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Cylindrical cavity expansion in nonlinear dilatant media
MECHANICS RESEARCH COMMUNICATIONS VO1.14(4), 219-227, 1987. Printed in the USA. 0093-6413/87 $3.00 + .00 Copyright (c) 1987 Pergamon Journals Ltd.
CYLINDRICAL CAVITY EXPANSION IN NONLINEAR DILATANT MEDIA
S. M. SAYED Department of Civil Engineering, University of New Orleans New Orleans, Louisiana, 70148, USA (Received 25 June 7986; accepted for print 29 May 1987)
Introduction
The study reported herein is concerned with the problem of expansion of long cylindrical cavity in nonlinear dilatant media. Theoretical solutions of the cavity expansion problem, in general, have been used in several applications in geomechanics [3,10,11,13,14]. The method of analysis described in this paper is a general approach. It treats the problem of expansion of long cylindrical cavity in nonlinear dilatant media, possessing both cohesion and friction in the Coulomb-Mohr sense. The solution takes into account the effect of volume changes in the plastic zone. This method of analysis enables a more reliable assessment of the determination of deformation parameters of porous media using the results of expansion (pressuremeter) test. Such assessment has until now been done by using the linear theory of elasticity. Also, new concepts are gained regarding the effects of the boundaries and the limitations of the linear elastic model.
Mathematical Formulation
The analysis presented here is based on axi-symmetric and plane strain conditions. Also cavity section remains circular during the expansion and unloading condition is not permitted. Fig. 1 shows a schematic representation of an expanding cylindrical cavity of initial radius ao. Prior to yield, the medium is idealized as a nonlinear isotropic elastic material. Upon yielding it behaves as a compressible plastic solid, defined by a hnear Coulomb-Mohr failure envelope as well as by an average volumetric strain ~ which can be determined form Vesi6's theory [14]. For the elastic phase, the constitutive relationships of the nonlinear response [6], material as well as geometric, are expressed as ,7, = ,,',o + [c,~, +
c~o]lJ
,,o = ,,'Oo+ [cf*o+ c2,+,.]IJ The radial equilibrium equation of a weightless medium has the following form d+,/dr
+ (<7, -
219
~o)/r = 0
(1) (2) (3)
220
S.M. SAYED
It is assumed that the relationship of the shear stress (~, - ~0)/2 versus the shear strain ~/2, as shown in Fig. 2, is represented by a hyperbola. This assumption was based on the fact that nonlinear stress-strain curves of both cohesive and frictional materials can be approximated by a hyperbola with a high degree of accuracy [7]. Accordingly (~, - o0)/2 versus 7/2 can be expressed as (o'r -- o ' 0 ) / 2
= (7/2)[ah + bh('y/2)]
(4)
in which ah = 0.5/G, and bh = 2/(~, - eo)~. = 2/(1 - N)[PL + c cotq~] where N = (1 - stn(O)/(1 + sm4~). G,, PL, c and ~ are referred to the initial shear modulus, limit radial stress, cohesion and angle of internal friction, respectively. The choice of such model is adopted to find a value for the scalar J which reflects the stress level developed in the cavity expansion.
( °'r- °-8)ult / 2
,
b~
/
Yielding Gi : 2Gi Shear strain, -~2= er-2 ea
FIG. 1 Expansion of Long Vertical Cylindrical Cavity
(o-, -
FIG. 2 ~o) vs.
%
~/2
Subtracting Eq. (1) from Eq. (2) and dividing the result side-by-side with Eq. (4), noting that 7/2 = (~, - c0)/2, the scalar J can be expressed as: J = [ah + bh(~, - ~ 0 ) / 2 ] [ ( ~ ,
- ~ 0 ) ( c l - c~.)]/[(~, - ~0) - A p o ( ~ h
+ bh)(~, - ~0)/2]
Ix)
APo = a,o - cr0o; O',o and Cr0o are in-situ radial and circumferential stresses; ~, and c,0 are radial and circumferential stresses; ~, and ~0 are Eulerian radial and circumferential strains; c~ and c2 are constants expressed as functions of Poisson's ratio u. The final formulation of the solution in the nonlinear elastic phase, based on Eq. (1) through Eq. (3) and Eq. (5), is expressed as a boundary value problem in the form of a set of three first-order differential equations expressed as du~/dr = F1
(fi)
dtr~/dr = F2
(7)
dz/dr = Fz
(8)
CAVITY IN NC~LINEAR DIIATANT MEDIA
221
In which z = du,/dr, u,=radial displacement, Fa, g2, and F8 are functionals. The boundaries of the solution domain encompass the cavity-medium interface and a radial distance that approximates an infinite boundary. The solution domain changes as the stress level (or strain level) changes at the interface and hence the boundaries should be adjusted continuously to reflect these changes. The following expression gives the radial distance r=,, defining infinity which reflects the material behavior, stress level and the error tolerance used in computing the stresses. r,,,.® = A,x/2Gt,/¢#',o a°; and ¢ = Aa',/o',.o at 1"= r=..
(9)
The circumferential strain ,° at the cavity-medium interface and dimensionless parameters ,L and ~ appearing in Eq. (9) introduce the dependence of r,~,. on the stress level, idealized material behavior and error tolerance, respectivdy, a is the shear modulus of the linearized stress-strain curve that closely approximates the nonlinear behavior. A complete discussion of the derivation of Eq. (9) can be found in Ref. [11]. At the radial distance defining infinity r = r=.., u, = 0 and ~, = ~,,°. At the cavity-medium interface r = a, u, = ~,. The numerical solution is unique and exists so long as r=.. # ~ and Poisson's ratio v ¢ 0.5. Eq. (3) along with a volume change parameter A [14] and the following Mohr-Coulomb yield criteria are used in the plastic phase ~,, - o-e = (,-, + *'Oin¢, + 2, co, O
(lo)
The solution in the plastic phase is expressed in a dosed form as P = [,%(1 + , i n ¢ ) + c(co,¢, + ca4,)][,~,(2~ - ~,)e°/,,o(2a - Uo)tp]e,p s i n e / ( 1 + , , n ¢ ) - c ca4,
(1])
For practical reasons, it may be assumed that an infinite expansion corresponds to the condition at which the cavity doubles its volume. In other words, the ultimate cavity radius g, = 1.4]ao. Accordingly, the radius of the plastic zone ~ can be computed from the following equation in which the modified-reduced rigidity index L, is expressed as L , = I,(~ + I, ZXsecO); I, = g,/2(1 + v)(c + ~,,otanO)
03)
The parameter ~ = E,/E, where E, and E, are the initial and secant moduli of deformation, respectively. The radial displacement u~ (,p is the corresponding Eulerian circumferential strain) at the boundary between plastic and elastic zones has the form up = ~(a + ,,,)g,(~,°oi,O + c c a ¢ ) / E,
(a4)
Once the values of Rp and ~ are computed, the relationship between the radial stress P and the radial displacement Uo at the cavity-medium interface can be established. The model presented in this paper can be used in applications assodated with geomechanics, such as expansion tests, and computation of induced stresses clue to plane strain and axi-symmetric conditions. The model parameters are Poisson's ratio v, cohesion c, angle of internal friction 0; volume change parameter A, in-situ stresses ~,,o and ~'eo, dimensionless
222
S.M. SAYED
parameters •, and (. These parameters can be obtained from conventional triaxial and field tests. It should be noted that in the case of an expansion test, the model parameters can be back-computed from the test results and then used to assess the adjustment needed to obtain reliable deformation characteristics so that the effect of finite length of the probe is accommodated. In this regard, the parameters to be used in the nonlinear model must be corrected as proposed by Laier et al [9]. This task requires a trial and error approach [11,12].
Applications
(I) Effect of L e n g t h / D i a m e t e r Ratio L/d of the Cavity on the Modulus of Deformation: The results of expansion tests in porous media are, in general, used to determine their deformation and strength characteristics. The computation of the modulus of deformation E is based on the theoretical solution of an infinitely long cavity in a linear elastic medium. This section is intended to explore any effect the finite length of the probe might have on the modulus of deformation by using the model presented herein. Most expansion (pressuremeter) tests are conducted without unloading/reloading cycles. However, cycled tests are often performed to allow the determination of rebound and reload moduli [2]. The cycled moduli are used in modeling the soil behavior under conditions simulating machine foundations, wave action and other repetitive loading situations. The analysis and discussion presented in this paper refer to the self-boring pressuremeter test and the use of its results in the pseudo-elastic range in computing the soil deformation modulus. The self-boring pressuremeter test is associated with very little disturbance, if any, as far as the borehole preparation is concerned. Therefore, the effect of disturbance on the modulus of deformation is not considered herein and discussed elsewhere by Sayed and Hamed [13]. A comparison is made between the results obtained from an expansion test with a probe of finite length and those obtained from the proposed model. Only the elastic range which typically ranges from ~o = 0% to 0.2% and 0% to 1.5% for dense and loose granular material, respectively, is considered in this discussion for its relevance in computing the modulus of deformation. Example 1: Expansion Test in Dense Granular Material: A simulated curve of the expansion test is compared with the experimental curve obtained by Jewell et al [8] in Fig. 3. The simulated curve in the elastic phase (theory) is different from the experimental curve (measurement). The experimental curve shows stiffer behavior and accordingly higher values of the modulus of deformation. Example 2: Expansion Test in Loose Granular Material: A similar comparison is made between the theoretical results of an expansion test obtained by the present analysis and the experimental results [4] as shown in Fig 4. As indicated in the case of dense granular material shown in Fig. 3, again the experimental data of the expansion test lie above the simulated data given by the nonlinear model. The experimental and theoretical results show a very important aspect concerning the shape of the data of the expansion test in the pseudo-elastic range and hence the deter-
CAVITY IN NONLINEAR DILATANT MEDIA
223
mmation of the modulus of deformation E from these data. Based on the linear theory of elasticity it has been shown [9] that the determination of the deformation modulus S from the expansion test results with a probe of finite length, assuming an infinite cavity, is accompanied by an error of about 5%. Such conclusion was drawn by comparing the linear elastic solution of infinite and finite cavities, however there is no guarantee that the multi-phase material is behaving linearly. The mathematical solution of the modulus of deformation E in an expanding infinite vertical cylindrical cavity in an ideal elastic-plastic media [14] is expressed as E = 2(1 +
r,)Vo(Ap/AV)
(is)
where Vo and AP/AV are the initial volume of the cylindrical cavity and the gradient of the pressure-volume curve, respectively. Since the strain increment ZXeois proportional to AV, E can be stated as E ~x Ap/A~o 22
2C
E
16
oo = 1.583 inch, L/d =4 ~o = 13 05 psi Gi= 4,496 psi
u:030
•
(16)
ao=O 60 inch, L/d = 16 ~ro = 8.68 psi 14 Gi= 137 psi
t
u=O 33 = 320 °
•
18
Z @
"
~
16
I0
0
14
MEASUREMENT; BHUSHAN (1970 THEORY; PRESENT ANALYSIS
• • MEASUREMENT; JEWELL ET AL
[E
( 1980 ) THEORY; PRESENT ANALYSIS
12 IO O0
I
I
OD5
O.IO
4
0 15
Hole strain, ~o' %
FIG. 3 Comparison of Experimental and Theoretical Results of Expansion Test in Dense Granular Material
0
I 0.5
I 1.0
~s
Hole strain, Go, %
FIG. 4 Comparison of Experimental and Theoretical Results of Expansion Test in Loose Granular Material
For the purpose of illustration, a schematic representation of experimental and theoretical (nonhnear model presented in this paper) expansion curves are shown in Fig. 5. These representations show that incorporating the slope Ap/Aeo (termed KL and K~ for the experimental and theoretical curves, respectively) with Eq. (16) will yield higher values for the modulus of deformation in the case of the experimental curve than the theoretical one; namely KL/K~, > 1.0. This is true whether the secant (Fig. 5-a) or conventional (Fig. 5-b) moduli are considered. Such finding shows that the determination of the modulus of deformation from the experimental data (finite cavity) along with the linear theory of elasticity of an infinite cavity is in doubt. The simulated curve in the elastic phase, obtained using the nonlinear general solution presented herein, is different from the experimental curve. The latter shows stiffer behavior and accordingly higher values of the modulus
224
S.M. SAYED
of deformation. This analysis indicates that the finite length of the pressuremeter probe ( L / d = 4 and 16 in Examples 1 and 2, respectively) is probably one of the most important causes for the discrepancy between the experimental and theoretical results. --
0-"
P
P
/
0..
Pf
Pf
~ -6 P0
/
--K:-
KN ~ KL
"0 I0 nr"
0::
Hole Stra=n, % , %
(a) Secant Modulus
Hole Strain,
%,%
(b) Conventional Modulus
FIG. 5 Schematic Representation of Experimental and Simulated Expansion Test Results The well documented comparison regarding the large discrepancy between the values of the modulus of deformation obtained from the self-boring pressuremeter and conventional triaxial tests is given by Amar et al [1]. Wroth [15] suggested that part of this discrepancy is due to the high strain rate used in the field and recommended that the results should be analyzed in terms of the engineering shear strain. Such recommendation reduces the difference between both sources by about 25%. Also, it has been thought [5] that the disagreement between the pressuremeter and triaxial tests is contributed to the disturbance as well as anisotropic effects. These effects still are some of the reasons contributing partially to such disagreement. However, the question regarding the effect of the finite length of the pressuremeter probe remains unresolved, if one uses the linear theory of elasticity along with the pressuremeter test results. Although there are several reasons to believe that the deformation moduli should not be the same from pressuremeter and triaxial tests, there is doubt behind the effect of the finite length of the probe on this issue. The validity of determining the deformation modulus from the pressuremeter test results using the linear theory of elasticity can not be justified by comparing the theoretical solution of finite and infinite cavities in a linear elastic medium. The results presented in this section indicate that the finite length of the probe has its effect on the modulus if the medium is nonlinear and such an effect is very pronounced. In other words, the theoretical results obtained from the cylindrical cavity expansion in a nonlinear dilatant medium based on infinite length are very much different from those of the actual test with a finite length probe. The experimental results show a much stiffer behavior than the one shown by the simulated results. It is very important. to keep in mind that the experimental results used in this study are based on high quality tests and, therefore, the effect of disturbance does not exist. Assuming that the nonlinear model is superior to the linear elastic model and more realistic in representing the soil behavior, the discrepancy between the actual and theoretical (nonlinear model) results is an indication of the effect of L / d ratio of the cavity. The values of E computed from the pseudo-elastic range along with Eq. (15) are higher for the experimental data than those for the theoretical results. In general, the determination of the
CAVITY IN NONLINEAR DILATANT MEDIA
225
soil deformation modulus from the self-boring pressuremeter test (finite cavity), assuming an infinite cavity, is accompanied by an error of about 200% to 300% (on the unsafe side) and not 5% as implied by the linear theory of elasticity [9]. In summary, the deformation modulus, determined from the pseudo-elastic phase of the self-boring pressuremeter test, should be elaborated to account for the limited length of the probe if plane strain condition is assumed to prevail. Therefore, it seems that the modulus must be divided by some factor (2 or 3) in order to estimate reasonable values for design. II. Theoretical Linear Elastic versus Elastic-Plastic Analysis: The theoretical results obtained from the linear elastic and nonlinear elastic- plastic model for an expansion test are given in Fig. ft. The results of nonlinear model show that the material is initially stiffer than the behavior assumed by the linear elastic model. The linear elastic idealization is a conservative model for porous media for very small strains (in the elastic range). However, when the yield stress approaches, the linear-model becomes unconservative. In other terms, the linear elastic model overestimates the real response, that response can be represented more realistically by the nonlinear model once the stress level is in the neighborhood of yielding. This finding shows that modeling the nonlinear geometric and material effects is essential in formulating constitutive relationships for porous media. III. Stresses in Linear Elastic and Nonlinear Dilatant Materials: The stress distribution in the mass surrounding the cavity in Section II is shown in Fig 7. The decrease in the radial stress increase with the radial distance for the nonlinear model is less rapid than the linear one. On the other hand, the reduction in the circumferential stress for the nonlinear model is smaller than the one of the linear model. A similar behavior has been reported in the literature for the linear dilatant material. Therefore, the radial stresses induced in porous media due to plane strain and axi-symmetric loading conditions that are based on the linear elastic model are on the unsafe side. t 19
v
r
~
I
6/
,4
~
/
/,,' f
/
n
,/
c
h
14 13 0
/
,/
"
,'
r
;
L~e~ ~ o• ~,,,,,o,
psi
G, = 2900psi
/
~
=48
u 0B~/7"f
°
/ L
----
LINEAR ELASTIC
--
NONLINEAR ELASTK I
003 Hole
.
i,
I
l
006
009
012
015
strain,~o%
FIG. 6 Comparison of Theoretical Linear Elastic with Theoretical Nonlinear Elastic-Plastic Analysis
.
.
.
Gi ,29oo.i
~/A u° F
t' = 0 3 3 ~) , 4 8 "
|
041 0
.
% " 1 585 inch
=,o ,13 p,~
||
/ -o o (2:
'
/
/
18 .~ 17
'
,,.o
t 4
I 0
,! 12
I J6
20
r/00
FIG. 7 Stress Distribution in Linear and Nonlinear Dilatant Granular Materials for ~o = 0.06 %
The results indicate that the finite distance (r,~a=) defining the infinite boundary, computed
226
S.M. SAYED
from Eq. (9) with ¢=0.001, affected the distribution of the circumferential and mean normal stresses. Naturally the assumption of rigid boundaries at relatively far distance from the cavity wall will prevent the soil expansion in the circumferential direction near these boundaries. Accordingly the circumferential stress increment will be of compressive nature. In this example, the assumed rigid boundary at r,,,~= introduces about 2% of the in-situ horizontal stress as compressive circumferential stress increment. It is clear that any finite distance used as an approximation to infinity will have its effects on the results obtained from any numerical solution. Since there is no other way to treat most of the complex problems having the infinity as one of its boundaries other than to approximate it, it is important to assess the limitations of such an approximation. The expression for rmo~, Eq. (9) in this paper, can be used to compute the distance defining infinity in plane strain and axi-symmetric problems.
Conclusions
The simulated initial portion of the expansion test results, obtained from the general solution presented in this paper, explains a very important aspect concerning the determination of the modulus of deformation of porous media. It illustrates the effect of the finite length of the probe on the computed values of the modulus. The solutions of finite cavities based on the linear theory of elasticity do not clarify such an effect. It is recommended that the soil deformation modulus obtained from the self-boring pressuremeter test must be divided by a factor of 2 or 3 to account for the limited length of the pressuremeter probe. The use of the linear elastic model as a constitutive law for multi-phase materials does not represent the real behavior of these materials accurately at approaching yield. The nonlinear elastic model gives significantly different results compared to those of the linear one. The former model indicates much less stiffness particularly near yielding. Because the nonlinear approach is genera3, it is superior to the linear one from a mathematical viewpoint. Also, the radial stresses in the medium surrounding the cavity, predicted by the nonlinear analysis, are much higher than the ones obtained by linear analysis. Thus the radial stresses predicted by linear elastic model are unconservative. For these reasons, it is very essential to incorporate the nonlinear geometric and material effects in modeling the behavior of porous media. The infinite boundaries of the solution domain of any boundary value problem represent obstacles in obtaining reliable numerical solutions. These boundaries of the solution domain need to be adjusted continuously as the stress level changes at the interface. An expression is given which reflects the effect of stress level, material behavior and error tolerance. It can be used to evaluate the distance at which the boundary can be assumed to imply the infinite boundary in plane-strain and axi-symmetric problems.
CAVITY IN NONLINEAR DILATANT MEDIA
227
References
1. S. Amar, F. Baguelin, J.-F. J4z4quel, and A. Le M4haut& "In-Situ Shear Resistance of Clays," Proceedings of the Speciality Conference on In-Situ Measurement of Soil Properties, Raleigh, North Carolina, ASCE, Vol. I, p. 22, (1975). 2. F. Baguelin, J.-F. J4z4quel, and D. H. Shields, The Pressuremeter and Foundation Engineering, Trans. Tech. Publications, Clausthal, Germany, (1978). 3. M. M. Baligh, "Cavity Expansion in Sands With Curved Envelopes," Journal of the Geotechnical Division, ASCE, Vol. 102, p. 1131, (1976). 4. K. Bhushan, "An Experimental Investigation into Expansion of Spherical and Cylindrical Cavities in Sand," Ph.D. Dissertation, Duke University, (1971). 5. G. M. Denby, "Self-Boring Pressuremeter Study of the San Francisco Bay," Ph.D. Dissertation, Stanford University, (1978). 6. D. C. Drucker, Introduction to Mechanics of Deformable Solids, McGraw-Hill Book Company, New York, U.S.A., (1967). 7. J. M. Duncan and C. Y. Chang, "Nonhnear Analysis of Stress Strain in Soils," Journal of the Soil Mechanics and Foundation Engineering Division, ASCE, Vol. 96, p. 1625, (1970). 8. R. J. Jewell, M. Fahey, and C. P. Wroth, "Laboratory Studies of the Pressuremeter Test in Sand," Geotechnique, 30, p 507, (1980). 9. J. E. Laier, J. H. Schmertmann, and J. H. Schaub, "Effect of Finite Pressuremeter Length in Dry Sand," Proc. ASCE Speciality Conf. on In-Situ Measurement of Soil Properties, Raleigh, North Carolina, Vol. I, p. 241, (1975). 10. J. H. Pr$vost and K. HSeg, "Analysis of Pressuremeter in Strain-Softening Soil," Journal of the Geotechnical Engineering Division, ASCE, Vol. 101, p. 717, (1975). 11. S. M. Sayed, "Expansion of Long Cylindrical Cavities in Nonlinear Dilatant Media," Ph.D. Dissertation, Duke University, (1982). 12. S. M. Sayed, "Cyhndrical Cavity Expansion in Nonlinear Dilatant Media," presented at the XVI International Congress of Theoretical and Applied Mechanics, Lyngby, Denmark, August, (1984). 13. S. M. Sayed and M. A. Hamed, "Expansion of Cavities in Layered Elastic System," International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 11, p. 203, (1987). 14. A. S. Vesi4, "Expansion of Cavities in an Infinite Soil Mass," Journal of the Soil Mechanics and Foundation Engineering Division, ASCE, Vol. 98, p. 265, (1972). 15. C. P. Wroth, "In-Situ Measurement of Initial Stresses and Deformation Characteristics," Proceedings of the Speciality Conference on the In-Situ Measurement of Soil Properties, Raleigh, North C_arohna, ASCE, Vol. II, p. 181, (1975).
CYLINDRICAL CAVITY EXPANSION IN NONLINEAR DILATANT MEDIA
S. M. SAYED Department of Civil Engineering, University of New Orleans New Orleans, Louisiana, 70148, USA (Received 25 June 7986; accepted for print 29 May 1987)
Introduction
The study reported herein is concerned with the problem of expansion of long cylindrical cavity in nonlinear dilatant media. Theoretical solutions of the cavity expansion problem, in general, have been used in several applications in geomechanics [3,10,11,13,14]. The method of analysis described in this paper is a general approach. It treats the problem of expansion of long cylindrical cavity in nonlinear dilatant media, possessing both cohesion and friction in the Coulomb-Mohr sense. The solution takes into account the effect of volume changes in the plastic zone. This method of analysis enables a more reliable assessment of the determination of deformation parameters of porous media using the results of expansion (pressuremeter) test. Such assessment has until now been done by using the linear theory of elasticity. Also, new concepts are gained regarding the effects of the boundaries and the limitations of the linear elastic model.
Mathematical Formulation
The analysis presented here is based on axi-symmetric and plane strain conditions. Also cavity section remains circular during the expansion and unloading condition is not permitted. Fig. 1 shows a schematic representation of an expanding cylindrical cavity of initial radius ao. Prior to yield, the medium is idealized as a nonlinear isotropic elastic material. Upon yielding it behaves as a compressible plastic solid, defined by a hnear Coulomb-Mohr failure envelope as well as by an average volumetric strain ~ which can be determined form Vesi6's theory [14]. For the elastic phase, the constitutive relationships of the nonlinear response [6], material as well as geometric, are expressed as ,7, = ,,',o + [c,~, +
c~o]lJ
,,o = ,,'Oo+ [cf*o+ c2,+,.]IJ The radial equilibrium equation of a weightless medium has the following form d+,/dr
+ (<7, -
219
~o)/r = 0
(1) (2) (3)
220
S.M. SAYED
It is assumed that the relationship of the shear stress (~, - ~0)/2 versus the shear strain ~/2, as shown in Fig. 2, is represented by a hyperbola. This assumption was based on the fact that nonlinear stress-strain curves of both cohesive and frictional materials can be approximated by a hyperbola with a high degree of accuracy [7]. Accordingly (~, - o0)/2 versus 7/2 can be expressed as (o'r -- o ' 0 ) / 2
= (7/2)[ah + bh('y/2)]
(4)
in which ah = 0.5/G, and bh = 2/(~, - eo)~. = 2/(1 - N)[PL + c cotq~] where N = (1 - stn(O)/(1 + sm4~). G,, PL, c and ~ are referred to the initial shear modulus, limit radial stress, cohesion and angle of internal friction, respectively. The choice of such model is adopted to find a value for the scalar J which reflects the stress level developed in the cavity expansion.
( °'r- °-8)ult / 2
,
b~
/
Yielding Gi : 2Gi Shear strain, -~2= er-2 ea
FIG. 1 Expansion of Long Vertical Cylindrical Cavity
(o-, -
FIG. 2 ~o) vs.
%
~/2
Subtracting Eq. (1) from Eq. (2) and dividing the result side-by-side with Eq. (4), noting that 7/2 = (~, - c0)/2, the scalar J can be expressed as: J = [ah + bh(~, - ~ 0 ) / 2 ] [ ( ~ ,
- ~ 0 ) ( c l - c~.)]/[(~, - ~0) - A p o ( ~ h
+ bh)(~, - ~0)/2]
Ix)
APo = a,o - cr0o; O',o and Cr0o are in-situ radial and circumferential stresses; ~, and c,0 are radial and circumferential stresses; ~, and ~0 are Eulerian radial and circumferential strains; c~ and c2 are constants expressed as functions of Poisson's ratio u. The final formulation of the solution in the nonlinear elastic phase, based on Eq. (1) through Eq. (3) and Eq. (5), is expressed as a boundary value problem in the form of a set of three first-order differential equations expressed as du~/dr = F1
(fi)
dtr~/dr = F2
(7)
dz/dr = Fz
(8)
CAVITY IN NC~LINEAR DIIATANT MEDIA
221
In which z = du,/dr, u,=radial displacement, Fa, g2, and F8 are functionals. The boundaries of the solution domain encompass the cavity-medium interface and a radial distance that approximates an infinite boundary. The solution domain changes as the stress level (or strain level) changes at the interface and hence the boundaries should be adjusted continuously to reflect these changes. The following expression gives the radial distance r=,, defining infinity which reflects the material behavior, stress level and the error tolerance used in computing the stresses. r,,,.® = A,x/2Gt,/¢#',o a°; and ¢ = Aa',/o',.o at 1"= r=..
(9)
The circumferential strain ,° at the cavity-medium interface and dimensionless parameters ,L and ~ appearing in Eq. (9) introduce the dependence of r,~,. on the stress level, idealized material behavior and error tolerance, respectivdy, a is the shear modulus of the linearized stress-strain curve that closely approximates the nonlinear behavior. A complete discussion of the derivation of Eq. (9) can be found in Ref. [11]. At the radial distance defining infinity r = r=.., u, = 0 and ~, = ~,,°. At the cavity-medium interface r = a, u, = ~,. The numerical solution is unique and exists so long as r=.. # ~ and Poisson's ratio v ¢ 0.5. Eq. (3) along with a volume change parameter A [14] and the following Mohr-Coulomb yield criteria are used in the plastic phase ~,, - o-e = (,-, + *'Oin¢, + 2, co, O
(lo)
The solution in the plastic phase is expressed in a dosed form as P = [,%(1 + , i n ¢ ) + c(co,¢, + ca4,)][,~,(2~ - ~,)e°/,,o(2a - Uo)tp]e,p s i n e / ( 1 + , , n ¢ ) - c ca4,
(1])
For practical reasons, it may be assumed that an infinite expansion corresponds to the condition at which the cavity doubles its volume. In other words, the ultimate cavity radius g, = 1.4]ao. Accordingly, the radius of the plastic zone ~ can be computed from the following equation in which the modified-reduced rigidity index L, is expressed as L , = I,(~ + I, ZXsecO); I, = g,/2(1 + v)(c + ~,,otanO)
03)
The parameter ~ = E,/E, where E, and E, are the initial and secant moduli of deformation, respectively. The radial displacement u~ (,p is the corresponding Eulerian circumferential strain) at the boundary between plastic and elastic zones has the form up = ~(a + ,,,)g,(~,°oi,O + c c a ¢ ) / E,
(a4)
Once the values of Rp and ~ are computed, the relationship between the radial stress P and the radial displacement Uo at the cavity-medium interface can be established. The model presented in this paper can be used in applications assodated with geomechanics, such as expansion tests, and computation of induced stresses clue to plane strain and axi-symmetric conditions. The model parameters are Poisson's ratio v, cohesion c, angle of internal friction 0; volume change parameter A, in-situ stresses ~,,o and ~'eo, dimensionless
222
S.M. SAYED
parameters •, and (. These parameters can be obtained from conventional triaxial and field tests. It should be noted that in the case of an expansion test, the model parameters can be back-computed from the test results and then used to assess the adjustment needed to obtain reliable deformation characteristics so that the effect of finite length of the probe is accommodated. In this regard, the parameters to be used in the nonlinear model must be corrected as proposed by Laier et al [9]. This task requires a trial and error approach [11,12].
Applications
(I) Effect of L e n g t h / D i a m e t e r Ratio L/d of the Cavity on the Modulus of Deformation: The results of expansion tests in porous media are, in general, used to determine their deformation and strength characteristics. The computation of the modulus of deformation E is based on the theoretical solution of an infinitely long cavity in a linear elastic medium. This section is intended to explore any effect the finite length of the probe might have on the modulus of deformation by using the model presented herein. Most expansion (pressuremeter) tests are conducted without unloading/reloading cycles. However, cycled tests are often performed to allow the determination of rebound and reload moduli [2]. The cycled moduli are used in modeling the soil behavior under conditions simulating machine foundations, wave action and other repetitive loading situations. The analysis and discussion presented in this paper refer to the self-boring pressuremeter test and the use of its results in the pseudo-elastic range in computing the soil deformation modulus. The self-boring pressuremeter test is associated with very little disturbance, if any, as far as the borehole preparation is concerned. Therefore, the effect of disturbance on the modulus of deformation is not considered herein and discussed elsewhere by Sayed and Hamed [13]. A comparison is made between the results obtained from an expansion test with a probe of finite length and those obtained from the proposed model. Only the elastic range which typically ranges from ~o = 0% to 0.2% and 0% to 1.5% for dense and loose granular material, respectively, is considered in this discussion for its relevance in computing the modulus of deformation. Example 1: Expansion Test in Dense Granular Material: A simulated curve of the expansion test is compared with the experimental curve obtained by Jewell et al [8] in Fig. 3. The simulated curve in the elastic phase (theory) is different from the experimental curve (measurement). The experimental curve shows stiffer behavior and accordingly higher values of the modulus of deformation. Example 2: Expansion Test in Loose Granular Material: A similar comparison is made between the theoretical results of an expansion test obtained by the present analysis and the experimental results [4] as shown in Fig 4. As indicated in the case of dense granular material shown in Fig. 3, again the experimental data of the expansion test lie above the simulated data given by the nonlinear model. The experimental and theoretical results show a very important aspect concerning the shape of the data of the expansion test in the pseudo-elastic range and hence the deter-
CAVITY IN NONLINEAR DILATANT MEDIA
223
mmation of the modulus of deformation E from these data. Based on the linear theory of elasticity it has been shown [9] that the determination of the deformation modulus S from the expansion test results with a probe of finite length, assuming an infinite cavity, is accompanied by an error of about 5%. Such conclusion was drawn by comparing the linear elastic solution of infinite and finite cavities, however there is no guarantee that the multi-phase material is behaving linearly. The mathematical solution of the modulus of deformation E in an expanding infinite vertical cylindrical cavity in an ideal elastic-plastic media [14] is expressed as E = 2(1 +
r,)Vo(Ap/AV)
(is)
where Vo and AP/AV are the initial volume of the cylindrical cavity and the gradient of the pressure-volume curve, respectively. Since the strain increment ZXeois proportional to AV, E can be stated as E ~x Ap/A~o 22
2C
E
16
oo = 1.583 inch, L/d =4 ~o = 13 05 psi Gi= 4,496 psi
u:030
•
(16)
ao=O 60 inch, L/d = 16 ~ro = 8.68 psi 14 Gi= 137 psi
t
u=O 33 = 320 °
•
18
Z @
"
~
16
I0
0
14
MEASUREMENT; BHUSHAN (1970 THEORY; PRESENT ANALYSIS
• • MEASUREMENT; JEWELL ET AL
[E
( 1980 ) THEORY; PRESENT ANALYSIS
12 IO O0
I
I
OD5
O.IO
4
0 15
Hole strain, ~o' %
FIG. 3 Comparison of Experimental and Theoretical Results of Expansion Test in Dense Granular Material
0
I 0.5
I 1.0
~s
Hole strain, Go, %
FIG. 4 Comparison of Experimental and Theoretical Results of Expansion Test in Loose Granular Material
For the purpose of illustration, a schematic representation of experimental and theoretical (nonhnear model presented in this paper) expansion curves are shown in Fig. 5. These representations show that incorporating the slope Ap/Aeo (termed KL and K~ for the experimental and theoretical curves, respectively) with Eq. (16) will yield higher values for the modulus of deformation in the case of the experimental curve than the theoretical one; namely KL/K~, > 1.0. This is true whether the secant (Fig. 5-a) or conventional (Fig. 5-b) moduli are considered. Such finding shows that the determination of the modulus of deformation from the experimental data (finite cavity) along with the linear theory of elasticity of an infinite cavity is in doubt. The simulated curve in the elastic phase, obtained using the nonlinear general solution presented herein, is different from the experimental curve. The latter shows stiffer behavior and accordingly higher values of the modulus
224
S.M. SAYED
of deformation. This analysis indicates that the finite length of the pressuremeter probe ( L / d = 4 and 16 in Examples 1 and 2, respectively) is probably one of the most important causes for the discrepancy between the experimental and theoretical results. --
0-"
P
P
/
0..
Pf
Pf
~ -6 P0
/
--K:-
KN ~ KL
"0 I0 nr"
0::
Hole Stra=n, % , %
(a) Secant Modulus
Hole Strain,
%,%
(b) Conventional Modulus
FIG. 5 Schematic Representation of Experimental and Simulated Expansion Test Results The well documented comparison regarding the large discrepancy between the values of the modulus of deformation obtained from the self-boring pressuremeter and conventional triaxial tests is given by Amar et al [1]. Wroth [15] suggested that part of this discrepancy is due to the high strain rate used in the field and recommended that the results should be analyzed in terms of the engineering shear strain. Such recommendation reduces the difference between both sources by about 25%. Also, it has been thought [5] that the disagreement between the pressuremeter and triaxial tests is contributed to the disturbance as well as anisotropic effects. These effects still are some of the reasons contributing partially to such disagreement. However, the question regarding the effect of the finite length of the pressuremeter probe remains unresolved, if one uses the linear theory of elasticity along with the pressuremeter test results. Although there are several reasons to believe that the deformation moduli should not be the same from pressuremeter and triaxial tests, there is doubt behind the effect of the finite length of the probe on this issue. The validity of determining the deformation modulus from the pressuremeter test results using the linear theory of elasticity can not be justified by comparing the theoretical solution of finite and infinite cavities in a linear elastic medium. The results presented in this section indicate that the finite length of the probe has its effect on the modulus if the medium is nonlinear and such an effect is very pronounced. In other words, the theoretical results obtained from the cylindrical cavity expansion in a nonlinear dilatant medium based on infinite length are very much different from those of the actual test with a finite length probe. The experimental results show a much stiffer behavior than the one shown by the simulated results. It is very important. to keep in mind that the experimental results used in this study are based on high quality tests and, therefore, the effect of disturbance does not exist. Assuming that the nonlinear model is superior to the linear elastic model and more realistic in representing the soil behavior, the discrepancy between the actual and theoretical (nonlinear model) results is an indication of the effect of L / d ratio of the cavity. The values of E computed from the pseudo-elastic range along with Eq. (15) are higher for the experimental data than those for the theoretical results. In general, the determination of the
CAVITY IN NONLINEAR DILATANT MEDIA
225
soil deformation modulus from the self-boring pressuremeter test (finite cavity), assuming an infinite cavity, is accompanied by an error of about 200% to 300% (on the unsafe side) and not 5% as implied by the linear theory of elasticity [9]. In summary, the deformation modulus, determined from the pseudo-elastic phase of the self-boring pressuremeter test, should be elaborated to account for the limited length of the probe if plane strain condition is assumed to prevail. Therefore, it seems that the modulus must be divided by some factor (2 or 3) in order to estimate reasonable values for design. II. Theoretical Linear Elastic versus Elastic-Plastic Analysis: The theoretical results obtained from the linear elastic and nonlinear elastic- plastic model for an expansion test are given in Fig. ft. The results of nonlinear model show that the material is initially stiffer than the behavior assumed by the linear elastic model. The linear elastic idealization is a conservative model for porous media for very small strains (in the elastic range). However, when the yield stress approaches, the linear-model becomes unconservative. In other terms, the linear elastic model overestimates the real response, that response can be represented more realistically by the nonlinear model once the stress level is in the neighborhood of yielding. This finding shows that modeling the nonlinear geometric and material effects is essential in formulating constitutive relationships for porous media. III. Stresses in Linear Elastic and Nonlinear Dilatant Materials: The stress distribution in the mass surrounding the cavity in Section II is shown in Fig 7. The decrease in the radial stress increase with the radial distance for the nonlinear model is less rapid than the linear one. On the other hand, the reduction in the circumferential stress for the nonlinear model is smaller than the one of the linear model. A similar behavior has been reported in the literature for the linear dilatant material. Therefore, the radial stresses induced in porous media due to plane strain and axi-symmetric loading conditions that are based on the linear elastic model are on the unsafe side. t 19
v
r
~
I
6/
,4
~
/
/,,' f
/
n
,/
c
h
14 13 0
/
,/
"
,'
r
;
L~e~ ~ o• ~,,,,,o,
psi
G, = 2900psi
/
~
=48
u 0B~/7"f
°
/ L
----
LINEAR ELASTIC
--
NONLINEAR ELASTK I
003 Hole
.
i,
I
l
006
009
012
015
strain,~o%
FIG. 6 Comparison of Theoretical Linear Elastic with Theoretical Nonlinear Elastic-Plastic Analysis
.
.
.
Gi ,29oo.i
~/A u° F
t' = 0 3 3 ~) , 4 8 "
|
041 0
.
% " 1 585 inch
=,o ,13 p,~
||
/ -o o (2:
'
/
/
18 .~ 17
'
,,.o
t 4
I 0
,! 12
I J6
20
r/00
FIG. 7 Stress Distribution in Linear and Nonlinear Dilatant Granular Materials for ~o = 0.06 %
The results indicate that the finite distance (r,~a=) defining the infinite boundary, computed
226
S.M. SAYED
from Eq. (9) with ¢=0.001, affected the distribution of the circumferential and mean normal stresses. Naturally the assumption of rigid boundaries at relatively far distance from the cavity wall will prevent the soil expansion in the circumferential direction near these boundaries. Accordingly the circumferential stress increment will be of compressive nature. In this example, the assumed rigid boundary at r,,,~= introduces about 2% of the in-situ horizontal stress as compressive circumferential stress increment. It is clear that any finite distance used as an approximation to infinity will have its effects on the results obtained from any numerical solution. Since there is no other way to treat most of the complex problems having the infinity as one of its boundaries other than to approximate it, it is important to assess the limitations of such an approximation. The expression for rmo~, Eq. (9) in this paper, can be used to compute the distance defining infinity in plane strain and axi-symmetric problems.
Conclusions
The simulated initial portion of the expansion test results, obtained from the general solution presented in this paper, explains a very important aspect concerning the determination of the modulus of deformation of porous media. It illustrates the effect of the finite length of the probe on the computed values of the modulus. The solutions of finite cavities based on the linear theory of elasticity do not clarify such an effect. It is recommended that the soil deformation modulus obtained from the self-boring pressuremeter test must be divided by a factor of 2 or 3 to account for the limited length of the pressuremeter probe. The use of the linear elastic model as a constitutive law for multi-phase materials does not represent the real behavior of these materials accurately at approaching yield. The nonlinear elastic model gives significantly different results compared to those of the linear one. The former model indicates much less stiffness particularly near yielding. Because the nonlinear approach is genera3, it is superior to the linear one from a mathematical viewpoint. Also, the radial stresses in the medium surrounding the cavity, predicted by the nonlinear analysis, are much higher than the ones obtained by linear analysis. Thus the radial stresses predicted by linear elastic model are unconservative. For these reasons, it is very essential to incorporate the nonlinear geometric and material effects in modeling the behavior of porous media. The infinite boundaries of the solution domain of any boundary value problem represent obstacles in obtaining reliable numerical solutions. These boundaries of the solution domain need to be adjusted continuously as the stress level changes at the interface. An expression is given which reflects the effect of stress level, material behavior and error tolerance. It can be used to evaluate the distance at which the boundary can be assumed to imply the infinite boundary in plane-strain and axi-symmetric problems.
CAVITY IN NONLINEAR DILATANT MEDIA
227
References
1. S. Amar, F. Baguelin, J.-F. J4z4quel, and A. Le M4haut& "In-Situ Shear Resistance of Clays," Proceedings of the Speciality Conference on In-Situ Measurement of Soil Properties, Raleigh, North Carolina, ASCE, Vol. I, p. 22, (1975). 2. F. Baguelin, J.-F. J4z4quel, and D. H. Shields, The Pressuremeter and Foundation Engineering, Trans. Tech. Publications, Clausthal, Germany, (1978). 3. M. M. Baligh, "Cavity Expansion in Sands With Curved Envelopes," Journal of the Geotechnical Division, ASCE, Vol. 102, p. 1131, (1976). 4. K. Bhushan, "An Experimental Investigation into Expansion of Spherical and Cylindrical Cavities in Sand," Ph.D. Dissertation, Duke University, (1971). 5. G. M. Denby, "Self-Boring Pressuremeter Study of the San Francisco Bay," Ph.D. Dissertation, Stanford University, (1978). 6. D. C. Drucker, Introduction to Mechanics of Deformable Solids, McGraw-Hill Book Company, New York, U.S.A., (1967). 7. J. M. Duncan and C. Y. Chang, "Nonhnear Analysis of Stress Strain in Soils," Journal of the Soil Mechanics and Foundation Engineering Division, ASCE, Vol. 96, p. 1625, (1970). 8. R. J. Jewell, M. Fahey, and C. P. Wroth, "Laboratory Studies of the Pressuremeter Test in Sand," Geotechnique, 30, p 507, (1980). 9. J. E. Laier, J. H. Schmertmann, and J. H. Schaub, "Effect of Finite Pressuremeter Length in Dry Sand," Proc. ASCE Speciality Conf. on In-Situ Measurement of Soil Properties, Raleigh, North Carolina, Vol. I, p. 241, (1975). 10. J. H. Pr$vost and K. HSeg, "Analysis of Pressuremeter in Strain-Softening Soil," Journal of the Geotechnical Engineering Division, ASCE, Vol. 101, p. 717, (1975). 11. S. M. Sayed, "Expansion of Long Cylindrical Cavities in Nonlinear Dilatant Media," Ph.D. Dissertation, Duke University, (1982). 12. S. M. Sayed, "Cyhndrical Cavity Expansion in Nonlinear Dilatant Media," presented at the XVI International Congress of Theoretical and Applied Mechanics, Lyngby, Denmark, August, (1984). 13. S. M. Sayed and M. A. Hamed, "Expansion of Cavities in Layered Elastic System," International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 11, p. 203, (1987). 14. A. S. Vesi4, "Expansion of Cavities in an Infinite Soil Mass," Journal of the Soil Mechanics and Foundation Engineering Division, ASCE, Vol. 98, p. 265, (1972). 15. C. P. Wroth, "In-Situ Measurement of Initial Stresses and Deformation Characteristics," Proceedings of the Speciality Conference on the In-Situ Measurement of Soil Properties, Raleigh, North C_arohna, ASCE, Vol. II, p. 181, (1975).