By M. L. Balinski, Eli Hellerman

ISBN-10: 3642007651

ISBN-13: 9783642007651

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This text discusses a few tools of describing and concerning mathematical items and of regularly and unambiguously signaling the logical constitution of mathematical arguments.

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65) is insensitive to plastic shears in global ideal crystal (κ). On the other hand, this deﬁnition is not always practical as we will see in forthcoming sections devoted to applications. Namely, if we use Hill’s logarithmic elastic and plastic strains (cf. 29)) then three constituents in Teodosiu’s deﬁnition make the analysis even more cumbersome. 1 Quasi-plastic strain and anholonomic coordinates Until now we have assumed that the instant deformed conﬁguration (χt ) contains dislocations and, eventually, disclinations.

For a correct constitutive theory a geometric description, able to describe properly the most important microstructural changes during an inelastic deformation process, is indispensable. Here only slight amendments to considerations of the ﬁrst chapter are made, whereas the last section of this chapter is reserved for formulation of the theory. 1 A kinematic addendum towards damage-thermo-plastic straining history For the sake of easier reading we ﬁrst repeat here brieﬂy some geometrical issues given in the ﬁrst chapter.

T) : (νt ) → (νtd ) obtained in such a way is again incompatible and may be termed the damage distortion tensor. Now, the quasi-plastic distortion has two components and reads Πω := Πθ Πd . , [Kro60, Sto62, Bil60]). , t) : (κ) → (νt ) is deﬁned as a plastic distortion, where (κ) is a global ideal crystal having the same intrinsic crystalline structure as (νt ) elements themselves. However, such a distortion is not unique since there are many indistinguishable conﬁgurations (κ) with various shapes but the same intrinsic structure.