- Email: [email protected]
A hypoelastic model for soils—I
ht. J, En.gg @ Pergmon
Sri, Vol. 19. pp. 511-520 Press Ltd., 1981. Printed
Olm-7U5/8I/c-iOI-O5I in Great
l/saz.00/0
Britain
A HYPOELASTIC MODEL FOR SOILS-I A CLASS OF HYPOELASTIC MATERIALS LUCIA DRAGUSIN Polytechnical Institute, Bucarest, Romania (Communicated by E. SOGS) Ahstraet-This paper refers to a certain class of hypoelastic materials for which work depends on the stress history. It is assumed that there are two potentials in the stress space whose time derivative are equal to the mechanical power necessary for the volume deformulation and with the total stress power, respectively.
1. INTRODUCTION THE CONSTITUTIVEequation
of hypoelastic materials (using Truesdell’s definition[ 11) can be regarded as a linear mapping H(T) from the vector space of the strain rates (Dll, &, Dj3, D12,Dz3,D13) to the vector space of Jaumann-No11 stress rates (G7 1R22, %, C*, &3, G)
‘i = H(T) [D]. Let C?3be a domain in R6, where det H(T) f 0. In 1958, Bernstein and Ericksen[2] defined two types of hypoelastic materials by imposing conditions upon the work done by a stress history T = T(T), T E [to, t] in the deformation of a material volume 0
[w(‘U.NlO)= j-i
(I,h(TD)do)dr=
j-(j:$tr(TD)dr)
duo,
(1)
v. = u(tO) marked the initial volume and p. = p(f,,), the initial mass density. Functions T = T(T) were assumed to take values in the simple connected subsets of 9 and to be continuous and smooth piecewise. From definition[2], a hypoelastic material is of I-type, if there is an initial stress value To, so that the functional [w(T(*))](t) be nonnegative for any 7 3 to, and for any stress history T = T(7) where T(t,,) = To. A hypoelastic material is of II-type, if for every t, 3 to, the work [w(T(*))](t,) vanishes, for any stress history T = T(T), where T(fo) = T(t,). The authors show that: (a) A hypoelastic material is of I-type, if and only if there is a stress value To E ~3 and a function y(T) defined on $3, so that
y(To) = 0,
Y =
$tr(TD) =$
0) 2 0, y(W))
&CT())](t)
*
(b) A hypoelastic material is of II-type, if and only if, there is a stress value T,, E $3and two functions A(T), 8(T), defined on 9, so that
Wo) = 0,
W’d = 0,
p(r) = exp [- A(T(t))l, w(t) =
@(T(t)) duo. 511
A(T) 3 0,
6(T) 3 0,
512
L.DRkX$IN
In other words, for the I-type materials, work w related to the present volume unit is not dependent on the stress path, whereas for the H-type materials, both the mass density values p as well as the values of work w are dependent only on time t but not on the stress path. In this paper, we’ll consider those hypoelastic materials, for which there are two potentials a(T), q(T), defined on 9, so that @(T(t)) = ; I&,
@(T(t)) = tr(TD),
where IT = trT, In = trD, and the superimposed dot means material time derivative.
2.HYPOELASTIC MATERIALS OF III-TYPE
Let 46 be a body and its motion
x: W(a)x [to, t11+R3, det Vx > 0, reported to a reference configuration X(9) C R3. A hypoelastic material was defined by Truesdell [ l] and Bernstein[3] (see also Truesdell and No11[4]), through (a) A constitutive equation
‘i’= H(T) [D],
(2)
where H(T) is an isotropic function in T of class C’ in 9. (b) A corresponding equivalence class of stress-configuration classes [T, F], determined by the initial condition [TO,FO]. The following symbols are used: T is the Cauchy’s stress tensor (defined with the aid of the internal normal); F= Vx is the deformation gradient; L = - fiF_’ is the spatial gradient of velocity with inverted sign; D = (1/2)(L + L’) is the deformation rate tensor, while LT denotes the transpose of L; W = (1/2)(L - LT) is the spin tensor; T = T + WT - TW is the Jaumann-No11 stress rate tensor. The continuity equation thus becomes
Let 1 be a set of motions consistent with the constitutive eqn (2), for which there is a continuous piecewise smooth function T(T), defined on an interval [to, tl] with values in a simply connected subset of 9. Definition. A hypoelastic material is of III-type, if for any motion M E A there exist two potentials a(T), T(T), defined on simply connected subsets of 9, so that @(T(t)) = f I&,,
‘@(T(t))= tr(TD).
(3)
Theorem. A necessary and sufficient condition for a hypoelastic material, having the constitutive eqn (2), to be of III-type, is the fulfilment of the following conditions identically in T
=
SiiH i’,pq- 6pqH-,‘ii,
A hypoelastic model for soils-1
513
in which 6,,= 11
1, for i=j fori#j’
I 0,
and H-r is the inverse of H. Proof. In the domain 9, the constitutive eqn (2) can be written as
(6) Since In = H ;;t’,ijfij = H ,‘,ij fij, tr(TD) = Tk,H i;ij ~j = Tk,H ~~ij$j, using the definition of the III-type material, we have (7) As the functions O(T), q(T) are potentials (see Vainberg[5]), in a simply connected subset of 9, we must have
conditions leading to the identities (4) and (5). Conversely, if the relations (4) and (5) are fulfilled, it follows that there are two potentials O(T), ‘P(T), defined on simply connected subsets of 9, so that (7) is fulfilled. Using the constitutive eqn (6) it is easy to deduce (3). Mention should be made that the relation (5) coincides with the existence condition of a free energy u, found by Mihailescu-Suliciu and Suliciu[6] for incompressible hypoelastic materials, when li = tr(TD). Proposition 1. A hypoelastic material of III-type is a hypoelastic material of I-type, if and only if, there exists a function y(T) defined on 9 and a stress value T,,, so that
Y(‘-M= 0,
y(T) 3 0,
Proof. According to [21, a hypoelastic material is of Z-type, if and only if, there exists a positive function y(T), defined on 9, and a stress value To, so that
y(To) = 0,
d r = ; tr(TD), dt 0P
namely + = y p + tr(TD).
514
L.DRAGUSIN
If the material is of III-type as well, we get
QY-p+4, T
leading to (8). Conversely, if, for a hypoelastic material of III-type, (8) is fulfilled, it follows that the material is of Z-type. Proposition2. A hypoelastic material of III-type is a hypoelastic material of II-type, if and only if, there exist two potentials A(T), O(T) defined on 9 and a stress value To, so that
NT,) = 0, 8
i!Lg.
” aTij
O(T,) = 0,
A(T) 3 0,
O(T) 2 0,
aa a9 aa aq a@ --=-__ ari aTpq aTpq aTi,*
iJaT,,’
(9)
Proof. According to [2], a hypoelastic material is of II-type, if and only if, there are two positive potentials A(T), e(T) defined on 9 and a stress value To, so that
No) = 0, P(T)= POexp[- MUI,
@(To) = 0, @(T(t)) = I,’ $ tr(TD) dz 0
If the material is of III-type as well, we must have aA 3 a@ -=-aT ITaT'
g = g exp [ - A(T)].
The functions A(T) and 8(T) are potentials. Then
It is easy to obtain (9), taking into consideration the fact that Cpand 9 are also potentials. Conversely, if the material is of III-type, and if (9) hold true, it follows that there exist two positive potentials A(T), 8(T), so that A(T(t)) = ;,
@(T(t)) = po tr (TD) P *
Hence, the material is of II-type.
3.A SUBCLASS OFZZZ-TYPE HYPOELASTIC MATERIALS
We shall further study the hypoelastic materials having the constitutive equation of the special form ‘F= - UZTZd+3aZ,D+ bZTZnT+ (6 - c)ZTtr(TD)I - 3ctr(TD)T
(10)
where a, b, c are strictly positive material constants. We also assume that b-c is strictly positive (see Dr&usin[7]). Proposition3. The hypoelastic materials having the constitutive eqn (10) admit two poten-
515
A hypoelastic model for soils-I
tials CJJ and $ on 9, depending on the invariants Z,, Z,* so that, for any motion M E A, we have
They are dzT,
IT*)=
h [z,212uzTbzT*)(b-2c)‘c] + kl, ,g,(bc_ c)
+tzT,IT*)= 6(b
IT2
1 _
c)
In(2uZT _
bZ,,j +
(11)
k2y
where kl, k2 are two arbitrary constants and IT*= tr(T*)‘, T* = T - 1/31~1. Proof. From the constitutive eqn (IO), we obtain ZT= bZ:Z, + 3(b &* = - 2( 01~ -
‘h)z,tr(TD),
bZT*)ZTZD+ 6(aZr - cZr*)tr(TD).
Hence ;
z,z,= 2(&-
cZT*)IT-(b -2c)ZT&* ’ 6(b - c)Z&UZT - bZ,*)
tr(TD) = @aIT - b&d, + biTiT= 6(b - c)Z&UZ, - bZ,*) ’
(12)
Since
a
2(aZ, - cZT*)
al,
- (b - 2c)Zr 6(b - c)Z&UZ, - bZ,.)
6(b - c)Zr(2uZr - bZ,t)
a al,
bZT 6(b - c)Z&aZ, - bZ,*)
2(aZ, - bZ,*) 6(b - c)Z,(hZ, - bZ,*) ’
’
It follows that there exist two potentials (p(Zr,IT*), $(Zr, IT*),defined on the set: {T E R6/Zrf 0 A 2aZr - bZTe# 0}, having the property that, for any motion M E A, we have
d(zT(f),
b*(f)) = f z&,,
hIT(f),
IT*(t))
=
tr(‘W.
(13)
The potentials P and II, may be determined apart from the arbitrary additive constants k,, respectively k2, integrating (12) on a path contained in a simply connected subset of their definition domain. Thus, one finds (11). We observe that substituting (13) on the right-hand side of (lo), we obtain ‘k= -3(uI-
bT)e +hzZ#+[(b
- c)ZrI-3cT]$,
or inverted:
D=a’
[f+3(uI-bT)+-[(b-c)ZTI-3eT]&].
Hence, in this case, the definition domain of constitutive eqn (10) can be inverted.
Q
(14)
and $ coincides with 9, domain where on the
516
L.DRjiGU$lN
Proposition 4. The constitutive eqn (10) can’t describe the properties of the Cauchy-elastic material. Proof According to Bernstein’s theorem[3], a necessary and sufficient condition for a hypoelastic material to be Cauchy-elastic lies in the fulfilment of the following integrability conditions identically in T
a&m B,ij - % aT IS
B,pq
+ BkmpjSiq- Bkmiqapj z 0,
(15)
,S
where B kwq =
1( Tkp&,,q+ Tmp8kq - Tkq&np - Tmq&p) + &npq. 2
For the constitutive eqn (10) we have B kwq --
’2 ( Tkp&,,q
+
+
Tmp8kq
-
bI,Tk,,$,, + (b -
Tkq&,,p
-
Tmqakp)
-
@kdpq
C)&Tpq8k,,,
- 3cTk,,,Tpq
c)IT&
+ cIT)+ [~j(~kp&q
+ y
lT@kpbq
+
akq&p)
The identity (15) becomes [Tpq~km&j
T&JpqI(b -
-
-
Tpq
(aki&j
•I
Tmq$
-
( Tmdpj
) akj •t
f
hdkj
•I
( Gpajq
Tmjspi
9a(b -c)
11
) akq
2
IT •If [(Tkphq
t
Tkqajp
J&i
t
-
( Tkiaqj
•t
Tkjaqi
( Tmpajq•t I &p
-
•I- &pakq) •I
Tkqapi
Tmqajp ( Tdqj
J&j
I •t
•I
( Tmpaqi
ski - ( Tkiapi Tmjaqi
I akp
•I
Tkj~pihq
1
- Tkm(Tpq&j-Ti$pq)3(b - c)‘IT=O. It is clear that these relation is not identically fulfilled. Indeed, if i = p = q = 1, k = 2, m = j = 3, we have ; T,2 t 3(b - c)~T,~T~~+ 0. Proposition 5. The material having the constitutive eqn (10) is a hypoelastic material of III-type. It isn’t of I or II-type. Proof. A hypoelastic material is of III-type if ther are two potentials (P(T), P(T), defined on 9, so that (3) holds. From (13) we obtain
Since
it results that Q(T), q(T) are potentials on 9. The material isn’t of I-type because for system of partial differential eqns (8) t$S,t2$T$, T
T'
A hypoelastic model for soils-1
517
isn’t fulfilled the integrability condition
a2Y - a5 aqaTp9
aTp,dTii
The material isn’t of II-type because the relations (9) are not fulfilled
It follows that, for the constitutive eqn (10) the work is not null on closed path in stress space, which is a normal phenomenon if mass density depends on the path travelled. Up to now we have studied only the properties implied by the first part of the definition for the hypoelastic materials, that is those connected to the constitutive equation. Point (b) from the definition refers to the stress-configuration classes corresponding to the constitutive equation and to the fixed initial conditions. Further, we use the following notations
Then In [p2)9up - bq21(b-2c)‘cl + kl, dh 4)= @,(;_ c) 1 P2 ti(p, q) = 6(b _ c) In Igap _ bq2/ + k2’ They are defined on the domain 9={T
E R6)pf0
A gap-bq2fO}.
The domain incorporates three simply connected subsets (Fig. 1) A,.p>O, A2.~>0,
9ap - bq2 > 0, 9ap - bq2 < 0,
A,.p
Fig. 1. The simply connected subsets of domain 9.
(16)
L. DRkU$IN
518
One may notice that a first class of materials corresponds to the initial condition T(&,)= 0. In this case, the constitutive equation has the singular solution T(t) = 0. We shall further assume T( to) f 0. We’ll show that there are three different classes of materials depending on whether the initial condition (pO,Q,) belongs to any of the subsets A,, Al, A3in particular. From the conditions (3) by using the continuity equation, we have
where d = (9ap - 2cq2)@- 2(b - 2c)p44 6(b - c)p(9ap - bq2) * It follows that the mass density rate is null if (9ap - 2cqQ - 2(b - 2c)pq4 = 0. By solving this differential equation for the initial conditions p(h) = po, q(to) = qo, one may say that the mass density remains constant for the stress path (17) Their plot is given in Fig. 2. The initial conditions were marked (PO, 4011, (PO, qoh, (PO, 40h depending on whether they belong to the subsets A,, A2,Aj. Further down we shall say that a material is of N-type if b -2c >O and of C-type if b-2c
Fig. 2. Stress paths which maintain the density p constant.
519
A hypoelastic model for soils-l
Proposition 6. Two stress-displacement pairs [TO,FO],[T,,F,] are equivalent if and only if (pO,Q,), (p,, sr) belong to the same connected subset. We used the symbols
PO =;
40 =;
IT(tO),
Mfo).
Proof. According to Bernstein’s definition[3], two stress-displacement pairs [To, Fo], [T,, F,] are equivalent if there are two continuous piecewise smooth functions T(t), F(t) defined on [lo, t,] so that [T(to), Wo)l
= PO,F-01,
[‘W,),F(h)1= [T,,F,l. It is easy to see that if (PO,qo) and (PI, q,) belong to the same subset then the pairs [To, Fo], [T,, F,] are equivalent. Let us assume that (PO,qo) E A,, (p,, q,) E AZ and that stress-displacement pairs [To, Fo], [Tr, Fr] are equivalent. Then there exists a continuous piecewise smooth function T(t) and a moment t*, so that
TOO)= To,
(9ap - bq2)(t *) = 0.
‘UfJ = TI,
Because, according to [4]
we shall study the behaviour of the improper integral
in the neighbourhood of t*. Let E > 0 be. Since curves (16) do not intersect curve 9up - bq2 = 0, there is a moment t* - c > to so that point (p(t* - e), q(t* - E)) be in that range of the plane (p, q) in which 9ap(t* - 4-bq2(t*-r)-($+-$@-2c)(9apo-bq;)<0. Using (15), we notice that the difference
dM* - l)v40* - l)) - dP0, qo)= 6b(bc- c) + b - 2c cln
9up(t* - c) - bq2(t* - c) 9a~o- bq:
has a positive value if b - 2c < 0 and has a negative value if b - 2c > 0. Because, for the materials of N-type and C-type, the product (l/p)+ has a constant sign, and because, there is always a number a 2 I so that lim (t*- d-
rrt* I
it results that the integral
P(7)
d(T) < + OJ I
520
L. DRAGUSIN
Fig. 3. Stress paths which maintain the work constant.
is divergent. In this case llm detF(T)= r/t*
00 forb-2c>O o forb_2c<0’
hence there is no function F acceptable for which
F(b) = R,,
F(tl) = F,,
and pairs [T,, FO],[T,, F,] are not equivalent. To conclude, four classes of materials correspond to the constitutive eqn (lo), depending on whether one has: To = 0, (p,,, 4,) E A,, (pO,qO)E A*,(p,,, q,,) E A3in the initial moment. We observe that the work done is given by the relation
1’;4d7
[w(P(.),d.)lU) = PO
(18) Acknowledgement-The author wishes to thank Dr. Eugen So& from INCREST, Bucarest, for her many helpful remarks and comments. REFERENCES [l] C.TRUESDELL, J. Raf. Mech. Anal. 4,83, 1019(1955). [2] B. BERNSTEIN and J. L. ERICKSEN, Arch. Rat. Mech. Anal. 1, 3% (1958). [3] B. BERNSTEIN, Arch. Rat. Mech. Anal. 6,89 (1960). [4] C. TRUESDELL and W. NOLL, Handbuch der Phisik, Vol. III/3. Springer-Verlag, Berlin (1965). [5] M. M. VAINBERG, Vr&rionnl Methods and Monotone Operators Method (in Russian). Moscow University Press (1972). [6] hf. MIIjAILESCU-SULICIU and I. SULICIU, Arch. Rat. Mech. Anal. 27, ! (1979). [7] L. DRAGUSIN, Contributions to the Mathematical Study of Deformation of Soils from R.S.R. Doctoral thesis (in Romanian). University of Bucarest (1979).
(Received 20 November 1979)
Sri, Vol. 19. pp. 511-520 Press Ltd., 1981. Printed
Olm-7U5/8I/c-iOI-O5I in Great
l/saz.00/0
Britain
A HYPOELASTIC MODEL FOR SOILS-I A CLASS OF HYPOELASTIC MATERIALS LUCIA DRAGUSIN Polytechnical Institute, Bucarest, Romania (Communicated by E. SOGS) Ahstraet-This paper refers to a certain class of hypoelastic materials for which work depends on the stress history. It is assumed that there are two potentials in the stress space whose time derivative are equal to the mechanical power necessary for the volume deformulation and with the total stress power, respectively.
1. INTRODUCTION THE CONSTITUTIVEequation
of hypoelastic materials (using Truesdell’s definition[ 11) can be regarded as a linear mapping H(T) from the vector space of the strain rates (Dll, &, Dj3, D12,Dz3,D13) to the vector space of Jaumann-No11 stress rates (G7 1R22, %, C*, &3, G)
‘i = H(T) [D]. Let C?3be a domain in R6, where det H(T) f 0. In 1958, Bernstein and Ericksen[2] defined two types of hypoelastic materials by imposing conditions upon the work done by a stress history T = T(T), T E [to, t] in the deformation of a material volume 0
[w(‘U.NlO)= j-i
(I,h(TD)do)dr=
j-(j:$tr(TD)dr)
duo,
(1)
v. = u(tO) marked the initial volume and p. = p(f,,), the initial mass density. Functions T = T(T) were assumed to take values in the simple connected subsets of 9 and to be continuous and smooth piecewise. From definition[2], a hypoelastic material is of I-type, if there is an initial stress value To, so that the functional [w(T(*))](t) be nonnegative for any 7 3 to, and for any stress history T = T(7) where T(t,,) = To. A hypoelastic material is of II-type, if for every t, 3 to, the work [w(T(*))](t,) vanishes, for any stress history T = T(T), where T(fo) = T(t,). The authors show that: (a) A hypoelastic material is of I-type, if and only if there is a stress value To E ~3 and a function y(T) defined on $3, so that
y(To) = 0,
Y =
$tr(TD) =$
0) 2 0, y(W))
&CT())](t)
*
(b) A hypoelastic material is of II-type, if and only if, there is a stress value T,, E $3and two functions A(T), 8(T), defined on 9, so that
Wo) = 0,
W’d = 0,
p(r) = exp [- A(T(t))l, w(t) =
@(T(t)) duo. 511
A(T) 3 0,
6(T) 3 0,
512
L.DRkX$IN
In other words, for the I-type materials, work w related to the present volume unit is not dependent on the stress path, whereas for the H-type materials, both the mass density values p as well as the values of work w are dependent only on time t but not on the stress path. In this paper, we’ll consider those hypoelastic materials, for which there are two potentials a(T), q(T), defined on 9, so that @(T(t)) = ; I&,
@(T(t)) = tr(TD),
where IT = trT, In = trD, and the superimposed dot means material time derivative.
2.HYPOELASTIC MATERIALS OF III-TYPE
Let 46 be a body and its motion
x: W(a)x [to, t11+R3, det Vx > 0, reported to a reference configuration X(9) C R3. A hypoelastic material was defined by Truesdell [ l] and Bernstein[3] (see also Truesdell and No11[4]), through (a) A constitutive equation
‘i’= H(T) [D],
(2)
where H(T) is an isotropic function in T of class C’ in 9. (b) A corresponding equivalence class of stress-configuration classes [T, F], determined by the initial condition [TO,FO]. The following symbols are used: T is the Cauchy’s stress tensor (defined with the aid of the internal normal); F= Vx is the deformation gradient; L = - fiF_’ is the spatial gradient of velocity with inverted sign; D = (1/2)(L + L’) is the deformation rate tensor, while LT denotes the transpose of L; W = (1/2)(L - LT) is the spin tensor; T = T + WT - TW is the Jaumann-No11 stress rate tensor. The continuity equation thus becomes
Let 1 be a set of motions consistent with the constitutive eqn (2), for which there is a continuous piecewise smooth function T(T), defined on an interval [to, tl] with values in a simply connected subset of 9. Definition. A hypoelastic material is of III-type, if for any motion M E A there exist two potentials a(T), T(T), defined on simply connected subsets of 9, so that @(T(t)) = f I&,,
‘@(T(t))= tr(TD).
(3)
Theorem. A necessary and sufficient condition for a hypoelastic material, having the constitutive eqn (2), to be of III-type, is the fulfilment of the following conditions identically in T
=
SiiH i’,pq- 6pqH-,‘ii,
A hypoelastic model for soils-1
513
in which 6,,= 11
1, for i=j fori#j’
I 0,
and H-r is the inverse of H. Proof. In the domain 9, the constitutive eqn (2) can be written as
(6) Since In = H ;;t’,ijfij = H ,‘,ij fij, tr(TD) = Tk,H i;ij ~j = Tk,H ~~ij$j, using the definition of the III-type material, we have (7) As the functions O(T), q(T) are potentials (see Vainberg[5]), in a simply connected subset of 9, we must have
conditions leading to the identities (4) and (5). Conversely, if the relations (4) and (5) are fulfilled, it follows that there are two potentials O(T), ‘P(T), defined on simply connected subsets of 9, so that (7) is fulfilled. Using the constitutive eqn (6) it is easy to deduce (3). Mention should be made that the relation (5) coincides with the existence condition of a free energy u, found by Mihailescu-Suliciu and Suliciu[6] for incompressible hypoelastic materials, when li = tr(TD). Proposition 1. A hypoelastic material of III-type is a hypoelastic material of I-type, if and only if, there exists a function y(T) defined on 9 and a stress value T,,, so that
Y(‘-M= 0,
y(T) 3 0,
Proof. According to [21, a hypoelastic material is of Z-type, if and only if, there exists a positive function y(T), defined on 9, and a stress value To, so that
y(To) = 0,
d r = ; tr(TD), dt 0P
namely + = y p + tr(TD).
514
L.DRAGUSIN
If the material is of III-type as well, we get
QY-p+4, T
leading to (8). Conversely, if, for a hypoelastic material of III-type, (8) is fulfilled, it follows that the material is of Z-type. Proposition2. A hypoelastic material of III-type is a hypoelastic material of II-type, if and only if, there exist two potentials A(T), O(T) defined on 9 and a stress value To, so that
NT,) = 0, 8
i!Lg.
” aTij
O(T,) = 0,
A(T) 3 0,
O(T) 2 0,
aa a9 aa aq a@ --=-__ ari aTpq aTpq aTi,*
iJaT,,’
(9)
Proof. According to [2], a hypoelastic material is of II-type, if and only if, there are two positive potentials A(T), e(T) defined on 9 and a stress value To, so that
No) = 0, P(T)= POexp[- MUI,
@(To) = 0, @(T(t)) = I,’ $ tr(TD) dz 0
If the material is of III-type as well, we must have aA 3 a@ -=-aT ITaT'
g = g exp [ - A(T)].
The functions A(T) and 8(T) are potentials. Then
It is easy to obtain (9), taking into consideration the fact that Cpand 9 are also potentials. Conversely, if the material is of III-type, and if (9) hold true, it follows that there exist two positive potentials A(T), 8(T), so that A(T(t)) = ;,
@(T(t)) = po tr (TD) P *
Hence, the material is of II-type.
3.A SUBCLASS OFZZZ-TYPE HYPOELASTIC MATERIALS
We shall further study the hypoelastic materials having the constitutive equation of the special form ‘F= - UZTZd+3aZ,D+ bZTZnT+ (6 - c)ZTtr(TD)I - 3ctr(TD)T
(10)
where a, b, c are strictly positive material constants. We also assume that b-c is strictly positive (see Dr&usin[7]). Proposition3. The hypoelastic materials having the constitutive eqn (10) admit two poten-
515
A hypoelastic model for soils-I
tials CJJ and $ on 9, depending on the invariants Z,, Z,* so that, for any motion M E A, we have
They are dzT,
IT*)=
h [z,212uzTbzT*)(b-2c)‘c] + kl, ,g,(bc_ c)
+tzT,IT*)= 6(b
IT2
1 _
c)
In(2uZT _
bZ,,j +
(11)
k2y
where kl, k2 are two arbitrary constants and IT*= tr(T*)‘, T* = T - 1/31~1. Proof. From the constitutive eqn (IO), we obtain ZT= bZ:Z, + 3(b &* = - 2( 01~ -
‘h)z,tr(TD),
bZT*)ZTZD+ 6(aZr - cZr*)tr(TD).
Hence ;
z,z,= 2(&-
cZT*)IT-(b -2c)ZT&* ’ 6(b - c)Z&UZT - bZ,*)
tr(TD) = @aIT - b&d, + biTiT= 6(b - c)Z&UZ, - bZ,*) ’
(12)
Since
a
2(aZ, - cZT*)
al,
- (b - 2c)Zr 6(b - c)Z&UZ, - bZ,.)
6(b - c)Zr(2uZr - bZ,t)
a al,
bZT 6(b - c)Z&aZ, - bZ,*)
2(aZ, - bZ,*) 6(b - c)Z,(hZ, - bZ,*) ’
’
It follows that there exist two potentials (p(Zr,IT*), $(Zr, IT*),defined on the set: {T E R6/Zrf 0 A 2aZr - bZTe# 0}, having the property that, for any motion M E A, we have
d(zT(f),
b*(f)) = f z&,,
hIT(f),
IT*(t))
=
tr(‘W.
(13)
The potentials P and II, may be determined apart from the arbitrary additive constants k,, respectively k2, integrating (12) on a path contained in a simply connected subset of their definition domain. Thus, one finds (11). We observe that substituting (13) on the right-hand side of (lo), we obtain ‘k= -3(uI-
bT)e +hzZ#+[(b
- c)ZrI-3cT]$,
or inverted:
D=a’
[f+3(uI-bT)+-[(b-c)ZTI-3eT]&].
Hence, in this case, the definition domain of constitutive eqn (10) can be inverted.
Q
(14)
and $ coincides with 9, domain where on the
516
L.DRjiGU$lN
Proposition 4. The constitutive eqn (10) can’t describe the properties of the Cauchy-elastic material. Proof According to Bernstein’s theorem[3], a necessary and sufficient condition for a hypoelastic material to be Cauchy-elastic lies in the fulfilment of the following integrability conditions identically in T
a&m B,ij - % aT IS
B,pq
+ BkmpjSiq- Bkmiqapj z 0,
(15)
,S
where B kwq =
1( Tkp&,,q+ Tmp8kq - Tkq&np - Tmq&p) + &npq. 2
For the constitutive eqn (10) we have B kwq --
’2 ( Tkp&,,q
+
+
Tmp8kq
-
bI,Tk,,$,, + (b -
Tkq&,,p
-
Tmqakp)
-
@kdpq
C)&Tpq8k,,,
- 3cTk,,,Tpq
c)IT&
+ cIT)+ [~j(~kp&q
+ y
lT@kpbq
+
akq&p)
The identity (15) becomes [Tpq~km&j
T&JpqI(b -
-
-
Tpq
(aki&j
•I
Tmq$
-
( Tmdpj
) akj •t
f
hdkj
•I
( Gpajq
Tmjspi
9a(b -c)
11
) akq
2
IT •If [(Tkphq
t
Tkqajp
J&i
t
-
( Tkiaqj
•t
Tkjaqi
( Tmpajq•t I &p
-
•I- &pakq) •I
Tkqapi
Tmqajp ( Tdqj
J&j
I •t
•I
( Tmpaqi
ski - ( Tkiapi Tmjaqi
I akp
•I
Tkj~pihq
1
- Tkm(Tpq&j-Ti$pq)3(b - c)‘IT=O. It is clear that these relation is not identically fulfilled. Indeed, if i = p = q = 1, k = 2, m = j = 3, we have ; T,2 t 3(b - c)~T,~T~~+ 0. Proposition 5. The material having the constitutive eqn (10) is a hypoelastic material of III-type. It isn’t of I or II-type. Proof. A hypoelastic material is of III-type if ther are two potentials (P(T), P(T), defined on 9, so that (3) holds. From (13) we obtain
Since
it results that Q(T), q(T) are potentials on 9. The material isn’t of I-type because for system of partial differential eqns (8) t$S,t2$T$, T
T'
A hypoelastic model for soils-1
517
isn’t fulfilled the integrability condition
a2Y - a5 aqaTp9
aTp,dTii
The material isn’t of II-type because the relations (9) are not fulfilled
It follows that, for the constitutive eqn (10) the work is not null on closed path in stress space, which is a normal phenomenon if mass density depends on the path travelled. Up to now we have studied only the properties implied by the first part of the definition for the hypoelastic materials, that is those connected to the constitutive equation. Point (b) from the definition refers to the stress-configuration classes corresponding to the constitutive equation and to the fixed initial conditions. Further, we use the following notations
Then In [p2)9up - bq21(b-2c)‘cl + kl, dh 4)= @,(;_ c) 1 P2 ti(p, q) = 6(b _ c) In Igap _ bq2/ + k2’ They are defined on the domain 9={T
E R6)pf0
A gap-bq2fO}.
The domain incorporates three simply connected subsets (Fig. 1) A,.p>O, A2.~>0,
9ap - bq2 > 0, 9ap - bq2 < 0,
A,.p
Fig. 1. The simply connected subsets of domain 9.
(16)
L. DRkU$IN
518
One may notice that a first class of materials corresponds to the initial condition T(&,)= 0. In this case, the constitutive equation has the singular solution T(t) = 0. We shall further assume T( to) f 0. We’ll show that there are three different classes of materials depending on whether the initial condition (pO,Q,) belongs to any of the subsets A,, Al, A3in particular. From the conditions (3) by using the continuity equation, we have
where d = (9ap - 2cq2)@- 2(b - 2c)p44 6(b - c)p(9ap - bq2) * It follows that the mass density rate is null if (9ap - 2cqQ - 2(b - 2c)pq4 = 0. By solving this differential equation for the initial conditions p(h) = po, q(to) = qo, one may say that the mass density remains constant for the stress path (17) Their plot is given in Fig. 2. The initial conditions were marked (PO, 4011, (PO, qoh, (PO, 40h depending on whether they belong to the subsets A,, A2,Aj. Further down we shall say that a material is of N-type if b -2c >O and of C-type if b-2c
Fig. 2. Stress paths which maintain the density p constant.
519
A hypoelastic model for soils-l
Proposition 6. Two stress-displacement pairs [TO,FO],[T,,F,] are equivalent if and only if (pO,Q,), (p,, sr) belong to the same connected subset. We used the symbols
PO =;
40 =;
IT(tO),
Mfo).
Proof. According to Bernstein’s definition[3], two stress-displacement pairs [To, Fo], [T,, F,] are equivalent if there are two continuous piecewise smooth functions T(t), F(t) defined on [lo, t,] so that [T(to), Wo)l
= PO,F-01,
[‘W,),F(h)1= [T,,F,l. It is easy to see that if (PO,qo) and (PI, q,) belong to the same subset then the pairs [To, Fo], [T,, F,] are equivalent. Let us assume that (PO,qo) E A,, (p,, q,) E AZ and that stress-displacement pairs [To, Fo], [Tr, Fr] are equivalent. Then there exists a continuous piecewise smooth function T(t) and a moment t*, so that
TOO)= To,
(9ap - bq2)(t *) = 0.
‘UfJ = TI,
Because, according to [4]
we shall study the behaviour of the improper integral
in the neighbourhood of t*. Let E > 0 be. Since curves (16) do not intersect curve 9up - bq2 = 0, there is a moment t* - c > to so that point (p(t* - e), q(t* - E)) be in that range of the plane (p, q) in which 9ap(t* - 4-bq2(t*-r)-($+-$@-2c)(9apo-bq;)<0. Using (15), we notice that the difference
dM* - l)v40* - l)) - dP0, qo)= 6b(bc- c) + b - 2c cln
9up(t* - c) - bq2(t* - c) 9a~o- bq:
has a positive value if b - 2c < 0 and has a negative value if b - 2c > 0. Because, for the materials of N-type and C-type, the product (l/p)+ has a constant sign, and because, there is always a number a 2 I so that lim (t*- d-
rrt* I
it results that the integral
P(7)
d(T) < + OJ I
520
L. DRAGUSIN
Fig. 3. Stress paths which maintain the work constant.
is divergent. In this case llm detF(T)= r/t*
00 forb-2c>O o forb_2c<0’
hence there is no function F acceptable for which
F(b) = R,,
F(tl) = F,,
and pairs [T,, FO],[T,, F,] are not equivalent. To conclude, four classes of materials correspond to the constitutive eqn (lo), depending on whether one has: To = 0, (p,,, 4,) E A,, (pO,qO)E A*,(p,,, q,,) E A3in the initial moment. We observe that the work done is given by the relation
1’;4d7
[w(P(.),d.)lU) = PO
(18) Acknowledgement-The author wishes to thank Dr. Eugen So& from INCREST, Bucarest, for her many helpful remarks and comments. REFERENCES [l] C.TRUESDELL, J. Raf. Mech. Anal. 4,83, 1019(1955). [2] B. BERNSTEIN and J. L. ERICKSEN, Arch. Rat. Mech. Anal. 1, 3% (1958). [3] B. BERNSTEIN, Arch. Rat. Mech. Anal. 6,89 (1960). [4] C. TRUESDELL and W. NOLL, Handbuch der Phisik, Vol. III/3. Springer-Verlag, Berlin (1965). [5] M. M. VAINBERG, Vr&rionnl Methods and Monotone Operators Method (in Russian). Moscow University Press (1972). [6] hf. MIIjAILESCU-SULICIU and I. SULICIU, Arch. Rat. Mech. Anal. 27, ! (1979). [7] L. DRAGUSIN, Contributions to the Mathematical Study of Deformation of Soils from R.S.R. Doctoral thesis (in Romanian). University of Bucarest (1979).
(Received 20 November 1979)