344 research outputs found
On Polymer Statistical Mechanics: From Gaussian Distribution to Maxwell-Boltzmann Distribution to Fermi-Dirac Distribution
Macroscopic mechanical properties of polymers are determined by their
microscopic molecular chain distribution. Due to randomness of these molecular
chains, probability theory has been used to find their micro-states and energy
distribution. In this paper, aided by central limit theorem and mixed Bayes
rule, we showed that entropy elasticity based on Gaussian distribution is
questionable. By releasing freely jointed chain assumption, we found that there
is energy redistribution when each bond of a molecular chain changes its
length. Therefore, we have to change Gaussian distribution used in polymer
elasticity to Maxwell-Boltzmann distribution. Since Maxwell-Boltzmann
distribution is only a good energy description for gas molecules, we found a
mathematical path to change Maxwell-Boltzmann distribution to Fermi-Dirac
distribution based on molecular chain structures. Because a molecular chain can
be viewed as many monomers glued by covalent electrons, Fermi-Dirac
distribution describes the probability of covalent electron occupancy in
micro-states for solids such as polymers. Mathematical form of Fermi-Dirac
distribution is logistic function. Mathematical simplicity and beauty of
Fermi-Dirac distribution make many hard mechanics problems easy to understand.
Generalized logistic function or Fermi-Dirac distribution function was able to
understand many polymer mechanics problems such as viscoelasticity [1],
viscoplasticity [2], shear band and necking [3], and ultrasonic bonding [4].Comment: 3 figure
Simulation of Wave in Hypo-Elastic-Plastic Solids Modeled by Eulerian Conservation Laws
This paper reports a theoretical and numerical framework to model nonlinear
waves in elastic-plastic solids. Formulated in the Eulerian frame, the
governing equations employed include the continuity equation, the momentum
equation, and an elastic-plastic constitutive relation. The complete governing
equations are a set of first-order, fully coupled partial differential
equations with source terms. The primary unknowns are velocities and deviatoric
stresses. By casting the governing equations into a vector-matrix form, we
derive the eigenvalues of the Jacobian matrix to show the wave speeds. The
eigenvalues are also used to calculate the Courant number for numerical
stability. The model equations are solved using the Space-Time Conservation
Element and Solution Element (CESE) method. The approach is validated by
comparing our numerical results to an analytical solution for the special case
of longitudinal wave motion.Comment: 34 pages, 11 figure
Prevention and Trust Evaluation Scheme Based on Interpersonal Relationships for Large-Scale Peer-To-Peer Networks
Peer reviewedPublisher PD
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