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84 5 Elastic and Plastic Models in Hyperplasticity 5.2.3 Rigid-plastic Models Rigid-plastic materials are special cases in which energy is dissipated only, and not stored. They can be described without use of internal variables because there is no elastic strain. The total strain is simply equal to the plastic strain. To de- scribe such materials without internal variables, it would be necessary to de- velop a theory in which the dissipation could depend on the strain rate, e. g. , ij ij dd HH . Rather than develop such a theory, we use instead the approach already adopted, but simply introduce a set of constraints 0 ij ij ij c H D . Using the f formulation, we start from 0f and, because of the constraints, write ij ij ff c c / and ij ij ij f c w V / wH (5.25) ij ij ij f c w F / wD (5.26) Then, as before, we use the dissipation function DD 2 ij ij dk together with the constraint 0 kk c D , so that 2S ij ij ij k c F D . Combining this with Equations (5.25), (5.26) and the constraint equation gives 2S ij ij ij k c V H , from which both the yield surface and the flow rule follow. Note that the mean stress kk kk kk V / F is undetermined by the constitutive law because of the 0 D kk c constraint, and appears purely as a reaction. The same result could also have been obtained by specifying the yield surface as 20 ij ij yk cc FF and deriving the expression for ij D and hence (from the constraint) ij H . If the g formulation is used, one simply starts from the expression ij ij g V D , from which it immediately follows that ij ij ij g H w wV D and ij ij ij g F w wD V . The rest of the formulation follows as above. Note that if the two starting functions are g and y, then no additional constraints need to be introduced. This is an example of a more general observation that the g and y formulation often offers the simplest route for deriving constitutive behaviour. 5.3 Frictional Plasticity and Non-associated Flow In the previous section, we demonstrated that hyperplasticity can reproduce sim- ple elastic perfectly plastic models where the size of the yield surface in the devia- toric plane does not depend on the stress state. This class of models describes a wide range of materials – from metals to saturated clays. However, most granu- lar materials exhibit a property called friction, characterized by dependence of 5.3 Frictional Plasticity and Non-associated Flow 85 shear strength on normal stress. As will be demonstrated below, the hyperplastic formulation can easily reproduce this behaviour with an interesting and impor- tant restriction – the flow rule for these models cannot be associated. 5.3.1 A Two-dimensional Model We shall start by introducing a simple frictional model defined in a two- dimensional stress space ,VW , where V and W are the normal and shear stresses. The corresponding normal and shear strains are H and J, respectively. Consider the model specified by 22 22 g KG HJ VW VD WD (5.27) d J PVD (5.28) complemented by the constraint 0c HJ D ED (5.29) This results in the modified dissipation function dd c J H c / PV/ED/D (5.30) with / as a Lagrangian multiplier. The standard procedures applied to d c pro- duce: S d d V H WJ J c w F / wD c w F PV/E D wD (5.31) which, by eliminating /, can be combined to produce the yield surface in the generalised stress space (see Figure 5.4): 0y WV F PVEF (5.32) with the flow rule S y y H V J W W w D O OE wF w D O O F wF (5.33) Substitution of g VV H w F F V wD and g WW J w F F W wD in (5.32) and (5.33) gives a yield surface identical to that derived from the conventional plasticity 86 5 Elastic and Plastic Models in Hyperplasticity model (2.41) in true stress space (with P* = P + E) and the flow rule (2.43) with the plastic potential (2.44), as shown in true stress space in Figure 2.5. 5.3.2 Dilation As seen from Figure 5.4, within the framework of the above model, when E > 0, plastic shearing leads to positive increments of normal plastic strains which cause an increase in volume. This is a well-known phenomenon in the behaviour of dense granular materials, called dilation. Experimental data, however, show that the associated flow rule would grossly overestimate the amount of dilation. An associated flow rule is obtained in this model only if P = 0. The ability of this hyperplastic model to accommodate non-associated flow by allowing E > 0 is important for realistic modelling of the behaviour of frictional materials. Fur- thermore, in this model, the purely empirical observation of P* = P + E is ex- plained for dense materials that exhibit dilation. As demonstrated in Chapter 4, the Second Law of Thermodynamics demands that mechanical dissipation be non-negative. Applied to Equation (5.28), this requirement yields P t 0. In the case P = 0, then P* = E, the dissipation is zero, and the flow is “associated” in the conventional sense. Clearly such a model of frictional behaviour is unrealistic because of the implication of zero dissipation. Figure 5.4. Yield surface and plastic potential for a frictional model 5.3 Frictional Plasticity and Non-associated Flow 87 There are no restrictions on the sign of E. Clearly, when E = 0, we observe in- compressible behaviour, which is achieved in the critical state. Choosing E < 0 allows for contraction, exhibited by some loose granular materials, at least for a limited range of strains. 5.3.3 The Drucker-Prager Model with Non-associated Flow Extension of the two-dimensional hyperplastic model into the general six-di- mensional stress-strain space is formulated through the following potential functions: 18 4 ii jj ij ij ij ij g KG cc VV VV V D (5.34) G cc D D 2 33 kk ij ij dM (5.35) complemented by the constraint 2 0 3 ii ij ij cB cc D D D (5.36) This results in the modified dissipation function, G/ cccc D D /D 3 2 33 kk ij ij ii MB d (5.37) with / as the Lagrangian multiplier. The standard procedures applied to d c produce 3 2 33 ij kk ij ij mn mn MB c D V/ F /G cc DD (5.38) From which we derive 3 ii F / (5.39) 2 33 ij kk kk ij mn mn MB c D VF c F cc DD (5.40) which can be rearranged to eliminate the rates of the plastic strains and give the yield surface in generalised stress space, expressed as 3 0 23 kk kk ij ij MB y VF §· cc FF ¨¸ ©¹ (5.41) which is closely analogous to the simple two-parameter model derived in Sec- tion 5.3.1. 88 5 Elastic and Plastic Models in Hyperplasticity 5.4 Strain Hardening 5.4.1 Theory of Strain-hardening Hyperplasticity The general hyperplastic framework was presented in Chapter 4. Here we revisit briefly some of its highlights, relevant for generation of strain-hardening hyper- plastic models. Potential Functions The constitutive behaviour of a dissipative material can be completely defined by two potential functions. The first function is either the Gibbs free energy ıĮș,, ij ij g or the Helmholtz free energy İĮș,, ij ij f . We shall omit the dependence on temperature in the following because we are not considering thermal effects. If g is specified, then the strain is obtained by , ij ij ij ij gH w VD wV and a “generalised stress” ij F from , ij ij ij ij gF w VD wD. For brevity in the following, we shall often omit the list of arguments where a function appears within a differential, so that we would write, for instance, simply ij ij g F w wD . By a suitable choice of ij D , it is possible to write the Gibbs free energy so that the only term which involves both ij V and ij D is linear in ij D : 123ij ij ij ij gg g g VDVD (5.42) Furthermore, if 3 g is also linear in the stresses, then it is always possible (again by a suitable choice of ij D ) to choose simply 3 ij ij g V V: 12 , ij ij ij ij ij ij gggVD V D VD (5.43) The second function required to define the constitutive behaviour is (assuming that g is the specified energy function) either a dissipation function ,, 0 g ij ij ij dd VDDt or a yield function ,, 0 g ij ij ij yy VDF . For a rate- independent material, dissipation is a homogeneous function of order one of the plastic strain rate tensor. In this case, dissipation and yield functions are related by a degenerate Legendre transformation. The dissipative generalised stress is defined as ,, g ij ij ij ij ij dF w VDD wD . Then from a property of the transforma- tion, it follows that ,, g ij ij ij ij ij yD Ow VDF wF , where O is an arbitrary non- negative multiplier. The formulation is completed by the orthogonality assump- tion of Ziegler (1977), which is equivalent to the assumption ij ij F F . 5.4 Strain Hardening 89 Link to Conventional Plasticity The strain hardening hyperplastic formulation, although based entirely on specification of two potential functions, intrinsically contains all the compo- nents of conventional plasticity theory. As mentioned above, the yield function, although in generalised stress space, is obtained as a degenerate Legendre trans- form of the dissipation function. When the Gibbs free energy can be written in the form of (5.43), no elastic-plastic coupling occurs, and it follows that 1 , ij ij ij ij ij ij ij ggwVD w V H D wV wV (5.44) The interpretation of the above is that ij D plays exactly the same role as the conventionally defined plastic strain p ij H (elastic strain is defined as 1 e ij ij ij ij gH HD wwV ). The generalised stress in this case also acquires a clear physical meaning: 2 , ij ij ij ij ij ij ij ggF w VD wD Vw D wD. The generalised stress simply differs from the stress by the term 2ij ij ij ij gUD w D wD, known as the “back stress”. In conventional kinematic hardening plasticity, the “back stress” would normally be associated with the stress coordinates of the centre of the yield surface. The flow rule is given by the expression ,, g ij ij ij ij ij ywVDF D O wF (5.45) where O is an arbitrary non-negative multiplier. When the yield (and dissipa- tion) function does not depend on stress explicitly, this flow rule is associated in both true and generalised stress spaces. Finally, the hardening rule is also specified by the two potential functions. Isotropic hardening is defined by dependency of the dissipation ,, 0 g ij ij ij dd VDDt or yield function ,, 0 g ij ij ij y VDF on the internal variable ij D . Kinematic hardening is defined by dependency of the “back stress” 2ij ij ij ij gUD w D wDon internal variable ij D , as defined by the second term of the Gibbs free energy function. The translation rule for the yield surface can be recognized in the expression 2 2 ij ij ij ij ij kl ij kl gwD UD VF D wD wD (5.46) 90 5 Elastic and Plastic Models in Hyperplasticity Dissipation and Plastic Work It is important to note the difference between the dissipation ij ij d F D , which we require to be non-negative on thermodynamic grounds, and the rate of plas- tic work p p ij ij W V H . Consider, for instance, uncoupled plasticity for which we can write 12 , ij ij ij ij ij ij gggVD V D VD. In this case, as discussed above, the internal variable plays the role of the plastic strain p ij ij D{H . The difference between the rate of plastic work and dissipation is therefore p p ij ij ij ij ij ij Wd VH FD UD . Using (5.46), we see that dissipation and plastic work are identical only if 2 2 0 kl ij kl gw D wD wD , which can only be true for general increments of the plastic strain if 2 2 0 ij kl gw wD wD . Thus the condition that the dissipation is equal to the plastic work reduces to a simple condition on the form of the energy function. Incremental Stress-strain Response In a way similar to conventional plasticity, two possibilities exist in the descrip- tion of the incremental response of a strain-hardening hyperplastic material. Consider a non-frictional material, either its state is within the yield surface ( DF ,0 g ij ij y ), in which case no dissipation occurs, and O = 0. If the material point lies on the yield surface ( ,0 g ij ij y DF ), then plastic deformation can occur, provided that 0 O t . In the latter case, the incremental response is ob- tained by invoking the consistency condition of the yield surface: ,0 gg g ij ij ij ij ij ij yy y ww DF D F wD wF (5.47) which is solved for the multiplier O: 2 2 g ij ij gggg ij ij kl kl ij ij y yyyy g w V wF O wwww w wF wD wD wF wD wF (5.48) Differentiation of Equation (5.44) and substitution of (5.45) in the result and in (5.46) give the incremental stress-strain response: 1 g ij kl ij kl ij y g w w H VO wV wV wF (5.49) 5.4 Strain Hardening 91 and the update equations for internal variable and generalised stress: g ij ij y w D O wF (5.50) 2 2 g ij ij ij ij ij ij kl kl y g w w F VU D VO wD wD wF (5.51) The multiplier O defined from Equation (5.48) is derived from the consistency condition ,0 g ij ij y DF . Therefore, it applies only when ,0 g ij ij y DF and 0 O ! . For all other cases (i. e. either when ,0 g ij ij y DF or when (5.48) results in a negative O value), then 0 O . 5.4.2 Isotropic Hardening One-dimensional Example Consider a model with constitutive behaviour completely defined by the follow- ing specific Gibbs free energy potential function ,g VD : 2 1 , 2 g E VD V VD (5.52) From (5.43), we can derive g E w V H D wV (5.53) g w F V wD (5.54) The second function required is a dissipation function: ,0dkDD D Dt (5.55) where 0k D! . The dissipative generalised stress F is obtained as ,SdkF w DD wD D D , so that the yield function is most conveniently expressed as 2 2 0yk F D (5.56) which defines the linear elastic range ;kkD D ªº ¬¼ for the generalised stress F . Differentiation of the yield surface gives 2 y w D O OF w F (5.57) 92 5 Elastic and Plastic Models in Hyperplasticity The incremental response of the hyperplastic model is found from Equations (5.48)–(5.51): E V H D (5.58) 2SkD O D D (5.59) F V (5.60) where we can obtain the solution, 2 2 2 2 ,when 0 2 0, when 0 k kk k V F D ° ° Dw wD O ® ° F D ° ¯ (5.61) The notation is used for Macaulay brackets (i. e. ,0xxx !; 0, 0xx d). Two cases now follow. Either 0 O d , in which case no dissipation occurs (“elastic” behaviour), 0D : EV F H (5.62) or 0 O ! , in which case dissipation occurs (“plastic” behaviour), 0Dz , and we can derive Ek Ek wwD V H w wD (5.63) which for linear hardening, kkHD D, yields S S EH EH D V H D (5.64) The constitutive behaviour described by these incremental equations is shown in Figure 5.5 for a first loading with positive (OAB) or negative (OCD) Figure 5.5. Behaviour of model with linear isotropic hardening: (a) in true stress space; (b) in gener- alised stress space 5.4 Strain Hardening 93 plastic strain, followed by an unloading. From Figure 5.5a, it is clearly seen that the initial linear elastic range >@ ;kk of the model undergoes expansion. For plastic loading, we observe strain hardening with the tangent modulus 1 EEHEH . In generalised stress space, where the effect of the elastic strain is eliminated, the behaviour appears as rigid-plastic with linear strain hardening/softening, as in Figure 5.5b. A further modification is required if this isotropic hardening model is to pro- vide realistic modelling on subsequent cycles of unloading and reloading. The expression kkHD D would imply softening if (after first having positive plastic straining) D were later to be reduced. The hardening should be a func- tion of the accumulated plastic strain. This can be achieved by introducing a second kinematic variable E, which is defined through the constraint equation 0c E D . The hardening is then expressed in the form kkHE E. For a first loading, this modification makes no difference to the model, but it gives realistic behaviour on unloading, as illustrated in Figure 5.6. Multidimensional Example (the von Mises Yield Surface) The following model is an example of isotropic hardening hyperplasticity in six- dimensional stress space. The constitutive behaviour of the model is again de- fined by two potential functions. In this case, these are supplemented by the plastic incompressibility condition, 0 kk D , which is introduced as a constraint (see Chapter 4). The first function is the specific Gibbs free energy: 11 , 18 4 ij ij ll kk ij ij ij ij g KG cc VD VV VVVD (5.65) Figure 5.6. Behaviour of unmodified and modified models with linear isotropic hardening on reversal of plastic strain direction [...]... most important cases of conventional plasticity In fact, generalisation of this framework allows derivation of even broader classes of modern plasticity models using the thermomechanical approach First, introduction of multiple kinematic internal variables leads to the thermomechanical formulation of kinematic hardening plasticity with multiple yield surfaces In the next step of generalization continuous... continuous internal functions instead of a discrete set of internal variables are introduced A direct application of this generalisation is the thermomechanical formulation of kinematic hardening plasticity with an infinite number of yield surfaces These applications will be considered in the following chapters Chapter 6 Advanced Plasticity Theories 6.1 Developments of Classical Plasticity Theory Chapter 2... is a single sudden transition from elastic to plastic behaviour as a single yield surface is encountered represents too dramatic a change in response The transition from elastic to plastic behaviour, especially for certain materials such as soils, can be more progressive Also, unload-reload loops tend to exhibit a certain amount of hysteresis in many materials, whereas the simple approach described above... Figure 5. 8 Cyclic stress-strain behaviour of the St-Venant model with linear kinematic hardening: (a) in true stress space; (b) in generalised stress space 98 5 Elastic and Plastic Models in Hyperplasticity Figure 5. 9 Schematic layout of the St-Venant model with linear kinematic hardening spring, i e the total strain is equal to the elastic strain After the stress reaches the value of slip stress k, the... original plasticity relations when the bounding surface is reached, we require F F , and so that reduced plastic strains are given within that h 0 H p ij ij the bounding surface h h 0 In practice, the form of h should be determined by calibration against experimental unloading-reloading curves A variety of forms have been suggested in the literature The advantage of the bounding surface approach is that,... is elastoplastic with a linear hardening characterized by the tangent modulus E1 from Equation (5. 93) Note that Martin and Nappi (1990) developed a version of kinematic hardening plasticity similar to that described here, but based their approach on the Helmholtz free energy rather than the Gibbs free energy We find the Gibbs free energy approach more attractive, because the link to conventional stress-based... g Dafalias and Herrmann, 1982), the yield surface f 0 plays a rather different role from the classical formulation It is renamed the bounding surface because strictly no plastic strain is allowed inside a yield surface To emphasize the change in role (and for consistency with the presentation of multiple surface plasticity below), we shall write the bounding 106 6 Advanced Plasticity Theories surface... conventional plasticity As an illustration of these features, the hierarchy of plasticity models within the thermomechanical framework is given in Table 5. 2 A Gibbs free energy formulation for isothermal conditions is taken as an example Let us start with classification with respect to elastic-plastic coupling A material is decoupled when the energy function can be presented as a sum of the following three... describes the approach used for basic plasticity theories, and Chapter 5 allows these theories to be reformulated within a thermodynamically consistent framework However, countless developments of classical theories will be found in the literature The developments are principally to allow more realistic modelling of behaviour in unload-reload cycles It is typically found that the model of material behaviour... incremental equations is shown in Figure 5. 8 From this figure, it is clearly seen that the linear elastic range k; k of the model does not undergo any expansion, just being translated along the stress axis following the current stress state In generalised stress space, where the effects of “back stress” and elastic strain are eliminated, the behaviour appears as rigid-perfectly plastic (Figure 5. 8b) This . Models The main features of hyperplasticity are the compactness of the formulation and a clear link to conventional plasticity. As an illustration of these features, the hierarchy of plasticity. the behaviour of dense granular materials, called dilation. Experimental data, however, show that the associated flow rule would grossly overestimate the amount of dilation. An associated flow. step of generalization continuous internal functions instead of a discrete set of internal variables are introduced. A direct application of this generalisation is the thermomechanical formulation
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