Fractal Dimensions in Elastic-Plastic Beam Dynamics

1995 ◽  
Vol 62 (2) ◽  
pp. 523-526 ◽  
Author(s):  
P. S. Symonds ◽  
J.-Y. Lee
Keyword(s):  
Short Pulse ◽  
Fractal Dimensions ◽  
Pulse Loading ◽  
Load Parameter ◽  
Beam Model ◽  
Self Similarity ◽  
Elastic Plastic ◽  
Intersection Points ◽  
Two Degree Of Freedom ◽  
Fractal Number

Calculations of two types of fractal dimension are reported, regarding the elastic-plastic response of a two-degree-of-freedom beam model to short pulse loading. The first is Mandelbrot’s (1982) self-similarity dimension, expressing independence of scale of a figure showing the final displacement as function of the force in the pulse loading; these calculations were made with light damping. These results are equivalent to a microscopic examination in which the magnification is increased by factors of 102; 104; and 106. It is found that the proportion and distribution of negative final displacements remain nearly constant, independent of magnification. This illustrates the essentially unlimited sensitivity to the load parameter, and implies that the final displacement in this range of parameters is unpredictable. The second fractal number is the correlation dimension of Grassberger and Procaccia (1983), derived from plots of Poincare intersection points of solution trajectories computed for the undamped model. This fractional number for strongly chaotic cases underlies the random and discontinuous selection by the solution trajectory of the potential well leading to the final rest state, in the case of the lightly damped model.

Fractal Dimensions in Elastic-Plastic Beam Dynamics

1993 ◽  
Author(s):  
P. S. Symonds ◽  
Jae-Yeong Lee
Keyword(s):  
Fractal Dimension ◽  
Short Pulse ◽  
Fractal Dimensions ◽  
Great Sensitivity ◽  
Beam Model ◽  
Transverse Loading ◽  
Self Similarity ◽  
Chaotic Vibrations ◽  
Two Degree Of Freedom ◽  
Initial Impulse

Abstract The final midpoint displacement of a two-degree-of-freedom beam model subjected to a short pulse of transverse loading may be either in the direction of the initial impulse or in the opposite (“negative”) direction, when moderately small plastic deformations occur. In the range where chaotic vibrations occur, the result depends with great sensitivity on the impulse magnitude. Considering a pulse of duration 0.5 × 10−3 sec, 100 calculations have been made for pulse forces P starting at 2500 N and increasing by increments of 2.0, 10−2, 10−4, and 10−6 N. It is found that the proportion and distribution of negative final displacements remain, on average, the same, independent of the size of the force increment. A fractal dimension representing a self-similarity property is calculated for the four choices of the force increment, and is found to be approximately 0.78 in each case. A correlation fractal dimension is also computed for undamped responses.

Chaotic Responses of a Two-Degree-of-Freedom Elastic-Plastic Beam Model to Short Pulse Loading

1992 ◽  
Vol 59 (4) ◽  
pp. 711-721 ◽  
Author(s):  
J.-Y. Lee ◽  
P. S. Symonds ◽  
G. Borino
Keyword(s):  
Short Pulse ◽  
Direct Method ◽  
Degree Of Freedom ◽  
Beam Model ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Arch Action ◽  
Time Histories ◽  
Energy Plane ◽  
Two Degree Of Freedom

The paper discusses chaotic response behavior of a beam model whose ends are fixed, so that shallow arch action prevails after moderate plastic straining has occurred due to a short pulse of transverse loading. Examples of anomalous displacement-time histories of a uniform beam are first shown. These motivated the present study of a two-degree-of-freedom model of Shanley type. Calculations confirm these behaviors as symptoms of chaotic unpredictability. Evidence of chaos is seen in displacement-time histories, in phase plane and power spectral diagrams, and especially in extreme sensitivity to parameters. The exponential nature of the latter is confirmed by calculations of conventional Lyapunov exponents and also by a direct method. The two-degree-of-freedom model allows use of the energy approach found helpful for the single-degree-of-freedom model (Borino et al., 1989). The strain energy is plotted as a surface over the displacement coordinate plane, which depends on the plastic strains. Contrasting with the single-degree-of-freedom case, the energy diagram illuminates the possibility of chaotic vibrations in an initial phase, and the eventual transition to a smaller amplitude nonchaotic vibration which is finally damped out. Properties of the response are further illustrated by samples of solution trajectories in a fixed total energy plane and by related Poincare section plots.

An Energy Approach to Anomalous Damped Elastic-Plastic Response to Short Pulse Loading

1989 ◽  
Vol 56 (2) ◽  
pp. 430-438 ◽  
Author(s):  
G. Borino ◽  
U. Perego ◽  
P. S. Symonds
Keyword(s):  
Strain Energy ◽  
Short Pulse ◽  
Elastic Strain Energy ◽  
Upper Bounds ◽  
Type Model ◽  
Load Parameter ◽  
Plastic Response ◽  
Lower And Upper Bounds ◽  
Peak Displacement ◽  
Elastic Plastic

In beams with full-end constraints, loaded transversely by short pressure pulses, the effect of extensional plastic deformation is to make possible instabilities related to snap buckling in the elastic-plastic recovery after the first peak displacement (Symonds and Yu, 1985). In the present paper we make use of a damped, Shanley-type model to study the calculation of the final displacement, reached asymptotically. We show that plots of the elastic strain energy and of the total energy as functions of the displacement help to guide thinking. They provide clarification of previously observed phenomena (Genna and Symonds, 1988) that appear complex at small damping, and lead to lower and upper bounds on the load parameter such that anomalous responses are observed. The response is calculable with the usual accuracy in problems where bifurcations are concerned.

scholarly journals Supramolecular Fractal Growth of Self-Assembled Fibrillar Networks

Gels ◽  
2021 ◽  
Vol 7 (2) ◽  
pp. 46
Author(s):  
Pedram Nasr ◽  
Hannah Leung ◽  
France-Isabelle Auzanneau ◽  
Michael A. Rogers
Keyword(s):  
Fractal Dimension ◽  
Crystallization Temperature ◽  
Cayley Tree ◽  
Fractal Dimensions ◽  
Self Similarity ◽  
Avrami Model ◽  
Cluster Aggregation ◽  
Box Counting ◽  
Self Assembled ◽  
Limited Diffusion

Complex morphologies, as is the case in self-assembled fibrillar networks (SAFiNs) of 1,3:2,4-Dibenzylidene sorbitol (DBS), are often characterized by their Fractal dimension and not Euclidean. Self-similarity presents for DBS-polyethylene glycol (PEG) SAFiNs in the Cayley Tree branching pattern, similar box-counting fractal dimensions across length scales, and fractals derived from the Avrami model. Irrespective of the crystallization temperature, fractal values corresponded to limited diffusion aggregation and not ballistic particle–cluster aggregation. Additionally, the fractal dimension of the SAFiN was affected more by changes in solvent viscosity (e.g., PEG200 compared to PEG600) than crystallization temperature. Most surprising was the evidence of Cayley branching not only for the radial fibers within the spherulitic but also on the fiber surfaces.

scholarly journals CLEAVAGE FRACTURE UNDER SHORT PULSE LOADING

1991 ◽  
Vol 01 (C3) ◽  
pp. C3-589-C3-596 ◽  
Author(s):  
H. HOMMA ◽  
Y. KANTO ◽  
K. TANAKA
Keyword(s):  
Cleavage Fracture ◽  
Short Pulse ◽  
Pulse Loading

SOFTENING BRANCHES OF A TWO-DEGREE-OF-FREEDOM SYSTEM INDUCED BY SPATIAL BUCKLING

2006 ◽  
Vol 06 (04) ◽  
pp. 493-512 ◽  
Author(s):  
NOËL CHALLAMEL
Keyword(s):  
Structural Model ◽  
Structural Parameters ◽  
Degree Of Freedom ◽  
Lateral Loading ◽  
Load Parameter ◽  
Imperfection Sensitivity ◽  
Softening Effect ◽  
Out Of Plane ◽  
Secondary Bifurcation ◽  
Two Degree Of Freedom

The aim of this paper is to show how geometrical non-linearity may induce equivalent softening in a simple two-degree-of-freedom spatial elastic system. The generic structural model studied is a generalization of Augusti's spatial model, incorporating lateral loading. This model could be used as a teaching model to understand the softening effect induced by out-of-plane buckling. The lateral loading in the plane of maximal stiffness is considered as the varying load parameter, whereas the vertical load is perceived as a constant parameter. It is shown that a bifurcation occurs at the critical horizontal load. The fundamental path becomes unstable, beyond this critical value. However, two symmetrical bifurcate solutions appear, whose stability depend on the structural parameters value. No secondary bifurcation is observed for this system. The presented system possesses imperfection sensitivity, and imperfection insensitivity, depending on the values of the structural parameters. In any case, for sufficiently large rotations, collapse occurs with unstable softening branches induced by spatial buckling.

scholarly journals Graph fractal dimension and the structure of fractal networks

2020 ◽  
Vol 8 (4) ◽  
Author(s):  
Pavel Skums ◽  
Leonid Bunimovich
Keyword(s):  
Complex Networks ◽  
Evolutionary Biology ◽  
Real Life ◽  
Fractal Dimensions ◽  
Network Models ◽  
Self Similarity ◽  
Descriptive Complexity ◽  
Network Communities ◽  
Fractal Properties ◽  
Continuous Objects

Abstract Fractals are geometric objects that are self-similar at different scales and whose geometric dimensions differ from so-called fractal dimensions. Fractals describe complex continuous structures in nature. Although indications of self-similarity and fractality of complex networks has been previously observed, it is challenging to adapt the machinery from the theory of fractality of continuous objects to discrete objects such as networks. In this article, we identify and study fractal networks using the innate methods of graph theory and combinatorics. We establish analogues of topological (Lebesgue) and fractal (Hausdorff) dimensions for graphs and demonstrate that they are naturally related to known graph-theoretical characteristics: rank dimension and product dimension. Our approach reveals how self-similarity and fractality of a network are defined by a pattern of overlaps between densely connected network communities. It allows us to identify fractal graphs, explore the relations between graph fractality, graph colourings and graph descriptive complexity, and analyse the fractality of several classes of graphs and network models, as well as of a number of real-life networks. We demonstrate the application of our framework in evolutionary biology and virology by analysing networks of viral strains sampled at different stages of evolution inside their hosts. Our methodology revealed gradual self-organization of intra-host viral populations over the course of infection and their adaptation to the host environment. The obtained results lay a foundation for studying fractal properties of complex networks using combinatorial methods and algorithms.

A conservation law, entropy principle and quantization of fractal dimensions in hadron interactions

2018 ◽  
Vol 33 (10) ◽  
pp. 1850057 ◽  
Author(s):  
I. Zborovský
Keyword(s):  
Conservation Law ◽  
Fractal Structure ◽  
Fractal Dimensions ◽  
The Self ◽  
Hadron Structure ◽  
Self Similarity ◽  
Similarity Variable ◽  
Variable Formula ◽  
Entropy Principle ◽  
Fragmentation Processes

Fractal self-similarity of hadron interactions demonstrated by the [Formula: see text]-scaling of inclusive spectra is studied. The scaling regularity reflects fractal structure of the colliding hadrons (or nuclei) and takes into account general features of fragmentation processes expressed by fractal dimensions. The self-similarity variable [Formula: see text] is a function of the momentum fractions [Formula: see text] and [Formula: see text] of the colliding objects carried by the interacting hadron constituents and depends on the momentum fractions [Formula: see text] and [Formula: see text] of the scattered and recoil constituents carried by the inclusive particle and its recoil counterpart, respectively. Based on entropy principle, new properties of the [Formula: see text]-scaling concept are found. They are conservation of fractal cumulativity in hadron interactions and quantization of fractal dimensions characterizing hadron structure and fragmentation processes at a constituent level.

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