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Tham khảo tài liệu 'a principles of hyperplasticity part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 136 8 Continuous Hyperplasticity 8.3.3 Dissipation Functional From the First Law of Thermodynamics the definition of the mechanical dissipation d Chapter 4 it follows that g Oij itii s0 _ d 8.7 Comparing with Equation 8.5 we can see that d ịxij n ẩ ij n r n dn 8.8 Y 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 dg ơij ảij 0 ầij J dg vij ảij b 0 ẳịj b n r n d 0 8.9 Y 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 d dg. 8.3.4 Dissipative Generalised Stress Function - . ddg By analogy with the definition Xij we define dãn j _ddg Xij 8.10 j Since dg will be first order in ấij for rate-independent materials it follows from Euler s theorem that tỉg c ij â ij b 0 a ij b b Xij yùồij b 8.11 and so dg ơij ảij 0 ầij Ịiij B ẩij- n r n dn 8.12 Y 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 f ii n -Xij- n ẩij- n r B dn 0 8.13 Y which compares with Equation 4.9 . At first glance Equation 8.13 would suggest ij b - Xij b 0 but strictly it implies the much weaker condition that 8.4 Legendre Transformations of the Functionals 137 iij h _Xij h and ẳịj are orthogonal functions. We shall adopt the strong condition Xij 5Cij- Co 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
