Impenetrability constraint

If the electrode material is assumed to be homogeneous, then the concentration gradient of lithium through the electrode is the only factor that drives lithium transport. Hence, lithium will enter/leave the planar electrode only at the electrode/ electrolyte interface, and cannot penetrate into the back of the electrode. Under such an impermeable (impenetrable) constraint, the electric current (I) can be expressed by Equation (5.18) during the initial stage of diffusion, and by Equation (5.19) during the later stage [45] ... 

We use the closest point projection technique (see Wriggers [21]) to determine contact between the membrane and the substrate surface. We consider a point which lies on the membrane surface, and find its projection on the substrate surface at x. The impenetrability constraint characterized by the gap between the two surfaces then reads,... 

As with the case for the impenetrability constraint, the tangential (stick-slip) constraints are also gradually enforced. The frictional slip is determined from a line search along the steepest descent direction M (Gr) (Pxr ), and the rate constraints are active during the Newton iterations while assuming sticking conditions, that is. 

Until recently, the theory did not allow the configuration of the positive state to be described, due to entry/exit DNAs interpenetration upon application of the positive constraint to the loop. A recent development [63] takes the DNA impenetrability into account and deals with the resulting DNA self-contacts, which were allowed to slide freely, following the needs of the energy minimization process. 

In order to proceed, I have to make some assumptions about the chain statistics and about the form of the crystal. The latter should be given again by the model sketched in (Fig. 2.2). In particular, all stems should have the same length. To start with a tractable model, I wU] further ignore excluded volume interactions between the segments of the amorphous fraction as well as the conformational constraints due to the impenetrable crystalline surface. Furthermore, I treat the chain statistics as Gaussian and ignore effects of finite flexibility of the chain. These relaxed conditions overestimate the entropy of the amorphous fraction. 1 will reconsider these approximations in the context of the exact solution for the idealized model. 

We now turn our attention to an important aspect of solving tire problems - the treatment of contact constraints. The contact problem to be solved is one of the two bodies contacting across their respective surfaces. The impenetrability of the two bodies (normal contact) will manifest itself as a set of unilateral constraints... 

Important and often challenging questions arise in the analysis of interfaces between different phases or different materials, possibly connected to the presence of geometrical constraints like the presence of an impenetrable region. The statistical mechanics modeling of interfaces often resorts to the so called reduced models, that is to models in which the complexity is reduced by imposing to the trajectories of the process to be functions. In two dimensions, this naturally leads to random walk based models, see Appendix C.l. 

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