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Soil Physics

Theoretical Analysis of Fluid Inclusions for In Situ Soil Stress and Deformation Measurements

^a Swiss Federal Research Station for Agroecology and Agriculture (FAL), Reckenholzstrasse 191, CH-8046 Zurich, Switzerland
^b University of Connecticut, Dep. of Civil and Environmental Engineering, 261 Glenbrook Road, Storrs, CT 06269-2037
^c Laboratory of Soil & Environmental Physics (LASEP), School of Architectural, Civil and Environmental Engineering (ENAC) Ecole Polytechnique Federale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

^* Corresponding author (markus.berli{at}alumni.ethz.ch)

	ABSTRACT

TOP ABSTRACT INTRODUCTION THEORETICAL CONSIDERATIONS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX 1 APPENDIX 2 REFERENCES

 
In situ determination of stress, strength, and deformation of soils for direct assessment of trafficability and susceptibility to compaction has been a long-standing problem in soil mechanics and agricultural engineering. Despite considerable progress in development of sophisticated probes, data interpretation remains in its infancy due to incomplete understanding of soil–probe interaction. In this study we developed a novel theoretical framework for describing pressure and deformation of fluid inclusions within an elasto-plastic soil matrix subject to anisotropic remote stresses that provides the basis for development of in situ probes for stress and deformation measurement. Results showed that for a compressible fluid inclusion (e.g., air) embedded in an elastic matrix, inclusion pressure is determined primarily by the matrix mean stress, Poisson's ratio, and the product of matrix bulk modulus and fluid compressibility. For incompressible fluids (e.g., water), inclusion pressure becomes independent of matrix stiffness. Differences in remote stress affect inclusion shape and influence probe pressure for large deformation due to increasing stress concentration. The solution for an elastic matrix also provides upper and lower bounds for inclusion pressure in an elasto-plastic matrix under isotropic stress. For the more common anisotropic remote stress, inclusion pressure and deformation differ considerably for elastic and elasto-plastic soil matrix. We found that elastic rubber membranes, often used to separate the fluid inclusion from the matrix, do not influence inclusion pressure or shape as long as membrane and soil stiffness are of similar magnitude. Pressure measurements from laboratory and field experiments using the so-called Bolling probe agreed well with inclusion pressures predicted by the proposed model.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORETICAL CONSIDERATIONS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX 1 APPENDIX 2 REFERENCES

 
IN SITU MEASUREMENTS of soil mechanical behavior can be determined with fluid inclusions such as the pressuremeter (Ménard, 1957; Gibson and Anderson, 1961) used in geotechnical engineering. The basic concept, as already introduced by Kögler (1933), consists of a liquid-filled, flexible, cylindrical membrane inflated in the soil within a vertical borehole (Fig. 1a ). The pressuremeter expands uniformly as a function of applied liquid pressure within the horizontally isotropic stress field of the soil while monitoring borehole deformation. The provided pressure-deformation information is used for estimating in situ stress and soil material properties employing a variety of analytical and numerical models (e.g., Nádai, 1937; Baguelin et al., 1972; Jardine, 1992; Hsieh et al., 2002; Pereira et al., 2003).

View larger version (27K): [in this window] [in a new window]   Fig. 1. Basic principal of (a) a pressuremeter to determine pressure-deformation relations of soil and rock in situ (after Mair and Wood, 1987) and (b) a Bolling probe to measure stress in soil under vehicle traffic (after Bolling, 1987).

The relation between measured probe pressure and actual stress in agricultural soil has been a longstanding subject of debate. In contrast to geotechnical engineering, however, little attention has been paid to the physical processes involved and the few available studies are limited to metal pressure cells (Weiler and Kulhawy, 1982; Kirby, 1999a, 1999b). Previously described models for the pressuremeter provide insights into the mechanics of soil body around an expanding cylindrical borehole driven by a fluid-filled membrane but do not necessarily explain the influence of a general stress state in the soil on an "entrapped" fluid inclusion. Typically, pressuremeter models describe static and isotropic stress conditions whereas stresses in soil underneath a surface load are typically anisotropic and time-dependent (e.g., passage of a tractor, see Fig. 2 ). Probe deformation due to stress anisotropy and subsequent effects of stress concentration along the probe surface are not captured by these pressuremeter models. The current pressuremeter models also treat the probe pressure as a (controlled) boundary condition while for the probes used in agricultural engineering the probe pressure is the result of the mechanical interaction between soil matrix, membrane and fluid and therefore not just influenced by the matrix but also by the membrane and fluid deformation behavior.

View larger version (37K): [in this window] [in a new window]   Fig. 2. Typical pressure profiles under the wheels of (a) a compact wheel loader (Volvo L30B ZS) and (b) steel track of an excavator (Liebherr 942) measured with Bolling-probes (Bolling, 1987) by Berli et al. (unpublished data, 2000).

	THEORETICAL CONSIDERATIONS

TOP ABSTRACT INTRODUCTION THEORETICAL CONSIDERATIONS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX 1 APPENDIX 2 REFERENCES

 
Motivated by the cylindrical shape of pressuremeter and Bolling probe (Bolling, 1987), we focus in this study on pressure and deformation of a cylindrical fluid inclusion within a soil matrix subjected to remote anisotropic compressive stresses (Fig. 3 ). The subsequent physical considerations are, however, also valid for fluid inclusions with shapes different than cylindrical.

View larger version (19K): [in this window] [in a new window]   Fig. 3. (a) Initially circular cylindrical fluid inclusion within a soil matrix subjected to remotely applied stresses 11 and 22, (b) inclusion pressure pI equal to the matrix stress normal to (c) the inclusion surface 1.

Deformation of a Fluid Inclusion within Elastic Soil Matrix—An Analytical Model
The analytical model presented here describes pressure and deformation of a fluid inclusion considering a homogeneous and isotropic linear elastic matrix. The effect of the hydrostatic inclusion pressure, p_I, on the soil matrix can be represented as a constant normal stress, _I, acting along the inclusion surface (Fig. 3b) with |p_I| = |_I|. The loading on the soil matrix can therefore be treated as a superposition of the inclusion pressure, p_I, uniaxial loading in the x₁ direction, ₁₁, and uniaxial loading in the x₂ direction, ₂₂. The solution for the soil matrix is derived using the following Airy stress function (for details see e.g., Timoshenko and Goodier 1970).

[1]

From Eq. [1], the radial and circumferential displacement components, u_r and u, within the soil matrix are

[2]

[3]

where k is Muskhelishvili's constant with k = 3 – 4 for plane strain, G is the shear modulus, and is Poisson's ratio of the matrix material. From these expressions, the volumetric strain of the fluid inclusion will be calculated. Knowing that the fluid inclusion deforms into an elliptical shape, it is necessary to calculate the radial displacement at the soil–fluid interface (r = r₀) at the two locations = 0 and = /2, denoted as Points B and A, respectively (Fig. 3a and 3b). From Eq. [3], the circumferential displacement at these two locations is zero. From Eq. [2], with k = 3 – 4, the radial displacement at Points A and B are

[4]

[5]

The semi-axes of the elliptical inclusion, a and b, can be written as a = r₀ + u_r^A and b = r₀ + u_r^B. The fluid inclusion pressure, p_I, is related to the volumetric inclusion strain, V/V₀, according to the following constitutive law for an elastic compressible fluid

[6]

with the fluid compressibility. Under plane strain conditions, V/V₀ becomes

[7]

Substituting Eq. [7] into Eq. [6] and substituting for the radial displacements given by Eq. [4] and [5], a quadratic equation for the inclusion pressure, p_I, is obtained. Solving the quadratic equation for p_I yields

[8]

For small deformations, the non-linear term, u_r^Au_r^B, in Eq. [7] can be neglected and the inclusion pressure can be approximated as

[9]

This approximation is of special interest since the underlying linear elastic constitutive law is strictly valid for small deformations. Also note that for the case of compressive remote stresses, ₁₁ and ₂₂ are defined negative, which yields a positive inclusion pressure. For an easier comparison of matrix stiffness and fluid compressibility, it is favorable to express the elastic matrix properties in terms of bulk modulus, K, and Poisson's ratio, . The bulk modulus, K, can be derived from shear modulus, G, and Poisson's ratio, , according to

[10]

(Kuchling, 1989). The pressure approximation from Eq. [9] together with Eq. [10] yields the following relation between remote stress, inclusion compressibility and matrix material properties

[11]

Note that for the case of a water inclusion in soils, K 10^–3 and Eq. [11] eventually simplifies to

[12]

Employing the inclusion pressure from Eq. [8] or [9], inclusion deformation can be easily expressed in terms of the aspect ratio, , of the inclusion semi-axes a, b

[13]

with u_r^A and u_r^B the radial displacements given by Eq. [4] and [5].

Measurements of different soils show that in general, stress–strain relationships are non-linear and elasto-viscoplastic rather than linear elastic (Vyalov, 1986; Ghezzehei and Or, 2001; Yong, 2003). Vyalov (1986) concluded that the soil matrix can be modeled as a viscoplastic Bingham material (Bingham, 1922), meaning the material flows as an incompressible Newtonian fluid (characterized by a plastic viscosity) when the applied stress exceeds a limiting shear stress referred to as yield stress (Vyalov, 1986). For stresses lower than the yield stress, the soil matrix acts as an elastic material. Although linear elastic solutions given by Eq. [4], [5], and [8] may be extended to describe purely linear viscous material behavior (see e.g., Berli et al., 2006), they are in general not valid for elasto-viscoplastic materials with complex yield behavior and for large deformations. For many practical applications (Fig. 1), fluid inclusion and solid matrix are also not in direct contact but separated by a membrane that may influence inclusion pressure and deformation behavior. To study the influence of more general material properties and the presence of an elastic membrane on inclusion pressure and deformation, a Finite Element model was introduced as outlined in the subsequent section.

Deformation of a Cylindrical Fluid Inclusion within an Elastic Membrane Embedded in an Elasto-Plastic Material–Finite Element Analysis
In a second step we investigate shape and size evolution of a cylindrical fluid inclusion with initially circular cross section area within a linear elastic membrane embedded in an elasto-plastic soil matrix using Finite Element analysis. For that purpose, we set up a quarter-symmetrical mesh, consisting of quadrilateral 4-node solid elements for soil and membrane and 4-node acoustic elements for the fluid using the ANSYS (ANSYS Inc., Canonsburg, PA) package (Fig. 4 ). The inclusion has an initial radius of 7.5 x 10^–3 m and the membrane was 5 x 10^–4 m thick. The overall length and height of the mesh is 0.1 m. The mesh around the pore is refined within a zone of three times the inclusion initial radius (Fig. 4). The mesh displacement is confined vertically at the bottom and horizontally at the left-hand edge. Static loads are applied in terms of principal stresses to the top, ₁₁, and the right-hand side of the mesh, ₂₂, simulating plane strain conditions under biaxial (remote) stress.

View larger version (67K): [in this window] [in a new window]   Fig. 4. Finite-element mesh (refined zone) used to calculate deformation and pressure of a cylindrical, initially circular liquid inclusion (plane strain situation) within a linear elastic membrane embedded in elasto-plastic material undergoing remote stress 11, 22.

Perzyna's model (Perzyna, 1966, p. 272) is based on the assumption that the total strain rate of an elasto-viscoplastic solid can be expressed as the sum of the elastic, _el, and the plastic strain rate component, _pl, expressed for uniaxial loading according to

[14]

In soils, we assume that elastic deformations are instantaneous and the elastic strain component, _el, can be expressed with Hooke's law as

[15]

with , the axial stress and E, Young's modulus of the material. The plastic strain rate, _pl, according to Perzyna (1966, p. 284), is related to the applied axial stress, , as

[16]

with ₀, the yield stress, ₀/, the plastic viscosity and m, the strain hardening parameter. The Finite Element program solves Perzyna's model of Eq. [14] to [16] (Perzyna, 1966) for multiaxial loading for the mesh given in Fig. 4 (for details see e.g., Bathe, 1996) and provides inclusion pressure, p_I, and deformation, u_r, u, as functions of time, applied remote stresses, ₁₁, ₂₂, and material properties (matrix Young's moduli, E, Poisson's ratio, , yield stress, ₀, plastic viscosity, ₀/, membrane Young's modulus, E_Me, membrane Poisson's ratio, _Me, and inclusion compressibility ). In this study, we focused on the transition from elastic to plastic matrix deformation behavior and reduced Perzyna's general elasto-viscoplastic model (Perzyna, 1966, p. 272) to a simpler elastic-perfectly plastic model, neglecting time-dependent viscous effects. The transition from elastic to plastic behavior is characterized by the von Mises yield criterion (von Mises, 1913). In a three-dimensional elastic-perfectly plastic body, plastic deformation occurs if the effective stress, _e, defined as

[17]

exceeds the material yield stress, _o, with _ij, i, j = x, y, z the components of the local stress tensor in Cartesian coordinates within the matrix material. For effective stress _e < ₀, the matrix deforms elastically. Note that the resulting inclusion pressure, p_I, and deformation u_ij, represent equilibrium conditions and are time-independent.

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION THEORETICAL CONSIDERATIONS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX 1 APPENDIX 2 REFERENCES

 
Pressure and Deformation of a Fluid Inclusion in Linear Elastic Material
In this section we present the influence of remote stress and material properties on inclusion pressure, volume, and shape. The matrix material is limited to the simple case of linear elasticity while the fluid is assumed to be compressible (e.g., an ideal gas). Simulation results show that inclusion pressure increases and volume decreases with increasing average stress in the soil matrix (Fig. 5 ). Under isotropic remote stress, ₁₁ = ₂₂, inclusion aspect ratio remains constant = 1, as expected. An increase in stress difference (at constant average stress) leads to a decrease in inclusion aspect ratio and a small increase in inclusion pressure (Fig. 6 ), directly associated with a small decrease in volume (see Eq. [6]). The difference in remote stress controls the change in inclusion shape and has also a small influence on inclusion pressure that increases with decreasing inclusion aspect ratio due to stress concentration effects along the inclusion surface.

View larger version (28K): [in this window] [in a new window]   Fig. 5. (a) Pressure pi and (b) relative volume, V/V0, of an initially circular inclusion within linear elastic material subjected to isotropic remote stress 11 = 22 for different products of fluid compressibility, , and matrix bulk modulus, K, for Poisson's ratio = 0.3.

View larger version (26K): [in this window] [in a new window]   Fig. 6. (a) Pressure pi and (b) aspect ratio of an initially circular inclusion within linear elastic material subjected to anisotropic remote stress |11| |22| (with (11 + 22)/2 = 100 kPa) for different products of fluid compressibility, , and matrix bulk modulus, K.

K

K

View larger version (33K): [in this window] [in a new window]   Fig. 7. Non-dimensional inclusion pressure –2pi/(11 + 22) as a function of fluid compressibility and matrix bulk modulus K for Poisson's ratio = 0.3.

K

K

K

View larger version (25K): [in this window] [in a new window]   Fig. 8. (a) Pressure pi and (b)relative volume, V/V0, of an initially circular inclusion within linear elastic material of Poisson's ratio subjected to remote stress 11 = 22 with fluid compressibility = 10–5 kPa–1 and matrix bulk modulus K = 103 kPa.

View larger version (24K): [in this window] [in a new window]   Fig. 9. (a) Pressure pi and (b) aspect ratio of an initially circular inclusion within linear elastic material of Poisson's ratio under anisotropic remote stress |11| |22| (with (11 + 22)/2 = 100 kPa) with fluid compressibility = 10–5 kPa–1 and matrix bulk modulus K = 103 kPa.

View larger version (36K): [in this window] [in a new window]   Fig. 10. Inclusion pressure within linear elastic perfectly-plastic material (Poisson's ratio = 0.3) for different yield stresses 0 under isotropic remote stress 11 = 22 with fluid compressibility = 10–5 kPa–1 and matrix bulk modulus K = 103 kPa.

View larger version (25K): [in this window] [in a new window]   Fig. 11. Pressure pi (a) and aspect ratio (b) of an initially circular inclusion within linear elastic perfectly-plastic material (Poisson's ratio = 0.3) for different yield stress 0 subjected to anisotropic remote stress |11| |22| (with (11 + 22)/2 = 100 kPa) with fluid compressibility = 10–5 kPa–1 and matrix bulk modulus K = 103 kPa.

View larger version (21K): [in this window] [in a new window]   Fig. 12. (a) Inclusion pressure pi and (b) aspect ratio within linear elastic material under isotropic (11 = 22 = 0 to 100 kPa) and anisotropic (11 = 100 kPa, 22 = 0...100 kPa) remote stress with fluid compressibility = 5 x 10–7 kPa–1 (water), soil bulk modulus K = 104 kPa and Poisson's ratio = 0.3 for different membrane stiffness (KMe, = 0.49).

[18]

Since Eq. [12] only requires the sum of the principal stresses, ₁₁ and ₂₂, one obtains for the inclusion pressure, p_I,

[19]

Considering the simplifying assumptions in the basis of our model calculations (linear elastic material, plane strain conditions, uniform stress distribution at the soil surface), measured and calculated pressures agree surprisingly well (Fig. 13).

View larger version (36K): [in this window] [in a new window]   Fig. 13. Measured and calculated pressure in a Bolling probe (Bolling, 1987) as a function of surface load and probe depth for constant contact area.

[20]

which introduces an empirical and non-dimensional factor, k_s, Bolling associated with the different stiffness of soil and probe. Assuming plane-strain conditions [i.e., no strain along the probe axis, _yy = (_xx + _zz)], the probe pressure in linear elastic material becomes (Bolling 1987)

[21]

From Eq. [11] and [27], k_s can then be calculated immediately as

[22]

(note: ₁₁ + ₂₂ = _xx + _zz) which shows that k_s is indeed a function of soil and probe stiffness but also of soil Poisson's ratio. Since for most soils, K < 10^–2, k_s becomes mainly a function of Poisson's ratio and can be estimated as

[23]

With Eq. [A3] and [A6], the theoretical upper and lower limits for probe pressure can be established as a function of soil mean stress assuming a range of possible Poisson's ratios, 0 0.5. Using k_s–values determined by Bolling (1987, Table 3.12), we found that these limits hold for a variety of probe pressure versus mean stress relations for both field and laboratory experiments on different soils (Fig. 14 ).

View larger version (31K): [in this window] [in a new window]   Fig. 14. Measured probe pressures for different field and laboratory experiments (Bolling, 1987) compared to predicted upper and lower limits for the probe pressure based on Eq. [26] and [29].

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION THEORETICAL CONSIDERATIONS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX 1 APPENDIX 2 REFERENCES

The analytical model showed that the inclusion pressure depends on the average stress in the matrix while the inclusion shape is controlled by the difference in matrix remote stresses. Besides the stresses, matrix Poisson's ratio and the product of matrix bulk modulus and fluid compressibility were identified to determine inclusion pressure and shape. We found that the elastic solutions provide upper and lower bounds for the inclusion pressure in elasto-plastic material for isotropic but not for anisotropic remote stress. Pressure measurements from laboratory and field experiments using the Bolling probe agreed well with predicted inclusion pressures for the model calculations. These preliminary results encourage us to expand the present model to capture time-dependent processes, run strain rate controlled experiments and develop a probe to determine soil elasto-viscoplastic material properties in situ.

	APPENDIX 1

TOP ABSTRACT INTRODUCTION THEORETICAL CONSIDERATIONS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX 1 APPENDIX 2 REFERENCES

 
Influence of a Rubber Membrane on Inclusion Pressure and Shape: Determine Rubber Material Properties
For practical applications, solid matrix and fluid inclusion are separated by an elastic membrane that prevents the inclusion fluid from flowing out into the matrix. Since Young's modulus, E_Me, and Poisson's ratio, _Me, of a rubber membrane are usually not known a priori, we present a model to determine the elastic properties from a simple inflation experiment and apply it to the case of a silicon rubber used for the Bolling probe (Fig. 15 ). The model relates the radial and axial deformation u(r) and w(z) of a thin linear elastic cylindrical membrane defined as

[A1]

where p is the fluid pressure, l the length, r the inner radius and t the thickness of the membrane at fluid pressure p, l₀ the initial length, r₀ the initial inner radius and t₀ the initial thickness of the membrane and E_Me the membrane Young's modulus and _Me the Poisson's ratio (Fig. 16a ). From equilibrium conditions, one can relate the fluid pressure, p, to the circumferential stress, (r) (Fig. 16b), and axial stress, _zz (r), (Fig. 16c) according to

[A2]

Where r is again the inner radius and t the thickness of the membrane. The calculation assumes the change in membrane thickness is small (and therefore can be neglected) compared to the change in membrane radius and length. Employing the two-dimensional constitutive law between stress, , _zz, and strain, , _zz, of a homogeneous, isotropic and linear elastic material (in cylindrical coordinates)

[A3]

and the relation between circumferential and axial strains, , _zz, and deformations, u,

[A4]

yields for the radial and axial deformations u and w,

[A5]

[A6]

Eq. [A5] and [A6] can be easily solved for Poisson's ratio, _Me, and Young's modulus, E_Me, as

[A7]

[A8]

To determine _Me and E_Me from the expansion experiment, it is favorable to write Eq. [A7] and [A8] in terms of initial (l₀, r₀) and current membrane length and radius (l, r), respectively

[A9]

[A10]

using the definitions of Eq. [A1]. Finally to compare membrane stiffness with soil stiffness, K, and fluid compressibility, , the membrane bulk modulus, K_Me, can be derived from Young's modulus, E_Me, and Poisson's ratio, _Me, according to

[A11]

(Kuchling, 1989).

View larger version (57K): [in this window] [in a new window]   Fig. 15. Cylindrical silicon rubber membrane (front) and assembled Bolling probe tip (back) (probe design: Institute of Terrestrial Ecology – ETH Zurich).

View larger version (34K): [in this window] [in a new window]   Fig. 16. (a) Cylindrical rubber membrane (length l, inner radius r, membrane thickness t) undergoing an internal pressure p resulting in circumferential and axial stresses, zz. Radial stresses were neglected due to the small membrane thickness.

View larger version (18K): [in this window] [in a new window]   Fig. 17. (a) Young's modulus and (b) Poisson's ratio of the silicon rubber membrane used for the Bolling probe (Bolling, 1987).

	APPENDIX 2

TOP ABSTRACT INTRODUCTION THEORETICAL CONSIDERATIONS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX 1 APPENDIX 2 REFERENCES

 

Symbol Description Unit E Young's modulus of the matrix kPa EMe Young's modulus of the membrane kPa G Shear modulus of the matrix kPa K Bulk modulus of the matrix kPa KMe Bulk modulus of the membrane kPa k Muskhelishvili's constant — ks Non-dimensional factor according to Bolling (1987) — m Strain hardening factor according to Perzyna (1966) — p Fluid pressure kPa pI Inclusion pressure kPa ur, u Radial and circumferential displacement components in soil m u, w Radial and axial displacement components in rubber membrane m ii, i = 1,2,3 Components of the remote stress tensor in principal directions, x1, x2, x3 kPa Viscosity factor according to Perzyna (1966) s–1 el Local elastic strain (uniaxial Perzyna model [Perzyna, 1966]) — Local total strain rate (uniaxial Perzyna model [Perzyna, 1966]) s–1 el Local elastic strain rate (uniaxial Perzyna model (Perzyna, 1966)) s–1 pl Local plastic strain rate (uniaxial Perzyna model [Perzyna, 1966]) s–1 zz, Axial and circumferential strain components in rubber membrane — Fluid compressibility kPa–1 Inclusion aspect ratio — Poisson's ratio of the solid matrix — Me Poisson's ratio of the membrane — Local stress (uniaxial Perzyna model [Perzyna, 1966]) kPa e von Mises effective stress kPa I Local normal stress within the matrix along the inclusion surface kPa 0 Matrix yield stress in compression kPa zz, Axial and circumferential stress components in rubber membrane kPa

Received for publication June 2, 2005.

	REFERENCES

TOP ABSTRACT INTRODUCTION THEORETICAL CONSIDERATIONS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX 1 APPENDIX 2 REFERENCES

 

• Baguelin, F., J.-F. Jezequel, E. Le Mee, and A. Le Mehaute. 1972. Expansion of cylindrical probes in cohesive soils. J. Soil Mech. Found. Div.–Proc. Am. Soc. Civil Eng. 98:1129-1142.
• Bathe, K.-J. 1996. Finite element procedures. 1st ed. Prentice Hall, Englewood Cliffs, NJ.
• Berli, M., M.L. Accorsi, and D. Or. 2006. Size and shape evolution of pores in viscoplastic matrix under compression. Int. J. Numerical and Analytical Methods in Geomechanics: (in press).
• Bingham, E.C. 1922. Fluidity and plasticity. 1st ed. McGraw-Hill, NY.
• Blackwell, P.S., and B.D. Soane. 1978. Deformable spherical devices to measure stresses within field soils. J. Terramech. 15:207–222.[CrossRef]
• Bolling, I. 1987. Bodenverdichtung und Triebkraftverhalten bei Reifen - Neue Mess- und Rechenmethoden. (In German.) PhD Thesis. Technische Universität München, München.
• Boussinesq, M.J. 1885. Applications des potentiels à l'étude de l'equilibre et du mouvement des solides élastiques Gauthier-Villars, Paris.
• Davis, J.R. 1995. Guide to materials selection. p. 107-154. In M.M. Gauthier (ed.) Engineered materials handbook. ASM International, Materials Park, OH.
• Ghezzehei, T.A., and D. Or. 2001. Rheological properties of wet soils and clays under steady and oscillatory stresses. Soil Sci. Soc. Am. J. 65:624–637.[Abstract/Free Full Text]
• Gibson, R.E., and W.F. Anderson. 1961. In-situ measure