110 6 Advanced Plasticity Theories when these two surfaces come in contact with each other. The modulus can, for instance, be specified by an expression of the form, 0 0 hH hH J §· G G ¨¸ G ©¹ (6.8) where as before PR G is the distance from the current stress point to the image point. The form ensures that 0hH and 00 hhG . This formulation is very close in concept to bounding surface plasticity. This form of plasticity, however, avoids the problem inherent in the bounding surface models of ratcheting for small cycles of unloading and reloading. The more sophisticated rules for defining the image point and the translation of the inner yield surface provide a more realistic way for describing hysteretic behav- iour and the effect of past loading history. This process can be taken a step further by introducing a set of nesting sur- faces within the domain between 0f and 0F . These surfaces can translate and expand or contract due to plastic straining. They are capable of encoding in a more subtle way the details of the past stress history. The relative motion be- tween each adjacent surface is defined by rules similar to (6.6) to ensure that they remain nested. The hardening modulus can be defined by an interpolation formula such that it is 0 h on the innermost yield surface, and equal to H on the outermost surface. For instance, it can be assumed if the stress point is in con- tact with n surfaces out of a total of N: 0 1 Nn hH h H N J §· ¨¸ ©¹ (6.9) The configuration of the nesting surfaces and therefore the subsequent stiff- ness for a particular stress path depends on the past history of loading. The ma- terial response for any loading history may be studied by following the evolution of the configurations of the nested surfaces. This model possesses a multi-level memory structure because, for cyclically varying stress, only a certain number of surfaces undergo translation; the other surfaces may change only isotropically. This approach can be extended to an infinite number of surfaces, although for practical computations, a finite number is necessary. 6.4 Multiple Surface Plasticity Although it has certain advantages, the translation rule for multiple yield sur- faces that requires that the surfaces remain nested is not strictly necessary (see Section 6.5). The principal advantage of the nested approach is that this allows the determination of a single plastic strain component, with its magnitude estab- lished by one of the above procedures for the hardening modulus. 6.4 Multiple Surface Plasticity 111 An alternative approach is to introduce multiple yield surfaces, but to treat each as independent, giving rise to a separate plastic strain component. The total strain is the sum of the elastic strain and the plastic strain components: 1 N epn ij ij ij n H H H ¦ (6.10) Each yield surface is specified in the form ıİ,0 pn n ij ij f , where for simplicity the yield surface depends only on the plastic strains associated with that surface, and is not coupled to other yield surfaces by dependence on their plastic strains. In principle, non-associated flow can be specified read- ily by defining plastic potentials distinct from the yield surfaces so that n pn n ij ij gw H O wV . Combining the elasticity relationship and the flow rule, one obtains: ıİ ı 1 n N n ij ijkl kl ij n g d §· w ¨¸ O ¨¸ w ©¹ ¦ (6.11) and the plastic multipliers are eliminated by the consistency conditions for each of the yield surfaces: 0 nn nnn pn n ij ij ij pn pn ij ij ij i j i j ff ffgww www V H V O wV wV wV wH wH (6.12) The analysis proceeds exactly as for the single surface model with the elastic- plastic matrix determined as 1 nn ijab mnkl N ep ab mn ijkl ijkl nn n pqrs pn pq rs rs gf dd dd f fg d ½ ww °° °° wV wV °° ®¾ §· °° www ¨¸ °° ¨¸ wV wV wH °° ©¹ ¯¿ ¦ (6.13) Strictly, the summations in the above equations are only for the “active” yield surfaces, for which 0 n f ; on the other surfaces, simply, 0 n O . It can be seen that the multiple surface method is simpler in concept than the nested surface method. It does not involve the plethora of ad hoc rules about translation and hardening of the inner surface, most of which are introduced simply to guarantee “nesting” rather than to reproduce any well-defined feature of material behaviour. The multiple surface models do, however, have all the advantages of nested surface models in modelling hysteresis and stress history effects. An example of this type of model was given by Houlsby (1999). This approach, too, can be extended to an infinite number of surfaces. 112 6 Advanced Plasticity Theories 6.5 Remarks on the Intersection of Yield Surfaces 6.5.1 The Non-intersection Condition The use of kinematic hardening plasticity with multiple yield surfaces has a his- tory of more than 30 years. It has proved a very convenient framework for mod- elling the pre-failure behaviour of soils and other materials, allowing a realistic treatment of issues such as non-linearity at small strain and the effects of recent stress history. The growing interest in modelling small strain behaviour of soils has recently resulted in the development of many so-called “bubble” models, such as those described by Stallebrass and Taylor (1997), Kavvadas and Amorosi (1998), Rouainia and Muir Wood (1998), Gajo and Muir Wood (1999), Houlsby (1999), Puzrin and Burland (2000), and Puzrin and Kirshenboim (1999). In all the above models, except Houlsby (1999), the “translation rules” are specified to avoid intersection of the yield surfaces, and it is commonly believed that this non-intersection condition must be met, but some publications express a contrary view. This subject is discussed here because the non-intersection condition leads to unnecessary complications in kinematic hardening hyperplasticity with multiple yield surfaces. As discussed above, the simple translation rules used by Ziegler or Prager correspond to simple forms of Gibbs free energy. If non-intersection is to be imposed, much more complex energy expressions are required, in which terms involving cross-coupling between different plastic strain components must appear. Puzrin and Houlsby (2001a) argue that the condition is not necessary, but is required only when an incrementally bilinear constitutive law is to be derived. Sometimes it is claimed that, even if not strictly necessary, the non-intersection condition should be accepted on pragmatic grounds. Incremental bilinearity (and hence non-intersection) certainly offers some advantage in computation. The main one is that, if an updated stiffness approach is taken in finite element analysis, the incremental stress-strain relationship is known (for plastic load- ing), reducing the need for iteration. At the opposite extreme, if an incremen- tally non-linear approach (e. g. as in hypoplastic theories) is used, the incre- mental stress-strain relationship cannot be determined without prior knowledge of the path during the increment. If intersection of yield surfaces is allowed, an intermediate case occurs: the response is incrementally multilinear (see the discussion below). In practice, this does not prove to be a significant disadvan- tage, since for most relatively smooth stress paths, the incremental plastic re- sponse can be determined in advance for each increment. 6.5.2 Example of Intersecting Surfaces To demonstrate that the non-intersection condition is not strictly necessary, we describe here a model with two yield surfaces that are allowed to intersect. It will 6.5 Remarks on the Intersection of Yield Surfaces 113 be seen that this model poses no theoretical difficulties. Consider the plasticity model with two kinematic hardening yield surfaces in a two-dimensional stress space, as shown in Figure 6.3. The yield surfaces are 2 11 1 1 2 22 2 2 0 0 T T fk fk V U V U V U V U (6.14) where ^ ` 12 , T VVV is the stress vector; ^ ` ȡȡ 11 1 12 , T U and ^ ` ȡȡ 22 2 12 , T U are the coordinates of the centres of the yield surfaces, and 1 k and 2 k are their radii. Plastic yielding and hardening are calculated using an associated flow rule: ȜȜ ȜȜ 1 1 111 2 2 222 2 2 p p f f w w w w H V U V H V U V (6.15) V 1 V 2 V ' U (1) U (2) F (1) F (2) Figure 6.3. Two-dimensional kinematic hardening plasticity model 114 6 Advanced Plasticity Theories where 1 O and 2 O are non-negative multipliers; ^ ` İİ 111 12 , T p pp H and ^ ` İİ 222 12 , p pp H are the plastic strain vectors associated with each of the sur- faces, so that the total plastic strain vector is given by 12 p pp H H H . Plastic hardening is calculated using Prager’s translation rule (which in this case is identical to Ziegler’s): 1 1 1 p E U H (6.16) and 2 2 2 p E U H (6.17) Finally, the elastic component of this model is defined by e E V H , where ^ ` 12 , T eee H HH is the elastic strain vector, so that the total strain vector is given by ep H H H . The model defined by Equations (6.14)–(6.17) is a particular case of the multi-surface model (7.38)–(7.39) which will be derived in Section 7.5 within the hyperplastic framework. Prager’s and Ziegler’s translation rules are known to violate the non- intersection condition. Consider as an example the case presented in Figure 6.4. During loading, the stress state P was reached, where the two surfaces touch each other (if they do not touch at only one point, they intersect and the proof is completed). Next a stress reversal took place and the stress state moved inside the yield surface 1 f such that the current stress state V was reached, which is on this yield surface but not on the outer yield surface. The next stress incre- ment dV is such that plastic response of the yield surface 1 f will occur, and will cause a strain increment 1 p dH directed along the vector 11 F V U , as prescribed by the associated flow rule (6.15). Then, according to Prager’s trans- lation rule (6.17), the instantaneous displacement 1 dU of the centre Q of the yield surface 1 f will also be directed along the vector 11 F V U . There- fore, if the current stress state V is located so that the angle D between the vectors F and 1 U is acute, the instantaneous displacement vector 1 dU will have a component directed along the ray QP. In this case, when the stress in- crement dV takes place, the point P on the yield surface 1 f moves into the exterior of the yield surface 2 f , and the surfaces intersect. 6.5 Remarks on the Intersection of Yield Surfaces 115 V 1 V 2 D U (1) F (1) d V P Q O V Figure 6.4. Example of a violation of the non-intersection condition 6.5.3 What Occurs when the Surfaces Intersect? There are no significant detrimental effects when yield surface intersect, pro- vided that the plastic loading and consistency conditions are applied separately to each yield surface [see, for example, de Borst (1986)]. In this case, the consti- tutive relationship simply becomes multilinear instead of bilinear. Consider, for example, the kinematic hardening model with two yield sur- faces described by Equations (6.14)–(6.17). The incremental stress-strain re- sponse of this model is derived by applying consistency conditions 1 0f and 2 0f separately to each surface as appropriate. For this case, the following incremental relationships can be obtained: ȜȜ 1122 22 E V H V U V U (6.18) where Ȝ 1 1 1 2 11 1 , when 0 2 0, when 0 T f Ek f ° ° ® ° ° ¯ V U V (6.19) 116 6 Advanced Plasticity Theories V 1 V 2 O Zone 1 Zone 3 Zone 4 Zone 2 Figure 6.5. Intersecting yield surfaces Ȝ 2 2 2 2 22 2 , when 0 2 0, when 0 T f Ek f ° ° ® ° ° ¯ V U V (6.20) and are Macaulay brackets (i. e. ,0; 0,0xxx x x ! d). Assuming that the surfaces intersect at the current stress state in Figure 6.5, four different types of behaviour are encountered, depending on which of the four possible zones the incremental stress vector is directed into. Zone 1: Ȝ Ȝ 1 1 1 2 2 11 2 2 2 22 0 0 T T E Ek Ek ! ° ® ° ! ¯ V U V V H V U V U V V U (6.21) Zone 2: Ȝ Ȝ 1 1 1 2 2 11 0 0 T E Ek ! ° ® ° d ¯ V U V V H V U (6.22) 6.6 Alternative Approaches to Material Non-linearity 117 Zone 3: Ȝ Ȝ 2 1 2 2 2 22 0 0 T E Ek d ° ® ° ! ¯ V U V V H V U (6.23) Zone 4: Ȝ Ȝ 1 2 0 0 E d ° ® ° d ¯ V H (6.24) Equations (6.21)–(6.24) represent an example of an incrementally multilinear constitutive relationship, as opposed to a bilinear one obtained when the non- intersection condition is satisfied. During loading, zones 2 and 3 would be en- countered only in rather rare circumstances which would involve rather con- torted stress paths. Many other recent developments in generalised plasticity, hypo-, and hyperplasticity are based on the use of incrementally non-linear and multilinear constitutive relationships. The main conclusion is that the non-intersection condition is necessary only when a bilinear constitutive law has to be derived. Intersection of yield surfaces, when treated properly, leads to multilinear constitutive relationships, which are consistent with recent developments in plasticity theory. Note that we make no case here that every model that allows intersection of yield surfaces may be theoretically consistent. It would be quite possible to for- mulate such a model so that it was either theoretically unacceptable or produced unjustifiable results. The case we present is simply that intersection of yield surfaces is allowable and on occasions may offer advantages. 6.6 Alternative Approaches to Material Non-linearity Plasticity theory is not the only method that has been used to model the irre- versible and non-linear behaviour of rate-independent materials. For complete- ness, two further alternatives should be mentioned. Endochronic theory [Valanis (1975); Bazant (1978)] enjoyed some popularity at one time, but has now largely fallen into disuse. Initially it was an attempt to model irreversibility within a thermodynamic context and without recourse to yield surfaces. It concentrated instead on the use of an “intrinsic time”, which was typically identified with some measure of plastic strain. Incremental rela- tions relating stresses, strains, and intrinsic time increment were proposed. Unfortunately, the main purpose of endochronic theory – to avoid yield surfaces – was the cause of its downfall. Real materials that exhibit rate-independent, irreversible behaviour also exhibit the phenomenon of a yield surface. Thus it became necessary to modify endochronic theory to include yield surfaces artifi- cially. The theories became increasingly contrived, and are now rarely used. Hypoplasticity is closely related to endochronic theory, although it does not employ an intrinsic time. Instead, rate equations are proposed specifying the 118 6 Advanced Plasticity Theories stresses in terms of the strain rates. These equations make much use of tensor analysis to identify the most general forms of first-order (but not necessarily linear) expressions for stress rate in terms of strain rate. For example, Kolymbas (1977) assumes a direct incrementally non-linear stress-strain relationship: ij ijkl kl ij kl kl LNV H HH (6.25) where ijkl L and ij N are linear operators. The early theories did not use yield surfaces, but (for the same reasons as encountered by endochronic theory) more recent theories have become increasingly complex to introduce the phenome- non of yield surfaces. The theories are still popular in some quarters, but in our view are unlikely to find long-term favour. 6.7 Comparison of Advanced Plasticity Models As seen from the above examples of different plasticity formulations, their com- mon feature is the existence of an outer or bounding surface 0F (in soils of- ten defined by the degree of material consolidation). In classical plasticity strain hardening models, this surface is assumed to be a yield surface, containing an entirely elastic domain. To incorporate plastic flow within this surface, bound- ing surface models, nested surface models, and multiple surface models have been developed. In bounding surface models, the 0F surface is treated as a bounding sur- face, and a loading surface passing through the current stress state is defined using a specific mapping rule. This mapping rule also defines the distance in the stress space between the stress state and the bounding surface, and the postu- lated hardening rule depends on this distance. The disadvantage of these models is the unrealistic “ratcheting” behaviour for small unload-reload cycles. In nesting surface models, the stress history of cyclic loading may be fol- lowed, and the ratcheting problem avoided by the more sophisticated rules for the evolution of the surfaces. Unfortunately, a number of ad hoc assumptions have to be introduced specifying the motion of the surfaces. True multiple sur- face models (without the nesting requirement) avoid these assumptions, and are simpler in concept than nested surface models. They can accommodate non- associated flow more easily. They too have a disadvantage. Since each surface acts independently, each must be checked for yield, whilst for nested surface models, it is known that the surfaces are engaged in order from the innermost to the outermost. All multiple surface models can in principle be extended to an infinite number of surfaces. There is no definitive choice between the more sophisticated plasticity mod- els. In the following, however, we shall develop hyperplasticity versions of mul- tiple surface models. It will be seen that these then lead naturally to a further extension into models with an infinite number of surfaces. Chapter 7 Multisurface Hyperplasticity 7.1 Motivation The purpose of this chapter is to present a more general framework for hyper- plastic modelling of the kinematic hardening of plastic materials. In Sec- tion 5.4.3, a simple example of a single kinematically hardening yield surface was presented. The elastoplastic stress-strain behaviour of this simple model was bilinear. The stiffness is controlled by elastic moduli within the yield surface and by the hardening moduli at the surface. The limitations of this simple model become clear when its behaviour is compared to that of some real materials, in particular soils. In soils, the true linear elastic region is often negligibly small, and plastic yielding starts almost immediately with straining. The behaviour therefore appears to be highly non-linear even within the large-scale yield sur- face. It also appears that soil has a memory of stress-reversal history within the large-scale yield surface. A simple single surface kinematic hardening model cannot simulate these features. In an attempt to solve this problem, Iwan (1967) and Mroz (1967) introduced the concept of multiple yield (or loading) surfaces, as discussed in Chapter 6. In multiple yield surface kinematic hardening models, the size of the true linear elastic region can be reduced, even to the limiting case in which it vanishes com- pletely. The stress-strain behaviour becomes piecewise linear and can follow more closely the true non-linearity of the material. Importantly, the model has a discrete memory of stress reversals, reflected in the relative configuration of the yield surfaces. Generalization of the multiple surface concept to an infinite number of yield surfaces produces models with a continuous field of yield sur- faces. These models allow the simulation of the true non-linear stress-strain behaviour and a continuous material memory, and will be the subject of Chap- ter 8. [...]... frameworks of generalised thermodynamics and rational mechanics, although we have not pursued that route One reason that the rational mechanics approach has not found favour in some quarters is that it requires the specification of a tensor-valued functional (the stress as a functional of the strain history) This is clearly a challenging task An advantage of the approach adopted here is that the functionals... loading, and the current response is expressed in terms of functions of the stress and/or strain state and the internal variables The rational mechanics approach [see, for example, Truesdell (1977)] instead expresses the response in terms of functionals of the history of the material (usually through the history of strain and temperature) Both approaches have advantages and drawbacks Rational mechanics... Continuous Hyperplasticity 8.1 Generalised Thermodynamics and Rational Mechanics As mentioned in Chapter 3, the theoretical approaches to the mechanics of inelastic materials can be divided into two main classes, which are often termed generalised thermodynamics and rational mechanics The generalised thermodynamics approach (which is used here) makes much use of internal variables to describe the history of. .. surfaces; modelling the memory of stress reversals; and approximation of a smooth transition from elastic to plastic behaviour The last of these purposes is perhaps the most important The use of internal variables (within the thermodynamic framework) is an extremely powerful method for describing the past history of an elastic-plastic material, but suffers from the disadvantage that it inevitably leads... Multisurface Hyperplasticity 7.2 Multiple Internal Variables For simplicity, in Chapter 4, we considered materials which could be characterised by a single kinematic internal variable ij , which was in the form of a second-order tensor The kinematic internal variable can often be conveniently identified with the plastic strain The significance of the single internal variable is that a single yield surface... to abrupt changes between elastic and elastic-plastic behaviour Although using multiple internal variables allows these changes to be divided into a number of smaller steps, a completely smooth 7.3 Kinematic Hardening with Multiple Yield Surfaces 121 transition can be achieved only by introducing an infinite number of internal variables Such an idea leads to the concept of an internal function rather... (Figure 7.2) An elongation of the E spring gives e , whereas an elongation of each of the Hn springs contributes elastic strain the plastic strain n to the total plastic strain; the sum of elastic and all plastic strains gives the total strain During initial loading, before the stress reaches the value of the first slip stress k1 , the behaviour is linear elastic and is governed by the elongation of the... particular, to incorporate plastic strains within a largescale yield surface and the material memory for modelling cyclic and transient behaviour The case of an infinite number of internal variables (i e an internal variable function) is considered in Chapter 8 It involves replacing energy and dissipation functions by equivalent functionals, resulting in a continuous hyperplastic formulation Chapter... is an abrupt change from elastic to elastic-plastic behaviour The generalisation of the results to some other cases is straightforward; for instance, a scalar internal variable can be obtained simply by dropping the subscripts from the variables ij , ij , and ij in Chapter 4 The generalisation to some more complex cases is marginally more complex For instance, N second-order tensor internal variables... possible transformations becomes enormous (for instance, there are 2 possible forms of the energy function) However, it is likely that only a small fraction of the possible forms would be of practical application, and so no systematic presentation of the forms with multiple internal variables is given here If any of the N internal variables are scalars rather than tensors, then all that is necessary is . that intersection of yield surfaces is allowable and on occasions may offer advantages. 6. 6 Alternative Approaches to Material Non-linearity Plasticity theory is not the only method that has. kinematic hardening of plastic materials. In Sec- tion 5.4.3, a simple example of a single kinematically hardening yield surface was presented. The elastoplastic stress-strain behaviour of this. Surfaces 121 transition can be achieved only by introducing an infinite number of internal variables. Such an idea leads to the concept of an internal function rather than internal variables.