Nonlinear straining

Thus, on the basis of facts discussed in this section, we may conclude that effective viscosity ct/x, in extension of some polymers, passes the maximum with increasing strain velocity x. The first time t measured reliably in the nonlinear strain region exceeds significantly the respective time 0 in the linear area. Dependency x decreases with further increase in t(x). 

The basic task in the computational portion of MM is to minimize the strain energy of the molecule by altering the atomic positions to optimal geometry. This means minimizing the total nonlinear strain energy represented by the FF equation with respect to the independent variables, which are the Cartesian coordinates of the atoms (Altona and Faber, 1974). The following issues are related to the energy minimization of a molecular structure ... 

M.H.Wagner, The nonlinear strain measure of pol3risobutylene melt in general biaxial flow and its comparison to Doi-Edwards model, RheolActa. 22 (1990), 594-603. 

M.H.Wagner, J.Schaeffer, Nonlinear strain measures for general biaxial... 

The first four terms of the function are commonly found in molecular mechanics strain energy functions, and they are modified Hooke s law functions. The last term has been added to insure the proper stereochemistry about asymmetric atoms. A model is refined by minimizing the highly nonlinear strain energy function with respect to the atomic coordinates. An adaptive pattern search routine is used for the strain energy minimization because it does not require analytical derivatives. The time necessary to obtain good molecular models depends on the number of atoms in the molecule, the flexibility of the structure, and the quality of the starting model. 

Porter [212] applied his group interaction modeling approach to both the nonlinear strain and the long-term mechanical properties of polymers. He showed that both problems can be explained qualitatively by his concept of discrete energy levels in a polymer. His key argument is that a fraction of rubberlike states coexists with the predominantly glassy state of a polymer below Tg is a function of the temperature, time and mechanical energy of deformation and... 

As for other changes, we have with considerable trepidation moved the description of deformation in materials from one to three dimensions. Perhaps the main impetus for doing this was to simplify the rather complex explanation in previous editions of the relationship between tensile and shear properties. As an admitted expense, we now have double-subscripted variables in several sections. However, we have refrained from including nonlinear strain theory, which should properly be left for more advanced courses. 

The meaning of strain hardening and strain thinning is more clearly seen, when the effects of the linear-viscoelastic spectrum of relaxation times and the nonlinear strain measure Q on the elongational viscosity are separated. In the tube model, the strain measure can be represented by the second rank orientation tensor (describing the orientation of tube segments) and a molecular stress function f [6],... 

M. H. Wagner and J. Schaeffer, Assessment of Nonlinear Strain Measures for Extensional and Shearing Flows of Polymer Melts Rheol. Acta 33, 506-516... 

NONLINEAR STRAIN MEASURES OF RUBBER NETWORKS AND POLYMER MELTS... 

Step-strain stress-relaxation measurements have been frequently used to determine Sr(X) for polymer melts > . Equation (6) shows that if separability of time and strain effects is possible for the melt under consideration, the stress after a step elongational strain can be factored into a time-dependent function, the linear shear relaxation modulus G(t), and a strain-dependent function, the nonlinear strain measure Sr(X). Also other types of experiment may be oerformed to obtain Sr(X), such as constant-strain-rate experiments "", creep under constant stress and constant-stretching-rate experiments but these methods require more involved analytical and/or numerical calculations. 

Figures 1 and 2 reveal some surprising facts. First, the nonlinear strain measures obtained from equilibrium measurements on a large number of crossllnked rubbers lie in the same range as those obtained from transient measurements on the same types of material. This indicates that the nonlinear relation between stress and elastic strain is the same for permanent and temporary structures in the network. The fact that time and strain effects are separable over a large time scale for a number of crossllnked rubbers, which is revealed by the stretching rate independence of the strain measure calculated for these networks, is in accordance with this conclusion.

We consider elastic insulators in the framework of quasi-magnetostatics. Of necessity all equations are written down in the material description so that all boundaries are fixed even for nonlinear strains. The equations consist of the equations of motion. Maxwell s equations and the associated boundary conditions together with the nonlinear, coupled, stress and magnetic-induction contitutive equations. For a material domain of regular boundary 5Dq equipped with unit outward normal components N, we have [5] ... 

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