Diss

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Hypoplastic and Viscohypoplastic

Constitutive Models for

Geotechnical Problems

zur Erlangung des akademischen Grades eines DOKTOR-INGENIEURS

von der Fakult¨at f¨ur

Bauingenieur-, Geo- und Umweltwissenschaften der Universit¨at Fridericiana zu Karlsruhe (TH)

genehmigte DISSERTATION

von Thomas Meier

aus Karlsruhe

Hauptreferent: em. Prof. Dr.-Ing. Dr.h.c. Gerd Gudehus, Karlsruhe

Korreferenten: Univ. Prof. Dr.-Ing. habil. Theodoros Triantafyllidis, Karlsruhe Univ.-Prof. Dr.-Ing. habil. Ivo Herle, Dresden

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Introduction

In geotechnical engineering practice in many areas purely empirical approaches are ap-plied until today, e.g. interpretation of cone penetration tests (CPT) or dimensioning and control of deep vibratory compaction measures. The difficulties this brings should be clear. On the other hand more and more complex constitutive models have been devel-oped in the academic world together with techniques for the solution to boundary value problems (e.g. finite element method, discrete element method, finite difference method). Generally, these more complex soil models and solution techniques require more input data to produce realistic calculation results (e.g. spatial displacements, density changes, earth pressure distributions), i.e. more material parameters and more information about the in-situ soil state in terms of the spatial distribution of the material, density, effective and total stresses and in some cases even of the most recent deformation history prior to construction. It is clear that if these requirements shall be met, this makes extensive in situ and laboratory investigations necessary, which is expensive. It is also true, that even if all the demanded data is provided, there are not only limitations entailed in the constitutive models but also many ”adjusting screws” in conjunction with the numeri-cal solution techniques. Using the finite element method for example, the choice of the spatial discretization (FE mesh), the element types, boundary conditions, the integration scheme and accuracy tolerances has an impact on the computation results and requires conscientious examinations.

Considering everything mentioned above, the author can understand the mistrust of many practical engineers with regard to the colourful images presented at practically every workshop or conference. But this is only half of the truth. The author had the privilege to participate in a ”Workshop on Nonlinear Modeling of Geotechnical Problems: From Theory to Practice” at Johns Hopkins University, Maryland in 2005 funded by the U.S. National Science Foundation. One statement of a practitioner was very revealing: ”If

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3

there are more than three parameters entailed in a model, I cannot use it in practice.” What does that mean? Practice wants easy to use, quick to learn, affordable and of course reliable tools. This may be comprehensible but the mathematical description of mechanical behaviour of soil simply is very complex. In some cases the application of simple elstoplastic models together with experiences from comparable projects is surely enough, but in general it is not.

It is the responsibility of the scientists to provide tools that meet the requirements of practice as far as possible on a sound physical basis and what is even more important to provide the knowledge required to use these tools in a comprehensible way. Of course this must include not only showing the strong points but also the shortcomings and limitations of a model and/or method. And even though science is a competitive business itself, the gained knowledge must be shared for the best of all parties involved in geotechnical engineering.

The main objective of the thesis at hand is to contribute to closing the gap between theory and practice. All projects described in Chapter 3 to 6 of thesis were cooperations with with major international foundation engineering companies, which shows that modern soil mechanics can provide new models, tools and techniques that can be beneficial for geotechnical practice, today and in the future. It is the hope of the author, that this humble contribution may be helpful for the work of geotechnical engineers in both theory and practice.

In Chapter 2 the two advanced constitutive models used by the author, i.e. hypoplas-ticity and viscohypoplashypoplas-ticity, are described in short. The corresponding procedures for the determination of the model parameters based on results of standard soil mechani-cal laboratory tests on disturbed soil samples are explained in detail. Parametric studies demonstrate the influence of each of the material parameters on the mechanical behaviour as predicted by the two constitutive models. In the case of viscohypoplasticity (for the mathematical description of the mechanical behaviour of clayey soils) eight different ma-terials were investigated extensively and correlations between index properties, such as the liquid limit, and material constants are proposed.

When applying hypoplasticity and viscohypoplasticity, e.g. for finite element analyses of boundary value problems, such as construction pits, it is important to know not only the material constants (e.g. the critical friction angle) but also the in-situ state of the ground in terms of density and the stress state, as accurate as possible. For this reason Chapter 3 and 4 are dedicated to cone penetration test interpretation methods (with respect to density) for cohesionless and clayey soils, respectively.

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bench-marked using the results of a series of CPT calibration chamber tests on quartz sands with different shell (calcite) contents. The same holds for two purely empirical methods, which are both applied in geotechnical practice (Schmertman and German standard DIN 4094). It is concluded that the two methods proposed by Cudmani and by Schmertman performed well in the case of the investigated materials, whereas DIN 4094 should be used with caution as it neglects for example the vital influence of the effective stress state on the penetration resistance.

For the interpretation of results of CPTs in clayey soils (Chap. 4) a new method based on finite element simulations using viscohypoplasticity is proposed. The applied FE model is validated using results of CPTU calibration chamber tests on two different clayey materi-als. It is shown that the FE results can be approximated with a simple equation containing two model parameters. These were determined for all cohesive soils investigated in the framework of this thesis and correlations with index properties and viscohypoplastic mate-rial parameters are proposed. The applicability and performance of the proposed method is demonstrated using field data from Vancouver, Canada.

In Chapter 5.1 a novel technique for the dimensioning and control of deep vibratory compaction measures is proposed based on numerical simulations using hypoplasticity (including the CPT interpretation method by Cudmani, Chap. 3). The simulation of the vibratory compaction process is divided into three steps. A non-linear soil–vibrator interaction model provides the ”’dynamic force”‘ of the vibrator, depending on vibrator parameters (e.g. geometry, frequency, unbalanced mass) and the surrounding soil. In the next step the one-dimensional dynamic cylindrical cavity expansion due to a sinusoidal cavity pressure in a hypoplastic continuum is simulated yielding radial and hoop stresses (amplitudes and mean values) at different distances to the vibrator axis. In the last step cyclic truly triaxial element test simulations are carried out yielding the sought-after change in density. A class A prediction is presented, which yielded a spacing of the com-paction points, which was close to the one carried out in-situ, despite the simplifications entailed in the procedure and the rough estimate of the material parameters.

In Chapter 6 two FE applications using viscohypoplasticity are presented. In the first example (Chap. ) buckling of slender pile-like structures in soft clayey soils is investigated. On behalf of Bauer Spezialtiefbau (Schrobenhausen/Germany) a finite element model using viscohypoplasticity was applied to examine under which circumstances buckling of an auger during the installation of granular soil stabilization columns in very soft soils can occur. On the basis of a parameter study it is concluded, that criteria for the necessity of a proof of safety against buckling in terms of the undrained shear strength may be misleading and that compressibility of the soil and geometrical imperfections of the structure (auger) are the decisive quantities.

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5

Chapter 6.2 presents a two-dimensional FE analysis of an open cut, realized downtown Chicago in the years 1947-1948 as reported by Wu and Berman [52]. Their description of subsurface conditions, construction procedure and the structural parts rendered possible a realistic finite element modeling. The cut was part of the western extension of the initial subway system. The agreement between measured and calculated strut forces is acceptable. Furthermore, it is shown that strut forces may increase with time due to creep relaxation. This effect, which was actually observed in-situ is also qualitatively obtained from the numerical simulations.

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Constitutive Modelling and

Parameter Determination

2.1 Introduction

A general description of the two constitutive models applied in this thesis ”Application of Hypoplastic and Viscohypoplastic Constitutive Models for Geotechnical Problems” is given in this chapter. The main attention is on the procedure for the determination of the material parameters from laboratory tests and on post test numerical simulations.

2.2 Hypoplastic Constitutive Equation

The mechanical behaviour of cohesionless soil is modelled by a hypoplastic constitutive equation. It describes the changes in stress of a simple grain skeleton due to rearrangement of the grains. A simple grain skeleton is characterized by the following conditions [16]:

1. The state is defined only by the grain stress tensor σ and density (void ratio e). 2. The grains are permanent, i.e. there is no abrasion and grain fracturing.

3. There is an upper (ei) and a lower limit (ed) of the void ratio – macropores are not

allowed for – depending on the effective mean pressure p0 _{= −trσ}0_{/3. The same}

holds for the critical void ratio ec, which is reached after large monotonic shearing.

4. Independently from the initial state proportional deformation paths lead asymptot-ically to proportional stress paths.

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2.2. Hypoplastic Constitutive Equation 7

5. The mechanical behaviour of the grain skeleton is rate-independent and the principle of effective stress holds.

6. Cementation and attractive/repulsive forces of the grain contacts are negligible.

A general form of a constitutive equation for a simple grain skeleton is

˙σ = f(σ, ˙ε, e) (2.1)

Thereafter, the stress rate ˙σ is a function of the stress state σ itself, density (void ratio e) and the strain rate ˙ε. The function f is 1st _{order homogenous with respect to the strain} rate (f(λ ˙ε) = λf( ˙ε), i.e. rate independence) and directionally homogenous w.r.t. stress (~f(λσ) = λ~f(σ)). Note that the strain tensor ε does not appear in (2.1). In this work, the version of the hypoplastic constitutive equation by von Wolffersdorff (2.1) is applied. For detailed descriptions of the model please refer to e.g. [51], [17], [18] or [38]. The complete mathematical formulation is given in Appendix A.1.1.

One advantage of this model is the rather simple determination of the material parameters (see next subsection) by means of standard laboratory tests, i.e. shear (triaxial, simple or direct shear) and oedometric tests. With one set of parameters the mechanical behaviour of a cohesionless granular material can be described realistically over a wide range of soil states relevant for geotechnical engineering purposes.

2.2.1 Procedure for the Determination of the Material

Param-eters

1. Limit Void Ratios ed0, ec0 and ei0

ed0 ≈ emin is a lower bound void ratio of a grain skeleton at zero pressure, ec0 ≈ emax

is the void ratio in a critical state at zero pressure. Both, emin and emax are determined

through standard index tests (e.g. according to ASTM D4254 and D4253). ei0is an upper

bound void ratio of a simple grain skeleton at vanishing pressure (without macropores).

ei0 ≈ 1.15·ecoand ed0≈ 0.6ec0are simple estimates [16] which have been used successfully

for more than a decade.

2. Critical Friction Angle ϕ_c

The critical friction angle ϕc determines the resistance of a granulate subjected to

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tr( ˙ε) = 0 hold. Appropriate for the determination of ϕcare drained or undrained triaxial

tests, simple shear or direct shear tests on initially very loose specimens (e ≈ emax).

The simplest, fastest (and therewith cheapest) and yet reliable estimate is the angle of repose (Fig. 2.1). A hopper is lifted slowly without loosing contact with the forming cone of dried granular material. On the surface of the cone the granulate is subjected to large monotonic shearing, therewith it may be assumed to be in a critical state. The inclination of the cone, and hence the critical friction angle, can be determined by measuring the height and diameter of the cone (tan ϕc = 2h/d) or with a set of stencils (increments of

0.5◦ _{are sufficient).}

ϕ

c

Figure 2.1: Angle of repose test.

3. Granulate Hardness hs and Exponent n

a) In the case of an isotropic compression of a very loose sample (e0 = emax) the

hypoplastic constitutive equation reduces to the compression law by Bauer [2]:

e = e0· exp " − Ã 3 · p0 hs !_n# (2.2)

where a referential pressure, the so-called granulate hardness hs, together with the

exponent n govern the compression for an increasing effective mean pressure p0_.

For simplicity hs and n are determined from conventional oedometric compression

tests (OCT). However, it is very laborious to install a sample with e0 ≈ emax by

moist placement [48]. Thus, hsand n obtained by curve fitting (least square method)

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2.2. Hypoplastic Constitutive Equation 9

b) The next step is to simulate the OCT with the aid of a hypoplasticity element test programme (ETP) and to adjust hs and n to match the test results.

Note: This pair of material constants also influences the curve progressions of triaxial compression test (TXC) simulation results.

4. Exponent α

The exponent α (cf. Eqn.(A.10) and (A.12)) controls the peak friction angle of the material, and hence also the dilatancy behaviour. Using the results of a drained triaxial test on an initially dense sample, α can be calibrated with the aid of the ETP. But as mentioned above, hs and n also influence the calculation results of the triaxial test

simulation, thus it might be necessary to adjust these values, too. 5. Exponent β

The stiffness of a grain skeleton with e < ec can be adjusted via the exponent β (cf.

Eqn.(A.11) and (A.12)). It can be calibrated with the aid of ETP simulations of an OCT on an initially dense specimen and the subsequent comparison with actual test results. It also influences the initial stiffness during TXC. As a first estimate or when lacking experimental data β ≈ 1.0 has often proved satisfactory.

Calibration steps 3b to 5 are interrelated, therefore the final set of parameters has to be found iteratively. The ”abort criterion” is the engineering judgement of the user and in general depends on the desired application. E.g. special attention may be paid to the correct reproduction of compressibility of dense specimen at very high pressures (p0 _{≥ 10 MPa) as in the case of the interpretation of CPTs in cohesionless soils (cf.}

Chap. 3), Example

Results of an OCT on a loose and a dense sample and one drained TXC on a dense sample of Dubai sand (M100, cf. Chap. 3) are used for the demonstration of the procedure developed for the determination of the hypoplastic material parameters. Furthermore, the limit void ratios, the grain size distribution, the grain density and the angle of repose were determined.

1. Parameters obtained directly from index tests:

ed0 ≈ emin = 0.762 ec0 ≈ emax= 1.223 ei0 ≈ 1.15emax= 1.406

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2. Angle of repose: ϕc= 37.7◦

3. (a) hs and n from curve fitting using (2.2) and the results of an OCT with e0 = 1.085 or ID0 = 27%, respectively (Fig. 2.2a).

(b) hs and n are slightly adjusted to better fit the ETP calculation results with α = 0.1 and β = 1.0 (Fig. 2.2b, the numbers #x specify sets of parameters

which are listed in Table 2.1 to demonstrate the calibration process and to show the effect of the changes in the parameters on the calculation results.)

0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 0.01 0.1 1 10 e p’ in MPa h_s=113.626 MPa n =0.544 I_D=27% fit 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 0.01 0.1 1 10 e p’ in MPa I_D=27% #1 #2 (a) (b)

Figure 2.2: Calibration of hs and n using OCT data: (a) first approximation (b)

calibra-tion with ETP.

4. After the simulation of the drained triaxial test n was slightly reduced to fit the curve progression in Fig. 2.3a (#3) with respect to the axial strain ε1 at the peak max q, then α is increased for a better approach of max q (#4).

5. Simulation of the OCTs on initially loose and dense samples, respectively, and calibration of β (Fig. 2.4): Parameter set #4 resulting from step 4 is used for the simulation of the two OCTs, and β is slightly increased in order to better reproduce the stiffness of the dense specimen for p0 _{> 10 MPa. With the new set #5 the test}

on the loose specimen is recalculated, and a satisfying result is obtained. The same holds for the TXC calculated with #5 (Fig. 2.5). Otherwise steps 3b to 5 would have been repeated until a satisfactory curve fit is achieved.

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2.2. Hypoplastic Constitutive Equation 11 0 100 200 300 400 500 600 700 800 900 1000 0 2 4 6 8 10 12 14 q in kPa ε₁ in % I_D=88% #2 #3 #4 −1 0 1 2 3 4 5 6 0 2 4 6 8 10 12 14 εV in % ε₁ in % I_D=88% #2 #3 #4 (a) (b)

Figure 2.3: Calibration of hs, n and α using TXC data: (a) q vs. ε1 (b) εv vs. ε1.

0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 0.01 0.1 1 10 e p’ in MPa I_D=27% I_D=79% #4 #5 0.50 0.60 0.70 0.80 0.90 1.00 1.10 0 2 4 6 8 10 12 e p’ in MPa I_D=27% I_D=79% #4 #5

Figure 2.4: Calibration of β using OCT data.

a better understanding of the impact of the parameters on the mechanical response as calculated by the hypoplastic model.

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0 100 200 300 400 500 600 700 800 900 1000 0 2 4 6 8 10 12 14 q in kPa ε₁ in % I_D=88% #5 −1 0 1 2 3 4 5 6 0 2 4 6 8 10 12 14 εV in % ε₁ in % I_D=88% #5 (a) (b)

Figure 2.5: Check of hs, n and α using TXC data: (a) q vs. ε1 (b) εv vs. ε1.

# hs n α β Test result in MPa used 0 113.6 0.544 — — OCT (loose) 1 113.6 0.544 0.10 1.0 OCT (loose) 2 95.0 0.544 0.10 1.0 OCT (loose) 3 95.0 0.500 0.10 1.0 TXC (dense) 4 95.0 0.500 0.13 1.0 TXC (dense) 5 95.0 0.500 0.13 1.1 OCT (dense)

Table 2.1: Sets of parameters during the calibration process (OCT – oedom. comp., TXC – triaxial comp.).

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2.3. Viscohypoplastic Constitutive Equation 13

2.3 Viscohypoplastic Constitutive Equation

For the description of the stress–strain–time behaviour of clayey soils a viscohypoplastic constitutive model by Niemunis [38] was employed. It is able to describe realistically viscous effects such as creep, relaxation and rate-dependence of incremental stiffness and shear strength. It has proven to yield realistic results in many FE simulations of boundary value problems, such as deformations due to open pit mining [21], construction pits [44], 3D analysis of an in situ cyclic horizontal pile loading test [5], 3D analysis of the bearing behaviour of a quay-wall at serviceability state [29], 2D and 3D buckling analysis of slender pile-like structures in soft soils [34] (cf. Chap. 6.1).

For a better understanding of the underlying concept, the one-dimensional version is described first. Conventionally, the equations (2.3) to (2.5) are used for the evaluation of oedometric compression tests:

² − ²0 = −λ ln σ/σ0 or ˙² = −λ ˙σ/σ for virgin loading, (2.3) ² − ²0 = −κ ln σ/σ0 or ˙² = −κ ˙σ/σ for unloading/reloading, (2.4) ² − ²0 = −ψ ln t + t0 t0 or ˙² = −ψ 1 t + t0 , (2.5)

where σ < 0 is the vertical stress and ˙² is the vertical strain rate, which in the case of oedometric compression equals the volumetric strain rate ˙²v. The material parameters λ

and κ belong to the compression law by Butterfield (2.14) [6] and must not be mistaken for the ones by Terzaghi (e − e0 = −λ∗ln (p/p0)). ψ is a coefficient of secondary compression. Furthermore, the logarithmic strain measure ²v = ln 1 + e_{1 + e}

0 is used instead of the small strain (Biot) formulation ²v = e − e_{1 + e}0

0. e0, σ0 and t0 are reference values of the void ratio, vertical stress and time. The shortcoming of the above equations (2.3) to (2.5) is that in (2.3) both points (σ0, ²0) and (σ, ²) must lie on the virgin compression line, and in case of (2.4) on the same unloading/reloading branch, and the meaning of t0 is not clear. When creep starts from the virgin compression line t0 = tp in (2.5) is assumed sometimes, where tp is the duration of the primary consolidation process. Even if (2.5) holds for that special

case, the general meaning of t0 should not depend on tp, i.e. on the height of the sample.

Furthermore, it is not possible e.g. to calculate creep from (2.5) after a small unloading. In practice a small overloading, unloading and reloading of ca. 5% is a frequently used precaution in order to avoid excessive creep settlements.

In order to overcome these shortcomings Niemunis proposed the following set of three equations

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˙σ = −σ κ ³ ˙² − ˙²vis´ _(2.6) ˙²vis _{= − ˙²} r µ_−σ σe ¶_1/I_v (2.7) σe = σe0 µ _{1 + e} 1 + ee0 ¶−1/λ or σ˙e= − σe˙² λ (2.8)

where λ and κ are again the compression and swelling index of Butterfield’s compression law. Iv is the so-called viscosity index introduced by Leinenkugel [26], which controls the

viscous behaviour, and σe (> 0) is the so-called equivalent stress introduced by Hvorslev,

he named

−σe/σ0

overconsolidation ratio OCR [19]. ˙²r(≡ Dr) is a reference creep rate, σe0 and ee0 are a

reference state on the virgin compression line corresponding to ˙²r. The triple (σe0, ee0, ˙²r)

defines a reference isotach. The conventional equations (2.3) to (2.5) are obtained from the above system by integration as special cases [38].

The general version of this model uses an extended definition of OCR (in place of −σe/σ).

It becomes a function of the entire stress state, density and an equivalent pressure pe

measured along the isotropic stress axis (−σ11 = −σ22 = −σ33). All equations of 3D viscohypoplasticity are summarized in Appendix A.2.1.

˚

T = fbL :ˆ

³

D − Dvis´ _(2.9)

In the basic equation (2.9) ˚T is the objective stress rate tensor, D is the strain rate tensor and Dvis is a viscous strain rate tensor. ˆL (A.21) is the linear part of the hypoplastic

stiffness matrix, which together with the barotropy function fb(A.19) controls the pressure

dependent hypoelastic stiffness and therewith volume changes in case of Dvis = 0. ˆT (cf. App. A.1.1, Eqn. (A.4)) is the stress obliquity and ˆT∗ (A.5) is the deviatoric part of ˆT. The material parameter a(ϕc) and the factor F , which is a function of the Lode angle and

incorporates the yield surface by Matsuoka and Nakai [30] are given in (A.6) and (A.7), respectively.

The intensity of the viscous creep ||Dvis_{|| is a function of stress and density (i.e. of OCR)} and controlled by the two material parameters Dr (reference creep rate) and the viscosity

index Iv (A.22). The direction (Dvis)→ (where t→ = t/|| t ||) is determined by the

unit vector ~B (A.23). The main difference to elastoviscoplastic models with a so-called overstress is that viscous flow takes place outside as well as inside the yield surface. There

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is no need for a switch function since ||Dvis_{|| decreases exponentially, i.e. very quickly,} with OCR−1/Iv for OCR > 1. (Example: With a typical value of I

v ≈ 0.02 for an OCR = 1.1 one obtains ||Dvis_{|| ≈ 10}−3_D

r and for OCR = 0.9 |Dvis| ≈ 100Dr.) Note that

an arbitrary choice of the reference isotach OCR < 1 is possible (Fig. 2.6), which does not effect the results if Dr is changed suitably.

e

p

e0

e

e0

(p /

_e+

p )

e

ln

ln(1+e)

,

Dvis || ||

D

r Dvis || ||

D

r Dvis || ||

D

r

1

e

p = p

e +

OCR=1

p e current size of the ellipse: current void ratio 1+e faste r co mp ressio n slo wer co mp ressio n refe ren_ce iso tach (λ_-lin e) inside outside 1 λ e

p < p

e +

OCR>1

p > p

e +

OCR<1

Figure 2.6: Reference isotach (OCR = 1) with two adjacent parallel isotachs with OCR < 1 and OCR > 1 due to different prescribed compression rates D = const [38].

The definition (A.24) of OCR is different from the one by Hvorslev [19], where only the vertical stress component is considered. The equivalent isotropic pressure pe, is calculated

with Butterfield’s compression law for a given void ratio e (Eqn. (A.25) and Fig. 2.7a). The reference point (pe0, ee0) defines the reference isotach with the creep rate Dr for isotropic

compression. Two of these three quantities can be chosen arbitrarily (the author uses

pe0 = 100 kPa and Dr = 10−6s−1), the third one (i.e. ee0) has to be calibrated. It is

noted, that (pe0, ee0) are true reference values, contrary to (e0, p0) in (2.3) and (2.4) they are valid for any kind of loading history.

An ellipse which is affine to the one of the modified Cam clay model [43] is used to determine p+

e as an intersection of the ellipse passing through the current (p, q) and the

isotropic axis p (Eqn. (A.26) and Fig. 2.7b). For each point on such an ellipse OCR =

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(p_e0,e_e0) 1+e p_e p_e (p,q) p_e+ OCR =1 OCR =p_e/p_e+ ln p ln(1+ e ) p q CSL ||Dvis||=D_r

(a)

(b)

1 λ

Figure 2.7: Definition of OCR in the framework of viscohypoplasticity.

In order to better reproduce stress paths during undrained shearing of normally consol-idated samples, the shape of the yield surface can be adopted as shown in Figure 2.8. Equation (A.26) is replaced by

p+new e =      p βR− 1 h βR q 1 + ¯η2_(β2 R− 1) − 1 i for ¯η < 1 p(1 + ¯η2_{)1 + β}R 2 for ¯η > 1 (2.10)

where ¯η = q/(Mp), with a further material parameter βR. In the modified Cam clay

model an analogous modification is used to improve the prediction of K0. Here, βR has

no effect on K0but controls the pore water pressure generation during undrained shearing.

2.3.1 Material Parameters and their Determination

The viscohypoplastic model described in the previous subsection contains five material parameters. In the following their determination is explained from a practical point of view, based on the results from either conventional oedometric compression tests or constant strain rate tests and undrained triaxial tests on remoulded samples.

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2.3. Viscohypoplastic Constitutive Equation 17 p q CLS β =0.99_R β =0.5_R p /2e+ βRp /2e + pe new + p_e+ _p q β =0.99_R β =0.5_R

(a)

(b)

Figure 2.8: (a) Modified shape of the yield surface, (b) resulting stress paths without drainage [38].

Critical Friction Angle ϕ_c: The critical friction angle can be easily determined from (2.11) and the inclination M of the critical state line in the p−q plane (cf. Fig. 2.9).

ϕc= arcsin

µ _3M 6 + M

¶

(2.11) Viscosity Index Iv: Principally, any type of test considering viscous effect, i.e. creep, relaxation and rate dependent shearing resistance can be used for the determination of Iv. Two methods are proposed here, which have been applied successfully for a

long time.

1. Stress-controlled oedometric compression tests:

When conventional oedometric compression tests are performed with creep phases after the equalization of excess pore water pressures of each loading step, Iv can be calculated with

Iv = Cα/Cc, (2.12)

where the coefficient of creep Cα = ∆e/∆ log10(t) = (e1− e2)/ log10(t2/t1) and the oedometric compression index Cc = ∆e/∆ log10(−σv/1 kPa). According

to [35] the values of Cα/Cc are in the range of 0.02 –0.10 for a broad variety

of natural soils including peats, organic silts, highly sensitive clays, shales as well as granular materials. This is in agreement with the experimental results gathered at the IBF. It is recommended to use a mean value ¯Cα obtained from

several loading steps. Especially in the case of low vertical pressures tilting of the top plate inside the oedometric ring can affect the results, even if the plate is guided.

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2. Undrained triaxial tests with alternating axial strain rates:

If the strain rate is changed abruptly during a CU test Iv can be calculated

using (2.13), where qa and qb are the deviatoric stresses before and after a

change from Da= ˙ε1, a to Db = ˙ε1, b (Fig. 2.9).

Iv = ln (qa/qb) ln (Da/Db) (2.13)

p

e1

q

_b

q

_a

fast

slo

w

1 M

CSL

D

_a _D a

q

D_b> D_a D

q

_b

q

_a

fast

slow

due to the jump of the

deformation rate

D_b> D_a

D

_a

Figure 2.9: Response during undrained triaxial shearing due to a change in the axial strain rate [38].

Reference Isotach with ee0, pe0, Dr, λ and κ: As described above the reference iso-tach is defined in the ln (1 + e)–ln p plane as an isotropic compression line passing through a reference void ratio ee0 and a reference effective mean pressure pe0. It

corresponds to the volumetric compression rate Dr·λ/(λ−κ). The inclination of the

reference isotach is λ, whereas κ (2.14) is the inclination of the unloading/reloading branches.

pe0 = 100 kPa is often taken by default. In this case the corresponding void ratio is

called e100. The reference creep rate Drcan be chosen arbitrarily, e.g. Dr = 10−6s−1.

Note that the value of OCR is somewhat arbitrary since it depends on the choice of Dr. The conventional definition of a normally consolidated soil, i.e. that it has

never been subjected to higher than the actual pressures (and therewith OCR = 1) is misleading. Soft soils densify with time due to ongoing volumetric creep. This is one aspect of what in the literature is sometimes referred to as ”ageing”. The consequence is that without a reference creep rate (e.g. the actual in situ rate of volumetric creep) there cannot be a unique relationship between OCR and density. When simulating a CU test on a normally consolidated sample and OCR= 1.0 D! r

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can be adjusted so that the resulting void ratio for the given OCR = 1.0 equals the measured one. ln µ_{1 + e} 0 1 + e ¶ = λ · ln Ã p p0 !

for virgin loading (2.14) ln µ_{1 + e} 0 1 + e ¶ = κ · ln Ã p p0 ! for unloading/reloading

In practice standard oedometric tests are preferred as against more expensive isotropic compression tests. The iterative calibration procedure is as follows:

1. Element test calculation with λ = λoedo_{, κ = κ}oedo_{, e}

100 = eoedo(p = 100 kPa) where p = −σv(1 + 2K0) and K0 = 1 − sin ϕc [20] [31], Dr = 10−6s−1, ϕc and Iv determined as described above (the missing parameter βR has no influence

on the simulation results of OCTs)

2. Adjustment of e100 and κ (since κ is slightly smaller than κoedo [38]).

Step 2 is repeated until a satisfying agreement of the measurements and the simu-lation results is achieved. κ has to be checked again when the intergranular strain parameters are calibrated (cf. Sec. 2.4).

Shape of the Yield Surface – β_R: The parameter βR influences the evolution of the

pore water pressure during undrained shearing and therewith the effective stress path (Fig. 2.8) and the undrained shear strength cu = max q/2. With the results

of an undrained TXC on an isotropically normally consolidated sample βR can be

calculated directly with (2.15).

βR = p0

0− p0(max q)

p0_{(max q)} (2.15)

A second possibility is a simple calibration consisting in the variation of βR until

max q from ETP calculations coincides with the measured value. Example:

Figure 2.10 depicts the results of an OCT and a CU triaxial compression test on material #159 (cf. Tab. 2.4). The triaxial test was performed with the following initial and boundary conditions:

– cell pressure σ3 = const. = 800 kPa, backpressure u0 = 700 kPa → σ0

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– void ratio e0 = 1.605 – ˙ε1 = −3.67/s                    ·10−6_; _{0.00% ≤ ε} 1 ≤ 2.18% ·10−7_; _{2.18% ≤ ε} 1 ≤ 2.48% ·10−6_; _{2.48% ≤ ε} 1 ≤ 3.95% ·10−5_; _{3.95% ≤ ε} 1 ≤ 10.98% ·10−6_; _{10.98% ≤ ε} 1 ≤ 13.00% 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 16 32 64 125 250 750 e −σ’_v in kPa lab. #1 #2 #3 0 10 20 30 40 50 60 70 80 90 100 0 20 40 60 80 100 q in kPa p’ in kPa lab. csl #2 #3 (a) (b) 0 10 20 30 40 50 60 70 80 0 5 10 15 q in kPa ε₁ in % lab. #2 #3 0 10 20 30 40 50 60 70 0 5 10 15 ∆ u in kPa ε₁ in % lab. #2 #3 (c) (d)

Figure 2.10: Laboratory and calculation results (material #158): (a) oedometric com-pression and (b-d) undrained triaxial shearing.

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2.3. Viscohypoplastic Constitutive Equation 21 1. ϕc= arcsin ³ _3M 6 + M ´ , M = 0.94 (Fig. 2.10b) → ϕc= 24◦ 2. λ = ln µ_{1 + e} 1 1 + e2 ¶ ln Ã p2 p1 ! ₌ ln µ_{1 + 2.17} 1 + 0.94 ¶ ln µ₇₅₀ 16

¶ = 0.127 (Fig. 2.10a, virgin compression)

3. κoedo ₌ ln µ_{1 + e} 1 1 + e2 ¶ ln Ã p2 p1 ! ₌ ln µ_{1 + 1.43} 1 + 1.22 ¶ ln µ₂₅₀ 16

¶ = 0.033 (Fig. 2.10a, unloading/reloading)

4. eoedo

100 ≈ 1.62 (Dr := 10−7s−1)

5. (a) Iv = ¯Cα/Cc = 2.56 · 10−3/0.139 = 0.018

( ¯Cα is the mean value of Cα, i from i = 15 creep phases)

(b) Iv = _{ln ( ˙ε}ln (qa/qb) a/ ˙εb) = ln (44/42)/ ln (10 −7_/10−6_{) = 0.02} _{(Fig. 2.10c)} 6. βR= p 0 0− p0(max q) p0(max q) = 100 − 52.652.6 = 0.9 (Fig. 2.10b)

7. Figure 2.10a also depicts ETP calculation results of the OCT with λ, κ, e100, Dr, Iv

and βR as determined above (Tab. 2.2, parameter set #1). The most striking

dis-crepancy between the calculation and the measurement is the shape of the hysteresis loop (unloading/reloading). This makes necessary an adjustment of the intergranu-lar strain parameter βχ (cf. Sec. 2.4). Furthermore, e100 and κ are slightly reduced (# 2).

8. Next, the post test calculation of the CU test is carried out with the current pa-rameter set #2 (Fig. 2.10b-d). The final set of papa-rameters (#3) is obtained with a small increase of Iv, which leads to a better reproduction of the changes in q due to

the changes in the axial strain rate ˙ε1.

9. With parameter set #3 from the previous step, the OCT is calculated again, result-ing in a satisfyresult-ing agreement with the experiment (Fig. 2.10a).

# ϕc λ κ e100 Dr in s−1 Iv βR βχ

1 24◦ _{0.127 0.033} _1.62 ₁₀−7 _{0.018 0.90} _0.10

2 24◦ _0.127 _0.03 _1.58 ₁₀−7 _{0.018 0.90} _0.03

3 24◦ _0.127 _0.03 _1.58 ₁₀−7 _0.02 _0.90 _0.03

Table 2.2: Set of parameters during the calibration process.

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2.4 Intergranular Strain

Both constitutive models were used with the so-called intergranular strain extension. The intergranular strain tensor δ stores the most recent deformation history and provides for an increase in a kind of incremental stiffness E = dT/dε in the case of a change in the direction of deformation (D)→_. ε

ε

ε 1 1 1

ε

* D 2 δ ε * D 2 δ ε * D 2 δ (a) (b) (c)

Figure 2.11: Direction of δ and D after a change in direction of (a) 180◦_{, (b) 90}◦ _{and (c)}

after large monotonic deformation [38].

ε d d ε ln ε SOM R E_R E= E₀ E₀ E_T 0 ε SOM E₀ T E₀ E 0.1( R- ) p q p q p q E = m E T T 0 0 R R E = m E * * * *

Figure 2.12: Increase of incremental stiffness due to change in the direction of deformation [38].

Suppose that density and stress are the same at point * despite different deformation paths. A 180◦ _{reversal (Fig. 2.11a) leads to E}

R= mR·E0 (Fig. 2.12), where the factor mR

is assumed to be a material constant and E0 is the stiffness after long monotonic shearing and the same state. A change in direction of 90◦ _{(Fig. 2.11b) leads to E}

T = mT· E0. With ongoing proportional shearing after a change in direction the intergranular strain tensor

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2.4. Intergranular Strain 23

δ gradually takes the direction of D. When the angle between D and δ vanishes, i.e. for ~

D : ~δ = 1, after a sufficient proportional deformation ²SOM the effect of the change in

direction is swept out of memory [14].

Altogether there are five additional material parameters. Apart from mR and mT there

is the quantity R which is related to (but not identical with) the elastic strain range, and there are two exponents βχ and χ which are used to adjust the decay of ER/T during

proportional shearing after a change in direction. The evolution equation for δ is given in Appendix A.1.1, Equation (A.15).

Without this extension both models produce excessive ratcheting in the case of cyclic loading, as shown in Figure 2.13b for oedometric compression and (d) for triaxial shearing. The intergranular strain extension removes this shortcoming (a and c).

a) (b)

c) (d)

Figure 2.13: Comparison between reference model without and the extended model with intergranular strain: (a),(b) oedometric compression (c),(d) triaxial shearing [38].

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2.4.1 Determination of the Parameters

Alongside with the intergranular strain model Niemunis and Herle [39] proposed a method for the determination of the additional parameters which is based on theoretical consid-erations. The set of parameters they gave for one sand was used successfully for many FE simulations of boundary value problems without further experimental investigations. However, the evaluation of dynamic (resonant column) and cyclic tests (oedometric com-pression and triaxial shearing) revealed that the parameter mR should depend on the

stress state and density. Figure 2.14 depicts the results of RC tests on Karlsruhe sand together with post test calculation results. If calibrated to match G/Gmeasured

max vs. ln γ

for the initial effective mean pressure p0

0 = 320 kPa an mR = 6 is obtained (Fig. 2.14a,

upper curve) and for p0

0 = 20 kPa an mR = 12 (Fig. 2.14b, lower curve). For each stress

state the calculation results are normalized with the corresponding Gmeasured

max . The results

show that the calculated Gmax and therewith G(γ) is underestimated for p00 less than the one used for the calibration of mr (i.e. Gmax/Gmeasuredmax ≤ 1) and it is overestimated for p0

0 greater than the one used for the calibration of mr (i.e. Gmax/Gmeasuredmax ≥ 1).

0 0.2 0.4 0.6 0.8 1 1.2 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 G/(G max ) measured γ p’₀ in kPa 20 40 80 160 320 0 0.5 1 1.5 2 2.5 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 G/(G max ) measured γ p’₀ in kPa 20 40 80 160 320 (a) (b)

Figure 2.14: Results of RC tests performed by S. Richter on Karlsruhe sand (e = 0.52 = const. compared with post test calculations with (a) mR = 6 and (b) mR= 12.

Cyclic oedometric compression tests on Zbraslav sand showed that mRshould also depend

on density. For the same (vertical) stress σv = −185 kPa an mR ≈ 4.3 was obtained from

tests on initially loose samples (e0 = 0.90) and mR ≈ 3.0 in the case of e0 = 0.62 [33]. Thus, a mean value should be taken, to best fit the experimental results in a relevant

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2.4. Intergranular Strain 25

R mR mT χ βχ

sands 10−4 ₅ ₂ _{1.0 0.10 – 0.60}

clays 10−4 ₇ ₇ _{1.0 0.03 – 0.10}

Table 2.3: Intergranular strain parameters.

range of stresses and densities.

It is conjectured that the stress and density dependence also holds for mT, but there

has not been a comprehensive experimental study with regard to the determination of the intergranular strain parameters. The parameters used in this study are given in Table 2.3. Experience has shown that R = 10−4 _{can be treated as a material independent constant,}

and that χ = 1.0 together with βχ= 0.1 are good starting values for calibration. Using the

results of e.g. oedometric compression tests with at least one unloading/reloading cycle

mR and βχ can be calibrated in order to represent realistically, the increase in stiffness

after a deformation reversal, and the range of influence ²SOM (in the sense of Fig. 2.12).

In the framework of this thesis mR and mT were taken as proposed in [39] for sands. For

clays mR(= mT) was calibrated as depicted in Figure 2.15a to best fit the laboratory test

results. For both groups of soils βχwas calibrated so that the shape of the hysteresis loop

´ıs reproduced realistically. Reducing βχ leads to an increase of ²SOM.

0.53 0.54 0.55 0.56 250 125 64 32 16 e −σ’_v in kPa m_R=5 m_R=7 mR=9 lab. 0.53 0.54 0.55 0.56 250 125 64 32 16 e −σ’_v in kPa βχ=0.01 β_χ=0.05 βχ=0.20 lab. (a) (b)

Figure 2.15: Oedometric compression test result (material #160) and post test calcula-tions (a) βχ= const. = 0.05 (b) mR= const. = 7.0.

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2.5 Interrelations between Parameters and/or Index

Quantities

According to the employed constitutive model the material constants are chosen in such a way that they control different aspects of the mechanical behaviour of the soil, and as such they are independent. However, some empirical correlations turn out to exist and are of use. It was found that both the index quantities wL, wP and IP and the

material constants (λ, κ, e100) depend linearly on the critical friction angle ϕc, which in

turn depends linearly on the clay content CC. From a physical point of view, this is not surprising, since the mechanical properties must depend on granulometric properties (mineral content, grain shape and size distribution, ion content of the pore water). The correlations between different parameters and/or index quantities given below may be helpful if parameters have to be estimated for comparable materials (cf. due to insufficient laboratory data. In order to enable a comparison the following tests were performed:

• Determination of clay minerals by means of roentgen diffractometry • Determination of the grain size distribution (cf. Fig. A.6)

• Determination of the plastic and liquid limit (wP and wL) • Determination of the calcite and dolomite content

• Determination of ignition loss (organic content)

• Oedometric compression tests with creep phases (Cc, Cs, ¯Cα, λ, κ) • CIU triaxial compression tests with changes in the strain rate (ϕc)

The results of the soil mechanical investigations is summarized in Table 2.4. The outcome of the roentgen diffractometry tests is given in Table 2.5). The content of swellable clay minerals of all materials was less than 25%.

The laboratory results (OCT and TXC) were used to calibrate the viscohypoplastic pa-rameters for every material (Tab. A.3).

Consistency limits wL and wP

In the undesirable case when only the clay content CC is known from a sedimentation analysis, wL and wP can be estimated by (cf. Fig. 2.16)

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2.5. Interrelations between Parameters and/or Index Quantities 27 Material #110 #154 #158 #159 #160 #162 #472 #578 Cla y Con ten t in % 30 50 68 85 78 19 29 42 CaCO 3 Con ten t in % 27.3 26.3 0.3 0.3 0.2 27.1 0.1 0.4 CaMg(CO 3 )2 Con ten t in % 7.6 5.0 0.0 0.0 0.0 7.7 0.0 0.2 Ignition Loss in % 2.3 3.1 6.0 7.5 4.1 1.1 3.0 2.3 Grain densit y ρs in g/cm 3 2.705 2.752 2.690 2.654 2.707 2.744 2.704 2.746 Plastic Limit wP in % 16.9 20.9 23.1 28.9 25.3 16.8 22.7 18.7 Liquid Limit wL in % 33.2 43.4 54.5 71.5 58.7 25.1 38.6 36.2 Plasticit y Index IP in % 16.3 22.5 31.3 42.6 33.4 8.3 16.0 17.5 Sym b ol (USCS) CL CL CH CH CH CL CL CL Critical friction angle ϕc 30.8 ◦ 26.1 ◦ 21.8 ◦ 24.0 ◦ 20.6 ◦ 35.0 ◦ 32.1 ◦ 27.0 ◦ Compression index Cc 0.213 0.332 0.529 0.735 0.656 0.165 0.291 0.300 Sw elling index Cs 0.033 0.071 0.108 0.175 0.152 0.015 0.027 0.063 Compression index λ 0.059 0.086 0.110 0.127 0.128 0.045 0.072 0.077 Sw elling index κ 0.009 0.018 0.024 0.033 0.031 0.004 0.007 0.017 Co efficien t of Creep ¯ Cα 4.5 ·10 − 3 4.6 ·10 − 3 6.2 ·10 − 3 1.4 ·10 − 2 1.4 ·10 − 2 1.7 ·10 − 3 1.7 ·10 − 3 3.0 ·10 − 3 T able 2.4: Summary of the lab oratory test results of the in vestigated materials.

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Material #110 #154 #158 #159 #160 #162 #472 #578 Chlorite 0 1 0 1 0 0 0 1 Sw ellable cla y minerals 15 10 10 8 3 9 25 3 Mica 11 20 6 17 19 16 22 50 Kaolinite 2 5 28 44 23 3 3 16 Gypsum N.D. N.D. N.D. N.D. N.D. N.D. N.D. N.D. Quartz 21 16 45 13 41 28 41 26 Anh ydrite N.D. N.D. N.D. N.D. N.D. N.D. N.D. N.D. F eldspar 5 3 1 2 1 8 7 2 Calcite 26 29 0 1 0 24 0 1 Dolomite 9 4 0 0 0 12 1 0 Sum non-la yered silicates 73 64 56 30 55 72 49 30 Chemically com bined H 2_O 5.14 6.57 10.31 13.13 10.60 4.09 6.65 7.75 CO 2 15.44 14.53 0.27 0.57 0.24 15.91 0.59 0.52 S 0.01 0.04 0.02 0.16 0.06 0.03 0.01 0.01 C* 0.15 0.41 0.19 0.39 0.01 0.19 0.09 -0.02 T able 2.5: Mineralogic con ten t in mass p ercen tage determined with the aid of ro en tgen diffractometry .

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2.5. Interrelations between Parameters and/or Index Quantities 29 Material ϕc λ κ e100 Dr in s−1 Iv βr 110 30.8◦ _{0.055 0.010 0.68} ₁₀−6 _{0.025 0.70} 154 26.1◦ _{0.085 0.018 0.81} _1.5·10−6 _{0.025 0.75} 158 21.8◦ _{0.106 0.024 1.05} ₁₀−6 _{0.020 0.55} 159 24.0◦ _{0.127 0.030 1.58} ₁₀−7 _{0.020 0.90} 160 20.6◦ _{0.127 0.027 1.29} ₁₀−6 _{0.020 0.66} 162 35.0◦ _{0.045 0.007 0.58} ₁₀−6 _{0.030 0.99} 472 32.1◦ _{0.067 0.010 0.75} ₁₀−6 _{0.020 0.99} 578 27.0◦ _{0.077 0.008 0.78} ₁₀−6 _{0.030 0.90}

Table 2.6: Viscohypoplastic parameters of the investigated materials.

wL = 0.6048 · CC + 0.1415 (s = 0.040) (2.16) wP = 0.1498 · CC + 0.1415 (s = 0.022)

where s is the standard derivation.

0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.00 0.20 0.40 0.60 0.80 1.00 wL clay content w_L=f(CC) 0.15 0.20 0.25 0.30 0.00 0.20 0.40 0.60 0.80 1.00 wP clay content w_P=f(CC) (a) (b)

Figure 2.16: Correlation between (a) wL and (b) wP and the clay content CC.

Critical friction angle ϕc

A linear correlation was observed between wL, wP, IP, CC and the critical friction angle ϕc in rad (cf. Fig. 2.17), viz

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ϕc = 0.6920 − 0.4819 · wL (s = 0.054) (2.17) = 0.7889 − 1.4515 · wP (s = 0.070) = 0.6326 − 0.6736 · IP (s = 0.049) = 0.6429 − 0.3360 · CC (s = 0.036) 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.20 0.40 0.60 0.80 ϕc in rad w_L ϕ_c=f(w_L) 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.15 0.20 0.25 0.30 ϕc in rad w_P ϕ_c=f(w_P) (a) (b) 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.00 0.10 0.20 0.30 0.40 0.50 ϕc in rad I_P ϕ_c=f(I_P) 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.00 0.20 0.40 0.60 0.80 1.00 ϕc in rad CC ϕ_c=f(CC) (c) (d)

Figure 2.17: Correlation between ϕc and (a) wL, (b) wP, (c) IP and (d) CC.

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2.5. Interrelations between Parameters and/or Index Quantities 31

The virgin compression index λ and the swelling index κ both show a linear dependency on wL, wP, IP, ϕc and CC (cf. Fig. A.7 and Fig. A.8):

λ = 0.2407 − 0.3257 · ϕc (s = 0.013) (2.18) = 0.1978 · wL− 0.0032 (s = 0.009) = 0.6699 · wP − 0.0590 (s = 0.015) = 0.2666 · IP + 0.0235 (s = 0.009) = 0.1268 · CC + 0.0226 (s = 0.005) κ = 0.0593 − 0.0896 · ϕc (s = 0.005) (2.19) = 0.0584 · wL− 0.0096 (s = 0.003) = 0.1942 · wP − 0.0253 (s = 0.005) = 0.0792 · IP − 0.0018 (s = 0.003) = 0.0366 · CC − 0.0016 (s = 0.003)

Reference Void Ratio e100

The reference void ratio e100also depends linearly on on wL, wP, IP, ϕcand CC (Eqn. (2.20)

and Fig. 2.20). The associated reference creep rates were Dr ∈ [10−7; 10−6] s−1. The

de-viations in e100 due to the choice of a different Dr are generally much smaller than the

inaccuracy of the given estimates and can therefore be neglected.

e100 = 2.3530 − 2.9783 · ϕc (s = 0.234) (2.20)

= 2.1919 · wL− 0.0496 (s = 0.075)

= 7.5744 · wP − 0.7012 (s = 0.133)

= 2.9347 · IP + 0.2507 (s = 0.087)

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0.040 0.050 0.060 0.070 0.080 0.090 0.100 0.110 0.120 0.130 0.140 0.20 0.40 0.60 0.80 λ w_L λ=f(w_L) 0.040 0.050 0.060 0.070 0.080 0.090 0.100 0.110 0.120 0.130 0.140 0.15 0.20 0.25 0.30 λ w_P λ=f(w_P) (a) (b) 0.040 0.050 0.060 0.070 0.080 0.090 0.100 0.110 0.120 0.130 0.140 0.00 0.10 0.20 0.30 0.40 0.50 λ I_P λ=f(I_P) 0.040 0.050 0.060 0.070 0.080 0.090 0.100 0.110 0.120 0.130 0.30 0.40 0.50 0.60 0.70 λ ϕ_c in rad λ=f(ϕ_c) (c) (d) 0.040 0.050 0.060 0.070 0.080 0.090 0.100 0.110 0.120 0.130 0.140 0.00 0.20 0.40 0.60 0.80 1.00 λ clay content λ=f(CC) (e)

Figure 2.18: Correlation between λ and (a) wL, (b) wP, (c) ϕc (d) IP and (e) the clay

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2.5. Interrelations between Parameters and/or Index Quantities 33 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.20 0.40 0.60 0.80 κ w_L κ=f(w_L) 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.15 0.20 0.25 0.30 κ w_p κ=f(w_P) (a) (b) 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.00 0.10 0.20 0.30 0.40 0.50 κ I_p κ=f(I_P) 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.30 0.40 0.50 0.60 0.70 κ ϕ_c in rad κ=f(ϕ_c) (c) (d) 0.005 0.010 0.015 0.020 0.025 0.030 0.00 0.20 0.40 0.60 0.80 1.00 κ clay content κ=f(CC) (e)

Figure 2.19: Correlation between κ and (a) wL, (b) wP, (c) ϕc (d) IP and (e) the clay

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0.40 0.60 0.80 1.00 1.20 1.40 1.60 0.20 0.40 0.60 0.80 e100 w_L e₁₀₀=f(w_L) 0.40 0.60 0.80 1.00 1.20 1.40 1.60 0.15 0.20 0.25 0.30 e100 w_P e₁₀₀=f(w_P) (a) (b) 0.40 0.60 0.80 1.00 1.20 1.40 1.60 0.00 0.10 0.20 0.30 0.40 0.50 e100 w_P e₁₀₀=f(I_P) 0.40 0.60 0.80 1.00 1.20 1.40 1.60 0.30 0.40 0.50 0.60 0.70 e100 ϕ_c in rad e₁₀₀=f(ϕ_c) (c) (d) 0.40 0.60 0.80 1.00 1.20 1.40 1.60 0.00 0.20 0.40 0.60 0.80 1.00 e100 w_P e₁₀₀=f(CC) (e)

Figure 2.20: Correlation between e100 and (a) wL, (b) wP, (c) ϕc (d) IP and (e) the clay

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2.5. Interrelations between Parameters and/or Index Quantities 35

Viscosity Index Iv

Different to what was expected, no clear correlation between Ivand any of other quantities

could be found. Krieg [24] e.g. found a correlation with wL, and Mesri and Castro [35] give Cα/Cc ≈ 0.04 ± 0.01 for inorganic soft clays and Cα/Cc≈ 0.06 ± 0.01 for highly organic

plastic clays. In this study, the values range between 0.02 and 0.03 for the investigated materials (CL to CH).

Undrained Strength Ratio (cu/p00)

The so-called undrained strength ratio cu/p00, where p00 denotes the initial in situ effective mean pressure, is often used in practice for the estimation of cu of normally consolidated

soils [25]. In order to determine this ratio for the different soft soils examined here, numerical simulations of CIU triaxial tests were performed for different initial isotropic mean pressures p0

0 and OCR and an axial strain rate ˙ε1 = 10−5/s.

0 50 100 150 200 250 300 0 50 100 150 200 250 cu in KPa p’₀ in kPa OCR=1.0 OCR=2.0 OCR=3.0 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 3 tan( α ) OCR (a) (b)

Figure 2.21: Results of numerical CIU tests (a) cu = f (p00, OCR), (b) tan (α) = cu/p00 vs. OCR (material #110).

Figure 2.21a depicts exemplarily the results for material #110. For each OCR the ratio

cu/p00 is constant. Figure 2.21b shows cu/p00 = f (OCR) with f (OCR) = a · OCRb as an appropriate fitting function. This yields

cu = p00· a · OCRb. (2.21)

The material constants a and b were determined for all materials (Tab. A.4). A linear correlation between those two parameters and ϕc in rad was found (cf. Fig. 2.22):

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a = 0.5558 · ϕc+ 0.0679 (s = 0.023) (2.22) b = 0.1670 · ϕc+ 0.8676 (s = 0.016) (2.23) 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.30 0.40 0.50 0.60 0.70 a, b ϕ_c in rad a b

Figure 2.22: Parameters a and b vs. ϕc.

The factor a, which in the case of OCR = 1 is equal to cu/p00, ranges between 0.262 and 0.408. This is in very good agreement with e.g. cu/p00 ∈ [0.3 ± 0.1] given by Lambe [25] ”for many remoulded clays”. These results are another indication, that the mechanical behaviour of clayey soils can be simulated realistically with the aid of the viscohypoplastic constitutive equation.

Equation (2.21) is also used in Chapter 4.3.3 for the determination of the so-called cone factor Nkt which relates CPT penetration resistance with cu.

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Chapter 3 Benchmarking of CPT Interpretation

Methods for Calcareous Sands

3.1 Introduction

When offering soil improvement measures such as vibro compaction geotechnical con-tractors often face the problem that clients demand cone penetration resistances after compaction, which sometimes cannot be achieved even if the material was well densified. For the performance control of deep vibratory compaction it is also of importance to develop interpretation methods based on scientific investigations in order to replace inap-propriate ones, which transfer empirical findings to other materials without an adequate physical base.

The cone penetration resistance depends not only on the state of the soil (density and stress) but also on the granulate properties (grain hardness, shape and size distribution). In practice calcareous sands often cause problems: Their grains are much softer and more breakable than quartz grains (calcite: Mohs’ hardness 3; quartz: Mohs’ hardness 7). Even for a comparable grain size distribution and grain shape and the same initial pressure and density, the penetration resistances of these materials can differ substantially [40]. On behalf of Keller Grundbau GmbH (Offenbach/Germany) CPT calibration chamber tests (Sec. 3.2) were performed at IBF in order to examine the influence of different mass fractions of a calcareous sand in a mixture with a quartz sand with respect to the attainable cone penetration resistances. The test facility allows the preparation of granular samples with a desired density in a calibration chamber. It is possible to simulate different stress states and densities and to get reproducible CPT results.

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Material CaCO3 ρs d10 d60 CU emin emax ϕc

content in g/cm3 _{in mm in mm = d} 60/d10

Dubai sand 90% 2.805 0.13 0.53 4.1 0.762 1.223 36.0◦

Karlsruhe sand ≈ 0% 2.647 0.14 0.31 2.2 0.531 0.875 31.0◦

Table 3.1: Index properties of the original materials.

The interpretation of the calibration chamber tests is performed applying a semi-empirical method after Cudmani [7], which is described in Section 3.4. It is based on a hypoplastic constitutive equation, which mathematically describes the mechanical behaviour of gran-ular materials [51]. All material constants are obtained from standard laboratory tests on disturbed samples. For a given stress state the only state variable is the void ratio (or the density) which can be indirectly determined from CPT results. However, one premise of hypoplasticty is the permanence of the grains. In reality CPT may lead to grain frac-turing even with quartz [3] [7]. Since calcite is more breakable than quartz, the purpose of the present investigation is to clarify how far the interpretation method after Cudmani is applicable to calcareous granular materials. In addition the tests are evaluated using a procedure proposed by the German code DIN 4094 as well as one by Schmertmann [45] (Sec. 3.3) for comparison. The performance of the different methods is benchmarked in Section 3.5. Section 3.6 addresses the influence of two different mass fractions of gravel and stones (d > 4 cm) in a quartz/shell sand mixture on the cone penetration resistance.

3.2 Calibration Chamber Tests

3.2.1 Test Material

The test material was Karlsruhe sand (quartz), Dubai sand (calcite) and mixtures of the two. The grain fraction d > 4 mm of the Dubai sand had to be removed, as it could not pass the diffusor sieve (cf. Sec. 3.2.2). Different mixtures of both sands were produced in order to examine the influence of an increasing calcareous sand content on index properties and the cone penetration resistance. Index properties of the two original materials are summarized in Table 3.1, those of the mixtures in Table 3.2. The grain size distributions are shown in Figure B.1. The label Mxx specifies a shell content of xx%.

The observed dependance of the bounding void ratios emin and emax and of the critical (residual) friction angle ϕc on the shell content is depicted in Figure 3.1. The quantities emin and emaxincrease with increasing shell content. This can be explained by the angular shape of the shell particles, which form grain skeletons with higher void ratios and better

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3.2. Calibration Chamber Tests 39

Mixture d50 CU ρs emin emax ϕc

[mm] = d60/d10 [g/cm3]

M15 0.51 2.83 2.671 0.579 0.859 31.1◦

M30 0.64 2.46 2.694 0.618 0.948 31.6◦

M60 0.62 3.65 2.742 0.653 1.014 32.2◦

Table 3.2: Index properties of the mixtures [4].

0.6 0.8 1 1.2 0 20 40 60 80 100 e Shell content [%] e_max e_min 28 30 32 34 36 0 20 40 60 80 100 ϕc [°] Shell content [%] (a) (b)

Figure 3.1: Evolution of emin and emax (a) and ϕc (b) with increasing shell content.

interlocking, which results in a higher critical friction angle compared to the rounded quartz grains of Karlsruhe sand.

3.2.2 Test Facility and Sample Placement

Cone penetration tests (CPTs) were performed in a calibration chamber, with d = 92 cm in diameter and h = 150 cm in height. The radial and vertical stress (σr and σv) can

be imposed independently to the sample by means of air cushions (Fig. 3.2). A standard 10 cm2 _{cone is pressed into the sample by means of a hydraulic jack with a penetration} rate of 2 cm/s and a force capacity of 400 kN. During the tests cone resistance, radial and vertical boundary pressure and penetration depth are collected automatically with a sampling rate of 10 Hz.

For the placement of granular specimens three different methods are applied, according to the desired density (loose, medium dense or dense):

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Figure 3.2: Cross section of the calibration chamber.

first brought into contact with the bottom of the chamber and then slowly pulled upward without loosing contact, with the material already flown out (Fig. 3.3a). 2. Uniform medium dense samples are realized using an air pluviation system (Fig. 3.4).

The distance of the lowest diffuser sieve to the sand surface and with it the height of fall is kept constant automatically.

3. In order to produce dense samples layers of approx. 40 cm height are filled into the chamber and densified by means of a vibratory plate (Fig. 3.3b). The resulting void ratios can be lower than emin from standard laboratory tests, due to the application of a higher compaction energy.

3.2.3 Influence of Boundary Conditions

Numerous authors [32] have pointed out the influence of the boundary conditions (BC) on the cone penetration resistance measured in calibration chambers. In this section only so-called soft BC are considered, i.e. BC with prescribed pressure. The influence of the BC increases with increasing density and decreasing diameter of the chamber.

According to Mayne and Kulhawy [32] a ratio η = DS/dc ≥ 70 (DS sample diameter, dc cone diameter) is sufficient to achieve ”free field” conditions. The IBF calibration

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3.2. Calibration Chamber Tests 41

(a) (b)

Figure 3.3: Placement of a loose sample using a hopper (a), and of a dense sample using a vibratory plate (b).

Diss

Hypoplastic and Viscohypoplastic

Constitutive Models for

Geotechnical Problems

Contents

Introduction

Constitutive Modelling and

Parameter Determination

2.1

Introduction

2.2

Hypoplastic Constitutive Equation

2.2.1

Procedure for the Determination of the Material

Param-eters

ϕ

2.3

Viscohypoplastic Constitutive Equation

p

e

(p /

p )

ln

,

D

D

D

1

p = p

OCR=1

p < p

OCR>1

p > p

OCR<1

(a)

(b)

2.3.1

Material Parameters and their Determination

(a)

(b)

p

q

q

q

fast

slo

w

1

M

CSL

D

q

q

q

q

fast

slow

due to the jump of the

deformation rate

D

2.4

Intergranular Strain

ε

ε

2.4.1

Determination of the Parameters

2.5

Interrelations between Parameters and/or Index

Quantities

Chapter 3

Benchmarking of CPT Interpretation

Methods for Calcareous Sands

3.1

Introduction

3.2

Calibration Chamber Tests

3.2.1

Test Material

3.2.2

Test Facility and Sample Placement