Method to Introduce the Cementation Effect into Existing Elastoplastic Constitutive Models for Soils

Yamada, Shotaro; Sakai, Takayuki; Nakano, Masaki; Noda, Toshihiro

doi:10.1061/(ASCE)GT.1943-5606.0002727

Open access

Technical Papers

Feb 22, 2022

Method to Introduce the Cementation Effect into Existing Elastoplastic Constitutive Models for Soils

Authors: Shotaro Yamada https://orcid.org/0000-0002-7024-6327 [email protected], Takayuki Sakai [email protected], Masaki Nakano [email protected], and Toshihiro Noda [email protected]Author Affiliations

Publication: Journal of Geotechnical and Geoenvironmental Engineering

Volume 148, Issue 5

https://doi.org/10.1061/(ASCE)GT.1943-5606.0002727

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Abstract

This study proposes a new method to introduce the cementation effect into existing elastoplastic constitutive models for soils. The mechanical properties of cement-treated soil are evaluated via element tests and compared with those of naturally deposited clay. The similarities and differences between cement-treated soils and naturally deposited clays are studied, focusing on two states, the undisturbed and remolded states. The effective stress for cement-treated soils incorporating an internal state variable representing the cementation effect is newly defined to describe the mechanical properties of cement-treated soils. Moreover, by applying this extended effective stress to the super-subloading yield surface (SYS) Cam-clay model, which is an elastoplastic model for soils based on the skeleton structure concept, the scope of this constitutive model is extended to include cement-treated soils. The cementation effect introduced by the proposed method allows reproducing the mechanical behavior of the cement-treated soil. Finally, a brittle behavior not described at the element level can be obtained, leading to a soil–water coupled finite deformation analysis incorporating the proposed constitutive model.

Introduction

A large amount of soil is deposited near the river mouths which is dredged in the ports and the harbors of Japan to secure shipping routes. Most of this soil is conveyed to an offshore sediment disposal site and reclaimed. The landfills remain unusable since it takes a long time for the dredged sediment to settle and for the convergence of the self-weight consolidation. The pneumatic flow mixing method (Coastal Development Institute of Technology 2008; Kitazume and Sato 2003) has been developed to construct sea-based facilities such as international airports by utilizing the dredged soil. This method pumps dredged soil, which has high water content, under high pressure along with mixing cement, in a tube. By depositing the end of the tube on the seabed, the suppression of the increase in the water content ratio is greater when compared to dumping sediment on the sea surface. Since the cement solidification of the pumped sediment occurs before the self-weight consolidation can progress, the landfill site can be utilized earlier. However, it is necessary to set the soil strength based on the intended use of the reclaimed ground in order to conserve the amount of cement because a large amount of cement is required. The numerical analysis methods, such as the finite-element method, are required to accurately predict the deformation and the failure of the artificial ground constructed by cement solidification methods. This study extends elastoplastic constitutive models for naturally deposited soils to describe the mechanical behavior of cement-treated soils. To this end, oedometer and undrained triaxial tests of cement-treated soil are presented. From an experimental perspective, this study discusses the material properties of cement-treated soil and compares them with those of naturally deposited clays. Further, the viewpoints of undisturbed and remolded states, which are familiar to naturally deposited clays, are employed for cement-treated soil. In addition, this study discusses similarities and differences in the mechanical properties between cement-treated soils and naturally deposited clays. The conclusion is that undisturbed cement-treated soil, i.e., the as-made cement-treated soil, has a large pore structure and can take a state in the impossible region (Atkinson and Bransby 1978) of the remolded soil. Besides, as the plastic deformation progresses, the cement-treated soil approaches its fully remolded state. These characteristics are similar to those of a well-known mechanical property occurring in naturally deposited clays (e.g., Schmertmann 1953; Asaoka et al. 2000; Noda et al. 2005b). While the effective stress path in the undrained shear process is similar to that of naturally deposited clays, the cement-treated soil differs significantly in that the plastic compression occurs above the critical state line (CSL) of the remolded soil. This behavior is unique to cement-treated soils, which is revealed through a comparison with naturally deposited clays. Another mechanical feature of cement-treated soils, which is easy to understand for our senses, is their high initial stiffness even under low confining pressure.

As the properties of cement-treated soils are similar to those of naturally deposited soils, the super-subloading yield surface Cam-clay model, namely, the super-subloading yield surface (SYS) Cam-clay model proposed by Asaoka et al. (2002), which can describe the mechanical behavior of naturally deposited soils, is considered in this study. The SYS Cam-clay model is an elastoplastic constitutive model based on the skeleton structure concept. The skeleton structure concept was first suggested by Mikasa (1964), who believed that the mechanical properties of soils are determined by the kind and state. He called factors other than density and water content as the skeleton structure among factors that determine the latter. Furthermore, Asaoka et al. (2002) considered the structure, overconsolidation, and anisotropy as independent factors in the formation of the skeleton structure for the development of an elastoplastic constitutive equation for naturally deposited soil. The SYS Cam-clay model is based on a modified Cam-clay model (Roscoe and Burland 1968; Muir Wood 1990) that introduces a superloading surface to realize the structure concept (Asaoka et al. 2000), a subloading surface (Hashiguchi 1978, 1989; Asaoka et al. 1997) to realize the overconsolidation concept, and rotational hardening (Hashiguchi and Chen 1998) to represent the induced anisotropy. This model describes the process of the transition from highly structured and overconsolidated states to the fully remolded and normally consolidated states by evolving three independent factors in association with each other through plastic deformation. Several research papers (e.g., Noda et al. 2005a, b, 2007; Takaine et al. 2010; Tashiro et al. 2011, 2015; Nguyen et al. 2015) have reported that the SYS Cam-clay model is highly capable of simulating the mechanical behavior of naturally deposited clays.

To represent differences between cement-treated soils and naturally deposited clays, the SYS Cam-clay model was extended in this study. The key points of the extension are: (1) the translation of each loading surface in the negative direction on the mean effective stress axis, (2) degradation of the translation associated with the plastic deformation, and (3) description of a core elastoplastic constitutive model (an existing model without cementation effect) by the extended effective stress. Key point (1) is a commonly used method for modeling cement-treated soils (e.g., Gens and Nova 1993; Matsuoka and Sun 1995; Kasama et al. 2000, Lee et al. 2004; Horpibulsuk et al. 2010). In this study, the amount of translation of each loading surface is treated as an internal state variable representing the cementation effect. Key point (2) is based on the experimental fact that undisturbed cement-treated soil asymptotically approaches its remolded state due to plastic deformation. This requirement is reflected in the evolution rule for the internal state variable representing the cementation effect. Similar efforts have been made by Lee et al. (2004), Hashiguchi and Mase (2007), Suebsuk et al. (2011), and Rahimi et al. (2016). Key point (3) is to define a new effective stress for the deformation of cement-treated soils using the presented internal state variables and to describe the core elastoplastic constitutive model using this extended effective stress. This serves as a countermeasure to the possibility that the mean effective stress may be negative due to the translation of the loading surface. The application of extended effective stress also provides a high initial stiffness under low confining pressure. The extended constitutive model returns to the core constitutive model in the limit of the degradation of cementation, which is achieved by applying the extended effective stress. Moreover, the application of the extended effective stress integrates the entire proposed method through a very simple procedure. It must be noted that key point (3) is the most significant feature of the proposed method, which has not been presented in previous studies.

The remainder of this paper is organized as follows: Section “Experimental Consideration: Similarities and Differences between Cement-Treated Soils and Naturally deposited Clays” describes the experiments and discusses the similarities and differences between cement-treated soils and naturally deposited clays. Section “Elastoplastic Constitutive Model for Soils Considering Cementation Effects” presents the method for expressing the cementation effect with the extended effective stress and introduces this method into the SYS Cam-clay model. In the section “Simulation of Element Tests with the Extended SYS Cam-Clay Model,” the results of element tests of cement-treated soil are reproduced using the proposed model. In the section “Simulation of Triaxial Tests as Initial Value and Boundary Value Problem,” a soil–water coupled finite deformation analysis of an undrained triaxial compression test is performed. This simulation shows that the brittle behavior, which is excluded from the modeling can be explained by the shear band formation. Section “Conclusions” concludes this paper. Even though a constitutive model is developed and validated from a standpoint that the triaxial test is considered as an element test in the sections “Experimental Consideration: Similarities and Differences between Cement-Treated Soils and Naturally Deposited Clays, Elastoplastic Constitutive Model for Soils Considering Cementation Effects, and Simulation of Element Tests with the Extended SYS Cam-Clay Model,” the triaxial test is considered as an initial and boundary value problem in the section “Simulation of Triaxial Tests as Initial Value and Boundary Value Problem.” This paper addresses the extraction of the mechanical properties of the cement-treated soils from the triaxial test results but does not aim to perfectly reproduce the result of the so-called element test by a single constitutive model. This paper aims to develop a constitutive model for cement-treated soil phenomenologically based on these two stand-points.

Experimental Consideration: Similarities and Differences between Cement-Treated Soils and Naturally Deposited Clays

Oedometer tests and undrained triaxial tests were performed to understand the mechanical properties of cement-treated soils. In the following, the mechanical behavior of a cement-treated soil is characterized by comparison with that of a naturally deposited clay.

Physical Properties of the Base Material and Blending Condition for Cement-Treated Soil

Clayey soil (60% clay and 40% silt) dredged from the Yuraku-cho Formation in the Tokyo harbor was used as a base material for the cement-treated soil. The dredged material in the Yuraku-cho formation has a liquid limit

w_{L} = 91.2 %

and a plastic limit

w_{P} = 39.0 %

. Assuming that the dredged soil is treated following the pneumatic flow mixing method, the target flow value was set to 90–100 mm, and the target strength of the unconfined compression tests after 28 days of curing was set to 100–200 kPa. These values are determined based on a manual of the technology on the pneumatic flow mixing method (Coastal Development Institute of Technology 2008). Table 1 shows the blending conditions satisfying the presented target values. The water content of the dredged material indicates the value before blending with cement. In the table,

S

,

W

, and

C

indicate the masses of soil particles, water, and cement in

1 m^{3}

of the mixed sample in saturation, respectively.

Table 1. Blending condition

Water content of dredged soil, $w_{0}$ (%)	Additive amount of cement, $C$ ( $kg / m^{3}$ )	Ratio of water to cement, $W / C$	Ratio of soil to cement, $S / C$
170	50	16.1	9.47

The samples mixed as presented were cured in water for at least 56 days after filling the molds. The specimens were cured under no confining pressure. An oedometer test specimen of 6 cm in diameter and 2 cm in height and a triaxial test specimen of 5 cm in diameter and 10 cm in height were prepared by trimming. The specimens prepared in this manner are referred to as undisturbed cement-treated soil. In contrast, the samples prepared in the presented manner and then fully disturbed into a paste are referred to as remolded cement-treated soil. For the oedometer test of the remolded cement-treated soil, a paste-like sample was filled into the consolidation ring and then loaded. The triaxial specimens of the remolded cement-treated soil were prepared by preconsolidation at 98.1 kPa for one week and then by trimming it into a cylindrical shape.

Oedometer Tests

Fig. 1 shows the oedometer test results for undisturbed and remolded cement-treated soil. The specific volume v is plotted over the vertical effective stress

σ_{v}^{'}

. As shown in Fig. 1, the state of the undisturbed sample can exist above that of the remolded sample. The experimental results indicate that the undisturbed sample is in a bulky state when compared to the remolded sample, i.e., the undisturbed sample has a larger void ratio than the remolded sample at the same vertical pressure. The undisturbed sample gradually approaches the remolded sample as the plastic deformation progresses beyond the consolidation yield stress. Fig. 2 shows the oedometer tests for the naturally deposited clays in undisturbed and remolded states sampled from Urayasu, Japan, by Nakai et al. (2014). As shown in the figures, the presented features are common to naturally deposited soils with highly developed structures.

Fig. 1. Oedometer test results of a cement-treated soil.

Fig. 2. Oedometer test results of a naturally deposited clay. (Data from Nakai et al. 2014.)

Fig. 1 includes the test results of the remolded base material (dredged soil) with the same water content as the liquid limit at the initial state for comparison. The compression line of the untreated soil is located far below the remolded cement-treated soil, implying that adding cement caused a change in the properties of the base material that did not disappear even if the treated soil is fully remolded. The undisturbed cement-treated soil is asymptotically closer to not the remolded base material, but to the remolded cement-treated soil with plastic deformation. Therefore, the reference condition of the cement-treated soil should be considered for its remolded state and not for the base material. The reference condition of a soil in the skeleton structure concept is a basic state to which the soil is asymptotically close because of plastic deformation. The compression behavior of the remolded cement-treated soils is considered different from that of the base materials themselves; however, its compression line is almost linear on the single logarithmic chart, as it is for the remolded sample of the naturally deposited clays.

Undrained Triaxial Tests

The undrained shear behavior of the remolded cement-treated soil is shown in Fig. 3.

p^{'}

and

q

in Fig. 3 are defined as follows:

p^{'} = - (1 / 3) T^{'}

,

q = \sqrt{(3 / 2)} ‖ S ‖

,

S = T^{'} + p^{'} I

, where

T^{'}

denotes the effective stress tensor defined as positive in tension,

‖ ‖

denotes the norm of a tensor

(‖ A ‖ = \sqrt{A \cdot A} = \sqrt{A_{i j} A_{i j}}, {}^{\forall}A)

, and the operator “

\cdot

” denotes the inner product of the tensor.

ε_{a}

is the conventional axial strain. The effective stress path for the confining pressure value of 98.1 kPa resembles that of the typical remolded and lightly overconsolidated clays because the preconsolidation pressure is also 98.1 kPa. The effective stress path for the confining pressure value of 294.3 kPa resembles that of the undrained shear of the typical remolded and normal consolidation clays. The CSL in the mean effective stress

p^{'} -

deviator stress

q

plane can be represented as a straight line through the origin. Moreover, the soil shows the same characteristics as soils without cement, where M denotes the slope of the CSL and is referred to as critical state constant. When combined with the results of the oedometer test in Fig. 1, the remolded cement-treated soils show the same mechanical properties as the remolded clays without cement even though they cannot return to the base material.

Fig. 3. Undrained triaxial compression tests of a remolded cement-treated soil.

Fig. 4 shows the results of undrained triaxial compression tests of undisturbed cement-treated soil. The CSL obtained from the undrained shear test of the remolded cement-treated soil is also shown in the figure. For comparison, the undrained shear behavior of the undisturbed sample of naturally deposited clay sampled at the depth of 32–41 m at Urayasu is shown in Fig. 5 (Nakai et al. 2014). There are similarities in the shear behavior of the cement-treated soil and the naturally deposited clay. In particular, the softening behavior of naturally deposited clays with plastic compression (a decrease in the mean effective stress under the undrained condition) due to structural degradation, which is the typical behavior of highly structured clay, and is observed in cement-treated soils. This indicates that the skeleton structure concept (Asaoka et al. 2000, 2002) is valid for cement-treated soils. The undisturbed cement-treated soil eventually approached the CSL of the remolded cement-treated soil. Note that the reference condition of the cement-treated soil is considered for its remolded state.

Fig. 4. Undrained triaxial compression tests of an undisturbed cement-treated soil.

Fig. 5. Undrained triaxial compression tests of an undisturbed naturally deposited clay. (Data from Nakai et al. 2014.)

Furthermore, Fig. 4 shows an unusual behavior of the cement-treated soil not seen in naturally deposited clays, which is the point where plastic compression (a decrease in the mean effective stress under the undrained condition) occurs above the CSL. The decrease in mean effective stress occurs only below the CSL in Fig. 5, while it also occurs above in Fig. 4. The tension cutoff line (

q = 3 p^{'}

) is shown in Fig. 4. This line represents the state where the lateral effective stress is zero. When the soil under test can actually reach above this line, the triaxial test cannot cope with this type of stress state. A characteristic of the cement-treated soil is that the stress ratio is high enough to reach this line when the confining pressure is low; therefore, the test results near this line should be carefully monitored.

Elastoplastic Constitutive Model for Soils Considering Cementation Effects

This section presents a new method to introduce the cementation effect into elastoplastic constitutive models that can represent the mechanical behavior of naturally deposited soils based on the presented experimental facts. The proposed method was applied to the SYS Cam-clay model, which describes the function of the skeleton structure of the soil, to construct a constitutive model that can reproduce the mechanical behavior of cement-treated soil.

Proposal for Modeling the Cementation Effect Using the Extended Effective Stress

Outline of the Proposed Method

The key points of the proposed method are divided into the following three parts.

The upper chart of Fig. 6 shows the three loading surfaces of the SYS Cam-clay model: normal yield surface, superloading surface (Asaoka et al. 1998, 2000, 2002), and subloading surface (Hashiguchi 1989; Asaoka et al. 1997). The first key point of the proposed method is the translation of each loading surface in the negative direction on the mean effective stress axis, as shown in the lower chart of Fig. 6. This method makes the core constitutive model withstand some tensile stress like cement-treated soils. The equation,

{\overset{⌣}{η}}^{2} = M_{a}^{2}

shown in Fig. 6 indicates the threshold line between the plastic compression and the plastic expansion of this model, and Area A, shown in the lower chart, indicates the plastic compression region that is newly added by the translation. The existence of this region enables plastic compression behavior above the CSL, which is observed in the undrained shear test of the cement-treated soil.

Fig. 6. Translation of the three loading surfaces.

The second key point of the proposed method is to solve the translation of each loading surface when plastic deformation occurs. The experiment shows that the undisturbed cement-treated soils asymptotically approached the remolded state with plastic deformation, and the remolded cement-treated soils showed the same mechanical properties as the remolded sample of naturally deposited soils. This experimental fact requires that by undergoing plastic deformation, the model considering the cementation effect should return to the Cam-clay model that describes the mechanical behavior of the fully remolded and normally consolidated clays. To achieve this requirement, the translation of each loading surface is solved with plastic deformation. The amount of translation is treated as an internal state variable

Ψ (\geq 0)

, and the degradation process of the cementation effect is described by its evolution rule.

Ψ

is referred to as cementation in this study.

The third key point of the proposed method is to newly define the effective stress for the deformation of the cement-treated soil, which is the stress obtained by isotropically subtracting the translation amount

Ψ

from the effective stress tensor

T^{'}

according to the general definition, and to set the SYS Cam-clay model as the core constitutive model. This defined stress is called extended effective stress because it can be regarded as an extension of the effective stress principle. The translation of each loading surface creates the possibility that the mean effective stress

p^{'}

becomes negative, which represents Area B shown in the lower chart of Fig. 6. As the Cam-clay model is based on the

v - \ln p^{'}

relationship, it is necessary to prevent

p^{'}

from becoming negative. Describing the core constitutive models using the extended effective stress can help solve this problem. Moreover, as the isotropic component of the extended effective stress is larger than the normal mean effective stress by

Ψ

, the confining pressure increases by this amount in the extended model. Thus, the initial shear stiffness of the proposed model becomes greater than that of the core constitutive model. As

Ψ

decreases monotonically with plastic deformation and the extended effective stress becomes identical to the normal effective stress at

Ψ = 0

, the model with the cementation effect can perfectly return to the core constitutive model at the ultimate state. Thus, the experimental fact that the cement-treated soil has the same mechanical properties as the naturally deposited soil samples can be expressed.

Specific Procedures for Introduction of Cementation Effect

Because of introducing the extended effective stress, the specific procedures for introducing the cementation effect into the core constitutive model can be summarized as follows:

Scheme 1. Converts the normal effective stress into the extended effective stress using the internal state variable

Ψ

.

Scheme 2. Describes the core elastoplastic constitutive model using the extended effective stress and obtains the constitutive relation between the rate of the extended effective stress tensor and the stretching tensor.

Scheme 3. Obtains the constitutive relation between the rate of the normal effective stress and the stretching by the inverse transformation from the extended effective stress into the normal effective stress in the rate-type formula.

The evolution rule for the internal state variable

Ψ

describing the degradation process of cementation is incorporated into the model in the last inverse transformation scheme. One of the advantages of the proposed method is that the cementation effect can be introduced without forcing changes in the core constitutive model. In other words, the existing constitutive models can be directly used in Scheme 2.

In this study, although the SYS Cam-clay model is used as the core constitutive model for the formulation of the above procedure, the proposed method can be adopted to another existing constitutive model. That is to say, the proposed method possesses high adaptability. However, it is also important to understand that the potential of the core constitutive model determines the performance of the extended model.

Transformation from Normal Effective Stress to Extended Effective Stress Considering the Cementation Effect

Definition of Extended Effective Stress

The extended effective stress tensor

\overset{⌣}{T^{'}}

considering the cementation effect is defined as

\overset{⌣}{T^{'}} = T^{'} - Ψ I

(1)

where

Ψ (\geq 0)

= objective internal state variable, providing the magnitude of the translation of each loading surface from the origin. Note that

Ψ

has the same dimension as the stress. Eq. (1) serves as the conversion formula from

T^{'}

to

\overset{⌣}{T^{'}}

using

Ψ

.

Description of Core Constitutive Models Using the Extended Effective Stress

In this section, the SYS Cam-clay model, which is the core constitutive model, is described using the extended effective stress tensor

\overset{⌣}{T^{'}}

.

Additive Decomposition of Stretching

The stretching tensor

D

(defined as positive in tension) is additively decomposed into elastic

D^{e}

and plastic components

D^{p}

D = D^{e} + D^{p}

(2)

Yield Function

The yield function is expressed in terms of the extended effective stress

\overset{⌣}{T^{'}}

as

F (\overset{⌣}{T^{'}}, β, R^{*}, R, ε_{v}^{p}) = f ({\overset{⌣}{p}}^{'}, {\overset{⌣}{η}}^{*}) + MD \ln R^{*} - MD \ln R - ε_{v}^{p} = 0

(3)

f ({\overset{⌣}{p}}^{'}, {\overset{⌣}{η}}^{*}) = MD \ln \frac{{\overset{⌣}{p}}^{'}}{{\tilde{p}}_{c 0}^{'}} + MD \ln \frac{M^{2} + {\overset{⌣}{η}}^{* 2}}{M^{2}}

(4)

In these equations, D = dilatancy coefficient, defined as

D = (\tilde{λ} - \tilde{κ}) / {Mv}_{0}

, where

v_{0}

denotes the initial specific volume. The conventional volumetric strain

ε_{v}^{p}

, which is defined as positive in compression, can be calculated as

ε_{v}^{p} = (v_{0} - v) / v_{0} = - \int_{0}^{t} J tr D^{p} d τ

. Here,

J

denotes the volume change ratio, and it is defined as

J = {v/v}_{0} = \det F

, where

F

denotes the deformation gradient tensor of soil skeleton.

{\tilde{p}}_{c 0}^{'}

denotes the initial size of the normal yield surface and is given such that Eqs. (3) and (4) are satisfied at the initial state.

\tilde{λ}

,

\tilde{κ}

, and

M

are material constants, called compression index, swelling index, and critical state constant, respectively.

{\overset{⌣}{p}}^{'}

and

{\overset{⌣}{η}}^{*}

are invariants of

\overset{⌣}{T^{'}}

and are defined as

{\overset{⌣}{p}}^{'} = - (1 / 3) \overset{⌣}{T^{'}}

,

{\overset{⌣}{η}}^{*} = \sqrt{3 / 2} ‖ {\overset{⌣}{η}}^{*} ‖

,

{\overset{⌣}{η}}^{*} = \overset{⌣}{η} - β

,

\overset{⌣}{η} = \overset{⌣}{s} / {\overset{⌣}{p}}^{'}

, and,

\overset{⌣}{s} = {\overset{⌣}{T}}^{'} + {\overset{⌣}{p}}^{'} I

Among these invariants,

{\overset{⌣}{η}}^{*}

is invariant to represent the rotation of a yield surface.

β

denotes a back stress ratio tensor (or rotational hardening variable tensor), and its magnitude is expressed as

ζ = \sqrt{3 / 2} ‖ β ‖

.

R^{*} (0 < R^{*} \leq 1)

is the similarity ratio of the normal yielding surface to the superloading surface;

R (0 < R \leq 1)

is the similarity ratio of the subloading surface to the superloading surface. The similarity centers of each loading surface are at the origin of the extended effective stress space. In consequence, in the normal effective stress space, each loading surface is translated to the negative direction of the

p^{'}

axis

Ψ

from the origin.

Eq. (3) represents the subloading surface. Note that the subloading surface shrinks at an unloading state following a change in the extended effective stress so that

F = 0

is always satisfied.

Elastic Constitutive Model

The pressure-dependent rate-type Hooke’s law is adopted as the hypoelastic constitutive model

\overset{\circ}{\overset{⌣}{T^{'}}} = (\overset{⌣}{K} - \frac{2}{3} \overset{⌣}{G}) (tr D^{e}) I + 2 \overset{⌣}{G} D^{e} = \overset{⌣}{E} D^{e}

(5)

\overset{⌣}{E} = (\overset{⌣}{K} - \frac{2}{3} \overset{⌣}{G}) I \otimes I + \overset{⌣}{G} (I + \bar{I})

(6)

\overset{⌣}{K} = \frac{J v_{0}}{\tilde{κ}} {\overset{⌣}{p}}^{'}, \overset{⌣}{G} = \frac{3 (1 - 2 ν)}{2 (1 + ν)} \overset{⌣}{K}

(7)

where

ν

= Poisson’s ratio;

{(I)}_{i j k l} = δ_{i k} δ_{j l}

;

{(\bar{I})}_{i j k l} = δ_{i l} δ_{j k}

; and

δ_{i j}

= Kronecker delta;

{()}^{°}

= corotational rate, and the Green and Naghdi (1965) rate is used in this study; and

\otimes

= tensor product. As the elastic modulus is not proportional to

p^{'}

but to

{\overset{⌣}{p}}^{'}

, the confining pressure virtually increases by

Ψ

.

Flow Rule

The associated flow rule in the extended effective stress space is adopted, and the subloading surface is used as a plastic potential. The plastic stretching tensor can be given as

D^{p} = Λ \frac{\partial f}{\partial {\overset{⌣}{T}}^{'}}

(8)

where

Λ (> 0)

= plastic multiplier that can be determined by the consistency condition (Prager 1949).

Evolution Rules of Structure, Overconsolidation, and Induced Anisotropy

The evolution rules of

R^{*}

,

R

, and

β

are defined as

{\dot{R}}^{*} = J U^{*} (R^{*}) {(1 - c_{s}) D_{v}^{p} + c_{s} \sqrt{\frac{2}{3}} ‖ D_{s}^{p} ‖}, U^{*} (R^{*}) = \frac{a}{D} R^{* b} {(1 - R^{*})}^{c}

(9)

\dot{R} = J U (R) ‖ D^{p} ‖, U (R) = - \frac{m}{D} \ln R

(10)

\overset{°}{β} = J \frac{b_{r}}{D} \sqrt{\frac{2}{3}} ‖ D_{s}^{p} ‖ ‖ {\overset{⌣}{η}}^{*} ‖ {\overset{⌣}{η}}_{b}, {\overset{⌣}{η}}_{b} = m_{b} \frac{{\overset{⌣}{η}}^{*}}{‖ {\overset{⌣}{η}}^{*} ‖} - β

(11)

where

D_{v}^{p} = - tr D^{p}

and

D_{s}^{p} = D^{p} - (1 / 3) (tr D^{p}) I

. The parameters

a

,

b

,

c

,

c_{s}

,

m

,

m_{b}

, and

b_{r}

are material constants used for defining the evolution rate of each internal state variable. For detailed descriptions, refer to Asaoka et al. (2002).

Eqs. (9)–(11) can be simplified as

{\dot{R}}^{*} = Λ J r^{*}, r^{*} = U^{*} (R^{*}) {(1 - c_{s}) (- tr \frac{\partial f}{\partial \overset{⌣}{T^{'}}}) + c_{s} \sqrt{\frac{2}{3}} ‖ {(\frac{\partial f}{\partial \overset{⌣}{T^{'}}})}^{*} ‖}

(12)

\dot{R} = Λ J r, r = U (R) ‖ \frac{\partial f}{\partial \overset{⌣}{T^{'}}} ‖

(13)

\overset{°}{β} = Λ J b, b = \frac{b_{r}}{D} \sqrt{\frac{2}{3}} ‖ {(\frac{\partial f}{\partial \overset{⌣}{T^{'}}})}^{*} ‖ ‖ {\overset{⌣}{η}}^{*} ‖ \overset{⌣}{η}

(14)

where

{(\partial f / \partial \overset{⌣}{T^{'}})}^{*}

= deviatoric part of

\partial f / \partial \overset{⌣}{T^{'}}

.

As the extended effective stress is always on the subloading surface,

R

changes during unloading. At the unloading state,

R

is updated according to

R = R^{*} \frac{{\overset{⌣}{p}}^{'}}{{\tilde{p}}_{c 0}^{'}} \frac{M^{2} + {\overset{⌣}{η}}^{* 2}}{M^{2}} \exp (- ε_{v}^{p})

(15)

which is derived from

F = 0

.

Plastic Multiplier

By substituting Eqs. (5), (8), (12), (13), and (14) into the consistency condition Eq. (3), that is,

\dot{F} = 0

, the plastic multiplier can be derived as

Λ = \frac{\frac{\partial f}{\partial {\overset{⌣}{T}}^{'}} \overset{⌣}{E} D}{J \frac{MD}{(M^{2} + {\overset{⌣}{η}}^{* 2}) {\overset{⌣}{p}}^{'}} (M_{s}^{2} - {\overset{⌣}{η}}^{2}) + \frac{\partial f}{\partial {\overset{⌣}{T}}^{'}} \cdot \overset{⌣}{E} \frac{\partial f}{\partial {\overset{⌣}{T}}^{'}}}

(16)

M_{s}^{2} = M_{a}^{2} + 3 {\overset{⌣}{p}}^{'} \overset{⌣}{η} \cdot b - (M^{2} + {\overset{⌣}{η}}^{* 2}) {\overset{⌣}{p}}^{'} (\frac{r^{*}}{R^{*}} - \frac{r}{R})

(17)

M_{a}^{2} = M^{2} + ζ^{2}

(18)

where

M_{s}

and

M_{a}

= threshold values of hardening/softening and plastic compression/extension, respectively. Hardening occurs when

M_{s}^{2} > η^{2}

, and softening occurs when

M_{s}^{2} < η^{2}

. Moreover, plastic compression occurs when

M_{a}^{2} > η^{2}

, and plastic expansion occurs when

M_{a}^{2} < η^{2}

. See Asaoka et al. (2002) for a detailed discussion of these aspects.

Elastoplastic Constitutive Model

The rate-type elastoplastic constitutive model is obtained by substituting Eqs. (2), (8), and (16) into Eq. (5) as

\overset{\circ}{{\overset{⌣}{T}}^{'}} = \overset{⌣}{E} D - Λ \overset{⌣}{E} \frac{\partial f}{\partial {\overset{⌣}{T}}^{'}} = {\overset{⌣}{C}}^{e p} D

(19)

{\overset{⌣}{C}}^{e p} = \overset{⌣}{E} - \frac{\overset{⌣}{E} \frac{\partial f}{\partial \overset{⌣}{T^{'}}} \otimes \overset{⌣}{E} \frac{\partial f}{\partial \overset{⌣}{T^{'}}}}{J \frac{MD}{(M^{2} + {\overset{⌣}{η}}^{* 2}) p^{'}} (M_{s}^{2} - {\overset{⌣}{η}}^{2}) + \frac{\partial f}{\partial \overset{⌣}{T^{'}}} \cdot \overset{⌣}{E} \frac{\partial f}{\partial \overset{⌣}{T^{'}}}}

(20)

Loading Criterion

Based on the assumption that the denominator of Eq. (16) takes a positive value (Asaoka et al. 1994), the loading criterion is provided by

\frac{\partial f}{\partial \overset{⌣}{T^{'}}} \cdot \overset{⌣}{E} D > 0 \Rightarrow D^{p} \neq 0 \frac{\partial f}{\partial \overset{⌣}{T^{'}}} \cdot \overset{⌣}{E} D \leq 0 \Rightarrow D^{p} = 0

(21)

Equation of Internal State Variables

The initial values of the stress, specific volume, and internal state variables should be given such that the following equation is satisfied

v = N - \tilde{λ} \ln \frac{{\overset{⌣}{p}}^{'}}{p_{r}^{'}} - (\tilde{λ} - \tilde{κ}) \ln \frac{R^{*}}{R} \frac{M^{2} + {\overset{⌣}{η}}^{* 2}}{M^{2}}

(22)

where N = specific volume on the normal consolidation line (NCL) of the remolded cement-treated soil at

{\overset{⌣}{p}}^{'} = p_{r}^{'}

(

p_{r}^{'}

denotes the reference stress). In addition, as already mentioned before,

{\tilde{p}}_{c 0}^{'}

should be determined from Eqs. (3) and (4) at the initial state. If the initial values are provided in this manner, each state variable satisfies Eq. (22). In this study, this equation refers to the equation of internal state variables.

Rewriting Constitutive Relationship by the Inverse Transformation of the Extended Effective Stress

Eq. (19) represents the relationship between

\overset{°}{\overset{⌣}{T^{'}}}

and

D

. As such, it is necessary to rewrite the equation as a relationship between

\overset{°}{T^{'}}

and

D

to apply it to a soil–water coupled finite-element analysis code. This is accomplished by transforming the extended effective stress into the normal effective stress using Eq. (1) and obtaining the correlation rate of normal effective stress. The cementation degradation process is described as the evolution rule of

Ψ

.

Evolution Rule of $Ψ$

From the aforementioned experimental results, the cementation effect is degraded with plastic deformation. Therefore, the following evolution rule is obtained such that

Ψ

asymptotically approaches zero due to plastic deformation

\dot{Ψ} = J V (Ψ) ‖ D^{p} ‖, V (Ψ) = - \frac{d}{D} Ψ

(23)

where

V (Ψ)

= monotonically decreasing function that satisfies

V (0) = 0

; and

d

= material parameter controlling the degradation rate of

Ψ

. Since

Ψ

is an objective scalar, its evolution rule, which is defined by Eq. (23), is satisfied with the principle of objectivity can be written as follows:

\dot{Ψ} = Λ J ψ, ψ = V (Ψ) ‖ \frac{\partial f}{\partial \overset{⌣}{T^{'}}} ‖

(24)

Elastoplastic Constitutive Model

The extended effective stress can be inversely transformed into the normal effective stress using Eq. (1). By substituting Eqs. (19) and (24) into the rate-type formula in Eq. (1), the following linear relationship for

\overset{°}{T^{'}}

and

D

is obtained:

\overset{°}{T^{'}} = \overset{°}{\overset{⌣}{T^{'}}} + \dot{Ψ} I = {\overset{⌣}{C}}^{e p} D + Λ J ψ I = C^{e p} D

(25)

where

C^{e p}

= continuum tangent modulus and is expressed as

C^{e p} = \overset{⌣}{E} - \frac{(\overset{⌣}{E} \frac{\partial f}{\partial \overset{⌣}{T^{'}}} - J ψ I) \otimes \overset{⌣}{E} \frac{\partial f}{\partial \overset{⌣}{T^{'}}}}{J \frac{MD}{(M^{2} + {\overset{⌣}{η}}^{* 2}) \overset{⌣}{p^{'}}} (M_{s}^{2} - {\overset{⌣}{η}}^{2}) + \frac{\partial f}{\partial \overset{⌣}{T^{'}}} \cdot \overset{⌣}{E} \frac{\partial f}{\partial \overset{⌣}{T^{'}}}}

(26)

Note that

C^{e p}

is asymmetric, as shown in Eq. (26). During unloading, the second term of Eq. (26) is zero, that is,

C^{e p} = \overset{⌣}{E}

.

Since the SYS Cam-clay model, which is the core constitutive model, and the evolution rule of

Ψ

are satisfied with the principle of objectivity, the expanded constitutive model is also satisfied with the principle.

\overset{⌣}{T^{'}}

is identical to

T^{'}

when

Ψ = 0

. In this case, the condition

C^{e p} = {\overset{⌣}{C}}^{e p}

is satisfied because

ψ

is also equal to zero. This implies that when the cementation effect completely degrades, the extended constitutive models perfectly return to the core constitutive model. However, the core constitutive model does not describe the mechanical behavior of the base material; instead, it describes that of the fully remolded cement-treated soil.

Simulation of Element Tests with the Extended SYS Cam-Clay Model

The proposed method is validated through simulations of the experimental results shown in the section “Experimental Consideration: Similarities and Differences between Cement-Treated Soils and Naturally Deposited Clays” using the extended SYS Cam-clay model performed under a uniform deformation field.

Analysis Conditions

Table 2 lists the material constants used in the calculations. The elastoplastic parameters were determined from the tests of the remolded cement-treated soil shown in Figs. 1 and 3. The evolution parameters were determined so that the model can simulate the tests of the undisturbed cement-treated soil shown in Figs. 1 and 4 adequately. The same material constants were used for the undisturbed and remolded cement-treated soil.

Table 2. Material constants

Parameter name	Symbol	Value
Elasto-plastic parameters
Compression index	$\tilde{λ}$	0.590
Swelling index	$\tilde{κ}$	0.050
Critical state constant	M	1.850
NCL intercept	N	4.700
Poisson’s ratio	$ν$	0.300
Evolution parameters
Degradation index of overconsolidation	$m$	0.7
Degradation indices of structure ( $b = c = 1.0$ )	$a$	0.5
Ratio of $D_{v}^{p}$ to $D_{s}^{p}$	$c_{s}$	0.5
Rotational hardening index	$b_{r}$	0.01
Limitation of rotational hardening	$m_{b}$	0.35
Degradation index of cementation	$d$	0.15

Table 3 shows the initial values of the state variables. Cases with and without cementation will be compared to evaluate the effect of the proposed method. The initial overconsolidation ratio was determined from other initial values using Eq. (22). For the triaxial tests, simulations were conducted, including the isotropic consolidation process.

Table 3. Initial values

Variable name	Symbol	With cementation	Without cementation	Remolded state
			Value
		Undisturbed state
Specific volume	$v_{0}$	5.1	5.1	5.1
Vertical effective stress (kPa)	$σ_{v 0}^{'}$	10.0	10.0	10.0
Stress ratio	$η_{0}$	0.0	0.0	0.0
Degree of anisotropy	$ζ_{0}$	0.0	0.0	0.0
Degree of structure	$1 / R_{0}^{*}$	10.0	10.0	1.0
Overconsolidation ratio	$1 / R_{0}$	2.9	57.8	5.8
Cementation (kPa)	$Ψ_{0}$	150.0	0.0	0.0

Simulation of Oedometer Tests

Fig. 7 shows the calculated and experimental results of the oedometer tests. The calculated and experimental results of the remolded cement-treated soil are also shown in the figure. Moreover, this figure includes graphs showing the evolution of the main internal state variables (

Ψ

,

R^{*}

, and

R

). The results can capture the experimental trends, such as the asymptotic behavior of the undisturbed cement-treated soil toward the remolded cement-treated soil. However, the presence or the absence of the cementation effect does not significantly affect the results. This is because the effect of the translation of each loading surface is difficult to achieve at low stress ratios and is relatively diminished for high confining pressure. Even in the no-cementation case, the difference in one-dimensional compression behavior between undisturbed cement-treated soil and remolded soil is well simulated, which is mainly due to the effect of the superloading surface. In other words, it is important to consider the structure concept to simulate the mechanical behavior of cement-treated soil. The effect of cementation will be revealed in the simulation of the triaxial test shown in the following section.

Fig. 7. Simulation of oedometer tests: (a) with cementation effect; and (b) without cementation effect.

Simulation of Undrained Triaxial Compression Tests

Fig. 8 shows the calculated and experimental results of the undrained triaxial compression tests. Two calculations have been performed for the same experimental results to show the effects of the expansion: (a) with cementation effect (

Ψ_{0} \neq 0

) and (b) without cementation effect (

Ψ_{0} = 0

). The figure also includes graphs showing the evolution of the internal state variables (

Ψ

,

R^{*}

,

R

, and

‖ β ‖ / m_{b}

). Softening behavior with plastic compression and hardening behavior with plastic expansion occur. These behaviors are represented by the effects of the superloading and subloading surfaces, which are instinct functions of the SYS Cam-clay model (Asaoka et al. 1994, 2002). The plastic compression (decrease in the mean effective stress under the undrained condition) occurs above the CSL in the

p^{'} - q

space for the case with the cementation effect. This is the effect of the translation of each loading surface. As the degradation of cementation occurs with plastic deformation, the effective stress path approaches the CSL of the Cam-clay model (

q = M p^{'}

). For the results of 100 kPa and 300 kPa, a discrepancy exists between the experiments and the numerical results. However, as described in the section “Undrained Triaxial Tests,” the triaxial test equipment under a constant cell pressure cannot reach values above the tension cutoff line (

q = 3 p^{'}

), and therefore, a simple comparison cannot be performed. The comparison between Figs. 8(a and b) indicates that the high initial stiffness at a low confining pressure is accurately reproduced in the case with cementation. This is because the confining pressure is practically increased by applying the extended effective stress. The decreasing rate of

q

after the peak in the deviator stress

q

–axial strain

ε_{a}

relationship is more rapid in the experiment than in the analysis. In the experiments, the shear band appeared in the specimen, and the loss of uniformity may have caused a sudden drop in the deviator stress (Asaoka and Noda 1995).

Fig. 8. Simulation of undrained triaxial compression tests: (a) with cementation effect; and (b) without cementation effect.

Simulation of Triaxial Tests as Initial Value and Boundary Value Problem

At a lower level of axial displacement, since the uniformity of a specimen is relatively preserved, the specimen can be considered to exhibit elemental behavior, but as the axial displacement increases, since internal strain localizes or shear bands develop, the nonuniformity of the specimen can be clearly observed. Therefore, it is more reasonable to consider the triaxial test as a solution to the initial and boundary value problem. In the stress-strain relationship shown in Fig. 8, the experiments showed a more rapid decrease in the load than in the analytical results. This section shows that this behavior can be explained as an effect of the formation of a shear band by solving a soil–water coupled boundary value problem for a triaxial test to validate the developed constitutive model from a different perspective than section “Simulation of Element Tests with the Extended SYS Cam-Clay Model.”

Analysis Conditions

The SYS Cam-clay model with the cementation effect was implemented in GEOASIA, which is a soil–water coupled finite deformation analysis code developed by Noda et al. (2008). The target is a cylindrical specimen of the same size (10 cm in height and 5 cm in diameter) as the specimens used in the experiments in the section “Experimental Consideration: Similarities and Differences between Cement-Treated Soils and Naturally Deposited Clays.” Fig. 9 shows the finite-element mesh used for the analysis. As shown in the figure, a three-dimensional analysis was performed. The surface of the specimen was set to an undrained boundary. The horizontal motion was fixed only at a node located in the center of the bottom end face, and the top and bottom end faces were set to no friction. A uniform vertical downward velocity was provided on the top end-surface. The axial strain rate was set at 20%/day, as in the experiment. However, as the analysis code considers the inertia force, the loading rate was linearly increased from zero to a predetermined axial strain rate for the first 100 s, and then, the condition is switched to a constant rate condition. The initial imperfection was given by shifting the fixed point on the bottom surface from the center. In the analysis code that considers inertial forces, this operation can induce asymmetric deformation and the formation of a shear band due to the loss of axisymmetry in the generated acceleration. The material constants are the same as those used in the simulations in the uniform deformation field. The confining pressure was set to 500 kPa, and the initial value was set as the same as that at the start of the simulation of the shear process in the uniform deformation field. The permeability coefficient was set to

2.0 \times 10^{- 6} cm / \sec

based on the value obtained from the oedometer test shown in Fig. 1. The gravity is not considered. The particle density of the cement-treated soil was set to

2.757 g / {cm}^{3}

.

Calculation Result

Fig. 10 shows the shear strain distribution. The strains were localized, and a shear band appeared. Fig. 11 shows the behavior of the specimen, which was calculated similarly as in the experiment, considering the specimen as an element. Deviator stress

q

was calculated considering the cross-sectional correction, and

ε_{a}

denotes the nominal axial strain. For the comparison, the response of the constitutive models in the uniform deformation field is also shown in Fig. 11. In the

q - ε_{a}

relationship that is obtained from the initial and boundary value problem, an imperfection-sensitive bifurcation occurred near the peak of

q

. Moreover, a clearer load drop occurred than that at uniform deformation. Although the decline rate of

q

does not exactly coincide with the experiment due to the dependency of the generated bifurcation patterns on the initial imperfections, the same tendency is observed in the experiment shown in Fig. 8(a). Fig. 12 shows an example of the shear bands observed in the triaxial test of the cement-treated soil. In the experiment, a shear surface similar to the numerical result was generated using the cement-treated soil. The calculation result shows that the loss of the uniformity of the specimen induces a more rapid decrease in the shear stress when compared to the constitutive model. The rapid load drop was not quantitatively modeled in this study since the possibility of the shear band formation is particularly high in materials, which can cause softening (e.g., Noda et al. 2007; Yamada et al. 2013). Careful consideration should be paid when describing such behavior at the level of the constitutive models.

Fig. 11. Effect of shear band formation on a triaxial test.

Fig. 12. Shear band in cement-treated soil specimen.

Conclusions

In this study, the mechanical properties of the cement-treated soil were studied by element tests, and a new method to introduce the cementation effect into existing elastoplastic constitutive models for naturally deposited soils was proposed. This method was applied to the SYS Cam-clay model, which is an elastoplastic constitutive model based on the skeleton structure concept. The effectiveness of the proposed method was validated by simulating element tests using this constitutive model. A soil–water coupled finite deformation analysis was performed for triaxial tests to confirm the effect of shear band formation on an element test. The conclusions of this study can be summarized as follows:

1.

The remolded sample of cement-treated soils had the same mechanical properties as the remolded sample of naturally deposited clay. However, the mechanical properties (material constants) of the remolded sample of cement-treated soils differed from that of the base material without cement.

2.

Undisturbed cement-treated soil has a large pore structure, and it can take a state in the impossible region of the remolded soil; however, as the plastic deformation progresses, it approaches the fully remolded state. This is the same mechanical property as found in naturally deposited clays.

3.

Undisturbed cement-treated soils exhibit plastic compression above the CSL. This behavior is unique to cement-treated soils, and it is not found in naturally deposited clays.

4.

To introduce the cementation effect into existing elastoplastic constitutive models that can describe the mechanical behavior of naturally deposited clays, loading surfaces were translated parallel to the negative direction of the mean effective stress axis. The amount of the translation

Ψ

was referred to as cementation, and it was treated as an internal state variable. In addition, the extended effective stresses tensor that is effective for the deformation of the cement-treated soils were newly defined using

Ψ

. The mechanical behavior of cement-modified soils can be described by applying the extended effective stress tensor to the existing elastoplastic constitutive models.

5.

The extended effective stress application enables us to describe the mechanical behavior of cement-treated soils such as the plastic compression above the CSL and high initial stiffness.

6.

The evolution rule of the internal state variable

Ψ

was proposed. This evolution rule describes the degradation process of the cementation due to plastic deformation. Further, it describes the asymptotic approach of undisturbed cement-treated soil to remolded cement-treated soil. The constitutive models extended by the proposed method can return to the core constitutive model when cementation completely degrades.

7.

To reproduce the mechanical characteristics of the cement-treated soils, the performance of core constitutive models is also required. In particular, introducing the skeleton structure concept of the superloading surface into the core constitutive model is useful to express that the undisturbed cement-treated soil is bulkier than the remolded cement-treated soil.

8.

A rapid decrease in the shear stress is often observed in element tests on cement-treated soils. It was confirmed that this behavior can appear because of the effect of the formation of a shear band caused by solving a soil–water coupled boundary value problem. Thus, careful judgment is required to describe such behavior at the level of constitutive models.

Although the cement-treated soil used in the experiments was artificially prepared in our laboratory, this method can be potentially applied to the naturally deposited soil that has acquired cementation due to certain natural chemical reactions. However, the scope of this paper is limited to modeling the degradation process of the cementation with the plastic deformation from the state where the solidification action is completed. The modeling of the solidification process will be addressed in the future.

Data Availability Statement

All data, models, and code generated or used during the study appear in the published article.

Acknowledgments

This study was supported by JSPS Grants-in-Aid for Scientific Research (Grant Nos. JP16H04408 and JP19H02402).

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Information & Authors

Information

Published In

Journal of Geotechnical and Geoenvironmental Engineering

Volume 148 • Issue 5 • May 2022

Copyright

This work is made available under the terms of the Creative Commons Attribution 4.0 International license, https://creativecommons.org/licenses/by/4.0/.

History

Received: Dec 17, 2020

Accepted: Sep 29, 2021

Published online: Feb 22, 2022

Published in print: May 1, 2022

Discussion open until: Jul 22, 2022

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Shotaro Yamada https://orcid.org/0000-0002-7024-6327 [email protected]

Associate Professor, Dept. of Civil and Environmental Engineering, Tohoku Univ., Sendai, Miyagi 980-8579, Japan (corresponding author). ORCID: https://orcid.org/0000-0002-7024-6327. Email: [email protected]

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Takayuki Sakai [email protected]

Assistant Professor, Dept. of Civil and Environmental Engineering, Nagoya Univ., Nagoya, Aichi 464-8603, Japan. Email: [email protected]

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Masaki Nakano [email protected]

Professor, Dept. of Civil and Environmental Engineering, Nagoya Univ., Nagoya, Aichi 464-8603, Japan. Email: [email protected]

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Toshihiro Noda [email protected]

Professor, Dept. of Civil and Environmental Engineering, Nagoya Univ., Nagoya, Aichi 464-8603, Japan. Email: [email protected]

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Cited by

Kang-fu Jiao, Chao Zhou, A Bounding Surface Model for Cemented Soil at Small and Large Strains, Journal of Geotechnical and Geoenvironmental Engineering, 10.1061/JGGEFK.GTENG-11604, 150, 2, (2024).
Abstract
Shotaro Yamada, Toshihiro Noda, Finite-Deformation Cam-Clay Model Considering Elastic Dilatancy and Soil Skeleton Structure, International Journal of Geomechanics, 10.1061/IJGNAI.GMENG-7117, 23, 2, (2023).
Abstract
Keitaro Hoshi, Shotaro Yamada, Takashi Kyoya, Finite Element Analysis of Expansive Bedrock Considering Electro-chemo-mechanical Coupling Phenomena in Crystal Layers of Clay Minerals and Internal Structural Degradation, Rock Mechanics and Rock Engineering, 10.1007/s00603-022-02997-3, 55, 12, (7387-7407), (2022).
Crossref

Abstract

Introduction

Experimental Consideration: Similarities and Differences between Cement-Treated Soils and Naturally Deposited Clays

Physical Properties of the Base Material and Blending Condition for Cement-Treated Soil

Oedometer Tests

Undrained Triaxial Tests

Elastoplastic Constitutive Model for Soils Considering Cementation Effects

Proposal for Modeling the Cementation Effect Using the Extended Effective Stress

Outline of the Proposed Method

Specific Procedures for Introduction of Cementation Effect

Transformation from Normal Effective Stress to Extended Effective Stress Considering the Cementation Effect

Definition of Extended Effective Stress

Description of Core Constitutive Models Using the Extended Effective Stress

Additive Decomposition of Stretching

Yield Function

Elastic Constitutive Model

Flow Rule

Evolution Rules of Structure, Overconsolidation, and Induced Anisotropy

Plastic Multiplier

Elastoplastic Constitutive Model

Loading Criterion

Equation of Internal State Variables

Rewriting Constitutive Relationship by the Inverse Transformation of the Extended Effective Stress

Evolution Rule of Ψ

Elastoplastic Constitutive Model

Simulation of Element Tests with the Extended SYS Cam-Clay Model

Analysis Conditions

Simulation of Oedometer Tests

Simulation of Undrained Triaxial Compression Tests

Simulation of Triaxial Tests as Initial Value and Boundary Value Problem

Analysis Conditions

Calculation Result

Conclusions

Data Availability Statement

Acknowledgments

References

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Evolution Rule of $Ψ$