Evaluation of fractal dimension of star polymers with different numbers of arms in computer experiment

2020 ◽  
Vol 63 (7) ◽  
pp. 39-45
Author(s):  
Oleg E. Sidorenko ◽  
◽  
Denis A. Rodnyansky ◽  
Keyword(s):  
Fractal Dimension ◽  
Molecular Modeling ◽  
Potential Energy ◽  
Constant Temperature ◽  
Thermodynamic Equilibrium ◽  
Computer Experiment ◽  
Star Polymers ◽  
Minimum Potential Energy ◽  
Fractal Properties ◽  
Minimum Potential

One of the important characteristics of a polymer molecule is its fractal dimension. Fractals are objects whose Hausdorff dimension is fractional and exceeds the topological dimension. The main distinguishing feature of such objects is self-similarity. The fractal characteristics of polymer macromolecules largely determine the chemical, physicochemical, and physical properties of these objects, such as the Mark-Kuhn-Houwink scaling parameters, toughness, tangent of the angle of mechanical losses, and dynamic modulus of elasticity. Today the fractal properties of topologically linear polymers are studied in detail, however, the fractal properties of practically significant star polymers are still poorly studied. This is probably due to the fact that computer simulation of polymer systems by methods of classical mechanics requires lengthy calculations even on supercomputers. In this regard, it is interesting to evaluate the possibility of using relatively simple software packages such as HyperChem in molecular modeling of polymers. The purpose of the research was to determine the fractal dimension for 3, 4, 5, and 6-arm star (-CH2-CH2-)n polymers in a state of minimum potential energy in an isolated system and in a state of thermodynamic equilibrium at a constant temperature, comparing the obtained values with the fractal dimension of linear polyethylene and to mfke an assessment of the appropriateness of using the HyperChem software package for macromolecular calculations. A computer experiment was conducted using the HyperChem package. To obtain isolated molecules in a state of minimum potential energy, a simulation by the conjugate gradient method was performed. To bring the macromolecules to a state of thermodynamic equilibrium at a constant temperature, the molecular modeling in the canonical ensemble was carried out. For the studied polymers, the values of the fractal dimension and critical Flory index were calculated. The data obtained confirm the relation between fractal dimension and conformation of the macromolecule and, within the accuracy achieved, indicate the absence of fractal properties for star macromolecules with a large number of arms.

Principles of minimum potential energy and complementary energy

2020 ◽  
pp. 228-248
Author(s):  
Lallit Anand ◽  
Sanjay Govindjee
Keyword(s):  
Boundary Condition ◽  
Potential Energy ◽  
Displacement Field ◽  
Mixed Boundary ◽  
Energy Functional ◽  
The Body ◽  
Minimum Potential Energy ◽  
Complementary Energy ◽  
Minimum Potential ◽  
Potential Energy Functional

With the displacement field taken as the only fundamental unknown field in a mixed-boundary-value problem for linear elastostatics, the principle of minimum potential energy asserts that a potential energy functional, which is defined as the difference between the free energy of the body and the work done by the prescribed surface tractions and the body forces --- assumes a smaller value for the actual solution of the mixed problem than for any other kinematically admissible displacement field which satisfies the displacement boundary condition. This principle provides a weak or variational method for solving mixed boundary-value-problems of elastostatics. In particular, instead of solving the governing Navier form of the partial differential equations of equilibrium, one can search for a displacement field such that the first variation of the potential energy functional vanishes. A similar principle of minimum complementary energy, which is phrased in terms of statically admissible stress fields which satisfy the equilibrium equation and the traction boundary condition, is also discussed. The principles of minimum potential energy and minimum complementary energy can also be applied to derive specialized principles which are particularly well-suited to solving structural problems; in this context the celebrated theorems of Castigliano are discussed.

Optimization for Minimum Potential Energy, Maximum Rigidity and Minimum Deformability

1984 ◽  
pp. 9-87
Author(s):  
Andrej M. Brandt ◽  
Wojciech Dzieniszewski ◽  
Stefan Jendo ◽  
Wojciech Marks ◽  
Stefan Owczarek ◽  
...  
Keyword(s):  
Potential Energy ◽  
Minimum Potential Energy ◽  
Minimum Potential ◽  
Energy Maximum

A Global Extremum Principle in Mixed Form for Equilibrium Analysis With Elastic/Stiffening Materials (a Generalized Minimum Potential Energy Principle)

1994 ◽  
Vol 61 (4) ◽  
pp. 914-918 ◽  
Author(s):  
J. E. Taylor
Keyword(s):  
Potential Energy ◽  
Problem Formulation ◽  
Extremum Problem ◽  
Equilibrium Analysis ◽  
Minimum Potential Energy ◽  
Problem Statement ◽  
Energy Principle ◽  
Potential Energy Principle ◽  
Minimum Potential ◽  
Minimum Potential Energy Principle

An extremum problem formulation is presented for the equilibrium mechanics of continuum systems made of a generalized form of elastic/stiffening material. Properties of the material are represented via a series composition of elastic/locking constituents. This construction provides a means to incorporate a general model for nonlinear composites of stiffening type into a convex problem statement for the global equilibrium analysis. The problem statement is expressed in mixed “stress and deformation” form. Narrower statements such as the classical minimum potential energy principle, and the earlier (Prager) model for elastic/locking material are imbedded within the general formulation. An extremum problem formulation in mixed form for linearly elastic structures is available as a special case as well.

Buckling Analysis of a Bi-Directional Strain-Gradient Euler–Bernoulli Nano-Beams

2020 ◽  
Vol 20 (11) ◽  
pp. 2050114
Author(s):  
Murat Çelik ◽  
Reha Artan
Keyword(s):  
Potential Energy ◽  
Gradient Elasticity ◽  
Buckling Analysis ◽  
Functionally Graded ◽  
Minimum Potential Energy ◽  
End Conditions ◽  
Graded Material ◽  
Minimum Potential ◽  
Gradient Elasticity Theory ◽  
Basic Equations

Investigated herein is the buckling of Euler–Bernoulli nano-beams made of bi-directional functionally graded material with the method of initial values in the frame of gradient elasticity. Since the transport matrix cannot be calculated analytically, the problem was examined with the help of an approximate transport matrix (matricant). This method can be easily applied with buckling analysis of arbitrary two-directional functionally graded Euler–Bernoulli nano-beams based on gradient elasticity theory. Basic equations and boundary conditions are derived by using the principle of minimum potential energy. The diagrams and tables of the solutions for different end conditions and various values of the parameters are given and the results are discussed.

scholarly journals On the continuum properties of repetitive beam-like pin-jointed structures

2002 ◽  
Vol 37 (5) ◽  
pp. 459-464
Author(s):  
N G Stephen ◽  
Y Zhang
Keyword(s):  
Potential Energy ◽  
Single Cell ◽  
Stiffness Matrix ◽  
Minimum Potential Energy ◽  
One Dimensional ◽  
Displacement Vectors ◽  
Repetitive Structure ◽  
Minimum Potential ◽  
Equivalent Continuum ◽  
The Continuum

The equivalent continuum beam properties of a one-dimensional repetitive structure have previously been determined through eigenanalysis of the transfer matrix of a single cell. A simpler procedure requires a knowledge of the stiffness matrix of the single cell, together with the ability to deduce the displacement vectors for tension, bending and shear. A once and for all application of the principle of minimum potential energy for tension yields the equivalent continuum Poisson's ratio, from which the remaining properties follow.

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