A Principles of Hyperplasticity Part 4 ppt

58 4 The Hyperplastic Formalism potential (the yield surface for the case of associated flow) with respect to stresses ij V . Collins and Houlsby also discuss the fact that a yield surface in stress space can be derived by elimination of generalised stresses from Equations (4.11). They also demonstrate that non-associated flow (in the sense of conventional plasticity theory) can be derived within this framework and is intimately linked to stress- dependence of the dissipation function. This issue is addressed in Section 4.10. 4.4.3 Convexity Since 0 ij ij d F D t  , it follows that the condition on e y is 0 e ij ij yw Ft wF (because 0 O t ). This has a straightforward geometric interpretation and is simply the condition that the surface 0 e y contains the origin in generalised stress space and satisfies certain convexity conditions. It does not require, however, that the yield surface should be strictly convex either in generalised stress space or in stress space. 4.4.4 Uniqueness of the Yield Function There are also relationships for each of the passive variables  ij x of the form:   e e ij ij y d xx w w O ww (4.13) where x stands for any of ,,, ij ij ij sHVD , or T. These relationships demonstrate that there is a close relationship between the functional forms of e d and e y . Note that, because of the nature of the singular transformation, the functional form of e y is not uniquely determined. In particular, the dimension of e y is not determined. However, the product e yO must have the dimension (stress) (strain rate)u . If, for instance, O is chosen to have the dimension of strain rate (i. e. the same dimension as ij D  ), then it follows that e y must be a homogeneous first-order function in stress. Note, however, that quantities with the dimension of stress might include the stresses ij V , generalised stresses ij F , and material properties with the dimension of stress. An alternative is that O could be chosen with the dimensions of (stress) (strain rate)u , in which case the yield function must be dimensionless. We place here no particular requirement on the form of the yield function. In Chapter 13, in which we express hyperplasticity in a convex analytical framework, we will find that it is possible to select a preferred form for the yield function, and we shall call this the canonical yield function. 4.6 A Complete Formulation 59 4.5 Transformations from Internal Variable to Generalised Stress For each of the functions e (u, f, h or g), a further transformation is possible, changing the independent variable from ij D to ij F in the form ij ij ee FD . Correspondingly, the relevant passive variable in e d or e y is changed from ij D to ij F . After the transformation, note the results ij ij ew D wF and   ij ij ee xx ww ww , where  ij x is any of the passive variables ,, ij ij sHV or T. This last result gives alternative forms for the differentiation to obtain the appropriate complementary variables. 4.6 A Complete Formulation Adopting the approach described above, the constitutive behaviour is entirely defined by the specification of two potentials. The first is an energy potential, and the second either a dissipation function or the yield surface. There are a total of 16 different possibilities, however, for the choice of the potentials, representing all permutations of the following possibilities: x choice of u, f, h or g or for the energy function x dissipation function e d or yield surface e y x transformation between ij D and ij F for the energy function The possibilities are illustrated in Table 4.1. In principle any of the 16 formulations could be used to provide a complete specification of the constitutive behaviour of a material. In each case, two potentials are specified. Technically, it would be possible to specify the energy potential from one of the 16 boxes and the dissipation or yield function from another, but presumably such a mixed form would be adopted only in rather special circumstances. The choice of formulation will depend on the application in hand. For instance, the four forms of the energy potential in classical thermodynamics are adopted in different cases (e. g. isothermal problems, adiabatic problems, etc.). On differentiating the energy function and dissipation or yield functions with respect to the appropriate variables, the relationships in Table 4.2 are obtained. Once the chosen two scalar functions have been specified, the entire constitutive behaviour can be derived from the differentials in the appropriate box in Table 4.2, together with the condition ij ij F F . 60 4 The Hyperplastic Formalism Table 4.1. The 16 possible formulations Energy function u or u f or f Dissipation function 0 e d t ij D  ,, ij ij usHD  ,,, u ij ij ij dsHD D   ,, ij ij f HDT  ,,, f ij ij ij d HDTD  ij F  ,, ij ij usHF  ,,, u ij ij ij dsHF D   ,, ij ij f HFT  ,,, f ij ij ij d HFTD  Yield surface 0 e y ij D  ,, ij ij usHD  ,,, u ij ij ij ysHD F  ,, ij ij f HDT  ,,, f ij ij ij y HDTF ij F  ,, ij ij usHF  ,,, u ij ij ij ysHF F  ,, ij ij f HFT  ,,, f ij ij ij y HFTF Energy function h or h g or g Dissipation function 0 e d t ij D  , , ij ij hsVD  ,,, h ij ij ij dsVD D   ,, ij ij g VDT ij ij ij g h w w H   wV wV ij F  , , ij ij hsVF  ,,, h ij ij ij dsVF D   ,, ij ij g VFT  ,,, g ij ij ij d VFTD  Yield surface 0 e y ij D  , , ij ij hsVD  ,,, h ij ij ij ysVD F  ,, ij ij g VDT  ,,, g ij ij ij y VDTF ij F  , , ij ij hsVF  ,,, h ij ij ij ysVF F  ,, ij ij g VFT  ,,, g ij ij ij y VFTF 4.6 A Complete Formulation 61 Table 4.2. Results from differentiation of energy and dissipation functions Energy function u or u f or f h or h g or g Dissipation function 0 e d t ij D ij ij uw V wH u s w T w ij ij uw F  wD u ij ij dw F wD  ij ij f w V wH f s w  wT ij ij f w F  wD f ij ij dw F wD  ij ij hw H  wV h s w T w ij ij hw F  wD h ij ij dw F wD  ij ij g w H  wV g s w  wT ij ij g w F  wD g ij ij dw F wD  ij F ij ij uw V wH u s w T w ij ij uw D wF u ij ij dw F wD  ij ij f w V wH f s w  wT ij ij f w D wF f ij ij dw F wD  ij ij hw H  wV h s w T w ij ij hw D wF h ij ij dw F wD  ij ij g w H  wV g s w  wT ij ij g w D wF g ij ij dw F wD  Yield surface 0 e y ij D ij ij uw V wH u s w T w ij ij uw F  wD u ij ij y w D O wF  ij ij f w V wH f s w  wT ij ij f w F  wD f ij ij y w D O wF  ij ij hw H  wV h s w T w ij ij hw F  wD h ij ij y w D O wF  ij ij g w H  wV g s w  wT ij ij g w F  wD g ij ij y w D O wF  ij F ij ij uw V wH u s w T w ij ij uw D wF u ij ij y w D O wF  ij ij f w V wH f s w  wT ij ij f w D wF f ij ij y w D O wF  ij ij hw H  wV h s w T w ij ij hw D wF h ij ij y w D O wF  ij ij g w H  wV g s w  wT ij ij g w D wF g ij ij y w D O wF  62 4 The Hyperplastic Formalism 4.7 Incremental Response In the numerical analysis of problems involving non-linear materials, the incremental form of the constitutive relationship is usually required. This, for instance, often forms a central part of a finite element analysis. Therefore, one of the most important criteria that needs to be applied to the formulation of any model is that the incremental form of the constitutive relationship should be derived solely by applying standard procedures, without the need to introduce either ad hoc procedures or additional assumptions. Within classical plasticity theory, more or less standardized procedures are adopted to derive incremental response [see for example Zienciewicz (1977)], although the mathematical treatment of the hardening behaviour tends to vary considerably. Differentiation of the energy expressions in Table 4.2 leads straightforwardly to the results in Table 4.3 where the (symmetrical) matrix > @ u cc is defined as >@ 222 222 222 2 ij kl ij kl ij ij kl ij kl ij kl kl uuu s uuu u s uuu ss s ªº www «» wH wH wH wD wH w «» «» www «» cc «» wD wH wD wD wD w «» «» www «» wwH wwD «» w ¬¼ (4.14) and the matrices > @ u cc , > @ f cc , f ªº cc ¬¼ , > @ h cc , h ªº cc ¬¼ , > @ g cc and > @ g cc are similarly defined with appropriate permutation of the energy functions and independent variables. These incremental relationships are true for both dissipation and yield function formulations. However, in general, the explicit stress-strain response can be obtained only for those formulations based on the yield functions and only of the Table 4.3. Incremental results obtained from energy expressions oruu orff orhh or g g >@ ij kl ij kl u s V ½ H ½ °° °° cc F D ®¾ ® ¾ °° °° T ¯¿ ¯¿       >@ ij kl ij kl f s V ½ H ½ °° °° cc F D ®¾ ®¾ °° °° T  ¯¿ ¯¿       >@ ij kl ij kl h s H ½ V ½ °° °° cc F D ®¾ ®¾ °° °° T ¯¿ ¯¿       >@ ij kl ij kl g s H ½ V ½ °° °° cc F D ® ¾®¾ ° °°° T  ¯¿ ¯¿       >@ ij kl ij kl u s V ½ H ½ °° °° cc D F ®¾ ®¾ °° °° T ¯¿ ¯¿       ij kl ij kl f s V ½ H ½ °° °° ªº cc D F ®¾ ®¾ ¬¼ °° °° T  ¯¿ ¯¿       ij kl ij kl h s H ½ V ½ °° °° ªº cc D F ®¾ ®¾ ¬¼ °° °° T ¯¿ ¯¿       >@ ij kl ij kl g s H  ½ V ½ °° °° cc D F ® ¾®¾ ° °°° T  ¯¿ ¯¿       4.7 Incremental Response 63 e y type. For each of these forms the incremental relationships can be written (noting that ij ij F F   ) in the following form: 222 222 222 2 ij kl ij kl ij ij kl ij kl ij kl ij kl ij kl kl eee bb b bz a b eee bz z x eee zb z z ªº www «» ww wwD ww «» ½ ½ «» °° °° www «» F D ®¾ ®¾ «» wDw wDwD wDw °° °° «» ¯¿ ¯¿ «» www «» ww wwD «» w ¬¼      (4.15) where substitutions for e, ij a , ij b , x, and z are to be taken from the appropriate column of Table 4.4. Equation (4.15) is used together with the flow rule: e ij ij y w D O wF  (4.16) The multiplier O is obtained by substituting the above equations in the consistency condition, which is obtained by differentiating the yield function: 0 ee ee e ij ij ij ij ij ij yy yy yb z bz ww ww DF wwDwwF    (4.17) Together with the orthogonality condition in its incremental form ij ij F F   , this can be used to derive eb ez ij ij ee A A bz BB O     (4.18) where for convenience, we define the notation, 2 ee eb ij ij kl kl ij yy e A bb ww w  wwFwDw (4.19) 2 ee ez kl kl yy e A zz ww w  wwFwDw (4.20) Table 4.4. Substitution of variables for different formulations e u f h g ij a ij V ij V ij H ij H ij b ij H ij H ij V ij V x T s T s z s T s T 64 4 The Hyperplastic Formalism 2 ee e e ij kl kl ij ij y yy e B §· ww w w  ¨¸ ¨¸ wD wF wD wD wF ©¹ (4.21) This leads to the following incremental stress-strain relationships: 22 22 22 22 2 22 22 eb ez mnkl mn ij kl ij mn ij ij mn ij eb ez mnkl mn kl mn mn ij eb ez mnkl mn ij ij kl ij mn ij ij mn e ijkl ee ee CC bb b bz b a ee ee CC x zb z z z ee ee CC bz C ww ww  ww wwD ww wwD ½ ww ww °°  °° ww wwD wwD w °° F ®¾ ww ww °°  D °° wDw wDwD wDw wDwD °° O ¯¿     kl bez ij eb e ez e ij b z C AB AB ªº «» «» «» «» «» ½ °° «» ®¾ «» °° ¯¿ «» «» «» «» «»  «» ¬¼   (4.22) Finally, this can be simplified to ebb ezb ijkl ij ij ebz ezz kl eb ez kl ij ijkl ij eb ez ij ijkl ij eb e ez e kl DD a DD x b DD z CC AB AB DD ªº ½ «» °° «» °° «» ½ °° °° «» F ®¾ ®¾ «» °° ¯¿ °° D «» °° «» °° O ¯¿ «»  ¬¼       (4.23) where 22 eb eb mnkl ijkl ij kl ij mn ee DC b E ww  wE w wE wD (4.24) 22 ezb eb kl mnkl kl mn ee DC zb z ww  ww wwD (4.25) 22 ez ez mn ij ij ij mn ee DC z E ww  wE w wE wD (4.26) 22 2 ez ez mn mn ee DC z z ww  wwD w (4.27) e eb eb kl mnkl e mn y A C B w  wF (4.28) e ez ez mn e mn y A C B w  wF (4.29) and E stands for either D or b. 4.7 Incremental Response 65 The first two rows of the matrix in Equation (4.23) describe the incremental relationships among the stresses, strains, temperature, and entropy. The third and fourth rows are the evolution equations for the generalised stress and the internal variable. The final row allows evaluation of the plastic multiplier O for the increment. The forms of the relationships, after the appropriate substitution of variables, are given in Table 4.5. The above solution applies only when plastic deformation occurs, i. e. 0 ij Dz  , and 0 O ! . If the above solution results in 0 O  , then it implies that elastic unloading has occurred. In this case, the consistency equation no longer applies but is simply replaced by the condition 0 O . For this case, it is straightforward to show that the above relations are replaced by 22 22 2 22 00 00 ij kl ij ij kl kl ij ij ij kl ij ee bb bz a ee x b zb z z ee bz ªº ww «» ww ww «» ½ «» °° ww «» °° «»  ½ ww °° °° w F « » ®¾ ®¾ ° ° «» ¯¿ °° ww D «» °° wD w wD w «» °° O ¯¿ «» «» «» ¬¼       (4.30) Table 4.5. Summary of incremental form of constitutive relations kl H  kl V  s  uus ijkl ij ij us us kl uus kl ijkl ij ij uus ijkl ij ij u us kl uu DD DD DD s CC A A BB HH H H DH D H H ªº «» V ½ «» °° «» T °° «» H ½ °° F «» ®¾ ®¾ ¯¿ «» °° D «» °° «» °° O ¯¿ «»  «» ¬¼       hhs ijkl ij ij hs hs kl hhs kl ijkl ij ij hhs ijkl ij ij h hs kl hh DD DD DD s CC A A BB VV V V DV D V V ªº «» H ½ «» °° «» T °° «» V ½ °° F «» ®¾ ®¾ ¯¿ «» °° D «» °° «» °° O ¯¿ «»  «» ¬¼       T  ff ij ijkl f ij f kl ff kl ij ijkl ij ff ij ijkl ij f f kl ff DD DD s DD CC A A BB HH HT TH T DH DT HT H T ªº «» V ½ «» °° «»  °° «» H ½ °° «» F ®¾ ®¾ «» T ¯¿ °° D «» °° «» °° O «» ¯¿  «» ¬¼       / gg ij ijkl g ij g kl gg kl ij ijkl ij gg ij ijkl ij g g kl gg DD DD s DD CC A A BB VV VT TV T DV DT VT V T ªº «» H ½ «» °° «»  °° «» V ½ °° «» F ®¾ ®¾ «» T ¯¿ °° D «» °° «» °° O «» ¯¿  «» ¬¼       66 4 The Hyperplastic Formalism The choice of formulation is determined by the application in hand, and to a certain extent by personal preferences. The u and h formulations are particularly convenient for problems where changes in entropy are determined ( e. g. adiabatic problems), whilst the f and g formulations are appropriate for those with prescribed temperature ( e. g. isothermal problems). The u and f formulations correspond to strain-space based plasticity models and are particularly applicable when the strains are specified. Conversely the h and g formulations correspond to the more commonly used stress-space plasticity approaches and are particularly convenient for problems with prescribed stresses. However, by appropriate numerical manipulation, it is possible to use any of the formulations for any application. For instance, the g formulation leads directly to the compliance matrix. This can be straightforwardly inverted to give the stiffness matrix. 4.8 Isothermal and Adiabatic Conditions Isothermal conditions can be imposed straightforwardly by the condition 0T  . They are most conveniently examined using either the Helmholtz free energy or Gibbs free energy forms of the equations. Thus the isothermal elastic-plastic stiffness matrix is f ijkl D HH and the isothermal compliance matrix is g ijkl D VV (both from Table 4.5). For elastic conditions, these reduce to 2 ij kl f w wH wH and 2 ij kl g w wV wV , respectively. Adiabatic conditions are slightly more complex. In reversible thermodynamics, the adiabatic condition (no heat flow across boundaries) is associated with isentropic conditions, but in the presence of dissipation, the adiabatic condition becomes 0 ij ij sd sT TFD   . Adiabatic conditions are most conveniently expressed using the internal energy or the enthalpy forms of the equations. Multiplying the fourth line of the appropriate matrix equations is Table 4.5 by ij F and substituting the adiabatic condition ij ij sT FD   gives uus ij ij ij ijkl kl ij ij CCss H FD F H F T   (4.31) or hhs ij ij ij ijkl kl ij ij CCss V FD F V F T   (4.32) 4.9 Plastic Strains 67 which can simply be rearranged to solve for s  in terms of either the stress or strain increment. Substituting in the first line of the appropriate matrix equation in Table 4.5 gives the adiabatic stiffness or compliance behaviour as  u uus mn mnkl ij ijkl ij kl us pq pq C DD C H HH H ªº F «» V  H «» TF «» ¬¼   (4.33) or  h hhs mn mnkl ij ijkl ij kl hs pq pq C DD C V VV V ªº F «» H  V «» TF «» ¬¼   (4.34) Similar substitutions for the entropy increment are necessary in the second to fifth lines of the equations to solve for the other incremental quantities. Note that for the elastic case, adiabatic and isentropic conditions are iden- tical, and the stiffness and compliance matrices are simply 2 ij kl uw wH wH and 2 ij kl hw wV wV , respectively. 4.9 Plastic Strains So far, no particular interpretation has been placed on the internal variable ij D . By a suitable choice of ij D , Collins and Houlsby (1997) showed that it is normally possible to write the Gibbs free energy so that the only term that involves both ij V and ij D is linear in ij D :    123ij ij ij ij gg g g VDVD (4.35) Furthermore, if 3 g is also linear in the stresses, then Collins and Houlsby (1997) showed that no elastic-plastic coupling occurs. In this case, it is always possible (again by suitable choice of ij D ) to choose 3 ij ij g V D . For this case, it follows that 1 ij ij ij g w H  D wV (4.36) 2 ij ij ij g w F V wD (4.37) The interpretation of the above is that ij D plays exactly the same role as the conventionally defined plastic strain  p ij H . It is convenient to define elastic strain [...]... the constraint equation can be added to d e without affecting Equation (4. 46) Again, we can express this by using d e d e c , where is an arbitrary constant, as the definition of d e Thus we obtain de c ij ij ij An example of this type of constraint is given in Section 10.3.1 (4. 47) 74 4 The Hyperplastic Formalism 4. 13 Advantages of Hyperplasticity The motivation for this work comes principally from... significance of f and –g is that Figure 5.1 One-dimensional elasticity 78 5 Elastic and Plastic Models in Hyperplasticity they represent the areas shown in the figure For the special case that the energy functions are quadratic functions, the areas f and –g are equal 5.1.2 Isotropic Elasticity The extension of the one-dimensional case to a continuum is straightforward and well known, and has already been... an example of a transformation from a yield function to a dissipation function for a non-trivial case 4. 12 Constraints The development of some models is most efficiently achieved by introducing constraints Typically these might be constraints on either the strains (e g incompressible behaviour) or on the rates of the internal variables (e g dilational constraints for granular materials) A full treatment... experimental observations on granular materials 4. 11 Conversions Between Potentials In the formulation described here, much emphasis has been placed on the concept that, once two scalar functions are known, then the entire constitutive behaviour of the material is determined Emphasis has also been placed on the fact that there are many possible combinations of functions that can be used, and that these are... Thermodynamics: u ij ij qk,k (4. 48) where ij is the stress that is work-conjugate to the strain rate and qk is the heat flux vector 4. 14 Summary 75 The second potential is the specific mechanical dissipation, a function of state and the rate of change of internal variable d d ij , s , ij , ij , satisfying the Second Law of Thermodynamics in the form, d 0 s qk,k (4. 49) where is the non-negative thermodynamic... quadratic in ij ), but can become extremely intractable for more complex forms All other transformations between stress and strain are possible, subject to analogous conditions 4. 11.3 Internal Variable and Generalised Stress Transformations between u, f, h, and g, in terms of the internal variable, and u , f , h , and g , in terms of the generalised stress, can be achieved under conditions that are analogous... Hierarchy of Isotropic Elastic Models An advantage of the hyperelastic approach (here extended to include thermal effects) is that models can easily be classified and placed in a hierarchy on the basis of energy functions Simpler models can be identified as special cases of more complex models For example, Table 5.1 shows the relationships among general thermoelasticity, elasticity without thermal effects,... ij (4. 53) 0 Assuming Ziegler’s orthogonality condition, u d ij ij 0 (4. 54) and that the processes of straining and change of entropy are mutually independent, we obtain from Equation (4. 53) u (4. 55) ij ij u s (4. 56) Equations (4. 54) (4. 56) are sufficient to establish the constitutive behaviour For convenience of the derivation of incremental response, we define generalised u d and dissipative generalised... give an example of a transformation from a dissipation function to a yield function for a non-trivial case 4. 11.5 Yield Function to Dissipation Function The transformation of the yield function to the dissipation function is also nonstandard because it involves a singular transformation The rate equations ye ij ij are first divided by to give n equations in n variables ij 2 e These equations can be... identified as the isothermal bulk modulus, G is the shear modulus (which is the same for isothermal and adiabatic conditions), is the coefficient of linear thermal expansion, c is the heat capacity per unit volume at constant strain, 0 the initial temperature and X 1 9K 2 0 c The value of X is typically very close to unity, and represents the magnitude of the difference between adiabatic and isothermal behaviour . instance, often forms a central part of a finite element analysis. Therefore, one of the most important criteria that needs to be applied to the formulation of any model is that the incremental. incremental response can be evaluated either by numerical differentiation or analytically, if a symbolic manipulation package is used. 4. 14 Summary It is convenient at this point to restate the. Internal Variable and Generalised Stress Transformations between u, f, h, and g, in terms of the internal variable, and u , f , h , and g , in terms of the generalised stress, can be achieved