Discretized kinematic hardening in cyclic degradation of rocks and soils

EngineeringFracrureMechanicsVol.21,NO.4, pp. 923-945,1985 Printedin the U.S.A.

0013-7944185 13.00t .oO PergamonPress Ltd.

DISCRETIZED KINEMATIC HARDENING IN CYCLIC DEGRADATION OF ROCKS AND SOILS T. HLJECKEL

PolishAcademy of Sciences; now ISMES, Bergamo,and Politecnicodi Milano, Italy Abstract-A discretized description is made of the process of kinematic hardening in elastoplastic solids in the range of infinitesimal strains. Applications of this description to the modelling of response to cyclic loading of rocks and soils are presented.

1. INTRODUCTION IT IS now common in the modelling of cyclic behaviour of metals and geological media to distinguish at least two types of domains of inelastic response [l-31.In this way, separate relationships are employed to model monotonic and cyclic loading response, associated with a “bounding” or “consolidation” surface on one hand and a yieid surface on the other. The latter surface enclosing an elastic domain is sometimes considered as degenerated to current stress point [4-61. Intrinsic model consistency conditions, however, often impose too strong constraints on material functions if the model is excessively complex. For example, the requirement of joint normals to yield surfaces at their contact points has been shown recently by Norris [6] to imply complicated evolution equations for back stress function. The requirement has been also indicated as a sufficient but not necessary condition to guarantee continuity of rate response at neutral path 161. Also, in the present model, two types of inelastic behaviour are distinguished. The aim of this paper is to simplify the description and to look for an alternative explanation of the complex effect of kinematic hardening within a “bounding surface”. The reference material in this context will be rock or soil. In the first type of inelastic response, the energy dissipation is supposed to occur mainly due to progressive rock fissuration (or damage) or cataclastic and plastic advanced flow. In the second type, the dissipation is attributed to local (i.e. without structural rearrangement) friction, e.g. on internal surfaces of existing voids. It gives rise to what is commonly called “elastic” hysteresis [7, 81. Phenomenological support for such a distinction comes, in the case of rocks, from microacoustic experimental evidence. In fact, microacoustic activity is characteristic for fracture progression whereas it is not observed in unloading or reloading [9, 10, 661. In what follows, the two domains are assumed to be separated by a variable fracture or yield locus. The stress-strain relationships for the two domains are formulated in different way. For the monotonic loading the material behaviour is supposed to be explicitly path dependent. The corresponding relationship is therefore assumed to be incremental. This part of constitutive law is not discussed in this paper. Extensive studies in this field has been made recently by Gerogiannopoulos and Brown [ 111, Dragon and M&z [ 121, Hashiguchi [ 131, Yamada et al. 1141, Batant ef al. [151, KrajcinoviE el al. [16], Nova 1171, Darve 1181, Chambon [19], Gudehus and Kolymbas [20], and many others. Considerations in what follows are restricted to processes internal to a current fracture or yield locus. The constitutive model proposed implies history independence of response by pieces. The pieces are defined on a stress trajectory by a set of its characteristic points (stress rate reversal points). Between the points the behaviour is path independent, All material memory effects are lumped in the discrete points (of reversals) in the form of a set of stepwise variables, following Hueckel and Nova [21-231. Description of cyclic behaviour by relationships valid for pieces has been adopted previously by Mritz and Lind 1241, Guelin and Boisserie 1251, Baiant and Kim [26] and Aubry et al. [27]; the last three approaches involving incremental laws. An analytical review of continuous kinematic hardening models and the present approach has been recently made by Norris [6].

924

T. HUECKEL 2. MAIN ASSUMPTIONS.

STRESS-STRAIN

RELATIONSHIP

Denoting the stress tensor by uij, the strain tensor, its rate, and rate of its plastic part by eij, 6ij and +’ respectively, and an incremental elasticity tensor by Eijk,, one may define a domain of validity of the following considerations as the interior of current fracture/yield locus, F = 0 (Fig. 1). The domain is restricted to stress states which, for a given plastic strain history expressed through usual hardening parameters, satisfy the following inequality:

F(uij,

K,

e>< 0,

(2.1)

and to these stress states among fulfilling the equality F(Ujj, K, e) = 0 for which strain rate kij satisfies the following unloading inequality

Eijkr

(2.2) [28]:

T$ kkl <0, lJ

(2.3)

where K and e are deviatoric and isotropic strain hardening parameters [17-29, 301. The response of geological materials to variable cyclic stress within the fracture/yield surface is in general path dependent. In many models which identify this range with elasticity, the behaviour is assumed to be path independent, which in reality is observed for a limited class of paths only. In the present approach a compromise hypothesis between the two above positions is made. In fact, the material behaviour is assumed to be piecewise path independent in the following sense. The stress trajectory is divided into segments by some characteristic points defined as stress rate reversals, or shortly reversals, &ij. The stress rate reversals occur at the stress points, at which a change of direction (not necessarily abrupt) of the stress trajectory takes place (Fig. 1). Between segment extremities the behaviour is path independent.

Ductile

Fig. 1. Stress trajecfory inside the yield fracture surface.

925

Cyclic degradation of rocks and soils

All path-dependent variables are supposed to be constant along segments. They change discretely at reversals, however, in the form of a Heaviside function. In order to satisfy the postulate of piecewise path independence, the behaviour along a single segment of the trajectory is first described in terms of differences, between a current variable value and its value at the most recent, say, Lth reversal point, &ij = Lati. Thus we define A Uij

Uij

=

-

1 Uij

. 9

‘Eij

Eij

=

-

(2.4)

tij.

The precise meaning of the term “most recent reversal” is specified in Section 3. The requirement of piecewise path independence of material behaviour is met by assuming a unique relationship between difference of stress and difference of strain. It is, in fact, a kind of non-linear, Cauchy, “one-way” elasticity, expressed, however, in differences rather than stress and strain themselves and at a given set of dead parameters p,k~,+. 6. These parameters are values of discrete variables, changing stepwise at reversals. Thus ‘eij

=

‘EijtAuklr

pkl,

$3

(2.5)

4)~

where t&r is a symmetric tensorial measure of rock damage, + is a scalar measure of the most recent reversal intensity, while 6 is a hysteretic strain accumulation parameter. The expression “one way” means that the above relationship holds only for path segments satisfying an appropriately formulated (Section 3) loading criterion. For an unloading, e.g. in a radial path, a different relationship has to be employed. The damage parameter is a function of stress and irreversible strain and is therefore referred to the most recent state at the fracture/yield surface at which non-zero irreversible strain rate took place, i$’ # 0. (2.6)

pij = I_Lij(Uij, EC’)*

A particular form of this variable will be specified later. The reason for introducing an irreversible strain variable comes from a phenomenological argument. Indeed, in the case of rocks, for example, deformation under cyclic loading consists mainly in frictional slidings which occur along previously formed crack surfaces, unless reloading does not exceed previously reached levels [7, 8, 65, 681. The two other parameters + and 6 are updated at each reversal. The former is employed in order to ensure continuity of response to neutral rate path at the reversals, while the latter reproduces a response sensibility to repeated strain accumulation. In a general case in which initial isotropy of undamaged material is assumed and ‘uij and /J-iiare symmetric tensors, the relationship (2.5) may be expressed as follows:

+

hbkAuklA@lj

+

$4f-bkpklAu!j

f

$6/hk~klA~hA~~,

(2.7)

where $I . . .c/I~ and IJJ,. . . (I~ are functions of invariants of stress difference, of damage tensor and mixed invariants of both tensors as well as of parameters 4 and 6. The invariants are defined in the following way:

I* =

AUijA Uij;

4

=

A

M4

=

bkpkjAujjIAUli;

YZ =

UijAUjkAUki;

Pij/J+ij;

M2 = /kikAUeAUjji;

=

M3

y3 MS

hjpjk(lki; =

=

kikAUkjAUjlAU[i;

(2.8)

/-hik/hkjAUji; Me

=

~ik~kjAUj,,,AU,lAU,i~

Note first that the principal axes of the tensors of stress difference and strain difference do not coincide, due to the introduction into relationship (2.7) of the tensorial damage parameter pij. This fact represents an anisotropy previously induced by oriented damage. It forces the material

926

T. HUECKEL

to deform also in the hysteretic range along privileged damage directions. An analogous phenomenon due to anisotropic preconsolidation is observed in soils, sands [31, 321 and clays [33, 341, and explains in part what is now broadly discussed in soil mechanics as “principal stress rotation effect.” In fact, irreversible deviatoric strain occurring during initial elastoplastic KO consolidation is supposed to change “elastic”, i.e. “overconsolidation” moduli of these materials as is assumed in elasto-plastic coupling models [35-371. Furthermore, the stress difference part in (2.7) described by terms $,t.~~~.+3kikky, has a direction determined exclusively by the damage parameter. The direction does not change in the range of validity of the equation. The multipliers $3 and +j depend, however, on stress difference. In the case of an initial state of trajectory segment which is strain difference free, one has to set equal to zero the functions +, 1 $I~ and & at “T;~ = 0. This assumption does not hold, however, for reactivation of a relationship in non-radial stress loops, as will be specified later on in Section 3. Description of hysteretic response by means of difference relationships valid piecewise was the earliest concept applied to one-dimensional cases. In fact, the archetype of cyclic rules known as the Masing rule [39], is an example of it. The most diffused model in soil mechanics (of Hardin and Drnevich [40]) is also path independent piecewise, if properly formulated through invariants. In a more general case, in which cyclic strain accumulation is included, the following observation can be made concerning difference relationships. Assume that the relationship (2.7) is such that the strain difference may be decomposed into two parts: A

Ejj

=

he!. + he!‘.

lJ

(2.9)

IJ*

where *E,$ is an odd function and ‘E;$ is an even function of stress difference at constant pij, + and 6. As will be shown, such a distinction allows us to analyse separately purely hysteretic effects and cyclic strain accumulation effects. The odd part, ‘E;, of the strain difference function, in terms of the variable *oij, is defined as satisfying the following equality:

*E&(*Uij) + “E;(The even part, *E;, of the strain difference *+(Auij)

*Uij)

=

0.

(2.10)

function should full51 -

A$(-

Auij)

= 0.

(2.11)

If the former part of the strain difference is assumed independent of 6 or if 6 is constant, it has the property of the polar symmetry with respect to the center K of the stress-strain loop ABCDA for any radial stress path and an identical return (Fig. 2). In fact, considering such a

C C’ F

Fig. 2. Perfect hysteresis

(a), and accumulation (b) strain in cyclic loading. Global behaviour (c).

927

Cyclic de~adation of rocks and soils

loop composed of two consecutive portions of a radial stress trajectory, for which AOijbecomes respectively AAuij = aij - U$ and ACaij = aii - I$ and taking such a - AAau that it is equal to Ac~U, one may easily see that (2.10) implies a complete similarity of the reloading branch to the unloading curve, In particular, it leads to an entire restoration of the initial loop compliance modulus after the reversal. For arbitrary (non-radial) closed stress cycles which include ony one reversal (with + = 1) the property (2.10) together with the path-independent postulate guarantees an entire recovery of strain after completion of such a stress cycle. This kind of cycle in the absence of the strain differences satisfying (2. If) will be referred to as perfect hysteresis. These cycles are a good approximation of stabilized cycles after saturation of cyclic strain accumulation, when A~G3 0, or when only energy damping is considered, accumulation effects being disregarded. If ‘E; depends on 6, which changes after any reversal, one obtains in a uniaxial case a form of rotation of a non-closed stress-strain loop with contrasting accumulation of strains at the loop extremities. This case will be discussed in Section 4. The even part of the strain difference function (2.11) is by its very definition insensitive to change of the sign of the stress difference. Therefore this part of strain corresponds to strain accumulation, as visualized in Fig. 2(b). Its progression may be mitigated by the parameter 6 updated after any reversal according, for example, to a rule presented in Section 4. Figure 2(c) shows a construction of the two contributions giving rise to a realistic description of ratcheting effects. Let us examine now the odd and even parts of strain difference separately. For the reason of simplicity a tensorially linear form of (2.7) will be discussed only by setting: $2

=

*3

=

$4

=

$*

=

qJ6 =

(b3 =

0,

(2.12)

and thus AEij =

$16,

+

(t)2'Uij

+

Jl*pij*

(2.13)

Such a hypothesis eliminates a number of deformation modes except one, that of dependence of the hysteretic strain directional prope~ies on the damage tensor. It allows, however, for a clear distinction and a separate analysis of stress-difference-dependent and damage-dependent hysteretic deformation components. Further restriction is made by assuming a polynomial representation of function 41, 42 and +i in such a way that stress enters the right-hand side of (2.13) in order no higher than 3. The odd part of (2.13) now reads

while the even part is

where

The coefficients ai, pi, yi are functions of invariants of the damage tensor and parameters Q, and 6. The set of equations (2.14) and (2.15) is a general form of polynomial representation of a tensor-My linear constitutive relationship (2.13). Therefore the 25 material functions encompass all possible deformational modes in the assumed range. Examples of constraints on the functions are given in Section 3.

928

T.HUECKEL

Let us discuss first the odd part of the strain difference. For radial unloading and reloading this strain forms perfect hysteresis. Note, that two first terms of the right-hand side of (2.14) do not depend on the direction of the damage tensor tag. The terms which contain coefficients OLIand a4 refer in the case of a hysteresis loop to an initial secant modulus, The modulus depends thus on the damage tensor invariants. In the absence of all other terms the effect of hysteresis disappears and the relationship (2.14) or (2.15) reduces to the damage-dependent elasticity as in elastoplastic coupling models [35, 361. Each non-linear component in stress forms separately a perfect hysteresis. Components which involve constant coefficient of d3, (Y6. cx7, dg form hysteresis loops which depend exclusively on stress differences (or stress amplitude). In other words, such hysteresis may occur without any presence of damage. Components which involve coefficients (x3, (Yg,d7, ds variable with l&l give rise to hysteresis, the curvature of which, together with its strain amplitude but not its direction, depends on invariants of damage tensor. Note, moreover, that the relative position of the stress deviator difference vector and deviatoric damage vector influences hysteresis contributions which contain multipliers y5 and y6. It implies that the strain amplitude is bigger when the stress difference direction is closer to the direction of previously included damage. In a case when the two vectors are orthogonal these terms give no contribution to hysteresis. Direction of the third part of the right-hand side of eqn (2.14) coincides with the direction of damage. As before, the term linear in stress determines the initial secant modulus, which explicitly depends on the projection of the stress difference on the direction of damage. Other terms may be interpreted in a similar manner as above. The same refers to the even part of strain differences. It is important to distinguish between hysteresis which is only stress dependent from hysteresis attributed to damage. A stress-difference-dependent hysteresis has been studied for sands and clays in refs. [21, 231, while a purely damage-induced hysteresis model has been proposed in ref. [45]. We shall now discuss a particular form of the piecewise independent description of hysteresis, which assures also a piecewise path independence of complementary energy. We shall therefore postulate the existence of a complementary energy potential V-a function of invariants of stress difference, damage tensor, and parameters + and 6 V =

V(Zi, Mi,

Furthermore, we postulate a hyperelastic differences, so that

relationship

A

-

(2.17)

Yiy $9 6).

between stress differences

dV

and strain

(2.18)

A particular form of this relation is identical with (2.13), for which the following constraints are satisfied: (2.19)

while the potential is a polynomial of third order in stress in terms of invariants Zi and Mi, and the multipliers ai, bi, ci are functions of the invariants Yi and discrete variables + and 6, as follows:

v = vo + v, = vo + all: +

b2M:

+

c7Z,M:;

+

+

azz:

b3M;

+

+

asz;l

+

u4zo

+

(c&

+

CZZ,ZD +

&ZlZD

+

&I$

c3Zo)M1

+

+

u7z:zD

(c4ZI +

+

c&

blil4:

+

c6ZD)M,

(2.20)

929

Cyclic degradation of rocks and soils

where V0 is a part of the energy referred to stressing of intact (with l.~kr= 0) rock, say perfectly elastic. If the coefficients al . . . c7 are expressed in an analogous way to the coefficients (~1 y13 it is easy to find restrictive conditions, which reduce eqns (2.14) and (2.15) to hyperelastic laws in differences, i.e. satisfying (2.18): (Ye= ci7 = 2a7;

&a9 = CQ = 2a5;

yg = y2 = 2C2;

y3

=

2C3;

f& = 4b3;

&j = 4a6;

~9

p2

=

3b2;

(Yl = 2Ul;

y6 = 2~;

=

72

=

2C2;

ylo = y7 = 2c7; y12

=

y4

=

C4;

PI = 2bl;

fy13 = yl = 2c,; 711

=

y3

=

c3;

ci3 = 4a3; ys

=

cx4 = 2a4;

(2.21)

cx2 = 3a2;

2C5.

Let us discuss now other restrictive conditions to the constitutive relation resulting from the requirement of positiveness of specific dissipated energy per cycle. In the present model it is possible to guarantee such positiveness a priori for closed, not necessarily radial, stress cycles involving one reversal between the cycle onset and its end. For simplicity, assume that the cycles begin at say Lth reversal point, and that + = 1. The work dissipated on the cycle L, L + 1, L + 2, Fig. 3, may be expressed in the following way: PA‘+I_..

D

=

,4+2

-

(J$+l)(,$+2

-

Et+‘) -

+

(ALcI)L+l

Jo “II(Eij

-

,AL.+z,U EC)daij - J0 (Eij - ES+‘) daij,

+

Fig. 3. Dissipated work in the stress cycle.

930

T. HUECKEL

or employing the complementary

work potential (2.17): (2.23)

where ‘V and ‘+’ V denote functions (2.17) referred to Lth and (L + 1)th reversal while the symbol ( )L +i denotes a value at (L + i)th reversal. Let us divide now (AL + 1EG)~+Z into its even and odd parts, defined by (2.10) and (2.12). Analogously one may introduce two parts V’ and V” which give rise to even and odd strain differences, correspondingly. Note that it has been assumed that the even part of strain difference is not sensitive to the accumulation parameter 6. Therefore, for a fixed stress amplitude (*aij = const.) this part of the strain difference is the same both for (L + l)th and (L + 2)th portions. Clearly, it refers also to the corresponding parts of the potential. Thus (2.23) may now be written: II =

(AL+‘UijAL+‘E;)L+2

+

(AL+‘UijAL+lE$)L+* - [LV)‘(82)]L+1 - [L+‘v”(*~)]~+z

- 2(L+‘V’)L+2.

(2.24)

In the case of steady accumulation, in which tiL = 6 L+ I (uniform ratcheting) both the evenly potential parts are equal, but of opposite sign. Therefore the dissipated work over cycle reduces to

. contributing

D =

(AL+‘UijAL+1Efj)J_+2

-

2(L+1V’)L+2e

(2.25)

In some applications it may interesting to require that the potential V be a homogeneous function of order n with respect to *CQ. In such circumstances by virtue of Euler’s theorem the expression (2.23) turns out:

D = (n - lv+lV)L+* or in the case of steady accumulation,

Pm+,,

6 = const,

D = (n - 2)(L+‘V’)L+*

+ n(=+W’)L+Z.

It often occurs that the influence of 6 is greatest at the beginning of a cycle series. Then it is enough to guarantee the dissipation non-negativeness for two first values of 6. The odd term contributions to the complementary energy potential V are positive, provided all ai, bi, ci 2 0, except the term c$rM?. The even strain difference contribution is not definite a priori. Positiveness of V is a highly desirable property since it at least ensures uniqueness of the response to a given stress difference. Positiveness of V in the case of a homogeneous potential guarantees positiveness of dissipated energy on closed cycles for perfectly hysteretic behaviour. Further constraints may be imposed on the functions ai, bi, ci, which ensures that V is positive both in the general case and in an assymptotic state of perfect hysteresis, when V + V’, as in the example of a potential in Section 4.1(b). Such constraints imply piecewise trajectory independence and uniqueness of segmental complementary energy together with invertibility of the constitutive relationship (2.18). The latter problem is not discussed here. A procedure similar to that proposed by Truesdell [40] or by Evans and Pister [41] may be adopted for this. Another property which is highly desirable in order to assure stability of stress-strain hysteretic behaviour is positive definitiveness of the tangential matrix of compliances d2V/ d *ai+3*ukl. A particular way to meet such a requirement is shown in Section 4.1.

3, STRESS RATE REVERSAL. DISCRETIZED

EVOLUTION OF REFERENCE

STATE

In order to define stress rate reversal for a generic trajectory, say ABCDE (Fig, 4) it is found useful to employ a variable locus which originates at a current reversal point, say A, and a

Cyclic degradation of rocks and soils

931

Fig. 4. Stress reversal loci.

series of loading functions. The locus is a function of the variable ‘uij and constant parameters, PH, 4 and 6. It expands isotropically from Aaij = 0 as the current stress point moves along an ongoing trajectory segment A, B:

Wz

W("Uij)

- WI)= 0,

(3.1)

where W. is a scalar value of the function W. The locus w = 0 delimits at a current stress between such further stress points, which (“outside” the locus) may be reached by employing the current constitutive relationship, and such stresses (“inside” the locus) to reach which a new relationship would have to be adopted. Instead of stress space locus a dual strain difference locus may be employed as in refs. [21, 221 to model cyclic behaviour of sands and clays. In [45] the locus function W has been identified with the complementary energy potential V giving rise to a form of normality rule. Norris [6] reviewing kinematical hardening applications to cyclic behaviour of soils found advantageous the lack of normality, W # V, in the piecewise path-independent models as less restrictive. Let us look now at reversal consequences. At a point B of a stress rate reversal a new difference relationship referred to that point starts to act together with a new locus which grows around the point from zero. The “old” family of loci is forgotten, except the last one, passing through the considered reversal B, WA*B = 0. This locus becomes inactive and determines now the range of validity of successive relationships, referring to possible successive reversals, e.g. B and C. Suppose now that shortly after reversal B, a new reversal takes place at C. Again, from the family of loci referring to B only that at C is not forgotten. A new relationship referring to C is then activated. Its validity is restricted to curvilinear figure BMCK. When the stress trajectory reaches a point D, belonging to locus WA*B = 0, the relationship referring to A is reactivated together with relative loci. The trajectory portion B, C, D is forgotten and only irreversible strain relative to it is remembered as a discrete number. In order to formalize the above procedure we shall first introduce an appropriate loading function 2. In contrast to the usual probing stress we shall employ a finite (in general) stress difference probing function BEaij, which is a two-point function, at a given stress difference at B defined as point,

BEu;j= (A*uij)E - (A”crij)B.

(3.2)

932

T.HUECKEL

The loading function is therefore (3.3) If the loading function at a generic point B is non-negative for the stress step to (‘*aij)E, %,,(W)

2 0

(3.4)

the acting constitutive relationship continues to be valid also at (‘*c;j)E and thus the reference point &ij, is still bij = US. For the negative loading function %V(~)

< 0

(3.5)

a reversal takes place. A new relationship should be formed with reference to the point &ij = (Uii)B. Clearly, it is often necessary that the probing stress difference must be infinitesimally small, so that the loading function becomes (3.6) The choice of (3.6) or (3.3) depends on accuracy adopted in constructing the stress path. Refer again, for example, to the stress path ABCDE from Fig. 4. An exact pursuit of the stress trajectory requires use of an incremental stress probe in order to detect the reversal at B. Suppose, however, that the loop BCD is not considered to be an important event in the material history. The property of segmental path independence permits us to avoid an exact description of the portion BCD without significant departure from an accurate description of the successive behaviour. For this purpose a finite stress probe (3.2), (3.3) may be adopted. It may lead either to complete neglect of the loop BCD, if a vector such as BE is chosen as the probing vector, or to inexact description of the loop if, for example, a vector BC’ is taken as the probing stress difference (Fig. 5). Note that employment of the finite stress difference loading function is consistent with the non-incremental approach to the stress-strain curve description. Moreover, it allows us to avoid a laborious updating procedure which is required at any incremental step, if adopted. Let us for completeness describe now briefly two constitutive rules already discussed at length elsewhere [23, 421; i.e. the discretized evolution rule for the reference stress 6U and discrete stress rate dependence of hysteretic compliance at reversals assumed in order to guarantee neutral path continuity. In order to describe formally the memory rules illustrated in Fig. 4, a set of loading functions

Fig. 5. Incremental

vs finite difference stress probe vector in loading condition.

Cyclic degradation of rocks and soils

933

is introduced. In what follows such functions are defined for incremental stress. Analogous functions are valid also for finite stress d~erences (3.2):

aWK9K+’ ’

aAKa__

*

Oijg

l
?I ZK

(3.7)

s

aWK * aAKa..

(%;

K = L,

V

where WK*K+l = 0 is an inactive locus referring to the Kth reversal passing through K + 1 reversal and which, if crossed by the trajectory, may be reactivated. Furthermore, introduce the contact index SK ISKCL-1 K = L,

(38

where Jr{ } is a step function defined as follows:

*cd = 1 44x1 = 0

for x < 0 for x 2 0.

(3.9)

The contact index is equal to one, if the stress point lies on or behind the considered locus. Otherwise it is zero. Finally, let us define the forgetting index: (3.10) where rl[ is the multiplication symbol. The above three functions allows us now to formulate an evolution equation for the variable 6ij, which is the reference state for a current segmental law: L Gij

=

C [O$ + (Oij - O$)*(eK}]~“<“.

(3.11)

K=l

Consider as an example a stress state u:j after the Lth reversal, so that 6ij = 4. Assume an incremental stress probe &ij. Suppose that the stress qii does not enter into contact with any inactive locus, so that for K = 1, . . . ,Llall[“=O,whereasforK=L c’= 1,and only this term may remain in (3.11). According to (3.10) corresponding CL = (1 - 4’). . . . .( 1 - eL) = 1. If &ij is external to the current locus, ZK is positive, so that Jl{Z”} = 0; thus 6ij = 4. For an internal &ij, + = 1; ciij becomes equal to the actual ufj at which the reversal occurs, Eifj = Uij. If however, the considered stress state is in contact with one or more (in an intersection point) inactive loci, e.g. those referred to as Mth, Nth and Jth, wKsK+’ = 0 for K = M, N, J, thus e”‘, SN, tJ = 1 and also .$’ = 1, while all 5’ = 0; i f M, N, J, L. Suppose now that J is the “oldest” amongst M, N, J, so that J < M < N, J < L. In consequence, CL = tM = ri” = 0, while only <-’= 1. In fact, for instance, tN = (1 - [J)$l - 5”) = 0 and cJ = (1 - 0), since there is no such K c J - 1 for which cK # 0. The value of +[I depends thus on the direction of ciUwith respect to WyaJ+l - Wo = 0. If 2’ is positive, ciij becomes equal to IY$ otherwise &ij becomes equal to Oii, which takes on the number J + 1, while all loci for K > J + 1 are cancelled. To conclude, note that the yield/fracture surface acts as the “oldest” amongst all reversal loci cancelling them all when reached. The above considerations were simplified by the assumption that the reversal intensity measure +, not yet specified, was set a priori as equal to one. It corresponds to a perfect reversal

934

T. HUECKEL

Fig. 6. Con~~ty

for neutral incremental path.

in which oij = -oaoij, where 01is a positive scalar multiplier. Experimental evidence for sand [40], both in thin-walled tube torsion experiments and in triaxial testing, for clays [44] and for rocks [7, 81 shows a total restoration of hysteresis compliance after any reversal. This is valid both for deviatoric hysteresis and for isotropic cycles if a logarithmic stress measure is adopted in the latter case [46]. On the other hand, a continuity condition for neutral incremental path [5, 48, 491 requires that a response to such a path be the same on both sides of the reversal locus in order to avoid numerical instabilities [52]. To meet these two requirements a hypothesis is made that initial compliance at reversal point depends on a measure of the intensity of the reversal, 4. Denoting by ob a unit stress tensor at the reversal, an appropriate reversal intensity is defined as follows (Fig. 6):

(3.12) where

(3.13)

Denote by ‘- ‘C& the tensor of initial compliance at (L - 1)th reversal and by Cijkl the current incremental compliance at ALoij. If at this stress a new, Lth, reversal occurs, new initial compliance at the point is postulated to be a linear combination of the two above compliances in the following way:

L&l = L-‘Czk( + where A C;jkl is the compliance

+ACijk[,

(3.14)

acquired along the portion L - 1, L is defined as ACijkf = Cfjkl - L- ’ C$kl.

(3.15)

The dependence of the tensor “C&t on $” through 4, and thus incremental sensibility to stress rate direction, is limited to isolated points of reversals, Such dependence clearly does not introduce viscous effects, because the function ft, (3.12) is homogeneous of degree zero with respect to time. Naturally, b, remains constant along the whole portion between the L and (L + 1)th reversals and thus enters the class of discrete variables. The question of continuity is common of all bi-operator laws. In classical plasticity the continuity is guaranteed by assuming insensibility of the plastic strain rate to the stress rate component tangent to yield surface. For pressure-sensitive materials it leads to serious departures from reality, as may be seen from experiments by Lewin et al. (150, 511; see also 1521). In non-linear laws additional rules are adopted in order to assure the continuity, often introducing explicitly stress rate direction dependence, see 118-20).

Cyclic degradationof rocks and soils

935

4. APPEICATIONS 4.1 Applications to cyclic degradation of rocks (a) Perfect hysteresis. Figure 7 presents a deviatoric and three kinds of volumetric strain hysteresis for a typical low confining pressure test at different stages of monotonic strain controlled process [54, 551. Figure 8 shows schematically the evolution of volumetric histeresis during repeated cycling [lo, 571. Note that the volumetric hystereses change both their inclination and circuit depending on conditions. Microstructural models [8, 54, 631 together with phenomenological and microacoustic experiments [9, 10, 661 indicate that hysteresis in rocks is strongly conditioned by previously induced damage, whereas no further damage is observed in cycling. This evidence induces us to consider hysteresis in rock as a damage tensor dependent, in the sense of, for example, eqn (2.7). Consider first evolution of perfect hysteresis with progressive inelastic damage without cyclic effects. A form of the potential V which seems to encompass essential features of such evolution is as follows [45, 531:

v =

;Bopr {($

+ 1) [In (&

where Bo, Lo and Do are hysteretic reference pressure

+ 1) - I]

+ 1} + i&lo

+ ~D.i#,

bulk, shear and dilatancy initial moduli, respectively.

pL = +L~kk - kTi;

k = const.

(4.1) The

(4.2)

is conceived so that in an isotropic, purely reversible and elastic test, extension isotropic stresses are included above extension strength, Ti. The strain difference corresponding to (4.1) reads (4.3) The two first terms of (4.3) represent purely elastic behaviour, whereas the third describes hysteresis. Accordingly, by previous assumption the hysteresis depends on induced damage, while hysteretic strain has the direction of damage-induced strain. In what follows, a damage second-order tensor ~~~has been employed, [45], defined as: (4.4)

0

c,+ 2E3

0 “,-c

3

Fig. 7. Typical volumetric and deviatoric hysteresis for different unloading levels, after [lo]

936

T. HUECKEL

Ql (kb)

Q3

=

70b , 20

/

N 10

Fig. 8. Schematic plot of volumetric hysteresis

evolution for repeated cyclic loading.

where Mijkl is a fourth-order symmetric tensor, p* is isotropic pressure at damage completion, & is deviatoric plastic/fracture strain, whereas &‘) and E{!$*)are irreversible, pressure-induced compaction and shear-induced dilation, respectively, at p*; y and l3 are positive constants, so that

Depending on the ratio r/P, pkk may become positive or negative. It is beyond the scope of this paper to develop a yield or fracture theory which should furnish rules for changes of EC;, I&‘), &*’ together with the specific meaning of the tensor Mijkr and constants y and p. In any case: it has been found necessary (for present purposes) to distinguish between the two contrasting components of the volumetric irreversible deformation because of their supposed different influence on the evolution of form of the volumetric hysteresis. A broader discussion of their phenomenological meaning is given in Ref. [.53]. An example of the variation of ukk with (Tkkfor a hypothetical granite, low confining pressure triaxial test is presented in Fig. 9. Plastic compaction is always positive, whereas dilatancy, e.g. in uniaxial process, is zero below so-called dilatancy onset C’, whilst above C’ it is negative. Thus tq& changes its sign from positive to negative above C’. Let us concentrate now on the volumetric hysteresis. From (4.3) one may derive

(4.6) Figure 10 presents all the three contributions to *eii. The first term corresponds to typical logarithmic volumetric elastic strains, which can, moreover, be extended in isotropic extension range. The first hysteretic term is insensitive to the sign of l&k, and essentially reproduces hysteresis part observed, for example, in purely isotropic or edometric test. The second hysteretic term is stress deviator dependent. Note that if the stress difference deviator is directed roughly as the damage deviator, for example, in uniaxial reloading, then for predominant compactive damage l.& > 0 and this term is positive. For predominant dilatancy, pkk < 0 this

937

Cyclic degradation of rocks and soils

hysteresis contribution is negative. Figure 10(b) and (c) show cases of moderate and advanced irreversible dilatancy giving rise to two forms of unaxial compression hysteresis of volumetric strain. A series of volumetric hysteresis for a sandstone evolving with advancing damage is presented after Ref. [53] in Fig. 11(a) calculated for the following set of constants: B. = 7.4 * 10e4, Lo = 0.5127 * lop4 MPa-‘, Ti = -0.52 MPa, kTi = -4 MPa, p = 0.3, Do = 0.289 * 10” MPa. Note that below a transition point, T, the hysteresis is very narrow and oriented clockwise, whereas it changes its direction to anti-clockwise above T and progressively its inclination and breadth become larger. In Fig. 11(b) are shown two series of consecutive cycles of loading with decreasing stress amplitude and common onset, for two different advances of damage for a hypothetical granite with following constants Ti = -0.627 MPa, k = 2.4, Bo = 1.22 * 10W3,L = 1.52 * lop5 MPa-’ and Do = 2.89 + 109MPa, after [53]. Clearly, no strain accumulation effect is reproduced. (b) Hysteresis evolution. In order to reproduce such an effect and in order to encompass hysteresis evolution shown in Fig. 8, let us complement eqn (4.9) with an even strain difference term depending on a degradation parameter 19and an additional odd strain difference term also depending on 6. A particular form of the potential V(Aau, pij), (2.20), is adopted, which guarantees its positiveness for any ‘uii both in “unloading” and “reloading”, and is proposed as follows R O‘Uij + 9

R -5:

6Mf~ij

I(

‘uij

4

+

6M: kij

7 >

(4.7)

where Bo, Lo, Do and R. are positive hysteretic bulk, shear, dilatancy and ratcheting modulus respectively, whereas +3is a discrete function of cyclic degradation (i.e. constant over portion), assumed as positive. Two first terms in (4.7) correspond to linear elastic (non-hysteretic) behaviour. A linear volumetric relationship has been employed for simplicity. The third-term strains form a perfect hysteresis, the direction and form of which depend on the damage tensor. The last term, written in the form of a quadratic binomial, is responsible for hysteresis evolution in cyclic loading. It may be rewritten in the following form corresponding to the potential (2.20) implying restrictions on functions al, a4, bz, b3:

R; 7 6

A

uijAuij + 2Ro f M: + a2 Y2Mf.

(4.8)

It is seen that this term, while positive as a whole, contains a non-definite pa& =hh

0

“hh

Fig. 9. Variation of p.& with au.

0

phh

2Ro 6M: ’

938

T. HUECKEL

corresponding to an even strain difference contribution. This contribution represents ratcheting strain in the direction of previously induced damage. The last term in (4.8) which is proportional to the perfect hysteresis term a00M;’ represents initial hysteresis stiffness degradation leading to a form of hysteresis loop rotation. The first term is again a cumulative term, linear in stress difference. Its numerical importance is negligible for higher values of its multiplier 6, whilst its role is to ensure formally positiveness of (4.8). The strain difference obtained through (2.18) from (4.7) reads: AEij = &+J&j

+ LOASij +

(Do + s*Y*)A!f: [ + 2R,$

Ml

1

kijkkIAuk/ + R2164Aaij.(4.9)

Before discussing the above equation let us note an important feature of the potential (4.7), i.e. a possibility of ensuring its convexity. It leads to positive definiteness of its incremental stiffness tensor. This property is equivalent to a sufficient uniqueness and stability condition (in Drucker’s sense) of the local and, under additional conditions, of global behaviour. In fact

Ql

OkkShk

1

0

Fig. 10. Decomposition

of volumetric hysteretic

strains.

Cyclic degradation of rocks and soils

1

939

I

I

I

=1 .\ \

\

\ \ \

/I

/I

/~, 1’

QV

0

x 10

-3

“1 MPo

.3

2

0

l

(a)

E

V

(b)

V

QV

m Fig. 11. Calculated volumetric hysteresis: (a) for different unloading onsets, (b) for different stress amplitude.

940

T. HUECKEL

the second-order work may be represented as drijijj = 4 BlJ& + LOS,S, + 3(D0 + 6’Yz)M: (lJ&ij)*

The above expression can be rewritten in a different form similar to (4.7), so that

Clearly, the last two terms are always positive, while the two first terms furnish a positiveness condition. This condition is not restrictive, however, since l/a4 is initially small and, moreover, rapidly decreasing with the number of cycles. The cyclic degradation function 9 is a discrete function. It may be seen as what is often called a fatigue function of number of cycles. In fact, B is assumed in the following form for a trajectory segment which follows after the Lth reversal: m

(4.12) The function L6 is thus proportional to the length square of a projection of the stress path on the preferred direction induced by previous damage. The stress path which enters (4.12) does not include the current trajectory portion. It does include, however, portions connected with forgotten loci. The empirical exponent nz is supposed to be 0 < m < 1; 9* is a constant. For uniform stress amplitude cycling the function La increases with the amplitude and with the number of cycles. For sufliciently small projected amplitudes an eventual stabilization of the hysteresis may be reached. In fact, with m < 1, the function L6 increases very little for higher L, so that the rotation term of hysteresis remains practically constant, whilst the ratcheting term decreased assymptotically with L6. The contribution of the last term of (4.6) has practical importance for few first semi-cycles only, which makes their strain accumulation markedly stronger than in successive ones, as observed experimentally [lo, 571. The form of fourth term of the potential (4.7) is a particularly chosen combination of four terms of (2.20) in order to ensure positive definiteness of the incremental stress-strain compliance. Clearly, several other more or less restrictive combinations may be proposed. The above has the advantage of being relatively simple and encompassing three essential features of volumetric cyclic behaviour of rocks. In Fig. 12 an evolution of volumetric hysteresis under constant stress amplitude is shown,

Fig. 12. Calculated volumetric hysteresis

evolution.

941

Cyclic degradation of rocks and soils

I 2.4 *I-=3 1.6

Fig. 13. Volumetric and deviatoric strain at constant pressure drained tests on Ostia sand, after

[621.

calculated for the following constants: B. = 0.1 * 10W3MPa-‘, Lo = 0.5125 * 10e5 MPa-‘, DO = 2.19 * lo9 MPa, p = 0.302, a* = 2.4 - lOI* MPa; R,, = 1.8, m = 8. Note that in the above formulation deviatoric and volumetric hysteresis undergo the same evolution described by the same degradation function and common constants 6* and Ro. Clearly, it is a simplification, which nevertheless permits us to show the principal possibilities of the model, with the use of a limited number of constants. A more realistic study would require, however, a larger experimental basis. 4.2 Application to sands and clays A prototype form of the model presented above has been applied to soils. A simulation of cyclic compaction and of cyclic mobility of sands [63, 651 and of cyclic effects in clays [64] such as undrained cohesion reduction has been shown to be tractable with such a model. Figure 13 presents an experimental stress-strain curve of Ostia sand for constant cyclic deviatoric stress amplitude at constant pressure in drained conditions. A remarkable feature of the above results is a compactive strain accumulation. This feature leads to cyclic mobility of sand if drainage is prevented, i.e. when sand deformation is practically isochoric. Cyclic mobility consists in cyclic reduction due to cyclic shear of effective pressure at constant volume, so that progressively the whole of the isotropic part of loading is carried by pore water. The decrease of the effective pressure may eventually lead to failure at constant shear stress amplitude. The constitutive relationship, which has been used to describe such behavior of sand, rests on the assumption that no damage occurs during cycling and that hysteresis is purely stress (or strain) induced. Following broadly the experimentally founded soil mechanics tradition, it is assumed that the hysteresis non-linearity depends on the strain amplitude ‘x [58] and on socalled “geotechnical stress variables ‘zij” which are generalizations of classical notions defined as follows:

‘x

= (AEijA,ij)l’2.

,

Azij =

!

3

ln

(4.13)

(4.14)

942

T. HUECKEL

The stress-strain

difference

relationship *Eij

cijkl

=

+(B

compliance

-

L)&jhkl

+

tensor,

Y(aikajjl

+

and +j$ is unit stress ratio tensor at current reference dilatancy moduli taken as linearly variable with *x. B = Bo(l + 0,*x);

(4.15)

CijdAX)*Zkl + t!iE*"XSij,

=

where Cti, is the secant hysteresis

may be now written as follows [62]:

&lDjk)

+

(4.16)

@$ok$ij,

point; B, L, and D are bulk, shear and

L = Lo(l + 0,*x);

D = &(I

+ ws*x)

(4.17)

Bo, Lo, Do, o,, w,, 08 are constants. The strain accumulation is limited to volumetric component only. The function [* is assumed, on the basis of the experimental findings of Ishihara and Okada [59, 601, to be a decreasing function of overconsolidation, plastic volumetric strain and of accumulated irreversible shear strains in cycling in preceding cycles.

c* =

-

II

L-l

1[

5 exp

A& + ‘p x

(A’~ - Aixe)

i=l

(4.18)

,

where 5, cp, A are constants, while Ai~ehas the meaning of the elastic part of the strain amplitude, i.e.

AiXe=

1

$(Bo IIl (T/&/&kk+ D~A’Ijijfi$)2 + LoATlijAT)ij

.

(4.19)

The stress reversal locus has been assumed as a locus of stress states of equal strain amplitude

w= w -

wo= 'x'(*qij,

ln u,&/&kk) -

w,, =

0.

(4.20)

Although it is not straightforward, the above constitutive relationships may be shown to be equivalent to a particular form of (2.13). In fact, the strain amplitude may be expressed through *& (see e.g. [61]) so that the right-hand side of (4.15) may be written in terms of stress dif-

ference . The existence of segmental complementary potential is not assumed. The third, non-symmetric term of the compliance (4.16) represents hysteretic dilatancy. In fact there are three contributions to volumetric strain difference: purely hysteretic, compactive in loading, isotropic stress induced, described by the modulus B; purely hysteretic, dilative in loading, shear induced and contrasting with the previous one (modulus D); and an accumulative one described by the function E*, which is always compactive. All the three contributions are / 100

50 ij % 2

0

k

-50

-.I

\

\

Fig. 14. Constant strain amplitude cyclic undrained stress path, after [63].

Cyclic degradation of

rocksand soils

943

visible in the experiments presented in Fig. 13. Often, however, it is sufficient to take into account some of the contributions only, as, for example, in simulation of undrained constant strain amplitude experiments (Ishihara et al. [59], Fig. 14) where DO has been set at zero. Numerical simulation of a variety of loading programmes for sand and for clays has been discussed in refs. [61, 63, 641. CONCLUSIONS Let us discuss now analogies to and differences from both elasticity and kinematic hardening plasticity of the present constitutive law. Note first that a linear elastic law in the presence of residual strain and stress of any origin should be written in terms of differences with respect to the residual state rather than in terms of strain and stress themselves. This holds true for the elastic potentials. A family of equipotentid loci growing isotropically around such residual stress may be made explicit. Its path independence is valid for the portion of the trajectory following the residual state. However, any re-entrance of the trajectory inside a current locus does not affect the validity of the acting law. Clearly, a further process may be alternatively described either with respect to the original residual state or with respect to the point of the re-entrance. The same is valid for the potential. Thus, although it is not explicit, an elastic process trajectory may be seen as a sequence of its portions accompanied by a discrete shifting of respective origin of difference equipotential growing isotropically around it. The behaviour is then both partly and totally path independent. Therefore, the present approach is a generalization of this property, by limiting path independence to single portions and by adding discrete variation of history-dependent variables at reversal points. Kinematic hardening in most theories of plasticity is combined with isotropic hardening. The yield locus undergoes thus an isotropic growth and a superposed rigid body shift which is simultaneous and thus continuous. Here, the rigid body shift is discretized motion of the locus origin or the so-called back-stress function (see also Norris 161). It is also interesting to compare the present model with the variable incremental stiffness approach often adopted in elastoplasticity. It is most visible in numerical methods. The process of loading is considered as consisting of a series of incremental (finite although small in computation) steps of loading. A response to such a step is calculated via an incremental stiffness which is incrementally path independent. The material properties at step onsets are updated after every increment. Therefore theoretically on an incremental level, but on the level of finite increments in numerical practice, an elastoplastic process may be seen as a sequence of historyindependent process segments and history-dependent extremities of segments where some variables undergo discrete changes. Thus, the effective difference between the variable tangential stiffness analysis of elastoplasticity and the present theory consists in the length of the historyindependent steps. Here, the steps are finite and the number of updating points is limited. In incremental plasticity the steps are infinitesimal and the number of updating points is infinite. In the numerical method, again, the steps are finite and the updating points are limited. An analogous comparison may be made to the loading functions. Note that the probing stress vector in numerical applications is also finite, as it may be in the present theory, Therefore, an inaccuracy in defining reversal is strictly connected with the length of this vector to the same extent as in the present approach. It is believed that in complex applications this model may appear more operative than incremental theories. For simple repeated cycling it often gives closed form solutions for stress path and/or stress-strain difference curves. Moreover, it is possible to adapt the model accuracy to the considered case by neglecting, for example, cumulative terms or employing finite instead of incremental loading function. REFERENCES [l] Z. Mr&, On the description of anisotropic hardening. Znt. J. Me&. Phys. Solids 15, 163-175 (1%7). [2] Y. F. Dafalias and E. P. Popov, A model of non-linear hardening materials for complex loadings. Acta Me&. 173-192 (1975).

21,

944

T. HUECKEL

[3] Z. Mr6z, V. A. Norris and 0. C. Zienkiewicz, Application of an anisotropic hardening model in the analysis of elasto-plastic deformation of soils. Geotechnique 29(l), l-34 (1979). [4] Y. F. Dafalias and L. R. Herrrnann, Bounding surface formulation of soil plasticity. In Soil Mechanics Transienr and Cyclic Loads (Edited by G. N. Pande and 0. C. Zienkiewicz), pp. 253-282. John Wiley, New York (1982). [51 Z. Mroz, On hypoelasticity and plasticity approaches to constitutive modelling of inelastic behaviour of soils. Int. J. Numer. Anal. Meths. Geomech. 4, 45-56 (1980). [6] V. A. Norris, Numerical modelling of soil response to cyclic loading using “stress reversal surfaces”. In Numerical Modelling in Geomechanics (Edited by R. Dungar. G. N. Pande and J. A. Studer). pp. 38-49 Balkema. Rotterdam

(1982). 171 J. B. Walsh, The effect of cracks on the compressibility of rock, J. geophys. Res. 70(2), 381-389 (1965). [8] D. J. Holcomb, A quantitative model of dilatancy in dry rock and its application to Westerly Granite. J. geophys. Res. 83b(lO), 4941-4950 (1978). 191 K. Hadley, The effect of cyclic stress on dilatancy: another look. J. geophys. Res. 81(14). 2471-2474 (1976). [lo] B. C. Haimson, Mechanical behaviour of rock under cyclic loading. In Advances in Rock Mechanics. Proc. 3rd Congr. Znt. Sot. Rock Mechs.. Denver, 1974, Vol. II. Part A, pp. 373-378. Nat. Acad. Sci., Washington. D.C. (1974). [ll] N. G. Gerogiannopoulos and E. T. Brown, The critical state concept applied to rock, fnt. J. Rock Mech. Min. Sci. 15, l-10 (1978). [12] A. Dragon and Z. Mroz, A continuum model for plastic-brittle behaviour of rock and concrete, Znt. J. Engng Sci. 17, 121-137 (1979). [13] K. Hashiguchi, Constitutive equations of elastoplastic materials with elasto-plastic transition. /. uppl. Mech. 102(2), 266-272 (1980). [I41 S. E. Yamada, J. F. Shatz, A. Abou-Sayed and A. H. Jones, Elastoplastic behaviour of porous rocks under undrained conditions, Znt. J. Rock Mech. Min. Sci. 18, 177-179 (1981). [I51 Z. P. BaZant and P. Bhat, Endochronic theory of inelasticity and failure of concrete. Proc. ASCE, EM3. 701722 (1976). [I61 D. Krajcinovic and G. U. Fonseka. The continuous damage theory of brittle materials. 1, II. J. appl. Mech. 48, 809-824 (1981). 1171 R. Nova and D. M. Wood, A constitutive model for sand in triaxial compression. Znt. J. Numer. Meths. Geomrch. 3(3), 255-278 (1979). [18] F. Darve, Une formulation

incrementale des lois rheologiques. Application aux sols. Doctoral thesis, Grenoble (1978). 1191 R. Chambon, Contribution a la modelisation numerique non-lineaire des soIs. Doctoral thesis, Grenoble (1981). 1201 G. Gudehus and D. Kolymbas, A constitutive law of the rate type for soils. Proc. 3rd In?. Conf Num. Meth. Geomech. (Edited bv W. Wittke). Vol. 1. DD. 319-330. A. A. Balkema. Rotterdam (1979). L211T. Hueckel and R. Nova. A mathematical mbhel of hysteresis in soils. In Materiuux er Struckres sous Churgements Cycliques (Edited by J. Zarka), pp. 107-110. Ass. Amic. E. N. Ponts chauss. Paliseau (1978). 1221 T. Hueckel and R. Nova, On paraelastic hysteresis of soils and rocks. Bull. Act. pa/on. Sci.. Ser. Sc,i. Tech. 27(l), 49-55 (1979). 1231 T. Hueckel and R. Nova, Some hysteresis effects of the behaviour of geologic media. Znt. J. Solids Struct. lS(8). 635-642 (1979). 1241 Z. Mrbz and N. L. Lind, Simplified theories of cyclic plasticity. Acta Mech. 22, 131-152 (1975). [25] J. M. Boisstrie and P. Guelin, Remarks on the tensorial formulation of constitutive laws describing mechanical hysteresis. Proc. SMIRT-4, San Francisco, 1977, paper L l/9, pp. I-15 (1977). [26] Z. BaZant and S. S. Kim, Plastic fracturing theory for concrete. Proc. ASCE EM 105, 429-446 (1979). [27] D. Aubry and J. C. Hujeux. A double memory model with multiple mechanism for cyclic soild behaviour. Numerical Models in Geomech. Proc. Znr. Conf., Zurich (Edited by R. Dungar, G. N. Pande and J. A. Studer). pp. 7-17. Balkema. Rotterdam (1982). [28] G. Maier and T. Hueckel, Non-associated and coupled flow rules of elastoplasticity for rock-like materials. Zm. J. Rock Mech. Min. Sci. 16, 77-92 (1979). [29] A. Dragon, On phenomenological description of rock-like materials with account for kinematics of brittle fracture, Archs Mech. 28(l), 13-30 (1976). [30] P. Wilde, Two invariants-dependent models of granular media. Archs Mech. 29(6), 799-809 (1977). . 1311 _ . J. R. F. Arthur. S. Bekenstein, J. I. Germaine and C. C. Ladd. Stress path tests with controlled rotation of principal stress direction. ASZSM Symp. Shearing Strength of Soil, Chicago (1980). [32] Y. Yamada and K. Ishihara, Yielding of loose sand in three-dimensional stress conditions. Soils Foundat. 3, 1531 (1982). [33] R. I. Mitchell, Some deviations form isotropy in a lightly overconsolidated clay. Geotechnique 22(3), 459-467 (1972). [34] J. P. Boehler, F. Lefevre and R. Bonaz. Relations quantitavies entre l’anisotropie de structure et I’anisotropie

mecanique dans une argile, evolutions au tours d’une deformation irreversible. Priv. comm. (1983). [35] T. Hueckel, On plastic flow of granular and rocklike materials with variable elasticity moduli. Bull. Pol. Acud. Sci.. Ser. Techn. 23, 405-414 (1975). [36] T. Hueckel, Coupling of elastic and plastic deformation

[37] Y. F. Dafalias, Elasto-plastic

of bulk solids. Meccanica 11, 227-235 (1976). coupling within a thermodynamic strain space formulation of plasticity. In?. J. Nun-

linear Mech. 12, 327-337 (1977). [38] J. P. Boehler, Lois de comportement

anisotropes des milieux continus. d. Met. 17(2). 153-190 (1978). [39] G. Masing. Wiss. Veroff. Siemens Konzern, III Band (1927). [40] B. 0. Hardin and V. P. Drnevich, Shear modulus and damping in soils: design equations and curves. Proc. ASCE 98/667-692, SM7 (1972). [41] R. J. Evans and K. S. Pister, Constitutive equations for a class of non-linear elastic solids. Znf. J. Solid Sfruct. 2, 427-445 (1966). [42] T. Hueckel, On discrete stress rate sensitivity in piece-wise modelling of cyclic loading of solids. J. Met. 20,365378 (1981).

Cyclic deg~dation

of rocks and soils

945

[43] C. Truesdell. The mechanical foundations of elasticitv and fluid dvnamic. J. ration Me&. Anal. 1. 125 (1952). ’ WI R. H. G. Parry, Strength and deformation of clay. Ph.D. thesis, Keys College, London (1956). [451 ‘I’. Hueckel, A model for dilatant rock behaviour under variable loading. In Numerical Models in Geomechanics (Edited by R. Dungar, G. N. Pande and J. A. Studer), pp. 193-2904. Balkema, Rotterdam (1982). T. Hueckel and R. Nova, Hysteresis behaviour of soils and rocks. Tran. SMIRT-S, Berlin vol. K(A), K l/7 (1979). t:;; G. H. Handelman, C. C. Lin and W. Prager, On the mechanical behaviour of metals in the strain hardening range. Q. Appl. Math. 397-407(1947). 1481 G. Gudehus, A comparison of some constitutive laws for soils under radially geometric loading and unloading. In Pro&. 3rd In?. Conf. Num. h4eths Geomech (Edited by W. Wittke). 319-330 Balkema, Rotterdam (1979). 1491 I. Nelson, Constitutive models for use in numerical computation. In Proc. Conf Dynnmic Meth. Soil Rock Mech. (Edited by G. Gudehus), Vol. 2, pp. 45-47. Balkema, Rotterdam (1978). WI P. I. Lewin and J. B. Burland, Stress probe experiments on saturated normally consolidated clay. Geotechnique 20(I), 38-56 (1970).

PII P. 1. Lewin, The influence of stress history on the plastic potential. Proc. Symp. The Role of Plasticity in Soil Mechanics, pp. 96-109. Univ. Press Cambridge (1973). WI D. RadenkoviC, Constitutive laws for granular media. In Problems of Plasticity (Edited by A. Sawczuk), pp. 333348. Noordhoff Int. Leyden (1972). L531T. Hueckel, On inviscid, strain recovery in rocks. Rep. No. 505, Civil Eng., University of Rome, 1982. In?. J. Rock Mech. Min. Sci. (submitted). [541 L. S. Costin and D. J. Holcomb, A continuous model of inelastically deformed brittle rock based on the mechanism of microcracks. Proc. Int. Conf. Tucson, pp. 185-195 (1983), preprint. I551 M. D. Zoback and J. D. Byerlee, The effect of cyclic differential stress on dilatancy in Westerly Granite under uniaxial and triaxial conditions. J. geophys. Res. 80(11) 1526-1530 (1975). WI V. Rajaram, Meehanic~ behaviour of granite under cyclic compression. In Int. Co& Rec. Adv. Geomech. ~arthq. Engng and Soil Dyn., Vol. 1, pp. 11-116. St Louis, MO (1981). C. Saint Leu and P. Sirieys, La fatigue des roches, Proc. Int. Symp. Rock. Fracture, Nancy, pp. 11-18 (1971). t:‘9; H. B. Seed, Soil liquefaction and cyclic mobility evaluation for level ground during earthquake. Proc. ASCE GT2, 201-255 (1979). [591 K. Ishihara and S. Okada, Effects of stress history on cyclic behaviour of sands. Soils Foundut. U(4), 31-45 (1978).

WI K. Ishihara and S. Okada, Yielding of overconsolidated sand and liquefaction model under cyclic stresses. Soils Foundat. 18(l), 51-66 (1978). Ml T. Hueckel, A model of soil behaviour in plastic and hysteretic ranges, Part II: cyclic loading. Proc. Inr. Workshop, Villard des Lans, 6-9 September (Edited by F. Darve, G. Gudehus, I. Vardoulakis), Balkema, Rotterdam (1984). WI L. Cavalera and T. Hueckel, Cyclic compaction and distortion of a saturated sand sample in drained conditions. In Proc. 15th itu~iun Nat. Co& Ceotechnics, Vol. I, pp. 93-103 (1983). WI R. Nova and T. Hueckel, On unified approach to the m~elling of liquifaction and cyclic modelling of sands. Soils Foundat. 21(4), 13-28 (1981). T. Hueckel and R. Nova, On cyclic degradation of clays, Riv. ital. Geotec. 1514,223-234 (1981). ;zz; D. J. Holcomb, Memory, relaxation and microfracturing in dilatant rock. J. geophys Res. 86(B7), 6235-6248 (1981). [Ml C. H. Sondet-feld and L. H. Estey, Acoustic emission study of microfracturing during the cyclic loading of Westerly Granite. J. geophys. Res. 86(B4), 2915-2924 (1981). [671 R. Nova and T. Hueckel, A geotechnical stress variable approach to cyclic behaviour of soils. In Int. Symp. Soils under Cyclic Transient Loading (Edited by G. N. Pande and 0. C. Zienkiewicz) Swansea, Vol. 1, pp. 301-314 (1980). I681 C. H. Scholz and S. H. Hickman, Hysteresis in the closure of a nominally flat crack. J. geophvs. Res. 88(B8), 6501-6504 (1983).