Inelastic evolution equation

The theory is initially presented in the context of small deformations in Section 5.2. A set of internal state variables are introduced as primitive quantities, collectively represented by the symbol k. Qualitative concepts of inelastic deformation are rendered into precise mathematical statements regarding an elastic range bounded by an elastic limit surface, a stress-strain relation, and an evolution equation for the internal state variables. While these qualitative ideas lead in a natural way to the formulation of an elastic limit surface in strain space, an elastic limit surface in stress space arises as a consequence. An assumption that the external work done in small closed cycles of deformation should be nonnegative leads to the existence of an elastic potential and a normality condition. 

The inelastic strain as an internal state variable is obtained from a linear evolution equation formulated with respect to the intermediate configuration ... 

Consider the p articular case of elastic (reversible) evolution correspwnding to stationary values of the internal variables V and plastic strains, with uniform tempreratures. We then have zero dissipration, retrieving the classic state equations (13). In the sequel it will be assumed that these state equations remain valid even under irreversible inelastic evolutions, so that the CD inequality becomes ... 

Also linked to the coming into operation of HERA we discuss in more detail the low x region in deep inelastic scattering. Here the usual evolution equations break down and one approaches the non-perturbative region of QCD, creating a tremendous challenge to theory. 

The third equation is the kinetic equation, which describes the evolution of the one-particle density matrix p(r, r, E) of the electron in the process of multiple elastic and inelastic scattering in a solid... 

The time evolution in Eq. (7.75) is described by the time-dependent Schrodinger equation, provided the molecule is isolated from the rest of the universe. In practice, there are always perturbations from the environment, say due to inelastic collisions. The coherent sum in Eq. (7.75) will then relax to the incoherent sum of Eq. (7.74), that is, the off-diagonal interference terms will vanish and cn 2 — pn corresponding to the Boltzmann distribution. As mentioned earlier, the relaxation time depends on the pressure. In order to take advantage of coherent dynamics it is, of course, crucial that relaxation is avoided within the duration of the relevant chemical dynamics. 

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