Int.I. Engng
Sci.Vol.16.pp.931-942
@ Pergamon
Press
Ltd..1978.Printed
inGreat
Britain
A zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
NONLINEAR MIXTURE THEORY REPRESENTATION
OF SATURATED SAND zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON
RACHEL
BECKER
Building Research Station, Technion. Haifa, Israel
and
I. K. McIVORt
The University
of Michigan, Ann Arbor, MI 48109. U.S.A.
Ah&act-Nonlinear
constitutive equations for saturated sands are proposed. They exhibit the interaction
of dilatation and shear stress observed in sands in its small strain range. The equations are based on a
continuum theory derived from a Newtonian approach to a mixture of a nonlinear solid and compressible
fluid. The determination of the constitutive parameters from common soil mechanics experiments is
discussed and illustrated by qualitatively valid numerical results.
I. INTRODUCTION
THE BEHAVIOR of saturated sands when subjected to shear in the small strain range is a
fundamental problem in analytical soil mechanics. The observed phenomenon is a change in
fluid pore pressure and effective solid stresses when the confining overburden is kept
constant [ l-51. The mechanism suggested to explain this phenomenon is the inherently nonlinear
material behavior of dry sand. Even in the small strain range dry sand exhibits dilatational
change when subjected to pure shear stress[l,5]. When the sand is saturated, the low
compressibility of the water prevents the dilatation with consequent changes in the pore
pressure and effective stress.
The purpose of this paper is to obtain constitutive equations for saturated sands which
exhibit this observed behavior. Continuum mixture theory provides a theoretical framework for
determining the form of these equations. In the next section a Newtonian approach is used to
derive the field equations of a two constituent mixture in a form appropriate for the present
application, The associated constitutive equations are derived in Section 3. For physical
applications it is necessary, of course, to have numerical values for the constitutive parameters.
For generalized continua their experimental determination is often a complex task. In the
present application there are a number of standard tests that are commonly employed in
experimental Soil Mechanics. In Section 4 these tests are interpreted in the context of
the proposed theory providing a basis for the determination of the constitutive parameters.
Finally, numerical results based on qualitative data in the literature are given in Section 5.
2. THE
FIELD
EQUATION
OF THE
MIXTURE
A formulation of mixture theory has been derived by Green and Naghdi[6, 71. Various
aspects and applications of the theory have been discussed by Green and Naghdi [B], Green and
Steel[9] and Steel[lO, 111. A linear theory of fluid-solid mixtures has been discussed by
Schneider[l2]. Garg et a1.[13, i4] have used various forms of mixture theory to examine wave
propagation in fluid saturated porous media. Here we are concerned with the application of
mixture theory to model the behavior of saturated sands. Anticipating future generalization to
include soil plasticity, the field equations are established from a Newtonian approach for a
mixture of a nonlinear solid and compressible fluid.. For simplicity attention is restricted to
chemically inert materials and to the case of zero heat flux through the boundaries. The
kinematic formulation follows Green and Naghdi[7] and is briefly summarized here. Coordinates in the undeformed reference configuration of the mixture are Xi for the solid constituent and Yi for the fluid constituent. Spatial coordinates in the deformed body are xi and y; for
the solid and fluid constituents respectively. It is assumed that each point in the mixture is
simultaneously occupied by both constituents. Thus in an Eulerian representation xi and yi are
equivalent and may be used interchangeably as the independent spatial variable.
Denoting the solid and fluid particle velocities by Ui and Vi, respectively, the kinematic
tkceased,
17 April 1978.
931
932
RACHELBECKER and
zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB
I.K.McIVOR
variables of interest in the sequel are the rate of deformation and rotation tensors for the solid
and fluid constituents. They are respectively zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH
(2)
(3)
=f(z-2).
Aij
(4)
It is necessary to introduce stress quantities prior to obtaining the equations of motion. When
regarding an element of the mixture, the common definition of stress is used leading to the well
known symmetry of the total stress tensor. We view the surface tractions per unit area of a
surface enclosing an arbitrary volume of the mixture as composed of tractions acting on the
solid constituent ti and tractions acting on the fluid constituent pi. Considering the translational
motion of an infinitesimal tetrahedron gives
ti + pi =
SijtIj
(3
where nj is the j component of the normal to the surface on which ti and pi are defined, Sijis the
symmetric total stress tensor and the summation convention holds. Partial stress tensors are
now introduced through the relations zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
tj = (Uij
Pj
'(Fj
+ aij)tIi
+Pij)ni
(6)
zyxwvutsrqpo
(7)
where oij is the symmetric part and ai] the antisymmetric part of the solid partial stress tensor.
Likewise nij and p/j denote the symmetric and antisymmetric part of the fluid partial stress.
Equations (5) and (6) imply zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
Clij+ flij = 0
(8)
tj + pj = (Uij+ T;j)fl;.
The motion of a material element of the total mixture is governed by Newton’s second law of
motion. It is
1 (t,+pj)dA,+]
A0
(p,F,+p2Gj)dVn=$f”O(plUI+p:uj)dVo
“0
(9)
where Fj and Gj are body forces per unit mass of the solid and fluid, respectively, A0 is the
surface enclosing the arbitrary volume element of the mixture VO,and p,, pz are the mass
densities of the solid and fluid constituents.
Through (8) and the divergence theorem, eqn (9) for small deformations and chemically inert
materials reduces to
(uij++ij),;+pIFj+pzGj=plOj+p?tii
(10)
where a comma denotes a partial space derivative, and a dot denotes a partial time derivative.
In (10) wj denotes the displacement of the solid constituent, i.e.
Wj = Xj - Xj.
(11)
933
A nonlinear
zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
mixture theory representation of saturated sand
We next consider the motion of the mass of each constituent separately. The solid equation of
motion is
(12)
where @j’ is the additional body force exerted on the solid constituent due to the presence of
the fluid. The fluid equation of motion is
(13)
where @j*is the additional body force exerted on the fluid constituent due to the presence of
the solid.
As in (9) above, eqns (12) and (13) reduce to
(14)
(Uij+aij),;+plFj+~j’=p,~j
For (10) to be compatible with (14) and (IS) it follows that zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO
(16)
cPj’+@ f =O.
Thus the constituent interaction can be represented by a unique body force term
tJrlj =
-q’
=
cq.
(17)
With this the equations of motion are
(U;j+a;j),i+plFj-~j==p,~j
(18)
This form is analogous to Green and Naghdi’s [7] equations of motion, the main difference being
the way apparent partial stresses and interaction forces are introduced. In [7] the nonsymmetric
partial stresses are denoted by oij and rij. The diffusive force term wj introduced in [7]
coincides with @j for chemically inert materials.
To the equations of motion it is necessary to add the fluid continuity equation since the
fluid’s constitutive eqns (Section 2) are expressed as functions of the density change. According
to Green and Naghdi, the continuity equation for small density changes reduces to
f) + piiU,,i= 0
(20)
where n is the change of the fluid’s density from its initial value pq to its current value p2.
3.
CONSTITUTIVE RELATIONS
TO complete the formulation constitutive relations are necessary for oii, rijrii,
“ii, pij and @j as
functions of displacement, change of density and velocity fields. As discussed above, the two
constituents considered are an elastic nonlinear solid and a compressible viscous fluid. Interactions are assumed to be of viscous and friction type, i.e. coupling terms in the constitutive
equations depend (1) upon rates of deformation of the other constituent, (2) upon differences in
the velocities, and (3) upon differences in the rotation rates. Thus, the assumed form of the
constitutive equations is
uij = Aij + Aij,fis
(21)
934 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
RACHEL BECKER and I. K. McIVOR zyxwvutsrqponmlkjihgfedcbaZYXWVU
Tj
“ij
=
- fiij
@j =
Kij(Ui
=
=
zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJI
Bij + Bijrs.frs + Cijrdm
(22)
Kle ijk(Uk
-
Vi)
+
-
Uk)
+
LZEjki(rki
LI(Tij
-
Aij)
-
+
Uj/Jki
A ki)
+
Eijkb k
(23)
+
(24)
aj
where zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
d,,, I’ ij, frs and Aij are defined in (l)-(4) and eijk is the permutation tensor. All the
coefficients are in general functions of the displacement gradient field axi/aXi, the fluid density
field pi, and initial composition of the mixture.
The functional dependence of the constitutive coefficients on the displacement and density
fields is obtained from thermodynamic considerations. According to Green and Naghdi[7], the
entropy inequality when no heat sources are present reduces to
05)
where U is the internal energy, A is the Helmholtz free energy, S is the entropy, all per unit
mass of the mixture, and T is the absolute temperature field. The material derivative for the
mixture is defined in [7] as
D’2’
D”’ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB
D
(26)
P&=PIDt+pzK
where D” ‘ /Dt and D” ‘ /Dt are the material derivatives for the solid and fluid constituents
resoectively.
The functional form of U is obtained from the conservation of energy equation
Through the divergence theorem, eqns (l4), (15) and (7) and the definitions of the rate of
deformation and rotation tensors, it reduces to
= @i(ui - Vi) + rijdij + Tiifii + aij(rij - Aij).
pg
The basic assumptiont for a mixture of solid and fluid is
A = A(eij,PZ, T)
(2%
S = S(e ij,
(30)
Pz,
T)
where
(31)
is the strain tensor and 6ij is the Kroneker Delta.
Substituting (28)-(30) and the constitutive eqns (2l)-(24) into (25) yields
+(
Ui-pl-
dA a&
apz aXi +P2
-I- Qijkbk(rij+
See,
e.g.
P4
1>
(Ui - Vi)+
Kij(Ui - Oi)(Uj- uj)
Aij) + Ll(rij - &j)(rij - Aij)
(Aij,s + C,ij)dijrs
[9].
$2
+ (Lz +
Kl)c jki(Uj
-
Uj)(rki
-
A ki)
+
ajkifki(uj
-
uj) ?O.
A nonlinear mixture theory representation
of saturated sand
93s
At a given state of deformation (eii and p2 specified), the system may have arbitrary rate of
deformation velocities and values of DT/Dt. The inequality (32) has to hold for all possible
choices of these fields. It follows that the coefficients of the linear terms must vanish. Thus,
(33)
(34)
(35)
aA
l?A
ai =P1-p2.i
(36) zyxwvutsrqpon
-Pzae,,epq,i
8P2
b; = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK
0.
(37)
With this the left hand side of (32) reduces to a quadratic form in the elements of the kinematic
variables. The elements of dii appear only in the term dij,,y. Consequently, the coefficient of this
term must vanish giving
Some additional restrictions on the constitutive parameters could be obtained by further
analysis of the quadratic form. Here, however, we restrict ourselves to an isotropic mixture.
Requiring that the form of (21)-(24) be invariant under arbitrary orthogonal transformations
leads to
Bijn
=
A&j&s
+
/J(Wjs
+
(39) zyxwvutsrq
&s&J
Lz=K,=O
(41) zyxwvutsrqp
(42) zyxwvutsrq
Kij = K26ij
aijk
=
(43)
0
where A, CL,y3, y4 and K2 are constants. With this it is readily verified that the inequality (32) is
satisfied.
The above results imply that for an isotropic mixture the antisymmetric part of the partial
stress tensors depend only on the relative rotation rate (Iii - Aii) whereas @j is independent
of this quantity. For simplicity in the present application we suppress dependence on the
rotation rate by choosing L, = 0 with the consequence that the antisymmetric partial stresses
vanish.
The Helmholtz free energy function A is equivalent to the strain energy function[9]. The
choice of its functional form determines uniquely the elastic properties of the mixture. The
properties of saturated sand are inherently nonlinear. The characteristic feature is dilatation
under shear. Including cubic terms involving the solid strains eij and fluid density change 77in
the free energy A is sufficient to model this nonlinear behavior. Thus we choose
A = a3emm + aqemmenn+
aSq2+ a6emmq+ alenmemleln
+
aSq3
+ a9q2enn
+ alOqemmenn+ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF
al rqemnenm + zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJ
a12emmenneb
+ a~3emmedeh
where a3, a4, . . .a13 are material constants still to be determined.
936
RACHELBECKERandI.K.McIVOR
Introducing the above results into the constitutive eqns (21)-(24) yields the general form of
the nonlinear stress-strain relations. They are zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJ
Uij
=
(2a4emm+a677
zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
+ 3a I?emmenn
+ a I%@/~Pi,
+~~$+2a~0?7e~~
+
zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
2a@;j + (4X4 + 2a13k;je m m+ (2C %+ 2a1 Ikij77
+(h3+
h7)eise.,j + o(e3)
+O(e’))}&j
+ Y3fmm&j
+ AfmmSij + 2/Lfij aij
=
+ 2')'4.f;,,
Y3dmm&,
-pii
=
zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM
(45)
(46)
- 2y4&
0
(47)
+a6q) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR
O(e3)
(48)
mm
e nn.i +
where O(e’) stands for terms of third order or higher in eij and b and p’, p? and pp are the initial
densities.
For application to soils we must determine numerical values of the constitutive parameters.
This is a formidable task for the general form of eqns (45), (46) and (48). Here we reduce the
number of parameters by requiring the coefficients of assumed negligible terms to vanish. For
example assuming that the pure fluid is a linear compressible material leads to the choice
1
as-p”o5=0.
(49)
In a similar manner the number of parameters necessary to account for solid-fluid interaction is
reduced to one static and one dynamic coefficient, In the fluid partial stresses we neglect
nonlinear terms in the solid dilatation and fluid density change (emmen”and e,,,,,,n). The latter
term is also neglected in the solid partial stresses as is the term eije,,. The latter assumption
confines the coupling of longitudinal and shear strain with shear stress in the solid to one term
governed by the parameter (4a3 + 3o,). Consequently, we choose
(50)
Finally, for application to soils the fluid considered is water. Moreover, interest is primarilyfocused on the effect of shear deformation and the fluid partial normal stresses. Thus, for the
present study we neglect viscosity, i.e.
A
=
/.L=
y3=
y4=0.
(51)
With this the final form of the constitutive equations proposed for a saturated sand is
Uij =
+
(
2ff4emm
a67
+%q’+
+ (4~x3 + 3a7)eimemj
7Tij
=
[
2P"
2$qe,,
+ 3a12emmenn
- a4em,en,
+ 2a3eij
2p%571-p4a6emm+ PZ
o(~+%)Gdh],ii
>
Sij
(52)
(53)
A nonlinear mixture theory representation of saturated sand
931
Characteristic features of these equations are illustrated by several simple deformation cases.
An imposed normal strain produces normal solid stresses and hydrostatic fluid stress but no
shear stress. Likewise, only normal stresses are produced by compressing the fluid constituent.
Pure shear deformation, however, generates normal stresses in both constituents in addition to
shear stresses in the solid. The normal stresses depend upon the square of the shear strain, thus
being unaffected by the shear direction.
Finally, the field equations in terms of solid displacements and fluid velocities and density
change are obtained by substituting (52~(54) into (18)~(20). They are
+I (2@a3-2a4
2
SW
P0
n.m
>
+ +
Wm.nWm.ni$4a3
+
3a7)(Wn.mWm.inWm .nWm
zyxwvutsrqponmlkjihgfedcbaZYXW
.in
+ W,,,Wi,m n + Wn,m nWm .i + Wm .nnWm ,i + Wn.mnWi.m + Wm.iv8Wi.m I= O.
w.r.mn zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
(53
6 + p2Ui.i
= 0.
4. ESTABLISHING
THE CONSTITUTIVE
(57)
COEFFICIENTS
EXPERIMENTS
FROM COMMON SOIL MECHANICS
in the constitutive relations developed for a saturated sand mixture seven constitutive
parameters are involved. Six of them (oj, ad, ok, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP
ah, a7, a,~) are associated with static
deformations; the seventh (Kz) is associated with the relative velocities. Of the first group five
parameters fa3, a4, as, a7, a12) relate to the behavior of the single materials in the absence of
the other; the other parameter (ag) relates to the static coupling between the two constituents.
In this section we briefly describe a number of common soil mechanics experiments which
permit determination of the parameters for the proposed mixture theory. The quantities a3, ad,
a7 and a12are obtained from dry sand test rest&s, and a5 is obtained from the compressib~ity of the
fluid. The parameter a6 is derived by considering the influence of one constituent’s static
deformations on the partial stresses of both, i.e. drained test results, and K2 is derived from the
permeability test. The mathematical representation implied by the proposed theory is given below
for a number of standard tests:
(a) Hy drostatic compression test zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB
of dry sand (pq = 0)
Under ideal test conditions
et1 = e22 =
UES Vo l. 16. No . 12- B
e33=
1
-jeO
(58)
938
RACHEL BECKER and 1. K. McIVOR
where e. is the compressional dilatation. The stresses are zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP
CT 11= CT 22= CT 33=
zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF
-po
(59)
where PO is the hydrostatic pressure applied to the specimen.
Using (52), (58) and (59) it follows that
(b) Confined compression test of dry sand (pq = 0)
The specimen is restrained from lateral movement. The strain components are given by zyxwvutsrqponmlkji
ell = -ei
eij = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG
0
for if I or j# 1.
(61)
The applied compressive stress q is related to the stress components by
-4.
(62)
2(03 + a4)el - (3a,? + 4c~)+ 3a7 - c~&f
(63)
UII
=
By (52) the relation of q to el is
q =
(c)
Undrained
confined
compression
test of saturated
sand
Under quasistatic load conditions the continuity eqn (57) yields by integration
(64)
-p!e,.
7) =
The external pressure q is applied to both constituents simultaneously, i.e.
UII
+
7rll
=
(65)
-4.
From (58) and (59) this yields
4 =(2a3+244-2pPa6+2p?a~)el-
-
(
1+24
>
[
3a12+
(
4+~
pn>
a3
ar+3a7+pI(l+$)as]ei.
(d) Drained confined compression test of saturated sand
Drained conditions are simulated by the requirement that no partial stresses are generated in
the fluid, i.e.
‘rrij
=
(67)
0
From (58) and (60) it follows that
q =
301?+4a3+3a7-a4
(68)
e:+O(e:)
A nonlinear mixture theory representation
of saturated sand
939
~ri~i~~ shear teft of dry sand @P= 0)
This is a pure shear test in the sense that the first stress invariant (mmris kept constant,
whereas the octahedral shear stress is varied.
For an initial hydrostatic compression state with initial dilatation of - eo and pressure zyxwvutsrqponmlk
PO the
additional strains are Ae,. From (52) and (60) it follows that during loading
(e)
ha4 + 2a3 - teo(4aJ + 3a7 - 3a4 + 27~~12)AeRm+ 9ffizAeim
I
+ (4a3+3a7-
3a4)(Ae:l +Ae&+A&)=O.
(69) zyxwvutsrqp
(gf Triaxial compression test of dry sand (~4= 0)
This test starts from a hydros~tic compression state with eo and POas above. Loading is
then performed along one axis’s0 that
Aa22
=
Aujg = 0
6)
where Avii are the additional stresses and Acr is the loading stress afong the axis.
Using (52), (60) and (70) yields
Au=
6a4 + 2a3 - ieo(4a, + 3~17- 3~ + zyxwvutsrqponmlkjihgfedcbaZYXWVUT
27ad de,,
I
+9ai2Ae~,+(4a3+3a7-3a4)(Ae:,+2Ae:2),
Au=
c
2aj - 5eo(4a~ + 3a7) (bell - Aez2)+ (4a, -t 3aT)(Ae:, - Ae&).
I
171)
(72)
(h) hydrostatic compression of a pure aced (~9= 0)
The hydrostatic stress p causes a fluid dilatation of E. The bulk modulus K, of the pure fluid
is defined by
fiV
p=K,r=K,
w
(73)
where A V,. is the change from the initial fluid volume VW Here
.
p is defined per unit area of the
fluid cross-section, whereas the definition of partial stresses used in the present formulation is
per total area of the mixture. This yields the relation
~?@I,
= 3BP
(74)
where /3 is the porosity factor defined as the ratio of fluid area to total area in a cross-section of
the mixture.
Integration of the continuity eqn (57) yields
V,
.AVw_
-AV,
li zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF
=- Pz
y.
(75)
- PwVw VW v.
where pNS
is the specific mass of the fluid and VOis the initial volume of the mixture. Introducing
the approximation j3 = V,/VO for an isotropic mixture, it follows from (73) to (74) that
This implies that the coefficient of 71in eqn (53) is a constant for all solid fluid mixtures with the
same fluid constituent.
940
RA~HEL~ECKERand~.K.McIVOR
(i) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
Pe~e~bili~y test of a su? u~a~e~ sand
Flow of the fluid constituent is generated in one direction relative to the solid. The velocity
of the fluid UI is related to the head gradient zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH
I H by Darcy’s Law. It is zyxwvutsrqponmlkjihgfedcbaZYXWV
VI
=
k&j
(77)
where kD is Darcy’s coefficient.
Assuming that at the steady state stage the solid strain field is uniform, eqns (19) and (54)
yield
K?V, =o.
Trl,,, -
(78)
The relation between the partial fluid stress gradient and the total stress in the fluid just outside
of the mixture is
q-J.,,.,
-.
po-pL
=
L
B
(79)
where PO- Pr is the pressure loss along L. The head gradient for water is
z
-PO--PL
H --
LPd
(80)
where g is the gravity acceleration. From (77) and (80) it follows that
K,
-
P,.@
kD ’
5.NUMERICAL APPLICATION
The coefficients a~, KZ have been directly related to commonly known fluid properties. The
remaining parameters must be determined from the standard soil mechanics experiments
described above.
The coefficients obtained from dry sand tests are established in the following manner:
I. From hydrostatic compression test results (6~ + 2a~) and (27a,~ + 4a3S 3a7 - 3c.x,)are
obtained by a parabolic least square fit.
2. With this the parameter au is found from a linear least square fit of (69) for the triaxial
shear test.
3. Finally, triaxial compression test results may be used to establish the individu~ values of
a3 and a7. if test results are available for both axial and lateral strains, this may be done
directly by fitting (72) to the data. Equation (71) then becomes a check on the validity of the
established coefficients. If lateral strain data is not available, eqn (71) is used to numerically
eliminate Ae::.
The value of o6 is now established by fitting (68) to quasistatic drained confined compression
test results. We note, however, that the load deformation relations of the dry and drained tests
differ only in terms that have a6 as a factor. According to Lambe and Whitman [5], drained tests
provide the same final load deformation relations as dry tests. It is thus concluded that for a
saturated sand
To carry out the above procedure requires that each of the postulated experiments be
conducted on a particular sand at a given void ratio. At the present time such a consistent set of
data is not readily available for any one sand, For the purpose of illustration here we use the
above procedure to obtain the constitutive parameters for a hypothetical sand at three different
void ratios using a qualitatively correct data set based on actual and extrapolated soil data
reported in Refs.[l, 51. The results are given in Table 1. The resulting load deformation
A nonlinear mixture theory representation
Table I. Numerical
data
Medium
Loose
Sand type
941
of saturated sand
Units
Dense
0.60
0.55
0.485
Void Ratio (V.R.)t zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
0.375
0.35
0.33
Porosity (8)
1.56x W4
I .62 x IO-’
1.67x W4
psi sec’/in.’
Solid Density @P)
0.35 x lo-’
0.33 x lo-’
0.3 I x w
psi sec’/in.’
Fluid Density @‘?)
7.5
7.0
6.5 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP
psi sec/ik2
K?
in.*/sec’
3.3 x IO9
3.3 x 109
3.3 x IO’
2pFas
0.9 x IO’
1.2x to’
1.5x IO’
psi
2a4
2.2 X 10’
2.6x 10’
3.2 x Id
psi
a3
-1.5x IO0
-1.8X 106
psi
-2.0x IO”
at ?
2.0x 10”
1.5x 105
3.0x lo”
psi
a7
tThe void ratios refer to the values used by Ko[l].
AG ,
(Psi)
50
-
o-DENSE
SAND
LOO zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
'-MEDIUM
300
.-LOOSE
SAND
SAND
200
0.2
0.4
0.6
0.8
1.0 e0 (%)
0.2
Fig. I. Generated
0.L
0.6
0.8
1.0 AC,,(%)
Fig. 2.
Fig. I.
data for hydrostatic
test of dry sands according to parameters
represented in Table I
(stress
vs strain).
Fig. 2. Generated data for triaxial compression test of dry sands (deviator stress vs axial strain).
AfZmm
(%)
0.12
t
I
0.10
0.08
0.10 0.08 -
0.06
0.06 -
/
0.04
0.02
0.2
0.4
0.6
0.8
1.0 At?,,P/.,
0.2
0.L
0.6
Fig. 3.
0.8
c
1.0 Ae,,(%)
Fii. 4.
Fig. 3. Generated data for triaxial compression test of dry sands (dilatation vs axial strain).
Fig. 4. Generated data for triaxial shear test of dry sands (dilatation vs strain).
behavior for the various test conditions is shown in Figs. 14 demonstrating the relative effect
of void ratio on stress-strain behavior. The characteristic behavior illustrated is in accordance
with typical experimental results [ 1, 51.
REFERENCES
[I] H. Y. KO, A Report for the NSF, CALTECH.
(1966).
[2] H. B. SEED, J. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
Soil Mechanics and FOWL Diu., ASCE, Vol. 94, SMJ (1968).
942
RACHEL BECKER zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC
and I. K. McIVOR
[31 H. B. SEED and K. L. LEE, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
1. Soil Mechanics and FOWL Div., AXE,
Vol. 92, SM6 (1%6).
141 W. D. L. FINN, D. J. PICKERING and P. L. BRANSBY, 1. Soil Mechanics and Foun. Div., ASCE, Vol. 97. SM4 (1971).
I51T. W. LAMBE and R. V. WHITMAN, Soil Mechanics. Wiley, New York (1%9).
[61 A. E. GREEN and P. M. NAGHDI. Q. 1. Mech. Appl. Mafh., 22,427 (1%9).
(71 A. E. GREEN and P. M. NAGHDI, Inf. 1. Engng Sci. 3,231 (1%5).
181 A. E. GREEN and P. M. NAGHDI, Acfa Mech., 9, 329 (1970).
[9] A. E. GREEN and T. R. STEEL, Inl. 1. Engng Sci. 4,483 (1966).
[IO] T. R. STEEL, Int. J. Solid Structures, 4, 1149 (1968).
[III T. R. STEEL, Q. 1. Mech. Appl. Math., 20,57 (1%7).
[I21 W. C. SCHNEIDER, Ph.D. Thesis. Rice University (1972).
[I9 S. R. GARG, A. H. NAYFEH and A. J. GOOD, J. Appl. Phys. 45, I%8 (1974).
[I41 S. R. GARG, D. H. BROWNELL Jr., J. W. PRITCHE’IT and R. G. HERRMANN, 1. Appl. Phys. 46,702 (1975).
[IS] R. BECKER, Ph.D. Thesis. University of Michigan (1975).
(Received 3 February
1978)