Three variations on a theme by Fibonacci

Several variants of the classic Fibonacci inflation tiling are considered in an illustrative fashion, in one and in two dimensions, with an eye on changes or robustness of diffraction and dynamical spectra. In one dimension, we consider extension mechanisms of deterministic and of stochastic nature, while we look at direct product variations in a planar extension. For the pure point part, we systematically employ a cocycle approach that is based on the underlying renormalization structure. It allows explicit calculations, particularly in cases where one meets regular model sets with Rauzy fractals as windows.

Evaluating Rigid and Semiflexible Fiber Orientation Evolution Models in Simple Flows

As American vehicle fuel efficiency requirements have become more stringent due to the CAFE standards, the auto industry has turned to fiber reinforced polymer composites as replacements for metal parts to reduce weight while simultaneously maintaining established safety standards. Furthermore, these composites may be easily processed using established techniques such as injection molding and compression molding. The mechanical properties of these composites are dependent on, among other variables, the orientation of the fibers within the part. Several models have been proposed to correlate fiber orientation with the kinematics of the polymer matrix during processing, each using various strategies to account for fiber interactions and fiber flexing. However, these all require the use of empirical fitting parameters. Previous work has obtained these parameters by fitting to orientation data at a specific location in an injection-molded part. This ties the parameters to the specific mold design used. Obtaining empirical parameters is not a trivial undertaking and adds significant time to the entire mold design process. Considering that new parameters must be obtained any time some aspect of the part or mold is changed, an alternative technique that obtains model parameters independent of the mold design could be advantageous. This paper continues work looking to obtain empirical parameters from rheological tests. During processing, the fiber–polymer suspension is subjected to a complex flow with both shear and extensional behavior. Rather than use a complex flow, this study seeks to isolate and compare the effects of shear and extension on two orientation models. To this end, simple shear and planar extension are employed, and the evolution of orientation from a planar random initial condition is tracked as a function of strain. Simple shear was imparted using a sliding plate rheometer designed and fabricated in-house. A novel rheometer tool was developed and fabricated in-house to impart planar extension using a lubricated squeeze flow technique, where a low-viscosity Newtonian lubricant is applied to the solid boundaries to minimize the effect of shearing due to the no-slip boundary condition. The Folgar–Tucker model with a strain reduction factor is used as a rigid fiber model and compared against a bead–rod model (a semiflexible model) proposed by Ortman. Both models are capable of predicting the data, with the bead–rod model performing slightly better. Orientation occurs at a much faster rate under startup of planar extension and also attains a much higher degree of flow alignment when compared with startup of steady shear.

Evaluating Rigid and Semi-Flexible Fiber Orientation Evolution Models in Simple Flows

As American vehicle fuel efficiency requirements have become more stringent due to the CAFE standards, the auto industry has turned to fiber reinforced polymer composites as replacements for metal parts to reduce weight while simultaneously maintaining established safety standards. Furthermore, these composites may be easily processed using established techniques such as injection molding and compression molding. The mechanical properties of these composites are dependent on, among other variables, the orientation of the fibers within the part. Several models have been proposed to correlate fiber orientation with the kinematics of the polymer matrix during processing, each using various strategies to account for fiber interactions and fiber flexing. However, these all require the use of empirical fitting parameters. Previous work has obtained these parameters by fitting to orientation data at a specific location in an injection-molded part. This ties the parameters to the specific mold design used. Obtaining empirical parameters is not a trivial undertaking and adds significant time to the entire mold design process. Considering that new parameters must be obtained any time some aspect of the part or mold is changed, an alternative technique that obtains model parameters independent of the mold design could be advantageous. This paper continues work looking to obtain empirical parameters from rheological tests. During processing, the fiberpolymer suspension is subjected to a complex flow with both shear and extensional behavior. Rather than use a complex flow, this study seeks to isolate and compare the effects of shear and extension on two orientation models. To this end, simple shear and planar extension are employed and the evolution of orientation from a planar random initial condition is tracked as a function of strain. Simple shear was imparted using a sliding plate rheometer designed and fabricated in-house. A novel rheometer tool was developed and fabricated in-house to impart planar extension using a lubricated squeeze flow technique, where a low viscosity Newtonian lubricant is applied to the solid boundaries to minimize the effect of shearing due to the no-slip boundary condition. The Folgar-Tucker model with a strain reduction factor is used as a rigid fiber model and compared against a Bead-Rod model (a semi-flexible model) proposed by Ortman. Both models are capable of predicting the data, with the Bead-Rod model performing slightly better. Orientation occurs at a much faster rate under startup of planar extension, and also attains a much higher degree of flow alignment when compared with startup of steady shear.

Comparison of Two Hyperelastic Models for Carbon Black Filled Rubber

Mooney-Rivlin model and Ogden model are frequently used by engineers for finite element analysis of rubber material. Before simulation, simple, biaxial and planar extension tests are always done to get the model parameters. In this paper, we compare these two hyperelastic models with experimental data produced under simple, biaxial extension and planar extension loading conditions. The ability of the two models to reproduce different deformation modes is analyzed. Both material parameters and the stretch range of validity of each model are determined.

ON THE TAXONOMY OF FLATTENED MOEBIUS STRIPS

The taxonomy of flattened Moebius strips (FMS) is reexamined in order to systematize the basis for its development. An FMS is broadly characterized by its twist and its direction of traverse. All values of twist can be realized by combining elementary FMS configurations in a process called fusion but the result is degenerate; a multiplicity of configurations can exist with the same value of twist. The development of degeneracy is discussed in terms of several structural factors and two principles, conservation of twist and continuity of traverse. The principles implicate a corresponding pair of constructs, a process of symbolic convolution, and the inner product of symbolic vectors. Combining constructs and structural factors leads to a systematically developed taxonomy in terms of twist categories assembled from permutation groups. Taxonomical structure is also graphically revealed by the geometry of an expository edifice that validates the convolution process while displaying the products of fusion. A formulation that combines some of the algebraic precepts of Quantum Mechanics with the primitive combinatorics and degeneracies inherent to the FMS genus is developed. The potential for further investigation and application is also discussed. An appendix outlines the planar extension of the fusion concept and another summarizes a related application of convolution.