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.

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Computational Practice in Mathematical Programming by M. L. Balinski, Eli Hellerman

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