Thông tin tài liệu
136 8 Continuous Hyperplasticity 8.3.3 Dissipation Functional From the First Law of Thermodynamics, the definition of the mechanical dissi- pation d (Chapter 4), it follows that ij ij gsd V H T (8.7) Comparing with Equation (8.5), we can see that ˆ ˆ ij ij dd 8 F KD K*K K ³ (8.8) In Chapter 4, the dissipation is defined as a function of state and rate of change of internal variable. Similary, we can write a dissipation functional in the form: ˆ ˆˆ ˆ ˆ ,,, , ,, , 0 gg ij ij ij ij ij ij dd d 8 ªº V D TD V D K TD K K* K Kt ¬¼ ³ (8.9) which can be compared with Equation (4.7). We use the superscript g simply to indicate that the arguments of the function are those used for the Gibbs free energy: numerically g dd . 8.3.4 Dissipative Generalised Stress Function By analogy with the definition g ij ij dw F wD , we define: ˆ ˆ ˆ g ij ij dw F wD (8.10) Since ˆ g d will be first order in ˆ ij D for rate-independent materials, it follows from Euler’s theorem that ˆ ˆˆ ˆˆ ,,,, g ij ij ij ij ij d VDKTDKK FKDK (8.11) and so, ˆˆ ˆ ˆ ,,, g ij ij ij ij ij dd 8 ªº V D TD F KD K*K K ¬¼ ³ (8.12) Equations (8.8) and (8.12) can be compared with Equations (4.6) and (4.8). It follows from (8.8) and (8.12) that ˆ ˆˆ 0 ij ij ij d 8 FKFKDK*KK ³ (8.13) which compares with Equation (4.9). At first glance, Equation (8.13) would sug- gest ˆ ˆ 0 ij ij FKFK , but strictly it implies the much weaker condition that 8.4 Legendre Transformations of the Functionals 137 ˆ ˆ ij ij FKFK and ˆ ij D are orthogonal functions. We shall adopt the strong condition ˆ ˆ ij ij FK FK here as a generalisation of Ziegler’s orthogonality. This condition is entirely consistent with the Second Law of Thermodynamics but is a more restrictive statement. In a similar way to Chapter 4, we shall adopt it here simply as a constitutive hypothesis: it defines a class of materials that satisfy thermodynamics. Wider classes of materials that satisfy thermodynamics but violate our constitutive hypothesis could exist. However, the class defined by this hypothesis proves very wide, encompassing realistic descriptions of many materials. Furthermore, these descriptions are very compact in that only two scalar functionals need to be defined. We have adopted the ˆ F and ˆ F notation to indicate the fact that these quantities are always equal in our formulation (by hypothesis), but are separately defined [Equations (8.6) and (8.10)]. 8.4 Legendre Transformations of the Functionals 8.4.1 Legendre Transformations of the Energy Functional In the original approach, a variety of Legendre transformations between energy functions were used, e. g. ,, ,, ij ij ij ij ij ij fgHDT VDTVH. Transformations that involve variables (as opposed to functions of internal coordinates) are simi- lar to the original form, e. g. ˆˆ ,, ,, ij ij ij ij ij ij fg ªºª º HDT VDTVH ¬ ¼¬ ¼ . Those involving the internal function and the generalised stress function are slightly more com- plex. Thus instead of the original ,, ,, ij ij ij ij ij ij ggVDT VFTFD we have ˆˆ ˆ ˆˆ ˆ ,,, ,,, ij ij ij ij ij ij ggV D KTK V F KTKF KD K (8.14) together with ˆ ˆ ˆ ij ij g w D wF (8.15) (See Appendix C for details of the Legendre transformation methods for func- tionals.) We can also define ˆˆ ˆ ,, , ,, ˆ ˆˆ ,, ij ij ij ij ij ij ij ij gg d g d 8 8 ªº VFT VF KTK*KK ¬¼ ªº V D T F KD K*K K ¬¼ ³ ³ (8.16) 138 8 Continuous Hyperplasticity 8.4.2 Legendre Transformation of the Dissipation Functional The only relevant transformation is the singular transformation from the dissi- pation functional to the yield functional. The original transformation was, for instance, of the form ,,, ,,, 0 gg ij ij ij ij ij ij ij ij ydO VDTF FD VDTD together with the result g ij ij yw D O wF . This now becomes (see Appendix C.7.3): ˆ ˆˆ ˆ ,,,, ˆ ˆˆ ˆˆ ,,,,0 g ij ij ij g ij ij ij ij ij y d OK V D K TF K K F K D K V D K TD K K (8.17) together with the result that ˆ ˆ ˆ ˆ g ij ij y w D OK wF (8.18) which is the analogy of the normality condition in conventional “associated” plasticity. Note, however, that in the conventional approach, the plastic strain rate (the internal variable rate) is given by the differential of the yield function with respect to the stress; here it is given by the differential with respect to the generalised stress. This allows the current formulation to encompass non- associated flow. It is possible to define a yield functional by the integration ˆ ˆˆ ˆ ˆ ˆ ,,, , ,, 0 gg ij ij ij ij ij ij yy d 8 ªº OVDTF OK VDKTFK*KK ¬¼ ³ , but it is unclear that this would serve any useful purpose. 8.5 Incremental Response Chapter 4 demonstrates how, given knowledge of the energy function and the yield function, it is possible to derive the entire incremental response for an elastic-plastic material within the adopted formalism. This is of particular im- portance because non-linear material models are frequently implemented in finite element codes for which an incremental response is required. The derivation of the incremental response begins with differentiation of the energy function, giving the results summarised in the sixth row of Table 8.1. Further differentiation gives the rates of the variables. This is set out for the single internal variable in general form as Equation (4.15), which, for the par- ticular case of the Gibbs free energy, takes the following form, easily obtained by double differentiation of the Gibbs free energy: 8.5 Incremental Response 139 222 222 222 2 ij kl ij kl ij ij kl ij kl ij kl ij kl ij kl kl ggg ggg s ggg ªº www «» wV wV wV wD wV wT «» H ½ V «» ½ °° www °° «» F D ®¾ ®¾ «» wD wV wD wD wD wT °° °° «» T ¯¿ ¯¿ «» www «» wTwV wTwD «» wT ¬¼ (8.19) In the new formulation, this becomes 22 2 ˆˆ ˆ ˆ ˆ ij kl kl ij kl ij kl ij gg g d 8 §· ww w H V D K T * K K ¨¸ ¨¸ wV wV wV wD wV wT ©¹ ³ (8.20) 22 2 ˆˆ ˆ ˆˆ ˆˆˆˆ ij kl kl ij kl ij kl ij gg gww w F K V D K T wD wV wD wD wD K wT (8.21) 22 2 2 ˆˆ ˆ ˆ ˆ kl kl kl kl gg g sd 8 §· ww w V D K T*K K ¨¸ ¨¸ wTwV wTwD wT ©¹ ³ (8.22) Table 8.1. Examples of comparisons between different formulations Single internal variable Internal function Variables ij V , ij H T , s ij D , ij F , ij F ij V , ij H T , s ˆ ij DK , ˆ ij FK , ˆ ij FK Typical ener gy function(al) ,, ij ij g VDT ˆ ˆ ,,, ij ij g gd 8 VDKTK*KK ³ Typical dissi- pation func- tion(al) ,,, g ij ij ij d VDTD ˆ ˆˆ ,,,, gg ij ij ij dd d 8 VDKTDKK*KK ³ Typical yield function ,,, 0 g ij ij ij y VDTF ˆˆ ˆ ,,,,0 g ij ij ij y VD KTF KK Typical de- rivatives ij ij g w H wV ij ij g w F wD g s w wT ˆ ij ij ij g g d 8 ww H *KK wV wV ³ ˆ ˆ ˆ ij ij g w FK wD K ˆ g g sd 8 ww * K K wT wT ³ Incremental response Equations (4.22) to (4.29) Equations. (8.33) to (8.41) 140 8 Continuous Hyperplasticity Equations (8.20)–(8.22) are used together with the flow rule, Equation (8.18), to derive 22 2 ˆ ˆˆ ˆ ˆ ˆˆ g ij kl ij kl ij kl kl ij y gg g d 8 §· w ww w H V O K T * K K ¨¸ ¨¸ wV wV wV wD wF wV wT ©¹ ³ (8.23) 22 2 ˆ ˆˆ ˆ ˆ ˆ ˆˆˆˆˆ g ij kl ij kl ij kl kl ij y g gg w ww w F K V O K T wD wV wD wD wF wD wT (8.24) 22 2 2 ˆ ˆˆ ˆ ˆ ˆˆ g kl kl kl kl y gg g sd 8 §· w ww w V OK T*K K ¨¸ ¨¸ wTwV wTwD wF wT ©¹ ³ (8.25) The multiplier function ˆ O is obtained by substituting the above equations in the consistency condition, which is obtained by differentiation of the yield function. Equation (4.17) results in the condition, 0 g ggg g ij ij ij ij ij ij yyyy y wwww VDTF wV wD wT wF (8.26) and for the functional approach, this now becomes ˆˆ ˆˆ ˆˆ ˆ 0 ˆˆ gg gg g ij ij ij ij ij ij yy yy y ww ww K V D K T F K wV wD wT wF (8.27) which leads immediately (on substitution of Equations (8.18) and (8.24) to 22 2 ˆˆˆˆ ˆ ˆˆ ˆˆ ˆˆ ˆ ˆ 0 ˆˆ ˆˆ ˆ ˆ g ggg ij ij ij ij gg kl ij ij kl ij kl kl ij yyyy yy gg g wwww V OK T wV wD wF wT §· ww ww w V OK T ¨¸ ¨¸ wF wD wV wD wD wF wD wT ©¹ (8.28) From this, we obtain ˆ ˆ ˆ ˆˆ g g ij ij gg A A BB V T K K OK V T KK (8.29) where 2 ˆˆ ˆ ˆ ˆˆ gg g ij ij kl kl ij yy g A V ww w K wV wF wD wV (8.30) 2 ˆˆ ˆ ˆ ˆˆ gg g kl kl yy g A T ww w K wT wF wD wT (8.31) 2 ˆˆ ˆ ˆ ˆ ˆˆˆˆˆ gg g g ij kl kl ij ij yy y g B §· ww w w K ¨¸ ¨¸ wD wF wD wD wF ©¹ (8.32) 8.5 Incremental Response 141 Note that Equations (8.29)–(8.32) are analogous to Equations (4.18)(4.21). Finally, Equation (8.29) is substituted in Equations (8.23)–(8.25) to obtain the complete incremental relationships, which can be expressed In a similar way to Equation (4.23): ˆˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆˆ ˆˆ gg ij ijkl ij g g kl gg kl ij ij ijkl gg ij ij ijkl g ggg ij DD sD D DD CC AB AB VV VT TV T DV DT VT V T ªº H ½ «» °° «» °° «» °° V «»½ F K KK ®¾ ®¾ «» T ¯¿ °° «» DK KK °° «» °° «» OK ¯¿ «» K KK K ¬¼ (8.33) where 22 ˆˆ ˆ ˆ gg ijkl mnkl ij kl ij mn gg DCd VV V 8 §· ww *KK ¨¸ ¨¸ wV wV wV wD ©¹ ³ (8.34) 22 ˆˆ ˆ ˆ g g mn ij ij ij mn gg DCd VT T 8 §· ww K*KK ¨¸ ¨¸ wV wT wV wD ©¹ ³ (8.35) 22 ˆˆ ˆ ˆ gg kl mnkl kl mn gg DCd TV V 8 §· ww K*KK ¨¸ ¨¸ wTwV wTwD ©¹ ³ (8.36) 22 2 ˆˆ ˆ ˆ gg mn mn gg DCd TT 8 §· ww K*KK ¨¸ ¨¸ wTwD wT ©¹ ³ (8.37) 22 ˆ ˆˆ ˆ ˆ ˆˆ gg ijkl mnkl ij kl ij mn gg DC DV V ww K K wV wD wV wD (8.38) 22 ˆ ˆˆ ˆ ˆ ˆˆ gg kl mnkl kl mn gg DC DT V ww K K wTwD wTwD (8.39) ˆ ˆ ˆ ˆ ˆ g g g kl mnkl g mn A y C B V V K w K wF K (8.40) ˆ ˆ ˆ ˆ ˆ g e g mn g mn A y C B T T K w K wF K (8.41) Equations (8.33)–(8.41) are analogous to Equations (4.23)–(4.29). 142 8 Continuous Hyperplasticity Thus we can see that the entire constitutive response of the material (ex- pressed through the incremental stress-strain relationships and the evolution equations for internal variables) can be derived from the original two thermo- dynamic functionals. In Chapter 4 we discuss a number of cases in which constraints are imposed (for example, on the rates of the internal variables). Constraints may also be necessary within this new formulation, but have not been addressed here. The purpose here has been to set out the basic theory of a new approach to plasticity theory with an infinite number of yield surfaces. The following chapters will pursue examples in detail. It is useful, however, to set out a simple example to demonstrate how the formalism can be used. In section 8.6 we develop the gen- eral equations for a kinematic hardening plasticity model, and in section 8.7 describe a particular model for the one-dimensional case. 8.6 Kinematic Hardening with Infinitely Many Yield Surfaces The advantage of the multiple surface models is clearly that they are able to fit the non-linear behaviour of certain materials more accurately across a wide range of strain amplitudes. This is important, for instance, in modelling geo- technical materials (Houlsby, 1999). The disadvantage is that a large number of material parameters (associated with each yield surface) are necessary. In this section, we take the modelling of non-linearity to its logical conclusion by intro- ducing an infinite number of yield surfaces. Paradoxically, this reduces the number of material parameters required to specify the models, although at the expense that certain functions also have to be chosen. 8.6.1 Potential Functionals The hyperplastic formulation for multiple yield surfaces (Section 7.3) can be further extended to describe a continuous field of kinematic hardening yield surfaces of the type originally suggested by Mroz and Norris (1982). The general formulation of continuous hyperplastic models is given in Sections 8.2–8.5. Be- low we show how the continuous hyperplastic models are capable of reproduc- ing decoupled associated kinematic hardening plasticity with a continuous field of yield surfaces. For this case, the specific Gibbs free energy is a functional (rather than function) of the stress and an internal variable function ˆ ij DK : 12 ˆˆ ˆ ˆ ,, ij ij ij ij ij ij g gdgd 88 ªº VD V V DK*KK DKK*KK ¬¼ ³³ (8.42) 8.6 Kinematic Hardening with Infinitely Many Yield Surfaces 143 where Y is the domain of K. The function *K is a weighting function, such that d*K K is the fraction of the total number of the yield surfaces having a dimensionless size parameter between K and d K K. The dissipation functional, which is a functional of internal variable function and its rate ˆ ij DK , is also required: ˆ ˆˆ ˆ ˆ ,, , , , 0 gg ij ij ij ij ij ij dd d 8 ªº VDD VD KD KK*KKt ¬¼ ³ (8.43) Furthermore, in the following, only dissipation functionals with no dependence on stress are considered. This automatically leads to models in which the flow rule (in the conventional sense in plasticity theory) is associated. 8.6.2 Link to Conventional Plasticity The field of yield functions is related to the function ˆ ˆˆ ,,, g ij ij ij d VDDK within the dissipation functional (8.43) by the Legendre transform (see Appendix C), where the rate of internal variable function ˆ ij DK is interchanged with the dissipative generalised stress function ˆ ij FK . Noting that here we are consider- ing only cases where the dissipation does not depend on the stress, the dissipa- tive generalised stress function ˆ ij FK is defined by ˆ ˆˆ ,, ˆ ˆ g ij ij ij ij dwDKDKK FK wD K (8.44) The transformation from the dissipation to the yield function is a degenerate special case of the Legendre transformation due to the fact that the dissipation is homogeneous and first order in the rates. Therefore, this transformation results in the following identity: ˆ ˆ ˆˆ ˆ ˆ ˆˆ ˆ ,, ,, 0 gg ij ij ij ij ij ij ydOK DFK FKDK DDK (8.45) where ˆˆ ˆ ,,0 g ij ij y DKFKK is the field of yield functions and ˆ OK is a non- negative multiplier. As seen from Equation (8.45), a complete field of yield functions is contained in the equation of the dissipation functional (8.43) in a compact form. The Gibbs free energy functional (8.42) allows the definition of the strain ten- sor: 1 ˆ , ˆ ij ij ij ij ij ij g g d 8 ªº wVD w ¬¼ H D K*K K wV wV ³ (8.46) 144 8 Continuous Hyperplasticity Here ˆ ij d 8 DK*KK ³ plays exactly the same role as the conventionally defined plastic strain p ij H . It is convenient also to define the elastic strain 1 e ij ij gH wwV. The flow rule for the field of yield surfaces is obtained from the properties of the degenerate special case of the Legendre transformation (8.45) relating yield and dissipation functions (see Appendix C): ˆ ˆ ˆ ˆ g ij ij yw DK OK wF (8.47) We restricted the dissipation function to exhibit no explicit dependence on the true stresses, so again it follows that the normality presented by Equation (8.47) in the generalised stress space also holds in the true stress space. The dependence of the dissipation functional on the internal variable func- tion ˆ ij DK is transferred to the field of yield functions by the Legendre trans- formation (8.45) where ˆ ij DK plays the role of a passive variable. Therefore, the strain hardening rule is obtained automatically through the functional de- pendence of the yield function on the internal variable (or plastic strain) func- tion ˆ ij DK . The generalised stress function is defined by Frechet differentiation of the Gibbs free energy functional (8.42) with respect to the internal variable function, resulting in: 2 ˆ ˆ ˆ ij ij ij g w FK V wD (8.48) Again it is convenient to introduce at this stage the “back stress” function ˆ ij UK associated with the internal variable function and defined as the differ- ence between the true stress and generalised stress function. By applying Ziegler’s orthogonality principle in the form ˆ ˆ ij ij FK FK , the back stress function can be expressed as 2 ˆ ˆˆ ˆ ij ij ij ij g w UK VFK wD (8.49) which, after differentiation, yields 2 2 ˆ ˆˆ ˆ ˆˆ ij ij ij kl ij kl gw UK VFK D K wD wD (8.50) Equation (8.50) is interpreted as the translation rule for the field of yield sur- faces when the dissipation function (and hence also the yield function) exhibits no explicit dependence on the true stresses. 8.6 Kinematic Hardening with Infinitely Many Yield Surfaces 145 8.6.3 Incremental Response Two possibilities exist for each value of K. If the material state is within the yield surface, ˆˆ ˆ ,,0 g ij ij y DKFKK, no dissipation occurs and ˆ 0OK . If the material point lies on the yield surface, ˆˆ ˆ ,,0 g ij ij y DKFKK , then plastic deformation can occur provided that ˆ 0OKt . In the latter case, the incremental response is obtained by invoking the consistency condition of the field of yield surfaces: ˆˆ ˆˆ ˆ ˆ ˆ ,, 0 ˆˆ gg g ij ij ij ij ij ij yy y ww DKFKK DK FK wD wF (8.51) Substitution of (8.47) and (8.50) in (8.51) leads to the solution for the multiplier ˆ OK function: 2 2 ˆ ˆ ˆ ˆˆˆˆ ˆ ˆˆˆ ˆ ˆˆ g ij ij g ggg ij ij kl kl ij ij y y yyy g w V wF OK wwww w wF wD wD wF wD wF (8.52) Differentiation of Equation (8.46) and substitution of (8.47) in both the result and in (8.50) gives the incremental stress-strain response, 1 ˆ ˆ ˆ g ij kl ij kl ij y g d 8 w w H VOK *KK wV wV wF ³ (8.53) and the update equations for the internal variable and generalised stress func- tions: ˆ ˆ ˆ ˆ g ij ij y w DK OK wF (8.54) 2 2 ˆ ˆ ˆ ˆˆ ˆˆ ˆ g ij ij ij ij ij kl kl y g w w FK VUK VOK wD wD wF (8.55) The multiplier ˆ OK is defined from Equation (8.52) when ˆˆ ˆ ,,0 g ij ij y DKFKK , and ˆ 0OK! . Otherwise, ˆ 0OK [when ˆˆ ˆ ,,0 g ij ij y DKFKK or when (8.52) gives a negative value of ˆ OK ]. Description of the constitutive behaviour during any loading requires a pro- cedure for keeping track of ˆˆ and , ij ij FK DKK8 , and this is achieved by using Equations (8.54) and (8.55). [...]... multi-dimensional kinematic hardening plasticity models All the above models are based on the concept of a number of elements, each consisting of a slider and a spring, arranged in series as in Figure 7. 2 The plastic strains in each of these elements are therefore additive An alternative way of generalising from a single plastic strain to many plastic strain components involves arranging the elements in parallel,... continuous hyperplasticity approach allows materials to be modelled in which the past strain history is in effect represented by an infinite number of internal variables but compactly represented as an internal function This approach, although rooted entirely within the methods of “generalised thermodynamics”, offers some of the advantages that are more often associated with the “rational mechanics” approach... proportional cyclic loading is similar to that presented in Figure 7. 1 The difference is that the stress-strain curves are smooth 8.11 Hierarchy of Multisurface and Continuous Models In Chapters 7 and 8, we have systematically presented a compact hyperplastic framework for kinematic hardening of plastic materials In Chapter 7, we generalize a model with a single yield surface to the case of multiple surfaces... variables common in soil mechanics These are particularly convenient for the analysis of the triaxial test, which is a compression test on a cylindrical specimen in which the axial stress is 1 and the radial stress is 3 The circumferential stress is assumed (in a uniform specimen) to be equal to the radial stress The so-called “triaxial” variables are the mean stress p and deviator stress q defined as... choice of a particular pair of functionals leads to a material model with a specific stress-strain curve In certain circumstances, it may be of value to reverse this process We may wish to calibrate the model by specifying the shape of the stress-strain curve and from this, deduce the form of the functionals In defining a model using the above approach, there is considerable freedom in the way the... multi-dimensional hyperplastic models are presented together with their conventional plasticity interpretations Because they are formulated using hyperplasticity, these kinematic hardening models are guaranteed to obey the First and Second Laws of Thermodynamics They can also be set within a simple hierarchical framework, as shown in Table 8.2, for one-dimensional kinematic hardening models, and in Table 8.3... stress-strain behaviour Elongation of the E spring gives e elastic strain , and elongation of the distribution of the H springs contributes the plastic strain ˆ to the total plastic strain; their sum gives the total strain It is assumed here that the elastic coefficients of all H springs are the same and equal to H 150 8 Continuous Hyperplasticity To complete the analogy between the hyperplastic formulation... that the plastic strain associated with this surface ˆ * 0 Then, from , and from Equation (8 .74 ) for the yield surface Equation (8 .75 ), ˆ * ˆ * k * Therefore, in this simple case, the parameter * can be interpreted as dimensionless current stress k , which makes the formulation (8 .79 ) identical to that of the continuous Iwan model during initial loading 8.9.3 Model Calibration Using the Initial Loading... Mises yield surfaces is described by ˆ kmin kmax kmin where kmin and kmax are Equation (8.91), with k the sizes of the smallest and the largest yield surfaces in the field This model simulates six-dimensional elastic-non-linear plastic stress-strain behaviour The elastic component of strain is calculated according to Hooke’s law An associated flow rule is implied, together with the plastic incompressibility... this chapter, we show how some special features of the mechanical behaviour of soils (such as the concept of effective stresses; frictional behaviour; non-associated flow; dependence of stiffness on pressure, density and loading history; stress induced anisotropy; critical state; and small strain non-linearity) can be expressed within the hyperplastic constitutive framework developed in earlier chapters . Surfaces The advantage of the multiple surface models is clearly that they are able to fit the non-linear behaviour of certain materials more accurately across a wide range of strain amplitudes important, for instance, in modelling geo- technical materials (Houlsby, 1999). The disadvantage is that a large number of material parameters (associated with each yield surface) are necessary choice of a particular pair of func- tionals leads to a material model with a specific stress-strain curve. In certain circumstances, it may be of value to reverse this process. We may wish to cali- brate
Ngày đăng: 10/08/2014, 11:21
Xem thêm: A Principles of Hyperplasticity Part 7 pdf
