Continuum mechanics

About: Continuum mechanics is a research topic. Over the lifetime, 5042 publications have been published within this topic receiving 181027 citations.

Instability of Nonlocal Continuum and Strain Averaging

Abstract: Nonlocal continuum, in which the (macroscopic smoothed‐out) stress at a point is a function of a weighted average of (macroscopic smoothed‐out) strains in the vicinity of the point, are of interest for modeling of heterogeneous materials, especially in finite element analysis. However, the choice of the weighting function is not entirely empirical but must satisfy two stability conditions for the elastic case: (1) No eigenstates of nonzero strain at zero stress, called unresisted deformation, may exist; and (2) the wave propagation speed must be real and positive if the material is elastic. It is shown that some weighting functions, including one used in the past, do not meet these conditions, and modifications to meet them are shown. Similar restrictions are deduced for discrete weighting functions for finite element analysis. For some cases, they are found to differ substantially from the restriction for the case of a continuum if the averaging extends only over a few finite elements.

From fractal media to continuum mechanics

Abstract: This paper presents an overview of modeling fractal media by continuum mechanics using the method of dimensional regularization. The basis of this method is to express the balance laws for fractal media in terms of fractional integrals and, then, convert them to integer-order integrals in conventional (Euclidean) space. Following an account of this method, we develop balance laws of fractal media (continuity, linear and angular momenta, energy, and second law) and discuss wave equations in several settings (1d and 3d wave motions, fractal Timoshenko beam, and elastodynamics under finite strains). We then discuss extremum and variational principles, fracture mechanics, and equations of turbulent flow in fractal media. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions and reduce to conventional forms for continuous media with Euclidean geometries upon setting the dimensions to integers. We also point out relations and potential extensions of dimensional regularization to other models of microscopically heterogeneous physical systems.

Introduction to the Foundations of Applied Mathematics

Abstract: Dimensional Analysis.- Perturbation Methods.- Kinetics.- Diffusion.- Traffic Flow.- Continuum Mechanics: One Spatial Dimension.- Elastic and Viscoelastic Materials.- Continuum Mechanics: Three Spatial Dimensions.- Newtonian Fluids.- Appendices.