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262 12 Behaviour of Porous Continua x large strains, x fluid flow in porous media, x heat flow in porous media, x viscous effects, x inertial effects. Apart from this generalization, the proposed framework places less stringent restrictions on the class of derived constitutive models in terms of the require- ments of the Second Law of Thermodynamics. Within the framework of Chap- ter 4, the fact that the mechanical dissipation had to be non-negative resulted in a condition that is more stringent than the Second Law. Within the present framework, it is the total dissipation (including dissipation due to heat and fluid fluxes) that has to be non-negative, which is equivalent to the Second Law. As in the standard hyperplastic approach, the entire constitutive behaviour is completely defined by specification of two scalar potential functions. However, in the generalised framework, these functions also include properties related to different phases of the media and their interaction. The fluid and heat conduc- tion laws are also built into these potentials, completing description of the con- stitutive behaviour of complex media. Chapter 13 Convex Analysis and Hyperplasticity 13.1 Introduction So far we have avoided use of the terminology of convex analysis in the presen- tation of hyperplasticity. The reason has been to make this book as accessible as possible to engineers, many of whom will not be familiar with the mathematical techniques of convex analysis. However, this terminology is the most natural and rigorous for the description of many of the concepts we require. So in this chapter, we re-present hyperplasticity in a convex analytical framework. This allows us to treat certain issues more rigorously where, in previous chapters, we have glossed over some problems. In general, we follow the terminology employed by Han and Reddy (1999) in their book in which they make much use of convex analysis for conventional plasticity theory. We acknowledge, too, that the French school of plasticity has made much use of this approach for many years. The principal motivation for adopting the convex analytical approach is that it allows us to deal more rigorously with the relationships between the dissipa- tion function and the yield function. It will be recalled that so far we have treated this relationship as a special degenerate case of the Legendre transform. In con- vex analysis, the Legendre transform is generalised to the Legendre-Fenchel transform (or Fenchel dual), and this allows more thorough treatment of the degenerate case. The alternative cases of elastic or elastic-plastic behaviour are also treated simply by convex analysis. The applicability of convex analysis to plasticity becomes so apparent that it seems highly likely that this will become the standard paradigm for expressing plasticity theory. Many of the concepts that have been given special names by plasticity theorists have parallels in the much more widely applied field of con- vex analysis. The advantages of expressing plasticity in this way are therefore twofold. Firstly, there is the extra rigour that is achieved; secondly, numerous 264 13 Convex Analysis and Hyperplasticity standard mathematical results can be employed, some of which give useful, new insights into plasticity theory. Further advantages come in the treatment of constraints that arise in (a) ex- treme cases such as incompressible elasticity or (b) dilation constraints in plas- ticity. These are treated by using indicator functions, which are one of the most simple and powerful concepts in convex analysis. Indicators can also be used to express unilateral constraints, which arise, for instance, in materials that are able to sustain compression but not tension. In hyperplasticity, the indeterminacy of the form of the yield function can be resolved by the use of a canonical yield function which is closely related to the gauge function of convex analysis. A brief introduction to the concepts of convex analysis and the terminology used here is given in Appendix D, and familiarity with the material there is as- sumed in the following sections. It is strongly recommended that any reader unfamiliar with convex analysis should study Appendix D in detail before pro- ceeding further with this chapter. Given that the notation of convex analysis is not entirely standardised, even a reader familiar with convex analysis may find it useful to study Appendix D, where our notation and terminology are defined. 13.2 Hyperplasticity Re-expressed in Convex Analytical Terms When potentials are not differentiable in the conventional sense, convex analy- sis serves as the framework for expressing constitutive behaviour, subject only to the limitation that the potentials must be convex. This does not prove too restrictive for our purposes. A complete exposition of hyperplasticity in convex analysis terminology would be lengthy, but suffice it to say (at least for simple examples) that each occurrence of a differential becomes a subdifferential. Thus instead of fV w wH we write fVw H . Thus the equations defining the “g formulation”, in which ,g VD and ,,d VDD are specified, may be expressed succinctly as g V Hw (13.1) g D Fw (13.2) d D Fw (13.3) F F (13.4) in which each of the variables may be a scalar or tensor, and the internal vari- ables (and generalised stresses) may be a single variable, multiple variables, or an infinite number of variables. 13.3 Examples from Elasticity 265 13.3 Examples from Elasticity Before going on to examine more complex problems in plasticity, it is useful to gain some familiarity with the techniques of convex analysis by looking at some problems in elasticity. As an example of the way convex analysis can be used to express constraints, consider some simple variants on elasticity. Linear elasticity (in one dimension) is given by either of the expressions, 2 2 E f H (13.5a) or 2 2 g E V (13.5b) Using derivations based on the subdifferential (which in this case includes simply the derivative, because both the above are smooth strictly convex func- tions) fVw H hence EV H , or gHw V hence EH V . Now consider a rigid material, which can be considered as the limit E of . The resulting f can be written in terms of the indicator function: ^` 0 fI H (13.6) which has the Fenchel dual 0 g . The subdifferential of f gives ^` 0 NV H , which gives > @ ,V f f for 0H , and is otherwise empty, so that there is zero strain for any finite stress. Con- versely, non-zero strain is impossible. The subdifferential of g (just consisting of the derivative) gives 0H directly, irrespective of the stress. In a comparable way, the limit 0E o , i. e. an infinitely flexible material, is obtained from either 0f or ^` 0 gI V . The above considerations become of more practical application as one moves to two and three dimensions. For instance, triaxial linear elasticity is given by 22 3 22 vs KG f HH (13.7a) or 22 26 p q g KG (13.7b) Incompressible elasticity ( K of ) is simply given by ^` 2 0 3 2 G fI v H (13.8a) 266 13 Convex Analysis and Hyperplasticity or 2 6 q g G (13.8b) without the need to introduce a separate constraint. Note that whenever it is required to constrain a variable x to a zero value, where x is one of the argu- ments of an energy function, one simply adds the indicator function ^` 0 Ix . In the dual form, the Fenchel dual does not depend on the variable conjugate to x. The above results can of course very simply be extended to full continuum models. Unilateral constraints can also be treated using convex analysis. A one- dimensional material with zero stiffness in tension (i. e. a “cracking” material) can obtained from 2 2 c E f H (13.9a) or >@ 2 ,0 2 c gI E f V V (13.9b) where we recall that the Macaulay bracket is defined such that xx if 0x t and 0x if 0x (where we use a tensile positive convention). Such a model might, for instance, be the starting point for modelling masonry materials, con- crete, or soft rocks. Another case, rigid in tension and with zero stiffness in compression (in other words, the “light inextensible string” found in many elementary textbooks) is given by > @ ,0 fI f H (13.10a) or > @ 0, gI f V (13.10b) In each of the above cases, elementary application of the subdifferential for- mulae gives the required constitutive behaviour, effectively applying the “con- straints” (unilateral or bilateral) as required. Table 13.1 gives the forms of both f and g required to specify a number of different types of “elastic” materials, to- gether with the derived stresses and strains. The table illustrates how the convex analytical framework can be used to express concisely the behaviour of materials with “corners” in the response, e. g. at the tension to compression transition. 13.3 Examples from Elasticity 267 Table 13.1. Some types of one-dimensional “elastic” materials Model Tension, compression moduli f H gV fVw H gHw V Linear elastic ,EE 2 2 EH 2 2E V EV H E V H Rigid ,ff ^` 0 I H 0 > @ ,V f f 0H Infinitely flexible 0,0 0 ^` 0 I V 0V > @ ,H f f Bilinear elastic , tc EE 22 22 tc EEHH 22 22 tc EE VV tc EEV H H tc EE VV H Elastic-cracking (no tension) 0, c E 2 2 c E H >@ 2 ,0 2 c I E f V V c EV H >@ ,0 c N E f V ½ H V ®¾ ¯¿ Elastic string (no compression) ,0 t E 2 2 t E H >@ 2 0, 2 t I E f V V t EV H >@ 0, t N E f V ½ H V ®¾ ¯¿ Rigid-cracking 0,f > @ 0, I f H > @ ,0 I f V > @ 0, N f V H > @ ,0 N f H V Inextensible string ,0f > @ ,0 I f H > @ 0, I f V > @ ,0 N f V H > @ 0, N f H V 268 13 Convex Analysis and Hyperplasticity 13.4 The Yield Surface Revisited The dissipation function (which is in this case the same as the force potential) dd z D D is a first-order function of D , and the conjugate generalised stress is defined by dFw D , which is the generalisation of dF w wD . The set X (capital F) of accessible stress states can be found by identifying the dissipation function as the support function of a convex set of F; hence applying Equation (D.21) from Appendix D, ^ ` ,,d& F FD d D D (13.11) Note that here the notation , is used for an inner product, or more generally the action of a linear operator on a function. The indicator and gauge functions of X can be determined in the usual way. Note that the indicator is the dual of the support function, so it is the flow poten- tial: 0, , Iw & F& F F ® f F& ¯ (13.12) where wN & Dw F F , which is the generalisation of wD w wF . It is useful at this stage to obtain the gauge function, ^ ` inf 0 & J F Pt FP& (13.13) The gauge may also be obtained directly as the polar of the dissipation: 0dom , sup d d & zD FD JF D (13.14) Furthermore, we define the canonical yield function (in the usual sense adopted in hyperplasticity) as 1y & F J F. Then, applying Equation (D.17), > @ ,0 IwIy & f F F F ª º ¬ ¼ (13.15) So that applying the usual approach, we obtain any of the following: Dw F w F F OwJ F Ow F XX X wI N y (13.16) where 0 O t [see Lemma 4.5 of Han and Reddy (1999)]. The above is the equiva- lent of the usual y D Ow wF . Clearly, ywF plays the role of ywwF, and O has its usual meaning. In particular, 0 O for a point within the yield surface (inte- rior of X) and takes any value in the range > @ 0,f for a point on the yield sur- face (boundary of X). 13.4 The Yield Surface Revisited 269 It can be seen, however, that the assumption made in developments in Chap- ter 4 that, because w y D w wF Ow wF with O an arbitrary multiplier, one could deduce that wy O was slightly too simplistic a step. Now we are in a position to address the process of obtaining either a yield surface from a dissipation function or vice versa. If we start with dzd D , then we apply (13.7) to find the set of admissible states X, and then use (D.15), together with the definition of the canonical yield function: ^ ` ^` ^` ^ ` inf 0 1 inf 0 , , dom 1 inf 0 , , dom 1 y dd dd F Pt FP& cc PtFPFFDdDD PtFDdPDD (13.17) so that y F can in principle be determined directly from d D . This is an important result. Then, yDOw F . Conversely, if we first specify the yield surface y F in the normal way, then X is easily obtained from ^ ` 0y& F F d , and the dissipation function is then the support function of this set: ^ ` ^ ` sup , sup , 0dy & D V D FD F& FD Fd (13.18) so that d D can in principle be determined directly from y F . This too is an important result, although it is more obvious than the transformation from dissipation to yield. It is not essential for (13.18), but there is a clear preference for expressing the yield surface in canonical form such that 1y & J F F is a homogeneous first-order function of F, so that it can be interpreted as the gauge function of the set X. Note that the yield function is not itself positively homogeneous, but it is, however, expressible as a positively homogeneous function minus unity. If it is chosen this way, then y is dimensionless, so that O has the dimension of stress times strain rate. If y is expressed in canonical form, then the dissipation function can be ex- pressed directly as the polar, 0 , sup 1 d y zF& FD D F (13.19) The results are summarised as follows: Option 1: start from the specified dissipation function dzd D : dFw D (13.20) 270 13 Convex Analysis and Hyperplasticity ^ ` 0dom inf 0 , , 1 , sup 1 d yd d zD F Ot FDdO D D FD D (13.21) Option 2: start from specified y F : > @ ,0 wI y f F F (13.22) wyDw F Ow F (13.23) ^ ` sup , 0dyD FD Fd (13.24) Note that if y is not expressed in canonical form, it cannot be readily con- verted to the gauge, and so the dissipation function cannot simply be obtained as the polar of the gauge. The function w (the flow potential) is the indicator of the set of admissible generalised stress states. If y F is in canonical form such that J F F X 1y is homogeneous of order one, then applying Option 2 to obtain d and then applying Option 1 to obtain y will return the original function. If this condition is not satisfied, then applying this procedure will give a different functional form of the yield function (the canonical form), but specifying the same yield surface. Thus if the yield surface is not originally defined in canonical form, it can be converted to ca- nonical form by first applying Equation (13.18) and then (13.17) (although in specific instances there may often be more straightforward ways of achieving the same objective). 13.5 Examples from Plasticity We first consider how indicator functions can be used to introduce dilation constraints. A plastically incompressible cohesive material in triaxial space, with cohesive strength (maximum allowable shear stress) c, can be defined by 22 26 s pq g q KG D (13.25) 2 s dc D (13.26) in which only a plastic shear strain is introduced. The canonical yield function can be obtained as 1 2 q y c F (13.27) 13.5 Examples from Plasticity 271 Alternatively both the plastic strain components are introduced, but the volumetric component is constrained to zero. This approach proves more fruit- ful for further development. In the past, this has been achieved by imposing a separate constraint, but now we do so by introducing an indicator function into the dissipation: 22 26 vs pq g pq KG DD (13.28) ^` 0 2 sv dc I D D (13.29) The yield function is unchanged for this case, and is again given by (13.23). This model is readily altered to frictional, non-dilative plasticity by changing the dissipation to ^` 0 sv dMp I DD (13.30) Note that we have introduced a Macaulay bracket on p which we did not use before, but strictly it is necessary to ensure that the dissipation cannot be nega- tive. The corresponding canonical yield function is 1 q y Mp F (13.31) The virtue of introducing the plastic volumetric strain is seen in that the model can now be further modified to include dilation by changing d to ^` 0 svs dMp I B DDD (13.32) The canonical yield function for this case becomes 1 qp B y Mp FF (13.33) This can be compared with the yield locus 0 qp yMpB F F used in the earlier example in Chapter 5. The above are some simple examples of the way in which expressions using convex analysis terminology can provide a succinct description of plasticity models for geotechnical materials. They may provide the starting point for using this approach in more sophisticated modelling. [...]... the literature of “continuum damage mechanics” (CDM) for the damage parameter, but here we use to emphasise that this is simply interpreted as an internal variable and is treated in the same way in the formulation as before Note, however, that although the role of is again associated with irreversible behaviour, it no longer has the physical interpretation of “plastic strain” It is straightforward to... involve a scalar internal variable, rather than a tensorial one It is worth noting that the damage parameter is dimensionless, so that the variables and here have the dimension of energy per unit volume (i e the same dimensions as f and g) Therefore, the terminology “generalised stress” is not appropriate for this application Noting that E 2 2 , the physical interpretation of these variables is that they... response that asymptotically approaches the fully plastic moment The correct response of the beam can be described by considering the axial strain at any point in the beam y R The free energy and dissipation expressions that express the elastic plastic behaviour of any element of the beam are E 2 and d Integrating the free energy and dissipation over f y 2 the area of the beam gives d2 f d2 E ˆ 2 ˆ 2 12. .. does not make f a function of x j The variables x j are the equivalent of the strains encountered in previous chapters, whilst the li play the role of the internal variables 278 14 Further Topics in Hyperplasticity We can express the compatibility of the structure through n compatibility equations expressing the length of each bar; each acts as a constraint on the kinematic variables: 1 2 ci x ai 2 x... relationship between axial stress , applied bending moment M, and curvature 1 R for an elastic beam bent about one of its principal axes: y M I E R (14.37) where y is the distance from the neutral axis, I is the second moment of area (equal to bd 3 12 for a rectangular beam of breadth b and depth d), and E is Young’s modulus It follows that the moment-curvature relationship for an elasEbd 3 1 and the stress... the point of departure They are intended to illustrate the fact that, once an engineer is familiar with this approach to constitutive modelling, it can provide a powerful technique capable of wide generality In these extensions of the applications of hyperplasticity, the two most important features to be borne in mind are (a) the emphasis on using two scalar functions to define material behaviour (different... instance, from the geometry shown in the diagram, we readily obtain H1 H 2 0, which describes the inability of the 1x1 3 x2 L mechanism to carry lateral load in this configuration On the other hand, it can carry arbitrary vertical loads 284 14 Further Topics in Hyperplasticity 14.4 Bending of Prismatic Beams Elementary beam theory, assuming that plane sections remain plane, gives the well-known relationship... the internal variable plays the role of the damage parameter in conventional isotropic damage models This development demonstrates the versatility and generality of the hyperplastic approach Consider a Helmholtz free energy, f , 1 E 2 2 (14.1) is effecwhere is a “damage parameter” such that 0 1 The factor 1 tively applied as a reduction factor on the elastic stiffness E The symbol D or is often used in... materials, we termed this approach hyperplasticity , although in Chapter 11, we showed that rate-dependent materials, too, can be described by this method In this chapter, we extend the hyperplasticity approach to modelling problems in a number of different areas The sections of this chapter are not connected, but represent different developments, each of which takes the hyperplasticity approach as... dy 12 d2 d d R ˆ b dy ˆ fd bd d (14.40) 12 12 y 12 12 2 ˆ 12 y d2 E 2 ˆ dd ˆ bd d (14.41) 12 where we have written ˆ and ˆ to emphasise that they are functions of the y d The internal internal coordinate, and furthermore have substituted coordinate has the physical interpretation of the dimensionless distance from the neutral axis 286 14 Further Topics in Hyperplasticity Application of the standard . French school of plasticity has made much use of this approach for many years. The principal motivation for adopting the convex analytical approach is that it allows us to deal more rigorously. So far we have avoided use of the terminology of convex analysis in the presen- tation of hyperplasticity. The reason has been to make this book as accessible as possible to engineers, many of. of elastic or elastic-plastic behaviour are also treated simply by convex analysis. The applicability of convex analysis to plasticity becomes so apparent that it seems highly likely that this