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186 9 Small Strain Plasticity, Non-linearity, and Anisotropy It can be shown that for transient and cyclic loading, the continuous hyper- plastic formulation of the model produces stress-strain behaviour identical to that of a classical plasticity model with an infinite number of kinematic harden- ing elliptical yield surfaces, each one with the associated flow rule and Ziegler’s translation rule. 9.5.5 Concluding Remarks The purpose of this section has been to demonstrate that a model for the small strain behaviour of soils, previously presented by Puzrin and Burland (1998) within classical plasticity concepts, together with some additional rules for han- dling stress reversals, can be expressed within a rigorous thermomechanical formulation. Modelling of the small strain non-linearity of soils is one of the key challenges of current theoretical soil mechanics. The establishment of a theoreti- cal framework within which such models can be achieved is regarded as an im- portant step towards a fuller understanding of soil behaviour. A particular chal- lenge is to combine the modelling of soil behaviour at small and large strains, where the latter has been very successfully achieved within the context of critical state soil mechanics. In the next chapter, we present a model which addresses this issue. Figure 9.5. Normalized stress-strain curves for uniform proportional cyclic loading Chapter 10 Applications in Geomechanics: Plasticity and Friction 10.1 Critical State Models Critical State Soil Mechanics (Schofield and Wroth, 1968) is arguably the single most successful framework for understanding the behaviour of soils. In particu- lar, within this approach, complete mathematical models have been formulated to describe the behaviour of soft clays. The most widely used of them is the “Modified Cam-Clay” of Roscoe and Burland (1968). Such models are highly successful in describing the principal features of soft clay behaviour: yield and relatively large strains under certain stress conditions and small, largely recov- erable, strains under other conditions. The coupling between volume and strength changes is properly described. The basic critical state models of course have their deficiencies. They do not, for instance, describe the anisotropy developed under one-dimensional consoli- dation conditions. Nor do they fit well the behaviour of heavily overconsolidated clays. Most importantly, they do not describe a wide range of phenomena that occur due to non-linearities within the conventional yield surface. Treatment of these phenomena within the hyperplastic framework will be presented in the next section; in this section, we show how the classical Modified Cam-Clay plas- ticity model can be derived. 10.1.1 Hyperplastic Formulation of Modified Cam-Clay In the following, we use the triaxial effective stress variables , p q c defined in Section 9.2. Because all stresses discussed are effective stresses, the mean effec- tive stress will be written simply p rather than p c . Conjugate to the stresses are the strains , vs HH . For the plastic strains, we use , p q DD , and the generalised stresses conjugate to them are , p q FF or , p q FF . 188 10 Applications in Geomechanics: Plasticity and Friction The Gibbs free energy is expressed as ,, , vs ggpq DD, and then it follows that v g p w H w (10.1) s g q w H w (10.2) p p g w F wD (10.3) q q g w F wD (10.4) If the dissipation function is used, it is expressed in the form ,, , , , p qpq ddpq DDDD , and then p p dw F wD (10.5) q q dw F wD (10.6) The dissipation function must be a homogeneous first-order function of the internal variable rates , p q DD . Alternatively, if the yield surface is specified in the form ,, , , , 0 pqpq yypq DDFF , then p p y w D / wF (10.7) q q y w D / wF (10.8) where / is an undetermined multiplier. We use / for the plastic multiplier here rather than O, because the latter is usually used as one of the material parame- ters in the Modified Cam-Clay model. Formally, these equations, together with the condition that ,, p qpq FF FF are all that is needed to specify completely the constitutive behaviour of a plastic material. The Modified Cam-Clay model can be expressed conveniently within the hy- perplastic approach by defining the following functions. The main parameters are defined in Figure 10.1, where V is the specific volume (total volume divided by volume of solids). The values of p, q, and x p at the reference (zero) values of strain and plastic strain are o p , 0, and xo p , respectively. The (constant) shear modulus is G. The following equations can be simplified by noting the definition exp xxo p pp DON . 10.1 Critical State Models 189 The Gibbs free energy is chosen as 2 log 1 exp 6 p pq xo o pq gp pq p pG D §· §· §· N D D ON ¨¸ ¨¸ ¨¸ ¨¸ ON ©¹ ©¹ ©¹ (10.9) The first two terms define the elastic behaviour (the unusual first term results in a bulk modulus proportional to pressure). The third term ensures that the internal variable plays the role of the plastic strain (see Table 5.2 in Section 5.5). The final term defines the hardening of the yield surface. From Equations (10.1) and (10.2), it follows that log vp o gp pp §· w H N D ¨¸ w ©¹ (10.10) 3 sq g q qG w H D w (10.11) exp p pxo p g pp D §· w F ¨¸ wD ON ©¹ (10.12) q q g q w F wD (10.13) The dissipation function may be specified as 222 exp p xo p q dp M D §· DD ¨¸ ON ©¹ (10.14) which leads [Equations (10.6)] to 222 exp p v pxo p pq d p M D §· D w F ¨¸ wD ON ©¹ D D (10.15) 2 222 exp p s qxo q pq M d p M D §· D w F ¨¸ wD ON ©¹ D D (10.16) q p M ln(p) ln(V) I s o t r op ic N CL O N p x p x C r it ica l st a t e lin e Figure 10.1. Definitions of modified Cam-clay parameters 190 10 Applications in Geomechanics: Plasticity and Friction These equations may be combined to obtain the yield function, which of course can alternatively be taken as the starting point: 2 2 2 2 exp 0 qp pxo yp M FD §· §· F ¨¸ ¨¸ ¨¸ ON ©¹ ©¹ (10.17) From this, it follows [Equation (10.8)] that 2 pp p y w D / /F wF (10.18) 2 2 qq q y M w / D / F wF (10.19) The derivation of the Modified Cam-Clay model from the above functions is not pursued here, but it can readily be verified that the above equations do define incremental behaviour consistent with the usual formulation of Modi- fied Cam-Clay. The only exception is that consolidation and swelling lines are considered straight in ln ,ln p V space rather than ln , p V space. The result is that O and N have slightly different meanings from their usual ones and that the (variable) elastic bulk modulus is given by Kp N rather than the more usual KpV N. Butterfield (1979) argues that the modified form is more satisfactory. Note that the above choice of the energy functions is not unique. This topic was addressed briefly by Collins and Houlsby (1997) and is discussed more fully in the following section. 10.1.2 Non-uniqueness of the Energy Functions Collins and Houlsby (1997) demonstrated that the modified Cam-Clay model can be derived from either of two different pairs of Gibbs free energy and dissi- pation functions. This raises the interesting concept that, because the same con- stitutive behaviour can be derived from different energy functions, then con- versely the energy functions are not uniquely determined by the constitutive behaviour. The energy functions are not therefore objectively observable quanti- ties. The case discussed by Collins and Houlsby is a special case of the following more general result. Consider a model specified by 1 ,gg VD and 1 ,,dd VDD . It follows that 1 gH w wV, 1 gF w wD, and 1 dF w wD . Using F F gives 11 0 g dwwDwwD . Now consider a model in which 12 ,gg g VDD and 12 ,,dd g VDDwwDD . In this case, again 1 gH w wV, but this time 12 g gF w wD w wD and 12 dgF w wDw wD . However, using F F again gives 11 0gdwwDwwD . Thus identical constitutive behaviour is given by the 10.2 Towards Unified Soil Models 191 two models. Of course, the models are acceptable only if both 1 ,, 0d VDD ! and 12 ,, 0dgVDD w wD D! for all D . For typical forms of the dissipation func- tion, it often proves possible to find a function 2 g that satisfies this condition. The particular model described above may alternatively be derived from the expressions, 2 log 1 6 pq o pq gp pq pG §· §· N D D ¨¸ ¨¸ ¨¸ ©¹ ©¹ (10.20) 222 exp p xo p p q dp M D §· DDD ¨¸ ON ©¹ (10.21) 10.2 Towards Unified Soil Models One of the major limitations of the basic critical state models is that they do not describe a wide range of phenomena that occur due to the irreversibility of strains and non-linearities within the conventional yield surface which may take place at a very small strain level. In Sections 9.5 and 10.1, we demonstrated how both large-scale yielding and small strain plasticity can be formulated (sepa- rately) within a hyperplastic and continuous hyperplastic framework. This sec- tion presents a unified formulation, where both large and small strain plasticity are described within a single, unified, continuous hyperplastic model. 10.2.1 Small Strain Non-linearity: Hyperbolic Stress-strain Law In Section 9.5, we presented a continuous hyperplastic formulation of small strain non-linearity based on the Puzrin and Burland (1996) logarithmic func- tion. It provides realistic fitting of the typical “S-shaped” curves of secant shear stiffness against the logarithm of shear strain observed for soils (see Fig- ure 10.2b). For the purposes of this section, however, the simple hyperbolic form is sufficient to illustrate the principles involved. As in the logarithmic model, a one-dimensional hyperbolic model can be defined by two potential functionals: >@ 11 2 2 00 ˆ ˆˆ ˆ , 22 h g dd E K V VD V D K K D K K ³³ (10.22) 1 0 ˆˆ dkd ªº D KDK K ¬¼ ³ (10.23) The dissipative generalised stress function ˆ FK is obtained as ˆ ˆˆ S ˆ d k w FK K DK wD K (10.24) 192 10 Applications in Geomechanics: Plasticity and Friction so that the field of yield functions is given by 2 22 ˆ ˆ 0ykK FK K (10.25) The generalised stress is defined by ˆ ˆ ˆ ˆ ˆ g h w FK V KDK wD K (10.26) and the strain is given by 1 0 ˆ g d E w V H D K K wV ³ (10.27) Combining Equations (10.24), (10.26), and (10.27) allows the stress-strain curve for monotonic loading (from zero initial plastic strain) to be expressed as * 0 ˆ k d E h K VVK H K K ³ (10.28) where *K specifies the largest yield surface that has yet been activated, such that ˆ ˆ ** *kFK K FK V . Differentiation of Equation (10.28) twice with respect to V (using standard re- sults for the differential of a definite integral in which the limits are themselves variable) leads to the important result 2 2 1 ˆ d kh k d H V V (10.29) so that the plastic modulus function ˆ h K is uniquely related to the second de- rivative of the initial “backbone” curve HV (see Section 8.10). The hyperbolic stress-strain curve (Figure 10.2a) is given by (see Section 8.7) 2 2 1 ka a Ek E Ek VV V V H V V (10.30) where 50 aEE , E is the initial stiffness, and 50 E is the secant stiffness to 2kV . This curve is generated by the function 2 2 23 21 1 ak d kh k d Ek H V V V , so that for the hyperbolic model, Equation (10.22) should be supplemented by 3 1 ˆ 21 E h a K K (10.31) 10.2 Towards Unified Soil Models 193 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0246810 H E / c / c a 1 ( a ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -3 -2 -1 0 1 2 log 10 ( H E / c ) E secant / E ( b ) Figure 10.2. (a) Hyperbolic stress strain curve (for a = 5); (b) typical stiffness - log strain curve for soil 10.2.2 Modified Forms of the Energy Functionals The continuous hyperplastic formulation presented in Chapter 8 considered a Gibbs free energy of the form, ˆˆ ˆ ,,, ij ij ij ij g gd 8 ªº VD VD KK*KK ¬¼ ³ (10.32) and a dissipation function of the form, 1 0 ˆ ˆˆ ˆˆ ,, , , ij ij ij ij ij ij dd d ªº VDD VDD KK*KK ¬¼ ³ (10.33) The following result is derived from the Frechet differential of the energy func- tional: ˆ ˆ ,, ˆˆ ˆ , ˆ ij ij ij ij ij ij ij g g ddd 8 wVDKK ªº c VD D D*KK ¬¼ wD K ³ (10.34) 194 10 Applications in Geomechanics: Plasticity and Friction where g c indicates the derivative with respect to the function ˆ ij D . By defining ˆ ˆ ij ij ij ij g ds 8 HV F KD K*K KT ³ (10.35) an expression for the generalised stress function is obtained: ˆ ˆ ˆ ij ij g w F wD K (10.36) Similarly, ˆ ˆ ˆ ij ij dw F wD K (10.37) In the cases considered below, g takes a more complicated form that can be writ- ten as 1 1234 0 ˆˆ ˆ ,, , ij ij ij ij ij ij ij ij ij gg ggdg ªº VDD V VD D D KKK D ¬¼ ³ (10.38) where for convenience, a variable 1 0 ˆ ij ij dD D K ³ is also introduced. From the above, it can be shown that 1 3 2 0 ˆ ˆ , ˆˆ ˆ ,, ˆ ij ij ij ij ij ij ij ij g g dg dd wDKK ªº c VDD D D D K ¬¼ wD K ³ (10.39) and 1 24 3 0 ˆ ˆ ij ij ij ij ij g gg gd ww w V D K K wD wD wD ³ (10.40) If the time rate of change of the Gibbs free energy is written 1 0 ˆ ˆ ij ij ij ij ij ij g ds H V F D F K D K K T ³ (10.41) and the definition of ij D as a constraint equation, 11 12 00 ˆˆ ˆ 0 ij ij ij ij cc c d d D DKK DDKK ³³ (10.42) 10.2 Towards Unified Soil Models 195 then it is possible to define the generalised stresses in terms of the derivatives of an augmented energy expression c gc/ , where c / is a multiplier to be de- termined: 32 2 ˆˆ ˆˆ , ˆ ˆˆ ij ij ij ij c ij ij gc g wDKK wDK F D / wD K wD K (10.43) 1 241 3 0 ˆ ˆ ij ij ij c ij ij ij g gc gd www F V D K K / wD wD wD ³ (10.44) The dissipation is considered of the form, 1 12 0 ˆˆ ˆˆ ˆ ,, ,, ij ij ij ij ij ij dddd ªº VDD VDK D K K ¬¼ ³ (10.45) From this, the Frechet differential leads to the result, 2 1 ˆ ˆ ˆ ,, ˆ ij ij ij ij d d w FK VDK wD K (10.46) and it also follows that 0 ij ij dw F wD (10.47) Given particular forms of the functions, the above equations are sufficient to determine the constitutive behaviour. 10.2.3 Combining Small-strain and Critical State Behaviour As mentioned in earlier sections, a major criticism of critical state models is the fact that they describe the behaviour of soils inadequately at small strains. Coupled to this is poor performance with respect to modelling cyclic loading, and no modelling of the effects of immediate past history. To remedy this situation, the benefits of the simple hyperplastic framework for describing kinematic hardening of an infinite number of yield surfaces is combined with the Modified Cam-Clay model. The following expressions are suggested for the thermodynamic potentials: 2 22 1 3 0 log 1 6 ˆˆ 3 1 21 2 pq o xpxq x pq gp pq pgp pgp dp a §· §· N D D ¨¸ ¨¸ ¨¸ ©¹ ©¹ ND D K K ON ³ (10.48) [...]... hyperplastic critical state model It was demonstrated how the specification of two potential functionals allows derivation of constitutive models that satisfy the laws of thermodynamics, and at the same time account for many important aspects of soil behaviour Houlsby and Mortara (2004) explore an approach to the behaviour of frictional materials based on continuous hyperplasticity The approach draws... on earlier work by Houlsby ( 199 2) Collins and Muhunthan (2003) explore the relationship between stress– dilatancy, anisotropy, and plastic dissipation for geotechnical materials Hyperplasticity is used to analyse the stress–dilatancy relation for shear deformations of frictional, granular materials Central to this approach is the recognition that in general, deformations of granular materials can involve... paths for clay at OCR of 2 after different immediate past stress histories (The irregularities in the curves are a result of numerical discretisation) The above examples illustrate that a model based on the continuous hyperplasticity approach can capture many of the important features of soil behaviour at small strains, whilst being consistent with the ideas of critical state soil mechanics for larger... hyperplasticity There are many ways this can be achieved by hyperplasticity For instance, alternative approaches within hyperplasticity are explored by Einav (2002) [see also Einav and Puzrin (2003, 200 4a, b) and Einav et al (200 3a, b)] and by Likitlesuang (2003) [see also Likitlersuang and Houlsby (2006)] 10.3 Frictional Behaviour and Non-associated Flow In Chapter 5, we gave examples of plasticity models... of stored plastic work are discussed The models automatically satisfy the laws of thermodynamics, and there is no need to invoke any stability postulates Some classical forms of the peak-strength/dilatancy relationship are established theoretically Some representative drained and undrained paths are computed Collins and Kelly (2002) and Collins (2003) propose a systematic, hyperplasticity based procedure... general than that traditionally used for materials with non-associated flow rules; in that plastic potentials are not needed and not presumed to exist In illustration, examples of families of models are given in which the critical state surface is either the Drucker-Prager or the Matsuoka-Nakai cone Einav and Puzrin (2003, 2004b) unify the hyperplasticity and continuous hyperplasticity formulations... triaxial conditions Once the form of the free energy and dissipation potential functions have been specified, the corresponding yield surfaces, flow rules, isotropic and kinematic hardening rules as well as the elasticity law are deduced in a systematic manner The families contain the classical linear frictional (Coulomb type) models and the classical critical state models as special cases The generalised... Non-associated Flow 205 The yield surface considered here is the generalised Houlsby ( 198 6) criterion, which is a generalisation of the criterion introduced by Matsuoka and Nakai ( 197 4) for describing yielding in geotechnical materials This case also serves as an example of the derivation of the yield function from the dissipation function and vice versa It illustrates the use of constraints and the fact... models for frictional materials possessing a critical state in a three-dimensional context The models involve a number of parameters, which can be interpreted in terms of micromechanical energy storage and dissipative mechanisms In most cases non-associated flow 210 10 Applications in Geomechanics: Plasticity and Friction rules are predicted, and in some cases, the yield surfaces have concave segments The... response At large strain, the response for case (b) is softest because of the swelling that has occurred during the unloading to point A The above behaviour is shown more clearly in Figure 10 .9, which shows the same data in terms of the normalised secant stiffness Gs G , where Gs q 3 s, against the deviatoric strain (on a logarithmic scale) Each of the tests shows the characteristic “S-shaped” curve . regarded as an im- portant step towards a fuller understanding of soil behaviour. A particular chal- lenge is to combine the modelling of soil behaviour at small and large strains, where the latter. Small-strain and Critical State Behaviour As mentioned in earlier sections, a major criticism of critical state models is the fact that they describe the behaviour of soils inadequately at. behaviour and critical state concepts in a single model based on hyper- plasticity. There are many ways this can be achieved by hyperplasticity. For instance, alternative approaches within hyperplasticity
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