Patterns, fabric, anisotropy, and soil elasto-plasticity

International Journal of Plasticity 21 (2005) 513–571 www.elsevier.com/locate/ijplas

Patterns, fabric, anisotropy, and soil elasto-plasticity E.T.R. Dean

*

37, Pitmedden Mews, Dyce, Aberdeen AB21 7ER, Scotland, UK Received in final revised form 22 February 2004 Available online 6 May 2004

Abstract This paper proposes a new method in the theory of soil plasticity – an advance on Hill [The Mathematical Theory of Plasticity, Clarendon Press, Oxford]. The method assumes that soil fabric consists of inter-locking, inter-twining, inter-laced, juxtaposed, and superposed elementary units called ‘‘patterns’’. A mechanics of patterns is developed. As well as elastic and plastic components, a third strain-increment component is deduced which helps explain nonassociated flow. The proposed method leads to explanations of critical states, anisotropy, sensitivity, the Bauschinger effect, and swept-out memory. All these appear in the method as near-inescapable features of plastic solids. Results are illustrated in detail for plane strain biaxial processes.
2004 Elsevier Ltd. All rights reserved. Keywords: Anisotropy; Bauschinger effect; Compatibility; Constitutive model; Critical states; Dilatancy; Dissipation; Elasticity; Energy; Fabric; Failure; Finite deformations; Flow rule; Fundamental assumptions; Hardening; Hvorslev surface; Internal variables; Liquefaction; Plastic potential; Plasticity theory; Proportional straining; Sensitivity; Softening; Soils; Steady states; Strain increments; Swept-out memory; Thermodynamics; Thermomechanics; Work; Yield

1. Introduction The concept of soil fabric and its relation to anisotropy has been discussed by Oda (1972, 1985), Matsuoka and Geka (1983), Hashiguchi (1985), Cambou (1990),

*

Tel.: +44-7958-152371; fax: +44-1224-774663. E-mail address: [email protected].

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2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2004.02.003

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Subhash et al. (1991), Kuganenthira et al. (1996), Bai and Smart (1997), Cottechia and Chandler (1997); Zlatovijc and Ishihara (1997), Masad and Muhunthan (2000), Dudoignon et al. (2001), Wan and Guo (2001), Chaudhary et al. (2002), and others. This paper proposes a new method of describing the anisotropic, elasto-plastic behaviour of soils, based on Dean’s (2003) proposal that fabric is built from ‘‘patterns’’. Anisotropy is an association between constitutive response and physical direction or orientation so that, for example, an undisturbed sample of clay taken from the ground will behave differently in a triaxial cell depending on its orientation in the apparatus (Saada and Bianchini, 1977; Molenkamp, 1998). Anisotropy for geotechnical and geological materials occurs in: • clay (e.g., Wood, 1973; Saada and Bianchini, 1977; Al-Tabbaa, 1984; Molenkamp, 1998); • silt (e.g., Zdravkovi
c, 2000, 2001), • sand (e.g., Casagrande and Carillo, 1944; Arthur and Menzies, 1972; Arthur et al., 1977; Elgamel et al., 2003); • gravel (e.g., Jiang et al., 1997; Tanaka, 2001); • sandstone (Benson et al., 2003), claystone (Chiarella et al., 2003), limestone (Cornet et al., 2002), and rock-like materials generally (Litewka and Debinski, 2003); • numerical models of granular assemblies (Shodja and Nezami, 2002). Anisotropy seems so common that one suspects plasticity in solids may be impossible without it. A brief review of recent publications shows that anisotropy is recognized in: • metals and metal forming (Barlat et al., 2003a,b; Br€ unig, 2003; Corradi and Vena, 2003; Lopes et al., 2003; Maudlin et al., 2003; Wu, 2003a,b; Wu et al., 2003; Bron and Besson, 2004; Green et al., 2004; Roos et al., 2004; Stroughton and Yoon, 2004; Tugcu et al., 2004; Yoon et al., 2004); • crystalline, polycrystalline, and fibrous materials (Bohlke et al., 2003; Castro and Ostoja-Starzewskib, 2003; Darrieulat and Montheillet, 2003; Dumont et al., 2003; Kobayashi et al., 2003; Langlois and Berveiller, 2003; Li et al., 2003; Carrere et al., 2004; Glazoff et al., 2004; Habraken and Duch^ene, 2004; Han et al., 2004; Hashiguchi and Proasov, 2004; Kalidindi et al., 2004; Kowalczyk and Gambin, 2004; Raabe and Roters, 2004); • metallic foams (Doyoyo and Wierzbicki, 2003); • viscoplastic and rate-dependent materials generally (Haupt and Kersten, 2003; H€ ausler et al., 2004; Voyiadjis et al., 2004). The existence of anisotropy across such a wide range of material types suggests that there may be a role for a concept that generalizes, and perhaps finds a common factor in, some of the many existing ideas. For soils, anisotropy and finite strains need careful consideration in relation to familiar concepts (Mroz, 1967; Hashiguchi, 1992; Gutierrez and Ishihara, 2000; Cleja-igoiu, 2003; Bruhns et al., 2003; Collins and Muhunthan, 2003; Tsakmakis, 2004). Noll’s (1958) Axiom of Local Action formalizes the notion that constitutive behaviour occurs ‘‘at a point in a continuum’’. In fact, soil consists of grains and voids and pore fluids. Lambe and Whitman (1979, p. 99) state that ‘‘when we talk about stresses at a point . . . we often must envision a rather large point’’. Other approaches include Desai (2001). Because

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anisotropy involves direction, the ‘‘large point’’ must have directional properties as well as position. The idea of ‘‘patterns’’ is a possible generalization of previous ideas and provides directional properties for points, as follows. Consider a litre of soil. Perhaps it is clay with particle sizes of around 10
3 mm, so the litre contains around 1015 particles. Perhaps it is coarse sand with particle sizes of 1 mm, so the litre contains a million or so particles. The particles are not organized like a crystal. There is much apparent randomness. In a finite element analysis of an embankment dam with dimensions of say 100 m, the litre might satisfy the concept of a ‘‘large point’’. Dean (2003) postulated that, if we look at all of the particles in the litre, we will see that the particle motions and distortions will fall into identifiable ‘‘patterns’’, with different patterns associated with different physical directions within the sample or in relation to the applied loads or deformations. For example, Cundall et al. (1982) found in numerical experiments that when a 2D assembly is sheared, some of the disk-particles form a web of ‘‘force chains’’. The chains support much of the imposed stress. They are supported against buckling by regions containing particles where inter-particle loads are smaller, but where much of the plastic work is done. Generalizing, one might define a ‘‘pattern’’ as organizational characteristic that has directional aspects and that repeats in some way in different parts of the large point. It involves a large collection of particles, inter-particle contacts and voids which act together in some coordinated way. Different co-existing patterns could intertwine and interlock in complex ways within the large point, creating a measurable fabric. They might also overlap so that, for example, a given particle might participate in more than one pattern. During a stress-increment or strain-increment, the patterns would interact with each other as well as with external boundary conditions. Co-existing patterns oriented differently in a soil element would experience externally applied conditions differently and so would evolve with different histories. The different histories would, for example, cause different amounts of elasto-plastic hardening or softening of different patterns, thus creating directional features of overall constitutive behaviour. These features would be interpreted at large scale as anisotropy. Patterns are thus postulated as ‘‘middle scale’’ or ‘‘mesoscale’’ entities between the macroscopic scale of engineering analyses and the microscopic scale of particle mechanics. In relation to a large analysis, the litre is a ‘‘large point’’ and a pattern would be a constitutive property assigned to the material in a point-wise fashion. The point would have a ‘‘state’’, which would be a description of the properties, orientations, and inter-relationships of all its patterns. Previous concepts of ‘‘states’’ in soil mechanics include: • critical states (e.g., Schofield and Wroth, 1968); • steady states (e.g., Poulos (1981); Been and Jefferies, 1985); • characteristic states (e.g., Luong, 1980); • intact and disturbed states (e.g., Desai, 2001). These states are very important in practice. However, there is a sense in which the constitutive laws that govern practical behaviours are determined by soil properties and are unaffected by their implications for engineers. This paper takes an egalitarian approach to the states of patterns and large points. All states are assumed equally

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valid and all histories are treated equally. Consistently with convention (Spencer, 1980; Baker and Desai, 1984), ‘‘isotropic’’ states are associated with symmetries. In the model developed below, isotropic states occur when the individual states of all patterns are the same, so patterns do not give rise to preferred directions. Anisotropic states occur when different patterns are in different states, so that the orientation of individual patterns relative to an external perturbation affects the overall material response. If we look inside the ‘‘large point’’, the patterns would appear as extended mechanical groupings or networks of particles. The paper proposes that some of the mechanical properties of those groupings can be represented as macroscopic continuum properties. A method is developed by which the properties can be collected into a constitutive model that ‘‘looks like’’ a conventional plasticity model, following principles such as those described by Hill (1950), Schofield and Wroth (1968), Hashiguchi (1985), Wroth and Houlsby (1985), Scott (1985), and others. Not surprisingly, a patterns model will have similarities to existing models that build overall behaviour from components, such as multi-laminates (Pande and Sharma, 1983; Pietruszczak and Pande, 2001), multi-mechanisms (Loret, 1990; Li and Dafalias, 2000, 2004; Fang, 2003), multiple yield functions (Koiter, 1960), multiple surfaces (Yoder and Iwan, 1981; Kiyama and Hasegawa, 1998; Li and Meissner, 2002), and multiple sub-structures (Bucher et al., 2004). However, the method for patterns proposed in this paper follows the different developmental path shown in Fig. 1. The path starts at the top and deduces features such as yield loci and plastic flow only at a relatively late stage in the development – these macroscopic features are seen as emergent properties, not fundamental. Concepts such as stress, work, and thermodynamics appear at intermediate stages. Because of the newness of this approach, and because patterns turn out to be rather complicated, it is felt useful to try to simplify the math. To achieve this, the following artificial limitations are made in this paper: (1) The paper considers only biaxial plane strain processes, in which a soil sample is subjected to principal strains in two fixed directions only. (2) The paper assumes that only two patterns exist in the soil element. The limitations allow the present paper to focus on the question of what types of logical structure and laws would be useful for the proposed generalized concept of patterns. Development for more complicated conditions is discussed at the end of the paper and details will be the subject of a later paper. A ‘‘bottom-up’’ constructive approach is used here, starting with the less complicated biaxial case, rather than a ‘‘top-down’’ approach which would start with a general theory. The following notes apply: (1) For initially isotropic materials, plane strain biaxial processes pre-define the possible directions of induced anisotropy as those of the principal strains. More generally, a model would need to specify how a material selects or evolves its directions of anisotropy. This is briefly discussed at the end of this paper.

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Basic concept - "Patterns" as elementary units of fabric

Geometry: Observation and material frames, scales, compatibility equations, pattern geometries

Stress: External work, workconjugacy, pattern stresses

Properties of individual patterns: Energy, elastic properties, energy limits, yield criteria

Energy interactions between patterns, thermodynamics, the third strain increment component

Mutual support, geometric fit, hardening and softening

Yield loci, anisotropy, dissipation equations

Plastic flow in proportional straining processes, critical states Fig. 1. Developmental path.

(2) Experiments suggest plane strain conditions enhance development of localized deformations like shear bands (Vardoulakis et al., 1978; Vardoulakis and Goldscheider, 1980; Vardoulakis, 1988; Drescher et al., 1990; Vermeer, 1990;

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Vardoulakis and Sulem, 1995; Mokni and Desrues, 1999; Saada et al., 1999; Alshibli and Sture, 2000; Iai and Bardet, 2001; N€ ubel and Gudehus, 2002; Gudehus and N€ ubel, 2004). Such bands may be special patterns. This paper does not explore this. In spite of these limitations, this paper will show that patterns can help explain induced anisotropy, sensitivity, the Bauschinger effect, critical states, non-associated flow, swept-out memory, and flow in proportional straining processes. This paper concludes with a brief discussion of the advantages of the approach proposed here, and a brief assessment of developments for general stress–strain conditions.

2. Pattern geometries It was noted above that we should expect patterns to have geometrical characteristics. To use these characteristics in a macroscopic model, we need to find a way to relate aspects of their changes to externally observable strains. Rowe (1962) developed an approach for spherical particles packed in regular array. Yimsiri and Soga (2000) develop a method for elastic responses. This paper proposes a new, more general, three-step method, as follows. The first step is to describe macroscopic geometry. This would involve the specific volume V and strain tensor or a zensor (Dean, 2001, 2004). For plane strain biaxial processes, it can be useful to make a simple visualization. Fig. 2 shows a soil element with initial dimensions B0  H0 , by 1 normal to the paper. The dimensions change to B  H  1. At the start of the process, we can draw a square in ink on the element, with dimensions nE0  nE0 , with E0 equal to the square-root of the specific volume V0 . The cuboid with dimensions nE0  nE0 by 1 unit into the paper contains a volume n2 of particles. It is convenient to call this cuboid an ‘‘observation frame’’. During a plane strain biaxial process, the frame deforms with the soil, with the square changing to a rectangle of dimensions nE1  nE2 . E1 and E2 are ‘‘specific lengths’’ (Dean, 1998). Their product will be the specific volume after the deformation. Plane strains in the directions of the edges of the observation frame are e1 ¼ lnðE0 =E1 Þ and e2 ¼ lnðE0 =E2 Þ. The second step starts by noting that strain is a kinematic variable whereas a pattern’s geometry needs to have a definite value at a given soil state. Some authors address this kind of problem by supposing the existence of special initial states such as ‘‘unstrained’’ states or indeed ‘‘intact’’ states (Desai, 2001). This paper resolves the issue by simply assuming the existence of non-kinematic descriptions of pattern geometries that have the same form as the descriptions used for the observation frame. No associated assumption about history is then needed. For the plane strain biaxial case, we consider a ‘‘material frame’’ with dimensions nE1  nE2 by 1 unit, where the over-barred quantities Ei are determined by the material’s patterns. In general, a material frame would be a visual model of a fabric tensor. Its relation to an observation frame would involve a transformation law. For the particular example here, we can characterize the transformation by defining scales S i as: E 1 ¼ S 1 E1 ;

ð1Þ

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519

soil element

Out-of-plane thickness 1 unit "Observation frame", with dimensions ξ.Eo x ξ.Eo x 1, marked on soil

Height H o

Breadth B o (a) soil element initially

ξ.E 2

observation frame follows the material strains

1

ξ.E 1

H

B (b) element during a plane strain biaxial process Fig. 2. Development of a description of macroscopic geometry for a plane strain biaxial process: (a) soil element initially and (b) element during a plane strain biaxial process.

E 2 ¼ S 2 E2 :

ð2Þ

The scales, and the transformation parameters in general, can be variables, because the geometrical behaviours of individual patterns need not be identical to the overall

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strain behaviour of a sample. For example, one pattern might densify while another dilates, with the overall sample deforming at constant volume. The third step of the proposed method is to resolve another issue. A pattern is unlikely to have the same topology or shape as a material frame. The present paper assumes each pattern has an orientation and that the ith pattern’s geometry is otherwise described by a single quantity V i . This would be like a specific volume. Its symbol illustrates a new consistent notation. A double over-bar will indicate a quantity for a pattern, either a description of geometry, stress, work, or energy. A single over-bar (as in Ei above) will indicate a quantity associated with the material frame. Quantities that are material constants, and factors that appear only in specific pattern equations, will generally not be over-barred. The idea of associating volumelike quantities with orientation is discussed by Pietruszczak and Krucinski (1989), Muhunthan and Chameau (1997), Masad et al. (1998), and others. This paper assumes: V i ¼ fi

ðgeometry of the material frameÞ;

ð3Þ

where f i is a function. This would be analogous to a compatibility equation in steel frame plasticity, because it relates a description of internal geometry, V i , to quantities that have the form of external geometry (e.g., Baker et al., 1965). For patterns, the function would be a conceptual model of a scalar function of a fabric tensor. In the present paper, two patterns are considered, with their specific volumes V i defined as: A

V 1 ¼ ðE2 Þ ðE1 Þ

2
A

;

ð4Þ

V 2 ¼ ðE1 ÞA ðE2 Þ2
A ;

ð5Þ

where the exponent A is taken as a material constant. The volumes V i are not functions of strain, but of the geometry of the material frame including effects of the kinematic scales of Eqs. (1) and (2). Two patterns give sufficient freedoms to model simplified plane strain biaxial processes. More are needed in general, as discussed at the end of this paper. In thermo-mechanical terminology (e.g., Maugin, 1999), the ith pattern volume V i might be classed as an ‘‘internal variable’’. An advantage of the present approach is that the concept of patterns provides this specific volume quantity with a clear, non-mysterious, physical meaning. An advantage of physical meaning is that it can also suggest related concepts. In the present case, we now consider stress in patterns.

3. Pattern stresses To develop the concept of stress in patterns, this paper adapts the method of work-conjugacy (Peric et al., 1990). The proposed new general method is to differentiate the compatibility equations, extract the increments that are associated with

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external strain increments, and use these in a work equation involving pattern stresses. The method may be illustrated by the simplified plane strain biaxial context as follows. Differentiating the compatibility Eqs. (4) and (5), and using the transformation Eqs. (1) and (2), we get equations that can be written in the following the form: dV i

¼

dV i;u

Vi

Vi

þ

dV i;s

ð6Þ

;

Vi

where the first component on the right, subscript u, is directly related to external observable strain increments, and the second component, subscript s, is directly related to increments of the transformation parameters. For the plane strain biaxial example: dV 1


¼

ð2
AÞ

dE1 dE2 þA E1 E2

ð2
AÞ

dE2 dE1 þA E2 E1

V1 dV 2


¼

V2






 dS 1 dS 2 þA þ ð2
AÞ ; S1 S2

ð7Þ


 dS 2 dS 1 þA þ ð2
AÞ : S2 S1

ð8Þ

The terms in the first large brackets are externally observable components, with dEi =Ei ¼
dei . Re-expressing these using volumetric and shear strain increments dv ¼
dV =V ¼ de1 þ de2 and de ¼ de1
de2 gives dE1 =E1 ¼
ðdv þ deÞ=2, dE2 =E2 ¼
ðdv
deÞ=2, and


dV 1u V1




dV 2u V2

¼ dv þ ð1
AÞ de ¼ dv1;u ; say;

ð9Þ

¼ dv
ð1
AÞ de ¼ dv2;u ; say:

ð10Þ

Solving for ðdv; deÞ gives       dv1;u dv2;u 1 1 dv þ : ¼ 1=ð1
AÞ
1=ð1
AÞ de 2 2

ð11Þ

Fig. 3 shows this graphically. The observables dv; de are plotted on the horizontal and vertical axes, respectively. The vector of observed increments, OB, is the sum of a component O1 due to the dv1;u for pattern 1, and a component O2 due to dv2;u . If we draw inclined axes for the pattern increments, the axis for dv1;u , defined by the condition dv2;u ¼ 0, would be in the direction of O1. The axis for dv2;u would be in the direction of O2. 0 Let P i be the effective stress in the ith pattern. According to the proposed general ^i for the ith pattern as follows: method, we define a quantity of work dW 0

0

^i ¼
P dV i;u ¼ P V i dvi;u ; dW i i

ð12Þ

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E.T.R. Dean / International Journal of Plasticity 21 (2005) 513–571

un-levered strain increment dv= 1u for pattern 1 1

observable shear strain increment dε

1– A 1 B

0 observable volume strain increment dv

2 1 1–A un-levered strain increment = dv 2u for pattern 2 Fig. 3. Relationships in strain-increment space.

^i would be the external work done on the ith pattern, per unit particle where dW volume taking account of all patterns. Its special symbol distinguishes it from other work quantities developed later for patterns. If the sum of the external works for all patterns is the net external work dW , then dW ¼

N X

^i ; dW

ð13Þ

i¼1

where N is the number of patterns. N ¼ 2 for the plane strain biaxial model of this paper. Using Eqs. (9), (10), (12) and (13), we can calculate dW in terms of the pattern stresses and volumes and the external strain increments: 0

0

dW ¼ P 1 V 1 ðdv þ ð1
AÞ deÞ þ P 2 V 2 ðdv
ð1
AÞ deÞ:

ð14Þ

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523

We can also calculate dW in the familiar way, using the strain increments and effective stresses (e.g., Schofield and Wroth, 1968; Wood, 1984; Houlsby, 1979; Gutierrez and Ishihara, 2000). For the plane strain biaxial case: dW ¼ s0 V dv þ tV de; 0

ðr01

ð15Þ

r02 Þ=2

ðr01

r02 Þ=2

þ is the mean in-plane effective stress, t ¼
is the inwhere s ¼ plane deviator stress, and r0i is the ith principal effective stress. Comparing this with the above equation, noting that both must apply independently of the signs and magnitudes of the strain increments dv and de, gives: 0

0

s0 V ¼ P 1 V 1 þ P 2 V 2 ;

ð16Þ

0

0

tV ¼ ð1
AÞðP 1 V 1
P 2 V 2 Þ:

ð17Þ

In steel frame plasticity (e.g., Baker et al., 1965), an analogous method is used to calculate the equilibrium relations between loads applied externally to a frame and internal loads and moments in the frame. By analogy, we might interpret the above results as equilibrium equations between the external stresses s0 ; t and the internal pattern stresses. Inverting and using the results to re-calculate the external works gives ^1 ¼ dW

s0 V ð1 þ gÞðdv þ ð1
AÞ deÞ; 2

ð18Þ

^2 ¼ dW

s0 V ð1
gÞðdv
ð1
AÞ deÞ; 2

ð19Þ

where g is a normalized stress ratio given by 0

0

t=s0 P V 1
P 2V 2 g¼ ¼ 10 : 1
A P V þ P0 V 1

1

2

ð20Þ

2

If the pattern stresses are restricted to being non-negative, then g will lie the range )1 to +1. The restriction on pattern stresses could correspond, for example, to an uncemented granular material which cannot sustain tensile principal effective stress (e.g., Schofield and Wroth, 1968). Fig. 4 shows Eqs. (16) and (17) graphically. The axes represent the Szalwinski (1983)-type effective stresses s0 V and tV . The equations can be written in vector form as  0      0 0 1 1 sV ¼ P 1V 1 þ P 2V 2 : ð21Þ tV 1
A
ð1
AÞ Hence, a stress vector OS is the sum of a component O1 due to stress in pattern 1, and O2 due to stress in pattern 2. The first pattern’s stress would be zero on a line of slope 0dt=ds0 ¼
ð1
AÞ. Hence,0 this line, where the normalized stress ratio is )1, is the P 2 V 2 axis. Similarly, the P 1 V 1 axis is a line at slope dt=ds0 ¼ þð1
AÞ, with normalized stress ratio +1.

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4. Preliminary results Comparing Figs. 3 and 4 shows that the strain increment axes are at right angles to the stress axes, giving a form of ‘‘normality rule’’, suggesting that compatibility and patterns may be the underlying cause of flow rules for plastic materials. To investigate further, consider the following possible simplified rules for the plane strain biaxial processes: 0 (1) The ith pattern yields plastically when P i V i reaches a particular value, say Yi . (2) The yield value Yi may be different for different patterns. (3) Some kind of problem will occur if a pattern stress reaches zero. (4) Each pattern strain increment dvi;u is the sum elastic and plastic components. Fig. 5 illustrates the following consequences: 0 (A) The locus of yield states for pattern 1, where P 1 V 1 ¼ Y1 , is a straight line AX . The locus for pattern 2 is AY . The two lines form a yield locus. If Y1 and Y2 are different, the two lines intersect at a point A which is not on the s0 -axis.

= = inclined axis for stress P' 1 V1 for pattern 1

in-plane Szalwinski shear or deviator stress t.V

1

1 1-A S

0 average in-plane Szalwinski effective stress s'.V

2

1-A 1

= = inclined axis for stress P'2 V2 for pattern 2

Fig. 4. Relationships in effective stress space.

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525

The consequent absence of symmetry between positive and negative t corresponds to an anisotropy in the material state. (B) The loci of stress states with zero pattern stresses are the lines OX and OY . The slope of the lines can be calculated from Eq. (21) as ð1
AÞ. These loci appear in the place where a failure surface would be located for a simple frictional material. Denoting the maximum as /0max ¼ sin
1ðt=s0 Þmax gives A ¼ 1
sinð/0max Þ. (C) Putting dvi;u ¼ dvi;u;e þ dvi;u;p into Eq. (11), with subscripts e and p for elastic and plastic parts, we get a plastic strain increment vector:       dv1;u;p dv2;u;p dvp 1 1 þ : ð22Þ ¼ dep 1=ð1
AÞ
1=ð1
AÞ 2 2 When only pattern 1 yields, only the first part applies, and the vector is normal to the part of the yield locus AX for pattern 1. Similarly, when pattern 2 yields, the first term is zero, and the plastic strain increment vector is normal to the part of the yield locus AY for pattern 2. The four rules giving these results are simplified and we use more realistic rules later. The rules serve to demonstrate the direct relations between compatibility and

Fig. 5. Consequences of patterns and work-conjugacy for linear compatibility equations – preliminary interpretations for yield loci, plastic flow, anisotropy, and limiting stress ratio.

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features of stress and strain-increment spaces. They also illustrate an important feature – that yield loci and plastic potentials are not fundamental properties of pattern models, but emergent ones.

5. Pattern energies and energy limits We noted earlier that patterns are composed of particles. Consequently, and since objects under stress will in general store ‘‘strain energy’’ (Spencer, 1980), it seems likely that a pattern will have an energy U i , say, which will be the sum of the energies of the particles in the pattern. It will be convenient to express this per unit overall particle volume. The amount of energy a pattern could store is also likely to have a limit, say U lim;i . This might be determined, for example, by crushing strengths of particles or inter-particle contacts (Ueng and Chen, 2000; Cheng et al., 2004; Coop et al., 2004), by frictional characteristics (Bolton, 1986), by the pattern’s geometry, and by support the pattern receives from other patterns. Energy and other thermodynamic aspects of plastic materials are discussed by Collins and Houlsby (1997), Maugin (1999), Han and Reddy (1999), Houlsby and Puzrin (2000), Collins and Kelly (2002), Tsakmakis (2004), and others. This paper assumes: Ui ¼

0 j P iV i; 1
j

U lim;i ¼

j exp ðC=kÞ ; 1
j ðV i Þð1
kÞ=k

ð23Þ ð24Þ

where 0 < j < k < 1 and C are material constants. Reasons for choosing these equations are: (1) Consider a simple process in which the scale increment component in Eq. (6) is 0 zero. In this special case, the external work of Eq. (12) is just
P i dV i . If this is all of the work for this pattern, then equating it to the change of elastic energy gives
 0 0 0 j dP i V i þ P i dV i :
P i dV i ¼ dU i ¼ ð25Þ 1
j Integrating gives 0

lnðV i Þ þ j lnðP i Þ ¼ constant;

ð26Þ

where the constant depends on the state of the pattern at the start of the integration. This is analogous to Butterfield’s (1979) empirical 1D swelling equation for clays. (2) It seems like that a pattern will yield if an attempt is made to increase its energy beyond its current limit. Putting U i ¼ U lim;i and considering the yield criterion used earlier gives:

E.T.R. Dean / International Journal of Plasticity 21 (2005) 513–571

Yi ¼

1
j U lim;i ; j

527

ð27Þ

0

lnðV i Þ þ k lnðP i Þ ¼ C:

ð28Þ

The first equation would represent an advance on the familiar association between yielding and energy limits which is associated with the von Mises yield criterion (e.g., Efunda, 2003). The second equation is analogous to Butterfield’s (1979) empirical yield equation for 1D compression of soft clays. These results show that patterns can relate directly to some experimental data. For purposes of later sections of this paper, we now develop some math. Using Eqs. (20), (21), and (23): 

s0 V tV

g¼



 1
j 1 ¼ 1
A j

U1
U2 U1 þ U2

1 A
1

"

# U1 ; U2

:

ð29Þ

ð30Þ

Thus, the Szalwinski (1983) stresses in the present formulation are directly related to the pattern energies. The normalized ratio g describes the energy difference. These results are particularly important because they allow us to work directly with energy and other thermodynamic quantities, making it possible to interpret observable stresses as consequences of more fundamental thermodynamic aspects of pattern mechanics. We will also find it useful later to define a reference energy U vol as U vol ¼

j exp ðC=kÞ ; 1
j V ð1
kÞ=k

ð31Þ

where V is the specific volume. In the special soil states for which V i ¼ V and U i ¼ U lim;i ¼ U vol , and denoting the average in-plane stress s0 for this case as s0iso , the equations lead to lnðV Þ þ k lnðs0iso Þ ¼ C þ k ln 2 ¼ constant ¼ Ciso;psb ; say:

ð32Þ

This gives an in-plane ‘‘isotropic’’ compression line in (s0 ; V ) space, where ‘‘in-plane isotropic’’ refers to equal strains in all in-plane directions with no strain normal to the plane.

6. Elastic processes In simple plasticity models for soils, elastic and plastic properties are specified separately. In contrast, the yield equation U i ¼ U lim;i shows that, for pattern models, equations involving yielding are likely to involve both properties. We already have

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E.T.R. Dean / International Journal of Plasticity 21 (2005) 513–571

almost enough information to deduce elastic properties for the plane strain biaxial model. Differentiating Eq. (29) gives:  1
j dU 1 þ dU 2 ; dðs0 V Þ ¼ ð33Þ j  1
j dU 1
dU 2 : ð34Þ dðtV Þ ¼ ð1
AÞ j If we assume that, for purely elastic processes, the increment dU i of a pattern’s ^i done on the pattern, then using Eqs. (9), (10), energy equals the external work dW (13), (16), and (17)  0   0   1
j s dv dðs V Þ t V : ð35Þ ¼ 2 0 de dðtV Þ t s ð1
AÞ j These results demonstrate the connection between compatibility, pattern energy, and elastic properties. The off-diagonal terms in the matrix produce a simple form of stress-dependent elastic anisotropy when the shear stress t is non-zero. A general solution can be developed by re-casting Eq. (35) in terms of stresses s01 ¼ s0 V ð1 þ gÞ and s02 ¼ s0 V ð1
gÞ, then integrating and solving for strains, giving pffiffiffiffiffiffiffiffiffiffiffiffiffi
 1
j s 0 V 1
g2 v ¼ 0 pffiffiffiffiffiffiffiffiffiffiffiffi2ffi ; exp ð36Þ j s 0 V 0 1
g0
exp

1
j ð1
AÞe j




¼

1þg 1
g

 


 1 þ g0 ; 1
g0

ð37Þ

where subscript 0 represents the initial state of an elastic path and v and e are the volumetric and shear strains from that state. The first equation represents a generalization of the in-plane isotropic swelling line, which is obtained when g and g0 are zero. The equations imply that a constant volume path is a hyperbola s01 s02 ¼ constant. A 1D unloading path (v ¼ eÞ is of the form ðs01 ÞA =ðs02 Þ1
A ¼ constant.

7. Energy interactions between patterns Because of geometric juxtapositions at microscopic scale between particles and voids of different patterns in a soil element, there will stresses that act across the interfaces between patterns. Consequently, if pattern boundaries move during a stress–strain process, patterns will be able to do work on each other. This paper denotes the increment of work done by pattern j on pattern i as dEji . The net internal work done on pattern i by all the other patterns is denoted as the internal ‘‘energy receipt’’ dEi , with dEi ¼

N X j¼1

dEji ;

ð38Þ

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529

where the sum is taken over all N patterns. Positive work done by one pattern on another is equivalent to negative work done by the second on the first, so dEij ¼
dEji . The work dEii done by a pattern on itself is zero and the sum of the receipts dEi for all N patterns is zero. ^i to give The receipt dEi for an individual pattern will add to the external work dW the net work for the ith pattern. Denoting this net work by dW i gives ^i þ dEi : dW i ¼ dW

ð39Þ

This paper assumes the net work can be separated into an elastic component dU i , which would be the change of the pattern’s energy in the increment, and a plastic component dW ip which would be the heat dissipated by the pattern in the increment dW i ¼ dU i þ dW ip

ð40Þ

(Some authors may prefer to interpret dU i as a change of the pattern’s Helmholtz free energy and dW ip the pattern’s dissipation function). The Second Law of Thermodynamics implies that heat cannot be converted into work or energy in a process at constant temperature. For such a process, dW ip P 0 is the dissipated or ‘‘irrecoverable’’ work. Solving the above two0 equations to eliminate dW ip and get the external work, then dividing the result by P i V i and using Eq. (12) gives: dvi;u ¼ dvi;u;e þ dvi;u;p þ dvi;u;x ;

ð41Þ

with: dvi;u;e ¼

dU i 0

ð42Þ

;

P iV i dvi;u;p ¼

dW ip 0

;

ð43Þ

:

ð44Þ

P iV i dvi;u;x ¼


dEi 0

P iV i These results show that an observable strain increment can be separated into three components, associated with elastic work (subscript e), plastic work (subscript p), and internal transfers of energy (subscript x). Since the sum of the internal transfers is zero, the third component does no external work. Using this to substitute into Eq. (11) for strain increments in the plane strain biaxial case gives:         dvp dv dve dvx ¼ þ þ ; ð45Þ dep de dee dex where the first elastic component is

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dve dee



    dv1;u;e dv2;u;e 1 1 þ : ¼ 1=ð1
AÞ
1=ð1
AÞ 2 2

ð46Þ

The second and third components satisfy similar equations with subscript e replaced by p (plastic) or x (workless). Fig. 6 illustrates an effect of the workless component. Consider a stress increment starting from a point S on the yield locus for pattern 1. Let SP and PQ be the plastic and elastic strain increment components, respectively. SP comes from dissipation in pattern 1 and can be normal to the part of the yield envelope associated with this pattern. PQ is the vector sum of elastic components from patterns 1 and 2. The workless strain-increment does no external work, so is normal to the stress vector, such as SR. The total strain-increment is SZ, which is the vector sum of SR, SP , and PQ. The non-elastic part is SY , the sum of SR and SP . In this way, the workless component can cause an impression of non-associated flow.

direction through S normal to the stress vector OS Z Y

tangent to yield locus for pattern 1

R

t.V

Q P

90˚ S

O s'.V

Fig. 6. Illustrating an effect of the workless strain-increment component. SP is the plastic component normal to the yield locus. PQ is the elastic component. SR is the workless component. The total is SZ. The non-elastic part is SY , which is not necessarily normal to the yield locus. Thus an impression of non-associated flow occurs.

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531

To develop explicit equations for the energy receipts dEi , this paper assumes that the doing of plastic work in a pattern is a disruptive process, altering the particle motions that would otherwise occur, and that the internal transfers of energy are one of the consequences of this. This suggests equations of the form: dE1 ¼ w1 dW 1p
w2 dW 2p ;

ð47Þ

dE2 ¼ w2 dW 2p
w1 dW 1p ;

ð48Þ

where the multipliers w1 and w2 will be determined later. These equations do not imply that the receipts are plastic or dissipative, just that they depend on the plastic works. This is consistent with equations developed earlier, because it implies the receipts are zero in elastic processes.

8. Mutual support – geometrical interactions between patterns Another way for patterns to interact with each other would be by providing mutual support in respect of yielding. For example, it seems likely that, if patterns were strongly interlocked, the yield limit for one pattern could depend partly on the state of the other pattern. One way to express the support mathematically could be to develop relations involving the energy limits. It can help to have an example. In this paper, we define a quantity F as  1þNM U lim;avg =U vol F ¼ h ð49Þ    iM ; U lim;1 =U vol U lim;2 =U vol    U lim;N =U vol where M is a material constant, U lim;avg is the average of the energy limits of the N patterns, and U vol is the reference energy from Eq. (31). The equation was developed from an analysis of an expected surface which will now be derived. For N ¼ 2 patterns, the denominator in Eq. (49) contains just two ratios. It proves useful to define a normalized limit ratio glim as: glim ¼

U lim;1
U lim;2 U lim;1 þ U lim;2

:

ð50Þ

So U lim;1 ¼ U lim;avg ð1 þ glim Þ;

ð51Þ

U lim;2 ¼ U lim;avg ð1
glim Þ:

ð52Þ

This limit ratio is not a limit on the normalized ratio g of Eq. (30). Instead, it is just the value of g calculated using energy limits instead of energies. Eq. (49) for the N ¼ 2 patterns of the plane strain biaxial model then simplifies to

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E.T.R. Dean / International Journal of Plasticity 21 (2005) 513–571

U lim;avg



M ¼ F 1
g2lim :

ð53Þ U vol If both patterns are at their energy limits, then glim ¼ g, and also the average limit equals the average pattern energy, which is related to the average in-plane stress s0 by the stress-energy equation (29). The reference energy U vol is related to the stress in Eq. (32) on the isotropic compression line for F ¼ 1. Denoting the latter stress as s0iso , the above equation becomes
0 

M s ð54Þ ¼ F 1
g2 : 0 siso Fig. 7 shows this expected surface for F ¼ 1. It is reminiscent of the Cam–Clay ‘‘state boundary’’ concept (e.g., Schofield and Wroth, 1968). However, the surface in the patterns model is not a boundary. The material state can go outside the F ¼ 1 surface if F > 1. Using Eqs. (24) and (31), the right-hand side of Eq. (49) reduces to an expression containing constants, the pattern geometries V i , and the specific volume V. The value of F is 1 if all the pattern geometries are the same as the specific volume. On this basis, F is interpreted as a measure of geometric fit between patterns. F ¼ 1 is taken as a ‘‘natural’’ fit. 9. Hardening and softening for patterns Early concepts of hardening for soils include Drucker et al. (1955), Roscoe et al. (1958), Schofield and Wroth (1968), and Roscoe and Burland (1968). For patterns, it seems natural to suppose that an attempt to increase a pattern’s energy beyond its current limit would cause yielding and plastic work, and the limit might also evolve. In the present paper, a pattern is said to ‘‘harden’’ if its energy limit increases and soften if its limit decreases. A ‘‘hardening law’’ for a pattern would govern the way the pattern’s energy limit changes in a stress–strain increment. It is convenient to develop some example equations for the purposes of exploration. First, differentiating Eqs. (51) and (52) gives: ! dU lim;1 dU lim;avg glim dglim dglim ¼ ; ð55Þ
þ 2 1
g 1
g2lim U lim;1 U lim;avg lim dU lim;2 U lim;2

¼

dU lim;avg U lim;avg

g dg
lim 2lim 1
glim

!


dglim : 1
g2lim

ð56Þ

The last term on the right sides provides differential hardening of one pattern relative to the other. Differentiating Eq. (53), with U vol obtained from Eq. (31), and combining the result with the above equation, gives ! dU lim;avg glim dglim 1
k dF g dg dv þ
ð1 þ 2MÞ lim 2lim : ð57Þ
¼ 2 k F 1
g 1
glim U lim;avg lim

E.T.R. Dean / International Journal of Plasticity 21 (2005) 513–571

533

t locus of largest shear stresses on the surface at a given specific volume intersection of surface and plane of constant volume

s'

asymptotic isotropic compression line, equation 30

V Fig. 7. The degree-of-fit surface for F ¼ 1, positive shear stress side. (The negative shear stress side would be obtained by reflecting the positive side in the t ¼ 0 plane.)

The first term on the right is a volumetric hardening component. The second term describes the evolution of the geometrical fit of the two patterns. The third term will turn out to be associated with induced anisotropy. The further derivation of suitable hardening equations is complicated and is done in Appendix A. The derivation is based on criteria of simplicity, symmetry, and unique solubility of the incremental equations. Unique solubility is discussed later in the main text and the discussion takes advantage of the results of Appendix A. The results are:
  1
k 1
FS  dU lim;1 ¼ U lim;1 dv þ ð58Þ R11 dW 1p þ R12 dW 2p þ H1 dW diff ; S k F dU lim;2 ¼

1
k U lim;2 dv þ k




  1
FS  dW dW R þ R
H2 dW diff : 21 1p 22 2p FS

ð59Þ

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E.T.R. Dean / International Journal of Plasticity 21 (2005) 513–571

The first term in each equation is volumetric component of hardening. The second term depends on mutual support or degree-of-fit, where the exponent S is a positive constant. This term is zero on the F ¼ 1 surface of Fig. 7. The coefficients Rij are: R11 ¼ R12 ¼

R ð1 þ glim Þ; 2

ð60Þ

R21 ¼ R22 ¼

R ð1
glim Þ; 2

ð61Þ

where R is a constant in the range 0–1. The last terms in the hardening equations (58) and (59) are sources of differential hardening. The coupled work component is: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
glim 1 þ glim dW 1p
dW 2p : dW diff ¼ ð62Þ 1 þ glim 1
glim This ensures that plastic work in one pattern causes hardening or softening in both. The ratios in the square-root signs have physical meaning through Eqs. (51) and (52). Appendix A relates the coefficients Hi in the hardening equations (58) and (59) to the coefficients wi in the receipts equations (47) and (48), with: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ glim H1 ¼ w1 ; ð63Þ 1
glim sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
glim ; H2 ¼ w2 1 þ glim

ð64Þ

and w1 ¼

K ð1
gð1 þ 2MÞÞ; 2

ð65Þ

w2 ¼

K ð1 þ gð1 þ 2MÞÞ; 2

ð66Þ

where K is a positive constant. The constant M comes from the equation for F, so the factors in large brackets are related to the degree-of-fit surface. Using Eqs. (55)–(59): U lim;avg

2dglim K ffi dW diff ; ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1
glim 1
g2lim

2U lim;avg

ð67Þ


  dF 1
FS  ¼R dW 1p þ dW 2p
Kð1 þ 2MÞ S F F ðg
glim Þ ffi dW diff :  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
g2lim

ð68Þ

E.T.R. Dean / International Journal of Plasticity 21 (2005) 513–571

535

The first equation turns out to be a law of induced anisotropy. It makes glim constant only when dW diff ¼ 0. It will be illustrated in calculation results later. The second equation is a law for the evolution of mutual support. The first part, proportional to R, is a restoring term: it is positive when plastic work occurs with F < 1 and negative when F > 1, so it tends to restore F to the value 1. The second part, proportional to K, is a ‘‘disturbing’’ term. It can cause F to move away from 1. Because of the volumetric hardening terms in Eqs. (58) and (59), pattern hardening can occur in elastic as well as elasto-plastic processes. If there is no plastic work, then dU lim;1

¼

U lim;1

dU lim;2

¼

U lim;2

1
k dv: k

ð69Þ

The simplified concepts that resulted in Fig. 5 took no account of these elastic changes of pattern limits. Consequently, the original concept that a yield locus is 0 represented by a given value of P i V i needs updating to incorporate pattern hardening. This will be done now.

10. Derivation of yield loci Let subscript o represent the current state, which will be either inside the yield loci for both patterns, or on one of the loci and inside the other, or on both at their intersection. Consider an exploration starting at the current state. Integrating the above Eq. (69) for elastic behaviour inside the yield locus gives U lim;1

¼

U lim;1o

U lim;2 U lim;2o


¼

V0 V

ð1
kÞ=k ;

ð70Þ

where subscript 0 represents the start of the exploration. From the elastic equation (36) for volume strain v ¼ lnðV0 =V Þ, pffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1
jÞ=j V0 s 0 V 1
g2 pffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ ð71Þ V s0A VA 1
g2A From Eq. (29), the pattern energies U 1 and U 2 are given by: U1 ¼

j=2 0 ðs V Þð1 þ gÞ; 1
j

ð72Þ

U2 ¼

j=2 0 ðs V Þð1
gÞ: 1
j

ð73Þ

Suppose the exploratory path reaches the yield locus for pattern 1. Then, U 1 ¼ U lim;1 . Using this and Eqs. (70) and (72), we can express the volume ratio V0 =V in terms of stresses. Then, using Eq. (71), we can eliminate the volume ratio, leaving

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s ¼ s00

1
j U lim;1o j=2 s00 V0 ð1 þ gÞ

!k=ðk
jÞ sffiffiffiffiffiffiffiffiffiffiffiffiffi!j=ðk
jÞ 1
g2 : 1
g20

ð74Þ

This is the stress-space yield locus for pattern 1. The locus for yield in pattern 2 is given by a similar equation with U lim;1o replaced by U lim;2o and g by
g. The two yield loci intersect at a point A where U lim;1o =ð1 þ gÞ ¼ U lim;2o =ð1
gÞ, implying that g ¼ glim;o at A, where glim;o is calculated from Eq. (50) with subscripts o. In the special case that the current stress state is point A, from Eq. (29) with U i ¼ U lim;i  1
j s00 V0 ¼ U lim;1o þ U lim;2o : ð75Þ j Putting this in the above equations and using Eqs. (50)–(52) with subscript 0 to simplify, then changing the subscript to A, gives
k=ðk
jÞ sffiffiffiffiffiffiffiffiffiffiffiffiffi2 !j=ðk
jÞ s 1  gA 1
g ¼ : ð76Þ s0A 1g 1
g2A The positive signs are for yield in pattern 1, the negative signs for yield in pattern 2. Figs. 8(a) and (b) show results calculated for j=k ¼ 0:2. Fig. 8(a) is for an isotropic soil state, with glim ¼ 0 at A. The combined yield locus OXAYO is symmetric about the average stress axes. Like the original Cam–Clay locus (Schofield and Wroth, 1968), the combined locus has a sharp point on the average stress axis. Fig. 8(b) shows the combined locus when the energy limit for pattern 1 is larger than for pattern 2. In this case, glim ¼ 0:3 at A. This gives an anisotropic soil state. The shape OXAYO is similar to shapes seen in laboratory and field data for clays, see for example Al-Tabbaa (1984), Graham et al. (1988), Koskinen et al. (2002), and others. These calculations show how the stress-space yield loci for a patterns model are derived and how the shapes are determined partly by elastic properties as well as by energy limit and hardening properties. This is different from other plasticity approaches. Yield functions for patterns models are not fundamental properties, but emergent ones. A similar result applies for dissipation, which is now explored.

11. Dissipation in patterns A patterns model will need a way to determine the dissipations characterized by the plastic works dW ip . As is the case for other plasticity approaches, e.g., Loret (1990), a question arises of how to ensure that, for a given incremental perturbation, a unique solution is obtained for the material response. In multiple surface models this can be manifested as a ‘‘consistency’’ condition or a ‘‘non-intersection’’ condition (Puzrin and Houlsby, 2001). The present paper assumes that a unique stress increment solution is required for any given strain increment perturbation. Limiting energy will be taken as the yield criterion for a pattern. To develop consistent dissipation equations, this paper adapts from Yoder and Iwan’s (1981) ‘‘relaxation’’ method. In their method, calculations are first done assuming elastic

E.T.R. Dean / International Journal of Plasticity 21 (2005) 513–571

537

Fig. 8. Some consequences of simple patterns – yield loci, elastic regions, isotropy, and anisotropy: (a) isotropic yield boundary and elastic region when glim ¼ 0 and (b) anisotropic yield boundary and elastic region when glim ¼ 0:3.

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E.T.R. Dean / International Journal of Plasticity 21 (2005) 513–571

responses. If the results violate one or more yield criteria, the stresses are ‘‘relaxed’’ to satisfy the criteria. Adapting this, we define an ‘‘energy excess’’ for each pattern, equal to the pattern’s energy less its energy limit. As a result of an increment of ^i of external work and an amount behaviour, the ith pattern receives an amount dW dEi of energy from other patterns. The pattern’s energy limit will also change, by dU lim;i say. The new excess X i at the end of the increment will be ^i þ dEi
dU lim;i ; X i ¼ ðU i
U lim;i Þ þ dW

ð77Þ

where the term in brackets is the excess at the start of the increment. This paper assumes that, if X i > 0, it is the plastic work dW ip for the ith pattern. Otherwise the pattern’s plastic work is zero. The right sides of the excess equation for each pattern can be expanded by using Eqs. (47), (48), (65) and (66) for the energy receipts and Eqs. (58)–(64) to compute the pattern hardenings. After re-arrangement this gives:   X 1 ¼ X 1b
Q1 dW 1p þ dW 2p
Q3 dW 2p ; ð78Þ   X 2 ¼ X 2b
Q2 dW 1p þ dW 2p þ Q4 dW 1p ;

ð79Þ

where X 1b is a ‘‘basic’’ excess for the ith pattern and ^i
X ib ¼ ðU i
U lim;i Þ þ dW

1
k U lim;i dv; k

ð80Þ


 R 1
FS Q1 ¼ ð1 þ glim Þ ; 2 FS

ð81Þ


 R 1
FS ; Q2 ¼ ð1
glim Þ 2 FS

ð82Þ

Q3 ¼ K

gð1 þ 2MÞ
glim ; 1
glim

ð83Þ

Q4 ¼ K

gð1 þ 2MÞ
glim : 1 þ glim

ð84Þ

Eqs. (78) and (79) show that the excess X i for the ith pattern consists of the basic excess X ib , less a term proportional to R and associated with mutual support or degree-of-fit, and a last ‘‘cross-coupling’’ term that depends on the plastic work in the other pattern. For a given strain increment, the values of the basic excesses can be calculated from Eq. (80) without needing to know the plastic works. The excess equations (78) and (79) can then be solved in the following step-wise manner: (1) assuming elastic behaviour, the excesses X 1 and X 2 would be calculated (i.e.. X i ¼ X ib . If both are negative, the plastic works are assumed to be zero. (2) If one of the excesses in step 1 is found to be positive, the relevant symbol X i is replaced by dW ip and the equations are again solved. For example, if step 1

E.T.R. Dean / International Journal of Plasticity 21 (2005) 513–571

539

indicates that X 1 is positive, the equations are re-written with X 1 replaced by dW 1p , giving: dW 1p ¼ X 1b
Q1 dW 1p ;

ð85Þ

X 2 ¼ X 2b þ ðQ4
Q2 ÞdW 1p :

ð86Þ

The first equation gives dW 1p ¼ X 1b =ð1 þ Q1 Þ. Putting this into the second, if the implied value of X 2 is negative, the calculation is complete. Otherwise step 3 is done: (3) If the implied excess from step 2 is found to be positive, both excess symbols are replaced by the plastic works. The equations then give: ð1 þ Q1 ÞdW 1p þ ðQ3 þ Q1 ÞdW 2p ¼ X 1b ;

ð87Þ

ð1 þ Q2 ÞdW 2p
ðQ4
Q2 ÞdW 1p ¼ X 2b :

ð88Þ

By substituting for Qi it may be verified that the determinant of this system of equations is positive if 0 < R < 1 and K > 0, and the equations can be solved to give positive plastic works. Fig. 9 shows the solution cases, plotted on a diagram whose axes are the basic excesses X 1b and X 2b . The diagram is analogous to Loret’s (1990) Fig. 1 and Mandel’s (1965) uniqueness conditions for plastic mechanisms. For patterns, if 0 < R < 1 and K > 0, there is always a unique solution for the plastic works. Once the solutions are obtained, the energy receipts can be calculated using Eqs. (47), (48), (65) and (66), the hardenings using Eqs. (58)–(64), energy increments using Eqs. (39) and (40), and the stresses at the end of the increment using Eq. (29).

12. Evidence for the dissipation equations In proportional straining, large compressive strains are applied to a sample such that the ratio of strain increments in different directions remains constant (e.g., Topolnicki et al., 1990; Chu and Lo, 1994). Examples include isotropic compression, where the ratio of shear to volume strain increment is zero; 1D compression, where strains in two directions are zero; and straining at a critical state, where the ratio of the volume to shear strain increment is zero (Schofield and Wroth, 1968). Typical experimental results indicate that the stress ratio tends to a fixed value as proportional strains accumulate, with stress ratio depending on the proportions for the strain increments. A ‘‘kind of flow rule’’ develops asymptotically and can be different from the rule in other elasto-plastic processes. In stress-volume space, the path becomes asymptotic to a line parallel to the isotropic compression line. It seems likely that all patterns will eventually yield during these processes, so case 3 above will apply. With both patterns at their energy limits, g ¼ glim , so that

540

E.T.R. Dean / International Journal of Plasticity 21 (2005) 513–571

Fig. 9. Solution regions.

if a constant stress ratio is achieved, dglim ¼ 0, and from Eq. (68), the geometric fit parameter F will tend asymptotically to 1. With F ¼ 1 and dglim ¼ 0, the hardening equations imply that only volumetric hardening occurs asymptotically in proportional strain processes, in agreement with data. Eq. (77) simplifies to: ^i þ dEi
U lim;i 1
k dv: X i ¼ dW ip ¼ dW k

ð89Þ

^i , Eq. (23) and Using Eqs. (9), (10), and (12) to substitute for the external works dW the relation U i ¼ U lim;i to simplify, the inverse of the stress-energy equation (29) to express the energies in terms of stresses, Eqs. (47) and (48) to substitute for the energy receipts, putting X i ¼ dW ip , and re-arranging, gives: ð1
w1 ÞdW 1p þ w2 dW 2p ¼

s0 V ð1 þ gÞðð1
BÞ dv þ ð1
AÞ deÞ; 2

ð90Þ

E.T.R. Dean / International Journal of Plasticity 21 (2005) 513–571

w1 dW 1p þ ð1
w2 ÞdW 2p ¼

s0 V ð1
gÞðð1
BÞ dv
ð1
AÞ deÞ; 2

541

ð91Þ

with 1
k B¼ k



1
j : j

ð92Þ

Since the stress ratio is constant and g ¼ glim , then dglim ¼ 0, so from Eq. (67), dW diff ¼ 0, so from Eq. (62), the ratio of the plastic works is ð1 þ glim Þ=ð1
glim Þ. Using the above equations to solve for the plastic works, then equating their ratio to this value, and noting that g ¼ glim in these processes gives 1
A de w2 ð1
glim Þ
w1 ð1 þ glim Þ ¼ : 1
B dv ð1
gglim Þ
gðw2 ð1
glim Þ
w1 ð1 þ glim ÞÞ

ð93Þ

This is a flow rule for asymptotic proportional strain processes, in terms of the total strain increments rather than the plastic ones. Using Eqs. (65) and (66) to substitute for w1 and w2 , and putting g ¼ glim , then gives 1
A de ð1
C 2 Þglim ¼ ; 1
B dv C 2
g2lim

ð94Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where C ¼ 1= 1 þ 2MK . This is broadly consistent with flow rules observed experimentally and discussed by Topolnicki et al. (1990), Chu and Lo (1994), and others. At a critical state, shearing continues at constant volume (Schofield and Wroth, 1968), so de=dv is infinite then. Hence, C is the value of jglim j at a critical state. If we define a potential surface by ds0
1 de ; ¼ 1
B dv dt

ð95Þ

then differentiating the identity t ¼ ð1
AÞgs0 (Eq. (20)), substituting for dt=ds0 , putting g ¼ glim and integrating, the potential surface is found
0 

MK s ; ð96Þ ¼ 1
g2 s0ref where s0ref is an arbitrary constant. Comparison with Eq. (54) shows the potential surface is congruent with the F ¼ 1 degree-of-fit surface if K ¼ 1.

13. Introduction to the detailed calculation results The model has been programmed in Excel. The objectives were to verify the expected behaviours and to identify whether any unexpected behaviours would emerge from the numerical calculations.

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It is convenient to re-express the material constants in terms that are directly measurable. In Table 1, the compatibility constant A is expressed in terms of a friction angle /0largest : A ¼ 1
sinð/0largest Þ:

ð97Þ

This can be interpreted as follows. For plane strain biaxial processes, the mobilized friction angle /0 satisfies j sin /0 j ¼ jtj=s0 (see for example Lambe and Whitman, 1979; Parry, 1995). From Eqs. (20) and (30), we can relate this directly to the pattern energies. If we impose the constraint that a pattern’s energy cannot be negative, we find that 1
A is the sine of the largest friction angle that can be achieved by a soil element in the present model. The energy constants j and k in Table 1 are directly measurable from in-plane isotropic processes (see the following section). The third energy constant is expressed in terms of the positional constant Ciso;psb for the in-plane isotropic compression line (Eq. (32)). The degree-of-fit constant M, in Eq. (49), is re-expressed as 0 1 !2 0 1 @ sinð/largest Þ
1A ; ð98Þ M¼ 2 sinð/0m Þ

Table 1 Material properties Constant

Description and limits

Values used here

Equations

/0largest

Largest achievable friction angle in plane strain biaxial processes, 0–90 Friction angle at the peak shear stress on the F ¼ 1 degree-of-fit locus 0 6 /0m 6 /0largest Friction angle at a critical state for plane strain biaxial processes, 0 6 /0cs 6 /0largest

36

Related by Eq. (97), to A in Eqs. (4) and (5)

24

Related, by Eq. (98), to M in Eq. (49)

30

1.57

Related, by Eq. (99), to K and wi and Hi in Eqs. (47), (48), (58), (59), (63), (64), (65), and (66) (32)

0.1

(24) and (32)

0.02

(23) and (102)

1

(58) and (59)

1

(58) and (59)

/0m /0cs

Ciso;psb

k

j

R S

Positional constant for the asymptotic in-plane isotropic compression line Slope constant for the asymptotic in-plane isotropic compression line, 0 < j 6 k < 1 Slope constant for an isotropic elastic (swelling) line, 0 < j6k < 1 Rate constant for recovery of geometric fit, 0 6 R 6 1 Rate exponent for recovery of geometric fit, S > 0

E.T.R. Dean / International Journal of Plasticity 21 (2005) 513–571

543

where /0m is the friction angle on the locus of largest shear stress in Fig. 7, representing the highest shear stress magnitudes on the F ¼ 1 degree-of-fit locus. (The above equation comes from differentiating Eq. (54) at constant s0iso , with g ¼ glim , and using Eq. (20) to find when dt ¼ 0.) Finally, instead of specifying the receipts coefficient K, the following relation is used: 0 1 !2 0 1 @ sinð/largest Þ K¼
1A ; 2M sinð/0cs Þ

ð99Þ

where /0cs is the friction angle at a plane strain critical state. This equation follows from the result derived earlier that the normalized stress ratio at a critical state is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi C ¼ 1= 1 þ 2MK for the model. This ratio is also ¼ sinð/0cs Þ=ð1
AÞ. The following notes apply: (a) The result ensures that, provided C < 1, the friction angle at the critical state will not exceed the largest friction angle /0largest for the model. Using Eq. (94), the difference between /0largest and /0cs is associated with dilation, since de=dv is negative when glim > C and positive when glim <
C. This is consistent with familiar concepts, see for example, Schofield and Wroth (1968), Bolton (1986), and Jefferies and Shuttle (2002). (b) Experimentally, friction angles that apply in plane strain tests can be different from those for triaxial and other test conditions, see, e.g., Rowe (1969) and Shantz and Vermeer (1996). A more general patterns model may explain why this occurs; such an explanation cannot be given yet because the present model considers only plane strain biaxial conditions. Table 2 lists the initial states for the calculations described below. For the purposes of classification, two types of over-consolidation ratio have been used. The isotropic over-consolidation ratio (IOCRA) is analogous to familiar ratio. We can also define an individual over-consolidation ratio POCRi for each pattern. Equations for the ratios are derived below.

14. In-plane isotropic processes Table 2a lists initial states for three isotropic samples P, Q, and R. All have the same initial effective stresses and specific volume. Each has a different initial degreeof-fit F. Fig. 10 presents calculation results for ‘‘in-plane isotropic’’ compression and unloading of the three samples. Equal principal strains are applied in all in-plane directions, with no out-of-plane strain. The initial states are represented by points P 1, Q1, and R1. From Eq. (53), U lim;avg ¼ F U vol when glim ¼ 0. Also, the energy limits for the patterns are the same, so they equal the average. Using Eqs. (16), (17), (24) and (31) with t ¼ 0, gives V i ¼ VF k=ð1
kÞ ;

ð100Þ

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Table 2 Initial states s0

Sample

F

3 3 3

0.5 1 1.5

2.7 2.7 2.7

23.1 25.0 26.2

3.24 3.00 2.87

1.1 2.7 4.5

(b) Figs. 11 and 12 S 20 U 100 V 100

3 3 3

1 1 1

8.5 1.14 1.14

10 50 50

3.00 3.00 3.00

8.5 1.14 1.14

(c) Fig. 13 H30 H100 H300

30 100 300

3 3 3

0.3 1 3

5.1 1.14 0.29

13.1 50 169.5

3.43 3.00 2.65

1.14 1.14 1.14

(d) Fig. 15 L05 L10 L20 L30

5 10 20 30

3 3 3 3

0.25 0.5 1 2

(e) Fig. 16 1 2

30 30

3 3

1 2

(a) Fig. 10 P Q R

0

Pi ¼

50 50 50

s0 =2

IOCRA

0

V

Pi

Vi

POCRi

48 20.2 8.5 5.1

2.14 4.63 10 15.7

3.50 3.24 3.00 2.87

8.5 8.5 8.5 8.5

5.1 5.1

15.0 16.2

3.00 2.78

5.1 12.2

ð101Þ

F k=ð1
kÞ

for these isotropic conditions. V i is larger than V when F > 1. Physically, this could occur, for example, if half of the soil particles were associated with each pattern and the patterns shared voids. The equations are the reason why the initial pattern geometries at points P 1, Q1, and R1 depend on F. The equations also apply throughout isotropic processes for the present model. Because the initial pattern states are different for the three samples, there are different observable behaviours. The initial responses are elastic. From Eq. (36) with v ¼ lnðV0 =V Þ and g ¼ 0, the samples follow an observable path in (V ; s0 Þ space of the form lnðV Þ þ j lnðs0 Þ ¼ constant:

ð102Þ

Since no plastic work is being done, Eqs. (62)–(68) imply F is constant, so the above 0 relations for V i and P i can be substituted to give: 0

lnðV i Þ þ j lnðP i Þ ¼ constant

ð103Þ

for these isotropic processes. Different constants apply for the different samples due to the different initial values of F.

E.T.R. Dean / International Journal of Plasticity 21 (2005) 513–571

Sample P, Fo = 0.5

Sample Q, Fo = 1

545

Sample R, Fo = 1.5

3.1 asymptotic line for isotropic compression, equation 32

P1,Q1,R1

Q1

Specific volume V

P1

R1

3.0 P2

P2 Q2

Q2 R2

R2

2.9 Q4 P4

Q4 R4

P4

R4

2.8 R3

R3

P3

P3 Q3

Q3

2.7 0

100

200

300

0

Average in-plane effective stress s' 3.3

2

pattern energy limit line, equation 26 P1

3.2

= Pattern volume V i

1

Degree of fit F

P2

3.1 Q1

3.0

Q2 2.9

R1 P4 Q4 R4

2.8

R2

P3

Q3 R3

2.7 0

50

100

150

= Pattern stress P'i

Fig. 10. Calculation results for in-plane isotropic processes.

For sample P, yielding begins simultaneously in both patterns at point P 2, when the pattern limit line of Eq. (28) is reached. This happens before the asymptotic compression line in (V ; s0 ) space is reached. Yielding continues along P 2 ! P 3 and the degree-of-fit F increases towards 1 as this occurs. The stress-volume path is asymptotic to the F ¼ 1 isotropic compression line of Eq. (32) and the average stress s0 at P 2 is equal to the product of the initial fit F0 ¼ 0:5 and the stress on the asymptotic line at the same specific volume. On unloading from P 3, elastic behaviour occurs along P 3 ! P 4. For sample Q, with F0 ¼ 1, yielding begins when the state reaches the asymptotic compression line at Q2 and continues along Q2 ! Q3. For sample R, yielding does not begin until R2 is reached. The average stress s0 at R2 is F0 ¼ 2 times the average

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stress on the asymptotic isotropic compression line at the same specific volume. F decreases towards 1 during the yielding of both patterns along R3 ! R4. This results in an initially unstable response, with reducing average stress. These calculations confirm that the model’s behaviours include fair approximations to familiar isotropic behaviours. Based on these results, it can be convenient to consider two measures of over-consolidation ratio. One, denoted as IOCRA, is the ratio of the stress at yield on the isotropic compression line, assuming F ¼ 1, to the average stress. The type of calculation needed is very similar to those of Butterfield (1979), Schofield and Wroth (1968), and others. Denoting the stress and specific volume at this yield by subscript C, Eqs. (32) and (102) give: lnðVC Þ þ k lnðs0C Þ ¼ C þ k ln 2;

ð104Þ

lnðVC Þ þ j lnðs0C Þ ¼ lnðV Þ þ j lnðs0 Þ;

ð105Þ

where V ; s0 represents the initial state. Eliminating VC gives IOCRA ¼

s0C ¼ exp s0




 C þ ln 2
ðln V þ k: ln s0 Þ : k
j

ð106Þ

For all three samples, the initial value of IOCRA is the average stress at Q2 in (V ; s0 ) space, divided by the average stress at P 1Q1R1. By analogy, we can also define an over-consolidation ratio POCRi for the ith pattern, in an analogous way: 0 POCRi ¼ exp @

0

1

C
ðln V i þ k ln P i Þ A : k
j

ð107Þ

The condition that a pattern’s energy cannot exceed its energy limit, which defines the limit line of Eq. (28), implies POCRi P 1. Using Eqs. (100) and (101) gives IOCRA ¼ POCRi F k=ðk
jÞ

ð108Þ

for isotropic states of the model. This is why the isotropic OCRs in Table 2 are greater than the pattern OCRs when F > 1 and less when F < 1. This also provides for the possibility of an observed OCR value being less than 1, which may be of interest for ‘‘under-consolidated’’ soils.

15. Constant volume processes for initially isotropic samples Table 2b lists the initial states of three initially isotropic samples S, U , and V . All have the same initial values of specific volume V0 ¼ 3 and degrees-of-fit F0 ¼ 1. Fig. 11 shows numerical results for constant volume compression applied to the first two samples. The initial states are points S1 and U 1.

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547

The effective stress paths and stress–strain results are broadly consistent with observed behaviours of clays. The normally consolidated sample shows little sensitivity, with average stress reducing as shearing continues. The initially over-consolidated sample shows dilative behaviour. Both samples show a little softening at large strain; softening will be discussed later. Both samples tend towards the same critical state C at large strains. These results confirm that patterns are consistent with critical states. The developments of the pattern states are shown in Figs. 11(c) and (d). Each diagram represents one sample and the two paths in each diagram are for the two patterns. For each sample, the patterns evolve in opposite ways. Yielding begins in pattern 1 at states S2 and U 2, but not until states S3 and U 3 for pattern 2. The critical state is not represented by a single point. Rather, it is represented by the pair of points Cð1Þ and Cð2Þ, giving a fixed relationship between the pattern states which remain constant at these points as the shearing continues. The points are related by Eqs. (51) and (52) with glim ¼ C, the normalized stress ratio at a critical state, and by the degree-of-fit equation (49) with F ¼ 1 and Eq. (31).

Fig. 11. Calculation results for constant volume processes of initially isotropic samples: (a) effective stress paths; (b) stress–strain curves and degree-of-fit; (c) pattern responses for sample S; (d) pattern responses for sample U; and (e) yield surface development for sample U.

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Fig. 11 (continued)

A thermodynamic analysis of critical state shearing, based on the current model, can be done as follows, with results summarized in Table 3. Denoting the p normalized ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi stress ratio at a critical state with t > 0 as C, putting g ¼ glim ¼ C ¼ 1= 1 þ 2MK , and Q1 ¼ Q2 ¼ 0 when F ¼ 1, Eqs. (87) and (88) give:

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549

dW 1p þ

1þC s0 V dW 2p ¼ ð1 þ CÞ ð1
AÞ de; C 2

ð109Þ

dW 2p


1þC s0 V dW 1p ¼
ð1
CÞ ð1
AÞ de; C 2

ð110Þ

where the right-hand sides are the external works of Eqs. (12)–(15) with g ¼ C and dv ¼ 0. Solving gives the plastic works listed in the penultimate line of Table 3. Putting the resultsp into Eqs. (47) and (48), using Eqs. (65) and (66) with g ¼ glim ¼ C, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi and using C ¼ 1= 1 þ 2MK , gives the energy receipts listed in Table 3. Based on these results, for critical state shearing at g > 0: • For pattern 1, the external work is partly used as plastic work and is partly transferred to pattern 2. This accounts for all of the work done in pattern 1, so the pattern’s energy does not change and its state remains constant at Cð1Þ as shearing continues,. • For pattern 2, the external work is negative. The energy transfer from pattern 1 offsets this negative work and the remainder is dissipated as plastic work. Again, there is nothing left, so the pattern’s energy does not change and its state remains constant at Cð2Þ. Since F, V, and glim are all constant at the critical state, the energy limits are also constant and equal to the respective pattern energies. By virtue of the stress-energy equation (29), and the fact that the volume is constant, the constancy of the energies implies that the effective stresses do not change as the shearing continues. The above calculations represent a new simplified proposed explanation of critical states and confirm that these states are consistent with patterns. One finds that the critical energy and work-flow balance is not very sensitive to details of the assumed hardening and receipts equations. Critical states seem almost inevitable in patterns models, as in reality. Fig. 11(e) shows the calculated development of the yield boundary during the constant volume compression of initially near-normally consolidated sample U. The results reflect the developments of F and glim . The initial yield locus is isotropic, with glim ¼ 0. After yielding begins at point U 2, the lower part of the yield curve, associated with pattern 2, reduces in size as the specific volume of this pattern increases (see Fig. 11(d)). The upper part of the curve, associated with pattern 1, expands

Table 3 Work and energy account at a critical state with g ¼ C > 0 ^i (RHS of Eqs. (109) External works dW and (110)) Energy receipts dEi Net pattern works dW i (using Eq. (39)) Plastic works dW ip Hence elastic works dU i (using Eq. (40))

Pattern 1

Pattern 2

ð1 þ CÞðs0 V =2Þð1
AÞ de


ð1
CÞðs0 V =2Þð1
AÞ de


ð1
C 2 Þðs0 V =2Þð1
AÞ de Cð1 þ CÞðs0 V =2Þð1
AÞ de Cð1 þ CÞðs0 V =2Þð1
AÞ de 0

ð1
C 2 Þðs0 V =2Þð1
AÞ de Cð1
CÞðs0 V =2Þð1
AÞ de Cð1
CÞðs0 V =2Þð1
AÞ de 0

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initially, due to small plastic compression of pattern 1. As strains continue, pattern 1 begins to dilate and the locus collapses. The combined yield locus shape at the critical state is thin, highly anisotropic, and passes through point C. Anisotropic critical states have also been considered by Dafalias (1987), Thevanayagam and Chameau (1992), and others. These results confirm that patterns are consistent with the development of induced anisotropy. Fig. 12 shows calculation results for a sample V that was initially identical to U, but which was unloaded from point VP after 3% compressive strain. Because of the anisotropic hardening that occurred during this small strain, the combined yield locus shape became unsymmetrical. Because of the lack of symmetry, the magnitude of the shear stress at yield in the reverse direction, at point VQ, is smaller than at the unload point VP . Consequently, a Bauschinger effect is seen in the calculated stress–strain results. These calculations again confirm that the existence of patterns is consistent with experimental data. Patterns lead to a link between the Bauschinger effect and induced anisotropy.

16. Further calculations at constant volume Table 2c lists initial states for three isotropic samples with various initial stresses and degrees-of-fit F. All samples have the same initial specific volume of 3 and the same initial pattern over-consolidation ratios of 1.14. Sample H100 is the same as sample U that was looked at previously. The initial state for H300 outside the asymptotic isotropic compression line in stress-volume space. This is possible because the initial value of F is greater than 1. Fig. 13 shows calculation results for constant volume compression. The high peak shear stress for the sample with an initial degree-of-fit greater than 1 occurs because the stress state is initially outside the F ¼ 1 surface. The final critical state point C has F ¼ 1, so F has to reduce. The reduction causes a collapse of stress after the relatively high peak shear stress is reached. The softening seen in these results, which is particularly strong for the sample with a high value of F initially, raises several issues including: (1) Softening has been associated with instability and strain localization (e.g., Vardoulakis and Sulem, 1995; Mokni and Desrues, 1999). In these calculations, would it cause strain localization, and what impact should that have on the patterns model? (2) Is it a material property in the model? What causes it? How does it relate to sensitivity and anisotropy? Is it thermodynamically admissible? Will the final critical state be actually attained in a real sample? For the first issue, one can only say at present that the model is essentially a ‘‘point’’ model. It seems feasible that strain localization and shear banding, for instance, may be treatable as the development of special types of patterns, requiring new compatibility equations. The topic of the stability of compatibility equations is outside the scope of this paper.

E.T.R. Dean / International Journal of Plasticity 21 (2005) 513–571

551

35 yield locus after 3% compressive axial strain

30

U3

C

VP

loading paths for samples U and V

25

in-plane deviatoric stress t, kPa

20 15 10

unloading path for sample V

U2

5

U1 0 0

20

40

60

80

100

120 s', kPa

-5 -10

VQ -15 path for unloaded sample V

-20 -25 -30

(a) -35

VC

VR

yield locus and elastic region when sample reaches critical state VC after large strain in extension

35 t, kPa

U3

VP

30 25

to critical state C in compression 20 15 U2

U1

10 5 0

-20

-15

-10

-5

0

5

-10

10

15

20

ε ax , %

-5

VQ

-15

to critical state VC in extension

-20 -25

VR

(b)

-30 -35

Fig. 12. Bauschinger effect – calculation results for constant volume compression followed by extension of initially isotropic samples: (a) effective stress paths and yield surface developments and (b) stress–strain results.

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in-plane deviatoric stress t, kPa

50 45

Sample H30, initial F=0.3

40

Sample H300, initial F=3

35 30

C

25 20 15

Sample H100, initial F=1

10 5 0 0

50

100

150

(a)

200

250

300

350

Average effective stress s', kPa 50

Sample H300, initial F=3

in-plane deviatoric stress t, kPa

45 40 35 30 25 20

Sample H100, initial F=1 15

Sample H30, initial F=0.3

10 5

εax , %

0 0

(b)

5

10

15

20

25

30

Axial compressive strain

Fig. 13. Some effects of initial fit for constant volume processes for initially isotropic samples – calculation results for initial pattern over-consolidation ratios POCRi ¼ 1.14: (a) effective stress paths and (b) stress– strain curves.

In respect of the second set of issues, Fig. 14 presents a simplified thermodynamic analysis for sample H300. Fig. 14(a) shows the work and energy flows that are part of the plane strain biaxial model. The external works are from Eqs. (9), (10), and (12) (or Eqs. (18) and (19)). Energy transfers are in Eqs. (47), (48), (65) and (66) and the net works dW i done on the individual patterns are in Eq. (39). In Eq. (40), each net work is partially dissipated in plastic work and partially used to change the stored energy of the pattern. Using these flow concepts and the evolutions shown in Figs. 14(b)–(d):

E.T.R. Dean / International Journal of Plasticity 21 (2005) 513–571 ^ External work dW 1 for pattern 1

^ External work dW2 for pattern 2

Energy transfer = dE 2 from pattern 1 to pattern 2

= Plastic work dW 1p for pattern 1

= Energy storage dU 1 in pattern 1

(a)

553

= Plastic work dW 2p for pattern 2

= Energy storage dU 2 in pattern 2

2.0

Value (dimensionless)

F 1.5

= =

U1 /U lim,1

1.0

η

= =

U2 /U lim,2 0.5

ηlim εax , % 0.0

(b)

0

5

10

15

20

Fig. 14. Thermodynamic analysis of softening for sample H300 of Fig. 13: (a) work flow paths and energy storage, (b) evolutions of some of the variables that control, and are controlled by, the patterns, (c) rates of external working, energy transfer, and net working, and (d) rates of net working, plastic working, and energy storing.

(1) From Eqs. (9) and (10), the increments dV iu are negative for pattern 1 and positive for pattern 2 for axial compression at constant volume. Hence, the rate of external working is positive for pattern 1 and negative for pattern 2. (2) During the initial elastic response, up to about 0.35% axial strain, the energy ratios U i =U lim;i are less than 1, so no yielding can occur. There are no transfers of energy, equations (47) and (48), and the external works go to changing the corresponding pattern energies. Hence, U 1 increases and U 2 decreases, so g increases, Eq. (30). (3) No hardening occurs in the constant volume elastic phase, Eqs. (58) and (59), so U 1 =U lim;1 increases, which means that this pattern is moving towads yield. U 2 =U lim;2 decreases and this pattern moves away from yield. (4) At 0.35% strain, U 1 reaches U lim;1 and yielding begins in pattern 1. Because g is still small, energy begins to be transferred from pattern 2 into pattern 1 (Eqs. (47) and (65)). However, because F is so large, the 1
F s term in the hardening Eq. (58) dominates, and pattern 1 experiences softening. Its state moves up the

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Work or energy rate per axial strain, kPa

800

(c)

=

net(1), dW 1 /d ε ax

600

^

external(1), dW 1 /d ε ax

400 transfer

=

200

dE 2 /d ε ax

0 0

5

-200

10

15

εax , %

20

^ /d ε external(2), dW 2 ax

-400

=

net(2), dW 2 /d ε ax

-600 -800

Work or energy rate per axial strain, kPa

1000

=

plastic(1), dW1p /d ε ax

800

=

net(1), dW 1 /d ε ax

600

=

10 x plastic(2), 10dW 2p /d ε ax

400 storing(1),

=

200

dU 1 /d ε ax

0 0

5

10

-200

15

ax ,

%

20

=

storing(2), dU 2 /d ε ax

-400

=

net(2), dW 2 /d ε ax

-600

(d) -800 Fig. 14 (continued)

0

limit line in ðP 1 ; V 1 Þ space. The positive rate of plastic working is larger than the rate of net working for this pattern, so the rate of storing of energy dU 1 =deax is negative, Eq. (39) and Fig. 14(d). (5) The rate of energy storage for pattern 2 isnegative because negative work is being done on this pattern and energy is being extracted from it. The crosscoupling work dW diff in Eq. (62) is positive, so comparing Eqs. (58) and (59) shows that pattern 2 will be softening faster than pattern 1 at this early stage of the process. This means U lim;2 reduces faster than U lim;1 , so from Eq. (50), glim increases, as shown in Fig. 14(b) (Eq. (67) leads to the same conclusion). In stress space this means that the intersection point A of the yield loci for the patterns is moving upwards, like in Fig. 11(e). (6) Because both energy limits are decreasing but the specific volume is not changing, the amount of mutual support represented by F in Eq. (49) decreases. This can also be seen from Eq. (68), since 1
F s and
ðg
glim ÞdW diff are both negative. Because of the reduction in F, the rates of softening reduce. The 1
F s terms in the hardening equations (58) and (59) become progressively less significant as F reduces towards 1.

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(7) At about 2% axial strain, g reaches the value 1=ð1 þ 2MÞ, and w1 in Eq. (65) changes sign, and the direction of energy transfers changes. After this point, dE2 =deax is positive, Fig. 14(c), and energy begins to be transferred from pattern 1 to pattern 2. The rate of softening for pattern 2 reduces, partly due to the reduction of F, partly to the change in energy flow, and partly because the increase in glim reduces the cross-coupling effect from H2 dW diff , Eqs. (59), (62), (64), and (66). This has the effect, around 5% strain, of reversing thedecline in the energy ratio U 2 =U lim;2 for pattern 2, Fig. 14(b). (8) After this, the energy ratio for pattern 2 increases, so that this pattern is now moving towards yield. The ratio reaches 1 at about 9.3% axial strain and the pattern begins to yield. Consequently, as seen in Fig. 14(d), positive plastic work begins to be done in pattern 2. Due to this, the rates of energy transfers reduce, Eq. (48), and this slightly reduces the rate of plastic work in pattern 1 – seen by the kink in the plastic work curve in Fig. 14(d) for pattern 1. At this stage, the net work for pattern 2 is still slightly negative. To comply with Eq. (39), the pattern’s energy continues to reduce. (9) Since both patterns are now yielding, g ¼ glim and the disturbing term in Eq. (68) is zero. This allows F to be restored ever closer to 1 as further straining occurs. (10) Also during the period after 5% strain, the rate of external working for pattern 2 becomes sufficiently small and negative that the energy transfer from pattern 1 dominates. The net work for pattern 2 begins to be positive around 9.4% axial strain. (11) Because both patterns are now experiencing positive net works, and because the degree of mutual support can settle to its asymptotic value represented by F ¼ 1, plastic work can be done in both patterns and the pattern states can mutually evolve towards dW diff ¼ 0, Eq. (62). As this happens, glim tends to a fixed value, Eq. (67), and the elastic energies settle towards constant values associated with the critical state. The above analysis represents a new thermodynamic analysis of softening and critical states. It confirms the central importance of the hardening equations (58) and (59). The following notes apply: • The role of the parameter F could be interpreted in terms of sensitivity, since the hardening equations ensure that strong softening would not have occurred if F had been less than 1. A more detailed interpretation, for instance in terms of structure or de-structurization (e.g., Cottechia and Chandler, 1997; Liu and Carter, 2000; Kavvadas and Amorosi, 2000; Desai, 2001; Rocchi et al., 2003) is not possible yet because the model does not start at the level of microstructure but at the level of the ‘‘large point’’ mentioned in Section 1. • Anisotropy is highly significant, as evidenced by the evolution of glim . The balance of work and energy flows that was summarized in Table 3 would not have been possible if the states of the different patterns had not been able to evolve differently. • There were no thermodynamic difficulties, essentially because the fundamental controlling variables are thermodynamic ones and the dissipation calculation developed earlier ensured that plastic work is always dissipative. By attending to

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these fundamental aspects first, the model has gained a number of freedoms, such as the freedom to soften. In these calculations, the final critical state is attained theoretically after infinite deformations. Practically, as seen in Fig. 13(b), an axial strain of 15–20% is sufficient to almost reach the final critical state. The present model is limited in the sense that it does not consider localization which may prevent the final critical state from being attained uniformly throughout a sample. Fig. 15 shows some further calculation results for samples with initial pattern OCRs equal to 8.5. The initial states are in Table 2d. Sample L20 is the same as sample S that

50

in-plane deviatoric stress t, kPa

45

Sample L30, initial F=1.5

40

Sample L05, initial F=0.25 Sample L20, initial F=1

35 30 25

C

Sample L10, initial F=0.5

20 15 10 5 0 0

10

20

30

40

50

60

70

80

90

Average effective stress s', kPa

(a) 50

in-plane deviatoric stress t, kPa

45

Sample L30, initial F=1.5

40

Sample L20, initial F=1

35 30 25 20 Sample L10, initial F=0.5

15 10

Sample L05, initial F=0.25 5

εax , %

0 0

(b)

5

10

15

20

25

30

Axial compressive strain

Fig. 15. Further effects of initial fit – calculation results for initial pattern over-consolidation ratios POCRi ¼ 8.5: (a) effective stress paths and (b) stress–strain curves.

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was considered previously. The initial value of F for sample L20 is larger than 1. The results for the highly over-consolidated samples L05 and L10 suggest a limit like the Hvorslev (1937) surface. These samples had initial values of F lower than 1, so the first terms in the hardening Eqs. (58) and (59) were positive once plastic work began. The largest stress ratios obtained by these samples were controlled by Eqs. (30) and (20), and the fact that pattern energies in the present model could not reduce below zero. In summary, these results demonstrate that patterns can produce a range of behaviours consistent with data for soils. The results suggest that anisotropy, and mutual support or degree-of-fit, may be relevant to sensitivity, liquefaction, and steady states (Castro, 1969; Poulos, 1981; Vaid and Chern, 1985), and the Hvorslev (1937) surface.

17. One-dimensional processes Table 2e lists initial states for two isotropic samples 1 and 2. They have the same effective stress and specific volume, but different initial fits F. Fig. 16 shows calculation results for 1D compression and unloading. Both samples start at point I. For sample 1, yield begins in pattern 1 at point J. Yield starts in pattern 2 at K. For sample 2, yield begins in pattern 1 at point N, to the right of the asymptotic line in Fig. 16(b). There is an initially unstable response after this. This is similar to the isotropic response of sample R in Fig. 10 and occurs for the same reason – the initial value of F is greater than 1. In Figs. 16(a) and (b), yield begins for sample 2 in pattern 2 at point P. Both samples have reached close to the asymptote well before point L. They are then unloaded elastically along LM. To calculate the asymptotic line, we can start with the asymptotic flow rule for proportional straining, Eq. (94). For 1D processes, de=dv ¼ 1. Solving for glim , using Eqs. (20) and (97) the relation C ¼ sin /0cs =ð1
AÞ, the asymptotic normalized ratio glim;1D and stress ratio ðt=s0 Þ1D for 1D processes in the model are found as t ¼ ð1
AÞglim;1D s0 1D sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0
2 1 2 sin /0cs 2 @ A: ð111Þ ¼ ð1
BÞð1
C Þ
1 þ 1 þ 2 ð1
BÞð1
C 2 Þ The coefficient of lateral earth pressure K0 ¼ ðs0
tÞ=ðs0 þ tÞ can then be expressed in terms of the critical state angle /0cs , the parameter C, and the ratio B of Eq. (92). Fig. 16(c) shows results. They compare well with Jaky’s (1944) simplified empirical relation K0 ¼ 1
sinð/0cs Þ which is widely used in engineering practice (e.g., Wroth and Houlsby, 1985). The results suggest that much of the empirical scatter around Jaky’s relation might be accounted for by different values of C for different soils, from 0.4 to 0.9. This is taken as further confirmation that the concept of patterns is good. To calculate the asymptotic line in stress-volume space, we note that the degree-of-fit will be 1 and the state point will be on the surface of Fig. 7.

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Using Eq. (54), we can then express the average stress s01D on the asymptotic 1D compression line as:  M s01D ¼ s0iso 1
g2lim;1D ; ð112Þ where s0iso is the average stress on the asymptotic isotropic compression line at the same specific volume. Hence using Eq. (32), and noting that the vertical effective stress r0v;1D on the asymptotic curve is given by s01D ð1 þ ð1
AÞglim;1D Þ,   lnðV Þ þ k lnðr0v;1D Þ ¼ Ciso;psb þ Mk ln 1
g2lim;1D þ k lnð1 þ ð1
AÞglim;1D Þ ð113Þ

¼ C1D ; say:

This line is plotted in Fig. 16(b) for the material properties used in the calculations. The calculated stress–volume curves for the two samples confirm that the line is an asymptote.

in-plane deviatoric stress t, kPa

250 L

Sample 1, F o =1. Path IJKLM Sample 2, F o =2, Path INPLM

200

150

100 N 50

J K

P

I 0 0

100

M

200

300

400

500

600

700

-50

(a)

Average effective stress s', kPa 3.2 Asymptotic line for 1D compression 3.1 I

Specific volume V

3.0

N 2.9

J P K

2.8 2.7

M

2.6 2.5 L

2.4 2.3 10

(b)

100

1000

Principal effective stress in the direction of compression, kPa

Fig. 16. Calculation results for 1D processes for initially isotropic samples: (a) effective stress paths, (b) stress–volume relation, (c) calculated and empirical relations for K0 for normally consolidated soils, (d) K0 during unloading – empirical relation and calculated results, and (e) calculated evolution of the yield locus during 1D compression for sample 1 (stress path I ! J ! K ! L).

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559

Ko for normally 1D-consolidated material

1.0 eqn.112 with B=0.2, C=0.7

0.9

eqn.112 with B=0.2, C=0.4

0.8 0.7 0.6

eqn.112 with B=0.2, C=0.9

0.5

eqn.112 with B=0.1, C=0.9

0.4 0.3 0.2

Jaky's simplified empirical relation 0.1 0.0 0

5

10

(c)

15

20

25

Critical state friction angle

30

35

40

φ' cs, degrees

10.0

Ko divided by value (K o,nc ) at L

Calculated result for model

Power relation K o = K

o,nc .VOCR

0.35

1.0 1

10

(d)

100

Vertical over-consolidation ratio

in-plane deviatoric stress t, kPa

250 L

anisotropic locus when point K is reached

200

150 initial isotropic locus

100

anisotropic locus when point L is reached 50 J

K

I 0 0

(e)

100

200

300

400

500

600

700

Average effective stress s', kPa

-50

Fig. 16 (continued)

Fig. 16(d) shows calculated values of K0 during unloading. The results demonstrate that the patterns model is consistent with the empirical finding that K0 increases with a power of the vertical over-consolidation ratio.

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Fig. 16(e) shows the evolution of the yield locus for sample 1 during the compression. The initial locus is isotropic. After first yield, which occurs at point J with yield in pattern 1, further plastic work results in strong differential hardening. Consequently, the apex of the yield envelope moves upwards, reaching point K. At this point, yielding begins in pattern 2. The disturbing term in Eq. (68), proportional to Kð1 þ 2MÞ, is zero because both patterns are yielding and g ¼ glim . Thereafter, F stabilizes towards the value of 1, the cross-coupling work dW diff tends to zero, and g tends to its asymptote. This calculated development of the yield boundary confirms that patterns are consistent with observed phenomena of induced anisotropy and swept-out memory (Topolnicki et al., 1990; Chu and Lo, 1994). The final yield locus is broadly consistent with triaxial data for a variety of undisturbed and reconstituted clays, see for example Al-Tabbaa (1984), Clausen et al. (1984), Crooks et al. (1977), and Graham et al. (1988).

18. Discussion and conclusions This paper started with Dean’s (2003) proposal that soil fabric is built from patterns. A simple plane strain biaxial model was developed. It showed that patterns can help explain anisotropy, yielding, plastic flow, flow in proportional straining, sensitivity, swept-out memory, liquefaction, critical states, the Hvorslev surface, the Bauschinger effect, and various effects of anisotropic consolidation and unloading. It is concluded that patterns offer good potential to explain and model soil behaviour. Significant advantages include: • Consistent inclusion of induced anisotropy and critical states. • Ability to include many other behaviours of practical engineering interest; for example, it is easy to see that including rate effects or sub-yield plasticity in the behaviour of an individual pattern will introduce those behaviours into macroscopic behaviour. • The fundamental nature of patterns, illustrated by the way the yield loci in the present biaxial model were deduced rather than assumed, and by the deduction of a flow rule for proportional straining, Eq. (94). Pattern models need not be limited to assumptions or conventions of macroscopic plasticity theory, • The inherent potential to link macroscopic and microscopic models through the intermediate concept of patterns. In addition to thermodynamic or thermomechanical principles, the need to produce a unique incremental response to a given external perturbation appeared as an important organizational principle for patterns. This was fundamental to the new calculation for dissipation, Eqs. (77)–(88), and Appendix A shows how it can help determine many of the factors involved in hardening. Plane strain biaxial processes are not trivial in soil mechanics. Nevertheless, the 2D context gives a mathematically simple arena in which to try out assumptions. For more complex processes, a key modelling task would be work out how patterns orient themselves physically to achieve a general modelling of anisotropy. Aspects of a new algebra that may help with this are presented by Dean (2001,

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2004). Fig. 17 outlines the following solution that will be proposed in a planned future paper: • The ‘‘observation frame’’ developed in the present paper will be further developed to represent a general 3  3 strain tensor incorporating six independent strain components. • A transformation law will relate the updated observation frame to an upgraded material frame which will represent a fabric tensor. • Compatibility equations will be involved, as generalizations of the compatibility Eqs. (4) and (5) of the present paper, to relate the specific dimensions of the material frame to the geometries of individual patterns. • In an incremental calculation, the incremental compatibility equations will provide un-scaled increments dV i;u , Eq. (6), which will be determined by strain increments and which will be involved in the definition of pattern stresses, Eq. (12), • A set of pattern hardening laws will determine the evolution of quantities, such as the energy limits U lim;i , which can be related to the pattern geometries V i (cf. Eq. (24)). The hardening laws will therefore lead to increments dV i , • As in the present model, the new externally observable stresses at the end of the increment will be calculated from the new pattern energies at that time. For the present model, the scale increments in Eqs. (7) and (8) are calculated from incremental changes of pattern energy limits. In a general model, the increments of the limits will allow incremental transformation parameters to be calculated. These will determine the incremental re-orientation of the material frame, thereby determining incremental evolution of the fabric tensor.

strain tensor or zensor (observation frame)

transformation law

fabric tensor or zensor (material frame)

incremental compatibility equations/laws

observable strain increments

incremental geometries = dV i

compatibility equations/laws

pattern hardening laws

unscaled increments = dV iu

incremental compatibility equations/laws

pattern geometries = Vi

Fig. 17. How patterns can orient themselves in a general stress–strain process.

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Thermo-mechanical aspects of Fig. 17 are discussed in Appendix B. Since there are six independent strains in a strain tensor, there will be at least six patterns in a general model, so six hardening laws and six transformation parameters. The six patterns would be expected to be active in less general processes, including plane strain biaxial ones. To this extent, the 2D arena is a simplification of what one may realistically expect in plane strain biaxial processes. In a wider context, it seems feasible that patterns might be relevant to the behaviours of metals, crystalline and fibrous materials, and visco-plastic and other plastic materials. Patterns might also be explored in wider applications, such as footing models like those of van Langen et al. (1997), Houlsby and Cassidy (2002), and others.

Appendix A. Derivation of the hardening equations Consider Eqs. (58) and (59) again, and suppose the differential dW diff is given by the following slightly more general equation compared to the previous equation (62): dW diff ¼ adW 1p
bdW 2p :

ðA:1Þ

We now determine values of a and b on the basis of simplicity, symmetry, and solubility of the equations. We also derive expressions for the coefficient Rij of Eqs. (58) and (59), the hardening coefficients Hi , and receipts coefficients w1 and w2 of Eqs. (47) and (48). The results will not be the only possibilities, but will be simple. Using Eqs. (55)–(59) to calculate dglim and dF gives:

 2dglim 1
FS H1 H2 U lim;avg ¼ ðA:2Þ dW F þ þ dW diff ; 1
g2lim FS 1 þ glim 1
glim dF ¼ 2U lim;avg F




 1
FS dW G þ ðA1 H1
A2 H2 ÞdW diff ; FS

ðA:3Þ

where 2Mglim ; 1 þ glim 2Mglim A2 ¼ 1
1
glim A1 ¼ 1 þ

and


dW F ¼


 R11 R21 R22 R12
dW 1p

dW 2p ; 1 þ glim 1
glim 1
glim 1 þ glim

dW G ¼ ðA1 R11 þ A2 R21 ÞdW 1p þ ðA1 R12 þ A2 R22 ÞdW 2p :

ðA:4Þ

ðA:5Þ ðA:6Þ

The factors A1 and A2 relate to the degree-of-fit surface (Fig. 7). It can be convenient to arrange that dW G is non-negative, because then the first term in Eq. (A.3) will tend

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to restore F to the ‘‘best-fit’’ value of 1. A simple way for the material to arrange this is by taking the values of Rij in Eqs. (60) and (61) of the main text. This gives dW F ¼ 0 and dW G ¼ RðdW 1p þ dW 2p Þ: ðA:7Þ Using Eqs. (58), (59), and (A.1) and the more general Eq. (77), we can construct dissipation equations that are slightly more general than those of the main text. For the case of both patterns experiencing plastic work, the new equations are: # " #  " 1 þ R11 f
ðw1
aH1 Þ R12 f þ ðw2
bH1 Þ dW 1p X 1b ¼ ; ðA:8Þ R21 f þ ðw1
aH2 Þ 1 þ R22 f
w2 þ bH2 dW 2p X 2b where f ¼ ð1
F s Þ=F s . It can be shown that unique solutions are obtained for all possible cases if the on-diagonal terms of the matrix are positive and the determinant is positive. A simple way that the material would be able to satisfy the first condition is by constraining R to be in the range 0 to 1, so that R11 f and R22 f are greater than )1 for all F, and arranging that: w1 ¼ aH1 ;

ðA:9Þ

w2 ¼ bH2 :

ðA:10Þ

The above equation then simplifies to: 

1 þ R11 f R21 f þ aðH1
H2 Þ

R12 f þ bðH2
H1 Þ 1 þ R22 f

"

dW 1p dW 2p

#

" ¼

# X 1b : X 2b

Using the equations for Rij , the determinant det of the matrix is:
 1
FS det ¼ 1 þ R þ abðH1
H2 Þ2
det 2 FS

ðA:11Þ

ðA:12Þ

with
det 2 ¼

 1
FS ðR21 b
R12 aÞðH1
H2 Þ: FS

ðA:13Þ

If R is in the range 0 to 1, a simple way for the material to ensure det P 0 would be to set: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
glim a¼D ; ðA:14Þ 1 þ glim sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ glim ; b¼D 1
glim

ðA:15Þ

where D is positive constant. This makes det 2 ¼ 0 and ab P 0. This paper takes D ¼ 1.

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We now consider the hardening coefficients H1 and H2 . They cannot be equal, because then the cross-coupling terms in the above matrix equations would vanish when F ¼ 1. This would be very unexpected given that patterns would realistically be in contact and interact with each other at microscopic scale. It is also found to produce a material without critical states, contrary to data for soils. Another approach is to use Eqs. (A.9) and (A.10), and select hardening coefficients on the basis of the receipts Eqs. (47) and (48). After some exploration one finds that Eqs. (65) and (66) of the main text have some advantages. First, they prevent a pattern from extracting more energy from another pattern than is stored in that other pattern, as follows: • If pattern 1 approaches low energy compared to pattern 2, such that g becomes more negative than
1=ð1 þ 2MÞ, then w1 > 0 and w2 < 0. Eq. (47) then ensures that energy will be transferred into the pattern if any plastic work occurs. • If the energy of pattern 2 gets sufficiently low compared to that of pattern 1, Eq. (48) ensures that energy is transferred into pattern 2 whenever any plastic work occurs. Using the above results and Eqs. (65) and (66), Eq. (A.3) leads to Eq. (68) of the main text. Appendix B. Thermomechanics of patterns Collins and Houlsby (1997) develop principles of thermomechanics for geotechnical materials, based on Ziegler (1983). Fig. 17 of the present paper can help illustrate how patterns may need a less restricted approach. Collins and Houlsby (1997) make small strain assumptions. Ignoring this, the strain tensor of Fig. 17 would correspond to their eij , and the fabric tensor to their aij . Since both are tensors, we can define a transformation tijkm between them: aij ¼ tijkm ekm : ðB:1Þ The analogues for tijkm in the present paper are the scales S i of Eqs. (1) and (2). More generally, differentiation gives an evolution equation: daij ¼ dtijkm ekm þ tijkm dekm : ðB:2Þ Based on Fig. 17, pattern hardening laws provide the evolution law for tijkm and require it to involve six variables. It is not clear whether Collins and Houlsby’s deductions for stresses and other results, based on their Eq. (2.8), include the possibility of this transformation and its evolution. If not, their consequent thermodynamics may be limited in applicability to patterns. References Alshibli, K.A., Sture, S., 2000. Shear band formations in plane strain experiments of sand. J. Geotech. Geoenviron. Engrg. 126 (6), 495–503. Al-Tabbaa, A., 1984. Anisotropy of clay. M.Phil Thesis, Cambridge University. Arthur, J.R.F., Menzies, T., 1972. Inherent anisotropy in a sand. Geotechnique 22 (1), 115–128.

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Arthur, J.R.F., Chua, K.S., Dunstan, T., 1977. Induced anisotropy in a sand. Geotechnique 27 (1), 13–30. Bai, X., Smart, P., 1997. Change in microstructure of kaolin in consolidation and undrained shear. Geotechnique 47 (5), 1009–1018. Baker, R., Desai, C.S., 1984. Induced anisotropy during plastic straining. Int. J. Numer. Anal. Methods Geomech. 8, 167–185. Baker, Sir.J., Horne, M.R., Heyman, J., 1965. The steel skeleton. In: Plastic Behaviour and Design, vol. 2. Cambridge University Press. Barlat, F., Ferreira Duarte, J.M., Gracio, J.J., Lopes, A.B., Rauch, E.F., 2003a. Plastic flow for nonmonotonic loading conditions of an aluminium alloy sheet sample. Int. J. Plasticity 19 (8), 1215–1244. Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Lege, D.J., Pourboghrat, F., Choi, S.-H., Chu, E., 2003b. Plane stress yield function for aluminium alloy sheets – Part 1, Theory. Int. J. Plasticity 19 (9), 1297–1319. Been, K., Jefferies, M.J., 1985. A state parameter for sands. Geotechnique 35 (2), 99–112. Benson, P.M., Meredith, P.G., Platzman, E.S., 2003. Relating pore fabric geometry to acoustic and permeability anisotropy in Crab Orchard Sandstone: a laboratory study using magnetic ferrofluid. Geophys. Res. Lett. 30 (19), 1976, doi: 10.1029/2003GL017929. Bohlke, T., Bertram, A., Krempl, E., 2003. Modelling of deformation-induced anisotropy in free-end torsion. Int. J. Plasticity 19 (11), 1867–1884. Bolton, M.D., 1986. The strength and dilatancy of sands. Geotechnique 36 (1), 65–78 and 1987, ‘‘Discussion’’, Geotechnique 37 (2), 225–226. Bron, F., Besson, J., 2004. A yield function for anisotropic materials. Application to aluminum alloys. Int. J. Plasticity 20 (5), 965–978. Bruhns, O.T., Xiao, H., Meyers, A., 2003. Some basic issues in traditional Eulerian formulations of finite elastoplasticity. Int. J. Plasticity 19 (11), 2007–2026. Br€ unig, M., 2003. An anisotropic ductile damage model based on irreversible thermodynamics. Int. J. Plasticity 19 (10), 1679–1713. Bucher, A., G€ orke, U.-J., Kreißig, R., 2004. A material model for finite elasto-plastic deformations considering a substructure. Int. J. Plasticity 20 (4), 619–642. Butterfield, R., 1979. A natural compression law for soils – an advance on e
log p0 . Geotechnique 29, 469–480. Cambou, B., 1990. A micromechanical analysis of the behaviour of granular materials. In: Darve, F. (Ed.), Geomaterials: Constitutive Equations and Modeling. Elsevier, pp. 263–282. Carrere, N., Valle, R., Bretheau, T., Chaboche, J.-L., 2004. Multiscale analysis of the transverse properties of Ti-based matrix composites reinforced by SiC fibres: from the grain scale to the macroscopic scale. Int. J. Plasticity 20 (4), 783–810. Casagrande, A., Carillo, N., 1944. Shear failure of anisotropic materials. Proc. Boston Soc. Civil Engrg. 31, 74–87. Castro, G., 1969. Liquefaction of sands. Ph.D Thesis, Harvard University, Harvard Soil Mechanics Series No. 81. Castro, J., Ostoja-Starzewskib, M., 2003. Elasto-plasticity of paper. Int. J. Plasticity 19 (12), 2083–2098. Chaudhary, S.K., Kuwano, J., Hashimoto, S., Hayano, Y., Nakamura, Y., 2002. Effects of initial fabric and shearing direction on cyclic deformation characteristics of sand. Soils Found. 42 (1), 147–158. Cheng, Y.P., Bolton, M.D., Nakata, Y., 2004. Crushing and plastic deformation of soils simulated using DEM. Geotechnique 54 (3), 157–163. Chiarella, A.S., Shao, J.F., Hoteit, N., 2003. Modelling of elastoplastic damage behaviour of a claystone. Int. J. Plasticity 19 (1), 23–45. Chu, J., Lo, S-C.R., 1994. Asymptotic behaviour of a granular soil in strain path testing. Geotechnique 44 (1), 65–82. Clausen, C-J.F., Graham, J., Wood, D.M., 1984. Yielding in soft clay at Mastemyr, Norway. Geotechnique 34 (4), 581–600. Cleja-igoiu, S., 2003. Consequences of the dissipative restrictions in finite anisotropic elasto-plasticity. Int. J. Plasticity 19 (11), 1917–1964.

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