
5.2 Bounds for \(\omega\) for models with structure
Comparison between \(\omega\) values used in the literature and the upper bound value proposed; with structure
References  \(\xi ^{*}_{i}\)  \(\chi _0\)  \(\omega\) used  \(\omega _{up}\) (Eq. 47) 

Yildiz et al. [26]  0.067  22  20  32 
Yildiz et al. [26]  0.033  30  20  65 
Yildiz et al. [26]  0.069  45  20  31 
Yin et al. [28]  0.057  77  12  37 
Grimstad et al. [5]  0.067  9  20  34 
Koskinen et al. [10]  0.066  14  20  33 
Koskinen et al. [10]  0.061  12  20  36 
Karstunen et al. [8]  0.079  8  25  29 
Karstunen et al. [8]  0.059  8  25  38 
5.3 Bounds for structure parameter a
Comparison of the values for parameter a of the literature with the proposed range of values
References  \(\xi ^{*}_{i}\)  \(\chi _{0}\)  b  \(\alpha _{K_{0}^{nc}}\)  \(M_{e}\)  a used  

Yildiz et al. [26]  0.067  22  0.2  0.44  0.83  8.0  6.4–12.4 
Yildiz et al. [26]  0.033  30  0.2  0.42  0.79  8.0  12.9–24.7 
Yildiz et al. [26]  0.069  45  0.2  0.41  0.79  8.0  6.1–11.7 
Yin et al. [28]  0.057  77  0.3  0.52  0.93  13.5  6.8–13.1 
Grimstad et al. [5]  0.067  9  0.2  0.44  0.83  10.0  6.7–13.3 
Koskinen et al. [10]  0.066  14  0.2  0.46  0.86  9.0  6.6–13.0 
6 Conclusions
An extended formulation of the CreepSCLAY1 model is presented that includes effects of structure. Most of the parameters required are easy to evaluate from experimental data. The structure parameter a and anisotropy parameter \(\omega\), however, need calibration with the type of tests that are not normally available. For the first time a fundamental approach to obtain a range for these parameters is presented. Although the equations in the paper have been derived for a particular model, the same principles can be adopted for any model that accounts for initial anisotropy and its evolution, and/or bonding and destructuration. The method is based on combining theoretical considerations with physically sound assumptions based on experimental observations.
A very good agreement is observed between the range proposed, and the reported values for these parameters after calibration. The range proposed for these two parameters will then be very useful for further optimisation of these parameters. The range for \(\omega\) for models which do not account for structure is given by Eq. 42, whilst the range for \(\omega\) for models which account for structure is given by Eq. 47. The lower bound value of a is given by Eq. 29, and finally the upper bound value for a is given by Eq. 24. For both parameters a and \(\omega\), the range of values is strongly dependent on the compressibility parameter \(\xi ^{*}\) (or \(\xi ^{*}_{i}\) if bonding effect are considered). The compressibility parameter and rate of destructuration have a great influence on the evolution of anisotropy during isotropic loading. We highlight that a quite widely used formula to estimate \(\omega\) in [11] has a sign error, and with a correct formula indeterminate values are avoided. Finally, the new formula in the case of a model with a Lode angle formulation of the critical state stress ratio is proposed in Eq. 35. Physical bounds for the m parameter in the Lode angle dependency formulation are proposed.
Notes
Acknowledgements
The financial support from Trafikverket in the framework Branch samverkan i Grund and EC/FP7 CREEP PIAGGA2011286397 is greatly acknowledged.
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