Jun 16, 1999 - 3 Invertibility, consistency and limit states. 14 .... classes of constitutive equations developed in Karlsruhe by Kolymbas and coworkers 56, 54 ...
A Review of Two Different Approaches to Hypoplasticity Claudio Tamagnini
Universita di Perugia, Italy
Gioacchino Viggiani, Rene Chambon Laboratoire 3S, Grenoble, France
June 16, 1999
Contents
1 Introduction 2 Mathematical structure
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 K{hypoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 CLoE hypoplasticity . . . . . . . . . . . . . . . . . . . . . . .
2 4
4 7 9
3 Invertibility, consistency and limit states
14
4 Strain localization and bifurcation analysis
23
5 Conclusions A Gudehus/Bauer K{hypoplastic model B von Wol ersdor K{hypoplastic model
34 43 43
3.1 General results . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Amorphous hypoplastic models . . . . . . . . . . . . . . . . . 16 3.3 Endomorphous hypoplastic models . . . . . . . . . . . . . . . 20 4.1 Introduction and historical background . . . . . . . . . . . . . 23 4.2 General results . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.3 The question of the shear modulus . . . . . . . . . . . . . . . 29
1
1 Introduction Non{linearity and irreversibility are striking features of soil behavior, affecting the response of any geotechnical \structure", be it, for example, a foundation, an excavation, an earth dam, or a natural slope. From a mathematical viewpoint (i.e., at the constitutive level), di erent strategies have been proposed to deal with such features of soil behavior, including: classical perfect or hardening plasticity with strain{dependent elastic moduli (non{linear elastic, or \variable moduli" models, see e.g. [47, 65]; generalized, or bounding surface plasticity, e.g. [24, 64]; incrementally non{linear theories, including incrementally non{linear elastoplasticity [24, 85], endochronic plasticity, e.g. [79], and incrementally non{linear rate{type models [11, 52, 26]). Although very popular in the geotechnical community (especially in the U.K.), variable moduli models su er from a number of important shortcomings, which have been thoroughly discussed in the literature (e.g.[60, 66]). In particular, it has been shown by Nelson already back in 1977 [60], that serious numerical instabilities may result from their use in FE computations when the direction of the stress increment approaches the boundary between the loading and unloading tensorial zones. As demonstrated by Gudehus [34], this is due to the lack of continuity of the relevant stress response envelope. Moreover, the use of pseudo{Hookean tangent sti ness tensors in the hypoelastic equations leads to coaxiality between stress and strain rates and to a complete volumetric{deviatoric uncoupling. Both features appear to be questionable on the basis of currently available experimental data. The origins of bounding surface plasticity as applied to soils can be traced back to the late `70s. Their development was mainly motivated by the need to accurately describe the irreversible and hysteretic soil behavior observed along cyclic paths. Di erently from classical elastoplasticity (with or without elastic non{linearity), in bounding surface plasticity, the constitutive equation is always characterized, for any particular loading mechanism, by more than one possible responses | plastic loading and elastic unloading | i.e., plastic ow can occur at any point in the domain of admissible stress states. This makes the material behavior not only non{linear (for nite loading paths), but also incrementally multi{linear. The particular way in which the di erent loading-unloading conditions are de ned guarantees the continuity of the incremental response at the 2
boundary between the tensorial zones (neutral loading conditions). Moreover, the derivation of plastic strain rates from a suitable ow rule allows for deviatoric{volumetric coupling as well as non{coaxiality of stress and strain rates. In spite of its merits, bounding surface plasticity has not found widespread application in geotechnical practice, and remains mainly con ned to research applications, mostly due to the relative complexity of its mathematical structure and the diculties associated to the calibration of the relevant parameters. In recent years, a completely di erent approach to soil non{linearity has been proposed, leading to a class of constitutive models generally referred to as incrementally non{linear models [25]. The distinctive features of such theories are: i) the absence of any kinematic decomposition of strain rates into reversible and irreversible parts; ii) the continuously non{linear dependence of the tangent sti ness tensor on strain rate direction. That soils are thoroughly non{linear, rather than bilinear or multilinear (multi{mechanisms plasticity), is in fact suggested by qualitative considerations on the mechanisms responsible for inelastic deformations at the microstructural level [75], as well as by the limitations shown by classical elastoplastic models with a single smooth yield surface in predicting the onset of shear banding, see, e.g. [70, 67, 49, 81], and by limited experimental evidence (Royis & Doanh 1998). Note that feature (ii), but not feature (i), is also present in some bounding surface models [24, 85], and in the endochronic models developed by Bazant [6] and by Valanis and coworkers (e.g., [79]). Although originally used for the rst time by Dafalias in the context of incrementally non{linear hardening plasticity [24], the term hypoplasticity has subsequently been adopted by Kolymbas & Wu [88] to refer to a class of incrementally non{linear, rate{type models developed as an extension of the classical theory of hypoelasticity [78]. A general outline of the theory was laid down by Kolymbas in 1991 [52], and several review papers followed thereafter (e.g., [56, 54, 57]). The objective of this paper is neither to give an outline of hypoplasticity (which can be found in the aforemetioned papers) nor to present one of the various approaches in detail. Rather, attention is focused on two particular classes of constitutive equations developed in Karlsruhe by Kolymbas and coworkers [56, 54, 57] | referred to in the following as K{hypoplastic models 3
| and those developed in Grenoble by Chambon and coworkers [11, 12, 16] under the general name of CLoE models. The problem of whether the latter can be termed \hypoplastic" | as they in fact should, according to the de nition given in [88] | is virtually irrelevant. What is more interesting, in the authors' opinion, is that although these two approaches share a number of similarities, the motivations for their | independent | development were di erent in various respects. As a result, major di erences are apparent in their original formulation, as well as in their respective subsequent developments. This paper is an attempt of critically reviewing the two classes of models with respect to a number of features which are of importance in the light of their application to both the fundamental understanding of granular soil behavior, and the analysis of practical problems in geotechnical engineering. It is the authors' hope that through such a comparative analysis, a better understanding can be obtained of both possibilities and limitations of the theory of hypoplasticity. In following, the usual sign convention of solid mechanics is adopted throughout. In line with the principle of e ective stress as stated by Terzaghi [74], all stresses are e ective stresses, unless otherwise stated. Both direct and indicial notations will be used to represent vector and tensor quantities according to convenience. In the description of stress states, use will sometimes be made of the three invariant quantities p (mean stress), q (deviator stress) and (Lode angle), de ned as follows: r p 1 p := ; 3 tr() q := 32 tr(s ) cos(3) := ; 6 tr(s ) = (1) [tr(s )] where s := + p1 is the deviatoric part of the stress tensor, 1 is the unit tensor ([1]ij = ij ), and tr(x) := x 1 for any 2nd order tensor x. 3
2
2
3 2
2 Mathematical structure
2.1 Preliminaries
A rigorous mathematical de nition of a hypoplastic material has been laid out by Wu & Kolymbas [88] and Kolymbas [52]. Herein, the attention will be focused on a particular subclass of hypoplastic constitutive equations, characterized by the following fundamental properties: 1. Let x = (X ; t), (X ; t) : B [0; T ] 7! St , denote the motion of a given body from its reference con guration B to the current con guration 4
St at time t, and F (X ; t) := @ =@ X the corresponding deformation gradient. Denoting with F Xt (s), s 0, the history of the deformation gradient up to time t at material point X , the response functional G ( )
entering in the general form of the constitutive equation for a simple material [78]: 1
(t) = sG F =0
satis es the identity:
1
(t) = sG F Xt (s) ( )
=0
t X (s ) ( )
1
= G
s=0
(2)
F Xt [(s)] ( )
(3)
for every monotonically increasing function (s) such that (0) = 0 and lim (s) = 1. s!1
2. There exist suitable tensor{valued functions h : A : S Q 7! L and b : S Q 7! S such that: 2
4
2
2
S2
S Q 7! S , 2
2
r = h (; d; q) = A (; q) d + b (; q) kdk
(4) where S is the space of symmetric second{order tensors; L is the space of fourth{order tensors; Q is the space of the additional state variables q which de ne the previous history of the material, and whose evolution is governed by suitable evolution equations in rate form; 2
4
r := _ + ! ; !
(5)
is the Jaumann{Zaremba stress rate; _ ; l := FF d := 21 ;l + lT ! := 12 ;l ; lT (6) are the spatial velocity gradient,pthe rate of deformation and the spin tensor, respectively; and kdk := d d is the Euclidean norm of d. REMARK 1. Property (1) states that the behavior of the material is not in uenced by any change in the time scale, i.e., the material is rate{independent. As shown by Truesdell & Noll [78], this property implies that the function h is positively homogeneous of degree 1 in d: h (; d; q) = h (; d; q) 8 > 0 (7) This hypothesis rules out some recent versions of argotropic hypoplastic models, which introduce in the basic formulation (4) some form of rate{ dependence, see, e.g., [52, 2, 38, 61]. 1
5
REMARK 2. According to the principle of material frame indi erence, the function h must be an isotropic tensor{valued function of all its arguments, see [78]. For example, if q 2 R 3 as in [87], this means that for any orthogonal tensor Q:
h ;QQT ; QdQT ; Qq = Qh (; d; q) QT
(8)
REMARK 3. A fundamental di erence between hypoplasticity and hypoelasticity [78] is represented by the term b kdk on the RHS of eq. (4), which renders the function h non{linear in d. This feature is necessary in order to correctly describe irreversible behavior upon stress reversal along a given loading path. However, as pointed out by Chambon [12], a possible limitation | to be evaluated through experimental veri cation | is that, due to the particular structure of the rate equation (4), for any given state and strain rate d:
r := h (d) ; h (;d) = 2b kdk
(9)
i.e., the di erence between the stress rates upon reversal of the rate of deformation is independent on the direction of d. REMARK 4. When Q = ;, i.e., when Cauchy stress is the only state variable, eq. (4) de nes an amorphous hypoplastic material according to the terminology laid out by Kolymbas [52]. A well{known limitation of amorphous hypoplasticity is related to cyclic loading, where strong ratcheting e ects are usually predicted for non{symmetric cyclic paths, see, e.g., [12, 3, 16, 62]. The opposite case of endomorphous hypoplastic materials is obtained when other additional state variables are present in the formulation. In particular, when one of the additional state variables is the porosity (or void ratio) eq. (4) de nes a pycnotropic hypoplastic model [52]. REMARK 5. The experimental observation that a proportional strain path starting from a nearly stress{free and undistorted state yields a proportional stress path [32, 40, 33] poses some additional restrictions on the function h. In the particular case of amorphous hypoplastic materials, it can be shown that a sucient condition to satisfy this requirement is that h be positively homogeneous with respect to :
h (; d; q) = mh (; d; q)
8 > 0
(10)
where m denotes the degree of homogeneity. This property, de ned barotropy in [52], implies that the behavior of the material can be normalized with respect to pm . 6
2.2 K{hypoplasticity
The development of K{hypoplasticity can be traced back to the pioneering work of Kolymbas on incrementally non{linear constitutive equations of the rate{type [48, 41, 50]. Although none of the aforementioned theories can be classi ed as K{hypoplastic, according to the de nitions laid out in the previous section, the approach followed in these works for the de nition of the constitutive function h is the same which lead to the subsequent development of K{hypoplasticity. The starting point in Kolymbas' approach is given by the condition of isotropy (8) for the constitutive functions to be de ned. As | in the particular case of amorphous materials | a representation theorem due to Wang [84] provides the most general form of isotropic tensor{valued function of two tensorial arguments, Kolymbas suggested to derive particular expressions for the function h by appropriately selecting some terms in Wang's general expression by trial and error (see, e.g., [53] for details). Employing this procedure, several particular amorphous K{hypoplastic models have been proposed and subsequently re ned, see e.g., [88, 86, 92]. As an example, the constitutive model developed by Wu & Bauer [92] is characterized by the following functional form for the tensors A and b:
A := tr()L L := C I + C ^ ^ 1
b := tr()N N := C ^ + C ^
2
3
2
4
2
(11) (12)
in which: I is the (symmetric) fourth{order identity tensor (2Iijkl = ik jl + il jk ); ^ := = tr() is a normalized stress tensor; ^ := ^ ; (1=3)1 its deviatoric component, and Ck , k = 1; : : : ; 4, are material constants. The model is characterized by a barotropy of degree one, and its calibration requires the determination of only 4 material constants, which can be obtained from standard laboratory tests. Subsequent developments of K{hypoplasticity have been aimed at improving the original formulation in the following aspects: i) Description of barotropy and pycnotropy e ects, in order to model the dependence of material response on density and e ective stress level, and to incorporate the concept of critical state [91, 1, 38, 95, 83]; ii) De nition of suitable calibration procedures, associated to a more straightforward physical interpretation of the various constants entering the model, linked to some fundamental properties of the solid skeleton as grain properties and grain size distribution [1, 45]; 7
iii) Extension of the space of state variables to some suitable structure tensor in order to: a) allow the modeling of cohesive powders [4]; b) improve the performance of standard formulations for cyclic loading [3, 62]; or, c) introduce inherent anisotropy [87]; iv) Extension of the range of application of the theory to multiphase granular materials [36, 39], and micropolar media [72, 73]. The aspects covered in the developments referenced in points (ii) to (iv) are beyond the scope of the present work, and thus will not be discussed further. However, some theoretical formulations referenced in point (i) deserve some more detailed comment, as they represent a natural evolution of the original amorphous hypoplastic models. In particular, two recent, advanced K{hypoplastic models proposed by Gudehus and Bauer [38, 1] and von Wol ersdor [83] are worth mentioning, as they can be considered the syntesis of the research work carried out in Karlsruhe on this subject over the last 25 years. For both models, the general constitutive equation (4) can be written as follows:
r = fbfeL (^) d + fbfefdN (^) kdk
(13)
where:
L := a I + ^ ^
N := a (^ + ^ )
2 1
1
(14)
for the Gudehus/Bauer model, and: L := ^ 1 ^ ;F I + a ^ ^ N := ^aF ^ (^ + ^ ) (15) for the von Wol ersdor model. The functions a , F , a, fe, fd and fb entering in eqs. (14){(15) are given for completeness in appendices A and B. As it is apparent from the comparison between eqs. (11) and (13), the fundamental progress from the older generation of amorphous K{hypoplastic models is represented by the introduction of barotropy and pycnotropy effects via the scalar factors fb, fe and fd, which allow a uni ed description of the behavior of granular materials for a wide range of pressures and densities. For any given rate of deformation, the functions fb and fe control the \directional sti ness" khk = kdk of the material. In particular, the barotropy factor fb accounts for the e ect of mean stress while the rst pycnotropy factor fe accounts for the e ect of densi cation (pycnotropy). The second pycnotropy factor fd , which operates only on the tensor N , controls the balance between the linear and the non{linear parts of the function h as the 2
2
1
8
density of the material is changed. A detailed explanation of the role played by the three factors is given in [1, 38]. An important consequence of the particular structure of eq. (13) is the possibility of an independent calibration of the di erent material constants entering in the constitutive functions (separability, see [38, 1]).
2.3 CLoE hypoplasticity
The origins of CLoE hypoplasticity | where the acronym CLoE stands for Consistance et Localisation Explicite | can be traced back to the pioneering work of Chambon and Desrues on strain localization in incrementally non{ linear materials [27, 14, 28]. In analyzing the problem of shear band analysis for incrementally non{ linear materials, Chambon and Desrues observed that a completely general non{linear bifurcation analysis could be carried out for the following \heuristic" model: r
s= d ; s kdk
(16)
in which and are material constants. Based on this observation, the basic structure of the heuristic model has been subsequently assumed as the starting point for the development of a class of incrementally non{linear constitutive equations sharing with eq. (16) a formally identical structure | i.e., eq. (4) | which are both capable of representing the actual behavior of real geomaterials, and allow a complete mathematical analysis of the non{ linear shear band localization problem. This particular feature of CLoE, as well as K{hypoplastic models, is discussed in detail in sect. 4. In the development of CLoE models, the following basic assumption have been introduced to derive speci c functional forms for the tensors A and b in eq. (4): 1. To keep the formulation as simple as possible, the set of state variables is limited to the Cauchy stress tensor. Therefore the general eq. (4) reduces to:
r = A () d + b () kdk
(17)
As already discussed in sect. 2.1, this prevents an appropriate description of cyclic behavior along some particular stress paths; 2. According to the experimentally observed behavior of granular material and the previous assumption (1), the domain of admissible states B = 9
[ @ B is bounded by a surface @ B , formally de ned through a isotropic tensor function () as: B
@ B :=
2S
2
() = 0
(18)
and referred to as limit surface. 3. The constitutive equation resulting from the de nition of the tensor functions A and b in eq. (17) is invertible in all points of B except on the limit surface, i.e.:
8 (; r ) 2 B S 9 d 2 S 2
2
r = A () d + b () kdk
REMARK 6. Due to assumption (1), the tensors A and b admit the following representation as a 6 6 matrix and a six{components column vector in the reference frame of the principal stress directions [9]: 2
A 6A 6 A = 66A 6 6 4
1111 2211 3311
0 0 0
A A A
1122 2222 3322
0 0 0
A A A
1133 2233 3333
0 0 0
A
0 0 0
1212
0 0
A
0 0 0 0
2323
0
A
0 0 0 0 0 3131
3 7 7 7 7 7 7 5
2
3
b 6b 7 6 7 b = 666b0 777 6 7 405 0 11 22 33
(19)
Note that this results holds also for K{hypoplastic models for which the stress tensor is the only tensor{valued state variable. However, while all K{ hypoplastic models previously discussed are de ned in such a way that A presents minor as well as major symmetries, this is in general not true for CLoE models, for which Aijkl 6= Aklij . REMARK 7. Assumptions (1) and (2) imply that for any stress state 2 @ B the stress rate predicted by eq. (4) is directed inside the limit surface, regardless of the direction of the rate of deformation, i.e.: @ r 0 (20) @ The above condition has a direct geometrical interpretation in terms of Gudehus' response envelopes [34]: the response envelope corresponding to a stress state on the limit surface must be tangent to () = 0 [11, 12, 16]. REMARK 8. As rst demonstrated in [11, 12], assumption (3) holds if and only if: 10
i) the tensor A is invertible, thus the constitutive equation (17) can be rewritten as:
r = A [d + B kdk]
ii) the tensor B is such that:
where:
kB k ; 1 < 0
B := A; b 1
(21) (22)
An alternative proof is given by Chambon in a contribution to this conference [13]. As the condition: ^() = kB k ; 1 = 0 (23) de nes a surface | referred to in the following as invertibility surface | which must include all admissible stress states in order for the constitutive equation to be invertible at all points, the assumption (3) implies that limit surface and invertibility surface must coincide: () ^() ) () = kB k ; 1 = 0 (24) Eq. (24) and (20) represent a fundamental constraint in the development of functional forms for the tensors A and b, known as consistency condition [11, 12, 16]. This point is detailed further in the following sect. 3. REMARK 9. A second important consequence of assumption (3) is more directly related to experimentally observed behavior of granular material. According to the de nition given by Chambon [12, 13], as loss of invertibility can be associated to the occurrence of material softening, all CLoE models are formulated in order to describe a non{softening behavior under homogeneous deformation, i.e. for such loading conditions as, e.g., drained TX compression and extension, the predicted stress{strain response is monotonic, and the maximum value of deviatoric stress is obtained asymptotically for large (theoretically in nite) deformations. The basic idea underlying this | admittedly strong | hypothesis is that the commonly observed response on such materials as, e.g., dense sands | showing a well de ned peak stress at a relatively small strain, and a subsequent reduction of deviatoric stress as axial deformation is increased | is a consequence of geometric, rather than material softening, due to strain localization into shear bands, or to other diffuse heterogeneous deformation modes (i.e., barrelling or bulging). Although true material softening can actually occur in geomaterials due to some kind of damaging processes at the microscopic level, some experimental observations | see e.g., [30] | suggest that this can be considered as a convenient 11
working hypothesis for such materials as uncemented sands. In this respect, it is also worth noting that this assumption is consistent with the main motivation for the development of CLoE hypoplasticity, i.e., the possibility of a complete non{linear analysis of shear band bifurcation processes. Di erently from K{hypoplastic models, the strategy adopted to develop speci c functional forms for the constitutive tensors A and b in the rst generation of CLoE models, i.e. CLoE v1.00{v1.02 and subsequent minor modi cations [43, 29, 16], is based on the following interpolation procedure. A number of special loading paths, known as basic paths, are de ned along which the material response to particular loading conditions is described via suitable response functions interpolating experimentally observed data (e.g., a ; r = f (a) for TX compression). For each stress state, a set of image points are then de ned along the basic paths, for which A and b can be easily determined by di erentiating the response functions. The actual values of A and b are then evaluated by interpolating the corresponding tensors at the image points according to the actual values of the Lode angle and the normalized deviatoric stress q=ql , where ql is the deviator stress on the limit surface corresponding to current values of p and . In performing the interpolation, the consistency condition at the limit surface, eq. (20), is enforced through a suitable rotation of tensor A. To ease the calibration procedure and to link material constants to commonly observed features of soil behavior, the basic paths are selected among those which are experimentally accessible by means of standard laboratory equipments | i.e., triaxial compression and extension; isotropic compression starting from isotropic and anisotropic stress states. These conventional laboratory tests can provide informations on all the unknown components of the constitutive tensors except for the \shear moduli" A , A and A . The question of shear moduli identi cation is discussed in more detail in the following sect. 4.3, in connection with shear band analysis. This approach presents the advantage of directly linking the constitutive functions to the observed material behavior along an as large as possible set of loading paths, and should be contrasted with common practice in the development of elastoplastic constitutive models, where the mathematical formulation of the constitutive equations is generally based on experimental investigation on a much more limited number of stress{paths. However, this advantage is paid in terms of a relatively large number of parameters required to describe the response functions, and of the complexity of the interpolation procedure, which does not allow to de ne an explicit form for the functions A() and b(). 1212
1
2323
3131
1
Here, A components are de ned in the principal stress space, see remark 6.
12
While recent developments of K{hypoplasticity aimed at extending the eld of application of older versions of the theory by introducing the modi cations already discussed in the previous section, the development of CLoE hypoplasticity followed a completely di erent path. In fact, starting from the basic requirements (1){(3), new versions of the theory have been developed by Chambon and his coworkers [17, 23, 19, 20] which represent drastic simpli cations of the original CLoE v1.00, and which, for this reason, will be referred to in the following as MiniCLoE models. In MiniCLoE models the concept of interpolation of the constitutive tensors from the image points along the basic paths is abandoned, and | in analogy to K{hypoplasticity | the tensors A and B in eq. (21) are de ned explicitly, taking into account the fundamental constraints of invertibility and consistency to the limit surface. An example of MiniCLoE model is given by the so{called \von Mises CLoE". As it is clear from its name, the limit surface is given by the von Mises yield function of classical perfect plasticity: r
3 ksk ; k = 0 (25) 2 where k is a material constant, representing the deviator stress on the limit surface. The expression for the tensor A is directly derived from the sti ness tensor for isotropic linear elasticity, except for the \shear moduli" A , A and A , while the tensor B is directly obtained by imposing the consistency and invertibility conditions, eqs. (20), (22). In the vector/matrix representation of eq. (19), the corresponding expressions for A and b are given by [23]: () := q ; k =
1212
2323
3131
2
K + 4G=3 K ; 2G=3 K ; 2G=3 0 0 0 6K ; 2G=3 K + 4G=3 K ; 2G=3 0 0 0 6 6K ; 2G=3 K ; 2G=3 K + 4G=3 0 0 0 A := 66 0 0 0 j ( q) 0 0 6 4 0 0 0 0 j (q) 0 0 0 0 0 0 j (q ) with:
q j (q) := G 1 ; ! k
13
3 7 7 7 7 7 7 5
(26)
(27)
and:
B
2p p3=2 (s11 =k )(2 6 3=2 (s22 =k )(2 6p 6 := 66 3=2 (s33 =k)(2 0 6 4 0
0
; q=k)3 ; q=k)77 ; q=k)77
(28)
7 7 5
where K and G represent the bulk and shear sti ness parameters, respectively, and ! is a material constant controlling the evolution of shear moduli with the normalized deviator stress q=k. Although MiniCLoE models do not allow an accurate description of the mechanical behavior of real geomaterials due to their relatively simple mathematical form, they have been speci cally developed in order to to develop a consistent interface constitutive equation to model post{localization behavior (Daphnis model, [18, 23]), and to test some fundamental results on: i) shear band localization, and, ii) existence and uniqueness of the general BVP in rate form [17, 7, 21, 19, 20].
3 Invertibility, consistency and limit states 3.1 General results
As already discussed in the preceding section, the assumption of invertibility of the rate constitutive equation for all admissible stress states 2 B represents one of the major di erences between CLoE and K{hypoplastic models. In order to discuss this aspect in more detail, it is useful to recall some fundamental results rst obtained by Chambon in [11, 12]. Let S be a unit tensor representing the direction of the stress rate such that:
r = S
(29)
where 0 is the stress rate norm. Provided that the tensor A is invertible, the constitutive equation (21) is invertible if, for any possible S , there exist: i) a unit rate of deformation tensor d, with kdk = 1, and, ii) a stress rate norm > 0, such that:
d = H ; B
with: 14
H := A; S 1
(30)
From eq. (30) and the condition kd k = 1, the following quadratic equation is obtained for the scalar unknown :
kH k ; 2H B + kB k ; 1 = 0 (31) When eq. (31) admits only a single real and positive solution, then the rate of deformation corresponding to the assumed stress rate direction is unique, and is given by eq. (30) . It is trivial to show that this condition is met if and only if the zero{order term in the quadratic equation (31) is negative, i.e., when (22) is satis ed. According to eqs. (11){(15), for the K{hypoplastic models discussed in sect. 2.2, the two second{order tensors appearing in eqs. (31) can be written as follows: H = fe1fb L; S B = fdL; N (32) where it is assumed that for amorphous constitutive equations as (11), the barotropy and pycnotropy functions fe, fd and fb are given by: fe = fd = 1 fb = tr() (33) 2
2
2
1
1
1
Following von Wol ersdor [83], a general expression for the tensor L; appearing in (32) is given by: L; = ;1 + 1 k^ k ;1 + k^ k I ; ^ ^ (34) in which: 1
1
1
=C =a = F = k^ k 1
1
2
1
= C =C = 1=a = a =F 2
2 1
1
2
2
2
2
2
(Wu & Bauer model) (Gudehus/Bauer model) (von Wol ersdor model)
1
2 1
2
2
2
2
2
2
(35) (36) (37)
Eq. (31) then specializes to K{hypoplasticity as follows:
(fefb ) 2
2
;1 2
L S
; 2 fffd e b
; ;1 ; ;1
L S L N
2
+ fd L; N ; 1 = 0 (38) 2
1
and the second invertibility condition, eq. (22) reads:
jfdj L; N < 1 1
15
(39)
3.2 Amorphous hypoplastic models
For amorphous hypoplastic materials with barotropy of the rst order | as all CLoE models of the rst generation, or Wu & Bauer model | the invertibility conditions (22) or (39) are met in the interior of the invertibility surface: (
; B = AL; Nb (CLoE) (40) (K{hypo) Due to the positive homogeneity of A, b, L and N with respect to , see
^ () = kB k ; 1 = 0 with:
1
1
remark 5, the invertibility surface de ned by eqs. (40) is a cone with the vertex at the origin. A direct inspection of eqs. (30) reveals that for stress states on the invertibility surface:
= 0 , r = 0 , d = ;B (41) that is, there exists a particular rate of deformation direction for which the stress rate vanishes. For this reason, and motivated by an analogous de nition introduced in the theory of hypoelasticity [77, 76], the surface ^ () = 0 has been also referred to in several works on K{hypoplasticity as yield surface, e.g. [52, 93, 54], or failure surface, e.g. [92, 95, 94, 96]. REMARK 10. Due to the fundamental assumptions (2) and (3), in all CLoE models the invertibility surface is assumed coincident with the limit surface (), and the constitutive tensors A and b are de ned in such a way to satisfy the consistency condition (20), see sect. 2.3. On the contrary, in amorphous K{hypoplastic models the invertibility surface does not represent a boundary for admissible stress states [52, 93, 96], due to the di erent strategy employed in the formulation of the relevant constitutive tensors. A geometrical explanation for this result can be obtained by considering in more detail the structure of eq. (38) which, for a state on the limit surface, reduces to:
L; S
; 2f ;L; S ;L; N = 0 (42) d fefb Apart from the solution = 0, the other possible solution of eq. (42) is subject to the condition that must be positive, as it represents the norm of the stress rate. Therefore, as fb , fe and fd are positive , the stress rate 2
1
2
1
1
2
A limit condition of vanishing stress rate is possible for a cryptoplastic state | i.e., when e = ed and fd = 0 | only if A is non{invertible. 2
16
direction is constrained by the following inequality: ; ;1 ; ;1
L S L N = S ;L;T L; N > 0 1
(43)
which means that the stress rate direction must remain con ned in the half{ space de ned by the plane:
(S ) = S
; ;T ;1
L L N =0
(44)
tangent to the six{dimensional stress{response envelope at the origin of the stress rate hyperspace. A schematic 2{d representation for the case of axisymmetric loading is shown in g. 1a. Now, when the stress rate space is superimposed to the actual stress space with the origin at the current stress state, the plane (S ) = 0 does not coincide, in general, with the tangent to the invertibility surface ^() = 0. As in this case the invertibility surface cuts the response envelope, there exist some particular rate of deformation directions for which the stress rate is directed outside the invertibility surface ( g. 1b). By appropriately selecting the loading path [93, 96], it is thus possible to reach stress states for which invertibility is lost and a homogeneous bifurcation condition occurs, as two di erent stress rates are generally associated to a given rate of deformation directions. This particular behavior requires a special consideration when solving practical boundary value problems, as in this case | as well as when shear band bifurcation occurs, see the following sect. 4 | special numerical techniques are required in order to obtain objective numerical solutions for those BVPs for which uniqueness of solution is lost. REMARK 11. The consistency condition for CLoE models, discussed in the previous sect. 2.3 (remark 7), can be interpreted geometrically in the same way, i.e., as a restriction imposed on the possible forms of the tensors A and b in order to have the plane (S) = 0 coincident to the tangent plane to the invertibility/limit surface ^() = () = 0. In mathematical terms this can be expressed as: @ = @ ^ = ; ;A;T A; b ( > 0) (45) @ @ Eq. (45) has been rst derived by Chambon in [11, 12]. In [96] and [57], it has been argued that a pathological consequence of the enforcement of consistency condition (45) in CLoE v1.00{v1.02 models is that the directional sti ness of the material tends to zero for any possible stress rate direction as the stress state approaches the limit surface, i.e., the response envelope shrinks to a point. This would imply that limit states 1
17
yˆ (s ) = 0
s11
S11
S11 p(S ) = 0
- L-T L-1 N Ñyˆ
p(S ) = 0
- L-T L-1 N
lS (d )
s
2S33
2S33
lS (d )
2s33
(a)
(b)
Figure 1: Schematic representation of a stress response envelope for kB k = 1. could not be left, once they are reached. In fact, this is not the case, as it can be easily shown by considering again eq. (31), written for a point on the limit surface:
kH k ; 2H B = 0 2
(46)
2
The response envelope can be considered shrinking to a point if, for any admissible stress rate directions, the solution = 0 is a double root of eq. (46), that is, if:
H B = S ;A;T A; b = 0 8S 2 S (47) Given the assumed regularity conditions for A, this can happen only for stress rate directions tangent to the plane (S ) = 0, but not for all stress 1
2
rate directions. A heuristic proof of the above result is given in g. 2, in which a number of response envelopes for axisymmetric loading have been drawn for CLoE v1.00, starting from di erent axisymmetric stress states with r = ;100:0 kPa, and assuming kdk = 1:0 10; s; . In deriving the response evelopes the set of material constants suggested in [71] for dense Hostun sand has been adopted. As it is clear from g. 2a, the envelopes reduce in size and rotate as the current stress state approaches the limit surface, but do not shrink to a point. From the enlargement in g. 2b it can be appreciated that the response envelope for the stress state on the limit surface (point F) is characterized 3
18
1
sa (kPa)
sa (kPa)
limit surface
-600
-625
F
F
-575
directional stiffness in unloading
-400
isotropic axis
-525 -100
(b)
-150
-200
2s r (kPa)
-200
sa (a)
0 0
-200
-400
2s r (kPa)
sr
sr
sa
Figure 2: Stress response envelopes for CLoE v1.00 under axisymmetric loading conditions. by a relatively high but nite aspect ratio, and that the directional sti ness for unloading at constant radial stress is, in fact, relatively large due to the rotation imposed by the consistency condition. REMARK 12. Although in amorphous K{hypoplastic models the invertibility surface does not represent, in general, a boundary to all the admissible stress states, the existence of a limit surface | or bound surface in the terminology adopted in [52, 93, 96] | must be predicted by the model to comply with the well{established experimental evidence that there exist some stress states which cannot be reached by any possible strain path. In other words, the region of the stress space accessible via the integration of the constitutive equation along any conceivable strain path must be bounded. Moreover, this region should be checked against experimental evidence in order to prevent the occurrence of some unreasonable stress conditions, e.g., with one or more positive (i.e. tensile) principal values. Unfortunately, the above requirements cannot be considered satis ed a 19
priori for amorphous K{hypoplastic constitutive equations, and have to be checked on a case{by{case basis. As a matter of fact, Wu & Niemunis [93] report that \[ : : : ] only a suitable choice of the tensorial expressions for L and N leads to the existence of the bound surface", while Gudehus & Kolymbas [42] observe that in some cases, the domain of admissible states for incrementally non{linear constitutive equation developed along the lines discussed in sect. 2.2 might include a non negligible region of tensile stresses. An analytical method for locating the bound surface has been given by Kolymbas in [52]. However, Wu & Niemunis [93, 96] observed that Kolymbas approach fails for non{axisymmetric stress states. To overcome this problem, they propose an alternative procedure, based on a expression formally equivalent to eq. (45). However, as in K{hypoplasticity the gradient @ =@ is not known in advance, the determination of the bound surface has to be carried out numerically. It is the writers' opinion that, in this respect, the explicit de nition of a limit surface in the mathematical formulation represents a de nite advantage of CLoE models as compared to equivalent, amorphous K{hypoplastic formulations. In fact, although this assumption is by no means necessary, it is certainly convenient from a practical point of view, expecially when the solution of boundary value problems with the nite element method is concerned, as:
i) it allows a straightforward check on the occurrence of physically impossible states due to inaccurate numerical integration at the stress{point level; and, ii) in conjunction with the consistency condition (24), it guarantees the invertibility of the rate equation, without which the well{posedness of some particular (rate) boundary value problem | and thus the con dence on the numerical results obtained | could be lost.
3.3 Endomorphous hypoplastic models
For endomorphous hypoplastic materials of the type discussed in sect. 2.2, the invertibility condition for the constitutive equation in rate form is given by eq. (39). Due to the pycnotropy function fd , the invertibility surface is de ned, in this case, in a state variable space including stress and void ratio: ^ (; e) = [fd (e)]
L; N
; 1 = 0 (48) 2
20
1
2
Similar considerations apply for the limit (or bound) surface, which | if existing | could be described by an analogous equation: (; e) = 0
(49)
However, in this case no analytical or numerical procedures are available for its determination. As for amorphous hypoplastic models, when loss of invertibility occurs, a limit state is reached characterized by a vanishing stress rate, and by an associated rate of deformation direction:
=0 ,
r = 0 , d = ;fdL; N 1
(50)
In addiction, the inclusion of void ratio in the set of state variables allows to embed the concept of critical states introduced by Casagrande [8]. According to Casagrande's de nition, critical states are those states (c; ec) for which the stress rate vanishes under continued isochoric deformation:
dc 1 = 0
e = ec = const.
L (^c) dc + fd;cN (^c) kdck = 0
(51)
Eqs. (51) and (51) pose an additional restriction to the possible choices for the tensors L and N , since at critical state: 2
3
dc = ;fd;cL; (^c) N (^c) ) L; (^c) N (^c) 1 = 0 kdck 1
1
(52)
The consequences of this restriction on the functional form of the second{ order tensor N are detailed in [83]. The projection of the critical state locus onto the (e; p) space, describing the evolution of critical void ratio with mean stress level, is given by the empirical relation (93). In order for the projection of the critical state locus onto the stress space to be represented as a single critical stress surface, the pycnotropy function fd;c at critical states must be independent of e. In particular, in both Gudehus/Bauer [38, 1] and von Wol ersdor [83] models the function fd has been selected in order to have fd;c = 1, see eq. (91). By taking the squared norm of eq. (52) , the equation of the critical stress surface is obtained as: 1
) =
L; N
; 1 = 0 (53) As eq. (53) is homogeneous of zero order in , it represents a cone in the c(
1
stress space with the vertex at the origin. 21
2
REMARK 13. According to eqs. (48), (53) and (91), when the material is sheared along a loading path with a prevalent deviatoric component (i.e., in drained or constant p strain controlled TX compression tests), the locus of limit states | in the sense de ned by eq. (50) | is found above the critical state cone when the material is denser than critical (dense sands, ep < ec and fd;p > 1), while peak and critical states generally coincide when the initial density of the material is lower than critical (loose sands, fd;p = fd;c), see [1]. This pattern of behavior appears in substantial agreement with experimental observations, as demonstrated by the comparison between measured and predicted peak friction angles reported in [1] for a medium quartz sand with various initial densities. REMARK 14. An interesting consequence of the mathematical structure of endomorphous K{hypoplastic models described in sect. 2.2 can be deduced by specializing eq. (50) to a particular constitutive equation. As an example, for the Gudehus/Bauer model [38, 1] eq. (50)3 reads: h i d = ;fd a ;a2 +1 k^ k2 ;a21 ; k^ k2 ^ + ;a21 + k^ k2 ^ (54) 1 1
from which:
d 1 = ;fd a ;a +1 k^ k ;a ; k^ k (55) since ^ 1 = 1 and ^ 1 = 0. To correctly recover a condition of isochoric
ow, at critical states k^ ck = a . Therefore, at peak states located outside the critical state cone; (i.e., when failure occurs at void ratios lower than critical) the quantity a ; k^ k is negative, and, according to eq. (55) the associated rate of deformation is dilatant (d 1 > 0). As the material dilates, 1
2
2 1
2 1
2
1
2 1
2
the void ratio is progressively increased and fd reduces accordingly, until a condition of critical state is reached at a lower stress ratio and higher void ratio. This behavior is not limited to a particular deviatoric path in stress space. According to Gudehus [37], for K{hypoplastic models of this type, critical states can be considered as asymptotic states (attractors) for all deviatoric paths leading to steady state ow. However, as the introduction of the concept of critical state in the mathematical structure of the constitutive equation implies the possibility of developing material softening for some particular initial states and loading paths, this development of K{hypoplasticity is not compatible with the assumption of invertibility on which CLoE hypoplasticity is based. 22
REMARK 15. Although for the aforementioned reason no direct comparison is possible between CLoE and endomorphous K{hypoplasticity, it is interesting to note that | similarly to all CLoE formulations | in K{hypoplastic models of this type, the shape of the critical state cone is assumed a priori. For example, in von Wol ersdor model [83], the various functions entering the constitutive tensors (see App. B) are de ned in order for the function c = 0 to be coincident with the Matsuoka/Nakai failure condition [59]. This is to be contrasted with earlier K{hypoplastic formulations, see e.g. [93, 92, 95], for which the invertibility (failure) surface is derived as by{ products of the particular assumptions for the constitutive tensors L and N.
4 Strain localization and bifurcation analysis 4.1 Introduction and historical background Cino's notes
The theoretical problem of the prediction of the onset of strain localization in non{standard, pressure dependent materials such as granular soils, considered as a assumed mode (shear band) bifurcation process in the sense of Rice [67], has received much attention since the early works of Rice and Rudnicki [70, 67, 68]. As a matter of fact, the inadequacy of classical ow theories of hardening plasticity with a single mechanism to provide a satisfactory description of experimentally observed shear banding processes was certainly one of the main motivations for the development of incrementally non{linear and hypoplastic constitutive theories, see, e.g. [49, 14, 10, 16]. The rst shear band analysis for incrementally non{linear materials | although not hypoplastic according to the de nition given in sect. 2 | dates back to the pioneering work of Kolymbas in 1981 [49]. A key point in Kolymbas' approach is the assumption of a vanishing rate of deformation outside the band. A few years later, in 1985, Chambon & Desrues [14] presented a general bifurcation analysis for the heuristic model de ned by eq. (16), which, as dicussed in sect. 2.3, can be considered the originator of the subsequent CLoE hypoplastic models. A key result in Chambon & Desrues [14] analysis is the fact that when bifurcation occurs, the ratio between the strain rate inside and outside the band goes to in nity, i.e., the material outside the 23
band stops deforming. Similar results have been subsequently obtained by Loret [58] for a slightly di erent incrementally non{linear model. A general shear band bifurcation criterion for hypoplastic materials has been subsequently given by Chambon and his coworkers [12, 22, 43, 15], starting from the assumption of a continuous variation of the tangent map between rate of deformation and stress rate spaces with the state of the material during the loading process (continuity in the large, according to the de nition given in [44]). See [15] for details. The extension of the aforementioned works to include the case of a discontinuous variation of material \sti ness" during the loading process has been recently discussed in [19]. Non{linear shear band analyses for homomorphic K{hypoplastic models have been presented by Kolymbas & Rombach in 1989 [55] (assuming kdk ' 0 outside the band), and by Wu & Sikora in 1991{92 [89, 90]. Recently, Bauer & Huang [5] have extended the aforementioned studies to K{hypoplastic formulations with barotropy and pyknotropy, discussing the dependence of stress ratio at bifurcation and shear band inclination on void ratio and mean stress.
4.2 General results
In the following, some general results of the non{linear shear band analysis for general hypoplastic materials discussed in [19] are brie y summarized. Note that, as the analysis is completely general, these results can be directly applied to both CLoE and K{hypoplastic models. The development of a shear band bifurcation in a homogeneous deforming body is completely de ned in terms of the following kinematic condition among spatial velocity gradients l := grad v:
l =l +g n 1
0
(56)
in which the indexes 0 and 1 denote the relevant quantities inside and outside the shear band, respectively; n is the unit vector normal to the shear band; and g is a vector de ning the velocity gradient jump in the direction n. The requirement of continuing equilibrium at the onset of localization also implies the following static condition:
_ n ; _ n = 0 1
0
(57)
in which _ and _ are the Cauchy stress rates inside and outside the band, respectively. The bifurcation condition is obtained by specifying in the static condition the hypoplastic constitutive equation (5), which for our purposes can be more 0
1
24
conveniently rewritten in terms of the material time derivative of Cauchy stress in the following general form:
_ = Ml + b kdk
where: for CLoE models, or:
(58)
M := A + C ;
b := AB
(59)
M := fsL + C ;
b := fsfdN
(60)
for K{hypoplastic models, and the fourth order tensor C in eqs. (59), (60), de ned as: Cijkl := 21 (ik lj ; il ki ; ik jl + il jk ) (61) accounts for corotational terms in appearing in the de nition of the stress rate, eq. (5). From eqs. (57), (58) it follows [12, 15, 19]:
d
;
d
bn = 0 (62)
As eqs. (56)-(58) are homogeneous in d , without loss of generality it can be assumed that d = 1, and:
d = (1 + r ) d = 1 + r ; r ;1 (63) [M (g n)] n +
;
1
0
0
0
1
0
Eq. (62) can therefore be written as:
[M (g n)] n + rbn = 0
(64)
subject to the condition: 1
l + l T + g n + n g
= 1 + r (65) 2 Bifurcation is possible if and only if the above system of nonlinear equations (64), (65) has at least one solution for g and r such that r ;1. De ne the second order acoustic tensor P associated with M as: 0
0
Pik := Mijklnj nl
(66)
the following results hold, depending on the invertibility of P [12, 15, 19]. 25
CASE 1. If P is invertible, then eq. (64) can be solved for g , with:
g = ;rP ; bn
(67)
d ; r sym P ; b (n n)
= 1 + r
(68)
1
Eq. (65) thus reads:
0
1
Shear band bifurcation is possible if and only if there exists some n for which eq. (68) has a solution r ;1. Eq. (68) can be written in the following compact form: ;
N ; 1 r + 2 (S ; 1) r = 0
where:
N := sym
2
(69)
2
;1
P b (n n)
;
S := d P ; b (n n) 0
1
(70)
The only non{trivial solution of eq. (69) is:
; S ) ;1 r = 2(1 N ;1
(71)
2
since for r = 0 the only possible solution of eq. (64) is g = 0 (no localization). According to eq. (71), it can be shown [19] that shear band bifurcation is possible if and only if:
N = sym 2
;1
P b (n n)
1 2
(72)
CASE 2. If P is not invertible, then the bifurcation condition reads [15, 19]:
det (P ) = 0
(73)
This second condition is apparently similar to the classical bifurcation condition for incrementally linear materials (see, e.g., [67]); however, in this case, the tensor P is derived from the linear part only of the constitutive equation. According to Chambon et al. [15], the bifurcation condition de ned by eq. (73) is practically never met before the \norm" criterion given by eq. (72) for CLoE models, for which the tensor A has the structure of an elastic tensor. The same conclusion is reached by Bauer & Huang [5] for the Gudehus{Bauer model [38, 1].
26
REMARK 16. As proven in [19], if and only if non{linear shear band bifurcation condition (72) holds, there exists a spatial velocity gradient l such that:
det [Q (dir l )] = 0
in which the tensor:
with: Qik = Mijkl + bij dkl nj nl kd k
:= M + b kddk
(74) (75)
represents the directional linearization of the non{linear rate equation (58). However, this results has only a theoretical interest, since dir l is not known a priori. REMARK 17. The limiting case N = 1 in the bifurcation condition (72) occurs if at the beginning of the loading process bifurcation is not possible | which seems a fairly reasonable assumption | and the response of the material is continuous in the large (see [15]). In this case, which is usually the rule for hypoplastic models discussed in this work, eq. (71) yields the two solutions r = 1 and r = ;1, according to which the ratio between the norms of the rate of deformation inside and outside the band is in nite. Note that the two cases of zero rate of deformation inside the band (r = ;1) or outside the band (r = 1) are perfectly symmetric from a mathematical point of view. From a physical standpoint, this means that at the onset of localization, strain rate concentrates inside the band, i.e., l is negligible with respect to l . This theoretical result appears to be con rmed by a number of experimental observations in both cohesive and granular soils [27, 82]. As observed in [15], the static condition (57) implies in this case: _ (n g) = 0 ) _ sym (g n) = _ d = 0 (76) 0
1
1
This means that, under the assumption of small strains (r ' _ ), localization implies the loss of positiveness of the second order work. REMARK 18. When material response changes discontinuously with the loading parameter, abrupt bifurcation is possible with N > 1. In this case the ratio between the norms of the rate of deformation inside and outside the band remains nite, and shear banding becomes possible for a fan of shear band orientations. As a direct consequence, in this case various possible bifurcation modes can be obtained from numerical simulation of the loading process for the same initial conditions, depending on the details of the numerical procedure employed, see [19]. This represents a clear drawback for this kind of constitutive models. 27
REMARK 19. The shear band analyses for K{hypoplasticity mentioned in the previous section [55, 89, 90, 5] di er from the one outlined above in that they all rely on a particular kinematic assumption, concerning: 1. the rate of deformation outside the band (in particular, d0 = 0) [55];
p
2. the dilatancy := tan; g =( 2g ) inside the band [89]; 3. the rate of deformation norm jump [90, 5]: 1
2
1
[ kdk] := d ; d 0
1
(77)
Interestingly, the result outlined in remark 17 provides an a posteriori theoretical argument in favor of the hypothesis made in the bifurcation analyses presented by Kolymbas and Kolymbas & Rombach [49, 55]. The hypotesis put forward by Wu & Sikora (1991) [89], on the contrary, does not appear to be necessary, as it represents an additional constitutive assumption for the material inside the band. Following a di erent approach, Wu & Sikora (1992) [90] and Bauer & Huang [5] solve the bifurcation problem posed by eq. (64) by assuming the absolute value of the deformation norm jump:
:= d ; d (78) In particular, they observe that the earliest bifurcation point (lower bound solution in [90]) is reached when: [ kdk] = k[ d] k (79) Eq. (79) implies the following: d = d ; 0 (80) 1
0
1
0
which represents a condition of continuous bifurcation ( = , see [28]) and includes the condition of vanishing rate of deformation outside (or inside) the band as a particular case (; = 0 or = 0). As a matter of fact, by comparing eqs. (63) and (80) it follows that: 1
0
1
=r+1 (81) and since for a continuous variation of the response with the loading parameter the only possible solutions for r at bifurcation are r = 1 and r = ;1, the lower bound solution of Wu & Sikora [90] and Bauer & Huang [5] coincides with the bifurcation condition obtained by Chambon and coworkers [14, 28, 43, 15, 19]. 28
4.3 The question of the shear modulus
The principal and most obvious use of the above results in practical applications is to de ne the conditions upon which uniqueness of the solution can be lost due to shear banding, either at the scale of laboratory tests (e.g., [49, 43, 15, 5] or in the analysis of full scale boundary value problems (e.g., [55, 31]). However, another possible application | motivated by the availability of a closed form solution for the nonlinear shear band problem | is the the use of experimental localization data to obtain informations on such details of the constitutive equations which cannot be directly de ned using experimental data from tests with xed principal stress and strain directions. In particular, since the pioneering works of Hill & Hutchinson [46], Rudnicki & Rice [70] and Rice [67], it is well known that the onset of the bifurcation and the orientation of the shear band can be strongly a ected by the way in which the constitutive model describes the response of the material for deformation paths which, upon bifurcation, change abruptly their direction. In the particular case of rectilinear plane strain compression or extension, this means that a major role in de ning bifurcation conditions is played by the shear modulus in the plane of deformation (see, e.g., [80, 81, 63] for a discussion on this topic in the framework of hardening plasticity). Following an initial idea by Vardoulakis [80], the possibility of using experimental shear band observations for the calibration of shear moduli, which are not accessible experimentally in standard triaxial or plane strain tests with xed principal stress and strain directions, has been rst suggested for CLoE models in [12], and then successfully exploited in [22, 43, 15, 31] for the particular CLoE model discussed in [16], and its subsequent modi cations. In this respect, the particular structure assumed for the fourth order tensor A represents an important di erence between CLoE and K{hypoplastic models. In fact, as the stress tensor is the only tensor{valued state variable in both CLoE and the K{hypoplastic models considered in sect. 2.2, the representation of the tensors A and b given by eq. (19) (see remark 6) in the reference frame of the pricipal stress directions holds in both cases. From eq. (19) is immediately apparent that the three components A , A and A of the tensor A represent the shear moduli relevant to the onset of localization | i.e., to a so{called loading to the side process. If, without loss of generality, only plane strain processes in the plane x : x corresponding to the principal directions of and are considered, then attention can be focused on the only relevant shear modulus A . Note that if the loading process is such that both principal axes of stress and strain remain xed up to the bifurcation point | as for example biaxial 1212
2323
3131
1
1
2
1212
29
2
compression tests | this shear modulus plays no role in determining the material response before the onset of localization. In all CLoE hypoplastic models, the shear modulus A is considered a function of the state variables according to the following general equation: A = j (1 ; !R) (82) In eq. (82), j is the shear modulus at the isotropic image point (q = 0), either assumed constant (von Mises and Mohr{Coulomb MiniCLoE models, see [17, 23]), or function of the mean stress p, to be determined starting from the response in triaxial compression and extension (see, e.g., [43] for details); ! is a material constant, and R is a normalized deviatoric stress, de ned as: 1212
1212
0
0
(
(CLoE v1.02; von Mises MiniCLoE) (83) R := q=ql tan m= tan (Mohr{Coulomb MiniCLoE) in which ql is the deviator stress at the image point on the limit surface, is the friction angle, and m represent a measure of the stress ratio in terms of mobilized friction angle. The term (1;!R) in eq. (82) introduces a deviatoric stress dependent stress{induced anisotropy in the shear modulus which is consistently recovered (A = j ) as the isotropic state is approached (R ! 0). See [9] for further details. The corresponding expressions for the shear modulus A for the three K{hypoplastic models discussed in sect. 2.2 are given by: 8 > (Wu & Bauer) fsa (84) (Gudehus/Bauer) : fsF = ( ) (von Wol ersdor ) The quantities appearing in the above equations have been de ned in sect. 2. From eqs. (82) and (84) it is apparent that, while in K{hypoplastic formulations the shear modulus is strongly related to the response of the material as observed in rectilinear compression or extension paths | as all the parameters entering in the di erent expressions in eq. (85) a ect the overall predicted behavior in such kind of loading | in CLoE models the use of the parameter ! introduces an additional degree of freedom which allows an independent calibration of the di erent components of material sti ness in rectilinear compression and shear. This feature of CLoE models is exploited in the analysis of bifurcation problems in the sense that experimental localization data can then be used to calibrate the shear modulus | through the parameter ! | without a ecting the response of the material in rectilinear compression or extension. 1212
0
1212
1
1212
2 1
2
30
In the following, a more detailed analysis of the e ects of the di erent constitutive assumptions for the shear moduli is presented, by comparing the predicted responses of the di erent hypoplastic models considered upon some suitable loading condition. Such a comparison has been carried out for the CLoE v1.03 [15] model, and for the K{hypoplastic models proposed by Wu & Bauer (WB, [92]), Gudehus/Bauer (GB, [38, 1]) and von Wol ersdor (vW, [83]). A series of isochoric plane strain compression tests has been simulated, starting from an isotropic initial stress p = 100:0 kPa. Note that the kinematic restraint imposed on testing conditions allows to rule out the e ect of pyknotropy, which is included in vW and GB models, but is not present in CLoE and WB formulations. In all CloE simulations, the values of material parameters suggested in [71] for dense Hostun sand have been adopted, except for the parameter !, which has been varied in the range [0:2; 1:0]. The material parameters and (where required) the initial void ratios entering in the three K{hypoplastic models have been determined in order to match the response predicted by CLoE. The resulting values are given in tab. 1. In the calibration of vW and GB models, the values suggested by Herle & Gudehus [45] for dense Hostun sand have been used as a starting point. It is worth noting that the only notable di erence between the parameters used in this work and those suggested in [45] is in the friction angle at critical state, c, for which a much larger value is needed to compensate the high friction angles assumed for CLoE in both TX compression ( C = 45) and extension ( E = 46). The results of the various simulations in terms of stress{strain curves on q:s plane, and of stress paths on q:p plane are shown in g. 3. As no attempt has been made in re ning the calibration of the di erent material parameters in order to obtain the best possible match, some di erences are observed in both stress{strain curves and stress paths. Nonetheless, the substantial overall agreement among all predictions still allows a meaningful comparison between the performance of the four di erent hypoplastic models, at least from a qualitative point of view. The evolution of shear sti ness with deviatoric strains for the four hypoplastic models considered is shown in g. 4 in terms of the non{dimensional normalized shear modulus: ; := A;n n (85) p pa in which pa = 100:0 kPa is the athmospheric pressure, and the barotropy constant n is either given in tab. 1 (vW and GB models), or equal to zero (CLoE and WB models, which assume a linear dependence of tangent sti 1212
(1
31
)
parameter Wu & Bauer [92] Gudehus [38] v.Wol ersdor [83] C 100.0 | | C 950.0 | | C 870.0 | | C -1450.0 | | e | 0.67 0.60 c | 40.0 40.0 hs (kPa) | 1.0e6 1.0e6 n | 0.3 0.3 | 0.13 0.13 | 2.0 1.0 ec | 0.96 0.96 ed | 0.61 0.61 ei | 1.09 1.09 1 2 3 4
0
0
0
0
Table 1: Material parameters adopted for K{hypoplastic models.
300
300 vW
200
deviatoric stress, q (kPa)
deviatoric stress, q (kPa)
CLoE
Gd WB
100
0 0.00
200
CLoE 100
vW Gd WB
0 0.01
0.02
0.03
deviatoric strain, es
0
100
200
mean stress, p (kPa)
Figure 3: Plane strain rectilinear compression tests results. 32
300
600
300
(b) K-hypo models
w= 0.2
norm. shear modulus, G
norm. shear modulus, G
(a) CLoE model w= 0.6
400
w= 1.0
200
0 0.00
0.01
0.02
vW Gd
200
WB
100
0 0.00
0.03
deviatoric strain, es
0.01
0.02
0.03
deviatoric strain, es
Figure 4: Plane strain rectilinear compression tests: evolution of normalized shear modulus with deviatoric strain. ness with mean stress p). The adopted normalization allows to account for the e ect of barotropy, as described by the di erent formulations. The three curves relative to CLoE model show that the normalized shear modulus reduces with increasing stress ratio q=p until, at deviatoric strains s larger than about 0.02, the stress ratio remains almost constant. The reduction of the normalized shear modulus is larger for larger values of !, as implied by eq. (82). A similar behavior is reproduced by the two advanced K{ hypoplastic models vW and GB, although in these two cases, the evolution of normalized sti ness with deviatoric strains is completely controlled by the parameters which de ne material behavior under the assumed testing conditions of rectilinear biaxial compression. On the contrary, a constant normalized shear sti ness | which appears a rather unrealistic response | is predicted by the WB model due to the relatively simple structure assumed for the tensor A. Note that the same response is given by CLoE when ! = 0. An indication of the level of stress{induced anisotropy in the shear sti ness developed along the loading process is given by the ratio: (86) := A ^ G between the current shear sti ness A and the corresponding equivalent isotropic shear sti ness G^ := 41 [(A ; A ) + (A ; A )] (87) de ned in terms of the components of the 3 3 submatrix of A containing the normal components of tangent sti ness. 1212
1212
1111
1122
33
2222
2211
3
2
CLoE model w = 0.0 w = 0.2
1
w = 0.6 w = 1.0 0 0.00
0.01
0.02
deviatoric strain, es
anisotropy ratio, Q
anisotropy ratio, Q
3
2
K-hypo models 1
Gd WB 0 0.00
0.03
vW
0.01
0.02
deviatoric strain, es
0.03
Figure 5: Plane strain rectilinear compression tests: evolution of anisotropy ratio with deviatoric strain. The evolution of with deviatoric strain is shown in g. 5. As expected, since the behavior of the material is isotropic under isotropic stress conditions, all the curves start with = 1. CLoE simulations are shown in g. 5a. For ! = 0:2, a rapid increase of is observed in the initial stage of the loading process, followed by a peak and by a subsequent reduction to a fairly stable value, as a condition of almost proportional loading with q=p ' const. is reached. However, as ! is increased a rather di erent behavior is observed, with much lower values of . In particular, for ! = 1:0 the anisotropy ratio is always less than unity, and reduces steadily as deviatoric strain increase. This rather complex behavior is due to the interplay between the di erent e ects of stress ratio on the three components of A entering in the de nition (86), which are controlled by ! according to eq. (82). The corresponding curves for the K{hypoplastic models are shown in g. 5b. For the WB model, the anisotropy ratio increases steadily up to a nal value ' 2:4, re ecting the decrease of the normal components A , A and A = A with increasing stress ratio. The behavior predicted by the two advanced GB and vW models is rather similar to that predicted by CLoE for small ! values. However, it is worth noting that, in this case, the anisotropy ratio remains always greater than unity. 1111
1122
2211
5 Conclusions To be written.
34
2222
References [1] Bauer E. (1996). Calibration of a comprehensive hypoplastic equation for granular materials. Soils and Foundations, 36(1), 13-26. [2] Bauer E., Niemunis A., Herle I. (1993). Visco{hypoplastic model for cohesive soils. In: Modern Approaches to Plasticity, Kolymbas ed., Elsevier, 365-384. [3] Bauer E., Wu W. (1993). A hypoplastic model for granular soils under cyclic loading. In: Modern Approaches to Plasticity, Kolymbas ed., Elsevier, 247-258. [4] Bauer E., Wu W. (1994). Extension of hypoplastic constitutive model with respect to cohesive powders. Proc. Computer Methods and Advances in Geomechanics, Siriwardane & Zaman eds., 1, Balkema, 531-536. [5] Bauer E., Huang W. (1998). The dependence of shear banding on pressure and density in hypoplasticity. Proc. Localization and Bifurcation Theory for Soils and Rocks, Gifu, Adachi et al. eds., Balkema, 81-90. [6] Bazant Z.P. (1978) Endochronic inelasticity and incremental plasticity. Int. Journal of Solids and Structures, 14, 691-714. [7] Caillerie D., Chambon R. (1994). Existence and uniqueness for B.V. problems involving CLoE models. Proc. Localization and Bifurcation Theory for Soils and Rocks, Aussois, Chambon et al. eds., Balkema, 35-40. [8] Casagrande A. (1936). Characteristics of cohesionless soils a ecting the stability of earth lls. J. Boston Soc. Civil Engineers, 23(1), 13-32. [9] Chambon R. (1981). Contribution a la modelisation numerique non{ lineaire des sols. These de Doctorat d'etat, Universite de Grenoble. [10] Chambon R. (1986). Bifurcation par localisation en bande de cisallement, une approche avec des lois incrementalement non lineaires. Journal de Mechanique Theorique et Appliquee, 5(2), 277-298. [11] Chambon R. (1989). Une classe de lois de comportement incrementalement nonlineaires pour les sols non visqueux, resolution de quelques problemes de coherence. C. R. Acad. Sci., 308(II), 1571-1576. [12] Chambon R. (1989). Bases theoriques d'une loi de comportement incrementale consistante pour les sols. Groupe C.O.S.M. Rapport de Recherche. 35
[13] Chambon R. (1999). Uniqueness, second order work and bifurcation in hypoplasticity. This Conference. [14] Chambon R., Desrues J. (1985). Bifurcation par localisation et non linearite incrementale: un example heuristique d'analyse complete. Proc. Plastic Instability, Paris, Presses ENPC, 101-113. [15] Chambon R., Desrues J., Tillard D. (1994). Shear moduli identi cation versus experimental localization data. Proc. Localization and Bifurcation Theory for Soils and Rocks, Aussois, Chambon et al. eds., Balkema, 101111. [16] Chambon R., Desrues J., Hammad W., Charlier R. (1994). CLoE, a new rate{type constitutive model for geomaterials. Theoretical basis and implementation. Int. J. Num. Anal. Meth. Geomech., 18, 253-278. [17] Chambon R., Crochepeyre S. (1995). Loss of uniqueness of the solution in a boundary value problem. Proc. NUMOG V, Pande & Pietrusczczak eds., 165-171. [18] Chambon R., Crochepeyre S. (1997). Incremental non linearity, bifurcation and shear banding. Proc. Localization and Bifurcation Theory for Soils and Rocks, Gifu, Adachi et al. eds., Balkema. [19] Chambon R., Crochepeyre S., Desrues J. (1999). Localization criteria for non{linear constitutive equations of geomaterials. Mech. CohesiveFrictional Materials, in print. [20] Chambon R., Caillerie D., Desrues J., Crochepeyre S. (1999). A comparison of incremental behavior of elastoplastic and CLoE models. Int. J. Num. Anal. Meth. Geomech., in print. [21] Chambon R., Caillerie D. (1999). Existence and uniqueness theorems for boundary value problems involving incrementally non linear models. Int. Journal of Solids and Structures, in print. [22] Charlier R., Chambon R., Desrues J., Hammad W. (1991). Shear band bifurcation and soil modelling: a rate type constitutive law for explicit localization analysis. Proc. 3 Int. Conf. on Constitutive Laws for Engineering Materials, Tucson, Desai & Krempl eds., ASME Press. [23] Crochepeyre S. (1998) Contribution a la modelisation numerique et theorique de la localisation et de la post localisation dans les geomateriaux. These de Doctorat d'etat, Universite de Grenoble. nd
36
[24] Dafalias Y.F. (1986). Bounding surface plasticity. I: Mathematical foundation and hypoplasticity. J. of Engng. Mech., ASCE, 112(9), 966-987. [25] Darve F. (1991a). The expression of rheological laws in incremental form and the main classes of constitutive equations. In: Geomaterials: Constitutive Equations and Modelling, Darve ed., Elsevier, 123-148. [26] Darve F. (1991b). Incrementally non{linear constitutive relationships. In: Geomaterials: Constitutive Equations and Modelling, Darve ed., Elsevier, 213-237. [27] Desrues J. (1984). La localisation de la deformation dans le materiaux granulaires. These de Doctorat d'etat, Universite de Grenoble. [28] Desrues J., Chambon R. (1989). Shear band analysis for granular materials: the question of incremental non{linearity. Ingenieur{Archiv, 59, 187-196. [29] Desrues J., Chambon R. (1993). A new rate{type constitutive model for geomaterials: CLoE. In: Modern Approaches to Plasticity, Kolymbas ed., Elsevier, 309-324. [30] Drescher A., Vardoulakis I. (1982). Geometric softening in triaxial tests on granular material. Geotechnique, 32(4), 291-303. [31] El Hassan N., Desrues J., Chambon R. (1997). Numerical modelling of borehole instability using a non linear incremental model with bifurcation analysis. Proc. Int. Symp. on Deformation and Progressive Failure in Geomechanics, Oka et al. eds., Nagoya, 677-682. [32] Goldscheider M. (1976). Grenzbedingung und Flieregel von Sand. Mech. Res. Comm., 3, 463-468. [33] Goldscheider M. (1984). True triaxial tests on dense sand. In: Constitutive Relations for Soils, Gudehus et al. eds., Grenoble, 11-54. [34] Gudehus G. (1979). A comparison of some constitutive laws for soils under radially symmetric loading and unloading. Proc. 3 Int. Conf. Num. Meth. Geomech., Aachen, Wittke ed., Balkema, 4, 1309-1324. [35] Gudehus G. (1985). Requirements for constitutive relations for soils. In: Mechanics of Geomaterials, Bazant ed., John Wiley & Sons., 47-63. rd
37
[36] Gudehus G. (1995). A comprehensive concept for non{staurated granular bodies. Proc. Unsaturated Soils, Alonso & Delage eds., Balkema, 2, 725-737. [37] Gudehus G. (1995). Attractors for granular storage and ow. Proc. Partec `95, 333-345. [38] Gudehus G. (1996). A comprehensive constitutive equation for granular materials. Soils and Foundations, 36(1), 1-12. [39] Gudehus G. (1996). Constitutive relations for granulate{liquid mixtures with a pectic constituent. Mechanics of Materials, 22, 93-103. [40] Gudehus G., Goldscheider M., Winter H. (1977). Mechanical properties of sand and clay and numerical integration methods: some sources of errors and bounds of accuracy. In: Finite Elements in Geomechanics, Gudehus ed., John Wiley & Sons, 121-150. [41] Gudehus G., Kolymbas D. (1979). A constitutive law of the rate{type for soils. Proc. 3 Int. Conf. Num. Meth. Geomech., Aachen, Wittke ed., Balkema, 1, 319-329. [42] Gudehus G., Kolymbas D. (1985). Numerical testing of constitutive relations for soils. Proc. 5 Int. Conf. Num. Meth. Geomech., Nagoya, Adachi et al. eds., Balkema, 1, 63-81. [43] Hammad W.I. (1991). Modelisation non lineaire et etude experimentale des bandes des cisallement dans les sables. These de Doctorat d'etat, Universite de Grenoble. [44] Hashiguchi K. (1993). A basic formulation of elastoplastic constitutive equations. In: Modern Approaches to Plasticity, Kolymbas ed., Elsevier, 39-58. [45] Herle I., Gudehus G. (1999). Determination of parameters of a hypoplastic constitutive model from grain properties. Mech. Cohesive-Frictional Materials, in print. [46] Hutchinson J.W., Hill R. (1975) Bifurcation phenomena in the plane tension test. Journal of the Mechanics and Physics of Solids, 23, 239264. [47] Jardine R.J., Potts D.M., Fourie A.B., Burland J.B. (1986). Studies of the in uence of non-linear stress-strain characteristics in soil-structure interaction. Geotechnique, 36, 377-396. rd
th
38
[48] Kolymbas D. (1977). A rate{dependent constitutive equation for soils. Mechanical Research Communications, 4, 367-372. [49] Kolymbas D. (1981). Bifurcation analysis for sand samples with a non{ linear constitutive equation. Ingenieur{Archiv, 50, 131-140. [50] Kolymbas D. (1984). A constitutive law of the rate type for soils | Position, calibration and prediction. In: Constitutive Relations for Soils, Gudehus et al. eds., Grenoble, 419-437. [51] Kolymbas D. (1987). A novel constitutive law for soils. Proc. 2 Int. Conf. on Constitutive Laws for Engineering Materials, Tucson, Elsevier, 1, 319-326. [52] Kolymbas D. (1991). An outline of hypoplasticity. Arch. Appl. Mech., 61, 143-151. [53] Kolymbas D. (1991). Computer-aided design of constitutive laws. Int. J. Num. Anal. Meth. Geomech., 15, 593-604. [54] Kolymbas D. (1994). Hypoplasticity as a constitutive framework for granular materials. Proc. Computer Methods and Advances in Geomechanics, Siriwardane & Zaman eds., 1, Balkema, 197-208. [55] Kolymbas D., Rombach G. (1989). Shear band formation in generalized hypoelasticity. Ingenieur{Archiv, 59, 177-186. [56] Kolymbas D., Wu W. (1993). Introduction to hypoplasticity. In: Modern Approaches to Plasticity, Kolymbas ed., Elsevier, 213-224. [57] Kolymbas D., Herle I. (1997). Hypoplasticity: a framework to model granular materials. In: Behaviour of Granular Materials, Cambou ed., Springer, Berlin, 239-268. [58] Loret B. (1987). Non{linearite incrementale et localisation des deformations: quelques remarques. Journal de Mecanique Theorique et Appliquee, 6(3), 423-459. [59] Matsuoka H., Nakai T. (1977). Stress{strain relationship of soil based on the `SMP'. In: Constitutive Equations of Soils, Specialty Session 9, IX ICSMFE, Tokio, 153-162. [60] Nelson I. (1977). Constitutive models for use in numerical computations. Proc. Plastic and Long Term E ects, DMSR `77, Karlsruhe, 2, 45-97. nd
39
[61] Niemunis A. (1996). A visco{plastic model for clay and its FE implementation. Proc. XI Coll. Franco{Polonais en Mechanique des Sols et des Roches Appliquee, Gdansk, 151-162. [62] Niemunis A., Herle I. (1997). Hypoplastic model for cohesionless soils with elastic strain range. Mech. Cohesive-Frictional Materials, 2, 279-299. [63] Papamichos E., Vardoulakis I., Han C. (1993). Noncoaxial ow theory of plasticity: shear failure predictions in sand. In: Modern Approaches to Plasticity, Kolymbas ed., Elsevier, 585-598. [64] Pastor M., Zienkiewicz O.C., Chan A.H.C. (1990). Generalized plasticity and the modelling of soil behaviour. Int. J. Num. Anal. Meth. Geomech., 14, 151-190. [65] Puzrin A.M., Burland J.B. (1998). Non{linear model of small{strain behaviour of soils. Geotechnique, 48(2), 217-233. [66] Pyke R. (1986). The use of linear elastic and piecewise linear models in nite element analyses. In: Geomech. Modeling in Engng. Practice, Dungar & Studer eds., Balkema, 167-188. [67] Rice J.R. (1976). The localization of plastic deformations. Theoretical and Applied Mechanics, Koiter ed., North{Holland, 207-220. [68] Rice J.R., Rudnicki J.W. (1980). A note on some features of the theory of localization of deformation. Int. Journal of Solids and Structures, 16, 597-605. [69] Royis P., Doanh T. (1998). Theoretical analysis of strain response envelopes using incrementally non{linear constitutive equations. Int. J. Num. Anal. Meth. Geomech., 22, 97-132. [70] Rudnicki J.W., Rice J.R. (1975). Conditions for the localization of deformation in pressure{sensitive dilatant material. Journal of the Mechanics and Physics of Solids, 23, 371-394. [71] Tamagnini C., Viggiani G., Chambon R., Desrues J. (1998). Evaluation of di erent strategies for the integration of hypoplastic constitutive equations. Application to the CLoE model. Mech. Cohesive{Frictional Materials, in print. [72] Tejchman J. (1994). Numerical study of localized deformation in a Cosserat continuum. Proc. Localization and Bifurcation Theory for Soils and Rocks, Aussois, Chambon et al. eds., Balkema, 257-274. 40
[73] Tejchman J. (1997). Modelling od shear localization and autogeneous dynamic e ects in granular bodies. Pubbl. Series of the Inst. of Soil Mech. and Rock Mech., University of Karlsruhe, vol. 140. [74] Terzaghi K. (1948). Theoretical Soil Mechanics. Wiley, New York. [75] Tobita Y. (1997). Importance of incremental nonlinearity in the deformation of granular materials. Proc. IUTAM Symp. on Mech. of Granular and Porous Materials, Fleck & Cocks eds., Kluwer, 139-150. [76] Tokuoka T. (1971). Yield conditions and ow rules derived from hypo{ elasticity. Arch. Rat. Mechanics and Analysis, 42, 239-252. [77] Truesdell C.A. (1956). Hypo{elastic shear. J. Appl. Physics, 27, 441-447. [78] Truesdell C.A., Noll W. (1965). The non{linear eld theories of mechanics. In: Encyclopedia of Physics, S. Flugge ed., Vol. III/3, Springer, Berlin. [79] Valanis K.C., Lee C.F. (1984). Endochronic theory of cyclic plasticity with applications. J. Appl. Mech., ASME, 51, 789-794. [80] Vardoulakis I. (1980). Shear band inclination and shear modulus of sands in biaxial tests. Int. Journal for Numerical and Analytical Methods in Geomechanics, 4, 103-119. [81] Vardoulakis I., Graf B. (1985). Calibration of constitutive models for granular materials using data from biaxial experiments. Geotechnique, 35(3), 299-317. [82] Viggiani G., Finno R.J., Harris W.W. (1994). Experimental observations of strain localisation in plane strain compression of a sti clay. Proc. Localization and Bifurcation Theory for Soils and Rocks, Aussois, Chambon et al. eds., Balkema, 189-198. [83] von Wol ersdor P.{A. (1996). A hypoplastic relation for granular materials with a prede ned limit state surface. Mech. Cohesive{Frictional Materials, 1, 251-271. [84] Wang C.{C. (1970). A new representation theorem for isotropic tensor functions, parts I and II. Archive for Rational Mechanics and Analysis, 36, 166-223. [85] Wang Z.{L., Dafalias Y.F., Shen C.K. (1990). Bounding surface hypoplasticity model for sand. J. Engng. Mech., ASCE, 116(5), 983-1002. 41
[86] Wu W. (1992). Hypoplasticity as a mathematical model for the mechanical behavior of granular materials. Publ. Series of the Institute of Soil and Rock Mechanics, University of Karlsruhe, n. 129. [87] Wu W. (1998). Rational approach to anisotropy of sand. Int. J. Num. Anal. Meth. Geomech., 22, 921-940. [88] Wu W., Kolymbas D. (1990). Numerical testing of the stability criterion for hypoplastic constitutive equations. Mechanics of Materials, 9, 243253. [89] Wu W., Sikora Z. (1991). Localized bifurcation in hypoplasticity. Int. Journal of Engineering Science, 29(2), 195-201. [90] Wu W., Sikora Z. (1992). Localized bifurcation of pressure sensitive dilatant granular materials. Mechanical Research Communications, 19(4), 289-299. [91] Wu W., Bauer E. (1993). A hypoplastic model for barotropy an pyknotropy of granular soils. In: Modern Approaches to Plasticity, Kolymbas ed., Elsevier, 225-245. [92] Wu W., Bauer E. (1994). A simple hypoplastic constitutive model for sand. Int. J. Num. Anal. Meth. Geomech., 18, 833-862. [93] Wu W., Niemunis A. (1994). Beyond invertibility surface in granular materials. Proc. Localization and Bifurcation Theory for Soils and Rocks, Aussois, Chambon et al. eds., Balkema, 113-126. [94] Wu W., Niemunis A. (1996). Failure criterion, ow rule and dissipation function derived from hypoplasticity. Mech. Cohesive{Frictional Materials, 1, 143-163. [95] Wu W., Bauer E., Kolymbas D. (1996). Hypoplastic constitutive model with critical state for granular materials. Mechanics of Materials, 23, 45-69. [96] Wu W., Niemunis A. (1997). Beyond failure in granular materials. Int. J. Num. Anal. Meth. Geomech., 21, 153-174.
42
A Gudehus/Bauer K{hypoplastic model The constitutive functions entering in the Gudehus/Bauer [38, 1] K{hypoplastic model, eqs. (13){(14), are given by: a := c + c k^ k 1[1 + cos(3)] (88) and: 1
1
2
(1
hs 3p ;n ei 1 + ei fb(p) := nh ec ei i hs e fe(e; p) := ec e ; e d fd (e; p) := e ; e c d )
(89)
0
0
with:
(90) (91)
n 3 p ei := ei exp ; h s n ec := ec exp ; 3hp s n ed := ed exp ; 3hp s
and:
0
(92)
0
(93)
0
(94)
1 1 1 e i ; ed p hi := c + 3 ; e ; e 3c c d r 3 3 ; sin 3 3 + sin c c c := 8 sin c := 8 sin c c 2 1
1
0
0
0
0
(95)
1
(96)
2
The quantities hs (granulate hardness), c (critical friction angle), ei , ei , ec , ed , , and n appearing in eqs. (89){(96) are material constant, to be determined from standard laboratory tests, or modi ed classi cation tests. 0
0
0
0
B von Wol ersdor K{hypoplastic model The functions de ning barotropy and pycnotropy factors (fb, fe and fd), as well as those de ning the characteristic void ratios (ei , ec and ed) which 43
enter in the von Wol ersdor [83] K{hypoplastic model are still given by eqs. (89){(91) and eqs. (92){(94), respectively. The two functions a and F appearing in eq. (15) are de ned as follows:
p c) a := p3 (3 ;sinsin c 2 2 = 1 2 ; tan F := 8 tan + p ; p1 tan 2 + 2 tan cos(3) 2 2 2
2
with:
p
tan := 3 k^ k
1 2
(97) (98) (99)
As in Gudehus/Bauer model, c represents the critical friction angle of the material.
44
