2 - Governing Equations for Motion and Deformation of Block Systems and Heat Transfer

Abstract

Publisher Summary This chapter aims to present a brief coverage of the governing equations most widely adopted in the present element methods (DEM) approaches. It assumes that the general reader is familiar with the basics of continuum mechanics, the finite element method (FEM) as well as tensor analysis techniques. It provides Newton's equations of motion for particles. A rigid body characterized by a domain of constant volume and mass does not deform. The distance between any two points in a rigid body remains unchanged. A rigid body is an idealization as all bodies deform under the action of external forces. However, this idealization is acceptable in many rock engineering problems, especially large-scale block movements under low stress conditions. Rigid body dynamics is governed by Newton's law of motion and Euler's rotations of rigid bodies. The equations of motion for the deformable bodies are acceptable descriptions if the ‘small displacement’ assumption is accepted. However, the small deformation or rigid body assumptions are just two extreme cases of uncoupled deformation-motion conditions, and are not necessarily universally valid. Under certain circumstances, deformable bodies may undergo large-scale displacements but have small strains that need to be taken into account. The chapter also presents the basic equations for conductive heat transfer and some key thermal properties.