Large-amplitude Deformations Research Articles

A weak-mode representation of floppy molecules. II. Kinetic energy and total Hamiltonian matrix

The total Hamiltonian matrix of a floppy molecule (i.e. in a bound state for which large-amplitude isomerization-like deformations are possible), is derived in an entirely analytical manner which rests on the basic idea that the weakest mode of internal deformation is separated from the other stronger modes. The latter are supposed to undergo throughout only small-amplitude vibrations with respect to equilibrium positions which vary adiabatically with the former. In using (i) for the potential energy function, a Taylor expansion in the strong-mode displacements around the weak-mode isomerization reaction pathway, and (ii) for the kinetic energy operator matrix representation, a basis that includes for the strong modes adiabatically displaced harmonic oscillator basis functions, the total Hamiltonian operator can be represented by a sparse matrix whose diagonal elements are very like analytical spectroscopic terms, and the off-diagonal elements are also analytical expressions in terms of the quantum numbers of the basis functions. This adiabatic representation, which is known to be efficient for numerical diagonalization, is also particularly interesting from the physical point of view insofar as it allows to bring very clearly to the fore the coupling scheme between the internal deformations. The J = 0 case (no angular momentum) is developed first. Then the case where J ≢ 0 is treated (also analytically), in using for the first time in dynamical calculations the principal-axis system as the body-fixed (BF) frame of reference. This last feature is important, since it allows to distinguish (at least formally, if not physically) the Coriolis contribution to the total energy from the rotational one.

Large Amplitude Cyclic Deformation of a Crosslinked Epoxy Resin in the Transition Zone

Abstract Absolute values of complex dynamic shear modulus at different frequencies and deformation amplitudes in the transition zone from glassy state to rubbery state were measured for an epoxy resin cured with diethylenetriamine. A sharp decrease of the modulus, truncation of the glassy state region and shifting of the transition zone to lower temperatures have been observed as an effect of large amplitude deformations above certain critical values. The maximum in the modulus values decrease and the minimum of the critical amplitude both take place at glass transition temperature. The values of the maximum and the minimum, as well as the temperature when they occur, depend upon frequency. The transition temperature, determined by the point of intersection of straight lines which express the modulus temperature relation in the glassy state and in the transition zone, can be frequency independent at large deformation amplitudes, contrary to the known behavior of the polymers at small deformation amplitude...

Viscoelastic Properties of Rubber Networks

Abstract The molecular theory of Rouse, Zimm, and Bueche correctly accounts for the viscoelastic properties of polymers in very dilute solution and, to a large extent, for those of polymers in bulk or in concentrated solution, as long as their mean molecular weight is below about 20 000. Above this MW limit, relaxation times appear which are longer than those provided for in this theory. The “viscoelastic plateau”, which then appears in the long relaxation time region of the dynamic spectrum, is ascribed to entanglements of molecular chains which behave like temporary crosslinks. An analogous phenomenon occurs in the same way in permanent polymer networks, such as rubber vulcanizates. In this case one finds abnormally slow relaxation or creep rates during the approach to equilibrium, as well as increased low-frequency mechanical energy losses under forced sinusoidal vibration. The presence of colloidal fillers, such as carbon blacks used to reinforce rubbers, also seems to increase this hysteresis within the polymer matrix, independent of thixotropic effects which result from the reversible rupture of filler particle aggregates under large-amplitude cyclic deformations. We propose to analyze here the results (obtained jointly at the Institut Français du Caoutchouc and at the laboratory of Professor J. D. Ferry, University of Wisconsin) of measurements over the entire rubbery spectrum of the dynamic properties and of stress relaxation on vulcanizates of natural rubber, cis-polybutadiene, and styrene-butadiene copolymer (SBR) in the absence of secondary crystallization or aging phenomena. Then we examine the interpretation of the behavior of these materials, both at low frequency and during the approach to equilibrium, by analogy with the theories of the “viscoelastic plateau” of linear polymers.

Electrohydrodynamically Induced Spatially Periodic Cellular Stokes-Flow

An experiment is described in which a spatially periodic, time-invariant electric field is used to create a spatially periodic cellular flow by the action of an electric shear stress in the region of a fluid-fluid interface. The cellular motion is closely related to large-amplitude interfacial deformations observed when an electric field stresses the interface between two fluids having nearly the same electrical conductivity. A simple theoretical model is developed for the cell streamlines, flow velocity, and distribution of electric potential, and this model is shown to be in agreement with the experimental observations.

The response of prestressed aluminum

Experimental data on the propagation of changes in strain associated with an unstable load are presented. These deformations frequently propagate at speeds which are less than 10 in/sec. These large amplitude deformations are the result of adding small increments of the load at long time intervals. The wave speed calculated from the Karman-Taylor wave theory and based on a smooth stress-strain relation for the material is 18 × 10 3 in/sec. One large change in strain ‘propagates’ at an average, speed of 3·7 × 10 −3 in/sec. These slowly propagating deformations occur only at certain discrete loads in prestressed specimens of annealed aluminum. Between these loads, the response is approximately that of a linearly elastic material. Some data on preloaded specimens tested in a torsional impact apparatus are also included.