a general formulation of hypoplasticity

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech., 2004; 28:1461–1478 (DOI: 10.1002/nag.394)

A general formulation of hypoplasticity

J. Lanier, D. Caillerie, R. Chambonn,y, G. Viggiani, P. B!eesuelle and J. Desrues

Laboratoire 3S, UJF, INPG, CNRS, Grenoble, France

SUMMARY

A general formulation of the theory of hypoplasticity is given, under the assumption that Cauchy stress isthe only state variable. Such a formulation allows to discuss the relation between the so-called out-of-axisand in-axis moduli. It is proved that, in general, the out-of-axis moduli are independent from the in-axismoduli, which allows the former to be calibrated making use of experimental shear band data, as it is donein CLoE-hypoplasticity. The implicit assumptions made in K-hypoplasticity are detailed for two particularmodels of this family. Copyright # 2004 John Wiley & Sons, Ltd.

KEY WORDS: constitutive equations; objectivity; shear moduli; hypoplasticity

1. INTRODUCTION

The theory of hypoplasticity is a relatively recent class of constitutive models which have beendeveloped to mathematically describe the non-linear, irreversible behaviour of geomaterials. Ascompared to elastoplasticity, its distinctive feature is the continuously non-linear dependence ofthe material response on the direction of strain rate. Moreover, material response is describedwithout introducing any decomposition of the strain rate tensor into a reversible and anirreversible part.

Plasticity with a single mechanism belongs to the class of incrementally bi-linear models,whereas in multi-mechanism plasticity, incremental non-linearity takes the form of multi-linearity (see Reference [1] or [2] for a general classification). In hypoplasticity, a thoroughlynon-linear relation is given between the stress and strain rate. More specifically, the stress rate iswritten as the sum of a first term which is linear with respect to the strain rate, and a second termwhich accounts for the non-linearity in the strain rate simply through the norm of the strainrate. A general outline of the theory of hypoplasticity was laid down by Kolymbas [3], andseveral review papers followed thereafter, the most recent of which are those by Wu andKolymbas [4] and Tamagnini et al. [5]. In fact, two different formulations of hypoplasticity havebeen given over the last 15 years with the specific objective of modelling the behaviour ofgranular materials. The first was developed in Karlsruhe after the pioneering work of Kolymbas(e.g. Reference [6]) and will be referred in the following as K-hypoplasticity. The secondoriginated in Grenoble from the work of Chambon and Desrues [7–9] under the general name of

Received 1 July 2003Revised 12 July 2004Copyright # 2004 John Wiley & Sons, Ltd.

yE-mail: [email protected]

nCorrespondence to: R. Chambon, Laboratoire 3S, UJF, INPG, CNRS, Grenoble, France.

CLoE-hypoplasticity. Indeed, these two approaches share the same basic mathematicalstructure, and therefore they are similar in various respects. However, some importantdifferences are apparent in their original formulation, as well as in their respective subsequentdevelopments. A thorough comparison between CLoE and K-hypoplasticity was recentlyprovided by Tamagnini et al. [5, 10].

Let us first emphasize that the aim of this work is neither to give simple examples ofhypoplastic models, nor to study the capabilities and limitations of hypoplastic constitutiveequations in modelling geomaterials, nor to evaluate the influence of the out-of-axis shearmoduli on the occurrence of shear banding. The reader interested in the first topic can study forinstance [11] for a simple Mohr–Coulomb hypoplastic model or Reference [12] for a simple VonMises hypoplastic model. Capabilities and limitations (for instance for small cycle loadingpaths) of such models have been clearly discussed elsewhere e.g. Chambon et al. [9]. As for theinfluence of the out-of-axis shear moduli, it has been extensively studied in References[10, 13, 14] or in Reference [5]. In Reference [10], in particular, the shear modulus forinfinitesimal deviations from a proportional loading path has been compared for the twofamilies of hypoplastic models.

In fact, the original motivation for this paper came from several, recurring discussionsbetween us (the authors of the paper) and Kolymbas and Herle. As a matter of fact, there is oneimportant difference between the two families of hypoplastic models, which is the very approachadopted in the construction of the two theories. On the one hand, K-hypoplasticity has beendeveloped in a deductive way, starting from the representation theorem for isotropic tensor-valued functions, which ensures the objectivity of the corresponding constitutive equation. Onthe other hand, CLoE-hypoplastic models have been obtained by rather following a constructiveapproach, using interpolation between experimentally known responses upon a set of basicloading paths, and using localization observations to fit the out-of-axis shear moduli. Usingobjectivity requirements, it can be proved that the out-of-axis shear moduli are in fact ‘free’ (i.e.independent from the in-axis shear moduli) except for isotropic or for axisymmetric stress states[9, 15]. Inspired by early works about the so-called deformation theory of plasticity [16], vertexmodels [17, 18] and other models [19, 20], such a freedom is extensively used in CLoE-hypoplastic models [13, 14]. This is a salient difference with K-hypoplastic models, for which theout-of-axis shear moduli are implicitly assumed to depend on the in-axis moduli. In a recentpaper [21], Kolymbas and Herle state that: ‘. . . the constitutive law of an isotropic simplematerial can be completely fixed (i.e. calibrated) with rectilinear deformation’. The mainobjective of this paper is to conclusively show, starting from the representation theorem, thatthis is not the case. That is, in general, the in-axis and out-of-axis shear moduli are independentfrom each other.

The structure of the paper is as follows. Starting from the general expression for a rate-independent constitutive equation in rate form where the Cauchy stress is the only statevariable, the objectivity requirement yields a general form of hypoplasticity. This is detailed inSection 2. Section 3 details the resulting expression of the moduli as a function of the stress bymeans of some arbitrary functions. This allows us to point out which are the restrictionsapplying for the choice of the out-of-axis moduli. Section 4 shows how to get the arbitraryfunctions involved in the general form of a hypoplastic model, assuming that the moduli areknown. In Section 5, the above results are further illustrated through a simple example. Then,both CLoE and K-hypoplasticity are discussed as special cases of the general formulation givenin this paper. In particular, it is detailed why the latter does not share the same freedom as

Copyright # 2004 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2004; 28:1461–1478

J. LANIER ET AL.1462

CLoE-hypoplasticity with reference to the out-of-axis moduli. Some concluding remarks arefinally made in Section 6.

As for the notation, a component is denoted by the name of the tensor (or vector)accompanied with tensorial indices. All tensorial indices are in lower position as there is no needin the following of a distinction between covariant and contravariant components. Thesummation convention with respect to repeated tensorial indices is adopted throughout. Inorder to avoid confusions, squares are systematically denoted with parenthesis. For any secondorder tensor A; the components of which are Aij ; the scalar quantity trðAÞ ¼ Akk denotes thetrace of A:

2. GENERAL FORMULATION OF HYPOPLASTICITY

2.1. Basic assumptions and objectivity requirements

We first assume that an objective (e.g. Jaumann) stress rate sr is a function of the (Cauchy) stresss and of the symmetric part of the velocity gradient D

srðtÞ ¼ FðsðtÞ;DðtÞÞ ð1Þ

where t is the current time, and

Dij ¼1

2

@vi@xj

þ@vj@xi

� �ð2Þ

where v is the velocity of a material point as a function of its co-ordinates x: Let us comment alittle this first assumption.

* In this framework the Cauchy stress is the only state variable, which is then assumed torepresent the whole history up to the current time. However, the following developmentsare still valid if we add to this basic description some scalar state variables such as thedensity or the void ratio. Adding other tensorial state variables, on the contrary, wouldrequire other developments.

* Constitutive equation (1) can be seen as a set of ordinary differential equations, giving thestress sðt1Þ at any time t1 provided that* the initial value of the stress sðt0Þ is known,* the kinematics history (which means the whole velocity gradient: its symmetric part as

well as its skew symmetric part) is known at any time t 2 ½t0; t1�:* In Equation (1), no initial configuration is explicitly mentioned, which means that time t0

can be chosen arbitrarily.

The principle of frame indifference (or objectivity) implies that function F in Equation (1)has to meet the following condition (see for instance Reference [22]):

QTFðs;DÞQ ¼ FðQTsQ;QTDQÞ ð3Þ

for any arbitrary orthogonal tensor Q:This implies that sr is the value of an isotropic function of both s and D:


GENERAL FORMULATION OF HYPOPLASTICITY 1463

2.2. General hypoplastic models

In order to define hypoplasticity, two assumptions are added.

2.2.1. First general assumption. It is first assumed that the material is rate independent.Consequently, function F has to be positively homogeneous of the degree one with respectto D: Let us define the direction d of the tensor D=0; as

d ¼D

jjDjjð4Þ

jjDjj; being the Euclidean norm of D

jjDjj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitr½ðDÞ2�

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiDijDij

pð5Þ

Note that d is a unit tensor, that is tr½ðdÞ2� ¼ 1; and jjd jj ¼ 1:This implies

Fðs;DÞ ¼ Gðs; dÞjjDjj ð6Þ

Clearly function G has to be like F an isotropic function of its arguments. The theoremsof representation for isotropic functions [23–27] yield the more general expression for thefunction G

Gðs; dÞ ¼ g0dþ g1d þ g2ðdÞ2 þ g3sþ g4ðsÞ2

þ g5ðdsþ sdÞ þ g6½dðsÞ2 þ ðsÞ2d�

þ g7½ðdÞ2sþ sðdÞ2� þ g8½ðdÞ2ðsÞ2 þ ðsÞ2ðdÞ2� ð7Þ

where d is the second-order identity tensor, and gi (i 2 0; . . . ; 8) are nine scalar functions of theinvariants and joint invariants of d and s; namely trðdÞ; tr½ðdÞ2� ¼ 1; tr½ðdÞ3�; trðsÞ; tr½ðsÞ2�;tr½ðsÞ3�; trðsdÞ; tr½ðsÞ2d�; tr½sðdÞ2�; and tr½ðsÞ2ðdÞ2�: Let us emphasize at this step that these ninefunctions are arbitrary and independent from each other. The nine tensors used in Equation (7),namely: d; d; ðdÞ2; s; ðsÞ2; dsþ sd; dðsÞ2 þ ðsÞ2d; ðdÞ2sþ sðdÞ2 and ðdÞ2ðsÞ2 þ ðsÞ2ðdÞ2 are calledgenerators. Incidentally, Wang has shown in Reference [25] that the last term can be skippedwithout loss of generality.

Equation (7) implies a number of properties, as detailed in References [9, 15]:

1. If the stress s is isotropic, then G is an isotropic function of d:2. If the stress s has two equal principal values, then G is transversely isotropic with the same

symmetry axis as the one of the stress tensor.3. Generally function G as a function of d is cross isotropic with respect to the principal

planes of the stress tensor.

These properties have been extensively used in the development of CLoE models which thenare objective [7, 9]. In Sections 3 and 4 using both the assumptions made in the present section,we will follow an other (but completely equivalent) route. This will allow us to see clearly therestrictive assumptions made in some hypoplastic constitutive equations, for instance inK-hypoplasticity.



2.2.2. Second general assumption. Assuming now that function G is affinez with respect to d;and generalizing the method used by Stutz [28] for hypoelasticity, implies the followingsimplifications:

* g2 ¼ g7 ¼ g8 ¼ 0 because all the corresponding generators involve d2 and consequentlycannot appear in the final expression.

* g1; g5 and g6 must only depend on trðsÞ; tr½ðsÞ2� and tr½ðsÞ3� since the correspondinggenerators are linearly dependent on d:

* g0; g3 and g4 have necessarily the following form 8i 2 0; 3; 4

gi ¼ ai þ bi trðdÞ þ ci trðsdÞ þ di tr½ðsÞ2d� ð8Þ

where ai; bi; ci and di are functions of trðsÞ; tr½ðsÞ2� and tr½ðsÞ3�:

Finally, the constitutive equation reads

srðtÞ ¼ ½LðsÞd þ BðsÞ�jjDjj ¼ LðsÞDþ BðsÞjjDjj ð9Þ

where the fourth-order tensor L and the second-order tensor B depend only on the stresstensor s:

Tensor valued isotropic function B has the general expression

BðsÞ ¼ a0dþ a3sþ a4ðsÞ2 ð10Þ

The linear part LðsÞD can be written as follows:

LðsÞD ¼fb0 trðDÞ þ c0 trðsDÞ þ d0 tr½ðsÞ2D�gd

þ fb3 trðDÞ þ c3 trðsDÞ þ d3 tr½ðsÞ2D�gs

þ fb4 trðDÞ þ c4 trðsDÞ þ d4 tr½ðsÞ2D�gðsÞ2

þ g1Dþ g5ðDsþ sDÞ þ g6½ðsÞ2DþDðsÞ2� ð11Þ

Let us emphasize here that clearly the general hypoplastic model written in Equations (9)–(11)depends on 15 arbitrary scalar functions of the stress invariants, namely a0; a3; a4; b0; b3; b4; c0;c3; c4; d0; d3; d4; g1; g5; and g6:

3. EXPRESSION OF THE MODULI AND THEIR DEPENDENCE

From Equation (10), it is easy to retrieve the properties of tensor B; see Section 4.3 ofReference [9].

Similarly to the above-mentioned properties of function G it can be proved (see References[9, 14] and) that the fourth-order tensor L is cross isotropic with respect to the principal planesof the stress tensor. Using the classical representation of symmetric second-order tensors as

zAnalytically, affine transformations are represented in the matrix form f ðxÞ ¼ Axþ b; where the determinant detðAÞ ofa square matrix A is not 0.



six-components vectors, a fourth-order tensor linking two symmetric second-order tensors canbe written as a 6� 6 matrix. Using this representation L becomes matrix ½L�; and writing ½L� inthe stress principal axes yields

½L� ¼

L1111 L1122 L1133 0 0 0

L2211 L2222 L2233 0 0 0

L3311 L3322 L3333 0 0 0

0 0 0 2L1212 0 0

0 0 0 0 2L2323 0

0 0 0 0 0 2L3131

2666666666664

3777777777775

ð12Þ

This defines the moduli of a hypoplastic model. L1111; L1122; L1133; L2211; L2222; L2233;L3311; L3322 and L3333 are called the in-axis moduli whereas 2L1212; 2L2323 and 2L3131 arecalled the out-of-axes moduli. Let us emphasize that they have been named moduli for simplicitybut they have nothing to do with elasticity. For instance, we will see in the following quiteclearly that matrix ½L� is a non-symmetric one. In the axes chosen sij are the components of thestress and s12 ¼ s23 ¼ s31 ¼ 0: In the following, s11; s22 and s33 are then the principal valuesof the stress tensor. Similarly Dij are the components of the symmetric part of the velocitygradient and

trðDÞ ¼ D11 þD22 þD33 ð13Þ

trðsDÞ ¼ s11D11 þ s22D22 þ s33D33 ð14Þ

tr½ðsÞ2D� ¼ ðs11Þ2D11 þ ðs22Þ

2D22 þ ðs33Þ2D33 ð15Þ

trðsÞ ¼ s11 þ s22 þ s33 ð16Þ

3.1. Expression of the moduli

Identifying the product of Equation (12) with D and Equation (11) yields the followingequations giving the 12 non-zero components of ½L� (i.e. the moduli) as linear functions ofthe 12 arbitrary scalar functions of the stress invariants b0; b3; b4; c0; c3; c4; d0; d3; d4; g1; g5

and g6:

L1111

L2222

L3333

2664

3775 ¼

1 s11 ðs11Þ2

1 s22 ðs22Þ2

1 s33 ðs33Þ2

26664

37775

b0

c0

d0

2664

3775þ

s11 ðs11Þ2 ðs11Þ

3


3


3

26664

37775

b3

c3

d3

2664

3775

þ

ðs11Þ2 ðs11Þ

3 ðs11Þ4

ðs22Þ2 ðs22Þ

3 ðs22Þ4

ðs33Þ2 ðs33Þ

3 ðs33Þ4

26664

37775

b4

c4

d4

2664

3775þ

1 2s11 2ðs11Þ2

1 2s22 2ðs22Þ2

1 2s33 2ðs33Þ2

26664

37775

g1

g5

g6

2664

3775 ð17Þ



L1122

L2211

L2233

2664

3775 ¼

1 s22 ðs22Þ2

1 s11 ðs11Þ2

1 s33 ðs33Þ2

26664

37775

b0

c0

d0

2664

3775þ

s11 s11s22 s11ðs22Þ2

s22 s22s11 s22ðs11Þ2

s22 s22s33 s22ðs33Þ2

26664

37775

b3

c3

d3

2664

3775

þ

ðs11Þ2 ðs11Þ

2s22 ðs11Þ2ðs22Þ

2

ðs22Þ2 ðs22Þ


2

ðs22Þ2 ðs22Þ


2

26664

37775

b4

c4

d4

2664

3775 ð18Þ

L3322

L3311

L1133

2664

3775 ¼

1 s22 ðs22Þ2

1 s11 ðs11Þ2

1 s33 ðs33Þ2

26664

37775

b0

c0

d0

2664

3775þ

s33 s33s22 s33ðs22Þ2

s33 s33s11 s33ðs11Þ2

s11 s11s33 s11ðs33Þ2

26664

37775

b3

c3

d3

2664

3775

þ

ðs33Þ2 ðs33Þ


2

ðs33Þ2 ðs33Þ


2

ðs11Þ2 ðs11Þ


2

26664

37775

b4

c4

d4

2664

3775 ð19Þ

2L1212

2L2323

2L3131

2664

3775 ¼

1 s11 þ s22 ðs11Þ2 þ ðs22Þ

2

1 s22 þ s33 ðs22Þ2 þ ðs33Þ

2

1 s33 þ s11 ðs33Þ2 þ ðs11Þ

2

26664

37775

g1

g5

g6

2664

3775 ð20Þ

It appears that if the arbitrary functions b0; b3; b4; c0; c3; c4; d0; d3; d4; g1; g5 and g6 areknown, then the 12 non-zero components of ½L� are known. Conversely assuming that 12 non-zero components of ½L� are given, Equations (17)–(20) can be seen as a system of 12 linearequations in 12 unknowns: b0; b3; b4; c0; c3; c4; d0; d3; d4; g1; g5 and g6:

After some algebra it is possible to get the determinant of this system, which reads

Det ¼ ½ðs11 � s22Þðs22 � s33Þðs33 � s11Þ�7 ð21Þ

This proves that it is possible to choose arbitrary values for the moduli except if two (or three)principal stresses are equal. A method to get in any case the values of the arbitrary functions,assuming that the moduli are known, is detailed in Section 4.

There exist some relations between the 12 moduli only for axisymmetric or isotropic stressstates, which are studied in the following Section 3.2.

However, for instance, some properties of symmetry must hold in the general case due tothe arbitrary numbering of the three principal stresses. For instance a permutation between the



values of s11 ¼ s1 and s22 ¼ s2 implies the following relations:

L1111ðs1;s2;s3Þ ¼ L2222ðs2; s1; s3Þ

L2211ðs1;s2;s3Þ ¼ L1122ðs2; s1; s3Þ

L1133ðs1;s2;s3Þ ¼ L2233ðs2; s1; s3Þ

L3311ðs1;s2;s3Þ ¼ L3322ðs2; s1; s3Þ

L2323ðs1;s2;s3Þ ¼ L3131ðs2; s1; s3Þ

ð22Þ

A circular permutation of s11 ¼ s1; s22 ¼ s2 and s33 ¼ s3 implies

L1111ðs1; s2; s3Þ ¼ L2222ðs2;s3;s1Þ ¼ L3333ðs3;s1;s2Þ




ð23Þ

3.2. Particular cases

If two principal stresses are equal, say s11 ¼ s1 ¼ s22 ¼ s2; then it is not possible to choosearbitrary values for the moduli Lijkl : By inspecting Equations (17)–(20), it appears clearly that inthis case

L1111 ¼ L2222

L1122 ¼ L2211

L3311 ¼ L3322

L1133 ¼ L2233

L2323 ¼ L3131

2L1212 ¼ L2222 � L1122 ¼ L1111 � L2211

ð24Þ

which are exactly the conditions proved in Reference [15] and given in References [7, 9], inthis case.

If the three principal stresses are equal, then Equations (17)–(20) yield

L1111 ¼ L2222 ¼ L3333

L1122 ¼ L2233 ¼ L3311 ¼ L2211 ¼ L3322 ¼ L1133

2L1212 ¼ 2L2323 ¼ 2L3131 ¼ L2222 � L1122

ð25Þ

which, also in this case, are exactly the conditions proved in Reference [15] and used inReferences [7, 9, 14].

From an algebraic point of view, Equations (24) and (25) are the compatibility conditionsnecessary to inverse the linear system of 12 equations detailed in Equations (17)–(20), when twoor three principal stresses are equal. Except for these particular cases, the out of axis moduli2L1212; 2L2323 and 2L3131 do not depend on the other moduli.



4. INVERTING THE RELATION BETWEEN THE MODULI ANDTHE ARBITRARY FUNCTIONS

Given the arbitrary functions b0; b3; b4; c0; c3; c4; d0; d3; d4; g1; g5 and g6; it is possible to getthe moduli through Equations (17)–(20), which implies that these moduli obey symmetryconditions (e.g. Equations (22) and (23)). Conversely, assuming that the moduli are known andobey all the symmetry conditions above-mentioned, it is possible to get the corresponding valuesof the arbitrary functions b0; b3; b4; c0; c3; c4; d0; d3; d4; g1; g5 and g6 using the following method.

4.1. First step

4.1.1. General case. Let us assume first that s11=s22=s33=s11:Starting from the values of 2L1212; 2L2323 and 2L3131; and using Equations (20) yields

g5 ¼ 2L1212½ðs22Þ

2 � ðs11Þ2� þ L3131½ðs11Þ

2 � ðs33Þ2� þ L2323½ðs33Þ

2 � ðs22Þ2�

ðs11 � s22Þðs22 � s33Þðs33 � s11Þð26Þ

g6 ¼ �2L1212ðs22 � s11Þ þ L3131ðs11 � s33Þ þ L2323ðs33 � s22Þ

ðs11 � s22Þðs22 � s33Þðs33 � s11Þð27Þ

g1 ¼ 23½L1212 þ L2323 þ L3131 � g5 trðsÞ � g6 trððsÞ2Þ� ð28Þ

Since the moduli obey the above-mentioned symmetry conditions, then g1; g5 and g6 asobtained by relations (26)–(28) are symmetric functions with respect to the three principalstresses s11; s22; and s33:

4.1.2. Particular cases. If two principal stresses are equal, say s11 ¼ s22 (the other two cases canbe easily extrapolated), then L2323 ¼ L3131: In this case there is an infinite number of solutionsbecause the second and third equations of (20) are exactly the same. Functions g1; g5 and g6 arenot completely defined, yet they can be computed using continuity requirements, depending onthe values of lims11!s22 ðL2323 � L3131Þ=ðs11 � s22Þ:

Similar arguments hold when s11 ¼ s22 ¼ s22: In this case the three equations of (20) areidentical.

4.2. Second step

4.2.1. General case. Let us define L01111; L

02222 and L0

3333 as

L01111 ¼ L1111 � g1 � 2s11g5 � 2ðs11Þ

2g6

L02222 ¼ L2222 � g1 � 2s22g5 � 2ðs22Þ

2g6

L03333 ¼ L3333 � g1 � 2s33g5 � 2ðs33Þ

2g6

ð29Þ



Using Equations (26)–(28), L01111;L

02222 and L0

3333 are known. We can rewrite Equation (17)as follows:

L01111

L02222

L03333

2664

3775 ¼

1 s11 ðs11Þ2

1 s22 ðs22Þ2

1 s33 ðs33Þ2

26664

37775

b0

c0

d0

2664

3775þ


3


3


3

26664

37775

b3

c3

d3

2664

3775

þ

ðs11Þ2 ðs11Þ

3 ðs11Þ4

ðs22Þ2 ðs22Þ

3 ðs22Þ4

ðs33Þ2 ðs33Þ

3 ðs33Þ4

26664

37775

b4

c4

d4

2664

3775 ð30Þ

In order to obtain b0; b3; b4; c0; c3; c4; d0; d3 and d4 as functions of L01111; L

02222; L

03333; L1122;

L1133; L2211; L2233; L3311 and L3322; let us introduce the three following functions:

X0ðxÞ ¼ b0 þ c0xþ d0ðxÞ2



ð31Þ

Equations (30), (18) and (19) can be rewritten as follows:

L01111 ¼ X0ðs11Þ þ s11X3ðs11Þ þ ðs11Þ

2X4ðs11Þ

L2211 ¼ X0ðs11Þ þ s22X3ðs11Þ þ ðs22Þ2X4ðs11Þ


ð32Þ



2X4ðs22Þ


ð33Þ




2X4ðs33Þ

ð34Þ

Starting from Equation (32) we get

X4ðs11Þ ¼ �L01111½s22 � s33� þ L3311½s11 � s22� þ L2211½s33 � s11�

ðs11 � s22Þðs22 � s33Þðs33 � s11Þð35Þ

X3ðs11Þ ¼L01111½ðs22Þ

2 � ðs33Þ2� þ L3311½ðs11Þ

2 � ðs22Þ2� þ L2211½ðs33Þ

2 � ðs11Þ2�

ðs11 � s22Þðs22 � s33Þðs33 � s11Þð36Þ

X0ðs11Þ ¼ 13½L0

1111 þ L2211 þ L3311 � X3ðs11Þ trðsÞ � X4ðs11Þ trððsÞ2Þ� ð37Þ

Similarly, it is possible to obtain X0ðs22Þ; X3ðs22Þ and X4ðs22Þ; using Equation (33) andX0ðs33Þ; X3ðs33Þ and X4ðs33Þ; using Equation (34).



Finally, writing Equation (31) for x ¼ s11; x ¼ s22 and x ¼ s33 yields

X0ðs11Þ ¼ b0 þ c0s11 þ d0ðs11Þ2



ð38Þ

which are similar to Equation (32). This consequently yields

d0 ¼ �X0ðs11Þðs22 � s33Þ þ X0ðs33Þðs11 � s22Þ þ X0ðs22Þðs33 � s11Þ

ðs11 � s22Þðs22 � s33Þðs33 � s11Þð39Þ

c0 ¼X0ðs11Þ½ðs22Þ

2 � ðs33Þ2� þ X0ðs33Þ½ðs11Þ

2 � ðs22Þ2� þ X0ðs22Þ½ðs33Þ

2 � ðs11Þ2�

ðs11 � s22Þðs22 � s33Þðs33 � s11Þð40Þ

b0 ¼ 13½X0ðs11Þ þ X0ðs22Þ þ X0ðs33Þ � c0 trðsÞ � d0 trððsÞ2Þ� ð41Þ

Functions b3; b4; c3; c4; d3 and d4 can be obtained in a similar manner, but the correspondingresults have not been given here for simplicity.

In order to prove that the solutions are symmetric functions with respect to the values of thethree principal stresses s11; s22; and s33; let us consider Figure 1, showing the deviatoric planecorresponding to a mean stress denoted p: The stress deviator s is defined as

sij ¼ sij � pdij ð42Þ

12

3

Zone 3

Zone 4 Zone 5

σ22> σ33 > σ11

σ33> σ22 > σ11 σ33> σ11 > σ22

Zone 6

σ11> σ33 > σ22

Zone 1

σ11> σ22 > σ33 Zone 2

σ22> σ11 > σ33

M4 M5

M3

M2 M1

M6

M

ϕ

Figure 1. The deviatoric plane and the Lode angle.



The second invariant of the stress, q is defined as

q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi23tr½ðsÞ2�

qð43Þ

Finally, the Lode angle denoted j is defined by

cosð3jÞ ¼ffiffiffi6

p tr½ðsÞ3�

ftr½ðsÞ2�g3=2ð44Þ

Then, we can choose

s11 ¼ pþ q cos j ð45Þ

and

s22 ¼ pþ q cos jþ2p3

� �ð46Þ

s33 ¼ pþ q cos j�2p3

� �ð47Þ

The deviatoric plane is divided into six zones, depending on the order of the three values s11;s22; and s33 as it is depicted in Figure 1. The six points Mi; i 2 f1; . . . ; 6g correspond to the sameprincipal stress values but numbered in different ways, allowing to get all the possiblepermutations of these three values.

Now, let us consider points M1 and M2; which correspond to a permutation between s11 ands22: This means that, in order to get the solutions for M2 starting from the solutionscorresponding to point M1; we have to make the following permutations: ½L2323; L3131; L1111;L2222; L2211; L1122; L3311; L1133; L3322; L2233� ! ½L3131; L2323; L2222; L1111; L1122; L2211; L3322; L2233;L3322; L2323�: Consequently, L0

1111 and L02222; are replaced by L0

2222 and L01111; see Equation (29).

Equation (18) becomes Equation (19) and conversely. Similarly, X0ðs11Þ is replaced by X0ðs22Þand conversely. Finally, by looking at Equations (39)–(41), one realizes that functions b0; b3; b4;c0; c3; c4; d0; d3 and d4; corresponding to pointM2; are the same as those corresponding to pointM1; except that we have performed a permutation between s11 and s22:

Similar arguments can be used to study the other permutations and finally it is proved thatfunctions b0; b3; b4; c0; c3; c4; d0; d3; d4 are symmetric functions with respect to the three stresscomponents s11; s22; and s33:

4.2.2. Particular cases. Let us study what happens if two principal stresses are equal, forexample if s11 ¼ s22 (the other two cases can be easily extrapolated). In this case, Equations (24)hold and moreover using definitions (29) and the results of Section 4.1 yields

L01111 ¼ L1111 � 2L1212 ¼ L2211 ¼ L2211 ¼ L2222 � 2L1212 ¼ L0

2222 ð48Þ

The first two equations of (32) become coincident and the set of Equations (33) is exactly thesame as (32), which means that it is possible to compute X0ðs11Þ ¼ X0ðs22Þ; X3ðs11Þ ¼ X3ðs22Þ;and X4ðs11Þ ¼ X4ðs22Þ: The first two equations of (34) are exactly the same, so it is possible tocompute X0ðs33Þ; X3ðs33Þ; and X4ðs33Þ: As in Section 4.1.2, it is possible to get these values usingcontinuity requirements. The results are depending on the values of lims11!s22 ðL1111 � L2222Þ=ðs11 � s22Þ and similar quantities.



Finally, the first two equations of (38) are identical and it is possible to compute b0; c0 and d0;using continuity requirements. The results are depending on the value of lims11!s22ðX0ðs11Þ � X0ðs22ÞÞ=ðs11 � s22Þ; which is known at this step.

Similar arguments can be used if s11 ¼ s22 ¼ s33; but are not detailed here for simplicity.

4.3. Conclusions

It is useful at this stage to summarize our results. Using either the general equations (10) and(11) or the moduli defined in (12) and the components of B is completely equivalent. It has beenproved in the above sections that it is always possible, provided that the symmetry conditionsare met, to obtain both the moduli from the scalar functions and vice versa.

5. APPLICATIONS

5.1. A simple example

In order to better understand the above conclusions, let us consider a simple constitutiveequation which belongs to the general class defined by Equation (11). The constitutive equationis obtained by giving to the 12 scalar functions the following (constant) values:

b0 ¼ 1; c0 ¼ 3; d0 ¼ 2

b3 ¼ 2; c3 ¼ 3; d3 ¼ 1

b4 ¼ 10; c4 ¼ 1; d4 ¼ 0

g1 ¼ 1; g5 ¼ 2; g6 ¼ 3

ð49Þ

Using Equations (45)–(47) and then substituting s11; s22 and s33 in Equations (17)–(20)allows to obtain the variation of the moduli with respect to j: This means that we can study theevolution of these moduli for stress states having the same mean stress and the same secondinvariant, differing only in terms of the Lode angle. Figures 2–4 show the variation of thedifferent moduli, clearly illustrating the symmetry conditions.

0

20

40

60

80

100

120

140

0 60 120 180 240 300 360

ϕ(ϕ(°)

L1111L2222L3333

Figure 2. Example of variation of the in-axis moduli L1111; L2222; and L3333 asa function of the Lode angle j:



5.2. The example of CLoE-hypoplastic models

All CLoE-hypoplastic models have been built up by specifying the moduli, rather than thescalar stress functions. However, since the corresponding equations meet all the symmetryproperties, these models are, as proved above, completely objective.

The aim of CLoE was to start from experimental data in order to be able to calibrate directlythe model. This was achieved by developing at the same time both a constitutive routine and acalibration routine (see Reference [9] for instance).

The moduli for a general stress state are obtained by interpolation from particular stressstates, corresponding to axisymmetric compression and extension states. For these last twostates, symmetries on the moduli are enforced to satisfy Equations (24). For a general stressstate, these conditions are not imposed.

Figure 3. Example of variation of the in-axis moduli L1122; L2211;L2233;L3322;L3311; and L1133 asa function of the Lode angle j:

0

5

10

15

20

25

0 60 120 180 240 300 360

ϕ(ϕ(°)

2L1212 =

2L2323 =

2L3131 =

Figure 4. Example of variation of the out-of-axis moduli 2L1212; 2L2323; and L3131 asa function of the Lode angle j:



Simplified versions of CLoE-hypoplasticity have been also developed. In these models(referred to as MiniCLoE, see Reference [11] for instance) the concept of interpolations of theconstitutive tensors from the image points along the basic paths is abandoned, and}in analogyto K-hypoplasticity}the tensors L and B in Equation (9) are defined explicitly.

A general characteristic of CLoE-hypoplastic models is the use of strain localization to getsome insight about the out-of-axis moduli. As proved above, this is possible provided somerestrictions are enforced. Clearly it remains a great freedom to choose these out-of-axis moduli,and depending on the studied materials we used different formulations (let us suggest to thereader to compare for instance the model used in Reference [13] and the one described inReference [14]). Although this approach can be also followed in the framework ofelastoplasticity [20], it is our opinion that the theory of hypoplasticity provides a moreconvenient and possibly simpler framework to develop such a possibility.

Usually the moduli are defined in one of the six zones of Figure 1 and the symmetryconditions (described in Section 4.2.1 and illustrated in Figures 2–4) are used to deduce thesemoduli in the other zones (see Reference [13]).

5.3. The example of K-hypoplastic models

They are many versions of K-hypoplastic models, however they all share some commonproperties. One of them which is of interest here, is the low number of parameters used inEquation (7). Multiplication by functions depending on the pressure and/or the void ratio, suchas it is suggested for instance in Reference [29], does not change the mathematical structure ofthe constitutive equation. It is then a priori clear that the 12 moduli cannot be ‘free’ and that inaddition to the relations coming from the symmetry conditions discussed above, there is someimplicit relations between the moduli. For instance, for models using only 4 materialparameters, 2 of them related with the first term of Equation (9), we will find a priori 10 relationsbetween the moduli.

In the following we consider two K-Hypoplastic models: the oldest one (to our knowledge)called in the following Kolymbas model [30], and one of the most used called in the followingthe Wu model (first detailed in Reference [31]).

5.3.1. Kolymbas model. This model reads

srðtÞ ¼ C1ðDsþ sDÞ þ C2 trðsDÞdþ C3sþ

C4

trðsÞðsÞ2

� �jjDjj ð50Þ

Identifying the two first terms of Equations (50) and (11) yields

b0 ¼ b3 ¼ b4 ¼ c3 ¼ c4 ¼ d0 ¼ d3 ¼ d4 ¼ g1 ¼ g6 ¼ 0

g5 ¼ C1

c0 ¼ C2

ð51Þ

Note that all the functions are constant, like in our simple example described inSection 5.1. From Equations (17)–(20) one can work out 10 independent relations betweenthe moduli implicitly assumed in this model. Firstly we find 6 relations, which are independent



on the stress

L2211 ¼ L3311

L1122 ¼ L3322

L2233 ¼ L1133

L1111 þ L2222 ¼ L2211 þ L1122 þ 4L1212

L2222 þ L3333 ¼ L2233 þ L3322 þ 4L2323

L3333 þ L1111 ¼ L3311 þ L1133 þ 4L3131

ð52Þ

Moreover, 4 other independent relations involve explicitly the stress

L2211

s11¼

L3322

s22¼

L2233

s33ð53Þ

L1212

s11 þ s22¼

L2323

s22 þ s33¼

L3131

s33 þ s11ð54Þ

Other sets of 10 independent relations can be obtained by combining Equations (52)–(54).Note that relations (53) have already been given in Reference [21].

5.3.2. Wu model. This model reads

srðtÞ ¼ E1 trðsÞDþ E2 trðsDÞ

trðsÞsþ ½E3ðsÞ2 þ E4ðsÞ2�jjDjj ð55Þ

Identifying the two first terms of Equations (55) and (11) yields

b0 ¼ b3 ¼ b4 ¼ c0 ¼ c4 ¼ d0 ¼ d3 ¼ d4 ¼ g5 ¼ g6 ¼ 0

g1 ¼ E1 trðsÞ

c3 ¼E2

trðsÞ

ð56Þ

Note that in this case some functions are dependent on the stress.From Equations (17)–(20) one can work out 10 independent relations between the moduli

implicitly assumed in this model. Firstly, we find 5 relations independent on the stress

L1122 ¼ L2211

L2233 ¼ L3322

L3311 ¼ L1133

L1212 ¼ L2323 ¼ L3311

ð57Þ

then 5 other independent relations explicitly involving the stress

L1122

s11s22¼

L2233

s22s33¼

L3311

s33s11¼

L1111 � 2L1212

ðs11Þ2

¼L2222 � 2L2323

ðs22Þ2

¼L3333 � 2L3131

ðs33Þ2

ð58Þ

Other sets of 10 independent relations can be obtained by combining Equations (57) and (58).Note that these relations are implicitly given in Reference [21].



6. CONCLUDING REMARKS

It has been proved in this paper that a general hypoplastic constitutive equation for whichCauchy stress is the only state variable (amorphous hypoplasticity according to Kolymbas [3])involves 15 arbitrary functions of the stress invariants. Amongst these functions, 12 are relatedwith the linear part of the model. This implies that the 12 non-zero components of theconstitutive tensor corresponding to the linear part of the model written in the principal axes ofstress are, for a general stress state, independent from each other. Clearly, these moduli arefunctions of the three principal stress values, and must obey some symmetry conditions. Allthese restrictions are embodied in CLoE-hypoplastic models, which conclusively proves theobjectivity of this family of hypoplastic models.

An important result of this work is the freedom of the choice for the out-of-axis moduli.Hypoplastic theory constrains these moduli only for axisymmetric or isotropic stress states. Theuse of shear band observations to calibrate some parameters related with these out-of-axismoduli is then not only legitimate but, most importantly it is very useful. Indeed, the use ofexperimental localization data allows to obtain information on those details of the constitutiveformulation which cannot be directly defined using experimental data from tests with fixedprincipal directions of stress and strain.

It is of course also legitimate to use more restricted versions of hypoplasticity. In this case,some implicit constraints are enforced for the out-of-axis moduli. It has been proved that theseconstraints depend on the chosen arbitrary functions (and consequently on the chosen numberof parameters). Even if two models are calibrated on the same loading paths, the shear bandpredictions of these models change from one specific model to another because the out-of-axismoduli critically depend on the arbitrary functions.

The physical pertinence of the chosen model is another, very important question, which ishowever beyond the scope of this paper.

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a general formulation of hypoplasticity

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