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Section 8.7 Solid Mechanics Part II Kelly 3148.7 Associated and Non-associated Flow Rules Recall the Levy-Mises flow rule, Eqn. 8.4.3, ijpijsddλε= (8.7.1) The plastic multiplier can be determined from the hardening rule. Given the hardening rule one can more generally, instead of the particular flow rule 8.7.1, write ijpijGddλε=, (8.7.2) where ijG is some function of the stresses and perhaps other quantities, for example the hardening parameters. It is symmetric because the strains are symmetric. A wide class of material behaviour (perhaps all that one would realistically be interested in) can be modelled using the general form ijpijgddσλε∂∂=. (8.7.3) Here, g is a scalar function which, when differentiated with respect to the stresses, gives the plastic strains. It is called the plastic potential. The flow rule 8.7.3 is called a non-associated flow rule. Consider now the sub-class of materials whose plastic potential is the yield function, fg =: ijpijfddσλε∂∂=. (8.7.4) This flow rule is called an associated flow-rule, because the flow rule is associated with a particular yield criterion. 8.7.1 Associated Flow Rules The yield surface ( ) 0=ijf σ is displayed in Fig 8.7.1. The axes of principal stress and principal plastic strain are also shown; the material being isotropic, these are taken to be coincident. The normal to the yield surface is in the direction ijf σ/∂ and so the associated flow rule 8.7.4 can be interpreted as saying that the plastic strain increment vector is normal to the yield surface, as indicated in the figure. This is called the normality rule. Section 8.7 Solid Mechanics Part II Kelly 315 Figu re 8.7.1: Yield su rface The normality rule has been confirmed b...